namely the diffraction limit. However, HHG provides an alternative to
achieving a smaller focal spot size by converting an optical wavelength to its
harmonic wavelength. Quere et al. [41] predict an intensity gain up to
$10^{6}$ for a 4-PW laser system. However, there are challenges to overcome
before it can be realized experimentally, including but not limited to
controlling the plasma mirror curvature and measuring extreme intensities.

###  4.3 Towards isolated attosecond electron bunches222This section co-
authored with Batson, T., Nees, J., Thomas, A. G. R., and Krushelnick, K.
(2020): Towards isolated attosecond electron bunches using ultrashort-pulse
laser-solid interactions. Scientific reports, 10(1), 1-11.

Driving high-intensity ultrashort laser pulses into overdense plasmas tends to
produce electron beams with higher charge number at lower peak energy compared
to those from underdense plasmas in laser-wakefield acceleration (LWFA). This
interaction between relativistic laser pulses and solids has been studied in
the past decades in different regimes of laser pulse characteristics and
plasma conditions. When the pulse duration is ultrashort ($\sim$ 30fs) and the
plasma density scale-length is short compared to laser wavelength
($L_{s}=n_{e}\cdot(\frac{dn}{de})^{-1}<0.1\lambda$), the incident radiation
drives a periodic motion of the critical surface to stretch and compress the
light reflected off the surface within each cycle. This is known as the
relativistic oscillating mirror model, which leads to the generation of
attosecond light sources through High-order Harmonic Generation (HHG) [149,
151, 153, 161, 39] as well as ejection of electrons from the surface. The
reflected laser field then accelerates these electrons to relativistic energy
through the Vacuum laser acceleration (VLA) mechanism [175, 176], usually
resulting in a ring-shaped electron beam structure [177, 178]. If the plasma
scale-length is longer than the laser wavelength, the interaction mainly
happens near the critical density and the acceleration mechanism becomes
extremely complex, including ponderomotive acceleration [179], surface
quasistatic fields [180] and direct laser acceleration. Varying the scale-
length from $<0.1\lambda$ to $5.5\lambda$, an optimal condition for generating
quasi-monoenergetic electrons is found at $L_{s}=0.5\lambda$ using
relativistic high repetition rate laser system with abundant statistics [66].
Scaling laws of electron temperature have also been studied [181]. A theory
for the electron acceleration mechanism in this regime is the standing-wave
acceleration. Superposition of the incident half and the reflected half of the
laser pulse results in a standing wave pattern, and the modulated electric and
magnetic fields in the standing wave can launch electrons in the forward and
backward directions [182, 183, 184]. Going to even shorter pulse duration
matching the plasma wavelength ($t=\lambda_{p}/2c$), wakefields can be
excited[70]. To have wakefield acceleration happen in laser-solid interaction,
a few-cycle pulse driver ($\tau\sim T_{0}/2cos\theta$) is required[70] so that
large-amplitude plasma waves can be generated along the density gradient up to
the effective turning point $n_{cr}cos^{2}\theta$.

Most of the reported laser-solid experiments were set up in normal[182, 62,
185] or oblique[66, 68, 178, 177] ($\sim 45^{\circ}$) incidence. However,
Naumova et al. [186] predicted the existence of attosecond electron bunches in
a grazing incidence setup. Since then there have been extensive theoretical
predictions and simulations for attosecond electron bunches using various
geometries, such as droplet target[187, 188], nanofilm[189] and transversely-
thin slice target[190], no experimental indication has been reported yet. In
this work, we confirm that generating attosecond electron bunches favors a
larger angle of incidence (measured from target normal) in both experiments
and simulations. We utilize PIC simulation and particle tracking to
investigate the formation mechanism of these attosecond electron bunches.
Furthermore, we generate isolated attosecond electron bunches using single-
cycle laser pulses in PIC simulations. We study some key parameters that
govern this process, such as preplasma density scale length, laser intensity,
carrier-envelope-phase (CEP) and focal-spot size.

####  4.3.1 Experimental and computational setup

Figure 4.8: Schematic of the experimental setup (a) and top view of the normal
incidence geometry (b) as well as the grazing incidence geometry (c). Electron
spectrometer: 1.15 kG magnetic spectrometer with a Fujifilm MS image plate
covered with lead and aluminum shielding; pellicle: 2-inch diameter, 2 $\mu m$
thick nitrocellulose pellicle; target: 4-inch diameter, 6 mm thick glass; OAP:
off-axis paraboloid. The spectrometer is placed at three different positions
in (c).

The experiments were performed at the University of Michigan, using the
$\lambda^{3}$ laser system with 30fs, 12mJ, $0.8\mu m$ pulses at a repetition
rate of 480 Hz. The experimental setup is shown in Fig. 4.8. A P-polarized
laser pulse was focused from a $90^{\circ}$ ($60^{\circ}$) f/1 gold off-axis
paraboloid onto the target at normal (grazing) incidence. Note that a hole was
drilled in the center of the $90^{\circ}$ OAP in order to capture signals on
the spectrometer. The geometry of normal and grazing incidence cases are drawn
in Fig. 4.8b,c, respectively. In such a tight focus setup, the light rays
cover a range of angles (approximately $\pm 20^{\circ}$). The angle of the
central ray in grazing incidence is estimated to be $\sim 70^{\circ}$, which
is as close to grazing as the beam could get without clipping. The focal spot
size (FWHM) was $1.5\mu m$, resulting in a peak intensity $I=\frac{0.94\cdot
E_{p}}{\tau_{p}\cdot\pi\cdot w^{2}/2}=1.6\times 10^{19}\;W/cm^{2}$ and
normalized vector potential $a_{0}\sim$ 2.6. The laser was focused through a
thin pellicle to protect the OAP from debris. A rotary target stage was used
to provide degrees of freedom in rotation as well as movement radially and
longitudinally. Most of the experiments were performed with glass targets
unless specified to be copper. An external prepulse was available by placing a
thin pellicle in the optical system to pick up $\sim 8\%$ of the pulse energy.
The prepulse intensity is around $5\times 10^{16}W/cm^{2}$. The generated
electrons were diagnosed by a Fuji MS image plate after being deflected by a
1.15 KG magnetic field. The image plate efficiency was included using the
published calibration data[191]. Lead and aluminum shieldings were used to
prevent noise from Bremsstrahlung radiation. A detailed description of the
spectrometer can be found in reference[66, 181].

PIC simulations were performed using the OSIRIS[122, 123] 4.4.4 framework in
2D3V Cartesian geometry. There are 100 macroparticles per cell and the grid
size is $\lambda/32\times\lambda/32=0.025\mu m\times 0.025\mu m$, where
$\lambda=0.8\mu m$ is the laser wavelength. The convergence of the simulation
is checked using up to $\lambda/64\times\lambda/64$ grid size. The
computational time-resolution is 0.027 fs or $\sim 100$ step per laser cycle.
The laser pulse is assumed to be Gaussian in both the longitudinal and
transverse direction with a FWHM pulse duration of $\tau=30fs$. The laser
pulse is continuously launched from the wall and focused down to beam waist
$w0=\acs{FWHM}\times 1.699/2=1.28\mu m$, $a_{0}=2.5$. The simulation box size
is $64\mu m\times 64\mu m$ in normal ($0^{\circ}$) incidence geometry, $48\mu
m\times 48\mu m$ in oblique ($45^{\circ}$) incidence geometry and $72\mu
m\times 28\mu m$ in grazing ($76^{\circ}$) incidence geometry. The reported
angles are those of the central rays. The simulations were run for 270 fs, 210
fs, and 300 fs, respectively, until the interesting electrons left the
simulation box. The initial plasma density profile is described by a uniform
glass-solid-density ($5.01\times 10^{22}cm^{-3}$) region plus an exponential
tail, as is illustrated in Fig. 4.9f.

####  4.3.2 Results

Fig. 4.9 demonstrates the similarities between the experimental and simulated
electron energy spectra and the bunch duration measurement in simulation. The
experiment was performed in a grazing incidence setup using a 20 ps prepulse
and the measured electron energy spectra are shown in Fig. 4.9a. The
experimental setup is shown in Fig. 4.8 in the previous section. All spectra
have been calibrated using published image plate efficiency [191]. Fig. 4.9b-f
are from PIC simulations, which were also conducted in grazing incidence
geometry with preplasma density scale-length matches the experimental
prepulse. This conversion from prepulse to preplasma scale-length is performed
in the 1D hydrodynamic code (HYADES[192]) assuming isothermal expansion.

Figure 4.9: Electron energy spectra from experiments (a) and simulations (b)
in grazing incidence setup. $45^{\circ}$, $90^{\circ}$ and specular reflection
in (a) correspond to position 1, 2 and 3 labeled in Fig. 4.8c, respectively.
(b): energy spectra of electrons labeled in (d). The bin size of the spectra
is 72 bins/MeV. Both energy spectra in (a) and (b) have dn/dE in arbitrary
units. (c): Electromagnetic energy density in the region of the reflected
pulse at time t=210 fs when peak laser intensity interacts with the critical
surface. (d): Total energy of particles in the simulation box beyond 1 MeV at
time t=300 fs before the attosecond electron bunches leave the simulation box.
(e): Particle energy of the attosecond electrons. It is the zoom-in of the
square region ”atto” in (d) with cutoff energy at 4 MeV. (f) Initial charge
density in unit $n_{0}=n_{cr}/4\pi^{2}$, where the scale-length is
$L_{s}=0.5\lambda$.

Comparing the experiment and simulation results, we notice a ”bump” feature in
the red curves: near 2 MeV in Fig. 4.9a and left to 3 MeV in Fig. 4.9b. This
feature does not show up in other curves in blue or green. In the simulations,
the ”bump” component of the electron spectrum is always observed along the
target surface and was always associated with the train of ultra-short
duration bunches of energetic electrons. Since the red spectra in Fig. 4.9b is
taken from the attosecond electrons (bunch duration measured to be
$\sim\lambda/8$ in Fig. 4.9e), it suggests the existence of attosecond
electron bunches in the red spectra in the experiment. The propagation
direction of the attosecond electron bunches is found to be in the specular
reflection direction through particle tracking, matching the experimental
observation.

Figure 4.10: (a): Experimental electron energy spectra from normal incidence
and grazing incidence cases using a 20 ps prepulse. (b-d): Simulated angular
energy distribution of electrons for grazing, oblique and normal incidence
using scale-length $L_{s}=0.5\lambda$. The laser pulse comes into the
simulation box from the left ($0^{\circ}$) and the target lies along the black
line. The snapshots were taken when peak laser intensity interacts with the
critical surface in each case.

While Fig. 4.9 presents the direction of the emitted attosecond electron
bunches, Fig. 4.10 investigates different incident angles of the laser pulses.
The unique ”bump” feature was observed only in the grazing incidence case but
not in the normal incidence case, as is shown in Fig. 4.10a. To understand it,
we ran simulations at different angles of incidence. Fig. 4.10b-d shows the
angular distribution of all electrons when peak laser intensity interacts with
the critical surface. This is also the time when the ultra-thin ”null” in the
electromagnetic energy density is formed in Fig. 4.9c. Due to the self-
intersection of electron trajectories, the electron concentration is abruptly
peaked[186] to initialize the bunching process. Fig. 4.10 shows that electrons
are more spreading out in (c) and (d) than in (b), confirming that the
bunching process favors grazing incidence.

It is worth pointing out that these observations are consistent with respect
to both experiments and simulations. From the 41 data shots we took at
different geometries, 25 were taken at the grazing incidence geometry that
encourages attosecond electron bunches where the laser pulse was at grazing
incidence and the spectrometer was positioned in the laser reflection
direction. We observed the ”bump” feature in 12 of the 25 spectra. We observed
no spectra with this feature in the other 16 shots taken at different
geometries. Energetic attosecond electron bunches with noticeable coherence
were also observed consistently in simulations as we scanned the preplasma
electron density scale-length from $0.1\lambda$ to $2\lambda$, and detailed
results will be presented in the following section.

#####  4.3.2.1 Preplasma effects

Figure 4.11: (a): Experimental electron energy spectra with and without an
external 20 ps prepulse using glass target. (b): Experimental electron energy
spectra using glass and copper target without external prepulse. (c)-(e):
Simulated spatial distribution of energetic electrons using preplasma scale-
length $L_{s}=0.25\lambda(c),\;0.5\lambda(d)\;and\;\lambda(e)$. Both
experiments and simulations were performed at grazing incidence geometry.

Preplasma density profile can affect the energy and propagation direction of
the attosecond electron bunches. Fig. 4.11a shows that including an additional
20 ps prepulse moves the ”bump” feature towards the right on the spectra,
suggesting bunching at higher energy. The same phenomenon is observed in Fig.
4.11b as copper has a lower ionization threshold and thus a more developed
preplasma profile with longer scale-length when the main pulse arrives. To
explore this effect in parameter space, we tune the preplasma density scale-
length from $0.1\lambda$ to $2\lambda$ in simulation. Energetic attosecond
electron bunches with noticeable coherence were observed in moderate scale-
length cases, as is highlighted in Fig. 4.11c-e, but not in the two extreme
cases where $L_{s}=0.1\lambda$ or $2\lambda$. Note that $L_{s}=0.5\lambda$
resulted in the most energetic attosecond electron bunches, matching the
optimal condition for producing quasi-monoenergetic electron beams at
$a_{0}\sim 2$ in previous experiments[66]. Besides, tuning the preplasma
profile affects the bunch propagation direction. As the scale-length
increases, the emission angle of the bunches becomes smaller as measured from
the target normal. It is due to the critical surface moving away from the
solid surface, as are the generated electron bunches.

#####  4.3.2.2 CEP effects

Figure 4.12: Single attosecond electron bunch and particle tracking using
CEP$=\;0,\;\pi/2,\;\pi$, and $3\pi/2$. (a)-(d): Spatial energy distribution of
bunched electrons at the last time-step. (e)-(h): Trajectory of the attosecond
electrons with emission angle labeled. (i)-(l): Perpendicular momentum vs.
parallel momentum, colors representing different particles. (m)-(p):
Perpendicular momentum change in the perpendicular axis.

