# Above-threshold ionization at laser intensity greater than $10^{20}$ W/cm2

A. Yandow  Center for High Energy Density Science, The University of Texas at
Austin, 2515 Speeday Stop C1600, Austin, TX 78712  Lawrence Livermore National
Laboratory, Livermore, CA 94551<EMAIL_ADDRESS>T. N. Ha  C. Aniculaesei  H.
L. Smith  C. G. Richmond  M. M. Spinks  H. J. Quevedo  S. Bruce  M. Darilek
C. Chang  D. A. Garcia  E. Gaul  M. E. Donovan  B. M. Hegelich  T. Ditmire
Center for High Energy Density Science, The University of Texas at Austin,
2515 Speedway Stop C1600, Austin, TX 78712

###### Abstract

We present the first experimental observation of above-threshold ionization
(ATI) electrons produced by ionization of the neon K-shell in a laser field
where intensity exceeds 1020 W/cm2. An array of plastic scintillating
calorimeter detectors was used to measure the high-energy electrons at four
angles in the laser forward direction. Coarse energy resolution was obtained
using aluminum filters of several thicknesses to block lower-energy electrons.
A threshold intensity around $2\times 10^{20}$ W/cm2 is observed for
production of energetic ATI electrons in the laser forward direction, with
maximum electron energy exceeding 10 MeV. L-shell electrons with energies
$<1.4$ MeV are scattered further forward along the laser direction than
expected. We present comparisons of the measured total electron energies to
the predictions of a Monte Carlo models employing the ADK-PPT ionization model
and the Augst barrier suppression ionization model.

Suggested keywords

††preprint: AAPM/123-QED

##  I Introduction

Above-threshold ionization (ATI) is the process by which an electron absorbs
many more photons than required for ionization during a laser-atom
interaction. Absorption of a single additional photon over the required
threshold was observed in 1979 by Agostini et al. [1]. The modern two-step
model of ATI was proposed by Corkum et al. to explain the absorption of
thousands of laser photons above the threshold [2], and it has been extended
to explain high-harmonic generation in gases [3][4] and nonsequential double
ionization (NSDI)[5][6][7]. The two-step model of ATI in a strong, near-
infrared laser field describes the ionization process as a quantum mechanical
tunneling model and predicts the subsequent electron dynamics by integrating
the classical Lorentz force equations. The two-step model of ATI has explained
a number of experimental observations, including relativistic electrons
gaining a momentum component in the laser forward direction[8] and the
preferential ejection of ATI electrons along the laser polarization direction
[9][10]. At present, ATI electrons with energies exceeding 1 MeV from argon
and xenon have been observed, corresponding to the absorption of one million
excess photons above the ionization threshold[10][11].

Tunneling ionization becomes the dominant ionization mechanism for near-
infrared laser fields when intensity exceeds $10^{14}$ W/cm2. Tunneling
ionization first observed in the pioneering experiments of Augst et al.
[12][13]. The Ammosov-Krainov-Delone and Perelomov-Popov-Terent’ev (ADK-PPT)
model of tunneling ionization [14][15] has been verified with precision
measurements of argon charge states at intensity exceeding $2\times 10^{19}$
W/cm2 [16]. The highest ion charge states observed experimentally are
Ar16+[16][17], Xe26+[17][18], and Kr24+[18]. The probability of tunneling
ionization is a strongly nonlinear function of laser intensity, leading to the
use of high ion charge states as a direct peak laser intensity diagnostic with
varying degrees of success [18][19]. With laser intensity estimates calculated
from indirect diagnostics exceeding $2\times 10^{22}$ W/cm2 [20][21] and
10-PW-class laser facilities in their final stages of development[22], there
has been considerable renewed interest in highly-charged ions as a direct peak
laser intensity diagnostic. Recent numerical studies of ionization have
developed a tunneling cascade ionization model for complex ions in a laser
field [23], demonstrated that K-shell ionization yields are the most robust
when considering different ionization models [24], and identified features of
the ionization yield curves that are robust when considering different
intensity distributions at the focal plane [25]. Monte Carlo simulations of
ionization that include ion motion in the laser field demonstrate the ions can
be accelerated to energies that make conventional time-of-flight ion yield
measurements impossible at intensity above $10^{21}$ W/cm2 [26], so we explore
the detection of the ATI electrons from these high charge states for future
ionization physics experiments and direct laser intensity diagnostics.

Modulations in ATI electron energy spectra and angular distributions
corresponding to ionization of different electron shells of the target atomic
species has been observed for the N, M, and L-shells of krypton and xenon
[11]. These modulations arise from the large difference in ionization
potential between the atomic shells, with ATI electrons produced by ionization
of a deeply-bound state accelerated to higher energies by a stronger laser
electric field than an outer shell state[11]. The large ionization potential
gap between the L-shell and K-shell of neon will result in a strong modulation
of the energy spectrum and angular distribution, with the K-shell electrons
gaining about an order of magnitude more energy than the L-shell electrons.
This strong modulation in both the energy spectrum and the angular
distribution raises the possibility of a novel direct laser intensity
diagnostic, where the production of highly-charged ion states can be inferred
by the detection of high-energy ATI electrons ejected in the laser forward
direction. Such a diagnostic will be relatively easy to execute experimentally
using a low-density stream of noble gas as a target and a magnetic
spectrometer to detect the ATI electrons. The ionization of the K-shell in
noble gases would allow for direct laser intensity measurement around 1020
W/cm2 (Ne9+), $3\times 10^{21}$ W/cm2 (Ar17+), and $10^{23}$ W/cm2 (Kr35+) in
any laser field, provided the ADK-PPT tunneling ionization model can be
verified experimentally on a laser system with reliable indirect diagnostics
at these intensities as well.

