Download Quantum control of laser induced dynamics of diatomic molecular

Quantum control of laser induced dynamics
of diatomic molecular ions using shaped
intense ultrafast pulses
A thesis presented upon application for
admission to the degree of
Doctor of Philosophy
in the Faculty of Engineering and Physical Sciences
by
Leigh Graham
MSci (Hons) 2008
School of Mathematics and Physics
Queen’s University Belfast
Northern Ireland
17/09/2013
Abstract
The beauty of ultrafast science lies inherently in the ability to induce and image dynamics on a timescale comparable to the fastest nuclear motion. In recent years, a plethora
of rich and fascinating phenomena involving the interaction of diatomic molecules with
intense femtosecond laser pulse has been unveiled. Such research is motivated by the
ambition to understand and optically drive chemical reactions to the highest degree
of specificity. In this work, the strategy employed toward achieving this goal relies
on the interaction of Hydrogenic ions and analytically shaped and well characterized
pulses. The ability to manipulate photodissociation dynamics using the instantaneous
00
000
frequency and temporal profile of pulses shaped with quadratic (ϕ ) and cubic (ϕ )
spectral phase functions was studied. A three-dimensional (3D) momentum imaging
technique was used to measure the kinetic energy release (KER) and angular distribution of the dissociation fragments. A significant enhancement in the dissociation
probability of non-resonant transitions from the low lying vibrational levels using the
sign and magnitude of the applied phase function as a control tool was demonstrated.
Furthermore, the tractability of Hydrogenic ions means a mechanistic explanation for
these observations can be theoretically determined.
Investigating the behavior of ions more complex than H+
2 in strong laser fields can
present many theoretical and experimental challenges. Laser-induced fragmentation
of CD+ was explored using the 3D momentum imaging technique in the longitudinal
field imaging mode. The high mass ratio (12:2) hinders the simultaneous measurement
of the two constituents, at all angles and kinetic energies. Alternatively, the recently
developed longitudinal and transverse field imaging technique was used to perform a
piecewise dissociation measurement. This allowed the branching ratio of the dissociation channels to be obtained. Furthermore, the fragmentation channels of CD+ were
identified and studied as a function of laser intensity and wavelength.
Contents
Abstract
1
1 Diatomic Molecular Structure and its Interaction with Light
2
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3.1
Molecular Orbital Theory . . . . . . . . . . . . . . . . . . . . . .
5
1.3.2
Molecular Kinematics and Potential Energy Curves . . . . . . . .
7
1.3.3
Molecular Term Symbols . . . . . . . . . . . . . . . . . . . . . . 10
1.3.4
Transition Dipole Moment and Selection Rules . . . . . . . . . . 12
1.4
1.5
Photodissociation Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.1
Dissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.2
Floquet Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.3
Bond Softening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.4
Above Threshold Dissociation . . . . . . . . . . . . . . . . . . . . 19
1.4.5
Vibrational Trapping . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4.6
Below Threshold Dissociation . . . . . . . . . . . . . . . . . . . . 22
1.4.7
Angular distribution of fragment ions . . . . . . . . . . . . . . . 23
Ionization Processes in Intense Laser Fields . . . . . . . . . . . . . . . . 25
1.5.1
Non Sequential Ionization . . . . . . . . . . . . . . . . . . . . . . 29
1.5.2
Charge Asymmetric Dissociation . . . . . . . . . . . . . . . . . . 32
CONTENTS
3
1.5.3
Coulomb Explosion and Multielectron Dissociative Ionization . . 34
1.6
Molecular Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.6.1
Adiabatic Alignment . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.6.2
Non-adiabatic Alignment . . . . . . . . . . . . . . . . . . . . . . 41
2 Ultrafast Measurement Techniques
2.1
2.2
2.3
2.4
42
Experimental Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.1.1
Neutral vs. Ionic Targets . . . . . . . . . . . . . . . . . . . . . . 43
2.1.2
Pulse Intensity Effects . . . . . . . . . . . . . . . . . . . . . . . . 45
2.1.3
Temporal Duration . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Experimental Imaging Techniques
. . . . . . . . . . . . . . . . . . . . . 48
2.2.1
Time of Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2.2
Covariance Mapping . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2.3
COLTRIMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2.4
Velocity Map Imaging . . . . . . . . . . . . . . . . . . . . . . . . 52
2.2.5
Pump-Probe Schemes . . . . . . . . . . . . . . . . . . . . . . . . 53
Laser Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.3.1
Time and Frequency Domains . . . . . . . . . . . . . . . . . . . . 54
2.3.2
Passive Optical devices; Materials, Prisms and Gratings . . . . . 57
2.3.3
Acousto-Optic programmable Dispersive Filters (AOPDF) . . . . 58
2.3.4
Masks in the Fourier Plane . . . . . . . . . . . . . . . . . . . . . 58
Coherent Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3 3D Momentum Imaging Technique
63
3.1
Neilson Ion Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2
Vacumn System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.2.1
Scroll Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2.2
Turbomolecular Pumps . . . . . . . . . . . . . . . . . . . . . . . 67
CONTENTS
4
3.2.3
Hot-Filament Ionization Gauges . . . . . . . . . . . . . . . . . . 68
3.3
Wien Velocity Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4
Ion Beam Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.1
Einzel Lens and Deflectors . . . . . . . . . . . . . . . . . . . . . . 72
3.5
Quadrupole Triplet Focusing Lens . . . . . . . . . . . . . . . . . . . . . 74
3.6
Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.7
Ion Beam Alignment Protocol . . . . . . . . . . . . . . . . . . . . . . . . 78
3.8
3.7.1
Collimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.7.2
Ion Beam Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.7.3
Ion Beam Current Measurement . . . . . . . . . . . . . . . . . . 81
Hexanode Delay-Line Detector . . . . . . . . . . . . . . . . . . . . . . . 82
3.8.1
3.9
Hexanode Delay-Line Detector Calibration . . . . . . . . . . . . 84
Data Acquisition and Electronics . . . . . . . . . . . . . . . . . . . . . . 87
3.9.1
Constant Fraction Discriminator (CFD) . . . . . . . . . . . . . . 87
3.9.2
Time to Digital Converter (TDC)
3.9.3
Timing Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
. . . . . . . . . . . . . . . . . 89
3.10 Measuring T 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.11 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.12 Femtosecond Laser System
. . . . . . . . . . . . . . . . . . . . . . . . . 92
3.12.1 Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.12.2 Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.12.3 Compressor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.13 Pulse Shaper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.13.1 Determining the Central Pixel . . . . . . . . . . . . . . . . . . . 97
3.14 Laser Pulse Characterization (GRENOUILLE) . . . . . . . . . . . . . . 99
3.15 Parabolic Mirror and Alignment and Imaging . . . . . . . . . . . . . . . 100
3.16 Z-Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
CONTENTS
5
4 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
4.1
4.2
104
Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses . . . . 105
4.1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.1.2
Fourier Transform Limited Pulses
4.1.3
+
+
Molecular structure of H+
2 , HD and D2
4.1.4
Linear Chirp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.1.5
Third Order Dispersion . . . . . . . . . . . . . . . . . . . . . . . 117
4.1.6
Combined Linear Chirp and Third Order Dispersion . . . . . . . 132
. . . . . . . . . . . . . . . . . 106
. . . . . . . . . . . . . 110
Conclusions and Future Outlook . . . . . . . . . . . . . . . . . . . . . . 136
5 Laser Induced Fragmentation of CD+
140
5.1
Laser-Induced Fragmentation of CD+
. . . . . . . . . . . . . . . . . . . 141
5.2
Longitudinal and Transverse Field Imaging Technique . . . . . . . . . . 143
5.3
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.4
5.3.1
Dissociation Channels . . . . . . . . . . . . . . . . . . . . . . . . 147
5.3.2
Branching Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.3.3
Ionization Channels . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.3.4
Charge Asymmetric Dissociation . . . . . . . . . . . . . . . . . . 158
Conclusions and Future Outlook . . . . . . . . . . . . . . . . . . . . . . 159
Research Publications
160
Research Presentations
162
References
164
Chapter 1
Diatomic Molecular Structure
and its Interaction with Light
Molecules are the fundamental building blocks of nature and constitute everything
from the air we breathe to the interstellar clouds of neighboring galaxies. Recent
advancements in laser technology has witnessed the onset of ultrafast bursts of electromagnetic radiation which can be delivered on a timescale shorter than the typical
molecular vibration and with a magnitude comparable to the binding energy of these
systems. Understanding how molecules interact with quantized light can provide a
direct insight into their internal structure and used as a tool to characterize and manipulate such systems. This chapter outlines the basic properties of molecules and
the nomenclature commonly used to describe them. Furthermore, a plethora of laserinduced dynamics surrounding photodissociation and ionization have been unveiled,
and dependence on various laser properties investigated. These dynamics are introduced in this chapter and form a platform to understanding the results presented in
chapter 4 and 5 of this thesis.
1.1 Introduction
1.1
3
Introduction
It was Lord Kelvin just before the turn of the 20th century who reputedly stated
“There is nothing new to be discovered in physics now. All that remains is more and
more precise measurement.” But of course, like much of Science and most definitely
in physics, many of the most interesting discoveries came shortly after these infamous
words. As physicists did indeed take more and more precise measurements and began
to probe deeper into the microscopic world. J.J Thomspon discovered the electron
and thus began the journey to understand and catalogue the mechanisms involved
in atomic and molecular physics. Through theory’s and postulates and experimental
trial and error the picture of the atom became clearer. The story of the atom is an
interesting one and won’t be covered here, however the knowledge gained from that
rich history forms the basis of understanding for the work carried out in this thesis.
So as all good physicists tend to ‘stand upon the shoulders of giants’, it is a good
starting point to begin by recapping some important information with regards to the
underlying mechanisms involved in the atom before moving on from this ‘simpler’ case
to the somewhat more complicated molecular picture.
1.2
Atomic Structure
An atom is the most fundamental unit of matter in existence and is made up of three
subatomic particles; protons, neutrons and electrons. In a classical picture, it can
be envisaged as a nucleus, consisting of the protons and neutrons, located amongst
orbiting electrons. It is the balance between the electrostatic attraction of these two
entities and the outward centrifugal force of inertia which binds them together. Since
the electrostatic attraction varies proportional to
1
,
r2
where r is the distance of separa-
tion between the nucleus and electron, the electrons can be depicted as bound to the
nucleus inside a potential well, with an energy corresponding to their distance from
the nucleus.
In a quantum mechanical model however, the wave-particle duality behavior of the
1.2 Atomic Structure
4
electron must be taken into account when describing the characteristics of the orbiting
electrons. This changes dramatically the conceptual image of a particle orbiting the
nucleus on a single path trajectory. Instead, we have a probability cloud inhabiting a
spatially confined region which can only exist at discrete energies [1]. Such regions are
known as atomic orbitals, energy levels or quantum states. These can be adequately
described by the quantum numbers contained within the solution to the Schrödinger
equation, a wavefunction Ψ. The square of the absolute value of the wavefunction, |Ψ|2 ,
is interpreted as a probability density, where |Ψ(x)|2 gives the probability density for
finding the particle at position x.
The conserved quantities of these orbitals can be eloquently expressed using four discrete sets of interrelated quantum numbers n, l, m and s, which can only take the
form of an integer or a half integer [2]. The principal quantum number, n, indicates
the total energy of the system. This can be physically interpreted as the most probable distance between the electron and the nucleus, therefore defining the size of the
orbital. An electron can transfer from one orbital to another through the emission or
absorption of a photon. The region of space delineated by each orbital is determined
by the wave-like properties of the electron, in particular the number of nodes created.
For more energetic systems, a greater number of nodes are created, and the angular
momentum quantum number l, which takes the form of an integer value is assigned to
quantify this. The values of l = 0,1,2 3 are then substituted by the letters s, p, d, f
respectively . The corresponding orbital shapes created by these differing values of l
are shown in figure 1.1.
Since angular momentum has directionality, the magnetic quantum number, m is used
to define the spatial orientation of each orbital. Each electron which inhabits an orbital
has an intrinsic spin s, which can pertain a value of s = ± 12 to define its orientation.
The electrons are then placed into orbitals abiding to the Aufbau Principle, Pauli
Exclusion Principle and Hunds rule to ensure the lowest possible energy configuration
is achieved, thus promoting stability [3].
1.3 Diatomic Molecules
5
Figure 1.1: The shape of the atomic orbitals resulting from an angular momentum
quantum number of l = 0,1 and 2 which corresponds to s, p and d respectively. Figure
adapted from [4].
1.3
Diatomic Molecules
If two atoms are in close enough proximity to form a bond then a molecule can be
created. Diatomic molecules are comprised of two atoms which can either be of the
same (homonuclear) or differing (heteronuclear) species, and are bound in a linear
arrangement. There are additional degrees of freedom associated with molecules which
not only increases the complexity of theoretical calculations, but also makes depicting
this information effectively a more intricate task.
1.3.1
Molecular Orbital Theory
For diatomic molecules the two nuclei are separated by an internuclear distance R, and
the electronic distribution no longer occupies the orbital for a specific atom. Instead
it extends throughout the entire molecular domain. A technique known as the linear
combination of atomic orbitals (LCAO) can be used to characterize properties such
as the respective energies and shapes of the molecular orbitals [5]. This procedure is
schematically depicted in figure 1.2 and is based on the superposition of constituent
atomic orbitals. In other words, it is simply a weighted sum of the contributing atomic
1.3 Diatomic Molecules
6
Figure 1.2: The respective energy levels and corresponding schematic orbital diagrams
for the two atomic 1s wave functions of hydrogen and the resulting molecular orbitals
obtained from the linear combination of atomic orbitals (LCAO) molecular orbital
(MO) method. The black circles represent both the constructive and destructive interference of the two atomic orbitals, where the wavefunction phases are designated
either (+) or (-) relative to their wave ‘up’ or wave ‘down’ displacements. The nodes
are regions in which there is zero probability of finding an electron. The diagrams on
the far right show the inversion through the geometric center for both orbitals where
(a) illustrates odd parity and (b) even parity.
orbitals.
The molecular hydrogen ion is a one electron system which contains two equivalent 1s
atomic orbitals within which the electron can reside. The probability of finding the
electron on either of the atoms, for a large internuclear distance R, can be determined
by squaring either of the atomic wavefunctions (ψa or ψb ). The stability of the system
is determined by the location of the electron. As the distance R between the two nuclei
is decreased, the wavefunctions of the two atomic orbitals begin to interfere [6].
The lowest energy solution can be modeled as the linear combination ψa +ψb which constructively overlap to form a high electron density region between the two nuclei. This
1.3 Diatomic Molecules
7
forms what is known as a bonding orbital, denoted σ. It is the electrostatic attraction
between the electron and the two independent positively charged nuclei which binds
them together. Conversely, in the ψa -ψb combination the two wavefunctions overlap
with opposing phases and the creation of nodes, which are otherwise known as the
annihilation of the overlap amplitude in the internuclear region, forcing the electron to
reside elsewhere. The lack of any electron density between the two nuclei causes them
to repel each other, and no bond can be formed. This unstable orbital is higher in
energy than the original two 1s orbitals and is referred to as an anti-bonding state σ ∗ .
The symmetry of the electronic part of the wavefunction from each state can also be
described by an important feature known as parity [3]. If after an inversion through
the geometric center of the system the wavefuntion remains the same, it is considered
to have even parity (see figure 1.2 (b)), and if any change occurs is denoted as odd
parity (see figure 1.2 (a)).
This model can be extended to incorporate heteronuclear diatomic molecules. It should
be noted however that unlike homonuclear diatomic molecules, the contribution from
each orbital is not equal and must be weighted. For the formation of a bonding or anti
bonding orbital to occur, sufficient overlap between the constituent atomic orbitals
must be achieved. This is more probable for atomic orbitals of similar symmetry and
energy.
1.3.2
Molecular Kinematics and Potential Energy Curves
Although the LCAO method can provide some insight into the shape of the orbitals, it
does not offer any quantitative analysis to support experimental findings. The relationship between the energy of a molecule and its geometry can be graphically represented
in a quantitative manner using potential energy curves (PEC). The kinematics surrounding the internal energy Eint of a diatomic molecule is more complex than that
of an atom. In addition to the electronic energy Eelec there are nuclear energy components as a result of the vibrational Evib and rotational Erot motions, and these must
be taken into account. The total energy of the system can therefore be expressed as
Eint = Eelec + Evib + Erot where Eelec Evib Erot .
1.3 Diatomic Molecules
8
In a classical picture, the motion of a diatomic molecule can be considered similar
to two balls on a spring which subsequently rotate. As this spring is compressed or
stretched through different configurations, the potential energy of the system will vary
accordingly. This motion resembles that of a simple harmonic oscillator which obeys
Hooke’s Law.
However, to explicitly describe the quantum state of a molecule the Schrödinger equation must be solved to calculate the quantized eigenenergies for a collection of mobile
electrons at various internuclear distances. This task is incredibly taxing from both
a mathematical and computational viewpoint. However, the Born-Oppenheimer approximation proposes that the electronic and nuclear motion can be decoupled due
to their significant mass difference. Since the timescales of an electron (10−18 s) and
the nuclear motion (∼ 10−15 - 10−12 s) differ by orders of magnitude, the electron is
capable of adapting to a modified nuclear position in an instantaneous fashion. In this
way, the electronic potential for a given separation can be determined solely by the
position of the nuclei. It is therefore assumed that the wavefunction can be separated
into its electronic and nuclear (Ψvibrational , Ψrotational ) components and treated independently. This reduces the problem of solving the Schrödinger equation to that of
electrons residing in a stationary potential. The potential energy curves of the nuclear
configuration as a function of internuclear distance for a hypothetical diatomic are
shown in figure 1.3. The origin signifies the center of mass of the system, and the
internuclear distance R is defined with respect to this.
The equilibrium internuclear separation, R0 , is at the minimum of the potential energy
curve, and the strength of the bond is indicated by the depth of the well. This potential
well, along with any other bonding orbital can support vibrational states so the population never resides at the very bottom of the well. The temperature of the molecule
determines which states are most likely populated and the energy required to fragment
each state is different. It is therefore important, in any situation, to understand which
states are initially populated, as this will influence the observations. The vibrational
energy associated with each individual level can be described by Evib = h̄ωv (υ + 21 )υ
where υ is the vibrational quantum number (which must be an integer), h̄ is Planck’s
1.3 Diatomic Molecules
9
Figure 1.3: (a) A schematic diagram of an anharmonic potential well created by the
bonding state σ which supports the vibrational levels (red) and the corresponding
rotational levels (black). The light blue box outlines the region of the continuum of
states within which the anti bonding state σ ∗ exists. The purple line highlights the
required dissociation limit from the ground vibrational level and R0 is the equilibrium
internuclear separation. (b) Illustrates the potential energy curves for a more complex
molecule which possesses a mixture of both singlet and triplet states and demonstrates
how the molecular nomenclature system is applied in practice.
constant and ωv is the vibrational frequency.
Since the strength of the electrostatic attraction between the electron and the nuclei
does not vary proportionally with an increase in internuclear distance, the potential
well is anharmonic and can be conveniently modeled using the Morse potential [7].
This causes the spacings of the vibrational energy to decrease as they approach the
dissociation energy, D0 . Furthermore, for each vibrational level there are numerous
associated rotational states. The rotational energy of a molecule is also quantized and
is given by Erot = J(J + 1)B where J is the rotational quantum number and B is the
rotational constant unique to that molecule.
The anti-bonding states σ ∗ (as shown in figure 1.3 (a)) do not form a potential well,
hence any population promoted to this state will experience a repulsive force and tra-
1.3 Diatomic Molecules
10
verse out to larger R values, until it is no longer bound. The shape of the potential
energy curves mimic the forces of electrostatic forces within the system. As the internuclear distance between the two nuclei approaches zero, the coulomb repulsion between
them tends to infinity. Conversely, the electrostatic attraction between the nuclei and
the electron diminishes with increasing internuclear distance, until no resultant binding
force exists.
For more complex molecules the wide range of different electronic configurations which
can exist within a molecule can be represented on the same potential energy curves
diagram as shown in figure 1.3 (b). Some of the states included may be ionic where
one or more electrons have been ejected from the molecule. Since a quanta of energy
must be absorbed or emitted for the population to move from one state to another,
these curves can be used to determine how much energy is required to pursue these
transitions. This enables the expected kinetic energy release Eker shared amongst the
dissociating fragments to be estimated from;
Eker (eV ) = nh̄ω − Evib
(1.1)
where n is the net number of photons absorbed, Evib is the energy of the initially
populated vibrational level υ of the molecule and ω is the angular frequency of the
laser. The potential energy curves can then be efficiently used to deduce the most
probable dissociation pathways of diatomic molecules.
1.3.3
Molecular Term Symbols
The spectroscopic labels for potential energy curves may give some insight into the
electronic configuration of a molecule in that electronic state. Since the four interrelated quantum numbers n, l, m and s used to characterize a particular atomic state
(see section 1.2) assume spherical symmetry, an additional set of operators are used
to describe diatomic states, where the internuclear axis provides a fixed reference coordinate.
The angular momenta of the individual electrons are vectorially added to give Λ, the
1.3 Diatomic Molecules
11
projection of the orbital angular momentum along the internuclear axis, which has a
magnitude Λh̄. The assigned integer values Λ = 1, 2, 3... correspond to the designated
spectroscopic terms Σ, Π, ∆, Φ respectively, which are analogous to the s, p, d and f
orbitals in atomic notation. The spin degeneracy of a diatomic is given by 2S + 1, akin
to atoms, where the vector projections are defined in terms of Σ, of magnitude Σh̄,
where Σ is a quantum projection for molecules, analogous to m for atoms. Finally the
sum of the projections Σ and Λ on the internuclear axis is defined as Ω, of magnitude
Ωh̄. All of these properties can be represented in the following molecular term symbol
shorthand,
2S+1
(+/−)
ΛΩ,(g/u)
(1.2)
except the Ω term which is normally omitted. For homonuclear molecules the symmetry property parity is an important feature (see section 1.3.1). Since the two nuclei are
equivalent the electron cloud distributions have point symmetry about the midpoint
of the internuclear axis. The corresponding orbitals are either symmetric with respect
to inversion about the midpoint, in which case they retain their original sign; or they
are antisymmetric in which case they change sign. Orbitals which are symmetric with
respect to inversion are designated g for gerade (even) and those which are antisymmetric are designated u for ungerade (odd). A symmetry property which is included
only for heteronuclear molecules for states Λ>0, is the reflection through a plane containing the internuclear axis. The symmetry in this plane can be either symmetric or
antisymmetric and is labeled ‘+’ or ‘-’ respectively.
Finally, an identity character which precedes the moleculeular term symbol is used to
differentiate between states of equivalent properties but situated at differing energies.
The ground state is always designated X (orX̃) and excited states of different multiplicity are generally labeled a,b,c,d... [8] in ascending order of energy (see figure 1.3
(b)), although, for historical reasons, this is not always the case.
1.3 Diatomic Molecules
1.3.4
12
Transition Dipole Moment and Selection Rules
If a molecule is exposed to a time varying electric field E(t) a transient, oscillating
electric dipole moment can be induced and is given by µ = −er, where e is the
electron charge and r is the location relative to the nucleus. There are two properties
which can be imparted from a photon to a molecule when absorbed. That is energy
EP hoton = h̄ω and a unit of angular momentum. The internal energy of the molecule is
subsequently raised and the excess energy is dissipated between different modes. This
can cause an electron to be promoted from its initial state ψi to one of higher energy
ψf .
Although molecules can exist in a multitude of states, not all transitions are allowed.
The probability of a certain transition occurring is proportional to the amplitude arising from the evaluation of the matrix transition dipole moment shown in equation 1.3
below where V (r, t) = µ · E(t) and hψf |V | ψi i is the matrix element of the transition
dipole moment.
Z
hψf |V | ψi i =
ψf∗ V ψi dr
(1.3)
This is an integral over all space and ultimately provides an estimate of the overlap
between the wavefunctions of the initial ψi and final states. For some particular combinations of quantum numbers the matrix element will yield zero, corresponding to
a forbidden transition. The relationship between the quantum numbers of the initial
and final states for which the transition matrix element is not zero are known as the
transition selection rules. In principle, these selection rules apply conservation arguments to the momentum and symmetry of the system. Table 1.1 below presents the
electronic transition rules for diatomic molecules and outlines a few possible pathways.
However, since the properties of angular momenta are added vectorially there are corresponding selection rules for the rotational quantum number J which must be taken
into consideration and are given in table 1.2.
In addition to the increase in angular momentum (unit momenta ±h̄) that ensures
1.3 Diatomic Molecules
13
Allowed Transitions
∆Λ = 0, ±1
∆Σ=0
+↔+
-↔−
g↔u
Examples
Σ ↔ Σ, Π ↔ Π, Σ ↔ Π, ∆ ↔ Π
1 Σ ↔1 Σ,3 Π ↔3 Π,1 Σ ↔1 Π,3 Σ ↔3 Π
Σ+ ↔ Σ+
Σ− ↔ Σ−
+
Σ+
g ↔ Σu , Σg ↔ Πu
Table 1.1: Selection rules for the electronic transitions in diatomic molecules for the
absorption of a single photon.
Electron Transition
Σ↔Σ
All Others
Allowed Transitions
∆J = −1
∆J = 1
∆J = 1
∆J = 0
Table 1.2: Selection rules for rotational quantum number J in electronic transitions
for the absorption of a single photon.
the parity of the initial and final states must be taken into account to ensure that the
overall parity of the interaction is even to prevent the resultant integral from tending
to zero. Parity is a multiplicative operator and since both the center of mass and
symmetry of a homonulear diatomic molecule are the same, the selection rules are
more stringent. The parity of the dipole moment µ is odd so for the absorption of an
even number of photons in this case, the initial and final states must differ in parity
g ↔ u. Conversely, this rule becomes redundant if an odd number of photons are
absorbed as here it is imperative that the two states are of equivalent parity g ↔ g
and u ↔ u. For heteronuclear diatomic molecules this inversion symmetry is broken
due to the coupling between the electronic motion with asymmetric rotational and
vibrational motion of the nuclei around the center of mass. This therefore defies the
parity transition rules described above rendering Σ+ ↔ Σ+ and Σ− ↔ Σ− as allowed
transitions.
Finally, the dipole moment operator does not incorporate any spin-orbit interactions
and the integral of transitions involving states of different spin multiplicities is zero.
In principal however these singlet-triplet transitions can occur, but the probability is
very weak.
1.4 Photodissociation Dynamics
1.4
14
Photodissociation Dynamics
In recent years, a plethora of laser induced dynamics has been unveiled, and various theoretical models have been developed in a bid to gain a more comprehensive
understanding of the principal mechanisms behind these scientific findings. Owing
to its simplicity, the molecular ion H+
2 has been subjected to extensive experimental
and theoretical scrutiny, details of which can be found in several references including [9, 10, 11]. The theoretical models supporting these laser induced dynamics can
be segregated into different regimes depending on the pulse characteristics, primarily
wavelength, intensity and temporal duration.
The polarisability of a light, diatomic molecule is small making it difficult to excite
with non-resonant radiation. The majority of studies involving H+
2 concentrate on
Ti:sapphire pulses centered at ∼ 795 nm, for which an appreciable transition dipole
moment 1 a.u. - 2.3 a.u [12, 13] can be created for the vibrational levels ν = 0 − 9,
respectively.
The strength of the radiative coupling of a system can be evaluated by considering the
Rabi frequency, ωR , which is given by;
h̄ωR = E0 · µ
(1.4)
where µ is the dipole moment and the magnitude of the electric field
E0 = E0 cos(θ)
(1.5)
which is related to the intensity of the pulse via
1
I = c0 E02
2
(1.6)
It is common practice to differentiate between the strong and weak coupling regimes
by comparing the Rabi frequency ωR and the vibrational frequency of the molecule
ωv [11]. If these two quantities are equivalent, then the field is considered to be intense.
1.4 Photodissociation Dynamics
15
At low intensities the perturbation theory can be acceptably used to describe the
transpiring dynamics. In this regime, the dissociation rate is proportional to the laser
intensity, and the transition rates from the initial to final states can be adequately
obtained from Fermi’s golden rule. For higher intensities where the irradiating field
is comparable to the binding energy of the system, the potential well in which the
electrons reside can become distorted. The onset of multiphoton processes introduces
several nonlinear effects, and subsequently ascertains a breakdown in the perturbation
theory and requires a more demanding time dependent treatment.
Diabatic processes must be modeled using time dependant calculations, as the temporal duration of the interaction is so short that the system has insufficient time to
adapt. This is particularly perceptible for the case of ultrashort pulses (1×10−15 s)
where the intensity varies on the same timescale as the molecular vibrational motion. Alternatively, the system is considered to be adiabatic if the temporal duration
of the pulse exceeds the timescale of the molecular vibrational motion, leaving the
molecule ample time to respond in the presence of the electric field. For the latter
case, time-independent methods such as the Floquet or molecular field dressed state
formalisms (see section 1.4.2) are suffice and offers a more intuitive representation of
the light-induced molecular potentials in strong fields.
