prediction at larger measurement errors. There are noticeable differences in
the last three table columns of the four models. Remember that the virtual
measurement errors are drawn from a perfect normal distribution, where the
percentage of values that lie within one, two, or three standard deviations
around the mean value are $68.3\%$, $95.5\%$, and $99.7\%$, respectively. As
is shown in the tables, RF and GP retain almost normal distributions in the
sample prediction, DNN gives normal-like distributions, and the results from
DJINN are far from normal distributions.

#####  5.3.3.3 Learning curve

Figure 5.16: Learning curves in the neural network model: including
measurement errors decreases overfitting. Training and test MSE without
measurement errors are plotted in black and red, while training and test MSE
with measurement errors are plotted in blue and yellow, respectively.

Overfitting occurs in machine learning when a model has learned the training
data so well that it also learns the statistical noise or random fluctuations
in the data. The learning curve is an intuitive tool to visualize the degree
of overfitting. Fig. 5.16 shows the learning curves in our Neural Network
model, which plots MSE at each epoch for both training data (black) and test
data (red). The training curve tells how well the model learns, while the test
curve tells how well the model generalizes. Since the red curve does not
decrease as much as the black curve does, the model overfits. In other words,
the overfitted model performs worse outside the training dataset. Fig. 5.16
also plots the learning curves considering measurement errors in blue and
orange to better reproduce experimental conditions. Measurement errors are
included in a similar way to the ones in the previous section, where the true
value is assumed to lie in the range of measured value$\cdot(1\pm 3\%)$.
Instead of generating 1000 copies of datasets at a time, here we generate only
one dataset with measurement errors and update the measurement errors at every
epoch in the learning process. Obtained learning curves are noisier but
demonstrate less overfitting, as is shown in Fig. 5.16. After applying
measurement errors, the training curve moves higher, but the test curve does
not shift much. Namely, the model finds it harder to learn, but it is still
able to make equally accurate predictions. Therefore, including virtual
measurement errors is beneficial as it not only represents practical
experimental conditions better but also decreases overfitting. Note that we
have also plotted the learning curve with and without measurement errors using
the DJINN model, and they almost overlap with each other. It is not surprising
since the DJINN model does not overfit as much as the neural network model
does.

#####  5.3.3.4 Feature importance

Model | Most important | $2^{nd}$ | $3^{rd}$ | $4^{th}$
---|---|---|---|---
Random Forest | $z_{0}$ | $z_{6}$ | $z_{1}$ | $z_{10}$
Gaussian Process | $z_{1}$ | $z_{10}$ | $z_{0}$ | $z_{6}$
Neural Network | $z_{0}$ | $z_{10}$ | $z_{1}$ | $z_{3}$
DJINN | $z_{1}$ | $z_{10}$ | $z_{0}$ | $z_{6}$
Correlation | $z_{0}$ | $z_{10}$ | $z_{1}$ | $z_{13}$
Table 5.3: Feature importance according to the test dataset and correlation raking. The Zernike coefficients (features) are ranked by their importance to producing high beam charge in the columns. Model | Most important | $2^{nd}$ | $3^{rd}$ | $4^{th}$
---|---|---|---|---
Random Forest | $z_{0}$ | $z_{6}$ | $z_{1}$ | $z_{12}$
Gaussian Process | $z_{10}$ | $z_{1}$ | $z_{0}$ | $z_{6}$
Neural Network | $z_{0}$ | $z_{10}$ | $z_{1}$ | $z_{8}$
DJINN | $z_{1}$ | $z_{10}$ | $z_{0}$ | $z_{6}$
Correlation | $z_{10}$ | $z_{0}$ | $z_{12}$ | $z_{13}$
GA subset 1 | $z_{0}$ | $z_{13}$ | $z_{9}$ | $z_{1}$
GA subset 2 | $z_{0}$ | $z_{2}$ | $z_{8}$ | $z_{4}$
GA subset 3 | $z_{0}$ | $z_{10}$ | $z_{14}$ | $z_{1}$
GA subset 4 | $z_{0}$ | $z_{8}$ | $z_{12}$ | $z_{6}$
Table 5.4: Feature importance according to the entire dataset, correlation
ranking, and wavefront of the optimized electron beam using genetic
algorithms. The Zernike coefficients (features) are ranked by their importance
to producing high beam charge in the columns. The genetic algorithm ran
$\sim$50 iterations in each subset.