To produce isolated attosecond electron bunches, we drove the interactions
with single-cycle Gaussian pulses in the PIC simulation, keeping the above
grazing incidence geometry but at higher normalized vector potential
$a_{0}=10$. This is to produce electron bunches at higher energy and more
separable from the background while keeping a comparable driving laser pulse
energy. It has been shown that CEP plays a role in the electron acceleration
process in LWFA [193] and in nanoplasma acceleration [194]. Fig. 4.12
demonstrates the generated single bunches using various carrier-envelope
phases. It is observed that changing the CEP can dramatically affect the
energy and shape of the electron bunch. It can also change the direction of
the generated bunch slightly, but not as effective as changing the preplasma
density profile. To better understand the effect of CEP on these electrons, we
select ten macroparticles in the high energy part in Fig. 4.12a-d and track
their trajectory from birth. The tracking results are shown in Fig. 4.12e-p in
a rotated axis perpendicular/parallel to the target surface. Electrons in the
CEP$=\;\pi/2\;or\;\pi$ case in Fig.4.12j,k experience a momentum loss in the
final acceleration stage. On the other side, electrons in the
CEP$=\;0\;or\;3\pi/2$ case in Fig.4.12i,l experience the momentum loss in
earlier stages but find the correct phase with respect to the reflected laser
field in later time to gain energy. This is in line with the difference in the
final energy of the accelerated electrons in Fig. 4.12a-d and e-h. Fig.
4.12e-h also provide the emission angle of the bunched attosecond electrons:
$\sim 68^{\circ}$ in CEP$=0\;or\;3\pi/2$ case and $\sim 58^{\circ}$ in
CEP$=\pi/2\;or\;\pi$ case. The emission angle is defined as the angle between
the electron bunch propagation trajectory and the target normal direction.
While the different CEP results in different emission angle, they are all
smaller than the incidence angle of $\sim 76^{\circ}$. It could be explained
by the magnetic field generated by the surface hot electrons. When a laser
pulse hit the target at grazing angle, it produces hot electrons traveling
along the target surface. A return current of cold electrons is therefore also
formed and travels in the other direction. The currents yield a magnetic field
where the emitted attosecond electrons feel the Lorentz force pointing away
from the target. Therefore, the emission angle of the attosecond electrons is
slightly smaller than the incident angle. A detailed investigation of such
magnetic field produced from surface electrons is presented by Becker et
al.[65] using a droplet target.

It is worth noting that electrons have negative initial perpendicular momentum
in Fig. 4.12m-o, moving towards the high-density target before the ejection.
Electrons which have positive initial perpendicular momentum and travel away
from the target cannot eject through the electromagnetic energy density gap
until pulled back towards the target, as is shown in Fig. 4.12p. However,
despite losing the initial momentum, these electrons are not slowed down once
they get ejected, as opposed to the momentum decrease of electrons in Fig.
4.12n,o at $x_{per}\sim 5\mu m$. Controlling CEP offers the ability to inject
electrons with various initial phases, which is important in the later stage
of phase change and acceleration.

#####  4.3.2.3 Focal-spot size

Figure 4.13: Spatial profiles of attosecond electron bunch generated from
different focal spot size, keeping the same pulse energy (a-c) or the same
$a_{0}$ (a,d). Cutoff energy in (a)-(d) is set at the high energy edge on each
spectra: 13 MeV, 9 MeV, 5 MeV and 18 MeV, respectively. (e) and (f) are the
particle tracking results of the thick and thin part of the bunch in (d).

Focusing the laser pulse to a larger focal spot can reduce the attosecond
electron bunch duration. Fig. 4.13a-c keeps the same pulse energy and varies
the focal beam waist, achieving a thinner bunch at $w_{0}=3\lambda$. The
numbers of electrons in these bunches are in the ratio 4:3:2. It is as
expected that larger focal spots yield shorter bunches. In our grazing
incidence setup using tight focus, the rays cover a range of angles
$\alpha\sim 1/(2\cdot f/\\#)$. When focusing the beam to a small focal spot
$w_{0}=\lambda$, this angle is around $20^{\circ}$. Given the incident angle
of the central ray is $76^{\circ}$, the rays cover from $56^{\circ}$ to
grazing. When the focal spot size is increased, this angle alpha decreases,
i.e., $\alpha^{{}^{\prime}}<\alpha$ and the rays cover from $\beta$
($\beta=76^{\circ}-\alpha^{{}^{\prime}}>56^{\circ}$) to grazing. Hence
overall, the rays are incident at larger angles. Note that large incident
angles are preferred to produce attosecond electron bunches, as is shown in
Fig. 4.10, and larger incident angles would lead to shorter electron bunch
duration, as is shown in Eq. 4.5. A sketch of the ray angles in the focal spot
geometry is attached below in Fig. 4.14.

Figure 4.14: Incidence angle increases with focal spot size. Blue and red rays
represent small and large focal spot.

If we keep $a_{0}=10$ and increase the focal beam waist, an extra thin bunch
is observed in addition to the thick bunch in Fig. 4.13d. Particle tracking
reveals that the thin bunch originates from earlier injection at
$x_{parallel}\sim 30\mu m$ in Fig. 4.13f, while the thick part (as well as the
bunch from the $w_{0}=\lambda$ case in Fig. 4.12e) originate from
$x_{parallel}>40\mu m$. Tracking the evolution of electron momentum along the
target surface direction in Fig.4.13e, thin-bunch electrons (in red) oscillate
within the laser field over a few cycles as it interacts with the dense
plasma. They then get accelerated without the momentum loss that the thick-
bunch electrons (in blue) experience, expected to reach even higher energy as
they propagate.

#####  4.3.2.4 Energy-wise bunch characteristics

Figure 4.15: Energy spectra and energy-wise bunch duration using $a_{0}=10$ in
(a)-(e), using $a_{0}=50$ in (f)-(j) and using $a_{0}=100$ in (k)-(o).

Duration of the isolated attosecond electron bunch and the electron
concentration have been measured against energy. Fig. 4.15a-e measure the
bunch in Fig. 4.12a ($a_{0}=10$, CEP=0, $w_{0}=\lambda$). Fig. 4.15f-o measure
the bunch using all the same parameters but $a_{0}=50$ and $a_{0}=100$. Energy
spectra in Fig. 4.15a,f,k again show the ”bump” feature, and the shape of
these most concentrated electrons are shown in Fig. 4.15d,i,n, respectively.
The low energy part in the $a_{0}=100$ case represents background electrons
that spatially overlap with the bunch at this time step, which goes away as
the bunch propagates. Movies that track the electron bunches can be found in
the Supplementary Video S1 in the associated publication [100]. Fig.4.15b-e,
g-j, l-o present the spatial profile of the electron bunch at different cutoff
energy, labeled respectively. Comparing them indicates that going to higher
$a_{0}$ leads to a thinner bunch at higher energy. Note that the optimal
scale-length for the energetic electron bunch decreases with $a_{0}$. We have
scanned multiple scale-lengths and present $L_{s}=0.25\lambda$ for the
$a_{0}=50$ case and $L_{s}=0.1\lambda$ for $a_{0}=100$. It is worth pointing
out that these electron bunches hardly see any radiation reaction effects even
when $a_{0}=100$. Classical radiation reaction effects become strong when the
classical radiation reaction parameter $R_{c}\sim a_{0}\chi>0.01$ and dominate
when $R_{c}>0.1$ [195], where $\chi$ is the relevant parameter for
supercritical fields. To consider radiation reaction effects when $a_{0}=100$,
one needs $\chi>0.01$. However, this is not available in our case even in the
frame of the electrons because the beam energy is not high enough. The fact
that the electron bunch is almost co-propagating with the reflected laser
field makes $\chi$ even smaller as $\chi\sim\gamma|E|(1-\beta\cos{\theta})$,
where $\theta$ is the angle between electron momentum and wave vector. In
fact, these attosecond electron bunches produced under this geometry should
see even weaker fields than the fields in the lab frame as $\theta\sim
10^{\circ}$ and $\cos{\theta}\sim 1$.

Another motivation to go to higher energy is to propagate the beam further as
attosecond bunches before being dispersed to longer duration. The spatial
dispersion between the fastest and slowest electron in a bunch is estimated
using Eq. 4.1, where $\gamma$ is the Lorentz factor.

$dL=0.5*\left(\frac{1}{\gamma_{\text{low}}^{2}}-\frac{1}{\gamma_{\text{high}}^{2}}\right)\cdot
L$ (4.1)

For few-MeV electron bunches as in Fig. 4.9e, the dispersion $dL/L$ is
calculated to be $4\times 10^{-3}$. The duration between the fastest and
slowest electrons would exceed attosecond levels (beyond 1fs) at 0.1mm away
from the source. We have also characterized the dispersion of the high-energy
electron bunches in Fig. 4.15. The estimated dispersion $dL/L$ is $4\times
10^{-4}$, $1\times 10^{-6}$ and $6\times 10^{-7}$ for the bunch in Fig.
4.15e,j,o, respectively. In other words, if the bunch in Fig. 4.150 propagates
for one meter in free space, the spacing between its front and back edge will
be 0.6 $\mu m$. The bunch can maintain duration below 1 fs until it propagates
0.45m. For direct measurements of the bunch duration, higher energy electrons
with less dispersion are clearly favored. Such electron beams will also
produce more optical transition radiation, which makes them easier to
characterize.

#####  4.3.2.5 Tilted laser pulses

Figure 4.16: Attosecond electron bunches generated using tilted pulses in
(a)-(c) vs. flat pulses in (d)- (f).

Apart from using single-cycle laser pulses, another approach to obtain
isolated attosecond electron bunches is using tilted laser pulses. The idea is
to spatially separate the electron bunches by rotating the wave vector within
a pulse. This concept of spatial-temporal couplings in ultrashort pulses was
first explored theoretically by Akturk et al.[196] and was experimented to
produce isolated attosecond pulses by Wheeler et al.[171]. We show that this
method also applies to the electron bunches as the tilted laser pulse
interacts with solid-density plasmas. Fig .4.16a presents the spatial profile
of energetic electrons ($>$10 MeV) produced by a tilted 3-cycle pulse, where a
perpendicular spatial chirp $d\omega/dx_{2}$ is included at the focus. It can
be achieved by introducing an angular dispersion in the laser beam, which
allows different frequency components of the laser pulse to disperse along the
perpendicular axis ($x_{2}$) and leads to the spatial chirp. As a result, the
laser cycle period ($T_{l}=c/\omega$) varies along the perpendicular axis.
This is equivalent to a perpendicularly varying spacing between laser field
cycles in the spatial domain, which leads to a rotating wave vector. An
intuitive illustration can be found in Fig. 1 in Wheeler et al.’s work[171].
The three cycles of the laser pulse go to focus at three different angles and
produce three separate bunches, as is shown in Fig .4.16a. Compared to the
central bunch generated during the second laser cycle, the first and third
bunch are shorter in length as the interacting position varies over time and
the phase of ”null” EM energy density lasts longest amid the second cycle.
Note that Fig .4.16a shows the electron bunches shortly after the laser pulse
interacts with the dense plasma, while Fig .4.16b presents the profile of
these electrons when they leave the simulation box. A movie that tracks the
propagation of these three bunches in every half-cycle time-step can be found
as Supplementary Video S2 online [100]. We also apply the tilted wavefront
concept to a 30fs laser pulse, which is more readily accessible in high-power
laser facilities. The resulted energetic electrons ($>$4 MeV) are presented in
Fig .4.16c with observable separation as well. For reference, fig .4.16d-f
show the energetic electron bunches from a flat wavefront without any spatial
chirp using a 3-cycle pulse and a 30fs pulse, respectively. In conclusion,
spatial-temporal couplings in ultrashort pulses can be inherited by the
electron bunches through interaction with dense plasmas. However, maintaining
these properties through propagation requires fine measures.

####  4.3.3 Discussion

Attosecond electron bunches are a unique product of ultra-short laser-solid
interactions at grazing incidence geometry: ”attosecond” comes from the ultra-
thin ”null” in the electromagnetic energy density shown in Fig. 4.9c while
”bunch” comes from the peak electron concentration shown in Fig. 4.10b.
Electrons oscillating in the fields at the correct phase can eject away from
the target through the ”null”, which is the cause of the ultra-short duration
of the bunched electrons. On the other side, the ”bunch” is formed by the
peaked electron density that these ejected electrons inherit[186], which
prefers a large laser incidence angle.

The electrons are accelerated in the reflected laser pulse after being ejected
from the target surface. This process is sensitive to the phase of the laser
pulse as well as the initial phase of the electrons. As is introduced in Sec.
2.3, the momentum change of an electron in a plane electromagnetic wave can be
described by: $p_{z}-p_{z0}=\gamma-\gamma_{0}$ and $p_{\perp}-p_{\perp
0}=\mathbf{a}$, where $p_{z}$ is the longitudinal momentum and $p_{\perp}$ is
the transverse momentum. Ejected electrons usually start with some non-zero
momentum, as is shown in the particle tracking, and obtains longitudinal
momentum by $\frac{a^{2}+2a\cdot p_{\perp
0}}{2\left(\gamma_{0}-p_{z0}\right)}$. There is a more generalized theory that
describes the energy gain in three-dimensional space with dephasing[197]. In
addition to the single electron model, the collective behavior of the plasma
is nontrivial. Depending on the direction of the incident electric field in a
single-cycle pulse, electrons move towards or away from the pulse to compress
or stretch it. The vector potential and the electric field of the reflected
pulse are[139]:

$A_{r}=2\pi\int_{0}^{t^{\prime}}d\tau
I_{y,e}(\tau)+I_{y,i}\xi_{r},\;\;I_{y}=-eN_{0}V_{y}$ (4.2)

$E_{r}=\frac{2\pi
N_{0}e\left(c\kappa_{y}+eA_{y}(\xi)\right)\mathcal{E}}{m_{e}^{2}c^{4}+\left(c\kappa_{y}+eA_{y}(\xi)\right)^{2}}+\frac{2\pi
N_{0}e}{c}V_{y}$ (4.3)

where $\xi$ is the initial phase of the laser pulse, $N_{0}$ is the plasma
density and $V_{y}=c\sin(\theta)$ shifts an oblique incidence angle $\theta$
to normal incidence frame[139]. Knowing the vector potential from Eq. 4.2 we
can determine the momentum gain $p_{z}$ in the modulated reflected fields:

$p_{z}=p_{z0}+\frac{a_{r}^{2}+2a_{r}\cdot p_{\perp
0}}{2\left(\sqrt{1+p_{z0}^{2}+p_{\perp
0}^{2}}-p_{z0}\right)},\;\;\;a_{r}=eA_{r}/m_{e}c^{2}$ (4.4)

This gives an estimation of the electron energy in the bunches. The electron
bunch duration can also be predicted by estimating the ”null” thickness in the
electromagnetic energy density. Setting $E^{2}+B^{2}=0$, the phase of the
reflected wave becomes $\xi_{r}=\int(1-a(2\pi\xi/\lambda)cot(\theta))d\xi$ ,
where $a(2\pi\xi/\lambda)$ is the vector potential of the incident wave. For
energy density $0^{+}$ and $0^{-}$,

$\Delta\xi_{r}\sim[a(2\pi\xi^{+}/\lambda)-a(2\pi\xi^{-}/\lambda)]cot(\theta)$
(4.5)

where $\xi^{+}$ and $\xi^{-}$ are the phases which correspond to density
$0^{+}$ and $0^{-}$. This agrees with the simulation and experimental results
that a larger incident angle is preferred. It also indicates that to get
shorter electron bunch duration, a slowly-varying vector potential is
preferred in phase space, or equivalently longer laser wavelength. Further
experimental studies are expected to verify this prediction.

