# Experimental estimates of the photon background in a potential light-by-
light scattering study

L Doyle1, P Khademi2,3, P Hilz3,4, A Sävert2,3,4, G Schäfer3,4, J Schreiber1
and M Zepf2,3,4 1 Fakultät für Physik, Ludwig–Maximilians–Universität München,
Am Coulombwall 1, 85748 Garching, Germany  2 Institute of Optics and Quantum
Electronics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743
Jena, Germany  3 Helmholtz-Institut Jena, Fröbelstieg 3, 07743 Jena, Germany
4 GSI Helmholtzzentrum für Schwerionenforschung, Planckstraße 1, 64291
Darmstadt, Germany<EMAIL_ADDRESS>
###### Abstract

High power short pulse lasers provide a promising route to study the strong
field effects of the quantum vacuum, for example by direct photon-photon
scattering in the all-optical regime. Theoretical predictions based on
realistic laser parameters achievable today or in the near future predict
scattering of a few photons with colliding Petawatt laser pulses, requiring
single photon sensitive detection schemes and very good spatio-temporal
filtering and background suppression. In this article, we present experimental
investigations of this photon background by employing only a single high power
laser pulse tightly focused in residual gas of a vacuum chamber. The focal
region was imaged onto a single-photon sensitive, time gated camera. As no
detectable quantum vacuum signature was expected in our case, the setup
allowed for characterization and first mitigation of background contributions.
For the setup employed, scattering off surfaces of imperfect optics dominated
below residual gas pressures of
$1\text{\times}{10}^{-4}\text{\,}\mathrm{mbar}$. Extrapolation of the findings
to intensities relevant for photon-photon scattering studies is discussed.

  * February 2022

##  1 Introduction

In contrast to the completely empty classical vacuum, quantum physics predict
the presence of virtual particle-antiparticle pairs constituting the quantum-
mechanical ground state of the vacuum. These virtual pairs of charged
particles act as the mediator of photon-photon-scattering that was proposed by
Euler in 1936, effectively making the vacuum a non-linear optical medium [1,
2]. Early investigations [3, 4] revealed first boundaries for the associated
scattering cross sections and advised on challenges for photon-photon-
scattering experiments in the optical regime. Other schemes probing e.g. the
vacuum birefringence due to external fields or crossing X-ray and laser pulses
are being investigated [5, 6, 7, 8, 9, 10]. In the high-energy regime, light-
by-light scattering could be observed in ultra-peripheral ion collisions
involving quasi-real photons [11, 12, 13]. To date, no direct light-by-light
interaction in pure vacuum has been observed experimentally, but the growing
availability of high power laser pulses has refueled the interest in this
topic (e.g. [14]) and experimental approaches to find, for example, direct
evidence of photon-photon scattering in the optical regime using high
intensity laser pulses have recently studied in more detail [15, 16, 17, 18,
19, 20]. In the most simple all-optical scheme, two laser pulses of same
frequency $\omega$ collide under an angle of 0 to $\pi$, giving rise to
_signal photons_ with spectra peaked around the frequencies $\omega$ and
$3\omega$, since the lowest order is essentially a four-wave mixing process
[2, 14, 15]. In order to maximize chances of detecting scattered photons, one
ideally chooses a parameter space (angular, polarization and spectral) that is
not occupied by photons from the driving pulses. The optimization of the ratio
of scattered photons over primary photons, often referred to as _discernible_
photons, has been addressed in a number of theoretical studies [15, 17, 21,
22, 23]. While a separation of emission directions and spectra can be achieved
for example by frequency doubling one of the beams, employing more than two
beams or measuring sum frequencies such as the third harmonic, the interaction
of two colliding laser pulses with fundamental frequency remains the simplest
experimental configuration and promises highest absolute yields.

It is therefore worthwhile to quantify the challenge posed by minimizing the
probability of photons scattering into the phase space by processes other than
vacuum photon-photon-scattering events - so called _background photons_. In
practice, the photon distribution of the driving beams is not the only
background that needs to be accounted for. Numerous other sources can
contribute, including scattering off rest gas atoms and electrons, surfaces of
optics, mechanical elements and walls of the experimental vacuum chamber.
Photons from some of these sources may be suppressed by appropriate filtering
techniques (spatial, temporal, polarization and, potentially, spectral) which
themselves have finite suppression rates.

With this study, we concentrate solely on investigating the background photon
signal resulting from a single high intensity, tightly focused laser pulse in
the residual gas pressures available with the standard high-vacuum technology
commonly in use in high power laser laboratories. No measurable quantum vacuum
signature is expected (see e.g. [24], a non-zero signal from a single beam
requires corrections for large beam cone angles or higher order focus modes.)
and any photons measured constitute the background. We determined the
contribution from various sources and scaling with background pressure and
draw conclusions on requirements for future measurements with Petawatt laser
pulses.

##  2 Experiment

###  2.1 Setup and Methodology

The experiments were carried out at the JETi-200 laser facility at the
Helmholtz Institute Jena, with a short pulse Ti:Sapphire system capable of
peak powers in excess of 200TW. In this campaign we are focusing predominantly
on photon background originating from linear scattering. Consequently the
laser was operated at reduced energy, providing approximately
$1\text{\,}\mathrm{J}\mathrm{/}\mathrm{s}$ (repetition rate of 5 Hz with
$175\text{\,}\mathrm{mJ}$ per pulse) in focus at $24\text{\,}\mathrm{fs}$
full-width-at-half-maximum (_FWHM_) pulse duration (with a peak power of
$\approx$7\text{\,}\mathrm{TW}$$) at a wavelength of
$800\text{\,}\mathrm{nm}$.

