19. (Part 1) The Use of Intense Femtosecond Lasers in Atomic Physics
Dr. Jason B. Greenwood
School of Maths and Physics, Queen’s University Belfast
Email: j.greenwood@qub.ac.uk
1. The Basics of Ultrafast Optics
1.1 Introduction
Femtosecond lasers are used in atomic and molecular physics mainly for two reasons. Firstly,
they provide ultrafast “shutter speeds” for imaging dynamics at the atomic and molecular
level as the movement of atoms in molecules due to vibration, rotation or in chemical
processes occurs on femtosecond timescales. This application led to Ahmed Zewail being
awarded the Nobel Prize for Chemistry
(nobelprize.org/nobel_prizes/chemistry/laureates/1999). Secondly, by squeezing energy into
such a short timescale, high intensities and therefore high electromagnetic fields can be
produced. By focussing these laser pulses, electric fields which are comparable to the binding
energy of an electron in a H atom can be routinely produced. For instance, if a 1 mJ pulse of
duration 25 fs is focussed down to a diameter of 20 μm = 2  10-3 cm, the intensity I is
0.001
I
= 1.2  1016 W cm-2
2

3
15


  25  10
4 2  10
To put this in perspective, the intensity of sunlight when it reaches Earth is about 0.1 Wcm-2.
You would need a magnifying glass of roughly diameter of the Earth to focus sunlight to a
similar intensity.
The electric field E can be found from
I  12  0 cE 2
E = 3.0  1011 V m-1
The atomic unit of electric field (the electric field experienced by an electron in an H atom in
its ground state) is 5.1  1011 V m-1.
It should also be noted that the use of femtosecond lasers to produce stable optical frequency
combs, currently provides the most accurate measurements of fundamental constants. The
Nobel prize was awarded for this work in 2005
(nobelprize.org/nobel_prizes/physics/laureates/2005).
1.2 Bandwidth of Femtosecond Laser Pulses
In order to produce ultrashort pulses, this necessarily means that the bandwidth of the laser
must be larger than the minimum determined by the uncertainty principle
E t  
 t  1
The time profile and spectrum of the pulse are related by a Fourier transform
Spectrum
Irradiance vs. time
Long pulse
FT

time
frequency
FT-1

Short pulse
time
frequency
Figure 1: The time-frequency relationship between pulse length and spectral bandwidth
For a pulse with a temporal Gaussian intensity profile, with t measured as the full width half
maximum (FWHM), the relationship is  t  2.76 . For a pulse with central wavelength of
800 nm and 25 fs FWHM this corresponds to a bandwidth of 37 nm.
1.3 Kerr Lens Mode Locking
A single pulse bouncing in a laser cavity results in the emission of an infinite train of pulses
with temporal separation given by the round trip time in the cavity. A Fourier transform of
this gives a comb of laser modes which are modulated by an envelope which depends on the
pulse width (Figure 2).
E(t)
Laser Cavity
Period T
t
F()
 FT
Laser modes
separated by 2 /T