If we consider the irradiation of H+
2 by a 795 nm pulse of photon energy 1.55 eV, we
need only be concerned with the two lowest electronic states: the bound state 1sσg and
the repulsive 2pσu , as the others are inaccessible due to being 11 eV higher in energy.
These two states are subsequently coupled and the population undergoes an oscillitary
motion which is temporally synchronized with the electric field. At an internuclear
distance of ∼ 4.7 a.u these two states become resonantly coupled for vibrational level
v=9. The probability of a molecule absorbing a photon during its time located near
the resonance is low if the Rabi transition frequency ωR is small compared to the
vibrational frequency ωv . However, if the reverse scenario ωR ωv is true, then it is
highly probable that the molecule will absorb a photon in this region. The molecule
may absorb and emit many times in a process known as Rabi oscillations. The outer
turning point of the population for the vibrational level υ = 9 is overlapped with the
1.4 Photodissociation Dynamics
16
resonant internuclear distance, thus creating a sizable Franck-Condon factor for this
transition. The vibrational energy of this level is ∼ 0.3 eV with a vibrational period
of 29 fs. The Rabi frequency approaches the vibrational frequency at an intensity of
1011 Wcm−2 [12], above which the non-perturbative approach must be employed to
describe the laser induced dynamics of H+
2.
1.4.1
Dissociation
It is now a widely accepted phenomena that diatomic molecules irradiated with intense
laser light may absorb an excess of energy which promotes the original population to
a dissociation continuum and subsequently fragments into its constituent atoms. Despite being the simplest molecular system to study, the photodissociation of H+
2 has
revealed remarkably rich dynamics and will be exemplified here to provide a comprehensive explanation of the underlying mechanisms of the dynamics reported. The
photdissociation of H+
2 into a neutral and an ion can be written in the form:
H2+ + nh̄ω → H + H +
(1.7)
Provided the selection rules are not violated (see section 1.1) the electronic states
for a particular transition can couple as a result of the induced polarisability of the
molecule. This charge displacement will occur along the direction of the internuclear
axis, provided the laser is aligned in the same direction. In the H+
2 picture the electron
is periodically driven back and forth between the two energetically accessible lower
lying electronic states, namely the 1sσg and the 2pσu , and this process is known as Rabi
flopping. The orbital composition of the bound well, 1sσg (see section 1.3) implies that
the electron is located midway between the two nuclei. During irradiation, once the
population is projected onto the 2pσu state, the electron can be physically interpreted
as having localized itself on one of the nuclei. At this instance, the attractive binding
force between the two constituents, (H+ ion and H atom) is very weak, and the laser
field can impel the H+ away, slowly (compared to electronic timescales). However, the
oscillatory nature of the laser field makes this a continuous process where the electron
becomes localized on alternate nuclei, and the internuclear distance increases up each
1.4 Photodissociation Dynamics
17
iteration. Eventually, the electron will be located on one of the nuclei, and the H+ be
driven away completely, leading to the breakup of the molecule.
1.4.2
Floquet Formalism
There are various techniques which can be employed to solve the time-dependent
Schrȯinger equation (TDSE). If the Floquet theorem [14] is adopted, the results provide an intuitive insight into the physical phenomena surrounding dissociation. In this
approach, at low intensities where the electronic states of a molecule are regarded as
diabatic, they are considered to be ‘dressed’ by ±n photons, meaning they are simply
shifted by multiples of photon energy [15, 16].
The field dressed states of H+
2 are given in figure 1.4 where it is apparent that the 1sσg
and 2pσu intersect at υ = 9 and υ = 3 for the 1ω and 3ω shifted curves respectively.
These locations indicate the resonant transitions and it is assumed the population
will transfer from the 1sσg to the corresponding 2pσu - nω curve and move outward,
achieving a greater internuclear separation before dissociating. Since each photon
absorbed invokes a parity change of the system, a direct two photon process for H+
2 is
forbidden as an overall odd function would be created, thus contravening the selection
rules [10] (see section 1.1). Alternatively, if three photons are initially absorbed and
one is emitted prior to reaching the 1sσg - 2ω and 2pσu -3ω interception, (highlighted
by the green circle in figure 1.4), the population can cross onto the 1sσg - 2ω curve.
The molecule can then dissociate via a net two-photon absorption in a process known
as above threshold dissociation ATD (see section 1.4.4).
The consequence of irradiating a molecule with a stronger electric field can be sought
through diagonalizating the TDSE matrix. The resulting adiabatic Floquet curves
qualitatively demonstrate the bond softening effect(see section 1.4.3) and its intensity
dependence is illustrated by the broken lines in figure 1.4, where only the pulse intensity
envelope has been considered. Despite having been a powerful tool to model the
coupling between molecules and monochromatic continuous wave (CW) fields (or at
least where the laser period is much shorter than the timescale of the nuclear motion),
1.4 Photodissociation Dynamics
18
Figure 1.4: Molecular potential energy curves for H+
2 dressed by 792 nm photons.
The solid lines represent the field-free curves which have been shifted by multiples of
photon energy (0ω, 1ω, 2ω and 3ω). This diagram elucidates three possible dissociation
pathways for H+
2 . Two of these are resonant transitions which require the absorption
of 1ω (red circle, υ = 9) or 3ω (blue circle, υ = 3) photons to proceed. In the
third scenario, three photons are initially absorbed thus promoting the electron onto
the 2pσu , but the emission of a photon prior to reaching the green circle causes the
electron to cross over onto the 1sσg , resulting in an overall net two-photon process.
For higher intensities these resonant transitions are avoided as the potential energy
surfaces are perturbed. The broken lines correspond to the modified potential surfaces
which are created for the range of intensities given on the right hand side. Adapted
from Frasinski et al [18]
it is only recently that the Floquet formalism has been suitably adapted for use with
ultrashort pulses (≤ 10 fs) [17].
1.4.3
Bond Softening
The vibrational state energy of a molecule and peak intensity of the pulse play a critical
role in determining the photodissociation dynamics most likely to occur. Molecules
with a vibrational state energy either at or close to the one-photon crossing are not
heavily influenced by the properties of the electric field as they dissociate early on in
the pulse, even for low intensities.
1.4 Photodissociation Dynamics
19
For high intensities, where the strength of the laser field is comparable to the binding force of the molecule, the potential surfaces can become distorted and figure 1.5
shows this effect in H+
2 for a range of intensities. The external field causes the two
potential curves to repel each other, creating what is commonly referred to in the
adiabatic picture as an ‘avoided crossing’. The height of the potential barrier can be
reduced to an energy below that of the vibrational state energy and the population
can subsequently flow out and dissociate via bond softening [19, 20]. Alternatively, the
molecule can tunnel through the finite width of the suppressed barrier, however the
likelihood of this is small compared to the dissociation over the barrier. The probability
of photodissociation for these levels increases nonlinearly with intensity.
Due to the oscillatory nature of the laser field, these avoided crossings are not static,
and will vary between the diabatic and adiabatic curves in sync with the intensity
profile of the pulse over time. The temporal characteristics of this mechanism cannot
be overlooked, as for a significant portion of the population to dissociate over the
barrier the potential barrier must be sufficiently suppressed on a timescale comparable
to the vibrational motion of the population within the bound well.
1.4.4
Above Threshold Dissociation
There are several different mechanisms than can lead to the fragmentation of a molecule,
but the resulting pathway largely depends on the initial vibrational state and the intensity and frequency of the laser field.
14 Wcm−2 ) the |2pσ − 1ω> chanIf H+
u
2 is irradiated with a strong enough field (5×10
nel becomes redundant as the |2pσu − 3ω> one opens up allowing the vibrational levels
at or below the three photon crossing to escape (see figure 1.4 (blue circle)). The dissociating portion of the wavepacket may proceed to dissociate along the |2pσu − 3ω>
curve. Alternatively, if an adequate intensity prevails until the wavepacket reaches the
|2pσu − 2ω> pathway, a photon can be re-emitted causing the molecule to dissociate
via a net two photon process. This mechanism is known as above threshold dissociation (ATD) and the molecule dissociates with a higher than expected kinetic energy
1.4 Photodissociation Dynamics
20
Figure 1.5: The two lowest lying electronic states of H+
2 in the absence of an electric
field (dashed line). In the presence of a strong enough electric field these electronic
surfaces can become perturbed. The extent to which the barrier is suppressed depends
on the time averaged intensity, where 1×1013 Wcm−2 (dotted line) and 5×1013 Wcm−2
(solid line) are exemplified here. Reproduced from Posthumus and McCann [21].
release (KER). Since an excess of the net minimum number of photons required to
break the bond energy were absorbed, the peaks in the KER spectrum are separated
by the photon energy [19]. Theory suggests that for a wide range of intensities this
could be the dominant dissociation mechanism [22]. At greater intensities even higher
order ATD process in H+
2 , such as the net 4ω can be invoked and even dynamically
controlled [23]. In general, ATD has a higher KER and lower probability of occurring
than BS because of the larger number of photons involved.
Despite the experimental evidence for the existence of ATD being manifested in the
KER spectra of H+
2 where the peaks are separated by the photon energy [19], these
results were considered ambiguous as the product fragments of the coulomb explosion
channel also span the same KER range. This issue was resolved by trapping and
cooling HD+ to its ground state and subsequently measuring the neutral fragments
[24]. Furthermore, clear evidence of a 2ω ATD channel in the experimental KER
spectrum of CO2+ was concurrent with theory at these high intensities [25].
1.4 Photodissociation Dynamics
21
[h]
Figure 1.6: Mechanism of vibrational trapping. (a) As the intensity increases on
the rising edge of the pulse, an avoided crossing is initiated, and a portion of the
wavepacket can traverse through. (b) Near the peak of the pulse the avoided crossing
is accentuated, creating a shallow potential well which essentially traps the traversing
population. (c) This shallow well becomes inverted as the intensity decreases on the
trailing edge of the pulse and the population can either dissociate or revert to the
bound potential well.[18]
1.4.5
Vibrational Trapping
If a molecule is exposed to an electric field of sufficient intensity, the adiabatic surface
of the bound state forms a maximum turning point which alleviates the liberation
of vibrational wavepackets through the bond softening effect (see section 1.4.1). The
adverse is true of the repulsive state as the forces act to bend the surface upwards
and form a concave turning point which is capable of confining the population. This
phenomenon is commonly referred to as bond hardening, vibrational trapping or ‘stabilization’ [26, 27, 18, 28].
Figure 1.6 presents a schematic diagram of the evolution of a light induced well and
hence the mechanism for vibrational trapping for H+
2 at distinct points in the pulse.
It is assumed in figure 1.6 (a) that the vibrational state υ = 4 is populated and as
1.4 Photodissociation Dynamics
22
the intensity increases on the rising edge of the pulse the population travels towards
the avoided crossing induced at the |1sσg − 0ω> and |2pσu − 3ω> intercept. At this
stage in the pulse, the gap at the avoided crossing is very small and the probability of
dissociation is relatively low. An escaping wavepacket from the υ = 4 or vibrational
states just above the intersection can cross onto the upper curve, which is materializing
towards a shallow potential well. As the peak intensity is approached, the depth of the
well increases, causing the population to be momentarily trapped (see figure 1.6 (b)).
Thus a reduction in the dissociation probability of highly excited vibrational states
has been experimentally observed [28]. As the intensity decreases on the tail end of
the pulse, the potential energy surface is inverted as it reverts back to its diabatic
configuration. This effectively creates a hill from which some of the population can
either return to the bound well or be repelled toward larger R and dissociate.
The mechanism of vibrational trapping is a controversial topic despite the experimental evidence showing that pulses of temporal duration in the range 45-500fs, obtained
by chirping, can be used to manipulate trapped wavepackets [18]. It was found that
the rise time of the pulse intensity plays a critical role in the kinetic energy release
(KER) of the fragments as this controls the height of the hill from which the dissociating wavepacket is ejected and hence a greater KER was measured for shorter pulses.
It has been suggested that this model is an artifact of an aligned one dimensional
quantum mechanical model and that this theory does not hold if nuclear rotations are
incorporated [29].
1.4.6
Below Threshold Dissociation
As the potential surfaces from the 3ω avoided crossing revert back toward their diabatic configuration on the trailing edge of the pulse, the light induced potential well
transforms in such a way that it can cause the vibrational level υ =4 to dissociate
along the 1ω pathway, despite this transition seemingly violating energy conservation.
As weaker intensities are approached, the shape of the repulsive state changes from
concave to convex (see figure 1.6) causing it to form the bottom part of the 1ω anti
1.4 Photodissociation Dynamics
23
crossing. During this process the trapped wavepacket is lifted up and about half
of it falls back into the bound 1sσg well, and the other half escapes along the 1ω
trajectory [18]. This process, since intuitively it is below threshold to occur is referred
to as below threshold dissociation (BTD). The fragments will emerge with a very low
kinetic energy release (KER), which is ultimately determined by the intensity profile
of the pulse, where fragments with higher KER are observed for a more rapid decrease
in intensity. This effect manifests itself in as a shift from 0eV to 0.3 eV in the low
energy proton peak when the pulses are shortened from 540 fs to 45 fs.
To comply with energy conservation this energy gain can be described in terms of
the dynamic Raman effect. Rabi flopping, as described in section 1.4, is a continuous
process in which photons from the field are continuously absorbed and re-emitted.
Owing to the bandwidth of the pulse, these photons differ slightly in energy. Thus
if a photon from the lower wavelength region of the spectrum, with higher energy, is
initially absorbed and a photon of lower energy is re-emitted then these small energy
gains are enough to enable this process to proceed. If this same mechanism occurs
at the 1ω crossing where the population will emerge via the 0ω trajectory then it is
known as zero-photon dissociation (ZPD).
1.4.7
Angular distribution of fragment ions
The direction in which the dissociating fragments are ejected is not random but depends upon the overlap between the wavefunctions of the initial and final states. For
convenience, in the laboratory frame, the reference axis is given by the polarization of
the laser and the angular distribution of the fragments is given with respect to this.
The probability of excitation is proportional to |E · µ| where E is the electric field vector and µ is the transition dipole moment. Provided the axial recoil approximation,
which assumes that the molecules dissociates in a period which is short compared to
the rotational period of the molecule, is valid, then the angular distribution (θ) of the
velocity v of the fragments with respect to the laser polarization is given by:
I(θ) α 1 + βP2 (cosθ)
(1.8)
1.4 Photodissociation Dynamics
24
Figure 1.7: (a) For pure parallel transitions in the axial recoil limit, the bond axis,
the transition dipole moment µ, and the recoil direction v are parallel, so that the
photofragments posses an axis of cylindrical symmetry about these vectors. (b) for a
pure perpendicular transition, µ is perpendicular to the bond axis and v, so that the
photofragments possess two orthogonal planes of symmetry. (c) For a mixed transition,
where µ is neither parallel nor perpendicular to the bond axis, the photofragments
posses only one plane of symmetry.
Where θ is the angle between the laser polarization and the velocity, υ, of the fragments,
the anisotropy parameter β indicates the correlation between the µ and υ and P2 is
the Legendre polynomial of order 2.
For transitions between two states with a similar angular momentum quantum number
Λ (Σ → Σ or Π → Π) in which ∆ = 0, the direction of the transition dipole moment
and internuclear axis must be aligned (see figure 1.7(a)). These are known as parallel
transitions and from equation 1.8 it can be shown that the angular distribution of the
fragments from a pure parallel transition is proportional to cos2 (θ).
Alternatively, a perpendicular transition is one in which the angular momentum quantum number Λ of the states change (Σ → Π or Π → ∆) and corresponds to a ∆± 1.
Since this transition stipulates that the laser polarization and internuclear axis must
be orthogonal (see figure 1.7(b)), the angular distribution of a pure perpendicular
transition is proportional to sin2n (θ), where n is the number of photons involved.
1.5 Ionization Processes in Intense Laser Fields
25
For a collection of heteronuclear molelecules, dissociation can proceed via a multitude
of possible pathways (see figure 1.7(a)), which can either be perpendicular or parallel
transitions. The angular distribution in this case will be a combination of cos2 (θ) and
sin2n (θ), and where an indication of the branching ratio between certain degenerate
pathways can be determined by the dominant contribution. The fact that both types of
transitions are observed shows the individual nature of the molecules electronic states
plays a role in the molecular dissociation.
1.5
Ionization Processes in Intense Laser Fields
The process of liberating an electron from a molecule is termed ionization. This phenomena is incredibly rich in dynamics as this procedure can be achieved through
various mechanisms which is ultimately determined by a combination of the properties
characterizing both the molecule and the radiating field.
The binding energy of the valence electron of a molecule is known as its ionization
potential Ip , and subsequently indicates the minimum amount of energy required to
remove this electron. If the energy E = h̄ω of the photons within the electromagnetic
field is adequate such that the following condition h̄ω>Ip may be satisfied, then the
electron can effectively be promoted to the continuum after absorbing a single electron.
This process is illustrate by the single arrow in the schematic diagram 1.8 (a), where
any excess energy manifests itself in the kinetic energy of the liberated electron.
However, since the Ip of the majority of molecules is quite high, the criterion above
is rarely met. In this case, it is not the photon energy which becomes important, but
instead the photon density, ρ which depends upon the intensity I and wavelength λ
of the laser field ρ =
Iλ3
hυc .
It was proposed by Göeppert-Mayer [31] that a sequence
of photons can be absorbed consecutively, causing the electron to undergo several
transitions through a series of virtual states, as it advances toward the continuum.
This multiphoton ionization (MPI) mechanism may proceed, nh̄ω>Ip , where n is the
integer number of photons, provided the uncertainty principal (∆E∆t ≥
h̄
2)
is not
violated. This condition stipulates that a successive photon must be absorbed within
1.5 Ionization Processes in Intense Laser Fields
26
Figure 1.8: Schematic diagram showing the three different ionization regimes. In (a)
at lower intensities < 1014 Wcm−2 , multiphoton processes are dominant whereas in
(b) at the highest intensity < 1015 Wcm−2 , one reverts to the tunnel ionization model.
In (c) the field completely suppresses the coulomb barrier (> saturation intensity)
over-the-barrier [30].
the lifetime of the preceding state. Thus a high photon density ρ is required. It is
also plausible that an electron can be promoted to an intermediate state which is a
real excited state. If this occurs, then progression to the continuum can cease and the
electron can remain in that state.
Analogous to dissociation processes (see section 1.4.1), ionization mechanisms can
be classified into different regimes depending on the intensity of the electric field. At
relatively low intensities < 1013 Wcm−2 where MPI processes are dominant, the system
can be described using a first order perturbation theory as the potential surface of the
double potential well remains unperturbed. The MPI probability P is given by:
P = σN I N
(1.9)
where N is the number of photons absorbed, I is the laser field intensity and σN is
the cross section for an N absorption process. Similar to above threshold dissociation
(ATD), the number of photons absorbed may surpass that required for the MPI process and this is termed above threshold ionization (ATI), where the excess energy is
transfered to the electron.
The system can no longer be treated as a perturbation once the electric field strength
becomes comparable to the binding energy of the electron, > 1013 Wcm−2 , as the
potential well within which the electron resides becomes distorted, namely suppressed,
1.5 Ionization Processes in Intense Laser Fields
27
and creating a barrier of finite width. The ionization process under such conditions
cannot be visualized as the the absorption of photons, but instead as a strong electric
field inducing a barrier suppression through with the electrons can tunnel through
toward the continuum. As the distortion to the barrier increases the resulting binding
potential experienced by the electron is reduced. As the field intensity is raised the
barrier suppression increases until the electron is no longer bound and can escape, this
is termed over the barrier (OTB) ionization (figure 1.8 (c)).
In the presence of a temporally changing electric field E(t), the physical width of
the barrier varies dynamically in accordance with the frequency of the sinusoidal oscillations of E(t). If the barrier is sufficiently suppressed, an electron can quantum
mechanically tunnel through the finite width and escape to the continuum (figure 1.8
(b)). The rate of these tunneling events can be calculated using the Ammosov, Delonoe and Krainov, the so called ADK, model [32]. For the tunnel ionization mechanism
to be effective the wavepacket must have enough time and energy to escape, hence the
tunneling time must be short compared to the period of the E(t).
The Keyldsh parameter, γ, is commonly used to distinguish between the MPI and TI
regimes, where γ is the ratio of the tunneling time to the time at which the electric
field remains quasi-static, or equivalently the ratio of the ionization potential to twice
the ponderomaotive potential of a laser pulse, which for the case of an atom is given
by:
s
γ =
Ip
2Up
(1.10)
where UP is the ponderomotive or cycle averaged quiver energy of a free electron, of
mass me , moving in the presence of the electric field of angular frequency ω and peak
amplitude E0 ., where the atomic core is neglected
Up (a.u.) =
e2 E02
4me ω 2
=⇒
Up (eV ) = 9.33 × 10−14 I(W cm−2 )λ(µm)2
(1.11)
The shape of the potential surface is not taken into consideration when calculating γ
using the formula above. It employs a zero range potential which over estimates the
1.5 Ionization Processes in Intense Laser Fields
28
value of γ for molecules. A molecular parameter, γM , which corrects the width of the
barrier through which the electron must tunnel by incorporating the extended length
of molecular electronic orbitals, can be more a appropriate representation, see [33] for
further details.
For any given system, the value of γ is used to categorize it into a specific regime,
according to the following condition;
γ 1 =⇒ TI and OTB
γ 1 =⇒ MPI
The dominant ionization mechanism is effectively depicted by the properties in equation 1.11. The probability of tunnel ionsiation (TI) is higher for lower values of γ. The
width of the potential barrier tapers with increasing intensity, and the temporal duration for which the barrier endures this suppression, prior to the reversal of the electric
field, is wavelength (frequency) dependent. Thus the preferred criteria for TI is high
intensity, E0 , (>1013 Wcm−2 ) and a long wavelength (low frequency). For alternative
conditions MPI will be favored, but it is unclear as to which mechanism will dominant
around the region for which γ = 1.
If during tunnel ionization the wavepacket gains insufficient drift momentum to escape
the attractive potential of the newly created ion, then it can subsequently be captured
into an excited Rydberg orbital of the atom, in a process termed frustrated tunnel
ionization (FTI) [34]. This procedure is thought to occur at the trailing edge of the
laser pulse in order to conserve both energy and momentum. The wavepacket is driven
by the oscillating electric field, as proposed in the the recollison model (see section),
which causes the wavepacket to gradually decelerate, over a series of cycles, toward
the trailing edge, and as a result it has insufficient energy to escape the coulomb field
of the atom and is subsequently recaptured into an excited Rydberg orbital. Since the
outcome of FTI and a direct excitation of an electron to an excited Rydberg state is
identical, this process has been studied as a function of the ellipticity of the incident
laser light. As the ellipticity of the light increased, the probability of the electron
returning to the ionic core decreases and so a reduction in the high KER excited
1.5 Ionization Processes in Intense Laser Fields
29
neutral fragments was measured [35] in accordance with that expected for an FTI
mechanism.
1.5.1
Non Sequential Ionization
It was realized that the theoretical models such as the ADK and its advanced single
active electron (SAE) version which use single ionization exclusively to predict the
ionization probability rate of the He → He+ → He2+ process were incorrect as not only
was the onset of this process much lower than expected, but that the theory did not
match experiment [36]. Furthermore, the presence of a ’knee’ structure was indicative
of a non-sequential ionization mechanism in which the transition proceeds as follows
He → He2+ and the conventional He+ step is evaded. Over recent years, multiple
photoionization processes have been under intensive scrutiny and the exact mechanism
underlying this non-sequential ionization process is a controversial topic since a photon
can only couple to a single electron and which results in the simultaneous ejection of two
or more electrons after the absorption of a single photon and is therefore thought to be
driven entirely by many electron correlations for which the following three mechanisms
have been proposed, shake-off, collective tunneling and rescattering model;
The shake − off model was proposed by Fittinghoff et al [37] as a source of nonsequential ionization in an intense laser field. It claimed that if the ionization of
the first electron occurred rapidly enough, then the remaining electrons would have
insufficient time to react adiabatically to the instantaneous change in their molecular
environment. This could cause the initial wavefunction to be projected onto the new
system whose unaltered potential may have some overlap with the continuum state,
resulting in one or more electrons being subsequently ’shaken off’. Furthermore, in an
analogous mechanism, the electrons may not be removed simultaneously but instead
have a finite probability of being ’shaken up’ onto a bound excited state, which decays
via further ionization.
It has also been suggested that whilst the barrier is suppressed, in the presence of
the electric field, more than one electron may simultaneously quantum mechanically
1.5 Ionization Processes in Intense Laser Fields
30
tunnel to the continuum in a Collective tunneling process [38].
Although these two proposed methods provide a viable explanation for the onset of
an increased yield in multiply charged ions, they are not compliant with the observation that this effect is strongly correlated with the ellipticity of the incident light. It
has been experimentally observed that non-sequential ionization processes are strongly
suppressed for circularly polarized light [39] [40]. However, the third proposed mechanism, electron rescattering is coherent with the observation that ionization has an
intrinsic dependence on beam ellipticity.
The three-step model proposed by Corkum [41] introduced a ‘plasma perspective’ on
strong field ionization processes, and has been effectively used to eloquently communicate non-sequential ionization mechanisms and high harmonic generation on a classical
level. Figure 1.9 provides a schematic illustration of this ‘simple man model’ and each
step will be referred to during the following explanation.
In the first step, a molecule undergoes tunnel ionization within the first few cycles of a
linearly polarized pulse, leaving the liberated electron at the disposal of the lingering
electric field. In the second step, the electron is driven by the laser field, which oscillates
away from the ionic core in the first instance. The accelerating electron accumulates
ponderomotive energy (see equation 1.11), and then when the oscillating laser field
changes direction, the motion of the electron also reverses, causing it to revisit the
ionic core.
The acceleration a, velocity v, and distance x, displaced from the ionic core (typically of
the order ∼ 10−10 m) are highly dependent on the release phase ϕ, and can described,
in a one-dimensional plane at an instantaneous time t, by the following expressions;
a(t, ϕ) = −
(1.12)
Eo
[cos (ωt + ϕ) − cos ϕ]
ω
(1.13)
Eo
[sin (ωt + ϕ) − sin ϕ − ωt cos ϕ]
ω2
(1.14)
v(t, ϕ) =
x(t, ϕ) =
eE0
[sin ωt + ϕ]
me
1.5 Ionization Processes in Intense Laser Fields
31
Ultimately, the trajectory the electron follows depends upon the phase at which it was
born in the field. The interaction between the ionic core and the returning electron
constitutes the third step, and may be scattering or recombinative in nature.
If an inelastic scattering process occurs, then the ponderomotive energy accumulated
by the electron (max 3.17 Up ) may be imparted to another electron, thus liberating it
via a collisional ionization event. Alternatively, a recollision induced excitation with
subsequent field ionization (RESI) event may occur whereby the returning electron
excites a bound electron to a higher energy state which is subsequently ionized later
in the pulse. Furthermore, from various photoelectron spectra, it is evident that the
returning electron can be elastically scattered from the charge cloud of the newly
formed ion. The cut-off energy for this process was found to be 10Up , where the
electron has been backscattered through 180◦ off the core.
If the electron recombines into the ground state of the parent ion, then a photon with
a frequency which is a high-order integer multiple of the fundamental frequency of
the incident light will be emitted. The energy of these photons is determined by the
ponderomotive energy of the electron Up and the ionization potential Ip of the state
to which the electron recombines. Since the maximum kinetic energy the electron can
acquire in the field is 3.17Up this cusps an upper limit for the energy of photons created
in the high harmonic generation (HHG) process.
h̄ω = Up + Ip
=⇒
(h̄ω)max = 3.17Up + Ip
(1.15)
The x-ray pulses produced from HHG are of the shortest possible temporal duration,
being in the attosecond regime (10−18 s), and the prospect of using these pulses as
an imaging tool for time resolved studies of electron dynamics is regarded an exciting
endeavor. For more details on HHG and its applications the reader is referred to [42]
and references therein.
1.5 Ionization Processes in Intense Laser Fields
32
Figure 1.9: Summary of the three step model. The electron is ionized in step 1 at some
particular phase of the electric field. It is then driven away from the parent ion in the
laser field (step 2). After sign reversal of the ac-field, the electron stops far from the
atom, possibly returns an recombines to emit a photon carrying a kinetic energy of
the electron plus its ionization potential (step 3). The kinetic energy of the returning
electron can be as high as 3.17Up , defining the so-called cut-off photon energy in the
harmonic spectrum, figure adapted from [42].
1.5.2
Charge Asymmetric Dissociation
It is widely accepted that a molecule can fragment via a charge-symmetric channel
(CSD):
AB+ → A+ + B or A + B+
However, in recent years, evidence to support the existence of charge-asymmetric
(CAD) dissociation has been reported [43]:
AB+ → AB2+ + e− → A2+ + B + e−
It has been proposed that the electronic states involved with CAD processes closely
resemble the charge-resonant states introduced by Muliken in 1935 [44]. These can
occur when there is a difference in charge state in the dissociation limit of a diatomic
molecule. They can exist as a gerade and ungerade pair which become asymptotically
degenerate at large R and possess a dipole coupling which increases proportional to
1.5 Ionization Processes in Intense Laser Fields
33
R. Energetically, these states lie significantly higher than the symmetric dissociation
limits, an example of this can be seen in figure 1.10, where the the N 2+ + N 0+
channel is situated 12.74 eV higher than the N 1+ + N 1+ , which equates to the
absorption of an excess of 9 photons, at 800nm. It is for this reason that the existence
and an understanding of the mechanism by which these states become populated has
remained, to a large extent, elusive.