ATI electrons hold significant promise as a direct laser intensity diagnostic
between $10^{20}$ and $10^{24}$ W/cm2, where ponderomotive expulsion of the
ions becomes a significant obstacle to direct ionization yield
measurement[26]. Recently proposed techniques using vacuum-accelerated
electrons and protons [27][28] will not yield an accurate intensity estimate
without a well-known pulse duration when prepulses scatter target electrons
[29] and the ions gain only a fraction of their ponderomotive potential energy
[28]. The strong nonlinearity of the tunneling ionization rate prevents the
K-shell electrons from being scattered by prepulses even when laser contrast
is as low as $10^{-3}$. The ceiling intensity of our method is about $10^{24}$
W/cm2, above which expulsion of highly-charged ions before the arrival of the
peak laser field strength would prevent K-shell ionization [26] and an ion
ponderomotive diagnostic such as that proposed in [27] would be most
appropriate.

We report in this paper the observation of multi-MeV ATI electrons produced by
the interaction of a low-density neon gas jet ($<3\times 10^{13}$ cm-3) with a
well-characterized laser pulse with intensity exceeding $10^{20}$ W/cm2. We
observe these ATI electrons on four plastic scintillating calorimeter
detectors positioned in the laser forward direction and along the plane of
laser polarization. We compare the observed integrated ATI electron energy
yields to the predictions of an ADK-PPT Monte Carlo model of ATI and a barrier
suppression model of ATI, using methods similar to those described previously
elsewhere[26]. The measurements have qualitative similarities with the models’
predictions, including the existence of a threshold intensity above which the
ionization probability increases rapidly with intensity and a saturation
intensity above which the ATI electron energy yield is dominated by the focal
volume rather than by the probability of ionization in the center of the
focus. However, we observe poor quantitative agreement with the modeling,
which significantly underestimates the observed threshold intensity by a
factor between two and three. We also observe electrons with energies
exceeding $10$ MeV for the first time [30].

##  II Experimental Design

Figure 1:  Diagram of the experimental setup, showing five detectors arranged
around the laser-atom interaction region. Three detectors were placed in the
plane of laser polarization at angles of $0^{\circ}$, $30^{\circ}$, and
$53^{\circ}$ from the laser forward direction. One additional detector was
placed $60^{\circ}$ out of the plane of polarization and $43^{\circ}$ from the
laser forward direction. A control detector was placed $110^{\circ}$ from the
forward direction out of the polarization plane.

A diagram of the experimental setup is given in Fig. 1. A low-density plume of
neon is introduced in vacuum using a flow-calibrated orifice with a diameter
of 100 $\mu$m held at a backing pressure of 60 Torr located 4 mm below the
laser focus. We estimate a gas density of $3\times 10^{14}$ cm-3 from a
steady-state Ansys-Fluent simulation of the gas expansion into vacuum [31].
Five scintillating calorimeter detectors were placed around the focus. Four
detectors were placed in the laser forward direction, with one oriented along
the direction of laser propagation, two lying in the plane of laser
polarization, and one outsize the polarization plane. The fifth detector was
placed out of the forward direction and polarization plane, where no ATI
electrons are expected as a control. We expect the higher-energy ATI electrons
to be ejected further towards the laser forward direction [8][11] and
preferentially ejected in the plane of laser polarization [9][10].

Each detector consisted of a 50 mm diameter, 40 mm long cylinder of long-
lifetime (285 ns) scintillating plastic (Eljen Technologies EJ240) coupled to
a photomultiplier tube (PMT) with a tapered voltage divider for optimal pulse
linearity. The plastic scintillator and PMTs were encased in a vacuum-
compatible PTFE housing that was made light-tight with colloidal graphite and
aluminum foil. The plastic scintillator was chosen to decrease sensitivity to
high-energy photons and a long scintillating lifetime allowed the detector to
function as a calorimeter, with the output signal producing an integrated
measurement of the energy of all electrons incident on the detector. The
relatively large solid angle subtended by the detectors ($\sim$ 0.03
steradians) allowed several hundred ATI electrons to hit each detector,
enabling relatively accurate calorimeter measurements with only a few shots at
each laser intensity. Information about the shape of the energy spectrum was
obtained by varying the thickness of aluminum shielding in front of each
detector, which is explained further in the next section.

The scintillating detectors were calibrated by using standard pulse-height
analysis techniques to measure absorbed energy spectra from two gamma
radiation sources, Co-60 and Cs-137, at high operating voltage. An MeV-
equivalent charge was obtained by measuring the location of the Compton edges
in the acquired spectra and comparing to absorbed energy spectra calculated
using G4beamline [32], a Monte Carlo particle transport software package based
on Geant4. The uncertainty in this MeV-equivalent charge is between 20% and
25%, and is the dominant source of uncertainty in the ATI electron energy
yields. The PMT gain curves as a function of bias voltage were characterized
by exposure to a Q-switched Nd:YAG laser light source.

The output current pulse from each PMT was recorded on a Tektronix TDS5054
oscilloscope and digitally filtered to eliminate noise from electromagnetic
pulse (EMP) on shot. The current pulse amplitude and integrated charge obey a
linear relationship during normal detector operation. The upper saturation
limit showed increasing amplitude without increasing charge, and corresponded
to $~{}7$ GeV of integrated ATI electron energy absorbed in the scintillator.
The lower charge limit corresponded to a regime where residual ringing
disrupted the linear relationship, with large amplitude current pulses
integrating to near-zero charge, and was chosen to prevent uncertainty induced
by detector ringing from dominating over the uncertainty from the charge
calibration. Measurements falling outside these limits are excluded from the
figures presented in this paper but the lowest detector charge threshold is
marked if appropriate.

