Download Atoms in strong laser fields

Atoms in strong laser
fields
Wavelength
Laboratory exercise
Atomfysik FK
vt 2003
Introduction
This laboratory exercise consists of two parts, in the first part you are going to study multiphoton ionisation by looking at the produced ions when an intense laser pulse is focused
into a gas. In the second part you will investigate high-order harmonics with a
spectrometer by looking at the radiation generated when the laser pulse is focused in a
gas-jet. The preparatory exercises should be solved before the laboratory exercise starts.
Theory
Strong field ionisation
An atom can generally become ionised by absorbing one photon with high enough energy
to overcome the ionisation potential. An atom exposed to the electric field of an intense
laser pulse could however get ionised even if the photon energy is too low to ionise the
atom by a single photon. If the field is intense enough to perturb or more strongly distort
the atomic potential, ionisation could still occur by other mechanisms. Depending on the
intensity and wavelength of the laser different types of ionisation could take place. The
Keldysh parameter is used to separate the different ionisation regimes. The Keldysh
parameter is defined as:
Ip
 
2U p
where Ip is the ionisation potential of the atom and Up is the ponderomotive energy, i.e.
the average energy an electron can gain in the electric field of the laser. The
ponderomotive energy is given by:
Up 
e2E02
4m 2
where
e
E0
m

= the electron charge
= the amplitude of the electric field
= the mass of electron
= the angular frequency of the laser
Note that the intensity, I, is proportional to E2
I 
1
 0cE 2
2
where 0 is the permittivity and c the speed of light. By studying the expression for the
Keldysh parameter we see that it is proportional to the intensity and also to the square of
the wavelength of the laser (1/2 = 2/(2c)2). When  >>1, the ionisation process is
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described as multiphoton ionisation (MPI). In order to be in this regime, the intensity
should be relatively low, and/or the wavelength long. For the wavelength used in this
laboratory exercise this means an intensity below 1014 W/cm2 in order to be in the
multiphoton regime. The ionisation process could be described according to figure 1. The
Coulomb potential, VC , experienced by the electron in the atom is given by:
VC (r )  
e
0
r
e
40r
= electron charge
= permittivity
= radial coordinate
The Coulomb potential is shown in a dotted line in figure 1.
Figure 1. Ionisation of the atom in the multiphoton model.
The atom or ion (A) is ionised by simultaneously absorbing N photons, where N is enough
to overcome the ionisation potential. n is the charge of the ion.
A n   N  A (n 1)   e 
The ionisation rate is proportional to IN according to perturbation theory. When  << 1 the
multiphoton description becomes less valid and the atom is mainly ionised by tunneling
ionisation. The intensity in this regime is typically 1014 - 1015 W/cm2 for our laser
wavelength. Figure 2 describes schematically the mechanism of tunneling ionisation. The
electric field of the laser is oscillating in time as:
E(t ) E 0 sin(t )
E0