Figure 1: Setup in vacuum chamber (top) and on air side (bottom). 1
$\lambda/2$-plate (only present for vertical polarization), 2 OAP, 3
collection optic and light-tight guiding and 4 4Picos single-photon camera.
Light paths and equivalent time to focal region a) for main pulse
($\approx$0.5\text{\,}\mathrm{ns}$$), b) nearest chamber wall (inside viewing
cone, $\approx$4\text{\,}\mathrm{ns}$$) and c) for nearest object outside of,
but close to viewing cone ($\approx$3\text{\,}\mathrm{ns}$$).

The beam was focused using a $None$ off-axis parabolic mirror (_OAP_) with
f/1.5 focusing, as shown in figure 1, thus providing a wide focusing cone
angle similar to that of a photon-scattering experiment. As achieving a high
peak intensity was not required, the FWHM focal spot diameter of
$2.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ achieved without compensating for
residual phase errors introduced by the parabola was deemed satisfactory.
Figure 2 shows the spatial focus distribution, the central spot contained 34%
of the pulse energy inside the $1/\mathrm{e}^{2}$ radius with a low intensity
pedestal at above 1% peak intensity containing 76% of the pulse energy and
extending $>$30\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ in the vertical and
$>$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ in the horizontal directions,
respectively. Overall this leads to an estimated peak intensity of
$5\text{\times}{10}^{19}\text{\,}\mathrm{W}\text{\,}{\mathrm{cm}}^{-2}$. The
pulse energy in focus was determined by integrating the beam brightness on
calibrated beam monitor cameras in the laser chain and subsequently corrected
for the measured transmission of remaining optics. The uncertainty of the
calibration is estimated to be less than the typical shot-to-shot fluctuations
of $<2\%$ rms. We note that the diamond machined copper OAP
($4.4\text{\,}\mathrm{nm}$ surface roughness, Kugler, Germany) scatters
$\approx 0.2\%$ of the laser light uniformly in $2\pi$, calculated via the
“total integrated scatter” due to surface roughness as described in [25],
affecting scattering from walls or other solid objects into the beam path.

Figure 2:  Spatial focus distribution with high dynamic range (HDR)
(logarithmic scale). 34% energy inside $1/\mathrm{e}^{2}$ area surrounded by
pedestal of $>$30\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ vertical extent (as
observed by collection optic) at >1% peak fluence. 76% of total energy in area
above 1% peak fluence. Close to diffraction limited performance via adaptive
optics is achievable at the JETi-200 for future studies.

The light scattered from the focal region was collected under an angle of
$None$ in the horizontal plane using a long working distance microscope
objective ($NA=0.40$, “M Plan Apo NIR B $20\times$”, Mitutoyo, Japan). The
polarization of the laser could be changed from horizontal (along observation
axis) to vertical (perpendicular to observation axis) by introducing a
$\lambda/2$-plate (uncoated mica, B Halle Nachfl., Germany) before the OAP. As
the data recorded at vertical polarization is more relevant in the following,
the pulse energy of $175\text{\,}\mathrm{mJ}$ is stated for this case. At
horizontal polarization, without the $\lambda/2$-plate, the energy was
increased by $\approx 11\%$ (calculated from the refractive index of the
uncoated mica plate). The collected light was guided by a light shielded tube
system outside of the vacuum chamber, where an achromatic lens
(“AC508-200-B-ML”, Thorlabs, US) formed a $20\times$ magnified image of the
focal region on the primary diagnostic, a temporally gated, single-photon
sensitive image-intensified CCD camera (4Picos, Stanford Computer Optics, US).
Considering the spectral characteristics of the optics and camera involved,
photons in a spectral range of $400900\text{\,}\mathrm{nm}$ were detectable.
The overall detection efficiency for light at the fundamental wavelength of
$800\text{\,}\mathrm{nm}$ is estimated to be 9%, while around
$400\text{\,}\mathrm{nm}$ wavelength (second harmonic) it is reduced to
$\approx 1\%$.

Special care was taken to keep the direct viewing cone free of any scattering
objects, since light scattered into this cone would be efficiently transported
onto the camera. Note that light scattered from objects outside the viewing
cone may still reach the camera through scattering in the collection optic and
imaging system, albeit at reduced efficiency. By introducing $\mathrm{He}$ gas
into the entire chamber, the residual gas pressure could be tuned from
$7\text{\times}{10}^{-6}\text{\,}\mathrm{mbar}$ to
$1\text{\times}{10}^{-3}\text{\,}\mathrm{mbar}$. Additionally, some data could
also be recorded at pressures up to $1\text{\,}\mathrm{mbar}$ while operating
the laser during pumpdown of the vacuum chamber. The pressure was determined
via a combined Pirani and cold-cathode gauge ($\pm 30\%$ precision, $\pm 5\%$
repeatability, PKR251, Pfeiffer Vacuum, Germany) situated
$<$1\text{\,}\mathrm{m}$$ from the focus point. When $\mathrm{He}$ is used, a
correction factor must be applied to the pressure reading. To account for the
background pressure of $>$7\text{\times}{10}^{-6}\text{\,}\mathrm{mbar}$$ of
residual air always present, the correction was applied assuming independent
partial contributions to the total pressure by $\mathrm{He}$ and residual air.
Since they may not be truly independent in reality, the error bars on any data
corrected were extended to include also the uncorrected value. During
pumpdown, a glass window was closed, separating the experimental chamber from
the evacuated beamline, leading to an overall reduction of peak fluence by 70%
compared to that shown in figure 2 (due to 15% energy loss by reflection from
the uncoated window and an increased focal spot size).

