Generation, Characterization and Applications of Femtosecond Electron Pulses by Christoph Tobias Hebeisen A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto Copyright c 2009 by Christoph Tobias Hebeisen
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Generation, Characterization and Applications of
Femtosecond Electron Pulses
by
Christoph Tobias Hebeisen
A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy
Graduate Department of PhysicsUniversity of Toronto
4.6 Time traces of the fit results . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.7 Probe electron deflection for different laser fluences . . . . . . . . . . . . 92
viii
Chapter 1
Introduction
Nunc age, quo motu genitalia materiai
corpora res varias gignant genitasque resolvant
et qua vi facere id cogantur quaeque sit ollis
reddita mobilitas magnum per inane meandi,
expediam: tu te dictis praebere memento. 1
Lucretius, De Rerum Natura 2:62-66
The concept that all matter is made up of small particles or “atoms” (�tomon = indi-
visible) was developed in the framework of religious and natural philosophies in ancient
India and Greece. The mythical founder of the V aise˙sika system of Hindu philosophy
Ka˙nada (possibly “atom eater”), who lived somewhere between the 6th and 2nd cen-
turies BC, taught that atoms exist, are spherical and have no cause [1]. Democritus
(Dhmìkrito ) of Abdera (*460 BC) and his teacher Leukippus (LeÔkippo ) of Miletus
thought that only atoms and the void are real and that different macroscopic substances
could be explained by different shapes and arrangements of the atoms [2]. Democritus’
atoms are solid without any internal structure. They are in constant motion and can
undergo “chemical reactions” when they collide, by forming clusters when hooks that
1“Now then, I will explain through which motion the seeds of substance produce the different thingsand dissolve the produced, further by what force they are compelled and through which vessels each isgiven the speed to travel through the vast void: remember to live by these words.”
1
Chapter 1. Introduction 2
they carry on their surface get entangled [3].
These early atomistic views were not supported by observations of any effects related
to actual atoms and even made some untenable assertions (e.g. the lack of internal struc-
ture), yet the basic concept of matter consisting of discrete particles inspired modern
atomism and is still valid even though the limits of what is considered indivisible have
shifted. In the early 1800s, John Dalton published his observation that in chemical re-
actions, elements combine in certain fixed proportions. He was able to derive atomic
weights in relation to hydrogen from these observed proportions [4]. This is generally
seen as the beginning of modern atomic theory, which underlies all modern natural sci-
ences and applied science and has become a pervasive scientific concept – even outside of
academia. Computer-generated renderings of ball-and-stick models of complex biological
molecules have become a staple not only in scientific publications but also in the science
section of newspapers, providing a concrete and intuitive (as well as aesthetically pleas-
ing) illustration to experts and laypeople of what a molecule or crystal structure “looks
like”.
The static structures of crystalline matter and of many molecules – even very large
ones – are known from electron diffraction, x-ray diffraction or nuclear magnetic reso-
nance spectroscopy. Crystal structures can also be observed by transmission electron
microscopy (TEM), and scanning tunnelling microscopy (STM) allows the detection of
even single atoms on a surface. Despite the seeming ease of observing atomic structure
and locating even single atoms, none of these techniques allows the observation of atomic
structure at arguably the most interesting time, i.e. during a chemical reaction in real
time. This limitation is mostly due to the timescales involved in structural changes on
the atomic level. Chemical bonds are made or broken on the timescale of 100 fs, the
approximate time it takes two atoms to move far enough apart to break a bond, or the
periods of the most highly damped motions along structural relaxation coordinates [5].
Chapter 1. Introduction 3
The time it takes to obtain structural data from any of the above-mentioned methods is
on the order of seconds to milliseconds at best, which is many orders of magnitude larger
than the timescale required to resolve motions involved in structural changes.
In order to record the progress of any process, one usually obtains a series of “snap-
shots”. These snapshots can be literal images (e.g. in the case of video rate microscopy),
or a single or multiple observables (e.g. reflectivity or absorptivity of a sample) that
are defined for each point in time. Cameras or other detectors fast enough to observe
changes on the 100 fs timescale are not available. However, if the measurement of a
quantity involves a “probe” (e.g. the illumination of the sample with a femtosecond laser
pulse), a short pulse probe can be used to obtain the same effect. In the absence of the
probe, the detector (which integrates the signal over a time much longer than the desired
timescale) does not receive any signal. A pulsed probe allows one to obtain a snapshot
of the process at the exact time of the probe pulse. Since the detector is much slower
than the process to be observed, only one snapshot can be taken at a time. To obtain
the time-series, the measurement has to be repeated many times with different delays
between the initiation of the process and the probe pulse. The initiation of the process
is accomplished by another pulse called the “pump” pulse. Hence, techniques based on
this principle are called time-resolved pump-probe techniques.
The duration of the probe pulse fundamentally limits the temporal resolution of any
pump-probe technique. Since the detector integrates the observable of interest over the
duration of the probe pulse, the measured quantity does not reflect the state of the
observed system at one point in time but its average state over the duration of the probe
pulse weighted with the intensity profile of the probe. This is analogous to the shutter
speed of a camera when trying to obtain a picture of a moving object (Fig. 1.1).
Since only one snapshot can be taken at a time, the experiment has to be repeated
for every time step. Beyond that, every time step may require multiple pump-probe
Chapter 1. Introduction 4
Figure 1.1: The probe pulse duration in a pump-probe experiment is analogous to the
shutter speed of a camera; when a changing state is recorded, the result is integrated
over the duration of the probe pulse / exposure time, respectively. The progress of the
state during the exposure time cannot be resolved. This becomes clear when comparing
these images. The left image was taken with a shutter speed of 1/80 s compared to 1 s
in the right image. The duration of a measurement causes an uncertainty in the position
measurement of moving objects.
cycles or “shots” in order to acquire enough signal. Hence, the number of pump-probe
cycles the sample has to undergo is the product of the number of time steps and the
number of shots needed to obtain a single measurement. This makes sample reversibility
an important consideration for pump-probe experiments. If the sample is fully reversible,
i.e. it can go through an arbitrary number of pump-probe cycles without changing any
of its properties, this is not a major concern. However, many if not most interesting
processes are not reversible or not fully reversible. Additionally, the probe pulse can
also damage the sample in some cases. Even transitions we normally consider reversible
may not be reversible for the purpose of these experiments. As an example, consider the
melting of a metal as a transition. While the metal will resolidify, there is no guarantee
that it will do so in the same crystal structure and shape. Since the available amount of
Chapter 1. Introduction 5
sample is often limited, it is important to maximize the signal that can be acquired with
a single shot, ideally to the limit where no integration beyond a single shot per time step
is necessary.
Laser pulses in the femtosecond range are now readily available and have been used
extensively to probe chemical reactions in a time-resolved manner. The wavelength of
the light used in these experiments is three orders of magnitude longer than typical
interatomic distances so atomic structure cannot be probed directly by optical means.
Spectroscopic methods can yield information about the potential energy surface in the
vicinity of a transient configuration; however, in all but a few relatively simple cases [6],
the atomic structure of the sample cannot be inferred from this information. Hence, if
we want to fully understand conformational changes on the atomic level, it is necessary
to combine the spatial resolution of a diffraction technique with the temporal resolution
of ultrafast optical spectroscopy. Both x-ray diffraction and electron diffraction can in
principle be used to achieve this aim. In the last few years, enormous strides have
been made in making such experiments possible, using ultrafast laser excitation pulses
to trigger structural changes and either x-ray or electron probes to follow the induced
atomic motions [7, 8, 9, 10, 11].
X-rays, as a form of electromagnetic radiation, almost exclusively interact with the
electrons of an atom making x-ray diffraction a probe of the electron density of the
sample. Electrons, on the other hand, are scattered by the electric potential in the
sample, which is formed by both the electrons and the nuclei. While x-ray diffraction is
by far the more prevalent technique for static structure determination, both can and have
been used successfully to determine structures. There are some important differences
[12] in the way x-rays and electrons interact with a sample. The elastic mean free
path for 1.5 A x-ray photons is 105 − 106× longer than for 80 − 500 keV electrons. In
transmission geometry, this means that solid samples for electron diffraction must be
Chapter 1. Introduction 6
thinner by this factor than samples for x-ray diffraction. This can be both an advantage
or a disadvantage depending on the affinity of the sample to form thin films or “large”
(≥ 10 µm) crystals, respectively. In experiments that involve an excitation beam, thin
samples are advantageous since it is easier to illuminate them evenly and thereby to obtain
even excitation conditions throughout the probed volume. Even metals, usually thought
of as completely opaque for visible light, have skin depths longer or roughly the same as
the thickness of thin film samples used in transmission electron diffraction (on the 10 nm
scale) for 10 − 100 keV electrons. On the other hand, ultrafast x-ray diffraction with its
thicker samples may suffer from “contamination” of the desired diffraction pattern from
the excited sample with the diffraction pattern of non-excited parts of the sample. This
so-called ground-state contamination must be subtracted from the recorded patterns,
decreasing the signal-to-noise ratio (SNR). Due to the large scattering cross-section,
electron diffraction is also naturally more suitable for low-density, i.e. gas-phase, samples
[13].
Both electrons and x-ray photons can scatter elastically (coherently) or inelastically
(incoherently) from the sample. While elastically scattered electrons / photons form the
diffraction pattern, inelastic scattering leads to a mostly featureless background. Also,
inelastic scattering events deposit much more energy into the sample than elastic scat-
tering events, leading to sample deterioration. The ratio of elastic vs. inelastic scattering
events is about three times higher for electrons while the deposited energy per inelas-
tic scattering event is 1000 times higher in the case of x-rays [12]. Sample damage of
biological specimens during x-ray diffraction is a well-known issue.
Ultrashort x-ray pulses can be generated in a number of different ways. Synchrotrons
produce very bright x-ray pulses in the 100 ps range. While these pulses have been used
for dynamic structure determination [14], they are three orders of magnitude longer than
the timescale of interest. Shorter pulses can be produced by so-called femtoslicing [15], at
Chapter 1. Introduction 7
the cost of the number of x-ray photons. This compromises the main advantage of syn-
chrotron sources, their brightness. Laser plasma sources [16] generate x-rays by focusing
a femtosecond high-intensity laser onto a solid. Fast electrons generated in the resulting
plasma produce x-rays by bremsstrahlung when they hit the solid sample and eject inner
shell electrons of the atoms in the target plasma, giving rise to characteristic-line radi-
ation [17, 18]. While laser plasma x-ray sources have been built using table-top lasers,
sources capable of producing a sufficient number of photons per pulse to follow structural
focused by a lens mounted on a three axis translation stage outside the vacuum chamber.
Translation of the lens along the laser propagation direction can be used to modify the
spot size of the laser on the sample and translation in the lateral directions allows moving
the laser spot on the sample to facilitate the overlap of laser pump and electron probe
beam on the sample.
The laser pulse in the probe arm drives a NOPA that produces a 500 nm pulse,
which is then compressed to 50 fs by a prism compressor. A computer controlled variable
Chapter 2. The Electron Diffractometer 14
delay stage provides the delay between pump and probe pulse and a variable attenuator
serves to control the number of electrons created per pulse. The laser pulse for the
probe arm then excites the photocathode to produce the electron pulse via two photon
photoemission.
The placement of the last mirror in the pump arm inside the chamber directing the
pump beam towards the sample is crucial for the temporal resolution of the experiment.
While the electron pulse hits the sample at normal incidence, the laser pulse arrives at an
angle α with the sample surface normal, therefore exciting different parts of the sample
at different times. The broadening effected by the non-normal incidence is τ = we
csinα.
Here, we is the width of the area probed by the electron beam in the plane spanned by
the sample surface normal and the pump beam. Note that α cannot be made arbitrarily
small because the mirror would have to be placed inside the cone of the diffraction pattern
recorded on the detector. Given the size of the detector (4 cm at 20 cm from the sample)
and practical problems such as the size of mirror mounts, in practise α ≈ 8◦. With a
150 µm diameter electron beam, τ ≈ 70 fs. For a 200 fs pump pulse, this causes only
about 6% broadening but for future experiments with shorter pump and probe pulses,
this broadening could become significant.
2.2 Ultra-Compact Femtosecond Electron Pulse
Source
The electron gun is the heart of the electron diffractometer. It is responsible for producing
the electron probe pulses and delivering them to the sample while keeping them as short
as possible. The electron gun also determines the beam quality of the electron beam and
therefore it is a deciding factor in the quality of the diffraction pattern.
Chapter 2. The Electron Diffractometer 15
Photocathode
Anode
Magnetic lens
E
GND
−V0
Laser pulse Electronpulse
Figure 2.2: Principle of a DC femtosecond electron pulse source.
The basic setup of a DC, photoactivated electron gun is shown in Fig. 2.2. A fem-
tosecond laser pulse hits the thin film photocathode and produces an electron cloud via
the photoelectric effect. The cathode is biased with a negative voltage with respect to
the anode. The resulting electric field between cathode and anode accelerates the elec-
tron pulse towards the anode. After acceleration, the electron beam is collimated by a
magnetic or electrostatic lens before it hits the sample.
