# Ultrafast Electron Diffraction: Visualizing Dynamic States of Matter

D. Filippetto  Lawrence Berkeley National Laboratory, Berkeley, California,
94720, USA  P. Musumeci  Department of Physics and Astronomy, University of
California at Los Angeles, Los Angeles, California 90095, USA  R. K. Li
Department of Engineering Physics, Tsinghua University, Beijing 100084, China
Key Laboratory of Particle and Radiation Imaging, Tsinghua University,
Ministry of Education, Beijing 100084, China  B. J. Siwick  M. R. Otto  Centre
for the Physics of Materials, Department of Physics and Department of
Chemistry, McGill University, Montreal, Quebec H3A 2T8, Canada  M. Centurion
J.P.F. Nunes  University of Nebraska-Lincoln, 855 N 16th Street, Lincoln,
Nebraska 68588, USA

###### Abstract

Since the discovery of electron-wave duality, electron scattering
instrumentation has developed into a powerful array of techniques for
revealing the atomic structure of matter. Beyond detecting local lattice
variations in equilibrium structures with the highest possible spatial
resolution, recent research efforts have been directed towards the long
sought-after dream of visualizing the dynamic evolution of matter in real-
time. The atomic behavior at ultrafast timescales carries critical information
on phase transition and chemical reaction dynamics, the coupling of electronic
and nuclear degrees of freedom in materials and molecules, the correlation
between structure, function and previously hidden metastable or nonequilibrium
states of matter. Ultrafast electron pulses play an essential role in this
scientific endeavor, and their generation has been facilitated by rapid
technical advances in both ultrafast laser and particle accelerator
technologies. This review presents a summary of the remarkable developments in
this field over the last few decades. The physics and technology of ultrafast
electron beams is presented with an emphasis on the figures of merit most
relevant for ultrafast electron diffraction (UED) experiments. We discuss
recent developments in the generation, manipulation and characterization of
ultrashort electron beams aimed at improving the combined spatio-temporal
resolution of these measurements. The fundamentals of electron scattering from
atomic matter and the theoretical frameworks for retrieving dynamic structural
information from solid-state and gas-phase samples is described. Essential
experimental techniques and several landmark works that have applied these
approaches are also highlighted to demonstrate the widening applicability of
these methods. Ultrafast electron probes with ever improving capabilities,
combined with other complementary photon-based or spectroscopic approaches,
hold tremendous potential for revolutionizing our ability to observe and
understand energy and matter at atomic scales.

###### Contents

  1. I Introduction
    1. I.1 Electrons as probes of matter
      1. I.1.1 The role of electron energy in electron scattering
      2. I.1.2 Scattering from gaseous targets
      3. I.1.3 Scattering from crystals
      4. I.1.4 Coherence length and reciprocal space resolution in UED
      5. I.1.5 Electron vs. X-ray scattering
    2. I.2 Electron beam brightness
      1. I.2.1 The electron beam concept
      2. I.2.2 Beam phase space and brightness definitions
      3. I.2.3 Quantum limit of beam brightness
    3. I.3 Different modalities of ultrafast electron scattering instrumentation: diffraction, imaging and spectroscopy
    4. I.4 Scientific drivers for ultrafast electron scattering
      1. I.4.1 Solid State: ordering, excitation and emergent phenomena in materials
      2. I.4.2 Gas Phase: Uncovering the structure-function relationship behind photochemical reactivity
  2. II Ultrafast probes for electron diffraction
    1. II.1 Overview of a general UED setup and operating modes
      1. II.1.1 Temporal resolution
      2. II.1.2 Electron packets: from single-electron to single-shot
    2. II.2 Generation of electron pulses
      1. II.2.1 Quantum efficiency
      2. II.2.2 Photocathode thermal emittance
      3. II.2.3 Response time of a photoemitter
      4. II.2.4 Source size and spatial resolution in UED
      5. II.2.5 Towards brighter photoemission sources
      6. II.2.6 Laser systems
    3. II.3 Electron dynamics
      1. II.3.1 The accelerating gap
      2. II.3.2 Electron acceleration via time-varying fields
      3. II.3.3 The effect of the cathode curvature
      4. II.3.4 Temporal beam evolution in simple systems: Vacuum dispersion
      5. II.3.5 Space charge effects
      6. II.3.6 Temporal compression
      7. II.3.7 Evolution of the beam energy spread
    4. II.4 Technologies for electron acceleration
      1. II.4.1 DC sources
      2. II.4.2 RF-based pulsed sources
      3. II.4.3 Continuous-Wave RF sources
      4. II.4.4 Advanced electron sources
    5. II.5 Control and measurement of ultrafast pulses of electrons
      1. II.5.1 Measuring the duration of ultrashort electron pulses
      2. II.5.2 Time-stamping
      3. II.5.3 Measuring time-zero
      4. II.5.4 Laser-to-RF synchronization
      5. II.5.5 Truly single shot measurements
      6. II.5.6 Control of lateral coherence and beam size
    6. II.6 Electron detection schemes
      1. II.6.1 Indirect electron detection schemes and efficiency
      2. II.6.2 Direct electron detection
  3. III Measuring dynamics of matter in a solid state with bright electrons
    1. III.1 Introduction
    2. III.2 Summary of theory results for time-resolved electron scattering from crystalline solids
      1. III.2.1 Scattering from crystals including phonon excitations
      2. III.2.2 Time-dependent factors in Bragg scattering: $I_{0}(\mathbf{s},t)$
      3. III.2.3 Time dependent factors in the diffuse intensity: $I_{1}(\mathbf{s},t)$
      4. III.2.4 Electron beam requirements and considerations
    3. III.3 Experimental requirements
      1. III.3.1 Sample preparation methods
      2. III.3.2 Laser excitation conditions
      3. III.3.3 Determinants of signal detection; shot-noise limits
      4. III.3.4 Heat dissipation and limitations in multi-shot experiments
      5. III.3.5 Data processing for solid-state scattering
    4. III.4 Examples from literature
      1. III.4.1 Following ultrafast evolution of irreversible processes with high brightness beams
      2. III.4.2 Exploring the dynamics of low-dimensional quantum materials
      3. III.4.3 Ultrafast electron diffuse scattering with high momentum resolution and SNR
  4. IV Techniques and challenges in gas-phase time-resolved electron diffraction
    1. IV.1 Introduction
      1. IV.1.1 Laser driven dynamical processes
      2. IV.1.2 Milestones in GUED
    2. IV.2 Pump probe requirements
      1. IV.2.1 Temporal resolution
      2. IV.2.2 Laser pump pulses
    3. IV.3 Sample delivery requirements
      1. IV.3.1 Continuous nozzles
      2. IV.3.2 Pulsed nozzles
      3. IV.3.3 Chamber design considerations
    4. IV.4 Signal analysis
      1. IV.4.1 Signal processing
      2. IV.4.2 Structural information retrieval methods
  5. V Opportunities and outlook for ultrafast electron diffraction
  6. VI Acknowledgements

##  I Introduction

The discovery of the wave nature of the electron at beginning of the 20th
century [67, 346, 66] marked the start of a new era in the human quest for an
atomic-level perspective on the architecture of the microscopic world. Since
then, the development of scientific tools exploiting the sub-Å imaging power
of electron waves and their strong interaction with matter has seen rapid
growth, starting with the invention of the transmission electron microscope
(TEM) by Ruska in 1932 [178]. Today, electron diffraction and microscopy are
primary enablers of research and development in many scientific disciplines
including chemistry, biology, physics and material sciences as well as in many
industries.

Over the years, continuous improvements in charged particle optics [17, 124,
296, 305, 123], detectors, and new algorithms, have culminated in spatial
resolution well below atomic spacing in matter and approaching the limit set
by lattice vibrations [44]. In diffraction mode, electron optics can form
beams able to illuminate areas well below 1 nm. These spectacular developments
indicate that there is less to gain from further improvements to spatial
resolution alone than there once was, and other frontiers in instrumentation
development are beginning to emerge or see renewed interests. These include
improving elemental contrast, _in-situ_ investigations in diverse sample
environments (liquid and gas) and under tunable conditions of temperature,
pressure, as well as enhanced time-resolution to interrogate systems far from
equilibrium [426]. At the temporal resolution frontier, the overarching goal
is to make the dynamic processes in materials across the sub-Å to micrometer
length scales directly accessible, “while they are occurring”, under non-
equilibrium conditions. This goal has become a reality by combining the
atomic-scale information that can be obtained using electrons, with the
femtosecond (10-15 s) time resolution afforded by ultrafast laser technology.
This review seeks to provide an account of the development of temporally-
resolved electron diffraction to date, with a focus on the fundamentals of
pulsed electron beams and their applications to visualizing dynamic, non-
equilibrium states of matter from the analysis of diffraction patterns.

Time-resolved electron scattering emerged first as a new scientific technique
for structural dynamics in the early 1980s [241]. The development of chirped
pulse amplification and ultrafast optical laser systems [338] enabled the
generation of short bursts of photo-electrons almost perfectly synchronized
with suitable pump pulses to initiate or trigger dynamics in a specimen. Prior
to the use of ultrafast laser-driven photoemission, beams used in time-
resolved electron microscopes were emitted via thermal or field emission.
Time-resolution in these instruments was determined by the switching speed of
mechanical or electronic shutters used to modulate the electron emission or
shorten the exposure times of detector cameras and was limited to the 100
nanosecond to microsecond scale or above [152, 24]. The absence of temporal
structure in the beam and the lack of fast triggers for pulsed electron
emission and specimen excitation, precluded access to the fastest time scales
restricting conventional electron scattering instrumentation to the study of
in-equilibrium systems by static images, diffraction patterns and spectra.
When technological developments provided direct access to the observation of
the most fundamental processes in materials as they occur, they naturally
ignited a revolution in research labs around the world [311, 174, 411, 234,
248]. Sub-picosecond time scales unlocked access to fundamental dynamical
processes in condensed matter and chemistry, such as nanoscale heat transfer,
phonon transport and chemical bond formation, while the sub-atomic electron
wavelength and the strong electron-matter interaction cross section enabled
atomic-scale recording of dynamical processes such as irreversible phase
transitions in solids [327], the formation of molecular bonds [151], and very
recently, hydrogen bond dynamics in liquids [394, 203].

Figure 1: Conceptual schematic for a pump-probe UED setup in transmission
geometry.

Ultrafast electron scattering is a rapidly growing cross-disciplinary field,
drawing from decades of instrument developments in the physical and energy
science areas, such as electron microscopy, particle accelerator and laser
technology, condensed matter physics and ultrafast chemistry. Atomic-level
information can be retrieved via different operating modes such as microscopy,
diffraction and spectroscopy, isolating specific electron-matter interaction
channels. Elastic and inelastic scattering processes encode sample information
respectively on the angle and energy of the scattered electrons, while the
specific electron optics setup determines the mapping of the electron
parameters into the detector plane, commonly energy, angle (momentum transfer)
or real space. Furthermore, the geometry of the interaction and the detector
collecting angle can be optimized for the study of surface structures in bulk
materials (reflection mode) or for the characterization of bulk structure in
thin films, liquids and gases (transmission mode). This review will mainly
focus on the technological and scientific advancements in transmission
ultrafast electron diffraction (UED), which has seen a very rapid increase in
interest over the last decade. Sustained by scientific discoveries of
increasing impact, UED is now considered an established technique in the
ultrafast sciences. However, it is worth noting that the vast majority of
techniques discussed here can be directly applied to the other operating modes
mentioned above. Throughout the manuscript, the topics are presented without
any assumption on the probe electron beam energy, whose dependence is
explicitly derived and discussed where needed. Such approach extends the
relevance of the treatment proposed to UED beamlines with probe energies in
the keV-to-MeV range. Low energy, eV-scale electron diffraction (LEED) are not
included, since they are not commonly used in transmission mode, and therefore
face a different set of challenges.

A conceptual schematic of the transmission UED technique in pump-probe
geometry is summarized in Fig. 1. A short (compared to the relevant
timescales) optical pulse impinges on the specimen at a time $t_{0}$,
initiating the process of interest over a selected region. A paired electron
pulse is spatially overlapped with the optical pulse at the sample and
illuminates the probed area at a time $t_{e}$, with a delay of $\Delta
t=t_{e}-t_{0}$. Diffraction patterns are acquired as $\Delta t$ varies from
negative to positive values, and provide temporal snapshots of the atomic
structural evolution from the initial equilibrium, through the transient, up
to a final equilibrium state, which may be identical to the initial state or
different.

A short summary of the structure of this review article follows. After
reviewing fundamental concepts in diffraction in Sec. I.1, we will define a
common metric for discussion and comparison of electron sources that will be
used throughout the article (Sec. I.2), and briefly compare the different
operating modes (Sec. I.3) in terms of electron beam requirements. The
scientific niche of UED setups will be discussed as introductory motivation to
the following Sec. II, which describes in details the state-of-the-art
techniques for electron generation (Sec. II.2), beam dynamics (Sec. II.3),
acceleration technologies (Sec.. II.4), and spatio-temporal control of
femtosecond electron beams including detection (Sec. II.5 and II.6). Sections
III and IV discuss respectively the case of solid state and gas-phase targets.
After an overview of the main processes of interest we clarify sample
requirements and describe the interaction geometry. We then review the main
techniques and challenges in data analysis, providing insightful information
on the requirement for source stability and reliability. We then conclude with
future prospects for UED techniques in Sec. V.

###  I.1 Electrons as probes of matter

The usefulness of electron diffraction stems from the large amount of
information about the sample atomic-scale structure that can be extracted from
a typical diffraction pattern. In order to understand the basic principles of
electron scattering, both particle and wave aspects of the nature of electrons
need to be considered [286, 336, 35]. Diffraction effects in particular result
from the scattering of electron waves of characteristic de Broglie wavelength
$\lambda=h/p$, where $h$ is the Planck constant, $p=mc\beta\gamma$ is the
electron momentum and $m$ and $c$ are the electron rest mass and speed of
light, respectively. $\beta=\sqrt{1-1/\gamma^{2}}$ is the electron velocity
normalized to $c$. In more quantitative terms, the de Broglie wavelength for 4
MeV (100 keV) electrons is $\lambda=0.277(3.701)$ pm, which highlights the
potential of using electrons to achieve atomic-scale spatial resolution.