It should also be pointed out that the carrier-envelope phase of single-cycle
pulses could affect the electric field strength directly. The intensity
profile of a Gaussian pulse is:

$I(t)=\epsilon
cn\cdot\frac{1}{T}\int_{t-\frac{T}{2}}^{t+\frac{T}{L}}|E(t^{\prime})|^{2}dt^{\prime}=\epsilon
cnE_{0}^{2}\cdot\frac{1}{T}\int_{t-\frac{T}{2}}^{t+\frac{T}{L}}e^{-(t^{\prime}/\tau)^{2}}\cos^{2}(\omega
t^{\prime}+\phi)dt^{\prime},$ (4.6)

In most cases when the pulse duration is much longer than one optical cycle,
$t>T=2\pi/w_{0}$, we can apply the slowly varying envelope approximation
(SVEA) and the intensity profile becomes: $I(t)=\epsilon
cnE_{0}^{2}e^{-t^{2}/\tau^{2}}\cdot\frac{1}{T}\int_{t-\frac{T}{2}}^{t+\frac{T}{L}}\cos^{2}(\omega
t^{\prime}+\phi)dt^{\prime}=I_{0}e^{-t^{2}/\tau^{2}}$, where
$I_{0}=\frac{1}{2}\epsilon cnE_{0}^{2}$ is the peak intensity. However, SVEA
is no longer valid when the pulse duration is comparable to one optical cycle.
In our case with a single cycle pulse, we have to integrate both the CEP term
and the exponential term in Eq. 4.6. The peak intensity scales with the
electric field amplitude as $I_{0}\sim 0.441\epsilon cnE_{0}^{2}$ for
CEP$=0\;or\;\pi$ and $I_{0}\sim 0.454\epsilon cnE_{0}^{2}$ for
CEP$=\pi/2\;or\;3\pi/2$, which are different from the commonly used formula
$I_{0}\sim 0.5\epsilon cnE_{0}^{2}$ under SVEA. Thus for a fixed laser
intensity, the carrier electric field is slightly stronger in the
CEP$=0\;or\;\pi$ cases. Note that this field is not exactly the electric field
that drives the plasmas: the absolute phase of the carrier wave slides
underneath the envelope as the beam focuses, especially in single-cycle
pulses. It is due to the Gouy phase shift generated by isodiffracting ultra-
braodband pulses through the beam waist[198]. Besides the vacuum focus,
plasmas act as another focusing lens as the ponderomotive force pushes the
electrons outwards. The phase change due to the ”plasma lens” is dependent on
its thickness and refractive index, which are essentially determined by the
preplasma density profile.

Experimental measurements of ultra-short electron bunch duration may be
accessible by measuring characteristics of coherent optical transition
radiation (COTR) produced by the electrons. When an electron bunch is incident
on an interface between two media with different refractive indices, coherent
electromagnetic fields are radiated and the electron bunch duration can be
inferred. For example, Lundh et al.[199] measured the COTR from an electron
beam generated in a colliding pulse injection regime, and compare the COTR
spectrum to analytic Gaussian COTR spectra for different bunch duration to
deduce the bunch duration is $\sim$few fs. In addition, Sears et al.[200]
measured the duration of electrons from inverse free-electron-laser process to
be $\sim 410$ attoseconds. While using overdense plasmas in a grazing
incidence setup, the electron bunch duration can potentially be shorter than a
tenth of a femtosecond. Since shorter electron bunches produce transition
radiation at shorter wavelengths, to measure the duration of such electron
bunches requires diagnosing the COTR further in the x-ray regime. These
attosecond electron bunches can potentially be applied to probe the temporal
evolution of dynamic systems such as magnetic field formation[201] and
attosecond electron microscopy and diffraction[202]. It would also be worth
trying to drive the interactions with laser pulses that carry orbital angular
momentum[203, 204] to further manipulate the electron bunch characteristics.

To sum up, we show the experimental electron energy spectra from driving 30
fs, 800 nm laser pulses at grazing incidence onto a 6 mm thick glass target.
The experimental energy spectra match the spectra of attosecond electron
bunches observed in simulations. The duration of the bunches is measured to be
$\sim$tenth of a micron ($\sim$100 attoseconds) in simulations. Direct
experimental measurement of the bunch duration was not performed, although it
can potentially be achieved by measuring characteristics of COTR produced by
the electrons into the x-ray regime. We find grazing incidence geometry is
necessary to produce attosecond electron bunches in simulations and
corresponding spectral features in experiments. We show that the generation of
energetic attosecond electron bunches favors a larger incident angle, higher
pulse energy, larger focal spot size, and moderately sharp preplasma density
profile. We obtain isolated attosecond electron bunches using single-cycle
pulses. Controlling CEP offers the ability to inject electrons with various
initial phases and to adjust the energy and shape of the bunch. Due to the
Guoy phase shift, CEP is changing as the single-cycle pulses focus through the
plasmas. Preplasma density profile governs the propagation direction of the
bunch and fine-tuning of the direction is accessible by tuning CEP. Higher
$a_{0}$ and larger focal spot size can result in an even shorter bunch
duration with less dispersion after propagation. When operating with much
higher pulse energy, a sharper preplasma profile is preferred to produce a
cleaner bunch. Using tilted laser pulses can pass angular properties to
electron bunches and spatially separate them. With a simplified analytic
model, we predict the momentum gain of electrons and we predict that shorter
electron bunch duration scales with larger incident angle and longer laser
wavelength.

###  4.4 Characteristic x-ray emission at different laser wavelengths

####  4.4.1 Introduction

When a short-pulse laser interacts with overdense plasmas from solid targets,
it accelerates electrons and generates radiation in the x-ray regime. For
solid targets like plasma mirrors having short density scale lengths, High-
order Harmonic Generation (HHG) is the most important mechanism for radiation
generation. An experiment on HHG has been discussed in Sec. 4.2. For metallic
targets with larger atomic numbers, other mechanisms are playing significant
roles in producing radiation, such as characteristic x-ray emission and
bremsstrahlung radiation.

Bremsstrahlung radiation is a common form of broad-band radiation when
collisional processes are involved [205]. When a high-energy charged particle
collides with another particle, it decelerates and loses energy via radiation
emission due to relativistic effects. As introduced in the theoretical
background in Sec. 2.4, a significant feature of physical processes associated
with overdense plasmas is the abundance of collisions. In short-pulse laser-
solid interactions where electrons are accelerated during the
$\sim$femtosecond pulse duration but ions are almost immobile, bremsstrahlung
radiation mostly results from the scattering of electrons from ions. The total
radiated power per unit area is given by:

$P_{Br}=1.54\times
10^{-38}g_{ff}Z^{2}n_{i}n_{e}\left(\frac{T_{e}}{eV}\right)^{1/2}\frac{W}{m^{3}}$
(4.7)

where Z is the number of electrons in the ion, $g_{ff}$ the gaunt factor for
quantum corrections, $T_{e}$ is the electron temperature, and $n_{i}$ and
$n_{e}$ are the ion density and electron density, respectively. From Eq. 4.7,
the energy radiated via bremsstrahlung increases with the energy of the
electron, and bremsstrahlung radiation becomes significant for relativistic
particles.

Radiation can also be generated through inner-shell ionization by electron
impact. When the laser field ionizes electrons from atoms, the emitted
electrons can collide with other atoms and ionize them to release more
electrons to trigger the avalanche. Details about the collisional ionization
mechanism have been introduced in Sec. 2.2. Although most of the electrons to
be ionized are in the outer shells of the atoms with lower binding energy,
electrons in the inner shells can also be ionized, but at a much smaller
probability. This process is known as inner-shell ionization. Note that the
ionization leaves a vacancy in the inner-shell in the transition state, and an
electron in the outer shell has to fall back to the inner shell to fill the
vacancy. Due to the conservation of energy, the energy difference between the
electron shells must be compensated, most probably through emitting radiation.
Since the photon energy of the radiation is exactly the energy difference
between the electron shells and purely dependent on the atom, it is called the
characteristic x-ray emission. Under rare circumstances, the energy difference
can be compensated by releasing an outer-shell electron via the Auger effect.
The probability of inner-shell ionization by electron impact in different
atoms has been studied extensively and is reviewed in Ref. [206, 207].

X-rays produced via this mechanism have unique features compared to those from
other mechanisms in short-pulse laser-solid interactions. The radiation
spectrum has distinct peaks at the characteristic emission lines of the atom,
unlike the broad-band spectrum from bremsstrahlung radiation. The peak energy
of the produced characteristic x-rays is independent of the driving laser
wavelength, unlike the harmonic dependence in HHG. Furthermore, characteristic
x-rays can be produced without strict vacuum condition [208, 209, 210], which
is usually required in HHG. Because of these remarkable properties, producing
characteristic x-ray emission using high-intensity lasers and solid targets
has drawn increasing attention and has been demonstrated in many successful
experiments [211, 212, 213, 214, 215, 216, 217, 218]. The source size,
directional property, spatial coherence, and laser to x-ray conversion
efficiency of the produced x-rays have been characterized and reported in Ref.
[219, 220, 221, 222]. Owing to its high spatial coherence, an useful
application of such x-ray sources is phase-contrast imaging for biological
objects [223, 224, 225]. To better understand the characteristic x-ray
emission from a laser-plasma point of view, the roles of some crucial laser-
plasma conditions have been investigated, such as the laser wavelength [226],
the laser pulse contrast ratio [227], and the hot electron refluxing [228].

In this section, we will present results of characteristic x-ray emission and
bremsstrahlung radiation when short-pulse lasers with different wavelengths
and pulse energies interact with the overdense plasma of various preplasma
profiles from a molybdenum target. The study is performed both experimentally
with hundreds of thousands of laser shots, and computationally with PIC
simulations scanning over the 4-dimensional parameter space consisting of
laser wavelength, pulse energy, preplasma profile, and x-ray emission
properties.

####  4.4.2 Experimental setup

The experiment was performed in the Lambda-cubed at CUOS at the University of
Michigan. Various laser-plasma conditions were used in the experiment. Laser
pulses at the near-infrared wavelength (0.8 $\mu m$, 6.1 mJ, 40 fs) and the
mid-infrared wavelengths (1.3 $\mu m$, 0.4 mJ, 80 fs, and 2 $\mu m$, 0.2 mJ,
67 fs) were used. While the Lambda-cubed outputs 0.8 $\mu m$ laser pulses, the
1.3 $\mu m$ and 2 $\mu m$ pulses are generated using an OPA, as is introduced
in Sec. 3.1. The pulse contrast ratio to the ASE is $>10^{8}$. A tunable
prepulse with intensity about two orders of magnitude lower than the main
pulse intensity was introduced using a 2 $\mu m$ thick nitrocellulose
pellicle. Details about the prepulse delay stage setup have been discussed in
Sec. 3.1 and shown in Fig. 3.3. The tunability of the prepulse covers 0 - 187
ps, which corresponds to a preplasma density scale length of 0.1 - 5.5
$\lambda$ from hydrodynamic simulations [66]. Various laser pulse energies
were also applied by adjusting the pump lasers.

Figure 4.17: Experimental setup for characteristic x-ray emission measurement.
The laser pulses are focused onto the molybdenum target at an incident angle
of $55^{\circ}$. The distance between the OAP and focal spot on the target is
75 mm. The CdTe detector is located at 530 mm away from the target, at
$65^{\circ}$ from the target normal.

The laser pulses were directed into a vacuum chamber at a pressure of tens of
mTorr, as is shown in Fig. 4.17. More information on the experimental chamber
operation can be found in Sec. 3.2. The 0.8 $\mu m$, 1.3 $\mu m$, and 2 $\mu
m$ pulses were focused by an f/1.3 OAP to 2.1 $\mu m$ FWHM, 4.3 $\mu m$ FWHM,
and 5 $\mu m$ FWHM, respectively. The peak laser intensity of the 2 $\mu m$
pulses and the 1.3 $\mu m$ pulses are on the order of $10^{16}W\cdot cm^{-2}$,
while that of the 0.8 $\mu m$ pulses are on the order of $10^{18}W\cdot
cm^{-2}$. Since the pulse energy is a control parameter in this study, a
specific laser intensity or $a_{0}$ will be reported associated with each
measurement. A 2 $\mu m$ thick nitrocellulose pellicle is inserted in the beam
path to prevent the OAP from the ablated debris from the target. The
molybdenum target sits on a rotary stage for high repetition operation.
Details on the target preparation and alignment have been described in Sec.
3.2.

A portion of the produced x-rays is measured, as is indicated in the blue path
in Fig. 4.17. A pair of magnet is placed next to the target to deflect high-
energy electrons from the x-ray path. Beryllium windows are used at the
chamber exit and at the detector entrance for high x-ray transmission
efficiency. A lead pinhole is placed in front of the detector to avoid pile-up
in the detection, and the size of the pinhole varies from $1.3\times
10^{4}\;\mu m^{2}$ or $6.7\times 10^{4}\;\mu m^{2}$. To align the components
along the x-ray path, a helium-neon laser is directed from the detector
backward into the chamber to overlap with the focal spot on the target
surface. The x-rays are captured by an Amptek XR-100 Cadmium Telluride (CdTe)
single-photon detector. The detector is calibrated using an americium-241
(am-241) source, as is discussed in Sec. 3.3.

####  4.4.3 Experimental results

Figure 4.18: An example of the measured x-ray spectrum from a molybdenum
target. The spectrum was integrated over 10,000 laser shots.

Fig. 4.18 shows the measured spectrum of the femtosecond x-ray source
generated using 2 $\mu m$, 0.2 mJ, 67 fs, horizontally polarized laser pulses
interacting with a molybdenum target. A prepulse arriving at 20 ps before the
main pulse was used in this measurement. The flux of the $K_{\alpha}$ and the
$K_{\beta}$ emission are $6.4\times 10^{6}$ photons per $2\pi$ steradian per
pulse and $1.8\times 10^{6}$ photons per $2\pi$ steradian per pulse,
respectively. The x-ray flux of the bremsstrahlung radiation is $6.2\times
10^{7}$ photons per $2\pi$ steradian per pulse.

The x-ray energy spectrum also tells the characteristic hot electron
temperature. When short-pulse lasers interact with solids, some of the
electrons are heated via the collisionless absorption mechanisms discussed in
Sec. 2.4 to energies much higher than the initial background plasma
temperature [103]. Zulick et al. [229] found an empirical relation to estimate
the hot electron temperature from the bremsstrahlung spectra using Monte Carlo
N-Particle eXtended (MCNPX) simulations:

$T_{h}=0.73\times T_{b}^{1.09}$ (4.8)

where $T_{b}$ is the bremsstrahlung temperature fitted from the slope of the
logarithm bremsstrahlung spectrum in Fig. 4.18. The electron temperature
associated with Fig. 4.18 is calculated to be 41.7 keV.

Figure 4.19: Characteristic hot electron temperature (a), $K_{\alpha}$
emission flux per pulse energy (b), bremsstrahlung emission flux per pulse
energy (c), and $K_{\alpha}$ to bremsstrahlung ratio (d) vs. laser pulse
energy. The prepulse delay is kept at 20 ps for the 2 $\mu m$ pulses (red),
the 1.3 $\mu m$ pulses (green), and the 0.8 $\mu m$ cases (blue).