We have trained models to predict the electron beam charge upon laser
wavefront modification represented by the first 15 Zernike coefficients. It is
natural to ask how sensitive the beam charge is to these features. We evaluate
the feature importance using our four models and compare them to the
correlation ranking, summarized in Tab. 5.3 and 5.4. In each row, we list the
four most important features decided by that model, while the numbers are the
orders of the Zernike coefficients. The importance of a feature is measured by
calculating the increase in the model’s prediction error (MSE) after setting
the feature values to a constant. A feature is considered important if the
prediction error increases significantly, and less important when the
prediction error does not change much. When evaluating each feature, the
values of this feature of all test data are set to their mean value. Note that
model training is performed prior to this process and the training data are
not modified. It is debatable whether the feature importance should be
computed on the test data or the training data. The former tells how much the
model relies on each feature for making predictions, while the latter tells
how much the feature contributes to the performance of the model on unseen
data [287]. We show the feature importance computed from both ways. In Tab.
5.3, we split the dataset to a training set ($80\%$) and a test set ($20\%$)
and evaluate the feature importance on the test set. In Tab. 5.4, we train the
model using the entire dataset and measure the feature importance also on the
entire dataset. For comparison, Tab. 5.4 also includes the laser wavefront
that optimized the electron beam charge using genetic algorithms, where each
subset contains $\sim$50 data points.

Depending on the model and the evaluation data, the feature importance
rankings in Tab. 5.3 and 5.4 are slightly different. Overall, the $0^{th}$,
$1^{st}$, $6^{th}$, and $10^{th}$ Zernike terms are generally believed to be
the more important ones according to the trained models, physically
representing the piston, tilt, vertical trefoil, and oblique quadrafoil in
wavefront aberration, respectively. It suggests that controlling these
features would be more effective at producing high electron beam charges in
this scenario. Similar to the machine learning models, the statistical
correlation and the genetic algorithm optimization emphasize on the low-order
($0^{th}$ and $1^{st}$) and high-order ($\geq 10^{th}$) Zernike terms.
However, they ca not recognize the $6^{th}$ Zernike term that repeatedly shows
up in the feature importance ranking among the ML models except the DNN. The
fact that the DNN is more overfitted and behaves differently from other models
in Fig. 5.14 adds validity to this finding on the $6^{th}$ Zernike term. It
indicates that machine learning can reveal information that can not be
observed in genetic algorithms or statistical correlation who do not involve
configuring models.

####  5.3.4 Discussion

When tuning the wavefront of the laser beam that drives an laser-wakefield
accelerator, the highest electron beam charge or the best electron beam
profile are not necessarily associated with a flat laser wavefront, even
though a flat wavefront produces the highest vacuum focal intensity. This has
been shown in previous experiments using genetic algorithms [73, 81]. As the
laser pulse propagates in the plasma, its wavefront can be altered by
nonlinear interactions such as self-modulation, self-focusing, Raman
scattering and even the ionization dynamics before it reaches the vacuum
focus. Moreover, a wavefront with aberration can move the focal point along
the plasma density gradient and thus affects the energy gain in the produced
electrons, which is dependent on the plasma density. Such a non-flat optimal
wavefront has also been observed in other relativistic laser-plasma
experiments when changing the wavefront, for example filament-induced
breakdown spectroscopy studies [288, 289, 83], which are sensitive to the
laser beam ellipticity and the ionization in the plasma.

There has been growing interest in the relativistic laser-plasma community to
determine what value machine learning can provide. Machine learning methods
are not expected to offer some generalized predictive models that save
experimentalists from carrying out every experiment. This is due to the lack
of reproducibility in high-power laser-plasma experiments, where laser systems
usually suffer from shot-to-shot fluctuations and a given plasma density
profile is hard to duplicate. However, using machine learning techniques can
help us better understand the experiment performance and improve the design of
next-step experiments. Tab. 5.4 has shown that machine learning can reveal
deeper information that are buried in the statistical correlation or the
optimization results from genetic algorithms. Moreover, it enables deeper
physical interpretation of the data since the predictive accuracy of the
regression models is determined by the data quality. For example, by ranking
the feature importance, we are able to identify the Zernike terms that are
most sensitive to noise. Some of these turned out to be the high order terms
(vertical trefoil and oblique quadrafoil) that located at the edge of the
wavefront. The ”importance” of these high order terms can be explained from an
experimental point of view as follows. Deformable mirrors are manufactured to
have actuators forming into a hexagon matrix, while a wavefront is usually
defined in a circular or rectangular shape, leading to lack of information on
the edge. If the wavefront is measured directly with wavefront sensors, it is
usually necessary to manually draw a circle that covers most lighted pixels on
the detector as the region of interest. As a result, uncertainty arises at the
pixels on the edges of the region of interest. If the wavefront is
reconstructed using the actuator displacement on the deformable mirror
surface, the phase in these unknown edges also needs to be defined manually.
Another possible source of noise is the imperfect overlap of the laser and the
deformable mirror surface: either the laser beam clipped off the mirror edge,
or it did not fulfill the whole mirror surface.

Including virtual measurement uncertainties can be useful even if the
experimental data come with some uncertainties. It may be able to narrow down
the range of uncertainty when the measurement uncertainty is large. For
instance, if a data point in the dataset has some considerable uncertainty and
we would like to know the true measurement value or at least narrow down its
range, machine learning regression methods can provide us a possible solution.
We can start by randomly sampling N points within the range of uncertainty of
this measurement value. The next step is to make N copies of the measurement
dataset and replace the point of large uncertainty with one of our sampled
points in each dataset. Therefore, we obtain N similar datasets with
difference only at one point. We then train and test the machine learning
models on each dataset. Those datasets that lead to less accurate predictions
(large test MSE) are less likely to contain true measurement at the uncertain
point if the models make sense.