Figure 2: Spectral modes of a short pulse in a laser cavity
However, if the laser modes have random phases, there is continuous wave emission. To
produce an ultrashort pulse, the modes must be in phase (mode-locked). This gives a high
intensity at the centre of the pulse which drops off sharply with time as the different
frequencies get out of phase.
There are a number of methods for achieving mode-locking. One of the most common is the
use of Kerr lensing. Kerr lensing is a non-linear effect in a material in which the refractive
index increases with intensity. The centre of an intense pulse experiences a larger phase delay
due to the increased refractive index and the medium acts like a lens causing self-focussing.
When the modes are locked, the intensity is high, and this is used to remove out of phase
modes which lead to a continuous emission. In Figure 3 an aperture is introduced to the
cavity so that only the pulsed, high-intensity modes are focussed through.
Figure 3: Kerr Lens Mode-Locking
1.4 The Titanium:Sapphire Laser
This is the femtosecond lasing medium of choice in research laboratories. The reason it is
favoured is due to its large gain bandwidth and good thermal stability properties. The peak
gain is around 800 nm but the medium will lase from 700 – 1000 nm (Figure 4). It is
typically pumped by the second harmonic of a Nd:YAG laser (532nm).
The Kerr lens mode-locked laser cavity is the basis of most Ti:Sapphire lasers and in
principal can produce pulses which are as short as 5 fs (2 optical cycles at 800 nm). In order
to achieve this, the dispersion in the crystal which tends to lengthen the pulse must be
accounted for. Typical pulse energies from the oscillator are in the nJ range.
Gain
700
800
Figure 4: Gain curve of a Titanium: Sapphire crystal
900
Wavelength
1000 (nm)
1.5 Dispersion
When the pulse passes through a non-absorbing medium, e.g. glass, it lengthens due to
positive dispersion caused by the wavelength dependent refractive index. This introduces a
positive “chirp” to the pulse, whereby longer wavelengths are at the front of the pulse and
shorter wavelengths at the rear.
Glass
Figure 5: The effect of dispersion on a femtosecond pulse
For instance, an unchirped pulse of duration 30 fs will be stretched to about 45 fs after
passing through 1 cm of fused silica. For an ultrashort 5 fs pulse, dispersion in air even needs
to be taken into account. A positive or negative chirp can also be achieved using the
dispersive properties of gratings and prisms.
1.6 Chirped Pulse Amplification (CPA)
Until the mid-1980s, the highest intensities needed for strong field physics could only be
produced at large laser facilities. When a short pulse is amplified, the increase in intensity
introduces non-linear self-focussing which damages the gain medium. The only way around
this was to expand the laser beam to very large diameters (meters as oppose to cm) to reduce
the intensity. The development of CPA enabled university labs to access intense laser pulses
using “table-top” devices.
In the CPA scheme (Figure 6), to reduce the intensity in the gain medium, the pulses
produced from the oscillator are expanded in time rather than space. In the stretcher, gratings
(or prisms) are used to create a shorter path length for longer wavelengths (red) than for short
wavelengths (blue) to generate a positive chirp. The reverse is achieved in the pulse
compressor.
Pulses are amplified either with a regenerative amplifier (a cavity with low gain and many
passes through the amplifying crystal) or a multi-pass amplifier. The high gain, multi-pass
amplifier shown in Figure 7 is favoured for producing short pulses as fewer passes through
the crystal introduces less dispersion. However, as the gain is not constant for all wavelengths
(Figure 4), the bandwidth of the amplified pulse is reduced, limiting the pulse length to more
than 20 fs. Typical pulse output energies will be in the region of 1 mJ at a repetition rate of 1
kHz. Higher pulse energies can be achieved with further amplification stages at lower
repetition rates.
1.7 Ultrashort pulses ( < 20 fs)
To produce even shorter pulses, it is necessary to broaden the frequency spectrum of the
pulse. This can be achieved by passing the pulse through a hollow capillary filled with a
noble gas at several atmospheres. The pulse polarizes the gas which generates a non-linear
response. As the refractive index becomes dependent on intensity, a frequency dependent
phase is produced which generates a frequency shift. The leading edge of the pulse becomes
“redder” while the trailing edge becomes “bluer”.
The spectrally broadened output of the capillary will be chirped but can be compressed down
to 5 fs using chirped mirrors. A chirped mirror possesses multiple dielectric mirrors which
reflect different wavelengths at different depths. These variations in optical path lengths for
different wavelengths are introduced for re-compression.
Figure 6: A Chirped Pulse Amplification schematic. Pulses from the oscillator are stretched by a
factor of more than 1000, amplified and then recompressed to produce a high-energy ultrashort pulse.
Figure 7: Multi-pass Amplifier: To limit the heat load on the crystal, most of the incident pulses are
rejected using a pulser picker - Pockel’s cell (PC). In this arrangement, the beam “walks” across the
mirrors while passing through the crystal about 10 times before being ejected.
1.8 Pulse Measurement
Knowledge of the laser pulse length is not only important for determining the intensity of the
radiation but the interaction timescale determines how the field couples to the system. For
instance in molecular dynamics, the fastest nuclear motion (10s of femtoseconds) occurs for
vibrations of covalently bonded hydrogen atoms. Therefore, when molecules are irradiated by
ultrashort ( < 10fs) pulses, the atoms can be considered frozen and molecular dissociation
during the pulse is suppressed. It may also be important to measure the pulse envelope as
variations in intensity can dynamically influence the outcome of the interaction.
For ns lasers, this can be performed using fast photo-diodes, but at femtosecond timescales
detectors are too slow to directly measure the pulse. The pulse itself is the only diagnostic
with which a fast enough measurement can be performed.
1.8.1 Autocorrelation
In autocorrelation, the pulse is split into two, passed through the arms of an interferometer
which introduces a delay  , overlapped and detected (Figure 8). If the electric fields of the
pulses are E(t) and E(t-), when they are overlapped the intensity is
I  E t   E t     E t   E t     2 E t E t   
2
2
2
The set-up shown in Figure 8 is an example of an intensity autocorrelator where the pulses
are overlapped at a slight angle in a non-linear crystal to generate second harmonic radiation
( I 2). Only the second harmonics emitted along the axis due to the temporal overlap of the
pulses (the final term in the above expression) are detected. Therefore, the signal is given by