2.2.1 Simulations
Space charge broadening of electron pulses is a well-established fact in electron accel-
erators and streak cameras. Intuitively, it acts to broaden the electron pulse in the
longitudinal as well as in the transverse direction. Obviously, the higher the number
density of electrons in the pulse, the stronger the broadening. The force acting on an
Chapter 2. The Electron Diffractometer 16
electron due to the presence of the other electrons is
~Fn =∑
m 6=n
e2
4πε0 |~rn − ~rm|3(~rn − ~rm) (2.1)
where ~rn is the position of the n-th electron in the pulse. To calculate this force, N − 1
evaluations of the Coulomb force are required for each of the N electrons. Since the force
electron n experiences due to electron m and the force electron m experiences due to
electron n have the same magnitude and opposite directions, the forces inside the pulse
can be calculated in N(N − 1)/2, i.e. O(N2), operations. This leads to computationally
very expensive simulations as the number of particles grows. While there are models that
allow to calculate the evolution of an electron pulse analytically [45] or by solving a set of
differential equations [46] by assuming a continuous charge distribution, these methods
are very pulse-shape specific. Simulations, on the other hand, allow calculations with
arbitrary pulse shapes. The obvious way to make this problem computationally tractable
is to lump the mass and charge of several electrons into a so-called macroparticle at the
position of the centre of mass of the electrons represented by the macroparticle. While the
quantization error introduced by this method may be small for 106 ( 3√
N = 100) particles,
it does not necessarily seem acceptable at 103 − 104 particles where 3√
N = 10 − 22.
The consolidation of multiple charges into one particle is obviously a problem for the
force calculation in the vicinity of the macroparticle. However, for charges that are
much further away than their distances from each other, i.e. that occupy a small solid
angle all at very similar distances, a single particle containing all their charge is a very
good approximation. This approach is taken by the Barnes-Hut tree algorithm [47],
first introduced to calculate N-body problems arising in astrophysics. This algorithm
reduces the computational complexity of the N-body problem to O(N log N) by sorting
the particles into a hierarchical tree of cubic cells. Cells containing multiple particles,
which are sufficiently far away, are represented by a single particle with the total mass and
Chapter 2. The Electron Diffractometer 17
charge of the particles contained in the cell. The tree is recreated for each integration
step. Thus, the composition of the groups of particles that are lumped together for
the force calculations changes in each step according to the evolution of the system, as
opposed to the fixed groupings imposed by macroparticles. The equations of motion are
then solved by a simple leapfrog integrator. While the code was written for gravitational
forces with gravitational mass equal to inertial mass, a particular choice of units allows
the use of the code with electrical charges, as long as the charge of all of them has
the same sign, once the sign in the force calculation is switched to turn the attractive
gravitational force into the repulsive Coulomb force between electrons [21].
The Barnes-Hut tree code uses classical mechanics without relativistic corrections.
At 55 keV electron energy, the electron velocity is 43% c and γ = 1.108. No attempt was
made to modify the code to accommodate relativistic corrections since such modifications
would have been extensive and would have required rewriting a large part of the code base.
While relativistic effects cause a significant deviation from non-relativistic mechanics at
the electron velocity, the simulations can still serve as a tool for estimating electron
pulse durations; relativistic effects act to slow the spread of the electron pulse, causing
the non-relativistic simulations to overestimate the pulse durations. In recent years,
commercial programs [48] have become available, which treat the N-body problem fully
relativistically.
This Barnes-Hut tree code is useful for calculating the free propagation of the electron
pulse, but in its original form, it is not able to simulate the electron pulse acceleration.
Space charge effects during this time are particularly important because at this time
the electron pulse is most dense. This is partly because at this time the pulse has the
shortest duration, and partly because it is moving much more slowly at the beginning
of the acceleration, i.e. the electron pulse is shorter in space for a given pulse duration.
Together, this leads to higher electron density and increased space charge. To allow the
Chapter 2. The Electron Diffractometer 18
simulation of an electron pulse in the acceleration region of a DC electron gun, I modified
the force calculation code to add a homogeneous electric field. Since the simulation
proceeds in discrete time steps, a problem arises at the injection into the region with
the field (i.e. at the photocathode) as well as at the anode, where the electrons leave the
electric field. At the photocathode, the electrons of the pulse are “created” at different
times over the duration of the laser pulse but in the same plane. Note that this situation
cannot be emulated by creating an electron distribution in space that mirrors the laser
temporal shape because the electrons at the leading edge have acquired more energy
from the electric field than electrons further behind. In order to construct an electron
distribution that corrects for this problem, we assume that an electron is to be emitted
from the photocathode at t = τ at ~rτ with a velocity of ~vτ while the time step of the
simulation starts at t = 0. Assuming a homogeneous electric field ~E, we get
~r0 = ~rτ − τ~vτ − τ 2 e
2me
~E and ~v0 = ~vτ + τe
me
~E (2.2)
for the position ~r0 and velocity ~v0 at t = 0 (note that e is the elementary charge i.e. the
charge of an electron is −e). Space-charge was disregarded in this calculation. An infinite
plane with a charge density of 1.3 × 108 electrons/cm2 (104 electrons per 100 µm circle)
gives rise to a field of 115 V/cm, which is about 1/1000 of the typical acceleration field.
Hence, for short τ , the influence of space-charge is negligible. A similar problem, which
arises at the anode where the electron pulse crosses from the acceleration region into the
field free drift region, requires an analogous correction of position and velocity of the
individual electrons. A pinhole in the beam path was implemented by stripping away
all electrons outside a certain radius and a mesh was modeled by removing all electrons
whose lateral positions are within the width of the grid wires. The magnetic lens was not
implemented for the simulations since in practise, it is usually placed close to the sample
in an effort to minimize the total electron propagation distance and therefore has only
Chapter 2. The Electron Diffractometer 19
0 mm9 mm
18 mm27 mm36 mm45 mm
−400 −300 −200 −100 0 100 200 300 400
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
−800 −600 −400 −200 0 200 400 600 800
Ele
ctro
n de
nsity
[arb
itrar
y un
its]
t [fs]
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
−800 −600 −400 −200 0 200 400 600 800
Ele
ctro
n de
nsity
[arb
itrar
y un
its]
t [fs]
N=2000N=3900N=7800
N=15000N=28000N=51000
Figure 2.3: Simulated electron pulse shapes. The main graph shows the temporal shape
of electron pulses containing different numbers of electrons (55 kV, 6 mm acceleration,
150 µm pinhole as anode, 23 mm drift region). The inset shows the evolution of the
temporal envelope of an N = 7700 electron pulse from the anode to a 45 mm drift.
minor influence on the pulse duration.
For practical reasons, such as electrical safety and cost, I decided to design the electron
source at a maximum of 60 keV. While it has reached and operated at the design voltage,
normal operation is performed at 55 keV for reasons of electrical vacuum breakdown
stability (vide infra). The current electron pulse source setup is constructed with a 6 mm
cathode-to-anode distance and with a pinhole as the anode. While further shortening of
the acceleration region is very difficult for DC electron guns because of breakdown, the
“drift region” between the anode and the sample can be optimized.
Simulated electron pulse shapes are shown in Fig. 2.3. As expected, the pulse duration
increases as the number of electrons per pulse increases and as the propagation time
Chapter 2. The Electron Diffractometer 20
0 20000 40000
0
200
400
600
800
1000
1200
1400
1600
1800
0 5000 10000 15000 20000 25000 30000
Pul
se d
urat
ion
[fs]
Electrons per pulse
electron pulse sourcePrevious 30 kV
45 mm
36 mm
23 mm
12 mm
0 mm
800
600
400
200
0
Figure 2.4: Pulse durations vs. number of electrons per pulse for various electron gun
configurations from simulations. The black dotted line describes the previous 30 kV
electron gun while the solid blue line describes the current setup (also shown in the inset
for higher electron numbers). The dashed lines stand for the same setup with different
drift region lengths. All configurations use a 150 µm pinhole.
increases. More surprisingly, the pulse shape also changes, becoming more and more
flat-top. This effect is also space-charge related since it occurs more strongly and earlier
in pulses with higher electron densities.
The most commonly used measure for the pulse duration is the full width at half
maximum with FWHM = 2√
2 ln 2σ ≈ 2.35σ for a Gaussian. This conversion technically
does not apply to other pulse shapes, but in practise it is a useful value to estimate the
effect of the pulse shape on the temporal resolution of a pump-probe experiment. We use
this conversion to calculate pulse duration from the simulation results. Electron pulse
durations for different lengths of the drift region are shown in Fig. 2.4.
Chapter 2. The Electron Diffractometer 21
It is obvious from Fig. 2.4 that even for relatively long drift regions the current setup
outperforms the old electron pulse source by a significant margin. In the previous electron
gun, the cathode as well as the anode were executed as Rogowski profiles (vide infra) at
a distance of 3 mm. The electric field was terminated on the anode side by a TEM mesh.
After 11 mm, the electron beam was stripped by a pinhole after which the electron pulse
propagated for another 28 mm through the magnetic lens before striking the sample. The
cathode was biased at −30 kV. The higher acceleration voltage (55 kV) in the current
setup reduces the time the electron pulse spends in the drift region and also increases
the spatial length of the electron pulse for the same duration, thereby reducing the effect
of space charge. Also, in the previous setup the mesh at the anode and the pinhole
11 mm downstream from it increased the number of electrons that were necessary at the
cathode to produce a given number of electrons in the pulse that reached the sample.
These improvements in combination with the shortened drift region allow the dramatic
improvement of the pulse duration evident in Fig. 2.4.
2.2.2 The Photocathode
The electron pulse is generated by a laser pulse impinging on the photocathode, creating
a cloud of photoelectrons. Ideally, the photocathode would be a single crystal bulk
material to guarantee a homogeneous work function across the size of the beam. Given
the geometry of the electron gun (vide infra), front-side illumination of the photocathode
is not possible. Instead, we use back-side illumination of a thin-film photocathode on a
transparent substrate. The substrate of the photocathode is a 12.5 mm diameter sapphire
disk. Since the electrical contact to the photocathode needs to be made from the back
side, a relatively thick layer (usually 500 nm) of chromium is deposited on that side of
the disk, overlapping one edge. A 20 nm gold layer is then deposited on the other side
Chapter 2. The Electron Diffractometer 22
of the photocathode, overlapping the same edge and therefore making electrical contact
with the chromium layer.
The electrons are produced by the photoelectric effect. If the energy delivered by a
photon is higher than the energy required to free the electron from the solid (the work
function), the electron is emitted with excess kinetic energy [49]. This initial kinetic
energy can lead to broadening of the electron pulse [21]. By choosing the wavelength of
the laser pulse so that the energy of the photons is very close to the work function, this
effect can be minimized. The work function of a freshly deposited, clean gold surface is
4.83 eV (λ = 257 nm) [50]. Our photocathodes are produced in a deposition chamber
which is used for various metals and which uses an oil pump. It is transferred to the
electron gun under normal atmospheric conditions. This treatment likely changes the
surface, and the operating wavelength needs to be determined empirically. The NOPA
in our setup produces visible light, so that two-photon photoemission has to be used to
produce electrons. In practise, the NOPA is operated at a wavelength just below the
two-photon cutoff (500 nm or 2.48 eV), at which no electron pulse is observed any more.
2.2.3 Electron Acceleration
As explained in section 2.2.1, space charge effects are strongest during the early stages
of the electron pulse acceleration and hence the electron pulse needs to be accelerated as
quickly as possible. Hence, the electric field between the photocathode and the anode is
maximized. The electric field at the surface of a charged conductor is increased where
the radius of curvature is small, which is particularly the case at any sharp corners.
These electric field spikes can lead to field emission and breakdown. Breakdown not
only disrupts the experiments by discharging the high voltage power supply but also
can do serious damage to the photocathode. The extreme current transients that occur
Chapter 2. The Electron Diffractometer 23
during breakdown may even lead to transient malfunction and permanent damage in
electronic equipment not related to the high voltage part of the experiment by inducing
damaging voltages in signal wires. These effects can be mitigated by careful shielding
and grounding. While macroscopic edges can be avoided in the design of the electrodes,
microscopic roughness can only be reduced by polishing of the electrodes. Despite all
measures to ensure smooth surfaces, some sparking will usually occur. Moderate sparking
can condition the surface by melting or evaporating the roughness on the surface.
In practise, the maximal practical electric DC field that can be obtained is on the order
of 10 kV/mm. Since the electrodes have finite size, some curvature is required. If we do
not want to compromise on the field strength in the centre of an electrode, the electrode
can be made according to a so-called Rogowski profile [51]. This shape ensures that
the electric field has its maximum value in the flat, centre part of the electrode. While
usually both the cathode and the anode are shaped to obey Rogowski profiles, the same
condition for the electric field can be satisfied when making one of the electrodes plane
and designing the Rogowski profile of the other electrode with different parameters. Since
the electrons have to travel through the anode, like all parts along the electron path, it
should be optimized for minimal length. Hence, the anode of the electron diffractometer
is flat while the cathode was originally designed to have the shape of a Rogowski profile.
However, this cathode design is not without problems: since the photocathode itself is a
thin sapphire disk and cannot be made in the shape of a Rogowski profile without undue
technical complexity, the photocathode is usually embedded into a metal piece that has
the shape of a Rogowski profile with a cut-out that fits the photocathode. The obvious
disadvantage of this solution is the seam between the photocathode and the metal profile.
It occurs necessarily in a high-field area and roughness is unavoidable. A different design
ultimately proved to be more practical and stable. The photocathode of the current
setup is mounted inside a macor tube with a lip at the end that covered the edges of
Chapter 2. The Electron Diffractometer 24
the photocathode disk. Macor is a vacuum-compatible, machinable ceramic with a high
bulk dielectric strength. In this design, only the flat part of the photocathode is exposed
while the edges are covered by the dielectric, which prevents breakdown. In practise, this
turned out to be much more stable and easier to handle for photocathode exchanges. In
contrast to the Rogowski profile design, which required gluing the photocathode into the
metal profile, the photocathode is held by a mechanical retaining ring in the macor tube
design.