We performed a series of control tests to confirm that the observed
scintillator signal was caused by high-energy electrons. We compare the signal
with minimal detector shielding at the $30^{\circ}$ and $110^{\circ}$
(control) positions and found the signal at the forward detector position was
at least two and a half orders of magnitude greater than the signal observed
at the control detector position. We swapped the scintillating detectors
between these two positions several shots after the experiment start to verify
that the observed signal in the laser forward direction was not an artifact of
that particular scintillating detector. Multiple control shots were taken with
no target gas to confirm the signal was not just electromagnetic pulse (EMP).
We also verified the effect was intensity-dependent and not energy dependent
by stretching the pulse to a length of 2 ps and observing the signal to
disappear. The observation of a much stronger intensity-dependent signal in
the laser forward direction means that the observed signal is generated by
high-energy electrons, as it cannot be attributed to a detector artifact. We
also included a number of helium control shots in this study to demonstrate
the contribution of any vacuum-accelerated L-shell electrons to the signal.

We used the Texas Petawatt Laser in rod shot mode (64 mm Nd:silicate amplifier
only) allowing an increased repetition rate of 2.5 shots per hour. We
installed a f/1.4 off-axis parabolic mirror (OAPM) to reach an intensity
exceeding $2\times 10^{20}$ W/cm2. Laser intensity was calculated using the
indirect Output Sensor Package (OSP) of the Texas Petawatt Laser, which
includes diagnostics to measure near field, equivalent far field (focal spot),
wavefront, pulse duration, and energy [20]. Pulse duration was deconvolved
from a second-order autocorrelation assuming a Gaussian pulse shape. Wavefront
was measured using a PHASICS SID4 wavefront sensor, and a deformable mirror
was used to optimize the laser wavefront before every shot.

The focal spot in the target chamber was measured using a Mitutoyo 50x plan
apochromatic long-working distance microscope objective (0.55 numerical
aperture), a 200 mm achromatic lens, and a vacuum-compatible CCD camera
mounted in a Thorlabs optical cage system. We estimate the central maximum of
the focal spot to have a full-width half-maximum (FWHM) of $2.6\pm 0.2$ $\mu$m
from Gaussian fittings of focal spot images. The far-field diagnostic plane
measured at OSP does not necessarily coincide with the plane of highest laser
intensity within the low-density gas jet due to defocus remaining in the
wavefront after correction, which can lead to a systematic underestimate of
the laser intensity. The focal spot profile used to compute the peak laser
intensity was calculated from the measured wavefront including all aberrations
except defocus. An inverted-field autocorrelator was used to diagnose pulse-
front tilt. We estimate a 42 fs pulse front tilt from the angular shift of the
far-field during grating optimization, for a total typical pulse duration of
170 fs. Intensity changes were achieved by inserting calibrated neutral
density filters (ND) before the rod amplifier in the TPW laser chain. The gain
of the rod amplifier remained fixed to ensure the amplified spectrum,
compressed pulse duration, and laser wavefront remain the same when the pulse
energy is decreased.

##  III ATI Model Description

The theoretical K-shell electron yields and energy spectra were calculated
using the two-step quasi-classical models of ATI. A Monte Carlo simulation of
tunneling ionization in the laser field using the ADK-PPT model of
ionization[15][14][33] predicted the initial conditions for K-shell electrons
in the laser field. The static tunneling ionization rate for a single electron
expressed in atomic units is given by:

$\displaystyle W_{ADK-
PPT}(t)=C_{n^{*}l^{*}}^{2}I_{p}\frac{(2l+1)(l+|m|)!}{2^{|m|}|m|!(l-|m|)!}\times$
$\displaystyle\bigg{(}\frac{1}{2}\tilde{F}(t)\bigg{)}^{1+|m|-2n^{*}}exp\bigg{(}-\frac{2}{3\tilde{F}(t)}\bigg{)}$
(1)

where the reduced field strength $\tilde{F}(t)$ is defined as

$\tilde{F}(t)=\frac{\sqrt{\textbf{E}^{*}(t)\textbf{E}(t)}}{(2I_{p})^{3/2}}$
(2)

where $I_{p}$ is the ionization potential and $l,m$ are the orbital quantum
numbers. The extension of the original PPT model by Ammosov, Krainov, and
Delone introduced an effective principle number $n^{*}$ and an effective
orbital quantum number $l^{*}$ given by

$\displaystyle n^{*}=\frac{Z}{\sqrt{2I_{p}}}$ (3) $\displaystyle
l^{*}=n^{*}_{o}-1$ (4)

where $Z$ is the residual charge ($Z=1$ for neutral atoms). The constants
$C_{n^{*}l^{*}}^{2}$ are expressed as

$C_{n^{*}l^{*}}^{2}\approx\bigg{[}\bigg{(}\frac{2exp(1)}{n^{*}}\bigg{)}^{n^{*}}\frac{1}{\sqrt{2\pi
n^{*}}}\bigg{]}^{2}$ (5)

The exponential factor in Eq. III dominates the scaling of ionization
probability with laser intensity, leading to an intensity threshold for the
appearance of high ion charge states. Ion motion, although it has no
significant effect on the ionization yield at $10^{20}$ W/cm2, was included
[26].