When such an electron wave is incident on a target, the scattered wave can be
described by the complex amplitude $f(\theta,\phi)$, which indicates the
probability of finding a scattered electron at angle $\theta$ and $\phi$ with
respect to the incident direction. Tying together particle and wave approaches
to electron scattering, this scattering amplitude depends on the detail of the
interaction between the electron and the target and is related to the the
differential scattering cross section as
$\frac{d\sigma}{d\Omega}=|f(\theta,\phi)|^{2}$. In the first Born
approximation (kinematic scattering), we can write the amplitude of the
scattered wavefunction in the direction $\mathbf{k^{\prime}}$, where
$\mathbf{k-k^{\prime}=s}(\theta,\phi)$ as the Fourier transform of the target
scattering potential, $V(\mathbf{r})$:

$f(\mathbf{s})=-\frac{m}{2\pi\hbar^{2}}\int\textup{d}\mathbf{r}V(\mathbf{r})\exp\left(-i\mathbf{s}\cdot\mathbf{r}\right).$
(1)

where the momentum transfer magnitude
$|\mathbf{s}|=\frac{4\pi}{\lambda}\sin\frac{\theta}{2}$.

In the case where the target is an atom, the largest contribution to the
elastic scattering amplitude will be the Rutherford scattering from the atomic
nucleus with a smaller contribution from the surrounding electrons. Following
Salvat et al. [301, 302], it is customary to express the (azimuthally
symmetric) elastic scattering from an atom with atomic number $Z$ in terms of
the momentum transfer $s$ as

$\frac{d\sigma}{ds}=\frac{4Z^{2}}{s^{4}a_{0}^{2}}\frac{1-\beta^{2}\sin^{2}\frac{\theta}{2}}{1-\beta^{2}}\left(1-F(s)^{2}\right)^{2}$
(2)

where $a_{0}$ is the atomic Bohr radius, and
$F(s)=\sum_{i}A_{i}\frac{\alpha_{i}^{2}}{s^{2}+\alpha_{i}^{2}}$ is a function
which depends on the approximation details of the screened atomic potential.
The sum over the index $i$ can include as many terms as desired for improved
accuracy. As an example for silver we have $A_{i}=[0.25,0.62,0.13]$ and
$\alpha_{i}=[15.59,2.74,1.14]$ Å-1.

####  I.1.1 The role of electron energy in electron scattering

It is instructive to plot (Fig. 2) the differential cross section vs.
scattering angle (a) and momentum transfer (b) for various electron energies
typically employed in UED beamlines [425]. The differential cross section vs.
momentum transfer increases proportionally to $\gamma^{2}$ for relativistic
electrons, essentially due to the scaling of the incident momentum of the
particles. To calculate how many electrons are scattered within a given
angular range, one needs to integrate the differential cross section over the
detector collection angle. Some care should be taken here as the angles
corresponding to a given $s$ depend on the incoming electron energy. So for
example if we are interested in the information around $s=$ 5 Å-1, we’d have
to collect the scattered intensity in an interval around 29 mrad for 100 keV
electrons and 2.2 mrad for 4 MeV electrons. The results of this integration
are shown in Fig. 2(c) which clarifies that the number of scattered electrons
(integrated over the entire solid angle, or even just in a small angular
interval around a region of interest) is nearly an order of magnitude smaller
for 4 MeV than for 100 keV.

Figure 2: Differential elastic scattering cross section vs. (a) scattering
angle and (b) momentum transfer for 100 keV, 750 keV and 4 MeV electrons using
Eq. 2. Panel (c) shows the resulting integrated cross section over the entire
solid angle (dashed) and over a small (0.1$\%$) interval around the momentum
transfer $s=$ 5 Å-1.

The total integrated cross section can be used to calculate the elastic mean
free path, i.e. the statistical average distance of propagation inside the
sample over which the electrons will undergo one scattering event as
$\frac{1}{n\sigma}$ where $\sigma$ is the integrated cross section and $n$ the
density of scatterers in the material under study. Directly resulting from the
scaling in Eq. 2, illustrated in the plots of Fig. 2, elastic mean free paths
for higher energy electrons are significantly longer than for lower energy
particles in the same material. For example, in an Al sample, the elastic mean
free path is 38 nm at 100 keV and 250 nm at 4 MeV. For higher energy
electrons, this allows the use of thicker samples, or alternatively yields
lower number of scattering events for equal thickness of materials.

In cases where the mean free path is shorter than the thickness of the
specimen, then it is likely that electrons would undergo more than one
scattering event. In order to quantitatively extract information from the
diffraction pattern, one must go beyond the simple kinematical approximation
(one scattering event per electron) and utilize the more complex dynamical
diffraction theory [429, 371].

####  I.1.2 Scattering from gaseous targets

If the sample is made up by a large number of scattering targets (atoms), the
total scattering amplitude will be the sum of the individual waves. The so
called scattering form factor $F$ can then be written using the independent
atom model as the sum of the atomic scattering factors $f_{j}$ from all the
atoms in the with atomic coordinates $\mathbf{r}_{j}=(x_{j},y_{j},z_{j})$
multiplied by a phase factor which takes into account the difference in phase
between the scattered waves in terms of the momentum transfer vector
$\mathbf{s}$

$F(\theta)=\sum_{j}f_{j}(\theta)e^{i\mathbf{s}\cdot\mathbf{r}_{j}}$ (3)

In gas phase electron diffraction, high energy electrons (keV to MeV)
elastically scattered from an ensemble of molecules produce an interference
pattern on a detector, from which structural information on the molecule can
be retrieved. The total scattering intensity can be obtained by the incoherent
sum of the scattering from each molecule since the transverse coherence of the
electron beam is typically smaller than the distance between molecules. For
randomly oriented molecules, averaging over all possible orientation results
in a scattered intensity only dependent on the polar angle (circular symmetry
diffraction pattern) and that can be written as a function of the momentum
transfer magnitude $s$ as
$I_{\mathrm{Total}}(s)=I_{\mathrm{A}}(s)+I_{\mathrm{MOL}}(s)$. We can separate
the contributions to the total scattering in two terms: the first is atomic
scattering term $I_{\mathrm{A}}(s)=\sum_{m=1}^{N}f_{m}^{*}(s)f_{m}(s)$ and
contains no structural information and only depends on the atoms present in
the molecule.; the second term, known as molecular scattering, can be written
as

$I_{\mathrm{MOL}}(s)=\sum_{m=1}^{N}\sum_{n=1,m\neq
n}^{N}f_{m}^{*}(s)f_{n}(s)\frac{\sin(sr_{mn})}{sr_{mn}}$ (4)

where $N$ is the number of atoms in the molecule, and $\mathbf{r}_{mn}$ is the
distance vector from atom $m$ to atom $n$ (assuming static molecular
structure), and contains the interference between all atom pairs in the form
of a sinusoidal modulation in the intensity of the diffraction pattern.

For ease of analysis and to compensate for the fast decrease in scattering
intensity with $s$, the modified scattering intensity is used:

$sM(s)=\frac{I_{\mathrm{MOL}}(s)}{I_{A}(s)}s$ (5)

The most straightforward method to extract structural information from
diffraction data is to Fourier (sine) transform the scattering intensity into
a Pair Distribution Function ($PDF$) [127]. The position of peaks in the PDF
reflects interatomic distances in the molecule, with peak amplitudes
proportional to the density (in the case where there are multiple atom pairs
with overlapping distances) and the product of the scattering amplitudes from
each atom in the pair, while it is inversely proportional to the distance $r$.
In practice, the diffraction pattern is only measured up to a maximum value
$s_{\mathrm{Max}}$, resulting in a truncated $sM(s)$. To avoid introducing
artifacts into the PDF from the sine transform of a truncated signal, a
damping factor $k$ is added as:

$PDF(r)=\int_{0}^{s_{\mathrm{Max}}}sM(s)\sin(sr)e^{-ks^{2}}ds$ (6)

where $r$ is the real space distance between atom pairs.

The spatial resolution of the measurement is strictly defined by the width of
the peaks in the $PDF$, and thus depends only on the value of
$s_{\mathrm{Max}}$. Note that this value determines whether two nearby
distances can be resolved in the $PDF$, but it does not determine the
precision with which any individual distance can be determined. Finding a
distance is equivalent to finding the center of the peak, which typically can
be done to a value much smaller than the width of the peak, and depends
strongly on the SNR of the measurement. Figure 3 shows the relative
contributions of the molecular and atomic scattering terms to the total
simulated scattering signal of CF3I and corresponding $sM(s)$ and PDF$(r)$.

Figure 3: Simulated gas-phase electron scattering for CF3I showing the a)
relative contributions of the each atom type to the atomic terms, and
contributions of the atomic and molecular terms to the total scattering, b)
simulated $sM(s)$ and c) PDF$(r)$ and d) a depiction of the inter atomic
distances in the molecular color-coded to the peaks in the PDF$(r)$.

####  I.1.3 Scattering from crystals

Consider the case of a beam of electrons with wavevector $\mathbf{k}$ incident
on a perfect, infinite single crystal consisting of periodically arranged unit
cells, which defines the smallest repeating atomic arrangement within the
material. The crystal can be described as a sum over all the $\mathbf{\alpha}$
atom positions within a unit cell, $\mathbf{r}_{\alpha}$, and an infinite sum
over all the unit cell coordinates $\mathbf{R}_{n}$. With these definitions
the scattering potential of the entire crystal can be written as [4, 371, 373]

$V(\mathbf{r})=\sum_{n}\sum_{\alpha}V_{\alpha}\left(\mathbf{r}-\mathbf{R}_{n}-\mathbf{r}_{\alpha}\right),$
(7)

where $V_{\alpha}$ is the potential of atom $\alpha$ in unit-cell $n$. The
periodicity of $V(\mathbf{r})$ ensures that the form of
$V_{\alpha}(\mathbf{r}-\mathbf{R}_{n}-\mathbf{r}_{\alpha})$ is identical for a
given pair of $n$ and $\alpha$ values.

Generalizing Eq. 1, we can write the scattering amplitude at wavevector
$\mathbf{k^{\prime}}$ in terms of the momentum transfer111In literature
focusing on solid-state samples, the momentum transfer is commonly denoted by
$\mathbf{q}$. The notation $\mathbf{s}$ is maintained here for consistency
with other sections of this review.
$\mathbf{s}=\mathbf{k}-\mathbf{k^{\prime}}$ in the single scattering (or
kinematic) limit as the Fourier transform of the scattering potential
$V(\mathbf{r})$:

$\displaystyle f(\mathbf{s})$ $\displaystyle=$
$\displaystyle\sum_{\\{\mathbf{G}\\}}\delta(\mathbf{s}-\mathbf{G})\sum_{\alpha}V_{\alpha}(\mathbf{s})\exp(-i\mathbf{s}\cdot\mathbf{r}_{\alpha}).$
(8)

which can be understood as the as the product of the structure form factor $F$
which contains the details of the unit cell atomic composition, and the
lattice or shape factor $G$ [286] which depends on the shape and external
structure of the crystal.

In writing Eq. (8) we assume infinite crystal structure, and therefore the
mathematical identity
$G=\sum_{n}\exp(-i\mathbf{s}\cdot\mathbf{R}_{n})=\sum_{\\{\mathbf{G}\\}}\delta(\mathbf{s}-\mathbf{G})$
has been applied. The reciprocal lattice vectors
$\mathbf{G}=h\mathbf{a}^{*}+k\mathbf{b}^{*}+\ell\mathbf{c}^{*}$ describe the
periodicity of the crystal in reciprocal space and satisfy
$\mathbf{G}\cdot\mathbf{R}_{n}=2\pi\times\textup{Integer}$ [4]. Eq. 8
demonstrates the well known Laue condition for single crystal diffraction,
which states that scattering amplitude is only non-zero when
$\mathbf{s}=\mathbf{G}$; the Bragg peaks of a diffraction pattern.

If the crystal is not infinite, the delta function must be replaced by the
finite sum over the unit cells. For example, considering a crystal with $N$
planes spaced by distance $d$, we have

$G=\left(\frac{\sin(s^{*}Nd)}{s^{*}d}\right)$ (9)

where $s^{*}=|\mathbf{s}-\mathbf{G}|$ is the deviation from the perfect Laue
condition (excitation error).

The amplitude of the lattice factor $G$ is particularly important. If
electrons are scattered by $N$ unit cells, at the Bragg peaks (i.e. $s^{*}=0$
in Eq. 9), the lattice factor $G$ is responsible for a $N$ times increase in
the scattered wave amplitude with respect to single atom case. The
corresponding scattered intensity increases by a factor of $N^{2}$. This Bragg
enhancement factor can be very significant (i.e. in excess of $10^{5}$ even
for small microcrystalline samples). In this simplified picture, the angular
width of the Bragg peaks just depends on the number of atomic planes in the
sample (i.e. the shape factor of the target). In practice, as we will see
below in the coherence length section, there are many other effects that must
be taken into account in the width of the Bragg peaks including the angular
distribution and energy spread in the probing electron wavepackets. For the
nanometer thick single crystal specimens used in UED, the measured width of a
Bragg peak in the direction of the film thickness is typically determined by
the finite size effects described above, while the measured width of a Bragg
peak in the plane of the thin specimen is typically determined by instrumental
broadening associated with the illuminating electron beam parameters.

In Eq. (8), $V_{\alpha}(\mathbf{s})$ is simply proportional to the _atomic
form factor_ $f_{\alpha}$ which is the normalized Fourier transform of the
atomic potential for an isolated (spherically symmetric) atom $\alpha$. While
the assumption of spherical symmetry often provides the starting point for
crystallographic calculations, it is important to keep in mind that chemical
bonding in the solid will modify the symmetry of the atomic scattering factors
somewhat and can lead to observable effects in diffraction experiments. The
crystal structure factor, defined as
$F_{0}(\mathbf{s}=\mathbf{G})=\sum_{\alpha}V_{\alpha}(\mathbf{G})\exp(-i\mathbf{G}\cdot\mathbf{r}_{\alpha})$
[112], determines the scattering amplitude into the Bragg peak located at
$\mathbf{s}=\mathbf{G}$, and depends sensitively on the relative position of
atoms in the unit cell.

