Download Free Clusters and Free Molecules in Strong, Shaped Laser Fields

Free Clusters and Free Molecules
in Strong, Shaped Laser Fields
im Fachbereich Physik
der Freien Universität Berlin
eingereichte Dissertation
vorgelegt von
Ihar Shchatsinin
Berlin 2009
ii
1. Gutachter: Prof. Dr. I. V. Hertel
2. Gutachter: Prof. Dr. L. Wöste
Tag der Disputation:
Die Arbeit wurde durchgeführt am
Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie (MBI)
iii
“Felix, qui potuit rerum cognoscere causas”
- Vergil [70 - 19 v. Chr]
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Überblick
Diese Dissertation erörtert die Wechselwirkung von großen, endlichen Systemen mit Femtosekunden Laserstrahlung bei moderaten Intensitäten bis zu 4 × 1014 W/cm2 . Die photoinduzierten Prozesses werden mit Hilfe von Photoionen- und Photoelektronenspektrometrie
analysiert. Als Untersuchungsobjekte wurden C60 Fullerene und Modellpeptide (Ac–Phe–
NHMe und Ac–Ala–NHMe) gewählt.
Das C60 Fulleren ist ein äußerst interessantes Modellsystem, um die Dynamik der Photoanregung, Photoionisation und Photofragmentation in komplexen Molekülen mit vielen Freiheitsgraden zu studieren. C60 mit seiner wohl definierten hochsymmetrischen Struktur und
der Vielzahl elektronischer und nuklearer Freiheitsgrade kann als ein ausgezeichneter Prototyp
eines großen endlichen molekularen Systems sowohl für theoretische als auch für experimentelle
Studien gesehen werden.
Um den Energieeintrag in das System von der Energieumverteilung in die verschiedenen
elektronischen und nuklearen Freiheitsgrade zu trennen, sind Ultrakurzlaserimpulse mit einer
Dauer von 30 fs bis hin zu 9 fs für die Untersuchungen verwendet worden. Die Anregungszeit
liegt damit deutlich unter der charakteristischen Zeitskala für die Elektronenrelaxation. Die
genaue Analyse der C60 -Massenspektren als Funktion wichtiger Laserparameter, wie beispielsweise Intensität, Impulsdauer und Zeitverzögerung zwischen Pump- und Probeimpulsen, führt
zu einem tieferen Verständnis der photoinduzierten Prozesse wie Ionisation und Fragmentation.
Die Beobachtung von vielfach geladenen Cq+
60−2n -Fragmenten (q > 1) sogar mit 9 fs Pulsen
läßt auf eine direkte nichtstatistische Fragmentation von Fullerenen auf einer Femtosekundenzeitskala durch nichtadiabatische Mehrelektrondynamik (NMED) schließen, während einfach geladene Ionen durch einen adiabatischen Ein-Elektronen-Prozess (SAE) erzeugt werden.
Lichtelliptizität abhängige Studien können zwischen SAE und NMED sehr empfindlich unterscheiden. Zwei Farben Pump-Probe Messungen geben direkte Information über die charakteristische Relaxationszeit. In allen Fällen spielt die resonante Mehrelektronenanregung des
t1g -Zustands eine Schlüsselrolle im Anregungsprozess.
Die Manipulation von Molekülen durch zeitlich geformte Laserimpulse hat sich zu einer
Standardtechnik für das Steuern und mögliches Analysieren von Reaktionspfaden in komplizierten Systemen entwickelt. Mit Hilfe einer geschlossenen Rückkopplungsschleife kann eine
optimale Anregung auch von sehr komplizierten Systemen gefunden werden, ohne dass eine
genaue Kenntnis der potenziellen Energieoberflächen notwendig ist. Als erstes wurde diese
Technik auf das C60 Fulleren angewandt, um den Fragmentationsprozesse zu studieren. Eine
optimale Pulsfolge führt zu einem deutlich erhöhten Energieeintrag und somit zu vermehrter
C2 -Emission, ein typischer Energieverlustkanal des schwingungsangeregten C60 , im Vergleich
mit der Reaktion auf einen einzelnen Laserimpuls derselben Energie und Dauer. Als zweites
wurde diese Technik für einen selektiven Bindungsbruch in den Modellpeptiden Ac–Phe–NHMe
und Ac–Ala–NHMe eingesetzt. Durch die starke Anregung im Laserfeld mit zeitlich geformten
Laserimpulsen können starke Bindungen im molekularen System mit hoher Selektivität gespalten werden während andere, schwächere Bindungen intakt bleiben. Diese Ergebnisse zeigen,
dass die adaptive Pulsformung in der Kombination mit hochauflösender Massenspektroskopie
einen ersten Schritt zur Entwicklung von effizienten Werkzeugen für die chemische Analyse
von großen Biomolekülen oder sogar Proteinen darstellen.
v
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Abstract
This thesis considers the interaction of large, finite systems with moderately intense femtosecond laser radiation up to 4 × 1014 W/cm2 studied by the methods of photoion and
photoelectron spectrometry. C60 fullerenes and amino acid complexes (Ac–Phe–NHMe and
Ac–Ala–NHMe) were chosen as objects of investigations.
C60 fullerene is an extremely interesting model system for studying the dynamics of photoexcitation, photoionisation, and photofragmentation in molecular systems with many degrees
of freedom. C60 with its well defined, highly symmetric structure and the large number of electronic and nuclear degrees of freedom can be seen as an excellent prototype for both theoretical
and experimental studies.
To separate energy deposition into the system from energy redistribution among the various
electronic and nuclear degrees of freedom ultrashort laser pulses with a duration from 30 fs up
to 9 fs have been utilised for investigations. The excitation time thus addressed lies well below
the characteristic time scale for the electron relaxation. From a detailed analysis of the C60
mass spectra as a function of important laser parameters e.g. intensity, pulse duration, and time
delay between pump and probe pulses insight into fundamental photoinduced processes such
as ionisation and fragmentation is obtained. The observation of multiple charged fragments
Cq+
60−2n (q > 1) even with 9 fs pulses indicates direct non-statistical fragmentation of fullerenes
on a femtosecond time scale through non-adiabatic multielectron dynamics (NMED), while
singly charged ions are generated by an essentially adiabatic single active electron mechanism
(SAE). Light ellipticity dependent studies can very sensitively distinguish between SAE and
NMED. Time resolved mass spectrometry in a two colour fs pump probe setup provides direct
information on the characteristic relaxation time. In all cases resonant multielectron excitation
of the t1g state was identified to play the key role in the energy deposition process.
The manipulation of molecules by temporally shaped laser pulses has become a standard
technique for controlling and possibly analysing reaction pathways in complex systems. A
closed feedback optimisation loop allows one to find optimal excitation schemes on potential
energy surfaces of very complex systems without prior knowledge of their structure. First, this
technique was applied to C60 fullerenes for studying the fragmentation processes. An optimal
pulse sequence results in significant enhancement of C2 evaporation, a typical energy loss
channel of vibrationally hot C60 , in comparison with the response to a single pulse of the same
energy and overall width. Second, the same technique was utilised for selective bond cleavage
in the amino acid complexes Ac–Phe–NHMe and Ac–Ala–NHMe that may be regarded as
model peptides. Strong field excitation with shaped laser pulses allows one to cleave strong
backbone bonds in the molecular system with high selectivity while keeping other more labile
bonds intact. These results show that pulse shaping in combination with high resolution mass
spectroscopy can be a first step towards creation of efficient tools for the chemical analysis of
larger biomolecules or even proteins.
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viii
List of Publications
I. Shchatsinin, H.-H. Ritze, C. P. Schulz, and I. V. Hertel, “Multiphoton excitation and ionization by elliptically polarized, intense, short laser pulses”, Phys. Rev. A (submitted)
I. V. Hertel, I. Shchatsinin, T. Laarmann, N. Zhavoronkov, H.-H. Ritze, and C. P. Schulz,
“C60 in elliptically polarized femtosecond laser fields: a challenge for theory”, Phys. Rev. Lett.
102 (2009), 023003
I. Shchatsinin, T. Laarmann, N. Zhavoronkov, C. P. Schulz, and I. V. Hertel, “Ultrafast
energy redistribution in C60 fullerenes: A real time study by two-color femtosecond spectroscopy”, J. Chem. Phys. 129 (2008), 204308
T. Laarmann, I. Shchatsinin, P. Singh, N. Zhavoronkov, C. P. Schulz, and I. V. Hertel,
“Femtosecond pulse shaping as analytic tool in mass spectrometry of complex polyatomic
systems”, J. Phys. B 41 (2008), 074005
T. Laarmann, I. Shchatsinin, P. Singh, N. Zhavoronkov, M. Gerhards, C. P. Schulz, and
I. V. Hertel, “Coherent control of bond breaking in amino acid complexes with tailored femtosecond pulses”, J. Chem. Phys. 127 (2007), 201101
T. Laarmann, I. Shchatsinin, A. Stalmashonak, M. Boyle, N. Zhavoronkov, J. Handt, R.
Schmidt, C. P. Schulz, and I. V. Hertel, “Control of giant breathing motion in C60 with temporally shaped laser pulses”, Phys. Rev. Lett. 98 (2007), 058302
T. Laarmann, I. Shchatsinin, M. Boyle, G. Stibenz, G. Steinmeyer, C. P. Schulz and I.
V. Hertel, “Nonadiabatic multielectron dynamics in (moderately) strong laser fields: C60 a
model case for large finite system”, Femtochemistry VII: Fundamental Ultrafast Processes
in Chemistry, Physics, and Biology, A. W. Castleman Jr. and M. L. Kimble eds. (Elsevier,
Amsterdam, 2006), 543-552
I. Shchatsinin, T. Laarmann, G. Stibenz, G. Steinmeyer, A. Stalmashonak, N. Zhavoronkov,
C. P. Schulz, and I. V. Hertel, “C60 in intense short pulse laser fields down to 9 fs: Excitation
on time scales below e-e and e-phonon coupling”, J. Chem. Phys. 125 (2006), 194320
M. Boyle, T. Laarmann, I. Shchatsinin, C. P. Schulz, and I. V. Hertel, “Fragmentation
dynamics of fullerenes in intense femtosecond-laser fields: Loss of small neutral fragments on
a picosecond time scale”, J. Chem. Phys. 122 (2005), 181103
ix
x
Contents
1 Introduction
1
2 Objects of Investigation
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2.1 C60 Fullerene and its Properties . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Model Peptides and Methods of Investigations . . . . . . . . . . . . . . . . . 12
3 Interaction of Strong Laser Fields with Matter
17
3.1 General Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Multiphoton Ionisation and Fragmentation . . . . . . . . . . . . . . . . . . . 19
3.3 Energy Redistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Experimental Apparatus
4.1 Vacuum Chamber . . . . . . . . .
4.2 Molecular Beam Source . . . . . .
4.3 Time of Flight Mass Spectrometry
4.4 Photoelectron Spectroscopy . . . .
4.5 Ion and Electron Detection . . . .
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5 Femtosecond Laser Pulses
5.1 Mathematical Description of Laser Pulses . . .
5.2 Generation of Femtosecond Laser Pulses . . .
5.2.1 Femtosecond Oscillators . . . . . . . .
5.2.2 Chirped Pulse Amplification . . . . . .
5.2.3 Multipass Amplification Laser System .
5.3 Generation of sub -10 fs Pulses . . . . . . . . .
5.4 Laser Intensity and Polarisation Control . . . .
5.5 Average Intensity of Elliptically Polarised Light
5.6 Pulse Shaping . . . . . . . . . . . . . . . . .
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6 Pulse Characterisation Techniques
6.1 Temporal Pulse Characterisation . . . . . . . . . . .
6.1.1 Methods of Temporal Pulse Characterisation
6.1.2 Autocorrelation Measurement . . . . . . . .
6.1.3 FROG Technique . . . . . . . . . . . . . . .
6.1.4 XFROG Method . . . . . . . . . . . . . . .
6.1.5 SPIDER Technique . . . . . . . . . . . . . .
6.2 Spatial Laser Beam Characterisation . . . . . . . .
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6.3
Pulse Intensity and Fluence . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7 Optimisation
7.1 Introduction to Mathematical Optimisation
7.2 Methods of Stochastic Optimisation . . . .
7.2.1 Simulated Annealing . . . . . . . .
7.2.2 Evolutionary Algorithms . . . . . .
7.3 Practical Implementation of Optimisation .
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8 Excitation of C60
8.1 Multiphoton Ionisation of C60 . . . . . . . . .
8.1.1 Different Ionisation Mechanisms . . . .
8.1.2 Saturation Intensity . . . . . . . . . .
8.1.3 Ionisation Energy and Model Potential
8.1.4 Calculation of Ionisation Rates for Cq+
60
8.2 Pulse Duration Dependent Study . . . . . . .
8.2.1 Experimental Observations . . . . . . .
8.2.2 Sequential Ionisation . . . . . . . . . .
8.2.3 Saturation Intensities . . . . . . . . .
8.3 Polarisation Dependent Study . . . . . . . . .
8.3.1 Photoion Spectroscopy . . . . . . . . .
8.3.2 Photoelectron Spectroscopy . . . . . .
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9 Dynamics of Ultrafast Energy Redistribution in
9.1 Energy Coupling . . . . . . . . . . . . . . . .
9.2 Single Pulse Study . . . . . . . . . . . . . . .
9.3 One Colour Pump-Probe Study . . . . . . . .
9.4 Two Colour Pump-Probe Study . . . . . . . .
9.5 Pulse Shaping Study . . . . . . . . . . . . . .
C60
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95
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113
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10 Mass Spectrometry of Model Peptides
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10.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
11 Summary and Outlook
179
Bibliography
183
Curriculum Vitae
223
Acknowledgements
225
xii
List of Figures
2.1
2.2
2.3
2.4
3.1
The structure of C60 fullerene . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Single electron energy diagram and electronic absorption spectra of C60 fullerene 10
Illustration of the energetics for plasmon enhanced multiphoton process in the
generation of Cq+
60 with different final charge state q (adopted from [HSV92]) . 11
Model peptides used in present work . . . . . . . . . . . . . . . . . . . . . . 14
3.2
3.3
3.4
Different photoionisation regimes as a function of the laser intensity and photon
energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Perturbative nonlinear regime of ionisation (γ > 1) . . . . . . . . . . . . . .
Strong field regime of ionisation (γ 1) . . . . . . . . . . . . . . . . . . .
Illustration of energy relaxation processes in large but finite systems . . . . .
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20
21
23
29
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
Photograph of the vacuum apparatus . . . . . . . . . . . . . .
Photograph of the oven . . . . . . . . . . . . . . . . . . . . .
Photoelectron and reflectron TOF mass spectrometer . . . . .
Experimentally measured shape and Gaussian fit of Xenon peak
Charged particles detection schemes . . . . . . . . . . . . . . .
Pulse height distribution . . . . . . . . . . . . . . . . . . . . .
The standard deviation of the integrated ion yield . . . . . . .
TOF mass and photoelectron spectra of C60 fullerene . . . . .
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32
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48
5.1
Pulses of different shapes (Lorentzian, Gaussian, and sech2 with 100 fs FWHM)
and their autocorrelation functions . . . . . . . . . . . . . . . . . . . . . .
Optical layout of the multipass amplification laser system . . . . . . . . . .
Laser system for the generation of sub -10 fs pulses and an example of the pulse
temporal profile produced by this system . . . . . . . . . . . . . . . . . . .
The action of λ/2- and λ/4-plates on linearly polarised light . . . . . . . . .
Scheme of the pulse shaping setup . . . . . . . . . . . . . . . . . . . . . .
The layout of SLM-S 640/12 . . . . . . . . . . . . . . . . . . . . . . . . . .
Calibration of LCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
5.3
5.4
5.5
5.6
5.7
6.1
6.2
6.3
6.4
6.5
6.6
6.7
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Schematic diagram of an autocorrelator/FROG setup . . . . . . . . . . . . . .
Pulse duration measurement by the autocorrelation technique for the pulse
produced by the Multipass laser system . . . . . . . . . . . . . . . . . . . . .
Temporal characterisation of the pulses produced by the Multipass laser system
Schematic diagram of a cross-correlator/XFROG setup . . . . . . . . . . . . .
Temporal characterisation of the pulses passed through the shaper setup . . .
Principle of SPIDER apparatus . . . . . . . . . . . . . . . . . . . . . . . . .
Spatial beam characterisation . . . . . . . . . . . . . . . . . . . . . . . . . .
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7.1
7.2
7.3
7.4
Diagram of the simulated annealing algorithm
Basic scheme of the genetic algorithm . . . . .
Scheme of the adaptive closed loop setup . . .
Results of the test optimisations . . . . . . . .
8.1
Typical mass spectra of C60 produced by laser pulses of 795 nm wavelength
with a pulse duration of 25 fs (top) and 5 ps (bottom) . . . . . . . . . . . . .
Ionisation potentials for different charge states of Cq+
. . . . . . . . . . . . .
60
Comparison between experimental data of C60 photoionisation and the theoretical ionisation rates derived from S-matrix theory by A. Becker and F. H. M.
Faisal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mass spectra obtained from photoionisation of C60 with 9 fs laser pulses centred
at 765 nm in intensity range between 4 × 1013 W/cm2 and 4.0 × 1014 W/cm2 .
Experimental yields of Cq+
60 ions measured as a function of intensity for 27 fs
and 9 fs laser pulses and their fits . . . . . . . . . . . . . . . . . . . . . . . .
2+
Yields of C+
60 and C60 calculated by solving the coupled differential equations .
Saturation intensities for Cq+
. .
60 ions as a function of the final charge state q
Mass spectra of C60 after excitation with linearly and circularly polarised laser
radiation of 27 fs pulses at 797 nm . . . . . . . . . . . . . . . . . . . . . . . .
Integrated yield of Xe+ measured at 797 nm normalised to the yield with linear
light polarisation plotted as a function of ellipticity angle β and laser intensity .
3+
Integrated yield of C+
60 and C60 measured at 797 nm normalised to the yield
with linear light polarisation plotted as a function of ellipticity angle and laser
intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3+
Integrated yield of C+
60 and C60 ions normalised to the yield with linear light
polarisation plotted as a function of ellipticity angle and laser intensity after
interaction of C60 with 399 nm laser pulses . . . . . . . . . . . . . . . . . . .
3+
Integrated yield of fragments ΣC+
60−2n and ΣC60−2n measured at 797 nm normalised to the yield with linear light polarisation plotted as a function of ellipticity angle and laser intensity . . . . . . . . . . . . . . . . . . . . . . . . . .
Examples of classical electron trajectories in the combined field of C+
60 and
linearly or circularly polarised light . . . . . . . . . . . . . . . . . . . . . . . .
Probability to find different recollision energies for electrons emitted from the
inner (0.65a) and outer radius (1.00a) of the C60 molecule in linearly or circularly polarised light at I = 4.3 × 1014 W/cm2 , 797 nm . . . . . . . . . . . . .
Photoelectron spectra of C60 obtained with laser intensity of 1.4 × 1014 W/cm2
for linear and circular light polarisations . . . . . . . . . . . . . . . . . . . . .
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
8.11
8.12
8.13
8.14
8.15
9.1
9.2
9.3
9.4
9.5
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Illustration of energy redistribution processes in laser excited C60 molecule monitored in the present work with two colour pump-probe spectroscopy . . . . .
Comparison between different parts of the C60 mass spectrum obtained with
9Pfs laser pulses
27 P
fs laser pulses for equivalent intensities . . . . . . . .
P and
2+
3+
+
C60−2n
fragment ion yield obtained with 27 fs and
C60−2n ,
C60−2n ,
9 fs laser pulses plotted as a function of the laser intensity on log-log scale .
Results of one colour pump-probe experiment with 9 fs laser pulses . . . . .
Mass spectra with blue (399 nm, 3.4 × 1012 W/cm2 ) and/or red (797 nm, 5.1 ×
1013 W/cm2 ) laser pulses . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
.
.
.
.
98
101
105
109
114
118
119
121
122
125
126
129
131
133
134
135
136
139
141
. 144
. 146
. 147
. 149
. 150
9.6
9.7
9.8
9.9
9.10
9.11
9.12
9.13
Total ion yield of different charge states of Cq+
60 , q = 1 − 4 as a function of the
time delay between 399 nm pump pulse (3.4 × 1012 W/cm2 ) and 797 nm probe
pulse (5.1 × 1013 W/cm2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . .
Total ion yield of different masses of C3+
60−4n , n = 0 − 3 as a function of the
time delay between 399 nm pump pulse (3.4 × 1012 W/cm2 ) and 797 nm probe
pulse (5.1 × 1013 W/cm2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . .
Relaxation times τel for highly excited electrons due to electron-electron and
electron-vibrational coupling . . . . . . . . . . . . . . . . . . . . . . . . . .
Ratios Hbr /Or , Hbr /Or , and Hbr /Mbr . . . . . . . . . . . . . . . . . . . .
Result of C+
50 optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . .
+
+
+
2+
2+
Correlation between C+
50 and C48 , C52 , C60 , C60 , and C50 ions during the
optimisation run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison between two-colour pump-probe experiment and pulse shaping optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Period of the ag (1) breathing mode as a function of the number of excited
electrons derived from NA-QMD simulations and as a function of charge state
derived from hybrid B3LYP level of the DFT method . . . . . . . . . . . . .
. 152
. 153
. 155
. 156
. 159
. 160
. 161
. 163
10.1 Mass spectra of the Ac–Phe–NHMe molecular system recorded with laser pulses
of 3.7 × 1013 W/cm2 intensity and 32 fs pulse duration centred at 797 nm . . . 166
10.2 Different possible fragmentation channels of the Ac–Phe–NHMe molecular system167
10.3 Fragment ion yields of mass 43 u, mass 162 u, and the parent ion (mass 220 u)
of the Ac–Phe–NHMe molecular system recorded with 34 fs laser pulses plotted
as a function of the laser intensity on the log-log scale . . . . . . . . . . . . . 168
10.4 Fitness f characterising the predominant formation of fragment mass 43 u as a
function of the generation in the adaptive feedback loop and SH-XFROG map
of the optimal pulse shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
10.5 Temporal envelope and corresponding mass spectra recorded for the unshaped,
stretched to 246 fs (FWHM), and optimal pulses . . . . . . . . . . . . . . . . 170
10.6 Distribution of individual optimal pulses grouped according to the similarities
in pulses structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
10.7 Calculated triple pulse sequence in the time domain illustrating the effect of
different phase parameters Φ on the pulse sequence . . . . . . . . . . . . . . 172
10.8 Ion yields plotted as a function of the relative phase shift in the parameterised
triple pulses and measured temporal envelope of the pulse sequence applied for
three different relative phase shifts Φ = 0, π/2, and 3π/2 . . . . . . . . . . . 173
10.9 Comparison between power spectra of the triple pulses assuming a third order
process and resonant two photon ionisation spectrum of Ac–Phe–NHMe . . . . 174
10.10Fitness f characterising the enhanced formation of fragment mass 77 u as a
function of the generation in the adaptive feedback loop . . . . . . . . . . . . 175
10.11Temporal structure and corresponding mass spectra of Ac–Ala–NHMe are
recorded with unshaped and optimal pulse . . . . . . . . . . . . . . . . . . . 177
xv
xvi
List of Tables
2.1
Optically active vibrational modes of C60 fullerene . . . . . . . . . . . . . . .
4.1
4.2
Distances and voltage settings for the reflectron TOF mass spectrometer . . . 37
Distances and voltage settings for the photoelectron spectrometer . . . . . . . 41
5.1
The mathematical description of some pulse shapes in the time and frequency
domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Deconvolution factors DAC and time-bandwidth products KT B for the various
shapes of laser pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Output parameters of oscillator and amplifier of the multipass amplification
laser system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2
5.3
7.1
8
Standard parameters for a free optimisation with the genetic algorithm . . . . 110
xvii
xviii
Chapter 1
Introduction
The interaction of strong laser fields with matter is a challenging topic in modern science
and has a long history. The possibility of multiphoton transitions was theoretically predicted
by Maria Göppert-Mayer in 1931 [Goe31]. The invention of the laser has opened new horizons
for experimental and theoretical studies on nonlinear interaction of laser fields with atoms,
molecules, and clusters. In 1965 G. S. Voronov and N. B. Delone investigated multiphoton
ionisation (MPI) of xenon with a ruby laser [VDe65]. In 1979 above threshold ionisation (ATI)
was observed [AFM79], where a photoelectron absorbs more photons than the minimum required for MPI. Under certain conditions the ionisation can occur by tunnelling of an electron
through the potential barrier created by an intense laser field. This process is known as “tunnelling ionisation” [GHa98]. If the laser intensity is high enough, the laser radiation suppresses
the potential barrier so far that the electron is able to escape freely from the atom. This
is called “over the barrier” ionisation (OTBI) or the “barrier suppression ionisation” (BSI)
[BRK93]. In addition, strong laser fields allow one to investigate intramolecular dynamics
for molecules and clusters. The dynamics takes place on the time scale of the atomic motion. Therefore, laser radiation with pulse durations on the fs time scale is required for such
investigations.
The aim of this thesis is to investigate the interaction of large but finite systems with
moderately intense femtosecond laser radiation using methods of photoion and photoelectron
spectrometry. In context of the present work “moderately intense radiation” implies laser
intensities strong enough to modify potential surfaces (in particular, if they are close to resonances) but far below the onset of relativistic effects. To separate energy deposition into
the system from energy redistribution among the various electronic and nuclear degrees of
1
2
CHAPTER 1. INTRODUCTION
freedom ultrashort laser pulses with a duration as short as 9 fs have been used. Combination
of different laser parameters i.e. intensity, pulse duration, and light ellipticity as well as the use
of time resolved measurements with the pump-probe technique provide powerful tools to study
fundamental photoinduced processes. Two different systems were studied: C60 fullerenes and
model peptides.
The discovery of carbon fullerenes by H. R. Kroto and co-workers [KHO85] opened for
science a new class of materials with very interesting physical and chemical properties. For
this discovery R. F. Curl, H. R. Kroto, and R. E. Smalley received the 1996 Nobel Prize in
Chemistry. The most stable member of fullerene family, C60 , has a highly symmetric structure
built out of 12 pentagons and 20 hexagons, it has 174 nuclear degrees of freedom, 60 essentially
equivalent delocalised π- and 180 structure defining, localised σ-electrons. The C60 molecule
can be considered as a model of a large but finite system. Experimental investigations of
isolated C60 have revealed both atomic properties, such as above threshold ionisation (ATI),
and bulk properties, such as thermionic emission (delayed ionisation) [CHH00]. The exploration
of the electronic and nuclear dynamics in C60 has a long history. One early observation was the
delayed ionisation on a microsecond time scale upon irradiation with nanosecond laser pulses
[CUH91] due to thermionic electron emission from vibrationally excited C60 . Strong electronphonon coupling leads to efficient heating of the nuclear degrees of freedom during laser
excitation, and the subsequent ionisation in turn is one important energy relaxation channel
[RHB01]. Another significant relaxation channel is the fragmentation due to the sequential
loss of C2 units [SDW96]. It has been found that the ionisation and fragmentation behaviour
depends sensitively on the excitation time scale, i.e. on the laser pulse duration τ [CHH00].
For pulse durations above 1 ps one observes a characteristic bimodal fragmentation pattern of
heavy fullerenes C60−2n separated by C2 units ranging down to 60 − 2n > 32 and a series of
small carbon clusters Cn below n < 28. In contrast, for laser pulse durations below 500 fs the
excitation energy tends to remain in the electronic system, and multiply charged clusters Cq+
60
are observed. Direct multiphoton ionisation dominates for very short laser pulses τ < 70 fs
[THD00]. As a fingerprint of the multiphoton process the photoelectron spectra of C60 exhibit
a characteristic ATI structure [CHH00] in which the active electron absorbs more photons
than necessary to overcome the ionisation potential. Additionally, sharp peaks attributed to
the population of several Rydberg series were found in the photoelectron spectra of C60 on
3
top of the ATI series [BHS01].
The study of the interaction of intense laser radiation with clusters is a rapidly expanding
field of research due to its fundamental and practical importance in many areas of physics,
chemistry, and material science. Nevertheless, knowledge about the energy absorption, ionisation and fragmentation pathways of large systems is still limited. Initially, the theoretical
description of the interaction with laser radiation has been considered through the perturbation
approach with a single active electron (SAE) [JBF07]. However, with increasing complexity
of molecules and increasing laser intensities, the limitations of this approximation become obvious, indicating that multiple active electrons (MAE) are necessary to explain the observed
phenomena [BCM01, TNE01, ZSG03]. The following questions are answered in this work: (i)
How does C60 absorb the incident laser radiation? (ii) How is the absorbed energy redistributed
among various degrees of freedom? (iii) Can these processes be controlled with temporally
shaped laser pulses?
Peptides, which are building blocks of proteins, are very interesting objects for investigations. So far, little is known about their interaction with laser pulses. Such investigations may
help to understand processes within a living cell which have a high importance for biochemistry.
Femtosecond, temporally shaped laser pulses may possibly open the door to selective peptide
bonds breaking and provide a new way compared to the well established methods of protein
sequencing such as chemical analysis by enzymatic digestion (Edman degradation reaction)
and ionic fragmentation in connection with mass spectrometry. Femtosecond pulse shaping as
an analytic tool for laser induced protein sequencing is considered in this work. Results of the
utilisation of ultrafast, shaped laser pulses picked by an evolutionary algorithm inside a learning
control loop to break specific, pre-selected bonds in the model peptides Ac–Phe–NHMe and
Ac–Ala–NHMe are discussed. These model peptides are not building blocks of proteins but
something similar to the real peptides in proteins which can be handled with relative ease.
This thesis has the following structure:
In Chapter 2, the physical properties of the objects of interest, namely, C60 fullerene
and model peptides are discussed. Chapter 3 discusses a general theoretical background
concerning the interaction of strong laser fields with matter. Direct multiphoton ionisation,
non-statistical fragmentation, energy redistribution, delayed ionisation, and statistical fragmentation are considered in detail. Chapter 4 consists of a record of the experimental apparatus
4
CHAPTER 1. INTRODUCTION
utilised in this work. Chapter 5 is dedicated to methods of the femtosecond as well as
sub -10 fs laser pulses generation, laser intensity and polarisation control, and a temporal manipulation of laser pulse shape with a pulse shaper technique. In Chapter 6, experimental
techniques used for the laser pulse characterisation both temporal and spatial are discussed.
Chapter 7 presents a description of optimisation methods, especially concentrates on the
method used in present work. Chapter 8 gives a detailed description and shows experimental
results on C60 excitation in moderate strong laser fields. Chapter 9 reports and discuss experimental data on the dynamics of ultrafast energy redistribution in C60 fullerene. In Chapter
10, the experimental results of detailed mass spectroscopic investigations of model peptides
with shaped laser pulses are presented. Finally, Chapter 11 outlines the most important
conclusions of this study.
Chapter 2
Objects of Investigation
C60 fullerenes and model peptides in the gas phase are the objects of interest in this work.
Due to its well defined highly symmetric structure, 174 nuclear degree of freedom, 240 valence
electrons (60 essentially equivalent delocalised π electrons and 180 structure defining localised
σ electrons), C60 may be considered as a model of large finite molecular systems for studying
of complex photo physical processes. Model peptides are an excellent example of a system
for studying selective bond breaking and rearrangement. Such investigations open the door to
new applications in biology and medicine. Gas phase studies of such objects provide a direct
access to isolated molecules free from interactions with the environment. Indeed, the dielectric
constant of protein powders is around 24 only [SBr96]. This contrasts with liquid water which
has a dielectric constant of ≈ 80 at room temperature and pressure. Therefore, physical
properties of large biological molecules studied in water solutions may be influenced strongly
by the interaction with the highly polarisable water molecules, while experiments in vacuum
may sometimes mimic the true behaviour of biological molecules in a protein environment
much better.
This chapter gives a general description of physical properties of these two many-body
systems and methods of their investigations in the gas phase.
2.1
C60 Fullerene and its Properties
Two main allotropic forms of bulk carbon are graphite and diamond. Carbon clusters, which
possess many unique properties, constitute another allotropic form. Small carbon clusters (Cn ,
n ≤ 9) exist in linear chains as the most stable form [RCK84]. Larger clusters (Cn , 9 < n < 30)
5
6
CHAPTER 2. OBJECTS OF INVESTIGATION
Figure 2.1: The structure of C60 fullerene.
are also found in ring structure of carbon atoms [HHK91]. Larger clusters (Cn , n > 30) have
a cage structure of fullerenes formed by rings of hexagons and pentagons [LCX99]. According
to the Euler’s theorem for polyhedra [FMR93] any fullerene must have 12 pentagonal faces
and an arbitrary number of hexagonal faces. Since the addition of an extra hexagon appends
two carbon atoms, all fullerenes have an even number of carbon atoms.
The C60 fullerene was discovered by Kroto, Smalley, Curl, and collaborators in 1985
[KHO85] and was rewarded by the Nobel prize in 1996. Due to the closed cage structure and
high symmetry C60 fullerene is much more stable in comparison to its neighbours [FMa92].
The structure of C60 fullerene is shown in Fig. 2.1. The cage is built out of 12 pentagonal and
20 hexagonal faces representing a truncated icosahedron. A regular truncated icosahedron has
90 edges of equal length and 60 equivalent vertices. Each carbon atom is trigonally bonded
to three other carbon atoms by sp2 bonds. The curvature of the C60 cage leads to some
admixture of sp3 bonding. The three bonds are not equivalent. Two bonds on the pentagonal
edges are electron-poor single bonds. While the bonds joining two hexagons are electron-rich
double bonds. The lengths of single and double bonds are 1.46 Å and 1.40 Å, respectively
2.1. C60 FULLERENE AND ITS PROPERTIES
7
[JMB90]. Since the lengths of single and double bonds are not exactly equal the C60 cage is
not a regular truncated icosahedron. Nevertheless, due to the small difference between the
single and double bounds, the cage can be considered as a regular structure. The diameter of
C60 fullerene based on the geometrical consideration and assuming the carbon atom as points
is 7.09 Å. The diameter of C60 experimentally measured with NMR technique is 7.10 ± 0.07 Å
[JBY92]. Taking into account the size of the π electron cloud associated with the carbon
atoms on the C60 cage, an estimate of the C60 outer diameter is 10.34 Å [DDE96].
A truncated icosahedron belongs to the Ih point symmetry group. It has 10 irreducible
representations and 120 symmetry operations which can be grouped into 10 classes [DDE96].
Many special properties of C60 fullerene are directly related to its highly symmetric structure.
C60 has 174 vibrational degrees of freedom which due to the icosahedral symmetry relate
to 46 distinct mode frequencies only. These modes have the symmetries
2Ag [1] + 3F1g [3] + 4F2g [3] + 6Gg [4] + 8Hg [5]+
(2.1)
+Au [1] + 4F1u [3] + 5F2u [3] + 6Gu [4] + 7Hu [5] ,
where the multiplicities indicate a number of distinct mode frequencies, the symmetry labels
relate to irreducible representations of the icosahedral symmetry group, the subscripts g and
u refer to the symmetry of the eigenvector upon the action of the inversion operator, and the
numbers in brackets indicate the degeneracy of each mode symmetry.
Carbon has two stable isotopes:
12
C with a natural abundance of 98.89% and
abundance of 1.11%. Thus, there is a high probability of finding one or more
13
13
C with an
C atoms in a
C60 molecule. The probability p(r) for r isotopic substitutions in a C60 fullerene is given by
60 r
p(r) =
x (1 − x)60−r ,
(2.2)
r
where x is the
13
C isotope abundance. Assuming x = 0.01108 [DDE96] the probabilities to
find 0, 1, 2, 3, or 4
13
C atoms in the C60 molecule are 0.5125, 0.3445, 0.1139, 0.0247, and
0.0039, respectively. The presence of
13
C isotope lowers the symmetry of the fullerene and
introduces various isotope effects such as, for example, the modification of the rotational levels
of C60 [HRe92].
Methods for studying the vibrations of C60 fullerene include Raman and infrared spectroscopy, inelastic neutron scattering, luminescence, and electron energy loss spectroscopy
[JMD92]. Among these methods Raman and infrared spectroscopy allow one to obtain the
8
CHAPTER 2. OBJECTS OF INVESTIGATION
Table 2.1: Optically active vibrational modes of C60 fullerene obtained with the first order
Raman and infrared spectroscopy (the values are taken from [DZH93, WRE93]).
Mode
Frequency, cm−1
Period, fs
Type of the activity
Ag (1)
497.5
67.0
Raman
Ag (2)
1470.0
22.7
Raman
Hg (1)
273.0
122.1
Raman
Hg (2)
432.5
77.1
Raman
Hg (3)
711.0
46.9
Raman
Hg (4)
775.0
43.0
Raman
Hg (5)
1101.0
30.3
Raman
Hg (6)
1251.0
26.6
Raman
Hg (7)
1426.5
23.4
Raman
Hg (8)
1577.5
21.1
Raman
F1u (1)
526.5
63.3
infrared
F1u (2)
575.8
57.9
infrared
F1u (3)
1182.9
28.2
infrared
F1u (4)
1429.2
23.3
infrared
most precise values for the vibrational frequencies. Only Ag and Hg symmetry modes are
Raman active. The infrared active modes have F1u symmetry. Therefore, C60 in the ground
state exhibits 10 Raman active (2Ag and 8Hg ) and 4 infrared active (4F1u ) modes. Vibrational frequencies and periods of the optically active modes derived from first order Raman
and infrared spectra are summarised in Table 2.1. The remaining 32 modes are not active
in first order Raman and infrared spectra. Nevertheless, many of these silent modes can be
observed in higher order Raman [DZH93] and infrared [WRE93] spectra, by inelastic neutron
scattering [CJB92], or by electron energy loss spectroscopy [GYP92].
Several vibrational modes are especially important in the excitation with an intense fem-
2.1. C60 FULLERENE AND ITS PROPERTIES
9
tosecond laser radiation namely the Ag (1), Ag (2), and Hg (1) modes. However, excitation at
high intensities implies mechanisms quite different from conventional, optical dipole induced infrared absorption or Raman processes: in fact, the excitation of these modes (at least at intensities range used in the present work) is due to non-adiabatic multielectron dynamics (NMED).
The Ag (1) “breathing” mode (497.5 cm−1 ) involves symmetric radial displacement of all 60
carbon atoms with an equal amplitude. The Ag (2) “pentagonal pinch” mode (1470.0 cm−1 )
corresponds to primarily tangential displacements with a contraction of the pentagonal rings
and an expansion of the hexagonal rings. The Hg (1) “prolate-oblate” mode (273.0 cm−1 ) involves an elongation of the cage along one axis. Theoretical simulations [TNE01] indicate that
the interaction of C60 with moderately intense laser radiation centred at 620 nm and a pulse
duration of 10 fs leads to the excitation of both Ag (1) and Ag (2) modes. The Ag (2) mode
dominates at the laser intensity of 0.56 × 1012 W/cm2 , while at higher intensities the Ag (1)
mode prevails over the Ag (2) mode. At the intensity of 1.5 × 1012 W/cm2 the Ag (1) mode is
found to be completely dominant. Recently, numerical results showed that the duration of the
exciting laser pulse may determine which mode is most prominently excited [ZGe04]. It was
found that for laser photon energies from 0.5 to 1.6 eV and pulse durations shorter than 40 fs
the Ag modes dominate, while for longer pulses the Hg (2) mode take over. The Hg (1) mode
dominates over the laser excitation of C60 with intense (∼ 3 × 1014 W/cm2 ) pulses centred
at 1500 nm with a duration of 70 fs [BCR03]. In this case the excitation mechanism is also
specific to strong field interaction but different to the one at shorter wavelengths [NKS07].
According to the theoretical prediction [SNS06, SNS06a] ionisation as well as excitation
of C60 weakens the bonds between the carbon atoms and has an influence on frequencies of
the vibrational modes. For example, at a laser intensity of 5 × 1013 W/cm2 nearly 31 valence
electrons are strongly excited resulting in an impulsive force that expands the C60 fullerene
dramatically up to 9.4 Å which is 130% of its ground state diameter [LSS07]. This leads
to a rise of the vibrational period which depends on the excited electronic configuration and
therefore on the absorbed energy.
In a molecular orbital (MO) picture the 180 σ binding electrons of C60 have energies 3−6 eV
below the Fermi level, while 60 π electrons with higher lying energy levels near the Fermi level
define the electronic structure. Many efforts were made to describe the electronic structure
of the fullerene molecule. Various theoretical methods have been suggested [DDE96] starting
10
CHAPTER 2. OBJECTS OF INVESTIGATION
Figure 2.2: Single electron energy diagram and electronic absorption spectra of C60
fullerene (the data and picture are taken from [CCC96, BAH98], and [HLS05], respectively).
from one electron Hückel calculations to first principle models [Had92]. The high symmetry of
the C60 molecule may even allow one to use simple phenomenological approaches for finding
the electronic structure based on symmetry considerations with satisfactory results. One may,
e.g. start with a spherical shape of the C60 molecule and introduce the icosahedral symmetry
of the C60 fullerene as a perturbation [SDD92].
In the most simple model, often used with success in dynamical calculations, one treats the
π electrons in a spherically symmetric shell potential (jellium model) [PNi93]. Then, according
to the Pauli principle the states with an angular momentum up to l = 4 are populated by 50 π
electrons, while the remaining 10 of π electrons occupy the states with l = 5 which can admit
2.1. C60 FULLERENE AND ITS PROPERTIES
N=5
7.4
7.8
8.2
N=5
hν
0
4
N=5
N=11
11
C60+ single photon
ion yield
N=8
8 12 16 20 24 28 32
total photon energy [eV]
+
C60 N=8
C2+
60
N=11
3+
C60
Figure 2.3: Illustration of the energetics for plasmon enhanced multiphoton process in
the generation of Cq+
60 with different final charge state q (adopted from [HSV92]). Dashed
vertical lines indicate photons energies N hν (hν=1.56 eV) required for the sequential ionisation.
up to 22 electrons in total. These 10 filled substates correspond to the completely filled hu
level leaving the t1u and t1g levels unoccupied. Thus, the highest occupied molecular orbital
(HOMO) is the hu level, while the lowest unoccupied molecular orbital (LUMO) corresponds
to the t1u level. The value of the HOMO-LUMO gap according to the density-functional
theory in the local-density approximation (DFT-LDA) is 1.79 eV [CCY97].
The energetic positions of these levels in the MO picture are indicated in the right part
of Fig. 2.2. In the left part of Fig. 2.2 these MO levels are correlated with the corresponding
ab initio calculated energies of the electronic states of C60 . This is compared with the optical
absorption spectrum which, in principle, gives the most direct information on the electronic
properties of C60 . Since the HOMO and LUMO states posses the same odd parity an electron
dipole transition between these single electron states is forbidden by parity considerations. The
first electric dipole allowed optical transition is between the HOMO and “LUMO+1” states.
The solid black line in the left part of Fig. 2.2 reproduces a measurement in the hexane solution
[BAH98], while the blue dotted line corresponds to gas phase studies [CCC96]. Unfortunately,
both spectra show essentially identical broad bands owing to the many internal degree of
freedom.
Due to the large number of valence electrons in a single C60 molecule multielectron effects
can play an important role. The evidence of multielectron effects was proven in one photon
12
CHAPTER 2. OBJECTS OF INVESTIGATION
ultraviolet excitation of the giant plasmon resonance centred at 20 eV using synchrotron radiation [HSV92]. This plasmon resonance is shown in Fig. 2.3 and is attributed to the collective
motion of the valence electrons of C+
60 . The excitation of multiple electrons may also be
possible with strong fs laser fields [LBR01].
Ionisation and fragmentation of C60 fullerenes investigated intensively by means of photoelectron spectroscopy and mass spectrometry. There is a large variety of experimental methods
based on the excitation of C60 by single photon [YRB92] or multiphoton [OHC88] absorption,
electron impact [FFM98], collisions with neutral particles [Tak92], atomic [WCV94] and molecular [VHC98] ions. Moreover, time resolved techniques provide a powerful tool for studying the
dynamics of relaxation processes within the C60 molecule [BHS04]. Photoinduced ionisation
and fragmentation of C60 fullerenes are discussed in Chapter 8 and Chapter 9 at great length.