The original design of the anode used a TEM mesh to allow the electron pulse through
the anode. Besides the limited transmission of the TEM mesh, meshes that separate two
regions of different electric field also distort the electric field around the grid lines, leading
to defocusing of the passing electron beam [52]. Since this effect is much less pronounced
when a pinhole is used instead, significantly improving the beam quality, we use a pinhole
as the anode (typically 150 µm). The size of the pinhole also limits the size of the electron
beam.
High Voltage Power Supply
The output of the high voltage power supply ultimately defines the kinetic energy of the
electrons and hence their velocity. At V0 = 55 keV, the time the electron pulse takes to
propagate from the photocathode to the sample is 271 ps and changes by 2 fs per 1 V
change in the high voltage. Therefore, the stability of the high voltage power supply
must be better than 0.1% in order to keep the timing error introduced by the power
supply below 100 fs. The power supply used in the electron diffractometer 1 guarantees
a drift of less than 0.01% per hour after warm up and 0.02% ripple.
1Glassman - FC60N2
Chapter 2. The Electron Diffractometer 25
2.2.4 Beam Conditioning
In order to obtain a high quality diffraction pattern on the detector, the electron beam
size at the detector needs to be minimized. The electron pulses created by photoemission
and acceleration in a DC field spread during their propagation. Without any measures to
control the electron beam divergence, the electron diffractometer would produce very poor
diffraction patterns. Electrostatic or magnetic solenoid lenses can be used to counteract
this spread and focus the electron beam. Since this focusing element needs to be placed
between the anode and the sample, it is critical to build it as short as possible to keep
the total pulse propagation time short.
The focal length f of the electron lens needs to be similarly short. The smallest elec-
tron spot on the detector is achieved by imaging the electron cloud on the photocathode
onto the detector. The necessary focal length is given by the thin lens equation
1
f=
1
lobject
+1
limage
(2.3)
where lobject is the distance from the object to the lens and limage is the distance from the
lens to the image or the detector in our case.
Since the distance from the lens to the detector is much larger than the distance
from the photocathode to the lens, the latter is crucial for the determination of the focal
length of the electron lens. Neglecting relativistic effects, electrons that emanate from a
point source at x = 0 in the plane of the photocathode with a lateral velocity vx at t = 0
arrive at the anode at t = 2d/vz where d is the distance between the cathode and the
anode and vz is the velocity of the electron resulting from the acceleration in the electric
field. Since the velocity in x-direction is unaffected by the acceleration in z direction,
the offset in x direction at the anode is twice the offset the electron would have had it
travelled at vz. Hence, the point source looks like a point source of accelerated electrons
at a distance 2d before the anode, twice the actual cathode-anode distance.
Chapter 2. The Electron Diffractometer 26
The size of the lens itself also increases the pulse propagation distance. As a rule of
thumb, stronger (i.e. shorter focal length) lenses are larger. The challenge is to build a
lens that will not unduly increase the pulse propagation distance but that will also be
strong enough to image the photocathode onto the detector from its position very close
to the anode.
Electrostatic Lens
A simple electrostatic lens is the so-called einzel2 lens. The lens consists of three di-
aphragms through which the electron beam passes. The first and the last diaphragms
are at ground potential while the middle diaphragm is biased at a high voltage. The lens
effects no net acceleration of the electrons but it first accelerates and then decelerates
(or vice versa, depending on the polarity of the centre diaphragm) the electrons passing
through it. The actual focusing is caused by the radial fields present due to the geom-
etry of the lens. The mathematical treatment of these electrostatic lenses can be found
in the literature [53, 54] and is beyond the scope of this thesis. The focal length for
the same geometry and voltage is shorter when the centre electrode is biased negatively.
However, this configuration is not desirable since it results in the electron pulse getting
compressed longitudinally, increasing space charge broadening. The distance between
the diaphragms is subject to the same limitations as the anode-cathode distance for a
given potential difference. Given these technical limitations, a focal length of less than
50 mm is very difficult to obtain with an electrostatic lens for 60 kV electrons.
Magnetic Solenoid Lens
The basic design of a magnetic solenoid lens is shown in Fig. 2.2. The magnet wire
windings (shown in black in the figure) are wrapped around the electron beam aperture.
2German: einzel = individual, single
Chapter 2. The Electron Diffractometer 27
They are shielded by a soft iron material with high magnetic permeability (µ > 200)
(shown in grey), which increases the magnetic field in the gap in the magnetic shield
placed inside the aperture and prevents the magnetic field from “leaking” outside the
magnetic lens. The magnetic solenoid lens is similar to the electrostatic lens in that the
focusing effect is also caused by the inhomogeneities of the field. The focal length of a
magnetic solenoid lens [54] is
f =4m2v2
z
e2∫
B2zdz
(2.4)
where the electron beam propagates along the z direction with a velocity of vz and Bz is
the magnetic field in z direction. The magnetic field in the gap is
B =µ0IN
d + lµ
(2.5)
where I is the current in the coil, N is the number of windings, d is the width of the gap
and l is the length of the shield from pole piece to pole piece.
Electrostatic lenses can be constructed from metal diaphragms and a vacuum com-
patible insulator, making them completely vacuum compatible. Magnetic lenses, on the
other hand, require large amounts of magnet wire. Magnet wire is not desirable inside
a UHV environment since it increases the area and the varnish used as insulation on
magnet wires is not considered vacuum compatible. The current in the magnet wire also
heats up the magnetic lens, which can become a serious problem in the absence of con-
vective cooling. For these reasons, magnetic lenses are often placed around a beam tube
so that the electromagnet is actually located outside the vacuum. However, this setup
requires additional vacuum seals, which would lengthen the anode-to-sample distance
by centimetres. It is therefore advantageous to build a magnetic lens for in-vacuum use
despite the obvious problems.
A large volume in the lens is taken up by the magnet wire windings. To keep the
electron beam path through the lens short I designed the magnetic lens so that the
Chapter 2. The Electron Diffractometer 28
0.0 A 0.2 A 0.4 A 0.6 A 0.8 A 1.0 A
2.2 A2.0 A1.8 A1.6 A1.4 A1.2 A
2.4 A 2.6 A 2.8 A 3.0 A 3.2 A 3.4 A
10 mm
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 0.5 1 1.5 2 2.5 3 3.5
Bea
m F
WH
M a
t det
ecto
r [m
m]
Magnetic lens current [A]
16000 electrons / pulse5000 electrons / pulse
Figure 2.5: Magnetic lens performance. Top: Diffraction pattern of single crystalline
(001)-oriented Si for different magnetic lens currents. At low lens currents, the “shadow”
of the supporting Si “mesh” is visible in the diffraction spots. Bottom: Width (FWHM of
Lorentzian) of the undiffracted electron beam at the detector vs. magnetic lens current.
Chapter 2. The Electron Diffractometer 29
bulk of the windings is placed away from the electron beam path, wrapped around the
electron gun. Note that because of the magnetic shield around the lens, the electron gun is
magnetic field free except for the small region around the gap in the magnetic lens shield.
This is where the actual focusing happens. Since the focal length is increased i.e. the
lens can be weaker when it is placed further from the electron source, the gap in the
shield was placed as far downstream as possible. The lens was constructed with a 5 mm
gap whose centre is placed 14.5 mm from the anode. With a gun-to-detector distance of
about 20 cm, the focal length of the lens needs to be f ≈ 23 mm. For the design electron
energy of 60 keV, the required magnetic field is B = 0.154 T = 1540 Gauss, produced by
700 A Turns.
The performance of the magnetic lens is demonstrated in Fig. 2.5. Obviously, the
best focusing occurs at around I = 2.4 A. Note that while the size of the focus for
the different electron densities is the same, when the beam is not properly focused, the
pulse with the higher number of electrons displays more lateral broadening due to space
charge. At the optimal focusing current, the voltage drop across the magnetic lens is
about 6.2 V. The dissipated power is about 15 W. To keep the magnet wire within
its operating temperature range, cooling is required. Water cooling inside a vacuum
chamber is problematic because of the potential of leaks in water cooling lines. This is
especially true in the presence of high voltages in the same vacuum chamber. I avoided
this problem by designing the magnetic lens to be bolted to a Conflat flange which can
be water-cooled without the need for in-vacuum water lines. The windings of the magnet
wire are wound on a copper yoke for better thermal conductivity. The yoke is Ni plated
for better vacuum compatibility. Layers of indium foil between the yoke and the magnetic
lens shield and between the shield and the flange could improve thermal contact but were
not implemented for convenience.
As mentioned above, the magnet wire used in the magnetic lens is technically not UHV
Chapter 2. The Electron Diffractometer 30
compatible. While magnet wire with Kapton insulation is sold as vacuum compatible
wire, it did not perform any better in our tests than off-the-shelf low-outgassing magnet
wire. On the contrary, pump-down time was increased with Kapton insulated wire over
standard magnet wire and the Kapton insulation seemed to severely hamper the heat
conduction between the wire and the yoke.
Heat can speed up the outgassing process of contaminants on surfaces such as water
or hydrocarbons. This is usually done in a process called “baking” in which the entire
vacuum chamber is heated to over 100◦ C. While this is not possible for the electron
diffractometer because the detector as well as the sample manipulation stages are heat
sensitive, the pump down process can be significantly accelerated by heating the magnetic
lens. This is achieved by running hot (55◦ C) water through the cooling lines for several
hours. A residual gas analyzer (RGA) indicated that the outgassing of the magnetic
lens is almost exclusively water vapour and should therefore not be harmful to other
components inside the vacuum chamber.
2.3 Sample Manipulation
The ability to move the sample accurately and quickly is particularly important for large
area irreversible samples in which every shot has to be performed on a separate spot on
the sample. The techniques we employ for overlapping the electron beam with the laser
beam as well as profile measurements of the electron pulse also require precisely controlled
in-vacuum motion. However, the magnetic fields associated with electrical motors are
not acceptable since even relatively small magnetic fields can deflect the electron beam.
Worse still, the magnetic field of motors that employ permanent magnets in the rotor
changes depending on the rotor position. Hence, the electron diffractometer is equipped
Chapter 2. The Electron Diffractometer 31
6.75" conflat flange
Cathode retaining ring
Cathode contact diskPhotocathode
Cathode holder
Anode
Pinhole
Magnetic lens shield
Yoke Anode mountCooling lines
Figure 2.6: The electron pulse source. Top: electron gun assembly with laser and electron
beams shown. Bottom: exploded electron gun.
Chapter 2. The Electron Diffractometer 32
with a pair of stainless steel translation stages driven by piezoelectric motors3. The stages
are equipped with optical encoders for closed loop operation and are controlled by the
experiment computer. These translation stages allow for the sample motion in the plane
normal to the electron beam propagation.
If the sample plane is not parallel to the motion of the stages, the sample is offset in
z-direction as it is moved to a new spot. This motion causes changes in timing because
it modifies the distance between the electron gun and the sample and it also changes the
distance the pump pulse has to travel. The change in the pump-probe delay due to a
change in z-position ∆z is
∆τ = ∆z
(1
vz
+1
c
). (2.6)
This problem is particularly serious for non-reversible samples. Experiments with these
samples require a large area sample to provide enough space for the many shots necessary
to obtain a complete time series of diffraction patterns. In order to keep the deviation
below 100 fs over a 2 cm long sample, the sample surface has to be parallel to the axis of
motion of the sample manipulation stages within 0.03◦. The sample holder is mounted on
a kinematic tip-tilt mount since this accuracy is far beyond regular machining precision.
The necessary corrections to the tip and tilt angles are determined by using the sample
frame as a mirror in one arm of an autocorrelator, and then measuring the change in
timing for different positions of the translation stage.
A condition for these corrections to be successful is the flatness of the sample. If
the sample is not plane but has some curvature, this curvature produces a local tip/tilt
even if the global orientation of the sample has been corrected. For this purpose, we
use silicon square meshes made from a commercial silicon wafer through Bosch etching
to support the thin sample film [38]. The sample surface is then defined by the surface
3Nanomotion - Yokneam, Israel
Chapter 2. The Electron Diffractometer 33
of the wafer which is very flat (2 nm flatness from wafer specification). Since these
sample supports could still deformed when mounted on a deformed sample frame, I
designed the sample frames to be in contact with these sample supports in three points
only, thereby preventing any stress on them. The sample frame is mated to the tip-tilt
stage by a specially built mount that allows rapid sample frame exchanges with high
reproducibility. In this mount, three spring-loaded ball bearings press the polished side
of the sample frame against a polished stainless steel surface. This setup allows the
sample frame to be inserted and removed easily. Through quick sample exchanges, the
exposure of the vacuum chamber to the atmosphere can be kept at a minimum.
2.4 Electron Detection
Spatially resolved electron detection is usually achieved by means of a phosphor or a scin-
tillator followed by detection of the produced photons by a CCD camera. The coupling
between the phosphor and the CCD camera can either be achieved by a free-space optical
system, in which a lens images the phosphor onto the CCD chip or by a fibre-optic plate,
which is placed directly between the phosphor and the CCD chip. While the former is
much simpler to implement, it suffers from poor optical efficiency
OE ≈ 1
16N2
(b
g
)2
(2.7)
with the f-number N of the lens used to image the detector on the CCD chip and
the magnification b/g of the imaging system. Given a lens with N = 2, the chip size
b = 13.3 mm and a detector size of g = 40 mm, the theoretical efficiency is OE ≈ 2×10−3.
This efficiency is further reduced by reflections on optical surfaces. Fibre optic systems,
on the other hand, can reach efficiencies in the 10−1 range (somewhat less for tapered
fibre plates). The high complexity and cost of a fibre-coupled detection system led to the
decision to use a lens-coupled system despite the obvious advantages of fibre coupling.