The experiment was performed at various laser-plasma conditions for a
parametric study. Fig. 4.19 investigates the x-ray properties at various laser
pulse energies. The $K_{\alpha}$ emission and the bremsstrahlung emission are
normalized to the laser pulse energy, reported in the unit of photon per
steradian per joule per pulse. The results from the 1.3 $\mu m$ pulses (green)
are substantially different from the others. This is possibly due to a
substantially thicker pinhole used for the x-ray measurements at 1.3 $\mu m$
cases, making the measurements much more sensitive to the pinhole placement if
the x-rays do not exactly incident from the normal direction. Results from the
2 $\mu m$ pulses and the 0.8 $\mu m$ pulses are in line with the simulation
results to be introduced in Fig. 4.22 and Fig. 4.23. The $K_{\alpha}$ flux per
pulse energy of the 2 $\mu m$ pulses (red) and the 0.8 $\mu m$ pulses (blue)
behave non-monotonically vs. $a_{0}$, as is shown in Fig. 4.19b, agreeing with
the simulation results in Fig. 4.22 where the normalized $K_{\alpha}$ and
bremsstrahlung flux peak at some $a_{0}$. The measured normalized
bremsstrahlung flux of the 2 $\mu m$ pulses (red) has a peak in Fig. 4.19c,
but that of the 0.8 $\mu m$ pulses (blue) shows a monotonic relation as laser
pulse energy increases. However, it has to be pointed out that the optimal
laser pulse energy moves to the left from Fig. 4.19b to Fig. 4.19c.
Analogously, the simulation results in Fig. 4.22 show that the optimal $a_{0}$
for the normalized $K_{\alpha}$ emission flux is 1 while the optimal $a_{0}$
for the normalized bremsstrahlung emission flux is 0.5. Therefore, it is
possible that the blue curve in Fig. 4.19c has a peak to the left, and more
experiments at smaller laser pulse energies are needed to draw any conclusion
on the monotonicity of the normalized bremsstrahlung flux vs. pulse energy.

Figure 4.20: Characteristic hot electron temperature (a), $K_{\alpha}$
emission flux per pulse energy (b), bremsstrahlung emission flux per pulse
energy (c), and $K_{\alpha}$ to bremsstrahlung ratio (d) vs. prepulse delay.
The pulse energy is 0.2 mJ, 0.4 mJ, and 0.6 mJ in the 2 $\mu m$ cases (red),
1.3 $\mu m$ cases (green) and 0.8 $\mu m$ cases (blue). The prepulse delay
scans over 0 - 187 ps, which corresponds to a preplasma density scale length
of 0.1 - 5.5 $\lambda$.

Fig. 4.20 investigates the preplasma effect while the pulse energy was kept
the same when tuning the prepulse. Results from 0.8 $\mu m$ pulses, the 1.3
$\mu m$ pulses, and the 2 $\mu m$ pulses are shown in blue, green, and red,
respectively. The results from the 1.3 $\mu m$ pulses are somewhat anomalous,
as is explained in the previous paragraph, while the trend of the 0.8 $\mu m$
pulses and the 2 $\mu m$ pulses agree with the simulation results to be shown
in the next section in Fig. 4.22 and Fig. 4.23. This is likely due to
variability in beam quality and focal spot structure among the three
wavelengths. It is observed that the measurement without a prepulse (0 ps) is
distinct from the measurements with a prepulse in all cases. An optimal
prepulse delay is suggested, and its value should be smaller than the shortest
prepulse we used at 20 ps ($L_{s}=\sim 0.5\lambda$). As the prepulse delay
passes the optimal value and extends to 187 ps, the normalized$K_{\alpha}$
flux drops more than seven times smaller from its peak value while the
normalized bremsstrahlung flux drops only $\sim$three-times.

####  4.4.4 PIC simulations

PIC simulations were performed using the OSIRIS[122, 123] 4.4.4 framework in
2D3V Cartesian geometry. Since the laser wavelength is a tuning parameter in
this work, which affects many other parameters in the simulation setup like
time-space resolution and critical density, these setup parameters will be
described universally rather than in real units. An example of the OSIRIS
input deck and the bash programming for parameter scan are appended in App.
6.0.2 and App. 6.0.2.

For the high-resolution simulations, there are 100 macroparticles per cell and
the grid size is $\lambda/64\times\lambda/64$. For the simulations for
parameter scan, there are 49 macroparticles per cell and the grid size is
$\lambda/32\times\lambda/32$. The convergence of the simulation is checked
using up to $\lambda/64\times\lambda/64$ grid size. The time step is chosen to
resolve both the Courant criterion (see Sec. 3.4) and the plasma density. The
laser pulse is assumed to be Gaussian in both the longitudinal and transverse
direction with a FWHM pulse duration of $\tau=50fs$. The laser pulse is
continuously launched from the wall and focused down to beam waist
$w_{0}=1.7\lambda$ ($FWHM=2\lambda$). The simulation box size is kept at
$40\mu m\times 40\mu m$ throughout the parameter scan. The simulations ran
until the interesting electrons left the simulation box, and the simulation
time varies as the laser wavelength changes. The initial plasma density
profile is described by a uniform solid density region plus an exponential
preplasma. The solid density is kept the same at the molybdenum density:
$1.46\times 10^{23}cm^{-3}$, while the critical density varies as the laser
wavelength changes according to Eq. 2.35.

Information of the electrons located within the solid density region are saved
from PIC simulations at every 15 fs, including their position and kinetic
energy. Further analysis of such information yields an estimation of the hot
electron temperature, the $K_{\alpha}$ emission, and the bremsstrahlung
emission.

The hot electron temperature is fitted from the energy spectra of the
electrons located within the solid density region, assuming a Boltzmann
distribution starting from a cut-off energy $E_{os}$ [230]:

$\frac{dN}{dE}\sim exp(-E/T_{h})\mbox{ if
}E>E_{os}=m_{e}c^{2}[\sqrt{1+a_{0}^{2}/2}-1]$ (4.9)

where the cut-off energy $E_{os}$ refers to the mean oscillation energy of the
electrons in the laser field. Electrons with energies higher than $E_{os}$ are
regarded as hot electrons, while electrons with lower energies are seen to be
cold and will be included in the curve fitting. Even though Eq. 4.9 is an
exponential function, a linear curve fitting is applied by taking the
logarithm of the spectra to reduce numerical errors. Note that the
mathematical description of the electron energy spectra remains an open
question. Another commonly used relation is
$\frac{dN}{dE}\sim\sqrt{E}exp(-E/T_{h})$, which assumes a Maxwellian
distribution on the electron velocity rather than the electron energy.
Liseykina et al. [230] have an in-depth investigation on this topic.

Figure 4.21: Hot electron temperature (shown in color) from various laser-
plasma conditions. Each dot represents a PIC simulation where the laser
wavelength $\lambda$ lies in between 0.4 $\mu m$ and 2 $\mu m$, the peak
normalized vector potential $a_{0}$ lies in between 0.1 and 2, and the
preplasma scale length $L_{s}$ lies in between 0.1 $\lambda$ and 5.5
$\lambda$.

The $K_{\alpha}$ emission (number of photons produced) is estimated given time
t, spatial position $\vec{r}$ of the electron, and electron energy denoted by
the Lorentz factor $\gamma$, assuming ions remain at rest:

$Yield=\int dt\int d^{3}\vec{r}\int d\gamma
f_{e}(\vec{r},t,\gamma)n_{i}(\vec{r},t)\sigma_{\mathrm{K}\alpha}(\gamma)\sqrt{1-1/\gamma^{2}}\cdot
c$ (4.10)

where $f_{e}(\vec{r},t,\gamma)$ is the electron distribution and
$n_{i}(\vec{r},t)$ is the ion distribution and $\sqrt{1-1/\gamma^{2}}c$ is the
velocity expressed in terms of $\gamma$. The spatial integral in Eq. 4.10 is
estimated by summing over all electrons in the solid-density region and
assuming the ion and electron density is approximately uniform. The temporal
integral in Eq. 4.10 is estimated by summing over the entire simulation time.
Instead of assuming $v\simeq c$, the particle velocity is calculated as
$\sqrt{1-1/\gamma^{2}}\cdot c$ to take into account contributions from non-
relativistic electrons. The cross-section for $K_{\alpha}$ emission
$\sigma_{\mathrm{K}\alpha}$ of an electron is proportional to cross section
for k-shell ionization $\sigma_{\mathrm{K}}$ [207]:
$\sigma_{\mathrm{K}\alpha}\propto\sigma_{\mathrm{K}}$. The analytical formula
to calculate the cross-section for inner-shell ionization by electron impact
is provided by Bote et al. in Ref. [206] given the kinetic energy of the
electron. Therefore, the $K_{\alpha}$ emission is estimated using Eq. 4.10 and
calculated regarding the laser pulse energy, then normalized to the largest
$K_{\alpha}$ emission in the dataset, as is shown in Fig. 4.22.

Figure 4.22: Normalized $K_{\alpha}$ emission per pulse energy shown in color
from various laser-plasma conditions. Each dot represents a PIC simulation
where the laser wavelength $\lambda$ lies in between 0.4 $\mu m$ and 2 $\mu
m$, the peak normalized vector potential $a_{0}$ lies in between 0.1 and 2,
and the preplasma scale length $L_{s}$ lies in between 0.1 $\lambda$ and 5.5
$\lambda$.

The radiation power from bremsstrahlung is calculated using Eq. 4.7 given the
fitted electron temperature in Fig. 4.23. The bremsstrahlung radiation per
pulse energy is calculated in $W/(m^{3}\cdot mJ)$ and then normalized to the
largest bremsstrahlung radiation in the dataset. The code to estimate
$K_{\alpha}$ emission, bremsstrahlung emission, and electron temperature from
PIC simulations are appended in App. 6.0.2.

Figure 4.23: Normalized bremsstrahlung radiation per pulse energy shown in
color from various laser-plasma conditions. Each dot represents a PIC
simulation where the laser wavelength $\lambda$ lies in between 0.4 $\mu m$
and 2 $\mu m$, the peak normalized vector potential $a_{0}$ lies in between
0.1 and 2, and the preplasma scale length $L_{s}$ lies in between 0.1
$\lambda$ and 5.5 $\lambda$.

The hot electron temperature, the normalized $K_{\alpha}$ emission (per pulse
energy), and the normalized bremsstrahlung emission (per pulse energy) are
plotted in Fig. 4.21, Fig. 4.22, and Fig. 4.23, with a multi-dimensional
parameter scan across the laser wavelength, the peak normalized vector
potential, and the preplasma density scale length. From Fig. 4.21, the hot
electron temperature clearly favors larger $a_{0}$ but has a more complicated
dependence on the other two parameters. The optimal normalized scale length
$L_{s}/\lambda$ increases when the wavelength decreases, and the optimal
absolute scale length $L_{s}$ lies in between 2 $\mu m$ and 3.25 $\mu m$
across the wavelength range. The highest electron temperature in Fig. 4.21
occurs at $\lambda=1.3\;\mu m$ and $L_{s}=3.25\;\mu m$. Although more data
points from more accurate PIC simulations are needed to draw a conclusion on
the exact optimal value, there does seem to be an optimal wavelength rather
than a monotonic dependence. It can be explained by the fact that the laser
pulse propagates deeper at a shorter laser wavelength (Eq. 2.35) while the
ponderomotive force favors a longer wavelength (Eq. 2.31).

The x-ray emission flux are normalized to the energy of the driving laser
pulses $E\sim a_{0}^{2}$. From Fig. 4.22 and Fig. 4.23, the laser wavelength
is playing a much more significant role in radiation generation (both
$K_{\alpha}$ and bremsstrahlung) than the preplasma density scale length and
the $a_{0}$. As wavelength increases, the critical density decreases as
$n_{cr}\propto\lambda^{-2}$ and the critical surface moves further away from
the solid density surface, making it less efficient to couple the laser energy
into the solid target. Therefore, the highest normalized emission flux occurs
when the laser wavelength is shortest at 0.4 $\mu m$. The normalized
$K_{\alpha}$ flux peaks at $a_{0}=0.5$ and $L_{s}=0.2\sim 0.3\lambda$. This
finding agrees with the observation in the experimental results in Fig. 4.20b
where the optimal prepulse is $<0.5\;\lambda$. It is also in line with the 0.8
$\mu m$ and 2 $\mu m$ experimental results in Fig. 4.19b where an optimal
pulse energy exists. The normalized bremsstrahlung radiation peaks at
$a_{0}=0.1$, which agrees with the experimental results in Fig. 4.19c where
the bremsstrahlung radiation per pulse energy monotonically decreases with
pulse energy. Similar to the experimental results in Fig. 4.20c, the
normalized bremsstrahlung in Fig. 4.23 favors a short plasma density scale
length.

To proceed further with the parametric study, it is worth looking at
parameters that are not directly controlled by the laser pulse or the plasma
profile. The similarity parameter $S=\frac{n_{e}/n_{cr}}{a_{0}}$ is proposed
by Gordienko et al. [170] for relativistic electrons in laser-plasma
accelerators, where $a_{0}$ is the peak normalized vector potential, $n_{e}$
is the electron density and $n_{cr}$ is the critical density. Tab. 4.3
summarizes three high-resolution PIC simulations performed with various laser
wavelength, peak normalized vector potential $a_{0}$, and normalized scale
length $L_{s}/\lambda$. The similarity parameter is kept the same among these
simulations by carefully choosing the tuning parameters, where the initial
electron density is constrained by using the same absolute scale length
$L_{s}$ = 0.8 $\mu m$.

Wavelength [$\mu m$] | 0.8 | 1.3 | 2
---|---|---|---
$n_{cr}\;[cm^{-3}]$ | 1.74$\times 10^{21}$ | 6.60$\times 10^{20}$ | 2.79$\times 10^{20}$
$a_{0}$ | 0.4 | 1.06 | 2.5
Scale length [$\lambda$] | 1 | 0.62 | 0.4
Scale length [$\mu m$] | 0.8 | 0.8 | 0.8
S | 1 | 1 | 1
Normalized $K_{\alpha}$ emission $[\\#/mJ]\cdot constant$ | 1 | 0.2 | 0.08
Normalized bremsstrahlung $[W/(m^{3}\cdot mJ)]\cdot constant$ | 1 | 0.04 | 0.002
$T_{h}$ [keV] | 42.7 | 237 | 514
Table 4.3: Comparison of $K_{\alpha}$ emission, bremsstrahlung radiation, and
hot electron temperature from three high-resolution PIC simulations at a fixed
similarity parameter. The $K_{\alpha}$ emission per pulse energy and
bremsstrahlung radiation per pulse energy are normalized to the largest values
among the three, and their dimensions are listed in the seventh and eighth
row.

It is observed that the bremsstrahlung emission and electron temperature vary
drastically among the three cases. The vast variation agrees with the trend
observed in Fig. 4.23 and Fig. 4.21 where the bremsstrahlung emission is
dominant by a short laser wavelength and the electron temperature favors a
high $a_{0}$. The variation of the normalized $K_{\alpha}$ emission among the
three cases is about an order of magnitude lower than that of the normalized
bremsstrahlung radiation. Since the similarity parameter S scales the dynamics
of relativistic electrons and given the fact that S works better for the
$K_{\alpha}$ emission, it suggests that the most energetic (relativistic)
electrons make substantial contributions to the $K_{\alpha}$ emission in our
parameter space. This agrees with the fact that making an assumption on the
particle velocity $v\simeq c$ in Eq. 4.10 only increases the $K_{\alpha}$
emission by $\sim 20\%$.