The intensity of electron scattering as a function of $\mathbf{s}$, the
quantity measured by an electron imaging detector, is [371]:

$I(\mathbf{s})\propto
G^{2}\left(\mathbf{s}-\mathbf{G}\right)\sum_{\alpha}\sum_{\beta}V_{\alpha}(\mathbf{s})V_{\beta}(\mathbf{s})\exp\left(-i\mathbf{s}\cdot(\mathbf{r}_{\alpha}-\mathbf{r}_{\beta})\right)$
(10)

The phase of the scattering amplitude is lost by intensity detection,
resulting in the well known phase problem of crystallography. The result in
Eq.(10) can be generalized in a straightforward manner to polycrystalline
samples by appropriate integration of Eq. (10) as described in detail by
Siwick _et al._ [328].

The Ewald sphere construction is often used to graphically represent the Laue
condition, describing which reciprocal lattice points (or diffraction peaks)
will be seen in a diffraction pattern in a specific scattering geometry (i.e.
crystal orientation with respect to the incident electron wavevector). We will
use this construction here to illustrate how the electron deBroglie
wavelength, $\mathbf{\lambda}$ (or beam energy), influences diffraction.
However, the impact of other beam parameters, like the spread in electron beam
energy and divergence angle, can also be understood using this construction.
The Ewald sphere is drawn on top of the crystal’s reciprocal lattice with a
radius of 1/$\mathbf{\lambda}$ and an orientation determined by the incident
beam angle with respect to the crystallographic axes. This is shown in a
simple geometry for a hypothetical simple cubic crystal at two beam energies
in Fig. 4. For elastic (Bragg) scattering both incoming and scattered beams
lie on this sphere, thus the Laue condition for diffraction is only satisfied
when the Ewald sphere cuts through a reciprocal lattice point. Note that the
curvature of the sphere is inversely proportional to the wavelength of the
incident radiation. Since the deBroglie wavelength of electrons is 3.88 pm at
100 keV, but only 0.39 pm at 10 MeV, the Ewald sphere at 100 keV has 10 times
higher curvature. The flatter the Ewald sphere, the larger the number of
reciprocal lattice points that can intersect with the sphere at large momentum
transfer (or scattering angle). This is an important advantage for MeV
electron probes in terms of the scattering efficiency for higher order Bragg
peaks, but even at 100 keV the Ewald sphere for electron scattering is already
approximately 25 times flatter than it is for hard xray scattering (using 100
pm xrays).

Figure 4: Ewald sphere construction for diffraction from a crystal using 100
keV and 4 MeV electrons. The reciprocal lattice spacing is set by the crystal
lattice constant. The volume of a reciprocal lattice ’point’ is determined by
the size of the crystal.

However, there is a practical consideration resulting from the scaling of the
de Broglie wavelength with electron energy and the resulting scattering angle
which is much smaller for relativistic electron energies. For example,
consider a set of crystalline planes separated by $d=2$ Å, the Bragg angle for
4 MeV (100 keV) electrons is 0.7 (9) mrad. This has strong implications on the
experimental setup of the distance from the sample to the detector or
diffraction camera length (which needs to be proportionally longer in the
relativistic case in order to allow for the scattered electrons to physically
separate from the unscattered ones, assuming no magnifying electron optics
between the sample and detector), but importantly bears no effect on the
attainable quality of the pattern as explained below.

####  I.1.4 Coherence length and reciprocal space resolution in UED

In order to form a diffraction pattern, a large number (a beam) of probe
electrons is used to illuminate the target. In Bragg scattering, if one wants
to distinguish the scattered particles from the undiffracted ones, it is
essential that the scattering angle $2\theta_{B}$ be much larger than the
uncorrelated spread of the divergence angles in the beam at the sample. In the
root-mean-square sense this can be expressed as $\sigma_{\theta}$ (i.e.
$\sigma_{\theta}\ll 2\theta_{B}$). Note that any angular divergence correlated
with position (for example due to a converging or diverging beam) can be
removed by the transport optics and does not play a role in the diffraction
contrast.

For polycrystalline or gas/liquid phase samples, where the diffraction pattern
is a series of concentric rings due to the random orientation of the grains,
it is customary to introduce as figure of merit for resolution
$\mathbb{R}=R/\Delta R$ where $R$ is the radius of the diffraction ring on the
detector screen and $\Delta R$ is the smallest distance between two
neighboring rings which can just be discriminated at the detector. Note that
the position on the detector screen is simply proportional to the scattering
angle so that $\mathbb{R}$ can also be interpreted as the inverse of the
relative reciprocal space resolution, i.e $\mathbb{R}=R/\Delta R=s/\Delta s$.
A typical TEM operating in diffraction mode achieves $\mathbb{R}>10^{2}$ or
more for static images. For UED, a resolving power of $\mathbb{R}>10$
guarantees a good quality diffraction pattern and provides enough resolution
to adequately resolve typical ultrafast structural rearrangements. The
experimental value of $\mathbb{R}$ is affected by multiple factors, such as
the electron beam angular and energy spread, and the spatial resolution of the
detector, as it will be discussed in detail in the following sections. In most
diffraction setups the uncorrelated beam divergence is the dominant limiting
factor in the resolving power of the diffraction camera [121], so one can
write $\mathbb{R}=\lambda/2d\sigma_{\theta}\approx\theta_{B}/\sigma_{\theta}$.
It is useful to note that the value of $\mathbb{R}$ is independent on the beam
energy, as both components of the ratio above are proportional to $\propto
1/\beta\gamma$. Note that the absolute reciprocal space resolution is simply
$\Delta s$. This quantity determines the longest range order which can be
observed in the diffraction pattern. In practice, this corresponds to
effectively how small the electron beam can be made on the detector screen.

The importance of the beam divergence at the sample in UED is encoded in the
concept of coherence length $L_{c}$ which is an equivalent figure of merit for
diffraction contrast. In standard optics the coherence length indicates the
extent of the coherent portion of the illumination (i.e. the spatial extent
over which the phase of the illuminating beam wavefunction is correlated). For
example, for an incoherent source, with no optics between the source and the
sample, the Van Cittert-Zernike theorem defines the coherence length as the
wavelength divided by the angle subtended by the source [21]. In a UED
beamline the definition must take into account that the beam from the electron
source is magnified and refocused before illuminating the sample. One can show
in this case that the the visibility of interference fringes from two
scattering centers (or planes) separated by a distance $d$, depends on the
ratio between $d$ and the transverse coherence length as
$L_{c}=\lambda/2\pi\sigma_{\theta}$ [176, 352] where $\sigma_{\theta}$ is the
uncorrelated beam divergence at the sample. This is important since, as we
have discussed above, the spatially periodic arrangement of the atoms in a
crystal allows for a large enhancement of the diffraction signal, but if the
beam phase front is not coherent over multiple unit cells of the structure
under study, then no constructive interference can be developed and the
visibility of the diffraction peaks is strongly reduced. In the limit that the
coherence length is smaller than a unit cell, the Bragg peaks disappear. Note
that this strong dependence suggests the use of diffraction pattern visibility
as a sensitive quantity to measure of the beam divergence [402]. The
visibility of the Bragg interference peaks also depends on the longitudinal
coherence properties of the beam, but in typical UED setups the longitudinal
coherence length i.e.
$L_{l}=\lambda/(2\pi\frac{\delta(\beta\gamma)}{\beta\gamma})$, even for energy
spreads as high as 1 $\%$, is often much longer than the differences in
optical path length for the diffracted beams and so hardly contributes to the
sharpness of the diffraction pattern.

To illustrate the impact of beam coherence on the quality of the diffraction
pattern, we show in Fig. 5 simulated diffraction patterns from salicylic acid
(aspirin) molecule for different coherence length values, ranging from 62.8 nm
to 0.628 nm. The unit cell vector lengths for this crystal lattice are [11.3,
6.5, 11.3] Å [377]. It is clear that much more detailed information on the
crystal structure can be extracted from the pattern to the left.

Figure 5: Simulated diffraction patterns of a Salicylic acid (aspirin) crystal
for electron probe beams having coherence lengths of (2$\pi$) 10 nm, 1 nm and
0.1 nm respectively.

To compare different electron beamlines, it is also useful to normalize the
coherence length to the electron beam size at the sample $\sigma_{x}$ and
define a relative coherence length

$l_{c}=\frac{L_{c}}{\sigma_{x}}$ (11)

Indeed beam divergence can be controlled by the electron optics before the
sample, and the coherence length can be adjusted, while the relative coherence
length is an intrinsic beam property and effectively can be thought as the
fraction of the beam which participates in coherent scattering.

A final point related to the study of sensitive materials is related to the
damage effects associated with the bombardment of the sample by high energy
electrons. The main mechanism involved is ionization damage (radiolysis), in
which valence or inner-shell electrons within the specimen are excited by
inelastic scattering events either directly breaking a chemical bond, or
indirectly by secondary electron emission [82]. In order to evaluate the
relative importance of these effects one needs to compare the elastic to
inelastic mean free path as well as the energy deposited per scattering event.
After taking all of this into account, it turns out that the overall damage is
not particularly sensitive to the electron energy. In addition, it should be
mentioned here the possibility for irreversible specimen damage associated
with the knock-on effect. This is a rare occurrence where collision between an
incident electron and an atomic nucleus create an atomic vacancy [81]. The
onset of this effect depends on the atomic species, but generally is above 80
keV. Due to the steep energy dependence, it had been one of the causes of the
progressive disappearance of high voltage (MeV) electron microscopy
(accelerated by the resolution improvements at lower voltage resulting from
aberration correction implementation). In high energy UED, the Bragg
enhancement effect (spatial averaging over the sample) allows to utilize a
much lower dose to acquire a diffraction pattern and significantly reduces
this problem. As an example, while to acquire a high-contrast nm-spatial
resolution image a dose of 100 e-/nm2 would be required, the typical doses for
high energy UED are 106 e-/ 10 $\mu$m2 which is 104 times smaller.
Furthermore, novel setups developed in the last few years hold the promise of
full diffraction signal acquisition faster than any structural change due to
damage (i.e. in few tens of fs), with an approach similar to the diffract-and-
destroy technique employed in 4th generation light sources [335].

####  I.1.5 Electron vs. X-ray scattering

It is useful at this point in order to better appreciate the opportunities
enabled by the development of ultrafast electron scattering to draw a
comparison with x-ray scattering techniques. In particular, there is often a
debate in the comparison of the effectiveness of probing with electrons or
x-rays, even though the information extracted from these different
technologies is mostly complementary.

Aside from significant difference in the size and cost of electron and x-ray
machines [34], there are two main differences in the interaction with matter.
The first one is that elastic scattering of X-rays from matter is relatively
weak due to the very small cross section for photon interaction with charged
particles (Thompson cross-section) [373]. To make a quantitative comparison,
considering the same momentum transfer $s$ = 10 Å-1, the Rutherford cross
section is more than 5 orders of magnitude larger than the x-ray cross section
for elastic scattering. This implies that 5 orders of magnitude less electrons
generate an equal diffraction signal when illuminating a target with the same
number of scattering centers. It is no surprise that electrons are then the
preferred choice anytime the number of scatterers in the target is small (gas
phase, membrane protein crystals, 2D and quasi-2D materials, etc.).

Owing to their higher cross section, electrons have significantly shorter
penetration depth than hard x-rays, with important consequences on the sample
thickness of choice and on the detector technology. The value of the probe
beam penetration depth is an important factor in designing pump-probe
experiments. An ideal excitation (absorbed fluence/layer) would have a uniform
profile throughout the sample thickness. On the other hand, perfect uniformity
is only reached with negligible absorption, i.e. negligible excitation.
Therefore a sample thickness roughly equal to one absorption length at the
excitation wavelength can be considered a good tradeoff between uniformity and
pumping efficiency. Typical electron elastic mean free path values limit
sample thickness for UED in the tens-to-hundreds of nanometers (depending on
electron energy and atomic composition).Such values are a good match for
optical radiation in a metal, while insulators and semiconductors can have
absorption depths up to cm-scale. For x-rays (non-resonant, hard and soft) the
penetration depth mostly depends on the form factor, i.e. how heavy the
elements are, but it is typically on the scale of cm or longer. For soft
x-rays, there is an additional situation when one goes into resonant
absorption. There, the elemental absorption becomes extremely strong and the
penetration depth short and in some cases comparable with visible light [205].
A different situation occurs when pumping in the THz regime of great interest
for material science where the pump penetration depth is significantly longer
[324].

Furthermore, the difference in wavelength of the probing particles leads to
key differences in the experimental data. An X-ray photon energy of 1-10 keV
corresponds to a wavelength in the range of 1-10 Å, while electrons with
energies typically used in UED exhibit wavelengths in the picometer-range,
with a dramatic difference in the curvature of the Ewald sphere between the
two cases. As a consequence, X-rays provide excellent momentum resolution in
reciprocal space within a narrow range, i.e.typically only few spots per
diffraction pattern. Conversely, each electron diffraction pattern typically
includes a large number of spots/rings/diffraction features from which more
information can be retrieved [401]. In addition, the technological development
of high quality X-ray optics significantly lags its electron counterpart, and
related to this, the focusability of X-ray and electron beams is very
different. While the latter can be easily focused down to spot sizes well
below 100 nm, typical spot sizes at state-of-the-art XFELs are still in the
micrometer range.

Another important difference relates to the amount of energy deposited in the
sample for a single inelastic scattering event. X-rays are fully absorbed,
depositing their entire energy into the sample, while electrons typically only
release a small fraction of their energy in a collision. In fact it has been
pointed out by Henderson [140] that per elastic scattering event electrons
deposit as little as 1/1000 of the energy of x-rays in the sample. Especially
for sensitive biology-relevant samples this might be an important advantage.
The same paper also points out that the inelastic scattering cross section of
soft x-rays has the same order of magnitude than the elastic cross section for
high energy electrons. This suggests the fascinating possibility of drawing
complementary information using potentially the same samples pairing up UED
and inelastic scattering techniques from soft x-ray beamlines.