2
S     E t E t    dt   I t I t   dt
By varying the delay  , the width of the initial pulse can be determined from this
autocorrelation function which is just a convolution of the pulse intensity with itself. For
instance, the autocorrelation of a Gaussian pulse is a Gaussian with a FWHM 2 wider than
the incident pulse.
Input Pulse
Beamsplitter
Aperture
Delayed
Pulse
Detector
Delay 
Lens
Non-linear
crystal
Figure 8: An intensity autocorrelator
1.8.2 FROG
The limitation of an intensity autocorrelation is that any structure in the pulse envelope is
effectively washed out in the convolution. In the most common implementation of a
Frequency Resolved Optical Grating (FROG), autocorrelation signals are obtained as a
function of the second harmonic frequency. Therefore, the detector in Figure 8 is replaced by
a spectrometer. This signal can be plotted in two dimensions as a spectrograph. In effect the
replica is used to sample the time dependent frequency of the pulse. An iterative algorithm is
used to extract the time dependent frequency and phase, and hence the intensity envelope of
the pulse (see www.physics.gatech.edu/gcuo/subIndex.html for more detail on pulse
characterization using FROGs).
Another, more complicated technique which can reproduce the electric field of the pulse
using a non-iterative algorithm for single pulses is Spectral Interferometry for Direct Electric
Field Reconstruction (SPIDER). This was developed by Ian Walmsley’s group in Oxford and
information on it can be found on their website ultrafast.physics.ox.ac.uk/spider/index.html.
1.9 Pulse Shaping
A FROG is useful in characterizing pulses which have complicated characteristics. Such
pulses are being used in the study of coherent control of quantum systems. Variations in the
pulse envelope can used to optimize the probability of the system emerging in a particular
final state. For instance, a molecule ABC might have dissociation channels AB + C or A +
BC. A strong laser pulse can be tailored to modify the potential energy surface dynamically
so that one or other of the channels is favoured [1].
Femtosecond pulses are usually shaped by introducing different phase delays for the
frequency components. A dispersive device such as a grating creates a spatial map of the
frequency components which can then be delayed independently. This delay can be
introduced through variations in refractive index as a function of position using an acoustooptical modulator or an array of liquid crystal modulators as shown in Figure 9. Electric
fields can be applied to individual pixels to vary both the phase and amplitude of each
frequency component. When recombined with another grating, the resulting pulse will have a
Amore
phase
mask
selectively
delays colors.
complicated
temporal
shape.
Figure 9: Frequency dependent phase delay introduced by a liquid crystal modulator array.
An amplitude mask shapes the spectrum.
1.10 A Focussed Gaussian Laser Pulse
To generate the high intensities necessary for experiments, the pulse is focused into the
atomic or molecular target. A drawback of this approach is that the interaction volume is not
irradiated with a uniform intensity. Indeed every atom or molecule will experience a variation
in intensity with time due to the pulse envelope. In an experiment this makes it very difficult
to observe processes which are strong functions of intensity, such as changes in ionization
rates enhanced by intensity dependent resonances.
For a beam with a Gaussian spatial profile, its diameter D is defined as the width at which the
intensity has dropped to e-2 of the central maximum. When focused by a lens or mirror of
focal length f, at a distance z from the focal point and r from the axis, the intensity is given by
I r , z , t  
I t 
1  z / z 0 
2