The gating window of the 4Picos camera was set to $1\text{\,}\mathrm{ns}$ and
the delay adjusted such that the camera recorded photons propagating through
the focal region simultaneously with the main pulse. We define this time as
$t\equiv 0$. The correct delay was determined by scattering attenuated pulses
off a glass needle placed in focus. The timing jitter of the trigger with
respect to the main pulse was measured to be below $100\text{\,}\mathrm{ps}$
(rms). Figure 3 shows an exemplary image taken at a pressure of
$1.3\text{\times}{10}^{-3}\text{\,}\mathrm{mbar}$ and vertical polarization,
clearly indicating the shape of the laser cone and focus as viewed from the
side, resembling a _bow tie_ shape. Since no other scattering objects are
close, the image is formed primarily by scattering off free electrons as
discussed below. Close to the image edges, transmission efficiency is reduced
by vignetting and consequently analysis concentrates on the central part of
the image.

Inside the non-vignetted area (inside dashed circle in figure 3), two regions
of interest (_ROI_) are compared. Firstly, a square box corresponding to an
area of $10\times$10\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$$ around the
waist of the beam is defined, large enough to capture all photons originating
from the focal volume. For a vacuum photon scattering experiment, the source
region for signal photons can be estimated as the intersection of all pulses’
focal volumes, approximated by cylinders with focal diameter $d_{FWHM}$ and
length of the Rayleigh range $z_{R}$. Outside this region, the spatio-temporal
overlap of the colliding pulses is negligible[14]. In addition to the bow tie,
a homogeneous scattering signal covering the entire non-vignetted image is
observed. To study this, a second ROI outside the laser cone is defined as the
_out-of-cone_ reference, highlighted in figure 3. The number of recorded
photons in this out-of-cone area is scaled down to
$100\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$ to match the focal ROI.

Figure 3:  Example shot of a single photon image at
$p=$1.3\text{\times}{10}^{-3}\text{\,}\mathrm{mbar}$$ and vertical
polarization. The laser cone (_bow tie_) is clearly visible, but surrounded by
a homogeneous signal only limited by vignetting toward the image edges. Green:
focus ROI of $10\times$10\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$$, blue:
out-of-cone ROI with margin to beam cone and clipped to vignette (dashed
circle). Red: nominal beam cone by f-number. Arrows: $z$ is laser propagation
direction, $y$ points upward in experiment. The rotation is due to the imaging
setup.

In a first step, this homogeneous background could be significantly reduced by
introducing a simple light shielding baffle, restricting the line of sight
from the OAP to the collection optic aperture. This is highlighted in figure
4, comparing the background signal in the initial setup and after a simple
light shielding baffle was installed. All other data shown was recorded with
the baffle installed.

Figure 4: Objective before (a) and after (c) covering with a light shielding
baffle, stray light rays from OAP sketched. (Focus diagnostic objective on
right removed during full power shot). Resulting raw images (b) without any
baffle at $1\text{\times}{10}^{-5}\text{\,}\mathrm{mbar}$ and (d) for same
settings with baffle, showing significantly reduced background.

To estimate the actual number of photons collected from a certain region by
the collection optic, the camera response was calibrated in a dedicated setup.
For a fixed gain of the image intensifier, the mean pixel count associated
with a single photon hit was determined by integrating the brightness in a
low-hit-count region and dividing by the number of hits determined with a peak
finding algorithm. For non-saturated images, the total number of photons
counted in a certain region is therefore the summed pixel brightness in that
region divided by the mean brightness of a single photon detection event. In
contrast to peak-finding, this method is also applicable at lower gain where
the images do not show single hits. The drawback is an increased sensitivity
to camera dark noise, since no thresholding is applied. This statistic
uncertainty is added to the error bars where applicable. The number of photons
on the detector were finally converted to the actual number of photons
collected by accounting for the quantum efficiency of the detector (16.6% at
$800\text{\,}\mathrm{nm}$ central laser wavelength) and the spectral
transmission characteristics of the collection optic setup.

The scattering signal was studied for a range of trigger delays, rest gas
pressures, laser energies and two polarizations . For every set of parameters
chosen, several shots were recorded.

###  2.2 Scattering from residual gas

Figure 5 shows the pressure dependence of the number of photons registered per
$1\text{\,}\mathrm{J}$ of laser energy in
$10\times$10\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$$ inside the focal
ROI and in the out-of-cone ROI in a time window of $1\text{\,}\mathrm{ns}$
around $t=0$. The number of photons was normalized to $1\text{\,}\mathrm{J}$
to correct for the energy differences due to the glass shutter and
$\lambda/2$-plate where relevant. During pumpdown of the chamber, the pressure
dropped continuously from atmospheric pressure. After the final pressure was
reached and the glass shutter was opened, $\mathrm{He}$ gas was introduced to
increase the pressure again.