####  4.4.5 Conclusion

In summary, a parametric study is reported for the interdependent relationship
between the tuning parameters (laser pulse wavelength, laser pulse energy, and
preplasma density gradient) and the critical features in characteristic x-ray
emission ($K_{\alpha}$ emission per pulse energy, bremsstrahlung emission per
pulse energy, and electron temperature). Overall, the laser wavelength is the
more dominant factor for the normalized $K_{\alpha}$ emission than the
preplasma density gradient and the pulse energy are. The wavelength dependence
is monotonic for the normalized $K_{\alpha}$ emission and for the normalized
bremsstrahlung emission, and both favor short laser wavelength. It is because
the critical surface is located closer to the solid density surface at short
laser wavelength, making it easier for the energetic electrons to reach the
atoms in the solid target. For the hot electron temperature, the wavelength
dependence is no longer monotonic since the laser pulse propagates deeper at a
shorter laser wavelength while the ponderomotive force favors a longer
wavelength. Both experimental results and simulation results indicate some
optimal preplasma profiles for the normalized $K_{\alpha}$ emission and for
the normalized bremsstrahlung emission, respectively, and this optimal
preplasma profile has a sharp density gradient ($L_{s}<0.5\lambda$). The
dependence on $a_{0}$ is monotonic for electron temperature but not monotonic
for the normalized x-ray emission, while the optimal $a_{0}$ is larger for the
normalized $K_{\alpha}$ emission than for the normalized bremsstrahlung
emission. Simulations using various control parameters but the same similarity
parameter is performed to reveal the contribution of relativistic electrons to
$K_{\alpha}$ emission.

Note that our results are contradictory to the results reported in [226]
regarding the wavelength dependence on $K_{\alpha}$ emission. However, there
are a few differences in approaching the problem. First, we report on the
normalized $K_{\alpha}$ flux ($K_{\alpha}$ emission flux per pulse energy)
rather than the $K_{\alpha}$ emission flux reported in their work. Moreover,
the calculation in [226] does not include the collective behavior of plasmas
so it is unlikely to be correct considering laser absorption and wave-particle
interactions.

##  Chapter V Applications of Statistical Methods at High Repetition Rates

###  5.1 Focus optimization at relativistic intensity using genetic
algorithms333This section co-authored with Easter, J. H., Krushelnick, K.,
Mathis, M., Dong, J., Thomas, A. G. R., and Nees, J. (2018): Focus
optimization at relativistic intensity with high numerical aperture and
adaptive optics. Optics Communications, 421, 79-82.

####  5.1.1 Introduction

To achieve relativistic intensities in the ultrafast regime using millijoule
laser pulse energies, ultrashort pulses are typically focused with high
numerical aperture optics, such as off-axis parabolic mirror (OAP)s. However,
wavefront aberrations resulting from these optics can be significant. In
addition, when approaching the diffraction limit, aberrations caused by laser
system distortions, thermal lensing, and self-phase modulation may also be
important. Due to the difficulty of direct measurements of the laser focus at
high intensity, wavefront correction and focus optimization are of significant
importance. The analogous problem of correcting wavefront distortions is well
known in the field of astronomy. To correct for atmospheric aberrations, high-
resolution telescopes utilize adaptive optics to reshape incoming wave fronts
[124]. Relying on reference beams to measure atmospheric distortions, adaptive
optics can be programmed to compensate for these and other distortions and
produce much higher quality images.

A typical adaptive optic is a deformable mirror (DM) with an array of
controllable actuators to deform the mirror surface. Such a mirror may also be
used to optimize the focal quality of a high-NA focusing optic in a high-
intensity, high-repetition-rate laser system. DMs together with genetic
algorithms (GAs) have already been utilized in several high-power laser
facilities. Traditional ways of determining the mirror shape include direct
measurement of the wavefront [231], optimization of Second Harmonic Generation
(SHG) in a nonlinear crystal [232] and in-situ optimization of surface SHG
[233]. However, these methods all require attenuation of the beam to avoid
damage to detection optics. An alternative method uses the Second Harmonic
(SH) signal generated in a plasma by the full power of the focused laser light
to provide feedback for the optimization process. It allows the DM to correct
for additional wavefront distortions that may not occur while the beam is
attenuated (e.g. thermal distortion of optics, thermal lensing, and self-
focusing). It also avoids possible wavefront aberrations introduced by the
attenuation optics. In addition, the optimization is no longer constrained to
a specific focal plane, which is a shared disadvantage of the three techniques
mentioned above. Since near-vacuum is maintained in between optimization and
experiment, shifting of optics due to pressure change in conventional methods
no longer exists. The use of optimization techniques to enhance other
phenomena generated from relativistic laser–matter interactions has also been
demonstrated recently at CUOS, such as laser filamentation [74], laser-
wakefield acceleration [73, 234, 81], THz generation [75], and high-order
harmonic generation [99].

Harmonic generation occurs in gaseous media when high-intensity laser ionizes
atoms and plasma is formed. In addition to the possibility of coherent X-ray
generation and nonlinear Thomson scattering, frequency doubling may also be
observed [235, 236, 237]. SHG in a plasma formed by a high-NA focus is due to
the formation of an electron density gradient by the ponderomotive force. The
electron gradient breaks the isotropy of the gas that would normally prohibit
the generation of even order harmonics. The second order polarization in an
underdense plasma is given by [236]:

$P(2\omega)=\chi\left[\frac{1}{2}\nabla
E^{2}+\frac{2}{\epsilon_{p}}E\left(E\cdot\nabla\ln n_{e}\right)\right]$ (5.1)

where E is the electric field, $n_{e}$ is the electron density, $\chi$ is the
susceptibility and $\epsilon_{p}$ is the relative permittivity. The first term
has zero curl and cannot radiate. Therefore, a better-corrected and smaller
focus will produce a stronger SH signal both by producing steeper electron
density gradients and by delivering higher intensity. Generation of SH signal
is a characteristic of laser–plasma interactions at high intensities using
both overdense and underdense plasma targets [238].

In this paper, we demonstrate a two-fold improvement in the focal quality at
f/1.4 using a genetic algorithm. Experiments have been performed with laser
pulses of both 800 nm and 2 $\mu$m wavelength, at $\sim$ mJ energy and with
pulse duration of a few tens of femtoseconds. The relationship between the
fundamental and SH signal is measured to be quadratic in both cases. The phase
distortion introduced by the back-filled gas in the experimental chamber is
determined to be negligible in both cases.

####  5.1.2 Experimental methods

Figure 5.1: (a) Diagram of the experimental setup for the 800 nm beam. DM:
Xinetics 37 channel deformable mirror; OAP1: 2-inch diameter f/1.4 off-axis
paraboloid; HR: 800 nm high reflector; OAP2: 1-inch diameter f/1 off-axis
paraboloid; M: flat mirror; BP: 340 nm $\sim$ 460 nm bandpass filter; PMT:
photomultiplier tube. (b) Diagram of the experimental setup for the 2 $\mu$m
beam. DM: Xinetics 37 channel deformable mirror; OAP: 2-inch diameter f/1.3
off-axis paraboloid; BP: 900 nm $\sim$ 1050 nm bandpass filter; PD: photodiode
detector.

The experiments were performed using the Lambda-cubed laser system at CUOS at
the University of Michigan. The experimental layout for the 800 nm beam is
shown in Fig. 5.1a. The deformable mirror is controlled by 37 programmable
piezoelectric actuators. The beam reflects off the DM at an incident angle of
$8^{\circ}$, and then propagates 2 m to a vacuum chamber through an anti-
reflection coated, 3 mm thick fused silica window. A 2- inch diameter gold-
coated f/1.4, $60^{\circ}$ OAP focuses the 30 fs, 3 mJ pulses inside the
chamber. Initial alignment of the paraboloid is performed by maximizing the
brightness of a visible spark generated in ambient air with attenuation to low
intensity. The chamber is then filled with 4 Torr of helium. A broadband
dielectric mirror reflects most of the fundamental light while passing the
second harmonic. The beam is recollimated by a $90^{\circ}$ f/1 OAP and
directed out of the chamber through a $M_{g}F_{2}$ window by a flat, silver
mirror. A second filter (Hoya B390) provides further discrimination against
the fundamental. The signal is then measured with a photomultiplier tube
(PMT). The PMT signal is fed into a boxcar integrator and the boxcar output is
read by the control computer via a standard data-acquisition device. The
computer runs a genetic algorithm [74, 125] to find the mirror configuration
which produces the maximum second harmonic. Each sample used in the genetic
algorithm optimization process is averaged over 25 laser shots.

The experimental setup for the focus optimization in the mid-infrared regime
is shown in Fig. 5.1b. An optical parametric amplifier [119] (OPA) with two
BBO crystals is used to generate a beam of 1 mJ energy, 2 $\mu m$ wavelength,
and $\sim$40 fs pulse duration. A mirror-based telescope expands the beam to
the size of the DM, which then directs the light into a vacuum chamber back-
filled with 40 Torr of air. The chamber window and initial alignment are all
the same as the experiment above. A 50 mm diameter f/1.4 OAP focuses the beam,
and a 20 mm diameter silicon photodiode with a 0.90 $\mu m$ to 1.05 $\mu m$
bandpass filter is used to collect the SH signal. A lock-in amplifier and a
control computer with data-acquisition cards are used to integrate, amplify,
and record the signal. The same genetic algorithm is used to find mirror
configurations that optimize the SH signal.

####  5.1.3 Results

Figure 5.2: Second harmonic signal of the 800 nm beam (a) and the 2 $\mu m$
beam. (b) Improvement charts against iteration number. Focal spot images are
taken for deformable mirror shape before (c) and after (d) correction for the
800 nm case.

Generational improvement charts for the genetic algorithm are shown in Fig.
5.2. The algorithm takes the SH signal as the figure of merit and produces 100
mirror figures for each generation from the ten best figure of merits in the
previous generation. A mutation rate of $5\%$ is applied to introduce
variation between generations. The algorithm starts from a fixed figure, which
has 30 Volts on all 37 actuators. Three best children are plotted in the 800
nm case in Fig. 5.2a while all ten children are plotted in the 2 $\mu m$ case
in Fig. 5.2b, indicating a higher noise level.

The improvement saturates after 20 generations in the 800 nm experiment,
taking 60 generations in the 2 $\mu m$ case. The SH signal is enhanced by
$70\%$ and $100\%$, respectively. To further evaluate the performance of the
technique, images of the focus before and after optimization are acquired with
a $60\times$ microscope objective and CCD camera at low power. A comparison of
the focal quality for 800 nm is also shown in Fig. 5.2c,d. The focal spot size
of the 800 nm beam is 1.7 $\mu m$, suggesting a peak intensity of $1.1\times
10^{18}wcm^{-2}$ and $a_{0}=0.7$. Due to the limitation of the camera’s
detection range, a focal spot image is not taken for the 2 $\mu m$ beam but
instead is approximated to be $\frac{2\mu m}{0.8\mu m}=2.5$ times larger than
the spot size measured with 800 nm beam. The estimated laser intensity of the
2 $\mu m$ beam is thus $5.9\times 10^{16}wcm^{-2}$ and $a_{0}=0.4$. The
corrected focus has a higher peak intensity, better circularity, more energy
above the noise level, and a larger fraction of energy in the main spot. These
features indicate that the technique indeed does optimize for the highest
focal intensity. A calculation of Strehl ratio is performed based on the
following definition: the ratio between the peak intensity of an image divided
by the peak intensity of a diffraction-limited image with the same total flux
[239]. The beam profile in the $\lambda^{3}$ laser system is measured to be
close to Gaussian. The Strehl ratio is improved from 0.65 before optimization
(Fig. 5.2c) to 0.95 afterwards (Fig. 5.2d).

Fig. 5.3 presents the data acquired to investigate the SH scaling laws for the
two cases. As is shown in Eq. 5.1, SH signal is strongly dependent on better
focus for higher intensity and steeper electron density gradient, both of
which scale with laser energy. In the 800 nm case, a wave plate is rotated to
control the input power, whereas, in the $2\mu m$ case, calibrated neutral
density filters are placed before the paraboloid to control attenuation. This
would vary the laser energy without changing the pulse duration, and thus the
SH signal is only modified by the fundamental laser power. Note that the
attenuation is performed only in the scaling measurement. The curves show a
good fit to the power relation that $P_{2\omega}\sim P_{\omega}$, which
confirms that the figure of merit in the genetic optimization process is the
second harmonic signal. A larger uncertainty is observed in Fig. 5.3b, showing
higher noise levels using longer wavelength and lower energy laser light. The
noise is mainly due to the fact that optical parametric amplification is a
sensitive process to the input laser beam, suggesting that shot-to-shot
fluctuations in pulse energy affect the efficiency of the optimization
process.

Figure 5.3: Second harmonic signal vs. the fundamental laser power, for the
$0.8\mu m$ case (a) and the $2\mu m$ case (b), respectively.

Furthermore, it is worth pointing out that the exponent in the SHG scaling
does not necessarily equal to 2, which could be counter-intuitive. For
example, SHG scales with an exponent of $1.49\pm 0.03$ in the nonlinear
response of 50fs pulses to gold nanorods [240]. The unexpected scaling in the
cited [240] study is explained as the effect of damping induced by a hot
thermal distribution of single-particle excitations, and agrees well with the
theoretically predicted value of 1.47 under this assumption. It would be
interesting to investigate this scaling with more complete measurements as
future work, which may reveal more about the underlying physics of SHG in a
rarefied gas.

####  5.1.4 Discussion

Since a rarefied gas is present in the chamber, it is important to be sure
that additional nonlinear effects (self-phase modulation, linear plasma
distortions, Etc.) are not significant. The phase shift can be quantified by
the B-integral:

$B(z)=\frac{2\pi}{\lambda}\int_{z_{0}}^{z}n_{2}I(z)dz$ (5.2)

where $n_{2}$ is the nonlinear index of refraction, and $I(z)$ is the
spatially dependent intensity. The beam profile can be approximated as
Gaussian in the Lambda-cubed. Therefore, the intensity is, with the maximum
amount of accumulated phase occurs on axis at the peak of the pulse:

$\left.I(z)\right|_{t=0,r=0}=I_{0}\frac{w_{0}^{2}}{w^{2}(z)}=I_{0}\frac{1}{1+z^{2}/z_{R}^{2}}$
(5.3)

Note that $I_{0}=4E/\left(w_{0}^{2}\tau\sqrt{\pi^{3}\ln 2}\right)$ is the peak
intensity in a Gaussian beam, and $z_{R}$ is the Rayleigh range. Integrating
the phase shift from the focusing optic at $z=z_{0}\gg z_{R}$ to the focal
point at z=0:

$B(z)=\frac{4E}{\lambda^{2}\tau}n_{2}\sqrt{\pi^{3}/\ln 2}$ (5.4)

In the case of the 800 nm, 3 mJ, 30 fs laser pulses, 4 Torr of helium is used.
The nonlinear refractive index of helium at atmosphere is $3.5\times
10^{21}wcm^{-2}$ [241], and it decreases linearly with pressure. From Eq. 5.4,
the B-integral is approximately 0.008 rad, or $\lambda/800$. On the other
hand, 40 Torr of air is backfilled when the 2 $\mu m$, 1 mJ, 30 fs pulses are
used. The nonlinear refractive index of $N_{2}$, $O_{2}$, and Argon at $2\mu
m$ wavelength are $7.3\times 10^{-20}cm^{2}w^{-1}$, $8.2\times
10^{-20}cm^{2}w^{-1}$, and $9.3\times 10^{-20}cm^{2}w^{-1}$, respectively
[242] . It yields a B-integral of 0.035 rad, or $\lambda/180$. Since the
intensity profile in the experiment is overestimated due to the Gaussian
approximations, the obtained nonlinear phase shift is indeed an upper bound.
Considering the chamber leaking, humid air can further eliminate the phase
shift. This can be inferred from the fact that water vapor has a lower
refractive index than air [243, 244] and presumably lower nonlinear refractive
index, which accumulates less phase shift.