At each timestep, Eq. III was used to predict the probability of ionization
and Monte Carlo methods were used to increment the ion charge state. Non-
sequential double ionization (NDSI), inelastic tunneling[34][35][36],
collective tunneling[37][35], and relativistic ionization
effects[38][39][40][41][42] were excluded from the calculations. From these
initial conditions, the electron trajectories were calculated by integrating
the Lorentz force equations using an adaptive-timestep Runge-Kutta (RK45)
numerical method. A maximum of $10^{5}$ test electrons were simulated at each
intensity, originating within an isointensity boundary where ionization
outside could be neglected due to the strong ionization rate dependence on
intensity. We chose a series of model intensities between $3\times 10^{19}$
W/cm2 and $4\times 10^{20}$ W/cm2 to demonstrate the model behavior above and
below the barrier suppression intensity of Ne9+. Within each volume, we chose
$2.5\times 10^{5}$ initial positions for neutral ions. From these ionization
events, we calculated the energy spectrum and angular distribution of at most
$10^{5}$ ATI electrons.

An additional ATI Monte Carlo model using the Augst barrier suppression
ionization (BSI) model [13] was developed as well. The simulations were
performed similarly, except the ionization event occurred at the timestep
$\tilde{F}>1/16n^{*}$ and did not occur otherwise. Figure 2 shows the K-shell
ATI electron yield predicted by the Monte Carlo simulation as a function of
laser intensity assuming a gas density of $3\times 10^{14}$ cm-3. Analytic
predictions of the ADK-PPT model, the Tong-Lin-Lotstedt model for tunneling
rate near the barrier suppression regime [43][44], and the Augst BSI model
compare favorably with the Monte Carlo modeling. The effect of barrier
suppression corrections on the tunneling ionization rate, which is predicted
to be significant with pulses shorter than 15 fs [45], can be safely neglected
for this relatively long pulse duration.

Figure 2:  Total number of K-shell ATI electrons predicted by different models
of ATI. Solid, dashed, and dotted curves are the predictions of the ADK-PPT
[14][15], Tong-Lin-Lotstedt model for helium-like ions [44], and barrier
suppression ionization [13]. Blue circles and green squares are from Monte
Carlo simulations using the ADK-PPT and BSI models, respectively. Gas density
is assumed to be $3\times 10^{14}$ cm-3 in the laser focus. Color figures
available online.

(a)

(b)

Figure 3: a) Simulated energy spectrum of the K-shell electrons at an
intensity of 1.06 $\times 10^{20}$ W/cm2 (blue) And 4.20 $\times 10^{20}$
W/cm2 (green, dashed). Inset figures shows same curves on a linear scale. b)
Simulated L-shell electron spectrum at the same two intensities. Color figures
available online.

The laser focal spot is computed for every shot but we lack information on the
exact structure of the phase fronts as the laser pulse propagates through the
focal plane, so we make a considerable number of simplifying assumptions when
modeling the laser focus. Ionization rate depends strongly on intensity, so we
assume the K-shell electrons are all produced in the central maximum at the
focal plane, and we do not consider laser energy scattered outside the central
maximum in the model. We also make the assumption that we can treat this
central maximum as a Gaussian laser focus with nonparaxial corrections
included up to fifth order in the diffraction angle [46]. We assume a focus
with a $1/e^{2}$ diameter of 2.25 $\mu$m, which we estimated from direct
measurements in the target chamber. During the experiment rod shots, we
estimate the 1/e2 spot of the central maximum was 2.2 $\pm$ 0.2 $\mu$m. We
incorporate the measured pulse front tilt of 42 fs by assuming a Gaussian
pulse shape with an intensity FWHM of 170 fs. Similar approaches have been
taken to model the laser fields at focus in previous experimental studies
[10][11][8]. Some particle-in-cell (PIC) methods have shown promise for
predicting the energy spectra of vacuum-accelerated electrons at an intensity
of $10^{19}$ W/cm2 [47], but no such methods have been applied to simulating
ATI electron dynamics in a laser field with a more complex spatial structure
than a Gaussian beam.

Figures 3a and 3b show the energy spectra of the K-shell and L-shell ATI
electrons, respectively, predicted by the Monte Carlo ADK-PPT modeling at two
intensities ($1.06\times 10^{20}$ W/cm2 and $4.2\times 10^{20}$ W/cm2). The
predicted angular distributions of the ATI electrons at the same intensities
are shown in Figure 4. The modeling predicts the ATI electron energy spectra
and angular distributions are be strongly modulated, with the higher-energy
K-shell electrons expelled at an angle around 25∘ from the laser forward and
the lower-energy L-shell electrons expelled an angles greater than
$60^{\circ}$ from the laser forward direction.

Figure 4:  Angular distribution of the K-shell electrons (no markers) and
L-shell electrons (square markets) at laser intensities of $1.06\times
10^{20}$ W/cm2 (solid, blue) and $4.2\times 10^{20}$ W/cm2 (dashed, green).
Color figures available online.