2.2
Model Peptides and Methods of Investigations
In recent years, great interest in studying of biological molecules has been demonstrated
[WHC03, JES04, JJC06, VHo07]. Among the different methods for analysis of structure, function, and electronic properties of these molecules spectroscopical methods in the gas phase
can be considered as the most powerful and promising techniques. Gas phase investigations
are not only ideally suited for studying isolated molecules and comparison with theoretical
predictions but also allow one to distinguish intrinsic properties of single molecules from their
properties in condensed phases. The low thermal stability of many biomolecules creates significant problems in their investigation in the gas phase. There are several practical techniques
to transfer biological molecules into the gas phase such as thermal evaporation in an oven
[SRK00], thermospray [RPL85], electrospray [YFe84] laser induced ablation [LLu88], or matrix assisted laser desorption (MALDI) [Lev94]. The molecules thus prepared usually possess
rather high vibrational and rotational energy. That can be a problem for a careful analysis of
their properties. Therefore, an isolation of biomolecules into large helium clusters [LTV99] or,
more generally, a combination with a seeded supersonic expansion of a rare gas [MAL92] is
applied to cool down the molecules. Collisions with rare gas atoms, which are cold due to the
adiabatic expansion, effectively cool the vibrational and rotational degrees of freedom of the
biomolecules, if carefully done without condensation.
2.2. MODEL PEPTIDES AND METHODS OF INVESTIGATIONS
13
Gas phase studies of biological molecules can be performed with mass, isomer, and state
selective spectroscopic methods. Moreover, modern femtosecond lasers provide a tool for
time resolved investigations of the various intermediates during photochemical reactions, protein folding, or other biochemical processes [ANO06, YKE07, YST07]. One may distinguish
two groups of methods: those based on optical spectroscopy and those based on the mass
spectrometry.
Classical method of optical spectroscopy mostly rely on laser induced fluorescence (LIF),
where the fluorescence yield is detected after excitation of (biological) molecules in the near
ultraviolet range [Kam05]. An assignment based on power saturation studies of vibrationally resolved LIF spectra provides information about stable conformations of these molecules present
in the gas pase at low temperature [Zwi01]. The main limitation of this method is imposed by
the weak quantum yield for many biological species. A second method of optical spectroscopy
applied to biomolecules is so-called “infrared cavity ring-down laser spectroscopy” (IR-CRLAS)
[CPP98]. The IR-CRLAS is an absorption spectroscopic technique combining high sensitivity
and high spectral resolution. It is based on measurements of the exponential decay of the
intensity of laser radiation trapped in a high reflective optical cavity with an absorbing sample
inside. The absolute absorbance of the sample is directly derived from the measured time
constant of the exponential decay.
To generate ions of biomolecules in mass spectroscopy one may e.g. use laser ionisation
or fast atom bombardment [BMa87, Bie92]. This allows one not only to determine molecular
weight of biological molecules but it may also help to distinguish between different structural
peculiarities of the molecules. More structural information can be obtained by utilising of a
tandem mass spectrometer, involving two ore more mass spectrometers [RYM95]. The first
spectrometer is used to select the ions of interest, while the second device detects all fragments
produced after additional excitation of the selected precursor ions. This method allows one
to analyse molecules mixtures without the necessity to separate individual components before
the analysis [Pap95].
Recently new combinations of spectroscopical techniques were successfully applied to map
conformational landscapes of small biomolecules and biomolecular clusters [SSP05, CPD06,
POo07]. If the object of investigation contains an aromatic chromophore, the combination
of infrared excitation with resonant two photon ionisation (IR/R2PI) is ideally suited to in-
14
CHAPTER 2. OBJECTS OF INVESTIGATION
Figure 2.4: Model peptides used in present work: (a) Ac–Phe–NHMe; (b) Ac–Ala–NHMe.
vestigate the infrared spectra of different neutral biomolecules in the ground electronic state
[WVN04]. This method is mass and isomer selective due to the R2PI process. Moreover,
the combination of the IR/R2PI method with ab initio, density functional theory (DFT),
or Hartree-Fock (HF) calculations allows one to identify different conformational structures
[BFG06]. The second powerful method is based on the combination of ultraviolet hole burning and R2PI [SMZ07]. This combination reveals the overlapping spectral features which
correspond to different conformers or clusters.
In the present work ionisation and fragmentation of two molecular systems Ac–Phe–NHMe
and Ac–Ala–NHMe after the excitation with intense, shaped laser pulses are studied with time
of flight mass spectrometry. The amino acid complex Ac–Phe–NHMe is obtained from the
amino acid phenylalanine by introducing an acetyl group to protect1 the base NH2 and by using
the amide group instead of the pure acid as shown in Fig. 2.4a. Thus, Ac–Phe–NHMe contains
1
A protecting group is any chemical moiety that replaces a highly reactive functional group in a
molecule to protect this functional group from unwanted chemical interactions.
2.2. MODEL PEPTIDES AND METHODS OF INVESTIGATIONS
15
a –CO–NH–CHR–CO–NH– moiety, the structural key element of all peptides. The second
amino acid complex Ac–Ala–NHMe has alanine instead the chromophore phenylalanine, while
the backbone structure with the protection groups is uncharged (Fig. 2.4b). Both molecules
systems contain CH–CO, CO–NH, and N–Cα bonds as well as side chains and therefore may
be regarded as models of small peptides.
16
CHAPTER 2. OBJECTS OF INVESTIGATION
Chapter 3
Interaction of Strong Laser Fields
with Matter
Ultrashort and intense laser radiation opens the door for a new class of nonlinear effects
including above threshold ionisation (ATI) [Cor89], high harmonic generation (HHG) [Cor93],
atomic stabilisation [PGa90], and molecular dissociation [SIC95]. Understanding ionisation
mechanisms plays an important role for the explanation of these and other strong field effects.
This chapter starts with a discussion of general aspects of strong field interactions. Then,
direct ionisation and non-statistical fragmentation are considered. After that, redistribution of
the absorbed energy in large but finite systems is discussed. Finally, the competing processes
of statistical ionisation and fragmentation are revisited.
3.1
General Aspects
In context of the present work the electrical field strength of laser radiation applied to an
object of interest is considered to be strong enough to induce significant modifications of its
energy landscape. One relevant quantity is the ponderomotive energy. It describes the average
oscillation energy that is acquired by a free electron in the radiation field of the laser pulse.
The ponderomotive energy is given by the following equation
UP =
q2
I
2me ε0 cω 2
,
(3.1)
where q is the electron charge, me is the mass of the electron, ε0 is the dielectric constant in
vacuum, c is the speed of light, ω is the angular frequency of the laser radiation, and I is the
laser field intensity. The ponderomotive energy depends on the square of the wavelength and
17
18
CHAPTER 3. INTERACTION OF STRONG LASER FIELDS WITH MATTER
is linearly dependent on the intensity, this can be numerically expressed as
UP [eV] = 9.34 × 10−20 × (λ[nm])2 × I W/cm2
.
(3.2)
UP can be used as a criterion to determine if a field is strong by comparison of the characteristic
ionisation energy EI of the system in question with UP . For the ionisation process the strong
field regime starts if
UP > EI
.
(3.3)
For example, the ionisation of C60 with EI = 7.58 eV [VSK92] by the laser radiation centred
at 800 nm would be strong in this sense if I > 1.3 × 1014 W/cm2 .
The interaction of strong laser radiation with complex many-body systems can lead to
substantial energy absorption. This process drives the system out of its equilibrium. After
the energy is deposited into the system, relaxation occurs by the release of the excess energy.
There are three relaxation mechanisms for a free molecule or cluster: ionisation, fragmentation,
and photon emission in a form of fluorescence (transitions on the ns time scale from an excited
singlet state to the ground state) or phosphorescence (transitions on the slower time scales from
an excited metastable, usually triplet, state to the ground state). Ionisation and fragmentation
are two most important mechanisms of excess energy release among them.
For ionisation the absorbed energy must be larger than the ionisation potential of the
system. Direct and delayed ionisation can be distinguished. Direct ionisation is the electron
emission which happens on the time scale of the laser pulse acting. Delayed ionisation is
observed after the laser pulse passed. The latter has a statistical nature and occurs after
equilibration among electronic degrees of freedom.
In principle, fragmentation happens when the internal vibrational energy is greater than
the dissociation energy of the system. Unimolecular fragmentation occurs when the system
separates into two or more parts. Usually, the initial system is called the “parent” and the
fragments are called “daughters”. Similar to ionisation, fragmentation may have non-statistical
and statistical origin. Non-statistical fragmentation occurs on a time scale before complete
energy equilibration among vibrational degrees of freedom. On the other hand, excitation of
the molecule to a repulsive potential energy surface may lead to non-statistical fragmentation.
Fragmentation due to multiple excitation of one particular vibrational mode until the amplitude
of the vibration leads to dissociation is a non-statistical process as well. Experimentally, it can
3.2. MULTIPHOTON IONISATION AND FRAGMENTATION
19
be achieved either by laser radiation tuned to the vibrational frequency or in the temporal
domain by exciting the system with a train of laser pulses.
3.2
Multiphoton Ionisation and Fragmentation
At low intensities, photoionisation can only occur if the energy of the absorbed photon h̄ω
is higher than the binding energy (or ionisation potential) EI of an electron. In multiphoton
ionisation (MPI) processes (see e.g. [PKK97, HSc08]) N photons of energy h̄ω are absorbed
and the energy balance is given by
EK + Eint = N h̄ω − EI
,
(3.4)
where EK is the kinetic energy of the ejected electron and Eint is the internal rotational and
vibrational energy of the system directly after the ionisation process. Here it is assumed that
N is the minimum number of photons needed for an MPI process
N h̄ω ≥ EI ≥ (N − 1)h̄ω
.
(3.5)
If the electric field strength becomes comparable with the atomic Coulomb potential, the
electron can tunnel through the potential barrier and leave the atom which is referred to as
the tunnelling ionisation [ASM89].
The dependence of the ionisation probability on the ionisation potential and properties of
the laser radiation (intensity I and frequency ω) was theoretically investigated by L. V. Keldysh
[Kel65]. It was found that the MPI and the tunnelling ionisation are two limiting regimes of
nonlinear ionisation. The Keldysh parameter γ was introduced to define the transition between
these two different ionisation regimes
r
γ=
EI
=
2UP
s
ε0 me cEI ω 2
q2I
.
(3.6)
Assuming h̄ω < EI , one can distinguish the MPI case if γ > 1 (ionisation with high frequency
and rather low intensity), while tunnelling ionisation occurs if γ 1 (ionisation with low
frequency and high intensity). The evolution from MPI to tunnelling ionisation was observed
experimentally [MBT93]. Often, these two limiting regimes are referred to as the the perturbative nonlinear regime and the strong-field regime, respectively. Alternatively, the Keldysh
20
CHAPTER 3. INTERACTION OF STRONG LASER FIELDS WITH MATTER
20
800 nm 400 nm
1x10
relativistic
18
2
intensity [W/cm ]
1x10
16
1x10
14
1x10
stabilisation
field
ionisation
tunnel
ionisation
MPI non-perturbative
12
1x10
MPI perturbative
10
1x10
1
10
photon energy [eV]
100
Figure 3.1: Different photoionisation regimes as a function of the laser intensity and
photon energy. The vertical dotted red and blue lines indicate photon energies related to
the laser wavelength of 800 nm and 400 nm, respectively. The dashed gray line shows case
of γ = 1 for C60 ionisation which separates the multiphoton regime (γ > 1) from the
tunnelling regime (γ 1).
parameter can be interpreted as the ratio between the time required for tunnelling and the
oscillation period of the laser radiation.
Different photoionisation regimes are schematically depicted in Fig. 3.1. For the rather low
laser intensities ionisation proceeds either by a multiphoton perturbative process for the lowest
intensities or a multiphoton non-perturbative process [MMa91]. If the laser frequency is quite
low but the laser intensity is moderately strong, ionisation occurs via tunnelling [ASM89].
Field ionisation happens at even higher laser intensity when the potential barrier is completely
suppressed by the laser field. Under certain conditions [PTV03] the stabilisation regime of
photoionisation can be achieved [Rei01]. Finally, extremely hight laser intensities open an
entirely new domain in atomic, plasma and nuclear physics, where relativistic effects play an
important role [JKy03, KFS03, SHH06]. For example, the laser electric field corresponding to
3.2. MULTIPHOTON IONISATION AND FRAGMENTATION
21
Figure 3.2: Perturbative nonlinear regime of ionisation (γ > 1): (a) multiphoton ionisation; (b) above threshold ionisation.
an intensity of 1019 W/cm2 is nearly 1011 V/cm, a value dramatically exceeding the intraatomic
field. The quiver energy of an electron reaches 1 MeV at 1019 W/cm2 and λ = 1053 nm. This
energy becomes comparable with the electron rest energy of ≈ 0.5 MeV. Hence, the electron
motion becomes relativistic.
The vertical dotted red and blue lines in Fig. 3.1 indicate photon energies of the laser
radiation used in the present work with wavelength of 800 nm and 400 nm, respectively. The
maximal achievable intensities are up to 4 × 1014 W/cm2 and 6 × 1013 W/cm2 at 800 nm
and 400 nm, respectively. Therefore, with such laser parameters both multiphoton and tunnel
regimes of ionisations are reachable. The dashed gray line shows case of γ = 1 for C60
ionisation which separates the multiphoton regime (γ > 1) from the tunnelling regime (γ 1).
A schematic diagram of the MPI process is depicted in Fig. 3.2a. The ionisation rate of
N -photon ionisation ΓN is given by
ΓN = σN I N
,
(3.7)
where σN is the generalised cross-section for N -photon absorption and I the laser intensity.
This expression leads to a linear dependence on a double logarithmic scale
log ΓN = N log σN I
,
(3.8)
where the slope N indicates the number of photons required for ionisation according to
Eq. (3.5). The ionisation rate in an MPI process strongly depends on the frequency of the
22
CHAPTER 3. INTERACTION OF STRONG LASER FIELDS WITH MATTER
laser radiation due to possible resonances between the energy of several absorbed photons and
the energy of some intermediate stationary state. In that case one speaks about resonance
enhanced multiphoton ionisation (REMPI). If the field strength is strong enough, REMPI can
be induced by the AC-Stark shift1 of electronic states into resonance. Eq. (3.7) breaks down at
some critical light intensity Is , above which a change in intensity dependence is observed due a
nearly 100% ionisation probability [LHM85]. This intensity Is is called “saturation intensity”.
At rather high intensities (but generally below Is ) an electron can absorb more photons
than the minimum number N required for ionisation (see Eq. (3.5)), say N + S, as shown in
Fig. 3.2b. In this case one speaks about “above threshold ionisation” (ATI) [AFM79]. In the
range of moderate laser intensities Eq. (3.4) and Eq. (3.7) can be generalised in the ATI case
to
EK + Eint = (N + S)h̄ω − EI
(3.9)
and
ΓN +S ∝ I N +S
,
(3.10)
respectively. An ATI photoelectron spectrum consists of several peaks separated by h̄ω. With
increasing laser intensity the ATI peaks corresponding to a higher number S of photons become
more probable and do not necessarily follow the power law Eq. (3.10) [YPA86]. At the same
time, the amplitude of low energy ATI peaks is reduced [KKM83].
At moderate laser intensities MPI can be well described by using of the lowest order perturbation theory (LOPT) with respect to the electron filed interaction [FPA82, PFA84]. This
theory predicts the power law for the ionisation rate described by Eq. (3.7). Near resonant
or resonant cases of MPI are the subject of an additional complexity for the theoretical description. However, the methods, which can handle these difficulties, have been developed
[Fai87]. The perturbation theory breaks down for higher laser intensities since the involved
atomic states undergo the AC-Stark shift and can not longer be considered as unperturbed
[FBM87]. Qualitative, the non-perturbative nature of ATI is understood by using a simple
approach [MAc89] based on the Keldysh model [Kel65].
In contrast, for γ 1 (the laser radiation with high intensity and/or low frequency) the
strong laser field distorts the potential energy surface of the atom such that a potential barrier
1
The dynamical shifting of energy levels in atoms and molecules due to an applied electromagnetic
field.
3.2. MULTIPHOTON IONISATION AND FRAGMENTATION
23
Figure 3.3: Strong field regime of ionisation (γ 1): (a) tunnelling ionisation; (b) above
barrier ionisation.
is formed. This is illustrated in Fig. 3.3a. The barrier divides a space into two regions where
the electron is bound or free. A local maximum of the barrier is called the “saddle point”. If
the laser frequency is low enough, the electron can leave the atom by tunnelling through the
potential barrier. This process is known as “tunnelling ionisation” [GHa98]. The rate of the
tunnelling ionisation ΓT has a simple exponential form
2(2EI )3/2
√
.
ΓT ∝ exp −
3 I
(3.11)
At even higher intensities, the laser radiation suppresses the potential barrier so far that the
electron is able to escape freely from the atom as depicted in Fig. 3.3b. It happens when the
saddle point reaches the electron binding energy. This is called “over the barrier ionisation”
(OTBI) or the “barrier suppression ionisation” (BSI) [BRK93]. The threshold intensity of
OTBI for hydrogen-like atoms is given by the following expression
IOT BI = 4 × 109
EI4
Z2
,
(3.12)
where IOT BI is the threshold intensity in W/cm2 , EI is the binding energy of the electron in
eV, and Z is the charge state of the relevant atom or ion. Since the orbital angular momentum
of the initially bound electron is not taken into account in Eq. (3.12), it does not give correct
results for complex atoms.
Assuming that the laser frequency is low it is possible to consider tunnelling within the
framework of the quasi-static approximation [SPD90]. It is based on the assumption that the
24
CHAPTER 3. INTERACTION OF STRONG LASER FIELDS WITH MATTER
tunnelling occurs in a fraction of an optical cycle at which the change of the electric field is
negligible. This framework allows one to calculate the ionisation rates in an altering laser field
using the static field ionisation rates. There are several theoretical approaches for an analytic
calculation of static field ionisation rates. The oldest among them is so-called the “Keldysh
theory” [Kel65, Fai73, Rei80]. The effect of the atomic Coulomb potential is neglected in the
Keldysh theory. Such assumption is also known as the “strong filed approximation” (SFA).
As a result, it underestimates the ionisation rates. In 1966 Perelomov, Popov, and Terent’ev
developed a new method to calculate the ionisation probability (PPT theory) [PPT65]. According to the PPT theory the rate of the tunnelling ionisation with the linear polarised light
for the hydrogen-like atom is
s
√
√
3n3 I (2l + 1)(l + |m|)!24n−2|m|−2 n−6n+3|m| exp [−2/(3n3 I)]
√
ΓT =
×
π
(n + l)!(n − l − 1)!(|m|)!(l − |m|)!
( I)2n−|m|−1
,
(3.13)
where n is the principal quantum number, l is the orbital angular quantum number, and m is
the magnetic quantum number. Eq. (3.13) is valid only if the strength of the external electric
field is weak compare to the atomic Coulomb potential. Twenty years later Ammosov, Delone,
and Krainov developed the so-called “ADK theory” [ADK86]. The ADK theory is an extension
of the PPT theory to the case of complex atoms and ions. The ADK theory gives the following
ionisation rate for linearly polarised light
s
√ √
2Z 3
3(n∗ )3 I ID2
lin
√
exp −
ΓADK =
,
πZ 3
8πZ
3(n∗ )3 I
∗
where n ≡
√Z
2EI
is the effective principal quantum number and D ≡
(3.14)
3
√4eZ
I(n∗ )4
n∗
. The
ionisation rates for the circular polarised light differ from the rates for the linear polarised light
and have a more simple form
√
Γcirc
ADK
ID2
2Z 3
√
exp −
=
.
8πZ
3(n∗ )3 I
(3.15)
Eq. (3.14) and Eq. (3.15) have a restricted region of validity. For the laser intensities which
are higher than IOT BI the ADK theory increasingly overestimates the ionisation rate. This is
due to the AC-Stark shift of the ground state. Indeed, the dynamic Stark effect caused by
the laser field which may shift atomic and molecular energy levels in a significant manner. For
molecules sufficiently strong fields can thus cause potential energy crossings and consequently
lead to laser field induced non-adiabatic transitions, a subject of very active research in many
3.2. MULTIPHOTON IONISATION AND FRAGMENTATION
25
experimental and theoretical studies [LBR01, KSF03, KST04, MRS04]. Such behaviour will
be most efficient if the laser frequency is close to a resonance of the system studied. Manifestations of the Autler-Townes splitting [ATo55] have long been known in studies of ATI
processes [Lag93] and definitively need to be considered also for molecules in intense laser
fields. According to estimations such splitting can be in the eV region at rather moderate
intensities of 1011 W/cm2 , providing the laser frequency is in resonance with a transition of
significant oscillator strength [HLS05].
All above described cases of nonlinear ionisation have been considered from the point of
view of the single active electron (SAE) approximation [KSK92, XTL92, SJW95]. The SAE
approximation assumes that only the single lowest bound electron interacts with the laser field
and it is bound by an effective potential which reproduces the ground state and singly excited
states. The dynamics of the remaining bound electrons is neglected in this approach. This
approximation supposes that multiple ionisation happens only as a stepwise process. In reality,
most of atoms and molecules have more than one electron. Nevertheless, the SAE approximation has been successfully used to describe experimental data in many cases. However,
multielectron effects are essential for the explanation of many physical phenomena. They
are especially important for the description of the photoinduced processes in large systems.
Both theoretical [FKM03] and experimental [LBI02] works demonstrated that multiple active
electrons (MAE) are excited in complex systems.
Several mechanisms of MAE response have been suggested. According to the collective tunnelling model there is an opportunity of the simultaneous tunnelling of two electrons
[EDM00]. But the tunnelling rates predicted by this model are too low for the explanation of
experimental data.
It was proposed that non-sequential ionisation may arise from the shake-off mechanism
[FBC92]. The escape of the first electron changes the effective potential of the atomic core.
The second electron can not adapt to this rapid change of the effective potential. It can
be shaken and displaced from the atom as well. Unfortunately, this model does not give any
dependence of ion yields on the laser polarisation and it fails to explain experimentally observed
the spatial distribution of ejected electrons [FBC94].
Another classical model based on electron electron rescattering was suggested [Cor93].
According to this model the electron after tunnelling through the potential barrier acts as a
26
CHAPTER 3. INTERACTION OF STRONG LASER FIELDS WITH MATTER
classical charged particle in the laser field. This electron is accelerated away from the ionic core
and after about one half (or multiple) of an optical cycle is driven back when the laser filed
changes its sign. The maximum value of the electron kinetic energy which can be acquired
from the laser field is 3.17UP 1 [MSt99]. The electron may be scattered inelastically by the
ion core, transferring part of its energy to another electron. Thus, a second electron may be
ejected from the ionic ground state to the continuum. This process is called “non-sequential
double ionisation” (NSDI) [WSD94, WGW00, WWS00]. Alternatively, the electron can be
elastically scattered by the ion core. In this case, it can acquire drift energies much higher
than otherwise2 . Such effect is referred to as high order above threshold ionisation (HATI)
[MPB03]. Finally, the electron can recombine with the ion emitting its energy in form of
a high energy photon. This process is known as the “high harmonic generation” (HHG)
[ZPM96, SKu97]. Presently, the rescattering model is commonly accepted as the mechanism
for the non-sequential ionisation [MFS00, FMF01, WHC01].
Multiphoton excitation may also lead to a decay process (fragmentation) of an excited
parent particle (molecule or cluster) to products which can be formulated as
N h̄ω
parent −→ daughter + ejected particle .
(3.16)
Usually, the daughter particle corresponds to the more heavy fragment, while the ejected
particle is the rest. The energy balance of a decay process is
Einit + N h̄ω = Eb + Ed + ε∗ + ε ,
(3.17)
where Einit is the initial internal energy of the parent particle before excitation, Eb is the
binding energy of the ejected particle, Ed is the internal energy of the daughter particle after
the decay, ε∗ is the internal energy of the ejected particle, and ε is the kinetic energy released in
the process. For fragmentation of a molecule (cluster) the excess energy Eint = Einit + N h̄ω
(see Eq. (3.4)) absorbed by the system must exceed the binding energy of the ejected particle
(Eint > Eb ). The released kinetic energy is distributed over the decay products in accordance
with conservation of momentum and energy. The kinetic energy release distribution provides
information concerning the system structure dynamics, and reaction energetics [Eng87].
1
The maximum energy which the electron tunnelled with zero velocity can return to the core according
to the classical model.
2
For the electron backscattered by 180◦ the maximum acquire energy is 10.007UP .
3.3. ENERGY REDISTRIBUTION
27
A multiphoton absorption in large, finite systems besides the direct processes also may lead
to delayed ionisation [OSi89, KCa98, LCP03] and fragmentation [SLe80, VCN96, GKD06].
Both processes demonstrate a statistical nature and occur on the long time scale after multiphoton excitation.
In addition to the excess energy, a so-called “kinetic shift energy” [Lif00] has to be deposited
to observe C60 fragmentation on the time scale of nanoseconds to microseconds. Assuming
that the total energy of the system is statistically distributed over the vibrational degrees of
freedom, the kinetic shift energy is required to allow for channelling the necessary dissociation
energy with sufficient probability into the coordinates involved in the bond breaking.
There are several theoretical approaches to describe statistical ionisation and fragmentation. The theory of the detailed balance [ABH02] takes its origin from the Weisskopf formalism
[Wei37]. This theory was successfully applied to describe the delayed ionisation of highly excited sodium clusters [SKI01], fullerene anions [AHN02], and neutral C60 [HHC03, LBo04].
The rate of a unimolecular decay can also be calculated using the Rice-Ramsperger-Kassel
(RRK) theory [RRa27, RRa28, Kas28a, Kas28b]. This theory assumes that the rate is proportional to the amount of the internal energy which is divided over all vibrational degrees
of freedom. The application of the RRK theory is known for wide range of systems, namely,
for polyatomic molecules [RKB81, SKS95], C60 fullerene [WLy92, EWC96], rare gas clusters
[HIN94], metallic clusters, their cations, and anions [CPL97, KSW99, SSE00].
The accuracy of all statistical models is limited because the distribution of the internal
energy as well as most of the parameters values used in the statistical models usually are
exactly not known. Moreover, the observations in many cases of discrepancies between the
statistical models and the experimental data [CAR93, YHW93] clearly indicate the evidence
for the breakdown of the statistical equilibrium hypothesis.
3.3
Energy Redistribution
The flow of absorbed energy in bulk materials is relatively well understood for solid state
systems [POg97, EPC00]. There is a number of relaxation processes such as electron-electron,
electron-phonon, electron-plasmon, electron-photon, electron-impurity interactions and diffusion and ballistic transport into the bulk. Generally, electron-electron and electron-phonon
28
CHAPTER 3. INTERACTION OF STRONG LASER FIELDS WITH MATTER
scattering are primary relaxation mechanisms [ONP97], therefore only these two relaxation
channels are considered below. The relaxation processes in large, finite systems have been investigated in detail for a few model systems [LBW99, HMo00]. It was found that the relaxation
processes in finite systems differ significantly from those in bulk materials.
The basic scheme of energy relaxation is depicted in Fig. 3.4. The laser energy is predominantly absorbed into the electronic system of a species which leads to a certain internal
energy Eint . The corresponding electron distribution has non-thermal character. The first
step of energy relaxation corresponds to the thermalisation of electrons by electron-electron
coupling. One may distinguish two different approaches to describe this mechanism depending
on the excitation density [LLB04]. For low excitation densities (< 10−4 e− /atom), when the
excited electrons interact predominantly with the equilibrium part of the electron distribution,
scattering rates can be estimated using the Fermi-liquid theory [EPC00]. For high excitation
densities (> 10−3 e− /atom), when rapid thermalisation via multiple collisions among the excited non-equilibrium electrons may establish a Fermi-Dirac distribution and the concept of
separate electron and phonon temperatures is valid, the well known “two temperature model”
can be applied [Gad00]. The second important relaxation mechanism is energy transfer of the
hot electrons to the lattice (in bulk materials) or coupling to the vibrational degree of freedom
(in finite systems). This process is called “electron-phonon coupling” and it is characterised by
a temperature exchange between electrons and phonons. Also surface effects have a significant
influence on the relaxation processes in large, finite systems. On the other hand, the relaxation
mechanisms described above can not be directly applied to very small clusters. In this case
the discrete structure of electronic states is considered. C60 is a large but still finite molecular
system, therefore the above mentioned approaches in the description of photophysical and
photochemical processes have to be taken with a grain of salt.
Time-resolved photoemission studies on bulk material show that the time scale for the
electron-electron coupling τel varies from 5 to ∼ 500 fs [POg97]. It is also sensitive to the
intensity and the frequency of the laser radiation used for investigations. The characteristic
times for the electron-phonon coupling τph lie on the time scale between 1 and 5 ps [KSR98].
In molecules, such as C60 , these processes occur on much faster time scales (τel < 70 fs, τph =
200 − 300 fs). However, both relaxation processes may be considered to occur sequentially.
The standard theoretical approach for the electron-electron coupling is the Landau theory
3.3. ENERGY REDISTRIBUTION
29
non-thermal
electrons
Eint
thermalised
electrons
Tel
Ĥ (Eint, Tph)
H (Tel, Tph)
phonons
Tph
Figure 3.4: Illustration of energy relaxation processes in large but finite systems.
of the Fermi liquid (LTFL) [PCa73]. This theory predicts characteristic time constants of
electron-electron scattering τel near the Fermi level in a free-electron gas based on its electron
density. τel is determined by the available phase space between the electron energy E and the
Fermi level EF and it is given by [SKB99]
τel = τ0
EF
E − EF
2
(3.18)
with
τ0 =
128
√
π 2 3ωp
,
(3.19)
where ωp is the plasmon frequency. At energies higher than the plasmon energy h̄ωp , electronplasmon interaction becomes important [XCM96]. Eq. (3.18) gives only the lower limit of the
electron-electron scattering time constant in real materials because the approximations used
to obtain this expression tend to overestimate the electron-electron interaction even in a free
electron metal, and since the plasma frequency in a real metal is screened by both virtual
valance band excitations and the ionic nuclear cores [POg97]. Moreover, Eq. (3.18) does
not include any band structure effects. For example, theoretical calculations of the electron
dynamics in metals demonstrate that band structure effects can lead to a substantial deviation
30
CHAPTER 3. INTERACTION OF STRONG LASER FIELDS WITH MATTER
of electron-electron scattering time constants from the prediction of the Landau theory of the
Fermi liquid [SKB99].
Experimental observations of the electron-electron interaction in non-metals and finite systems demonstrated the mismatch with the Landau theory. In this case the following empirical
expression for τel was suggested [XCM96, HMo00]
τel = (E − EF )−n
,
(3.20)
where the constant n is 1 as it was found for graphite [XCM96] or 1.5 for carbon nanotubes
[HMo00]. Such deviation arises from the reduction of the dimensionality and increasing influence of the electron-plasmon interactions.
The electron-phonon coupling can be described within the framework of the two temperature model [HWG00]. In this model, the electron and phonon baths are coupled by the
electron-phonon coupling constant and the energy flow between these baths is given by two
differential equations. The classical two temperature model is valid only after full electron
thermalisation. For the correct description of the dynamics on the time scale before electron
thermalisation completed the two temperature model can be extended by taking into account reduced electron-phonon coupling of non-thermalised electrons (thin arrow in Fig. 3.4)
[LLB04].
C60 fullerene is a large but still finite molecular system with discrete energy levels and a
well defined number of 274 modes for nuclear motion. Therefore, parameters such as “electron temperature”, “phonon temperature”, or “heat capacity” and the two temperature model
wholly have a restricted application for the description of photophysical and photochemical
processes in C60 . Moreover, one would rather tend not to attribute experimentally observed
relaxation times in C60 to electron-electron coupling exclusively. Rather, highly excited electrons formed during an intense laser pulse will exchange energy by the combined action of
electron-electron scattering and electron coupling with the various nuclear degrees of freedom
of the neutral C60 .
Chapter 4
Experimental Apparatus
All experimental results presented in this thesis are obtained with ion and electron time of
flight (TOF) spectroscopy. These techniques combine a rather simple mechanical setup with
low cost and simplicity of operation. It is able to provide complete mass or photoelectron
spectra for each ionisation event without the necessity for scanning through voltage settings
to select certain intervals of masses or kinetic energies. At the same time, fast electronic data
acquisition schemes allow one to increase the signal to noise ratio in a short analysis time by
instantaneous averaging of individual spectra obtained in every event of ionisation, even in
the case of high repetition rates, typically 1 kHz. Another advantage of TOF spectrometry is
wide detection range of masses or kinetic energies. This makes it well suited for gas phase
investigations of large molecules and clusters [PMi90].
This chapter begins with a description of the vacuum chamber and the molecular beamsource. After that, the general principles of TOF spectroscopy as well as the spectrometer
and the detection schemes used during this work are presented.
4.1
Vacuum Chamber
The vacuum chamber is shown in Fig. 4.1. It consists of three separately pumped parts:
a laser ionisation chamber, an electron TOF chamber, and an ion TOF chamber. The laser
ionisation chamber contains the molecular beam source and a cold trap. A laser beam comes
through the entrance window (diameter D = 10 mm and thickness h = 250 µm) and crosses
the molecular beam and the spectrometer axis perpendicularly. The vacuum in this ionisation
chamber is maintained by a large turbo-molecular pump (TMU 1001P, Pfeiffer Vacuum Tech31
32
CHAPTER 4. EXPERIMENTAL APPARATUS
Figure 4.1: Photograph of the vacuum apparatus consisting of ion TOF chamber, laser
ionisation source chamber, and electron TOF chamber (from left to right).
nology). The ion TOF chamber is evacuated by a smaller turbo-molecular pump (TMU 260,
Pfeiffer Vacuum Technology). The third and smallest turbo-molecular pump (TPH 180H,
Pfeiffer Vacuum Technology) is used to keep vacuum inside the electron TOF chamber.
A pair of Bayard-Alpert type ionisation tubulated gauges (model 274003, Granville-Phillips)
controls the pressure inside the vacuum chamber. By using three turbomolecular pumps the
background pressure is maintained between 1.3 × 10−7 mbar and 3.3 × 10−7 mbar. One needs
around 24 hours to reach such pressure after the chamber has been vented. Longer times are
required for the reduction of the water residual gas in the chamber. This can be very important
for experiments using high laser intensities. The cold trap filled with liquid nitrogen allows
one to further decrease the pressure approximately by one order of magnitude. The presence
of the low density effusive molecular beam in the ionisation chamber does not influence the
background pressure. For purposes of calibration and alignment a constant flow of Xenon is
supplied in most experiments raising the pressure in the ionisation chamber to a range between
0.7 × 10−6 mbar and 1.3 × 10−6 mbar (depending on the used laser intensity).
4.2
Molecular Beam Source
For gas-phase investigations of molecules and clusters one needs to have suitable beam
4.2. MOLECULAR BEAM SOURCE
33
sources. Different methods of beam generation can be applied. Most important and commonly
used among them are: effusive beam source [Ste86, BGM93] and adiabatic beam expansion
in a jet nozzle [KGr51, AFe65]. The first method involves a gas effusion from an oven or
a source cavity through an aperture into a high vacuum chamber. The gas in the oven or
the source cavity is at a density sufficiently low so that molecules undergo no collisions when
passing through the aperture. Any effusive beam has to satisfy the following conditions: i)
collisions between particle inside the beam are negligible and ii) mean free path Λ is much
larger than diameter of the source aperture d.
The main idea of the second method is a supersonic jet expansion from a high pressure
gas source into a chamber maintained at high vacuum by continuous pumping. In this case
the thermal energy of the gas transforms directly to kinetic energy of molecular motion predominantly ordered in the direction of a supersonic flow. As a result, a molecular beam with
narrow velocity and angular distributions as well as a high particle density is formed.
In many cases, mass spectroscopic studies with intense laser fields do not require large
particle densities in a molecular beam and effusive beam sources are more preferable than jet
nozzles. In the present studies an effusive molecular beam is produced by the evaporation of
sample powders in a resistively heated oven. A microprocessor controller (JUMO iTRON 32,
JUMO GmbH) is used to stabilise the oven temperature with an accuracy better than 0.2%.
In the case of experiments with C60 fullerenes the temperature range is 500 ± 10 ◦ C, except
for some experiments with photoelectrons done at 640 ◦ C. Experiments with model peptides
are performed at oven temperatures between 70 ◦ C and 130 ◦ C. The temperature values used
will be specified when the corresponding data are presented.
A photograph of the oven is shown in Fig. 4.2. The molecular beam is formed by expansion
of the heated gas into free space through the source aperture with diameter dA = 1 mm.
A diaphragm with diameter dD = 1 mm is placed to collimate the molecular beam in the
interaction region. The distances between the source aperture and the diaphragm and between
the source aperture and the interaction region are LD = 1 cm and LI = 5 cm, respectively. A
pure geometrical approximation of the molecular beam diameter Dm at the interaction region
assuming the absence of the collisions between particles and the diaphragms can be done with
the following formula
Dm =
LI
dD
LD
.
(4.1)
34
CHAPTER 4. EXPERIMENTAL APPARATUS
Figure 4.2: Photograph of the oven.
For the data given above the molecular beam diameter is Dm = 5 mm.
To check the validity of an effusive beam assumption (Λ dA ) one has to calculate the
C60 mean free path Λ which depends on the collisional (geometrical) cross-section σg and the
particle density n
1
Λ= √
2σg n
.
(4.2)
Assuming ideal gas conditions and using n = 4.1 × 1019 m−3 at the temperature of 500 ◦ C
[JKa00] and σg = 336 Å2 [DDE96] the mean free path for C60 is Λ = 5.2 mm. It is larger
(but not considerably larger) than the source aperture diameter. Therefore, under current
experimental conditions the assumption of an effusive beam flow is still possible.
The flow dN/dt of the particles emerging from the oven into a solid angle dΩ with velocities
lying within dv is given according to [Est46] by
dN
P
dΩdv =
f (v)v cos(θ)d2A dΩdv
dt
16kT
,
(4.3)
where v is the velocity of an emerging particle, f (v) is the velocity distribution in the source,
P is the gas pressure in the source, T is the gas temperature, θ is the angle with respect to
the source aperture normal, and k is the Boltzmann constant. Integration over all possible
velocities gives the total flow I(θ)
I(θ) =
P hvid2A
cos(θ) ,
16kT
(4.4)
4.3. TIME OF FLIGHT MASS SPECTROMETRY
35
where hvi is the average velocity in the source gas. The angular distribution of the emerged
particles follows a cosine law with a maximum in forward direction (θ = 0). The particle
density in the interaction region nL can be calculated using the formula [Ram90]
1
nL = √
12 π
dA
LI
2
n ,
(4.5)
where n is the particle density in the source.
The particles velocities v in a molecular beam are described by the Maxwell-Boltzmann
distribution
f (v) = 4π
r
m 3 2 − mv2
v e 2kT
2πkT
,
(4.6)
where m is the mass of a particle. The most probable velocity vp , the average velocity hvi,
and the root mean square velocity vrms of the Maxwell-Boltzmann distribution are
r
2kT
vp =
,
m
r
8kT
,
hvi =
πm
(4.7)
(4.8)
and
r
vrms =
3kT
m
.
(4.9)
According to Eq. (4.7)-(4.9) vp , hvi, and vrms for C60 at the temperature of 500 ◦ C are
133.6 m/s, 150.8 m/s, and 163.7 m/s, respectively. On a typical time scale of the pumpprobe experiments (500 fs) such velocities correspond to a C60 drift of about 0.75 Å only.
Thus, the C60 can be considered space fixed during the interaction time with the laser.
4.3
Time of Flight Mass Spectrometry
Time of flight (TOF) mass spectrometry is based on the measurement of the time that
it takes for an ion to reach a detector after an ionisation event while travelling over a known
distance [Gui95, Mam01, GVa03]. In the most simple arrangement the ions created are accelerated by an electric filed over a finite distance (acceleration region) to a well defined kinetic
energies. They then fly through a field free (drift) region and, finally, are registered by a
detector. The arrival time t at the detector for a fixed kinetic energy of all ions starting from
36
CHAPTER 4. EXPERIMENTAL APPARATUS
MCP-detector
G1 G2 G3
e-
G5
las
ion+
er
G4
R1 R2
double µ-metal shield
oven
R3
MCP-detector
Figure 4.3: Scheme of reflectron TOF mass spectrometer (right part) and photoelectron
spectrometer (left part). For the distances and applied voltages see Table 4.1 and Table 4.2.
the same point at the same time (so-called “time of flight”) depends on their mass m to
charge q ratio
r
t∝
m
q
.
(4.10)
Thus, by measuring this time t, the mass to charge ratio m/q of the ion can be determined.
The resolution R of a mass spectrometer is defined as the ability to distinguish two ions
of different masses
R=
m
∆m
,
(4.11)
where ∆m is the mass difference of the two ions. Since m ∝ t2 and ∆m/∆t ∝ 2t, the
resolution R can by written as
R=
t
2∆t
,
(4.12)
where ∆t is usually the full width at half maximum (FWHM) of a mass peak. It is clear that
the mass resolution R can be improved by increasing t (low acceleration potential or/and long
drift length) or narrowing the peak width ∆t.
A schematic picture of the reflectron TOF mass spectrometer employed in the present work
is shown in the right part of Fig. 4.3. Distances between grids and potentials applied to them
are illustrated in Table 4.1. Ions are originated inside the dual stage space focusing system of
three electrodes (G1 , G2 , and G3 ) forming two electric fields. The magnitudes of both fields are
4.3. TIME OF FLIGHT MASS SPECTROMETRY
37
Table 4.1: Voltage settings used in this work and distances between grids of the reflectron
TOF mass spectrometer.
Region
Voltage, V
Distance, mm
G1 - G2
2470 - 2076
20
G2 - G3
2076 - GND
60
G3 - R1
GND - GND
1530
R1 - R 2
GND - 1670
26
R2 - R 3
1670 - 2800
234
R1 - MCP
GND - GND
1086
chosen to meet well known Wiley-McLaren spatial focusing conditions [WMc55, SRe01]. The
extraction electric field with strength of 1.97 × 104 V/m pushes the ions out of the ionisation
region into the acceleration region. Here the uniform electric field with strength 3.46×104 V/m
accelerates the ions into the first field free drift region. The length of this region is 1.53 m.
The drift through this field free region allows the ions to be separated depending on their mass
to charge ratios. Then the ions reach the reflectron [MKS73]. This is a device consisting of
three electrodes (R1 , R2 , and R3 ) which produce two electric fields: decelerating (for incoming
ions)/accelerating (for outgoing ions) and reflecting. In order to improve the homogeneity of
the electric field in the reflectron a number of equally spaced ring electrodes is placed between
the end electrodes, connected by a chain of resistors. The first electrode is grounded, while
the second one is at a voltage which creates the decelerating or retarding electric field slowing
the ions to approximately 75 % of their velocity. The electric field between the second and the
third electrodes reflects the electrons back at an angle of 2◦ with respect to the TOF axis.
Finally, the same electric field that was used for the deceleration now accelerates the ions back
out of the reflectron in the direction of the detector so that they leave with the same velocity
they have entered it. The ions with higher kinetic energy penetrate the reflecting field further
than the ions with lower kinetic energy, hence, the faster ions have to travel larger distances
inside the reflectron and spend more time there. By choosing the appropriate geometry and
38
CHAPTER 4. EXPERIMENTAL APPARATUS
in te n s ity [a r b . u n its ]
1 .0
m e a s u re d
G a u s s ia n fit
0 .8
0 .6
0 .4
0 .2
0 .0
4 6 .0 0
4 6 .0 5
T O F [ µs ]
4 6 .1 0
4 6 .1 5
Figure 4.4: Experimentally measured shape and Gaussian fit of Xenon peak with mass
129 u. The width of the peak (FWHM) is 18.7 ns or 0.1 u.
voltages the shortest flight time of the high energy ions in the drift region is compensated by
their longer residence time within the reflectron. After the reflection the ions drift through a
second field free region for further mass separation and, finally, reach a detector. The total
distance between the interaction region and the detector is ≈ 3.2 m. This length together
with the actual voltage settings give flight times in the range between ≈ 5 µs (for H + ) and
≈ 160 µs (for C+
60 ). Assuming that the average thermal velocity of molecules in the C60
beam is 150 m/s, the corresponding perpendicular drift during the flight time on the way to
the detector is 24 mm. Such drift is not much smaller than the detector size (see Sec. 4.5).
Therefore, some loss of resolution for heaviest ions due to missing the detector is possible.
A peak shape of Xenon ion with mass 129 u as well as a Gaussian fit is presented in
Fig. 4.4. The width of the peak (FWHM) is 18.7 ns or 0.1 u. This corresponds to a resolution
of ≈ 1.2 × 103 . The peak shape is slightly asymmetric with respect to the fit. This is a result
of not a full spatial focusing, where the ions produced far from the centre of the interaction
region have larger flight times than the ions formed in the mean part of the interaction region.
4.3. TIME OF FLIGHT MASS SPECTROMETRY
39
Peak broadening and, hence, resolution losses are caused by a spread in ions arrival times.
The spread arises due to three major reasons: i) the distribution of initial kinetic energies
(velocities) of ions, ii) the distribution of positions where, and iii) the distribution of times when
ions are formed in the interaction region prior the acceleration. Of course, there are several
minor additional factors for resolution losses such as space charge effects, ions collisions, and
different kinds of instrumental effects. These factors play, however, a minor role and will not
be discussed here.