It is also worth discussing the physics interpretation induced from the kernel
functions in the Gaussian Process method. The common way of kernel selecting
is either to have expert knowledge about the dataset or to compare candidate
kernels for the best performance. In this project, we have tried different
kernels and decide that a combination of Matern kernel and Rational Quadratic
kernel works best. We can thus infer some knowledge about our dataset based on
the rationale of these kernels. Matern is a generalization of the popular
Gaussian radial basis function (RBF) with tunable smoothness [134]. The
smoothness of the model can be controlled by a parameter $\mu$ while the $\mu$
value in our case is as low as 0.3, suggesting that the resulting function is
far from smooth. The other kernel that fits our model, the Rational Quadratic
kernel, is a sum of RBF kernels with different length scales. Note that the
length scale decides the safe distance to extrapolate when modeling, and the
discontinuity can be handled with a short length scale. According to the
optimized parameters in these two kernels, our model function is neither
smooth nor continuous, which is not surprising as the high-dimensional dataset
was taken from a highly-nonlinear physical process.

In summary, we demonstrate several applications of machine learning in
relativistic laser-plasma experiments beyond optimization purposes. We have
built four supervised learning regression models to predict the electron beam
charge using the laser wavefront change in an LWFA experiment. All four models
present similar statistics in the evaluation matrix, although Random Forest
performs slightly better and Gaussian Process performs slightly worse. To
justify the data quality affected by the irreproducibility in experiments, we
characterize the model prediction on every single data point. Three of the
models show similar performance, providing a potential way of recognizing
outliers without repeated measurements. The Deep Neural Network is easy to
overfit our dataset and thus not the best candidate for analyzing data
quality. We include virtual measurement errors to the measured electron beam
charges, where Gaussian Process and Random Forest are found to be less
sensitive to measurement fluctuations. Having virtual measurement errors is
beneficial as it not only represents experimental conditions better but also
decreases overfitting. The significance of the Zernike coefficients in terms
of generating high electron beam charge is analyzed using the trained models,
which reveals more information than the genetic algorithms and the statistical
correlation can provide. The Deep Neural Network requires the most
computational cost, followed by DJINN, while Gaussian Process and Random
Forest consume the least. Therefore, Random Forest is recommended when working
with datasets from similar relativistic laser-plasma experiments.

##  Chapter VI Conclusions and Outlook

####  6.0.1 Summary

In this dissertation, six experiments are presented on electron acceleration
and radiation generation using relativistic laser-plasma interactions at high
repetition rates. This work demonstrates radiation produced via surface HHG
and via characteristic x-ray emission, and relativistic electrons accelerated
via LWFA and from solid-density plasmas. The main objects of this work are to
investigate the impact of long-wavelength laser pulses in the interactions,
and to explore the application of statistical methods in the experiments given
the high-repetition-rate capability.

The three experiments presented in Chap. IV were performed with bulk solid
targets at the Lambda-cubed at CUOS. In Sec. 4.2, HHG spectra and harmonic
divergence were measured experimentally when 2 $\mu m$ laser pulses interacted
with silicon and glass targets. The harmonic efficiency scales with harmonic
order in a power law as $I(\omega)\propto(\omega/\omega_{L})^{-2.752}$, which
is close to the frequently quoted value of -8/3 predicted by the ROM model.
The scaling law of the (third) harmonic efficiency vs. laser intensity is also
studied in a power-fit $I_{3\omega}\propto I_{L}^{n}$, which suggests a
nonlinear scaling (n$\sim$2) predicted by the ROM model. The intensities of
harmonics polarized in horizontal and vertical directions are characterized
when the driving laser pulses are polarized in horizontal, vertical, left-
circular, and right-circular directions. For linearly-polarized driving
pulses, the measurements do not display the distinct feature in even and odd
harmonics predicted by the unique selection rule of the ROM model. For
circularly-polarized laser pulses, both even and odd harmonics were observed,
which is in line with the ROM selection rule. Overall, generating
horizontally-polarized harmonics with P-polarized interactions yields the
highest laser-to-harmonic efficiency.

In Sec. 4.3, MeV-level attosecond electron bunches are studied when driving a
laser pulse onto a glass target at grazing incidence. The experimental energy
spectra match the spectra of attosecond electron bunches observed in PIC
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. It is found that the generation of energetic attosecond
electron bunches favors larger incident angle, higher pulse energy, larger
focal spot size, and moderately sharp preplasma density profile. Single-cycle
pulses are used to obtain isolated attosecond electron bunches in PIC
simulations. 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 changes as the single-cycle pulses get focused and
propagate through the plasmas. The propagation direction of the electron bunch
is dominated by the preplasma density profile, while fine-tuning is accessible
by varying the CEP. Higher $a_{0}$ and larger focal spot size can result in
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. A
simplified analytic model is used to predict the momentum gain of electrons,
and short electron bunch duration is found to favor large incident angle and
long laser wavelength.