Chapter 2. The Electron Diffractometer 34
A 55 kV electron striking a suitable phosphor screen produces on the order of 103
photons. The noise floor (dominated by readout noise for typical exposure times at a
detector temperature of −20◦ C) of the CCD camera4 is about 10 electrons and the
detection efficiency is > 90%. Hence, the detection efficiency of the entire system is
significantly below the goal of every-electron detection. The detection system currently in
use employs a chevron pair of microchannel plates (MCPs) together with a P20 phosphor
screen5. The MCPs act as an array of electron multipliers with an amplification of up
to 104 per plate [55]. In the electron diffractometer, the MCPs are typically operated
at 1.3 kV and the phosphor screen is biased at 4.6 kV, resulting in an electron energy
of 3.3 keV at the phosphor screen. While this detector configuration certainly increases
the intensity so that every primary electron is detected far above the noise level, the
efficiency of the MCPs in the detection of primary 55 keV electrons is estimated to be
only about 10%, i.e. only 10% of the primary electrons trigger an avalanche of secondary
electrons in one of the electron multiplier channels. This efficiency dictates the overall
detection system efficiency.
While every-electron detection has been reached in lens coupled systems for electron
microscopes [56], such systems operate on the margins of every-electron detection. For
future electron diffractometers, fibre optic coupling between the phosphor screen and
the CCD chip is recommended owing to the robust every-electron-detection capability of
such a setup without any additional amplification. More exotic detection systems, such
as direct bombardment of radiation-hardened CMOS active pixel sensor chips [13] have
so far failed to materialize but may play a role in UED in the future.
In order to measure the electron beam current, the vacuum chamber also contains a
Faraday cup, which can be moved in front of the detector by a linear motion feedthrough.
4Apogee Instruments - Alta U475Photonis - APD-30-40-12/10-8-FM-I-60:1-P20
Chapter 2. The Electron Diffractometer 35
The electron beam currents that need to be measured are very low; a fairly typical
configuration of 5000 electrons/pulse at a laser repetition rate of 500 Hz correspond to
a beam current of 0.4 pA. To measure these beam currents reliably, the Faraday cup is
connected to a sensitive electrometer6.
2.5 Vacuum Chamber Design
The vacuum chamber (see Fig. 2.7) consists of two almost completely separate sections,
each of which is pumped out by a separate turbomolecular pump. One section contains
the electron gun, while the other section contains the sample, sample manipulation and
the pump beam optics. The electron gun is very sensitive to the quality of the vacuum
– arcing is much more likely at higher pressures. By venting the magnetic lens to the
sample chamber, the surface area in the gun chamber can be kept quite small and the
gun chamber can reach 10−8 Torr after a few hours of pumping. The pressure in the
sample chamber drops much more slowly due to the larger surface area and the magnetic
lens. However, for the reliable operation of the electron diffractometer, a pressure of
10−6 − 10−7 Torr in the sample chamber is sufficient. This pressure is reached after
6 − 8 h of pumping.
The separation of the vacuum chamber into two sections also shields the electron gun
from pressure increases due to the pump laser, which had caused significant difficulties
in the previous 30 kV electron diffractometer. On the other hand sensitive parts inside
the sample chamber, such as the MCPs and the optical encoders are protected from the
electromagnetic pulse produced by the electron gun during an arc.
6Keithley - Model 6514
Chapter 2. The Electron Diffractometer 36
Laser windows
Ion gauges
Phosphor screento turbo pumps
ViewportSample chamber
60 kV feedthrough
Electron gun section
Faraday cup manipulator
Electron gun flange
Figure 2.7: The vacuum chamber of the electron diffractometer. The gun flange with
the electron gun (shown in detail in Fig. 2.6) separates the vacuum chamber into two
almost completely separate sections. The laser window on the electron gun section is for
the laser pulse that drives the electron gun. The pump beam in UED experiments enters
through the laser window in the sample chamber closer to the detector. The nominal
size of the top and bottom flanges of the sample chamber is 10”.
Chapter 3
Characterization of Femtosecond
Electron Pulses
The duration of the probe pulse is the dominant contribution to the time resolution of
the electron diffractometer. Simulations of the electron pulse propagation provide useful
estimates for the electron pulse duration but they need to be verified experimentally. The
experimental verification of the electron pulse durations is particularly important since
the actual electron pulse duration depends on some experimental parameters (e.g. the
number of electrons before the anode), which are difficult to verify directly.
The conventional method for measuring electron pulse durations is a streak camera.
In a streak camera, a pair of deflection plates are positioned along the electron pulse
path and a voltage ramp is applied to them as the electron pulse passes between them
(see Fig. 3.1). The resulting time-dependent electric field deflects electrons by a different
amount depending on their arrival time. When the electron pulse hits the detector, it is
Parts of this chapter have previously been published in the following articles: [42] Christoph T.Hebeisen, Ralph Ernstorfer, Maher Harb, Thibault Dartigalongue, Robert E. Jordan and R. J. DwayneMiller, Optics Letters 31, 3517 (2006). Copyright 2006 by the Optical Society of America. [43] ChristophT. Hebeisen, German Sciaini, Maher Harb, Ralph Ernstorfer, Thibault Dartigalongue, Sergei G. Kruglikand R. J. Dwayne Miller, Optics Express 16, 3334-3341 (2008). Copyright 2008 by the Optical Societyof America.
37
Chapter 3. Characterization of Femtosecond Electron Pulses 38
Voltage rampGenerator
vz
Unstreakedspot
Streakedspot
Streaking plates
lDetector
E(t)
L
Figure 3.1: Principle of a streak camera.
streaked across the detector i.e. the temporal dimension of the electron pulse is projected
into a spatial dimension, which is easily measured.
In order to calculate the length of the streak on the detector, we assume an electron
initially moving in z direction with velocity vz. The electron enters the streaking field
~E(t) = (0,−st, 0) at t = 0. The velocity in y direction vy(t) is then
vy(t) =
t∫
0
−eEy(T + τ)
medT =
es
me
(t2
2+ τt
). (3.1)
I introduced a time offset τ in the electric field ramp in order to account for the different
arrival times of the electrons in the pulse. When the electron leaves the field at z = l,
vy
(l
vz
)=
esl
mevz
(l
2vz
+ τ
)(3.2)
and
y
(l
vz
)=
l
vz∫
0
vy(t)dt =esl2
2mev2z
(l
3vz
+ τ
). (3.3)
If the detector is placed at distance L after the end of the streaking plates, the offset of
the electron at the detector is
Y (τ) = y
(l
vz
)+
L
vz
vy
(l
vz
)=
esl
mev2z
[l
2vz
(L +
l
3
)+ τ
(L +
l
2
)]. (3.4)
Chapter 3. Characterization of Femtosecond Electron Pulses 39
The derivative of this offset at the detector with respect to the arrival time of the electron
at the streaking plates is called streaking speed
us =dY (τ)
dτ=
esl
mev2z
(L +
l
2
). (3.5)
The streaking speed is an important measure of the performance of a streak camera
because it determines directly how long the streak of a given electron pulse is on the
detector screen. The intensity profile of the streak along the streaking direction is a
convolution of the lateral electron pulse intensity profile and the temporal pulse profile.
Hence, the instrument response of a streak camera is given by w/us where w is the lateral
size of the electron beam. While a sub-picosecond system response has been achieved
for special streak camera designs [57], there are severe problems with streak cameras
that limit their ability to measure the high number density electron pulses used in FED.
The measurement of femtosecond electron pulses requires a very high streaking speed.
According to Eq. 3.5, the simplest way to increase the streaking speed is to increase the
length of the streaking plates. Usually, streaking plates for electron pulses in the tens of
keV range are a few centimetres long. Such a setup has been used e.g. by Cao et al. [58].
However, as shown by simulations in the previous chapter, the electron pulse duration
would drastically change over such a length. The measured temporal profile would be
ill-defined and only poorly reflect the duration of the electron pulse that would arrive at
a sample. An increase in the length of the drift region L leads to a corresponding increase
in the lateral electron beam size at the detector and hence does not improve the streak
camera resolution. The ramp speed of the electric field s is limited by technical properties
like the capacity of the streaking plates. An RF cavity could be used instead of the
streaking plates but such a setup requires extensive (and expensive) RF synchronization
circuits and amplifiers and only partially solves the problems with respect to the required
interaction length and the changing electron pulse duration.
Chapter 3. Characterization of Femtosecond Electron Pulses 40
The problem of characterizing electron pulses is analogous to earlier problems in
determining optical pulse shapes. With the introduction of passive mode-locking [59],
laser pulses became too short to be characterized by fast photodiodes or photoactivated
streak cameras. The only probe short enough to probe laser pulses were the laser pulses
themselves. In response to this problem, autocorrelation [60] and cross-correlation of
laser pulses were widely adopted to characterize femtosecond laser pulses. Given the
problems in characterizing femtosecond electron pulses with streak cameras, it is clear
that a similar solution is necessary for electron pulses.
Optical correlation techniques rely on nonlinear effects in a material e.g. sum fre-
quency generation (SFG). In cross-correlation, two different laser pulses are overlapped
inside a non-linear crystal. The non-linear signal is a function of the overlap integral of
the envelopes of the two laser pulses. By scanning the relative delay between the laser
pulses, a cross-correlation function can be obtained. In the case of auto-correlation, the
two pulses are copies of the same pulse.
An interaction between an electron pulse and another short pulse would enable a
similar cross-correlation scheme for electron pulses. We proposed a laser-electron cross-
correlation scheme based on electron scattering by the laser ponderomotive force [61] and
demonstrated a technique based on this proposal in a proof-of-principle experiment [42].
We also developed a related practical technique with improved performance that can be
adopted in any UED lab [43]. These two techniques are described below in sections 3.2
and 3.3, respectively.
Several other techniques for measuring electron pulse durations have been demon-
strated. A method developed by Wang et al. [62] uses an RF cavity with the electric
field along the pulse propagation direction. The field is timed so that the pulse passes
the cavity at the zero-crossing of the field i.e. when the time derivative of the field is
at its maximum. The integrated electric field the electrons are exposed to during their
Chapter 3. Characterization of Femtosecond Electron Pulses 41
traversal of the cavity is proportional to their longitudinal (temporal) position in the
electron bunch. The cavity is followed by a spectrometer which spreads the electrons in
the pulse according to their energy. This approach is closely related to the principle of
a streak camera described above and suffers from similar limitations. Another technique
uses the coherent transition radiation emitted when an electron passes an interface be-
tween two media of different dielectric constants [63]. This method works best with large
numbers (N ≈ 107) of relativistic (107 eV) electrons and requires a priori knowledge of
the temporal pulse shape. Electro-optical sampling of the electron pulse shape [64] does
not suffer from this limitation since it determines the pulse shape through the measure-
ment. This method exploits the fact that the electric field of a highly relativistic electron
bunch is almost completely perpendicular to its motion. This electric field can induce
birefringence in an electro-optical crystal near the electron beam via the Pockels effect.
The shape of the electron pulse envelope is then encoded in the polarization of a chirped
laser pulse co-propagating parallel to the electron pulse through the crystal. While this
method, too, is suitable only for highly relativistic electrons and therefore unsuitable for
UED setups, it is useful for x-ray free electron lasers and time resolution down to 60 fs
has been shown [65].
Two electron pulse characterization techniques for electron pulses used in UED have
been proposed but not demonstrated. An electron pulse-electron pulse cross-correlation
scheme has been proposed by Baum et al. [33]. This scheme relies on the space-charge
interaction between two electron pulses and therefore its effectiveness strongly depends
on the number of electrons per pulse. It also requires extensive modelling to extract
the electron pulse duration and modifications of the electron beam path, such as an
additional magnetic lens to focus the electron beam in the sample position. Another
laser-based measurement whose principle is based on impact-ionization induced Auger
electron emission has recently been proposed [66].
Chapter 3. Characterization of Femtosecond Electron Pulses 42
To date, electron pulse-laser pulse cross-correlation by means of the ponderomotive
force is the only proven method to measure femtosecond electron pulses with pulse pa-
rameters as used in UED.
3.1 Laser Pulse-Electron Pulse Temporal Overlap
Determination
In order to take meaningful and accurate measurements of time constants of processes
observed with UED, which are on the same order of magnitude as the instrument response
of the electron diffractometer, the exact coincidence of the pump and probe pulses or
t = 0 has to be determined with good precision [5] independently of any changes of the
observed diffraction patterns. In gas-phase studies, photoionization-induced lensing has
been used for this purpose [67]. In these measurements, a distortion of the electron beam
due to the charge separation caused by the photoionization of the gas target by the laser
beam is observed if the electron pulse arrives after the laser while no such distortion is
observed if the electron pulse passes the target before it is excited by the laser pulse.
A similar technique can be used with a solid target instead of the molecular beam.
Electrons emitted from a needle tip [68, 69, 13] or a TEM mesh [5] after femtosecond
laser illumination can deform the electron beam, allowing for a t = 0 determination.
These measurements are performed by replacing the sample in a UED experiment by
a fine TEM mesh and increasing the pump pulse intensity so that it is just above the
ablation threshold. The physics of the electron emission from surfaces after intense laser
pulse illumination, which underlies this effect is discussed in chapter 4.
Beam images from such a t = 0 determination are shown in Fig. 3.2. Before taking
an image with the pump laser pulse present, we take a reference image under the same
Chapter 3. Characterization of Femtosecond Electron Pulses 43
−0.5 0.5 1.0 1.5 2.0 2.50.0
0
0.2
0.4
0.6
0.8
1
−10 −5 0 5 10
RM
S d
iffer
ence
Pump−probe delay [ps]
Figure 3.2: Top: Difference images of the pulse shapes with and without laser illumina-
tion in a laser pulse-electron pulse temporal overlap measurement. The number in the
lower right corner of each image indicates the delay time in picoseconds. Bottom: RMS
difference signal between the before and during images with some example during images.