The simplest peak broadening reason arises from ions formed at slightly different times.
Such ions, even when produced at the same position with equal kinetic energies, will reach
the detector with a spread of arrival times. Therefore, the resolution will be reduced. This
temporal distribution can arise, for example, from the finite pulse duration of the laser used for
the ionisation. In the case of Xenon ion shown in Fig. 4.4 this peak broadening factor along
even on a typical time scale of the pump-probe experiments (≈ 500 fs) corresponds to the
resolution of ≈ 4.6 × 107 . Therefore, this effect is completely irrelevant for the experiments
presented here. Nevertheless, the resolution can be further improved by the enlarging of the
flight time using either a longer field free region or a lower acceleration potential. A second
possibility to improve the resolution is the application of more complicated and sophisticated
techniques such as “time-lag-focussing” [WMc55], “impulse-field-focussing” [BMT81], “rapid
field reversal” [MHo84], or the combination of TOF mass spectrometry with momentum
focusing by a magnetic sector [Pos71].
The next peak broadening factor comes from the ions spatial distribution, when ions are
produced at different space points, for example, due to a finite size of the interaction region
of an ionising laser. This distribution creates a spread of ion kinetic energies after acceleration
because ions formed far from the detector obtain larger final kinetic energies by travelling longer
distances through the electric field than ions formed closer to the detector. For Xenon ion
shown in Fig. 4.4 assuming the waist of the interaction region is 72.3 µm (see Sec. 6.2) only this
peak broadening factor leads to the resolution of ≈ 1.1 × 104 . Since ions formed close to the
detector have shorter flight distances than ions formed far from the detector, it is possible by
adjusting the electric field strength to find conditions of space focusing [WMc55, SRe01], when
this spatial distribution and resulting spread of kinetic energies after acceleration compensate
each other. As a result, ions of any given mass will arrive to the space focus plane at the same
40
CHAPTER 4. EXPERIMENTAL APPARATUS
time.
The last peak broadening factor is due to the initial velocities and, hence, kinetic energies
distribution, when ions with different initial velocities arrive to the detector at different times.
There are ions which initially move towards the field free region as well as ions which move in
the opposite direction. The latter have to be decelerated first, then accelerated back and will
pass through their initial positions after the so-called “turn-around time”. Two ions starting
with the same velocity but in opposite initial directions, thus, obtain the same final velocity.
Their arrival times at the detector are separated by the turn-around time. This can be used
to extract information about the initial kinetic energies of ions. But in most cases this effect
is undesirable and it leads to the lost of a resolution. For example, for Xenon ion shown in
Fig. 4.4 assuming the initial velocity of 240 m/s the resolution is ≈ 0.7 × 103 only. Hence,
this is the most significant factor of resolution loses in the present experimental setup. The
simplest way to improve the resolution is the operation with longer field free region. But the
best solution for correcting the initial kinetic energy spread is to use a reflectron. Utilisation
of the reflectron allowed to improve the resolution for Xenon ion up to ≈ 1.2 × 103 .
4.4
Photoelectron Spectroscopy
The scheme of the photoelectron spectrometer is presented in the left part of Fig. 4.3.
Distances between grids and potentials applied to them are summarised in Table 4.2. To obtain
the highest possible temporal resolution and to prevent the corruption of the photoelectrons
original kinetic energy distribution no extra electric field is applied to the grids.
It is important to shield the electrons flight region from the influence of the earth magnetic
field or magnetic fields induced by currents. A double µ-metal shield is used for that. This
shield consists of two concentric tubes made from nickel-iron alloy with diameters 130 mm and
110 mm, respectively. A copper tube with an inner diameter of 48 mm and a length of 450 mm
is put inside. The inner surface of this copper tube is coated with graphite to avoid charging
effects. The double µ-metal shield system provides the magnetic field shielding around four
orders of magnitude [Boy05].
Since the photoelectrons are emitted into the full solid angle 4π and no electric extraction
field is applied, only a very small fraction of the photoelectron reaches the detector. This
4.5. ION AND ELECTRON DETECTION
41
Table 4.2: Voltage settings used in this work and distances between grids of the photoelectron spectrometer.
Region
Voltage, V
Distance, mm
G2 - G1
GND - GND
20
G1 - G4
GND - GND
10
G4 - G5
GND - GND
∼ 445
G5 - MCP
GND - GND
10
fraction can be estimated from the inner diameter of the internal flight tube (48 mm) and the
distance between the interaction region and the photoelectron detector (475 mm). For the
present design the acceptance angle is ∼ 2.9◦ corresponding to a fraction of the photoelectrons
reaching the detector of ∼ 0.7%.
Unfortunately, the simultaneous detection of electrons and ions can not be achieved with
the present experimental apparatus. But it is possible to switch between photoelectron and
mass spectrometer operation very fast without breaking the vacuum and carry out the measurements under almost identical conditions.
4.5
Ion and Electron Detection
Ions and photoelectrons are registered using a microchannel plate (MCP) detector. Such
detectors are widely used in a variety of applications for detecting and imaging of ions [KZT05],
electrons [KWK86], protons [FSH96], ultraviolet [FHD83] and X-ray photons [Shi97] providing
high spatial and temporal resolution, high gain, low background signal rates, and stability
against the magnetic field influence.
The individual MCP used in this work (MCP-56-15, TOPAG GmbH) represents a lead
glass plate of 0.8 mm thickness and round shape with a diameter of 50 mm. The front and
back surfaces of the MCP is coated with metal. It has a pore structure and contain in one
square centimeter up to one million of separate channels of 15 µm nominal diameter coated
with semiconducting lead oxide (PbO). All channels have 10◦ tilt with respect to the MCP
42
CHAPTER 4. EXPERIMENTAL APPARATUS
surface normal, so-called “channel bias angle”. In the present work a pair of MCPs is used in
chevron-type (or V-stack) configuration for the detection both ions and photoelectrons. In this
configuration the tilt of the channels in the two plates with respect to the spectrometer axis is
opposite. The channel tilt obviates the passing of incident particles through the MCP without
interaction. Since particles falling on the MCP between the channels can not be detected, the
MCP detection sensitivity depends on the open area ratio(OAR). It is a ratio of the MCP
open area (integrated pore area) to the MCP total effective area. The OAR can be calculated
as
π
OAR =
2
2
d
P
,
(4.13)
where P is a period of the pore structure. For MCP-56-15 OAR is 63%. According the
specification typical operation voltage at 104 gain is 1100 V per plate.
The detection mechanism of incident particles by the MCP has been studied in detail
[Wiz79]. An incident particle hits a channel of the MCP detector producing secondary electrons
by the emission from the channel wall. Each secondary electron is accelerated further into
the cannel by an applied electric field and strikes the channel wall generating more and more
secondary electrons and, finally, forming a cloud of cascade electrons. This electron cloud hits
the anode and it is detected by the electronics.
Fig. 4.5a illustrates the ion detection scheme used in this work. Before entering the MCP
ions can be further accelerated by the acceleration voltage UAC applied between the grounded
grid and the MCP front surface. The power supply scheme is realised in such way that it
allows one to hold the bias voltage UM CP between the front and back surfaces of the MCPs
constant while varying UAC in a wide range. In the present work UAC and UM CP are set to
−2.0 kV and −2.2 kV, respectively.
Detection efficiency is one of the most important characteristics of MCPs. The MCP
detection efficiency for different kinds of incident particles, their masses and kinetic energies has
been investigated by many groups [GSe00, YNK01, KZT05]. The highest detection efficiency
is observed for electrons, the lowest – for UV photons and X-rays [Wiz79]. For positive ions
the detection efficiency varies between 5 % and 85 % [Wiz79] depending on their composition
and velocities [GSe00]. Two different physical mechanisms determine the detection efficiency
for positive ions: kinetic and potential emission of secondary electrons [Kre68]. The kinetic
emission is induced by direct collision of the ions with an MCP material and it depends on
4.5. ION AND ELECTRON DETECTION
43
Figure 4.5: The schemes used for the detection of: a)ions and b)electrons.
ions velocities and masses. The yield of secondary electrons caused by this process rises as
a function of ion impact energies. If the work function of the MCP material is less than one
half of the ion ionisation potential, the potential emission can play an essential role [GSe00].
This process is independent of ion impact energies but it depends on an ion internal energy.
Therefore, it is sensitive to the charge of ions. Usually, the potential emission dominates over
the kinetic emission at low ion impact energies.
The detection efficiency for C60 fullerenes is well known and described as follows [ITM99]
F = 1 − e−
1.48E+2.18E 2
100
,
(4.14)
where F is the detection efficiency and E [keV] is the ion impact energy. For the experiments
involving the comparison of the absolute abundances for the different charge states of C60
(for example, intensity dependent studies) measured ion yields are rescaled with the detection
efficiency described by Eq. (4.14). In the present work the ion impact energies are 6.47, 12.95,
and 19.42 keV for single, double, and triple charged C60 ions at UAC = −2.0 kV (taking into
account UM CP and voltages from Table 4.1) giving rescaling factors 0.64 and 0.98 for C+
60 and
C2+
60 , respectively. The rescaling factor for the higher charged ions is 1.00.
The acquisition of mass spectra is done using either a fast multiscaler card (P7886, FAST
ComTec GmbH) or an analogue analyser card (AP240, Acqiris SA). The detector output is
always connected with inputs of the data acquisition cards through an amplifier to decouple the
cards from the detector and protect the expensive devices against possible electronic damages.
For the detection of ions the amplifier with an amplification factor of 5 (model 352c, Novelec)
is utilised, while the electron detection is done with an amplification of 10 (wideband amplifier
44
CHAPTER 4. EXPERIMENTAL APPARATUS
model 6954, Phillips Scientific).
The multiscaler P7886 card is based on a multistop time-to-digital converter (TDC)
[PJB80, FST85]. Stop event pulses are generated by particles arriving at the detector. Such
devices simply count and store time intervals between a start event and stop events. Then
these data are converted into a histogram showing the number of events as a function of
discrete time intervals (bins). The maximal resolution or time bin width is 500 ps. There is no
dead time between time bins. The card is capable of counting only one event in every time
bin. For all experiments presented here time bins are grouped by 32 and, hence, the card is
used with the resolution of 16 ns. The start event is generated by a fast photodiode installed in
front of the entrance window of the vacuum chamber. Each arriving ion produces a NIM peak
through the discriminator (model 7011, FAST ComTec GmbH) which is used as a stop event.
The major advantage of this counting method is the possibility to suppress the noise arising
from the background ions and from the electronics by a proper setting of the discriminator
level [SIm99]. This is especially important in the case of extremely weak signals. To determine
the proper discriminator level the pulse height distribution (PHD) for the signal in front of the
discriminator must be measured. Fig. 4.6 shows PHD for the different charged states of C60
ions obtained with the usual voltage settings: UM CP = −2.2 kV, UAC = −2.0 kV and the following laser parameters: I = 5.2 × 1013 W/cm2 , τ = 27 fs, and λ = 797 nm. The very strong
signals below 0.04 V correspond to noise. The signals above 0.04 V are due to the detection
of C60 ions. They have asymmetric shapes with maxima in the range of 0.25 − 0.35 V. The
maximum of the peak distributions shifts to higher values with increasing C60 ion charge. This
illustrates the fact that the ion kinetic energy after acceleration by the same voltage is larger
for the higher ionised ions. Each signal curve has a local minimum (valley) around 0.04 V.
Fortunately, the position of valley does not depend on the ion charge state and the trigger
level can be set to 0.04 V in order to cut off the noise pulses below this value and to count
C60 ionisation events independently of the ion charge state. Furthermore, the PHD contains
important information about the relative detection efficiency of different ions [FHa02].
In the case of high ion yields detection methods based on the counting technique fail
due to the possibility of two or more ions arriving within the same time interval (bin). In
such situation it is better to employ analogue signal detection. The AP240 card is based on
an analogue-to-digital converter (ADC) and allows one to digitise signal with 8-bit amplitude
4.5. ION AND ELECTRON DETECTION
45
n u m b e r o f p u ls e s
1 0 0 0 0
1 0 0 0
C
+
C
2 +
C
3 +
6 0
6 0
6 0
1 0 0
n o is e
1 0
1
0 .0 1
0 .1
1
p u ls e h e ig h t [V ]
Figure 4.6: The distribution of pulse height for the different charged state of fullerenes.
resolution in real time at a maximal sampling frequency up to 2 GHz and then average it for up
to 65536 laser shots [Rub05]. In addition to the simple averaging mode this card can be used
for more complicated data acquisition procedures. In order to avoid overflow in the summed
data, a fixed noise base can be subtracted from each signal value before the summation is done.
Furthermore, there is an ability to set a threshold level and to detect only signals above this
threshold. With the threshold the signals can be detected in counting or summation modes. In
the counting mode the card counts events corresponding to the appearance of the signal above
the threshold. In this mode the card runs as a counter possessing all advantages of the counting
techniques. In the summation mode the card accumulates digitalised amplitudes of the signal
if the signals are above the threshold level. This mode is especially interesting because of
the possibility to combine advantages both the counting techniques and the techniques based
on the analogue-to-digital converting. The noise can be minimised by the proper settings of
the threshold, at the same time digitalisation of the incoming signal amplitude can overcome
46
CHAPTER 4. EXPERIMENTAL APPARATUS
the problem when more than one ion is detected in the same time bin. But these additional
modes are not used in the present work and the card is only utilised for signal averaging
because the averaging mode is most simple in operation and best suitable among others for
quick measurements.
It is important to inspect stability and reproducibility of the mass spectra acquired. For
that one can repeat the measurements of mass spectra many times and then calculate the
standard deviation of the integrated ion yield for particular ions. Since the laser system stability
influences the reproducibility of the mass spectra a test was performed with the laser used
2+
3+
mostly in the present experiments. The outcome of such an examination for C+
60 , C60 , and C60
is shown in Fig. 4.7. The standard deviation in each point is computed using 100 separate mass
spectra measured under identical conditions. Between 100 and 50000 laser shots were used
for data accumulation to produce an individual mass spectrum and to obtain results plotted in
Fig. 4.7. The standard deviation drops sharply down with an increasing number of a laser shots.
Mass spectra obtained with more than 1000 laser shots demonstrate good reproducibility (the
standard deviation < 5%) and can be used for further data evaluation. Usually, individual mass
spectra were acquired by averaging over 50000 laser shots. In pump-probe and polarisation
dependent studies mass spectra were obtained by the averaging typically over 5000 − 10000
laser shots. The mass spectra in the pulse shaping experiments were obtained by the averaging
over 3000 − 5000 laser shots.
The measured TOF mass spectrum s(t) can be transformed into the mass spectrum de pending on the mass-to-charge ratio s mq as
s
m
q
= s A[t − t0 ]2
,
(4.15)
where t is the measured ion arrival time, m/q is its mass-to-charge ratio, t0 is the time delay
between the ionisation event in the vacuum chamber and the electronics triggering time, and
A is a scaling factor. A and t0 can be determined from the mass-to-charge ratios m1 /q1 and
m2 /q2 of two specific, well known ion peaks in the TOF mass spectrum which have the arrival
times t1 and t2
qm
A=
1
q1
−
q
t1 − t2
m2
q2
2

(4.16)
s ta n d a r d d e v ia tio n [% ]
4.5. ION AND ELECTRON DETECTION
47
C
+
C
2 +
C
3 +
6
6 0
6 0
6 0
4
2
0
0
5 0 0 0
1 0 0 0 0
5 0 0 0 0
n u m b e r o f la s e r s h o ts
Figure 4.7: Dependence of the standard deviation of the integrated ion yield on the number
of laser shots used for the acquisition.
and
q
m1 m2
/ q2 t2
q1
t0 = q
m1 m2
/ q2
q1
− t1
.
(4.17)
−1
3+
Normally, mass spectra of C60 fullerenes are calibrated by the arrival times of C+
60 and C60
ions. The conformation of peak assignment can be done by the inspection of different peaks
or/and Xe+ ). To keep the integrated ion
positions in the mass spectrum (for example, C2+
60
yield per mass-to-charge interval s mq ∆ mq equal to the integrated ion yield per time
interval s (t) ∆t, the mass spectrum has to be recalculated taking into account the following
Jacobian
s
m
q
∆
m
q
∝ t s(t)∆t .
(4.18)
One example of a TOF mass spectrum for C60 fullerenes obtained with the laser parameters
I = 1.4 × 1014 W/cm2 , τ = 27 fs, and λ = 797 nm is presented in Fig. 4.8a. Exactly the
48
CHAPTER 4. EXPERIMENTAL APPARATUS
C60
2+
(a )
0 .8
C60
0 .6
+
C60
0 .4
4+
0 .2
C60
3+
0 .0
7 5
1 .0
TOF [µs]
1 0 0
1 2 5
0 .2
0 .8
4+
C60 C60
0 .4
0 .2
3+
m/q [u]
3 0 0
4 0 0
1
0 .4
TOF [µs]
+
2 0 0
0 .6
0 .4
C60
0 .6
0 .0
0 .8
0 .0
2+
0 .8
(c )
0 .0
C60
(b )
1 5 0
y ie ld [a r b . u n its ]
y ie ld [a r b . u n its ]
1 .0
y ie ld [a r b . u n its ]
y ie ld [a r b . u n its ]
1 .0
1 .2
5 0 0
6 0 0
7 0 0
(d )
0 .1
0 .0 1
0
2
kinetic energy [eV]
4
6
8
1 0
1 2
1 4
1 6
Figure 4.8: (a) TOF mass spectrum of C60 fullerene; (b) the same spectrum as Fig. 4.8a
but plotted as a function of m/q; (c) photoelectron spectrum of C60 fullerene; (d) the same
spectrum as Fig. 4.8c but modified accordingly to Eq. (4.22) and plotted on the logarithmic
scale as a function of EK . In all cases the laser parameters are: I = 1.4 × 1014 W/cm2 ,
τ = 27 fs, and λ = 797 nm.
same mass spectrum but plotted on m/q scale (without Jacobian transformation) is shown in
Fig. 4.8b.
The photoelectron detection scheme is very similar to one operated for the ions detection.
This scheme is depicted in Fig. 4.5b. The main difference is the absence of any extra voltage
between the grounded grid and the MCP front surface. Only the bias voltage UM CP = 2.5 keV
is applied. Such settings correspond to voltage of 1135 V per MCP plate. It is slightly more
than voltage recommended by the specification (1100 V) but with lower MCP voltage a good
signal to noise ratio in photoelectron spectra can not be achieved.
Photoelectron spectra are acquired with 0.5 ns temporal resolution using AP240 analyser
platform connected to the MCP detector output through the wideband amplifier model 6954
with an amplification factor of 10.
4.5. ION AND ELECTRON DETECTION
49
The transformation of the measured TOF photoelectron spectrum s(t) into the photoelectron spectrum depending on the kinetic energy of photoelectrons s(EK ) can be written
as
1
s(EK ) = s A
,
(4.19)
[t − t0 ]2
where t is the measured TOF of a photoelectron, EK is its kinetic energy, t0 is the time delay
between the ionisation event in the vacuum chamber and the electronics triggering time, and
A is a scaling factor. One way of determination of A and t0 for photoelectron spectra is
more complex than for mass spectra. For that one has to make an assumption about kinetic
energies EK1 and EK2 for at least two peaks in the TOF photoelectron spectrum with flight
times t1 and t2 . It can be fulfilled by either the kinetic energy assignment of Rydberg peaks
in C60 photoelectron spectra [BLH05] or the resonance structure inspection of ATI peaks in
photoelectron spectra of xenon measured at several well known laser intensities [SLH98]. If
this is done, A and t0 can be determined with the following expressions

2
t2 − t1 
A = q
EK1
EK1
−
1
EK2
and
q
K1
t1 E
− t2
EK2
t0 = q
EK1
−1
EK2
.
(4.20)
(4.21)
The energy resolution of the photoelectron spectrometer within the range of interest (0.5 −
20 eV) varies with EK from 1 meV at EK = 0.5 eV to 92 meV at EK = 20 eV.
The integrated photoelectron yield per time interval ∆t is s(t)∆t and per kinetic energy
interval ∆EK is s(EK )∆EK . The time interval ∆t in any TOF photoelectron spectrum is a
constant. At the same time, after the transformation of the TOF photoelectron spectrum s(t)
into the photoelectron spectrum depending on the kinetic energy of photoelectrons s(EK )
according to Eq. (4.19), the kinetic energy interval ∆EK depends on EK . Hence, to keep
the integrated photoelectron yield per kinetic energy interval identical with the integrated
photoelectron yield per time interval for the whole range of EK , the photoelectron spectrum
can be recalculated with the following formulas
s(EK )∆EK ∝ t−3 s(t)∆t
(4.22)
or
3/2
s(EK )∆EK ∝ EK s(t)∆t .
(4.23)
50
CHAPTER 4. EXPERIMENTAL APPARATUS
This modification can by done by the applying either Eq. (4.22) before the TOF scale transformation or Eq. (4.23) after the transformation. Fig. 4.8c-d show the TOF photoelectron
spectrum of C60 fullerenes and the photoelectron spectrum depending on EK , respectively.
The laser parameters are identical to the parameters used for the measurements of the mass
spectrum presented in Fig. 4.8a-b. The intensity of the photoelectron spectrum depicted in
Fig. 4.8d is modified accordingly to Eq. (4.22) and plotted on the logarithmic scale.
Both the multiscaler P7886 card and the AP240 analyser platform are capable to be run
with LabVIEW software. It allows one to automate the acquisition of mass or photoelectron
spectra in the experiments requiring many repetitions of spectral measurements, as needed for
laser intensity or/and polarisation dependent studies, optimisations with a pulse shaper, and
pump-probe experiments.
Chapter 5
Femtosecond Laser Pulses
Most of the experiments described in this work were performed using femtosecond laser
pulses delivered from a commercial multipass amplification laser system [Ste03]. Generally, the
name “femtosecond laser pulses” is related to laser pulses with a duration on the femtosecond
(1 fs = 1 × 10−15 s) time scale. But in this work the term “femtosecond laser pulse” is used
for a pulse with a duration around 30 fs. Shorter pulses with a duration below 10 fs are called
“sub -10 fs pulses”.
This chapter starts with a mathematical representation of an electromagnetic field needed
for the description of laser pulses. Then, the laser system together with methods and techniques of femtosecond and sub -10 fs laser pulses generation as well as additional equipment
employed in the experiments for the operation and manipulation of laser pulses are described.
This description is essentially based on Refs. [Rul05, DRu06, HSc08]. After that, the “classical” intensity averaging theory for elliptically polarised light [SRS09] is given. Finally, the
pulse shaping technique is presented.
5.1
Mathematical Description of Laser Pulses
In the general case, the electric field strength of an electromagnetic wave is a function of
spatial coordinates and time. Often, such functional dependence can be factorised and the
electric field can be represented as a product of two functions depending on spatial coordinates
and time, respectively. For any given point in space the strength of the electric field is only a
function of time. It is convenient to write the real electric field E(t) of linear polarised light
51
52
CHAPTER 5. FEMTOSECOND LASER PULSES
(e.g. E k x) as
E(t) =
1
E + (t) + E − (t)
2
,
(5.1)
where E + (t) and E − (t) are the complex electric field and its complex conjugate. The complex
electric field can be presented by a product of an amplitude function A(t) and a phase term
E + (t) = A(t)e−i[ω0 t−φ(t)]
(5.2)
and
E − (t) = A(t)ei[ω0 t−φ(t)]
,
(5.3)
where ω0 is a carrier or central frequency, and φ(t) is a temporal phase. In quantum mechanics
the E + (t) field is responsible for absorption, the E − (t) field for emission. The intensity of a
pulse I(t) is related with the pulse amplitude as
I(t) = ε0 cA2 (t) cos2 (ω0 t − φ(t)) ,
(5.4)
where ε0 is the dielectric constant in vacuum and c is the speed of light. If A(t) varies slowly,
the period averaged intensity is
hI(t)i =
ε0 c 2
A (t) .
2
(5.5)
The temporal amplitude and the phase contain all information about the laser pulse. The
complex electric field can be expressed by inverse Fourier integrals
Z ∞
1
+
E (t) =
Ẽ + (ω)eiωt dω
2π −∞
(5.6)
and
1
E (t) =
2π
−
Z
∞
Ẽ − (ω)eiωt dω
,
(5.7)
−∞
where Ẽ + (ω) and Ẽ + (ω) are a complex representation of the electric field in a frequency
domain (spectral representation) and its complex conjugate. By analogy, the complex electric
field in a frequency domain
Ẽ(ω) =
1 +
Ẽ (ω) + Ẽ − (ω)
2
(5.8)
can be presented by a product of an spectral amplitude Ã(ω) and a phase term
Ẽ + (ω) = Ã(ω)eiϕ(ω)
(5.9)
5.1. MATHEMATICAL DESCRIPTION OF LASER PULSES
53
and
Ẽ − (ω) = Ã(ω)e−iϕ(ω)
,
(5.10)
where ϕ(ω) is a spectral phase. The spectral representation is related to the temporal representation by the Fourier transformation
∞
Z
+
Ẽ (ω) =
E + (t)e−iωt dt
(5.11)
E − (t)e−iωt dt .
(5.12)
−∞
and
−
Z
∞
Ẽ (ω) =
−∞
˜
The period averaged spectral intensity (spectrum) of a pulse I(ω)
is related to the spectral
amplitude as
ε0 c 2
˜
I(ω)
=
à (ω) .
(5.13)
4π
Both representations either in the time domain using a temporal amplitude and a phase or in
the frequency domain using a spectral amplitude and a phase are equivalent.
The mathematical description of the temporal amplitude and intensity as well as the spectral amplitude and intensity distribution for some commonly used pulse shapes are summarised
in Table 5.1. These pulse shapes are standard for the analytical calculations and they are often
used to approximate experimental data. A rectangular pulse does not have wings, therefore
it is a very poor approximation of reality but most simple for calculations. A Gaussian pulse
2
is also quite simple for calculations. It has quickly falling wings (∝ e−t ) which is not always
true in a real experiment. The sech2 pulse has wings which are slowly falling (∝ e−t ). This
pulse shape is often more appropriate to describe experimental situations precisely. And finally,
Lorentzian pulses with very large wings (∝ 1t ) may also be useful sometimes.
Usually, pulse durations τ are given as full width at half maximum intensity (FWHM) as
done so in Table 5.1. Sometimes the pulse duration is determined at the intensity levels 1/e
(τ1/e ) or 1/e2 (τ1/e2 ). For a rectangular pulse durations determined at the different intensity
levels are identical. For Gaussian, sech2 , and Lorentzian pulses these durations are related as
follows
√
Gaussian: τ =
r
ln 2
τ1/e2 ,
2
= 0.532τ1/e2 ,
ln 2τ1/e =
sech2 : τ = 0.812τ1/e
(5.14)
(5.15)
and
Lorentzian: τ = 0.763τ1/e = 0.396τ1/e2
.
(5.16)
54
CHAPTER 5. FEMTOSECOND LASER PULSES
Table 5.1: The mathematical description (taken from [DRu06]) of some pulse shapes in
the time and frequency domains. τ and ω0 are temporal FWHM and carrier frequency,
respectively. Frequency amplitudes and intensities are given without normalisation factors.
Shape
Temporal amplitude
Temporal intensity
Rectangular
1 for τt ≤ 12 , 0 else
1 for τt ≤ 12 , 0 else
h
exp −2 ln 2
Gaussian
i
h
exp −4 ln 2
√ sech 2 ln 1 + 2 τt
sech2
h
Lorentzian
Shape
1+
i−1
t 2
τ
4√
1+ 2
τ sinc
p
Gaussian
π
τ
2 ln 2
h
1+
√
π
√
1+ 2
τ
2
τ (ω−ω0 )
2
sech
i
4√
1+ 2
i−2
t 2
τ
Frequency intensity
i
τ 2 sinc2
i
h 2
τ (ω−ω0 )2
exp − 8 ln 2
π √
τ
2 ln (1+ 2)
sech2
h
t 2
τ
√ sech2 2 ln 1 + 2 τt
Frequency amplitude
Rectangular
Lorentzian
t 2
τ
πτ (ω−ω0 )
√
4 ln (1+ 2)
√ √
1+ 2τ |ω−ω0 |
exp −
2
π
τ2
2 ln 2
π √
2 ln (1+ 2)
√
π 2 (1+ 2) 2
τ
4
The intensity autocorrelation function
Z ∞
I(t)I(t − δ) dt
S(δ) =
h
τ (ω−ω0 )
2
i
i
h 2
τ (ω−ω0 )2
exp − 4 ln 2
2
2
τ 2 sech
πτ (ω−ω0 )
√
4 ln (1+ 2)
i
h p
√
exp − 1 + 2τ |ω − ω0 |
(5.17)
−∞
plays an important role in temporal pulse characterisation techniques. It is determined as the
time integral of temporal pulse intensity function I(t) multiplied by its shifted replica I(t − δ).
Here the pulse replica is used as a gate to scan the same pulse. The autocorrelation function
is always symmetric and centred around δ = 0. The width of the autocorrelation function τAC
is determined by the pulse width τ :
τAC =
1
τ
DAC
,
(5.18)
5.1. MATHEMATICAL DESCRIPTION OF LASER PULSES
L o r e n tz ia n
G a u s s ia n
(a )
0 .8
s e c h
0 .6
2
1 0 0 fs
0 .4
0 .2
0 .0
-3 0 0
-2 0 0
-1 0 0
0
1 0 0
2 0 0
3 0 0
1 .0
in te n s ity [a r b . u n its ]
in te n s ity [a r b . u n its ]
1 .0
55
L o r e n tz ia n
G a u s s ia n
(b )
0 .8
s e c h
2
0 .6
0 .4
0 .2
0 .0
-3 0 0
-2 0 0
-1 0 0
0
1 0 0
2 0 0
3 0 0
d e la y [fs ]
tim e [fs ]
Figure 5.1: (a) Pulses of different shapes, (Lorentzian, Gaussian, and sech2 with 100 fs
FWHM) and (b) their calculated autocorrelation functions.
where DAC is the deconvolution factor depending on the pulse shape. The values of these
factors for different commonly used pulse shapes are summarised in Table 5.2. Calculated
intensity autocorrelation traces for the pulses with different shapes but with constant width of
100 fs FWHM are plotted in Fig. 5.1. While all three pulses have the same FWHM, the width
of the autocorrelation traces differs according to the values of DAC from the Table 5.2.
If a spectral phase ϕ(ω) varies slowly with the frequency ω, it can be expanded into the
Taylor series around the carrier frequency ω0
∞
X
ϕ(k) (ω0 )
(ω − ω0 )k
ϕ(ω) =
k!
k=0
(5.19)
with
∂ k ϕ(ω) ϕ (ω0 ) =
∂ω k ω=ω0
(k)
.
(5.20)
The first term ϕ(ω0 ) in Eq. (5.19) describes the absolute phase of the pulse in the time
domain. The first derivative ϕ0 (ω0 ) = Tg (ω0 ) is the so-called “group delay” (GD). GD leads
to a shift of the pulse envelope in the time domain. The second derivative ϕ00 (ω0 ) = D2 (ω0 )
is the “group delay dispersion” (GDD) (also sometimes called “second order dispersion”) .
ϕ000 (ω0 ) = D3 (ω0 ) and ϕ0000 (ω0 ) = D4 (ω0 ) are the “third (TOD) and fourth (FOD) order
dispersion”, respectively. Since these higher order derivatives (GDD and higher) describe the
frequency dependence of the GD
∂Tg (ω) D2 (ω0 ) = ϕ (ω0 ) =
∂ω ω=ω0
00
,
(5.21)
56
CHAPTER 5. FEMTOSECOND LASER PULSES
2
∂
T
(ω)
g
D3 (ω0 ) = ϕ000 (ω0 ) =
∂ω 2 ω=ω0
,
(5.22)
and
∂ 3 Tg (ω) D4 (ω0 ) = ϕ (ω0 ) =
,
(5.23)
∂ω 3 ω=ω0
they are responsible for dispersive effects and changes in temporal structure of the pulse
0000
envelope.
By analogy, the temporal phase φ(t) can be expanded into the Taylor series around time
zero for small derivations
φ(t) =
∞
X
φ(k) (0)
k=0
k!
tk
(5.24)
with
∂ k φ(t) φ (0) =
.
∂tk t=0
The time derivative of the temporal phase defines the instantaneous frequency ω(t)
(k)
ω(t) = ω0 −
dφ(t)
dt
.
(5.25)
(5.26)
The first term in the Taylor series (Eq. (5.24)) φ(0) is the absolute phase of the pulse. The
absolute phase gives the temporal relation of the pulse envelope with respect to the underlying
carrier oscillation. The first derivative φ0 (0) in the second term, which is linear with time,
describes a shift of the carrier frequency ω0 . The term with φ00 (0) corresponds to the linear
motion of the instantaneous frequency. Such term is named a linear chirp. The frequency
rising (decreasing) with time is referred to as up-chirp (down-chirp). The next terms are the
quadratic, cubic, and so on chirp.
If the instantaneous frequency for a pulse does not depend on time (ω(t) = const), such
pulse is called a “bandwidth limited” or “Fourier transform limited” pulse. In this case the
pulse duration τ is determined by its spectral width only
τ ∆ω = 2πKT B
,
(5.27)
where τ and ∆ω are FWHM of intensity profile in the time and frequency domains respectively.
KT B is a constant which depends on a pulse shape. The values of KT B for some commonly
used pulse shapes are given in Table 5.2. Usually, Eq. (5.27) is known as a “time-bandwidth
product”. It indicates that for the achievement of the shorter pulse duration one needs to
apply a greater frequency bandwidth. Because real laser pulses are mostly not bandwidth
limited, Eq. (5.27) provides only the lower limit for estimations of the pulse duration.
5.2. GENERATION OF FEMTOSECOND LASER PULSES
57
Table 5.2: Deconvolution factors DAC and time-bandwidth products KT B for the various
shapes of laser pulses.
Pulse shape
DAC
KT B
Rectangular 1.0000 0.443
√
Gaussian
1/ 2 0.441
5.2
5.2.1
sech2
0.6482 0.315
Lorentzian
0.5000 0.142
Generation of Femtosecond Laser Pulses
Femtosecond Oscillators
The standard approach to produce femtosecond laser pulses includes pulse generation by a
mode-locked oscillator followed by chirped pulse amplification (CPA), in which an initial laser
pulse is stretched by a prism or grating stretcher, then amplified in regenerative or multipass
schemes, and finally recompressed [SMo85, ZHM95, BDM98].
Generally, the construction of a femtosecond oscillator is rather straight forward and consists of the same elements as any laser: a resonator cavity with (partially) reflecting end mirrors
and a gain medium inside [SSK94, CSM96]. The key difference in comparison with lasers or
long pulsed lasers is (aside from a medium with broad band amplification) the presence of a
dispersive element (a pair of prisms or chirped mirrors) in the cavity to control the spectral
dispersion. This element allows one to compensate the dispersion introduced by the gain
medium and other optical parts. Typically, an oscillator is pumped by a continuous wave (cw)
laser. The pumping beam is focused into the active medium collinear with the mode of the
cavity itself.
For a given wavelength λ a discrete number of longitudinal modes can exist inside the
oscillator cavity for which
L=n
λ
2
,
(5.28)
where L is the cavity length and n an integer describing one longitudinal mode. At the same
58
CHAPTER 5. FEMTOSECOND LASER PULSES
time, only such longitudinal modes for which the gain of the pumped medium exceeds the
laser threshold on the gain curve of the pumped laser medium can oscillate in the cavity. The
output radiation field consists of the sum over all fields in the oscillating modes. If there is no
constant phase relation between the modes, the intensity of the output radiation fluctuates
randomly as a function of time without any regular temporal structure. The so-called “mode
locking” holds all longitudinal modes in phase and constructive interference between the phase
locked modes generates a single pulse circulating in the cavity. Each reflection of this pulse
from a partly transmitting cavity mirror produces a pulse leaving the cavity. It forms the
output laser radiation composed of a laser pulses train. The separation between the pulses in
the train is equal to the round trip time TRT of the cavity
TRT =
2L
c
,
(5.29)
where c is a speed of light.
All techniques of mode locking can be divided into two groups: active and passive mode
locking [HBe87]. Active mode locking involves either a periodic modulation of the cavity
losses with the period corresponding to the cavity round trip time or a periodic round trip
phase change. It can be achieved by an acousto-optical modulator (AOM) or an electro-optical
modulator (EOM) placed inside the cavity [KSi70]. The first device is used to introduce a
periodic modulation of the cavity loses, while the second one produces a time dependent phase
variation. Active mode locking has two main disadvantages. First, the modulating frequency
of AOM or EOM has to match the reciprocal of the cavity round trip time. Even very small
cavity length drifts can lead to frequency mismatching and a strong timing jitter or even to
chaotic behaviour. This problem can be solved using the so-called “regenerative mode locking
technique” [SES91]. This technique uses a feedback circuit for automatic adjustment of
the modulating frequency to keep the synchronisation intact. Second, the pulse compression
mechanism introduced by active mode locking itself is quite weak. An active modulator
initiates a strong compression as long as the duration of the compressed pulse is larger than
the modulating function. But when the pulse duration reaches the width of the modulating
function further compression becomes very weak. Hence, active mode locking can be used to
produce laser pulses with a duration on the picosecond time scale. Shorter pulse durations
can be achieved by application of active mode locking driven at high harmonics of the round
trip frequency (harmonic mode locking) [JLK98] or with passive mode locking.
5.2. GENERATION OF FEMTOSECOND LASER PULSES
59
In passive mode locking a fixed phase relationship is created by a saturable absorber,
often thin semiconductor films or organic dyes. Their absorption coefficient decreases with
increasing intensity of the light passing through. Because the transmission of the absorber
follows the incoming intensity, a pulse propagating through the saturable absorber undergoes
weak interaction with the absorber around its peak intensity, while the less intensive wings are
exposed to larger losses. Therefore, the pulse is shortened during each propagation through
the saturable absorber till the pulse duration reaches the absorption recovery time [Pen86].
This is a regime of a fast saturable absorber, where the losses induced by the absorber follow
the pulse intensity profile. If the pulse duration is shorter than the absorption recovery time,
the saturable absorber is slow. In the slow regime the induced losses do not follow the pulse
intensity profile. The slow saturable absorbers can be utilised for the generation of femtosecond
pulses combining this method with the saturable amplification [GSS90] or in colliding pulse
mode locking [FGS81].
The most important type of passive mode locking, especially for the generation of femtosecond pulses, is the so-called “Kerr lens mode locking” (KLM) [BSC92]. The name KLM
comes from the Kerr type nonlinearity, in which the refractive index n of a medium is altered
by the applied optical intensity I according to
n = n0 + n2 I
,
(5.30)
where n0 is the normal and n2 the nonlinear refractive index. n2 has a very small value, and
therefore the nonlinearity of the refractive index is negligible under normal conditions. But
inside a laser cavity it is possible to reach intensities at which nonlinear effects start to play a
significant role. A pulse propagating through the Kerr active medium experiences the highest
refractive index for the central and most intense part. It thus produces an intensity sensitive
phase modulation. It also leads to self-focusing and creation of a “Kerr lens”. This effect
can be used to induce instantaneous intensity dependent losses inside the cavity. They can
be weak for a short intensive pulse, while less intensive pulses or cw radiation are exposed
to larger losses. The action of the Kerr type nonlinearity is analogue to very fast saturable
absorbers providing a modulation function which can follow even femtosecond pulse profiles.
This type of mode locking is successfully used for the generation of laser pulses with a duration
down to 10 fs [CMK94, ZTH94]. Its main disadvantage is a poor long-term stability of the
laser because operation close to the cavity stability limit is required [CSM95].
60
CHAPTER 5. FEMTOSECOND LASER PULSES
The temporal profile of a pulse generated by a femtosecond oscillator is determined by the
Fourier transformation of its spectrum (Eq. (5.6)). If all modes are locked with a constant
phase difference between them, the resulting pulse has a minimal pulse duration for the given
spectrum. This leads to so-called “bandwidth-limited” or “transform-limited” pulses. For such
a pulse the product of the pulse duration and its spectral width (time-bandwidth product) is
a constant depending on the pulse shape only. The values of the time-bandwidth products
for different commonly used pulse shapes have been given in Sec. 5.1. In the situation when
the modes are locked incompletely and the spectral phase depends on the frequency, the pulse
duration is longer than the duration of the transform-limited pulse with an identical spectrum.
In this case the temporal profile can be derived from the spectrum only if the spectral phase
is known.
It is clear that very short pulses can be only generated using broad spectral bandwidth.
This bandwidth can be achieved by special crystals doped with a transition metal (titanium,
chromium) as gain material [SCB94]. Such materials combine broad fluorescence emission
spectra with good thermal conductivity and are very commonly used for the generation of
femtosecond pulses. Most important and widely used among them is a sapphire crystal doped
with titanium atoms (Ti:sapphire) [Mou86]. It has a gain bandwidth centred around 800 nm.
Usually, Ti:sapphire oscillators are pumped by a frequency doubled YAG or argon ion laser
[PXK98, MKC99]. Another material is a chromium doped lithium strontium aluminium fluoride
(Cr:LiSAF). It generates radiation at the wavelength of 850 nm and can be pumped by a red
diode laser [HVA02]. Finally, chromium doped forsterite (Cr:forsterite) and chromium doped
YAG (Cr4+ :YAG) lasers operated at 1.2 µm and 1.5 µm, respectively, have many applications
in the field of microscopy of living systems and metrology [SMB98, AMP05].
5.2.2
Chirped Pulse Amplification
Typical values of the pulse energy delivered by Ti:sapphire femtosecond oscillators lie in the
nanojoule range. The task to amplify such pulses to the millijoule level and above is difficult
due to the extremely high peak power involved. Stretching the pulse in time by a factor of
103 − 104 before amplification solves this problem. There are two ways of the pulse stretching
and recompression: prism sequences and grating arrangements [Fre95].
A prism pair can be employed to introduce adjustable GDD and, hence, to stretch or to
5.2. GENERATION OF FEMTOSECOND LASER PULSES
61
recompress a pulse [FMG84]. The angular dispersion of the first prism creates the spatial
separation of spectral components of the incoming beam. The second prism is assembled in
such a way that the longer wavelength spectral components travel longer distances through
the prism material than the shorter spectral components of the beam. This leads to a negative
GDD. Recovering of the original beam can be done with either a second pair of prisms or with
a reflecting mirror and the same prism pair. In addition, a positive GDD is introduced by the
material dispersion when the beam propagates through the prisms. By translation of one prism
along its axis of symmetry it is possible to change the thickness of the material transverse by
the beam, and therefore achieve a positive GDD. The prism pair allows one to tune the GDD
continuously from negative to positive values without beam deviation. Operating the prisms
at Brewster angle can minimise the reflection losses for the correctly linearly polarised beam.
A pair of diffraction gratings in a parallel configuration creates a negative GDD in a similar
manner as prisms [Tre69]. But the amount of the negative GDD produced by a grating pair
can be much larger compared with a prism pair. To minimise the spatial chirp of the output
beam parallel alignment of the gratings has to be done with extremely high accuracy. While
diffraction gratings are free from material dispersion, optical losses of a grating sequence are
quite high even if they are utilised in the Littrow configuration. A pair of diffraction gratings in
the antiparallel configuration with a telescope between the gratings is more flexible because of
the possibility to introduce both positive and negative GDD [Mar87]. GDD can be controlled
by variation of the distance between the second grating and the image of the first grating.
After being stretched, a pulse can be amplified using either a regenerative [WRH94, BKR96]
or a multipass [BPH95] amplifier. A regenerative amplifier is a cavity with a gain medium
inside. A time gated polarisation devices such as a Pockels cell and a thin film polariser are
used to put a pulse into the amplifier. The incoming pulse makes several round trips to gain
energy and then it is coupled out of the cavity. Regenerative amplification is the most common
technique for the femtosecond pulse amplification because of its reliability, pulse-to-pulse and
pointing stability. An alternative and possibly more promising design for femtosecond pulse
amplification is based on the multipass scheme. In this configuration an incoming pulse passes
through the gain medium several times at slightly different points without using a cavity. Main
advantage of the multipass scheme is the ability of generating much shorter pulses than with a
regenerative amplifier because this scheme allows one to avoid the large gain narrowing in the
62
CHAPTER 5. FEMTOSECOND LASER PULSES
active medium [LSW95]. Moreover, it has a higher gain per pass, and therefore only few passes
through the gain medium are required. Thus, the introduced higher order dispersion is rather
small and, hence, shorter pulses can be obtained after recompression. Both regenerative and
multipass amplifiers are usually pumped either with a frequency doubled, Q-switched Nd:YAG
laser at 532 nm [BGL94] or with an intracavity doubled cw Nd:YLF laser at 527 nm [BDM97].