In Sec. 4.4, the key features in characteristic x-ray emission ($k_{\alpha}$
emission efficiency, bremsstrahlung emission efficiency, and electron
temperature) are investigated at various tuning parameters in the laser-plasma
experiment (laser pulse wavelength, laser pulse energy, and preplasma density
gradient). The parametric study is performed both experimentally and
computationally with PIC simulations. Among the three controlling parameters,
the laser wavelength is the most dominant one regarding the $k_{\alpha}$
emission efficiency. The wavelength dependence is monotonic for both
$k_{\alpha}$ efficiency and for bremsstrahlung efficiency, while both favor
short laser wavelengths. Such a dependence of the x-ray efficiency is
determined by the fact that the critical surface is located closer to the
solid density surface at a short laser wavelength, making it easier for the
energetic electrons to reach the atoms in the solid target. The hot electron
temperature, however, no longer has a monotonic dependence on laser wavelength
since the laser pulse propagates deeper at a shorter laser wavelength while
the ponderomotive force favors a longer wavelength. The other two tuning
parameters have smaller impact on the $k_{\alpha}$ emission efficiency but are
crucial for the bremsstrahlung efficiency and the hot electron temperature.
The electron temperature favors a larger $a_{0}$ but sees an optimal preplasma
density scale length, while the bremsstrahlung efficiency sees an optimal
preplasma density scale length and an optimal $a_{0}$. Moreover, simulations
using various control parameters but the same similarity parameter is
performed to reveal the contribution of relativistic electrons to $k_{\alpha}$
emission.

The three experiments presented in Chap. V were performed with gaseous plasma
targets assisted by statistical methods. In Sec. 5.1, a deformable mirror and
a genetic algorithm are used to optimize the high numerical aperture focal
spot of a 800 nm, 30 fs, 3 mJ pulse and of a 2 $\mu m$, 67 fs, 1.6 mJ pulse.
The focus optimizations are performed at relativistic intensity without
attenuation. The laser pulses were directed into rarefied gas to produce
second harmonic signal, which was found to be a convenient and effective
feedback for the genetic algorithm. This technique provides significant
experimental convenience compared to other methods that require attenuating
the laser or breaking the vacuum condition in between the focus optimization
and the laser-plasma experiments.

In Sec. 5.2, the first experiment to optimize the quality of the electron beam
from mid-IR ($\lambda=3.9\mu m$) laser pulses interacting with near-critical
density plasmas is presented. Electron beam charge, energy spectrum, beam
pointing and fluctuation have been improved by manipulating the laser
wavefront via an evolutionary algorithm and a deformable mirror. 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.

In sec. 5.3, applications beyond optimization purposes of machine learning in
LWFA are demonstrated. Four supervised learning regression models are built to
predict the electron beam charge using the laser wavefront change caused by a
deformable mirror. The quality of the measurement is examined by evaluating
the model performance on every measured data point, showing 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,
the model robustness is characterized against a range of virtual errors
assigned to the data. Feature importance analysis using the trained models
shows that specific aberrations in the laser wavefront are favored in
generating higher beam charges, which reveals more information than the
genetic algorithms and the statistical correlation do. Overall, Random Forest
works best for this experimental dataset considering the model performance and
the computational cost.

####  6.0.2 Future work

The reported results in this dissertation may guide several follow-up
experiments. In Sec. 4.3, the duration of the ultrashort electron bunches is
measured only in PIC simulations. Future experiments could measure the bunch
duration by diagnosing the COTR of the produced electron bunches. The
simplified model suggests that a longer wavelength laser would produce
electron bunches of shorter duration. It would be interesting to verify that
experimentally using mid-infrared laser pulses. In Sec. 4.4, the parametric
study on characteristic x-ray emission can be further investigated both
experimentally and computationally. Future experiments could endeavor to
employ more laser wavelengths to enrich the validity of the parametric study,
such as 400 nm pulses from frequency doubling the 800 nm pulses and 1.3 $\mu
m$ pulses from the idler of the 2 $\mu m$ OPA. Future simulation work should
take into account the ionization physics to differentiate the difference in
driving laser wavelengths. The current OSIRIS framework only provides a field
ionization module, which is not the main source of ionization in solid-density
plasma. Collisional ionization is to be included in future PIC simulations. In
Sec. 5.1, two-fold optimization has been presented using the second harmonic
signal of the laser pulses when focusing in a rarefied gas. Despite the
exceptional convenience, this method is not as efficient as other methods that
require attenuating the laser or breaking the vacuum condition [105]. To
better understand the physics process that leads to the SHG in such
conditions, a power-law scaling for the SHG at various laser intensities is to
be investigated with more complete measurements. In Sec. 5.2, the optimization
of the LWFA electrons driven by a mid-IR pulse is achieved by a deformable
mirror and a genetic algorithm. Future experiments could extend it to
optimization using temporal control of the pulse with a Dazzler.