Inset: Sketch of the experimental setup. The electron pulse (blue) passes through the
TEM mesh, which is excited by the laser pulse (red).
Chapter 3. Characterization of Femtosecond Electron Pulses 44
conditions but with the pump laser beam blocked. We refer to these images as “before
images” and the images with the pump beam present are called “during images”. The
images are usually analyzed by subtracting the before image of each time step from its
corresponding during image and calculating the RMS value of each difference image. The
choice of the RMS of the difference image is somewhat arbitrary and was chosen as a
convenient measure of the deformation the probe pulse has undergone.
The time zero position is usually taken to be the onset of the rise of the RMS signal.
Obviously, this choice is somewhat sensitive to the signal-to-noise ratio (SNR). It could
also be skewed by the pump pulse duration if e.g. the leading edge of the pump pulse
already causes an appreciable change to the detector image. Although it was not known
if the observed changes in the electron beam shape started exactly at the maximum
overlap of the laser pulse, this technique has been used in several UED studies. The
dynamics of this effect are rather complex and are known to evolve on the 100 fs to ps
timescale in a highly surface-dependent and excitation-dependent manner. In contrast,
the scattering of electrons by the ponderomotive force is basically instantaneous and
therefore provides a perfect marker for the t = 0 position of the laser pulse-electron pulse
overlap in time. The beam path for this measurement is different from that of a UED
experiment and hence a calibration of the path length difference is necessary. The much
simpler measurement using a TEM mesh can be performed in both geometries. This can
be used to quantify the delay between the actual laser arrival time as determined by the
ponderomotive electron characterization and the observed change in beam shape in the
mesh-based experiment. This comparison can then serve as a calibration for the t = 0
measurement.
Chapter 3. Characterization of Femtosecond Electron Pulses 45
3.2 Laser Pulse-Electron Pulse Cross-Correlation
Charged particles naturally interact with electromagnetic fields through the Lorentz force
~F = q(
~E + ~v × ~B)
. (3.6)
Here, ~E is the electric field, ~B the magnetic field, ~v the velocity of the particle and q its
charge. In the oscillating field of a plane wave, the Lorentz force leads to a quiver motion
of the particle that does not lead to any net displacement over a full cycle. The electric
and magnetic fields of a plane wave in vacuum are
~E =1
2
(~E0e
iωt + ~E∗0e
− iωt)
with ~E0 = Ee− i~k·~r and (3.7)
~B =1
2
(~B0e
iωt + ~B∗0e
− iωt)
with ~B0 = Be− i~k·~r , (3.8)
respectively, where ~k is the wave vector of the plane wave, ω =∣∣∣~k∣∣∣ c and E and B with
∣∣∣B∣∣∣ = 1/c
∣∣∣E∣∣∣ are the amplitudes of the fields where E, B, ~k are mutually orthogonal
and form a right handed system. For light intensities I ≪ 1018 W/cm2 the quiver motion
of an electron is much slower than the speed of light and we can neglect the magnetic
field for the calculation of the quiver motion:
m~r =q
2
(~E0e
i ωt + ~E∗0e
− i ωt)
. (3.9)
We integrate this to yield
~r = − i q
2mω
(~E0e
i ωt − ~E∗0e
− i ωt)
and ~r = − q
2mω2
(~E0e
i ωt + ~E∗0e
− i ωt)
. (3.10)
Charged particles undergoing an accelerated motion lose energy by radiation. However,
the energy classically radiated by an electron undergoing an oscillation corresponding
to the peak laser intensity used in this experiment for the entire laser pulse duration is
much less than the energy of a single photon (0.04 eV vs. 1.55 eV). Therefore, radiation
losses are neglected in this derivation.
Chapter 3. Characterization of Femtosecond Electron Pulses 46
If we replace the constants E and B by slowly varying functions (i.e. add an envelope
to the light intensity), there is an additional slow drift component ~R in the particle
motion [70, 71]. We rewrite Eq. 3.6 as
m(
~R + ~r)
=q
2
{E0
(~R)
eiωt + E∗0
(~R)
e− iωt+
(~r · ∇)[E0
(~R)
eiωt + E∗0
(~R)
e− i ωt]
+ ~r ×[B0
(~R)
ei ωt + B∗0
(~R)
e− iωt]}
. (3.11)
Since E and B vary slowly and the amplitude of the quiver motion is small (18 nm for
1017 W/cm2 at 800 nm), we expanded E0 to first order and used only one term for B0.
Substituting Eq. 3.10 into Eq. 3.11 and we obtain an equation for ~R
m~R = − q2
4mω2
{[(~E0e
i ωt + ~E∗0e
− i ωt)· ∇] (
~E0eiωt + ~E∗
0e− i ωt
)
+ iω(
~E0ei ωt − ~E∗
0e− i ωt
)×(
~B0ei ωt + ~B∗
0e− iωt
)}. (3.12)
Since ~R describes the slow drift motion, we can discard the terms oscillating at 2ω:
m~R = − q2
4mω2
[(~E0 · ∇
)~E∗0 +
(~E∗0 · ∇
)~E0 + i ω
(~E0 × ~B∗
0 − ~E∗0 × ~B0
)]. (3.13)
Using Maxwell’s equations we find ~B0 = i /ω∇× ~E0. Hence, after applying ( ~E0 ·∇) ~E∗0 +
( ~E∗0 · ∇) ~E0 = ∇( ~E0 · ~E∗
0) − ~E0 × (∇× ~E∗0) − ~E∗
0 × (∇× ~E0) we get
m~R = − q2
4mω2∇∣∣∣ ~E0
∣∣∣2
(3.14)
or, written as a function of the light intensity I
~F = − q2λ2
8π2mε0c3∇I . (3.15)
~F is called the ponderomotive force, λ is the wavelength of the light and ε0 is the per-
mittivity of free space. Given the mathematical form of Eq. 3.15, it can be rewritten as
the result of an effective scalar potential ~F = −∇Φ where the ponderomotive potential
is
Φ =q2λ2
8π2mε0c3I . (3.16)
Chapter 3. Characterization of Femtosecond Electron Pulses 47
Neutral atoms experience a similar force in inhomogeneous laser fields, which can
be used to trap cold atoms [72]. For a free electron, the instantaneous position of the
quiver motion (see Eq. 3.10) is 180◦ out of phase from the electric field. The phase of the
oscillation of the dipole moment of an atom, on the other hand, changes from in phase
at frequencies far below resonance to out of phase above resonance. Hence, a laser beam
whose wavelength is red-shifted with respect to the resonance produces an attractive
potential, i.e. the resulting force on the atom is directed towards higher laser intensity.
In the case of a blue-shifted laser, the force is directed towards lower intensity [73], as in
the case of a free electron. The potential for an atom in an inhomogeneous laser field is
given by the ac Stark shift [73].
3.2.1 Experimental Setup for the Laser Pulse-Electron Pulse
Crosscorrelation Experiment
The objective of this measurement is to determine the electron pulse duration of the
electron diffractometer as it impacts the system response function. Hence, it is important
to perform the measurement as closely to the normal UED sample position as possible.
Since the ponderomotive force interacts with the electron pulse in vacuum, the beam
paths of the laser and electron pulses just need to be crossed at the position at which the
electron pulse duration is to be measured. The ponderomotive force of the laser pulse will
then deflect electrons that traverse the high intensity laser pulse, modifying the shape of
the electron beam at the detector. The relative delay between the electron pulse and the
laser pulse can then be scanned, similar to a pump-probe experiment, to probe different
parts of the electron pulse sequentially.
One of the factors contributing to the system response of the measurement is the
time that the electron pulse takes to cross the laser beam (see section 3.2.2). For this
Chapter 3. Characterization of Femtosecond Electron Pulses 48
reason and to maximize the intensity gradients of the laser pulse, the laser beam needs
to be focused. For 55 keV electrons, there is a transit-time broadening of 100 fs for
a laser beam diameter of 13 µm. However, since the lateral electron beam size at the
intersection between the two beams is much larger than the laser beam waist, only a
small percentage of the electrons would interact with the laser pulse, leading to a large
background of unscattered electrons and a poor SNR. Furthermore, the crossing time
of the laser beam through the electron beam also broadens the system response of the
measurement. To mitigate these problems, a pinhole is introduced in the laser beam
path to strip the electron pulse of its outer electrons just before its crossing point with
the laser beam. We used a 30 µm diameter pinhole to limit the time broadening due to
the laser pulse transit time through the finite size of the electron beam to 100 fs. To
prevent damage to the pinhole by the intense laser beam, the laser focus was moved 5 mm
downstream from the usual sample position along the electron propagation direction.
Calculations show that for reasonable laser pulse parameters (100 fs, 20 µm beam
waist), a laser pulse energy on the order of millijoules is needed to deflect an electron at
the detector by the size of the electron beam. Simulated electron beam shapes at the
detector are shown in Fig. 3.3. The simulations were performed by using the Barnes-
Hut tree code (see section 2.2.1) to produce an electron pulse and to propagate it to
a position just before its interaction with the laser beam. For about 1 mm along the
electron beam path centred around the laser beam-electron beam crossing, the electron
pulse propagation was calculated without Coulomb interaction between the electrons
by velocity Verlet integration under consideration of the ponderomotive force of the
crossing laser pulse. After that, the Barnes-Hut tree code was used again to propagate
the electrons to the detector.
The required pulse energies are not available from table-top laser systems and hence
experiments were performed using the beam line 2B of the Advanced Laser Light Source
Chapter 3. Characterization of Femtosecond Electron Pulses 49
1 mm
4 mm
−800 −600 −400 −200 0 200 400 600 t [fs]
Figure 3.3: Simulated electron beam images on the detector after interaction with a
10 mJ, 100 fs laser pulse. Top row: The electron beam (100 µm) is being scattered
by a laser pulse with 25 µm beam waist. The image intensity is scaled to make the
depletion of electrons along the laser beam path visible. Bottom row: the electron beam
is stripped by a 30 µm pinhole before interacting with a 15 µm laser beam waist. The
image intensity is scaled to make the scattered electrons visible.
(ALLS) at the Institut National de la Recherche Scientific (INRS) in Varennes, QC. The
beam line consists of a Ti:Sapphire oscillator followed by a regenerative amplifier and two
multi-pass amplifiers which produce laser pulses with up to 40 mJ with a bandwidth of
40 nm at a repetition rate of 100 Hz. For the electron pulse duration measurements, we
detuned the compressor to produce 90 fs pulses. The scattering laser pulse had a pulse
energy of 14.9 mJ at the focus and the electron gun was operated at V = 55 kV. Due
to beam time limitations, the pulse characterization was only performed for one electron
number density (3100 electrons per pulse, 100 µm extraction pinhole).
The experimental setup is shown schematically in Fig. 3.4. The optical setup is very
similar to a UED setup that uses the fundamental laser wavelength as its pump pulse.
However, the beam geometry was changed so that the laser beam crosses the electron
beam at 90◦. Due to the high pulse energy of the laser, a beam dump was necessary to
Chapter 3. Characterization of Femtosecond Electron Pulses 50
PhotocathodeV= −55 kV
ElectronPulse
AnodeV=0
Laser Pulse
Pinhole
Detector
Laser267 nm
(a) (b)
RailDelay f=250 mm
Prism Comp.
SHG SFG
Beam Splitter
Attenuator
Telescope (8:1)
10%
90%
Figure 3.4: (a) Optical setup of the laser pulse-electron pulse crosscorrelation experiment.
The incoming laser pulse is split into two arms with 90% of the pulse energy focused to
scatter the electron beam. The remaining 10% is further attenuated, sent through a
variable delay line, tripled and then compressed before entering the electron gun. (b)
Schematic view of the experiment geometry. The electron pulse is stripped of its outer
electrons by a 30 µm diameter pinhole before it is scattered by the ponderomotive force
of the laser pulse.
absorb the laser pulse after its interaction with the electron pulse to prevent damage to
the vacuum chamber and background signal in the recorded beam images.
At the time these experiments were done, a UV pulse was used to drive a silver
photocathode by single photon photoemission. For this purpose, the third harmonic
(267 nm) of the fundamental Ti:Sapphire laser pulse was produced by two BBO crystals.
They were cut for type I second harmonic generation (SHG) and type II sum frequency
generation (SFG), respectively. They were followed by a prism compressor to compresses
the resulting UV pulse. The spatial separation of the different wavelengths in the prism
compressor also facilitated the extraction of the tripled light from the collinear beams of
Chapter 3. Characterization of Femtosecond Electron Pulses 51
fundamental, second harmonic and third harmonic light.
One of the main challenges of this experiment was to obtain overlap in space and time
between the laser pulse and the electron pulse. In order to facilitate this task, I designed
an alignment tool consisting of a thin stainless steel foil with two micromachined features.
The first feature was a long, narrow horizontal slit (1 cm × 30 µm), the second was a
rectangular pinhole, machined to appear circular (30 µm diameter) when viewed at 45◦
which was in line with the slit. By moving this tool with respect to the electron beam
or the laser beam with respect to the alignment tool in vertical direction, the long slit is
located, by then moving in the horizontal direction, the pinhole can be found. Through
this procedure, the time consuming two-dimensional search for a pinhole can be turned
into two one-dimensional searches. This alignment tool was mounted so that it could be
inserted at the intersection of the electron and laser beams at 45◦ with respect to both
the electron beam and the laser beams. Rough temporal overlap was established by using
the method described in section 3.1.
3.2.2 Results and Data Analysis
A series of electron distribution maps for different time delays between the electron pulse
and the scattering laser pulse is shown in Fig. 3.5. In the images without a stripping
pinhole, the propagation of the laser pulse through the electron pulse (right to left) can
be directly observed, strikingly demonstrating the broadening of the observed signal by
the lateral width of the electron pulse. Also note that while the scattered electrons are
spatially well-separated from the undisturbed beam in the lower row, they are mixed
with the largely undeflected outer parts of the beam in the case without the pinhole.