Finally, with the advent of X-ray lasers [85], fully transversely coherent
ultrashort x-ray pulses can be available enabling coherent diffraction imaging
algorithms to replace the role of the optics in retrieving real-space images
of the sample [231]. In short-pulse electron scattering instrumentation, as it
will be discussed below, this limit is still very far from reach and only
partially coherent electron beams have been used to date.

###  I.2 Electron beam brightness

In this section we introduce a metric for measuring the ability of a specific
setup to deliver high density electron beams, and for comparing different
instruments. The definitions introduced below will be used throughout the
article to elaborate on the capability of an electron beam to perform specific
experiments or provide the required spatial and temporal resolution.

####  I.2.1 The electron beam concept

Adding temporal resolution to electron scattering experiments requires the
formation of an electron bunch, i.e. a three-dimensional charge distribution
well defined and limited in space and time. Such electron beam can be defined
by the sum of isolated electrons correlated in time by periodic emission
(stroboscopic approach) [13], or by a set of electrons tightly packed in a
small volume (single-shot setups), traveling together along a preferred
direction. In both cases, the level of confidence by which one can describe
the temporal contours of the beam will set the basis for the definition of
temporal resolution $\tau_{res}$ in a ultrafast experiment. In conventional
continuous sources electrons are emitted at random times and, therefore, no
temporal information can be extracted without further manipulation of the
electron stream. A quality metric for such sources is provided by the five-
dimensional beam brightness $\beta_{micro}=\frac{4i_{e}}{(\pi
d_{0}\alpha_{0})^{2}}$ [379], a measure of the average current $i_{e}$ per
unit of source size $d_{0}$ (full beam diameter at crossover) and solid angle
of emission $\alpha_{0}$ (semi-angle of emission at crossover). In absence of
downstream beam acceleration, $\beta_{micro}$ is a constant of the motion
along the electron beamline/column, that is if one desires a smaller spot
size, a larger beam divergence is unavoidable.

If the beam spatial and angular distributions are not uniform, a more general
definition of beam diameter and angular spread is needed. Using the
statistical framework, we introduce the generalized standard deviations of the
beam along a specific direction, also known as root mean square moments of the
distribution about its mean, rms hereafter [291].

For the case of pulsed electron sources, a distinction between average and
peak current needs to be made, the latter describing a local property of the
individual bunch of electrons in a longer bunch train, and defined as the
instantaneous rate of change of the beam charge. The resulting peak and
average brightness values will bear different information, the former
describing the ability of a particular setup of performing single-shot
measurements, and the latter providing information on experiment recording
times. Unless specified otherwise, the quantities defined in the following of
this section will relate to isolated bunched beams.

####  I.2.2 Beam phase space and brightness definitions

The key distinction between pulsed and continuous wave beams is the role
played by the longitudinal parameters in the experiments. Similarly to
transverse focusing in which beam size and divergence can be trade-off for
each-other, longitudinal compression allows to manipulate pulse length and
energy spread to achieve optimal resolution. A modified metric for pulsed
source quality which includes both the transverse and longitudinal degrees of
freedom, is obtained by introducing the concepts of six-dimensional phase
space and six-dimensional brightness.

From a classical mechanics standpoint a set of $N$ particles represents a
system with a total of 6N degrees of freedom, including each particle
coordinates in space $r_{i}$ and their relative conjugate momenta $p_{i}$. In
most cases of interest the temporal evolution of such system can be described
by an Hamiltonian which, in turn describes the evolution of a unique
trajectory in the 6N-dimensional space defined by the full system degrees of
freedom. The number of dimensions can be reduced back down to six if particle-
particle interactions can be neglected or described by a mean field
approximation, resulting in a description of the electron beam as a clustered
set of points in the hyper-volume $\mathcal{V}_{6}$, called 6D phase space for
each instant in time. A key concept in this description of electron beams is
represented by the phase space charge density
$\rho_{6}(\mathbf{r},\mathbf{p},t)$, also called microscopic six-dimensional
brightness [292], defining the charge distribution in the phase space
$dQ=\rho_{6}\delta\mathcal{V}_{6}$.

Although the shape of the distribution changes with time, the Liouville
theorem states the invariance of its total volume during motion under the the
assumption of Hamiltonian evolution. The six-dimensional beam brightness is
therefore a constant of motion.

In the special but not uncommon case of decoupled motion between the different
planes, the 6D volume can be written as
$\mathcal{V}_{6}=\mathcal{A}_{x}\mathcal{A}_{y}\mathcal{A}_{z}$, where
$\mathcal{A}_{i}$ is the phase space area in the $(i,p_{i})$ plane
($i=x,y,z$). If we use second order moments of the distribution to describe
the area enclosed by the beam, then $\mathcal{A}_{i}$ takes on the meaning of
normalized rms emittance $\epsilon_{n,i}$.

It is often convenient to express the beam properties in terms of the angle of
the particle trajectory with respect to the propagation direction $z$,
$x^{\prime}=\frac{p_{x}}{p_{z}}$. Considering a beam waist at a position
$z_{0}$ as shown in Fig. 6(a), the normalized transverse rms emittance in the
$(x,x^{\prime})$ plane can then be written as
$\epsilon_{n,x}=\gamma\beta\sigma_{x_{0}}\sigma_{x^{\prime}_{0}}$, where
$\beta$ and $\gamma$ are the relativistic Lorentz factors. In the more general
case depicted in Fig. 6(b), the emittance calculation at a plane $z$ will need
to account for correlations $\sigma_{xx^{\prime}}$ in the plane, and the
equation becomes:
$\epsilon_{n,x}=\gamma\beta\sqrt{\sigma^{2}_{x}\sigma^{2}_{x^{\prime}}-(\sigma_{xx^{\prime}})^{2}}$.
Introducing the uncorrelated transverse rms spread in divergence
$\sigma_{x^{\prime}_{u}}$ simplifies the general equation back to the product
of two terms, $\epsilon_{n,x}=\gamma\beta\sigma_{x}\sigma_{x^{\prime}_{u}}$.
Figure 6(c) clarifies the physical meaning of uncorrelated divergence at a
position $z$ along the beam path, equivalent to $\sigma_{\theta}$ introduced
in Sec.I.1.4. The uncorrelated divergence is a key parameter in UED
experiments, determining the beam transverse coherent length and the
reciprocal space resolution.

Figure 6: Schematic visualization of rms beam properties and emittance. The
elliptical contours represents the beam density in phase space (a) and (b). In
(c) the concept of uncorrelated beam divergence (and its relation with the
beam emittance) is clarified.

In case of uncoupled dynamics, the rms six-dimensional brightness can be
written as:

$B_{6D}=\frac{Ne}{\epsilon_{n,x}\epsilon_{n,y}\epsilon_{n,z}}=\frac{I_{rms}}{\epsilon_{n,x}\epsilon_{n,y}\frac{\sigma_{E}}{mc}}$
(12)

where we assume no time-energy correlation in the bunch and
$\epsilon_{n,z}=\sigma_{z}\frac{\sigma_{p_{z}}}{mc}\approx
c\sigma_{t}\frac{\sigma_{E}}{mc^{2}}$, and $I_{rms}=Ne/\sigma_{t}=\eta
I_{peak}$, with $I_{peak}$ being the maximum current within the pulse, $N$ the
number of electrons in the bunch, and $\eta$ a numerical value depending on
the shape of the temporal distribution ($\eta=\sqrt{2\pi}$ for a gaussian
temporal profile).

Depending on the specific application, it is common to introduce different
brightness definitions which better capture the key beam properties. In
typical ultrafast electron diffraction experiments electron beam’s transverse
emittance rather than the energy spread dominates the minimum beam size at the
sample and the resolution in reciprocal space. In this case we can then
consider the five-dimensional brightness to be more representative of the
effectiveness of the electron beam to carry out an experiment,
$B_{5D}=\frac{I_{rms}}{\epsilon_{n,x}\epsilon_{n,y}}$. This parameter is
directly proportional to the $\beta_{micro}$ defined above and used in
microscopy. The proportionality factor depends on the details of the charge
distribution (for example uniform, gaussian, parabolic). There is also an
additional factor $(\gamma\beta)^{2}$ which is used to make $B_{5D}$ invariant
under particle acceleration. On the other hand, this value can be increased by
longitudinal beam compression, which increases the beam peak current at
expenses of energy spread.

Lowering further the number of dimensions, one can define a brightness in the
transverse planes, called four-dimensional brightness and defined as:

$B_{4D}=\frac{Ne}{\epsilon_{n,x}\epsilon_{n,y}}$ (13)

This metric results particularly useful when balancing trade offs between
temporal and spatial resolution in time-resolved electron scattering. Larger
values of $B_{4D}$ result in better diffraction pattern contrast and higher
spatial resolution. One simple way to increase $B_{4D}$ is by starting with a
longer pulse length, which would increase the charge at expenses of temporal
resolution. Assuming no coupling between longitudinal and transverse planes,
the four-dimensional brightness is set at emission and remains constant during
transport and acceleration.

####  I.2.3 Quantum limit of beam brightness

The fermionic nature of the electrons limits the number of electrons that can
occupy the same phase space area through the Pauli exclusion principle. This
sets a value for the maximum phase space electron density which can be derived
starting from the uncertainty principle, stating that
$\sigma_{x}\frac{\sigma_{p_{x}}}{mc}\geq\frac{\lambda_{c}}{4\pi}$, providing
the volume of a coherent state in phase space [28, 428]. Here $\lambda_{c}$ is
the Compton wavelength of the electron. The final quantum limited rms
brightness can be written:

$B^{q}_{6D}=2e\left(\frac{2\pi}{\lambda_{c}}\right)^{3}$ (14)

The ratio between the beam six-dimensional brightness and the quantum limited
brightness defines the beam degeneracy parameter
$\delta=\frac{B_{6D}}{B^{q}_{6D}}$, a measure of the source quality with
respect to the ultimate physical limit. In the case of a unpolarized source,
$\delta_{max}=1$. Typical values of $\delta$ for state-of-the-art electron
sources range from $10^{-2}$ of single-atom emitters to $10^{-6}$ of large-
area photo-emitters.

When normalized by the quantum-limited transverse brightness
$B^{q}_{4D}=2e(\frac{2\pi}{\lambda_{c}})^{2}$, the four-dimensional brightness
provides a direct measure of the source lateral coherence. Using the
definition of beam normalized emittance, the relative coherence length (Eq.
11) can be rewritten as $l_{c}=\lambda_{c}/(2\pi\epsilon_{n})$, and the
normalized transverse brightness for a round beam (same emittance in $x$ and
$y$ planes) then is:

$\frac{B_{4d}}{B^{q}_{4d}}=\frac{N}{2l_{c}^{2}}=\frac{N_{c}}{2}$ (15)

where $N_{c}=N/l_{c}^{2}$ is the number of electrons per coherent area in the
beam.

###  I.3 Different modalities of ultrafast electron scattering
instrumentation: diffraction, imaging and spectroscopy

As an electron beam interacts with matter, a wealth of information related to
the lattice and electronic structures, as well as their dynamics, gets encoded
in the beam phase space, corresponding to changes of the momentum, energy, and
intensity of the beam. Over the past century, various specialized methods and
instruments utilizing electron probes have been developed, focusing on one or
few types of changes in phase space, giving rise to various modalities of
electron scattering instrumentation such as diffraction, imaging and
spectroscopy [379, 286, 336].

Figure 7: Different modalities for ultrafast electron scattering
instrumentation, including (a) diffraction, (b) imaging, and (c) spectroscopy.
Exemplary images are adapted from references Kogar _et al._ [179], Barwick _et
al._ [7], and Feist _et al._ [95], respectively.

Referring to Fig 7, the imaging modality allows the formation of real space
images of an illuminated sample area using electron optics and apertures (TEM
mode). Contrast in the image is introduced by filtering out the scattered
electrons with momentum and/or energy changes using apertures to only transmit
a certain region of the beam phase space (dark and bright field imaging).
Imaging is most powerful to observe and track non-periodic features of
interest.

The quality of the image depends on many parameters, including those related
to illumination (electron dose, relevant to the so called Rose criterion
[295]) and contrast (intrinsic beam divergence and energy spread). These
quantities also contribute to the image blurring through the spherical and
chromatic aberrations of the electron lenses. Considering the definition of
the 6D beam brightness (Eq. 12), $Ne$ is proportional to the total scattered
and recorded signal and $\sigma_{x,y}$ are the transverse rms spot sizes of
the probe. The spatial resolution and contrast are encoded in the rms beam
divergence $\sigma_{x^{\prime},y^{\prime}}$ and the rms energy spread
$\sigma_{E}$. The rms bunch length $\sigma_{t}$ sets the limit for the
temporal resolution, so that very high beam brightness is required for this
application.

Accessible time scales in ultrafast electron imaging range from ns for single
shot full field images [184, 23, 271] to fs in stroboscopic mode [411, 269,
94, 56, 146, 29]. Aiming at reaching enhanced capabilities, ultrafast imaging
using electron beams with higher energy (MeV level) and potentially higher
brightness is an area under intense development [194, 385, 399, 366, 39, 213,
198], which drives innovative approaches to electron sources, beam optics, and
operation schemes. A separate operating mode for imaging can be achieved by
scanning a focused electron probe across the sample and recording the
scattering signal for each position (STEM, 4D STEM, ptychography, and
ultrafast nanodiffraction [155]). The advantage in this case is the
opportunity to identify correlations between the material domains (easily
identifiable in imaging mode) and the ultrafast changes in the unit cell
(accessible in diffraction mode).

Adding an energy filter at the end of the electron column enables observation
of time-dependent changes in the electron energy loss spectrum (EELS) [7, 33,
95]. The EELS signal is directly correlated to chemical and electronic
properties of the specimen. EELS requires nearly monochromatic illumination,
i.e. the beam energy spread $\sigma_{E}$ must be smaller than the energy
feature to be resolved (from single eV to meV level, depending on the
process).