The modeled K-shell energy spectra in Figure 3a also show that the number of
high energy electrons ($>15$ MeV) produced can increase by more than an order
of magnitude as the laser intensity increases toward the maximum intensity
used in the experiment. However, other features of the K-shell electrons are
relatively stable, with the peak of the energy spectrum in Fig 3a (inset)
increasing from 3.5 MeV to 4.7 MeV and the angular distribution in Figure 4
nearly unchanged. The modeling predicts that the energy yield attributed to
the highest-energy electrons, which are observed on the $0^{\circ}$ detector,
demonstrate a stronger scaling with laser intensity than the other detectors.
The energy yield at this position increases with intensity due to both the
larger number of electrons generated in the focus, as shown in Figure 2, and
the larger number of electrons in the high-energy tail of the spectrum. The
energy yields detected at the $30^{\circ}$ position, where the electron
energies and angular distribution change little with increasing intensity,
will display a scaling dominated by the total number of electrons produced by
the K-shell ionization.

The L-shell electrons, shown in Figure 3b, are predicted to obtain energy less
than 1 MeV and be ejected from the focus at an angle around $70^{\circ}$, and
can therefore be filtered from the K-shell electrons in energy and angle, but
we must treat these model predictions with caution. Vacuum acceleration of
electrons demonstrates very strong sensitivity to initial position in the
laser focus [48], leading to the possibility that the simulation method may
under-sample initial positions in the focus that yield L-shell electrons that
are accelerated to higher energies or ejected further towards the laser
propagation direction. The last study of vacuum acceleration of electrons from
ionized helium in this intensity regime ($\sim 3\times 10^{20}$ W/cm2) found
disagreement between the measured angular distribution of vacuum-accelerated
electrons and the angular electron distributions predicted by particle-in-cell
modeling. The authors suggested this discrepancy may be caused by poor
sampling of the focal volume in their simulations [49], and there is evidence
of L-shell electrons scattered as far forward as $30^{\circ}$ from the laser
propagation direction from the helium control shots.

The simulated ATI electron yields and energy spectra incident on each detector
are used to calculate the energy deposited in the plastic scintillator.
Several thicknesses of aluminum shielding were used in the experiment to block
lower-energy electrons. The detector efficiencies for each shielding thickness
were calculated using G4beamline [32], a Monte Carlo particle transport
software package based on Geant4, that includes energy deposited in the
plastic scintillator by electrons, positrons, and high-energy photons
generated in the interaction. The detector efficiencies are shown in Figure 5.
The detector efficiency at each energy and shielding thickness is calculated
from the simulated visible energy deposited in the scintillating plastic by a
monoenergetic beam of $10^{4}$ electrons with a divergence similar to the
incident ATI electrons.

Figure 5:  Detector efficiencies at different aluminum thicknesses (right
axis) alongside a simulated K-shell ATI electron spectrum using the ADK-PPT
model at intensity of $1.06\times 10^{20}$ W/cm2 (left axis). Color figures
available online.

The predicted energy yield in the scintillators can be computed by combining
the electron yields and energy spectra from the ATI modeling with the
calculated detector efficiencies. The predicted ATI electron energy yield is
given by

$Y_{ATI}(\theta,\phi,Z)=\int_{0}^{E_{max}}w_{V}E^{\prime}p(E^{\prime},\theta,\phi)\eta_{Al}(E^{\prime},Z)dE^{\prime}$
(6)

where $p(E,\theta,\phi)$ is the energy spectrum (count) at the detector
position, $\eta_{Al}(E,Z)$ is the detector efficiency with an aluminum
shielding thickness of Z, and $w_{V}$ is a volume weighting factor
corresponding to the number of real K-shell electrons produced per simulated
ATI electron. The volume weighting factor is calculated by using a gas density
of $3\times 10^{14}$ cm-3 and a confocal volume estimated by integrating a
Gaussian beam volume bounded by the same isointensity shell used in the model
simulations.

We can gain some information about the energy spectrum of electrons at a given
intensity by varying the shielding thickness $Z$ and measured the energy
deposited. Although the energy integration cannot be uniquely inverted to give
an electron spectrum, we can compare the predicted energy deposited to the
observed energy deposited and search for energy ranges where the ATI model
spectrum either overestimates or underestimates the electron number. From the
efficiency curve shapes, we conclude this method has poor resolution for
electron energies above $10$ MeV but can yield some energy information in the
0.3-6 MeV energy range.

##  IV ATI Electron Energy Yields

A number of laser intensity scans were performed using different shielding
configurations. On all unshielded detectors in the laser forward direction the
integrated electron energy increase up to almost three orders of magnitude.
Helium control shots show that some of the electrons accelerated toward these
detectors are low-energy L-shell electrons ($<$ 1.4 MeV) scattered in the
laser forward direction by vacuum acceleration.

Figure 6:  Measured electron energy deposited in a scintillating detector
located at $30^{\circ}$ from the laser forward direction in the polarization
plane. Unshielded and shielded (2.6 mm aluminum) intensity scans are given by
blue circles and green triangles, respectively. Helium control shots in the
unshielded configuration are given by purple crosses and the detector charge
floor for helium control shots in the shielded configuration is marked. Monte
Carlo simulations using ADK-PPT (solid) and BSI (dashed) models shown for
comparison. Color figures available online.

Figure 6 shows the energy absorbed in the scintillating detector at
$30^{\circ}$, where the number of K-shell ATI electrons is expected to be the
highest. Intensity scans with the unshielded scintillator (blue circle) and a
shielded configuration (green triangles) are shown to compare the total
integrated electron energy with the integrated energy from electrons with
energy greater than 2.8 MeV. Helium control shots (purple crosses) demonstrate
that the L-shell electrons contribute some of the deposited energy in the
unshielded configuration. The helium control shot energy yield is about an
order of magnitude lower than the neon yield at $2.5\times 10^{20}$ W/cm2,
demonstrating that the neon L-shell electrons account for $\sim 1/2$ of the
observed energy yield when accounting for the difference in electron density
at the focus. Two helium control shots taken in the shielded configuration
with the same backing pressure ($n_{a}\sim 3\times 10^{14}$ cm-3 yielded no
repeatable signal, with the dynamic range floor for these control shots marked
on Figure 6. The helium control shots establish an upper limit of 2.8 MeV for
vacuum-accelerated, which is slightly lower than the maximum energy of vacuum-
accelerated electrons by Kalashnikov near this angle in this intensity regime
[49].