Machine learning as a tool for relativistic laser-plasma experiments has been
preliminarily explored in Sec. 5.3, and more ambitious applications are
expected in the future. In this study, in order to avoid the complexity of
image processing, the laser wavefront is converted to the Zernike polynomials
and the electron beam charge is used instead of the electron beam profile.
Future work could employ a convolutional neural network (CNN) to address this
issue. Instead of quantifying the image into a number (FOM) in traditional
methods or analyzing the raw image in neural networks, a CNN can extract much
more information from the image, such as recognizing a specific pattern. Its
ability to deal with complex images should find wide applications in
diagnosing the particles and radiation produced in relativistic laser-plasma
experiments. The concept of reinforcement learning (RL) is another category of
machine learning to be explored to assist high-intensity laser facilities.
While most of the previous optimization experiments were performed with high-
repetition-rate lasers $\sim$kHz, a number of laser systems with lower
repetition rates $\sim$Hz but much higher intensities are emerging, for
example. the ZEUS laser system at CUOS. To run optimization experiments on
these laser systems, algorithms with higher efficiency are needed. Receiving
growing interests in the past few years in the machine learning community, RL
should also be able to integrate with high-intensity laser via adaptive
optical systems. There is potential for RL to take over evolutionary
algorithms to enable more powerful and more efficient real-time interpretation
and optimization. For lasers with the highest power and record intensities,
machine learning could also find usefulness despite the fact that such lasers
usually fire only a few shots per day. Future work could utilize transfer
learning to narrow the gap between expensive experimental results and low-cost
simulation results, which allows to construct a predictive model using the
latter and to transfer the learned knowledge to a model constructed by the
former.

APPENDICES

APPENDIX A

Example OSIRIS input deck

⬇

\par! 2um solid target exp - Molybdenum

! scale length = 0.5*lambda

! laser wavelength = 2um, pulse duration = 50fs, spot size = 2*lambda FWHM
diameter

! In Vacuum

! written on: 2/23/2021 by Jinpu Lin<EMAIL_ADDRESS>
! —————————————————

! Units normalised to laser units

! t0 to To, 6.67 fs, which is 2um/3e8m/s

! x0 to lambda0, 2 um

! omega0 = 2pi, 6.283

! Grid size [1/64, 1/64] lambda0

! 20ps prepulse - 0.5lambda

! —————————————————

\parsimulation

{

n0 = 7.06e18, ! [cm^-3], n0 = nc/4pi^2

}

\par!———-the node configuration for this simulation———-

node_conf

{

node_number(1:2) = 4,4,

if_periodic(1:2) = .false., .false.,

}

\par\par!———-spatial grid———-

grid

{

nx_p(1:2) = 1280, 1280, ! (20) * 64 = 1280

coordinates = ”cartesian”,

}

\par\par!———-time step and global data dump timestep number———-

time_step

{

dt = 0.0034739, ! dt = 0.995 * sqrt[1/(c^2/dx^2 + c^2/dy^2)] / T0 ;
dt<0.5/sqrt(20716), 20716*no in density profile, choose the smaller dt

ndump = 1079, ! 1079 iterations * 0.0034739 * T0 = 25 fs. The time step for
output is 25 fs, pulse duration is 50fs

}

\par!———-restart information———-

restart

{

ndump_fac = 0,

if_restart = .false.,

}

\par!———-spatial limits of the simulations———-

!(note that this includes information about

! the motion of the simulation box)

space

{

xmin(1:2) = 0.000d0, 0d0,

xmax(1:2) = 20d0, 20d0, !20*2um=40um in x1 and x2

if_move= .false., .false.,

}

\par!———-time limits ———-

time

{

tmin = 0.0d0, tmax = 53, ! total simulation time is 53*T0=350fs

}

\parel_mag_fld

{

\par}

\par!———-boundary conditions for em-fields ———-

emf_bound

{

type(1:2,1) = ”open”, ”open”,

type(1:2,2) = ”open”, ”open”,

}

\par\par!———-diagnostic for electromagnetic fields———-

diag_emf

{

ndump_fac = 1,

reports = ”e1”, ”e2”, ”ene_emf”,

}

\par!———-number of particle species———-

!———-number of particle species———-

particles

{

num_species = 2,

num_neutral = 0,

interpolation = ”cubic”,

}

\par\par\parspecies

{

name = ”preplasma”,

num_par_max = 1.0d9,

rqm = -1.000, ! Means ’electron’

num_par_x(1:2) = 10, 10,

add_tag = .true.,

}

\par!———-inital proper velocities—————–

udist {

!uth(1:3) = 0.014, 0.014, 0.014, !100eV. 1 keV calculated as
sqrt(1/rqm)*uth_e, where uth_e = sqrt(T/m_ec^2)

uth(1:3) = 0.0014, 0.0014, 0.0014, !1eV

ufl(1:3) = 0.0d0 , 0.0d0 , 0.0d0 ,

}

profile

{

density = 1,

\parprofile_type(1:2) = ”math func” ,

\parmath_func_expr =
”39.48*(x2+sqrt(2)*0.4*log(20716/39.48)>=x1)*exp(((x1-x2)/sqrt(2))/0.4)”,
!L1=0.4*log(20716/39.48) is distance between critical sueface and solid
surface. ncr=39.48*n0.