It is believed that the mechanism by which these states are populated is caused by
the fact that in the presence of a strong enough field a large induced dipole represents
an asymmetric displacement of charge which would lead to a natural asymmetry in
charge states of dissociating products. This explanation concurs with the experimental
findings that the yield of the asymmetric channels increases with increasing intensity
[45]. Furthermore, it was found that for 600ps interaction of 248 nm with N2 , that
the CAD processes dominated [43]. It was suggested that in a system behaving in this
manner that the preferred fragmentation mechanism does not depend on the energetics
of the system, but instead selects the dynamical mode of interaction with the strongest
coupling.
Initially CAD was not considered to be a ubiquitous process as it was only observed
+
for N+
2 and I2 , both of which were assumed to be special dications with accessible
metastable states and that CAD process arises from a special precursor state. However,
the single ionization charge-asymmetric (2,0) channel for a series of molecules N+
2,
CO+ , NO+ and O+
2 for the interaction with 790nm laser pulses for a range of intensities
up to 7×1015 has been recent;y reported [45]. They note that these channels are
considerably weaker, with a < 10% yield than the symmetric channel (1,1) and a
higher appearance intensity [45]. The kinetic energy release (KER) for the CAD (2,0)
channel was found to average at ∼ 3eV which is less than half that of the CSD (1,1)
channel at ∼7 eV. It was proposed that the Coulomb repulsion between the (1,1)
charged centers will result in a higher KER.
1.5 Ionization Processes in Intense Laser Fields
34
Figure 1.10: A partial potential-energy diagram of N2 indicating the ground N2 X1 σg+
and the coulombic contributions to several ionized species. Assuming vertical excitation, the ion energies observed will be one-half the amount indicated
1.5.3
Coulomb Explosion and Multielectron Dissociative Ionization
The ionization mechanisms discussed in the previous section 1.5 are applicable to both
atoms and molecules as they are non-dissociative in nature, implying the removal of
one or more electrons results in the creation of a cation of a higher multiply charged
state p, as outlined, for the absorption of n photons, in the process below;
AB + + nh̄ω
=⇒
AB p+ + (p − 1)e−
(1.16)
However, owing to the additional degrees of freedom of a molecule, dissociative ionization may occur, whereby in addition to the removal an electron, the molecule can
break apart.
AB + + nh̄ω =⇒ A2+ + B + e− or AB + + nh̄ω =⇒ A + B 2+ + e−
(1.17)
1.5 Ionization Processes in Intense Laser Fields
35
In the presence of a strong enough field, a multielectron dissociative ionization (MEDI)
event may proceed, which can effectively be described as ionization followed by coulomb
explosion.
AB + + nh̄ω =⇒ Ap+ + B q+ + (p + q − 1) e−
(1.18)
The removal of multiple electrons from a molecule, as outlined in equation 1.18, results
in a mutual repulsion between the two charged fragments causing them to coulomb
explode apart. The kinetic energy release (KER) of these fragments is higher than
that obtained from other dissociation pathways and can be determined from equation
1.19 below;
KER1 = µ
q1 q2
4π0 R
where
µ=
m2
m1 + m2
(1.19)
where qi and mi are the residual charge and mass of the two fragments respectively,
R is the internuclear separation at the moment of coulomb explosion and µ is the
reduced mass to ensure the KER is shared between the fragments so that momentum
is conserved. However, the experimentally measured KER was found to be lower than
expected at Ro , indicating that the MEDI occurred at a larger internuclear distances
than expected.
Some of the various mechanisms which have been identified to explain multielectron
dissociative ionization of a diatomic molecules are illustrated in figure 1.11, where each
step can be accomplished through either tunneling or multiphoton absorption. Each
different pathway can be considered as either a direct or indirect ionization process,
depending on the evolution of the internuclear separation R throughout the ionization
procedure.
Direct ionization processes [46] occur around the equilibrium separation Ro , meaning
the molecule is not stretched in the field and that the nuclei can be considered as
effectively frozen. A conventional direct ionization via TI or MPI is illustrated in
figure 1.11 (a), where the KER of the fragments produced in this way is higher as
the repulsive forces of two nuclei is distributed over a shorter distance. In addition to
this, figure 1.11 (b) shows a schematic representation of electron rescattering [47], as
discussed in section ??. From this it is apparent that there is insufficient time for the
1.5 Ionization Processes in Intense Laser Fields
36
Figure 1.11: Schematic diagrams illustrating the different mechanisms for the multiple
ionization of a typical molecule, AB+ , in an intense laser field (a) direct ionization,
(b) electron rescattering (nonsequential ionisation, NSI), (c) enhanced ionization, (d)
stretch only in dissociation and (e) stairstep process [45]
molecule to stretch within the return time of the electron, which is approximated to
be ∼ 3/4 of a laser cycle, and so the the ionization will occur at an R similar to that
of the initial ionization step.
Conversely, in indirect ionization mechanisms the molecule will undergo a stretch, prior
to the ejection of the electrons, a widely accepted example of which is charge-resonance
enhanced ionization (CREI) [21], [48]. In this instance, the internuclear distance R,
is increased to some critical value Rc , at which the electron becomes localized at one
of the potential wells and subsequently the removal of an electron becomes easier.
Hence the ionization rate at Rc is enhanced (see figure 1.11 (c)). Since Rc >Ro , this
accounts for the lower KER which has been experimentally observed, and several
quantum mechanical treatments of this mechanism exist to fully quantify the process
(see discussion below).
In addition to CREI, other possible MEDI mechanisms include the molecule being promoted to the continuum after having being stretched to an internuclear distance in the
range Ro <R<Rc , at the initial dissociation stage (see figure 1.11 (d)). Alternatively,
MEDI may proceed via a stairstep process in which the electrons advance toward the
continuum via a series of sequential transitions which are preceded by a small increase
1.5 Ionization Processes in Intense Laser Fields
37
in internuclear separation (see figure 1.11 (e)) [10], [49].
Figure 1.12: Schematic diagram of an I+
2 molecule ion in a laser field. The full curve
represents the potential energy of the outermost e− ina combined field of two point like
atoms I+
2 and the laser. The internuclear distance and the laser field Fc have critical
values such that the electron energy level EL touches both the inner potential barrier
Ui and the outer potential barrier Uo [21]
In the classical field ionization (FICE) model proposed by posthumus et al [21], the
molecular double-well potential U, for a molecule aligned with an electric field of amplitude Fc is given by;
Q1
Q
− 2 R − Fc x
U = −
R
x + x − 2
2
(1.20)
where Q1 and Q2 are the residual charges on the two fragment ions, x is the distance
along the internuclear axis and R is the internuclear separation and all entities are in
atomic units.
The outer valence electron of the molecule is regarded as a freely moving entity which
can span the entire double potential well, defined by U, at a specific electronic energy
level EL given by equation 1.21 below, whereas the remaining electrons are confined
to their constituent atomic orbitals (see figure 1.12).
EL =
(−E1 −
Q2
R )
+ (−E2 −
2
Q1
R )
(1.21)
1.5 Ionization Processes in Intense Laser Fields
38
Figure 1.13: Double potential wells for I+
2 at three different internuclear separations in
a strong external field, illustrating the calculation of the classical appearance intensity
of I+ + I+ . The numbers in the figure denote the laser intensity, eg. 7.9e12 represents
7.9 × 10 12 W cm−2 . Adapted from [50]
where E1 and E2 are the ionization potentials of the two constituent ions and EL
is considered as the average ionization potential distributed across the internuclear
separation, R.
In the CREI model the critical internuclear distance Rc , of a diatomic system is defined
as the separation at which the energy of the valence electron, EL and the height of both
the inner (U1 ) and outer (UO ) potential barriers are equivalent, such that ionization via
over the barrier (OTB) mechanism can prevail. Figure 1.12 illustrates these conditions
for the (1,1) channel of I2+ which occur at an internuclear distance of 7 a.u. and an
applied intensity, Fc , of 4 × 10
13
Wcm−2 [21].
The appearance intensity of each ionization channel (Q1 , Q2 ) is highly dependent on
the internuclear separation, as the inner potential barrier Ui , between the two atomic
potential wells, heightens with increasing R. Thus attaining the scenario depicted in
figure 1.12 becomes an interplay between the appropriate R and corresponding intensity of the laser field, such that both barriers are adequately lowered. This effect is
illustrated in figure 1.13 which shows how the appearance intensity of the (1,1) channel
of I+
2 varies with R.
1.5 Ionization Processes in Intense Laser Fields
39
Figure 1.14: Classical appearance intensities of the fragmentation channels of I2 (full
curves) and classical trajectories (broken curves). The laser parameters corresponding
to each of the trajectory calculations are as follows [21]
Around the equilibrium internuclear separation Ro , the inner potential barrier Ui poses
no hindrance to the motion of the valence electron as it can occupy the entire region
between the two atomic wells. The electronic energy level EL in this case is located at
a relatively low energy, and the only stipulation in this scenario is that the electric field
is strong enough to suppress the outer potential barrier enough to enable ionization to
occur (similar to R = 5 in a.u. in figure 1.13 (a) and (d)). As the nuclei are moved
further apart the inner potential barrier Ui will eventually rise above the electronic
energy, EL causing the electron to become localized in one of the atomic potential
wells. However, the onset of stark shifts can cause the electronic energy EL in the
upper potential well to be raised above the inner potential barrier Ui . These shifts
combined with a more effective lowering of the outer potential barrier Uo caused by
the increased distance over which the electric field acts creates a minimum in the
appearance intensity of this ionization process. Hence the critical separation in this
model is when the stark shifted EL and both the Ui and Uo are equal (see figure 1.13 (b)
and (e)). If the internuclear separation is increased further still, then the continually
rising inner potential becomes insurmountable for the electron to pass and an increase
in laser intensity is required (see figure 1.13 (c) and (f)).
The appearance intensities as a function of internuclear separation R, as obtained
from the FICE model, are plotted in figure 1.14 for various ionization channels (Q1 ,
1.6 Molecular Alignment
40
Q2 ) of I2 . The distinct minimum in all curves around ∼ 10 a.u. correspond to the
critical internuclear separation Rc and illustrate explicitly that the required equilibrium
between the the physical entities described above occurs over a very small range of R.
1.6
Molecular Alignment
Laser-induced alignment is defined as the molecular axis being oriented at an angle
(θ=0) parallel to the direction of the laser polarization [51]. The response of a molecule
to an electric field can differ according to this angle as many dynamics demonstrate a
strong alignment dependence. Alignment can be considered to be either a geometric
or dynamic mechanism, ultimately however a combination of both are expected. In
geometric alignment the molecules are initlally aligned parallel to the field. For geometric alignment the laser polarization couples with the molecule dipole to exert a
torque on non-aligned molecules.
Irradiating a molecule with an electric field of insufficient intensity to ionize, polarizes
the molecule creating an induced dipole moment:
Vµ = −µ · E(t)
(1.22)
where µ is the the dipole operator and E(t) = E0 cos(ωt) the electric field. The fieldfree molecular Hamiltonian Ho of the molecule then evolves in the electric field into
the eigenstates of the new Hamiltonian Hmol [52]:
Hmol = Ho + Vµ + Vα
(1.23)
where the induced dipole interaction potential is given by:
1
Vα = − E 2 (t)(∆α cos2 θ + αper ) and ∆α = αpar − αper
2
(1.24)
The αpar and αper are the components of the polarizability tensor parallel and perpendicular to the molecular axis, respectively. The induced dipole moment interacts with
1.6 Molecular Alignment
41
the electric field and rotates the molecule to a position that minimises its energy. It is
the interference from Rabi-type osciallations and the continual exchange of a number
of units of angular momentum with the field which populate the rotational wavepackets. The degree of rotational excitation is determined by either the pulse duration or
the balance between the laser intensity and the detuning from resonance.
1.6.1
Adiabatic Alignment
If the temporal duration of the pulse is greater than the rotational period of the
molecule (τ >τrot ) then the alignment is considered to be adiabatic. The electric field
oscillates on a femtosecond timescale which is too rapid for the molecule to follow on
its picosecond response time. This condition requires that the interaction be averaged
over several laser cycles. This in turn leads to the permanent dipole term being negated
and the induced dipole dominates. As the electric field is adiabatically switched on the
field-free eigenstates H0 evolve to eigenstates of the complete Hamiltonian Hmol . These
so-called ‘pendular’ states librate about the polarization axis [53]. As the electric field
is adiabatically switched off, the eigenstates return to the field-free Hamiltonian H0 .
1.6.2
Non-adiabatic Alignment
Laser pulses which are shorter than the rotational period of the molecule (τ <τrot ) can
be used to induce non-adiabatic alignment. Within a quantum mechanical framework,
a coherent superposition of rotational states (rotational wavepacket) is created via a
series of Raman excitations and deexcitations. All the states within the wavepacket
acquire phase at integer multiples of the fundamental angular frequency. It is the
dephasing and rephasing of the individual components of the rotational wavepacket
that act to align the molecule along the direction of polarization. Thus ‘field-free’
alignment can be achieved shortly after the peak of the pulse. Furthermore, a ‘revival’
can occur at a time commensurate with the ground state rotational period of the
molecule, even after the field has passed.
Chapter 2
Ultrafast Measurement
Techniques
The development of highly sophisticated spectrometric methods has enabled the various laser-induced dynamics described in the previous chapter to be eloquently explored.
This chapter gives an overview of the experimental properties which inevitably make
interpreting the data complicated. Some potential solutions to overcome these predicaments and extract the relevant information are then provided. The nature of these
experiments make measuring all the resulting fragments simultaneously and unambiguously extremely challenging. A few widely used experimental techniques to investigate
topical areas in this field are outlined.
In addition to the advancements in detection systems, the development of laser technology has propelled the ideology of moving from passive to active quantum control
methods. The realization of using laser pulses to drive chemical reactions is now tangible due to the onset of pulse shaping. The principal behind pulse shaping and a few
commonly used techniques are also discussed in this chapter.
2.1 Experimental Work
2.1
43
Experimental Work
There has been impressive progress made towards gaining a comprehensive understanding of how atoms and molecules behave in the presence of a strong laser field.
However, the experimental observations are a convolution of various parameters concerning the target molecules and the laser. These include the central wavelength and
spectral bandwidth of the laser, peak intensity and temporal duration, the focusing of
the laser beam and the interaction volume, the vibrational and rotational population
of the target ions and the resolution of the detector.
2.1.1
Neutral vs. Ionic Targets
There are two experimental methods used to prepare ions for laser induced fragmentation. In this discussion, H+
2 is used as an example to elucidate the two processes.
The H2 molecules can be ionized on the rising edge of the pulse to form H+
2 , and
then fragmented around the peak of the same pulse [54, 55]. One drawback with this
method is that it restricts analysis to intensities greater than that required to ionize
the neutral H2 (> 1×1014 Wcm−2 ). Hence obtaining any information on low intensity
dynamics is inaccessible. It was found that the lower vibrational states [56] are predominantly populated using this method. Moreover, the exact distribution was found
to be wavelength and intensity dependent [57].
Alternatively, the development of a fast ion beam target [58] has permitted the dy14 Wcm−2 ) to be thoroughly investigated.
namics of H+
2 at lower intensities (< 1×10
Thus any ambiguities introduced in the intermediate ionization stage are eradicated.
The most powerful aspect of this technique is the initial translational velocity of the
ions which means that the neutral dissociation fragments have enough energy to be
efficiently detected. Ions formed in an electron impact ion source have their vibrational
states populated according to a Franck-Condon distribution [59, 60, 61]. The nature
of the ion production however means that coherence between the vibrational states of
separate ions cannot be achieved, unlike with the laser. Furthermore, heteronuclear
molecules can undergo internal transitions during the flight time from the ion source
2.1 Experimental Work
44
Figure 2.1: Profile of a focused laser pulse showing the isointensity contours bounded
by the saturation intensity of ion species A+ , A2+ etc. The slit (ion beam) therefore
probes a particular intensity and volume region. Figure adapted from [62].
to the interaction region and so the vibrational population of such molecules upon interaction with the laser can be ambiguous. A major drawback of this technique is the
low target density of ion beams which limits the rate of measurements and promotes
time consuming experiments.
Focal Volume Effect
The spatial intensity distribution of a focused Gaussian laser pulse is extremely complicated. It varies in a Lorentzian manner along the direction of propagation z, and
Gaussian along the radial direction, r. This effectively creates a series of isointensity
contours, as shown in figure 2.1. The intensity experienced by an ion depends upon
the position at which it interacts with the pulse and can be determined from equation
2.1. This makes extracting information for a particular intensity a non-trivial task.


I(r, z) = 

1+


I0


2  exp 
−2r2

2 
ω0 1 + zz0
z
z0
(2.1)
The dissociation yields measured are therefore a convolution of the dissociation rate
and the intensities enclosed within the signal producing volume. High intensity regions
2.1 Experimental Work
45
occupy a lesser volume than those of lower intensity. This leads to a problem known as
the ‘volume effect’. By exposing the entire laser pulse to the detector, the dissociation
signal becomes dominated by the large-volume, low-intensity regions which tend to
an I 3/2 trend at saturation. These contributions can be reduced by employing the
intensity selective scan (ISS) technique [63, 64]. The interaction volume exposed to
the detector can be restricted by the FWHM of the ion beam (see the slit in figure 2.1).
A series of cylindrically symmetric thin slices of width ∆z are created and vary only in
the radial direction. The laser pulse can then be translated along the direction of laser
propagation z by translating the focal lens and effectively probe different intensity
regions.
The volume effect can be removed from an angular-resolved kinetic energy release
(KER) spectra (eg., see figure 2.2 (A)) using a technique known as the intensitydifference-spectrum (IDS). In principal IDS is the difference between two KER-cosθ
distributions obtained under the same conditions but with different peak intensities.
This allows the intensity dependence of a feature to be qualitatively investigated.
The spatial and temporal distribution of the laser beam are kept constant and beam
splitters and neutral density filters are used to attenuate the pulse energy. A sequence
of measurements are made for various intensities, and the appropriate subtractions
allow exactly the contribution from the intensity range between the two peak intensities
to be extracted. This method stipulates that the size of the ion beam must be larger
than the laser focus, but much smaller than the Rayleigh range of the laser beam.
Thus the interaction region is actually in a 2D configuration [65].
2.1.2
Pulse Intensity Effects
An insight into the dynamics of photodissociation has been attained from studying its
intensity dependence. Obtaining a quantitative comparison between published data is a
non-trivial task due to the numerous techniques employed to change the peak intensity
probed. Neutral density (ND) filters can be used to decrease the peak intensity and
have the added advantage of maintaining a constant spatial and temporal profile.
Alternatively, the intensity selective scan (ISS) technique described in section 2.1.1 can
2.1 Experimental Work
46
Figure 2.2: (A) The KER-cos θ distributions for the laser-induced dissociation of H+
2
for 135 fs pulses of peak intensity I0 = 2.4×1014 Wcm−2 . The peak intensities are
then labeled in each respective panel. (B) The KER distribution of (a) the filled
circles and (c) the open circles. The arrows mark the KER from the v=8 level at each
peak intensity. Only the dissociation inside |cosθ| >0.9 is taken into account. Figure
adapted from [66].
be used at the expense of a variation in the interaction volume for different intensities.
Furthermore, if the temporal duration of the pulse is increased, the peak intensity
decreases but the overall pulse energy can be kept constant. The latter method is
usually achieved by chirping the pulse (see section 4.1.4 ). However, recent studies
have shown that the dissociation dynamics can be influenced by the instantaneous
frequency distribution of the pulse and care should be taken when interpreting such
data [67].
The intensity dependent dissociation phenomena for H+
2 is qualitatively illustrated in
figure 2.2 (A). The various KER-cosθ distributions were measured for 135 fs pulses
starting at a peak intensity of I0 = 2.4 ×10 14 Wcm−2 and reduced using ND filters.
Molecules that dissociate via bond-softening demonstrate a continuous shift toward
lower kinetic energy release (KER) for increasing intensity. This effect is better observed in the vibrational structure of the one-dimensional KER distribution shown in
figure 2.2 (B). Only molecules dissociating inside |cosθ| >0.9 around the laser polarization direction are taken into account and the arrows indicate the shift in the v=8
2.1 Experimental Work
47
Figure 2.3: The KER-cos θ distributions for the laser-induced dissociation of H+
2 for
14
−2
a peak intensity I0 = 2.4×10 Wcm . (a) The intensity averaged spectrum, for a
pulse duration of 135 fs. (b) The IDS spectrum for (a). (c) The intensity averaged
spectrum for a pulse duration of 45 fs. (d) The IDS spectrum for (c). Figure adapted
from [66].
vibrational level. This shift can be explained by the widening of the gap at the avoided
crossing for higher intensities. Furthermore, above threshold dissociation (ATD) is seen
to peak at around 1.2 eV. This is observed only for the higher intensities as this effective two-photon dissociation feature is the result of the opening of the 3 photon curve
crossing and subsequent emission of a photon.
2.1.3
Temporal Duration
The effect of the temporal duration τ on the dissociation of H+
2 was qualitatively investigated by comparing the KER-cos θ distributions for 135 fs and 45 fs pulses of
equivalent peak intensity. Considering the laser-induced avoided crossing changes dynamically with intensity and pulse duration, interpreting the data can be problematic
due to the interdependence of these two parameters. The measured spectra and corresponding IDS spectra (see section 2.1.1) shown in figure 2.3 demonstrate the dramatic
differences observed.
The IDS spectra in figure 2.3 (b) and (d) demonstrate that dissociation induced by
2.2 Experimental Imaging Techniques
48
shorter pulses is dominated by ATD, whilst bond-softening prevails for the longer
pulses. Although vibrational structures can be observed for the long pulses (135 fs)
this is not true of the shorter pulses. This is a consequence of the vibrational period
for the vibrational levels from which the H+
2 molecule dissociates (20 fs) approaching
the pulse duration (45 fs). Furthermore, the narrow bond softening channel below 0.4
eV observed for the shorter pulse does not appear for the longer pulses.
2.2
Experimental Imaging Techniques
There are various experimental techniques used to study the behavior of molecules
in strong laser fields. These all contribute some additional information or clarity to
established scientific findings. In principal, to study ultrafast dynamics it is necessary
to identify the resultant fragments and measure their kinetic energy. There are various
experimental methods capabable of delivering this criteria but they vary drastically in
complexity, sophistication and cost.
2.2.1
Time of Flight
Arguably the easiest and most effective way to analyze a full mass range of resulting fragments simultaneously is using time-of-flight (TOF) mass spectrometry. Such
devices operate on the principle that ions can effectively be temporally and spatially
separated unambiguously using their mass to charge m/q ratio.
A linear mass spectrometer is comprised of two adjacent regions. The first acceleration
region contains a uniform electrostatic field E of potential U and length S. The second
is a field free ‘drift’ region of length D as shown in figure 2.4 (a). Ions of charge q are
produced at some potential Ep = U q in the acceleration region and gain some kinetic
energy Ek = 12 mv 2 . The ions then traverse the ‘drift’ region with a constant velocity
v. The ions TOF is governed by the m/q ratio as higher charged states are accelerated
to higher energies in the extraction region, thus they have a shorter TOF. Similarly,
heavier ions have a lower velocity and longer TOF. The measured TOF for the ions
2.2 Experimental Imaging Techniques
49
Figure 2.4: (a) Schematic diagram of the principle behind a linear time-of-flight (TOF)
device, where an ion of charge q, is ‘born’ (purple circle) amidst a homogeneous electric
field Es and accelerated over a distance S before traversing a field free region D and
striking the detector at a corresponding t time later. (b) Typical TOF spectra from a
neutral xenon target showing charge states Xe+ → Xe8+ and illustrating the concept
that higher charge/mass states have a shorter flight time and hence will arrive at the
detector first (adapted from [68]).
incorporates the distance from the position of formation to striking the detector. The
sequential detection of the different ions leads to a temporal spectrum, that may easily
be converted to a plot of signal intensity versus ion mass-to-charge ratio, an example
of which is shown in figure 2.4 (b).
The TOF technique is advantageous as theoretically it has an unlimited mass range and
a high transmission. Furthermore, it enables a complete m/q spectrum for molecular
fragmentation and is relatively cost effective. The resolution of this device however is
hindered by ions formed at non-zero velocity. This is a consequence of the focal volume
2.2 Experimental Imaging Techniques
50
of the laser producing fragments at slightly different potential energies U q within the
electric field. To compensate for the initial distribution of energies and increase the
resolution, a Wiley-McLaren spectrometer comprised of two acceleration regions was
developed, see [69] for details.
2.2.2
Covariance Mapping
Figure 2.5: Two dimensional covariance map of carbonyl sulphide measured using 790
nm light 55 fs pulses at an intensity of 2 × 1014 Wcm−2 (adapted from [68]).
The covariance mapping technique was initially used to unambiguously identify the
fragmentation channels of the multi-electron dissociative ionization (MEDI) process in
intense sub-picosecond laser fields [70]. The TOF of fragments for each laser shot is
measured but not averaged over a number of shots as in conventional averaging procedures. While averaging TOF results improves the signal-to-noise ratio, any inherent
statistical fluctuations from shot-to-shot caused by the laser pulse can become lost.
Instead, covariance mapping compares the changes in one measurement with another
measurement, by way of a shot-by-shot analysis. Covariance is defined as a measure of
association between two random variables. The covariance between two points on the
TOF spectrum is determined by the average vector minus the vector product of the
2.2 Experimental Imaging Techniques
51
averages (for mathematical details see [71]). A typical spectra is shown in figure 2.5
where the fragment ions which are statistically correlated with themselves appear on
the diagonal line and are meaningless. The real events corresponding to the forwardbackward and backward -forward pairs are situated off-diagonal. The momenta of
each pair of ions can be determined and used to identify the coulomb explosion (CE)
channels, as labeled on the right hand side of figure 2.5. False coincidences can occur
if the target gas pressure is high enough to allow more than one CE event to occur per
laser shot. An example of this is highlighted by the red circle in 2.5 where a correlation
between C+ and C2+ from an OCS molecule is unfeasible.
2.2.3
COLTRIMS
Figure 2.6: Schematic diagram showing the layout of a COLTRIM.
The ability to perform kinematically complete laser-ion interaction measurements with
no limitations on the collection angle of the reaction fragments can be achieved using
COLTRIMS (Cold Target Recoil Ion Momentum Spectroscopy). This imaging technique has the capability of measuring the momentum vectors of all charged fragments
(electrons and ions) over a 4π solid angle.
The device is essentially comprised of an ion and electron TOF spectrometer located
at opposing ends of the interaction region, as shown in the schematic diagram in figure
2.2 Experimental Imaging Techniques
52
2.6. The copper plates generate a weak uniform electrostatic field and following the
interaction, are used to extract the charged fragments in a direction corresponding to
their charge. Then after traversing their respective acceleration region, the fragments
traverse a field free drift region before impinging a 2D position sensitive channel plate
detectors with multi-hit capability.
Due to the considerably large kinetic energy of the electrons, achieving a high acceptance of these particles is challenging. The pair of Helmholtz coils situated along the
spectrometer axis act to confine the electron motion in space by enforcing them to
traverse in cyclic trajectories. The resolution and acceptance angle of the ion and electron can be changed by modifying the magnetic field strength and the spectrometer
voltage. These instruments are highly versatile as this can be done independently for
each particle.
The nature of these experiments stipulates that to increase the resolution the target
ions must be cold (below thermal motion at room temperature). Usually they are injected into the interaction region using a cooled supersonic gas jet. Thus only a small
initial momentum (≤ 0.05 a.u.) is projected along the laser polarization axis. From
the measured position and TOF information the fragments trajectories can be reconstructed and the initial momentum vector calculated unambiguously. This technique
is particularly useful when studying three-body breakup as the momentum vectors of
all components can be determined.
2.2.4
Velocity Map Imaging
The velocity map imaging (VMI) technique is a powerful tool used to study molecular
dissociation as it provides the resolution needed to determine the structure and energy
levels of the ion. The energy imparted to a molecule, in excess of the dissociation
threshold, is partitioned between the kinetic and internal energies of the constituent
atoms. Following dissociation the ejected fragments from a single vibrational level lie
spatially distributed on a Newton sphere with a radius R, defined by the conservation of momentum. This sphere then expands at a rate determined by the velocity
2.2 Experimental Imaging Techniques
53
v0 of the fragments as it traverses towards the detector. The three-dimensional velocity distribution of the fragments is projected onto the detector. Thus creating a
circularly shaped pattern as shown in figure 2.7 for the neutral dissociation fragments
of H+
2 . The detector consists of a multichannel plate (MCP) mounted in front of a
phosphorus screen. As the localized electron avalanche exiting the MCP impinges the
phosphorus screen, luminous photons are emitted and their positions recorded using
a charge-couple device (CCD) camera. The full three-dimensional information can be
reconstructed from the projected two-dimensional image using a mathematical transformation called an Abel inversion. It should be noted that this method does not
provide a means of completely distinguishing between the dissociation and ionization
processes due to an overlap in their kinetic energy release (KER) distributions.