The shielded measurements show a threshold intensity around $2\times 10^{20}$
W/cm2, above which the probability of electron production with energy $>$ 2.8
MeV increases rapidly with intensity. A scaling transition around $3\times
10^{20}$ W/cm2 marks the saturation intensity where the scaling transitions
from an ionization probability scaling dominated by the exponential term in
Eq. III to a focal volume scaling. The threshold-like behavior and scaling
transition are features of ATI that are mirrored both Monte Carlo models,
although neither model correctly predicts the threshold or saturation
intensities and both overestimate the ionization yield.

Similar laser intensity scans at two additional positions are presented in
Figures 7 and 8, corresponding to positions $53^{\circ}$ from the laser
forward direction (in polarization plane) and $43^{\circ}$ from the laser
forward direction ($60^{\circ}$ out of the polarization plane), respectively.
These positions were chosen due to space restrictions in the target chamber
and experimental setup. The helium control shots with no shielding installed
are comparable to the deposited energies measured with neon in both cases,
showing the L-shell electrons will contribute to the signal substantially.
Installing a 1 mm aluminum shield, which blocks electrons with energy $<1.4$
MeV, decreases the electron energy yields an order of magnitude at each
detector. The electron energy yields in the shielded configuration show the
same characteristic ATI features, the threshold and saturation intensities,
seen in 6, with the saturation effect somewhat more exaggerated because the
K-shell ATI electrons will be ejected further forward in the laser direction
as laser intensity continues to increase. The unshielded measurements
dominated by lower-energy electrons do not display the clear scaling change
visible in the shielded measurements, and so they are likely L-shell electrons
scattered in the laser forward direction by a vacuum acceleration process.

Figure 7:  Measured energy deposited in a scintillating detector located at
$53^{\circ}$ from the laser forward direction in the polarization plane.
Unshielded and shielded (1 mm aluminum) intensity scans are given by blue
circles and green triangles, respectively. Helium control shots in the
unshielded configuration are given by purple crosses. Monte Carlo simulations
using ADK-PPT (solid) and BSI (dashed) models shown for comparison. Color
figures available online. Figure 8:  Measured energy deposited in a
scintillating detector located at $43^{\circ}$ from the laser forward
direction in the polarization plane. Unshielded and shielded (1 mm aluminum)
intensity scans are given by blue circles and green triangles, respectively.
Helium control shots in the unshielded configuration are given by purple
crosses. Monte Carlo simulations using the ADK-PPT ionization model (solid)
shown for comparison. Color figures available online.

At these larger angles, the ADK-PPT simulations tended to overestimate the
electron energy yields while the Augst BSI model tended to be an
underestimate, instead predicting a greater proportion of higher-energy ATI
electrons that would scatter further forward in the focus. The BSI model also
exhibited an unexpectedly strong polarization dependence for low-energy ATI
electrons because the probability of being “born” into the field off a laser
cycle peak is higher for ATI electrons produced by the rising edge of the
laser focus and scattered out before the arrival of peak laser intensity. An
insufficient number of test electrons in the BSI simulations were scattered
toward the $43^{\circ}$, so only the ADK-PPT model is shown in Fig. 8.

Figure 9:  Measured electron energy deposited in a scintillating detector
located at on the laser propagation axis. Two shielded (1 mm and 2.6 mm
aluminum) intensity scans are given by blue circles and green triangles,
respectively. Lowest detector charge floors are marked over the intensity
range where shots were taken in each shielding configuration. Monte Carlo
simulations using ADK-PPT (solid) and BSI (dashed) models shown for
comparison. Color figures available online.

Figure 9 shows the measured electron energy deposited in the scintillating
detector oriented in the laser forward direction, shielded with a minimum of 1
mm of aluminum to block electrons with energy $<$ 1.4 MeV. We observe a
threshold appearance intensity of $2\times 10^{20}$ W/cm2 for high-energy
electrons in the laser forward direction, which are found to be in the 10-16
MeV range [30]. The measured ATI electron energy yields along the laser
forward direction fall nearly two orders of magnitude lower than the ADK-PPT
and BSI simulation predictions. While a scaling transition is not obvious in
the measurements, it is important to note that the average energy of these
electrons is much higher than at other detector positions. A single 15 MeV
electron incident on this detector would yield a $\sim 500$ MeV/Sr response,
so some of these measurements between 2-3 $\times 10^{20}$ W/cm2 represent a
single-digit number of electrons, and uncertainty due to sampling statistics
obscures the scaling transition. Measurements falling below the instrument
dynamic range floor (hollow markers) at $10^{20}$ W/cm2 show that not even a
single one of these ATI electrons exceeding 10 MeV was detected below the
threshold intensity.