A potential disadvantage of the setup with the stripping pinhole is the high sensitivity
of the stripped beam setup to laser beam pointing instability. In the series of images
Chapter 3. Characterization of Femtosecond Electron Pulses 52
Figure 3.5: Series of images of the beam on the detector without (upper row) and with
(lower row) the stripping pinhole (30µm) in place. An image of the undisturbed beam
has been subtracted from each of the images to highlight the scattering of electrons due
to the ponderomotive force of the laser pulse. Hence, dark regions are in the image are
depleted of electrons due to the interaction with the laser while bright regions receive
more electrons than in the undisturbed image. The images in the upper row are integrated
over 720 pulses while the ones in the lower row are integrated over 4000 pulses due to the
much lower number of electrons that reaches the detector when the pinhole is in place.
Both series of images are taken at steps of 300 fs between adjacent images.
Chapter 3. Characterization of Femtosecond Electron Pulses 53
Figure 3.9: Intensity profiles of (a) a single laser pulse, (b) two counterpropagating laser
pulses, not overlapping in time and (c) two counterpropagating pulses overlapping.
result of the spatial envelope of the laser pulse. In the standing wave case, an intensity
grating is formed with a distance of λ/2 between the maximum and the minimum of the
intensity. Obviously, this leads to an enhancement in the ponderomotive force. Under
otherwise identical conditions, the peak force of two counterpropagating waves along the
propagating direction is about 100× stronger than that of a single propagating pulse
in the lateral direction if we assume half of the intensity of a single beam in each of
the counterpropagating beams, a 10 µm FWHM Gaussian beam and λ = 800 nm. This
situation is schematically shown in Fig. 3.9. Each of the laser pulses in the two pulse case
has half the energy of the pulse in the case of a single pulse. The drastically enhanced
intensity gradients along the propagation direction are obvious in the standing wave case
(c).
According to the van Cittert-Zernike theorem, the transverse coherence width of the
electrons is λe−/θsource where λe− is the de-Broglie wavelength of the electrons and θsource
is the angle the electron source (the laser spot on the photocathode) subtends at the
sample [78]. In our current setup, the coherence length is therefore ≈ 1 nm i.e. much
shorter than the periodicity of the intensity grating ≈ 400 nm. Hence, we do not expect
diffraction of the electrons off the laser intensity grating but purely classical scattering
of the electrons off the ponderomotive potential.
For two counterpropagating Gaussian laser pulses, each with a peak intensity of I0/2,
Chapter 3. Characterization of Femtosecond Electron Pulses 62
propagating in the positive and negative x-directions, the intensity equivalent is
I(~r, t) =I0
2
∣∣∣∣∣exp
(i(−ωt + kx) −
(t − x
c
)2
4w2t
)+
exp
(i(−ωt − kx) −
(t + x
c
)2
4w2t
)∣∣∣∣∣
2
exp
(−y2 + z2
2w2f
), (3.28)
where I0 is the peak intensity of each laser pulse, k = 2π/λ, ω = ck, and 2√
2 ln 2wt and
2√
2 ln 2wf are the FWHM laser pulse duration and the beam waist, respectively. The
x-component of the ponderomotive force takes the following form:
Fx(x, y, z, t) = − I0e2λ2
16π2meε0c3exp
(−y2 + z2
2w2f
)×
∂
∂x
[exp
(−(t − x
c
)2
2w2t
)+ exp
(−(t + x
c
)2
2w2t
)+
2 exp
(− t2
2w2t
)exp
(− x2
2w2t c
2
)cos(2kx)
]. (3.29)
The first two terms of the expression to be derived describe the laser pulses travelling
in opposite directions and the last term describes the standing wave produced where the
two pulses overlap. Note that the standing wave has the same duration as the original
pulses. Its spatial envelope in x-direction has the width 2√
2 ln 2wtc. Since λ/2 ≪ is
much smaller than this envelope, the derivative is dominated by the former.
Fx(x, y, z, t) ≈
− I0e2λ
2πmeε0c3exp
(−y2 + z2
2w2f
)exp
(− t2
2w2t
)exp
(− x2
2w2t c
2
)sin(2kx) , (3.30)
It is obvious from Eq. 3.30 that the force varies from a maximum in one direction to its
maximum in the opposite direction within λ/4. Since the diameter of the electron beam
is much larger than λ/2, we expect equal numbers of electrons deflected in the positive
and negative x-directions. This is confirmed by simulations of the electron beam shape
at the detector shown in Fig. 3.10. The simulations were performed analogously to the
Chapter 3. Characterization of Femtosecond Electron Pulses 63
2 mm
0 fs−200 fs
−400 fs−600 fs−800 fs
200 fs
400 fs 600 fs 800 fs
Figure 3.10: Simulated beam images at the detector after interaction with the standing
wave produced by two counterpropagating 200 fs laser pulses (175 µJ per pulse, 30 µm
beam waist.
simulations of the single beam case described in section 3.2.1. Note that the laser pulse
energy in this simulation is 2× 175 µJ while 10 mJ were required to produce good signal
in the previous measurements.
3.3.1 Grating Enhanced Electron Pulse Characterization Setup
The setup is shown in Fig. 3.11. The optical setup of the arm of the laser beam path,
which drives the electron gun is identical to the standard UED setup. The beam path that
delivers the pump pulse to the sample in a UED setup has been modified to accommodate
the pulse duration measurements. The setup requires two counterpropagating laser pulses
to collide at the overlap of their beam paths with the electron beam path. For this
Chapter 3. Characterization of Femtosecond Electron Pulses 64
PhotocathodeV= −55 kV
AnodeV=0
ElectronPulse
Laser Pulses
f=50 cm
Detector
BS
(b)
Pinhole
BS
NOPA
(a)
1:3 telescope775 nm210 fs
50 fs500 nm
z
x
y
Figure 3.11: (a) Optical setup of the grating enhanced electron pulse characterization
experiment. (b) Schematic view of the experiment geometry.
purpose, the laser beam is split by a 50/50 beam splitter. One of the resulting beams is
sent through a delay stage to permit balancing of the two arms. Due to the long beam
path of one of the beams inside the vacuum chamber, a relatively long (50 cm) focal
length lens has to be used to focus the scattering pulses. In order to still obtain a small
beam waist, the laser beam first had to be expanded from its usual diameter (2.7 mm)
by a 1:3 telescope. In practise, the size of the laser spot at the electron beam position
was 41 ± 10 µm.
Alignment
The two laser pulses used to create the ponderomotive force profile need to overlap
in space as well as in time with each other and with the electron pulse to produce a
scattering signal. Since the scattering laser pulses are propagating in opposite directions
at the overlap position, beam pointing variations in the laser cause them to move in
opposite directions if the sum of the numbers of reflections the two beams undergo after
Chapter 3. Characterization of Femtosecond Electron Pulses 65
the beam splitter (i.e. not counting the beam splitter itself) in the horizontal or vertical
plane is odd. Hence, we use three mirrors in the horizontal plane on the relative delay
stage to make the setup self-compensating for small beam pointing instabilities. The
spatial and temporal alignments of this experiment are somewhat challenging. Note that
there is no observable signal even very close to perfect spatial and temporal overlap of
both laser pulses and the electron pulse, as long as there is not at least significant partial
overlap of all three pulses. In order to facilitate the alignment, an alignment tool similar
to the one described in section 3.2.1 with slit and pinhole widths of 50 µm instead of
30 µm was used.
The alignment of the setup was performed as follows:
• The focus of the scattering laser pulse transmitted through the beam splitter (pri-
mary beam) is optimized by scanning the laser focus with the top edge of the
alignment tool using the in-vacuum translation stages. The focus is adjusted by
moving the lens along the beam propagation direction.
• The focus of the scattering laser pulse reflected by the beam splitter (secondary
beam) is optimized in an analogous fashion but the focus is now changed by moving
the relative delay stage. Since the two arms are focused by the same lens, the
coincidence of the focus also indicates the coincidence in time. This method is
accurate to about ±1 mm (±3 ps).
• The pinhole is centred on the electron beam using the in-vacuum x/y stages.
• The primary beam is aligned on the pinhole by moving the focusing lens in the
lateral plane.
• The secondary beam is overlapped with the transmitted primary beam on the last
mirror in the secondary beam path by adjusting the last mirror on the relative
Chapter 3. Characterization of Femtosecond Electron Pulses 66
translation stage.
• The last mirror in the secondary beam path is used to align the secondary beam on
the pinhole. The electron beam, and both laser beams are now spatially overlapped.
• A TEM copper mesh is moved to the intersection of the electron beam and the
laser beams. By blocking the secondary beam, rough temporal overlap between the
electron pulse and the primary laser pulse is established using the “grid effect”.
• The primary beam is blocked and the temporal overlap between the electron pulse
and the secondary laser determined in the same manner. The relative delay stage
is then moved to compensate for the differences in the temporal overlap of the
primary and secondary beams.
While this procedure should in principle establish perfect spatial and temporal overlap,
in practise there were unforeseen problems. A minor and somewhat predictable problem
is that the last step of the alignment procedure may lead to the loss of spatial overlap of
the laser beams because the mirrors on the delay stage are not perfectly aligned. This
can easily be corrected by repeating the spatial alignment procedure for the secondary
beam. A more severe problem is that the use of the alignment tool in practise led to a
systematic deviation in the spatial overlap of the laser beams. This problem was solved
by using the primary beam to burn a small hole into the edge of the copper TEM mesh
in situ and using that to correct the spatial alignment. While the exact cause of the
deviation is unclear, it is likely related to the thickness of the stainless steel foil that was
used to make the alignment tool since the problem did not appear when using the hole in
the much thinner TEM mesh. Once the signal has been established, it can be optimized
in real time while watching the electron beam image on the detector.
Chapter 3. Characterization of Femtosecond Electron Pulses 67
3.3.2 Results and Data Analysis
A series of images of the electron beam on the detector is shown in Fig. 3.12. As expected
from the simulations, symmetric scattering along the horizontal (laser pulse propagation)
direction is observed. Fig. 3.13 shows beam profiles through the electron beam images
are shown in. The horizontal beam profile at t = 0 is clearly a sum of the unbroadened
electron beam and a significantly broadened part. This highlights the fact that only the
(temporal) centre section of the pulse has been scattered while the electrons at the front
and back of the pulse continue on a nearly unaltered trajectory. The vertical line profile
through the electron beam image only displays a reduction in amplitude while changes
to its profile due to the vertical envelope of the laser pulses are small since the laser beam
waist is wider than the electron beam after the stripping pinhole.
In analogy to section 3.2.2, the images taken at different electron pulse-laser pulse
delays τ are analyzed to give a time trace
S(τ) =
∫|X|Dτ (X,Y ) dX dY , (3.31)
where X and Y are the horizontal and vertical coordinates on the detector screen (origin
at electron beam centre) and Dτ (X,Y ) is the number density of electrons detected in
the image at time delay τ . A number of these time traces for different electron numbers
per pulse are shown in Fig. 3.14.
To illustrate that this time trace is a direct cross-correlation of the electron and laser
pulses, we assume that the displacement that electrons experience due to the pondero-
motive force is negligible over the size of the laser focus and therefore the force acting
on an electron can be integrated along a straight line. This condition will be further
discussed below.
The distance from the beam crossing point to the detector is much longer than the
Chapter 3. Characterization of Femtosecond Electron Pulses 68
µ400 m
0 fs−160 fs
−320 fs−480 fs−640 fs
160 fs
320 fs 480 fs 640 fs
Figure 3.12: Detector images of the electron beam at different electron pulse-laser pulse
delays. The black spot in the electron beam is caused by detector damage. Conditions:
10,000 electrons per pulse, 135 µJ laser pulse energy, images are integrated over 4000
pulses.
X [ m]−400 −200 0 200 400
Y [ m]µ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
−400 −200 0 200 400
Inte
nsity
[A.U
.]
µ
Figure 3.13: Horizontal (left) and vertical (right) line profiles through the detector images
at −640 fs (solid line) and at 0 fs (dashed line).
Chapter 3. Characterization of Femtosecond Electron Pulses 69
τS
( )
28800
56001750
15700
electrons per pulse:
[fs]τ−1000 −500 0 500 1000
Figure 3.14: Time traces with Gaussian fits. Vertical offset for clarity. While a Gaussian
appears to be a good fit for traces of pulses with relatively few electrons, there are
systematic deviations between the data and the fit for pulses with more electrons. The
fits have widths of 1005 fs, 757 fs, 546 fs and 404 fs FWHM, respectively.
laser spot size and therefore, the deflection of an electron at the detector is given by its
velocity after its interaction with the laser:
X =Td
me
∫Fx(x, y, z, t) dt . (3.32)
Td is the propagation time of the electron pulse from its intersection with the laser to the
detector. According to Eq. 3.30 with the lateral envelope of the laser beam, Fx can be
written as a product of a constant F0 and of four functions, each of which only depends
Chapter 3. Characterization of Femtosecond Electron Pulses 70
on one of the variables t, x, y, z:
ft(t) = exp
(− t2
2w2t
), fx(x) = exp
(− x2
2w2t c
2
)sin(2kx),
fy(y) = exp
(− y2
2w2f
), fz(z) = exp
(− z2
2w2f
). (3.33)
Eq. 3.31 can be interpreted as the sum of |X| over all electrons in a pulse. Using the elec-
tron density in the electron pulse at the beam crossover position ρ(~r, t) = ρxy(x, y)ρt(t +
τ − z/v) with v being the velocity of the electrons in z-direction and Eqs. 3.32 and 3.33,
we rewrite 3.31 as
S(τ) =
∫F0ft(t)fx(x)fy(y)fz(z)×
ρxy(x, y)ρt
(t + τ − z
v
)dtdx dy dz
∝∫
fz(z)ft(t)ρt
(t + τ − z
v
)dz dt . (3.34)
S(τ) can therefore be identified as a convolution of the temporal profiles of the electron
pulse and the laser pulse and of the transverse spatial profile of the laser pulse as it is
crossed by the electron pulse. As opposed to the earlier experiments (see section 3.2.2),
the scattering potential is not moving through the electron beam, thereby eliminating
the transverse spatial profile of the electron pulse from the convolution.