Figure 2.7: Two-dimensional momentum projection of the neutral photofragments at
a pulse energy of 1.0 mJ and a wavelength of 785 nm. (a) τ = 575 fs, I0 = 3.5 ×1013
Wcm−2 (b) τ = 135 fs, I0 = 1.5 ×1014 Wcm−2 . Adapted from [72].
2.2.5
Pump-Probe Schemes
An optical technique which can be coupled with the experimental set-ups described
above to initiate and image time-resolved vibrational and rotational dynamics of
molecules is a pump-probe scheme. For these experiments two (or more) laser pulses
are required. The pump pulse creates a molecular wavepacket, and the probe pulse
images how this wave packet evolves with time. Usually the variable in these studies
is the the delay τ between the pulses.
2.3 Laser Pulse Shaping
54
For this scheme to work, the effective duration of the probe and the pump pulse has
to be shorter than the time scale of the process of interest. Furthermore, the temporal
resolution of the scheme is increased for shorter laser pulses. A major advantage of this
scheme is that following the pump pulse, the system evolves by the natural eigenstates
of the field-free system. Also, addition information can be obtained from so-called ‘two
color’ pump-probe measurements where two synchronized sources of short pulses are
used.
2.3
Laser Pulse Shaping
Pulse shaping has played a critical role in the quest to control chemical reactions using
laser pulses. There are two types of pulse shaping schemes which can be employed.
The closed-loop system can be implemented using a programmable pulse shaper and
a learning algorithm to find the specific pulse shape suitable for optimizing a reaction.
The signal from a particular process is fed within a feed back loop into a computer and
analyzed using an evolutionary algorithm. The pulse shape is then modified based on
the outcome of the algorithm. This iterative or adaptive process is then effectively used
to find the pulse shape required to maximize the yield of the predefined target product.
Although this is a powerful tool, obtaining any insight into the physical mechanism
induced by that particular pulse is a formidable task. Alternatively, in an open-loop
scheme the user theoretically predicts the desired pulse shape and then creates and
tests the hypothesis. This is a more intuitive method and relies on prior knowledge of
the molecular system in question.
2.3.1
Time and Frequency Domains
The electromagnetic waves of linearly polarized femtosecond (1×10−15 s) laser pulses
can be fully described by the time and space dependent electric field. A complete
description of the pulse can be given in either the time or the frequency domain.
e + (t) and its complex
The real electric field E(t) is composed of a complex electric field E
2.3 Laser Pulse Shaping
55
e − (t) contain the positive and negative values respectively, where E(t) is
conjugate E
given by:
e + (t) + E
e − (t)
E(t) = E
(2.2)
The spectral amplitude of the pulse is centered around a carrier frequency ω0 , and the
complex electric field can be represented by the product of an amplitude function A(t)
and a temporal phase term φ(t):
E + (t) = A(t)e−i[ω0 t−φ(t)]
(2.3)
E − (t) = A(t)ei[ω0 t−φ(t)]
(2.4)
and
Where the latter two properties contain all the information about the laser pulse. The
complex electric field is given by the Fourier integral:
1
2π
Z
1
E (t) =
2π
Z
E + (t) =
∞
e + (ω)eiωt dω
E
(2.5)
e − (ω)eiωt dω
E
(2.6)
−∞
and
−
∞
−∞
e + (ω) and E
e − (ω) are a complex representation of the electric field in the
Where E
frequency domain. Similar to equation 2.2 the complex electric field for the frequency
domain can be written as:
e
e + (ω) + E
e − (ω)
E(ω)
=E
(2.7)
where each of these spectrum components can be expressed as:
iϕ(ω)
e + (ω) = A(ω)e
e
E
(2.8)
−iϕ(ω)
e − (ω) = A(ω)e
e
E
(2.9)
and
e
Where A(ω)
is the spectral amplitude and ϕ(ω) the spectral phase. The relation
2.3 Laser Pulse Shaping
56
between the spectral and temporal representations of the pulse is then given by the
Fourier integral:
Z
+
∞
E (ω) =
e + (t)e−iωt dt
E
(2.10)
e − (t)e−iωt dt
E
(2.11)
−∞
and
Z
−
∞
E (ω) =
−∞
e and
The description of laser pulses in the time domain using a temporal amplitude A
e and a phase
a phase φ(t) or in the frequency domain using a spectral amplitude A
ϕ(ω) are equivalent.
If the spectral phase ϕ(ω) of a pulse varies slowly with frequency ω, it can be expanded
in the frequency domain into a Taylor series around the carrier frequency ω0 :
1 00
1 000
0
ϕ(ω) = ϕ(ω0 ) + ϕ (ω0 )(ω − ω0 ) + ϕ (ω0 )(ω − ω0 )2 + ϕ (ω0 )(ω − ω0 )3 + . . .
2
6
(2.12)
By analogy, this Taylor expansion can be performed in the time domain and the time
derivative of the temporal phase φ(t) defines the instantaneous frequency ω(t):
ω(t) = ω0 +
Pulses with
dφ(t)
dt
dφ(t)
dt
(2.13)
= 0 are known as Fourier transform limited and contain no phase
φ(t) variation. However, for
dφ(t)
dt
= f (t) the carrier frequency varies with time and
the corresponding pulse is said to be frequency modulated.
The terms in equation 2.12 can be used to characterize laser pulses as follows:
• Absolute Phase ϕ(ω0 ): The carrier envelope phase describes to the phase
between the envelope of the electric field and the carrier frequency. This is
particularly important for extremely short pulses as they only have a few cycles
within the pulse.
0
• First Derivative ϕ (ω0 ): The Linear delay term or so-called “group delay”
generates a constant group delay. This shifts the entire pulse in time with respect
to an arbitrary origin.
2.3 Laser Pulse Shaping
57
00
• Second Derivative ϕ (ω0 ): The group delay dispersion (GDD) is a quadratic
spectral function. A frequency sweep, commonly known as linear chirp is created by the corresponding linear group delay. Each frequency component ω(t)
experiences a linearly increasing delay as the spectral components of the pulse
are scanned.
000
• Third Derivative ϕ (ω0 ): Third order dispersion (TOD) is a cubic function
which causes the spectral components of the pulse to be redistributed according
to a quadratic group delay. The interference between the high and low frequencies
causes beating phenomena which manifests itself as post/pre pulses in the time
domain.
Shaping femtosecond pulses directly in the time domain is difficult due to the response
time of electronic devices. Instead, pulse shaping can be more readily achieved by
multiplying the spectral components in the frequency domain by a transfer function
M (ω) of a device or medium.
2.3.2
Passive Optical devices; Materials, Prisms and Gratings
Femtosecond pulses can be considered as being composed of several groups of quasi
monochromatic waves. Each group can be envisaged as a wavepacket with a narrow
spectrum, all of which can be added together coherently. In vacuum, the group velocity
of the wavepackets are constant and equal to the speed of light. In reality however,
the refractive index of each group of quasi monochromatic waves is wavelength (hence
frequency) dependent. This causes each group to acquire a different group velocity
and subsequently broadens the temporal profile of the pulse.
As the pulse traverses transparent media in the optical domain, positive GDD can be
introduced. This occurs as the higher frequencies travel faster through the medium
than the low frequencies. This creates what is known as an ‘up-chirped’ pulse. The
time reversal of this situation known as a ‘down chirped’ pulse is induced for negative
GDD and can be created by the angular dispersion of prisms or gratings. Although this
technique can be used to chirp pulses, any higher order spectral terms (see equation
2.3 Laser Pulse Shaping
58
2.12) introduced in this process cannot be eliminated. Furthermore, the intensity of
the pulses cannot be too high or else new frequencies are generated due to self phase
modulation.
2.3.3
Acousto-Optic programmable Dispersive Filters (AOPDF)
A method used to control the phase and amplitude of an ultrashort laser pulse is an
acousto-optic programmable dispersive filter (AOPDF), often referred to as a dazzler
(see the schematic diagram in figure 2.8). It relies on an acousto-optic interaction
between an acoustic wave, whose spatial and temporal shape is controlled by a radio
frequency transducer, and a laser pulse [73]. The laser pulse traverses the ordinary
(fast) axis of the acousto-optical crystal alongside the acoustic wave. The phase matching between the two results in diffraction which causes the laser pulse to transit from
the ordinary (fast) to the extraordinary (slow) axis. The frequency components will
experience a delay if the velocity of the laser pulse on these two axes differs.
Figure 2.8: Principle of operation of a dazzler. In an acousto-optic interaction the phase
matching between the two waves cause the optical wave to transit from the ordinary
(fast) axis to the extraordinary (slow) axis. A time delay is introduced between each
frequency component if the velocity along the two axes differ. (adapted from [73]).
2.3.4
Masks in the Fourier Plane
The conventional 4-f pulse shaper configuration used for Fourier transform pulse shaping is shown schematically in figure 2.9. The incoming pulse Ein (t) is dispersed by
2.3 Laser Pulse Shaping
59
Figure 2.9: Standard design for the conventional 4-f setup for Fourier-transform femtosecond pulse shaping. Adapted from [74]
the first grating creating several groups of spatially separated quasi monochromatic
waves. These waves are then focused to a minimum beam waist at the Fourier plane
– focal plane of a converging optical lens – by a lens (or curved mirror to eliminate
chromatic abberations) of focal length f. Each individual spectral component can then
f(ω)) located at the Fourier plane. After
be manipulated by the pulse shaping mask (M
traversing the mask, the laser pulse is reconstructed by performing an inverse Fourier
transformation back into the time domain.
The most representative spatial light modulator pulse shaping masks used to modify
the optical path of each group of quasi monochromatic waves are described below;
Liquid Crystal Spatial Light Modulators (LC-SLM) This programmable pulse
shaping tool is comprised of a linear array of independently controlled pixels. The
applied voltage changes the refractive index of each pixel. The spectral components are therefore retarded with respect to one another by the frequency dependent phase added by the specific pixel they are spatially mapped to. To a first
approximation a phase-only LC-SLM does not change the spectral amplitudes
and the integrated pulse energy remains constant for different pulse shapes.
Acousto-Optic Modulator (AOM) The AOM crystal is oriented at the brag angle
to the Fourier plane in the 4-f setup. An acoustic (sound) wave created using
2.4 Coherent Control
60
a radio-frequency (RF) electrical signal to drive the piezoelectric transducer is
propagated through the crystal. This induces a change in the refractive index of
the crystal, effectively creating a grating, and light passing transversely through
the medium is then diffracted. The diffracted beam is shifted in frequency by an
amount equal to the electrical drive frequency (typically in the one hundred MHz
range), ideally with an amplitude and phase that directly reflect the amplitude
and phase of the RF drive.
Flexible Membrane Mirrors The front plate (reflective side) of a mirror composed
of flexible thin metal coated silicon nitride can be reshaped by the voltage applied
to the electrodes beneath the surface. A movement accuracy of a few microns
is achieved using electrostatic actuators to control the voltage. The different
spectral components are dispersed as they are reflected off the mirror, and the
different distances traversed by different spectral components acts to shape the
pulse. This device is widely used a corrective optic within the laser system.
2.4
Coherent Control
Research into laser-ion interactions is motivated by the ambition to understand and
optically drive chemical reactions to the highest degree of specificity [75, 76, 77]. Over
the years various active coherent control strategies have been developed and exemplified. They largely rely on effects such as coherence, interference and time-delays
between laser pulses [78, 79, 80]. The realization of pulse shaping, and in particular
learning algorithms [81, 82, 83, 84, 85, 86], has expedited the control of molecular
dynamics such as the bending and stretching of molecules (e.g., for CO2 [87]) and
electron localization [88]. Furthermore, the ability to perform selective-bond cleavage and rearrangement as well as optimize branching ratios has been readily achieved
[85, 89]. Although learning algorithms are relatively simple to implement and require
no prior knowledge of the molecules Hamiltonian, obtaining a mechanistic explanation
is a formidable challenge.
In recent years, our understanding of the behavior of small molecules in the presence of
2.4 Coherent Control
61
strong laser fields has been acquired through varying laser parameters such as the peak
intensity, duration, central frequency, bandwidth and the carrier envelope (see section
2.1). Our strategy to achieve and understand coherent control of photodissociation
relies on the interaction of the theoretically tractable H+
2 , with analytically shaped,
well characterized pulses.
The spectral phase function φ(ω) used to shape the pulses discussed in this thesis can
be expressed by certain terms of the Taylor expansion described in equation 2.12. Using
an LC-SLM pulse shaper (see section 2.3.4) the magnitude and sign of these terms were
used as a tool to control the dissociation yield of the low lying vibrational levels v≤6
of H+
2 and its isotopic variants. Furthermore, the combination of experimental results
and theoretical calculations elucidate the underlying mechanism responsible for these
findings.
2.4 Coherent Control
62
Chapter 3
3D Momentum Imaging
Technique
The experimental results presented in chapters 4 and 5 of this thesis were carried out at
two different institutes; the Weizmann Institute of Science, Rehovot, Israel (WIS) and
Kansas State University, Manhattan, Kansas, USA (KSU). Although the respective
experimental setups were used to explore different aspects of laser-induced dissociation,
the 3D momentum imaging technique employed to measure the momentum components
of the fragments was the same. The two set-ups differ only in ion optic configurations
and the laser systems used.
Such experiments involve generating, extracting and focusing ions to form a well collimated ion beam. The laser beam is then perpendicularly focused onto these ions
amidst a longitudinal spectrometer located in the interaction region. The longitudinal
dc electric field of the spectrometer acts to accelerate the charged fragments relative
to their neutral counterparts. This allows homogeneous dissociation channels to be
unambiguously identified. The position and time of flights of the dissociation fragments is measured. From this information the 3D momentum components of each
fragment can be calculated and used to determine the kinetic energy release (KER) of
the dissociation process. Furthermore, the angle between the molecular axis and laser
polarization at the time of dissociation can be obtained. The experimental apparatus
64
can be considered as two separate sections; the ion beam and the laser system. A
schematic diagram of the whole experimental layout is shown in figure 3.1. The following chapter describes the various experimental components and outlines the setup
procedure followed at WIS.
Figure 3.1: A schematic diagram of the experimental system.
3.1 Neilson Ion Source
3.1
65
Neilson Ion Source
The molecular hydrogenic ions (H2 + ,HD+ and D2 + ) were generated simultaneously
in a Neilson-type ion source [90]. The construction of the ion source is shown in a
schematic diagram in figure 3.2. A tungsten filament is mounted coaxially within a
hollow, cylindrical anode which is subsequently enclosed within a magnetic field which
acts parallel to the axis of the anode. The ion source region was maintained at a base
pressure of ∼ 7×10−8 mbar. Then the target gas mixture (composed of 45% H2 , 45%
D2 and 10% Ar) was admitted through a long, thin walled gas inlet into a region close
to the filament to achieve a pressure of ∼ 3×10−6 mbar.
During operation, the filament is incandescently heated so that the chamber is maintained at a constant temperature of ∼ 800◦ . The anode is held at a slightly positive
potential relative to the filament (∼ 100 volts) and the surge of electrons, emitted
via thermionic emission of the filament, are accelerated towards it. These energetic
electrons can then collide with the background gas with sufficient energy to knock
electrons from the neutral target atoms and produce positive ions. The presence of the
magnetic field causes the positively charged ions and electrons to gyrate in opposite
directions. This increases the frequency of collisions and the concentration of electrons, which in turn augments the ionization rate. When collecting data from these
experiments the ion beam is expected to maintain stability for a number of consecutive
days. Maintaining a constant pressure is a critical aspect of achieving beam stability
as under certain operating conditions a plasma can be created and the ionization rate
becomes proportional to the gas density.
All the apparatus was grounded with the exception of the ion source which is floated
at some potential (typically 5keV). The ions were extracted and accelerated between
the anode and the conical shaped extraction electrodes and then traverse the beam
line with an energy equivalent to the source potential. The anode current was used to
monitor the stability of the plasma conditions. Any minor fluctuations were corrected
by adjusting the filament current or anode voltage to maintain the pressure stability
within the anode. This method will not affect and focusing and steering properties of
3.2 Vacumn System
66
Figure 3.2: A schematic diagram of the Neilson-type Ion Source as viewed from (a)
above (b) the side.
the beam.
It should be noted that the electron impact ionization mechanism generating the ions in
this source involve vertical transition where the vibrational population approximately
follows the Franck-Condon factors [59].
3.2
Vacumn System
A ‘perfect vacumn’ is a space which is completely empty of matter, however these conditions cannot be experimentally achieved. The term vacumn is therefore considered
to be the number of particles contained within a certain volume and can be calculated
using the Ideal Gas Law
P V = nRT
(3.1)
where P is pressure in Pa, V is volume in m3 , n is the number of moles of gas where
1 mol is 6×1023 particles, R is the universal gas constant = 8.314J kg−1 mol−1 and
T is the temperature in Kelvin. In these experiments the entire beam line is maintained under vacumn to prevent unwanted interactions and background collisions from
being measured, to maintain clean working surfaces and to prevent any chemical re-
3.2 Vacumn System
67
actions. Depending on the capability of the vacuum technology, the lowest possible
pressures achieved are generally classed as either vacuum (∼ <10−3 mbar, V), high
(∼ <10−8 mbar, HV) or ultrahigh (∼10−12 mbar, UHV) vacuum.
There are four gate valves along the beam line effectively separating slightly different
vacumn regions. During operation the ion source is maintained at a relatively high
pressure (∼ 3×10−6 mbar) compared to the rest of the beam line and differential
pumping is required. A small pipe (conductance limiting apertures) creates a transition
region between between the two chambers (Wien filter to adjacent chamber see figure
3.1) so that the pressure ratio across this connecting stage is manageable. This strategy
is possible due to the longer mean free path in the lower pressure region. Two (name)
scroll pumps (see section 3.2.1) are used to initially back out the system to a pressure
of <10−3 mbar. Then four turbo pumps (see section 3.2.2) are used to reduce the
pressure further along the beam line until UHV (7∼10−9 mbar) is achieved in the
detection region. Several ion gauges are used to obtain pressure readings (see section
3.2.3).
3.2.1
Scroll Pumps
Before the turbomolecular pumps can be switched on the chamber must be evacuated
to a rough vacumn or to achieve the ’molecular flow’ and such that the molecules don’t
interact with each other(∼1×10−3 mbar). A scroll pump is an oil free backing pump
composed of two interleaved archemidean spiral shaped scrolls. One scroll is fixed
whilst the other encompasses an elliptical path without rotating. The air becomes
trapped and compressed between the scrolls and is eventually successively compressed
out through a central exhaust.
3.2.2
Turbomolecular Pumps
To achieve and maintain HV/UHV turbomolecular pumps can be used to reduce the
pressure from the ‘molecular flow’ (10−3 mBar) to the ‘viscous flow’ (10−7 mBar)
region. Turbo pumps consist of a series of rotors with angled blades. Inbetween these
3.2 Vacumn System
68
rotors are fixed stators with blades oriented in the opposite direction. As the rotor
begins to spin momentum is imparted to those gas molecules hit by the rotating blades.
A net gas flow is then created in a particular direction due to the angle of the blades
and the molecules are driven toward the exhaust where they are collected by a backing
pump. The volume flow rate (pumping speed) of these momentum transfer pumps is
constant. The throughput and mass flow rate of the system decreases exponentially
as less mass mass becomes available for evacuation after pumping has commenced.
However, the constant throughput into the system introduced from real and virtual
leakage, evaporation, sublimation, desorption of materials and back streaming rates
means the turbo pump will reach a maximum compression ratio and the pressure will
become constant.
3.2.3
Hot-Filament Ionization Gauges
Several hot-filament ionization gauges are located in different regions of the beam
line to monitor the pressure in the chamber (the one located in the detection region
should be switched off during data collection). These sensitive pressure gauges can
be effectively operated in the region ∼10−10 - ∼10−3 mBar. The electrons generated
from thermionic emission of the filament are accelerated (by ∼700 volts) with sufficient
kinetic energy to ionize any background molecules. The ions created are then attracted
to and measured at an appropriately biased electrode. Hence for a fixed temperature
the current measured is proportional to the number of incident ions and the pressure
as shown in equation 3.2.
I + ∝ I −P
(3.2)
However, hot-filament ionization gauges must be carefully calibrated as each different
molecular species in the chamber has a different ionization cross section which could
bias the measurement.
3.3 Wien Velocity Filter
69
Figure 3.3: (a) Top view of the Wein filter illustrating the operation principle, where
m is the mass of the target ion [?]. (b) Wein filter viewed from the beam entrance.
3.3
Wien Velocity Filter
A Wien filter is a mass selection device comprised of a magnet and a pair of electrostatic
deflection plates. They are assembled so that the electric and magnetic field lines are
directed perpendicular to each other, as shown in figure 3.3 (b). It operates on the
principle that an ion beam of velocity v, passing through the center, will be deflected
in one direction by the electrostatic field, and in another by the magnetic field.
The force experienced by an ion traveling perpendicular to a magnetic field, causing
it to deflect, is given by,
FB = Bq × v
(3.3)
FE = Eq
(3.4)
and similarly for an electric field,
Where B and E are the magnetic and electric field strengths, and v and q are the
velocity and charge of the ion respectively.
However, if FB and FE are of the same magnitude, the two forces will cancel and
3.3 Wien Velocity Filter
70
the ion will experience no net force and traverse unperturbed through the two crossed
fields and straight through the filter. It follows from equation 3.5
qE = qvB
⇒
v0 =
E
B
(3.5)
that this condition is achieved for an ion with a velocity v0 equal to the ratio of the
electric to magnetic field. However, ions with a velocity v0 which is not equal to this
ratio experience a non-zero force and are deflected away from the exit of the filter.
The kinetic energy of the extracted ions is defined by the source potential, but the
velocity of each species differs according to equation 3.6 below:
1
(2qE) 2
v=
m
(3.6)
During operation the electric field is kept constant and the magnetic field is tuned to
allow the desired ion to pass through the aperture. This is an important characteristic
for the work described in this thesis as switching between hydrogenic isotopes is an
integral part (see section 4.1.5). The H2 + , HD+ and D2 + ions are created simultaneously in the source with the same energy. A small electric field (16 V) can be applied
to spatially separate the isotopes and the desired species can then be alternated by
switching the magnetic field of the Wien filter.
During the experiment the magnetic field is automatically changed periodically using
a Labview program. This is timed in accordance with the pulse shaper (see section
3.13) so it can be assured that each isotope are interacted with the same pulses, thus
accounting for any long term laser drifts. These experiments are typically run over a
duration of days and the magnet may undergo some hysteresis. This acts to reduce
the ion beam current, and slight modifications to the magnetic field may be required
to counteract this effect and maintain maximal ion current.
The major advantage of using a Wien filter as a mass selection device as opposed
to a magnet in this setup, is that the transmitted ions are not subjected to spatial
dispersion and exit through the aperture on the same trajectory. It follows that since
the remaining ion optic elements are electrostatic, it is possible to focus all three
3.4 Ion Beam Manipulation
71
Figure 3.4: (a) Schematic of an operating Einzel lens. (b) Simion simulation of the
saddle potential created by an Einzel lens.
isotopes into the interaction region using the same conditions.
3.4
Ion Beam Manipulation
To achieve a measurable rate of dissociation events the extracted and mass selected
ions are manipulated using a series of electrostatic ion optic elements to form a well
collimated, pulsed beam of sufficient current density in the interaction region (see
section 3.6). Although roughly 300 nA of each hydrogenic species exits the Wien filter,
subsequent to traversing the beam line optics described below, typically only ∼0.5 nA
remains.
3.4 Ion Beam Manipulation
3.4.1
72
Einzel Lens and Deflectors
An einzel lens is an electrostatic focusing element comprised of three cylindrically
symmetric electrodes (cylinders, rectangular prisms or plates) assembled in series, as
shown in figure 3.4 (a).
The two outer electrodes are held at the same potential (usually grounded) whilst a
different (positive or negative) voltage is applied to the central electrode. The electric
field lines disseminate from the central electrode to the two outer ones and create a
saddle like potential energy surface (see figure 3.4 (b)). The ions traverse the center
of these electrodes and those on the outer edges will experience a greater deflection
towards the central axis. In this way, all the ions will converge on the axis at a focal
distance f, which is determined by the applied voltage and energy of the beam. The
symmetry of the device ensures that the ions will regain their initial energy as they
exit the lens, hence the translational energy of the ions is unaltered.
Located directly after the Einzel lens are a set of horizontal and vertical deflector
plates. These act to guide as many ions as possible into the entrance aperture of the
90◦ quadrupole deflector (see section 3.4.1). In this configuration the horizontal and
vertical deflector plates act independently. One plate in each direction is grounded,
whilst a voltage (+ve or -ve) is applied to its constituent counterpart to generate an
electric field which will deflect the ions by an appropriate amount in a given direction.
Ion Beam Chopping
A pulsed ion beam is used to reduce the rate of scattered particles from the undissociated beam hitting the detector. It also reduces the likelihood of exposing the detector
to a DC beam and causing permanent damage.
An ion beam chopper (deflector) composed of two vertical parallel plates is used to
generate bunches of ions (typically 350 µs). When the DC voltage is applied the ions are
deflected away from the entrance of the 90◦ quadrupole deflector (see 3.4.1). However,
the voltage is switched off at a time toff synchronized with the laser (photdiode).
3.4 Ion Beam Manipulation
73
Figure 3.5: The SIMION geometry configuration of a 90◦ quadrupole deflector and
the simulated trajectory of an ion beam as it propagates through the two-dimensional
electrostatic quadrupole field.
This enables an ion bunch which is temporally overlapped with the laser within the
interaction region to propagate along the beam line (see section 3.9.3).
Two-Dimensional 90◦ Quadrupole Deflector
The ion beam must be deflected through 90◦ with respect to the initial propagation
direction to reach the interaction region. This is done using a two-dimensional 90◦
quadrupole deflector. This device is composed of four circular electrodes arranged
in a square (see figure 3.5). The shim electrodes are installed to produce hyperbolic
equipotentials and are used to control the fringing fields which may cause significant
abberations to the ion beam profile.
The arrangement of +ve and -ve voltage pairs creates a two-dimensional electrostatic
quadrupole field between the four rods. An ion beam of energy Ubeam enters the deflector on a straight trajectory between two adjacent rods through a grounded rectangular
3.5 Quadrupole Triplet Focusing Lens
74
aperture strategically placed to reduce any fringing effects (see figure 3.5). The combination of electrostatic and centrifugal forces acting on the ion beam cause it to bend by
90◦ around one of the circular electrodes in a trajectory of radius R0 given by equation
3.7 below.
qE =
mv 2
R0
or
E(V m−1 ) =
2Ubeam (eV )
R0 (m)
(3.7)
Where E is the electric field strength, and v, m and q are the velocity, mass and charge
of the ion respectively. Furthermore, the curvature of the two-dimensional electrostatic
quadrupole field lines causes the ions to experience a focusing effect in the y direction
and can lead to an astigmatic beam profile.
This electrostatic device will not be subjected to hysteresis and can be used to deflect
all isotope beams of equivalent energy along the same trajectory. It is also particularly advantageous as it facilitates differential pumping and removes any neutrals
transmitted from the ion source.
3.5
Quadrupole Triplet Focusing Lens
A single quadrupole lens is comprised of four parallel, equi-spaced, hyperbolic rods
arranged in a square, with each adjacent pair held at the same potential but alternate
polarity. The ions traverse through the center of these rods and the magnitude of the
electric field varies proportially to the distance from the central axis (V =0) (see figure
3.6 (a)).
This configuration effectively creates a linear lens for a cylindrical geometry. The
potential at any location within this field can be determined from V (x, y) =
V0
(x2 −y 2 ),
b2
where b is the radius of the rod, and V0 its potential. The force exerted on the ions
is perpendicular to the equipotential lines and the velocity components given to the
ions cause them to focus in either of the two planes of symmetry (y-z and the x-z).
Hence, the quadrupole acts as a perfect converging lens in one symmetry plane and
a diverging lens in the other. To overcome this problem and achieve a symmetrically
focused beam, two or more of these devices, with reversed polarities, can be arranged
3.5 Quadrupole Triplet Focusing Lens
75
Figure 3.6: (a) Electrodes and equipotential lines in an electrostatic quadrupole.
Adapted from [91]. (b) Focusing of a positive ion by a single quadrupole, where f
is the focal length given by the z value at which the ion crosses the axis (y = 0).
Adapted [92].
in close succession. The focal length of such a configuration can be found from transfer
matrices.
In a quadrupole triplet, the two outer lenses are of the same length, L1 , with a slightly
longer middle component L2 (see figure 3.7 (a)). As the ion beam traverses the first
quadrupole it will undergo diversion in one plane which will subsequently be counteracted by the restoring force in the next lens, resulting in an overall converging effect.
Furthermore, an additional steering force can be incorporated if the voltage difference
between the two positive and negative electrode pairs of the last element are not the
same.
3.6 Spectrometer
76
Figure 3.7: (a) Longitudinal section through an electrostatic triplet quadrupole lens.