Linear plasma dispersion also contributes to the wavefront distortion.
Consider the phase difference of a wave in vacuum and in oscillating plasma:

$\displaystyle\Delta\phi$ $\displaystyle=\int\Delta k\cdot
dz=\frac{z}{c}\left(\sqrt{\omega^{2}-\omega_{p}^{2}}-\omega\right)\cong\frac{z}{c}\cdot\frac{\omega_{p}}{\omega}\cdot\left(\omega-\frac{\omega_{p}}{2}-\omega\right)=\frac{z\omega_{p}^{2}}{2c\omega}$
(5.5)

where the plasma frequency $w_{p}$ is much smaller than the laser frequency
$w_{l}$ in low-pressure gas. The atomic density at 4 Torr is $2.6\times
10^{17}cm^{-3}$, which corresponds to a plasma frequency of 29 rad/ps. The
distance z in dispersive plasma starts from the point where the pulses reach
helium double ionization intensity at $3\times 10^{15}wcm^{-2}$ [245]. An
f/1.4 optic focuses the 800nm, 3 mJ, 30 fs beam to this ionization intensity
at 91 $\mu m$ before the focus. The phase shift of an 800 nm beam is
calculated to be 0.05 rad, or $\lambda/116$ from Eq. 5.5. Air at 40 Torr has
an electron density of $2.6\times 10^{18}cm^{-3}$, and has considerable
ionization at intensities around $3\times 10^{14}wcm^{-2}$ [246]. For the
$2\mu m$, 1 mJ, 40 fs beam focused by an f/1.3 optic, the accumulated phase
shift is approximately 0.9 rad or $\lambda/7$. This higher distortion is
mainly due to the higher operating pressure, but it is still smaller than the
distortion one would expect from strongly attenuating filters, which are
necessary for other alternative methods to perform focus optimization.
Furthermore, the second level ionization rate of air is lower than its first
level ionization rate for orders of magnitudes at intensities below
$10^{15}wcm^{-2}$. Recall that our focal intensity is around $5.9\times
10^{16}wcm^{-2}$, suggesting that for most of the propagation distance, the
distortion is dominated by single ionization. The accumulated phase shift
through singly ionized air is 0.5 rad, or $\lambda/13$. Note that both self-
phase modulation and linear plasma distortion can be significantly reduced by
operating at lower pressure, where the nonlinear index of refraction and
plasma frequency are correspondingly lowered. Gas with a higher ionization
threshold can also decrease the effect of wavefront distortion.

####  5.1.5 Conclusion

A new method of optimizing the surface figure of a deformable mirror for high
numerical aperture focusing to relativistic intensity using second harmonic
signals generated in rarefied gas is presented. SH signal is demonstrated to
be a convenient and effective feedback for the genetic algorithm. Optimization
at full intensity corrects for all linear and nonlinear wavefront distortions
present in the laser and focusing systems without introducing additional
distortions due to attenuation. Another advantage of the method is that the
optimization is not constrained to a fixed plane or dependent on fine
alignment of diagnostics. This technique simplifies access to the relativistic
intensity regime, and should be applicable to higher intensity systems and may
employ other kinds of gas at lower gas pressures. For a 1 J, 40 fs, 800 nm
laser beam with fast focus of f/1.4, the wavefront distortion experienced in
this focus optimization technique with helium pressure down to 1 Torr is
estimated to be $\lambda/125$. The B-integral would be more significant,
reaching f/10 but still small compared to the distortion from thick optical
filters. The benefits of applying full-power optimization methods with
adaptive optics may be even more significant in ultra-relativistic
experiments, owing to the additional difficulty in attenuating beams. For
example, in multi-joule systems, amplifiers must be disabled to achieve
sufficient attenuation. In addition, this technique also provides significant
experimental convenience and consistency: near-vacuum can be maintained in
between optimization and experiment, which eliminates potential alignment
errors caused by shifting of optics during pressure change. The number of
optics required to collect the second harmonic signal is limited, and they can
be controlled remotely. However, to employ this technique as a robust pre-
experiment routine in more relativistic high repetition rate laser facilities,
there are still questions to be answered, such as a power scaling for the SHG
of these laser pulses in rarefied gas.

###  5.2 Adaptive control of a laser-wakefield accelerator driven by mid-IR
laser pulses444This section co-authored with Ma, Y., Schwartz, R., Woodbury,
D., Nees, J. A., Mathis, M., Thomas, A. G. R., Krushelnick, K., and Milchberg,
H. (2019): Adaptive control of laser-wakefield accelerators driven by mid-IR
laser pulses. Optics express, 27(8), 10912-10923.

####  5.2.1 Introduction

Coherent control of dynamic processes through systematic optimization of the
phase of laser has been applied to a variety of systems, such as quantum dots
[247, 248], qubits [249], two-photon transitions [250], photocurrent
generation [251] and chemical reactions [252]. In the field of intense laser-
matter interactions, deformable mirrors (DMs) controlled by genetic algorithms
(GAs) taking in feedback measurements have already been utilized widely in
high-power laser facilities. These adaptive optical systems have been
implemented to control THz generation [75], multi-filament configuration [74],
high order harmonic generation [253] and optimization of the focal spot [232,
231, 76]. Plasma waves produced from the interaction process can also be
controlled via this phase shaping technique, suggesting that a particular
laser wavefront can steer the plasma wave to a final state using an optimal
electric field structure [73]. Plasma waves produced by high-power lasers, in
particular relativistic electrons from Laser-wakefield accelerators (LWFAs),
have been studied extensively [16, 21, 22, 23, 19, 24, 254] as it has
extremely large accelerating gradients and consequently the short accelerating
distance compared to conventional accelerators. There remain issues with beam
pointing, stability control, energy spread and dark current for the use of
such beams. In LWFA, plasma electrons are expelled from the relativistic laser
pulse and form a cavity, or a void of electrons, behind the pulse. The
cavity’s spatial extent is close to a plasma wavelength in length, a laser
focal spot size in width and close to the speed of light in phase velocity.
Background electrons can get captured in the cavity and get accelerated to
high energy. However, when the laser power is high enough ($P>P_{cr}$),
relativistic self-focusing modifies the laser wavefront to overcome
diffraction limit and focuses the laser pulse to higher intensity or guides
the beam, depending on the ratio $P/P_{cr}$. If the pulse length is long
relative to the plasma wavelength, it also overlaps multiple plasma buckets
and the laser pulse can modulate and break up into a train of short pulses
with pulse length around the plasma wavelength. Operating in this Self-
modulated laser-wakefield acceleration (SM-LWFA) regime with higher density, a
large amplitude wakefield approaching the wave-breaking limit is generated to
trap background electrons, but the acceleration length is limited. SM-LWFA has
recently aroused interest in betatron radiation using picosecond duration
laser pulses at large laser facilities [255].

It is worth pointing out that both the critical power (Eq. 2.47) and the
ponderomotive force (Eq. 2.31) scale with the square of the laser wavelength.
For laser wavelengths at $\lambda=0.8\mu m$ and $3.9\mu m$, the critical
densities are $2\times 10^{21}cm^{-3}$ and $7\times 10^{19}cm^{-3}$,
respectively. Assuming an electron density of $3\times 10^{19}cm^{-3}$, the
critical powers are 987 GW and 42 GW, respectively. Over the past decade,
remarkable progress has been made in the generation [120, 256] and application
of mid-infrared (MIR) laser pulses, showing their superiority in generating
high order harmonics [257, 165, 99], electromagnetic pulses [258], filaments
[259] and x-rays[226]. For instance, it has been found that the characteristic
$k_{\alpha}$ flux of hard x-rays from $3.9\mu m$ laser driver is much greater
than that from the 800 nm driver [226]. Towards reaching longer wavelengths in
LWFA has also drawn attention, not only for its lower critical power threshold
and higher $a_{0}$ but also for being less difficult in achieving near-
critical density interactions, which enables the generation of MeV-scale
electrons with moderate laser intensity. These MeV-scale electron sources from
high repetition rate laser systems have demonstrated their use in electron
radiography [260, 261]. SM-LWFA at near-critical density can be approached
using cryogenically cooled, high-density gas jets with 800nm Ti:Sapphire
lasers [262, 54, 263], and more recently using moderate density gas jets with
a mid-infrared laser at $\lambda=3.9\mu m$[264].

It is natural to consider coherent control of the LWFA dynamics with mid-
infrared (IR) lasers. In this work, we present the first experiment to
optimize the quality of the electron beam from mid-IR ($\lambda=3.9\mu m$)
light interacting with near-critical density plasma. Beam charge, energy
spectrum, beam pointing and fluctuation have been improved by controlling the
laser wavefront via an evolutionary algorithm. Wavefront reconstruction and
PIC simulations illustrate that changes on laser wavefront lead to different
laser focusing and self-guiding in plasma. Filamentation has been observed in
the case of a flat laser wavefront, and can be corrected by the adaptive
control system for better electron acceleration. This work also demonstrates
the ability to have regular deformable mirrors with 4 $\mu m$ full stroke to
properly function in a mid-IR laser system, and the ability to reconstruct
wavefronts without the presence of a mid-IR wavefront sensor.

####  5.2.2 Experimental setup

Figure 5.4: Schematic of the setup: Deformable mirror: AOA Xinetics 37-channel
2 inch; OAP: f/2.7; Gas jet: 150$\mu m$ orifice diameter nozzle; CCD: The
Imaging Source DMK41BU02.H Charged Particle Device (CCD) camera; Lanex: LANEX
Regular screen.

The experiment was conducted at the University of Maryland, using a hybrid
optical parametric amplifier/optical parametric chirped-pulse amplifier
(OPA/OPCPA) laser system which generates $25\pm 1$ mJ, 87 fs, $3.9\mu m$
pulses at a repetition of 20 Hz [120]. A deformable mirror (DM) controlled by
the evolutionary algorithm was used to adjust the laser wavefront based on the
diagnostic feedback. The experiment setup is shown in Fig. 5.4. An f/2.7 focus
was achieved from the paraboloid and the 37.5mm diameter laser beam was
focused to $15\mu m$ at the beam waist, measured using a knife-edge scan. The
pulse energy on target was measured to be 15 mJ, resulting a peak intensity
$I=\frac{0.94\cdot E_{p}}{\tau_{p}\cdot\pi\cdot w^{2}/2}=4.6\times
10^{16}\;W/cm^{2}$ and $a_{0}\sim$ 0.7. Plasma density up to $3\times
10^{19}cm^{-3}$ (or $40\%$ of the critical density at $\lambda=3.9\mu m$) can
be easily reached by the gas jet without cryogenic cooling. The jet was
mounted onto a 3-D translation stage to adjust the position of the laser focus
throughout the hydrogen gas target. A LANEX regular screen with a shield of
100 $\mu m$ thick aluminum was placed 9 cm from the jet and imaged onto the
CCD camera. The camera was synchronized to the 3.9 $\mu m$ pulse and
integrated over 2 ms. In each iteration, the genetic algorithm analyzed 50
electron beam profiles corresponding to 50 deformable mirror surfaces, and the
median of 10 shots was used to evaluate a figure of merit function for each
mirror surface. While the system repetition rate was limited to 1 Hz due to
radiation safety requirements, this was still adequate for averaging over the
shot-to-shot fluctuation while keeping the data acquisition period reasonable.
The starting point of the optimization process was chosen at the condition of
minimum phase changes, where the DM was initialized to a flat mirror surface.
It took the evolutionary algorithm $\sim 30$ iterations to find an improved
mirror surface where the generation curve approached convergence, as is shown
in Fig. 5.5.

####  5.2.3 Results

#####  5.2.3.1 Optimizing the total electron beam charge

Figure 5.5: Improvement charts using different figure of merit (FOM)
functions: (a) total charge collected in the region of interest on CCD image
after background subtraction; (b) the fitness function defined in Eq. (5.6)
with n=2. Both optimizations started from initializing the deformable mirror
to a flat surface. The shaded area refers to the variation of 5 best genes in
each iteration. The number of iteration was limited by the experimental time
considering the system repetition rate was as low as 1 Hz. The figure of merit
values were calibrated to real units taking into account the geometry and the
efficiency of the optics, the LANEX screen [265] and the CCD camera. Figure
5.6: Comparison of electron beam energy spectra: (a)-(c) before and after
optimization; (d),(e) at front, center and back of the gas jet. 20 consecutive
images were taken in each case. The shot-to-shot variation is shown in (a)-(c)
while the statistic mean was shown in (d),(e). Examples of raw spectra with
non-thermal peak features are shown in (f), in which the optimal laser
wavefront found by the genetic algorithm was focused at back, center and front
of the gas jet. Note that peaks do not occur on all shots.

Fig. 5.5(a) illustrates the improving curves using total charge as FOM with
laser focus at different positions on the Gaussian gas density profile (FWHM
$\sim 250-1000\mu m$ [264]). Note that the low energy electrons ($<$500 keV)
were filtered by the aluminum foil and the total charge collected was
$\sim$450 pC. The gas jet was moved in 10-$\mu m$-step along the laser
direction. A Nomarski interferometer with 515 nm probe light indicated that
the plasma density was $37\%$, $35\%$ and $29\%$ of critical density at the
front, center, and back side of the gas jet.

Electron energy spectra were compared at different focusing positions before
and after optimization in Fig. 5.6(a)-5.6(e), while examples of raw energy
spectra are shown in Fig. 5.6(f). A high-energy bump around 3 MeV shows up in
some individual shots for the back focus, and gets lower and weaker as the
focus moves towards the front of the density ramp. Improvements are observed
at all three focal positions and focusing at the center ($0.35n_{cr}$) gives
the best energy spectrum. Focusing on the back ($0.29n_{cr}$) gives very high
beam charges, shown in Fig.5.5(a), but worse energy spectra. It could be
caused by the electron beam missing the slit of the spectrometer, due to
inferior beam collimation and pointing stability. The trend in Fig. 5.6
indicates the existence of an optimal plasma density for electron acceleration
in the mid-IR regime, as was mentioned by Woodbury [264].

#####  5.2.3.2 Optimizing the electron beam profile

To further improve the quality of the electron beam profile, the image moment
function, as is defined in Eq. (5.6), was applied to the genetic algorithm.