t
= amplitude of the electric field
= angular frequency
= time coordinate
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From this we get the laser field potential, VE:
VE (t )  E0 r sin(t )
The laser field potential is shown in a dashed line in figure 2. Note that the strength and
the sign of the potential will depend on time, figure 2 shows an instant in time where the
strength is at maximum. The sum of the two potentials, VTOT = VE + VC is marked in a
solid line. Since the electric field strength is so large, it strongly distorts the Coulomb
potential and lowers the potential barrier, increasing the probability for the electron to
tunnel out and the atom becomes ionised.
Figure 2. Schematic picture of tunnel ionisation.
Harmonic generation
Harmonic generation is merely another aspect of a strong field interacting with a medium.
Instead of looking at ions or electrons, we study the radiation that is emitted when the
intense laser field interacts with the gas. This radiation is emitted at odd multiples of the
laser frequency and can extend up to very high orders (>200) without decreasing in
amplitude. Figure 3 shows schematically a typical harmonic spectrum. The first harmonics
drop off rapidly in amplitude, they are then followed by a long so called "plateau" of
harmonic peaks ending in an abrupt "cut-off". The frequency spacing between two
harmonics are 2L, where L is the laser frequency.
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Figure 3. Schematic harmonic spectrum.
Model
The behavior of the first few harmonic peaks could be understood with perturbative
theory, which is valid for low intensity laser fields. Perturbative theory predicts a rapid
decrease in harmonic intensity with harmonic order, q. This is illustrated to the left in
figure 4. The atom is excited by multiphoton excitation to a virtual level, then it decays to
the ground state by emitting one high-energy photon. In this regime, the dependence of
harmonic intensity, Iq, on laser intensity, IL, follows a power law: I q  I L .
Figure 4. Model of harmonic generation. To the left we see the perturbative model valid
for lower intensities, and to the right, the model describing high-order harmonic
generation.
Higher-order harmonic generation, which occurs at stronger laser intensities, could be
understood with a semi-classical model shown to the right in figure 4. In this model the
atom can be approximated as having only one state, the ground state and only one
electron. The harmonic generation occurs in three steps:
1) The atom is ionised by tunnel ionisation as described above.
2) The electron is accelerated by the force from the laser field away from the atom.
When the electric field changes sign the electron will experience a force in the
opposite direction that decelerate it and then accelerate it again towards the atom.
3) When the electron pass by the atom it has a probability to recombine.
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When the electron recombines, the energy it has gained from the acceleration in the field
is emitted as a photon. What energy the electron could gain, and hence the emitted photon
will get, depends on the time of tunneling and the time of recombination (i.e. the phase of
the electric field at those times). The maximum kinetic energy the electron theoretically
can gain from the laser field is 3.2UP. The total maximum photon energy is then
IP + 3.2UP, where IP is the ionisation potential of the atom. All possible photon energies
up to the maximum energy have approximately equal probability, leading to the long
"plateau" of peaks of almost equal amplitude. The reason why we get peaks at the
harmonic frequencies and not just generate a continuum is because the process is periodic
in time since the electron tunnels out when the electric field is close to maximum. Because
the process is periodic in time it will also be periodic in frequency. Further, the period
time of this process is actually T/2 where T is the laser period. Since the gas is isotropic
there is no difference in the case the electron tunnels out when the electric field is -E0 or
+E0. This leads to the periodicity of 2 in the frequency domain ( is the laser frequency),
and we only observe odd harmonics.
Influence of medium
High-order harmonics are traditionally generated in the noble gases Xenon, Krypton,
Argon, Neon and Helium. From the expression for the cut-off energy it is quite obvious
that it is possible to generate much higher order harmonics in Helium than in Xenon
because Helium has a much higher ionisation potential, IP. On the other hand the
conversion efficiency is much higher for the gases with lower ionisation potentials. This
is because the polarizability, how easy the electron cloud is polarized by the electric field
of the laser, increases with atomic number. The electron cloud is more loosely bound in
Xenon than Helium. It could be added that the conversion efficiency for harmonic
generation generally is very low.
Gaussian beams
A laser beam could be described as having a Gaussian intensity distribution transverse to
the direction of propagation. The intensity at a distance r from the center is expressed as
I  I0 exp(
2r 2
)
w2
where I0 is the peak intensity and w is the spot size radius of the beam, i. e. where the
intensity decreased to 1/e2 of the peak intensity. The Gaussian beam is rotational
symmetric. A Gaussian beam that is focused remains its Gaussian shape. The following
expression could be used to calculate the beam diameter in focus (d) from the beam
diameter at the focusing lens (D):
d
4f
D
This is valid when d<<D.
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Experimental set-up
Laser system
The laser used in this laboratory exercise is a frequency doubled, mode locked and
Q-switched 10 Hz Nd:YAG laser. The laser pulses generated in the oscillator are 80-90 ps
long with an energy of 3 mJ/pulse at the fundamental laser wavelength of 1064 nm. These
pulses are amplified in a first amplifier and divided into two beams in a beam-splitter. The
pulses in the first beam-path are frequency doubled, giving an output pulse of ~20
mJ/pulse at 532 nm. Since the pulses in the second beam path have a lower energy these
pulses are amplified in a second amplifier before frequency conversion. The amplified
pulses are frequency doubled or frequency tripled depending on the current experiment
and have an energy of 5-10 mJ/pulse. The reason for splitting the laser pulse in two is that
in some experiments you need two laser pulses synchronised in time. However in, this
laboratory exercise we will use one beam-path for the first part of the experiment; when
we are looking at multiphoton ionisation and the other beam-path for the high-order
harmonic generation during the second part of the experiment, thus we avoid re-aligning
when changing from one set-up to the other.
Time-of-Flight spectrometer
The output laser pulse is focused in the middle of a time-of–flight spectrometer (TOFspectrometer). Krypton is introduced into the TOF-spectrometer by a leak valve giving us
a static pressure. The pressure inside the TOF-spectrometer is controlled by opening or
closing the valve. In the focus of the laser beam the intensity is high enough to ionise the
krypton atoms. The generated ions are separated in mass in a time-of-flight tube (TOFtube) and detected by an electron multiplier (EMT). The registered signal is shown as an
inverted peak on the oscilloscope. A pair of acceleration plates are placed directly above
and under the interaction region of the TOF-spectrometer (se Figure 5). By applying a
positive voltage on the lower plate the ions are accelerated in the electric field between the
plates. The TOF-tube is field free so the ions fly with constant velocity through the tube.
Since the applied acceleration voltage is known the ions get a known kinetic energy,
depending on mass and charge, which is used to identify them. This is done by measuring
the flight time in the TOF-tube for the specific ion and from that calculate the
corresponding mass for the ion. The flight time depends on the velocity the ions have
acquired in the extraction region.
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Figure 5. Time-of-flight.
High-order harmonic generation chamber and spectrometer
A lens with a focal length of 20 cm is used to focus the laser pulse just below the orifice
of a synchronised gas-jet nozzle. The gas-jet nozzle can be moved in the direction of the
optical axis, sideways and up and down, this makes it possible to optimise the harmonic
yield. To further improve the harmonic yield it is possible to change the backing pressure
to the gas-jet nozzle and to change the pulse length of the gas and the time between laser
shot and the gas pulse.
The generated, odd harmonics are separated with a 1200 grooves/mm normal-incidence
spherical grating with a 1 m radius of curvature. The harmonic spectrum can be registered
with a computer program that turns the grating and register the EMT-signal for every new
position of the grating. A boxcar integrator is used to select and integrate the signal from
the EMT. The boxcar is simply a tool that lets you look in a certain time gate for the signal
of interest. The gate can be moved in time and the time width of the gate can be changed.
The signal within the time gate is integrated and sent to the computer.
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TOF
Photodiod
Focusing
lens
Laser beam
Gas jet
Figure 6. Experimental set-up for multiphoton ionisation.
Figure 7. Experimental set-up for harmonic generation.
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Tasks during the laboratory exercise
Multiphoton ionisation
1) Focus the laser beam into the Krypton gas. Look at the ion spectrum with the time-offlight. Change the voltage on the time-of-flight plate and the gas pressure and see what
happens with the ion peaks. Why? Try to identify some of the different peaks, and
figure out which one is the Kr+ - peak. Print out the ion spectrum and mark the
identified peaks.
2) Vary the energy of the laser and record the intensity dependence of the Kr+ - ions by
simultaneously measure the Kr+ ions and the laser energy with a photo diode (using a
special program). Fit the points to IN and obtain N. Make another fit with N fixed to
the number of photons required to inonize Kr. Obtain a third fit to the tunneling
ionisation rate:
P
1