Charged particles move in the z direction from the entrance plane (position 0) to the
exit plane (position 5) and beyond adapted from [93]. (b) Focusing of a positive ion
by a single quadrupole adapted from [92].
3.6
Spectrometer
The breakup momentum of the ions in the transverse and longitudinal directions cause
their relative positions and time of flight to the detector to differ, respectively. The
laser beam is focused perpendicularly onto the ions within a spectrometer which creates
a weak, uniform static electric field along the ion beam direction (z). This field acts
to accelerate the ions with respect to their neutral counterparts. This enhances the
temporal separation (z axis) between the fragments according to their mass-to-charge
ratio and reduces the distance from the center of mass in the spatial axes (x,y) for
the faster fragment. In this way, the ionization and dissociation channels can be
measured simultaneously and unambiguously identified, even if an overlap in kinetic
energy release of the two channels occurs.
The spectrometer is composed of 14 concentric ring electrodes (1 mm thickness) arranged in series with a 5 mm separation, as shown in figure 3.8. The first and last
electrodes are grounded and a voltage Vs (typically 600V for H+
2 to create a ∼150
3.6 Spectrometer
77
Figure 3.8: Schematic diagram of the spectrometer (ariel view) where the ion beam
propagation direction and the direction of the electric field are shown by the blue and
purple arrows respectively. Also marked is the location of the two circular apertures
(yellow rectangles) and the interaction point z0 where V = 0.85Vs and the laser beam
is perpendicularly focused ont to the ions.
fs temporal separation) is applied to the fourth electrode. A resistor chain serves to
connect the electrodes either side of Vs and divide the voltage such that a uniform
potential drop per unit length is created. Two thin foil circular apertures which are 1.5
mm and 2 mm in diameter are mounted on the first and fourth electrode rings respectively. These not only act as a beam alignment tool, but help reduce scattering and
create a more uniform electric field. The interaction point z0 (origin of z-axis(z=0)) is
selected to be midway between electrodes 6 and 7 (see figure 3.6) where the voltage is
equal to 0.85 Vs and the electric field is uniform. In reality however, due to the finite
aperture of the last electrode the effective electric field extends beyond the physical
spectrometer dimensions and does not drop linearly to zero. Furthermore, the radial
electric field behaves as a magnifying glass causing dispersion of the ions from the
spectrometer axis but is compensated for in the analysis procedure.
3.7 Ion Beam Alignment Protocol
3.7
78
Ion Beam Alignment Protocol
The kinematics of the dissociation process are calculated from the time and position
information of the the ion and neutral fragments. This imaging technique is limited
to measurements of KER above some minimum value (∼ 0.1 eV) as fragments with
low transverse velocity cannot be cleanly separated from the undissociated beam and
are captured by the small Faraday cup FC4 (see figure 3.6). The formation of a
well collimated beam is imperative to reduce the rate of scattered particles hitting
the detector and increase the signal-to-noise ratio (SNR). This in turn decreases the
probability of false coincidences.
A histogram of the time of flight (TOF) of the photodissociation fragments is recorded
as a live image during the ion beam alignment procedure, as shown in figure 3.9 (a).
This is used to verify that the timing sequence (see 3.9.3) has been set correctly and
spatial overlap between the ion beam and the focused laser has been achieved. Furthermore, the coincidence TOF density plot (see figure 3.9 (b)) can be used to estimate
P
Counts
the SNR and the dissociation rate
for a specific channel. Consequently, the
Time
overlap between the ion beam and laser can be obtained by scanning the laser in the
y direction to find the highest dissociation rate.
3.7.1
Collimation
The ion beam current is initially optimized on Faraday cup 3 (FC3 see figure 3.1) which
is located after the spectrometer. It is positioned along the same line as the center
of the detector and the two circular apertures of the spectrometer. The apertures are
used as a tool to collimate the ion beam. Once the current has been optimized the
spatial profile is reduced in a symmetrical manner to a cross section of ∼0.6 × 0.6
mm2 using the set of four jaw slits denoted slits 1 (see figure 3.1).
To improve the signal to noise ratio the scatter incident on the detector must be
minimized. The ion beam viewer described in section 3.7.2 can be used to visualize
the position and extent of the scatter. Most importantly, the small Faraday cup (FC4 )
3.7 Ion Beam Alignment Protocol
79
Figure 3.9: (a) Histogram of the time of flight (TOF) of H+
2 photodissociation fragments. (b) Coincidence TOF spectra for the different fragmentation channels of CD+
induced by intense laser pulses. Where t1 and t2 is the TOF of the first and second
fragment respectively.
located in front of the detector must be positioned correctly. Following this, slits 2 can
be closed to collect any scatter induced from the closure of slits 1. The two ways this
can be done is outlined below, the latter of which is usually the most effective.
1. Each jaw on slits 2 was closed independently until they cut into the ion beam.
They were then slowly retracted to the point that the initial current can be
retrieved. If required these positions can be adjusted by monitoring the scatter
on the ion beam viewer. Any charging effects can be corrected by opening the
corresponding jaw further. An indication of how well collimate dthe ion beam is
can be obtained by comparing how symmetrically slits 2 are closed around the
center of slits 1.
2. By observing the coincidence TOF density plot, each jaw can be fully closed and
then opened very slowly to the point where the compromise between the SNR
and dissociation rate is operational.
3.7 Ion Beam Alignment Protocol
80
Figure 3.10: Schematic diagram showing the honeycomb structure of an MCP and the
electron avalanche created by an electron propagating through one of the individual
glass capillary electron multipliers.
3.7.2
Ion Beam Imaging
A multi-channel plate (MCP) is a specially fabricated plate which can amplify an
incoming signal due to particles or radiation. It is composed of an array of several
individual electron multipliers arranged in a honeycomb structure as shown in figure
3.10. Each electron multiplier is effectively a thin glass capillary of very small diameter
(roughly 10 micrometers) internally coated with a high resistance, low work function
material. If an ion of sufficient energy strikes the inner wall of the capillary it will
liberate secondary electrons. These electrons are then accelerated towards the exit
due to the potential difference (approx ∼ 1.8 kV ) applied along the length of the
capillary. As the electrons propagate toward the exit they repeatedly hit adjacent
walls creating a charge cloud of approximately 106 electrons at the outlet.
The capillaries are mounted with a slight angle of impact (roughly 8◦ to the plate) and
one method of increasing the degree of electron multiplication is to produce a chevron
(v-like) shape by positioning two plates back to back. The electron cloud which exits
the first plate subsequently initiates the cascade in the next. The overall amplification
achieved depends on the applied voltage and geometry of the micro-channel plate.
The structure of the MCP allows the position of the incident ion to be transformed
into a spatially well defined charge cloud that can be accurately measured using a
phosphorus screen or delay-line anode. Overall, MCP’s are highly desirable as they
can provide high gain signals with great spatial and temporal resolution. Furthermore,
despite a dead time being associated with each individual channel electron multiplier,
3.7 Ion Beam Alignment Protocol
81
Figure 3.11: Schematic diagram of ion beam viewer technique
the multiplicity of this device makes it capable of handling multihit events.
The ion beam viewer is composed of a 40 mm diameter MCP positioned in front of a
phosphorus screen readout device and mounted at 45◦ to the ion beam axis (see figure
3.11). The phosphorus screen is biased (+ 3.6 kV) and the electron clouds emerging
from the outlet of each individual electron multiplier are accelerated towards it. The
kinetic energy of each impinging charge cloud releases an intense green photon from
the phosphor screen. This fluorescence can be captured by a charge coupled device
(CCD) connected to a television screen enabling live images to be obtained. The whole
readout device is mounted on a verticle manipulator which when lowered terminates
behind FC4 allowing it to be accurately positioned.
3.7.3
Ion Beam Current Measurement
During the experiment the undissociated ion beam is collected and measured using a
metal receptacle known as a Faraday cup (FC). The FC is composed of an electrically
conducting hollow cup (diameter 2mm) which is isolated from the thin metallic rod
which acts to hold it in place. In this way, only the ions in the cup itself contribute
to the measurement. The whole FC device is located at a distance of 20 cm from the
front of the detector and is mounted at a 45◦ angle to the vertical direction using a
movable x,y and z manipulator.
3.8 Hexanode Delay-Line Detector
82
Figure 3.12: The HEX80 delay-line detector by RoentDek Gmbh (a) Schematic of the
delay-line wire array construction, illustrating the 60◦ angle between each dimension
(u, v and w), which consists of a pair of parallel wires (reference (red) and anode
(green)), between which is a potential difference (+ 90 V). (b) The delay-line detector
assembled for use in vacuumn.
An ammeter can be used to measure the electrical current induced for a DC ion beam.
However, the substantially lower ion densities for a bunch of ions in a pulsed ion
beam means that a boxcar integrator must be used instead. The boxcar integrator
is a sampling instrument which integrates an amplified input signal for a predefined
gatewidth after an applied trigger. This integrated signal is then averaged over a
number of cycles.
3.8
Hexanode Delay-Line Detector
There is currently no single device which can provide both timing and position information simultaneously. However, the combination of a multichannel plate (MCP) and
a helical delay-line anode can be used to perform kinematically complete measurements
of dissociating molecules.
If a multichannel plate (MCP) is used to amplify the signal of an incident fragment
then the spatial information is preserved and the localized charge cloud which exits
the back side of an MCP can be collected on a helical delay-line wire arrangement and
3.8 Hexanode Delay-Line Detector
83
used to obtain two dimensional position information.
The RoentDek hexanode delay line detector consists of three layers (u, v and w) of
wire pairs (reference and anode) wound in parallel around two ceramic rods (see figure
3.12). The layers are oriented at 60◦ relative to each other and separated vertically
by 1mm. The charge cloud emerging from the MCP is distributed equally between
each layer and the position x (from the center of the wire) of the incident fragment
for that respective dimension, is defined by the center of the charge distribution. The
inter-wire voltage difference ensures the anode (+ 90 V) acts as the collector, whilst the
reference serves as a background indicator. The difference between these two signals is
determined using an RF pulse transformer (AMP TP-104) resulting in a comparatively
clear signal.
The induced signals will traverse parallel to the wire with velocity vpar which is close
to the speed of light in vacuum. However, due to the helical construction of each layer
of wire pairs, an effective propagation speed vper is introduced as the signal is required
to traverse 250 mm of wire around a loop in order to cover a 1 mm distance in the
transverse direction (typical values of signal on the anode
0.7 mm/ns). Since vper
is geometry dependent, a calibration procedure is required to determine the correct
value for each respective dimension (see section 3.21).
The delay line technique relies on the relative delay experienced by a propagating
signal as it traverses to adjacent ends of a wire ((u1 u2 ), (v1 v2 ), (w1 w2 )). The time
required for a signal to travel from position x to one end of the wire is given by
u1 =
lx + x
vper
(3.8)
and similarly the time taken to reach the opposite end of the wire (where lx is the
half-length of the wire).
u2 =
lx − x
vper
(3.9)
Therefore the position of a signal in the hexagonal frame, along a particular wire (in
3.8 Hexanode Delay-Line Detector
84
this case u) can be determined from:
u=
vper × (u1 − u2 )
2
(3.10)
The two-dimensional information can be achieved by reading a second wire in coincidence. Despite having three wire pairs, only two combinations are required to convert
from the hexagonal frame to the Cartesian coordinate system. This can be done using the equations given below, where a(u), b(v) and c(w) are the calibration factors
described in section 3.21.
xuw = u.a(u)
yuw =
√1 (2w.c(w)
3
− u.a(u))
xuw = u.a(u)
yuw =
√1 (2w.c(w)
3
− u.a(u))
xvw = (v.b(v) + w.c(w))
yvw =
√1 (w.c(w)
3
− v.b(v))
In practice, the signals from the third wire are a redundant source of information that
can be used to reconstruct signals lost due to electronic dead-time and inadequate
electronic threshold conditions. The variables measured for each dissociation event are
the timing ( t1 , t2 ) and spatial (x1 , y1 , x2 , y2 ) components for both fragments.
3.8.1
Hexanode Delay-Line Detector Calibration
The exact geometry of each Hexanode Delay-Line Detector manufactured may differ
slightly. For this reason, each detector must be independently calibrated to determine
the effective propagation speed vper for each respective dimension (u, v and w). It is
imperative that this value is determined correctly as it scales the timing signals to a
well defined spatial coordinate system and plays a critical role in the quality of the
resolution of the images.
An image of a calibration mask mounted ∼ 5 mm in front of the MCP of the detector,
comprising of 0.25 mm holes arranged linearly with 5 mm spacing, is generated by
irradiating it with a uniform distribution of H+
2 ions (see figure 3.13). If a sufficiently
3.8 Hexanode Delay-Line Detector
85
Figure 3.13: Image of the calibration mask, as measured using the uw wire pair of the
delay line detector and the calibration parameters as defined in equations 3.12 and
3.14. The dashed lines are to placed to guide the eyes to straight lines.
high voltage (+ 4 keV) is applied to the Einzel lens (see section 3.4.1) the traversing
ions will be very tightly focused at some focal length f, on the exit of the 90◦ quad
deflector. These ions will then disperse as they propagate through the unobstructed,
field free beam line (the deflector and spectrometer were removed from the beam line)
and strike the detector. The 2 m distance traversed is adequate to minimize the
divergence angle of the impinging ions.
Since the physical length (∼ 20 m) and propagation velocity (close to the speed of
light) through each wire is invariant, the overall travel time of a signal within each
wire (∼ 100 ns) is a constant Tsum and given by the ‘sum rule’ below:
Tsum = tright + tlef t − 2tmcp
(3.11)
Where tright and tlef t are the signals measured on adjacent ends of each wire respectively, and tmcp is measured from the MCP. In practice, Tsum was found to be position
3.8 Hexanode Delay-Line Detector
86
Figure 3.14: The‘sum rule’ as calculated from equation 3.12 for the measured calibration data for u, v and w wires corresponding to a, b and c respectively.
dependent, as shown in figure 3.14. The ‘sum rule’ can be used to correlate ‘good’
signals and eradicate any undesirable background data introduced as a result of ringing and electronic noise. Any data which lies outside the two parallel, horizontal, red
dashed lines in figure 3.14 is excluded.
To determine the vper for each respective dimension (u, v and w) two calibration procedures were completed. One relies on an unconstrained, non-linear optimization fitting
procedure (in Matlab) to optimize the cross-correlation of a background reduced image of the mask measured from the detector and an accurately scaled replica. A linear
correction term was introduced to the calibration parameters as Tsum is position dependent. The scaling and linear correction parameters of the experimental image were
then varied to align these two matrices. The algorithm converged on the minimized
value for the cross-correlatation once the appropriate calibration parameters had been
obtained. In the second procedure a 2D Gaussian was fitted to the signal from each
hole of the background reduced image of the mask measured from the detector and a
non-linear least squares fitting procedure was used to match it to an accurately scaled
replica.
3.9 Data Acquisition and Electronics
87
The optimal calibration factors for each dimension were found and are substituted
into equations 3.11 to convert from the hexagonal frame to the Cartesian coordinate
system.
3.9
a(u) = 0.3660 × (1 − u · 3.0 × 10−4 )
(3.12)
b(v) = 0.3791 × (1 − v · 3.0 × 10−4 )
(3.13)
c(w) = 0.3537 × (1 − w · 3.0 × 10−4 )
(3.14)
Data Acquisition and Electronics
The time of flight (TOF) measured for each fragment is initiated at the dissociation
event and terminated as the fragments strike the MCP. Such measurements can only be
achieved using a sophisticated timing sequence. In this setup, all internal and external
triggers are controlled using a labview program and are initiated every laser shot (see
section 3.9.3 for details). This timing scheme ensures temporal overlap of the target
ions and the laser in the interaction region. Furthermore, it facilitates the the relevant
data to be recoded with minimal background contributions.
The signals obtained from the Hexanode delay-line detector are temporal readouts.
They are initially amplified (Ortec Fast Amp) before being processed by a constant
fraction discriminator (CFD Ortec 845) and then measured using a multi-hit time to
digital converter (TDC) which is then read using a labview program.
3.9.1
Constant Fraction Discriminator (CFD)
Discriminators can be used as a means of reducing background and non desirable effects
such as electronic ringing. They operate by generating precise logic pulses in response
to input signals exceeding a given threshold. In a leading edge discriminator the output
pulse corresponds to the time at which the input pulse crosses a given threshold voltage.
This method however can lead to a shift known as a ‘time walk’ between the timings
of pulses with equivalent rise times but differing amplitudes. Figure 3.15 (a) illustrates
3.9 Data Acquisition and Electronics
88
this effect where it is evident that the pulse of smaller amplituted crosses the threshold
at a later time.
Where accurate timing signals are imperative a constant fraction discriminator (CFD)
can be used to alleviate this problem. The amplified input signal is split into two and
one half is attenuated to a certain fraction of the original amplitude while the other is
inverted and then delayed by an amount determined by the cable length. The addition
of these two constituent parts creates a bipolar signal which produces an output pulse
as it crosses the baseline. This zero crossing time tcross is effectively given by equation
3.15 below,
tcross =
td
(1 − f )
(3.15)
where td is the delay and f the fraction by which it has been attenuated, hence tcross
is always independent of amplitude (see figure 3.15 (b)).
Figure 3.15: (a) In a leading edge discriminator two input pulses (dotted lines) of
equivalent rise time but differing amplitudes will trigger at dissimilar times tHigh (blue)
and tLow (green) as they cross the threshold (solid yellow line). This time difference
is known as a time walk and can be alleviated through use of a constant fraction
discriminator (CFD). This operation causes each pulse (solid black line) to cross the
baseline tcross (solid red line) and hence trigger simultaneously, regardless of amplitude.
(b) Outlines the principle of the CFD by illustrating the input (dash line), delayed
(dotted) and attenuated and inverted (dash-dot) pulses and then finally the resulting
bipolar pulse (solid line) created upon addition of the latter two pulses. The output
pulse of the CFD occurs at tcross (solid red line) as the bipolar pulse crosses the baseline.
3.9 Data Acquisition and Electronics
3.9.2
89
Time to Digital Converter (TDC)
For each event, the output pulses from each channel of the CFD (t1 , u1 , v1 , w1 and t2 ,
u2 , v2 , w2 ) are fed into a separate channel of the 16-channel time to digital converter
(TDC). This device provides a digital representation for the time of arrival of an
incoming pulse with respect to a well defined reference. In this case the laser is used as
the start trigger, effectively defining the interaction as t=0 and then the time denoted
by the TDC is given by the the time of flight of the fragment to the detector (i.e.
from CFD) plus some time associated with the electronics (td see section 3.10). This
information is then read and saved by a labview program.
3.9.3
Timing Sequence
The timing sequence for the electronics and data acquisition associated with this experiment is illustrated in a schematic diagram in figure 3.16. A signal from each incoming
laser pulse is obtained by focusing the leakage from a dielectric mirror onto a photodiode (see figure 3.1). This is then used to initiate a labview program which provides
the required delays (tdelay ) and gating widths (tgate ) for all components involved. Each
incident pulse redefines the time as t=0 and effectively signifies the time of the interaction. The chopper delay is syncronised with the previous pulse. The repetition rate
of this timing sequence is 1 kHz. There are three components for which the delay and
timing gate width is controlled using the labview program:
Chopper The delay between the trigger from the laser and the chopper is such that
the subsequent laser pulse and ions are temporally overlapped and reach the
interaction region simultaneously. A continuous voltage is applied to the chopper
deflecting the ions elsewhere until it is switched off at a time tdelay later for a
time duration tgate permitting a well defined bunch of ions to proceed to the
interaction region.
CFD The CFD is vetoed so that the only signals processed are those which arrive
after tdelay but within tgate . Thus the relevant species are incorporated within
3.9 Data Acquisition and Electronics
90
Figure 3.16: A schematic diagram of the timing sequence for the data acquisition.
Every incoming laser pulse activates the labview program which redefines a zero time
(t=0) and assigns the correct time delay to each component. The first laser pulse will
not be recorded as an event as the ions and laser pulses will not be simultaneously
present in the interaction region. However, the subsequent timing sequence will be
repeated on a 1kHz rate for the duration of the experiment.
tgate due to their expected time of flight (ToF) with respect to the trigger. In
this way the surrounding background signals recorded are minimized.
TDC The output signals sent from the CFD to the TDC are no continually read.
Instead the TCD is triggered after all events from each pulse is collected and
sent to the VME computer to be read by a labview program.
3.10 Measuring T 0
3.10
91
Measuring T 0
The measured times of flight (TOFi ) are shifted from their actual values (ti ) by a
constant delay (t0 ) caused by the signal propagation time through their associated
electronics (mainly cable lengths). Thus the actual time of flight of each fragment is
determined by:
ti = T OFi − t0
(3.16)
In practice (t0 ) is measured at the end of each experimental run. The parabolic mirror
is moved parallel to the ion beam (z-direction) using a translation stage. Eventually
the incident photons strike one of the spectrometer electrodes and they are reflected
toward the detector. The value for t0 can then be determined from the histogram of
the time of flight (TOF) of the photodissociation fragments (see section 3.7).
3.11
Resolution
The accuracy of the measured timing signals from the opposite ends of each delay-line
wire are position dependent. The length of each wire is ∼ 20 m which equates to
an overall signal propagation time of ∼ 100 ns. Those signals which travel a greater
distance along the wire may undergo a broadening effect. Since the CFD relies on
pulses of similar shape, this may have a slight adverse affect on the times measured.
There is however no alternative solution to providing exact timing signals independent
of amplitude (see section 3.9.1). The precision of the calibration is also a critical factor
as it scales the signals to a well defined spatial coordinate system whilst correcting for
any non-linear effects. The resolution obtained directly from the multi-channel plates
and the delay-line wire signals is ∼ 0.3 ns and ∼ 0.5 ns respectively. This results in
an overall energy resolution of 25 meV in the measured kinetic energy release (KER)
spectra.
3.12 Femtosecond Laser System
92
Figure 3.17: The Femtosecond Laser System (adapted from Femtolasers Gmbh).
3.12
Femtosecond Laser System
The Fourier transform limited femtosecond pulses used for the dissociation experiments
are 33 fs in duration with a spectral FWHM of 40 nm centered around 795 nm. The
pulses have an energy of approximately 1.5 mJ and are delivered at a repetition rate of
1 kHz. Such pulses are created using a Ti:Sapphire based multi-pass amplifier (Femtolasers Gmbh) system consisting of three main components; an oscillator, amplifier
and compressor.
These pulses are shaped using a conventional 4-f phase-only pulse shaper with a programmable liquid crystal spatial light modulator (Jenoptik Phase SLM-640) situated
in the Fourier plane (see section 3.13). The laser polarization is rotated to the vertical
direction and the laser beam is focused using a 20cm off-axis cylindrical mirror (section
3.15) through an anti-reflective (AR) coated glass window onto the ion beam target.
3.12 Femtosecond Laser System
3.12.1
93
Oscillator
Typically, an oscillator configuration (see figure 3.17 (a)) is comprised of a gain medium
(with a specific emission spectrum) positioned between an end mirror and an output
coupler. The gain medium is then optically pumped causing a large range of frequencies
to resonate within the cavity. A specific phase relationship between the stable modes
(mode-locking) must then be established to amplify the signal and create a Fourier
transform limited pulse (minimum temporal length). Other than the achievable pulse
width, stability and reproducibility are the most important demands on a practical
femtosecond source.
The femtosecond pulses at WIS are generated in a Kerr lens mode-locked Ti:Sapphire
oscillator [94] which is pumped by a diode-pumped solid-state laser (Spectra-Physics
Millenia V). The seed pulses created are 10 fs and centred around 795 nm with a
spectral FWHM of 100 nm. They are produced at a repetition rate of 75 MHz with
average output power of ∼400 mW when the medium is pumped with 4.2 W, thus
creating 5 nJ pulses.
3.12.2
Amplifier
This seed beam is then passed through the amplifier (see figure 3.17 (b)) amplified
using the chirped pulse amplification (CPA) technique [95]. To avoid generating very
high peak powers which can damage the optics and introduce nonlinear distortions
to the spatial and temporal profile of the beam during the amplification process, the
pulses are first temporally stretched from 10 fs - 10ps. The power can be reduced by
three orders of magnitude by using dispersive optics and third order (TOD) mirrors to
create a frequency-chirped pulse and introduce high-order phase terms. This elongated
beam then enters a multi-pass configuration. It is amplified via a simulated emmison
process at each pass through the Ti:Sapphire crystal which is pumped by 11W from
a Nd:YLF laser (Spectra-Physics Empower 30). After traversing the gain medium 4
times, a single pulse from the 75-MHz pulse train is picked off by a Pockels cell every
millisecond, a rate which is in unison with the pump laser (1kHz). These selected pulses
3.13 Pulse Shaper
94
then undergo 5 more passes of the Ti:Sapphire crystal to achieve a typical energy of 1.5
mJ at the exit of the multipass stage. The spectral bandwidth of the pulse is reduced
from 100 nm to 50 nm due to gain narrowing and some wavelength mismatch of the
optics in the amplifier stage.
3.12.3
Compressor
Optical pulse compression relies on accurately reversing the stretch factors induced
in the amplifier. The objective is to create an overall zero group-delay dispersion
and create Fourier transform limited pulses (FTL). There are 4 low-dispersion Schott
prisms located in the compressor which applies negative TOD to counteract the positive
TOD which is introduced by previous optical components in the system (materials and
mirrors)[96]. The compressed and amplified beam consists of pulses 33 fs in duration
with a repetition rate 1 KHz around 795 nm.
3.13
Pulse Shaper
The pulses are shaped using a 4-f phase-only pulse shaper with a programmable liquid
crystal spatial light modulator situated in the Fourier plane. This configuration is
comprised of a combination of gratings and lenses (or mirrors) assembled as shown in
figure 3.18.
In the 4f-line configuration the spectral components of the incident pulse are angularly
dispersed by the first set of gratings (10001/mm). This creates several spatially separated groups of quasimonochromatic waves coupled into a given direction. The first
20 cm cylindrical lens is then used to focus each spectral group to a small diffraction
spot at a specific position on the Fourier plane (i.e. creating a spatio-temporal coupling). Any mask located in the Fourier plane can be used to manipulate the spectral
components of the pulse. These components are then recombined and recollimated by
the second lens and grating combination in the second half of the 4f-line, fabricating
the desired pulse shapes.
3.13 Pulse Shaper
95
Figure 3.18: A schematic diagram illustrating the principle of the 4f-line. The first
grating disperses each frequency in a given direction and the lens maps it to a given
position in the Fourier plane (i.e. creating a spatio-temporal coupling). In the second
half of the 4f-line all the components are recombined.
An aperture placed in the Fourier plane can be used to reduce the spectral bandwidth
of the pulse, creating what is termed a ‘narrowband’ pulse. The interdependence
of laser pulse parameters means this increases the temporal duration. For a Fourier
transform limited pulse the temporal duration is inversely proportional to the spectral
width. These ‘narrowband’ pulses demonstrate an enhanced resolution in the data
obtained (see section 4.1.2).
Alternatively, to accurately introduce higher orders of dispersion a programmable liquid crystal spatial light modulator was used as a mask. The pixels are constructed
from a thin layer (9 µm) of nematic liquid crystals enclosed within two glass substrates.
One substrate is coated with a thin layer of transparent ITO (indium tin oxide) as this
material is electrically conducting. The entire LC-SLM contains 640 of these pixels arranged parallel to each other. An independent voltage can be applied to each individual
pixel which causes the rod-like molecules to orient themselves with the corresponding
electric field (see figure 3.20 (a)). This results in a change in the refractive index for
light polarized in the y direction. This retards the spectrally dispersed components
3.13 Pulse Shaper
96
Figure 3.19: Schematic diagram showing how the rod-like molecules in each pixel of a
liquid-crystal spatial light modulator (LC-SLM) reorient themselves along the direction
of the applied electric field.
according to wavelength. The performance and accuracy of this configuration depends
on careful alignment, and the quality of the phase voltage and frequency φ(ω, U ) calibration. A correct calibration ensures an appropriate retardation is applied for the
corresponding wavelength. The 50 nm bandwidth of the incident pulses means only
150 pixels are actually active during the experiment. The pixel resolution is 0.34 nm
for frequency domain pulse shaping.
In principle, linearly chirped pulses (frequency sweep) can be generated by inserting
additional glass into the optical path. However, this technique provides no means of
compensating for higher orders of dispersion. To accurately shape the pulses which
will propagate the interaction region are first aligned through a replica AR coated
window into a GRENOUILLE (see section 3.14). The corresponding frequency-time
relation is used as a tool to eliminate any undesirable higher order dispersion terms
introduced due to self phase modulation from various downstream optical components
and create an FTL pulse. The desired spectral phase was then applied to this FTL.
Furthermore, the programmable nature of the LC-SLM allows the pulse shape to be
alternated periodically throughout the experiment (typically every 2 minutes). This
allowed drift biases in the laser or the ion beam pointing stability to be compensated
3.13 Pulse Shaper
97
Figure 3.20: Schematic illustration of shaping the temporal profile of an ultrashort
laser pulse by retardation of the spectrally dispersed individual wavelength components
in a phase only LC-SLM.
for.