Figure 5:  Dependence of the scattered signal on the residual gas pressure for
both polarizations inside the focal ROI and area-equivalent for the out-of-
cone ROI. Scaled to $1\text{\,}\mathrm{J}$ laser energy for comparability.
Horizontal error bars: measurement uncertainty and $\mathrm{He}$-correction
where applicable. Vertical error bars: standard deviation, at low counts
dominated by camera dark noise. Vertical bar at
$1.3\text{\times}{10}^{-3}\text{\,}\mathrm{mbar}$: pressure at which figure 6
and 7 were evaluated. Blue line: Thomson scattering estimate from a free
electron density equal to the rest gas particle density.

For both horizontal and vertical polarization, the signal inside the focus
region increases approximately linearly with pressure, while the out-of-cone
contribution stays constant also for higher pressures. This indicates that
outside the nominal beam cone, scattering from the rest gas is negligible
compared to other, pressure-independent scattering sources. Inside the focal
ROI, this contribution adds a constant offset to the number of photons
measured. Below $\approx$1\text{\times}{10}^{-4}\text{\,}\mathrm{mbar}$$, the
amount of scattered light in focus and outside the laser cone each converge to
the same value for each polarization, namely $(62\pm 7)$ photons per
$1\text{\,}\mathrm{J}$ per $10\times$10\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{2}$$ for vertical and $(43\pm
4)${\mathrm{J}}^{-1}$($10\text{\,}\mathrm{\SIUnitSymbolMicro m}$)^{-2}$
photons for horizontal polarization.

The approximately linear dependence of photon number on gas pressure is
related to Thomson scattering of the laser pulse by free electrons. Field
ionization dominates already at intensities beyond
$10^{15}$\mathrm{W}\text{\,}{\mathrm{cm}}^{-2}$$[26]. One hence expects
multiply ionised residual gas atoms and correspondingly high electron density.
This is in line with the observed difference of total scattering for the same
number density of gas particles for $\mathrm{He}$ or residual air dominated
data points. In the former case, an electron density up to twice the gas
particle density $n_{0}$ can be expected, while in the latter case e.g.
$\mathrm{N}_{2}$ or $\mathrm{O}_{2}$ can contribute several electrons each.
The blue solid line in figure 5 represents the Thomson scattering signal from
an electron density that equals the gas particle density at the respective
pressure, corresponding to single ionization. This yield was calculated from
integrating the emission pattern of a dipole directed perpendicular to the
observation axis over the acceptance angle of the collecting objective. Both
for $\mathrm{He}$-dominated as well as for residual air dominated data points
the signal in focus for vertical polarization lies below this estimate and
indicates that the electron density is significantly reduced within the focal
spot volume. This will be analysed in more detail below. The same simple
calculation at horizontal polarization (where the fictive dipole points
perfectly into the direction of observation of the objective) predicts a
reduction of the signal by a factor of 12, while the experimental data (orange
symbols in figure 5) shows a ratio of $\approx 6\times$ between horizontal and
vertical polarization. This discrepancy can be expected since the short focal
length of the OAP leads to dipole orientations that deviate considerably from
the assumption. The scattering cross sections for bound atoms and electrons
(Rayleigh scattering) and charged nuclei are orders of magnitude lower at the
relevant wavelengths and can be neglected.

In addition to the analysis of the focal region, the effective scatterer
density along the laser propagation axis was determined for groups of shots at
similar settings.

Figure 6:  a) Average number of photons per shot $N_{p}$ per
${\mathrm{\SIUnitSymbolMicro m}}^{2}$ over 242 shots at
${\approx$1.3\text{\times}{10}^{-3}\text{\,}\mathrm{mbar}$}$ and
$175\text{\,}\mathrm{mJ}$ pulse energy each. The out-of-cone background was
subtracted. b) Image a) integrated along vertical extent
($\pm$70\text{\,}\mathrm{\SIUnitSymbolMicro m}$$), giving number of photons
per $\mathrm{\SIUnitSymbolMicro m}$ slice along the optical axis. A reduction
of integrated intensity is clearly visible in the center region. For the image
shown, the reduction factor between minimum in focus and maximum peak is
$5.6\times$ (black triangles). Laser propagation from right to left. c)
Numerical simulation of Thomson scattering by electrons ionized from
$\mathrm{He}$ by an ideal Gaussian laser pulse. Details see text. d) Image c)
integrated along vertical extent ($\pm$70\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$), giving number of photons per $\mathrm{\SIUnitSymbolMicro m}$ slice
along the optical axis. The reduction of integrated intensity in focus is even
more significant compared to b), most likely due to ideal focusing of the
simulated Gaussian.

Figure 6a shows the averaged number of detected photons per
${\mathrm{\SIUnitSymbolMicro m}}^{2}$ per shot over 242 shots recorded at
$1\text{\times}{10}^{-3}1.5\text{\times}{10}^{-3}\text{\,}\mathrm{mbar}$ and
$175\text{\,}\mathrm{mJ}$ at vertical polarization. The signal level
determined in the out-of-cone ROI was subtracted from each image before
averaging. Figure 6b shows the integral of this image along the entire
vertical extent and yields the total number of photons scattered per
$\mathrm{\SIUnitSymbolMicro m}$ slice along the optical axis. The scatter
yield drops significantly towards the focus by a reduction factor of $5.6$
compared to the maximum at $>$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
($>46$ Rayleigh lengths) upstream. More than
$250\text{\,}\mathrm{\SIUnitSymbolMicro m}$ away from focus, the integrated
intensity drops at the image edges due to vignetting.