The last stage is the pulse recompression. A simplest method involves an identical pair of
prisms or gratings as used for pulse stretching. But the amplification process itself leads to extra
dispersion, both of second and higher orders. One of the major problem of pulse recompression
is to compensate all these contributions. GDD can be totally balanced by a proper compressor
alignment. The easiest way to correct the TOD is to adjust the incident angle between
stretcher and compressor because of the sensitivity of the third order contribution to this
angle [FSR94]. The fact that the third order contribution from a prism pair has the opposite
sign compared to a grating pair can be used to compensate the GDD, TOD, and FOD terms
as well [FBB87, WRH94]. Another method employs gratings with a different groove density
in compressor and stretcher [SBS98]. GDD and TOD can be also corrected by using chirped
mirrors [SFS94]. Such mirror consists of multiple dielectric layers with a variable thickness. The
penetration depth of the incident radiation depends on its frequency, and therefore reflection
from this mirror introduces frequency dependent dispersion. Modern chirped mirrors have a
high reflectivity and exhibit dispersion control over a broad spectral range [PKA07].
The pulse duration delivered by CPA systems is mainly limited by two factors. First,
the finite bandwidth of the gain medium leads to the spectral narrowing during the pulse
amplification and therefore to broadening of the pulse. Second, and more importantly, the
higher order dispersion induced in the amplifier results in an additional pulse broadening. A
proper compensation of higher order dispersion is a challenging task for laser physics. Many
different designs were suggested to overcome this problem [LBa93, WPC93, CRS96, ZMB99].
Up-to-date CPA schemes can provide laser pulses at 1 kHz repetition rate with up to 15 mJ
energy and a duration below 20 fs [NKT98].
5.2.3
Multipass Amplification Laser System
A schematic diagram of the multipass amplification laser system utilised in this work is
shown in Fig. 5.2. This system consists of the Ti:sapphire oscillator (Femtosource Scientific
5.2. GENERATION OF FEMTOSECOND LASER PULSES
63
Figure 5.2: Optical layout of the multipass amplification laser system.
PRO, Femtolasers GmbH) pumped by the frequency doubled 5 W Nd:YVO4 laser (Millennia V
, Spectra-Physics Lasers Inc.) and the multipass amplifier (Femtosource Omega PRO, Femtolasers GmbH) pumped by a frequency doubled 13 W Nd:YLF laser (JADE, Thales Laser SA).
This system produces laser radiation of 27 fs pulse duration (FWHM) centred at a wavelength
of 797 nm with a bandwidth of 45 nm (FWHM) and a repetition rate of 1 kHz. Maximal available pulse energy is 800 µJ. The main parameters of the oscillator and the multipass amplifier
are summarised in Table 5.3.
A spatial and directional stabilisation of the output laser beam is implemented using an
additional stabilisation system placed behind the amplifier. The stabilisation system is based
on the analysis of two beam images created by a pair of CCD cameras which are connected to
piezoelectrically controlled mirrors for the motorised correction of the beam position and the
propagation direction. This system provides an accuracy of the stabilisation up to 0.5 µrad
[SZH06].
Since the beam diameter after the multipass amplification laser system is larger than the
standard size of laboratory optics (1 inch), the laser beam size is decreased by a telescope.
It is built by a 2 inch concave mirror with a curvature radius of −2000 mm and a 1 inch
convex mirror with a curvature radius of 500 mm. The telescope alignment is done by beam
size controlling in the far field. The telescope in the actual configuration reduces the beam
diameter four times.
64
CHAPTER 5. FEMTOSECOND LASER PULSES
Table 5.3: Output parameters of oscillator and amplifier of the multipass amplification
laser system.
Parameters
Oscillator
Amplifier
Central wavelength, nm
∼ 785
797
Spectral bandwidth (FWHM), nm
110
45
Pulse duration (FWHM), fs
< 12
27
Pulse repetition rate, Hz
7.5 × 107
1.0 × 103
Pulse energy, µJ
4 × 10−3 − 6 × 10−3
800
Pulse-to-pulse energy stability, % rms
< 0.1
< 1.5
For experiments requiring the second harmonic such as two colour pump-probe or intensity
and polarisation dependence studies the second harmonic is generated using a type I of a
beta-barium borate (BBO) crystal with a thickness of 50 µm. The thin crystal has only
moderate conversion efficiency for second harmonic generation. The maximal energy of the
second harmonic yield is limited to 100 µJ. On the other hand, time the additional dispersion
introduced by this thin crystal can be neglected so that a dispersion compensation scheme is
not required and the shortest pulses are available directly behind the crystal. The duration
of the second harmonic pulses has been measured by a self-diffraction autocorrelator [ZKo02]
and is around 25 fs.
5.3
Generation of sub -10 fs Pulses
One of the commonly used and well known techniques for generating sub -10 fs laser pulses
is based on spectral broadening in a hollow fiber filled with a noble gas [NSS97, DBK99,
SSV04]. The key idea of this technique involves the Kerr type nonlinearity of a noble gas
which induces a modulation of the optical phase (Sec. 5.2).
Phase modulation of an optical signal as a consequence of refractive index changes generated by the optical signal itself is called “self phase modulation” (SPM) [KKe95]. SPM leads
to a nonlinear shift of frequency components resulting in spectral broadening. The physical
5.3. GENERATION OF SUB -10 FS PULSES
65
Figure 5.3: (a) Laser setup for the generation of sub -10 fs pulses. (b) Intensity and phase
of a pulse with a duration of 8.2 fs measured with the SPIDER technique.
mechanism of this effect is well known [YSh84]. Noble gases exhibit weak nonlinear properties,
but long interaction lengths up to 1 m inside the hollow fibre is enough to induce a considerable
spectral broadening.
A main inherent restriction of this technique is the rather low output energy due to possible
beam distortions and even damage of optical components caused by self-focusing. Typically,
hollow fibre compressors deliver laser pulse energies limited to a few hundreds of µJ [KMS00].
Therefore, various alternatives were proposed. To obtain a larger energy with a hollow fibre
compressor a so-called “pressure-gradient method” was developed [NSH03]. It provides a
better energy transmission through the fibre and was successfully applied for the generation
of pulses with a duration of 9 fs at an energy of 5 mJ [SHN05]. Alternatively, fibre free
filamentation can be utilised for the generation of sub -10 fs pulses [BKL95, SZS06]. Selfdefocusing due to plasma formation in a filament can equilibrate the self-focusing effect from
the Kerr nonlinearity. Moreover, this method is less sensitive to experimental parameters such
as beam alignment, input pulse duration, or gas pressure [HKH04].
The setup for pulse compression used in this work has been built by the group of Dr. G.
Steinmeyer at the MBI and it is depicted in Fig. 5.3a. Laser pulses with a pulse energy of
700 µJ and a duration of 30 fs at 1 kHz repetition rate from a multipass Ti:sapphire amplifier
system are focused into a hollow capillary of 48 cm length and 250 µm inner diameter which is
filled with 500 mbar of argon to broaden the pulse spectrum. The GDD of the output pulse is
then compensated by 4 reflections from specially designed double chirped mirrors that provide
a GDD of −60 fs2 per reflection which leads to the pulse temporal compression. Pulses of up to
300 µJ energy and with a duration down to 10 fs are available after compression [SSt06]. The
66
CHAPTER 5. FEMTOSECOND LASER PULSES
temporal profile of compressed pulses is monitored by the SPIDER technique (Sec. 6.1). An
example is shown in Fig. 5.3b. The pulse compressor is tuned for the generation of single pulses
with suppressed satellites of a shortest possible duration. For example, the pulse presented in
Fig. 5.3b has a low satellite content (less than 5% of the pulse peak value) and a duration
of 8.2 fs (FWHM) only. All dispersive optical elements between the pulse compressor and the
interaction region are doubled in the pulse characterisation arm in order to simulate the real
experimental conditions. This allows one to characterise pulses which are identical to those in
the interaction region.
5.4
Laser Intensity and Polarisation Control
While carrying out experiments it is often necessary to control and/or change the laser
intensity. It is done by the variation of the laser power with two different devises: i) a linear
neutral density reflection filter (31W00ML.1, Newport Corp.) or ii) an attenuator consisting
of a
λ
-plate
2
plate coupled with two polarisers (Continuously Variable Attenuator, Altechna
Co. Ltd.).
The neutral density reflection filter is a plate with a gradient of a reflectivity across it.
The reduction of the laser power is achieved by the reflection of the laser beam from different
parts of the plate. The gradient of the reflectivity allows one to reduce the pulse power in the
range from 7% to 90%. This reflection method does not introduce any additional dispersion
and therefore avoids pulse broadening. It is extremely helpful for the manipulation of sub -10 fs
laser pulses. But the neutral density filter has two disadvantages. First, a sharp change of the
reflectivity across the plate requires a very precision calibration and reduces the accuracy of the
power attenuation. Second, the filter introduces a gradient in the spatial energy distribution
of the reflected laser beam.
The second method of the power alteration involves an attenuator which consists of a λ2 plate followed by a pair of Brewster type polarisers. The λ2 -plate rotates the linear polarisation
of the incoming laser beam. While the polarisers divide the beam into two orthogonally
polarised components. It is preferable to use the reflected vertical polarised part due to the
absence of beam transmissions through the polarisers for this component. Two polarisers are
utilised to increase a polarisation contrast as well as to keep the output beam parallel to the
5.4. LASER INTENSITY AND POLARISATION CONTROL
67
input one. The power variation due to the rotation of the λ2 -plate is
P = P0 sin2 (2θ)
,
(5.31)
where P0 is an input power, P is an output power, and θ is an angle between the horizontal
plane and the fast axis of the λ2 -plate. This method can not be used for the power attenuation
of sub -10 fs pulses because the extra dispersion created by the λ2 -plate. The second limitation
is that this attenuator introduces rather large astigmatism.
The polarisation dependent studies on C60 require the polarisation variation and control
with λ4 - and λ2 -plates. A detailed mathematical description of the different optical elements
acting on polarised light is given below.
Since light is a transverse electromagnetic wave, the vector of the electric field strength
lies in a plane perpendicular to the wave propagation direction. In this plane the electric field
vector can be considered as a sum of two orthogonal components in an arbitrary oriented
coordinate system. According to [HSc09] Eq. (5.1) for arbitrary light polarisation at the point
described by a position vector r can be presented in a vectorial form as
i E(r, t) = A0 e e−i[ω0 t−kr−φ(r,t)] − e∗ ei[ω0 t−kr−φ(r,t)]
2
,
(5.32)
where e is an unit polarisation vector and
k=
ω0
c
(5.33)
is an angular wave number. The polarisation vector can be described in two different basis sets:
the Cartesian basis ex , ey , ez and the spherical basis e+1 , e0 , e−1 . Assuming propagation of the
light along the z-axis (with the x- and y-axes pointed horizontally and vertically, respectively),
they are related to each others as
1
e+1 = − √ (ex + iey ) = −e∗−1
2
e0 = ez
,
(5.34)
,
(5.35)
and
1
e−1 = √ (ex − iey ) = −e∗+1 .
(5.36)
2
According to Eq. (5.32) the linear polarisation along x-axis of the electromagnetic wave is
described by
E(r, t) = A0 sin [ω0 t − kr − φ(r, t)]ex
,
(5.37)
68
CHAPTER 5. FEMTOSECOND LASER PULSES
while the linear polarisation along y-axis leads to
E(r, t) = A0 sin [ω0 t − kr − φ(r, t)]ey
.
(5.38)
Circular polarisation is described by
1
E(r, t) = − √ A0 (sin [ω0 t − kr − φ(r, t)]ex − cos [ω0 t − kr − φ(r, t)]ey )
2
(5.39)
1
E(r, t) = − √ A0 (sin [ω0 t − kr − φ(r, t)]ex + cos [ω0 t − kr − φ(r, t)]ey )
2
(5.40)
and
for left and right circular polarised light, respectively. In the case of circular polarisation the
unit polarisation vector is
eel = e−iδ cos βe+1 − eiδ sin βe−1
,
(5.41)
where β is the so-called ellipticity angle. Combining Eq. (5.32) and Eq. (5.41) one can write
the electric field for elliptically polarised light as a column vector


sin β sin [ω0 t − kr − φ(r, t) − δ] + cos β sin [ω0 t − kr − φ(r, t) + δ]
1
E(r, t) = √ A0  sin β cos [ω0 t − kr − φ(r, t) − δ] − cos β cos [ω0 t − kr − φ(r, t) + δ] 
2
0
(5.42)
Often, a polarisation ellipse is used to portray the light polarisation. The polarisation ellipse
is depicted by a trace of the electric field vector in a plane perpendicular to the direction of
the electromagnetic wave propagation. The two semi axes of the ellipse are
a = A0 sin (β + π/4)
(5.43)
b = A0 cos (β + π/4) .
(5.44)
and
The ratio of the two semi axes of the ellipse (the ellipticity)
ε = cot (β + π/4)
(5.45)
together with the angle between the x-axis and the major semi axis (the azimuthal angle)
φ = δ are an alternative parametrisation of the polarisation state of the light.
.
5.4. LASER INTENSITY AND POLARISATION CONTROL
(a)
(b)
e
69
e
α
α
ϑ
o
o
λ/2 - plate
λ/4 - plate
Figure 5.4: (a) The action of the λ/2-plate on linearly polarised light. The vertical and
horizontal dashed blue lines show the extraordinary (optical) and ordinary axes, respectively. Light polarisation (E-vector) is aligned at an angle α with respect to the optical axis
of the plate. (b) The action of the λ/4-plate on linearly polarised light whose polarisation
(E-vector) is aligned at an angle ϑ with respect to the optical axis of the plate. For details
see text.
λ
4
and λ2 -plates are phase retarding devices. They introduce a phase shift Γ between two
orthogonally polarised light components (ordinary and extraordinary beams)
d
Γ = ω0 (|no − ne |)
c
,
(5.46)
where d is the thickness of the plate and no , ne are ordinary and extraordinary refractive
indexes, respectively. Usually, such plates are produced from a birefringent crystal which is cut
so that the extraordinary (optical) axis of the crystal is parallel to the surface of the plate. In
particular, λ4 - and λ2 -plates create the phase difference of
π
2
and π, respectively. Their action
on linearly polarised light is shown in Fig. 5.4.
Linearly polarised light passing through the λ2 -plate whose polarisation (E-vector) is aligned
at an angle α with respect to the optical axis of the plate (as it is depicted in Fig. 5.4a) remains
linearly polarised, but the direction of the polarisation is rotated by an angle 2α.
The transmission of linearly polarised light through a λ4 -plate whose polarisation (E-vector)
is aligned at an angle ϑ with respect to the optical axis of the plate is shown in Fig. 5.4b. The
result is elliptically polarised light for arbitrary values of ϑ with circular and linear polarisation
as limiting cases for some particular values of ϑ. This angle is related to the ellipticity angle
β as
β=
π
−ϑ .
4
(5.47)
70
CHAPTER 5. FEMTOSECOND LASER PULSES
The alignment of the polarisation ellipse is independent of ϑ. Its principal axes point along
the optical axis of the λ4 -plate and perpendicular to it. Therefore, the ellipse azimuth angle φ
is 0. If ϑ = πN or ϑ = π2 (2N + 1) (N is an integer) the light is linearly polarised along the
optical axis or perpendicular to it, respectively. ϑ =
π
(2N
4
+ 1) corresponds to the circular
polarisation.
5.5
Average Intensity of Elliptically Polarised Light
The expression for the instantaneous intensity I(t, β) of the elliptically polarised light
according to Eq. (5.32) is
I(t, β) = I0 1 − sin (2β) 1 − 2 sin2 (ω0 t + ϕ)
,
(5.48)
where I0 is the laser averaged intensity, ω0 is the light carrier frequency, and ϕ is the phase.
Eq. (5.48) can be used to describe ellipticity dependence of the ion yield as a function of the
laser intensity.
One key difference between linearly and circularly polarised light is the temporal behaviour
of its instantaneous intensity I(t, β): for linearly polarised light (sin (2β) = 1) it varies as
0 ≤ I(t, π/4) ≤ 2I0 , while for circularly polarised light (sin (2β) = 0) it is constant with time
I(t, 0) = I0 .
The excitation probability in a multiphoton process does not directly follow the instantaneous intensity changes with time described by Eq. (5.48) during a single period of the
light oscillation. From a classical viewpoint one expects that the overall transition rate in an
N -photon process will depend on the time average of the intensity I N (t, β) which is
ω0
I (t, β) =
2π
N
Z
2π/ω0
[I(t, β)]N dt .
(5.49)
0
Using tabulated expressions for definite integrals Eq. (5.49) can be rewritten as
N
1
N
I (t, β) = [I0 cos (2β)] PN
,
cos (2β)
(5.50)
where PN (x) is the Legendre polynomial. Thus, the generalised multiphoton cross-section
(see Eq. (3.7)) is expected to depend on the ellipticity angle β. In a direct N -photon process
this β-dependent cross-section σN (β) will be largest for linear polarisation (β = π/4) since
5.6. PULSE SHAPING
71
the instantaneous intensity variation is largest; and it will become a minimum for circular
polarisation (β = 0 or π/2), where it is constant with time. For comparison the generalised
multiphoton cross-section is normalised to linear polarisation
N
I (t, β)
σN (β) = σN N
,
hI (t, π/4)i
(5.51)
which replaces σN in Eq. (3.7) in a case of elliptically polarised light [SRS09].
In contrast, the time averaged intensity hI(t, β)i = I0 in a single photon absorption is
independent on the polarisation state. Therefore, the excitation probability, which depends
linearly on intensity, is independent on polarisation for isotropic targets.
5.6
Pulse Shaping
Femtosecond pulse shaping technique in which practically arbitrary shaped ultrashort laser
pulses can be generated is one of the hot topics in modern laser science due to its importance
in many areas of physics, material science, chemistry, and biology [JLH07]. Since state of arts
in the technique of optical engineering is too slow to directly manipulate an electric field on
the femtosecond time scale, pulse shaping can be realised by modulation of the electric field
in the frequency domain
Ẽout (ω) = H(ω)Ẽin (ω) ,
(5.52)
where H(ω) is a modulation function, Ẽin (ω) and Ẽout (ω) are the spectral representation
of the input and output electric field, respectively. The modulation function can modulate
spectral amplitude, phase, or polarisation [PWW07]. The resulting temporal structure is
determined according to Eq. (5.6) by the inverse Fourier transformation of the modulated
spectral representation
1
E (t) =
2π
+
Z
∞
−∞
+
Ẽout
(ω)eiωt
1
dω =
2π
Z
∞
+
H(ω)Ẽin
(ω)eiωt dω
.
(5.53)
−∞
This is the so-called “Fourier transform pulse shaping”.
There are three basic techniques of Fourier transform pulse shaping using deformable mirrors [RWW04], acousto-optical modulators (AOM) [DTW97], and liquid crystal modulators
(LCM) [WZL01]. A deformable mirror is a membrane mirror driven by a set of electric actuators. The reflective surface of the membrane is displaced by actuators. Therefore, a pulse
72
CHAPTER 5. FEMTOSECOND LASER PULSES
shaping with deformable mirrors allows one to make a phase modulation only. Restricted
deformations which can be induced on the membrane together with a limited number of actuators constrain the phase variation range. The second technique employs an AOM. Usually,
it is a TeO2 (for the visible light) or InP (for the infrared light) crystal, where modulations of
the refractive index are created by the propagation of acoustic waves. Since the transit time
of a laser pulse in the crystal is much shorter than the propagation time of an acoustic wave
this wave produces a spatial grating which can be considered as a stationary structure on the
time scale of the laser pulse propagation through the crystal. The laser pulse diffraction by
this structure leads to its spectral modulation. Pulse shaping with an AOM makes available
both phase and amplitude modulations. The efficiency of an AOM is limited by 10 − 15% due
to the diffraction nature of this technique. The latter technique is based on a liquid crystal
modulator (LCM). An LCM consists of an array of liquid crystal cells (pixels). The refractive
index of each pixel can be controlled by applying the voltage across the pixel which generates position dependent either phase or amplitude modulation. The main disadvantage of an
LCM is undesirable pulse distortion resulting from the pixellation effects and gaps between the
neighbouring cells [Hae04].
Practical implementation of the Fourier transform pulse shaping includes a modulator positioned at the Fourier plane of a 4f - zero-dispersion stretcher (compressor) [Mar88]. Basically,
the 4f - zero-dispersion stretcher consists of a dispersive element (usually a grating) which
spectrally separates the input laser beam and a lens or mirror which images the spectrum into
the Fourier plane, where the modulator is installed. Subsequently, the second optical arm
which is symmetric with respect to the Fourier plane is used to perform the transformation
back to the temporal domain. Normal spherical optics are replaced with a cylindrical one that
focuses the spectrum in a vertical plane only to avoid a damage of a modulator. The distance
between the dispersive elements is equal to four times the focal length of the focusing lens or
mirror. In that case the output laser beam has the same characteristic as the input one. If the
distance between the focusing elements deviates from 2f , a temporal dispersion is introduced.
Moreover, different spectral components remain spatial distributed across the profile of the the
output beam (spatial chirp) if a position of dispersive elements moves aside the focal planes
of focusing elements.
Many different schemes for a 4f - zero-dispersion stretcher have been suggested [PWA03].
5.6. PULSE SHAPING
73
Figure 5.5: Scheme of the pulse shaping setup: A – aperture, G – spectral grating, PM –
plain mirror, CM – cylindrical mirror, and LCM – liquid crystal modulator.
The optical layout of 4f - zero-dispersion stretcher employed in this work is shown in Fig. 5.5.
To minimise beam distortions the laser beam propagates within one horizontal plane only.
The input laser beam is diffracted on a spectral grating with 1200 lines/mm. The spectrally
separated beam is focused after the reflection from a plane mirror by a cylindrical concave
mirror with f = 40 cm. An LCM (SLM-S 640/12, Jenoptik GmbH) [SHF01] is placed in a
focal plane, where the spectral modulation can be carried out. Next, the optical elements
which are symmetric with respect to the position of the LCM reconstruct the output beam.
Because lenses introduce chromatic abberations and possess some material dispersion, focusing
mirrors are a good alternative for pulse shaping of short laser pulses. The incident angle on
the cylindrical mirror has to be as small as possible to minimise astigmatism. All mirrors
in this setup have a silver coating. Dielectrically coated mirrors are avoided because of the
large phase distortions produced upon a reflection when the incidence angle deviates from a
specified value.
Unfortunately, the energy losses of the present pulse shaper setup are rather large. The loss
on the diffraction grating even in near the Littrow configuration is 20 %. Each silver coated
mirror and the LCM add losses of 7 % and 15 %, respectively. Hence, the total transmission
74
CHAPTER 5. FEMTOSECOND LASER PULSES
Figure 5.6: (a) The active area of the LCM. (b) Cross-sectional view of the LCM without
an electric field applied (top) and with an electric field applied (bottom).
of the pulse shaper setup is around 40 % only.
The alignment procedure of a 4f - zero-dispersion stretcher consisting of multiple optical
elements with many degree of freedom is complex. Nevertheless, it is well known and described
in the literature [Wei00] and has been used also for aligning the present pulse shaper setup.
Directly after the alignment, the duration of the output pulse (≈ 31 fs) is slightly longer than
the duration of the input pulse (≈ 27 fs). Temporal profiles of the pulses measured before
and after the pulse shaper setup will be reported in Chapter 6 (see Fig. 6.3c and Fig. 6.5c,
respectively). A pair of apertures is installed behind and in front of the pulse shaper setup
to control the propagation of the laser beam through the setup. Due to the drift of the laser
beam periodical adjustment of the propagation direction is required.
The construction of the LCM is shown in Fig. 5.6a. The optical area of the LCM consists of
a thin layer (9 µm) of a nematic liquid crystal placed between two parallel rubbed glass plates.
The plates are coated inside with a indium tin oxide (ITO) which is an electrically conductive
but optically transparent material. One coated plate acts as a ground plate, while the coating
of the second plate forms an array of 640 electrodes (each 96.52 µm wide, separated by gaps
of 3.05 µm) that defines the pixels. The active area of the LCM has a height of 7.0 mm and a
width of 63.7 mm. Each pixel can be addressed electronically by applying a voltage between 0
and 8 V with a 12-bit resolution (overall 4096 values). The liquid crystal in the gaps can not
be controlled.
5.6. PULSE SHAPING
75
The parallel rubbing of the glass plates causes an orientation of long rod-like molecules
of the liquid crystal along the rubbing direction (x-axis in Fig. 5.6). This substrate possesses
optical anisotropy and acts as a birefringent uniaxial crystal with its optical axis parallel to the
molecular orientation. For light propagating in z-direction the refractive index in the xz-plane
(extraordinary refractive index ne ) differs from the refractive index in the yz-plane (ordinary
refractive index no ). When the voltage is applied between the electrodes, the molecules of
the liquid crystal have a tendency to be oriented in the direction of the applied electric field
(z-direction) and the extraordinary refractive index is changed. Thus, the applied voltage
controls the birefringence and modifies the optical path length which induces a phase delay
(retardation)
Γ(U, λ) =
2πd
(nθ (U, λ) − no (λ))
λ
,
(5.54)
where nθ (U, λ) is the voltage dependent extraordinary refractive index (nθ (0, λ) ≡ ne (λ)), d is
the thickness of the liquid crystal layer, U is the applied voltage, and λ is the wavelength of the
light passed through the liquid crystal layer. The dependence of the extraordinary refractive
index on the applied voltage is
1
cos2 (θ(U )) sin2 (θ(U ))
=
+
n2θ (U, λ)
n2e (λ)
n2o (λ)
,
(5.55)
where θ(U ) is an angle of the liquid crystal molecules tilt induced by the applied voltage
[Jen02]. This angle increases monotonous as a function of the applied voltage.
The calibration of the LCM requires knowledge about the phase retardation. It can be
determined from the measurement of a light transmission T (U, λref ) at some wavelength λref
with crossed polarisers [Hen03]. The LCM inside the 4f - zero-dispersion stretcher is placed
between two crossed polarisers in xy-plane oriented at +45 ◦ and −45 ◦ with respect to xaxis (see Fig. 5.6) and transmission at λref as a function of the driver voltage value can be
measured by a spectrometer. This transmission depends on the phase retardation according
to [Jen02] as
T (U, λref ) = sin
2
Γ(U, λref )
2
.
(5.56)
The transmission curve measured at λref = 800 nm and utilised for the LCM calibration is
shown in Fig. 5.7a. By measuring transmission at different wavelengths information about the
wavelengths λ propagating across each pixel of the LCM PN can be obtained. This dependence
p h a s e r e ta r d a tio n [r a d .]
CHAPTER 5. FEMTOSECOND LASER PULSES
tr a n s m is s io n [n o r m a lis e d ]
76
(a )
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
0
5 0 0
1 0 0 0
1 5 0 0
w a v e le n g th [n m ]
∆n ( λ) = n e ( λ) - n o ( λ)
0 .2 2 5
0 .2 2 0
0 .2 1 5
1 5
1 0
7 5 0
7 7 5
8 0 0
0
0
8 2 5
5 0 0
1 0 0 0
1 5 0 0
2 0 0 0
d r iv e r v a lu e [a r b . u n its ]
(d )
8 5 0
m e a s u re d
lin e a r fit
8 2 5
8 0 0
7 7 5
7 5 0
7 2 5
7 2 5
5
8 7 5
(c )
c a lc u la te d
fit
2 0
2 0 0 0
d r iv e r v a lu e [a r b . u n its ]
0 .2 3 0
(b )
2 5
0
1 2 8
2 5 6
3 8 4
5 1 2
6 4 0
p ix e l n u m b e r
w a v e le n g th [n m ]
Figure 5.7: Calibration of LCM: (a) measured transmission function T (U, λref ) (λref =
800 nm); (b) recalculated phase retardation Γ(U, λref ) and its fit; (c) wavelength dependence of the dispersion ∆n(λ); (d) wavelength dependence on the pixel number of the
LCM.
depicted in Fig. 5.7d is clearly linear
λ = −0.17644PN + 854.10441 ,
(5.57)
while coefficients (-0.17644 and 854.10441) are determined by the alignment of the 4f - zerodispersion stretcher and a position of the LCM. The phase retardation recalculated from
Eq. (5.56) as
q
T (U, λref ) ± 2πk
Γ(U, λref ) = arcsin
(5.58)
is presented in Fig. 5.7b. The phase retardation Γ(U, λ) at an arbitrary wavelength can be
converted to the retardation Γ(U, λref ) at the known wavelength as
Γ(U, λref ) = Γ(U, λ)
λ ∆n(λref )
λref ∆n(λ)
,
(5.59)
5.6. PULSE SHAPING
where ∆n(λ) = ne (λ) − no (λ). ∆n(λ) is approximated by the following expression
r
0.04008
∆n(λ) = λ
,
2
λ − 0.10722
77
(5.60)
where λ is measured in µm [Jen02]. This dependence is shown in Fig. 5.7c. The recalculated
phase retardation Γ(U, λref ) together with Eq. (5.59) and Eq. (5.60) can be used for the LCM
calibration.
A voltage needed to obtain the required phase retardation at some arbitrary pixel of the
LCM can be calculated using the following procedure. First, the wavelength λ corresponding to
this pixel has to be determined using Eq. (5.57). Second, the required phase retardation at this
wavelength has to be converted to the known retardation at the wavelength λref according to
Eq. (5.59) and Eq. (5.60). Third, the driver voltage value can be calculated using the following
expression
Γ(U, λref )
Γ(U, λref )
U = −71.6 + 1483.5 exp −
+ 389.0 exp −
.
2.1
19.7
(5.61)
This expression is derived from a fit of the known data Γ(U, λref ) and presented in Fig. 5.7b.
In spite of the LCM capability to retard the phase up to approximately 5π at 800 nm, the
maximal phase retardation is restricted by 2π during the calibration procedure. Possible values
of the phase retardation lies in the range between 2π and 4π.
78
CHAPTER 5. FEMTOSECOND LASER PULSES
Chapter 6
Pulse Characterisation Techniques
A detailed investigation of complex molecular systems interacting with femtosecond laser
radiation is only meaningful if one possesses all information about the laser pulses used in the
experiment. Measurements of the beam energy, spectral distribution, spatial profile, and wave
front can be easily done even in the case extremely short laser pulses. But the measurement
of both, amplitude and phase, of very short laser pulses in the time domain is not a trivial
problem.
First, methods for temporal pulse characterisation are described in this chapter. After that,
spatial characterisation techniques of laser beams are presented. And finally, the energetic
properties of laser pulses are described.
6.1
6.1.1
Temporal Pulse Characterisation
Methods of Temporal Pulse Characterisation
Usually, a short event can be characterised by another, even shorter event. Classical electronic techniques using fast streak cameras for temporal pulse width characterisation have,
at best, a resolution of ∼ 0.5 ps [Kim03]. Hence, femtosecond pulse characterisation in the
time domain requires other methods. These are based either on spectral interferometry or
optical correlation techniques that make use of the short pulse itself. The simplest method
is an “autocorrelation measurement”. It provides minimum information namely the pulse
duration[SGR87, TDF97]. There is some information on the pulse shape as well albeit symmetrised. The second method is the “frequency-resolved optical gating” (FROG) technique.
79
80
CHAPTER 6. PULSE CHARACTERISATION TECHNIQUES
FROG involves a relatively simple experimental setup for determining the spectrally resolved autocorrelation function. Coupled with a mathematical algorithm this allows one to retrieve both
the pulse amplitude and phase [Ktr93, DFT96, TDF97]. If a well characterised reference pulse
is available, one may employ the so-called “cross-correlation frequency resolved optical gating”
(XFROG). It involves measuring the spectrally resolved cross-correlation (sum frequency generation) of the unknown pulse with a full characterised reference pulse [LGK98, YFK01, DGX02].
Another method, based on spectral interferometry, is “spectral phase interferometry for direct electric-field reconstruction” (SPIDER) [IWa99, Dor99]. SPIDER is ideally suited for the
characterisation of ultrashort single pulses [GSM99]. It combines a rather simple experimental
setup with a robust and non-iterative mathematical algorithm for fast and noise insensitive
pulse reconstruction. Moreover, the reconstructed pulse is free from ambiguities of time direction even when the second order nonlinearity is used. Combination of FROG and spectral
interferometry leads to a new method which is known as “temporal analysis, by dispersing
a pair of light electric fields” (TADPOLE). This technique does not use a nonlinear optical
medium and therefore has an extremely high sensitivity [LCJ95, FBS96, DBL00]. But it requires that the spectral width of the unknown pulse is completely covered by the spectral
width of the reference pulse.
Alternatively, different methods for temporal pulse characterisation can be applied. The
preference of one or another method depends on a wide range of circumstances such as required
accuracy of measurements, complexity of pulse structure, pulse duration, available pulse energy,
or accessibility of a fully characterised reference pulse. In our case intensity autocorrelation
and FROG technique are used for characterisation of the pulses directly delivered by the the
laser system. The alignment of the shaper needs the possibility to control both amplitude
and phase on-line. That is the reason why a SPIDER is employed for the shaper alignment.
The same technique is used for the characterisation of sub -10 fs pulses. And finally, in pulse
shaper experiments, where a reference pulse is available, the resulting complicated pulses are
characterised by the XFROG method. A detailed description of all these techniques is given
below.
6.1.2
Autocorrelation Measurement
Fig. 6.1 shows the setup for the intensity autocorrelation measurement used in the present
6.1. TEMPORAL PULSE CHARACTERISATION
81
Figure 6.1: Schematic diagram of an autocorrelator (if the detector is a photodiode) or of
a FROG setup (if the detector is a spectrometer).
work. The setup is designed to minimise temporal and spectral aberrations. The unknown
pulse is split into two parts by a 50% : 50% thin (thickness of 1 mm) beamsplitter. One pulse
is variably delayed with respect to the other pulse by passing through a computer controlled
motorised translation stage. Then these pulses are focused by a lens with a focal length of
50 mm and overlapped under a crossing angle into a nonlinear optical medium, as a second
harmonic generating (SHG) crystal. In the actual setup the type I of a beta-barium borate
(BBO) crystal with a thickness of 50 µm is used. Such crystal thickness is small enough to
minimise temporal smearing and to fulfil the conditions of phase matching [Wei83, OKG00].
It is important to mention that the crossing angle must be as small as possible to avoid losses
of temporal resolution [TKS96, BPW99].
The SHG crystal produces a signal at twice the frequency of the input pulse in the direction
of the bisector of the two incoming pulses. The autocorrelation signal is related to the intensities of the two incoming pulses by Eq. (5.17). An aperture lets pass only the autocorrelation
signal. One extra reflection is used to remove the remaining radiation at the frequency of
the input pulse. A 100 mm focal length lens focuses the autocorrelation signal onto a photodiode interfaced to a LabVIEW computer. To achieve better statistics the measurement of
the autocorrelation signal (Eq. (5.17)) at each time delay δ is done for 1000 laser shots and a
complete cycle of measurements is usually repeated at least 10 times. The minimal step of the
82
CHAPTER 6. PULSE CHARACTERISATION TECHNIQUES
in te n s ity [a r b . u n its ]
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
-1 5 0
-1 0 0
-5 0
0
5 0
1 0 0
1 5 0
d e la y [fs ]
Figure 6.2: Pulse duration measurement by the autocorrelation technique for the pulse
produced by the Multipass laser system. The measured autocorrelation is depicted by black
open squares and fitted with Gaussian, sech2 , and Lorentzian functions as it is shown by
full red, dashed blue, and dotted olive lines, respectively. The pulse durations (FWHM)
are 32.77 ± 0.46 fs, 29.09 ± 0.34 fs, and 21.89 ± 0.22 fs according to Gaussian, sech2 , and
Lorentzian fits, respectively.
time delay available from the motorised translation stage is 4 fs. The intensity autocorrelation
shows a maximum at δ = 0 and is always symmetrical. Autocorrelation measurements provide
only a limited amount of information about an unknown pulse. Specifically, it contains no
information about the phase of the pulse and the symmetry of any autocorrelation trace is
a serious disadvantage for pulse form determination. To obtain information about the pulse
width one has to make an intelligent guess about pulse shape. Generally, the full width at
half-maximum (FWHM) of the unknown pulse τ is proportional to the FWHM of the measured intensity autocorrelation function τAC . But the proportionality factor (or deconvolution
factor) varies significantly for different pulse shapes (see Sec. 5.1 and Eq. (5.18)).
Fig. 6.2 shows the autocorrelation measurement of the pulse produced by the Multipass
laser system. The measured autocorrelation is depicted by black open squares and fitted
6.1. TEMPORAL PULSE CHARACTERISATION
83
with Gaussian, sech2 , and Lorentzian functions as it is shown by full red, dashed blue, and
dotted olive lines, respectively. The pulse durations (FWHM) obtained by Gaussian, sech2 , and
Lorentzian fits are 32.77 ± 0.46 fs, 29.09 ± 0.34 fs, and 21.89 ± 0.22 fs, respectively. It is clear
that the laser pulse duration determined in a such type of measurements strongly depends
on the assumption about the pulse shape. Among these three fit functions the sech2 one
describes the measured data better excepting the region of rather large wings coming from a
small pre-pulse which is not resolved in this measurement. Therefore, the sech2 pulse shape is
always assumed later on for the pulse duration estimations in autocorrelation measurements.
Also it is difficult to determine the presence or absence of some systematic or random errors
in the measured autocorrelation function. Nevertheless, the autocorrelation measurement can
be a powerful tool for the pulse width determination of simple pulses with an a priori known
shape.
6.1.3
FROG Technique
As described above, the intensity autocorrelation measurements do not give sufficient information for full temporal characterisation of arbitrary shaped pulses. The FROG technique is
free from the drawbacks of an autocorrelation measurement. This technique operating in the
“time-frequency domain” involves both temporal resolution and frequency resolution simultaneously. FROG does this by spectrally resolving the signal pulse in any type of autocorrelation
measurement performed in an instantaneously responding nonlinear medium. Depending on
the nonlinear optical effect used for the signal pulse generation there are several FROG geometries such as polarisation gating (PG), self diffraction (SD), transient grating (TG), third
harmonic generation (THG), and second harmonic generation (SHG) [TDF97].
SHG FROG was chosen in this work as a standard measurement technique for the characterisation of the pulses produced by the laser system because it is the most sensitive and
simple FROG geometry. It involves only a second order nonlinearity. The main disadvantage
of SHG FROG is the symmetry with respect to the time delay and, hence, SHG FROG has
an ambiguity in the direction of time. This ambiguity can be removed in one of several ways
[TDF97]. Practical implementation of this technique is done using the setup shown in Fig. 6.1,
where the SHG signal is spectrally resolved by the spectrometer (AvaSpec-2048, Avantes BV).
The spectral resolution of the spectrometer is 0.17 nm and it is enough for these purposes.
84
CHAPTER 6. PULSE CHARACTERISATION TECHNIQUES
4 1 0
1 .0
(a )
in te n s ity [a r b . u n its ]
w a v e le n g th [n m ]
4 1 5
4 0 5
4 0 0
3 9 5
3 9 0
3 8 5
3 8 0
-1 5 0
m a r g in a l
a u to c o n v o lu tio n
(b )
0 .8
0 .6
0 .4
0 .2
0 .0
-1 0 0
-5 0
0
5 0
1 0 0
3 7 0
1 5 0
3 8 0
1 0
= 2 7 .9 + 0 .8 fs
0 .8
8
0 .6
6
0 .4
4
0 .2
2
0 .0
-1 5 0
0
-1 0 0
-5 0
0
5 0
1 0 0
1 5 0
tim e [fs ]
1 .0
in te n s ity [a r b . u n its ]
F W H M
(c )
p h a s e [r a d ia n ]
in te n s ity [a r b . u n its ]
1 .0
3 9 0
4 0 0
4 1 0
4 2 0
4 3 0
w a v e le n g th [n m ]
d e la y [fs ]
F W H M
(d )
= 2 8 .9 + 1 .2 fs
0 .8
0 .6
0 .4
0 .2
0 .0
-1 5 0
-1 0 0
-5 0
0
5 0
1 0 0
1 5 0
d e la y [fs ]
Figure 6.3: Temporal characterisation of the pulses produced by the Multipass laser system:
(a) recorded FROG map; (b) frequency marginal and autoconvolution; (c) reconstructed
temporal intensity and phase from the FROG map; (d) measured autocorrelation function.
The spectral response of the spectrometer is calibrated using a source of well-characterised
emission spectrum.
The spectrogram of the signal pulse in SHG FROG is given by
2
Z ∞
SHG
−iωt
IF ROG (ω, δ) = E(t)E(t − δ)e
dt
.
(6.1)
−∞
In contrast to the autocorrelation measurements described by Eq. (5.17) this spectrogram
provides an unknown pulse representation as a two dimensional function of frequency and delay
time and comprises essential information required for full pulse characterisation. One example
of a spectrogram or “SHG FROG map” is shown in Fig. 6.3a. This is the spectrogram of the
pulses produced by the Multipass laser system. Any FROG map is a type of time-frequency
distribution that contains all relevant information about the pulse. An iterative pulse retrieval
algorithm extracts the pulse intensity and phase from a FROG map by finding the electric
6.1. TEMPORAL PULSE CHARACTERISATION
85
field that best reproduces the map. It must be mentioned before discussing the algorithm
that FROG provides consistency checks not available with autocorrelation measurements. An
electric field sampled at N points has 2N degrees of freedom (N points of both magnitude
and phase), but it corresponds to a FROG map with N 2 points. The fact that the FROG
map contains redundant data allows for such checks. This is accomplished by comparing the
”frequency marginal” of the FROG map with the independently measured pulse spectrum
[DFT96, TDF97]. The frequency marginal is the FROG map integrated with respect to the
delay
MFf ROG (ω)
Z
∞
IFSHG
ROG (ω, δ) dδ
=
.
(6.2)
−∞
The frequency marginal has to be equal to the autoconvolution of the fundamental pulse
spectrum S(ω)
MFf ROG (ω)
Z
∞
S(ω 0 − 2ω0 )S(ω 0 − ω)dω 0
=
,
(6.3)
0
where ω0 is the carrier frequency. A systematic experimental error such as incorrect wavelength
or temporal calibrations of FROG data, insufficient doubling crystal bandwidth, spatial or
temporal distortions of the pulses at the focus can be a reason of non-agreement of the
SHG FROG frequency marginal with the autoconvolution of the pulse spectrum. Fig. 6.3b
shows an example of how the frequency marginal can be used to check the validity of SHG
FROG measurements. There is rather good agreement of the frequency marginal with the
autoconvolution of the measured spectrum. Small narrowing of the frequency marginal is
explained by insufficient doubling crystal bandwidth or/and other bandwidth losses in the
FROG setup. Thus, this marginal provides a powerful check of the experimental apparatus.
Moreover, it is possible to correct the measured FROG map by multiplying its with a ratio of
the fundamental pulse spectrum autoconvolution and the frequency marginal. There is also a
“delay marginal”
MFd ROG (δ)
Z
=
∞
IFSHG
ROG (ω, δ) dω
,
(6.4)
0
which is essentially the same as an autocorrelation function and can be used for a quick
estimation of τ .
As already mentioned, the information contained in the FROG map is sufficient for a full
determination of amplitude and phase of the pulsed electric field. The goal of pulse retrieval
algorithm is to find an E(t) that satisfies Eq. (6.1). Several modification of FROG pulse
retrieval algorithms were developed and can now be used for electric field reconstruction.
86
CHAPTER 6. PULSE CHARACTERISATION TECHNIQUES
The simplest of them, but a very powerful one, is the method of generalised projections
[DFT94, TDF97]. This algorithm starts the first iteration with an initial guess of the pulsed
field E(t) and reconstructs the FROG map using Eq. (6.1). Then the FROG error G is
calculated as an rms difference between the measured FROG map IF ROG (ωi , δj ) and the
(k)
reconstructed FROG map IF ROG (ωi , δj )
v
u
N 2
u 1 X
(k)
t
G=
IF ROG (ωi , δj ) − αIF ROG (ωi , δj )
2
N i,j=1
,
(6.5)
where N is a size of the FROG map array, α is the real number for the renormalisation, and k
is the number of the iteration step. The algorithm analyses the FROG error G and generates
a new guess for E(t) using the method of generalised projections. Then the initial guess
is replaced with a new one and the whole procedure is repeated in a cycle. The goal of the
algorithm is to find such a pulsed field E(t) that minimises the FROG error G. For our purposes
of pulse characterisation the commercial program (FROG 3.0.9, Femtosoft Technologies) is
used. This program allows one to use several retrieval algorithms. The measured FROG map
together with the retrieved amplitude and phase of the pulses delivered by the laser system is
shown in Fig. 6.3a and Fig. 6.3c, respectively.
6.1.4
XFROG Method
For the temporal characterisation of shaped pulses with complicated structure XFROG is
applied. In this technique a fully characterised reference pulse is overlapped with the unknown
pulse to be characterised in the BBO crystal. The spectrally resolved sum frequency signal is
measured as a function of the delay between the reference and the unknown pulse and gives
the XFROG map
SHG
IXF
ROG (ω, δ)
Z
= ∞
−∞
−iωt
Eunknown (t)Eref (t − δ)e
2
dt
.