3.13.1
Determining the Central Pixel
00
000
The quadratic (linear chirp, φ ) and cubic (TOD, φ ) spectral phase functions used to
produce the shaped pulses described in this thesis can be expressed by certain terms of
the Taylor expansion given in equation 3.17 and is discussed in detail in section 2.12.
1 00
1 000
0
φ(ω) = φ(ω0 ) + φ (ω0 )(ω − ω0 ) + φ (ω0 )(ω − ω0 )2 + φ (ω0 )(ω − ω0 )3 + . . .
2
6
(3.17)
Equation 3.17 is incorporated into an algorithm in Labview and used to supply the
correct voltage to each pixel in the 640 LC-SLM arrangement. This ensures that the
induced changes in the refractive index of the pixel arrangement retards the spectrally
dispersed individual wavelength components of the pulse with respect to each other
according to the applied spectral phase function (as illustrated schematically in figure
3.20 (b)). The spectral phase function is applied with respect to some central pixel
of the LC-SLM, and it is imperative that the central wavelength of the pulse matches
3.13 Pulse Shaper
98
Figure 3.21: Calibration showing the pixel number to wavelength correlation.
this pixel. This is particularly important when applying asymmetric spectral phase
functions so the frequency components are temporally redistributed around the central
frequency of the pulse, and not an arbitrary one.
The central pixel is defined as the one through which the central wavelength of the pulse
propagates. This value can be experimentally determined by applying a pi gate (delta
function) to a known pixel. The transmitted spectrum (wavelength and intensity) was
measured using a highly sensitive spectrometer (ANDO) at the exit of the shaper.
All wavelength components of the pulse traverse the LC-SLM unaffected, with the
exception of the pi gated pixel. This component is highly dispersed and a clear dip
in the spectrum can be observed. This procedure was repeated for several different
pixels, and a linear relation with a gradient of 0.34 nm (corresponding to the width
of each pixel δλ nm) was obtained (see figure 3.21). The central wavelength of our
spectrum was measured as ∼795 nm, which from figure 3.21, implies the central pixel
should be set to ∼405, for this particular alignment.
To quantify the sensitivity of this effect consider an asymmetric function applied 10
pixels offset from the central wavelength. This corresponds to the phase function
3.14 Laser Pulse Characterization (GRENOUILLE)
99
being ∼3.4 nm off center. Experimental data illustrating the effect an off-centered
TOD phase function has on the dissociation of H+
2 can be seen in section REF.
3.14
Laser Pulse Characterization (GRENOUILLE)
An inherent problem with measuring femtosecond pulses is that to measure an event in
time, a shorter event is required to measure it. To combat this problem autocorrelation
and frequency resolved optical gating (FROG) techniques use the pulse to gate itself.
The full time-dependent intensity and phase of the shaped pulses can be retrieved using a GRENOUILLE (grating-eliminated no-nonsense observation of ultrafast incident
laser light electric fields) a schematic diagram of which is shown in figure 3.22 [97].
Figure 3.22: (a) Schematic diagram of the GRENOUILLE components and configuration. (b) Schematic diagram illustrating the two pulse replicas (different colors) being
focused and crossed into the SH crystal by the Fresnel biprism. (c) Crossing the beams
at an angle maps delay onto transverse position.
The input pulse is split into two replicas and focused at a mutual angle (crossed) into
a thick second harmonic (SH) crystal by a Fresnel biprism (a prism with an apex angle
close to 180, see figure 3.22(a)). This causes the light at the center of the laser beam
3.15 Parabolic Mirror and Alignment and Imaging
100
to undergo a greater retardation than the light at the edges (see figure 3.22 (b)). This
spatiotemporal overlap creates a relative delay τ between the input E(t) and gated
pulse E(t- τ ), and is mapped in the x-direction. The second harmonic signal generated
by the overlapped components of the electric field (2E(t)E(t-τ )) exit the crystal on
axis and are measured as a function of delay. The signal can be described by following
equation,
Z
2
iωt
S(τ, ω)α E(t)G(t − τ )e dt
(3.18)
Since the crystal has a relatively small phase-matching bandwidth, the phase matched
wavelength produced by the crystal varies with angle and can be used as a spectrometer. The cylindrical lens positioned after the SHG crystal maps the position of each
wavelength to a location on the y-axis of the CCD camera. The final trace (i.e., spectrogram) is a 2D map of pulse intensity as a function of delay time and frequency. The
advantage of GRENOUILLE devices is that they are compact, affordable and contain
zero degrees of alignment.
3.15
Parabolic Mirror and Alignment and Imaging
The laser beam must be focused within the interaction region to achieve intensities high
enough to induce dissociation. This was done using a 20 cm off-axis parabolic mirror
as opposed to a conventional glass lens to avert the onset of any further distortions
due to chromatic abberations.
Alignment of the off-axis parabolic mirror was initially conducted using a HeNe laser
to accurately mark the relevant reference points. The laser beam is aligned through the
center of the first two irises (to set the height and tilt of the beam) onto the center of the
dielectric mirror (see figure 3.23). It is crucial that the incident laser beam is parallel
to the table and centered on the off-axis parabolic mirror. The laser beam is then
reflected 90◦ through the center of the AR coated entrance window (the abberations of
which have been recompensed using the GRENOUILLE see section 3.14) and focused
amid two spectrometer electrodes. The beam then exits the interaction chamber and
should terminate (concentric with a closed iris) on a pre-aligned (using HeNe) cross
3.15 Parabolic Mirror and Alignment and Imaging
101
Figure 3.23: (a) Schematic diagram showing the laser being focused into the interaction
region. The dashed red line is where the laser beam was diverted into the CCD for
imaging purposes. (b) The focusing dimensions of the laser beam corresponding to
certain translations of the off-axis parabolic mirror.
on the wall.
However, unlike a convex lens, the parabolic mirror could potentially introduce astigmatisms if not aligned properly. A glass plate inserted before the chamber entrance
window was used to deflect an attenuated beam (ND filters inserted as shown in figure
3.23) into a CCD camera to image the focus. A high neutral density filter (ND = 8)
was needed to protect the camera when locating the focus and creates an overall higher
sensitivity to the beam profile. To achieve a non-astigmatic beam, the parabolic mirror
can be adjusted in the θ and φ directions to correct profiles which are tilted or appear
horizontal/vertical, respectively. However, such modifications can cause undesirable
changes to the trajectory of the laser beam through the interaction region. To combat
this issue the position of the focal image should be kept at a constant position on the
3.16 Z-Scan
102
Figure 3.24: (a) The area of the focused laser beam at various z positions obtained
from fitting an ellipse (red line) or a contour (blue line) to the CCD images. (b) The
corresponding peak intensity for these z positions and fit type.
screen (thus essentially implying constant directionality). In practice this is accomplished by focusing using θ and φ in the appropriate directions according to figure 3.23
(b) and then using x and y manipulators to bring the image back to the same position
on the screen. After each iteration the new focus must be found and the procedure
repeated until the best focusing conditions are reached.
Images of the beam profile are obtained for a range of z positions (where z= 0 is the
focus) and an ellipse and contour are fitted around each profile to determine the area.
The area is then used to provide an estimate of the intensity of the laser across the
various z positions (see figure 3.24).
3.16
Z-Scan
The profile of a focused laser pulse varies both spatially and in intensity along the axis
of propagation (z-axis) depending on the strength of the focusing lens. The spatial
contour defining each specific intensity region In can be described as a ‘peanut’. The
volume vn occupied by In is vn minus the volume of all lesser intensities VP i , as
shown schematically in figure 3.25 (a). As In increases, the volume occupied by the
3.16 Z-Scan
103
Figure 3.25: (a) Profile of a focused laser pulse showing the isointensity contours
bounded by the saturation intensity of ion species A+ , A2+ etc. The slit (ion beam)
therefore probes a particular intensity and volume region. Figure adapted from [62].
(b) The measured dissociation rate of H+
2 as a function of laser position.
In contour decreases and tends towards a finite point as I tends to Io. Therefore, at
a given position along the focal axis, the laser- ion interactions are unique to that z
coordinate.
The finite ion beam can effectively be used as an aperture to restrict the interaction
region to a cylindrically symmetric thin slice of width ∆z (see figure 3.25 (a)). The
laser focus can then then be translated with respect to this aperture enabling ion
interactions unique to specific z positions to be probed. The signal measured at each z
increment is a compromise between the volume of interaction and range of intensities
exposed. This gives rise to the z-scan shown in figure 3.25 (b). In practice, this z-scan
is carried out to establish the position of the focus and subsequently determine the
z-position of the experimental overlap region to achieve a particular peak intensity.
Although more than a single intensity is probed at each z increment, this method
allows the lower intensity regions to be appropriately masked from the detector for
high intensity measurements. Typically, the pulse shaping experiments were carried
out at a 19 mm distance from the laser focus. This region is still within the Rayleigh
radius of the laser where the volume is still less than double the beam waist.
Chapter 4
Photodissociation of Hydrogenic
Ions Using Shaped Laser Pulses
The long-outstanding ambition of scientists to understand and optically drive chemical
reactions to the highest degree of specificity has lead to extraordinary advancements in
coherent control strategies. Ultimately, the extent to which molecules can be controlled
is governed by how precisely the electric field can be manipulated. The development
of generic algorithms has proven a useful technique in finding the specific pulse shape
required to optimize a specific chemical process. The complexity of the electric fields
created however, makes extracting a mechanistic explanation a formidable task. This
chapter presents a systematic study into the interaction of the theoretically tractable
H+
2 and well characterized pulses which can be described analytically. Specifically,
00
Fourier transform limited pulses are shaped via the application of quadratic (ϕ ) and
000
cubic (ϕ ) spectral phase functions described by the corresponding terms of the Taylor
expansion given in equation 2.12. The magnitude and sign of these parameters are
varied and the ability to use them as a tool to control the photodissociation of H+
2 is
demonstrated and the underlying control mechanisms are identified and supported by
theory.
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
4.1
105
Photodissociation of Hydrogenic Ions Using Shaped
Laser Pulses
4.1.1
Motivation
Momentum imaging techniques can be eloquently used to study how a particular pulse
shape influences the fragmentation of a molecule. One form of pulse shaping involves
redistributing the frequency components in time according to the group delay imposed
by an applied spectral phase function. In turn, the modified temporal profile can
be obtained from the inverse Fourier transform (see section 2.3.1). With only two
energetically available potential energy curves at 800 nm, H+
2 is highly theoretically
accessible. The interaction of H+
2 and analytically shaped pulses therefore plays a
critical role in gaining an insight into how the instantaneous frequencies and temporal
profile of shaped pulses can be used to control dissociation dynamics. Furthermore,
00
000
experimentally the quadratic (ϕ ) and cubic (ϕ ) spectral phase functions used are
accurately applied and the shaped pulses well characterized.
Polarization-Gate Frog
Various geometries of the frequency resolved optical gating (FROG) technique can be
used to fully characterize ultrafast pulses. However, the retrieved FROG traces from
each variant contain non-trivial differences [98, 99, 73]. The Polarization-Gate (PG)
technique is the most intuitive method as the traces do not contain a direction of time
ambiguity. Therefore the sign of the spectral phase function can be clearly determined.
For this reason PG FROG traces have been simulated to elucidate the time-frequency
relationship for the different pulse shapes used in an unequivocal manner. A measured
spectrum of the pulse is incorporated and the phase function is applied with the same
resolution as the pulse shaper in the experiment.
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
106
Figure 4.1: (a) Simulated Polarization-Gating FROG traces showing a 33 fs FTL pulse
with a 40 nm spectral bandwidth. (b) The corresponding Gaussian temporal profile is
normalized to 1 (black line).
4.1.2
Fourier Transform Limited Pulses
The shortest, most intense laser pulses a particular laser system can produce are termed
Fourier Transform Limited (FTL). The shorter the pulse duration, the larger the spectral width of the pulse. This relationship is given by the time-bandwidth product
which can be described in terms of the frequencies of the pulse [74, 73] and is given in
equation 4.1 below for an ideal Gaussian pulse:
∆t∆ν =
2ln2
= 0.441
π
(4.1)
Where ∆ν is the spectral frequency width and ∆t is the temporal duration of the pulse.
In an FTL pulse the spectral phase ϕ(ω) is constant and each frequency component
ω(t) has an equal probability during the pulse duration (see the PG FROG trace in
figure 4.1 (a)). The temporal intensity profile is Gaussian and defined by the full width
half maximum (FWHM) of the intensity distribution (see figure 4.1(b)). The addition
of any non-linear phase terms results in the inequality ∆t∆ν ≥ 0.441 which stretches
the pulse in the time domain and acts to reduce the peak intensity.
Before the behavior of H+
2 in a shaped electric-field can be understood, its response
to these simple FTL pulses must be explicated. In the three-dimensional momentum
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
107
Figure 4.2: The KER-cosθ density plots for FTL pulses with (a) I0 ∼ 1×1012 Wcm−2
and 18 nm bandwidth measured off-focus. (b) I0 ∼ 1×1014 Wcm−2 and 40 nm bandwidth measured off-focus. (c) I0 ∼ 3×1014 Wcm−2 and 40 nm bandwidth measured
on-focus. (d)-(f) Projection of the corresponding KER (|cosθ| > 0.9) spectra.
imaging technique employed for the purpose of these experiments the time-of-flight
(TOF) and impact position of the photofragments from each laser pulse were measured
in coincidence. Thus the full 3D momentum components of both fragments can be
retrieved. Subsequently, the KER of the dissociation process as well as the angle θ
between the dissociation velocity and the laser polarization were determined. These
experiments are therefore very sensitive to the precise experimental conditions.
The factors which are known to influence the measured kinetic energy release (KER)
spectra are a convolution of the vibrational and rotational population of the target
molecular ions, the vibrational levels which can be accessed by the bandwidth of the
laser, the resolution of the detector, the laser beam focusing and the interaction volume.
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
108
Figure 4.3: Angular distribution for the dissociation events of H+
2 for fragments with
KER 0.54 eV and 0.82 eV (blue and red data points respectively). The corresponding fitted functions (dashed line) demonstrate a cos2 and cos8 distribution indicating
dissociation via a near-resonant transition or bond softening, respectively.
Dissociation of H+
2 Induced by FTL Pulses
The angular distributions and kinetic energy release (KER) spectra for the dissociation of H+
2 using FTL pulses of different bandwidths and intensities are presented in
figure 4.2. Each spectrum can be categorized into two different regimes depending
on the KER and angular distribution of the fragments. Since perturbative transitions
demonstrate a linear intensity dependence and are not obligated to geometric or dynamic alignment, a broad cos2 θ angular distribution signifies their occurrence. This
is illustrated by the cos2 θ fit to the angular distribution of the dissociated fragments
with KER = 0.82 eV in figure 4.3.
The bandwidth of the laser defines the frequency range of the pulse and the vibrational levels that can be accessed by a particular h̄ω transition. The condon point is
defined as the internuclear separation where the energy spacing between two electronic
states is equivalent to the incident photon energy. For H+
2 this occurs at Rc =4.8 a.u.
corresponding to a vibrational level v=9 for a pulse centered at 795 nm. The cross
section for the one photon transition is maximal at this location, and these results are
compliant with previous studies [66]. The main KER peak therefore appears around
0.82 eV for an FTL pulse in figure 4.2, as expected.
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
109
The contributions from the narrow angular distribution from vibrational levels v≤7
with KER <0.61 eV have no overlap with the laser bandwidth and dissociate via bond
softening (see section 1.4.3). The cos8 θ fit to the angular distribution for fragments
with KER = 0.54 eV verifies that these are multiphoton processes (red dashed line in
figure 4.3). The dissociation rate of these lower vibrational levels are strongly enhanced
with increasing intensity (also shorter pulses) as the potential barrier is suppressed
further.
Changing the Bandwidth of the Pulse
The spectral bandwidth of the laser can be manipulated by placing an amplitude mask
in the Fourier plane of the shaper. This reduces the spectral content of the pulse with
respect to the 795 nm central frequency, see section 3.13 for details. This procedure
was used to transform 33 fs, ∆λF W HM = 40 nm bandwidth (∆E∼77 meV) FTL
pulses of peak intensity I0 ∼ 8×1013 Wcm−2 into 120 fs ∆λF W HM = 18 nm (∆E∼36
meV) ‘narrowband’ pulses of I0 ∼ 5×1012 Wcm−2 . Their effect on H+
2 dissociation is
discussed below.
Narrow Spectral Bandwidth (18 nm)
The contribution from bond-softening events for ‘narrowband’ ∆λF W HM =18 nm (∆E∼36
meV) pulses is minimal as I0 is substantially lower (see figure 4.2 (a)). The most salient
feature however, is the apparent enhancement in energy resolution compared to the
pulses with a greater spectral bandwidth. Since the emanating photons across the full
spectral bandwidth of an FTL pulse are distributed with equal probability (see PG
FROG trace figure 4.1 (a)), the vibrational resolution can be obscured by their energy
range. It is therefore expected that vibrational structure becomes indiscernible with
increasing bandwidth (shorter pulses) and increasing laser intensity (i.e., field strength)
[100]. These differences are compliant with previously published KER spectra which
also demonstrate a higher energy resolution involving FTL pulses of longer temporal
duration, predominantly due to a smaller spectral bandwidth [72, 101] .
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
110
Full Spectral Bandwidth (40 nm)
The bandwidth ∆λF W HM =40 nm (∆E∼77 meV) of the original FTL pulses is smaller
than the vibrational energy spacing of H+
2 . However, the vibrational structure (see
figures 4.2 (e) and (f)) is no longer as clearly resolved due to the greater energy spread
in the bandwidth of the laser. Intensity inhomogeneity of the laser pulses as a function
of spatial position is inevitable, and the observed spectrum is hence the sum of the
physical phenomena occurring at different intensities ranging from zero up to the peak
intensity at the focus. This is commonly referred to as intensity averaging and becomes
more pronounced for increased intensity and its effect is largely dependent on the zposition (3.16). Figure 4.2 (b) - (f) demonstrate the differences in the KER spectra
for the same FTL pulses but probed 19 mm off-focus at I0 ∼ 5×1013 Wcm−2 and
I0 ∼ 8×1013 Wcm−2 on-focus (see figure 3.15). In both cases, the v=9 contribution
arises from the larger volume, weaker intensity confocal shells of the pulse (see figure
3.25) which have a larger overlap with the ion beam. The contribution of low KER
fragments related with strong field phenomena is substantially higher and extends to
lower vibrational levels for an increased intensity, as expected. It should however be
noted that the lowest KER measurable is limited to 0.1 eV due to the small Faraday
cup collecting the on-axisundissociated ion beam.
4.1.3
+
+
Molecular structure of H+
2 , HD and D2
+
+
The isotopes H+
2 , HD and D2 share the same Born-Oppenheimer potential energy
curves and dipole matrix elements. However, the vibrational energy spacing for the
heavier molecules are smaller, as the vibrational frequency depends inversely on the
reduced mass. The vibrational states are populated according to the Franck-Condon
distribution. This is determined by the ground vibrational states of their respective
neutral counterparts. Essentially, the vibrational state distribution as a function of the
vibrational energy is the same. The KER spectra for these isotopes is therefore similar
in shape, but the peak positions change with the respective vibrational spectra. When
dressed with a 795 nm laser, the condon point for each isotope will be associated with
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
111
Figure 4.4: Schematic diagram illustrating the Gaussian fit to the measured power
00
spectrum I(ω) normalized to 1. The applied quadratic phase function ϕ(ω) = 500 fs2
(red dash) results in the frequencies being redistributed in time according to a linear
group delay Tg (ω) (blue dashed), which subsequently leads to a frequency sweep within
the pulse. Figure adapted from [100].
+
+
a different vibrational level (H+
2 ; v=9 τ =29 fs, HD ; v=11 τ =36 fs, D2 ; v=13 τ =41
fs).
Any results presented in this chapter comparing two or more isotopes was obtained
by alternating the pulse shapes (see section 3.13) and the ion beam (see section 3.3)
periodically. This kept experimental conditions such as the ion source temperature
(vibrational population), ion beam current, interaction volume, laser focusing, peak
intensity, pulse shape and any long term laser drifts consistent. Otherwise, these
parameters can fluctuate significantly between measurements.
4.1.4
Linear Chirp
A chirped pulse contains a linear increase in frequency with time (frequency sweep).
00
00
This is achieved by applying a quadratic spectral phase function ϕ(ω) = 12 ϕ (ω0 ) · (ω − ω0 )2
to a pulse of central frequency ω0 . The chirp rate is defined by the group dispersion
00
delay (GDD) parameter ϕ . The frequency components of the pulse are temporally
redistributed according to a linear group delay. This concept is illustrated in figure
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
112
Figure 4.5: Simulated Polarization-Gating FROG traces of a 33 fs FTL pulse shaped
00
00
with (a) φ = -1050 fs2 and (b) φ = 1050 fs2 . Figures (c) and (d) show the temporal
profile of the 33 fs FTL pulse (black line) normalized to 1 and the relative temporal
00
00
profiles of the chirped pulse (red dashed line) obtained for φ = -1050 fs2 and φ = 1050
fs2 respectively.
4.4 where the blue dot-dashed line indicates that the higher the frequency component,
the greater the Tg (ω) acquired, for positive chirp.
The PG FROG traces shown in figure 4.5 (a) and (b) can be used to elucidate the
frequency-time relation of the redistributed frequencies of the chirped pulses. Heuristically speaking, we can envisage this as the ‘red’ detuned frequencies of the pulse
00
leading the ‘blue’ for a positive chirp (+φ , see figure 4.5 (a)) and the time reversal of
00
this for a negative chirp (-φ , see figure 4.5 (b)).
The temporal profile of a chirped pulse is Gaussian, as indicated by the red dashed
line in figure 4.5 (c) and (d). The temporal duration τ is elongated compared to the
FTL and the peak intensity reduced. The final temporal duration ∆τout for an FTL
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
GDD [fs2 ]
0
780
1040
1300
∆RM S [fs]
33
78
101
125
113
Relative Intensity
1
0.38
0.28
0.22
Table 4.1: Calculated relative peak intensity and pulse duration for the various GDD
pulses used in the experiment.
00
pulse of duration ∆τ , chirped by a magnitude φ can be calculated from the formula
below [74]:
s
∆τout =
00 2
φ
∆τ 2 + 4ln2
∆τ
(4.2)
00
For the range of GDD used φ(ω) =± 1300 fs2 , the 33 fs FTL pulses were elongated
to a maximal duration of 125fs. The fluence and bandwidth of the pulses were kept
fixed, but the peak intensity decreased from ∼8×1014 Wcm−2 to ∼3×1012 Wcm−2 for
the highest GDD value used, see table 4.1.4.
The instantaneous frequency ω(t) of linearly chirped pulses has been used as a tool to
control different dissociation dynamics (see review for details [102]). To name a few
examples, it has been shown that the directionality of the chirp can be used to manipulate population transfer for transitions which are closely spaced within the chirped
pulse bandwidth [103]. The ability to enhance a particular dissociation product by
shaping the nuclear wave packet using chirped pulses was demonstrated by Pastirk
et al [104]. In addition to this, control of the Landau-Zener (LZ) transitions in NaI
predissociation using chirped pulses was studied theoretically. It revealed an enhancement or suppression in the transitions between the excited and the ground state for
chirped pulses [105]. The group at WIS recently reported that the KER spectra for
the higher vibrational levels of H+
2 v≥7 close to the Condon point can be manipulated
using chirped pulses [106, 67]. The near-resonant transitions for the vibrational levels
v≥7 of H+
2 were shown to dissociate well before the peak of the pulse and are therefore driven by the instantaneous frequency of the pulse E(ω) [67]. This observation
manifests itself as shifts in the positions of the KER peak positions to higher or lower
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
114
KER values for negative and positive chirp respectively. It should be noted however
that many of the observed effects depend on the intensity, chirp rate and frequency of
the laser.
Figure 4.6: The cos θ vs KER density plots of the dissociation of HD+ for FTL pulses
shaped with (a) -1300 fs2 and (b) 1300 fs2 creating 125 fs pulses with I0 ∼3×1013
Wcm−2 . The black dotted lines represent the expected field-free vibrational levels of
HD+ .
00
The density plots of the HD+ dissociation signal for φ(ω) = ±1300 fs2 pulses as a
function of cos θ and KER are shown in figure 4.6, and the differences are striking. A
significant increase in the contribution from the low lying vibrational states is observed
for positive GDD. The narrow distributions can be expressed as higher cosine powers
which suggest geometric or dynamic alignment has taken place. To quantify the dissociation probabilities a yield analysis is presented in figure 4.7. The signal along the
axis of the laser polarization in a cone with an angle of 25◦ (|cos θ| > 0.9) from figure
4.6 was integrated over certain KER ranges and normalized to the FTL pulse. The
results show the relative photodissociation yield for the one-photon transitions located
close to the Condon point with KER ≥ 0.75 eV. This corresponds to vibrational levels
+
v≥9 for H+
2 and v≥11 for HD and the yield proves independent of pulse shape. This
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
115
Figure 4.7: Dissociation probabilities comparing H+
2 and the two dissociation channels
of HD+ for various GDD pulses normalized to the yield of the FTL of peak intensity
5 ×1014 Wcm−2 pulse for (a) One-photon transitions (KER ≥ 0.75 eV) and (b) bondsoftening driven dissociation (KER ≤ 0.41 eV).
is expected as it is a one-photon absorption process and does not require any intricate
dissociation mechanism.
A maximal 20% enhancement in the dissociation rate for low lying vibrational levels
(v≥8) of H+
2 with KER ≤ 0.41 eV was recently reported for positively chirped pulses
[107]. These observations were attributed to the manipulation of the avoided crossing
using the instantaneous frequency of the pulse E(ω) [107]. The Condon point, which
defines the position of the avoided crossing, was found to be dependent on E(ω). The
frequency sweep of a chirped pulse can therefore be used to displace the internuclear
distance at which the avoided crossing is induced RAC , and the dynamics vary according to the chirp rate. The ‘red’ detuned frequencies will shift RAC to larger values
with a higher potential barrier. As a consequence the dissociation rate relative to the
FTL pulse is suppressed. For the ‘blue’ frequencies RAC is displaced in the opposite
direction and experiences a shift toward smaller RAC . This results in an increased
reduction of the potential barrier and thus promotes bond-softening. This concept is
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
116
Figure 4.8: Modified light induced potential energy curves of H+
2 (black solid line)
illustrating the initiation of an avoided crossing which allows vibrational levels 7 (dotted orange) to dissociate via bond softening. The position (internuclear separation R)
of the crossing is defined by the dressed diabatic potential curves (grey). Inset: For
positively chirped pulses, the adiabatic potential curves are dynamically modified from
red (dashed red) to blue (solid blue) shifted potential curves, according to the direction
of the frequency sweep. A larger gap size for blue detuned frequencies develops as the
temporal alignment evolves close to peak intensity. Figure adapted from [107].
illustrated pictorially in the inset of figure 4.8. Furthermore, the gap at the avoided
crossing is opened wider for higher intensities and if the light-molecule interaction is
adiabatic, the molecule can align along the laser polarisation direction.
This study was extended to explore any differences in the dissociation rate of H+
2 and
its isotopic variant HD+ using chirped pulses. The main discerning feature between
these two molecular ions is the permanent dipole moment of HD+ . This means that
+
the conservation of parity which applies to H+
2 can be violated in HD . Two-photon
dissociation is therefore possible and has been observed at intensities around 5.0 × 1012
W cm−2 . As the intensity is increased to 1.5 × 1015 W cm−2 four-photon absorption
begins to dominate [108]. Furthermore, the nuclear mass correction to the BornOppenheimer approximation means that the 1sσ state lies 3.7 meV below the 2pσ state
at the dissociation limit. The ability to use a chirped pulse to control the difference
between these two channels would allow realization of chemical control. The H+
2 and
the HD+ ion beams and the pulse shapes were alternated periodically every few minutes
(see section 4.1.3) allowing a direct comparison of the isotopes to be accurately made
and thus any long term drifts compensated for.
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
117
+
Figure 4.7 shows that the dissociation rate for the H+
2 and HD fragments with KER
≥ 0.41 eV is enhanced by ∼20%. This is in agreement with that previously reported
for H+
2 [107] and demonstrates the ability to manipulate the dissociation of low lying
vibrational levels for HD+ using linearly chirped pulses. However, no difference in the
+
relative dissociation yield between H+
2 and HD (or either of its two dissociation chan-
nels) was observed for chirped pulses in this range. This supports the idea [108] that
permanent dipole transitions play an insignificant role in the strong-field dissociation
of HD+ .
Recently, HD+ has attracted a lot of attention as a benchmark molecule to explore
the role of a permanent electric dipole moment of heteronuclear molecules and their
dissociation dynamics. Although no difference in the branching ratio of the H & D+
and D & H+ dissociation channels was observed in this experiment, the group of Prof.
Ben-Itzhak at KSU have observed slight differences due to better statistics [109]. Owing
to the sensitivity of these experiments, the higher peak intensities, laser bandwidth or
even the chirp parameters studied could be contributing factors for this discrepancy.
Furthermore, the effects of a weak third pulse and the carrier envelope phase of the
dissociating pulse in the asymmetry in the branching of dissociated fragments of HD+
has recently been reported and several control schemes for above threshold dissociation
in HD+ .