The observed reduction of scattered photons in the volume of highest laser
intensity with increasing laser intensity is expected. Ponderomotive
cavitation occurs for [27]
$I_{18}\lambda_{\mu}^{2}>1/20n_{18}\sigma_{\mu}^{2}$ ($I_{18}$ intensity in
$10^{18}$\mathrm{W}\text{\,}{\mathrm{cm}}^{-2}$$, $n_{18}$ electron density in
$10^{18}cm^{-3}$, $\lambda_{\mu}$ wavelength in $\mathrm{\SIUnitSymbolMicro
m}$ and $\sigma_{\mu}$ the $1/e$ radius in intensity in
$\mathrm{\SIUnitSymbolMicro m}$). In the above example for
$1.3\text{\times}{10}^{-3}\text{\,}\mathrm{mbar}$ the electron density of
singly ionized Helium is $n_{18}=$3\text{\times}{10}^{-5}$$ and the peak
intensity in focus is $I_{18}=50$, hence, we exceed this threshold by far.
Assuming a perfect Gaussian pulse with $\sigma_{\mu}=0.7$ (beam waist
$w_{0}=\sqrt{2}\sigma=$1.05\text{\,}\mathrm{\SIUnitSymbolMicro m}$)$,
cavitation on axis is expected as long as
$I_{18}\frac{\sigma_{0}^{2}}{\sigma(z)^{2}}\lambda_{\mu}^{2}>1/20n_{18}\sigma(z)^{2}$,
that is for
$z<(20I_{18}\lambda_{\mu}^{2}/(\sigma_{\mu}^{2}n_{18}))^{1\textfractionsolidus
4}z_{R}\approx 74$ times the Rayleigh length $z_{R}$ or
$320\text{\,}\mathrm{\SIUnitSymbolMicro m}$, larger than the field of view of
our microscope.

On the other hand, the laser pulse duration
$\sigma_{t}=\tau_{FWHM}/\sqrt{4\mathrm{ln}2}$ is too short for establishing an
equilibrium situation on which the above estimate is based, that is the plasma
period $1.1ps/\sqrt{n_{18}}\gg\sigma_{t}$. This also means that the force of
the ionic background on the electrons can be ignored for early times of
electron motion. We therefore can attempt to explain the photon scatter image
and vertical integral of figure 6a and b by a simple model. $V\cdot n_{0}$
Helium atoms are randomly distributed within a predefined simulation volume
$V=\Delta x\Delta y\Delta z$, where
$n_{0}=$3.2\text{\times}{10}^{13}\text{\,}{\mathrm{cm}}^{-3}$$ is the particle
(atom) density corresponding to a residual gas pressure of
$1.3\text{\times}{10}^{-3}\text{\,}\mathrm{mbar}$. We consider a Gaussian
laser intensity distribution
$I_{L}(t,r,z)=I_{0}w_{0}^{2}/w(z)^{2}\cdot\exp[-2r^{2}/w(z)^{2}-(ct-z)^{2}/(c\sigma_{t})^{2}]$,
where $r=(x^{2}+y^{2})^{1/2}$ and $w(z)=w_{0}(1+(z/z_{R})^{2})^{1/2}$ with
$w_{0}=$1.05\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
$z_{R}=$4.3\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
$\sigma_{t}=\tau_{FWHM}/\sqrt{4\mathrm{ln}2}=$14.4\text{\,}\mathrm{fs}$$ and
$I_{0}=$4\text{\times}{10}^{20}\text{\,}\mathrm{W}\text{\,}{\mathrm{cm}}^{-2}$$,
corresponding to an ideal Gaussian pulse with same $d_{FWHM}$ and $I_{peak}$
as the theoretical, diffraction limited best focus in the experiment.
Electrons are considered free when the intensity exceeds the 50% threshold
intensity for field ionization, $I_{k}$ [26]. The electron $i$ will hence
start moving in the laser field for times
$t>t_{0i}=z_{i}/c-\sigma_{t}(\ln[(I_{0}/I_{k})(w_{0}^{2}/w^{2}(z_{i})]-2r^{2}/w(z_{i})^{2})^{1/2}$.
We then solve the equation of motion for each electron, considering only the
transverse ponderomotive force,
$\dot{\vec{p}}_{i}(t)=-1/(2cn_{c})\nabla_{\perp}I(t,r_{i}(t),z_{i})$ and
$\dot{\vec{r}}_{i}(t)=c\vec{p}_{i}(t)/((mc)^{2}+p_{i}^{2}(t))^{1/2}$, where
$n_{c}=$1.74\text{\times}{10}^{21}\text{\,}{\mathrm{cm}}^{-3}$$ is the
critical density and $z_{i}$ the initial (random) position of the electron
along laser propagation direction. The numerical solution yields the
trajectories $\vec{r}_{i}(t)=(x_{i}(t),y_{i}(t),z_{i})$. We then calculate for
each electron the number of photons that it scatters via Thomson scattering
while moving along its trajectory, $\Delta
N_{i}(t,x_{i},z_{i})=\sigma_{T}/(2\hbar\omega_{L})(I_{L}(t,x_{i}(t),z_{i})+I_{L}(t+\Delta
t,x_{i}(t+\Delta t),z_{i}))\Delta t$. This contribution is distributed onto a
predefined $(x_{k},z_{l})$ grid. Finally, all contributions in a single cell
$x_{k},z_{l}$ are summed to yield a calculated side view image of the focal
volume (figure 6c) and the corresponding integrated yield along propagation
direction (figure 6d). The reduction is clearly visible and stronger than in
the experimental case. This is likely due to the fact that the pulse in the
experiment is not a perfect Gaussian. Overall, this simple model is in good
agreement with the measured effective scatterer density considering the
idealized assumptions made.