$FOM=\sum\limits_{(i,j)}^{r_{ij}\neq r_{0}}\frac{I_{ij}}{|r_{ij}-r_{0}|^{n}}$
(5.6)

Where $I_{ij}$ is the intensity collected at pixel position $r_{ij}$ on the
camera. The beam center $r_{0}$ is determined in each shot from the center of
mass calculation after background subtraction. It quantifies not only the
total charge but also beam collimation and pointing. The improvement chart is
shown in Fig. 5.5(b) and the optimization performance is shown in Fig. 5.7.
The genetic algorithm started from initializing the DM to a flat surface. It
is observed that the raw images in Fig. 5.7(a) are divergent and have
significant pointing fluctuations while the ones in Fig. 5.7(b) are collimated
and directional. Detailed analysis can be found in Fig. 5.7(c)-5.7(f).
Averaging over 30 shots, the optimization was able to increase the total beam
charge by $\sim 40\%$ and the peak charge density to threefold in 35
iterations. In Fig. 5.7(e) the beam divergence $\theta_{x}$ and $\theta_{y}$
in transverse directions were reduced from $206\pm 64$ mrad and $228\pm 69$
mrad to $128\pm 21$ mrad and $110\pm 20$ mrad, respectively. The pointing
instabilities, defined as the standard deviations of beam pointing
$\delta\theta_{x}$ and $\delta\theta_{y}$ in Fig. 5.7(f), were reduced from
25.2 mrad and 45.7 mrad to 14.5 mrad and 20.6 mrad.

Figure 5.7: Electron beam profile optimization using the image moment as
figure of merit, defined in Eq. (5.6). (a) and (b): 30 consecutive raw images
before and after optimization. The circular edge, due to a collimation tube in
front of the LANEX, corresponds to a solid angle of 550 mrad. (c) - (f) are
the visualization of beam quality in terms of total beam charge, peak charge
density, divergence angle and beam pointing, respectively. Each dot represents
one shot.

#####  5.2.3.3 Wavefront reconstruction

The laser wavefront was measured $ex\;situ$ by applying the recorded voltage
distribution on the deformable mirror and subsequently measuring the wavefront
with visible light. An imaging system involving the deformable mirror and a
Shack-Hartman wavefront sensor was set up using a helium-neon laser after the
experiment. The FrontSurfer wavefront analyzer (Version 1.4.7, OKO
Technologies), consisting of a high-precision lenslet array and a CMOS
UI-2210M CCD camera, can describe the wavefront in Zernike polynomials up to
$200^{th}$ order:

$\Delta\phi=\sum\limits_{j=1}^{200}A_{j}Z_{j}$ (5.7)

where $A_{j}$ and $Z_{j}$ represent the $j^{th}$ coefficient and base of the
Zernike polynomials, respectively. Knowing the voltages on the deformable
mirror, the coefficients could be obtained from the influence matrix $C_{ij}$:

$A_{j}=\sum\limits_{i=1}^{37}C_{ij}V_{i}$ (5.8)

Figure 5.8: Reconstruct the laser wavefront propagation for three cases.
(a-c): wavefront for the optimal electron beam, (d-f): wavefront from a flat
mirror surface, and (g-i): perfect Gaussian wavefront. Wavefronts leaving the
DM, propagating 4.5m and focused by the OAP are shown in the first, second and
third column. First 50 Zernike coefficients are included in the
reconstruction.

The influence matrix method is useful for real-time analysis of the wavefront,
but here only the optimized wavefront is to be analyzed. Instead of measuring
each element in the influence matrix, an alternative way to restore the
wavefront change due to the DM ($\phi_{DM}$) is to directly apply the voltage
recorded in the experiment and measure the mirror surface. A reference mirror
surface with a known wavefront would be necessary to reconstruct the 3.9$\mu
m$ laser wavefront. Analogous to the setup in Fig. 5.4, the laser beam was
attenuated and focused onto an AGS crystal to generate second harmonic (SH)
signal. The genetic algorithm was run to improve the SH signal to threefold
till convergence, which suggested the highest peak intensity available [232].
It corresponds to the smallest focal spot and the flattest wavefront
available. A knife-edge scan showed the focal spot was decreased from $\sim
25\mu m$ to $\sim 15\mu m$ after the second harmonic optimization. This
voltage map on the DM was recorded and afterwards applied to the wavefront
analysis system with visible light to obtain the phase $\phi_{SHG}$. Assuming
the laser wavefront going into the DM was $\phi_{laser}$, the wavefront after
the SHG optimization would be almost Gaussian:

$\phi=\phi_{laser}+2\times\phi_{SHG}\simeq\phi_{Gaussian}$ (5.9)

On the other hand when the DM was set to optimize the electron beam, the :

$\phi_{opt}=\phi_{laser}+2\times\phi_{DM}$ (5.10)

Subtracting Eq. (5.9) from Eq. (5.10) would give the laser wavefront leaving
the deformable mirror during the experiment, $2\times(\phi_{DM}-\phi_{SHG})$,
as is shown in Fig. 5.8(a). Fresnel diffraction was taken into account to
propagate the wavefront 4.5 meters to the OAP, as is shown in Fig. 5.8(b), and
the phase change was calculated with LightPipes [266] using direct integration
approach. Fig. 5.8(c) shows the wavefront at 500 $\mu m$ before the geometric
focus, which was used in the PIC simulation. Fig. 5.8(d)-5.8(f) present the
laser wavefront before the evolutionary algorithm was run where the DM was
initialized to a flat surface. The propagation of a perfect Gaussian beam is
included in Fig. 5.8(g)-5.8(i) for comparison. Note that the validity of this
whole reconstruction process is dependent on a list of factors, including the
stability of the voltage on the DM actuators, the accuracy of measurement
using the visible wavefront sensor, and mostly the flatness of the wavefront
after the second harmonic optimization, or the validity of Eq. (5.9).

#####  5.2.3.4 Particle-in-cell simulations

Figure 5.9: Laser field evolution and electron beam qualities with different
laser wavefronts in PIC simulations. (a) Evolution of peak laser field
strength with different wavefronts in vacuum and plasma, respectively. (b) and
(c) Electron spectra and angular distributions at the end of the simulation
($t=2$ ps) with different laser wavefronts and the same plasma profile showing
in (a).

The effect of wavefront changes on wakefield acceleration was further
investigated with two-dimensional PIC simulations in the EPOCH framework
[267]. The simulation box with moving window is 200 $\mu\mathrm{m}\times$ 160
$\mu$m with grid size of $1/32$ and $1/16~{}\lambda_{L}$ in $x$ and $y$, where
$\lambda_{L}=3.9~{}\mu$m is the laser wavelength, $x$ is the laser propagation
direction, $y$ is the transverse direction and $z$ is the laser polarization
direction. There are 64 macro-particles per cell. The laser pulse is Gaussian
in both transverse and longitudinal directions with a FWHM pulse duration
$\tau=100$ fs, a $1/e^{2}$ spot size $w_{0}=13\mu$m and a normalized vector
potential $a_{0}=0.7$. The plasma density distribution along the laser
propagation direction was fitted from interferometric measurements which
indicate a Gaussian distribution with a peak density of $0.4~{}n_{c}$ and a
FWHM of 505 $\mu$m, as shown in Fig. 5.9(a). The focus position of the laser
pulse was initially set at 374 $\mu$m, which corresponds to the “Center” case,
for a perfect Gaussian laser beam. In PIC simulations, we compared three
different cases, namely “Optimized”, “Flat mirror” and “Gaussian”, with
wavefronts shown in Fig. 5.8(c), 5.8(f) and 5.8(j), respectively.

Figure 5.10:  Snapshot for plasma density and electron beam distribution with
(a) Gaussian, (b) flat mirror and (c) optimized wavefront at the same time,
$t=1$ ps. Self-injection has occurred with optimized wavefront in (c) while
not in the other two cases.  Figure 5.11:  Snapshots of PIC simulations with
different wavefronts. Laser field distribution at $t=1.5$ ps for (a) Gaussian,
(c) Flat mirror and (e) Optimized wavefront, respectively. Plasma density
distribution at $t=1.64$ ps for (b) Gaussian, (d) Flat mirror and (f)
Optimized wavefront, respectively. Spatial distribution of accelerated
electron macro-particles are overlaid on the plasma density distribution where
color scale represents longitudinal momentum of the particles.

The propagation of laser pulses with different wavefronts in both vacuum and
plasma were examined. As shown in Fig. 5.9(a), the peak laser field strength
reaches its maximum much earlier in the optimized wavefront case than in the
other cases. The LWFA process starts as the laser field reaches its maximum
during the self-focusing, and almost the whole acceleration happens within the
density up-ramp region. The laser pulse with optimized wavefront initiates the
acceleration earlier, as is shown in Fig. 5.10, and thus experiences a lower
plasma density. Since the maximum energy gain in LWFA [18] scales as $\Delta
E_{max}\propto n_{p}^{-2/3}$, the relative lower plasma density for the
“Optimized” case would result in higher final energy gain. This has been
confirmed by the electron spectra at the end of the simulations, as shown in
Fig. 5.9(b). The electron spectra from PIC simulations agree with the
experimental results in Fig. 5.6 qualitatively.

Moreover, the laser pulses in the “Gaussian” and “Flat mirror” cases suffer
more from the transverse self-modulation, leading to the self-filamentation
shown in Fig. 5.11(a) and (c). These filaments are intense enough to drive
LWFA on their own, which eventually lead to the wing structure of the electron
beam in Fig. 5.11(b) and (d). Consequenctly, the electron beam collimation in
these two cases is worse than that in the “Optimized” case, as is presented in
Fig. 5.9(c), which also agrees with the experimental results.

####  5.2.4 Discussion

Understanding the phase front condition of the laser to the plasma is crucial
to the success of wakefield acceleration. The non-Gaussian features of laser
wavefront in experiments can strongly affect the acceleration mechanism and
betatron sources [268]. It is, therefore, of great interest to control the
phase front in LWFA. Optical steering of the electron beam direction [269],
enhancement of betatron radiation [270] and spectral control of the x-rays
produced in the process [271] have been achieved by modifying the laser
wavefront. Here we have demonstrated the ability to coherently control the
relativistic electron beam from wakefield acceleration by mid-IR laser pulses
in near-critical density plasma. Electron total charge, energy spectrum, beam
pointing and fluctuation are improved and the effect of wavefront changes on
the acceleration process are studied with PIC simulation. The optimal
wavefront initiates the acceleration earlier on the density up-ramp and thus
experiences a lower plasma density, which leads to higher energy gain during
the interaction. It also sees less filamentation from the transverse self-
modulation, which would be responsible for the wing structure and divergence
of the electron beam. With this improved wavefront, better electron beam
collimation and energy spectra are observed in both experiment and simulation.
The computer modeling is based on wavefront reconstruction using the voltage
applied to the deformable mirror, in the absence of a mid-IR wavefront sensor.

Improvement in electron beam quality is independent of the improvement in
laser focus since the highest intensity laser focus produced an order of
magnitude lower electron charge. This behavior, together with the intensity
wings from optimized wavefront, has been observed in previous work [73] with
$\lambda=800nm$ as well. Analogously, looking at the x-rays producing by the
wakefield acceleration, a wavefront with coma aberration generates more high-
energy photons than a flat wavefront [271] does. Modifying the phase of the
light can cause strong optical nonlinear effects in the plasma interactions,
which can affect the plasma wave dynamics in a complex but deterministic
manner.

This work opens a new window to the study of coherent control of relativistic
mid-IR laser-plasma interactions. It is worth noting that the full stroke of
the deformable mirror surface is 4 $\mu m$, or a wavelength of the mid-IR
driver. Namely, without upgrading the DMs to deeper stroke or the wavefront
sensors to a longer wavelength range, current adaptive optical systems are
capable of conducting experiments using mid-IR lasers. Our work shows the
potential for the use of long-wavelength lasers in LWFA in near-critical
density plasma which would be difficult to achieve using near-IR lasers.
Recently, pulse shaping implemented into the system algorithm [78] has been
validated using near-IR lasers and can be extended to the mid-IR critical-
density regime.

###  5.3 Beyond optimization - supervised learning applications in a laser-
wakefield accelerator

####  5.3.1 Introduction

High-repetition-rate laser systems have been widely used with evolutionary
algorithms to solve optimization problems in the field of relativistic laser-
plasma interactions, including laser wakefield acceleration [73, 80, 81], ion
acceleration [82, 77], x-ray production [78], terahertz generation [75], laser
filamentation [74, 79, 83], and laser focus optimization [72, 76]. A detailed
review of high-repetition-rate laser-plasma experiments has been given in Sec.
1.2. However, evolutionary algorithms usually provide little information other
than a local optimum, which can be difficult to interpret. Instead, machine
learning (ML) methods can generate predictive models that reveal more
information in the dataset to help understand the physical processes.

The broader discipline of plasma physics has adopted various machine learning
methods in recent years. For instance, supervised learning regression
algorithms have been applied to inertial confinement fusion (ICF) experiments
with growing interests, such as Deep Jointly-Informed Neural Networks [85, 89,
87, 272] and Gaussian Process regressor [88]. Another popular machine learning
technique called Random Forest has found success in magnetic confinement
fusion experiments for both classification and regression problems [273, 274].
In space physics, Gaussian Processes are used to classify solar wind plasmas
into categories [275], and deep neural networks are used to predict solar
flares from sunspot data [276, 277]. Beyond supervised learning, plasma
physicists have utilized in other powerful and increasingly popular machine
learning methods, such as transfer learning [91] and reinforcement learning
[278, 94]. The laser-plasma community is starting to embrace machine learning
techniques as well. Artificial neural networks are employed to analyze
features in high-order-harmonic spectra [95] and laser-induced-breakdown
spectra [96]. Our work explores the capability of machine learning techniques
in the field of laser-wakefield acceleration using all the supervised learning
methods mentioned above.

Laser-wakefield accelerators (LWFAs), first proposed by Tajima and Dawson
[16], provide a possible alternative to conventional particle accelerators at
a substantially smaller size and cost. Taking advantage of the electric fields
in laser-produced plasmas, LWFA can reach acceleration gradients of tens of
GeV/m, which are many orders of magnitude greater than those produced in
conventional accelerators. Extensive experiments have been performed to
understand LWFA mechanisms and to generate energetic electron beams [20, 21,
22, 23], and the highest electron energy achieved so far is 7.8 GeV [14].
While the highest energy laser facilities usually fire a few shots a day,
there have been rapid developments in high repetition rate laser systems with
lower peak power [55, 54, 47, 45, 58, 279]. One of the rationales to increase
the repetition rate is that many applications demand higher repetition rates
but at only moderate beam energies. MeV-level electrons from LWFAs above 1 Hz
have been used in transmission electron radiography [261], picosecond electron
diffraction [280], and generating $\gamma$-ray sources through bremsstrahlung
conversion for imaging [281]. In addition, having higher repetition-rate
allows meeting the statistical requirements to have better control over
experiments [73, 80, 81, 282, 283, 284, 93].