Combining spectroscopy with diffraction one would provide access to momentum-
resolved EELS carrying a great deal of information on the electronic structure
of the sample. An important additional benefit of using ultrafast sources for
EELS is that the time structure of the beam allows the possibility for more
accurate energy measurements [360] by taking advantage of beam control
techniques in the longitudinal phase space (for example using RF cavities as
time-domain lenses) . Time-of-flight electron spectroscopy [359] is also
enabled by having short electron bunches at the sample.

In comparison with these other modalities, diffraction requires no further
electron optical elements between the sample and the detector. The signal is
generated by the interference of elastically scattered electrons, i.e. those
electrons with modified transverse momentum by negligible energy changes .
This signal encodes the structural information averaged over the entire probed
area. Benefiting from its simplicity, diffraction usually also allows for most
quantitative correlation between the measured pattern and the structure of
matter.

For these reasons, UED has been the first modality of time-resolved electron
scattering to receive attention, but advances in ultrafast electron beams from
photoemission sources establish exciting new capabilities for imaging and
spectroscopy as well. In this review we will focus on the recent developments
in UED, with the understanding that the other modalities will likely take
advantage of much of the technical progresses described below. In addition,
mixed-modalities instruments, for example setups where ultrafast electron
microscopy and UED can take place in the same modified TEM column [96, 34,
339] are becoming more widely available for scientific discoveries.

###  I.4 Scientific drivers for ultrafast electron scattering

####  I.4.1 Solid State: ordering, excitation and emergent phenomena in
materials

Many of the central questions of materials physics relate to the complex
interplay between charge, spin, orbital and lattice-structural degrees of
freedom that give rise to the emergent macroscopic properties and ordered
phases of materials [10, 349]. Since electron diffraction provides a ‘map’ of
the electrostatic potential of a crystal in reciprocal space [112], as
discussed above in Sec. I.1, the intensity of diffraction peaks are profoundly
sensitive to the details of the lattice, charge and orbital order present in a
material. Only spin-specific ordering is relatively hidden from view with
high-energy electron beams (even spin polarized ones) due to the relatively
small differential scattering cross section between aligned and anti-aligned
spins at high energies. Magnetic structure peaks are not present in a UED
pattern as they are in neutron scattering, however, rich information on
magnetism in materials can be obtained with electron beams via imaging.
Magnetic domain structure Park _et al._ [266] and magnetic texture Eggebrecht
_et al._ [83], Huang _et al._ [147] dynamics are accessible to UEM when
operated in Lorentz microscopy mode.

In addition to the static ordering of charge, spin, orbital and lattice
degrees of freedom in materials, an understanding of the elementary
excitations that are present –both collective and single particle– and how
these excitations couple/interact with one another is required for a
fundamental understanding of the diverse phenomena and properties found in
condensed matter. The interactions between collective excitations of the
lattice system (phonons) and charge carriers, specifically, are of particular
relevance and easily studied by UED. These interactions are known to lead to
superconductivity, charge-density waves, multi-ferroicity, and soft-mode phase
transitions. Carrier-phonon interactions are also central to our understanding
of electrical transport, heat transport, and energy conversion processes in
photovoltaics and thermoelectrics. Phonons can themselves be intimately mixed
in to the very nature of more complex elementary excitations, as they are in
polarons or polaritons. Further, the coupling of spin and lattice systems can
also be studied from the lattice perspective with UED.

By tuning the excitation wavelength in the mid to far IR and THz (see for
example [324]), UED tools can be used to follow the linear and non-linear
behaviour of selectively driven phonon modes [106, 143], and their coupling
with other degrees of freedom. The development of bright ultrafast electron
beams has opened up an enormous space for experimentation on the structure,
dynamics and nonequilibrium properties of materials. In some of its earliest
manifestations, UED was used to probe strongly driven melting (order –
disorder) transitions in materials, thanks to the ability of obtaining high
quality diffraction patterns in a single shot. More recently, strongly-
correlated or quantum materials have been the target of study (see for example
[179], [76] and [323]). The non-equilibrium properties of quantum materials
are particularly interesting because the interactions between lattice, charge,
orbital or spin degrees of freedom are typically on par with electronic
kinetic energy. The presence of a ‘soup’ of competing and collaborating
interactions on similar energy scales tends to result in a complex free-energy
landscape that can show many nearly degenerate ground states that each exhibit
different ordering and properties. Mode-selective excitations that modify the
interplay between these DOF have been shown to result in dramatic
transformations (Fig. 8 a)). The associated changes in lattice, orbital and
charge order can be followed directly with UED (Fig. 8 b)). The manipulation
and control of material properties far from equilibrium with light offers
almost completely untapped and unexplored possibilities for discovering novel
states and phases of materials with exotic and transformative behaviours (see
for example [285], [334], and [236]). This new ’properties on demand’ frontier
[10] is a _Grand Challenge_ for the fundamental sciences [104] and complements
the conventional means of materials discovery, which has been to explore the
structural and compositional phase space that is accessible at thermodynamic
equilibrium in the search for desirable properties example [235]). Ultrafast
pulsed electron beams provide the sophisticated tools of structural
characterization on femtosecond timescale that are a basic requirement of such
work.

Figure 8: Properties on demand: controlling the structure and properties of
Quantum Materials with light. a) Laser excitation can lead to a photoinduced
phase transition on the material’s free energy landscape, steering the system
to a competing ground, metastable or transient state with dramatically
different ordering and properties. Some photoinduced phases can be completely
inaccessible at thermal equilibrium. Adapted from Zhang and Averitt [414] b)
Schematic of a UED experiment on Manganite, which exhibits crystalline/lattice
(Bragg), orbital (OO) and charge order (CO). Since the diffraction patterns of
Manganite show separated peaks associated with each order, UED can follow
their time-depedence and provide deep insignts into photoinduced phase
transitions like that shown schematically in a). Adapted from Li _et al._
[191].

####  I.4.2 Gas Phase: Uncovering the structure-function relationship behind
photochemical reactivity

Knowledge of how molecules responds to the incidence of light is essential to
our understanding of nature and its fundamental processes, e.g. photosynthesis
[47], vision [275], DNA photo damage [308], as well as the technological
development of light harvesting and storage devices [220]. The absorption of
ultraviolet (UV) light by a molecule leads to its promotion to an
electronically excited state. The absorbed photon energy may be redistributed
through the breaking of chemical bonds leading to photolysis, or through the
coupling between Franck-Condon active and inactive modes leading to new
vibrations. Alternately, structural rearrangement may result in a new
molecular geometry in which the excited electronic state becomes degenerate
with another electronic state. These geometries represent conical
intersections, which provide an efficient pathway for radiationless decay
between electronic states.[70] Electron scattering is perfectly suited to
capture structural changes, as electrons interact with the Coulomb potential
of the target system,[228] and thus are sensitive to both changes in the
position of the nuclei and the redistribution of electron density. UED
experiments in the gas-phase have resolved coherent nuclear motions of
vibrational wavepackets along both ground and excited states [395] and
captured the photolysis [378, 208] and ring-opening dynamics on the atomic
scale [384] $i.e.$ with angstrom spatial resolution and temporal resolution
approaching 100 fs. The main scientific driver for UED is to capture the
structural dynamics that takes place as the photoexcited molecule returns to
the ground state by following the coherent motion of nuclear wavepackets and
redistribution of energy. The focus of the work so far has been on a)
Investigating coupled-nuclear electronic motion in the excited state, b)
Capturing relaxation dynamics: resolving reaction paths during the relaxation
of molecules to the electronic ground state, and determining the structure and
vibrational motions of intermediates and end products, c) Direct retrieval of
three-dimensional structure from diffraction measurements.

The observation of coupled electronic and nuclear rearrangements, arising from
conical intersections, are key to understanding the conversion of light into
mechanical and chemical energy. Many important photochemical processes, such
as photosynthesis, retinal isomerization in vision, ultraviolet-induced DNA
damage [57], and formation of vitamin D [144] are governed by non-adiabatic
processes taking place at conical intersections. The first spatially resolved
observation of a wavepacket traversing a conical intersection was a recent
landmark UED study of the photodissociation dynamics of trifluoroiodomethane,
by Yang $etal.$, [401], however, much remains to be learned, particularly in
more complex molecules. While most UED experiments have focused on capturing
nuclear motion, a recent studied has shown that electronic changes can also be
retrieved from electron diffraction signals [400], which enables UED
measurements to capture both electronic and nuclear changes, and measure time
delays between electronic and nuclear motions.

The non-radiative relaxation of a system relies on the redistribution of
internal energy into nuclear degrees of freedom as the molecule returns to the
ground state. By spatially resolving the nuclear wavepacket motion from its
inception in the excited state to its vibrational dephasing in the ground
state, UED experiments can glean information into the mechanisms mediating the
dissipation of internal energy. A recent UED experiment probing the
photoinduced ring-opening dynamics of 1,3-cyclohexadiene, CHD, a model for the
photosynthesis of previtamin D3, using UED, revealed a coherent oscillatory
rotation of the terminal ethylene groups in the ground state photoproduct
1,3,5-hexatriene on the ground state [384]. UED has also successfully
investigated structural dynamics triggered by dissociation in
1,2-diiodotetrafluoroethane, C2F4I2 [378] and 1,2-diiodoethylene, CH2I2 [208].
Knowledge of the structure of a transient state in a reaction is key to the
rationalization of chemical reactivity. The photodissociation reaction of
C2F4I2 produces the intermediate state C2F4I before dissociation of the second
iodine atom to produce C2F4. The structure of the intermediate was determined
first with picosecond resolution [150], and later with femtosecond resolution
[378].

In gas-phase UED, the random orientation of molecules in the target volume
results in the loss of structural information, which prevents the retrieval of
three-dimensional structural information directly from the diffraction pattern
alone. Controlling the angular distribution of the target molecules, more
specifically alignment along a single axis, increases the information content
of the diffraction patterns [37, 393] and has been shown to be sufficient to
retrieve 3D structures from a combination of multiple diffraction patterns
from molecules aligned by a femtosecond laser pulse [141, 397, 392]. In
principle, by alignment of the molecules before excitation, it should be
possible to retrieve the full time-dependent three dimensional structure of
the evolving molecules, at least for simple structures [257]. This capability
could greatly enhance the information content of UED experiments.

Advances in the UED sources have, and will undoubtedly continue to be
reflected in great strides in our understanding of photochemistry and
photobiology. The technique has demonstrated its enormous impact in providing
complementary information to laser-based spectroscopic methods that probe the
electronic structure, and in combination with other methods can help to build
a complete picture of the electronic and nuclear dynamics. Technological and
methodology developments in gas-phase UED will soon allow for the study of
large and more complex model systems and the study of classes of reaction
across multiple systems. These will enable the rationalization of general
rules for reactivity with the goal that molecules can be designed from first
principles to fulfill a particular function.

##  II Ultrafast probes for electron diffraction

###  II.1 Overview of a general UED setup and operating modes

The consolidation of ultrafast electrons as probes of matter providing high
spatial and temporal resolution is the result of concerted advancements in
multiple scientific and technological areas. To start, the widespread adoption
of photoemission for particle accelerator sources has revolutionized the field
of high brightness electron beams which had already seen a leap forward with
the invention of field-emission electron guns in the late 60s [58] with
respect to thermal emission sources used earlier. For field-emission based
guns, higher beam quality is achieved by minimizing the effective source size
rather than by increasing the total current. In case of photoemission, the
laser pulse triggers prompt emission of densely packed electron pulses. In
this case, the temporal duration of emission is limited by the laser pulse
length, thus reducing the effective duty cycle (ratio between emission time on
and time off) by orders of magnitude when compared to continuous field or
thermal emission sources. To compensate the ensuing reduction in average
current, UED instruments commonly generate pulses with many electrons per
bunch via emission from macroscopic flat photocathode surfaces, with typical
sizes ranging from micrometers to millimeters. Here the angular spread of the
emitted electrons is a key factor that sets the limit on the achievable beam
brightness [71] and the large area enables the extraction of Ampere-scale
instantaneous currents [100].

After extraction, preserving high beam quality to the sample becomes of upmost
importance. The interactions of the electron beam with the environment and
within itself via Coulomb forces can indeed broaden the pulse temporal
distribution effectively resulting in degradation of the instrument temporal
resolution. Progress in understanding the latter (Coulomb-broadening or space
charge effects) in these instruments [329, 284] has led to identifying the
most efficient ways of avoiding or managing temporal stretching, i.e. limiting
the propagation distance to the sample and/or rapidly accelerating the
electrons to higher energies. The accelerating electric field and the final
kinetic energy have then turned into key parameters of electron guns for UED
applications. Cross-fertilization with the neighboring field of high
brightness electron sources for high energy particle accelerators promoted the
introduction of a variety of beam manipulation methods and technologies,
expanding the parameter space and tailoring the beam phase space around the
particular application. Examples include the use of radio-frequency (RF)
accelerating cavities where electric fields approaching 100 MV/m can be used
to quickly boost the energy of the electrons to the MeV range [368, 370]. RF
fields can also be employed to reverse the space charge induced temporal
expansion to retrieve very short bunches at the sample plane [263, 43, 114,
260, 118]. RF-based deflecting cavities have been used as ultrafast streak-
cameras [251, 264], high-speed beam blankers [360], or in high resolution
time-of-flight spectrometers [359]. A more recent example is the adoption of
achromatic beam transport lines originally developed for synchrotron x-ray
sources, to passively reverse the space charge induced expansion and at the
same time reduce the time-of-arrival jitter of electron bunch at the sample
[171, 278].