\par}

\parspe_bound

{

type(1:2,1) = ”open”, ”open”,

type(1:2,2) = ”open”, ”open”,

}

\pardiag_species

{

ndump_fac = 1,

reports = ”charge”,

\parndump_fac_pha = 1,

ndump_fac_raw = 1,

!raw_math_expr = ”t > 22”, !select particles at time > 30*T0, right before
focus

!raw_math_expr = ”step(g-1.03)*step(t-22)”,!>15kev, gamma>1.03

raw_math_expr = ”step(g-1.03)*step(x1-x2)”,!all electrons in target higher
than k_alpha energy

\parps_gammamin = 1.0,

ps_gammamax = 20.0,

ps_ngamma = 1000,

\par\parphasespaces = ”g”,

}

\par\parspecies

{

name = ”eTarget”, !ionized electrons

num_par_max = 1.0d9,

rqm = -1.000, ! Means ’electron’

num_par_x(1:2) = 10, 10,

add_tag = .true.,

}

\par!———-inital proper velocities—————–

udist {

uth(1:3) = 0.00626, 0.00626, 0.00626, !20 eV.

ufl(1:3) = 0.0d0 , 0.0d0 , 0.0d0 ,

}

\parprofile

{

\pardensity = 1,

\parprofile_type(1:2) = ”math func” ,

\parmath_func_expr = ”20716*(x1-sqrt(2)*0.4*log(20716/39.48)>x2)”,

\par}

\par\parspe_bound

{

type(1:2,1) = ”open”, ”open”,

type(1:2,2) = ”open”, ”open”,

}

\pardiag_species

{

\parndump_fac = 1,

reports = ”charge”,

\parndump_fac_pha = 1,

ndump_fac_raw = 1,

!raw_math_expr = ”t > 22”, !select particles at time > 30*T0, right before
focus

!raw_math_expr = ”step(g-1.4)*step(t-22)”,!>200kev, gamma>1.4

raw_math_expr = ”step(g-1.03)*step(x1-x2)”,!all electrons in target higher
than k_alpha energy

\parps_gammamin = 1.0,

ps_gammamax = 20.0,

ps_ngamma = 1000,

\parphasespaces = ”g”,

}

\par\parzpulse_wall

{

a0 = 2.5,

omega0 = 6.283,

pol = 0,

propagation = ”forward”,

tenv_type = ”gaussian”,

tenv_duration = 18.5, ! fwhm = 50fs

tenv_range = 74, ! total length of the laser pulse. ~ 4*fwhm

\parper_type = ”gaussian”,

per_center = 10,

per_w0(1:1) = 1.7, !with a spot size (radius) of ?*lambda. fwhm = 2*lambda.
w0=fwhm*1.699/2. keep this the same for all wavelengths

per_focus = 10, !position of focus. depth. focused at critical density

}

\par\par! ——————— end of osiris input file —————

\par

APPENDIX B

Iterative bash submission for parameter scan

⬇

#!/bin/bash

\par# Dec 2020, by Jinpu Lin

#This script will run all of my simplified simulation files

#Need an ”inputDeck” for the software (OSIRIS, etc), and a ”runscript.sh” for
the computing system, both under the same root path

#Edit file and save: sed ’s/”p”/0/g’ file.txt > fileNew.txt. ”g” stands for
global

# remove file: rm

\par#Loop over a0

for i in {1..17}

do

#Loop over scale length

for j in {1..55}

# for i in {1..56..5}

# for j in 0.1 0.5 1

do

((x=$i/10))

((y=$i-10*$x))

# sed s/a0value/printf sed s/a0value/$x.$y/g inputDeck>temp

# echo $x

# echo $y

((c=$j/10))

((d=$j-10*$c))

sed s/scaleLength/$c.$d/g temp>a_${x}.${y}_L_${c}.${d}

# sed s/scaleLength/$c.$d/ temp>a_$x.$y_L_$c.$d

sed s/deckname/a_${x}.${y}_L_${c}.${d}/g
runscript.sh>runscripta_${x}.${y}_L_${c}.${d}.sh

sbatch runscripta_${x}.${y}_L_${c}.${d}.sh

rm temp

rm runscripta_${x}.${y}_L_${c}.${d}.sh

# rm a_${x}.${y}_L_${c}.${d}

done

done

\parclear

\par

APPENDIX C

Characteristic x-ray emission analysis algorithm

⬇

\par# -*- coding: utf-8 -*-

”””

Created on Wed Feb 3 20:10:43 2021

\par@author: linji

”””