4.1.5
Third Order Dispersion
An increase in the number of molecules dissociating via the bond softening mechanism
was observed for positively chirped pulses (see section 4.1.3). Furthermore, laser induced alignment was considered to be an important aspect of these scientific findings.
Alignment in a strong laser field can be achieved through various mechanisms depending on the nature of the electric field. If the temporal duration of the pulse is longer
than the rotational period of the molecule, a pendular state which liberates around
the polarization vector is created and adiabatic alignment occurs. However, once the
field is switched off, the molecule will return to its isotropic state. Alternatively, if
a molecule encounters a laser pulse whose duration is less than the rotational period
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
118
Figure 4.9: Schematic diagram illustrating the Gaussian fit to the measured power
000
spectrum I(ω) normalized to 1. The applied cubic spectral phase function φ = 40500
fs3 (red dash) results in the frequencies being redistributed in time according to a
quadratic group delay Tg (ω) (blue dashed). Figure adapted from [100].
of the molecule, the momentum imparted acts to ‘kick’ the molecular axis toward the
laser field vector. This process is known as non-adiabatic alignment and the system
exhibits field-free, post-pulse alignment and rotational revivals [10, 51]. It follows that
the combination of these two effects, in the form of a temporally asymmetric pulse
where the electric field is turned on slowly and switched off quickly could potentially
be used as a tool to enhance molecular alignment.
TOD Pulses
Recall from section 2.3.1 that the fourth term of the Taylor expansion of the spectral
phase function, which is commonly referred to as third order dispersion (TOD), is cubic
000
000
φ(ω) = 16 φ (ω0 ) · (ω − ω0 )3 . This leads to the frequencies being redistributed in time
according to a quadratic group delay. This concept is illustrated in figure 4.9 where the
000
φ(ω)
is applied with respect to the central frequency, ω0 of an FTL pulse. The blue
dot-dashed line demonstrates that whilst the central frequencies experience no group
delay, the high and low frequencies of the pulse acquire a greater delay. The PG FROG
traces shown in figure 4.10 (a) and (b) can be used to elucidate the frequency-time
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
119
Figure 4.10: Simulated Polarization-Gating FROG traces of a 33 fs FTL pulse shaped
000
000
with (a) φ = -40500 fs3 and (b) φ = 40500 fs3 . Figures (c) and (d) show the temporal
profile of the 33 fs FTL pulse (black line) normalized to 1 and the relative TOD
000
000
temporal profiles (red dashed line) obtained for φ = -40500 fs3 and φ = 40500 fs3
respectively.
relation of the redistributed frequencies of the TOD pulses. Heuristically speaking,
000
for negative TOD (-φ ) the ‘red’ and ‘blue’ detuned (high and low) frequencies from
the edge of the spectrum are shifted towards the leading edge of the pulse (see figure
000
4.10 (a)). For positive TOD (+φ ) the time order of this process is reversed and
these frequencies are shifted towards the trailing edge of the pulse, so that the central
frequencies arrive first. This effectively creates a narrower FTL bandwidth on the
rising edge of the pulse (see figure 4.10 (b)). In both cases the detuned frequencies
arrive simultaneously and the interference between them leads to beating in the time
000
000
domain. This creates a sequence of pre (-φ ) or post (+φ ) pulses depending on the
TOD sign (see figure 4.10 (c) and (d)).
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
TOD [fs3 ]
0
13500
27000
40500
∆RM S [fs]
33
54
97
142
120
Relative Intensity
1
0.71
0.48
0.39
Table 4.2: Calculated relative peak intensity and pulse duration for the various TOD
pulses used in the experiment.
In the experiment, the fluence and bandwidth of the pulses was kept fixed, but the
peak intensity decreased from about 8×1014 W cm−2 for the FTL pulse to 3×1014
W cm−2 for the highest TOD value used. In the latter case, the amount of energy
redeployed to the post or pre pulses is increased, and can contain up to half of the
overall pulse energy. The highly asymmetric temporal profile of TOD pulses implies
that the full width at half maximum (FWHM) is not the most meaningful quantity to
characterize the pulse duration. Instead, we use a statistically defined width given in
equation 4.3 [74]:
s
2σ =
∆τ 2
+ 8(ln2)2
2ln2
φ000
∆τ 2
2
,
(4.3)
Where ∆τ is the FWHM of the Gaussian FTL pulse before the spectral phase φ(ω)
000
is applied. According to this definition, the 33 fs FTL pulses were elongated to about
50-140 fs for the range of TOD values used, see table 4.1.5.
Near-Resonant Transitions
The two-dimensional (2D) density plots of H+
2 dissociation as a function of KER and
cos θ for pulses shaped with TOD in the range φ
000
= ± 40500 fs3 are shown in figure
4.11 (a)-(g). The most prominent feature is the significant differences observed in
the structure of the v=9 peak. Its evolution can be traced by following the field-free
vibrational line (black dashed) from positive to negative TOD values. The initially well
resolved peak for large positive TOD values broadens and then splits into two distinct
peaks. These changes are more clearly illustrated in figure 4.12, which shows the KER
spectrum (for figure 4.11 (a), (d) and(g)) integrated along the laser polarization in a
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
121
Figure 4.11: The 2D KER-cosθ density plots of H+
2 dissociation induced by TOD
000
3
3
pulses in the range φ = ± 40500 fs (with 13500 fs increments) as marked on each
individual panel. The black dashed lines signify the expected peak position for the
field-free vibrational levels at 795 nm.
cone with an angle of 25◦ (|cos θ| > 0.9).
The KER peak for the FTL pulse appears as expected around 0.82 eV for a central
wavelength of 795 nm (see figure 4.12 (b)) but the energy resolution deteriorates because of the 40 nm spectral bandwidth of the pulse (see section 4.1.2). Recall the
argument in section 4.1.2 implying that an enhanced resolution can be obtained as a
consequence a smaller spectral bandwidth. In figure 4.11 a narrower v=9 peak is observed for increasingly positive TOD despite maintaining a constant bandwidth. This
feature and the peak splitting induced by negative TOD can be explained heuristically by considering the temporal behavior of H+
2 dissociation within the pulse and the
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
122
limited range of spectral components of the pulse within this time.
Time-dependent dissociation probabilities for the v=7–9 vibrational levels of H+
2 were
calculated by the group of Prof. Esry at Kansas State University by solving the timedependent Schrödinger equation (TDSE) numerically. The method used is described in
detail elsewhere [66] and assumes that the nuclei have insufficient time to rotate during
the pulse and that any rotation after the pulse is negligible. The internuclear axis was
kept fixed along the laser polarization allowing nuclear vibration on the coupled 1sσg
and 2pσu channels. The calculations were performed for a 30 fs pulse linearly chirped
(both in the positive and negative direction) to 120 fs with a peak intensity of 2× 1013
Wcm−2 and the results are shown in figure 4.13. In reality, where rotations play a role,
the dissociation probability will saturate at a lower intensity. However, these relatively
simple calculations are used to demonstrate that the v=7-9 states are depleted before
the peak of the pulse. The exact time window where saturation occurs is not critical
for our qualitative model explaining the difference in the structure of the v=9 peak.
This early saturation implies that a limited range of frequencies, situated on the rising
edge of the pulse are responsible for dissociation. The enhanced energy resolution
observed for positive TOD is a consequence of the effective ‘narrow bandwidth’ FTL
pulse created on the leading edge. Alternatively, the combination of ‘red’ and ‘blue’
frequency components contained within the rising edge of negative TOD pulses cause
the molecules to dissociate at slightly lower or higher KER respectively. Thus the
v=9 peak is split into two distinct components. This effect is sensitive to specific
pulse parameters such as the bandwidth of the original FTL pulse and the TOD
magnitude applied. Since a lower magnitude of TOD will redistribute a smaller range
of frequencies around ω0 , a smaller split is observed, as observed in figure 4.11.
Non-Resonant Transitions
In order to quantify the dissociation probabilities the same yield analysis discussed in
section 4.1.3 was performed on the KER and cos θ density plots in figure 4.11. The
photodissociation yield for the one-photon transitions located close to the Condon
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
123
Figure 4.12: Kinetic energy release (KER) spectra (|cos θ| > 0.9) of H+
2 dissociation
000
000
000
for 33 fs FTL laser pulses shaped with (a) φ = 40500 fs3 (b) φ = 0 fs3 (c) φ = -40500
fs3 . Note the change in the width and the splitting of the v=9 peak for different TOD
parameters.
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
124
Figure 4.13: Calculated time-dependent dissociation probability for a few vibrational
levels of H+
2 , using a 30 fs FTL laser pulse positively and negatively chirped to 120
fs. Negative (dashed) and positive (solid) chirps and plotted for the peak intensity ∼2×1013 Wcm−2 with the intensity envelope depicted by the short dashed line.
Adapted from [100].
point and emerging from vibrational levels v≥9 were obtained from integrating those
counts along the laser polarization in a cone with an angle of 25◦ (|cos θ| > 0.9) with
KER ≥ 0.75 eV. The flat red line in figure 4.14 indicates that these transitions are
independent of pulse shape. This again is expected as they proceed via a one photon
absorption process and do not require any intricate dissociation mechanism.
The highly aligned contribution from the low lying vibrational levels v≤6 of H+
2 in the
KER region ≤ 0.41 eV dissociate via bond-softening and are represented by the black
curve in figure 4.14. It is evident that the TOD sign is of paramount importance to the
mechanism driving the dissociation. A remarkable 50% increase in dissociation yield
was measured for the optimum range of negative TOD ( -27000 to -13500 fs3 ) compared
to that of an FTL pulse. This enhancement can be explained in conjunction with the
temporal profile of a negative TOD pulse (see figure 4.10). Although the pre-pulses
are relatively low in intensity, they can contain up to half of the pulse energy for a
peak intensity of 4×1013 Wcm−2 . This is sufficient to induce some degree of alignment
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
125
Figure 4.14: Dissociation probabilites of H+
2 for various TOD pulses normalized to
the yield of the FTL of peak intensity 5 ×1014 Wcm−2 pulse for (a) One-photon
transitions (KER ≥ 0.75 eV) and (b) bond-softening driven dissociation (KER ≤ 0.41
eV). Imperceptible error bars are smaller than the symbols
+
of H+
2 and the dissociation is known to be more efficient for H2 aligned with the laser
polarization axis as this increases the dipole coupling.
Table 4.2 shows how certain pulse parameters vary when TOD of different magnitudes
are applied to pulses of fixed fluence. It is clear that increasing the TOD magnitude results in a greater temporal duration. Furthermore, the relative peak intensity
decreases from 1 to 0.39 for the highest TOD value used. The overturn in the dissociation yield in figure 4.14 can be explained by considering the two critical factors
for inducing dissociation of H+
2 . This is the compromise between the pulse duration
required to align the molecules, and a sufficiently high peak intensity to induce bondsoftening. The overturn in the dissociation yield in figure 4.14 (b) for more negative
TOD values therefore suggests the conditions for which molecular alignment becomes
the predominant mechanism. Contrary to this augmentation, a dramatic 35% reduction in dissociation yield relative to the FTL pulse is observed for positive TOD. This
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
126
can be attributed to the absence of pre-pulses and a lesser peak intensity than the
FTL pulses (see figure 4.10).
A recent study demonstrates theoretically an enhancement in the alignment of N2 for
TOD pulses [110], but theses calculations were limited to the rigid rotor regime. The
group of Prof. Brett Esry at KSU performed calculations to solve the time-dependent
Schrödinger equation (TDSE) for H+
2 in an intense laser field, replicating the experimental condititions. The method used is described elsewhere [111] and includes radial
dynamics (vibrations and in particular dissociation). The time-dependent electric field
for these calculations was obtained by adding TOD to the amplitude of the measured
power spectrum and performing a Fourier transform. Reproducing the experimental
results quantitively requires a non-trivial intensity averaging process and is not within
the scope of this work. Instead, the purpose of the calculations is to support the experimental interpretation of the data. Therefore, the theoretical results were obtained
by separately propagating all the J=0 bound vibrational states of the 1sσg channel of
H+
2 . It has been shown that the results of such calculations are independent of the
initial J state used [111].
The results presented in figure 4.15 (a) show the calculated time-dependent bound
state population dynamics of v=6 for H+
2 in an intense laser field for an FTL pulse
shaped with the optimum TOD magnitude φ
000
= 28800 fs3 but opposing signs. The
bound state exhibits fast (∼ 1 fs) oscillations which occur with the carrier frequency.
The slower oscillations which are locked to the envelope (or sub-pulses of the TOD
pulse). A small delay is observed between the slow bound-state population dynamics
and the laser field envelope. This delay depends on the pulse parameters, in particular
the intensity. The resulting dissociation yield is determined from one minus the bound
state population at the end of the pulse. It is clear that the depletion of the vibrational
level v=6 is more efficient for the φ
000
= -28800 fs3 pulse, which is congruent with
experimental findings. It is interesting to note that qualitatively the difference between
the final bound state population for the FTL and φ
the FTL and the φ
000
000
= 28800 fs3 is less than that of
= -28800 fs3 pulses, similar to that observed in the experiment.
To understand how the use of TOD accomplishes photodissociation control, the align-
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
127
Figure 4.15: Time-dependent calculations for various TOD parameters. The FTL
pulse has a peak intensity of 5×1013 Wcm−2 to match the experiment. It is clear
the increasing negative TOD leads to a greater pre-pulse alignment and leads to an
increase the dissociation for for vibrational level v=6 of H+
2.
ment dynamics have to be considered. The duration of the pre-pulses for negative
TOD are < 5 fs and the revival period of H+
2 is 560 fs. Thus the induced alignment is
highly non-adiabatic [112]. Although the separation time between the prepulses is not
tailored to ‘kick’ the molecule at the precise revival times [113], each successive pulse
acts to gradually increase the degree of alignment [114]. It follows that the degree of
alignment achieved for TOD may not be as effective as an accurately timed sequence
of pulses. On the other hand, TOD provides a coherent application tool which is
experimentally and theoretically accessible.
+
+
H+
2 and its Isotopic Variants HD and D2
To experimentally explore the concept that prepulses play a principal role in the underlying mechanism responsible for the enhancement observed in the dissociation rate,
+
+
the isotopic variants of H+
2 (HD and D2 ) were studied simultaneously as a function
of TOD (see figure 4.16). For this data set, it is known that the phase of the FTL pulse
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
128
Figure 4.16: Dissociation probabilities for fragments produced with KER ≤ 0.41 eV
+
+
corresponding to v= 6 of H+
2 (black), v= 8 of HD (red) and v= 9 of D2 (blue) for
various TOD pulses normalized to the yield of the FTL.
000
is not completely flat as it contains some residual TOD of the order of φ ∼ 5000 fs3 .
This creates a substantial uncertainty on the x-axis. In the experiment, the ion beam
and pulse shapes are alternated periodically (see section 4.1.3) allowing a direct comparison of the hydrogenic ions to be accurately made. The only discernible differences
in the dissociation yields occurs in the region of the optimal TOD values as found for
3
H+
2 (-27000 to -13500 fs ), where alignment was considered to be the dominant control
mechanism (see section 4.1.5).
The nuclei in heavier molecules move more slower than in lighter molecules. This
reduced nuclear motion of heavier molecules allows their evolution to be traced with
femtosecond pulses. A qualitative picture has been proposed to suggest that due to the
lower velocity of heavier molecules, they effectively experience a shorter pulse duration
[115, 116]. This therefore implies that heavier molecules are subject to a lesser degree
of laser-induced alignment for the same pulse. This in turn can lead to the molecule
experiencing a lesser peak intensity. It is these two contributing factors combined
which lead to the reduction in the dissociation yield for the heavier D+
2.
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
129
Figure 4.17: Time-dependent laser intensity calculations demonstrating the bound
+
state population and alignment for the vibrational level v=6 of H+
2 and v=9 of D2 for
000
000
TOD pulses shaped with φ = -28800 fs3 and φ = -26800 fs3 , respectively.
The time-dependent laser intensity calculations (described in section 4.1.5) for the
+
vibrational levels v=6 of H+
2 and v=9 of D2 which are located at roughly the same
vibrational energy were calculated and compared. Figure 4.17 shows that the alignment
and consequently the dissociation is predicted to be significantly lower for D+
2 , as is
experimentally observed. Although the TOD is slightly different this is not believed
to be the main cause of the effect. Despite the dissociation efficiency apparently
decreasing for more massive molecules, it was theoretically determined that TOD could
be used to enhance the alignment of N2 [110]. This supports the concept that TOD
can be used as a general application tool to increase the molecular alignment and
subsequently molecular dissociation. The effect of changing certain properties of the
original FTL pulse which is subsequently shaped with TOD was not explored and
could potentially change its efficiency as a tool to enhance bond-softening of heavier
molecules.
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
130
Figure 4.18: Dissociation probabilities of H+
2 for various TOD pulses normalized to the
yield of the FTL pulse of peak intensity 8 ×1014 Wcm−2 for (a) One-photon transitions
(KER ≥ 0.75 eV) and (b) bond-softening driven dissociation (KER ≤ 0.41 eV).
TOD at Higher Intensity
The TOD pulses must maintain a sufficient peak intensity to induce dissociation of the
low lying vibrational levels v≤6 of H+
2 . However, the efficiency of the TOD as a control
tool to enhance dissociation decreases relative to FTL pulses for higher intensity. The
relative dissociation yield presented in figures 4.14 and 4.18 were measured under the
same experimental conditions with the exception of the peak intensity, where 5 ×1014
Wcm−2 and 8 ×1014 W cm−2 was used respectively. As expected, the near-resonant
transitions are independent of TOD magnitude or peak intensity, as they are governed
by the pulse energy. The same shape of trend line was observed for the bond-softening
events for both intensities. However, the quantitative analysis demonstrates that the
augmentation in the dissociation yield dramatically decreased from 50% to 10%, for the
lower and higher intensities respectively. Recall from figure 4.2 that the dissociation
rate of the vibrational levels v≤6 can be increased for FTL pulses of higher intensity.
Since the relative dissociation yield is determined with respect to the yield of the FTL,
this suggests that intensity is the prevailing factor. Furthermore, this implies that
TOD is a more efficient tool for inducing dissociation in the lower intensity regime
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
131
Figure 4.19: Relative dissociation probabilities of H+
2 for various TOD pulses where
the spectral phase function applied is offset by 10 nm from the central frequency of the
laser. Each yield is normalized to that of the FTL pulse for (a) One-photon transitions
(KER ≥ 0.75 eV) and (b) bond-softening driven dissociation (KER ≤ 0.41 eV).
around 5×1013 W cm−2 .
Sensitivity of Wavelength-Pixel Correspondence
The experimental technique used to match the central wavelength of the laser to the
central pixel of the spatial light modulator (SLM) in the pulse shaper is discussed in
section 3.13.1. This procedure is particularly important when applying spectral phase
functions which are asymmetric, such as TOD, but not critical for GDD as the linear
group delay is monotonically increasing. The results shown in figure 4.14 demonstrate
the dissociation yield analysis for TOD pulses where the frequencies are temporally
redistributed according to a quadratic group delay around the central frequency, ω0
of the pulse (shown schematically in figure 4.9). However, if the spectral phase is
applied around a frequency other than ω0 , an effective additional linear chirp is added.
Figure 4.19 shows the dissociation yield analysis for a TOD pulse where the spectral
phase is applied 10 nm offset from ω0 . These results contradict the theoretical and
experimental findings discussed above disclosing that negative TOD pulses are more
efficient for dissociation due to non-adiabatic alignment induced by the pre-pulses.
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
132
Figure 4.20: Schematic diagram illustrating the Gaussian fit to the measured power
spectrum I(ω) normalized to 1 (black line). The applied combination of a quadratic
00
000
φ = 1040 fs2 and cubic φ = 13500 fs3 spectral phase function (red line) results in the
frequencies being redistributed in time according to the group delay given by the blue
line.
Instead, due to the dominance of the higher order of the effective additional linear
chirp, the results are similar to those in figure 4.7 for GDD pulses as they suggest that
positive TOD is ∼20% more effcient.
4.1.6
Combined Linear Chirp and Third Order Dispersion
The ability to enhance the dissociation rate of the low lying vibrational levels v≤6 of H+
2
using GDD and TOD as independent control tools has been demonstrated in sections
4.1.4 and 4.1.5 above. The underlying control mechanisms were identified as being
manifested in the temporal order of the instantaneous frequencies within the pulse and
the asymmetric temporal profile respectively. A pertinent question however is how the
00
000
combination of these two terms φ(ω) = 12 φ (ω0 ) · (ω − ω0 )2 + 16 φ (ω0 ) · (ω − ω0 )3 can
be used to manipulate the dissociation of H+
2.
Since GDD is the higher-order term it dominates the spectral phase. The addition of
TOD enforces an asymmetry to this parabolic spectral phase, as shown by the red line
00
000
in figure 4.20 for φ = 1040 fs2 and φ = 13500 fs3 combined. The corresponding group
delay becomes a mildly curved line as represented by the blue line in figure 4.20.
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
133
Figure 4.21: Simulated Polarization-Gating FROG traces of a 33 fs FTL pulse shaped
00
000
00
000
with (a) φ = -1040 fs2 and φ = -13500 fs3 and (b) φ = 1040 fs2 and φ = 13500
fs3 . Figures (c) and (d) show the temporal profile of the 33 fs FTL pulse (black line)
normalized to 1 and the corresponding relative TOD temporal profiles (red dashed
line) respectively.
For greater TOD values, the asymmetry of the parabola increases and the function as it
tends towards a more cubic phase. The corresponding PG FROG traces and temporal
profiles in figure 4.24 demonstrate that these pulses contain a frequency sweep and
triangular temporal profile. For these particular GDD and TOD parameters there is
no beating phenomena on the leading or trailing edge of the pulse.
The temporal duration of the GDD and TOD pulses can be defined using equations 4.2
and 4.3 respectively. To estimate the width of the majority of the intensity when these
√
two parameters are combined, the intensity variance ∆RM S = 2 < t2 > − < t >2
was calculated. Various pulse parameters for 33 fs FTL pulses shaped with several
combinations of GDD and TOD used are given in table 4.3.
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
GVD [fs2 ]
0
780
1040
1040
1040
TOD [fs3 ]
0
13000
27000
13000
40500
∆RM S [fs]
33
126
100
110
100
134
Relative Intensity
1
0.31
0.33
0.32
0.24
Table
4.3: Calculated relative peak intensity and pulse duration (∆RM S =
√
2 < t2 > − < t >2 ) for the various GVD and TOD combinations used in the experiment.
The same yield analysis described above (and presented in figure 4.14) was performed
for this data and is shown in the crossword (color map) in figure 4.22. The white
squares contain no data and the color bar indicates the strength of the dissociation
yield for that particular pulse shape relative to the FTL. The photodissociation yield
for the one-photon transitions located close to the Condon point (v≥9) with KER ≥
0.75 eV are orange, (representing a value of 1), for all pulse shapes (see figure 4.22
(a)). Thus indicating, as described before, that these transitions are independent of
pulse shape.
However, dissociation via bond-softening from the vibrational levels (v≤6) with KER ≤
0.41 eV exhibit a multi-colored dependence, see figure 4.22 (b). The two optimal pulse
000
shapes for this process were found to be (a) φ = -13500 fs3 and (b) a combination of
00
000
φ = 780 fs2 and φ = -13500 fs3 (see figure 4.24). Both these pulses are of comparable
peak intensity and are asymmetric on the rising edge. This again implies that alignment
is the dominant mechanism responsible for the dissociation enhancement. It is known
that the sufficiently intense sequence of pre-pulses for TOD in pulse (a) can effectively
induce non-adiabatic alignment and consequently enhance the dissociation (see section
4.1.5). Pulse (b) however contains no pre-pulses and a rising edge of comparatively
lesser intensity. The enhancement could therefore be due to either adiabatic alignment
or the temporal order of the frequencies on the rising edge of the pulse. Since the ‘blue’
frequencies are situated on the leading edge they can displace the avoided crossing to
lower RAC which creates a lower potential barrier, and increases dissociation (see
section 4.1.3). Both pulse shapes appear to be equally efficient, so determining the
dominant mechanism requires further theoretical support.
4.1 Photodissociation of Hydrogenic Ions Using Shaped Laser Pulses
135
Figure 4.22: Crossword for the dissociation probabilities of H+
2 for pulses shaped with
various combinations of GDD and TOD parameters normalized to the yield of the FTL
pulse for (a) One-photon transitions (KER ≤ 0.75 eV) and (b) bond-softening driven
dissociation (KER ≤ 0.41 eV). The colourmap gives an indication of the dissociation
yield relative to the FTL pulse. The white squares contain no data.
4.2 Conclusions and Future Outlook
136
Furthermore, the complicated interplay between these two mechanisms is highlighted
in figure 4.24. These two pulses are shaped with the same magnitude but opposite
00
000
00
000
sign with (a) φ = -780 fs2 and φ = -13500 fs3 and (b) φ = 780 fs2 and φ = 13500
fs3 . Where we would expect pulse (a) to be more efficient due to the asymmetry on
the rising edge, this is not the case. Since it is the ‘blue’ frequencies which lead the
pulse, this suggests that it is the temporal order of the frequencies which dominate for
these parameters. It follows from the above discussion that although the mechanisms
underlying the enhanced dissociation yield for GVD and TOD independently has been
unveiled, it is difficult to predict which will dominate for certain conditions.
Figure 4.23: Crossword illustrating the relative dissociation probabilities for bond000
3
softening of the vibrational level v≤6 of H+
2 for pulses shaped with φ = -13500 fs
00
000
and (b) φ = 780 fs2 and φ = -13500 fs3 . The simulated Polarization-Gating FROG
traces of these pulses are shown in (a)(ii) and (b)(ii) and the corresponding temporal
profiles are outlined by the red line in (a)(iii) and (b)(iii) respectively. The blue line
in figures (a)(iii) and (b)(iii) show the original 33 fs FTL pulse normalized to 1.
4.2
Conclusions and Future Outlook
In summary, a three-dimensional (3D) momentum imaging technique was used to measure the KER and angular distributions for the dissociation fragments of hydrogenic
4.2 Conclusions and Future Outlook
137
Figure 4.24: Crossword illustrating the relative dissociation probabilities for bond00
softening of the vibrational level v≤6 of H+
2 for pulses000shaped with (a)(i) φ = -780
000
00
fs2 and φ = -13500 fs3 and (b)(i) φ = 780 fs2 and φ = 13500 fs3 . The simulated
Polarization-Gating FROG traces of these pulses are shown in (a)(ii) and (b)(ii) and
the corresponding temporal profiles are outlined by the red line in (a)(iii) and (b)(iii)
respectively. The blue line in figures (a)(iii) and (b)(iii) show the original 33 fs FTL
pulse normalized to 1.
molecules using analytically shaped pulses. The pulses were shaped using the quadratic
00
000
(ϕ ) and cubic (ϕ ) spectral phase terms of the Taylor expansion given in equation
2.12 and are referred to as GDD and TOD respectively. The magnitude and sign of
these terms was then altered in a systematic manner and any differences in the spectra
observed and a yield analysis relative to the FTL performed.
The scientific findings reported in this thesis conclude that the KER structure of the
resonant transitions can be manipulated using the temporal order of the frequencies
of the pulse. This manifests itself as either the resonant peak (v=9 for H+
2 ) splitting
into two distinct features or an apparent increase in the energy resolution for negative
and positive TOD pulses respectively. The ability to increase the dissociation yield of
the low lying vibrational levels of H+
2 using pulses shaped with GDD and TOD was
experimentally demonstrated. It was proposed that the 20% enhancement in bondsoftening events observed for positive GDD was a consequence of the position and
4.2 Conclusions and Future Outlook
138
size of the avoided crossing being dynamically shifted in accordance with the direction
and rate of the frequency sweep. Since the blue detuned frequencies are located on
the rising edge of the pulse for positive GDD, the avoided crossing will be induced at
smaller internuclear separtations and hence the potential barrier will be lower, leading
to an increase in dissociation.
Similarly, a remarkable 50% increase in the bond-softening events observed for negative TOD pulses is attributed to the asymmetry of the temporal profile. The series of
pre-pulses created are intense enough to induce non-adiabatic alignment, which subsequently leads to an increase in the dissociation rate. It is also interesting to note
that the relative dissociation enhancement observed for negative TOD is stronger for
lower intensities. Although both the instantaneous frequency and the temporal profile
of the pulse can be used as a tool to control the dissociation of hydrogenic molecules,
it is unclear which mechanism dominates when both GDD and TOD are combined.
Further experiments using a single stronger pre-pulse, instead of the series of pulses or
an asymmetric temporal profile which contains no frequency sweep could be used to
elucidate the conditions under which control mechanism prevails.