In this fertile research environment different technological approaches
sprung, with the shared ultimate goal of achieving ever improving spatio-
temporal resolution. In many cases, custom instruments have taken the form of
compact accelerator beamlines with flexible designs, equipped with a mix of
electromagnetic, electrostatic and magnetostatic optical elements and
insertable diagnostics stations [249, 192, 245, 221, 131, 374, 425, 43, 101,
217, 364, 31, 109]. A parallel technological approach utilizes modified
electron microscope columns to effectively take advantage of the unsurpassed
lateral beam quality and electron optics of these setups [94, 289, 272, 183,
187, 146, 411, 30, 405]. Such systems usually work in the single-electron
emission mode to achieve sub-picosecond resolution and necessitate coupling
with high repetition rate optical excitation of the sample to maintain an
acceptable signal-to-noise ratio. In TEM-column instruments, it is relatively
easy to achieve nanometer-scale spot sizes at the sample plane, and the large
flux density (electrons/s/$m^{2}$) allows for the collection of nanoscale
information from heterogeneous specimens. This approach has demonstrated
successful, especially in the area of time-resolved electron nano-diffraction
and microscopy [65, 355].

Figure 9 provides a general schematic of a UED beamline with all its
components. The electron source consists of a photocathode and subsequent
accelerating gap. Its geometry also provides an optical path for an ultrafast
laser pulse to reach the photocathode, either by back or front illumination.
Acceleration can be provided by static or time-varying electric fields
II.4).Electron optics and collimation are used to tune sample illumination and
reciprocal space resolution, and time-varying fields can be used for temporal
beam compression (bunching). After the passage of the electron probe beam
through the sample, the diffracted signal is detected downstream the sample
plane.

In its most general configuration, a UED setup includes a timing and
synchronization system, as schematically shown in Fig.9. The generation of an
electron pulse is temporally coordinated with downstream beamline subsystems
via a timing distribution system consisting of opportunely generated and
delayed trigger pulses. Such signals, electronically or optically distributed,
initiate or terminate synchronous actions along the line, such as image
acquisition or pulsed sample delivery systems.

####  II.1.1 Temporal resolution

The overall temporal resolution is probably the single most important
parameter in a UED setup, and it is described as a combination of multiple
uncorrelated terms, including the excitation pulse length $\tau_{pump}$, the
electron beam pulse duration $\tau_{probe}$, the velocity mismatch $\tau_{VM}$
(if applicable, see Sec.IV), and the fluctuations $\tau_{\Delta pp}$ in
temporal delay ($\Delta t$) between the laser pump and the electron probe (see
Fig. 1). A generally accepted metric for calculating and reporting the
instrument temporal resolution of an instrument using Eq. 16 is that of Full-
Width-Half-Maximum (FWHM hereafter).

Accelerating and bunching field amplitude and relative phase fluctuations
cause shot-to-shot fluctuations of $\tau_{\Delta pp}$ (see Sec. II.3.4 and
II.3.6), and require precision phase synchronization between the different
sources is required (Sec. II.5.4). In the assumption that the same laser
system is used to both generate photo-electrons and to excite the sample, we
then have $\tau_{\Delta pp}=\tau_{\Delta_{eTOF}}$, where
$\tau_{\Delta_{eTOF}}$ is the electroh time-of-flight (TOF) from cathode to
sample. If instead more than one laser system is used in the experiment, a
similar synchronization system is required between the different optical
oscillators, and the jitters in arrival time of the laser to the cathode and
to the sample plane would need to be taken into account separately.

$\tau_{res}=\sqrt{\tau^{2}_{pump}+\tau^{2}_{probe}+\tau^{2}_{VM}+\tau^{2}_{\Delta
pp}}$ (16)

Figure 9: Schematic of a general UED setup.

In the following we will provide an in-depth review of the state-of-the-art of
each of the subsystems introduced above.

####  II.1.2 Electron packets: from single-electron to single-shot

The number of electrons interacting with the specimen required to obtain
structural information varies by orders of magnitude, depending on the
modality and on the specimen details. As an example, electron microscopy
provides real-space local information, and therefore it requires high dose at
the sample (10-100 electrons/(spatial resolution)2). The requirement for
number of electrons $N^{I}_{e}$ illuminating the sample is usually in the
range of $10^{8}$ to $10^{9}$. In electron diffraction on the other hand, the
signal at the detector carries reciprocal space information integrated over
the entire illuminated sample area. For solid-state specimens the signal is
concentrated in few areas of the detector, usually spots or rings, as a
consequence of the highly ordered atomic structure of the sample. Typically
less than $N^{ED}_{e}\approx 10^{6}$ electrons are sufficient to obtain high
quality (multiple Bragg spots) diffraction patterns from a thin (one elastic
mean free path) solid-state sample [327]. The sample material (high Z atoms
scatter more efficiently) and thickness (dynamical scattering effects can
lower the signal on the Bragg peaks), play a role in the definition of
$N^{ED}_{e}$, such as the density of the material itself. For electron
diffraction on gas-phase targets, the value of $N^{ED}_{e}$ is usually many
orders of magnitude larger, depending on the gas density and types of atoms in
the molecules.Furthermore, in UED experiments the transient signal are usually
retrieved from the difference image between diffraction pattern before and
after excitation. Hence the value of $N^{ED}_{e}$ will also depend on the
magnitude of the signal to be detected. If the goal is to resolve 1$\%$-level
changes in peak intensity, then Poisson statistics dictates at least 10000
electrons in the Bragg peaks analysed.

When evaluating the feasibility of an experiment it is instructive translating
electron diffraction requirements into constrains for the beam four-
dimensional brightness. Electrons must be tightly confined spatially within
the specimen boundaries, while maintaining a small angular spread for
achieving good resolution in reciprocal space (and a large enough spatial
coherence length). Using the definition of $\mathbb{R}$ from I.1.4, the
minimum required value for the 4d brightness (Eq. 13) is equal to:

$B^{min}_{4d}=eN^{ED}_{e}\left(\frac{2\pi\mathbb{R}_{min}}{sd_{s}\lambda_{c}}\right)^{2}$
(17)

where $\mathbb{R}_{min}$ is the experimental target for resolving power at
momentum transfer $s$, $d_{s}$ is the illuminated specimen lateral size
(assuming circular symmetry for simplicity), and $\lambda_{c}$ is the Compton
wavelength. Figure 10 reports calculated values of four-dimensional brightness
assuming $N^{ED}_{e}=10^{6}$ needed to obtain diffraction patterns with
adequate SNR, using $\mathbb{R}=10$ for different diffraction momentum
transfer values. The illuminated sample size strongly affects the requirements
on the electron beam, and can ultimately drive instrument design choices.

Figure 10: Requirements of 4D brightness as function of sample size for
different values of momentum transfer, from 0.1 to 5 Å-1. The calculations
assume that the electron beam fully illuminates the sample area,
$N^{ED}_{e}=10^{6}$ and $\mathbb{R}=10$. Different curves relate to different
momentum transfer values.

Experiment acquisition modalities can be separated in two broad categories:
single-shot and multi-shot (stroboscopic) modes.The choice of the modality is
often dictated by the details of the phenomenon under study. In a reversible
process, the excited specimen can be cycled between identical initial and
final states by a very large number of times, undergoing exactly the same
dynamical process and allowing data integration over many shots. Other samples
show enhanced sensitivity to the excitation and damage or a modified initial
state develop after a finite number of pulses, limiting the total number of
excitation events (partial reversibility). Finally, if the excitation pulse
drives the system to an irreversible final equilibrium state different from
the initial one, only the paired probe pulse will be able to capture the
transition before the sample is permanently altered.

In line with the different types of processes, UED operation modalities span
from single-electron to high-charge per bunch, and from one/few shots per
second to millions, with fundamental impact on the instrument technology used,
starting from the choice of the laser system and repetition rate, the electron
source size and geometry, the transverse and longitudinal compression schemes,
and the overall footprint of the setup.

A key difference between the single and multi-electron beam modalities is the
role of the beam self-fields (see Sec. II.3.5) in the beam dynamics. The so-
called space-charge fields effect the bunch duration, the beam energy spread,
and the total beam emittance of a multi-electron bunched beam, while single-
electron pulses are only constrained by transverse and longitudinal emittance
at emission [1]. Upon RF compression, for example, single-electron wavepackets
can theoretically be squeezed down to well below 1 fs [13]. Since the
longitudinal emittance is conserved, temporal compression does come at
expenses of energy spread, but typical UED experiments can tolerate this.
Another advantage of single-electron ”beams” operations, is that the emission
source can be arbitrarily small (and correspondingly higher beam brightness),
due to the absence of external field screening from other electrons. As it
will be more clear in Sec. II.2.4, such beams can be focused down to
nanometer-scale sizes at the specimen maintaining good transverse coherence
length.

Note that the concepts of beam size and angular spread in single-electron mode
take the meaning of moments of distribution of the statistical ensemble
represented by many single-electron beams, generated and transported through
the beamline at different times. Although for an isolate electron one could
define and measure angle and position to a better degree, a visible
diffraction pattern is only formed upon accumulation of many electrons, and
the overall resolution will still depend on the moments of the ensemble
distribution. This issue could potentially be minimized via the combined used
of fast single-electron detectors and time-stamping, although high precision
non-invasive time-stamping methods for single-electron beams are still out of
reach. Finally it is also worth pointing out that, as a direct consequence of
the statistical nature of photo-emission, the beam current in this
configuration is in practice limited to much less than 1 electron per shot.
Indeed, in order to maintain the spatio-temporal characteristics of the beam
shot by shot, the generation of beams with more than one electron should be
avoided. The photo-emission probability is described by Poisson statistics
and, in order to ensure that the overwhelming majority of pulses contain only
one electron, the average value of the distribution needs to be below 0.5
[13].

###  II.2 Generation of electron pulses

Although a continuous electron stream can be temporally chopped or bunched by
(a series of) RF cavities (see for example Sec. II.5), most UED electron
sources use short pulse lasers for generation of electron bunches by
photoemission. When a laser beam impinges on a photocathode surface, single or
multiphoton absorption can cause electrons in the material to gain enough
energy to overcome the potential barrier at the interface and escape into the
vacuum. The spatio-temporal format of the exciting laser pulse is nearly
preserved in the photoemission process offering the opportunity to shape the
initial electron beam distribution by controlling the properties of the
illuminating laser.

Photocathodes are evaluated by a few key parameters: the quantum efficiency
$QE$, the mean transverse energy of emitted electrons $MTE$ [161], response
time, and effective emission lateral size. The geometry of the emitting
surface is also of importance. A small radius of curvature can be used locally
enhance the external fields amplitude (DC, RF or optical). Larger radius of
curvatures would not produce significant enhancement, but introduce transverse
focusing or defocusing fields in the cathode vicinity, which would modify the
downstream beam dynamics(Sec. II.3.3).

####  II.2.1 Quantum efficiency

The cathode quantum efficiency $QE$ is defined as number of emitted electrons
per number of photons incident on the material, i.e.
$QE=\frac{\hbar\omega}{e}\frac{Q}{E_{ph}}$, where $Q$ is the electron beam
charge and $E_{ph}$ is the laser pulse energy. A theoretical expression for
$QE$ in metals can be found by following the three step model model [19], and
the QE can be directly related to the difference between laser photon energy
$\hbar\omega$ and material work function $\Phi_{W}$ (i.e. to the electrons
excess energy $E_{ex}=\hbar\omega-\Phi_{W}$). For photo-emission to happen,
the electron first absorbs one (or more) photon, then travels to the surface
avoiding scattering with other electrons, and lastly reach the vacuum
interface with enough energy in the normal direction to overcome the potential
barrier. Typical metals used as photocathode materials (Cu, Ag) have work
function in the $4.5-5eV$ range, with $QE$ values upon UV pulse illumination
ranging between $10^{-5}-10^{-4}$. As a numerical example, using a Cu cathode
with $10^{-5}$ QE, a laser pulse with 80 nJ energy at 266 nm (third harmonic
Ti:Sa laser) would suffice to generate $10^{6}$ electrons.

In presence of an externally applied electric field $E_{0}$ on the cathode
surface, the total potential barrier is modified by the Schottky potential
$\Phi_{Schottky}$ [307]. The resulting effective potential therefore becomes
$\Phi_{eff}=\Phi_{W}-\Phi_{Schottky}$, where
$\Phi_{Schottky}=\frac{e}{2}\sqrt{\frac{eE_{0}}{\pi\epsilon_{0}}}$. In the
approximation of constant electron density of state close to the Fermi level
(where electrons are emitted from), and approximating the material temperature
to zero, it can be shown that for small excess energies
$QE\propto(\hbar\omega-\Phi_{eff})^{2}$ [72]. Besides lowering the work
function, the applied field at the cathode plays an important role in
determining the maximum charge and current density that can be extracted, as
we will discuss later.

####  II.2.2 Photocathode thermal emittance

The mean transverse energy (MTE) of the emitted electrons determines the beam
emittance and, therefore, plays a relevant role in determining the beam
brightness. The beam normalized rms emittance at emission can be written as
$\epsilon_{n}=\sigma_{laser}\sqrt{\frac{MTE}{mc^{2}}}$ [161]. Using the same
approximations for the density of states and the Fermi-Dirac distribution used
above to calculate the $QE$, we can integrate the standard deviation of the
particle transverse momentum leading to the value for the
$MTE=\frac{\hbar\omega-\Phi_{eff}}{3}\propto\sqrt{QE}$ [72], clarifying the
trade-off between larger $QE$ values and smaller transverse beam emittance. As
an example, using longer laser wavelengths to decrease $E_{ex}$ is a clear
path to smaller emittance values and larger brightness, but it also decreases
rapidly the cathode QE, requiring more laser energy [132]. Also, in the limit
of $E_{ex}$ similar or smaller than the thermal energy $K_{B}T$ (where $K_{B}$
is the Boltzmann constant and $T$ is the temperature of the cathode) the
approximations used in the calculation of electron transverse momentum spread
fails as the tails of the Fermi-Dirac distribution dominate the spread,
limiting the mininum achievable $MTE$ to $K_{B}T$ [97]. The same behavior has
been measured in semiconductor cathodes, as shown in Fig. 11.

Figure 11: MTE versus excess energy for different cathodes, and compared with
theory. The minimum of MTE measured corresponds to ambient temperature (26
meV) adapted from [247].