We do not observe the complete extent of the volume that is affected by this
dynamic behavior of the scattering electrons. Nevertheless it is interesting
to state that the reduction was observed for a variety of other experimental
settings. Table 1 summarizes the reduction factors between the focus
brightness and maximum brightness observed in each graph for the cases that we
analysed. For highest pulse energies, a reduction factor of 4.6 to 10.5 was
determined for both polarizations at best focusing. For pressures
$>$2\text{\times}{10}^{-3}\text{\,}\mathrm{mbar}$$, the reduction factor was
limited to $1.7-4.3\times$, likely due to the degraded focus with the
transmissive glass shutter present. The reduction is also moderated to
$2.2\times$ ($1.7\times$) for $20\text{\,}\mathrm{mJ}$
($4\text{\,}\mathrm{mJ}$) of pulse energy, respectively. This trend supports
the importance of the ponderomotive cavitation with increasing intensity once
more.

Table 1:  Summary of scattering reduction factors as exemplified in figure 6
for other parameters. Note that the peak fluence was reduced when glass
shutter was closed and increased for horizontal polarization, where the
$\lambda/2$-plate was removed. (**) Signal low.

  * Glass shutter | Closed | Closed | Open | Open | Open | Open
---|---|---|---|---|---|---
p[mbar] | $>$2\text{\times}{10}^{-3}$$ | $1.3\text{\times}{10}^{-3}$ | $1.3\text{\times}{10}^{-3}$ | $3\text{\times}{10}^{-4}$ | $1.3\text{\times}{10}^{-3}$ | $1.3\text{\times}{10}^{-3}$
E[mJ] for V pol. | 148 | 148 | 175 | 175 | 20 | 4
V. polarization | 3.7 | 4.3 | 5.6 | 10.5 | 2.2 | 1.7
E[mJ] for H pol. | 164 | 164 | 194 | 194 | 22 | 4
H. polarization | 1.7 | 3.3 | 4.6 | ** | ** | **

We stress that the model contains several simplifications that might affect
the results. In particular, although the equation of motion is
relativistically correct, we ignore the force in laser propagation direction.
We also do not resolve the oscillation period and hence have no measure of
describing coherent effects or scattering in frequency ranges other than
covered by the laser. Nevertheless, because in our experiments the signal
recorded was dominated by light at the fundamental frequency (this was tested
by inserting a broadband dielectric mirror optimized for reflection of light
around $800\text{\,}\mathrm{nm}$ wavelength before the camera), we are
confident that the effect of ponderomotive cavitation in sub-relativistic
intensity regions dominates the reduction of Thomson scatter background.

###  2.3 Scattering from static scatterers

Toward a potential quantum vacuum experiment, in addition to the scattering by
the rest gas it is also crucial to understand and subsequently minimize the
static, pressure-independent scattering sources. To study this contribution
further, the time dependence of the signal can provide some additional
insight.

By varying the delay between the laser pulse and the camera exposure trigger,
a temporal scan of the photon signal was obtained. Figure 7 shows a time scan
of the photon number in a $1\text{\,}\mathrm{ns}$ window recorded at
$p=$1.3\text{\times}{10}^{-3}\text{\,}\mathrm{mbar}$$ for vertical
polarization, again scaled to $1\text{\,}\mathrm{J}$ of laser pulse energy.
The number of photons in the focus ROI around $t=0$ is $(800\pm
300)${\mathrm{J}}^{-1}$($10\text{\,}\mathrm{\SIUnitSymbolMicro m}$)^{-2}$ and
is equal to that shown in figure 5 (uncertainty due to photon statistics and
camera dark noise). Notably, the signal inside the focal ROI and the out-of-
cone region only differ around $t=0$, i.e. all the photons registered at other
times must have a pressure-independent origin.

Figure 7: Time delay scan. Number of photons recorded in a
$1\text{\,}\mathrm{ns}$ gate width in the focus ROI and area-equivalent from
out-of-cone ROI for vertical polarization at
$1.3\text{\times}{10}^{-3}\text{\,}\mathrm{mbar}$, per $1\text{\,}\mathrm{J}$
of pulse energy. Error bars show standard deviation. The signal at $t=0$ is
equal to that shown in pressure scan. Due to log-scale, data points $\leq 0$
not visible.