\parimport os

import numpy as np

import h5py as h5

import scipy.constants as const

from scipy.optimize import curve_fit

import math

\parpathRoot =
’/Users/jimlin/Desktop/umich/CUOS/OSIRIS/dataFile/solidXrays/scan3param/1.6um/all/’

pathResult =
’/Users/jimlin/Desktop/umich/CUOS/OSIRIS/results/solidXrays/scan3param/1.6um/postProcess/’

\par\par# calculate kalpha emission caused by electron impact of all energies

def IonizeXsec(ene):

ene=ene*1000 #input energy in kev, converts into ev

z=42;en=1.997e4;a00=5.29177210903e-11 #Bohr radius

u=ene/en

if ene<16*en:

a1=1.281e-3;a2=4.105e-5;a3=-1.410e-3;a4=1.450e-3;a5=-1.642e-3

xsec=4*math.pi*a00**2*(u-1)*u**(-2)*(a1+a2*u+a3/(1+u)+a4/(1+u)**3+a5/(1+u)**5)**2

else:

bm=2.692e-1;anj=2.647e-8;g1=7.942e-1;g2=5.981;g3=-2.59;g4=1.369

beta=math.sqrt(ene*(ene+2*const.m_e*const.c**2))/(ene+const.m_e*const.c**2)

X=math.sqrt(ene*(ene+2*const.m_e*const.c**2))/(const.m_e*const.c**2)

pwba=4*math.pi*a00**2*anj*beta**(-2)*((math.log(X**2)-beta**2)*(1+g1/X)+g2+g3*(1-beta**2)**0.25+g4/X)

xsec=pwba*ene/(ene+bm*en)

xsec=xsec*1e25

return xsec

\pardef getXrays(pathJob, a0, ls):

path_rawI = pathJob + ’/MS/RAW/eTarget/’

path_raw = pathJob + ’/MS/RAW/preplasma/’

path_gammaI = pathJob + ’/MS/PHA/gamma/eTarget/’

path_gamma = pathJob + ’/MS/PHA/gamma/preplasma/’

x0 = 1.6

n0 = 1.103e19

ns = 5800

\parkalpha=[];

for i in range(0, 22):

f_raw=h5.File(path_raw+”RAW-preplasma-ene = f_raw[’ene’][()]; #local kinetic
energy

x1 = f_raw[’x1’][()];

x2 = f_raw[’x2’][()];

\parf_rawI=h5.File(path_rawI+”RAW-eTarget-eneI = f_rawI[’ene’][()]; #local
kinetic energy

x1I = f_rawI[’x1’][()];

x2I = f_rawI[’x2’][()];

x1T=np.hstack([x1,x1I])

x2T=np.hstack([x2,x2I])

eneT=np.hstack([ene,eneI])

x1T = x1T * x0;

x2T = x2T * x0;

\parenetarget=eneT[(x1T-x0*np.sqrt(2)*ls*np.log(ns/39.48)>x2T)]

\parxsec=enetarget+1; #define an array of the same size as eneT

kyield=enetarget+1; #define an array of the same size as eneT

for j in range(0,enetarget.size):

xsec[j]=IonizeXsec(enetarget[j]*1e3)

kyield[j]=xsec[j]*np.sqrt(1-(enetarget[j]+1)**(-2)) #x-ray yield ~ corss
section * particle velocity

\parkyieldArray = np.array(kyield)

kyieldArray[np.isneginf(kyieldArray)] = 0

kalpha.append(sum(kyieldArray))

\par\par# fit Te

allTemp=[];

allTempErr=[];

allBrem=[];

def log_func(E,a,T):

return (a-E/T)

\parfor i in range(14, 22):

f_gamma=h5.File(path_gamma+”gamma-preplasma-
f_gammaI=h5.File(path_gammaI+”gamma-eTarget-gamma = f_gamma[’gamma’][()];

xstart = f_gamma[’AXIS’][’AXIS1’][(0)]

xend = f_gamma[’AXIS’][’AXIS1’][(1)]

gammaI = f_gammaI[’gamma’][()];

gammaT = gamma + gammaI

x = np.linspace(xstart, xend, np.size(gammaT), endpoint=True)

E=(x-1)*511; #keV

y=abs(gammaT)

for j in range(0, len(y)):

if y[j]==0:

y[j]=1e-10

\parf_raw=h5.File(path_raw+”RAW-preplasma-ene = f_raw[’ene’][()]*1e3; #local
kinetic energy in kev

\parlow=((1+a0**2/2)**0.5-1)*511

idxLow = (np.abs(E - low)).argmin() + 1 #avoid extreme indices like 0

Elow=E[idxLow]

yLow=y[idxLow]

yHigh=yLow*1e-2

idxHigh=(np.abs(y - yHigh)).argmin()

Ehigh=E[idxHigh]

lny=np.log(y)

\parif ene.max()>low and lny[(E<Ehigh)&(E>Elow)].size>3: #avoid cases where
idxHigh-idxLow <= 2

popt, pcov = curve_fit(log_func, E[(E<Ehigh)&(E>Elow)],
lny[(E<Ehigh)&(E>Elow)], p0=(5, 100))