Given the low laser energy needed to obtain typical electron charges for UED
setups (Sec. II.2.1), it may seem natural to trade quantum efficiency for
better beam quality. On the other hand, a large increase in laser beam energy
compensating lower $QE$ approaching the work-function threshold may have
detrimental effects. First, when coupled with a small focus at the cathode, it
can lead to values of optical fluence approaching the material damage
threshold. Furthermore, high intensity beams can increase the temperature of
the transient electronic distribution within a material by orders of
magnitude. For sub-ps photoemission, electrons do not have time to thermalize
with the lattice, since typical electron-phonon coupling constants are in the
few-picosecond range. This can lead to photo-emission of hot electrons,
contributing to the beam momentum spread and ultimately limit the achievable
MTE [226].

####  II.2.3 Response time of a photoemitter

Most photocathode materials have response times in the few femtoseconds to
sub-picosecond range, dominated by the travel time of electrons from the bulk
to the vacuum interface, which is determined either by penetration depth of
the optical pulse and/or the photocathode film thickness. In certain materials
(negative electron affinity semiconductor cathodes) the surface is chemically
prepared to energetically boost the bottom of the conduction band above the
vacuum level. Upon photon absorption electrons will reach the conduction band,
and some of them will slowly relax to the bottom of the band via scattering
with the lattice, while traveling toward the surface. Once there, they will
escape into the vacuum thanks to the negative electron affinity, forming long
temporal tails (up to 100 ps) with close-to-zero excess energy [14]. This
effect is more visible when using small excess energies, while tends to
disappear with increasing photon energies.

####  II.2.4 Source size and spatial resolution in UED

The size and shape of the photo-emitting area has an direct impact on
scientific breadth of the instrument, influencing paramters as the transverse
brightness and spatial resolution, and driving technological choices, such as
the accelerating field, illumination geometry and repetition rate.

Single-shot ultrafast experiments require large peak currents $I_{peak}$,
which are achieved using mm-scale source sizes. Such dimensions are only
acceptable when the heterogeneity of the sample is not a concern. When probing
reversible dynamics in stroboscopic mode, the real space resolution of UED can
be greatly increased by decreasing the spot size and optimizing for the
current density $J_{peak}$ rather than for $I_{peak}$. At its limit, the
stroboscopic modality could provide combined nanometer-femtosecond spatio-
temporal resolution, ultimately enabling nano-UED [155, 96] and ultrafast STEM
experiments. The sample illumination area for a fixed transverse coherence
length scales linearly with the source size. Indeed, using similar
considerations to the one that led to Eq. 17, and assuming transverse
emittance conservation along the beamline, the initial emitter diameter can be
directly related to the spot size at the sample:

$\frac{\sigma^{min}_{x,sample}}{\sigma_{x,cath}}\geq\frac{\pi\mathbb{R}_{min}}{\lambda_{c}s}\sqrt{\frac{MTE}{\
m_{e}c^{2}}}$ (18)

For typical MTE values of 0.5 eV and a resolving power above 10 at around 1
Å-1, we get $\sigma^{min}_{x,sample}>1.3\sigma_{x,cath}$. This simple result
puts in evidence the need of nanoscale emitters to reach nanometer-scale
spatial resolution. Alternatively, transverse collimation downstream of the
cathode can be performed, a common practice in static electron microscopes,
with the result of decreasing the initial emittance at expenses of beam
current [155](see Sec. II.5.6).

In photo-emission, two main factors limit the initial spot size: the numerical
aperture (NA) of the optical delivery system, describing the laser beam
convergence angle $\theta$ ($NA=sin(\theta)$ in vacuum), and the wavelength
$\lambda_{ph}$ of the laser used for photoemission. Even in the ideal case of
$NA\approx 1$, the laser beam waist $w_{0}$ is limited to
$w_{0}=\frac{\lambda_{ph}}{\pi}$. In practical circumstances the geometry of
the setup may even prevent using large values of $NA$, due to physical
constrains on the minimum distance of the last focusing lens from the cathode
plane. Solutions to this issue have been investigated, for example by
developing photocathodes operating in transmission geometry, allowing the last
optical element to be positioned right to the back of the photoemission
material [209].

In the last decade, laser-assisted electron emission from tips has been
extensively explored to overcome the light diffraction limit. Selecting
electrons emitted by the apex of the tip upon laser illumination provides
nanoscale sources of femtosecond pulses. Laser triggered emission can be
achieved via control of the temperature and the voltage applied to the tip,
modulating the Fermi distribution tail and the potential barrier,
exponentially suppressing electron emission in absence of laser. Linear
photoemission from tips using near UV laser pulses can also be achieved upon
coating of the tungsten apex with ZrO layer [52] [390] [94].

The shape of the source has a strong impact on the amplitude and direction of
externally applied field in the vicinity of the cathode plane [379], which in
turn has an impact on the magnitude of electron emission and on the beam
dynamics in the accelerating gap (see Sec. II.3.3). Curved surfaces enhance
the amplitude of the electric field at the surface, lowering the work function
through the Schottky effect. On flat surfaces, such effect usually accounts
for a decrease in the work function of no more than a few tenths of eV for all
practically achievable accelerating fields. If a sharp tip with large aspect
ratio $L/R$ is used instead, where $R$ is the tip radius and $L$ is the tip
height, the field at the tip surface is greatly enhanced ($E_{enh}\propto
E_{in}L/R$ [274]), leading to a dramatic change in effective work function (up
to more than 1 eV) and in photoemission yield. Note that in the limit of
extreme electric fields (in excess of $10^{9}$ V/m), field emission rather
than photoemission would dominantly contribute to the output current [107]. If
the tip radius is comparable or smaller than the wavelength of the laser pulse
used for photoemission, optical field enhancement takes place. Depending on
the amplitude of the enhanced laser field, either weak or strong
photoionization regimes can be achieved (measured by the Keldysh parameter
$\gamma$ [166]), leading to multiphoton photoemission [294] and/or optical
field emission [145].

At a first glance, the demands for high peak current and large current density
may seem to be simultaneously achievable. As will discuss in the following,
the two requirements instead drive the size of photoemission area in opposite
directions. The number of electrons emitted from a flat cathode in a given
time and from a given area is limited by cumulative image-charge fields at the
cathode interface. As electrons get emitted from the material into the vacuum,
charge at the surface promptly re-distribute to screen the bulk material from
the external field. The total electric field $E_{tot}$ in the vacuum region
between the emitted electrons and the cathode surface is therefore the sum of
the externally applied electric field and the opposing image-charge field. In
the limit of very short pulses the electron beam aspect ratio $A$ shows a
”pancake-like” format, with $A=\frac{2mR}{\Delta t^{2}eE_{0}}\gg 1$, $E_{0}$
is the external accelerating electric field, and $R$ and $\Delta t$ are the
laser beam radius and pulse duration, respectively (considering for simplicity
a uniformly charged cylinder). In this case the electron density in vacuum can
be approximated as an infinitely wide sheet of charge, and the emission will
stop when $E_{tot}=0$, leading to a maximum charge of $Q=\epsilon_{0}E_{0}\pi
R^{2}$, where $\epsilon_{0}$ is the vacuum permittivity [15]. As an example,
for an accelerating field of 20 MV/m, an emitter area larger than
$17\leavevmode\nobreak\ \mu m$ in radius would be required to extract $10^{6}$
electrons.

Decreasing the source size to sub-micrometer changes the beam aspect ratio,
eventually leading to cigar-like formats ($A<1$). In this case the finite
transverse extension of the beam plays a dominant role in the extraction
process, changing the functional form of the scaling laws for current density
and brightness [100]. It is worth reporting the 4D brightness scaling for the
case of large and small aspect ratio:

$B_{4D}^{m}ax\propto\begin{cases}\frac{E_{0}}{\textrm{MTE}}&\textrm{for $A\gg
1$ (pancake beam)}\\\ \frac{E_{0}^{3/2}}{\textrm{MTE}}\frac{\Delta
t}{R^{1/2}}&\textrm{for $A<1$ (cigar beam)}\end{cases}$ (19)

In the case of cigar aspect ratios, decreasing the source size will cause a
smaller change in the maximum charge extracted than in the corresponding
emittance (squared), with the important and often overlooked consequence of
introducing a dependence between maximum 4D brightness and the source size.
Therefore a possible scheme to achieve larger brightness values would include
starting from a cigar-shaped electron beam, and then perform temporal
compression downstream the electron gun. Indeed, as it will be discussed more
in detail in Sec. II.3.6, the electron beam can be temporally compressed with
minimal implications on the transverse emittance (see for example [101]). This
setup allows smaller initial spot sizes and disentangles spatial and temporal
resolution. The drawback is an increased longitudinal emittance, that would
ultimately limit the shortest pulse length achievable [224].

####  II.2.5 Towards brighter photoemission sources

Relevant research directions aim at increasing the brightness of electron
sources by decreasing the cathode MTE or decreasing the photoemission source
size and at the same time increasing the acceleration field. As shown in Fig.
11, an effective way to reduce the MTE is to decrease the excess energy, up to
the limit where the residual MTE of the emitted electrons is limited by the
cathode temperature. Values of MTE as low as 26 meV have been measured at room
temperature [97], while more recently measurements as low as 5 meV have been
demonstrated by cooling single crystal Cu (100) surface to cryogenic
temperatures [160], about two orders of magnitude lower than typical MTE
values from metal photocathodes. One of the drawbacks of working close to the
work function threshold is the strong reduction in QE, which complicates the
use of such cathodes, especially for applications targeting large peak
currents (See Sec. II.2.2). Recently it has been shown that using ordered
crystal surface structures can partially reverse the dependence between QE and
MTE [162]. Here the values of electron transverse energy can be constrained by
a careful choice the energy band structure, decreasing the MTE of the emitted
electrons even for relatively large excess energy values.

Alternatively, semiconductor cathodes can provide low MTEs and very large QE,
on the order of few to few tens of percent, mostly thanks to the suppression
of electron-electron scattering leading to a much more efficient transport of
excited electrons from the bulk to the vacuum interface. In such materials
electrons occupy states up to the top of the valence band, while the
conduction band is empty. The energy barrier to overcome in this case is the
sum of the material band gap and the electron affinity, often enabling linear
photoemission with visible or infrared photons (see for example [59, 60, 61].
The photoemission surface of such materials is often chemically very reactive,
and contamination from the external environment rapidly lowers the QE by
orders of magnitude [71, 102]. Special high vacuum load-lock chambers are
typically employed to transfer these photocathodes from the growth chambers to
the electron guns.

A possibility to reduce the emission area to be much smaller than that
achievable by direct lens focusing is offered by laser field impinging on
nano-structured metallic surfaces that can excite traveling waves confined at
the metal-dielectric interface, called surface plasmon polaritons (SPP).
Mediated by SPP, whose wavelength can be much shorter than that of the
excitation pulse, the optical field energy can be transported and concentrated
in areas of sub-wavelength size, leading to large local field enhancement.
This concept has been lately used to enhance absorption on metal tips [242].
More recently, the same idea has been studied to induce large enhancement
factors on nanoscale flat surfaces [79], which could be extremely useful if
the cathode is immersed in high field areas, where tips may not be ideal due
to large amounts of field-emitted current and short lifetimes.

####  II.2.6 Laser systems

A critical element in any UED setup is the ultrafast laser system which is
used to provide pulses to the cathode and excite the sample. It should not
come as a surprise that the chirped pulse amplification was a critical
technological step in enabling the development of UED technique. Additional
laser applications in UED setups include ponderomotive scattering for time-
stamping the time-of-arrival of electrons with the photon beam [138](see Sec.
II.5.2), generating THz waves can be used to compress the beam or streak it
[90]. In most advanced cases, lasers provide the energy for actual electron
acceleration (ACHIP, LPA) [137].

Typical architectures for UED laser systems include a modelocked oscillator
cavity followed by a chain of amplifiers to bring up the energy to the
required levels. In cases in which RF is used to manipulate, control or
diagnose the electron beam, it is important to choose the oscillator cavity
length that can be easily synchronized with the RF frequency used in the
experiment. Typically, a intra-cavity piezo-mirror is used to close a feedback
loop to maintain phase-locking to an external signal. More on this is
discussed in Sec. II.5.4. State-of-the-art systems are also able not only to
lock the envelope of the laser pulse to an external signal, but also to lock
the phase. CEP-phase locked phases so far have not been employed in UED
setups, but this might change as attosecond electron pulses are starting to be
used to probe attosecond dynamics [238].

While most of the UED instruments up to now have operated using the Ti:Sa
technology due to the large gain-bandwidth and clear advantage in the
generation of ultrashort pulses of this crystal, the limitations associated
with the poor efficiency and associated low average power as well as the
progress in other competing laser technologies such as Yb:based lasers are
increasing the diversity of the laser systems out in the field. As discussed
in Sec. II.1.2, one of the main characteristics of any setup is the targeted
operation mode (ranging from single shot to stroboscopic). For the latter,
being able to increase the repetition rate beyond 50 KHz greatly affects the
laser technology choice. An important issue that requires a compromise in fact
is the longer pulse length typical of the higher repetition rate and higher
average power laser systems. Ti:Sa systems routinely generate $<$ 40 fs
pulses, while the pulse length in Yb-based systems is 5-6 times longer. An
open question is how to get ultrashort pulses at high repetition rates.
Different technologies are being pursued ranging from OPCPA [77] to employment
of non linear compression techniques [159].

Precise control of the laser distribution illuminating the cathode has been
shown to improve the beam brightness especially in space charge dominated
beamlines [250]. Both transverse and longitudinal shaping of the laser pulse
before photocathode illumination have been employed. In the transverse
dimension, predetermined schemes like imaging an overfilled aperture, or
refractive shapers, compete with adaptive computer-controlled approaches based
on liquid crystal mask [225] or digital micro-mirror arrays [199]. On the
longitudinal size, the temporal profile can be controlled with dispersive
crystals [421], acousto-optic [200] and mechanical [49] spectral shaping. For
oblique cathode illumination, the technique of pulse front tilt [139], which
is also used to velocity match the pump and the probe on the sample as
discussed later (see Sec. IV.2.2), can be also applied.