We do not observe good quantitative agreement between the predicted ATI energy
yields of either Monte Carlo model and the measured energy yields, although
the measurements demonstrate self-consistent qualitative features of tunneling
ionization between the four detector positions. All show an appearance
intensity for a population of high-energy electrons above $2\times 10^{20}$
W/cm2 and three of the four detector positions show a consistent saturation
intensity. The ADK-PPT tunneling ATI model predicts these features will appear
on all detectors at about the same intensity, although the model intensity
underestimates the experimental intensity by a factor of 3-4. The BSI ATI
model predicts a narrower angular distribution of ATI electrons that broadens
as the intensity increases, the focal volume grows, and a broader range of
electron initial conditions over the focal volume and pulse duration are
sampled. This broadening of the ATI electron angular distribution as intensity
increases leads to the higher predicted appearance intensity at 53∘.
Therefore, the measured ATI energy yields are more consistent with some form
of tunneling process that allows for electrons to originate from a wider range
of initial conditions below the saturation intensity than it is with a true
intensity threshold process.

##  V Electron Energies

The limited number of laser shots and the low density of target gas necessary
to avoid collective plasma effects prevented measurement of the electron
spectrum using a magnetic spectrometer. We placed a series of aluminum filters
of different thicknesses in front of the scintillating plastic to gain
spectral information. While such a method provides only crude information
about the energy spectrum, it can be used to show the maximum ATI electron
energy is between 10-16 MeV. A comparison to the maximum energies predicted by
ATI models shows the maximum ATI energy range consistent with the measurements
falls between the predictions of relativistic and nonrelativistic
ponderomotive models [30].

Figures 10a and 10b show the measurements of integrated electron energy at the
$30^{\circ}$ positions at two average laser intensities. The predictions of
ADK-PPT Monte Carlo model and BSI model at several intensities are marked on
Figures 10a and 10b, respectively. Figure 11 shows the measured electron
energy yield along the laser forward direction, and the predictions of the
ADK-PPT (solid) and BSI (dashed) models, and is consistent with a maximum ATI
electron energy in the 10-16 MeV range [30].

Both models only show that quantitative agreement with the electron energy
yield measurements is only possible when the laser model intensity is taken to
be significantly less than the laser intensity computed using indirect laser
diagnostics. As with the laser intensity scans discussed in Section IV, we
observe the ADK-PPT ATI model provides a more consistent description of the
measurements between different detector positions, even though the model
intensity is four times lower than the estimated laser intensity in the
experiment. The BSI model does not make predictions that are consistent
between the on-axis and $30^{\circ}$ detectors, with the intensity that is
most consistent with the electron energy yields for the on-axis detector in
Figure 11 ($1.55\times 10^{20}$ W/cm2) underestimating the measurements at
$30^{\circ}$ by a factor of $\sim 5$. The ADK-PPT model shows a more
consistent model intensity around $10^{20}$ W/cm2 between the two detector
positions.

Some qualitative statements about the shape of the spectrum can be gathered by
comparing the measurements in Figure 10a to the detector efficiency curves in
Figure 5. We cannot draw many conclusions about the unshielded measurement
because of the evidence of forward-scattered L-shell electrons shown by the
control shots in Figure 6, so we cannot make a statement about the population
of electrons with energy $<2.8$ MeV. The Monte Carlo ADK-PPT model predicts a
steeper electron energy yield drop-off than the measurements, corresponding to
a model overestimate of the proportion of electrons with energy between
2.8-4.7 MeV and an underestimate of the number of electrons with energy $>6.5$
MeV. The BSI model predictions in Figure 10b show a decrease in electron
energy yield with shield thickness that is more consistent with measurements,
which could indicate an ionization process with higher onset intensity than
predicted by the ADK-PPT model.

(a)

(b)

Figure 10: Electron energy yields measured at the $30^{\circ}$ position with
varying shield thicknesses at two average intensities, $4.1\pm 0.4\times
10^{20}$ W/cm2 and $2.2\pm 0.4\times 10^{20}$ W/cm2 compared to closest
predictions of a) the ADK-PPT Monte Carlo model (solid lines) and b) the Augst
BSI Monte Carlo model(dashed lines). Color is added for clarity. Color figures
available online. Figure 11:  Electron energy yields measured along the laser
forward direction with varying shield thicknesses at an average intensity of
$4.1\pm 0.4\times 10^{20}$ W/cm2. Solid curves (open circles) are ADK-PPT
model predictions and dashed curved (open squares) are BSI model predictions.

The electron energy yields predicted by the modeling at the on-axis position
shown in Figure 11 should be interpreted with care because the Gaussian focus
assumption made in the model is not a realistic description of the laser
fields. Higher-order spatial modes experience increased Guoy phase shifts as
the beam passes through the focus, which will limit the distance a
relativistic electron can stay in phase with the peak of the paraxial laser
electric field to a fraction of a Rayleigh range, which should substantially
decrease the maximum ATI electron energy. The fields of higher-order spatial
modes will also scatter high-energy electrons over a larger range of angles
than expected from a Gaussian model, which could explain why the ADK-PPT model
underestimates the electron energy yield at the $30^{\circ}$ detector with 7.6
mm of shielding in Figure 10a. Further development of ATI simulation
techniques to incorporate a more realistic model of the laser fields is
necessary to further study ATI electrons and develop laser intensity
diagnostics using ATI electrons.

We performed a similar analysis for the detectors at the $53^{\circ}$ and
$43^{\circ}$, and found that the installation of 1 mm aluminum shield
decreased the energy deposited more than an order of magnitude as seen in
Figs. 7 and 8. No repeatable signal was observed when 2.6 mm of shielding was
used, limiting the maximum ATI electron energy to below 2.8 MeV at these two
angles.