(6.6)
Both XFROG and SHG FROG use the same nonlinear process. But the SHG FROG involves
only one replica of an unknown pulse, while the XFROG involves two pulses that are not
identical. This leads to additional advantages of the XFROG technique. The main one is the
absence of a time ambiguity in XFROG maps. XFROG maps are more accessible to intuition
than FROG maps. Also this technique is better suited for characterisation of weak pulses.
The XFROG setup is similar to the FROG setup as shown in Fig. 6.4. A thin glass plate with a
6.1. TEMPORAL PULSE CHARACTERISATION
87
Figure 6.4: Schematic diagram of a cross-correlator (if the detector is a photodiode array)
or of an XFROG setup (if the detector is a spectrometer).
reflectivity of only 0.4 % is inserted into the laser beam path just in front of the shaper to create
the reference pulse. Phase and amplitude distribution of this pulse are determined by a FROG
as discussed above, and then used as the fully characterised reference. An additional delay line
is introduced to equalise the optical paths of both pulses, the reference pulse and the pulse
emerging from the the shaper which is to be characterised. The reference pulse is overlapped
with the unknown pulse in a BBO crystal. The sum frequency signal is measured for different
delays between the reference and the unknown pulse, thus generating the XFROG map. This
XFROG map then is used to reconstruct phase and amplitude of the unknown pulse, using
commercial XFROG software (FROG 3.0.9, Femtosoft Technology). Fig. 6.5a and Fig. 6.5c
show an XFROG map and the reconstructed amplitude and phase for the pulse propagated
though the shaper without applying any extra phase on it. Some pulse lengthening and pulse
shape distortions for the pulse propagated though the shaper as compared to the incoming one
are caused by the bandwidth losses in the shaper setup and unavoidable optical aberrations
induced in the shaper.
6.1.5
SPIDER Technique
During the shaper alignment a temporal pulse characterisation is performed using the
88
CHAPTER 6. PULSE CHARACTERISATION TECHNIQUES
in te n s ity [a r b . u n its ]
4 1 0
1 .0
(a )
4 0 5
4 0 0
3 9 5
3 9 0
3 8 5
3 8 0
-1 5 0
-1 0 0
-5 0
0
5 0
1 0 0
8
0 .6
6
0 .4
4
0 .2
2
-1 0 0
-5 0
0 .8
8
0 .6
6
0 .4
4
0 .2
2
0 .0
-1 5 0
-1 0 0
-5 0
0
tim e [fs ]
5 0
1 0 0
1 5 0
1 .0
0
in te n s ity [a r b . u n its ]
= 3 1 .2 + 1 .1 fs
1 0
p h a s e [r a d ia n ]
in te n s ity [a r b . u n its ]
F W H M
(c )
0
5 0
1 0 0
1 5 0
0
tim e [fs ]
d e la y [fs ]
1 .0
1 0
= 3 0 .4 + 1 .5 fs
0 .8
0 .0
-1 5 0
1 5 0
F W H M
(b )
p h a s e [r a d ia n ]
w a v e le n g th [n m ]
4 1 5
F W H M
(d )
= 3 1 .7 + 1 .6 fs
0 .8
0 .6
0 .4
0 .2
0 .0
-1 5 0
-1 0 0
-5 0
0
5 0
1 0 0
1 5 0
d e la y [fs ]
Figure 6.5: Temporal characterisation of the pulses passed through the shaper setup: (a)
measured XFROG map; (b) temporal intensity and phase measured with SPIDER technique; (c) reconstructed temporal intensity and phase from an XFROG map; (d) measured
cross-correlation.
SPIDER technique (APE Spider, APE GmbH). The same technique is also employed for the
temporal characterisation of the sub -10 fs pulses. This interferometric technique is based on
nonlinear conversion of two temporally delayed replicas of an unknown pulse with a stretched
pulse[GSM99, SWW99].
The principle of a SPIDER is shown in Fig. 6.6a. An unknown pulse is split into two
replicas with a constant time delay δ between each another using a glass etalon or Michelson
interferometer. These replicas are with a stretched, chirped pulse in a nonlinear medium. The
chirped pulse is produced by stretching a part of the unknown pulse itself (alternatively, one
may use a pulse from an external source) in a dispersive delay line. This can be a block of
dispersive material (glass) as well as a pair of prisms or gratings. The width of the chirped
pulse has to be much larger than the time delay δ. Since there is a time delay between the
replicas, each pulse replica is upconverted in the nonlinear medium with a different frequency
6.1. TEMPORAL PULSE CHARACTERISATION
89
Figure 6.6: Principle of SPIDER apparatus: (a) SPADER optical layout; (b) algorithm
for the spectral phase reconstruction [IWa98]. For details see text.
slice of the chirped pulse, and consequently, the two upconverted pulses have slightly different
frequencies. The result of the mixing is two temporally delayed and spectrally shifted replicas
which interfere in a spectrometer producing an interferogram
S(ωc ) = |E(ωc )|2 + |E(ωc + Ω)|2 + 2 |E(ωc )E(ωc + Ω)| ×
(6.7)
× cos [φ(ωc + Ω) − φ(ωc ) + ωc δ] ,
where ωc is the central frequency of the pulse, δ is the fixed delay time between replicas, Ω
is the the frequency shift or the amount of spectral shear, E(ωc ) is the pulse electric field in
frequency domain, φ(ωc ) is the pulse phase. From this spectrogram it is possible to calculate
the spectral phase of the unknown pulse. For exact phase reconstruction the spectral shear
must satisfy the Whittaker-Shannon sampling theorem [Goo96]. The value of the spectral shear
is adjusted by changing the second order dispersion of the stretched pulse. In addition, the
interferogram sampling interval has to be not greater than the Nyquist limit [Goo96]. With the
calculated spectral phase, an additional measurement of the unknown pulse spectrum allows
one to completely characterise such pulse.
A robust non-iterative mathematical procedure can be used for the spectral phase extraction
from the measured spectral interferogram [TIK82]. This procedure is schematically shown in
Fig. 6.6b. First, the interferogram is Fourier transformed into the time domain. The result of
this transformation gives three peaks centred at t = 0, +δ. The peak near zero time (t = 0)
is due to the constant terms in Eq. (6.7), while the peaks at t = +δ are due to the cosine
90
CHAPTER 6. PULSE CHARACTERISATION TECHNIQUES
term. Each of these two peaks possesses all phase information. The peaks at t = −δ, 0 are
removed by filtering and the remaining peak at t = δ is inverse Fourier transformed back to
the frequency domain. After subtraction of the linear term ωδ that is due to the delay δ, the
remaining part is the phase difference between two frequencies separated by the spectral shear
Ω: φ(ωc + Ω) − φ(ωc ). Finally, the spectral phase of the unknown pulse for a discrete set of
frequencies is reconstructed by concatenation of the phase in steps of Ω.
A result of the SPIDER technique applied to characterise a pulse propagated through the
shaper is shown in Fig. 6.5b. This result is rather similar to the one obtained with the XFROG
technique (Fig. 6.5c). Mainly, the difference between these two results comes from a fact
that these measurements were done in two different days. Larger distortions of the pulse
structure in the case of the XFROG technique are due to the iterative nature of the pulse
reconstructive algorithm. Nevertheless, both techniques found characteristic features of the
temporal pulse structure and determined essentially the same τ . Therefore, both SPIDER and
XFROG techniques can be employed for the temporal characterisation of femtosecond laser
pulse.
6.2
Spatial Laser Beam Characterisation
An idealised Gaussian beam can be considered as good approximation for the simplest
types of beams provided by laser sources. Its spatial parameters and propagation properties
are well known [BWo99]. But real beams deviate more or less from this idealised description.
The closest physical approach to the idealised Gaussian beam is the rotationally symmetric
beam from a low power TEM00 laser. For the sake of simplicity, only such type of beams is
analysed below.
Spatial characterisation of a laser beam at the focal point is very important for the determination of the interaction volume, laser fluence, and intensity. The essential information lies
in the knowledge of the spot size of a focused laser beam. Usually in laser physics the spot
size is depicted by w (a radius at which the intensity drops down to 1/e2 with respect to its
maximum value).
There are many techniques of measuring laser beam sizes [AHM71, SWe81, KLa83, HKo91].
If the radius of the focal spot is larger than the laser wavelength (w λ), the intensity profile
6.2. SPATIAL LASER BEAM CHARACTERISATION
91
can be imaged by a CCD camera [MMR01] or simply be recorded by measuring the intensity passed by a knife-edge scanning the laser beam in the focal plane [AHM71, FHS77].
Smaller spots can be characterised using different modifications of the knife-edge scan technique [STa75, CLL84, CGC86] or Ronchi ruling method [CLL84, CGC86].
Under the action of a focusing mirror (lens) with a focal length f and assuming a parallel
monochromatic Gaussian input beam, one can calculate the beam waist at the focal plane wf
with the following formula
wf =
fλ
πw0
,
(6.8)
where w0 is the waist of the unfocused beam. Both waists have to be measured at 1/e2 level.
The measurement of the unfocused laser beam waist w0 is much simpler than the measurement
of wf at the focal plane. Therefore, one can measure the waist of the unfocused laser beam
and estimate the beam waist at the focal plane using Eq. (6.8). However, Eq. (6.8) assumes the
perfect focussing conditions which can not be always satisfied for real laser systems. Hence,
the direct measurement of the beam waist at the focal plane is more accurate and desirable.
In the present studies such measurement is done using the knife-edge scan at the focal
plane of the focusing mirror [FHS77], where a thin metallic plate is translated across the laser
beam and the intensity is recorded as a function of the plate position. For this measurement
the pulse energy has to be diminished by the several reflections from uncoated glass for the
reduction of the pulse intensity at the focal plane below the damage threshold of the metallic
plate used for the scanning. The spatial intensity distribution I(x, y) of a Gaussian beam at
the focal plane is
I(x, y) =
2
2P0 −2 x2 +y
w2
e
πw2
,
(6.9)
where P0 is the total laser power, x and y are coordinates in the plane perpendicular to the
beam axis. The origin of this coordinate system is taken on the beam axis. If the beam is
partly blocked by a plate aligned parallel to the y axis, the transmitted laser power is
√
Z ∞Z ∞
Z ∞
2
2P0 −2 x2 +y
2
1 − 2x22
2
w
√
P (x0 ) =
e
dydx
=
P
e w dx =
0
2
π
−∞ x0 πw
x0 w
!
√
(6.10)
1
2
= P0 erfc
x0
,
2
w
where x0 is the current position of the plate edge on the x axis. By scanning the x coordinate
with the plate and recording the transmitted laser power P (x) as a function of x, the beam
92
CHAPTER 6. PULSE CHARACTERISATION TECHNIQUES
Figure 6.7: Spatial beam characterisation: (a) measured knife-edge scan of the laser beam
for the Multipass laser system, its derivative, and Gaussian fit; (b) calculated spatial
intensity distribution using the waist obtained in Fig. 6.7a. The laser propagates in z
direction. The horizontal black dotted lines shows the beam waist w. The vertical black
dotted lines give the Rayleigh region. The red dashed lines indicate the molecular beam
interacting with the laser beam.
profile along the x axis is obtained by the differentiation of the recorded signal. To get the
beam profile in the perpendicular direction one has to repeat this procedure for the scanning
along y axis. Fig. 6.7a illustrates the result of a typical scan at the focal plane of a concave
mirror with f = 50 cm for the Multipass laser system. The derivative of this scan is fitted by
a Gaussian function with w = 72.3+1.1 µm.
Close to the focal plane the beam collimation along the direction of propagation is characterised by the Rayleigh length zR
zR =
πw2
λ
,
(6.11)
defined as the distance between the focal plane and the plane at which the beam waist increases
√
by a factor of 2. For the waist measured in Fig. 6.7a w = 72.3 µm, the corresponding
Rayleigh length is zR = 20.5 mm. The distance between the points ±zR about the waist is
called the confocal parameter
b = 2zR
(6.12)
of the beam. Fig. 6.7b shows a (calculated) map of the spatial intensity distribution around
the focal plane for a focused laser beam centred at the wavelength of 800 nm. The beam
propagates along the z axis and has radial symmetry with respect to this axis at x, y = 0.
6.3. PULSE INTENSITY AND FLUENCE
93
The focal plane is perpendicular to both the picture plane and z axis. It is located at z = 0.
The beam symmetry axis and the focal plane are indicated by solid lines. The beam waist
w = 72.3 µm and the confocal parameter b = 41 mm are shown by the black dotted lines. As
discussed in Sec. 4.2, the diameter of the effusive molecular beam used in this work is only
DM = 5 mm at the point, where the molecular beam crosses the laser beam. The red dashed
lines in Fig. 6.7b represent the molecular beam. Since the Rayleigh length is larger than the
molecular beam diameter it is straightforward to assume for the experimental geometry used
in the present work that the interaction region is a cylinder with the radius w and the length
Dm . Intensity variations along the cylinder axis in this case are negligible small.
6.3
Pulse Intensity and Fluence
Intensity and fluence are two most important attributes of laser radiation when describing
laser-matter interaction. Mainly, intensity is considered in the present work for this characterisation.
The intensity I(r, t) of an ideal Gaussian laser pulse at a given time t and a distance r
from the beam axis can be written as
r
2
t
2
I(r, t) = I0 e−2( w ) e−4 ln 2( τ )
,
(6.13)
where I0 is the pulse peak intensity, τ is a temporal FWHM, and w is a beam waist (a radius
at which the intensity drop down to 1/e2 with respect to its maximum value). Sometimes, the
beam radius is measured at levels, where the intensity decreases to 1/e (w1/e ) or 1/2 (w1/2 ).
These two quantities relate to w as
√
w1/e =
2
w
2
(6.14)
and
r
w1/2 =
ln 2
w
2
.
(6.15)
Both spatial and temporal integration of Eq. (6.13) leads according to [HSc09] (see Eq. (13.67))
to the pulse energy
Z
W = I0
∞
2
e
0
r
−2( w
)
Z
∞
2
e
−∞
−4 ln 2( τt )
√
π π 2
drdt = I0 √
w τ
4 ln 2
.
(6.16)
94
CHAPTER 6. PULSE CHARACTERISATION TECHNIQUES
The pulse energy can be quite easy measured by a photodetector. In this work, it is done
using the pyroelectric laser energy metre (TPM-300CE, Gentec Electro-Optics, Inc.). The
peak intensity is obtained from Eq. (6.16) by term rearrangement
√
√
√
4 ln 2 W
2 ln 2 W
2 ln3 2 W
√
I0 = √
= √
=
2
2
τ
τ
π π w2 τ
π π w1/e
π π w1/2
.
(6.17)
Intensity is commonly measured in units of [W/cm2 ]. Only spatial integration of Eq. (6.13)
gives the time dependent pulse power (the amount of energy per unit of time)
Z ∞
2
r 2
t 2
π
−4 ln 2( τt )
e−2( w ) dr = I0 w2 e−4 ln 2( τ ) .
P (t) = I0 e
2
0
(6.18)
The peak power P0 is obtained from Eq. (6.18) at t = 0
π
π 2
2
P0 = P (0) = I0 w2 = I0 πw1/e
= I0
w
2
ln 2 1/2
.
(6.19)
The peak intensity is simply equal to the pulse peak power divided by the waist area measured
at 1/e level as mentioned in [HSc09].
Integration of Eq. (6.13) over time only gives the fluence (the amount of energy per unit
of area)
2
F (r) = I0 e
r
−2( w
)
Z
∞
2
e
−4 ln 2( τt )
−∞
√
r 2
π
dt = I0 √
τ e−2( w )
2 ln 2
.
(6.20)
The fluence is usually measured in units of [J/cm2 ]. The peak fluence F0 is obtained from
Eq. (6.20) at r = 0
√
π
F0 = F (0) = I0 √
τ
2 ln 2
.
(6.21)
Using Eq. (6.16) the expressions for peak power and peak fluence can be rewritten as
√
2 ln 2 W
P0 = √
(6.22)
π τ
and
F0 =
2W
π w2
.
(6.23)
Since the pulse energy, the pulse duration (Sec. 6.1), and the beam waist (Sec. 6.2) are
measurable, the intensity, the power, and the fluence can be calculated using Eq. (6.17),
Eq. (6.22), and Eq. (6.23), respectively.
Chapter 7
Optimisation
The possibility of fast and practically arbitrary laser pulse modification with pulse shaping
techniques described in Sec. 5.6 provides a very effective tool for an active control of molecular
processes. This control can be obtained through the interaction of shaped laser pulses with a
molecular system, where a closed loop between the pulse shaper and the resulting response of
the molecular system is utilised together with a self-learning algorithm [JRa92]. Such scheme
of an adaptive closed loop allows one to control different photoinduced processes such as
bond dissociation and rearrangement [LMR01], selective fragmentation [ABB98], laser induced
fluorescence [BYW97], or high harmonic generation [BBZ00].
This chapter presents concepts of adaptive closed loop optimisation. It begins with a
general overview of excising optimisation methods. Then methods based on stochastic search
are described in detail. Finally, the practical implementation of the optimisation algorithm
employed in the present work is described.
7.1
Introduction to Mathematical Optimisation
The aim of an optimisation is to maximise or minimise some quantity by systematically
modifying the values of parameters within an allowed range of variation. The quantity which
has to be optimised is called an objective or cost function. A set of parameters (unknowns
or variables) constitutes the optimisation variable. An ensemble of optimisation variables
constitutes the search space. The restrictions on allowed parameter values are known as
“constraints”. The constraints allow the unknowns to have only certain values and exclude
others. Generally speaking, constraints are not essential.
95
96
CHAPTER 7. OPTIMISATION
Mathematically, the optimisation problem is formulated as
maximise f (x), subject to ci (x) ≤ bi , i = 1, ..., m ,
(7.1)
where the vector x = (x1 , ..., xn ) ∈ Rn is the optimisation vector of n independent variables,
f (x) : Rn → R is the objective or cost function, ci (x) : Rn → R are the constraint functions,
and bi are the bounding conditions. The constraints in the form of ci (x) = bi are termed
equality constraint functions, while those in the form of ci (x) < bi are inequality constraint
functions. Generally, a minimisation of a function f (x) can be regarded as a maximisation of
−f (x). The purpose of an optimisation is to find the global maximum – i.e. a vector xG that
maximises the objective function f (x) over all possible vectors x
f (xG ) > f (x) ∀x ∈ D(x), x 6= xG
,
(7.2)
where D(x) is set of feasible values of the vector x. Obviously, for an unconstrained problem
D(x) is infinitely large.
In the general case several local maxima can exist in the search space. A local maximum
is defined as a vector xL which satisfies the following condition
f (xL ) > f (x) ∀x ∈ N (xL , δ), x 6= xL
,
(7.3)
where N (xL , δ) is defined as the set of vectors x contained in the neighbourhood of xL (within
some arbitrarily small distance δ of xL ).
A large variety of mathematical methods was developed to solve optimisation problems.
Many methods are suitable only for certain types of problems. Therefore, it is important to
recognise the problem type and find an appropriate solution technique. Complexity of an optimisation problem depends on relationships between the objective and constraint functions.
The simplest class comprises linear optimisation problems. An optimisation problem is linear if
the objective and all constraints are linear functions of the decision variables. If the objective
or at least one of the constraints is a smooth nonlinear function, the corresponding optimisation problem is called a smooth nonlinear optimisation problem. The most general class of
optimisation problems is a non-smooth optimisation problem.
Linear optimisation problems can be solved using the simplex method [CLR01]. Alternatively, the interior point method (also referred to as barrier method) was developed to solve
this type of optimisation problem [Wri97]. There is no single method which is the best for all
7.2. METHODS OF STOCHASTIC OPTIMISATION
97
smooth nonlinear optimisation problems. The most widely used and effective methods are the
gradient projection and reduced gradient methods [WXi00]. Non-smooth optimisation problems can have multiple locally optimal points. Therefore, since gradient information can not
be used to determine the direction in which the function is increasing (or decreasing), methods
based on either the systematic or random search are preferable for the solution of such type
of optimisation problems. The classical group of methods based on the systematic search is
called branch and bound methods [Moo91]. These methods are methods for global optimisation. Their efficiency depends critically on the effectiveness of the branching and bounding
algorithms used. There is no universal bounding algorithm that works for all problems. Wrong
choices can lead to a slowdown of the method. In the worst case the computation time
grows exponentially with the problem size. Random search methods are non-deterministic
and stochastic. Therefore, they can provide different solutions on different runs, even when
starting from the same point on the same model. Generally, these methods are not able to
prove that the solution obtained is the optimal solution. However, these stochastic optimisation methods are very fast. The application area for these methods comprises tasks, where
systematic search methods fail. Many problems of practical interest fall into this category, for
example, those where the search space is too large or the underlying mathematical model is
not known exactly. The next section describes several examples of optimisation methods from
this group in detail.
7.2
7.2.1
Methods of Stochastic Optimisation
Simulated Annealing
A very popular stochastic algorithm which can be used efficiently for a wide range of
optimisation problems is simulated annealing (SA) [KGV83]. The original motivation of this
method comes from the statistical physics by an analogy with the cooling of fluids into a
crystalline structure of minimal energy.
The SA is just an iterative improvement [DFF63] incorporating with the Metropolis criterion
[MRR53] for accepting or rejecting of a randomly generated trial move. The main advantages
of SA over simple iterative improvement lies in the ability to avoid trapping into locally optimal
98
CHAPTER 7. OPTIMISATION
set initial temperature T
define primary state Ci
evaluate this state f(Ci)
generate new state Cj
evaluate new state f(Cj)
change temperature T
stop optimisation
yes
final state
no
f(Cj) > f(Ci)
yes
replace Cj → Ci
no
no
p ≤ exp(δf/T)
yes
replace Cj → Ci
Figure 7.1: Diagram of the simulated annealing algorithm. For details see text.
points.
The structure of the SA algorithm for maximisation is depicted in Fig. 7.1. It starts with
setting an initial system temperature T and the definition of a primary (initial) system configuration Ci . It is important to mention that the temperature here is not a real temperature but
a control parameter which is called temperature by analogy with statistical physics. Then the
primary system configuration is evaluated resulting in a number of a scalar objective function
f (Ci ). After that, a new system configuration Cj is generated by random changes of the old
configuration. The changes have to be rather small, but at the same time they must allow to
7.2. METHODS OF STOCHASTIC OPTIMISATION
99
reach any possible configuration of the system within a finite number of iterations. Then the
new configuration is evaluated by calculating f (Cj ). After that, the temperature of the system
is changed. Now, if the value of the objective function for the new system configuration is
larger than for the primary configuration (f (Cj ) > f (Ci )), the old configuration is replaced
by the new one (Cj → Ci ). Otherwise the replacement can occur with some probability p
depending on the temperature
p = exp
δf
T
,
(7.4)
where δf = f (Cj ) − f (Ci ) is the increase of the objective function and T is a current
temperature. This “temperature” is a control parameter measured in the same units as the
objective function. The procedure is repeated in the loop until stopping conditions or a final
temperature is reached. The SA algorithm for minimisation has only insignificant differences.
Namely, the old configuration is replaced by the new one (Cj → Ci ) if f (Cj ) < f (Ci ),
otherwise the replacement probability is given by Eq. (7.4) with δf = f (Ci ) − f (Cj ). For fast
and successful performance of the algorithm, it is very important to judiciously set the initial
temperature and the rules for reducing it. An appropriate initial temperature can be found using
the following procedure [Kir84]. The SA algorithm is started a few hundred times with some
arbitrary temperature to determine a fraction of accepted system configurations according
to Eq. (7.4). If this fraction is less than 80 %, the initial temperature has to be doubled.
This process is repeated until the fraction overcomes this threshold. During optimisation the
temperature can be decreased either in an exponential cooling scheme
Tj = αTi
(7.5)
Tj = Ti − ∆T
(7.6)
with α < 1 or in a linear cooling scheme
with a small ∆T . The final temperature is either predetermined by setting the total number
of iterations or, alternatively, the optimisation can be interrupted if no further progress is
observed.
SA has advantages and disadvantages compared to other global optimisation techniques.
Among its advantages are the relative ease of implementation and the ability to provide reasonably good solutions for many combinatorial problems. Its drawbacks include the need for a
great deal of computer time for many runs and carefully chosen tunable parameters [ECF98].
100
7.2.2
CHAPTER 7. OPTIMISATION
Evolutionary Algorithms
The next group of optimisation algorithms (genetic algorithm [Hol73], evolution strategy [Rec73], and genetic programming [Koz92]) attempts to simulate principles of biological
evolution processes. These are so-called evolutionary algorithms.
The genetic algorithm (GA) is based on the phenomena of natural evolution and survival
of the fittest. Its terminology is drawn from the evolution theory of Darwin [Dar01]. The
main difference between GA and SA is that the SA works with one solution, while the GA
operates with a set of solutions (a population of individuals in the evolution terminology).
The process of transforming one population to another is described by the term “generation”
or “iteration”. The construction of a new generation involves specific operators of selection,
crossover, and mutation. Usually, an objective function for maximisation with GA is referred
to as “fitness function” to accentuate the evolutionary nature of this algorithm.
The basic structure of the GA algorithm is presented in Fig. 7.2. The optimisation begins
with building of an initial population of N individuals. This can be done either by a random
generator or by some problem specified guess if possible. The GA employs a binary representation of individuals, where each individual is encoded by a bit string (chromosome). The
chromosome length determines the accuracy with which an individual can be encoded.
Then all individuals are evaluated by applying the fitness function. This function returns
a scalar number (fitness) depending on the fitness of the individual which is taken as the
function input. Assuming that the aim of the optimisation is maximisation, higher values of
the fitness function correspond to more successful individuals.
After this evaluation some individuals have to be selected according to their fitness and
used for reproduction to create a new generation of individuals. There are several methods of
selection. All methods imply that the individuals with a higher fitness have a larger probability
to be chosen for the reproduction. One of the commonly used and most simple among them
is the roulette wheel selection (proportional selection) method [Bak87]. The probability of any
individual to be selected for reproduction in this method is proportional to its fitness value.
Unfortunately, the roulette wheel selection method is affected by a scaling problem when
the selection probabilities become strongly dependent on the scaling of the fitness function
[BHo91]. Another popular selection method is called tournament selection [GDe91]. Here the
best individual according to the fitness chosen from two randomly taken individuals is used
7.2. METHODS OF STOCHASTIC OPTIMISATION
101
initial population
evaluation
stop optimisation
yes
optimal solution
no
yes
elitism
no
crossover
mutation
Figure 7.2: Basic scheme of the genetic algorithm. For details see text.
for reproduction. This method can be generalised involving a larger number of the individuals
to pick up the best one. The selection method, when some fraction of the individuals with
the highest fitness are selected and then these individuals utilised for the reproduction with
the same probability, is known as the “truncation selection” [BTh95]. Usually, the truncation
threshold is set to select between 10% and 50% of the population for the reproduction. For
the ranking selection the individuals are ranked according to their fitness values. The highest
rank corresponds to the best individual. Then the individuals are selected for the reproduction
with the probability proportional to the their rank. The case of the probability linearly assigned
to the rank corresponds to the linear ranking selection [BTh95]. For the exponential ranking
selection [BTh95] the probability has to be exponentially weighted with the rank. Sometimes
an individual is characterised not only with a single value of the fitness function, but additional
criteria are involved in order to evaluate the quality of the individual. Then, the “multi-objective
fitness assignment” has to be applied [Fon95].
A new generation of individuals is created by crossover and mutation using individuals
selected with one of the above described methods. In the simplest form of the one point
102
CHAPTER 7. OPTIMISATION
crossover a chromosome part of one individual is exchanged with a chromosome part of another
individual producing an offspring. The split point of both chromosomes is defined randomly.
Generally, up to n − 1 points crossover can be performed (n is a chromosome length) but
the optimisation performance is degraded when many points are employed for the crossover
[Jon75]. The idea of the multi-point crossover is that parts of the chromosome that contribute
most to the fitness of a particular individual may not necessarily be contained in adjacent
substrings [Boo87]. The next type of crossover is the so-called “uniform crossover” operator
which creates offsprings by picking each bit from either of the two parent chromosomes [Sys89].
The uniform crossover operator produces on the average n/2 crossings of a chromosome. The
“shuffle crossover” is similar to the uniform crossover [CES89]. The variables are randomly
shuffled in both parents before the exchange. After one point crossover, the variables in the
offsprings are unshuffled in reverse. Then new individuals created by the crossover undergo
the mutation. Each bit of the chromosome can be flipped from 0 to 1 or from 1 to 0 with
some probability. Since the initial population of individuals may not contain enough variability
to reach the optimal solution via the crossover alone, the mutation is used to introduce extra
variability. There are several modifications of the GA with respect to the crossover and the
mutation. For example, the crossover can be applied with some probability and then the
mutation takes place only for individuals for which the crossover fails. Sometimes a number
of possible mutations is restricted to only one per an individual. Commonly, a probability of
the mutation is very low, while the crossover is applied with a high probability.
If all individuals of a new generation are created by crossover and mutation, such process
is called the “total replacement”. Every individual lives one generation only. But it can lead to
the situation when the best individuals are replaced with worse offspring, and therefore good
information is lost. To avoid this, a number Ne of the best individuals can be transferred
into a new generation without any modifications. This is so-called “elitism replacement”.
The rest N − Ne individuals are created by the crossover and the mutation. Assuredly, these
Ne of the best individuals in the original population are also available for the crossover and
the mutation. The elitism guarantees that the best chromosomes will be always presented in
succeeding generations. The proper rate of the individuals Ne /N involved in the elitism affects
the success of an optimisation. If the output of the fitness function is noise free, it is enough
to send only one individual to the next generation. But, if the fitness function contains also
7.2. METHODS OF STOCHASTIC OPTIMISATION
103
an experimental noise, at least several individuals have to be used to be sure that the best
individual is selected among them. A too large rate can create a predominance of particular
type of chromosomes in a population. This reduces the diversity of individuals required for an
adequate search and even can be a reason of the premature convergence to a local optimum.
The high selection probability exclusively given during the selection procedure to the individuals
with large fitness values especially at the initial stage of the optimisation process can again
be a reason of the premature convergence to a local optimum. Sometimes, the “fitness based
replacement” is applied. In this case the new generation is built from individuals with the
highest fitness taken from both offspring and the old generation.
Then a new generation is evaluated again and the optimisation loop is run until stop
conditions, such as a preselected number of generations or some desired fitness value, are
reached. Alternatively, a stop point can be determined according to the evaluation of the
fitness function. For example, if the difference between the best fitness value and the average
fitness value over the same generation is rather small.
The “evolution strategy” (ES) is similar to GA [Rec73]. In the general case λ offspring
individuals are created from µ parent individuals. Then the offspring individuals are modified by a mutation operator. Usually, µ and λ are small integers. As with the GA, an
individual represents a possible solution of an optimisation problem. But the ES uses the
parameter representation in a forms of vectors of integer or real numbers. This vector is called
“object-parameter”. This vector together with another vector of real numbers (the “strategyparameter”) defines the data-structure for a single individual. The data-structure is usually
referred to an ES-chromosome. The object-parameters contain the variables which have to be
optimised, while the strategy parameters control the mutation of the object-parameters .
The “genetic programming” (GP) brings the idea of the GA one step further and evolves
computer programs [Koz94]. Not only parameters of a problem but also the algorithm for
the problem solving itself is subject of evolutionary changes. Most of the theory behind GP
is the same as that behind the GA. The main difference between GP and the GA lies in the
representation of a solution. The GA operates on a string of bits that represents the solution.
While GP creates computer programs as the solution. The main disadvantage of GP is the
huge computing resources required to solve any real world problem.
For the optimisation of complex systems the combination of several optimisation methods
104
CHAPTER 7. OPTIMISATION
can be a promising approach. If several different evolutionary algorithms are combined, it
is called the “application of different strategies”. The “competing subpopulations” involves
simultaneous using of different optimisation strategies. Both methods open new dimensions
to the application of evolutionary algorithms and make one step towards the development of
powerful tools for the solution of complex problems.
7.3
Practical Implementation of Optimisation
This section describes the optimisation program which was developed by the author of this
thesis to perform experiments presented in this work.
A schematic of the adaptive closed loop setup is depicted in Fig. 7.3. It consists of three
main parts: a reflectron time of flight (Re-TOF) mass spectrometer, a pulse shaper, and an
evolutionary algorithm. The pulse shaper is controlled by the evolutionary algorithm and creates pulses with a temporal structure depending on the applied phase mask. The shaped laser
pulses are focused with a mirror onto a molecular beam inside an interaction region of the ReTOF mass spectrometer which detects produced ions. The response of the molecular system
is used as feedback signal for the evolutionary algorithm. The algorithm adapts the temporal
structure of the shaped pulses in a loop to optimise a specified process. The optimisation is
carried out until the desired experimental output is received.
A genetic algorithm implemented with LabVIEW is used in the present work. This algorithm was chosen because it outperforms other evolutionary algorithms in noisy environments
[TLo94].
A population of individuals is realised as an array of structures consisting of two elements.
A first element of the structure is a vector of integer numbers for the encoding of an individual
(the phase mask used in the shaper), while a second element presented by a single real
number characterises a fitness value of the corresponding individual (an expression derived
from a combination of measured experimental signals).
The vector of 10 bits integer numbers (optimisation parameters) represents a spectral
phase distribution (phase mask). Of course, the integer numbers have to be rescaled to the
interval between 0 and 2π before be applied to the shaper. The maximal length of this vector
is 640 and is limited by the numbers of pixels constituting the LCM. The vector length and
7.3. PRACTICAL IMPLEMENTATION OF OPTIMISATION
evolutionary
algorithm
105
pulse shaper
Re-TOF
feedback signal
Figure 7.3: Scheme of the adaptive closed loop setup.
the maximal value of each parameter determine a size of the search space which in this case
is 1024640 . This number is extremely huge. Therefore, to obtain reasonable convergence
times the size of the search space must be limited. For that purpose the range of possible
values of the optimisation parameters (often referred to as the resolution r) can be restricted
to some number below 1024. Moreover, the vector length can also be reduced either by
grouping of pixels or by parametrisation. In the first case a whole phase mask is reconstructed
by using one parameter for a group of several pixels or by interpolation (linear or spline) of
parameters to get a phase mask. In the second case a phase mask is described by some
mathematical function and parameters of this function are optimisation parameters. The first
case of optimisation is called “free optimisation”, while the second case is corresponded to the
parameterised optimisation. In the program used in this work, the genetic algorithm operates
on 32 parameters only with 100 possible values of each parameter during a free optimisation.
These two restrictions allow one to reduce a size of the search space down to 10032 . Taking
into account that only the central fraction of the LCM is effective due to the narrow laser
106
CHAPTER 7. OPTIMISATION
bandwidth the real size of the search space is even smaller. But this is still a gigantic number.
Thus, only random search methods, which are non-deterministic and stochastic, can effectively
handle such large search space.
The number of individuals in the population has a great influence on the efficiency of the
optimisation. The large number provides many degrees of freedom to explore the search space
and minimise the risk of being trapped in a local maximum. This is especially important in
the case of the complex search space with multimodal topology, such as encountered in the
present work when e.g. optimising the fragmentation of model peptides. But the convergence
time increases with growing population size as well. Generally, a larger population size is
required if the optimisation problem has a high complexity. In this work, the population
size of 20 individuals is determined from the condition that one optimisation run must not
exceed two hours. This limitation mainly comes from the maximal time during which a stable
laboratory environment can be maintained. Approximately the same size of the population is
successfully used in many applications of pulse shaping even with a much larger search space
[BER07, Mer07]. The two best individuals (survivors) are transferred into the next generation
without any modifications (elitism). Generally, the population of individuals are initialised with
random numbers. But the optimisation programm provides the possibility to include one or
more preselected individuals in the initial population. Alternatively, the programm can try to
build the initial population out the individuals which exhibit a fitness above some threshold.
This strategy can be useful when, for example, an average fitness of a random phase mask is
comparable with a noise level.
As described above, there are several methods to select individuals for the crossover. These
selection schemes have various characteristics and therefore can influence the optimisation efficiency differently. For example, the proportional selection is not translation invariant [BHo91]
and therefore the selection probabilities strongly depend on the scaling of the fitness function.
The truncate selection excludes some number of individuals from selection. Moreover, the
efficiency of a selection method can depend on the type of an optimisation problem. Here
the ranking selection is employed as the main method because it behaves in a more robust
manner [Why89]. After evaluation of the whole population of N individuals they are ranked
according their fitness values starting from the best individual. A special parameter ν (nonlinearity) allows one to change the “nonlinearity” of this selection method and thus to regulate
7.3. PRACTICAL IMPLEMENTATION OF OPTIMISATION
107
the probability of better individuals being selected compared to the average probability of selection for all individuals. The probability of a m-th individual to be selected for the crossover
is given by
pm =
1
(N − m)[1−lg ν] ; m ∈ [0, N − 1] ,
K
where K is a normalisation coefficient which is determined from the condition of
(7.7)
PN −1
m=0
pm = 1.
The parameter ν can be altered from 0 to 1. ν = 1 corresponds to the situation of the linear
ranking selection scheme. The reduction of ν shifts the selection probability to the direction
of the individuals with the higher fitness. Finally, only one best individual is selected at ν = 0.
The crossover probability pc itself is defined as
pc =
n−1
n+1
,
(7.8)
were n is a chromosome length. This manner of the crossover probability determination leads
to very large values of the probability. For example, the crossover probability of ≈ 94%
corresponds to chromosome length of 32.
The non-binary representation of individuals is a key difference of this algorithm over the
GA described in Sec. 7.2. Mainly, this difference effects on the implementation of the mutation.
The mutation, which is applied with some probability pm to each optimisation parameter of a
chromosome, can modify a value of a parameter x according to the following expression
x ± xmax P [1−lg σ]
,
(7.9)
where P ∈ (0, 1) is a uniform distributed random number, xmax is a maximal possible value of
x, and σ is a parameter which characterises the mutation step size. If the value of x obtained
after the mutation is larger than xmax (or less than 0), it has to be rescaled to the range
between 0 and xmax by the subtracting (or adding) xmax . The case of σ = 0 corresponds to
the absence of mutation. The mutation variability increases with larger σ and it reaches the
maximum at σ = 1 when any x is simply replaced with a random number.
The fitness function f is determined by the response of the molecular system upon excitation with shaped laser pulses. Depending on the aim of an experiment in the simplest case
the fitness function can be equal to an integrated yield of some particular ion. Generally, the
fitness is a function of several parameters. For example, if the yield of one particular ion a
must be maximised, while at the same time the yield of different ion b must be suppressed, it
108
CHAPTER 7. OPTIMISATION
was found that an appropriate fitness function is
f = [a − a0 ] × [b0 − b] ,
(7.10)
where a0 and b0 are the respective ion yields obtained with an unshaped laser pulse. Such
definition of the fitness function allows one to maximise the formation of a particular ion and
to minimise the formation of a different one simultaneously. After the evaluation of the current
generation the best fitness fbest and the worst one fworst in the generation can be found. The
fitness averaging over the whole generation gives a mean fitness fmean . An optimisation is run
while fbest still increases. If no further growth of fbest is observed at least during 5 generations,
the optimisation can be terminated. Usually, only a small number (between 20 and 30) of
generations are required to find an optimal solution.
The choice of the parameters used in the optimisation program has a significant impact
on its performance. Hence, it is very important to find a proper set of these parameters. The
best way of doing so is to perform optimisations of real physical systems and investigate the
influence of the parameters on the optimisation results. Since this requires a lot of time, a set
of test optimisations was made instead. The main idea of the test optimisation is to imitate the
pulse shaper by applying the Fourier transformation to a Gaussian spectrum (45 nm FWHM)
modulated with a phase mask defined by the program. The goal of the test optimisation
is to find a phase mask which minimises the deviation between the calculated shaped pulse
and some predefined shaped pulse with a complex temporal structure. The pulse shown in
Fig. 9.12d was chosen as a target. The fitness function f in this test optimisation is defined
as
Z
∞
|Itarget (t) − Icalc (t)| dt ,
f=
(7.11)
−∞
where Itarget (t) is the temporal structure of the target pulse and Icalc (t) is the the temporal
structure of the calculated pulse. The test optimisation was repeated 100 times for each set of
the parameters to find the average values of the best fitness fbest , the mean fitness fmean , and
the worst fitness fworst . The mutation probability pm , the nonlinearity ν, and the mutation
step size parameter σ were tested very carefully. First, the influence of pm on the final fitness
was investigated in a range of pm between 0.001 and 0.9 with two kind of optimisations: a
long and a short one. The long optimisation was performed with 1000 generations to see the
maximal fitness achievable in such kind of experiment, while the short optimisation was done
7.3. PRACTICAL IMPLEMENTATION OF OPTIMISATION
2 .5
2 .5
fitn e s s [a r b . u n its ]
(a )
2 .0
2 .0
1 .5
1 .5
1 .0
1 .0
0 .5
1 E -3
p
0 .0 1
m
0 .1
1
0 .5
2 .5
2 .0
2 .0
1 .5
1 .5
1 .0
1 .0
1 E -7
1 E -3
1 E -5
0 .0 1
p
(c )
0 .5
(b )
[a r b . u n its ]
2 .5
1 E -3
0 .1
0 .5
109
0 .1
1
[a r b . u n its ]
m
(d )
1 E -3
0 .0 1
σ[a r b . u n its ]
0 .1
1
ν [a r b . u n its ]
fb
e s t
fm
e a n
fw
o rs t
Figure 7.4: Results of the test to determine optimal values of the parameters used in the
optimisation program: (a) test of the mutation probability pm within 1000 generations;
(b) test of the mutation probability pm within 30 generations; (c) test of the mutation step
size parameter σ within 30 generations; (d) test of the selection nonlinearity parameter ν
within 30 generations. The best fitness fbest , the mean fitness fmean , and the worst fitness
fworst are represented by red squares, green circles, and blue triangles, respectively. For
details see text.
with 30 generations only to get the feeling about the fitness which can be received within a
time limited real optimisation experiment. The results of these two optimisations are presented
in Fig. 7.4a and Fig. 7.4b, respectively. The two important conclusions originates from these
test optimisations. First, fbest , fmean , and fworst exhibit a strong dependence on pm . fbest
growths with increasing of pm from 0.001 to ∼ 0.3. At the same time the spread between
fbest , fmean , and fworst is dramatically increased. fbest has a maximum around 0.3 ± 0.1
and drops sharply for larger values of pm . Second, the difference between fbest obtained
within 1000 and 30 generations is negligible small. Therefore, 30 generations are enough
for the algorithm convergence. There is a recommendation to employ only one mutation
110
CHAPTER 7. OPTIMISATION
Table 7.1: Standard parameters for a free optimisation with the genetic algorithm.
Parameter
Value
Population size, N
20
Number of survivors, Ne
2
Number of optimisation parameters, n
32
Resolution, r
100
Selection nonlinearity, ν
1
Crossover probability, pc
0.94
Mutation probability, pm
0.2
Mutation step size parameter, σ
0.0001
per an individual [Bae93]. In case of 32 optimisation parameters the corresponding mutation
probability is ≈ 0.03 only. Therefore, the mutation probability of 0.2, which is slightly less than
the value from the test optimisations, is utilised in real optimisations. Then the mutation step
size parameter σ and the selection nonlinearity parameter ν were tested within 30 generations.
These results are shown in Fig. 7.4c and Fig. 7.4d, respectively. The influence of σ is very weak
on the best fitness. The optimal value of σ can be seen around 0.0001. The effect of ν is not
visible in the test optimisation. For the real optimisation this parameter is set to 1. Finally, the
values of the parameters found with the test optimisations were verified experimentally with
the optimisation of the second harmonic yield [Boy00]. Table 7.1 summarises all parameters
used in the present work for a free optimisation with the algorithm described above.
In this chapter different optimisation methods were described. The practical implementation of the genetic algorithm used in the present work was given. Proper values of the most
important parameters involved in the optimisation algorithm (namely, the mutation probability
7.3. PRACTICAL IMPLEMENTATION OF OPTIMISATION
111
pm , the nonlinearity ν, and the mutation step size parameter σ) were determined using the
test optimisation and summarised together with other parameters in Table 7.1.
112
CHAPTER 7. OPTIMISATION
Chapter 8
Excitation of C60
The response of C60 fullerenes upon excitation with intense pulsed laser radiation can be
investigated by photoion and photoelectron spectroscopy. A laser pulse duration of down to
9 fs allows one to study excitation mechanisms on a time scale below the characteristic time
scales for electron-electron and electron-phonon coupling. Thus, it is possible to separate the
energy deposition into the system from the energy redistribution among the various electronic
and nuclear degrees of freedom in time. Moreover, single pulse experiments as a function of
laser parameters such as pulse duration, intensity, and/or ellipticity provide a powerful tool for
to explore the energy flow in C60 at moderately strong laser intensities (< 1015 W/cm2 ).