2r 2
exp   2
2
 w0 1   z / z 0 






where I(t) is the intensity envelope at the focus (r = 0, z = 0), w0 is beam waist diameter, and
z0 is the Rayleigh length defined as the distance at which the intensity drops by a factor of 2
along the beam axis
w02
2f
w0 
, z0 
D

The variation in intensity at the laser focus is shown in Figure 10. When focused into a gas
jet, there is only a very small volume of gas being irradiated at the highest intensities. If
ionization is being measured, although the rate depends strongly on intensity, the greatest
contribution to the signal may be coming from the lower intensity “shells” which have a
larger interaction volume.
Figure 10: Isointensity Contours for an 800nm laser of 1mJ pulse energy, 50fs pulse length focused
by a 250mm focal length lens with a beam size at the lens of 10mm.
2. Atoms in Intense Femtosecond Laser Fields
2.1 Ionization
Excitation or ionization of an atom can be induced if the coupling between the ground state
and an excited or continuum state is sufficiently strong. In a weak field excitation only occurs
if the frequency of the radiation is resonant with the frequency difference of the states and
selection rules are obeyed. In a strong field it is possible to ionize an atom even though the
energy of one photon in insufficient, but may proceed by the absorption of many photons
through Multi-Photon Ionization (MPI). But how can a photon be absorbed if it is not
resonant with an intermediate level?
It is possible for an atom to absorb a photon of energy  to form a virtual state, for a time
interval t governed by the uncertainty principle
.t  
t 
1

For an 800 nm photon (T = 2.5 fs,  =1.55 eV, 0.057 a.u.), t  0.5 fs. By squeezing a large
number of photons in time and space, the probability of this virtual state absorbing another
photon is high. If the numbers of photons N absorbed is greater than the ionization potential,
an electron is liberated with a kinetic energy
Ee  N  I P
Since the probability of absorbing the “next” photon is proportional to the intensity of the
laser, the ionization rate is proportional to IN (can be derived from perturbation theory).
Electron
Yield
0



Electron energy Ee
Figure 11: Electron energy spectrum obtained from multi-photon ionization
An electron energy spectrum produced from this MPI process is shown in Figure 11. You
will notice that there are a series of peaks in the spectrum which indicates that it is possible to
absorb more photons than needed for ionization in a process known as Above Threshold
Ionization (ATI). By conservation rules, a free electron cannot absorb energy from an applied
field, but in this case an electron ionized to the continuum is not strictly free as it is still under
the influence of the Coulomb field of the ion it has left behind.
Although MPI proceeds when the field is “strong”, the electric field of the laser is still
relatively weak compared to that binding the electron to the atom. Eventually, perturbation
theory breaks down and energy levels are modified by AC stark shifting. The ionization limit
of the atom is effectively raised as the total energy of the electron in the continuum must
include the classical quiver energy of an electron in an oscillating field (the ponderomotive
energy UP). UP can derived from the classical equations of motion. If initially the electron is
at rest
mx  eE cos t
eE
sin t
m
eE
x
cos t  x0
m 2
The ponderomotive energy UP is the cycle averaged kinetic energy
2 / 
e2 E 2
U P   12 mx 2 
4m 2
0
In terms of intensity
I  12  0 cE 2
x 
UP 
e2 I
e 2 I2