In this work, we demonstrate the use of machine learning in LWFA beyond
optimization purposes. We build four regression models using supervised
learning algorithms: Random Forest, Neural Network, Deep Jointly-Informed
Neural Network, and Gaussian Process. The models are trained to predict the
electron beam charges in an LWFA given the laser wavefront modification caused
by a deformable mirror, while Random Forest turns out to be the best
considering the model performance and the computational cost. The trained
models help examine the quality of the measurement by evaluating the model
performance on every measured data point. Three of the models show similar
performance, providing a potential way for anomaly detection without repeated
measurements. To investigate if the ML models can make accurate predictions
when the measured data have uncertainty, we characterize the model robustness
against a range of virtual error bars assigned to the data. Gaussian Process
and Random Forest are found to be more resistant to measurement uncertainties.
We rank the important Zernike terms that lead to high electron beam charge
according to the ML models, compare them to the results from the genetic
algorithms and from the statistical correlation. The feature importance from
the ML models is found to reveal more information than the latter methods do.

####  5.3.2 Data and Methods

#####  5.3.2.1 LWFA precise control

The physics mechanism of LWFA has been introduced in Sec. 2.5. Over the past
decade, advances in theoretical analysis [19, 18] and experimental
demonstrations [14, 285] of LWFA has pushed our understanding of its
mechanisms to the forefront. The critical questions to address in the next
decade within the LWFA community [286] have moved towards precise control of
LWFA experiments to serve as bright sources of relativistic particles [14, 55]
and high energy photons [27]. While unprecedented stability of laser-plasma
accelerators with many hours of operation has been demonstrated in recent
studies [59, 60], challenges remain to precisely control the highly nonlinear
physical processes in LWFA. As an effective tool for data analysis in
nonlinear and high-dimensional problems, machine learning is gaining attention
[73, 80, 93].

#####  5.3.2.2 Experimental

The experiments were conducted using the Lambda Cubed laser system in the
Gerard Mourou Center for Ultrafast Optical Science at the University of
Michigan, which produces 35 fs, 20 mJ laser pulses at 800 nm wavelength. The
laser pulses were focused by an f/2 off-axis paraboloidal mirror to a vacuum
spot size of 2.5 $\mu m$ FWHM, resulting a peak intensity of $3\sim 4\times
10^{18}W\cdot cm^{-2}$. The experiments were operated at a plasma density
roughly a Gaussian with a 120 $\mu m$ FWHM and a peak of $6.5\times
10^{18}cm^{-3}$. The laser focus was generally placed on the down-ramp of the
target density profile. The target is free-flowing argon gas from a capillary
with inner-diameter of 100 $\mu m$, and the optimal backing pressure was in
the range of $21\sim 23$ psi. Details of the experimental setup and electron
acceleration mechanisms are described in reference [53]. The laser wavefront
change was induced by a deformable mirror (AOA Xinetics 37-channel 2 inch
diameter) and recorded as the 37 actuator voltages that control the mirror
surface. The electron beam was captured via a scintillator screen that was
imaged by an electron-multiplying charge-coupled device (CCD) camera (Andor
Luca-R, 14-bit). The LWFA electron beam data were recorded from four separate
experimental days when running optimization algorithms. The optics that
deliver the laser to the experimental chamber were unchanged throughout the
four days. The target is placed with respect to the laser focus in the same
position, and the alignment procedures were routinely carried out the same way
each day. An example of the raw electron beam image is shown below.

Figure 5.12: A sample image of the measured electron beam from the LWFA. Color
scale indicates the intensity of the electron signal.

#####  5.3.2.3 Data pre-processing and correlation

The dataset is pre-processed before the regression modeling. The dataset
contains the information of the optical wavefront change of the driving laser
caused by a deformable mirror, as well as the electron beam charge of the
accelerated electrons. The wavefront changed by the deformable mirror has 37
dimensions in space and can be described mathematically by a polynomial, known
as the Zernike polynomial [239], to reduce the dimensionality. In this case,
the coefficients of the first five layers (15 terms) in the Zernike polynomial
can accurately reproduce the wavefront. The 15-dimensional vectors consisting
of the Zernike coefficients are used as the input to our supervised learning
models, while the electron beam charges integrated from the beam image are the
output, normalized to the range [0, 1]. In the context of machine learning,
the input is called a feature and the output is called a label. We have 208
data samples in total, while each data point was averaged over $\sim 120$
laser shots. The dataset is split into two subsets: $80\%$ of the data points
are used to train the models while $20\%$ are for testing. The feature matrix
in the training set is sphered so that its rows have zero sample mean and
unity sample variance. The feature matrix in the test set is updated
accordingly with the same transformation.

The statistical correlation coefficient measures the dependence between two
variables, and its value falls in the range of [-1,1]. The absolute value
represents the strength of the dependence while the sign represents the
direction. We calculate the correlation between each feature (Zernike
coefficients) and the electron beam charge in the test dataset, as is
illustrated in Tab. 5.1. Among all the Zernike coefficients, while $z_{10}$
has the largest magnitude of correlation. The correlation matrix will be
compared to the machine learning model predictions in Tab. 5.3, 5.4 in
Section. 5.3.3.

$z_{0}$ | $z_{1}$ | $z_{2}$ | $z_{3}$ | $z_{4}$ | $z_{5}$ | $z_{6}$ | $z_{7}$
---|---|---|---|---|---|---|---
0.51 | -0.33 | -0.28 | 0.29 | 0.14 | 0.21 | -0.15 | 0.37
$z_{8}$ | $z_{9}$ | $z_{10}$ | $z_{11}$ | $z_{12}$ | $z_{13}$ | $z_{14}$ | charge
-0.42 | 0.20 | -0.55 | 0.35 | 0.47 | -0.42 | -0.019 | 1
Table 5.1: Statistical correlation between Zernike coefficients ($z_{0}$ \-
$z_{14}$) and the electron beam charge.

#####  5.3.2.4 Machine learning methods

Four supervised learning regression methods are used to predict the electron
beam charges based on laser wavefront changes: Random Forest (RF), Gaussian
Process (GP), Deep Neural Network (DNN), and Deep Jointly-Informed Neural
Network (DJINN). Supervised learning is a branch of machine learning, which
learns a function that maps an input (feature) to an output (label) based on
example input-output pairs in a training sample. In each of the supervised
learning methods, the model is trained on the training dataset recursively
until it can accurately predict the labels using the features. The model
performance is then characterized by the test dataset. Details about the
supervised learning methods have been introduced in Sec. 3.5.

In the RF method, the algorithm is implemented using the RandomForestRegressor
library in Scikit-learn [128]. The hyper-parameters to tune are the number of
trees, the maximum depth in a tree, and the maximum number of features when
splitting. In the DNN method, a fully-connected five-layer DNN is built using
the Tensorflow.Keras library [131] based on Google’s deep learning software
TensorFlow [132]. When constructing the network, we use the rectified linear
unit (ReLU) function and the Sigmoid function as the activation functions for
different layers. The activation function for the first, the fourth, and the
fifth layer is the rectified linear unit (ReLU) function, and that for the
second and the third layer is the Sigmoid function. The cost function is the
mean squared error loss governed by the Adam optimizer [133] to update the
network weights. A $L_{2}$ norm regularization is added to the loss function
to reduce overfitting. The main tuning parameters are the number of layers,
the number of neurons in each layer, the epoch size, and the initialization of
the weight matrix. The DJINN regression source code is accessible at the
LLNL/DJINN github directory. The main tuning parameters in the DJINN method
are the maximum depth of trees, and the number of trees (nets) in ensemble. In
the GP method, the algorithm is implemented using the
$Sklearn.gaussian\\_process$ library in Scikit-learn [128] with a combination
of Matern kernel and Rational Quadratic kernel. The hyper-parameters to tune
are the smoothness, the length-scale, and the scale mixture parameter in the
kernels.

####  5.3.3 Results

Figure 5.13: Predicted electron beam charges using DNN, RF, GP, and DJINN vs. measured electron beam charge in the test dataset. Model | MSE | MAE | $R^{2}$ | ExVar
---|---|---|---|---
Random Forest | 0.00132 | 0.0268 | 0.986 | 0.987
Neural network | 0.00162 | 0.0292 | 0.983 | 0.984
DJINN | 0.00154 | 0.02741 | 0.98403 | 0.98404
Gaussian Process | 0.00185 | 0.0305 | 0.981 | 0.981
Table 5.2: Evaluation matrix: the mean-square-error, the mean-absolute-error,
$R^{2}$, and the explained variance of the predictions in test dataset using
four models.

All codes are written in Python. After training the models, we predict the
electron beam charge using the laser wavefront change in the test dataset.
Predicted electron beam charges using the above models are shown in Fig. 5.13
against measured electron beam charges. A reference line at $45^{\circ}$ is
included, and data points closer to the reference line are considered better
predictions. The bottom left corner of the plot is magnified and shown to the
right. Detailed statistical evaluations are summarized in Tab. 5.2, in which
we report the mean-square-error (MSE), mean-absolute-error (MAE), R-squared
($R^{2}$), and explained variance score (ExVar) based on the predicted charge
and the measured charge. MSE measures the average squared difference between
the predictions and the real values, which contains information of both
variance and bias. It is the most popular metric when evaluating machine
learning models and we use MSE as the target for the hyperparameter tuning
process. The problematic aspect of MSE is that it can be sensitive to
outliers, which MAE handles better by measuring the absolute error instead of
the squared error. $R^{2}$ is the proportion of variance of the measured value
from the prediction. It tells how likely a new sample (out of the dataset) can
be predicted by the model. Explained variance considers bias on top of
$R^{2}$. It is the same as $R^{2}$ if the mean of error is 0. In general, one
would like to have MSE and MAE close to 0 while $R^{2}$ and ExVar close to
unity.

All four models demonstrate similar statistics in Tab. 5.2, though RF performs
slightly better and GP gives the largest MSE and MAE scores while the smallest
$R^{2}$ and ExVar scores. It also predicts negative values when the electron
beam charges are small. However, it does not necessarily mean that Random
Forest is the best model and Gaussian Process is the worst. The results in the
evaluation matrix are sensitive to the way we split the training set and test
set. We will show in the next section that training and evaluating the model
on different data points yield different results. We will also present more
analyses, such as the model consistency against measurement errors and
overfitting-related issues.

#####  5.3.3.1 Data quality

Figure 5.14: Prediction error when testing ML models with every data point.
The test data point is presented in an order where the DJINN prediction error
is monotonically increasing. (a) compares the results from DJINN, GP and RF,
and (b) compares the results from DJINN and DNN.

Experimental measurements in LWFA can suffer from a lack of reproducibility
and may have outliers in the dataset. Possible sources include shot to shot
fluctuations in the beam pointing and pulse energy in high power laser
systems, as well as the irreproducibility in the plasma density profile from
gas jets. A natural question to ask is how much we can trust each of the
measured data points. In this section, we train machine learning models to
justify the quality of our measured electron beam charge. Instead of splitting
the dataset into $80\%$ for training and $20\%$ for testing, we test only one
data point while all the other data points are used to train the model. We
then compare the predicted beam charge to the measured electron beam charge of
this particular data point and calculate their difference. This process is
looped over the entire dataset. Fig. 5.14a plots the prediction error
($\sigma=|y_{predicted}-y_{measured}|$) at each data point from GP, RF, and
DJINN. The prediction errors from DNN are shown in Fig. 5.14b. Fig. 5.14 is
presented in a monotonic order of the DJINN prediction error. It is observed
that the three models in Fig. 5.14a have a similar trend while the prediction
errors from DNN in Fig. 5.14b behave differently. We interpret three messages
from these two plots. 1. Prediction errors vary across seven orders of
magnitude, i.e., some data points can be accurately predicted ($\sigma\sim
1e-8$) while some data points can hardly be predicted ($\sigma\sim 0.1$).
Therefore, selecting different data points into the training or test set can
lead to different evaluation matrices from the one in Tab. 5.2. 2. This huge
variation can be caused by either the inconsistency of the model across data
points, or this specific data being very different from all the other data
points in the dataset that are used to train the model. We are satisfied with
the reliability of the models, as is characterized in Tab. 5.2. In addition,
the similarity among the three models’ performance in Fig. 5.14a suggests that
the models are less likely to be inconsistent at the same time since they
behave in a similar manner regarding the data points. Thus it provides a
potential characterization of the quality of each data point, i,e., a data
point that has a very large prediction error in all three models can be
considered as having poor data quality and may be dropped as an outlier. 3.
DNN performs differently from the other models, suggesting that it overfits
the data set and it is less reliable in this scenario. It matches the learning
curves in Fig. 5.16, and a detailed discussion on the overfitting issue can be
found later.

Evaluating the prediction error on every data point can assist anomaly
detection. For example, if we drive the same laser pulse into the laser plasma
accelerator twice, we might observe different electron beam charges due to a
lack of reproducibility in LWFA. A typical solution would be to calculate the
mean value and the error bar. However, it could include misinformation if one
of the measurement is an outlier due to technical glitch and should be
dropped. By performing the above analysis, we would be able to tell which one
of the measured beam charges is more reliable. Moreover, it can help identify
outliers not only in repeated measurements but anywhere in the parameter space
that the experiment scans across. The data points in the top-right corner in
Fig. 5.14a are examples of possibly poor data quality, where all three models
have large prediction errors.

#####  5.3.3.2 Robustness against measurement errors

Figure 5.15: Model performance against virtual measurement errors. The figures
on the left show the distribution of test MSE using RF (a), GP (b), DJINN (c),
and DNN (d). The color indicates the amount of virtual measurement error
applied. The tables on the right list the mean value, standard deviation
($\sigma$), and the percentage of points that fall within one, two, or three
standard deviations around the mean value.

Another fundamental question to ask about the data is if they are sufficient
for the machine learning models to make accurate predictions. It is crucial
for experimental data considering all experimental measurements are associated
with some degrees of uncertainties. In this section, we investigate the
performance of these models against measurement errors of the electron beam
charges. Since measurement errors were not recorded during our experiments, we
include various virtual error bars to every measured electron beam charge. At
each measurement, the true value is assumed to lie in the range of measured
value $\cdot(1\pm X\%)$, where $X=1,3,5,10)$, and 1000 points are drawn
randomly from a normal distribution within this range. Therefore we get 1000
copies of the original dataset with the same wavefront but different electron
beam charges. The reason to have 1000 datasets is to generate enough
statistics to justify the model performance against unsure measurements.
Results are presented in Fig . 5.15.

Fig. 5.15a shows the distribution of the test MSE using RF. Each colored line
is generated from 1000 MSEs. During the training process, the model
configuration was kept the same among the 1000 datasets but the weight
learning was updated in each dataset. Measurement errors that define the range
of the dataset fluctuation are $1\%,\;3\%,\;5\%$ and $10\%$, while the
corresponding test MSE distributions are plotted in red, green, blue, and
black, respectively. Statistical analysis is summarized in the adjacent table
to the right, illustrating the mean value and standard deviation ($\sigma$) of
the 1000 test MSEs as well as the distribution within one, two, or three
standard deviations around the mean value. Fig. 5.15b-d present the results
using GP, DJINN, and DNN. The four models share some common performances. The
mean test MSE value increases with the measurement error, which means it is
more likely to make a less accurate prediction when the measurement itself is
less accurate, as expected. The standard deviation also increases with the
measurement error, suggesting a less consistent or less precise model