The wavelength selectivity of the gain mediums does not cover all the possible
wavelengths. For example in photocathode drivers it is useful to be able to
tune the photon energy to the cathode work-function, and similarly when
pumping a material one wants to excite certain optical modes and steer away
from high reflectivity regions. Non linear frequency generation, both directly
in crystals as well in optical parametric amplifier setups are usually added
to the main laser system. While the price to pay in pulse energy is
significant, the continuous wavelength tunability they offer allow exploration
of new physics. For longer wavelengths, either difference frequency generation
options in the OPA [103] or optical rectification [111] can be used to
generate THz which can be used for compression/diagnostics and also directly
for pumping.

###  II.3 Electron dynamics

In the final step of the photoemission process, electrons escape the cathode
surface and enter vacuum with a residual kinetic energy typically in the range
of a few to 100s meV. Transport and control at these low energies is quite
challenging, and electrons are therefore accelerated to higher kinetic
energies, ranging from 100 eV[122, 6, 362, 243] for surface science and low-
dimensional materials in reflection geometry, to keV and MeV levels, more
typical for transmission modes.

As explained in Sec. II.2.4 larger accelerating fields at the cathode surface
allow to extract larger current densities, and thus enable higher beam
brightness for a given cathode MTE. In this section we will review the
electron beam dynamics downstream the cathode plane, including acceleration
and compression, which allow tailoring of the beam phase space to the specific
application, but may also lead to potential degradation of initial beam
brightness due to nonlinear forces, time-varying fields and/or self-forces
within the electron bunch.

####  II.3.1 The accelerating gap

The most mature and widely used acceleration technologies use DC and RF fields
[282]. The schematics of the geometry and field profiles of a DC, multi-cell
RF, and very-high-frequency (VHF) quarter-wave resonator RF gun are shown in
Fig. 12. The geometry for these electron guns is essentially cylindrically
symmetric, and the acceleration electric field can be written as:

$E_{z}(z,r,t)=\begin{cases}E_{0}e_{z}(z,r)&\textrm{for DC fields}\\\
E_{0}e_{z}(z,r)\sin(\omega t+\phi)&\textrm{for RF fields}\end{cases}$ (20)

where $z$ and $r$ are the axial and transverse coordinates, $z=0$ corresponds
to the cathode position, $E_{0}$ is the peak electric field, $e_{z}$ is the
normalized profile of the field distribution, and $\omega$ and $\phi$ are the
RF angular frequency and phase, respectively. Maxwell equations relate the
longitudinal field component $E_{z}$ with the transverse component
$E_{r}=-\frac{r}{2}\frac{\partial E_{z}}{\partial z}$, which is important in
the transverse evolution of the beam in the gun. In presence of time-varying
fields, the ensuing magnetic field has also to be taken into account in the
transverse dynamics, but bears no effect on the kinetic energy.

Figure 12: For a (a) DC, (b) multi-cell RF gun, and (c) VHF quarter-wave
resonator RF gun used for UED, the schematics of gun geometries are shown with
field contours lines (equal-potential lines for DC gun and field lines for RF
guns). The longitudinal ($E_{z}$, red solid) and transverse ($E_{r}$, blue
dashed, small offset from the axis) field profiles are also shown.

In a static accelerating gap with a flat cathode (Fig.12(a)) electric field
lines are normal to the surface and therefore only contribute to the increase
of the longitudinal component of the particle momentum. Longitudinal single-
particle dynamics is straightforward, and the final beam kinetic energy is
simply the integral of the field $E_{z}$ over the longitudinal position $z$
multiplied by the electron charge. The kinetic energy of electrons accelerated
by static fields is limited to $\approx$ 350 keV or lower by electrical
breakdown, still in the non-relativistic regime. Here a variation of output
energy has a strong effect on the final particle velocity, and hence the time
of arrival of the beam at the sample (see Sec. II.3.4).

The anode aperture is a perturbation from the ideal parallel plate geometry,
which bends field lines outwards at the gap exit. The net effect on electron
dynamics is transverse defocusing, typically requiring an optical element to
re-capture the diverging beam after the gun. The strength of the electrostatic
lens scales with the accelerating gradient, but at first order does not
degrade the beam quality. As we will see later, in a time-dependent
accelerating field the defocusing kick (visible by the magnitude of $E_{r}$ in
Fig. 12(b) and (c) )) will also be time dependent, leading to an increase of
the total emittance.

####  II.3.2 Electron acceleration via time-varying fields

Particle dynamics is more complicated in time-varying fields. In this section
we describe the electron behavior in RF fields as an example. Most of the
treatment can be extended to different frequency ranges. The longitudinal and
transverse motion of electrons in an RF gun can be treated analytically [173],
by modeling the fields as a standing wave of frequency $\omega$ with a given
on axis amplitude profile $E_{0}e_{z}(z)$. Approximating $e_{z}$ as a
sinusoidal function with a wave number $k=\omega/c$, the longitudinal
acceleration field can be expressed as $E_{z}(z,t,\phi_{0})=E_{0}\cos
kz\sin(\omega t+\phi_{0})$. The longitudinal equations of motion can then be
rewritten, decomposing the standing wave into forward and backward traveling
wave components, as

$\frac{d\gamma}{dz}=\alpha k[\sin\phi+\sin(\phi+2kz)],$ (21)

where $\phi(z,t)=\omega t-kz+\phi_{0}$ is the so-called synchronous phase. The
use of the phase coordinate is particularly convenient for RF linacs, as
particles reaching relativistic energies move along constant $\phi$
trajectories. The dimensionless parameter $\alpha=\frac{eE_{0}}{2m_{e}c^{2}k}$
is a normalized measure of the strength of the accelerating field. In order to
capture electrons from rest $\alpha$ must be larger than 0.5 [297], implying
that higher frequencies require larger peak fields.

Particles are released from the photocathode with low speed, and quickly fall
behind the synchronous phase until they reach relativistic energies. Due to
the rapid acceleration of RF guns ($\alpha\approx$1), most of the phase
slippage occurs in close vicinity to the cathode, where electrons are much
slower than wave phase velocity. The final synchronous phase depends on the
launch phase $\phi_{0}$ for a given gun geometry and operation field strength.
This dynamics is an intrinsic feature of particle acceleration with time-
varying fields. If $\alpha$ is too small, the electrons do not gain enough
energy during the accelerating phase and keep slipping back in phase until
they start experiencing a decelerating field, like a surfer with not enough
initial speed to catch the incoming wave. The implications of such dynamics on
the bunch length and time of flight of electrons will be discussed more in
depth in Sec. II.3.6.

The kinetic energy at the gun exit is a function of $\phi_{0}$. An example of
$\gamma$-$\phi_{0}$ correlation is shown in Fig. 13(a). The example
corresponds to the SLAC-UCLA-BNL 1.6 cell S-band RF gun, one of the most
widely used sources for relativistic UED applications, operating at a peak
field of $E_{0}$ = 100 MV/m ($e_{z}$ profile shown in Fig. 12(b)). Such
correlation translates to RF-induced energy spread in an electron beam. For a
finite laser pulse length illuminating the cathode, electrons emitted at
different times will experience a different instantaneous accelerating field
amplitude $E_{0}e_{z}\sin(\phi_{0})$. Neglecting space charge effects (see
Sec. II.3.5), the launch phase providing maximum energy gain
$\phi_{0}=\phi_{m}$ is also the phase which minimizes the total energy spread.
For the S-band gun example above, $\phi_{m}\sim$ 30∘ (blue circle in Fig.
13(a)). The accelerating field experienced by the particles at photo-emission
in this case is roughly $50\%$ ($\sin 30^{\circ}$) of the peak acceleration
field. The inset shows the evolution of the beam energy inside the gun. Higher
peak fields or different gun geometries can be exploited to obtain larger
values of optimal injection phase, increasing the accelerating field at
emission. For example a 1.4 cell S-band RF gun can shift $\phi_{m}$ to 70∘ or
higher, increasing the field at photo-emission to $\sim$95% of the maximum,
and leading to higher beam brightness [194].

In the VHF range (30-300 MHz) [304],the phase slippage become negligible and
the launch phase is therefore much closer to 90∘, allowing to take full
advantage of the maximum accelerating field.

Figure 13: For a 1.6 cell S-band RF gun operating at 100 MV/m, (a) the beam
energy $\gamma$ at gun exit as a function of the launch phase. (inset) the
evolution of $\gamma$ in the gun at the maximum energy launch phase, indicated
by the blue circle. (b) Relative time-of-flight from the photocathode to
$z=15$ cm, and bunch length compression ratio $C$ as a function of launch
phase.

Transverse RF fields (see Fig. 12)act as time-dependent focusing/defocusing
lenses. The variation in focusing strength experienced by different
longitudinal beam slices, (i.e. the head, center and tail of the beam), causes
the transverse phase space distribution to fan out in correlation with the
longitudinal beam coordinate. This increases the area of the beam transverse
phase space and induces RF-emittance growth. This RF-induced effect is
minimized by choosing the initial launching phase so that the beam exits the
gun at the maximum energy [173]. In this case the contribution to the
transverse emittance is
$\epsilon_{x}^{\textrm{rf}}=\frac{1}{4mc^{2}}eE_{0}\sigma^{2}_{x}\sigma^{2}_{\phi}$,
where $\sigma_{x}$ is the rms transverse beam size and $\sigma_{\phi}$ is the
rms longitudinal beam size in radians of RF phase. In UED applications where
spot sizes are less than 100 $\mu$m and $\sigma_{\phi}$ is 0.1 degrees or
smaller, this effect can be often neglected.

####  II.3.3 The effect of the cathode curvature

The profile of the photocathode surface has an impact on the output electron
beam parameters and dynamics in the gap. In general, the area can either have
a flat or curved profile. In the case of the flat profile, the field lines
will be normal to the surface and all the acceleration will be in the
longitudinal direction with no effects on the transverse plane. In the case of
a curved surface, three different cases can be distinguished, comparing the
radius of curvature $R$ with the laser spot size $r$ used for photo-emission.
If the surface radius of curvature is large, the main effect is a distortion
of the field in the cathode vicinity, adding transverse components and leading
to transverse focusing (concave) or defocusing (convex) effects. The cathode
is an equipotential surface, with $\Phi(\mathbf{r})=0$. Expanding the electric
potential in $r$ and $z$ to the second order, under the assumption of $R\gg r$
one finds aberration components due to curvature to be proportional to $1/R$
[134].

As the cathode radius of curvature gets smaller and becomes comparable to the
laser spot size, both transverse and longitudinal effects need to be
considered. The electric field enhancement along a curved surface discussed
earlier, can be used to increase the accelerating field in the cathode area
while keeping a large transverse emission size, obtaining at the same time
extraction of multi-electron beams and ultrashort pulses from setups with
otherwise modest accelerating gradients, typically DC guns [267]. A similar
approach can be taken in cathodes for RF guns. In this case the time-varying
nature of the field can be used at one’s advantage for beam temporal
compression. By fabricating curved cathodes, a radial-temporal correlation is
established by means of two related effects: the delay of the outer region of
the laser pulse in reaching the cathode surface with respect to the central
area, and the different accelerating field amplitude experienced by the
particle at birth. Such concave shape can be optimized to pre-compensate for
the non-isochronicity of the following focusing elements, leading to shorter
final electron pulse [212].

For tip-like cathodes, the radius of curvature is orders of magnitude smaller
than the illuminating laser, and in most of cases even smaller than the laser
wavelength. The main advantage of a tip is that the source size is now
determined by the physical extension of the tip and not by the laser spot
size. Accelerating field at the tip apex can be enhanced by factors exceeding
100, with a longitudinal extensions comparable to the tip radius. While this
may locally increase the maximum brightness achievable, it also increases
field non linearities and, in order to obtain a high brightness beam, heavy
collimation is needed downstream the accelerating gap, selecting only
electrons emitted from the tip apex (see Sec. II.5.6).

####  II.3.4 Temporal beam evolution in simple systems: Vacuum dispersion

In this section we will review the role of key instrument parameters on the
beam longitudinal dynamics in absence of space charge. The evolution of the
beam center of mass is unaffected by self-fields, and we are therefore able to
provide approximate analytical equations that can be used to accurately
predict the final energy and arrival time of the electron(s) at the sample. At
the same time, for an accurate prediction of the final pulse length at the
specimen both longitudinal emittance and the eventual contribution of space-
charge forces to the dynamics need to be accounted for. This requires solving
self-consistently the Maxwell equations coupled to the equations of motion for
the beam, and it is generally achieved through the use of sophisticated
simulation codes (see for example Fig. 16).

In the simplest setup, which includes a static accelerating field within a gap
and a downstream drift to the sample, accelerating field fluctuations and beam
energy spread at emission (the electron excess energy) contribute to shot-to-
shot energy and time-of-flight variations. Variation in electron energy
translates in time of flight fluctuations through vacuum dispersion. To
quantify the impact of such effect on the instrument performance we first
consider only the accelerating gap starting from the photocathode surface, and
then we include the transport from the output of the gun to the sample.

Figure 14: (a) Dependence of vacuum dispersion broadening $\tau_{vd,gap}$ on
the acceleration field $E_{0}$ inside an DC accelerating gap for several
different initial energy bandwidth of photoelectrons $\Delta E$. (b)
Broadening of the bunch length in a drift transport channel, for different
beam energy and energy spread values.

We define $\tau_{vd,gap}$ as the temporal distance at the output of the
accelerating gap between two electrons photo-emitted from the cathode at the
same time with respect with the laser arrival time. Depending on the operation
mode, $\tau_{vd,gap}$ represents the shot-to-shot TOF fluctuations (single-
electron mode) or the final temporal spread of the beam (assuming negligible
space charge effects). The value of $\tau_{vd,gap}$ at the output of the
static gap is mainly affected by the energy bandwidth of the photo-electrons
at emission $\Delta E$, which depends on the detail of the photocathode
material and driving laser. Fluctuations in the amplitude of the accelerating
fields are generally below $1e-4$ for state-of-the-art high-voltage power
supplies, and can be ignored inside the short accelerating gap, while they
will need to be included when calculating the TOF to the sample. For an