 9.3  10 6 I2 eV
2 0cm 2 8 0c3m 2
For an intensity of 1013 Wcm-2 at 800nm, UP  0.6 eV. As UP increases, it is possible to
suppress some of the ATI peaks in a process known as channel closing. At an intensity of
1015 Wcm-2, UP  60 eV and the amplitude of a free electron is much larger than the size of
the atom
1/ 2
xmax
eE
e2  2 I 




m 2 4 2 mc2   0c 
Weak field MPI
 1.36  10 6 2 I 1 / 2  2.7  10 9 m  50 Bohr radii
Strong field TI
Figure 12: The atomic potentials for different ionization regimes
Figure 12 shows the effect of the classical field of the laser on the atom in this regime at a
particular instant of the laser cycle. Ionization can now proceed via tunnelling of the electron
and is most likely to occur at the peak of the field. To quantify the boundary between MPI
and tunnel ionization (TI), we can use the Keldysh parameter  which is the ratio of the
orbital rate of the electron to the tunnelling rate. It is given by
1/ 2
 I 
   P 
 2U P 
As a general rule MPI ( > 1), TI ( < 1). For  << 1, the barrier may be completely
suppressed so that the electron can move over the barrier (OB). For an H atom in a 800 nm
field
Intensity (W cm-2)

1012
10.6
Ionization
Mechanism
MPI
1013
3.4
MPI
1014
1.1
MPI/TI
1015
0.34
TI
1016
0.11
OB
2.2 Electron Spectroscopy Using Time of Flight
As the interaction with a short pulse laser gives a well defined temporal and spatial
interaction region, electron energy spectroscopy is often performed using time of flight
techniques. The technique has been used in mass spectrometry for many years, but its use in
electron spectroscopy requires much faster detectors and associated electronics to resolve the
arrival times of light, fast electrons.
The principal of a field free time of flight is very simple. A clock is started when the laser is
fired, and the time t taken by the ionized electron to travel along a flight tube of length L is
measured.
1/ 2
 m 
t  L

 2E 
where L is the length of the tube and E is the kinetic energy of the ejected electron. The
resolution is given by
L m 
t    3 
2  2E 
E 1  8E 
 

E
L m 
1/ 2
E
1/ 2
t
For modern detectors and electronics, time resolutions of t < 1ns are routinely achievable. If
t = 0.5ns, E = 50 eV, L = 0.4m then E  0.5 eV.
Electron spectra can be more efficiently obtained using a magnetic bottle mass spectrometer
[2] shown in Figure 13. This device uses a strong magnetic field for 2 collection (up to 4 if
an electric field is used) of the ejected electrons. The magnitude of the magnetic field is
adiabatically reduced as the electrons move towards the detector so that the electron
experiences a small change in field strength for each cyclotron orbit it undergoes. For this
condition, if the electron is initially emitted at an angle i to the time of flight axis, to
conserve its angular momentum the angle just before it is detected f is given by
sin  f
1/ 2
 Bf 