This chapter begins with a brief discussion of multiphoton ionisation of C60 fullerenes.
Then the experimental results are reported, starting with single pulse excitation at different
pulse duration and intensities, followed by ellipticity dependent studies using photoion and
photoelectron spectroscopy.
8.1
8.1.1
Multiphoton Ionisation of C60
Different Ionisation Mechanisms
Various systematic studies on multiphoton ionisation (MPI) of C60 fullerenes with different
laser parameters have been performed recently [THD00, CHH01, CHR01]. It was found that
the ionisation behaviour of C60 depends strongly on the laser pulse duration τ [CHH00].
Depending on the excitation time scale three different ionisation mechanisms were identified.
For very short pulses (τ < 70 fs) the excitation energy tends to remain in the electronic system
113
CHAPTER 8. EXCITATION OF C60
ion signal [arb. units]
114
m/q [u]
m/q [12u]
m/q [u]
m/q [12u]
Figure 8.1: Typical mass spectra of C60 produced by laser pulses of 795 nm wavelength with
a pulse duration of 5 ps (top) and 25 fs (bottom) (the picture is taken from [HLS05]). The
spectra were recorded at roughly equal laser fluences, the corresponding intensities being
3.2 × 1012 W/cm2 and 1.0 × 1015 W/cm2 , respectively.
with relatively little vibrational heating of nuclear motion. Thus, multiply charged C60 ions
formed by direct multiphoton ionisation are observed. For longer laser pulses (τ < ps) the
absorbed energy is coupled among electronic degrees of freedom and the ionisation becomes
of thermal nature. Finally, for very long pulses (τ > ps) electron-phonon coupling leads to
massive fragmentation and delayed ionisation. As an example Fig. 8.1 illustrates the substantial
differences observed in mass spectra when the laser pulse duration is changed from 25 fs
to 5 ps. Both spectra were recorded at roughly equal laser fluences of ≈ 22 J/cm2 which
corresponds to intensities of 1.0 × 1015 W/cm2 (τ = 25 fs) and 3.2 × 1012 W/cm2 (τ = 5 ps).
The mass spectrum obtained with a pulse duration of 25 fs exhibits a strong contribution of
multiply charged Cq+
60 ions and from corresponding large, fullerene-like fragments which are
formed by sequential C2 losses. However, only a limited number of singly charged fragments
is detected, as shown in the enlarged insert. On the other hand, the mass spectrum recorded
upon interaction with 5 ps pulses resembles very much the typical bimodal mass distribution
8.1. MULTIPHOTON IONISATION OF C60
115
observed in ns experiments [KHO85]. Here only singly charged ions and various fragments are
observed. In addition, the enlarged insert shows a significant tail on the higher delay time side
of the C+
60 mass peak, originating from delayed ionisation explained in see Sec. 3.2. Such a
tail is not visible in the case of excitation with 25 fs pulses. The fraction of delayed ionisation
increases almost linearly with laser pulse duration [HLS05].
In both cases (excitation with 25 fs and 5 ps laser pulses), peaks originating from metastable
statistical decay of highly vibrationally excited fullerene-like fragments on a µs-ms time scale
[CLe00] are marked with asterisks. In this context some questions still remain open: how is
the energy ≥ 40 eV needed for metastable fragmentation initially deposited into the system?
Does the single active electron (SAE) dynamics, which prevails in strong field interactions with
atomic systems, still dominate in a large molecular system over the intuitively more plausible
many active electrons (MAE) response [LBI02, MSR03]?
8.1.2
Saturation Intensity
In a simple multiphoton picture the nonresonant ionisation rate WN of C60 depends on the
generalised multiphoton cross-section σN according to Eq. (3.7) as
WN = γN I N
,
(8.1)
where γN = σN /(hν)N . Lacking any better knowledge1 , for simplicity in the following discussion σN is assumed to be independent of the laser intensity. In accordance with the considerations on the Rayleigh length (see Sec 6.2) the interaction with a cylindrically symmetric
parallel Gaussian laser beam is assumed
r
2
t
2
I(r, t) = I0 e−2( w ) e−4 ln 2( τ )
,
(8.2)
where I(r, t) is the beam intensity at a given time t and distance r from the beam axis, I0 is
the peak pulse intensity, τ is the temporal FWHM, and w is the beam waist (radius at which
the intensity drops down to 1/e2 with respect to its maximum value). The change of the
target density dn(r, t) at any given r and t is
dn(r, t) = −n(r, t)γN I N (r, t)dt .
(8.3)
1
However, the ADK theory (see Sec. 3.2), with a number of shortcomings such as wavelength independence [TZL02], can be considered as an unproven alternative for such a molecular system.
116
CHAPTER 8. EXCITATION OF C60
Integration of Eq. (8.3) over the pulse durations gives
r
r 2 π
N
n(r) = n0 exp −γN I0
τ exp −2N
4 ln 2N
w
,
(8.4)
where n0 is the initial target density. By introducing the following substitutions β 2 = 2N/w2
p
and v N = γN I0N 4 lnπ2N τ Eq. (8.4) can be rewritten in the form
n(r) = n0 exp{−v N exp[−β 2 r2 ]} .
(8.5)
Obviously, v is a dimensionless intensity parameter which can be expressed by
v=
I0
Isat
,
(8.6)
where a “saturation intensity”1 is defined as
!1/N
r
1
4 ln 2N
Isat =
γN τ
π
.
(8.7)
The saturation intensity Isat determines the intensity at which the ionisation probability in the
beam center [n0 − n(0)]/n0 becomes sufficiently large, i.e. 1 − 1/e or 63% (note, n(r)/n0 can
never becomes unity). Any further increase of the ion yield with intensity essentially originates
from an increase of the interaction volume in which saturation is reached as I0 is further
increased. Isat contains information about the multiphoton cross-section, the ionisation order,
and its value depends on the laser pulse duration. Therefore, from Eq. (8.7) the following
scaling law for saturation intensities as a function of the laser pulse duration can be deduced
1
Isat ∝ √
N
τ
.
(8.8)
The measured total ion signal S is determined by the decrease of the initial target density
n0 − n(r). The signal dS originating from a ring between r and r + dr and a length l at a
detection efficiency F is given by
dS = 2πn0 F lr 1 − exp −v N exp −β 2 r2
dr
.
(8.9)
The total measured signal S is derived by integration of Eq. (8.9) over the whole Gaussian
beam
S = αn0 [Γ + ln v N + E1 (v N )] ,
(8.10)
1
Note that this definition differs slightly from that of [HVC00]. In contrast to that work, however,
an analytical expression for Isat (based on the assumption of a strict power law according to Eq. (8.1))
is used here rather than a somewhat difficult to justify linear extrapolation of the semilogarithmically
plotted experimental data.
8.1. MULTIPHOTON IONISATION OF C60
117
where α is the proportionality constant, Γ = 0.5772156649 is the Euler constant, and E1 (v N )
is the integral exponential function defined as
Z ∞
(e−u /u)du .
E1 (x) =
(8.11)
x
In the present work Eq. (8.10) is used to fit the experimental data and to derive saturation
intensities for different charge states q of Cq+
60 . The approach of [HVC00] and [HVC01] already
mentioned above to extract saturation intensities from experimental data leads to similar
results. There the saturation intensity is obtained from the intersection of the recorded ion
signal in a lin-log plot (linearly depending on ln I) with the abscissa.
8.1.3
Ionisation Energy and Model Potential
The saturation intensities can be derived using a model potential of Cq+
60 in the description
(q+1)+
of the ionisation process Cq+
60 −→ C60
in the laser field. Here saturation is assumed to
occur when the potential barrier falls below the ionisation energy of the respective charge
state. Thus, values of the saturation intensities become sensitive to the ionisation potential
(IP) and the model potential of the ionic core.
The available experimental data on the IPs for different charged states of C60 demonstrate a
good reliability only for q = 1 [HSV92, SVK92, VSK92], while for q = 2 the experimental value
is known with less accuracy [WDS94]. Also a number of theoretical works was dedicated to
the IPs of Cq+
60 [YLa94, TBF95, SVS96]. Recently, a linear dependence between the ionisation
potential EIP and the charge state q has been extracted from density-functional calculation
[DAM05]
EIP = 7.108 + 3.252q
.
(8.12)
This dependence is indicated by the dashed line in Fig. 8.2 together with IPs from the above
mentioned experimental data (symbols). The theoretical calculation underestimates the experimental values. Therefore, a different linear equation with a somewhat larger slope and
much better agreement to the experimental data is suggested, which is presented by the solid
line in Fig. 8.2
EIP = 7.6 + 4.248q
.
(8.13)
The second therm of Eq. (8.13) corresponds to the energy qe2 /(4πε0 RC ) needed to remove
an electron from the Cq+
60 surface at radius RC to infinity. The value RC of 3.39 Å is used in
118
CHAPTER 8. EXCITATION OF C60
io n is a tio n p o te n tia l [e V ]
4 0
3 0
2 0
1 0
0
0
1
2
3
4
5
in itia l c h a r g e s ta te q
Figure 8.2: Ionisation potentials for different charge states of Cq+
60 . Experimental values (symbols) are adopted from [HSV92, SVK92] (red circles), [WDS94] (green triangle),
and [MER97] (blue squares). The data from [MER97] were originally reported for Cq+
70 ,
therefore, these values have been corrected by a factor of 1.08 to account for the smaller diameter of C60 . The theoretical data taken from [DAM05] (grey dashed line) and calculated
results according to Eq. (8.13) (black solid line) are also shown.
Eq. (8.13). This value is in reasonable agreement with the known mean radius 3.55 Å of C60
[DDE96].
For estimating saturation intensities the potential of the ionic core is modelled. Different
models are discussed in the literature. Corkum et al. [BCR03] used a model based on charged,
conducting, and polarisable sphere for Cq+
60 . Alternatively, a more realistic jellium type potential
was used in Ref. [PNi93]. An extended potential V (r, q) based on the jellium type model,
which is used in the present work, is given by the following expression

−2 exp (−r0 |r − r0 |) − |r − r0 |−1 − 0.14978(q − 1)




−2 exp (−r0 |r − r0 |) − |r − r0 |−1 − h(r, q)



−1.5 (1 + exp (|r − r0 | − 1.5580) /0.006)−1 + g(r, q)
V (r, q) =
−1.5 + 0.012(r − r0 )4 + g(r, q)





−1.5 (1 + exp (|r − r0 | − 1.5580) /0.006)−1 + g(r, q)


−2 exp (−r0 |r − r0 |) − |r − r0 |−1 + g(r, q)
if
if
if
if
if
if
r < 2.75
2.75 ≤ r < 5.13
5.13 ≤ r < 5.15
5.15 ≤ r ≤ 8.23
8.23 < r ≤ 8.2497534
8.2497534 < r
(8.14)
,
8.1. MULTIPHOTON IONISATION OF C60
+
C
io n s ig n a l [a r b . u n its ]
C
2
6 0
+
C
+
3
C
( b ) τ=
5 0 0 fs
0 .5
6 0
0 .0
1 .0
2 +
C
1 .0
(a )
2 +
C
+
C
119
+
C
6 0
6 0
1 0 0 fs
0 .5
3 +
6 0
0 .0
1 .0
+
C
C
6 0
2 5 fs
0 .5
2 +
6 0
0 .0
1 .0
C
0
2 4 0
4 8 0
7 2 0
th e o ry
e x p e r im e n t
+
6 0
0 .5
9 6 0
0 .0
0
9 fs
1
2
C
m /q [u ]
6 0
3
4
5
6
c h a rg e s ta te
Figure 8.3: Comparison between experimental data of C60 photoionisation and the theoretical ionisation rates derived from S-matrix theory by A. Becker and F. H. M. Faisal:
(a) mass spectra obtained with 797 fs laser pulses as a function of pulse durations at
4 × 1013 W/cm2 (the upper three spectra are reproduced from [CHH00]); (b) the abundance
of different charged states of C60 is compared with rates derived from S-matrix theory
[JBF07].
where r0 = 6.69. The functions h(r, q) and g(r, q) for the final charge state q are
h(r, q) = (q − 1) 0.143 + 0.00424r − 0.00063r2
and
"
g(r, q) = (q − 1) −1/r + e−αr
4
X
(8.15)
#
Ai ri
(8.16)
i=0
with the parameters: α = 1.66194, A0 = 484.16696, A1 = −152.88756, A2 = 12.47619,
A3 = −3.83926, and A4 = 1.04143.
8.1.4
Calculation of Ionisation Rates for Cq+
60
The ionisation rates for Cq+
60 can be calculated in the SAE picture applying the S-matrix
120
CHAPTER 8. EXCITATION OF C60
theory [JBF07]. This theory provides quantitative estimations of transition probabilities in
complex atoms, molecules, and clusters interacting with strong laser fields [JBF04]. Here the
lowest order S-matrix theory, namely, the well known “Keldysh-Faisal-Reiss” (KFR) model
[Rei80] is utilised to calculate total ionisation rates of C60 as a function of laser pulse duration.The total ion yield for the individual charge states is determined by summing over the
contributions from all points in the laser focus. The comparison of the theoretical results
with experimental data is shown in Fig. 8.3a. The calculated data are normalised to the maximum signal of C+
60 . Both experimental and theoretical data are plotted in Fig. 8.3b. The
experimental ion yields of different charge states are scaled by the detection efficiency of the
microchannel plates which depends on the impinging energy of Cq+
60 .
The experimental data show the general trend of increasing abundance of multiply charged
ions with increasing pulse duration. This trend is in a qualitative agreement with the theoretical
modelling. Especially, good agreement is observed in the short pulse limit, where energy
redistribution processes such as electron-electron scattering and coupling to the vibrational
degrees of freedom, which are not included in the S-matrix theory, play a minor role. It indicates
that the SAE assumption may be an appropriate model for the description of photoinduced
processes in the limit of ultrashort laser pulses.
8.2
8.2.1
Pulse Duration Dependent Study
Experimental Observations
Here the experimental results for C60 photoionisation obtained in an intensity range between
3 × 1013 W/cm2 and 4 × 1014 W/cm2 with different laser pulse durations are presented. A
detailed analysis of the experimentally determined ion yields measured as a function of laser
intensity provides important information about photoinduced processes such as ionisation.
First, experiments with laser pulses centred at 765 nm at a pulse duration of 9 fs are discussed. Fig. 8.4a-e illustrates the characteristic dependence of charge states and the fragment
mass distribution on laser intensity. At highest intensity of 3.7 × 1014 W/cm2 one observes
charge states of Cq+
60 up to q = 5, as well as a significant amount of multiply charged, fullerenelike fragments which have lost up to 8 units of C2 as shown by the enlarged inset in Fig. 8.4a
8.2. PULSE DURATION DEPENDENT STUDY
121
Figure 8.4: Mass spectra obtained from photoionisation of C60 with 9 fs laser pulses centred
at 765 nm. The laser intensity decreases from 3.7 × 1014 W/cm2 (a) to 4 × 1013 W/cm2
(e).
(fragmentation of C60 is discussed in Chapter 9). With decreasing intensity the formation of
ions decreases, especially so for the highest charge states of C60 . Finally, at 4 × 1013 W/cm2
only C+
60 is noticeable in the mass spectrum (Fig. 8.4e). For comparison a set of mass spectra is
measured as a function of intensity for laser pulses with a duration of 27 fs centred at 797 nm.
Fig. 8.5a-b compare the ion yields for several final charge states of Cq+
60 as a function of the
laser intensity recorded with 27 fs and 9 fs pulses, respectively. The standard log-log plot is
used to illustrate the power law (Eq. (8.1)) at intensities below saturation. The experimental
data exhibit the characteristic saturation behaviour at high intensities for all charge states.
122
CHAPTER 8. EXCITATION OF C60
2
1 0
1
io n s p e r s h o t
1 0
(a )
C
+
τ= 2 7 fs
C
2 +
C
3 +
C
4 +
0
1 0
6 0
6 0
6 0
N = 5
N = 8
-1
1 0
N = 1 1
1 0
-2
1 0
-3
N = 8
N = 1 1
0
1 0
C
+
C
3 +
C
4 +
C
5 +
C
6 +
(b )
6 0
τ= 9 fs
io n s p e r s h o t
6 0
1 0
-1
1 0
-2
N = 5
1 0
-3
1 0
-4
1 0
-5
C
2 +
6 0
6 0
6 0
6 0
6 0
N = 1 1
N = 1 1
N = 6
N = 8
N = 1 0
N = 9
4 x 1 0
1 3
N = 1 1
1 .9 x 1 0
in te n s ity [W /c m
2
1 4
3 .4 x 1 0
1 4
]
Figure 8.5: Experimental yields of Cq+
60 ions measured as a function of intensity (symbols)
and their fits (lines): (a) laser pulse duration is 27 fs; (b) laser pulse duration is 9 fs.
It is determined by the absolute magnitude of the MPI cross-section and the experimental
geometry.
To extract values for the saturation intensity Isat and the order (slope) N of the MPI at
low intensities the data are fitted according to Eq. (8.10). The slopes obtained for the Cq+
60
ion yield at low intensities (full lines in Fig. 8.5) for q = 1, 2, and 3 are N = 5, 8, and 11,
respectively. For higher charge states the slopes remain essentially constant at N = 10 ± 2.
However, the statistics of the 9 fs experimental data does not allow one to make a fit with
one unique slope for q = 2 and 3 over the whole intensity range (as indicated by the dashed
2+
3+
lines in Fig. 8.5). For strictly stepwise MPI processes: C60 −→ C+
60 −→ C60 −→ C60 −→ . . .
8.2. PULSE DURATION DEPENDENT STUDY
123
one would expect N = 5, 8, and 11 for q = 1, 2, and 3, respectively, and even higher values
of N for larger charge states. In the SAE picture, N is the number of photons required to
remove subsequently an additional electron after saturation of the precursor charge state. The
remarkable fact is that the slopes derived meet the expectations only for q = 1, 2, and 3. But
a stepwise MPI model does not explain constant or even falling values of N for higher charge
states. Even more interesting is the observation that the slopes for q = 2 and 3 appear to
(q−1)+
reflect the number of photons one would need for ionisation C60
q+
in an intensity
−→ C60
range, where the respective precursor is far from being saturated.
These observations clearly indicate the limitations of the SAE picture with sequential ionisation for such complex systems as C60 fullerenes. This may be due to the resonant pre-excitation
to “doorway” state [LSS07], and/or to high lying electronic states close to ionisation threshold,
possibly the Rydberg states [BLH05], and/or even due to collective excitation which will be
discussed in the following. The theoretical calculations predict multielectron excitation even at
intensities far below the presently observed saturation values [TNE01, ZSG03, ZGe04]. Consequently, many electrons are excited prior the ionisation and can possibly support simultaneous
multielectron ejection.
The largest slope (N = 11) observed in the present experiment corresponds to a total
photon energy of 17.8 eV with laser pulses of 9 fs duration at 765 nm. This energy is enough
to excite the first maximum of the σ-part of the C60 giant plasmon resonance (see Sec. 2.1)
[HSV92]. Fig. 2.3 illustrates this energetics for the single photon ion yield. According to this
picture, even for 8 photons there is substantial oscillator strength in the ionisation continuum.
Therefore, if the plasmon resonance is sufficiently excited, any further excitation is very efficient
due to the absorption of photons by the hot electron-hole plasma. Thus, the transition
saturates and no further increase in the slope is observed, while at the same time multiple
ionisation can occur. Indeed, it might explain both the observed slopes N = 8 and 11 for
q = 2 and 3, respectively, at low intensities and the absence of slopes increasing for higher
charge states. After the excitation of the giant plasmon the system is bound to undergo
non-adiabatic multielectron dynamics (NMED) [MRS04]. Then, sequential or non-sequential
non-adiabatic excitation is driven by the strong laser field which mixes the dense manyfold of
electronic states in this energy range.
124
8.2.2
CHAPTER 8. EXCITATION OF C60
Sequential Ionisation
To further critically interrogate the usually assumed concept of sequential ionisation, the
coupled differential equations for single and double ionisation of C60 are treated for such a
stepwise process
+
C60 + N hν −→ C60
+ e− , EIP = 7.6 eV
(8.17)
+
2+
C60
+ M hν −→ C60
+ e− , EIP = 11.4 eV .
(8.18)
and
From Eq. (8.3) the reduction of the initial (neutral) C60 density n0 during the laser excitation
is given by
dn0 (r, t)
= −n0 (r, t)γN I N (r, t) .
dt
(8.19)
The densities of singly and doubly charged species (n1 and n2 , respectively) are obtained from
dn1 (r, t)
= n0 (r, t)γN I N (r, t) − n1 (r, t)γM I M (r, t)
dt
(8.20)
n2 = n0 (r, 0) − n0 (r, t) − n1 (r, t) .
(8.21)
and
For a given peak pulse intensity I0 the coupled rate Eqs. 8.19-8.21 have been solved numerically
by integrating over the Gaussian beam profile and the cylindrical interaction volume using the
experimental parameters reported in Chapter 6. The results plotted on a log-log scale are shown
in Fig. 8.6. The calculation was performed for 800 nm laser pulses with a duration of 27 fs,
setting the power laws according to the ionisation potentials (7.6 eV and 11.4 eV) to N = 5 and
M = 8 for stepwise ionisation. Two different sets of saturation intensities were used for these
+
2+
model calculations: Isat (C60
) = 1 × 1013 W/cm2 , Isat (C60
) = 1 × 1014 W/cm2 as shown in
+
2+
) = 1.39 × 1014 W/cm2 as
Fig. 8.6a and alternatively Isat (C60
) = 1.00 × 1014 W/cm2 , Isat (C60
shown in Fig. 8.6b (the latter set of Isat represents the experimentally derived values reported
later).
The C+
60 ion yield shows a reasonable behaviour for intensities below saturation with a
slope of N = 5 in agreement with the corresponding power law. However, the yield of C2+
60
exhibits the expected slope M = 8 expected for sequential ionisation only for intensities above
+
the saturation of C+
60 . Hence, the saturation intensity for C60 had to be chosen a factor of
10 below that for C2+
60 as illustrated in Fig. 8.6a. On the other hand, according to the rate
8.2. PULSE DURATION DEPENDENT STUDY
125
1
+
calc. ion yield [arb. units]
10-2
+
C60
C60
10-4
10-6
M=8
N=5
10-8
N=5
++
++
C60
10-10
10-12
C60
M+N
=13
M+N
=13
10-14
1013
1014
1013
1015
intensity [W/cm2]
1014
1015 1016
2+
Figure 8.6: Yields of C+
60 and C60 calculated by solving the coupled differential equations
2+
13
2
assuming two different sets of saturation intensities for C+
60 and C60 : (a) 1 × 10 W/cm
and 1 × 1014 W/cm2 , respectively; (b) 1.00 × 1014 W/cm2 and 1.39 × 1014 W/cm2 , respectively.
equation model using the measured values of the saturation intensities, the yield of C2+
60 follows
a power law with a slope of N + M = 13 and overshoots the C+
60 yield at higher intensities
by more than one order of magnitude (Fig. 8.6b).
8.2.3
Saturation Intensities
The saturation intensities derived from the fits of the experimental data presented in
Fig. 8.5a-b are plotted as a function of the final charge state q in Fig. 8.7. The values of
saturation intensities obtained with 9 fs and 27 fs laser pulses are indicated by magenta squares
and red circles, respectively. For comparison saturation intensities measured with 70 fs pulses at
1800 nm (brown triangles) [BCR03] and with 45 fs pulses at 395 nm (blue diamonds) [THD00]
are also shown. The brown dashed line in Fig. 8.7 shows the theoretical prediction based
on the classical over the barrier ionisation mechanism for the conducting sphere model of
126
CHAPTER 8. EXCITATION OF C60
765 nm, 9 fs
797 nm, 27 fs
[JBF06]
[BCB05]
[BCR03]
[THD00]
Figure 8.7: Saturation intensities for Cq+
60 ions as a function of the final charge state q.
Experimental data obtained with 70 fs pulses at 1800 nm (triangles) and with 45 fs pulses
at 395 nm (diamonds) are taken from [BCR03] and [THD00], respectively. The present
data obtained with 765 nm, 9 fs and 797 nm, 27 fs are indicated by squares and circles,
respectively. The experimental data are compared to theoretical models: over the barrier
ionisation using the conducting sphere model [BCR03] and the model potential (Eq. (8.22))
including polarisation screening with and without centrifugal barrier as well as theory from
[BCB05] and [JBF06].
C60 with polarisation screening for a wavelength of 1800 nm from [BCR03]. In contrast to
the original publication the ionisation potentials described by Eq. (8.13) are used here. This
model describes the experimental data for the higher charge states (q = 3 − 6) quite well,
but large deviations between theory and experiment are observed for q = 1 and 2. Saturation
intensities from the same ionisation model, but using a more realistic potential described by
Eq. (8.14), are also given. In this case the overall interaction potential U (r) including the laser
field, the field induced dipole, and the centrifugal barrier for an ejected electron in a spherical
8.2. PULSE DURATION DEPENDENT STUDY
127
approximation is given by
U (r) = U (r, q) + E0 r cos θ − α/r2 + l(l + 1)/ 2r2
,
(8.22)
where U (r, q) is the jellium type potential described by Eq. (8.14), E0 is the laser field amplitude, α is the polarisability of C60 , and l is the angular momentum. The red dotted line
in Fig. 8.7 shows the result using this potential, but without the centrifugal barrier. The experimental saturation intensities are underestimated for all charge states by nearly one order
of magnitude in this model. Taking into account the centrifugal barrier with the angular momentum l = 5, which corresponds to the angular momentum of the hu ground state electrons,
gives results (red lines with dots) which are very similar to the theoretical values obtained with
the above described conducting sphere model. While on the other hand, the conducting sphere
model with the centrifugal barrier gives the saturation intensities (brown line with dots) which
are much higher than the experimental values. Based on these results, one can concludes
that the centrifugal barrier is important for understanding the high saturation intensities for
Cg+
60 in terms of the above barrier model. This conclusion is supported by recent theoretical
work [BCB05], where it was found that the width and the shape of the momentum distribution in complex systems demonstrate strong deviations from conventional tunnelling ionisation
theories.
However, important experimental parameters such as laser pulse duration and central wavelength do not enter into the above barrier model at all. Therefore, to explain the observed
dependencies of the saturation intensities on laser parameters a more detailed description is
required. A key result is that the saturation intensities for low charge states depend significantly on the laser wavelength in direct contrast to the quasi-static classical tunnelling and
over the barrier ADK theory. A special case is the ionisation with 395 nm laser pulses which is
very likely affected by the t1g resonance in the neutral C60 . This can explain the significantly
lower experimental saturation values. In this context, it is also interesting to compare the
saturation intensities measured with 9 fs and 27 fs laser pulses. For the lowest charge state
(q = 1) the 9 fs value is about 2.5 times higher than the result with 27 fs. While the 27 fs data
show a weak but monotonic increase of the saturation intensity as a function of the charge
state, the 9 fs data even slightly decrease with the charge state. This observation can not be
explained by any simple above barrier stepwise picture. According to Eq. (8.8) on can compare
equivalent intensities I1 and I2 for different durations τ1 and τ2 of laser pulses with similar
128
CHAPTER 8. EXCITATION OF C60
shapes using the following expression
I1
=
I2
τ2
τ1
1/N
.
(8.23)
This formula may be utilised to compare saturation intensities measured with different laser
pulse durations. Unfortunately, the difference between the saturation intensities obtained with
9 fs and 27 fs pulses can not be simply expressed by Eq. (8.23) indicating complex nature of
C60 ionisation mechanisms.
8.3
8.3.1
Polarisation Dependent Study
Photoion Spectroscopy
Investigations of effects induced by the laser light ellipticity play a significant role in studying
ionisation mechanisms. The influence of the light ellipticity on single and double ionisation
and high order harmonic generation (HHG) is quite good known for atoms and small molecules
[FKM02, SLW03]. In this case a reduction of ion and HHG yields is attributed to the recolliding
electron being driven away from its origin by circularly as opposed to linearly polarised light.
At the same time, apart from [BCR04] and one pioneering study at particularly high intensities
[RKM03] only little is known about the role of ellipticity in the energy deposition process during
the interaction of intense laser pulses with complex molecular systems. To get additional
information about C60 ionisation mechanisms and, especially, about the influence of electrons
recollision on ionisation, the effect of the laser light ellipticity on C60 ionisation was investigated
in details. Key questions are whether here recollision effects can be identified and/or whether
a particular signature of the doorway state can be found.
Resent studies at longer wavelength (1500 nm) have demonstrated the importance of the
recollision process in fragmentation from higher charged states of Cq+
60 (q = 3, 4) [BCR04].
With long wavelength one purely goes into the tunnelling ionisation regime at lower laser
intensity. Indeed, the Keldysh parameter γ is ≈ 0.2 for the experimental conditions described
in [BCR04], implying a ponderomotive potential UP = 95 eV which is proportional to the
square of the laser wavelength (Sec. 3.1). Hence, energies of the rescattered electrons of
up to 3.17UP = 300 eV are encountered in this process that can safely be described by a
8.3. POLARISATION DEPENDENT STUDY
+
C60
C60-2n
+
}
4+
C60
}
0.02
2+
2+
C60
C60
C60-2n
ion yield [arb. units]
0.04
2+
(b)
3+
60-2n
+
C60
(a)
}C
0.2
0.1
129
3+
0.00
0.2
0.1
C60
(c)
(d)
0.04
0.02
0.00
180
360
720
180
m/q [u]
360 720
Figure 8.8: (a), (b) Mass spectra of C60 obtained with linearly polarised laser radiation
of 27 fs pulse duration centred at 797 nm with intensity of 9 × 1013 W/cm2 and 4.1 ×
1014 W/cm2 , respectively. (c), (d) Mass spectra obtained with circularly polarised laser
radiation with the same intensities as in (a) and (b), respectively. The small horizontal
lines in (c) and (d) indicate the peak maxima in (a) and (b), respectively.
single active electron (SAE) [NKS07]. For shorter wavelengths, which are used in this work,
Up is typically much smaller and the ionisation process is more complex due to competition
between multiphoton ionisation, tunnelling, excitation of intermediate electronic states and
non-adiabatic multielectron dynamics (NMED).
Generally, the ionisation rate may be affected by the polarisation of the laser field for various
√
reasons: (i) maximum field amplitude for linear polarisation is 2 times the constant circularly
polarised field for any given average intensity; (ii) if multiphoton absorption dominates, special
angular momentum selection rules may also be significant; (iii) the efficiency of the recollision
is determined by the trajectory of the electron being driven away from its origin by circularly
130
CHAPTER 8. EXCITATION OF C60
as opposed to linearly polarised light. More efficient ionisation with circular polarised laser
radiation at higher intensities was theoretically predicted for H atoms in ab initio quantum
mechanical calculations [PLK97]. Recently similar results have been experimentally observed
for ionisation of NO with 800 nm laser pulses [GGi01] and qualitatively described with ADK
model [ADK86]. However, the ADK theory may not give an accurate description of ionisation
of complex molecules.
Fig. 8.8 shows mass spectra recorded with 27 fs pulses centred at 797 nm and intensity
of 9 × 1013 W/cm2 for linear (Fig. 8.8a) and circular (Fig. 8.8c) light polarisation as well as
with intensity of 4.1 × 1014 W/cm2 for linear (Fig. 8.8b) and circular (Fig. 8.8d) polarisation.
Fig. 8.8a-b reproduce the well known C60 mass spectra obtained with short linearly polarised
laser pulses. At intensities below 1 × 1014 W/cm2 intact Cq+
60 ions are observed with charge
states q ≤ 3, while for higher intensities fullerene-like fragments (Cq+
60−2n ) appear in the mass
spectrum especially for q ≥ 2. At first glance, the mass spectra shown in Fig. 8.8c-d taken
with circularly polarised laser pulses look very similar. However, a closer inspection shows
small but significant differences as indicated by the small horizontal bars on top of the mass
peaks in Fig. 8.8c-d, representing the respective signals in Fig. 8.8a-b, respectively. At lower
intensities the ion signals decrease by approximately 25%, when changing the polarisation
from linear to circular. On the other hand, by comparing Fig. 8.8b and Fig. 8.8d one finds that
for high intensities the ion signals of the fullerene-like fragments even increase up to 20%,
when circularly polarised light is used, while the ion signals of the intact Cq+
60 ions are almost
unchanged. These first qualitative inspections already hint at interesting effects of the laser
intensity and ellipticity on the excitation and ionisation process in complex systems.
Before discussing the C60 results the Xe+ ion yield, which can be viewed as a supposedly
simple test case, will be considered. The overall dependence of the Xe+ ion yield on laser
pulse intensity (linearly polarised) is well known from the literature [LTC98]. With an ionisation potential of 12.13 eV at least 8 photons are needed to generate the lowest ionic state
Xe+ (2 P3/2 ), but a number of intermediate resonances obscure the expected power law (see
Sec. 3.2) somewhat. Thus, the log-log plot of ion yield as a function of intensity shows a slope
of 8 only for intensities below 1013 W/cm2 , while between 3 × 1013 W/cm2 and saturation intensity of Is ' 1014 W/cm2 a slope of 5 is observed, giving evidence for an effective 5 photon
process.
8.3. POLARISATION DEPENDENT STUDY
ellipticity ε
+
relative Xe ion yield
β = 45º
Ex(0)
1.25
1.00
0.75
0.50
0.25
0.00
0.0
Elin(0)
Ey(0)
45º 30º 15º 0º
ellipticity angle β
131
0.5
1.0
β = 0º
Ec(0)
Ec(�/2)
5 2]
4 /cm
W
3 014
1
coherent
2 ity [
s
model
1 nten
----- N = 5
ri
e
–––– N = 8
las
Figure 8.9: Integrated yield of Xe+ measured at 797 nm normalised to the yield with
linear light polarisation plotted as a function of ellipticity angle β and laser intensity.
The polarisation changes from linear (β = 45◦ ) to circular (β = 0◦ ). The full and dashed
red lines at I = 0.65 × 1014 W/cm2 correspond to coherent excitation in an 8- and 5photon processes, respectively.
Two inserts illustrate that the maximum field amplitude
√
for linear polarisation is 2 times the constant circularly polarised field for any given
average intensity.
Ellipticity dependence of the ion yield as a function of the laser intensity can be considered
using the “classical averaging” model described in Sec. 5.5. Eq. (5.51) predicts a drop of
the effective multiphoton cross-section between linear and circular polarisation as a direct
consequence of the fact that the electric field vector for circularly polarised light is constant
√
but only 1/ 2 of that for the maximum of linearly polarised light.
Fig. 8.9 shows a 3D plot of the integrated Xe+ yield as a function of both, laser intensity
and ellipticity angle β which is related to the often used ellipticity ε according to Eq. (5.45).
To compensate for the strong intensity dependence of the ion yield the present experimental
data are normalised to their values for linear polarisation. When the polarisation changes
from linear to circular, a dramatic drop of the signal is observed at all intensities, but most
significantly at the lower intensities. While for higher intensities, where the saturation for the
ion signals has been reached, some reduction of this trend is recognised. The dashed red
132
CHAPTER 8. EXCITATION OF C60
line in Fig. 8.9 represents the results of the classical averaging model described by Eq. (5.51)
with N = 5 at an intensity of I = 0.65 × 1014 W/cm2 which has been chosen significantly
below the saturation intensity for Xe+ ion formation. This theoretical model (N = 5) roughly
reflects the experimentally observed trend. An even better fit is found, if N = 8 is assumed,
as shown by the full red line in Fig. 8.9. Interestingly, ionisation of Xe requires indeed at least
' 8h̄ω0 (λ = 797 nm). There is no conclusive explanation for this finding. However, the slope
I 5 intensity dependence is usually attributed to “population trapping” in high lying states
[TCC96] which is obviously not reflected in the β-dependence. In that sense, the polarisation
dependent studies probe the order of the photon absorption process in a much more sensitive
manner than ion yield measurements.
3+
Now the main subject will be considered. The measured ion yield for C+
60 and C60 is shown
in Fig. 8.10. In both cases a small, but very clear reduction of the signal for circularly polarised
light is seen at low intensities (< 1 × 1014 W/cm2 ). Here the ponderomotive potential is only
' 5 eV and recolliding electrons can not play a significant role. However, a nearly perfect match
of the observed β-dependence (low intensity) is found with an average intensity distribution
hI 2 (t, β)i for a coherent two photon process as indicated by the full red line in Fig. 8.10. This
remarkable result gives clear evidence to the importance of the t1g doorway state which can be
populated by such a coherent two photon excitation. At higher intensities the β-dependence
decreases and the ion yields become almost independent of ellipticity. This can be attributed
to saturation of the ionisation process.
3+
N
For the current laser parameters the ion yields of C+
with N = 5 and
60 and C60 are ∝ I
3+
11, respectively. In case of C+
60 and C60 the corresponding dependencies (5 and 11) are not
observed as a function of ellipticity (in contrast to the Xe+ case). This gives strong evidence
for genuine multielectron processes dominating the ionisation and energy deposition in C60 :
once the doorway state is reached, many electrons can absorb energy through transitions in
a quasi-continuum of states which in fact explains the high number of photons absorbed. If
more or less loosely coupled chromophore electrons each absorbs one photon, one expects
no polarisation dependence. For a process which requires an energy equivalent of at least 5
5
photons (such as C+
60 formation), the intensity dependence will nevertheless be ∝ I since
5 photons must be absorbed during the interaction with the laser pulse. These processes
are, however, no longer coherent since several electrons are involved independently. The
8.3. POLARISATION DEPENDENT STUDY
133
0.0
0.5
1.0
relative ion yield
3+
C60
1.25
1.00
0.75
0.50
0.25
0.00
4
2
5
3
1
45º 30º 15º
0º ellipticity ε
0.0
0.5
1.0
relative ion yield
C60+
1.25
1.00
0.75
0.50
0.25
0.00
4
2
1
45º 30º 15º 0º
ellipticity angle β
n
ri
e
las
ten
14
3
sit
y
0
[1
5
2
/cm
]
W
3+
Figure 8.10: Integrated yield of C+
60 and C60 measured at 797 nm normalised to the yield
with linear light polarisation plotted as a function of ellipticity angle and laser intensity.
The full red lines correspond to coherent excitation in a 2-photon process.
corresponding statements also hold for higher charge states.
As a crucial test the ellipticity dependence of the Cq+
60 signals at λ = 399 nm for laser
intensities between 0.2×1013 W/cm2 and 4.0×1013 W/cm2 was measured. The corresponding
plots presented in Fig. 8.11 are absolutely flat to within 1%, while the ion yields at this
3+
wavelength were found to be ∝ I 3 and ∝ I 6 for C+
60 and C60 , respectively [THD00]. This
strongly confirms the key role of the doorway state, being excited by absorption of a single
399 nm photon so that its population depends only on I0 and not on β. Again, subsequent
photons are deposited via different chromophore electrons of which each absorbs one photon.
134
CHAPTER 8. EXCITATION OF C60
blue
relative ion yield
3+
C60
1.1
1.0
1
2
3
4
1.020
0.995
0.9 45º 30º 15º 0º -15º 0
+
relative ion yield
C60
2
1.1
3
1.0
4
2 ity
s
1
en
t
in
0 ser
0.9
45º 30º 15º 0º -15º
la
13
[
10
/cm
]
W
ellipticity angle β
3+
Figure 8.11: Integrated yield of C+
60 and C60 normalised to the yield with linear light
polarisation plotted as a function of ellipticity angle and laser intensity after interaction
of C60 with 399 nm laser pulses. Here complete independence on ellipticity at all intensities
is found, in contrast to corresponding measurements with 797 nm pulses.
The behaviour of the fullerene-like fragments Cq+
60−2n is even more surprising. Fig. 8.12
shows the singly and triply charged ion yield summed over all fragments n ≥ 1. For the
lower intensities again the coherent two photon signature hI 2 (t, β)i (at least for C3+
60−2n , while
C+
60−2n fragments are very weak and the noisy signal does not allow one to make a clear
conclusion) can be recognised. However, at higher intensities a significant enhancement of
the fragment signal is observed with circularly polarised light. At first sight, this appears to
be against all common wisdom: typically the signals decrease with circularly polarised light.
One potential explanation could be complicated multiphoton processes with absorption and
induced emission steps leading to constructive interference. In view of the fact that at high
8.3. POLARISATION DEPENDENT STUDY
135
ellipticity ε
0.0
0.5
∑C
1.0
3+
relative ion yield
60-2n
1.25
4
1.00
1
45º 30º 15º
∑C
3
2
0.75
5
0º
0.0
0.5
1.0
+
relative ion yield
60-2n
1.25
4
1.00
2
3
0.75
45º 30º 15º 0º
ellipticity angle β
1
las
i
er
y
e
nt
it
ns
14
[
10
5
2
/cm
W
]
3+
Figure 8.12: Integrated yield (enlarged scale) of fragments ΣC+
60−2n and ΣC60−2n measured
at 797 nm normalised to the yield with linear light polarisation plotted as a function of
ellipticity angle and laser intensity. The full red lines correspond to coherent excitation
in a 2-photon process.
intensities a multitude of intermediate states could play a role, in particular the intense giant
resonance between the first ionisation threshold and about 35 eV. This is just pure speculation
and the question is, of course, why this should be specific to generate fragments rather than
higher charge states.
However, taking into account the potential influence of rescattered electrons, alternative
explanations can also be discussed. Even in the classical trajectory picture one might imagine
loops of recolliding electrons in a circularly polarised electric field (especially so, if the electrons
ejected have initial kinetic energy [SGP08]). Rescattering for extended atomic systems has
recently also been discussed in the context of Xe and Ar cluster ionisation [SRo08]. In contrast
136
CHAPTER 8. EXCITATION OF C60
10 a
(a)
(b)
y
(c)
(d)
10 a
- e 0 E ci
- e0 Elin(t = 0)
(t =
rc
0)
x /a
-20
-15
-10
-5
0
5
Figure 8.13: Examples of classical electron trajectories in the combined field of C+
60 and lin14
early (dashed blue lines) or circularly (full red lines) polarised light. I = 4.3×10 W/cm2 ,
797 nm , electron initial kinetic energy is 1 eV (a,b) or 10 eV (c,d), starting point of the
trajectory is 0.65a (b,d) or 1.00a (a,c), Φ = π/4, Θ = π/2 (for details see text). The
initial field phase at t = 0 is ϕ = 162◦ in all cases shown, implying that the force on the
electron at t = 0 points into the direction of −72◦ for circularly and into the direction
of 0◦ for linearly polarised light as indicated by the vectors. The thick full gray circles
indicate the C60 shell radius 0.65a . . . 1.00a.
to single atoms, in C60 the photoionised (or tunnelling) electrons emerge from a sphere with
radius ≥ a ' 8.12 a.u., the C60 shell radius [DDE96]. They may even be ejected with
relatively high initial kinetic energy due to strong field absorption of many photons. If e.g. the
C60 plasmon resonance supports the photo absorption process in the continuum [SLS06], one
expects initial kinetic energies of up to 30 eV. Assuming the electrons to be ejected radially this
will lead most trajectories in a linearly polarised field to miss the C60 on return. In contrast,
one may find a number of trajectories in a circularly polarised field which return, even several
times, as exemplified in Fig. 8.13 for a few specific initial parameters.
8.3. POLARISATION DEPENDENT STUDY
137
Using classical mechanics to describe the electron motion r(t) one can write
r00 (t) =
F(r)
e0 E(t)
− s(ρ)
me
me
,
(8.24)
where me is the electron mass, e0 is the electron charge, E(t) is the electric field of the laser
radiation, and |F(r)| = −e0 dV (r)/dr is the force derived from a model potential by which
the C60 core acts on the electron.
For the present classical trajectory studies a commonly used model potential V (r) for
the Cq+
60 ions is employed [PNi93]. It is parameterised similarly to that given in Sec. 8.1. It
−10
describes C+
m
60 essentially as a potential well with outer and inner radius a = 4.2916 × 10
and b = 0.65a, respectively, approaching asymptotically a Z/r Coulomb potential for r a.
For the present classical trajectory calculations the model potential in its most recent form
[MCR08] is approximated by smoothing the very sharp (and physically unrealistic) edges over
a width of w = 0.15a using sigmoid functions, while the potential depth was adjusted for
Z = 1 to V0 = 1.633 so that the binding energy of the additional electron corresponds to the
experimental values −7.56 eV. The model potential (in a.u.) is
a−r
qa e w 1 + Z
Z −1
1
1
1
V (r) = −
V0 +
×
−
a−r −
r−a +
b−r − 1
r
a+b
1+e w
1+e w
1+e w
r−b
qi e w 1 + q0
1
−
r−b
a
1+e w
(8.25)
with w1 = 1.59, qa = 4.5, qi = 2.68, and q0 = 0.73.