The temporal width of this first peak is determined by the convolution of the
laser pulse temporal shape ($\tau_{FWHM}\approx$24\text{\,}\mathrm{fs}$$) and
the gating function of the camera ($\tau_{exposure}=$1\text{\,}\mathrm{ns}$$),
thus yielding a $\approx$1\text{\,}\mathrm{ns}$$ peak. After
$t>$1\text{\,}\mathrm{ns}$$, the signal in focus drops again to less than one
photon per Joule, indicating very few photons crossing the focal region after
the main pulse passed. The signal next rises at $t>$2.5\text{\,}\mathrm{ns}$$
where $(6300\pm
1300)${\mathrm{J}}^{-1}$($10\text{\,}\mathrm{\SIUnitSymbolMicro m}$)^{-2}$
photons are recorded in $1\text{\,}\mathrm{ns}$. This temporal delay matches
the additional path length travelled by stray light of the main pulse bouncing
off objects located in the experiment chamber as shown by path c) in figure 1.
Between $t=$5\text{\,}\mathrm{ns}$$ and $t=$10\text{\,}\mathrm{ns}$$, the
signal decays but does not drop to zero again, which is in accordance with
temporally smeared out scattering from the large walls with a minimum path
delay highlighted as b) in figure 1. Finally, at $t>$10\text{\,}\mathrm{ns}$$
the signal increases drastically and the detector saturates (even with lower
gain applied). This temporal delay matches the additional path length
travelled by the main pulse bouncing off the (bare aluminium) chamber walls
and back to the focal region, giving rise to a signal several orders of
magnitude larger than the signal at $t=0$.

This behaviour can be understood as follows. The laser pulse carries the
majority of its energy in a time window much shorter than the
$1\text{\,}\mathrm{ns}$ gate width around the main peak. Most of this light
travels inside the cone expected from geometrical optics. Diffraction effects
can be neglected, except close to the focal region
($\ll$1\text{\,}\mathrm{mm}$$). This major part of the pulse can only
contribute light around $t=0$ via scattering directly from free electrons and
residual gas atoms or molecules. Further downstream, photons from the main
beam hit the chamber walls, get scattered in all directions and can enter the
imaging system after a minimum round trip of
$\approx$3.3\text{\,}\mathrm{m}$$, giving rise to the large signal after
$11\text{\,}\mathrm{ns}$. One possibility for additional background signal
around $t=0$ therefore arises by light travelling
$\approx$11\text{\,}\mathrm{ns}$$ ahead of the main pulse. Although high power
laser pulses are often accompanied by pre-pulses and _amplified spontaneous
emission_ (ASE) on the picosecond and nanosecond time scale, a contribution
over $11\text{\,}\mathrm{ns}$ ahead is assumed to be negligible, supported by
the observation that immediately before $t=0$ (corresponding to ASE at
$-12\text{\,}\mathrm{ns}$ ahead of the main pulse) the background signal is
below detection threshold. A possible contribution of ASE at shorter
timescales ahead of $t=0$ will be discussed below.

Apart from the light travelling inside the expected path, a small but relevant
portion of the energy will be scattered by imperfect surfaces of the beam
transport optics. With hardly any delay, photons escaping the main cone can
potentially enter the collection optic at a large angle, where they further
scatter in the lens media and off the tube walls. The importance of this
contribution is evidenced by the images shown in figure 4, where a simple
shielding of the exposed entrance aperture reduces the background
significantly. When also considering additional delay, the light scattered off
beam optics can illuminate an object inside or close to the observation cone,
increasing the background signal at later times, for example giving rise to
the peak at $t=$2.7\text{\,}\mathrm{ns}$$. With the same argument as above,
light preceding the main pulse could therefore contribute to the background
signal around $t=0$ via the delayed paths. On this shorter time scale, the
contribution of ASE or pre-pulses may not be negligible. The JETi-200 laser
exhibits a temporal intensity contrast better than $1\text{\times}{10}^{-10}$
at times earlier than $50\text{\,}\mathrm{ps}$ before the main pulse.
Integrating the peak intensity over $24\text{\,}\mathrm{fs}$ and the ASE
intensity level over the gating time of $1\text{\,}\mathrm{ns}$ yields a ratio
of $4\text{\times}{10}^{-6}$ in terms of energy/ photon number. As the main
pulse contributes $\approx
6300${\mathrm{J}}^{-1}$($10\text{\,}\mathrm{\SIUnitSymbolMicro m}$)^{-2}$
photons at $t=$2.7\text{\,}\mathrm{ns}$$, the ASE could therefore contribute
$0.03${\mathrm{J}}^{-1}$($10\text{\,}\mathrm{\SIUnitSymbolMicro m}$)^{-2}$
photons at $t=0$. For a prepulse to contribute 1 photon per Joule at $t=0$, it
would have to be at a power fraction of
$1/6300\approx$2\text{\times}{10}^{-4}$$ compared to the main pulse. Typical
high power laser systems exhibit prepulses at lower than
$1\text{\times}{10}^{-6}$ to $1\text{\times}{10}^{-8}$ level on the picosecond
to nanosecond time scale, the contribution from prepulses can thus be
neglected. Finally, light arriving at later times can still contribute to the
signal at $t=0$ if the detector does not turn off completely, i.e. the high
voltage on the image intensifying micro-channel plate takes some time to
return to $0\text{\,}\mathrm{V}$. This was tested in a separate setup by
shining attenuated pulses directly onto the camera and comparing the number of
photons registered for times $13\text{\,}\mathrm{ns}$ before $t=0$ to those
registered around $t=0$. This results in an upper bound for the gating
suppression ratio of $1\text{\times}{10}^{-4}$ at $13\text{\,}\mathrm{ns}$
before $t=0$. Light of the peak around $t=$2.7\text{\,}\mathrm{ns}$$ ($\approx
6300${\mathrm{J}}^{-1}$($10\text{\,}\mathrm{\SIUnitSymbolMicro m}$)^{-2}$) can
therefore contribute no more than
$0.6${\mathrm{J}}^{-1}$($10\text{\,}\mathrm{\SIUnitSymbolMicro m}$)^{-2}$
photons at $t=0$ via camera “leakage”.