 
sin  i  Bi 
where Bi and Bf are the initial and final field strengths. If Bi/Bf is large, all the electrons have
f close to zero and have similar path lengths in the spectrometer. The magnetic field is
typically generated using a permanent magnet close to the interaction region, while the time
of flight tube is surrounded by a coil which produces a weak uniform field along the axis and
by -metal shield which excludes any external fields.
Figure 13: Magnetic Bottle Mass Spectrometer used at the Free Electron Laser at DESY (see Chapter
9, part 2 for details of this experiment)
2.3 Electron Re-collision
When an electron is ionized, it is still under the influence of the Coulomb potential of the
residual ion. As the electric field of the laser is periodic, the force on the electron will change
direction which accelerates the electron back towards the ion. A purely classical analysis of
the trajectory shows that depending on the phase of the field at which the electron is “born”,
it will either quiver away from the core or return with an energy dependent on the phase[3].
The maximum return energy (3.17UP) is produced when the electron is liberated just after the
field has gone through a maximum.
In this re-collision process the electron can be elastically scattered, cause ionization leaving a
multiply charged ion, or recombine generating an energetic photon. Signatures of these
processes can be found in experiments which study the electrons, ions and photons produced
from the laser interaction. For instance, if the ion charge state yield is measured as a function
of the laser peak intensity, the behaviour shown in Figure 14 is typical.
At low intensity, single ionization increases very steeply due to multi-photon ionization. As
the intensity is increased, tunnel ionization takes over, until saturation is reached where there
is a 100% probability of ionizing an atom in the laser focus. Above this intensity the ion yield
still increases in proportion to I1.5 due to the expansion of the iso-intensity surfaces of the
laser focus (Figure 10). The doubly charge ion can be produced if the intensity reaches a
sufficiently high value during the remainder of the laser pulse (sequential ionization).
However, a direct double ionization process can proceed via the re-collision mechanism
which contributes to the enhanced signal at lower intensities than expected for sequential
ionization (the “shoulder” in the data).
Ion Yield
106
Saturation
OB
X X+
 I 1.5
TI
105
Re-collision
X  X2+
104
Sequential ionization
X  X+X2+
103
102
MPI
 IN
101
100
1012
1013
1014
1015
Intensity I
(Wcm-2)
Figure 14: Single and double ionization of an atom as a function of intensity
This process has been extensively studied experimentally using the technique of COLd
Target Recoil Ion Momentum Spectroscopy (COLTRIMS) [4,5]. In Figure 15 a cold atomic
or molecular beam is crossed with the focussed laser pulse. Perpendicular to both, an electric
field is applied to extract the positive ions produced towards a 2D imaging detector while
electrons move in the opposite direction towards a second detector. To ensure complete
collection of the electrons an additional weak magnetic field is applied parallel to the electric
field. From the hit position and time of flight of the electrons and ions, the momentum of the
ejected electrons and recoiling ions can be determined giving a kinematically complete
picture of the interaction.
Figure 15: COLTRIMS experiment for studying the ionization and dissociation of atoms and
molecules by intense laser fields see reference [5]
2.4 High Harmonic Generation
A re-colliding electron can alternatively recombine to form an atom again with the emission
of a photon at a harmonic frequency (High Harmonic Generation HHG). As the atom is
initially centro-symmetric, by parity conservation the emitted harmonic must be odd. Using
the classical picture, the maximum energy of an emitted harmonic is IP + 3.17UP. This
coherent light source has a number of potential applications such as lithography and
biological imaging. At energies just above the carbon K shell edge at 284 eV (harmonic order
183), the photons are relatively transparent to water but are strongly absorbed by biological
structures (the “water window”). Emission of photons with energies greater than 1 keV has
already been demonstrated in the laboratory [6].
Closed shell noble gases like He and Xe do not have a permanent dipole moment, but when
an electric field is applied, a dipole is induced which can become large if the electron is
drawn away from the atom as in the re-collision process. The expectation value of the
induced dipole yields the time evolution of emitted radiation field, and the square of its
Fourier transform yields the intensity spectrum. As Xe has a larger polarizability than He, the
harmonic emission is stronger for Xe, but He can produce higher harmonics as it is more
resistant to depletion through ionization and can experience a higher intensity.
The HHG generation process is inefficient as the phase properties of the emission depend
strongly on the intensity, which varies across the laser focus. Therefore, only a small part of
the interaction volume results in coherent emission. To maximise the emission, a pulsed gas