The function s(ρ) describes polarisation screening [BCR03] of the laser field outside the
Cq+
60 and in addition screening of the field inside so that it can not penetrate into the C60 core.
A cutoff function, which serves this purpose sufficiently well, is used for simplicity
s(ρ) =
1
1+e
1.5a−ρ
0.4a
(8.26)
p
with ρ(t) = x2 (t) + y 2 (t) (assuming the electric field vector to lie in the xy-plane). Setting
p
|r(t)| = x2 (t) + y 2 (t) + z 2 (t) Eq. (8.24) can be rewritten as
s(ρ)e0
x F (r)
E0 sin (β + π/4) sin (ω0 t + ϕ) +
me
r me
y F (r)
s(ρ)e0
y 00 (t) =
E0 cos (β + π/4) cos (ω0 t + ϕ) +
me
r me
z F (r)
z 00 (t) =
.
r me
x00 (t) =
(8.27)
138
CHAPTER 8. EXCITATION OF C60
E0 , ω0 , ϕ, and β are the electric field amplitude, frequency, phase, and ellipticity angle,
respectively. Assuming the electron to be ejected radially with a total initial kinetic energy
Win from the C60 molecule at a position r(0) = {r0 sin Θ cos Φ; r0 sin Θ sin Φ; r0 cos Θ} the
following initial conditions have to be implemented
x(0) = r0 sin Θ cos Φ
y(0) = r0 sin Θ sin Φ
z(0) = r0 cos Θ
p
x0 (0) = 2Tin /me sin Θ cos Φ
p
y 0 (0) = 2Tin /me sin Θ sin Φ
p
z 0 (0) = 2Tin /me cos Θ
(8.28)
Tin = Win + |V (r0 )| .
(8.29)
with
To derive r(t) as well as the electron kinetic energy W (t) a set of the differential equations
Eq. (8.27) with the initial conditions Eq. (8.28) is solved using Mathematica. The recollision
on return occurs if the following condition is satisfied
rrec ≡ |r(trec )| = r0
,
(8.30)
where trec (trec > 0) is the time when recollision happened. The kinetic energy of the recolliding
electron is
Wrec
me (r0 (trec ))2
=
− |V (rrec )| .
2
(8.31)
Experimental observations together with the trajectory studies at long wavelength of
1500 nm documented that fragmentation is significantly amplified by recollision in linear polarised laser fields [BCR04]. In contrast, the present classical trajectory calculations for the
wavelength of 800 nm demonstrate that electron trajectories in a circularly polarised field starting on the C60 radius a or at the inner shell radius 0.65a return significantly more often as
illustrated in Fig. 8.13.
To quantitative investigate influence of the numerous parameters entering in Eq. (8.27)
and Eq. (8.28) on the kinetic energy of the recolliding electrons statistical test using a Monte
Carlo approach is performed. Wrec is calculated 100000 times starting with uniform random
recollision probability [% eV-1]
8.3. POLARISATION DEPENDENT STUDY
1.5
139
circular (15.1 %)
1.0
3.17 UP
0.5
linear (10.8 %)
0
0
10
20
70
electron recollision energy [eV]
80
Figure 8.14: Probability to find different recollision energies for electrons emitted from
the inner (0.65a) and outer radius (1.00a) of the C60 molecule in linearly or circularly
polarised light at I = 4.3 × 1014 W/cm2 , 797 nm. Initial conditions (at t = 0) are taken
statistically: laser field phase 0 ≤ ϕ < 2π, electron starting radius 0.65a or 1.00a, electron
kinetic energy 0 ≤ Win ≤ 10 eV (radial emission) at 0 ≤ Φ < 2π and 0.95π/2 ≤ Θ ≤ π/2.
distribution of the following parameters: laser field phase 0 ≤ ϕ < 2π, electron starting radius
0.65a or 1.00a, electron kinetic energy 0 ≤ Win ≤ 10 eV (radial emission) at 0 ≤ Φ < 2π
and 0.95π/2 ≤ Θ ≤ π/2. The results generated from this statistical test are represented
in Fig. 8.14. This histogram shows electron recollision probability plotted as a function of
its kinetic energy at the moment of recollision for circularly (red squares) and linearly (black
circles) polarised radiation at I = 4.3×1014 W/cm2 and 797 nm. About 15% of the trajectories
return in circularly polarised light, while only 11% recollide for linear polarisation. On the other
hand, there is still a small, but non negligible number of trajectories in linearly polarised light
which return with much higher energies – up to the well known value of 3.17UP as indicated
in Fig. 8.14, while trajectories in circularly polarised light only harvest up to 25 eV.
It should be pointed out that this trajectory study is not intended to give a quantitative
or even qualitative explanation for the experimentally observed enhancement of fragments
140
CHAPTER 8. EXCITATION OF C60
created by circularly polarised light over those excited by linearly polarised light at high laser
intensities. This is not to be expected from a single electron trajectory study in a situation,
when multielectron dynamics dominates. A key point here is just to illustrate that the general
believe about circular polarisation being unable to lead to energy deposition due to recollision
is not necessarily true for extended systems and strong laser fields. A more sophisticated
calculation in the spirit of those done for rare gas clusters [SSR06] may lead to interesting
results, when the ionising field is circular rather than linear. In view of the examples of the
trajectories shown in Fig. 8.13 one may very well imagine a vehement collective motion and
collisions in a many electron system driven by circularly polarised strong laser fields. This
certainly could lead to substantial heating of the electron gas and subsequently of the nuclear
backbone which is then finally probed by fragmentation long after the laser pulse is over (see
Sec. 9.4).
8.3.2
Photoelectron Spectroscopy
At high laser intensities multiphoton ionisation occurs via above threshold ionisation (ATI),
when an atom or molecule exposed to intense femtosecond laser radiation absorbs more photon
than required for ionisation (see Sec. 3.2). The corresponding photoelectron spectrum has
to exhibit peaks separated by the laser photon energy h̄ω0 . The envelope of ATI peaks in
the photoelectron spectrum does not decrease regularly, but shows plateau region, where its
intensity stays roughly constant or even increases with peak number [PNX94] due to recollision
of photoelectrons on their parent ion [BLK94]. Such recollisions can produce electrons with
kinetic energies above 2UP [PBN94, WSK96]. Therefore, if the recollision process is influenced
by the laser light ellipticity, this must be reflected in plateau region of mass spectra measured
with different ellipticity.
To see recollision fingerprints in photoelectron spectra of C60 several spectra were recorded
as a function of the laser light ellipticity. As an example, Fig. 8.15 shows two photoelectron
spectra obtained with the laser intensity of 1.4 × 1014 W/cm2 for linear and circular polarised
light. Both photoelectron spectra show ATI peaks up to the kinetic energy of ' 13 eV
(' 1.6UP ). The observed Rydberg structure [BHS01] on top of the first ATI peak is shown
in the enlarged insert. This structure arises from the AC-Stark shift (see Sec. 3.2) of Rydberg
states producing multiphoton resonances with the laser frequency. Each peak appears at a fixed
p h o to e le c tr o n
y ie ld [a r b . u n its ]
p h o to e le c tr o n y ie ld [a r b . u n its ]
8.3. POLARISATION DEPENDENT STUDY
0 .0 1
141
0 .0 0 9
0 .0 0 6
0 .0 0 3
0 .3
1 E -3
0 .6
0 .9
1 .2
k in e tic e n e r g y [e V ]
1 .5
1 E -4
lin e a r p o la r is a tio n
c ir c u la r p o la r is a tio n
1 E -5
0
3
6
9
1 2
1 5
1 8
2 1
2 4
k in e tic e n e r g y [e V ]
Figure 8.15: Photoelectron spectra of C60 obtained with laser intensity of 1.4×1014 W/cm2
for linear (black line) and circular (red line) light polarisations. Insert demonstrates
Rydberg states on top of the first ATI peak. Dashed vertical line indicates 2UP = 16.1 eV.
kinetic energy corresponding to the precise intensity at which this level shifts into resonance
with the laser. There is no any essential difference between these two photoelectron spectra
in the photoelectron energy range above 2UP = 16.1 eV. But the photoelectron signals in
this range are very weak and noisy and ATI structure is not visible at any light ellipticity.
Therefore, the experimental data obtained under present experimental conditions can not
definitely indicate presence or absence of recollisions.
142
CHAPTER 8. EXCITATION OF C60
Chapter 9
Dynamics of Ultrafast Energy
Redistribution in C60
Energy redistribution processes in C60 after the excitation with ultrashort laser pulses are
the focus interests in this chapter. First, a general overview on energy redistribution processes
important for the fragmentation of C60 fullerenes is given. Next, the experimental observations
of ultrafast energy redistribution obtained with a single pulse are discussed in detail. Then, one
and two colour pump-probe techniques are applied to study the energy redistribution dynamics.
Finally, the control of this dynamics with judiciously tailored laser pulses is presented.
9.1
Energy Coupling
There are two dominant processes of energy redistribution in large systems: electronelectron coupling and electron-phonon coupling. These processes correspond to the energy
flow within the electronic subsystem and its exchange with the nuclear degrees of freedom,
respectively. However, one has to keep in mind that C60 is a large, but still finite molecular
system with discrete energy levels and well defined modes of nuclear motion.
In a photoinduced excitation process (involving typically many absorbed photons), electronic and nuclear motion are intrinsically connected through the Franck-Condon (FC) principle. There is one notable exception in the case of C60 : the direct single multiphoton ionisation.
There, the initial ground state potential hypersurface of the neutral C60 and that of the ionic
ground state have a very similar structure and thus by virtue of the FC principle very little
vibrational excitation is expected. All ionisation processes into higher charge states may involve the plasmon resonance which strongly couples with vibrational motion. Consequently,
143
144CHAPTER 9. DYNAMICS OF ULTRAFAST ENERGY REDISTRIBUTION IN C60
e--e- and e--phonon relaxation τel ≤ 60-400 fs
q+
+ ℓ hνpr
+ (ℓ-j) hνpr
q+
C60 (very high Eint ) +qe-
C60 (high Eint ) +qe-
electronic and vibrational thermalisation
in the ion (100 fs to several ps)
C60 (low Eint ) +qe-
direct MPI
direct MPI
+ nhνpu
C60 → C60(e- highly excited) → C60(e- medium excited)
q+
q+
C60 (vibrationally hot)
C60 (low Eint ) +qe-
sequential unimolecular fragmentation
(ns to µs)
q+
C60-2n + n C2
Figure 9.1: Illustration of energy redistribution processes in laser excited C60 molecule
monitored in the present work with two colour pump-probe spectroscopy.
one can not attribute experimentally observed relaxation times to electron-electron coupling
exclusively. Rather, highly excited electrons formed during an intense laser pulse will exchange
energy by the combined action of electron-electron scattering and electron coupling with the
various vibrational degrees of freedom of the neutral molecule. After electron-electron coupling
statistical emission of electrons can occur from the excited electronic subsystem. Statistical
energy redistribution among vibrational degrees of freedom leads to the observation of delayed
ionisation (thermionic emission for a completely statistical process) accompanied by considerable fragmentation of the excited Cq+
60 .
The fs pump-probe technique provides a direct access to the dynamics of the energy coupling. The relaxation pathways of laser excited C60 in a pump-probe experiments are very
schematically depicted in Fig. 9.1. Considering a relatively weak (in a sense described in
Sec. 3.2) pump pulse, one expects the energy to be deposited into the electronic and vibrational system of the neutral species, thus forming a highly excited, non-equilibrium population
of states without significantly ionising the system. The following energy redistribution, i.e.
“cooling” within the electronic system and coupling this energy into the nuclear degrees of
9.2. SINGLE PULSE STUDY
145
freedom of the neutral system is then monitored directly by a delayed probe pulse. Inelastic electron-electron scattering and electron-phonon coupling occurs on time scales τel from
some tens to some hundreds of fs [CHH00, HHC03]. In previous studies such information
was extracted rather indirectly from pulse duration dependent experiments. In contrast, fs
pump-probe technique offers a more direct view into the dynamics with a temporal resolution
q+
of nuclear motion by analysing the time dependent signals of parent Cq+
60 and fragment C60−2n
ions. As schematically indicated in Fig. 9.1, these ions are generated by the probe pulse via
MPI (i.e. immediately) or, respectively, after sequential evaporation of C2 units on a time scale
of ns to several µs.
9.2
Single Pulse Study
The mass spectra obtained with 9 fs laser pulses which were presented in Fig. 8.4a-e illustrate very interesting aspects of C60 fragmentation. Even for the highest intensity beyond
saturation for C+
60 , the ion yield shows almost no singly charged fragments – in contrast to
rich fragmentation observed for higher charge states. More insight into C60 fragmentation is
gained from comparing the results measured with laser pulses of 9 fs and 27 fs duration, respectively. There, an intensity of 2.9 × 1014 W/cm2 for 9 fs pulses was used. This intensity roughly
corresponds to the the saturation intensity for C+
60 . For comparison mass spectra for 27 fs
have been measured at equivalent intensities according Eq. (8.23), assuming N = 5, 8, and
11 for singly, doubly, and triply charged ions, respectively. The results are shown in Fig. 9.2
illustrating the dramatic influence of the laser pulse duration on the formation of fullerene-like
fragments. Much less fragments are formed with 9 fs which clearly indicates that for shorter
pulse duration the SAE picture becomes more and more applicable. Nevertheless, fragmentation is not negligible for higher charge states, especially not for q > 2. A similar trend has been
seen in fast collision experiments [SHM99], where such interaction is regarded as a “δ-kick”
on the electronic system by a strong transient Coulomb field leading to instantaneous electron
emission and very little fragmentation.
Also, it is interesting to study the fragment ion yield with both pulse durations as a function
of laser intensity. Fig. 9.3a-b present a log-log plot of the measured fullerene-like fragment ion
P
yield Cq+
60−2n summed over all observed fragments, usually 1 ≤ n ≤ 6, for final charge states
146CHAPTER 9. DYNAMICS OF ULTRAFAST ENERGY REDISTRIBUTION IN C60
Figure 9.2: (a), (b), (c) Different parts of the C60 mass spectrum obtained with 9 fs laser
pulses at 2.9×1014 W/cm2 . (d), (e), (f ) The same parts of the mass spectrum but obtained
with 27 fs laser pulses at equivalent intensities of 2.7 × 1014 W/cm2 , 2.4 × 1014 W/cm2 ,
and 2.2 × 1014 W/cm2 , respectively. The equivalent intensities are calculated according to
Eq. (8.23). The data have been corrected for changing detection efficiencies of different
charge states (see Sec. 4.5) and are normalised to the C+
60 ion yield for each pulse duration.
q = 1−3. These data can not be fully interpreted in analogy to the parent ions using the same
fit procedure applied there. However, a qualitative slope analysis of the log-log plot leads to
slopes smaller than those for the parent ion yields of the same charge. This is quite interesting
and may possibly indicate that ions are generated in such a way that they can easily absorb
further so that the nuclear core can be heated efficiently. Saturation of fragment formation is
reached in all cases between 2 × 1014 W/cm2 and 3 × 1014 W/cm2 . This is approximately 2 or
3 times higher than for the respective parent ions, indicating that indeed many more photons
are absorbed to heat the system prior to fragmentation.
In order to validate the above discussion quantitatively the fraction of fullerene-like frag-
9.2. SINGLE PULSE STUDY
147
1·1014
2·1014
4·1014
(a)
ions per shot
102
++
ΣC60-2n
27 fs
10
+
ΣC60-2n
3+
ΣC60-2n
1
10-1
(b)
3+
ΣC60-2n
ions per shot
10-1
10-2
+
ΣC60-2n
10-4
(c)
q+
ΣC60-2n / [C60+ ΣC60-2n]
27fs
9fs
10-1
27fs
9fs
27fs
q=3
q=2
q=1
q+
q+
++
ΣC60-2n
10-3
1
fraction of fragment ions
9 fs
10-2
9fs q=1
1·1014
2·1014
4·1014
intensity [W/cm2]
P +
P 2+
P 3+
Figure 9.3:
C60−2n ,
C60−2n ,
C60−2n fragment ion yield plotted as a function of
the laser intensity on log-log scale: (a) obtained with 27P
fs and (b) 9P
fs laser pulses; (c)
q+
q+
q+
comparison of the fraction of fullerene-like fragment ions
C60−2n
/
C60−2n
+ C60
as
a function of laser intensity for different charge states and laser pulse durations.
ment ions
P
Cq+
60−2n /(
P
q+
Cq+
60−2n + C60 ) is plotted as a function of laser intensity for different
charge states and laser pulse durations. Such graph is shown in Fig. 9.3c. The fraction for
9 fs data is negligible (< 1%) for singly charged fragments. In contrast, the fraction of triply
charged fragments becomes as large as 60%. The fractions are 20% and about 80% for singly
charged and triply charged fragments, respectively, when 27 fs laser pulses are used. Thus,
two different pulse durations have fairly different impact on the C60 fragmentation. Moreover,
it is clear that very different physical processes take place for q = 1 and q = 3. The simple
message of Fig. 9.3 is that Cq+
60 fragmentation is still significantly affected by changing the laser
pulse duration from 9 fs to 27 fs. This is particulary true for final charge states q > 1, while
148CHAPTER 9. DYNAMICS OF ULTRAFAST ENERGY REDISTRIBUTION IN C60
for q = 1 fragmentation is already reduced to a negligible amount at all intensities. Thus, for
the formation of fragments with q > 1 the electron-electron interaction and as discussed in
Sec. 8.2 the plasmon excitation [MCR08] play a key role in the energy redistribution.
9.3
One Colour Pump-Probe Study
To directly observe fingerprints of multielectron effects (MAE/NMED) in the initial excitation step of C60 one colour pump-probe mass spectrometry with 9 fs pulses is applied. The
energy of the ultrashort pump pulse with an intensity of 7.9×1013 W/cm2 is deposited into the
electronic system during the ultrafast interaction. The delayed probe pulse of 6.8×1013 W/cm2
is used to follow (probe) the energy redistribution among the electronic degrees of freedom.
Fig. 9.4a shows the normalised C+
60 yield of the time resolved pump-probe measurement.
Since the individual action of both pump and probe pulses on the C60 molecule leads to some
ionisation at these intensities, the sum of single pulse signals has been subtracted from the
resulting pump-probe yield. The simultaneously measured Xe+ yield, which is also given in
Fig. 9.4a, can be considered as an autocorrelation function because the ionisation of Xe atoms
is a direct MPI process probably with a SAE determining the system ionisation.
+
The broadening of the C+
60 trace at the bottom with respect to the Xe signal indicates
the presence of the MAE/NMED response [ZGe07]. As shown in Fig. 9.4b the total ion
yield can be deconvoluted into two parts: a contribution from the direct MPI of C60 from
the ground state (dark gray shaded) which essentially follows the Xe+ signal and a significant
contribution exhibiting dynamics on a sub -100 fs time scale (light gray shaded) which is slightly
shifted towards positive time delays. The deviation of the C+
60 pump-probe signal from the
autocorrelation function may be interpreted as multielectron excitation of C60 during the
laser interaction. According to recent theoretical calculations [TNE01, ZSG03, ZGe04] this is
tentatively attributed to a finite probability to resonantly excite two or more electrons into the
t1g state which acts as a doorway to ionisation. The ionisation rate of C60 is determined by the
excited electron density in the doorway state. This density depends on the laser intensity. Since
the pump and probe pulses have slightly different intensities (1.16 : 1), the ion distribution
due to the excitation of the doorway state is shifted to positive time delays. The rough fit with
two functions in each delay direction for the undelayed direct SAE/MPI process (proportional
9.4. TWO COLOUR PUMP-PROBE STUDY
ion signal [arb. units]
ion signal [arb. units]
1.0
0.8
Xe
149
(a)
+
+
0.6
0.4
C60
acf
0.2
0.0
-150
1.0
0.8
0.6
0.4
-100
-50
0
50
time delay [fs]
100
+
C60
150
(b)
total fit
direct MPI
NMED/MAE
0.2
0.0
-150
-100
-50
0
50
time delay [fs]
100
150
Figure 9.4: Results of one colour pump-probe experiment with 9 fs laser pulses. (a) C+
60
yield (open triangles) as a function of the time delay between pump (7.9 × 1013 W/cm2 )
and probe (6.8 × 1013 W/cm2 ) pulses. The ion yield is normalised to the maximum signal
and zero delay is defined by the autocorrelation function (acf ) (dotted line) derived from
a fit of the simultaneously measured Xe+ signal (closed circles). (b) Contributions from
direct SAE/MPI (dark gray shaded) and MAE/NMED (light gray shaded).
to the auto correlation signal) and the MAE/NMED (taken as an exponential delay) is used
to estimate magnitudes of these two contributions. These contributions to the total C+
60
pump-probe signal are ∼ 65% and ∼ 35% for the SAE and NMED processes, respectively.
9.4
Two Colour Pump-Probe Study
Further studies of the energy redistributions pathways in C60 were performed using a two
colour pump-probe technique. Results of time resolved mass spectrometry in a two colour
pump-probe setup provide direct access to characteristic relaxation times. Mass spectra are
150CHAPTER 9. DYNAMICS OF ULTRAFAST ENERGY REDISTRIBUTION IN C60
(a)
(c)
(b)
(d)
+
+
C60
0.8
2+
C60
C60
0.4
0.24
0.8
0.24
2+
0.00
180 240 360
672 720
2+
C60-n
0.20
0.16
0.12
3+
C60
4+
C60
2+
+
C60
3+
×100
0.08
C60 fragments
0.12
3+
3+
C60
C60-n
0.16
+
C60
fragments ×100
C60
×100
C60
+
C60
fragments ×100
3+
0.20
0.04
1.2
0.4
C60 fragments
ion signal [arb. units]
1.2
+
C
2+ 60
C60
4+
C60
180 240 360
180 240 360
672 720
672 720
180 240 360
0.08
0.04
0.00
672 720
m/q [u]
Figure 9.5: Mass spectra with blue (399 nm, 3.4 × 1012 W/cm2 ) and/or red (797 nm, 5.1 ×
1013 W/cm2 ) laser pulses: (a) only the blue pulse is active; (b) the red pulse leads, blue
follows, the time delay is 524 fs; (c) the blue pulse leads, red follows, the time delay is
29 fs corresponding to maximum signal; (d) only the red pulse is active. The small inserts
for the C+
60−2n fragments illustrate that these signals are extremely weak.
taken at varying delay times ∆tbr between blue (399 nm) and red (797 nm) laser pulses ranging
from −530 to 530 fs in step sizes of 6.7 fs. From these, transient ion signals are derived by
q+
integration over the full area of the individual mass peaks for parent Cq+
60 and fragment C60−2n
ions of charge states 1 ≤ q ≤ 4 (for parents up to q = 5). For positive ∆tbr the blue pulse
leads, the red pulse follows – and vice versa for negative delays.
Fig. 9.5a-d show typical mass spectra in these pump-probe experiments. The intensity of
the blue laser pulse is chosen such that the ion signals are nearly vanishing. Only a very weak
C+
60 signal is seen in the mass spectrum obtained with the blue only pulse as shown in Fig. 9.5a.
In contrast, C60 interaction with only the (significantly more intense) red laser pulse leads to
substantial ion signals, dominated by parent ions Cq+
60 (q = 1 − 4) as presented in Fig. 9.5d.
The pattern changes significantly when red and blue pulses are combined. If the red pulse hits
9.4. TWO COLOUR PUMP-PROBE STUDY
151
C60 prior to the blue pulse, the parent signals decrease in favour of the smaller fragments down
to Cq+
44 (q ≥ 2), i.e. fragmentation is clearly enhanced by the additional blue pulse (Fig. 9.5b).
However, the most dramatic change in the mass spectra is detected when the blue pulse comes
first. In this case, as it is shown in Fig. 9.5c, all parent ions are significantly enhanced and the
fragment Cq+
60−2n signals increase by a factor of up to 15.
Fig. 9.6a-d show the integrated ion yield for parent ions Cq+
60 (q = 1 − 4) as a function of
time delay ∆tbr between 399 nm pump and 797 nm probe pulses. Fig. 9.7a-d show the analogue
transients for some selected fullerene-like fragments C3+
60−2n , n = 0 − 3. The transient spectra
show ion yields as a function of time delay from −530 fs to 530 fs with a step size of 6.7 fs.
For positive delays the blue pump pulse leads and the red probe pulse follows and for the
negative delays vice versa. As already mentioned, the relatively weak blue pulse intensity of
3.4×1012 W/cm2 was adjusted in such a way that the ultrashort radiation excites the electronic
system via the resonant t1g state without inducing significant ionisation and fragmentation as
seen by the nearly negligible “blue-only” contributions (navy-blue long dashed lines) in Fig. 9.6
and Fig. 9.7. Only for C+
60 shown in Fig. 9.6a it can be distinguished from zero for ∆tbr 0.
On the other hand, the red probe pulse of 5.1 × 1013 W/cm2 already causes some ionisation
and fragmentation even without pre-excitation by the blue pump pulse. This is indicated in
Fig. 9.6 and Fig. 9.7 by red long dashed lines for ∆tbr 0.
The pump pulse resonantly pre-excites the electronic system via the first dipole allowed
LUMO+1 (t1g ) state. This induces a highly non-equilibrium distribution of excited electrons
in neutral C60 . Efficient population of the LUMO+1 (t1g ) “doorway” state is obviously a rate
limiting step for depositing energy into the system. A laser pulse of relatively moderate intensity can initiate multielectron excitation when tuned into resonance [ZSG03, ZGe04]. This
excitation prepares the neutral C60 molecules in a specially favourable state from which absorption of probe pulse photons into highly excited vibrational states of the ions may proceed.
The thus prepared sample of excited neutral C60 molecules consists of a multitude of microcanonical ensembles with different total energies which initially are not in thermal equilibrium.
Each of these microcanonical ensembles will have a specific characteristic relaxation time τel
and a particular preference to finally end as a specific product, such as a parent Cq+
60 or fragment ion Cq+
60−2n . The second delayed probe pulse monitors relaxation process of the coupling
to electronic and vibrational degrees of freedom. The combined action of the weak blue and
152CHAPTER 9. DYNAMICS OF ULTRAFAST ENERGY REDISTRIBUTION IN C60
(a) C60+
200
200
τel = 90fs
Or
ions per 50 shots
100
τel = 86fs
100
Or
Mbr
0
-500
40
(b) C602+
Hbr h(Δtbr)
-250
0
250
Hbr h(Δtbr)
0
-500
500
0
(d) C604+
τel = 81fs
(c) C603+
-250
250
Mbr
500
τel = 79fs
4
20
Hbr h(Δtbr)
0
-500
-250
0
250
Or
Mbr
Hbr h(Δtbr)
2
0
500
-500
time delay Δtbr [fs]
-250
0
250
Mbr
Or
500
Or o(Δtbr , τb , ξb)
Hbr h(Δtbr , τb , τel)
Mbr m(Δtbr , τb , τ el)
Ob o(-Δtbr , τr , ξr)
Hrb h(-Δtbr , τr , τ*el)
Hrb m(-Δtbr , τr , τ*el)
2+
3+
4+
Figure 9.6: Total ion yield of (a) C+
60 , (b) C60 , (c) C60 , and (d) C60 as a function of
the time delay ∆tbr between 399 nm pump pulse (3.4 × 1012 W/cm2 ) and 797 nm probe
pulse (5.1 × 1013 W/cm2 ). Zero delay has been determined from MPI in Xe and indicated
by the vertical black dashed line. At positive delay times the blue pulse comes first, the
red follows, while the opposite holds for negative delay times. The transient ion signals
are fitted (full black line) according to the fitting function with individual contributions
indicated in the legend and described in the text.
strong red pulses has mainly tree effects on the formation dynamics of parent and fullerene-like
fragment ions: (i) The red probe pulse only can already create significant ion signals. However, if the blue pulse follows (∆tbr ≤ 0), it reduces the ion signals due to excitation of these
ions, thus initiating subsequent ion fragmentation on the ns and µs time scale. (ii) The pump
pulse generates highly excited electrons in the neutral system. They thermalise rapidly through
inelastic scattering among the electronic degrees of freedom and coupling to the nuclear motion on a time scale τel (the relaxation time of the respective highly excited neutral parent).
When the red probe pulse comes the system is ionised from highly excited states particularly
efficiently (large transition dipole moments or/and Franck-Condon factors). Hence, the ion
yields increase dramatically and maximum ionisation is found at ∆tbr ' 30 − 70 fs. (iii) With
the same time constant τel a thermalised (medium hot) electron population builds up and can
9.4. TWO COLOUR PUMP-PROBE STUDY
40
(a) C603+
153
3+
(b) C56
τel =
81fs
τel =
115fs
20
ions per 50 shots
20
Hbr h(Δtbr)
0
-500
15
-250
0
250
Mbr
Or
Mbr
500
τel =
138fs
3+
(c) C52
Hbr h(Δtbr)
10
0
-500
4
Mbr
5
0
-500
Hbr h(Δtbr)
-250
0
250
Or
500
τel =
151fs
3+
(d) C48
Mbr
2
Hbr h(Δtbr)
Or
-250
0
250
500
time delay Δtbr
0
-500
[fs]
Or
-250
0
250
500
3+
3+
3+
Figure 9.7: Total ion yield of (a) C3+
60 , (b) C56 , (c) C52 , and (d) C48 as a function of the
time delay ∆tbr between 399 nm pump pulse (3.4 × 1012 W/cm2 ) and 797 nm probe pulse
(5.1 × 1013 W/cm2 ). Otherwise as Fig. 9.6.
also be ionised by the red probe pulse. In this situation the ionisation probability is somewhat
lower for these medium energy electrons but still higher though than for direct ionisation from
the ground state.
For a quantitative comparison of the observed transients a fit function based on three
components corresponding to the above described effects is used. The first contribution
o(∆tbr , τprobe , Opump , ξprobe ) takes into account ions that are formed in the single pulse interaction with pump and probe only illumination, as well as the effect of probe pulses reducing
the ion yield compare to the pump only situation
" √
#
2 ln 2∆tbr
1
+
o(∆tbr , τprobe , Opump , ξprobe ) = (Opump − ξprobe ) erf −
2
τprobe
1
+ (Opump + ξprobe )
2
(9.1)
,
where Opump is the experimentally determined signal produced by the pump pulse alone and
Opump − ξprobe describes the ion reduction due to the action of the probe pulse. The second
contribution h(∆tbr , τpump , τel ) to the ion signal arises from the “highly excited” electron
154CHAPTER 9. DYNAMICS OF ULTRAFAST ENERGY REDISTRIBUTION IN C60
distribution. This signal is given by a convolution of the pump Gaussian pulse of τpump
duration (FWHM) with an exponential decay function
r
2
τpump
1
π
∆tbr
h(∆tbr , τpump , τel ) =
×
τpump exp −
+
4 ln 2
τel
16 ln 2τel2
" √
#!
2 ln 2∆tbr
τpump
× 1 + erf
− √
,
τpump
4 ln 2τel
(9.2)
where τel denotes the electron relaxation time constant. The third contribution m(∆tbr , τpump , τel )
arises from the “thermalised” electron distribution which builds up with the same time constant
τel . This signal follows the convolution of the Gaussian with the corresponding exponential
rise function according to
1
m(∆tbr , τpump , τel ) =
4
r
π
τpump
ln 2
" √
#!
2 ln 2∆tbr
1 + erf
− h(∆tbr , τpump , τel ) . (9.3)
τpump
Finally, assuming that the blue and red pulses are pump and probe, respectively on positive
time delays and vice versa on negative time delays, the fitness function can be written as
f (∆tbr ) = o(∆tbr , τr , Ob , ξr ) + Hbr h(∆tbr , τb , τel ) + Mbr m(∆tbr , τb , τel )+
+o(−∆tbr , τb , Or , ξb ) + Hrb h(−∆tbr , τr , τel∗ ) + Mrb m(−∆tbr , τr , τel∗ ) ,
(9.4)
where the parameters Hbr , Mbr , Hrb , and Mrb are giving the fraction of the different processes,
τb and τr are the duration of the blue and red pulses, respectively, τel and τel∗ are the electron
relaxation time constant, when the blue and red pulses are used as pump, respectively, Ob
and Or are the experimentally determined signals produced by the blue and red pulses alone,
respectively, Ob − ξr and Or − ξb describe the ion reduction due to the action of the probe
red pulse and the probe blue pulse, respectively.
The contributions o(∆tbr , τr , Ob , ξr ), h(∆tbr , τb , τel ), and m(∆tbr , τb , τel ) describe the dynamics at positive time delays, while o(−∆tbr , τb , Or , ξb ), h(−∆tbr , τr , τel∗ ), andm(−∆tbr , τr , τel∗ )
– at negative delays. The laser pulse durations of τb = 25 fs and τr = 27 fs for the blue and
red pulses, respectively, are used for the fitting. The quantities Hbr , Mbr , Hrb , Mrb , τel , τel∗ ,
ξr and ξb are fitted to the experimental data. τel∗ is found to be very similar to τel but it
is determined with much less accuracy due to the statistic of the data. Also, Ob and ξr are
essentially zero and Hrb as well as Mrb are significantly smaller than Hbr and Mbr , respectively.
In the following only the dynamic behaviour for positive time delays, where the relative weak
blue pulse exclusive excites the electronic system, will be considered. In this case, the energy
9.4. TWO COLOUR PUMP-PROBE STUDY
155
τel [fs]
400
fragments
300
fragments
200
fragments
fragments
100
0
parent
parent
parents
5+
C60
4+
C60
3+
C60
parents parents
2+
C60
C60+
Iblue= 6.4×1012 Wcm-2, Ired = 1.3×1013 Wcm-2
Iblue= 3.4×1012 Wcm-2, Ired = 5.1×1013 Wcm-2
Figure 9.8: Relaxation times τel for highly excited electrons due to electron-electron and
electron-vibrational coupling as determined from fitting the transient signals shown in
Fig. 9.6 and Fig. 9.7. Two sets of data for different pump and probe laser intensities
(Iblue and Ired , respectively) have been evaluated as noted in the legend. The dotted line
(with error bar) indicates the electronic relaxation time for C60 derived in previous studies
[CHH00, HHC03, SLS06], while dashed and dash-dotted lines are drawn to guide the eye.
redistribution process is directly monitored by the recording ion yields, whereas in the case of
negative time delays, it is a priori not clear if initially formed charged species are further ionised
and/or dissociated in the probe step. The discussion on the physics to be gleaned from these
data focusses on two quantities: on the relaxation time τel and on the relative contribution
from highly excited and thermalised, medium energy electrons, specifically on Hbr /Mbr .
The results obtained from fitting the transient signals of parent ions and of fullerenelike fragment ions are summarised in Fig. 9.8 and Fig. 9.9 for two different combinations of
pump and probe pulses intensities. The first set of laser intensities corresponds to Iblue =
3.4 × 1012 W/cm2 and Ired = 5.1 × 1013 W/cm2 . Under these conditions the blue only
signal was practically zero, while the red only signal was quite substantial. The second set of
156CHAPTER 9. DYNAMICS OF ULTRAFAST ENERGY REDISTRIBUTION IN C60
signal ratios
fragments
(a)
10
1
fragments
Mbr / Or
C605+
ratio Hbr / Mbr
Hbr / Or
C604+
fragments
C603+
C602+
C60+
fragments
10
1
(b)
fragments
fragments
parent
parent
C605+
C604+
fragments
parents
C603+
parents parents
C602+
C60+
Iblue= 6.4×1012 W/cm2, Ired = 1.3×1013 W/cm2
Iblue= 3.4×1012 W/cm2, Ired = 5.1×1013 W/cm2
Figure 9.9: (a) Ratios Hbr /Or and Hbr /Or of additional ion yields from highly excited
(∝ Hbr ) and thermalised (∝ Mbr ) medium energy electrons to the red only signal (∝ Or ).
(b) Ratios Hbr /Mbr of ion yields from highly excited to those from thermalised electrons.
Otherwise as Fig. 9.8.
parameters is Iblue = 6.4 × 1012 W/cm2 and Ired = 1.3 × 1013 W/cm2 . At these conditions the
blue only signals were still small but not negligible, while no red only signals were observed.
Fig. 9.8 shows the characteristic electron relaxation time constants τel for the various parent
and fragment ions of different charge states. Several trends may be identified here: (i) The
higher the pump pulse intensity and (ii) the higher the detected charge state of parent ions
Cq+
60 (q = 1 − 5), the faster is the relaxation process. The lowest value observed here is
12
2
13
2
τel ' 60 fs for C3+
60 with Iblue = 6.4×10 W/cm and Ired = 1.3×10 W/cm . (iii) Fragments
with an increasing number of evaporated C2 units originate from microcanonical ensembles
with increasing τel , i.e. from those with the smallest amount of excess electron energy. (iv)
9.4. TWO COLOUR PUMP-PROBE STUDY
157
Singly charged fragments – if detected at all – originate from microcanonical ensembles with
particularly long thermalisation times. The longest relaxation time τel ' 400 fs is observed for
C+
54 .
To understand these trends one may refer to the relation between excess electron energy
and electron-electron collision time in the Landau theory of Fermi liquid (LTFL) (see Sec. 3.3).
The general trend found in the LTFL can be applied to free the C60 molecule: the higher
the excess electron energy, the faster the relaxation of very highly excited electrons. The
formation of multiple charged parent ions requires a precursor with as little vibrational energy
as possible. These ensembles with the lowest vibrational excitation correspond to the highest
electronic excitation and thus to the fastest relaxation. Therefore, τel are smallest for the parent
ions. The longer relaxation times corresponding to smaller fragments can be explained by the
same argument. The corresponding precursor ensembles are born with increasing vibrationally
excitation, hence, with decreasing electronic excitation and thus have longer relaxation times.
The trend with blue pulse intensity just emphasises these observations: higher blue intensity
corresponds to more electronic at the cost of vibrational excitation, hence, faster relaxation
of the precursor ensemble. The derived coupling time constants below 125 fs in the formation
12
2
of multiply charged fragments Cq+
60−2n , q > 1 at Iblue = 6.4 × 10 W/cm are significantly
different from those related to singly charged fragments which reach values up to 400 fs.
Obviously, the formation of multiply charged fragments proceeds in a considerably different
way. The very long relaxation times observed for C+
60−2n fragments indicate that in this case
the precursor has a particularly low electronic energy. Hence, the probability for accessing the
plasmon state by the red probe photon is particular low. All other neutral ensembles with
some excess electronic energy content just have a very high chance to be transferred through
the plasmon state into multiply charged ions. This is in agreement with the experimental
results discussed in Sec. 8.2, where the importance of the C60 giant plasmon resonance for the
formation of multiple charged species was demonstrated.
Additional information about the ionisation and fragmentation processes can be gained by
analysing also the ion yield parameters derived from the measured pump-probe transient ion
signals. Fig. 9.9 illustrates the relative importance of the signal enhancement due highly excited
(Hbr ) and medium energy electrons (Mbr ). For simplicity of comparison Fig. 9.9a shows their
ratio to “the red only” signal (Or ). The ratio Hbr /Mbr is presented in Fig. 9.9b. According
158CHAPTER 9. DYNAMICS OF ULTRAFAST ENERGY REDISTRIBUTION IN C60
to Fig. 9.9a the higher the precursor excitation is the more pronounced for the enhancement
of ionisation and fragmentation by blue pre-excitation in comparison to the “red only” signal:
highly excited microcanonical ensembles obviously have a larger ionisation cross-section which
enhances both fragments and parents. Fig. 9.9b shows the relative magnitude Hbr /Mbr of the
relaxation which is obviously higher if the initial energy and density of the excited electrons
are higher, i.e. for the more intense blue pulse. Fragments arise again from those ensembles
with a high average vibrational energy. Thus, there is less electronic excitation density to relax
and the ratio Hbr /Mbr naturally gets smaller. This supports the conclusion that fragmentation
originates from species which are vibrationally excited but electronically rather cold. Therefore,
this amplitude ratio drops as more and more C2 units can evaporate since they belong to an
initially higher vibrational energy content.
9.5
Pulse Shaping Study
This section is dedicated to the active control of the molecular response with judiciously
tailored laser pulses. This powerful technique allows one both to maximise a specific, selected
reaction and to learn about the the excitation and relaxation processes in molecules itself.
Since the formation of higher charged fragments is strongly intensity dependent (see
Fig. 9.3), single charge fragments are used for the optimisation procedure. The maximisation of C+
50 formation is chosen as prime aim of present study because this process can be
considered as a measure for the temperature of the nuclear backbone. The optimal criterion
(fitness function f ) is simply defined as
+
f = S(C50
) ,
(9.5)
+
where S(C50
) is the yield of C+
50 ion. Fig. 9.10a illustrates a typical learning curve for the
maximisation of C+
50 formation. The algorithm of finding the optimal pulse shape is described
in Chapter 7. For free optimisations the spectral phase was specified at 32 equidistant wavelengths between which a spline interpolation was used to build all 640 points. This turned
out to be a good compromise for approximating the global maximum in solution space, while
keeping the number of free parameters small and the convergence time short. The optimisation starts with 20 randomly chosen phase masks (1st generation in Fig. 9.10a). Usually,
(a)
24
20
16
12
random pulse
(1st generation)
SH-wavelength [nm]
C5+0 signal [arb. units]
28
159
410
400
optimal pulse
leading
edge
390
380
-300 0 300
unshaped pulse
time [fs]
(0th generation)
0
10
20
30
generation
ion signal [arb. units]
9.5. PULSE SHAPING STUDY
3 (b)
2 R=0.25
1
0
16 (c)
8 R=0.035
0
120 (d)
R=0.0078
60
0
0
360
1
C5+0
1
C5+0
1
C 5+0
720
m/q [u]
Figure 9.10: (a) C+
50 signal as a function of generation of the evolutionary algorithm.
The inset shows the SH-XFROG map of the optimal solution. On the right, mass spectra
recorded with optimal pulse (b), stretched to 340 fs pulse (c), and unshaped pulse of 31 fs
+
FWHM (d) are plotted. The derived ratios R = C+
50 /C60 are given.
convergence of the optimisation algorithm is achieved after 30 − 40 generations. During the
optimisation the ion signal of C+
50 increased by a factor of ' 2.4 compared to the signal
recorded with unshaped pulse (0th generation in Fig. 9.10a). In this particular case the energy
of the 31 fs laser pulse at 797 mn central wavelength was 220 µJ. The final, optimal pulse
is characterised by the cross-correlation frequency resolved optical gating (SH-XFROG). The
SH-XFROG map of the optimal pulse is shown on inset in Fig. 9.10a. The leading edge of
the optimal pulse (the part which interacts with the C60 molecule first) is on the left. This
map clearly shows that a sequence of several pulses is best suited for the formation of C+
50 .
Although the shape of the optimal pulse change slightly from one optimisation to another,
the pulse presented in Fig. 9.10a is representative of the majority (∼ 90%) of optimal pulse
shapes resulting from different optimisation runs.
To prove whether the structure of the optimal pulse is significant or simply acts as a
stretched pulse, a mass spectrum obtained with a stretched pulse of the same overall energy
and duration as the optimal pulse was recorded. This is illustrated in Fig. 9.10c. The yield of
C+
50 with optimal pulse as shown in Fig. 9.10b is more than two times stronger compare to the
+
yield with the stretched pulse. The ratio R = C+
50 /C60 with the optimal pulse even increase by
factors of 7 and 32 compared to those of the stretched and the unshaped pulse, respectively.
160CHAPTER 9. DYNAMICS OF ULTRAFAST ENERGY REDISTRIBUTION IN C60
5 0
c o r r e la tio n w ith C
+
1 .0
0 .9
0 .8
0 .7
0 .6
0 .5
0 .4
0 .3
C
+
4 8
C
+
5 2
C
+
6 0
C
2 +
6 0
C
2 +
5 0
+
+
+
2+
2+
Figure 9.11: Correlation between C+
50 and C48 , C52 , C60 , C60 , and C50 ions during the
optimisation run. Each point is obtained by the averaging over the 16 individual optimisation runs.
During each optimisation run correlations between the formation of C+
50 and other ions
+
+
+
2+
are recorded. Fig. 9.11 illustrates the correlations between C+
50 and C48 , C52 , C60 , C60 , and
C2+
50 ions during the optimisation run. Each point is obtained by the averaging over the 16
individual optimisation runs. A strong correlation with the neighbouring fragments C+
48 and
C+
52 is observed. On the other hand, a correlation with parent ion as well as with doubly
charged ions is much weaker if at all present. This is a clear indication that the formation
of singly charged fragments and the formation of highly charged ions have different nature.
The yield of highly charged ions is mainly affected by the applied laser intensity, while singly
charged fragments are formed as a result of sequential evaporation of C2 units from highly
vibrationally excited C60 in a statistical process. Thus, present optimisation scheme allows one
to maximise the process of energy flow into vibrational modes of C60 .