In this derivation, we used the assumption that an electron propagates through the
standing wave with only a negligible deviation from the z direction. This condition is
much stricter than in the case of a single laser pulse since the ponderomotive force in this
experiment changes drastically over very short distances, defined by λ/2 (the wavelength
of the intensity profile of the standing wave) as opposed to the spatial envelope of the
laser beam. A “short” distance, for this purpose, is therefore a distance ≪ λ/2. If
an electron moves a significant fraction of this distance in x-direction while still inside
the standing wave, the linearity of S(τ) with respect to the laser intensity could be
compromised. Electrons that experience the strongest force also experience the strongest
Chapter 3. Characterization of Femtosecond Electron Pulses 71
deflection while crossing the standing wave. These electrons would then encounter a
weaker ponderomotive force as they are deflected towards the intensity minimum of the
standing wave and may even enter a region in which the force is inverted with respect
to the force experienced initially. This would lead to a reduced cumulative acceleration
along the trajectory, primarily for strongly deflected electrons. This “saturation” effect
would flatten the signal trace around its maximum and broaden time trace, leading to
an overestimation of the pulse duration. Hence, even if slight saturation occurs, a pulse
duration determined in the measurement is still guaranteed to be an upper limit to the
actual pulse duration.
To obtain an estimate for the upper limit of the deflection an electron might experience
while still inside the standing wave, we calculate the deflection it would experience if it
were exposed to the maximum possible force
Fmax =I0e
2λ
2πmeε0c3. (3.35)
While the duration of the standing wave is determined by the laser pulse duration wt,
the electron experiences it for a shorter time we since it traverses the beam spatially and
therefore the intensity envelope experienced at the electron position is the product of the
temporal envelope and the spatial envelope crossed at v:
1
w2e
=1
w2t
+
(v
wf
)2
. (3.36)
For the parameters used in this experiment, the FWHM of the resulting profile is 170 fs.
If the electron were exposed to Fmax for this duration, its offset from its initial trajectory
would be
Fmax
2me
(2√
2 ln 2we
)2
≈ 150 nm . (3.37)
Since this is on the same order of magnitude as λ/2, we verified the linearity of the signal
vs. the laser pulse energy experimentally. The measurement of the signal amplitude S(0)
Chapter 3. Characterization of Femtosecond Electron Pulses 72
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Sig
nal [
A. U
.]
Scattering Laser Pulse Energy [mJ]
Figure 3.15: Laser power dependence of the ponderomotive scattering signal vs. scattering
laser pulse energy. A straight line is a good fit and demonstrates the linear dependence
of the effect on the laser intensity over the entire accessible range.
vs. the laser pulse energy (see Fig. 3.15) did not show any deviations from the expected
linear response within the range of pulse energies accessible in our setup.
As expected, the width of the time traces shown in Fig. 3.14 and hence the electron
pulse duration increases with the electron number density. Particularly for pulses con-
taining large numbers of electrons i.e. long electron pulses whose shape dominates the
corresponding time trace, the Gaussian fits shown in the graph are not a suitable fit
function. The centre portion of the pulse is flatter and the edges somewhat steeper than
expected for a Gaussian pulse, as expected from the simulations described in section
2.2.1.
The actual temporal shape of the electron pulse cannot be directly deconvolved from
noisy experimental time-traces unless the other contributions to the convolution are small.
Chapter 3. Characterization of Femtosecond Electron Pulses 73
However, since the variance σ2 of the convolution of multiple functions is the sum of their
individual variances, the variance σ2ρ of the temporal shape ρt(t) of the electron pulse can
be extracted from the experimental trace without prior knowledge of the non-Gaussian
pulse shape by
σ2ρ = σ2
trace − σ2t − σ2
f/v2 (3.38)
where σ2trace, σ2
t and σ2f are the variances of S(t), ft(t) and fz(z), respectively.
The most common measure for pulse durations is the full width at half maximum.
For a Gaussian, FWHM = 2√
2 ln 2σ. This conversion technically does not apply to other
pulse shapes but in practise, it is a useful value to estimate the effect of the pulse shape on
the temporal resolution of a pump-probe experiment. Therefore, we calculate the electron
pulse duration T as T = 2√
2 ln 2σρ. The contributions that need to be deconvolved from
the trace are the temporal laser pulse shape (210 ± 10 fs FWHM sech2) and the spatial
profile of the laser pulse at the focus in z direction (41 ± 10 µm = 310 ± 80 fs FWHM
Gaussian for 55 keV electrons). Fig. 3.16 shows the deconvolved electron pulse duration
as a function of the number of electrons per pulse.
The measured pulse durations are shown together with simulations performed with
the general particle tracer [48] software package. The parameters used in the simulations
correspond to the setup used in these pulse duration measurements: electrons are emitted
from a gold photocathode by two-photon photoemission with a 50 fs, 500 nm, 200 µm
FWHM diameter laser pulse. In order to account for the two-photon process, the effective
pulse duration used in the simulations is 35 fs and the diameter is 141 µm. An excess
energy distribution of 0.1 eV width is assumed. The electrons are accelerated over a 6 mm
gap to 55 keV. The anode is a 150 µm diameter pinhole and the electrons propagate
freely for 26 mm after that. A 30 µm diameter pinhole strips the electron pulse followed
by another 7 mm of free propagation before the laser interaction. Note that this setup
adds 9 mm in total to the propagation length of the electron pulses over the setup used
Chapter 3. Characterization of Femtosecond Electron Pulses 74
GPT simulationsmeasurement
0
200
400
600
800
1000
0 5000 10000 15000 20000 25000 30000
T [f
s]
number of electrons per pulse
Figure 3.16: Electron pulse duration T vs. number of electrons per pulse; general particle
tracer (GPT) simulations and measurements obtained in the described experiments.
in UED experiments. This is not a limitation of this pulse characterization technique
but was chosen to minimize the complexity of the setup caused by the introduction of
the stripping pinhole.
3.3.3 Discussion
The observed electron pulse durations are in reasonable agreement with calculations
both with respect to duration and dependence on electron number. The systematic
deviation seen in the Fig. 3.16 may have been caused by imperfections of the experiment,
e.g. pedestals in the temporal profile of the laser pulse or stem from assumptions used
in the simulations such as the initial momentum distribution of the photoelectrons at
the photocathode, which have not been experimentally verified. The experimental errors
could be reduced and the system response could be improved by using shorter, cleaned
Chapter 3. Characterization of Femtosecond Electron Pulses 75
up laser pulses to scatter the electron pulse. Currently, the intrinsic system response is
determined mostly by the electron transit time across the focus (310 fs presently). A
10 µm focus and 20 fs laser pulse would give a system response of 80 fs and a similar
scattering strength as presented here, with only 10 µJ laser pulse energy.
The grid measurement for t = 0 determination together with the presented pondero-
motive force electron pulse characterization methods constitute a comprehensive set of
practical tools for the characterization of electron pulses for UED and for the performance
analysis of femtosecond electron pulse sources.
Chapter 4
Femtosecond Electron Deflectometry
Given their high charge-to-mass ratio, electrons are easily deflected when they traverse
electric or magnetic fields. This property makes them a sensitive probe to measure these
fields. Hence, femtosecond electron pulses can be used as probes for rapidly changing
fields. This fact is exploited in the grid t = 0 determination (see section 3.1) on a
day-to-day basis but little attention had been paid to the physics behind the observed
electron beam deflections. Using the electrometer connected to the faraday cup in the
vacuum chamber, we had observed negative as well as positive currents during these t = 0
measurements and hence we were aware that electrons as well as ions were being emitted
from the target i.e. laser ablation was taking place. As demonstrated in section 3.2.3,
there is not appreciable delay between the arrival of the laser pulse and the onset of the
deflection of the electrons. Hence, these measurements are a sensitive time-resolved probe
of the electric fields produced during the earliest stages of femtosecond laser ablation.
Femtosecond laser ablation of solids has attracted much attention due to its applica-
tions in laser micromachining [79], laser surgery [80] and matrix assisted laser desorption
Parts of this chapter have previously been published in [44] Christoph T. Hebeisen, German Sciaini,Maher Harb, Ralph Ernstorfer, Sergei G. Kruglik, and R. J. Dwayne Miller, Physical Review B 78,081403 (2008). Copyright 2008 by the American Physical Society.
76
Chapter 4. Femtosecond Electron Deflectometry 77
[81]. There are many different processes that lead to the removal of material after ex-
citation with an intense laser pulse. These processes largely fall into two groups [82].
Thermal processes happen after energy relaxation between the electrons and the lattice
has taken place. Melting, evaporation, phase explosion and thermal plasma generation
are examples of thermal processes. In non-thermal ablation processes, disintegration
of the lattice occurs before the electrons and the lattice have reached thermal equilib-
rium. Non-thermal processes include non-thermal melting and Coulomb explosion. For
a non-thermal process to take place, the energy required to drive the process needs to be
deposited within less than the relaxation time. Non-thermal processes are therefore only
possible with ultrashort excitation pulses. Laser pulse parameters and properties of the
material being ablated determine which process ultimately dominates.
Transient electric fields play an important role in laser ablation processes. When an
intense laser pulse hits the surface of a solid, multi-photon photoemission, avalanche and
field ionization lead to the emission of large numbers of electrons and charging of the
surface. Electron escape, recapture, ion emission and transport processes within the solid
modify the spatial charge distributions and the resulting electric fields over time. The
trajectories of electrons and ions emitted during ablation are subject to these fields. If
the charge density in the lattice is high enough, Coulomb repulsion between the ions can
even cause the lattice to disintegrate, possibly aided by the additional pull of the cloud
of emitted electrons above the surface, a process referred to as Coulomb explosion [83].
Even below the ablation threshold of a given material, large numbers of electrons
can be emitted from a sample. Over time, the positive counter charge on the sample is
compensated by recapture of some of the emitted electrons and influx of electrons from
non-excited parts of the sample [84]. Because of electron recapture, the magnitude of the
initially emitted charge is often underestimated when the total charge of the irradiated
sample is measured in the long time limit. However, a transient electron deficit can
Chapter 4. Femtosecond Electron Deflectometry 78
lead to equally transient changes in the material properties of the excited sample, a
fact often ignored in ultrafast high excitation experiments which could lead to incorrect
interpretations of measurements. These surface charging effects can also be detected with
the method presented here.
Usually, ablation is studied by analyzing the energy, charge and direction of the pro-
duced electrons, ions and neutral particles using time-of-flight or other mass spectrometry
methods [85, 83, 86, 82]. These methods yield valuable data allowing identification of dif-
ferent ablation regimes. Assuming a model for the ablation process, certain parameters
of the actual ultrafast process can be extracted. However, these measurements inherently
integrate over the ultrashort timescales of the microscopic processes during ablation and
as a result, the dynamics have to be implied from a model or need to be determined us-
ing a different method. Alternative all-optical methods like time-resolved shadowgraphy
[87, 88] can detect ablation plumes in a time-resolved manner but are not sensitive to
electrical charge. Okano and coworkers [89] used electron pulses to detect electric fields.
The temporal resolution of their study was limited by the 64 ps duration of the electron
probe pulses. By using a femtosecond electron pulses to probe the electric fields during
femtosecond laser ablation, we achieve subpicosecond resolution on the visualization of
transient electric fields [44]. This technique opens the door to a more detailed view of the
ablation process since it provides insights into the spatial distributions and behaviour of
charged particles at the earliest stages of ablation. After we had completed and submit-
ted this work for review, we learnt of a study [90], which used electron pulses to study
plasma generation by a femtosecond laser pulse in a nitrogen gas jet. The temporal reso-
lution these experiments is limited to several picoseconds by the duration of the electron
pulses and the velocity mismatch between the laser pump and electron probe pulses.
Chapter 4. Femtosecond Electron Deflectometry 79
Figure 4.1: The experiment geometry. The sample is placed inside the electron beam
and the laser pump pulse hits the sample surface within the trace of the electron beam.
4.1 Experimental Setup for the Observation of Tran-
sient Charge Distributions
The setup of this experiment is based on the basic UED setup described in chapter 2.
However, the sample in this experiment is not a thin film and is hence completely opaque
for the electron pulse. In order to probe the electric fields near the sample surface, the
sample is placed inside the beam path of the electron pulse so that it partially blocks the
beam (Fig. 4.1) and that the surface on which the ablation will take place is parallel to
the electron propagation. The laser pump pulse is aligned to strike the sample surface
at normal incidence inside the electron beam trace on the sample surface.
The probe electrons in the unblocked part of the electron beam pass the excited
sample surface at close distance. Transient electric fields present when the probe pulse
passes the sample impart momentum on the probe electrons, which impacts their spatial
distribution at the detector. Electrons contained at different positions within the electron
Chapter 4. Femtosecond Electron Deflectometry 80
Silicon
Silicon nitride Photoresist
StrippingExposing,developing
Spin coating,baking etching
Dry (plasma) Wet (KOH)etching
Bufferedoxide etch
Silicon strip
Figure 4.2: Fabrication of the silicon strip samples (layer thicknesses not to scale).
beam probe the field at different positions and are therefore deflected differently.