jet is injected at high pressure into the vacuum chamber. Conversion efficiencies of around
10-5 have been achieved.
2.5 Attosecond Pulses
In the HHG process, the highest harmonics can only be emitted for a small range of phases of
the laser cycle where the re-collision energy is at its highest. For a laser pulse with many
cycles, if the lower energy harmonics are removed, the emission is a train of short attosecond
(< 1 fs) pulses separated by half the period of the driving frequency (Figure 16). As the
classical orbital time of an electron in the ground state of a H atom is about 150 as, this opens
up the possibility for direct imaging of the motion of electrons in atoms, molecules and
surfaces.
However, to use attosecond pulses to generate “movies” of electronic phenomena, it is
necessary to use a single attosecond pulse at a time. To generate one attosecond pulse per
laser pulse the emission of the high harmonics must be limited to a single laser half-cycle.
This can be achieved by either making the laser pulse very short, or by dynamically altering
the laser polarization during the laser pulse so that it varies from circularly polarized to linear
to circular. In the latter case, elliptically polarization causes the electron to orbit the core
rather than collide and high harmonics are suppressed. For the former, producing pulses as
short as 5 fs (approximately 2 cycles long) as described in section 1.6 is necessary but not
sufficient for the production of a single attosecond pulse.
Electron “born”
18 after peak
2.67 fs @ 800 nm
Phase 0
180
360
Electron returns 235
after “birth”
E field
of highest
harmonics
Time
Figure 16: Electron trajectories yielding maximum re-collision energies and hence highest harmonics
The phase  of the carrier frequency with respect to the pulse envelope must be considered.
In Figure 17(a), the electric field is strongest at the centre, resulting in the highest harmonics
being produced for the trajectory shown. In Figure 17(b), an electron can be accelerated by
the maximum electric field at two points in the laser pulse producing two attosecond pulses.
Stabilizing this carrier-envelope phase is an essential part of the attosecond generation
process. When a phase of zero (or 180) is achieved, as the pulse produced is very short it has
a very broad bandwidth and indeed can no longer be considered a harmonic. The signature of
the attosecond pulse is a continuum in the spectrum (Figure 18).
By isolating the continuum emission with a filter, attosecond pulses in the extreme ultraviolet (XUV) of energy around 100 eV, bandwidth 10 eV, and duration about 200as, have
been produced [7].
Carrier-envelope phase  = 0
Single attosecond pulse
(a)
Carrier-envelope phase  = 90
2 attosecond pulses produced
(b)
Figure 17: 5 fs pulses with different carrier-envelope phases with electron trajectories which produce
the highest harmonics
Intensity
Lower harmonics
removed by filter
Continuum emission
Attosecond radiation
Frequency
Figure 18: High harmonic spectrum obtained from an ultrashort pulse with  = 0
2.6 Attosecond Physics
This field is in its infancy, but the first task is to measure the length of the attosecond pulse.
An autocorrelation process is not possible as there are no efficient non-linear processes in the
extreme ultra-violet. Alternatively, a cross correlation with the parent laser pulse can be
performed. Although the parent pulse is about 5 fs long and the laser cycle is 2.6 fs, the
electric field changes significantly on the attosecond timescale and can used as a diagnostic.
The non-linear process at work is the photoionization of an atom in the presence of this
electric field.
The apparatus used to perform the measurement is shown in Figure 18. The ultrashort, phase
stabilised ( = 0) laser pulses are focussed into an atomic gas jet to produce the attosecond
XUV pulses. The XUV beam has a smaller angular divergence than the laser beam. A filter
which transmits XUV but absorbs optical photons is used to generate a hollow laser beam copropagating with the XUV beam at the centre. Both are focussed by a multi-layer mirror into
another gas jet. The mirror has two components with the central portion capable of being
driven in 30 nm steps to introduce delays to the XUV pulses at 100as intervals. The energy
spectrum of the electrons ejected along the polarization direction is obtained by s field-free
time of flight spectrometer (TOF).
Figure 18: Experimental apparatus used to produce attosecond pulses and perform pump-probe
measurements of ionization [8], see www.attoworld.de
A free electron in a laser field will gain no momentum, unless it is injected into the field
instantaneously. This is the situation when the attosecond XUV pulse ionizes an electron
from an atom. In the field free case the electron has momentum
p0  2m X  I P 
where IP is the binding energy of the electron and X is the energy of the XUV photon. With a
time-dependent electric field present E0 f t ' cost ' , where E0 is the peak electric field,
f(t’) is the pulse envelope function (-1 > f(t’) > +1)and  is the carrier-envelope phase, the
electron will gain additional momentum p along the direction of polarization of the laser
pulse. If the electron is born at a time t,