To find additional evidence of the results of the optimisation experiments, a two-colour
pump-probe study was performed with a relatively weak 399 nm pump pulse. The dynamics
of the energy redistribution is then probed by a time delayed 797 nm probe pulse. Fig. 9.12a
25
25
C3+
48
20
(a)
15
10
5
20
(metastable)
red leads
0
-200
-100
20
(b)
0
100
200
300
3+ 3+
C58
C56 ... C3+
48
average
¨7a 80+6f s
(c)
¨72 = 116 ± 20fs
10
5
Ep=220mJ
0
-200
20
0
200
400
600
¨71 = 97 ± 17fs
15
10
¨72 = 127 ± 12fs
(d)
5
0
-200
¨71 = 84 ± 9fs
15
blue leads
40
data minus fit
161
XFROG projection [arb.un.]
ions per 1000 laser pulses
9.5. PULSE SHAPING STUDY
-100
0
100
time delay [fs]
200
300
0
Ep=280mJ
-200
0
200
400
separation [fs]
600
Figure 9.12: (a) Metastable C3+
48 ion signal as a function of time delay between blue pump
and red probe pulses. An exponential decay is fitted to the data. (b) A modulation is found
for all fullerene-like fragments C3+
60−2n by subtracting the fits from the measured transients.
(c) Optimised temporal shape with 220 µJ laser pulses and (d) 280 µJ pulses.
shows the metastable C3+
48 ion signal as a function of the time delay between 399 nm pump
(1.7 × 1013 W/cm2 ) and 797 nm probe pulse (7.3 × 1013 W/cm2 ) with pulse durations of 25 fs
and 27 fs, respectively. Triply charged, fullerene-like fragments were chosen because (i) they
are most abundant using fs pulses as visible in Fig. 9.10d and (ii) metastable fragmentation
(on a µs-ms timescale) is a particularly sensitive probe of the temperature of Cq+
60 generated
in the initial photo absorption process [HLS05]. At negative time delay, when the 797 nm
pulse leads, almost no signal from C3+
48 is observed. Once pump and probe pulse overlap, the
ion yield increases strongly and a maximum fragment signal is found at a positive delay of
50 fs. It can be concluded from this observation that the resonant pre-excitation of the t1g
doorway state by the 399 nm laser pulse significantly enhances multiple ionisation and massive
fragmentation both induced by the subsequent 797 nm pulse.
Closer inspection of the pump-probe transient reveals a weak modulation on top of the C3+
48
ion signal. By fitting the data to the laser cross-correlation function convoluted with a single
162CHAPTER 9. DYNAMICS OF ULTRAFAST ENERGY REDISTRIBUTION IN C60
exponential decay function and subtracting this from the measured signal, this modulation is
found in all pump-probe data from multiply charged fullerene-like fragments with a periodicity
of 80 ± 6 fs. This is shown in Fig. 9.12b. Obviously, nuclear rearrangement upon electronic
excitation by the 399 nm pump pulse via the t1g state occurs. The oscillation is then probed
by the 797 nm probe pulse, assuming that the absorption cross-section for further energy
deposition depends on the C60 oscillation.
The comparison of the pump-probe modulation with results from pulse shaping experiments
(Fig. 9.10a) gives evidence that these different spectroscopic techniques probe very similar
dynamics. Figs. 9.12(c) and 9.12(d) show the optimal temporal shapes for excitation with
220 µJ and 280 µJ pulses, respectively, derived by projecting the SH-XFROG maps onto the
time axis. The key findings reproduced in several optimisation runs are the follows: (i) Each
pulse shape consists of two distinct regimes with periodicity T1 and T2 . (ii) The periodicity is
smaller on the leading edge of the pulse than on the trailing edge (T1 < T2 ). (iii) It increases
with increasing pulse energy. The observed values range from T1 = 84 ± 9 fs at 220 µJ up to
T2 = 127±12 fs at 280 µJ. All observed times including the pump-probe result are much larger
than the well known radially symmetric breathing mode ag (1) of neutral C60 molecule which
has an experimentally determined period of 67 fs [DZH93]. On the other hand, the observed
periods are in general shorter than the lowest prolateoblate mode hg (1) (122 fs [DZH93])
recently suggested as the dominantly excited mode of C60 due to the strong laser induced
dipole forces acting in intense 1500 nm pulses [BCR03].
To get information on the nuclear motion excited in strong 400 nm and 800 nm laser fields
theoretical calculations were performed using the so-called “non-adiabatic quantum molecular dynamics” (NA-QMD) [KSc03], developed recently. In this approach, electronic and
vibrational degrees of freedom are treated simultaneously and self-consistently by combining
time dependent density functional theory (TDDFT) in basis expansion with classical molecular
dynamics.
The NA-QMD theory has already been successfully applied to excitation and fragmentation
mechanisms in ion-fullerene collisions [KSc01] and laser induced molecular dynamics [HKS06].
Such kind of calculations is limited due to the computational effort by using the frozen core
approximation and only a minimal basis set and thus describing the ionisation mechanism not
very realistically. However, in principle it is possible to include the full ionisation process into
9.5. PULSE SHAPING STUDY
163
c h a rg e s ta te
0
5
1 0
1 5
2 0
2 5
3 0
3 5
0
5
1 0
1 5
2 0
2 5
3 0
3 5
T b r e a th in g [ f s ]
1 0 0
9 0
8 0
7 0
6 0
e x c ite d e le c tr o n s
Figure 9.13: Period of the ag (1) breathing mode as a function of the number of excited
electrons (black full circles) derived from NA-QMD simulations (laser parameters: λ =
370 nm, τ = 27 fs, I = 3.4 × 1012 − 5.0 × 1013 W/cm2 ) and as a function of C60 charge
state (red open circles) derived from hybrid B3LYP level of the DFT method [SNS06].
NA-QMD, demanding however many more basis functions and a reliable absorber potential
[UKS06].
In Fig. 9.13 (black full circles) results for exciting C60 by an intense laser field (λ = 370 nm,
τ = 27 fs, I = 3.4 × 1012 − 5.0 × 1013 W/cm2 ) are presented. The wavelength of 370 nm is
close to the experimentally used 399 nm pump pulse and matches the first optical resonance
calculated in the local-density approximation (LDA). This calculation predicts an efficient
excitation of many electrons by the laser field. At the highest laser intensity (5 × 1013 W/cm2 )
nearly 31 valence electrons are strongly excited resulting in an impulsive force that expands
the molecule dramatically up to 9.4 Å which is 130% of the normal C60 diameter, orders of
magnitude larger than expected for any standard harmonic oscillation. The new equilibrium
position as well as the oscillation period of the ag (1) mode depend on the excited electronic
configuration and thus on the absorbed energy. Fig. 9.13 shows a strong increase of the
oscillation period with increasing number of absorbed electrons. The calculated oscillation
period of highly excited C60 is in a good agreement with the results of the pump-probe
164CHAPTER 9. DYNAMICS OF ULTRAFAST ENERGY REDISTRIBUTION IN C60
experiment presented in Fig. 9.12b as well as with the first time regime (T1 ) of the optimally
shaped laser pulse given in Fig. 9.12c-d. The longer periods seen experimentally in the second
time regime (T2 ) in Fig. 9.12c-d are not reproduced by the NA-QMD theory, possibly, because
this time regime corresponds to the ionised C60 .
Indeed, the theoretical data derived from hybrid B3LYP level of the DFT method [SNS06]
predict that the breathing mode period of the ionised C60 is increased in comparison with the
neutral molecule. This is shown in Fig. 9.13 (red open circles), where the oscillation period
of the ag (1) breathing mode is plotted as a function of different charge states q of Cq+
60 .
Coulomb repulsion between the charged carbon atoms may weaken the bonds in Cq+
60 , hence,
the oscillation period of the ag (1) breathing mode increases.
Chapter 10
Mass Spectrometry of Model
Peptides with Shaped Laser Pulses
This chapter describes experimental results obtained with femtosecond pulse shaping as
an analytic tool in mass spectrometry of complex polyatomic molecules. As examples the
molecules Ac–Phe–NHMe and Ac–Ala–NHMe were studied in this work. They may be considered as model peptides as described in some detail in Sec. 2.2. Both peptides may be seen
as prototypes for trying selective bond breaking with shaped strong field pulses in an adaptive
feedback loop. Strong field excitation with shaped laser pulses allows one to cleave the strong
backbone bonds in the model peptides preferentially, while keeping other, more fragile bonds
intact.
10.1
Experimental Results
First, Ac–Phe–NHMe was studied. The molecular beam is produced by evaporation of
Ac–Phe–NHMe powder in an oven heated to 395K. Fig. 10.1 presents a typical mass spectrum
recorded with laser pulses of 3.7 × 1013 W/cm2 peak intensity and 32 fs pulse duration centred
at 797 nm. Rather than the parent ion peak with mass of 220 u, fragments with masses 162 u,
120 u, 87 u, and 43 u dominate the mass spectrum. A molecular rationalisation of the most
relevant fragmentation channels is given in Fig. 10.2.
As one has seen in the previous chapters, mass spectra recorded as a function of laser
intensity can provide additional information about photoinduced processes in strong fields.
Fig. 10.3 shows the ion signal for the parent ion (220 u) and its fragments (43 and 162 u)
in log-log scale as a function of laser intensity in the range from 2.5 × 1013 W/cm2 and
165
166
CHAPTER 10. MASS SPECTROMETRY OF MODEL PEPTIDES
io n y ie ld [a r b . u n its ]
0 .0 2 5
8 7 u
0 .0 2 0
1 2 0 u
0 .0 1 5
1 6 2 u
0 .0 1 0
2 2 0 u
4 3 u
0 .0 0 5
5 8 u
1 0 4 u
9 1 u
1 2 9 u
0 .0 0 0
0
5 0
1 0 0
1 5 0
2 0 0
m /q [u ]
Figure 10.1: Mass spectra of the Ac–Phe–NHMe molecular system recorded with laser
pulses of 3.7 × 1013 W/cm2 intensity and 32 fs pulse duration centred at 797 nm.
1.5 × 1014 W/cm2 for a laser pulse duration of 34 fs. While a fitting procedure as applied
in Chapter 8 to C60 would obviously be difficult to perform for these data, the slopes give,
nevertheless, evidence of a highly nonlinear behaviour of fragmentation and ionisation of this
molecule. Especially, the steeper slope for the 43 u fragment indicates that indeed several
photons are needed to ionise the system and cleave this particular bond. All fragment ions
tend to saturate eventually as typical for the interaction of strong filed short pulse laser
radiation with larger molecules.
To exclude that the observed changes in the fragmentation pattern induced by the optimal
pulse is simply an effect of the laser peak intensity, the energy of the unshaped pulse in
optimisation experiments was set to the onset of fragmentation at 3.7 × 1013 W/cm2 .
A free optimisation is performed in a first set of experiments. The maximisation of one
peptide bond CO–NH (Sec. 2.2) cleavage resulting in a strong enhancement of the CO–CH3
10.1. EXPERIMENTAL RESULTS
167
Figure 10.2: Different possible fragmentation channels of the Ac–Phe–NHMe molecular
system.
ion (43 u) is chosen as a target for the adaptive control loop, while keeping other bonds intact
at will. For that the following fitness criterium
f = S(43) − S 0 (43) × S 0 (162) − S(162)
(10.1)
is defined, where S(43) and S(162) are the ion yields of mass 43 u and 162 u, respectively, and
the superscript “0” denotes the ion yields obtained with unshaped laser pulses. For individuals
returning S(43) < S 0 (43) or/and S(162) > S 0 (162) the fitness is set to 0 (f = 0) and such
individuals are excluded from the further evaluation.
A typical learning curve for maximising the CO–NH peptide bond breaking is shown in
Fig. 10.4a. The SH-XFROG of the final optimal pulse shape is plotted in Fig. 10.4b. The
SH-XFROG map clearly shows that a sequence of 4 − 5 pulses with a separation of 156 ± 15 fs
is best suited for selective cleavage of this particular peptide bond. Information about the
temporal envelope of the laser pulse obtained by projecting the SH-XFROG map onto the
time axis is illustrated on the left side of Fig. 10.5. During the optimisation process the signal
ration R = S(43)/S(162) increased by a factor of 7.5 compared to the signals recorded with
168
CHAPTER 10. MASS SPECTROMETRY OF MODEL PEPTIDES
1 0 0
io n y ie ld [a r b . u n its ]
4 3 u
1 0
1 6 2 u
2 2 0 u
1
0 .1
0 .0 1
2 .0 x 1 0
1 3
7 .0 x 1 0
in te n s ity [W /c m
1 3
2
1 .2 x 1 0
1 4
1 .7 x 1 0
1 4
]
Figure 10.3: Fragment ion yields of mass 43 u, mass 162 u, and the parent ion (mass
220 u) of the Ac–Phe–NHMe molecular system recorded with 34 fs laser pulses plotted as a
function of the laser intensity on the log-log scale. The vertical black dashed line indicates
the laser intensity utilised in the optimisation experiment.
unshaped pulses (0th generation in Fig. 10.4a) as visible in the corresponding mass spectra on
the right side of Fig. 10.5.
For comparison R was measured with a stretched pulse of the same overall duration and
energy as the optimal pulse by applying a quadratic spectral phase function to the pulse
shaper. The ratio for the optimal pulse is more than 4.5 times larger. Furthermore, in this
case the fitness f has a negative value indicating that the signal of both, mass 43 u and mass
162 u increased upon the interaction with elongated pulse. It is known that laser induced
fragmentation of large finite systems is mainly determined by the interaction time scale due to
efficient energy coupling to nuclear motion, while the excitation pulse is still active as described
in Chapter 9. Therefore, the observed general enhancement of fragmentation in model peptides
for stretched pulses can be due to efficient heating of nuclear degrees of freedom.
It should be noted that the overall fragmentation pattern with additional major peaks at
1.8
(a)
169
optimal pulse
(b)
400
random pulse
1.5 (1st generation)
395
1.2
0.9
390
43 u
0.6
162 u
385
0.3
unshaped pulse
(0th generation gives S 0)
0.0
0
5
10
leading
edge
15
20
generation
-400
SH wavelength [nm]
f = [S (43)-S 0(43)]×[S 0(162)-S(162)]
10.1. EXPERIMENTAL RESULTS
380
-200
0
200
400
time [fs]
Figure 10.4: (a) Fitness f characterising the predominant formation of fragment mass
43 u as a function of the generation in the adaptive feedback loop. Fluctuations of f are
due to experimental noise. The solid line is added to guide the eye. The insert sketches
the chemical structure of the Ac–Phe–NHMe molecular system. Both, the optimised bond
cleavage resulting fragment mass 43 u and the suppressed one (162 u) are indicated by the
straight and dotted lines, respectively. (b) SH-XFROG map of the optimal pulse shape.
mass 87 u and 120 u observed with intense femtosecond pulses – an in particular with the
optimally shaped pulse – is quite a bit different from that which has been found in an early
single photon photofragmentation study, using UV excitation with 12.1 eV photons utilising a
hydrogen discharge tube and a monochromator [OVM74], what is shown for comparison in the
upper right panel of Fig. 10.5. The appearance of mass 43 u in the current experiment clearly
indicates that strong field effects need to be taken into account to describe the photophysical
and photochemical mechanism under fs irradiation.
Due to its stochastic nature each optimisation run leads to a slightly different optimal
pulse. Therefore, to see how the optimal results are changed from one optimisation to another
24 individual optimisation runs were performed. The resulting optimal pulse shapes sorted
according their shapes of all 24 optimisation experiments is presented in Fig. 10.6. Each
column consists of optimal pulses with similar structure and separation between peaks. The
colour code indicates values of the fitness f obtained in each optimisation experiment. Often,
very similar shapes of optimal pulses result in significantly different values of the fitness f .
Hence, not only the temporal pulse structure and therefore intensity is responsible for the
observed results, but also the pulse phase and initial conditions of the system as well.
170
CHAPTER 10. MASS SPECTROMETRY OF MODEL PEPTIDES
1 .0
U V e x c ita tio n
1 2 .0 8 e V
0 .5
0 .0
1 6 2 u
le a d in g e d g e
in te n s ity [a r b . u n its ]
0 .4
0 .0
0 .8
(b )
2 4 6 fs
le a d in g e d g e
0 .4
0 .0
0 .8
( a ’)
R = 0 .5
f = 0
0 .0 2
u n s h a p e d
3 4 fs
(c )
o p tim is e d
0 .0 1
io n s ig n a l [a r b . u n its ]
0 .8
(a )
0 .0 0
0 .2
4 3 u
R = 0 .8
f = -1 0 3
( b ’)
R = 3 .6
f = 1 .1
( c ’)
0 .1
0 .0
0 .0 4
le a d in g e d g e
0 .4
0 .0 2
0 .0
0 .0 0
-5 0 0
-2 5 0
0
2 5 0
5 0 0
0
5 0
1 0 0
tim e [f s ]
1 5 0
2 0 0
2 5 0
m /q [u ]
Figure 10.5: Temporal envelope and corresponding mass spectra recorded for the unshaped
(a), stretched to 246 fs (FWHM) (b), and optimal pulses (c) are presented. The signal
ratios R = S(43)/S(162) and fitness values f are given. For comparison a photofragmentation pattern using a hydrogen discharge tube reconstructed from [OVM74] is plotted in
the upper right corner.
To investigate the influence of the pulse phase on the present optimisation in a second
set of experiments an optimisation with a restricted, parameterised phase function ϕ(m) was
performed. Since results of the free optimisation mainly led to a temporal structure with a
periodic pulse sequence (see Fig. 10.6), the phase function was chosen to give essentially an
equispaced triple pulse. The following sin function [WPS06] is used
ϕ(m) = A sin (a · m + Φ) + π
,
(10.2)
where m is the pixel index (0 ≤ m ≤ 639) in the shaper which is proportional to the wavelength
λ. The bias π in Eq. (10.2) just keeps the phase function within the calibration range 0 − 2π
10.1. EXPERIMENTAL RESULTS
171
1 .0
f = 2.52 – 3.11
f = 1.92 – 2.51
f = 1.32 – 1.91
f = 0.72 – 1.31
f = 0.11 – 0.71
f = 1 .5 2
k = 1 4 .2 8
0 .8
0 .6
0 .4
0 .2
0 .0
-6 0 0 -3 0 0
0
3 0 0
6 0 0
1 .0
f = 1 .4 5
k = 1 2 .4 3
0 .8
0 .6
0 .4
0 .2
0 .0
-6 0 0 -3 0 0
0
3 0 0
6 0 0
1 .0
1 .0
f = 1 .0 7
k = 8 .6 4
0 .8
0 .6
0 .4
f = 0 .9 7
k = 9 .4 7
0 .8
0 .6
0 .6
0 .4
0
3 0 0
6 0 0
f = 1 .1 8
k = 1 0 .0 8
0 .8
0 .6
0 .4
3 0 0
6 0 0
0 .6
0
3 0 0
6 0 0
0 .0
3 0 0
6 0 0
f = 0 .7 6
k = 1 1 .7 9
0 .8
0 .0
0
3 0 0
6 0 0
6 0 0
3 0 0
6 0 0
0
3 0 0
6 0 0
3 0 0
6 0 0
-6 0 0 -3 0 0
0
3 0 0
6 0 0
1 .0
f = 1 .6 7
k = 1 1 .9 1
0 .8
0 .6
0 .0
3 0 0
6 0 0
0 .6
0 .4
0 .2
-6 0 0 -3 0 0
0
3 0 0
0 .0
6 0 0
f = 0 .1 1
k = 8 .5 5
0 .6
-6 0 0 -3 0 0
0
3 0 0
6 0 0
-6 0 0 -3 0 0
0
3 0 0
6 0 0
1 .0
f = 1 .1 1
k = 1 2 .2 4
0 .8
0 .6
0 .0
f = 1 .3 6
k = 8 .8 8
0 .8
f = 1 .0 1
k = 9 .5 8
0 .8
0 .6
0 .4
0 .2
0 .2
0
0 .8
0 .0
0 .0
0 .4
-6 0 0 -3 0 0
0 .2
0
0 .6
1 .0
f = 0 .4 4
k = 7 .8 9
0 .6
0 .4
-6 0 0 -3 0 0
f = 1 .9 9
k = 1 1 .6 1
0 .8
0 .2
-6 0 0 -3 0 0
1 .0
f = 0 .7 4
k = 1 0 .5 9
0 .6
6 0 0
0 .4
0 .8
0 .0
3 0 0
1 .0
f = 0 .6 2
k = 1 0 .9 1
0 .6
0 .2
0
0 .8
0 .0
3 0 0
0 .4
-6 0 0 -3 0 0
0 .2
-6 0 0 -3 0 0
0
0 .8
0 .0
0
0 .2
-6 0 0 -3 0 0
1 .0
1 .0
0 .4
0 .2
-6 0 0 -3 0 0
0
-6 0 0 -3 0 0
0 .4
0 .2
-6 0 0 -3 0 0
1 .0
f = 0 .9 2
k = 1 1 .3 3
0 .8
0 .4
0 .2
0 .0
0
0 .6
0 .0
0 .0
1 .0
f = 0 .9 7
k = 9 .9 8
0 .8
0 .4
0 .2
-6 0 0 -3 0 0
1 .0
f = 1 .7 9
k = 1 0 .7 3
0 .6
0 .0
6 0 0
1 .0
0 .6
0 .4
0 .2
0 .8
6 0 0
f = 0 .8 3
k = 1 2 .1 4
0 .8
0 .6
0 .4
1 .0
3 0 0
0 .2
-6 0 0 -3 0 0
1 .0
0 .0
0
0 .4
0 .2
0 .0
-6 0 0 -3 0 0
1 .0
f = 2 .5 7
k = 1 0 .7 4
0 .8
3 0 0
0 .2
0 .2
1 .0
0
0 .4
0 .4
0 .0
0 .2
-6 0 0 -3 0 0
1 .0
1 .0
0 .6
0 .4
0 .2
0 .0
f = 3 .1 1
k = 1 3 .1 9
0 .8
-6 0 0 -3 0 0
0
3 0 0
0 .0
6 0 0
-6 0 0 -3 0 0
0
3 0 0
6 0 0
1 .0
1 .0
f = 0 .9 5
k = 1 0 .5 8
0 .8
0 .6
0 .6
0 .4
0 .4
0 .2
0 .2
0 .0
0 .0
-6 0 0 -3 0 0
0
3 0 0
1 .0
f = 0 .9 3
k = 7 .3 9
0 .8
6 0 0
f = 0 .6 4
k = 1 1 .2 6
0 .8
0 .6
0 .4
0 .2
-6 0 0 -3 0 0
0
3 0 0
6 0 0
0 .0
-6 0 0 -3 0 0
0
3 0 0
6 0 0
Figure 10.6: Distribution of individual optimal pulses grouped according to the similarities
in pulses structure (leading edge on positive delays). Each column consists of optimal
pulses with similar structure and separation between peaks. Parameter k is determined
as: k = [S(43)/S(162)] / [S 0 (43)/S 0 (162)]. Colour code corresponds to a value of the
fitness f .
of the LC array and has no influence on the resulting pulse structure. Due to the small
spectral bandwidth across the LC compared to the laser carrier frequency ω Eq. (10.2) can be
approximated as
ϕ(ω) ' A sin (∆ω · T + Φ) + π
,
(10.3)
where ∆ω = ω0 − ωref is the difference between the carrier frequency ω0 and the reference
frequency ωref of the LC calibration, T is the temporal separation between pulses in the pulse
sequence (determined experimentally by the parameter a).
Only one free parameter a is used for parameterised optimisation. The parameter Φ = π/2
1.0
=0
E+-field
phase
=
0.5
0.0
E+-field
phase
-0.5
-1.0
-200
-100
-N×T0
0
100
N×T0
200
time [fs]
-100
-N×T0
0
100
electric field phase [ ]
CHAPTER 10. MASS SPECTROMETRY OF MODEL PEPTIDES
{
{
{
electric field amplitude [arb. units]
172
200
N×T0
Figure 10.7: Calculated triple pulse sequence in the time domain illustrating the effect
of different phase parameters Φ on the pulse sequence. Note that the pulse distance and
intensity ratios are not affected by the changing of the parameter Φ. The absolute value
of the phase is unknown. Zero phase for the maximum (middle) pulse is assumed.
is utilised without loss of generality because the absolute phase is unknown anyhow and is
simultaneously varied as the parameter a is optimised. To mimic the ratio of the side bands
of the optimal solution from the free optimisation the parameter A ' 2π × 0.17 is chosen. In
this parameterised optimisation experiment the cleavage of the peptide bond resulting in the
fragment with mass 43 u is again maximised. In this case the convergence is reached after
few iterations only. The parametrisation of the phase function speeds up the convergence
by an order of magnitude which could be of significant advantage for mass spectroscopic
applications. The results lead to a value of the parameter a which corresponds to a pulse
separation of T = 156 ± 7 fs. This separation is essentially identical to the pulse separation
obtained in the free optimisation experiment.
To gain additional information about effects involved in the observed control mechanism,
the relative phase of the triple pulses is varied by changing the phase parameter Φ. This change
shifts the maximum of the phase modulation function across the LC array. Such procedure has
first been introduced in atomic excitation [MSi98] and was recently investigated in great detail
for the photoionisation of potassium using photoelectron spectroscopy [WPS06]. The effect
10.1. EXPERIMENTAL RESULTS
0 .8
0 .4
(a )
1 .0
in te n s ity [a r b . u n its ]
io n s ig n a l [a r b . u n its ]
1 .2
173
4 3 u o p tim is e d
1 6 2 u n o t o p tim is e d
(b )
0 .8
0 .6
0 .4
0 .2
2 2 0 u p a r e n t io n
0 .0
0 .0
-0 .5 0
-0 .2 5
0 .0 0
0 .2 5
0 .5 0
s h i f t o f o p t i m i s e d p h a s e p a r a m e t e r [ 2 π]
-3 0 0
-1 5 0
0
1 5 0
3 0 0
d e la y [fs ]
Figure 10.8: (a) Ion yields of mass 43 u, 162 u, and the parent 220 u are plotted as a
function of the relative phase shift in the parameterised triple pulses. The signal of mass
43 u varies by about 33% by changing the relative phase. In contrast, the parent ion as
well as the suppressed fragment mass 162 u show no distinct relative phase dependence.
(b) Measured temporal envelope of the pulse sequence applied for three different relative
phase shifts Φ = 0, π/2, and 3π/2, illustrating that the pulse shape remains uninfluenced
by changing of the relative phase.
of the phase modulation introduces a corresponding shift in the side pulses as is illustrated for
two extreme cases Φ = 0 and Φ = π, respectively, in Fig. 10.7. The variation of the parameter
Φ in a symmetric manner allows one to monitor continuously the fragmentation dynamics as
well as the both spectral and phase distribution over the full range of 2π. The absolute phase
is not known, and it varies somehow from pulse to pulse without phase stabilisation. However,
the relative phase ∆ωT + Φ between the maximum and the side bands can be can be varied
from −π and π during the experiment.
As clearly visible in Fig. 10.8a the ion yield of fragment mass 43 u depends significantly
on the the phase shift among the triple pulses. The phase shift of the optimised triple pulse
sequence found by varying only T was arbitrarily defined as zero. Quite remarkably, the
yield of parent ions, as well that of the suppressed 162 u fragment exhibits no distinguishable
dependence on the phase shift. Projections of recorded SH-XFROG maps onto the time axis
corresponding to values of the parameter Φ = 0, π/2, and 3π/2 demonstrate no measurable
174
CHAPTER 10. MASS SPECTROMETRY OF MODEL PEPTIDES
Figure 10.9: Comparison between power spectra of the triple pulses and resonant two
photon ionisation spectrum of Ac–Phe–NHMe. The power spectra are calculated assuming
a third order process and using phase parameters Φ = 0 and π (blue solid line and orange
dashed line, respectively). The resonant two photon ionisation spectrum of Ac–Phe–NHMe
in the range of 37200 − 37800 cm−1 is taken from [GUG04]. The transitions are labelled
with the number of the corresponding isomer (for details see text). The bands at 22 and
44 cm−1 belong to isomer I, the bands at −18 and −8 cm−1 are hot bands of isomers
I and III, respectively. The band marked by an asterisk arises from a fragment of a
AcPheNHMe/water cluster.
effect on the temporal pulse envelope and consequently does not affect its overall intensity
as shown in Fig. 10.8b, while the 43 u fragment yield drops by about 33% for Φ = ±π in
comparison to the optimised measurement with phase parameter Φ = 0.
Possibly, the experimentally observed effect of phase sensitivity can be explained considering
such effect in the frequency domain, rather than in the time domain and taking into account
the spectroscopic properties of the model peptide studied. Fig. 10.9 shows the resonant two
photon ionisation spectrum of Ac-Phe-NHMe in the range of 37200 − 37800 cm−1 (taken
from [GUG04]) together with a pair of power spectra which were calculated by the Fourier
transformation of the idealised triple pulses illustrated in Fig. 10.7 and assuming a third order
10.1. EXPERIMENTAL RESULTS
175
o p tim a l p u ls e
0 .0 0 8
0 .0 0 6
0 .0 0 4
S H - w a v e le n g th [n m ]
0
0
f = [S (7 7 )-S (7 7 )]x [S (4 3 )-S (4 3 )]
0 .0 1 0
r a n d o m p u ls e
s t
( 1 g e n e r a tio n )
0 .0 0 2
0 .0 0 0
u n s h a p e d p u ls e
th
( 0 g e n e r a tio n g iv e s S
0
5
1 0
0
)
1 5
g e n e r a tio n
2 0
2 5
3 0
4 0 0
le a d in g
e d g e
3 9 5
3 9 0
3 8 5
-5 0 0
-2 5 0
0
2 5 0
tim e [fs ]
5 0 0
Figure 10.10: SH-XFROG map of the optimal solution and fitness f characterising the
enhanced formation of fragment mass 77 u as a function of the generation in the adaptive
feedback loop. Large fluctuations of f are due to experimental noise. The solid line is
added to guide the eye. On the right, both, the optimised bond cleavage resulting in mass
77 u as a fragment and the suppressed one (43 u) are indicated in the sketch of the Ac–
Phe–NHMe molecular structure by the straight and dotted lines, respectively. In addition,
SH-XFROG map of the optimal solution is also presented.
process for phase parameters Φ = 0 and π (blue solid line and orange dashed line, respectively).
Fig. 10.9 clearly indicates that the variation of the phase parameter Φ leads to the different
efficiencies of the laser power absorption by various Ac-Phe-NHMe isomers1 . Hence, the
distinction in Ac-Phe-NHMe fragmentation may be a result of this selective excitation of the
isomers.
Fig. 10.10 demonstrates the potential usefulness of the optimal control technique as an
analytic tool in mass spectrometry. Here an optimisation run is shown aiming at keeping
the peptide bond intact, i.e. the fragment mass 43 u is suppressed in the mass spectra, while
splitting the Ac–Phe–NHMe model peptide directly at the benzene ring system. This bond
cleavage forms mass 77 u as shown by the sketch in Fig. 10.10. The corresponding learning
curve is given as well as the SH-XFROG characterisation of the optimal pulse picked by the
algorithm. Although the enhancement is rather limited it is sufficient to obtain a special motif
1
Isomers are molecules with the same chemical formula, but in which the atoms are arranged differently.
They differ in properties because of differences in the arrangement of atoms.
176
CHAPTER 10. MASS SPECTROMETRY OF MODEL PEPTIDES
as optimal pulse shape. Small but specific photoinduced changes in the complex fragmentation
pattern relative to that of the unshaped pulse may be sufficient to identify chemical constituents
in future.
The intensity dependent fragmentation pattern shown in Fig. 10.3 already suggests that
strong field effects, such as multielectron excitation of the π-electron ring system and nonadiabatic coupling to nuclear degrees of freedom, are the key for understanding the photophysical and photochemical processes under investigation. Although coherent control and phase
sensitivity for atomic systems are now well established and understood, it is by no means
evident that the same schemes can explain the high phase sensitivity observed in the present
experiment for such a relatively large molecular system. During the strong field and highly nonlinear excitation of the presumably neutral molecule with shaped laser pulses selective cleavage
of the N1-C3 peptide bond in Ac-Phe-NHMe is increased. This effect may indicate that the
critical step in the fragmentation process is the excitation mechanism in the neutral molecules
which initiates the propagation of specific wave packets enhanced by the pulse sequence with
optimised phases. These wave packets will eventually lead to fragmentation of the system,
most probably in the ionic state. It is important, however, that this explanation for the experimental observations at the time being as a very tentative, handwaving interpretation. It is clear
that for the elucidation of the detailed molecular reaction dynamics many more experimental
evidence is needed and the interplay between experiment and theory will be crucial.
Such further experimental studies will of course involve also different model peptides and in
particular different chromophores. As a first step to modify the conditions for energy deposition
the chromophore phenylalanine is replaced by alanine while keeping the backbone structure
with the protection groups unchanged. Such chemical change is achieved in the model peptide
Ac-Ala-NHMe with mass of 144 u.
The aim is to break the same bonds as in Ac-Phe-NHMe. The target chosen for the
adaptive control loop is again to maximise the cleavage of the peptide bond (N1-C3) forming
the fragment with mass 43 u while keeping the neighbouring bond save, i.e. to suppress the
corresponding mass 86 u. The optimal pulse shape as a result of such study is shown in
Fig. 10.11. It significantly changes the fragmentation pattern by a factor of 3 and its temporal
structure differs very clearly from that found for the Ac–Phe–NHMe model peptide. This again
indicates that the photochemical process is presumably driven during the photoexcitation and
10.1. EXPERIMENTAL RESULTS
(a )
u n s h a p e d
le a d in g
e d g e
0 .4
0 .0
0 .8
8 7
( a ’)
io n s ig n a l [a r b . u n its ]
in te n s ity [a r b . u n its ]
0 .8
177
(b )
o p tim is e d
le a d in g
e d g e
0 .4
0 .2
R = 0 .0 5
4 3
8 6
4 4
0 .0
( b ’)
R = 0 .2 5
0 .2
0 .0
0 .0
-6 0 0
-4 0 0
-2 0 0
0
2 0 0
4 0 0
6 0 0
0
tim e [f s ]
5 0
1 0 0
1 5 0
m /q [u ]
Figure 10.11: Temporal structure (a), (b) and corresponding mass spectra of Ac–Ala–
NHMe (a’), (b’) are recorded with unshaped and optimal pulse, respectively. The insert
shows the molecular structure. Straight and dotted lines indicate the optimised and suppressed bond cleavage, respectively.
not as kind of an afterthought by secondary processes.
Another interesting observation is that the maximum peaks in the mass spectrum do not
correspond to the masses expected from a naive look at the chemical structure (43 u and 86 u)
which are utilised to define the fitness function as described above. Rather, the protonated
species (44 u and 87 u) appear as dominant fragments in the mass spectrum. As one may have
expected, the photochemistry of the molecular system sensitively depends on the chromophore
even if similar bonds of the backbone structure are taken as target for the adaptive feedback
loop. In principle this is good news since mass spectrometry on biomolecular systems with
shaped pulses may look for individual pulse forms for individual amino acids. Furthermore,
even the dynamics of intramolecular chemical processes, such as proton respectively hydrogen
transfer reactions may be investigated by means of optimal control or coherent strategies.
This would be another highly relevant application in biophysical research keeping in mind that
such type of reactions are assumed to play a key role for the photostability of DNA base pairs
[SSR04].
178
CHAPTER 10. MASS SPECTROMETRY OF MODEL PEPTIDES
Chapter 11
Summary and Outlook
The main goal of this work was to investigate the interaction of moderately intense femtosecond laser pulses with free clusters and molecules. Processes of energy absorption and
redistribution within various electronic and vibrational degrees of freedom in large, complex
systems have been studied within the framework of this thesis.
First, the response of C60 fullerenes upon excitation by intense laser pulses of 9 fs duration
has been investigated with photoion spectroscopy. Using a laser pulse duration below the
characteristic time scale for electron relaxation allows one to separate the energy deposition
processes into the electronic system from the energy redistribution among the various electronic
and vibrational degrees of freedom. From an analysis of the resulting mass spectra as a
function of laser intensity information on the dynamics of the ionisation and fragmentation
has been obtained. The experimental data suggest that ionisation processes do not occur
sequentially and that the relevant mechanisms can not be considered in a single active electron
(SAE) picture. Both, the slopes of the ion yields for the final charge states q as well as the
unexpected behaviour of the saturation intensities appear to indicate that for q > 1 the giant
plasmon resonance may be involved in the energy deposition processes, while for q = 1 the
results may be viewed as essentially adiabatic and dominated by SAE behaviour. A barrier
suppression ionisation model with an appropriate, jellium type potential and centrifugal term
can qualitatively account for the observed saturation intensities. Nevertheless, many important
details of their relative magnitude at different laser parameters and charge states remain
unexplained and contradict such quasi-static model. Time resolved pump-probe experiments
with 9 fs pulses also indicate the presence of multielectron effects. In addition to the direct
multiphoton ionisation process, a small contribution (< 25%) to the total ion yield is attributed
179
180
CHAPTER 11. SUMMARY AND OUTLOOK
to energy stored intermediately in an excited doorway state (t1g ) which relaxes on a time scale
of about 50 fs. A more rigorous theoretical effort is needed taking into account multiphoton
excitation of the giant plasmon as well as possible other intermediate resonances, non-adiabatic
multielectron excitation (NMED/MAE), and dynamic polarisation.
A significant influence of ellipticity on ionisation and fragmentation processes in C60
fullerene in intense fs laser pulses was observed. The decrease of ion yields at lower intensities gives strong evidence for the crucial role of the t1g state as a doorway state for energy
deposition, followed by efficient multielectron dynamics. In contrast, at higher intensities a
remarkable increase of fullerene-like fragments is observed in circularly polarised light. This
might be caused by closed loops of recolliding electrons in circularly polarised light due to the
particular structure of C60 .
The detailed study of energy redistribution dynamics in C60 has been performed with time
resolved two colour pump-probe mass spectrometry. The role of intermediate excited states
as rate limiting step in energy deposition into the molecular system was addressed by resonant
pre-excitation of C60 fullerene with 399 nm laser pulses of 25 fs duration via the first dipole
allowed t1g state. Characteristic electron relaxation times are derived by probing the initially
neutral many-body system with delayed 27 fs pulses at 797 nm. Detailed information on the
relaxation process is obtained from the analysis of singly and multiply charged parent and
fragment ion signals formed by the probe pulse as a function of pump-probe delay and pulse
intensities. Three different trends were identified and characterised : (i) Electron relaxation
times in the neutral precursor decrease with increasing charge state q of the eventually detected
3+
Cq+
60 , q = 1 − 5. The lowest relaxation time observed for C60 is τel ' 60 fs. (ii) Fragments
that have evaporated the largest number of C2 units arise from pre-excited species showing
the slowest electronic thermalisation. These fullerenes are vibrationally hot but electronically
rather cold. (iii) Electronic relaxation times observed in the fragment channels Cq+
60−2n increase
with decreasing charge states and with increasing number of C2 losses. This corresponds to
more and more vibrational excitation at the cost of electronic excitation energy. The longest
relaxation times τel ' 400 fs are observed in the singly charged fragment ion channels. C+
60−2n
can apparently only be formed when the electron energy is particularly low, thus avoiding multiple ionisation via quasi-resonant multiphoton ionisation (MPI) through the plasmon resonance.
This also solves the long standing puzzle why only spurious amounts of singly charged large
181
fragment ions are observed in mass spectra created by very short laser pulses. The particularly low electronic excitation combined with substantial vibrational excitation as intermediate
neutral precursor required for the formation of C+
60−2n occurs only with a very low probability.
Ensembles with higher electronic pre-excitation are much more abundant and lead to multiply
charged ions by MPI via the plasmon resonance.
To control the enhancement of C60 fragmentation the femtosecond pulse shaping technique
has been combined with an evolutionary algorithm. A characteristic pulse sequence was found
to excite large amplitude oscillations by coherent heating of nuclear motion and to enhance
a formation of large, singly charged fullerene fragments C+
60−2n . The resulting pulse form
concurs with the two colour pump-probe results, where the t1g state was identified to play the
key role in the energy deposition process. With the help of time dependent density functional
theory (TDDFT) the experimentally observed periods can be connected to the calculated, laser
induced giant vibrational motion. A strong optical laser field excites many electrons in C60 via
the t1g doorway state. The almost homogenously distributed excited electron cloud couples to
the radially symmetric ag (1) breathing mode. The observed periodicity (80 − 127 fs) depends
on the number of excited electron (deposited energy) and the degree of ionisation. Despite
various electronic and vibrational degrees of freedom, this essentially one dimensional motion
prevails for up to 6 cycles with an oscillatory amplitude of up to 130% of the C60 diameter.
This work has also generated a lot of new questions which have to be investigated in
future. These further investigations require new high power lasers and new experimental
methods such as a two colour (400 nm/800 nm) pump-probe techniques with sub -10 fs laser
pulses to continue the investigation of energy redistribution pathways in C60 , a two colour
(800 nm/VUV) pump-probe technique to study electronic structure of excited C60 , measurements of electron-ion coincidence which would allow to gain important information about
photochemical reactions. In future strong field pulse shaping technique could also be applied
to optimise the population of the Rydberg states, the formation of small fragments, and even
spatial distribution of ions.
Finally, this work presents first experimental data using pulse shaping technique embedded
in an adaptive feedback loop to provide an additional dimension for mass spectrometric studies
of the primary structure of peptides. It was found that specific pulse sequences are able to break
pre-selected bonds in the amino acid complexes Ac–Phe–NHMe and Ac–Ala–NHMe. During
182
CHAPTER 11. SUMMARY AND OUTLOOK
the strong field and highly nonlinear excitation of the presumably neutral molecule with shaped
laser pulses cleavage of the N1-C3 peptide bond in Ac-Phe-NHMe is enhanced. Monitoring
the fragmentation pattern with the parameterised phase function indicates sensitivity of the
formation of fragment mass 43 u as a function of the phase parameter. The observed mass
spectra in the optimisation experiment on Ac–Ala–NHMe with additional peaks of protonated
species suggest that this technique may even be useful to elucidate intramolecular reaction
channels such as excited state hydrogen transfer which plays an important role in efficient
deactivation of DNA base pairs irradiated by UV light. The plans for further investigations
should include pulse shaping experiments with larger polypeptides and hopefully even small
proteins.
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Curriculum Vitae
Personal information
Name:
Ihar Shchatsinin
Date of Birth:
23.05.1980
Place of Birth:
Gomel, Belarus
Citizenship:
Belarus
Marital Status:
married, 1 child
Education
09.1997-06.2002
Diploma in Physics, Department of Physics, Belarussian
State University, Minsk, Belarus
09.1994-06.1997
Lyceum No. 1, Gomel, Belarus
09.1986-05.1994
School No. 44, Gomel, Belarus
Employment
since 09.2003
Ph.D. student in the group of Prof. I.V. Hertel at MaxBorn-Institute, Berlin, Germany
08.2002-05.2003
physicist, Institute of Molecular and Atomic Physics,
Minsk, Belarus
223
224
Acknowledgements
Finally, I would like to say many thanks to all people who helped me and gave their support
during the last five years of my life.
First of all I would like to thank my supervisor Prof. I. V. Hertel for the opportunity to make
this work within his group at the Max-Born-Institute. His enthusiasm and helpful suggestions
increased my motivation of going even deeper into the essence of my work.
I am also indebted to Prof. L. Wöste for accepting to be my second supervisor, his interest
to this work, and providing the opportunity to present the obtained results at his group seminar.
I am very grateful to Dr. C. P. Schulz and Dr. T. Laarmann for being the direct supervisors,
their guidance and knowledge, answering not only scientific questions but also multiple personal
questions, and of course for their suggestions, criticism, and corrections to improve the text
of this thesis.
I would especially like to thank Dr. N. Zhavoronkov for opening me the door to the laser
world and for the help with the experimental equipment.
I also thank Dr. G. Steinmeyer and Dr. G. Stibenz for the organisation of sub -10 fs
experiments.
Many thanks to all my colleagues from Department A2, both past and present, for the
support and cooperation. I sincerely acknowledge my office colleagues Pushkar Singh and
Yuliya Rulyk for the pleasant working atmosphere; Mr. A. Hentschel for his help with computers
and electronics; Mrs. H. Gromilovich for her help with administrative duties; Mrs. S. Winter,
Mrs. K. Damm, and Mr. B. Kinski for their help on all bureaucratic and personal aspects of
my work.
I am happy to thank my friends in Germany and abroad for their friendship, loyalty, and
staying with me when I have needed them.
Lastly special thanks to my whole family: my wife Alena, daughter Zlata, parents, grand225
mothers, and brother for their love, support throughout the years, and help me to achieve my
goals. In memory of grandfathers who will always be in my heart.
This work has been financially supported by the German Science Foundation (DFG) via the
collaborative research centre 450 “Analysis and Control of Ultrafast Photoinduced Reactions”,
project A2.
226