# for small a0 and sharp gradient, temperature can be << 100kev

if popt[1]<0:

popt, pcov = curve_fit(log_func, E[(E<Ehigh)&(E>Elow)],
lny[(E<Ehigh)&(E>Elow)], p0=(4, 20))

perr = np.sqrt(np.diag(pcov))

\parallTemp.append(popt[1])

allTempErr.append(perr[1])

Z = 42; ne = n0*4*math.pi**2;

Te = popt[1];

brem = 1.54*1e-38*Z**2*ne**2*(Te*1e3)**0.5 # is the total bremsstrahlung power
per unit volume

allBrem.append(brem)

\parif not kalpha: #if list is empty

totalKalpha=0

else:

totalKalpha=sum(kalpha)

\parif not allTemp: #if list is empty

avgTe=0

else:

avgTe=np.mean(allTemp)

\parif not allTempErr: #if list is empty

avgTeErr=0

else:

avgTeErr=np.mean(allTempErr)

\parif not allBrem: #if list is empty

totalBrem=0

else:

totalBrem=sum(allBrem)

\parxrayDiag = [totalKalpha, avgTe, avgTeErr, totalBrem]

xrayDiagNorm = [totalKalpha/a0**2, avgTe, avgTeErr, totalBrem/a0**2]

\par\parreturn xrayDiag, xrayDiagNorm

\par\parresultJob = []

resultJobNorm = []

jobList = os.listdir(pathRoot)

\parfor x in jobList:

if not x.startswith(’.’):

jobContent=os.listdir(pathRoot+x)

print(pathRoot+x)

for xx in jobContent:

if xx.startswith(’a’):

a0str=(xx[2:5:1])

a0=float(a0str)

lsstr=(xx[8:11:1])

ls=float(lsstr)

\parxrayEmission = getXrays(pathRoot+x, a0, ls)[0]

resultJob.append(os.listdir(pathRoot+x)[0])

resultJob.append(xrayEmission)

\parxrayEmissionNorm = getXrays(pathRoot+x, a0, ls)[1]

resultJobNorm.append(os.listdir(pathRoot+x)[0])

resultJobNorm.append(xrayEmissionNorm)

\parnp.savetxt(pathResult+’XrayDiag.txt’,resultJob,
fmt=’np.savetxt(pathResult+’XrayDiagNorm.txt’,resultJobNorm,
fmt=’\par\par\par\par

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  *[HHG]: High-order Harmonic Generation
  *[LWFA]: laser-wakefield acceleration
  *[PIC]: particle-in-cell
  *[CPA]: Chirped pulse amplification
  *[SFQED]: strong-field quantum electrodynamics
  *[QED]: quantum electrodynamics
  *[LINACs]: linear accelerators
  *[XFELs]: X-ray free-electron lasers
  *[LCLS]: Linac coherent light source
  *[PWFA]: plasma-wakefield acceleration
  *[FELs]: free-electron lasers
  *[LWFAs]: Laser-wakefield accelerators
  *[Lambda-cubed]: relativistic lambda cubed laser facility
  *[CUOS]: Center for Ultrafast Optical Science
  *[ML]: machine learning
  *[ICF]: inertial confinement fusion
  *[DJINN]: Deep Jointly-Informed Neural Network
  *[RL]: reinforcement learning
  *[AWAKE]: Advanced Wakefield
  *[LIBS]: laser-induced breakdown spectroscopy
  *[OPA]: Optical parametric amplifier
  *[ASE]: amplified spontaneous emission
  *[OPCPA]: optical parametric chirped pulse amplification
  *[MPI]: multi-photon ionization
  *[REMPI]: resonance-enhanced multi-photon ionization
  *[ATI]: above-threshold Ionization
  *[SM-LWFA]: Self-modulated laser-wakefield acceleration
  *[DLA]: direct laser acceleration
  *[AOPDF]: acousto-optic programmable dispersive filter
  *[MIR]: mid-infrared
  *[WLC]: white light generation
  *[OAP]: off-axis parabolic mirror
  *[SHG]: Second Harmonic Generation
  *[CCD]: charge-coupled device
  *[am-241]: americium-241
  *[AOS]: adaptive optical system
  *[DM]: deformable mirror
  *[GA]: genetic algorithm
  *[FOM]: figure of merit
  *[RF]: Random Forest
  *[DNN]: Deep Neural Network
  *[GP]: Gaussian Process
  *[CWE]: coherent wake emission
  *[ROM]: relativistic oscillating mirror
  *[FWHM]: full-width half-maximum
  *[VLA]: Vacuum laser acceleration
  *[CEP]: carrier-envelope-phase
  *[COTR]: coherent optical transition radiation
  *[MCNPX]: Monte Carlo N-Particle eXtended
  *[SH]: Second Harmonic
  *[PMT]: photomultiplier tube
  *[IR]: infrared
  *[CNN]: convolutional neural network