t
t
p   mx dt '   eE0 f t ' cos t ' dt '
If the rate of change of the pulse envelope is much less than that of the carrier wave (a good
approximation even for ultrashort pulses), then
eE f t 
p  0
sin t

Only electrons ejected along the polarization direction, which is parallel to the axis of a time
of flight spectrometer, are detected. The measured energy of these electrons is given by
2
p 2  p 0  p 
T  T0  T 

2m
2m
where T0   X  I P
T 
p 0 p
m
T eV  
2T0 E 0 f t 
sin t
m

T eV   2.5  10 10 T0 eV  E 0 f t sin t
For a peak intensity of 6.5  1014 Wcm-2, E0 = 7  109 Vm-1 and T0 = 71.5 eV,
T  15 f t sin t eV
Therefore the time evolution of the electric field f t cos t can be directly re-produced by
differentiation the electron energy spectrum. This demonstration of the fastest measurements
ever made is shown in Figure 19(a). The reconstructed electric field shown in Figure 19(b)
can be seen to have the zero phase expected for the production of the single attosecond pulse.
Figure 19: (a) Electron energy distribution produced from ionization of Ne with a 93 eV attosecond
pulse in the presence of an ultrashort laser pulse. (b) Electric field of the laser pulse reconstructed
from differentiation of (a), see reference [9]
These techniques have already been used to directly measure the lifetime of an M-shell
vacancy in Kr [10]. This shows the future potential for measuring electronic quantum
phenomena with unprecedented time resolution.
2.7 Femtosecond and Attosecond Resources in the UK
The UK is at the fore-front of the study of ultra-fast processes. Attosecond sources are under
development through an RCUK Basic Technology programme led by Imperial College(IC)
with Reading, Oxford, University College London (UCL), Birmingham and the Rutherford
Appleton Laboratory (RAL) as partners www.attosecond.org. The Central Laser Facility in
UK is located at RAL and intense, femtosecond pulses are available at the ASTRA laser.
Current specifications of this laser system are given below and through involvement in the
attosecond programme, will be upgraded to include carrier-envelope phase stabilized pulses
and an attosecond XUV source.
Laser
ASTRA
TA2
ASTRA TA1
Repetition Rate
1kHz
1kHz
10Hz
10Hz
10 Hz
Compressor
Prism
Prism then
Fibre
Grating
Grating
then Fibre
Grating
Pulse Length
30fs
10fs
35fs
10fs
40 fs
Energy per Pulse
0.8mJ
0.3mJ
16mJ
0.5mJ
500 mJ
Max Intensity
Wcm-2
1016
1016
1017
1016
1019
Current specifications of the ASTRA laser at RAL
http://www.clf.rl.ac.uk/Facilities/AstraWeb/AstraMainPage.htm
There are a number of other University based high power femtosecond laser systems in the
UK. These include (not an exclusive list)

IC
www.imperial.ac.uk/research/qols

Strathclyde
tops.phys.strath.ac.uk/main.html

Reading
www.ull.rdg.ac.uk/
UCL and Queen’s University Belfast also have systems in development.
References
[1] Quantum control of gas-phase and liquid-phase femtochemistry
T. Brixner and G. Gerber, ChemPhysChem 4, 418 (2003)
[2] Magnetic field paralleliser for 2 electron-spectrometer and electron-image magnifier
P. Kruit and F.H. Read, J. Phys. E, 16, 313 (1983)
[3] Ionization Dynamics in Strong Laser Fields
L. F. DiMauro and P. Agostini, Adv. At. Mol. Opt. Phys., 35, 79 (1995)
[4] Multiple Ionization in Strong Laser Fields
R. Dörner et al., Adv. At. Mol. Opt. Phys., 48, (2002)
[5] Multiple fragmentation of atoms in femtosecond laser pulses
A. Becker, R. Dörner and R. Moshammer, J. Phys. B,. 38, S753 (2005)
[6] Source of coherent kiloelectronvolt X-rays
J. Seres et al., Nature, 433, 596 (2005)
[7] Attosecond Metrology Comes of Age
R. Kienberger and F. Krausz, Physica Scripta., T110, 32 (2004)
[8] X-ray Pulses Approaching the Attosecond Frontier
M. Drescher et al., Science 291, 1923 (2001)
[9] Direct Measurement of Light Waves
E. Goulielmakis et al., Science, 305, 1267 (2004)
[10] Time-resolved atomic inner-shell spectroscopy
M. Drescher et al., Nature, 419, 803, (2002)