1

exp(  ) 
(
)
Eo
E0
I
I
Which model is the best fit to the data?
3) Measure the laser energy with a power meter and the photodiode at the same time for
one laser energy. Estimate the laser intensity in focus. The beam diameter is 1 cm, and
the pulse duration 80 ps. (Calibrate the values measured with the photodiode to
intensity.)
Harmonic generation
4) Focus the laser beam into the Argon gas-jet. Send the generated harmonic radiation
into a spectrometer. Record the harmonic spectrum. Measure the position of the peaks
using the program and identify the order of the harmonic peaks. Whart are the different
part of the spectrum? Do you see the cut-off?
5) Measure the laser energy witn a power meter. Estimate the laser intensity in focus.
Calculate the cut-off energy, which correspond to the highest order harmonic that
could be obtained and compare to the experimental value.
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Preparatory exercises
1) Estimate the laser intensity in focus by following the steps (a), (b), and (c).
a) The laser is running in 10 Hz (10 laser pulses/second). The average power is measured
to 100 mW. What is the total energy in one laser pulse? Hint: 1 W = 1 J/s
b) A Gaussian beam, with a diameter of 1 cm is focused with a lens with 20 cm focal
length. What is the spot size diameter in focus?
c) What is the average intensity in focus for a laser pulse of 80 ps.
2) A beam is focused in the middle between the plates of the TOF according to figure 5
and ions are produced. The voltage between the plates is 1000 V. Calculate the total flight
time to the detector for the following ions: Kr+, Kr2+, He+, H2O+, N2+, H2+ .......
Hints:
-Calculate the velocity at the top plate by knowing the kinetic energy.
-What is the acceleration voltage experienced by the ion?
-Don't take the acceleration time into account.
3) How many laser photons are required to ionise Helium, Krypton and Argon according
to perturbation theory? The laser wavelength is 532 nm.
4) Calculate the cut-off energy in the three gases: Krypton, Argon, and Helium. The laser
intensity is 1014 W/cm2 and the wavelength 532 nm.
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Potential hazards to be aware of during
the lab
Laser safety
The light from the laser used in this laboratory exercise is very intense and may cause
severe eye damage if the laser beam or a reflected beam hit the eye. Be very careful with
the laser beam and follow the advice below.
Hold your head above the laser beam during the laboratory exercise. Never sit down or
bend yourself down to pick up something while the laser is running.
Take off watches and rings during the laboratory exercise, they might accidentally reflect
the beam.
Never insert or remove optical components in the laser beam while the laser is running.
After inserting an optical component, be sure you know where the reflexes from it is going
before turning the laser on again.
High Voltage
Several pieces of the equipment are operated with high voltage, e.g. the laser, the time-offlight and the detectors. You will not risk getting in contact with the high voltage when
following the normal procedures of this lab. However, when not handled carefully it might
cause damage to the equipment.
You will be using sensitive and expensive research equipment, handle it with care.
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