There was no experimental evidence found in this data to support the concept that
the frequency sweep of a GDD pulse could be used as a tool to control the branching
ratio of HD+ . The difference in energy between the dissociation limits of HD+ are very
small (3.7 meV) and such a control scheme could potentially be demonstrated more
clearly using a molecule with a larger energy separation between the two dissociation
channels. TOD lacks the flexibility of an accurately time sequence of pulses tailored to
kick molecules at specific revival times and is therefore not the best pulse shaping tool
for optimizing molecular alignment. However, it does provide a robust, coherent control
tool which should not be highly isotope dependent and moreover is both experimentally
and theoretically accessible. The underlying mechanisms induced by GDD and TOD
have been identified and can act as a platform to understanding the interaction of
these shaped pulses with more complex molecules. Perhaps when interacted with
polyatomic molecules alignment along a preferred direction of orientation will occur
due to a specific property of the molecule and subsequently enhance the dissociation of
4.2 Conclusions and Future Outlook
139
a specific product. In a recent study, the most efficient temporal profile for silicon wafer
micro machining was found to be asymmetric [117]. It follows that the combination
of GDD and TOD could be explored in a similar manner. Furthermore, such pulse
shapes could be used to explore laser ablation of metals and areas of semi-conductor
science. Consequently, subject to further extensive investigation, such pulse shapes
could potentially find a use in industrial and healthcare applications.
Chapter 5
Laser Induced Fragmentation of
CD+
The electron density of heteronuclear molecules is shared unequally between the two
nuclei. Therefore the inversion of charge symmetry that exists between the electronic
states of homonuclear molecules is violated and electronic transitions which would nominally be forbidden are allowed. Heteronuclear molecules contain a complex plethora
of electronic states and an abundance of possible fragmentation pathways. Owing to
the different electron affinity and ionization energy of the two constituent atoms, the
energy required to reach these pathways can differ greatly. It is therefore interesting
to explore how the branching ratio of these channels change as a function of laser pulse
parameters.
However, measuring the molecular fragmentation of mass asymmetric molecules using
the 3D momentum imaging technique is experimentally challenging. For this reason, a
longitudinal and trasverse field imaging (LATFI) technique has been developed by the
group of Prof. Ben-Itzhak at the J.R.M laboratory, at Kansas State University, USA
(KSU). As part of a collaboration it was used to study the laser-induced fragmentation
of the highly asymmetric CD+ molecular ion. The principles of the LATFI method
and the various fragmentation channels and branching ratios measured for CD+ are
outlined in this chapter.
5.1 Laser-Induced Fragmentation of CD+
5.1
141
Laser-Induced Fragmentation of CD+
The development of short pulse lasers and the study of ultrafast dynamical processes has recieved tremendous attention in recent years, leading to a new realm of
physics coined ‘femtochemistry’. Following technological advancements, unprecidented
progress in this emergent field has been achieved with the ‘holy grail’ being able to
control and direct chemical reactions of complex molecules. However, interpreting the
laser-induced dynamics of molecular ions with a more complicated internal structure
than that provided by the two dominant potential energy curves (PEC) of H+
2 can
prove problematic. The various theoretical and experimental challenges associated
with studing more complex molecules are daunting, but have inspired scientists to
explore this regime more extensively.
In this work CD+ , a multielectron heteronuclear diatomic, has been chosen as an
exemplar molecule for extending the study of homonuclear simple molecules. The
multitude of potential energy curves (PEC) of differing spin degeneracy, symmetry
and quantum number, as denoted by the term symbols (see section 1.3.3), of CD+
are shown in figure 5.1. Interpreting the laser-induced dissociation pathways of such
a molecule is an important step towards understanding more complex systems. In the
prescence of the laser field, the spin-orbit coupling between the singlet and triplet states
is negligible. Hence from the optical transition rules (see section 1.1) these states can
be treated independently as the required spin flip of an electron is highly improbable.
Provided reliable PES of a molecule can be obtained and ideally an estimation of
the initial vibrational population, the Floquet picture (see section 1.4.2) can be used
to gain an understanding of possible dissociation pathways. Previous studies have
reported the use of angular distributions to limit possible dissociation pathways of O+
2
[118], the and kinetic energy release (KER) to find the initial and final states in double
ionisation [119]. Furthermore, in a more comprehensive study all of the information
was combined using a 3D momentum imaging technique and used to determine the
laser-induced dissociation pathways of O+
2 [120].
From an intense field perspective, CD+ is a particularly interesting target to probe
5.1 Laser-Induced Fragmentation of CD+
142
Figure 5.1: The potential energy curves of CD+ where the singlet curves are represented
by the red lines and the triplet represented by the black. Where the Π states are dashed
and the Σ states are solid.
as the two lowest dissociation limits C(3 P) + D+ and C+ + D(1 S) are separated by
an energy of 2.4 eV which is accessible by either one 785 nm or two 381 nm photons. Investigating the intensity and wavelength (381 nm and 785 nm) dependence
of the possible dissociation pathways may elucidate dynamics associated with each
spin state. Furthermore, the branching ratio of these two dissociation channels is
particularly interesting and timely given the long out-standing goal of achieving and
controlling chemical reactions. Previous studies show a strong intensity dependence
in the branching ratio of for dissociation of ND+ and DCL+ [121, 122]. However the
energy separation of the two studied dissociation limits was less than one photon in
both cases. Thus CD+ presents unique characteristics for further study.
Although tremendous progress has been made in understanding laser-induced dynamics, measuring the dissociation products unambiguously has in the past proven difficult. The advantage of using a coincidence 3D-imaging technique (as described in
sectio [REF]) is that the initial velocity of the target allows both neutral and charged
fragments to be detected. Furthermore, the longitudinal electric field accelerates the
5.2 Longitudinal and Transverse Field Imaging Technique
143
charged fragment with respect to its neutral counterpart creating a temporal separation. Thus when measured in coincidence the fragmentation channels can be clearly
separated. However, the large mass ratio (12:2) of CD+ makes measuring both of the
dissociation channels simultaneously using this method impractical. The slow fragment can get blocked by the Faraday cup positioned to collect the primary ion beam.
Alternatively, the fast fragment may travel outside the detector face. In these first measurements of intense field dissociation of CD+ , a piecewise measurement procedure to
determine the dissociation branching ratio was developed.
5.2
Longitudinal and Transverse Field Imaging Technique
As part of a collaboration with the J. R. M Laboratory at Kansas State University
(KSU), USA, the fragmentation of the CD+ molecular ion was studied in the lab
of Prof. Itzik Ben-Itzhak. The 3D momentum imaging setup used at KSU is very
similar to that at WIS and described in detail in chapter 3 and shown schematically
in figure 3.1. It is operated as a longitudinal field imaging (LFI) technique in the
format described. In these experiments the dissociation velocity of the fragments
are measured and the kinetic energy release (KER) and angle θ between the laser
polarization and the molecular dissociation axis calculated. The molecular breakup is
symmetrical about the azimuthal angle φ and this can be used to reconstruct losses
in measured data if required. The limitations imposed by operating the technique in
this mode is the loss of low energy fragments caused by the Faraday cup which collects
the primary ion beam. To overcome this restriction the group at KSU developed the
longitudinal and transverse field imaging (LATFI) technique. This method requires an
additional deflector strategically positioned between the spectrometer and the detector
(see figure 5.2 (a)). The fragments are then deflected by the transverse static E-field
separating low energy KER fragments from the primary ion beam. The fragments are
then spatially separated on the detector according to their energy to charge ratio. This
technique enables fragments with zero KER to be measured.
Figure 5.2 (b) shows a schematic diagram of the relative spatial separations for the
5.2 Longitudinal and Transverse Field Imaging Technique
144
centre of mass of the CD+ fragments on the detector for various theoretical deflector
voltage conditions. For the traditional longitudinal field imaging (LFI) technique where
zero voltage is applied to the deflector (see Figure 5.2 (b)(i)) there are substantial losses
due to the Faraday cup. For the low voltage conditions (see Figure 5.2 (b)(ii)) the
primary ion beam is deflected and collected by an off-axis Faraday cup. The neutrals
strike the detector where the primary ion beam would have been in the absence of
the deflector. The centre-of-mass (COM) of only two CD+ fragments (D+ and C2+ )
can be cleanly measured under these conditions as the COM of the C+ is lost to the
Faraday cup. For the high voltage settings (see Figure 5.2 (b)(iii)) the COM for the
C+ is sufficiently deflected away from the Faraday cup. However, as a consequence the
COM of the D+ is deflected completely off the detector and theoretically only about
half of the C2+ distribution can be measured.
A quantitative illustration of where the COM of each fragment lies relative to each
other with respect to the deflected CD+ ion beam is shown in figure 5.3. If the primary
ion beam is deflected by 4-5 mm from the centre of the detector then correspondingly
the COM of the D+ fragments will be located at a distance 28-35 mm from the centre
(green line). A separation distance of 2-2.5 mm between the C+ and the Faraday cup
centre (light blue line) is necessary to ensure the C+ COM can be measured. The
primary beam should therfore be deflected by 12-15 mm. The voltages required to
achieve these relative distances can then be determined theoretically. Although the
size of the detector is effectively reduced by the deflection of the CD+ beam, some
additional detection space can be obtained by initially dog-legging the CD+ beam.
To determine the dissociation branching ratios a piecewise method was used. The
deflector voltage settings were optimized to measure each dissociation channel independently. Ideally, under all voltage conditions a common fragmentation channel would
be measured and used to normalize the data sets to each other. In principal, the charge
asymmetric dissociation (CAD) channel C2+ +D would have been an ideal candidate
for this as it suffers from minimal geometric losses in both cases. However, in practice
this CAD channel was not observed for the 391 nm pulses and appears only in the
high intensity 795 nm measurements.
5.2 Longitudinal and Transverse Field Imaging Technique
145
Figure 5.2: (a) A schematic diagram of the longitudinal and transverse field imaging
(LATFI) setup developed at Kansas State University (KSU), USA. The neutrals are
unaffected by the presence of the static transverse electric field and the charged species
are deflected according to their Energy/Mass ratio. (b) The relative positions of the
CD+ fragments on the detector are illustrated for (i) 0 V (ii) low voltage and (iii) high
voltage settings applied to the transverse deflector respectively.
5.3 Results and Discussion
146
Figure 5.3: The relative positions of the fragments on the detector with respect to the
deflected primary CD+ ion beam.
To further minimize geometric losses the polarization of the laser can be used to increase the angular acceptance of the detector. Also the spectrometer voltage and ion
beam energy can be altered. The physical size of the momentum distribution of the
fragments on the detector can be reduced be decreasing the travel time to the detector.
Although this may retain a greater contribution of the lighter fragments to the confines
of the detector, the distribution of heavier fragments are more likely to be lost in the
Faraday cup.
Another reason for using a higher ion beam energy is to ensure that all fragments are
efficiently detected. For a CD+ ion beam energy of 21 keV and spectrometer voltage
1400 V the D neutral fragment was detected with an energy 2.8 keV. It was experimentally verified that the measurement was obtained on the platau of the efficiency
curve. For the same experimental conditions the brancing ratio of the two dissociation
channels was found to be independent of MCP voltage.
5.3 Results and Discussion
147
Figure 5.4: Coincidence time of flight spectra showing the laser-induced fragmenation
channels measured for the interaction of 30 fs pulses at 785 nm and I0 = 1 ×1016 W
cm−2 with CD+ .
5.3
Results and Discussion
The 30 fs and 55 fs pulses at a central wavelength 785 nm and 391 nm respectively, were
generated by the PULSAR laser (10 kHz) at KSU. The laser beam was then focused to
achieve an intensity range 1.3 × 1013 − 1.6 × 1016 W cm−2 and 2.2 × 1011 − 2.5 × 1015
W cm−2 respectively. The coincident time of flight spectra for the laser induced fragmentation of CD+ is shown in figure 5.4. There were five fragmentation channels
identified, two dissociation channnels, two charge symmetric ionization channels and
a charge asymmetric dissociation (CAD) channel.
5.3.1
Dissociation Channels
The potential energy curves for the field free electronic states for CD+ are given in
figure 5.1. In the experiement CD+ is produced from methane by electron impact in
an electron cyclotron resonance (ECR) ion source. The length of the C-D bond in
methane (1.085 a.u.) is the same as that of the ground state of the CD+ molecular
ion (1.09 a.u.). A Franck-Condon projection of the the ground state wave function
5.3 Results and Discussion
148
onto the CD+ states can then be used to estimate the vibrational population. The
population is likely to be relatively low-lying in a mixture of both the singlet X1 Σ+
and triplet a3 Π states, as there are no optical transitions between the two. However,
the exact ratio of the population between the two states is unknown.
To determine the most probable dissociation pathways the dressed states Floquet approach (see section 1.4.2) is employed to visualise the dissociation routes (see figure
5.5). The excitations between states are indicated as vertical transitions, resonant with
an integer number of photons (nh̄ω). Thus all the electronic curves are shifted downwards (or upwards) in energy by the number of absorbed (or emitted) photons. For
example X1 Σ+ − 2ω indicates that the X1 Σ+ state has been shifted down in energy
by two photons. Each state then repeats itself periodically with an energy separation
equal to multiples of photon energy. The singlet and triplet CD+ potential energy
curves (PEC) dressed with 785 nm photons are shown in figure 5.5 (a) and (b) respectively. When considering the second harmonic (391 nm photons) one may still refer to
5.5, but should note that only transitions involving an even number of photons at 785
nm are applicable.
From the abundance of potential energy curves for CD+ determining the possible dissociation pathways seems an insurmountable task. However, there are four recommended
guidelines to follow in the process of elimination [120]. The transition probability stipulates that the most plausible pathway requires the fewest number of photons. This
means that contributions from highlying excited states are unlikely to make major
contributions. The angular distribution of the fragments are a signature of the type
and number of contributions from transitions favoring internuclear alignment parallel (∆Λ=0, leading to a cosn θ distribution) and perpendicular (∆Λ=±1, giving sinn θ
distribution) to the laser polarization (see section 1.4.7). Furthermore, the position
and shape of the KER spectra can be used to gain an insight into the general shape
of the potential barrier over which the vibrational wavepacket dissociates and hence
the PECs comprising it. Each curve crossing should be considered in the adiabatic
picture as an avoided crossing. The electric field must be turned on for long enough
to allow sufficient time for the wavepacket to traverse the series of potential energy
5.3 Results and Discussion
149
Figure 5.5: The potential energy curves of CD+ dressed with 795 nm photons for the
(a) singlet and (b) triplet states.
5.3 Results and Discussion
150
Figure 5.6: Angular distributions (where only half of the distribution is measured and
the other half reflected) and KER spectra for the C+ + D dissociation channel of CD+
for pulses of (a) 30 fs at 795 nm and I0 = 3.2 ×1015 W cm−2 and (b) 50 fs at 391 nm
and I0 = 2.5 ×1015 W cm−2 .
curves and escape. In addition to all of the above, the molecular dipole selection rules
must be obeyed which elimnates numerous ineffectual crossings. Finally, it should be
noted that an inherent problem with the volume effect (see section 2.1.1) is that any
intensity dependent transitions may be obscured and sensitivity to any fine structures
can become lost.
C+ + D Pathway
The lowest dissociation limit of CD+ is the C+ + D(1 S) channel which can dissociate
only dissociate via two different pathways, from either a singlet X 1 Σ+ → A1 Π or a
triplet a3 Π+ → c3 Σ+ state (see figure 5.1). The angular distributions and KER spectra
for this channel at 795 and 391 nm are shown in figure 5.6 (a) and (b) respectively.
The KER spectra for the 391 nm pulses demonstrates a sharp peak at ∼0.1 eV. The
5.3 Results and Discussion
151
broad angular base in the distribution shown in 5.6 (b) indicates a mixture of both
parallel and perpendicular transitions. Only one possible transition from a singlet state
can result in KER distribution peaked close to zero:
X 1 Σ+ → A1 Π − 2h̄ω
(5.1)
This transition requires only one (381 nm) photon and relies on the low lying vibrational levels of the ground state being populated making it a highly feasible option.
However, equation 5.1 suggests that perpendicular transitions dominate and this would
result in a distribution concentrated at cosθ=0. This is conflicting with the strongly
peaked contribution observed along the laser polarization, as shown in figure 5.6 (b).
Nevertheless, for the intensity range studied dynamic alignment could be the reason
for the observed angular distribution. If the population undergoes a resonant Raman
transition to the bound part of the A1 Π state at the inner turning point where the
transition moment is highest, then an aligned and highly rotationally excited molecule
is more likely to dissociate.
For the 785 nm pulses, the KER spectra is peaked around 0.4 eV and the angular distribution is highly aligned along the laser polarization with a sharp cos2 θ distribution.
Thus suggesting that parallel transitions dominate. The same singlet state pathway
and mechanism described in equation 5.1 but instead using two 785 nm photons are
also relevant. From figure 5.6 it is clear that the contribution for the 785 nm is more
highly aligned than for the 391 nm. This could reflect the higher intensities used for
the 795 nm as this would lead to an increase in alignment.
From the dressed triplet state potential energy curves in figure 5.5 (b) it is clear that
the molecule can dissociate through the following pathway:
a3 Π → b3 Σ− − 2h̄ω → d3 Π − 3h̄ω → a3 Π − h̄ω
(5.2)
However, this combination of parallel and perpendicular transitions requires six 785
nm photons. It is therfore expected to demonstrate a strong intensity dependence and
considered unlikely.
5.3 Results and Discussion
152
Figure 5.7: Angular distributions and KER spectra for the D+ + C dissociation channel
of CD+ for pulses of (a) 30 fs 795 nm and I0 = 3.2 ×1015 W cm−2 and (b) 50 fs at 391
nm and I0 = 2.5 ×1015 W cm−2 .
D+ + C Pathway
The alternative dissociation channels D+ + C are situated at an energy of 2.4 eV,
3.65 eV and 5 eV above the C+ + D(1 S) dissociation limit. The KER spectra and
angular distributions shown in figure 5.7 peak around 0.5 eV and do not span as
low as zero KER, contrary to the observation for the C+ + D channel. The angular
distributions are peaked along the direction of the polarization for both wavelengths
and their similarity in structure suggest similar dissociation pathways.
There are no feasible pathways for the 381 nm photons starting from the triplet states
as the corresponding dissociation limit in the dressed state picture is shifted to an
energy greater than 1 eV from the bottom of the a3 Π curve (see figure 5.5). From the
KER spectra and consideration of the singlet dressed states, an energetically possible
5.3 Results and Discussion
153
pathway for both wavelengths is given by:
X 1 Σ+ → A1 Π − 2h̄ω → 31 Σ+ − 6h̄ω
(5.3)
However, this involves two perpendicular transitions and is therefore expected to peak
at cosθ=1 and not along the polarisation as was experimentally observed. For the 785
nm pulses an additional pathway consisting of parallel and perpendicular transition is
also possible:
X 1 Σ+ → A1 Π − 2h̄ω → 21 Π+ − 5h̄ω
5.3.2
(5.4)
Branching Ratio
Each dissociation channel was measured independently using the optimal longitudinal
and transverse field imaging (LFTI) conditions to maximize the solid angle collected
for each fragment (see section 5.2). The number of counts from each channel [C+ + D]
and [D+ + C] was then normalize to ensure the number of molecules exposed to the
laser was constant. A correction factor to account for the fraction of the solid angle
efficiently collected was calculated using a moving average of the counts in the φ plane.
The overall counts in an ideal flat distribution was then estimated and the percentage
of the φ distribution experimentally measured was determined.
The branching ratio is defined as the ratio of the number of dissociation counts from
one channel to the total number of dissociation counts:
Branching Ratio =
[C +
[C + + D]
+ D] + [D+ + C]
(5.5)
This can be used to examine the intensity dependence of the two dissociation channels.
Figure 5.8 shows the branching ratio as a function of intensity for the 391 nm and the
785 nm pulses. The error bars calculated reflect the statistical errors. For the 391
nm pulses a linear increase in the C+ + D(1 S) channel with intensity was observed.
This is in agreement with the one photon pathway X 1 Σ+ → A1 Π − 2h̄ω suggested
as such processes are expected to scale linearly with intensity. At higher intensities
5.3 Results and Discussion
154
Figure 5.8: The branching ratio for the two dissociation channels of CD+ as a function
of intensity for 391 nm (red dots) and 785 nm (black dots). The error bars calculated
reflect only statistical errors.
the gradual flattening of the line may be an indication of the dissocaiation dynamics
being complicated by the opening of new dissocaition pathways or even the onset of
ionisation. The branching ratio for the 785 nm pulses shows no significant intensity
dependence over the range of intensities studied.
5.3.3
Ionization Channels
The charge symmetric ionization channels are the hardest to measure. There are no
optimal combination of voltage conditions that can accomodate the efficient collection
of the two charged constituent fragments of CD+ . Hence substantial losses are inevitable for this data. The angular distributions for the single and double ionization
channels for an intensity range 1 × 1014 Wcm−2 − 1 × 1016 Wcm−2 at 785 nm are shown
in figures 5.10 and 5.11 respectively.
5.3 Results and Discussion
155
Figure 5.9: The Potential Energy Curves of CD+ including the ionisation limits.
Single Ionization
The single ionization channel of CD+ results in the two fragment ions C+ and D+ .
The angular distribution of the kinetic energy released in the single ionization channel
for 785 nm pulses at a range of intensities are shown in figure 5.10. The cut along
the direction of the polarisation axis in angular distribution is an experimental artifact
caused by the loss of the slow C+ to the Farady cup. The observed structure is
a broad angular base with a highly aligned central feature. This is a signature of
stepwise ionisation and has been reported for other diatomics [45]. Furthermore, the
contribution from each feature demonstrates an intensity dependence. If ionisation
proceeds in a stepwise manner, the molecule is first excited to a dissociative state and
then subsequently ionized as the molecule stretches. Thus some characteristics of the
dissociation channels are retained and reflected in the angular distributions as is the
case here.
5.3 Results and Discussion
156
Figure 5.10: The angular distribution for the single ionisation channel C+ + D(1 S) of
CD+ at 785 nm for an intensity range 1 × 10 14 Wcm−2 to 1 × 10 16 Wcm−2 .
5.3 Results and Discussion
157
Figure 5.11: The angular distribution for the double ionisation channel C2+ + D+ of
CD+ at 785 nm at an intensity of (a) 2.5 × 10 14 Wcm−2 and (b) 1 × 10 16 Wcm−2
Double Ionization
The multiple ionization of CD+ can lead to the fragment ions C2+ and D+ . The
angular distribution of the kinetic energy released in the double ionization channel for
795 nm pulses are shown in figure 5.11. A strongly aligned contribution along the laser
polarization can be observed, but it should be noted that the ability to measure low
cosθ values in this KER range is limited, and the angular distribution may be broader.
The peak of the KER is seen to shift from 10-14 eV with increasing intensity. The
contribution from electron rescattering (see section 1.5.1) was experimentally tested by
using circularly polarized light. This way the initially ionized electron cannot return to
the parent ion and release another electron through ineleastic scattering. No significant
reduction in the multielectron ionisation was observed for circulary polarised light.
Thus ruling any significant contribution from this ionisation mechanism out.
The proposed mechanism for this double ionisation channel is the stair-step ionisation process described in section 1.5.3. In this sequential ionization procedure, the
molecule begins to traverse either of the two dissociation channels before being ionized. Following this, the molecule then stretches along the single ionisation channel
before being ionized again. Since these series of events occur consecutively at later
times in the pulse, the laser field has more time and intensity to populate a broader
higher rotational state. This means that the angular distributions emerginging from
higher ionization stages are likely to be more aligned. Furthermore, the increasing
5.3 Results and Discussion
158
Figure 5.12: The angular distribution and KER spectra for the charge asymmetric
dissocaition channel C2+ + D of CD+ at 785 nm at an intensity of (a) 1 × 10 16
Wcm−2 .
number of events acts to obscure any KER structure from the inital stages.
5.3.4
Charge Asymmetric Dissociation
The loss of an electron from CD+ can also result in charge asymmetric dissociation
(CAD) as described in section 1.5.2. This refers to the charge being shared unevenly
between the fragments. For CD+ this can result in population dissociating into a
C2+ +D state. The charge symmetric dissociation (CSD) channels are usually favored
as their limits are generally energetically lower. The calculations of such highly charged
CAD potential energy curves are extreemely demanding and have been attempted on
several occassions with variations in the assignment of states.
The PEC’s presented in figure 5.9 show the 10 eV energy separation between the single
ionisation C+ (2 D)+D+ and the CAD channels. Thus contributions from the CAD
channel are substantially weaker by comparison. Furthermore, the authors suggest the
opening of an avoided crossing between the CSD (D+ + C+ (2 D)) and CAD channels
at 10.5Å, with a 1.5 eV energy separation.
The wavepacket can only feasibly reach this avoided crossing via excitation to either
5.4 Conclusions and Future Outlook
159
the 22 Σ+ or the 22 Π potential curves. From the angular distribution and KER spectra
for the C2+ +D dissociation channel of CD+ (see figure 5.12) it can be concluded that
the molecule must be stretched to approximately 4 - 8 a.u. in order to emerge from
either of these two channels with a KER within the range 0-3 eV. According to the
Landau-Zener formula, the transition probability of the wavepacket at the avoided
crossing is governed by its velocity and it is this property that will determine the final
fragmentation pathway of the molecule.
Interestingly, the CAD channel was only observed at 785 nm (30 fs) and not 391 nm
(50 fs) for equivalent intensities. Intuatively, one would expect the latter to be more
probable with the higher energy photons, given the location of the dissociation limit.
A possible explaination is that the temporal duration of the pulse plays an integral
role in the fragmentation dynamics [123].
5.4
Conclusions and Future Outlook
the use oof short pulses.... would allow how far the molecule strtches during the
stairstep ionisation process to be monitored.
Research Publications
Refereed Publications
A. Natan, J. A. Levitt, L. Graham, O. Katz and Y. Silberberg
Standoff detection via single-beam spectral notch filtered pulses.
Appl. Phys. Lett. 100, 051111 (2012)
J. B. Greenwood, O. Kelly, C. R. Calvert, M. J. Duffy, R. B. King, L. Belshaw,
L. Graham, J. D. Alexander, I. D. Williams, W. A. Bryan, I. C. E. Turcu,
C. M. Cacho and E. M Springate
A comb-sampling method for enhanced mass analysis in linear electrostatic
ion traps.
Rev. Sci. Instrum. 82, 043103 (2011)
J. D. Alexander, L. Graham, C. R. Calvert, O. Kelly, R. B. King,
I. D. Williams and J. B. Greenwood
Determination of absolute ion yields from a MALDI source through calibration of an image-charge detector.
Meas. Sci. Technol. 21, 045802 (2010)
J. D. Alexander, C. R Calvert, R. B. King, O. Kelly, L. Graham, W. A. Bryan,
G. R. A. J. Nemeth, W. R. Newell, C. A. Froud, I. C. E Turcu, E. Springate,
I. D. Williams and J. B. Greenwood
Photodissociation of D+
3 in an intense, femtosecond laser field.
J. Phys. B. 42, 141004 (2009)
5.4 Conclusions and Future Outlook
161
P. Bruggeman, L. Graham, J. Degroote, J. Vierendeels and C. Leys
Water surface deformation in strong electrical fields and its influence
on electrical breakdown in a metal pin-water electrode system
J. Phys. D. 40, 4779 (2007)
L. Graham, P. J. Van der Burgt, J. Alexander, T. L. Merrigan, C. A. Hunniford, C.
J. Latimer, I. D. Williams and R. W McCullough
Fragmentation of Acetonitrile in collisions with H− and O− negative ions
In Preparation
M. Zohrabi, L. Graham, U. Lev, U. Ablikim, B. D. Bruner, J. J Hua, J. McKenna, K.
J. Betsch, B. Jochim, A. M. Summers, B. Berry, D. Strasser, O. Heber, Y. Silberberg,
D. Zajfman, K. D. Carnes, B. D. Esry and I. Ben-Itzhak
Dissociation of H+
2 by broad bandwidth laser pulses: selective molecular
response causing effective bandwidth narrowing
In Preparation
U. Lev, L. Graham, B. D. Bruner, J. J Hua, V. S. Prabhudesai, A. Natan, C. B.
Madsen, B. D. Esry, I. Ben-Itzhak, D. Schwalm, I. D. Williams, O. Heber, Y. Silberberg and D. Zajfman
Quantum control of H+
2 photodissociation using femtosecond pulses shaped
with third order dispersion
In Preparation
Research Presentations
September 2012
Quantum Control of Photodissociation Using Shaped Ultrafast Pulses
Quantum Atomic, Molecular, and Plasma Physics, (QuAMP), Belfast, UK
July 2011
Fragmentation of Acetonitrile in collisions with H− and O− negative ions
XXVII International Conference on Photonic, Electronic and Atomic Collisions
(ICPEAC), Belfast, UK
January 2010
The Ionization and Fragmentation of Acetonitrile by Low-Energy NegativeIon Impact
Atomic and Molecular Interactions Group (AMIG), Milton Keynes, England
May 2009
Absolute Calibration of an image-charge detector
4th Annual ITS-LEIF Meeting 2008, Girona, Spain
May 2009
Ultrafast Laser Driven Recombination in Excited ions
Physics at EBITS and Advanced Research Lightsources (PEARL), Dublin, Ireland
Dec 2008
Ultrafast Laser Driven Recombination in Excited ions
High Power Laser Science Meeting, Abingdon, England (3rd poster prize)
5.4 Conclusions and Future Outlook
163
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