We obtain a time-series of the resultant probe electron distributions by repeating this
measurement for different pump-probe delays. Each of the recorded images is a snapshot
containing information about the electric field present at the moment when the electron
pulse passed the excited part of the sample. For reference, an electron beam image
without the laser beam is taken before each frame with the laser beam present under
otherwise identical circumstances. We refer to these images as “before images”.
The fabrication process of the samples is shown schematically in Fig. 4.2. The silicon
strips were produced starting from a commercial wafer ((100) orientation) with a 40 nm
silicon nitride layer deposited on both sides. Stripe features were patterned in the silicon
nitride layer using photolithography and reactive ion (plasma) etching. This was followed
by wet etching in KOH to produce strips of Si. Finally, the nitride layer was removed
using 10 : 1 buffered oxide etch and the sample surface was cleaned with methanol. The
resulting Si strips were 330 µm thick.
As the excitation laser pulse, we used the frequency-doubled (λ = 388 nm) laser
pulse. The pulse duration was 150 fs and the beam size was 17 µm FWHM at the sample
position. The electron pulse duration was 300 fs. Each beam image was integrated
over 40 shots. Inspection of the sample after the experiment revealed that ablation had
taken place at all investigated fluences. Hence, potential surface contamination has no
Chapter 4. Femtosecond Electron Deflectometry 81
30 ps 100 ps 300 ps 1000 ps9.5 ps
3 ps
100 mµ
0.5 ps 1 ps 2 ps0 ps
Figure 4.3: Probe electron density maps at the detector. Each frame is averaged over 40
laser shots. In these images, the laser beam is incident from the right side, the sample
surface is indicated by the yellow line. The scale indicates size in the sample plane, the
image on the detector appears 5.9× larger due to electron beam divergence. Peak laser
fluence: 5.6 J/cm2.
significant effect on these experiments since the surface contaminants would be removed
after the first or very few laser shots. The sample was moved vertically to a new position
for each image to prevent deep crater formation and to ensure identical conditions for
the images taken at each time step.
4.2 Experimental Results and Analysis
Selected images from the time series are shown in Fig. 4.3. The detected number of
electrons was constant throughout the time series i.e. no time-dependent absorption of
probe electrons by material ejected from the sample or high angle deflection outside the
detector range was observed.
Qualitatively, we interpret the beam shapes shown in Fig. 4.3 as follows: The probe
Chapter 4. Femtosecond Electron Deflectometry 82
electrons that pass near the excited part of the sample are initially strongly deflected
away from the sample surface due to the emission of a large number of electrons from the
sample. These emitted electrons are initially very close to the surface and form a dipole
with the countercharges at the surface. Despite the attraction of the countercharge on
the surface, electrons that are sufficiently far away from the sample surface experience a
net force in the opposite direction due to space charge of the emitted electron cloud [84].
By 9.5 ps, this self-acceleration has caused the electron cloud to expanded so far that the
probe beam is split into two lobes. At this time, the probe electrons above and below the
excitation spot are visibly deflected into the sample shadow by the attractive field of the
counter charge. This deflection towards the sample becomes more and more pronounced
at 30 ps and 100 ps. However, a localized electron cloud still partially shields the Coulomb
attraction to the center part of the probe beam, causing a persistent indentation in the
middle of the electron beam profile. As more of the emitted electrons continue to escape
or get reabsorbed, the negative charge density near the surface reduces. At 300 ps and
later, the attractive force of the ions clearly dominates. Even at that time, the positive
charge on the sample surface is strongly localized, leading to a slight contraction of the
probe beam in y-direction.
Probe Electron Deflection for a Given Charge Distribution
In order to extract quantitative information about the charge distributions we calculate
the effect of a given charge distribution on the probe electron beam. The momentum
change a probe electron experiences due to a charge Q at the origin is
∆~p =
∫ −Qe
4πε0 |~r(t)|3~r(t)dt . (4.1)
Here, ~r(t) = (x, y, z) is the position of the probe electron, e is the elementary charge and
ε0 is the permittivity of free space. We assume a “frozen” charge distribution, i.e. that
Chapter 4. Femtosecond Electron Deflectometry 83
the charge Q is not moving in the period during which the probe electron traverses the
region of significant electric field caused by Q. Given the size of the excitation beam
(17 µm) and the kinetic energy of the probe electrons (55 keV), the probe electron
passes the excited region of the sample in 130 fs. At least at early times i.e. before the
charge distribution has any time to spread significantly beyond the excited region, the
assumption of a frozen charge distribution is therefore justified. If the angle of deflection
and the longitudinal momentum transfer are small, we can rewrite Eq. 4.1 as
∆~p =−Qe
4πε0vz
∫~r
|~r|3dz (4.2)
for an electron initially propagating parallel to the z-axis with velocity vz. The deflection
∆(X,Y ) of the probe electron at the detector is then
∆(X,Y ) =∆ (px, py) l
mevz
=−Qel
2πε0mev2z
(x, y)
(x2 + y2), (4.3)
where me is the mass of the electron and l is the distance between the sample and
the detector. If the charge distribution is concentrated in a short section along the
electron beam path, we can integrate Eq. 4.3 over the charge distribution to calculate
the cumulative effect that the charge distribution has on the probe electrons i.e. a three-
dimensional charge distribution ρ(x, y, z) can be collapsed into an area density
ρ′(x, y) =
∫ρ(x, y, z) dz . (4.4)
For such a charge distribution, Eq. 4.3 can then be rewritten as
∆(X,Y ) =−el
2πε0mev2z
∫ ∫ρ′(x′, y′)(x − x′, y − y′)
((x − x′)2 + (y − y′)2)dx′ dy′ . (4.5)
Eq. 4.5 allows us to numerically calculate the positions of probe electrons on the detector
for a given charge distribution at the sample, but we still cannot calculate the charge
distribution from the detected electron beam shapes because the “origin” (x, y) of an
electron detected in a given detector position (X,Y ) is not known. If this information
Chapter 4. Femtosecond Electron Deflectometry 84
were available, the electric field integrated along z experienced by an electron while
passing the sample at (x, y) could be directly calculated.
Another factor complicating the analysis of the acquired images further is the imper-
fect correlation between the radial position and momentum in the electron pulse even
before it reaches the sample. If the electron beam emanated from a point source or were
perfectly parallel, the radial momentum of paraxial electrons would be directly propor-
tional to their radial position. However, in practise the electrons are produced by a finite
size laser beam and with an initial lateral momentum distribution. Other imperfect ele-
ments of the electron gun may further contribute to this problem through inhomogeneous
electric and magnetic fields. The result is image blur. This can be easily seen in Fig. 4.3:
the left edge of the electron beam is created by the sharp edge of the sample and without
blur would therefore be a step function, not a smooth (albeit steep) gradient as seen
in the measurement. However, if a distribution of deflecting charges is known from a
physical model, the resulting image can be relatively easily calculated and compared to
the experimental results using a blurring function to imitate the imperfections of the
electron beam.
Fitting to a Model Charge Distribution
A full quantitative model of the ablation process treating free electrons and carriers in
the sample could be used to predict the probe beam images. Given such a model, these
experiments could serve as a direct qualitative as well as quantitative test. Lacking a
detailed microscopic model, we fit the experimental data using a phenomenological model
for the charge distribution, which makes no assumptions about dynamics. This model
charge distribution consists of two parts: positive charges on the sample surface and a
cloud of emitted electrons above the surface (see Fig. 4.4). Both parts of the charge
distribution are radially symmetric around the x-axis and the model assumes that the
Chapter 4. Femtosecond Electron Deflectometry 85
Excitationlaser
Electronprobe pulse
Emittedelectrons
z
SampleSurface charge
Figure 4.4: Schematic view of the charge distribution model used to fit the data.
radial distributions are Gaussian. The positive charge on the sample is contained in
a plane on the sample surface while the electron density in the charge cloud near the
sample falls off exponentially with increasing distance from the surface.
The parameters of the model were the amount of positive charge on the sample
surface Qs, the radial size (FWHM) of that charge distribution Hs, the (negative) charge
contained in the electron cloud above the sample surface Qc, the thickness (1/e) of
the electron cloud Wc and the radial size of the electron cloud Hc. For the fit, the
vertical position of the two charge distributions Ys had to be introduced as an additional
parameter to compensate for electron beam pointing instability. We also set Qc = −Qs
because in fits we performed with Qc as an independent fit parameter, Qs + Qc was
consistent with 0 for the range of delay times for which the fits delivered meaningful
results. However, the fits with the additional independent parameter were slower and
their outcome was noisier.
We calculated the probe beam shapes resulting from such a charge distribution as
Chapter 4. Femtosecond Electron Deflectometry 86
follows. The electron distribution is simulated on a grid where each vertex contains a
certain amount of charge. For convenience, the pitch of the grid was chosen to coincide
with the size of one pixel on the detector CCD image i.e. 12.6 µm on the phosphor screen.
Owing to the divergence of the electron beam, there is a scaling factor between the size
of the buckets at the sample position and their size at the detector. First, the lateral
distribution of the probe pulse had to be determined by fitting the before images. As a
functional shape of the electron beam just before the sample, I assumed
ρ′(r) =
(1 − r
R
)A r < R
0 r ≥ R
with r2 = (x − xb)2 + (y − yb)
2 (4.6)
where A and R are parameters that describe beam amplitude and size, respectively and
xb and yb are the centre position of the beam. The electrons that hit the sample do not
reach the detector. Hence, the charge on each vertex with |x− xs| < ws/2 is set to 0 (xs
is the sample position and ws is the sample width). Finally, a two-dimensional Gaussian
blur is applied on the array. This model was used to fit the before images using a simplex
downhill [91] algorithm with the parameters A, R, xb, yb, xs and ws. The parameter ws
gives the width of the “sample shadow” on the detector. By using the actual thickness of
the sample, the scaling factor for the bucket size at the sample position vs. the detector
position was determined to be 5.9.
While the functional form of the electron beam described in Eq. 4.6 gives only a
very crude desription of the electron pulse envelope, it works well because only the outer
part of the tail of the electron beam passes the sample and only that part needs to be
approximated.
Each of the frames taken with the pump laser pulse present was then fit using the
charge distribution model and the electron probe beam parameters determined from the
before images. A genetic search algorithm (similar to [92]) was used to perform the fits.
Chapter 4. Femtosecond Electron Deflectometry 87
In this algorithm, an “individual” is defined by a set of parameters (called “genes”) of the
model and the “population” is the set of n individuals currently being considered. For
each individual, the probe electron map at the detector is simulated according to Eq. 4.5
followed by a Gaussian blur function. The fitness of an individual is determined as the
inverse of the RMS difference between the simulated and the detected probe electron
maps. After the fitness of all individuals has been calculated, the m fittest individuals
are selected as the “mating pool” (m is called the “mating pool size”). From these m
individuals, a new population, the next “generation”, is created. There are several ways
by which a new individual is created from one or two parent individuals:
• “Cloning:” an exact copy of an individual in the mating pool is created. While this
method obviously does not lead to any improvements of fitness, it may sometimes be
desirable to keep the fittest individual(s) of the mating pool in the next generation
to ensure that the best combination of genes found so far is not lost.
• “Mutation:” similar to cloning except that every gene has a certain probability
Pmut. of mutating during the copy process. If gene number i mutates, a random
variable with a Gaussian probability distribution with width σi is added to the
value of the gene. While Pmut. is constant throughout the fit and identical for each
gene, ~σ is a vector containing a value for each gene and changes from generation to
generation.
• “Recombination:” each gene is chosen with equal probability from one of the two
parents.
After the production of the next generation, the fitness of each individual in the new
population is calculated. The fitness of all individuals created by mutation is compared
to the fitness value of their parents. The percentage of “successful” mutants (i.e. mutants,
whose fitness is higher than that of their parent) is compared to a threshold value ηc.
Chapter 4. Femtosecond Electron Deflectometry 88
If the number of successful mutants is smaller than ηc, the mutation width vector ~σ is
multiplied by a factor q (this corresponds to a decrease since 0 < q < 1), otherwise
it is divided by q. After the selection of the new mating pool, the next generation is
produced and the process begins anew. The convergence criterion for this algorithm is
the reduction of ~σ below a given fraction of its initial value.
For our fits we used a population size of n = 64 and a mating pool size of m = 8.
New individuals were exclusively created by mutation, i.e. each individual in the mating
pool was used to create 8 new individuals by mutation. The probability for each gene to
mutate was Pmut. = 0.2, the threshold for the increase of ~σ was ηc = 0.6 and the factor
q = 0.8.
These parameters were found by experimentation. The main aim was to speed up
convergence and to prevent runaway solutions and local maxima (this search maximizes
fitness). Recombination of individuals did not produce high fitness individuals because
the different genes (parameters) of the model are not orthogonal and the fitness surface is
complicated. Hence, no recombination of genes is used. Cloning was abandoned because
it favours local maxima in the fitness surface.
The charge distribution model we used obviously places restrictions on the shapes of
the charge distributions and ignores escaping electrons, emitted ions and charge refilling
from the bulk. While all of these effects modify the charge distribution after some time,
the model can still deliver an estimate of the size of the charge distributions as well as the
amount of charge for times shortly after excitation. Up to about 10 ps, the experimental
results are reproduced well by the fits to this model. Measured and fitted beam images
are shown in Fig. 4.5. The deviations for later times are likely caused by the escape
of some of the emitted electrons well beyond the envelope of the probe beam as well as
shortcomings of the assumptions about the charge density distributions in our model. We
attempted to add a separate escaping charge distribution to the above model, however,