##  VI Conclusion

We report the first observation of ATI electrons with energies exceeding 10
MeV as well as the first indirect evidence of the ionization of helium-like
neon in an intense laser field to the best of our knowledge. We measured the
energy deposited in an array of scintillating detectors by high-energy ATI
electrons and performed scans of laser intensity in several configurations and
presented a comparison with the two Monte Carlo models of neon K-shell
ionization.The ADK-PPT ATI model predicted roughly consistent appearance and
saturation intensities between the four detector positions, a qualitative
prediction consistent with the experimental measurements, although the ADK-PPT
model significantly underestimated these intensities. These qualitative
features were not predicted in the BSI Monte Carlo modeling because a
probabilistic tunneling process allows for a much less restricted range of
electron initial positions in the focal volume and laser phases at ionization.

ADK-PPT model-derived intensities derived from ionization yield measurements
in prior studies have not always demonstrated consistency with laser intensity
calculated from indirect diagnostic measurements or self-consistency when
different atomic species are used. Ionization of lithium-like argon (Ar16+)
has been demonstrated to occur in an intensity range from $1-2\times 10^{19}$
W/cm2 in two different studies [16][17]. Ionization yields of xenon in the
same laser field were found to give an ADK-PPT model intensity of $3.5\times
10^{18}$ W/cm2, much lower than the indirectly estimated intensity of
$2.6\times 10^{19}$ W/cm2 or argon-yield ADK-PPT model-derived intensity of
$1.3\times 10^{19}$ W/cm2 [17]. The authors emphasized the repeatability of
their results but were not able to provide a theoretical explanation for the
systematic decrease of model-derived intensity with atomic number. Chowdhury
et al. similarly calculated a model intensity from precision measurements of
argon charge states and found a similarly low model intensity, although it was
within the lower bound of their experimental intensity uncertainty [16]. An
ADK-PPT model intensity shift factor of $\sim 4$ was not expected for
ionization of helium-like neon given the simplicity of the electronic shell
structure and given how precisely helium ionization yields agree with the ADK-
PPT model [6].

Some recent modifications to the ADK-PPT model have been proposed to account
for barrier suppression effects for helium-like ions [44], but they are
typically more relevant for pulses much shorter than 170 fs [45] and L-shell
or M-shell orbitals [24], which we confirmed in the calculations presented in
Figure 2. Relativistic corrections that suppress the ionization rate are
predicted to be negligible at an intensity of 1020 W/cm2 [38][39]. The
spectral information we were able to obtain by increasing the shielding
thickness at $30^{\circ}$ may be consistent with a higher threshold intensity
accelerating electrons ejected at this angle to higher energies but the model
of the laser fields is not realistic enough to demonstrate this agreement
conclusively. Our observation of a neon K-shell ionization intensity above
$10^{20}$ W/cm2 may be a reason why it has not been reported in previous
studies, but no study has explicitly stated that neon charge states were not
observed in this intensity range. Momentum conservation during the ionization
process will accelerate the ions to energies on the order of tens of eV, so
spectrometer design in previous studies may have been a factor as well.

Our observation of forward-scattered L-shell electrons is unexpected from the
simplified model of the laser focus used in this paper but is consistent with
other experiments reported. Kalashnikov et al. report vacuum accelerated
electrons from helium over a similar laser intensity range and angular
distribution [49]. They also found disagreement with the angular distributions
of vacuum-accelerated electrons predicted by both their particle-in-cell
modeling and analytical calculations, which predicted a local maximum around
$20^{\circ}$ for forward-scattered electrons. Instead they observed the
electron number to increase monotonically as angle increased from 5∘ to 70∘,
which they attribute to poor sampling of initial conditions in the focal
volume. A comprehensive model of the L-shell electrons in the detected energy
range ($>$ 0.3 MeV) will likely have to into account pulse shape [29], focal
spot asymmetry [50], and a more realistic model of laser fields at the focal
plane to match experiment.

At laser intensity exceeding 1021 W/cm2, ATI electrons from the K-shell of
argon ($>3\times 10^{21}$ W/cm2 and krypton ($>10^{23}$ W/cm2) are predicted
to exceed energies of 100 MeV and 1 GeV, respectively. These ATI electrons
will be ejected very nearly in the laser forward direction and hold promise as
a low-dose ultrafast radiation source and as a direct laser intensity
diagnostic. Measurement of the ATI electron spectrum would be more
straightforward than the measurements presented in this paper, as the high
energy and low electron divergence would enable the use of a large-aperture
magnetic spectrometer located outside the vacuum chamber and along the laser
forward direction. Similar scintillating detectors could be placed behind the
magnet to detect ATI electrons in different energy ranges. Vacuum acceleration
of the L-shell electrons to comparable energies can be suppressed by
engineering a $\sim 10^{-2}$ pre-pulse that arrives a few pulse durations
before the main laser pulse [29].

###### Acknowledgements.

A. Y. acknowledges helpful conversations with E. Chowdhury regarding the
design of this experiment. This work was supported by the DOE, Office of
Science, Fusion Energy Sciences under Contract No. DE-SC0021125: LaserNetUS: A
Proposal to Advance North America’s First High Intensity Laser Research
Network, the Air Force Office of Scientific Research through Awards No.
FA9550-14-1-0045 and No. FA9550-17-1-0264, and the National Nuclear Security
Agency (NNSA) through Award No. NA0002008. This work was also performed under
the auspices of the U.S. Department of Energy by Lawrence Livermore National
Laboratory under Contract DE-AC52-07NA27344. A. Y. gratefully acknowledges the
generous support of the Jane and Michael Downer Fellowship in Laser Physics in
Memory of Glenn Bryan Focht.

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