Taken together, these temporal effects can therefore explain no more than
$\approx 0.6${\mathrm{J}}^{-1}$($10\text{\,}\mathrm{\SIUnitSymbolMicro
m}$)^{-2}$ photons recorded at $t=0$. The majority of the pressure-independent
background contribution must originate from directly scattered light of the
main pulse, arriving within $\ll$1\text{\,}\mathrm{ns}$$ with respect to the
pressure-dependent signal from the rest gas scattering. Primarily, this is
expected to originate from light scattered directly off the focusing optic.
However, any upstream optic can also contribute, since light scattered off
these optics is concentrated by the focusing optic. This can be easily
understood by tracing the rays of a point source imaged by the OAP to a finite
imaging distance behind the focus. The closest upstream optic is the
$\lambda/2$-plate (only present for measurements at vertical polarization) at
a distance to the OAP of $\approx$30\text{\,}\mathrm{cm}$$. A real image with
$1.5\times$ magnification is then formed at $45\text{\,}\mathrm{cm}$ behind
the OAP. Part of the rays forming the image intersect the opening aperture of
the collection optic, hence light scattered off the $\lambda/2$-plate can
contribute significantly to the background signal. For an object distance
$>$61\text{\,}\mathrm{cm}$$ no more imaging rays intersect the opening
aperture at the right angle, so optics further upstream will have a smaller
impact.

In summary, assuming other scattering sources (e.g. optics
$\gg$61\text{\,}\mathrm{cm}$$ upstream) are much less significant than the OAP
and the $\lambda/2$-plate, the total number of observed background photons can
be split between the contributions identified. As all effects are based on
scattering from imperfect surfaces, no significant polarization-dependence of
each contribution is expected and difference between the total scattered light
for horizontal and vertical polarization can be attributed to the presence of
the $\lambda/2$-plate. Table 2 summarizes this result.

Table 2:  Summary of estimated pressure-independent background photons per
$1\text{\,}\mathrm{J}$ per $100\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$
in focus attributed to different sources.

  * Polarization | Horizontal | Vertical
---|---|---
Total | 43 | 62
ASE | 0.03 | 0.03
Gating suppression | 0.6 | 0.6
OAP | 42 | 42
$\lambda/2$-plate | n.a. | 19

##  3 Extrapolation to Petawatt scale

To provide insights and potential guidelines for future photon-photon
scattering experiments, the results need to be put into context of higher
laser peak power and potentially improved detection schemes. To this end, the
recorded data is scaled to $175\text{\,}\mathrm{J}$ pulse energy and the
pressure-independent background of the out-of-cone ROI is subtracted from the
signal in the focus ROI. Hereby we can provide estimates for the parameters
required to reach single photon level inside the focal ROI. This corresponds
to the conditions with a peak power of $\approx$5\text{\,}\mathrm{PW}$$ as
expected e.g. at the “Extreme Light Infrastructure” (ELI)[28]. Figure 8
visualizes the estimated background contributions for these parameters.

Figure 8:  Simple estimate for the expected background signal contributions at
$175\text{\,}\mathrm{J}$ laser energy ($\approx$5\text{\,}\mathrm{PW}$$). The
out-of-cone background was subtracted from the focus ROI data to show
pressure-dependence more clearly. To reach the single photon level (red
ellipse), a reduction of scattering by $10^{3}\times$ is necessary for the
given setup, while at a pressure of
$1\text{\times}{10}^{-7}\text{\,}\mathrm{mbar}$ below 100 photons are
expected, potentially reduced further by electron cavitation due to higher
peak intensities.

At a residual gas pressure on the order of
$1\text{\times}{10}^{-7}\text{\,}\mathrm{mbar}$ the rest gas will contribute
$<100$ photons per $100\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$ on
average assuming linear scaling. The ponderomotive electron cavitation will
play a vital role to reduce scattering from the focal volume further and might
make other (electron) cleaning methods obsolete. While the scattering cross
sections of the nuclei will remain negligible, the electric fields of these
nuclei could affect the quantum vacuum state and hence one needs to consider
removing them dynamically.

Assuming similar components and detectors are used as in this setup, the
static background would have to be reduced by 3-4 orders of magnitude to reach
the single photon level. By shortening the gating time from
$1\text{\,}\mathrm{ns}$ to the pulse duration scale, e.g. based on optical
switching, the ASE contribution can be suppressed by
$\frac{$1\text{\,}\mathrm{n}\mathrm{s}$}{$24\text{\,}\mathrm{f}\mathrm{s}$}=$4\text{\times}{10}^{4}$$
times to a negligible level. The gating suppression of the detector discussed
will also be improved in a similar order. For a prepulse to contribute to the
signal at $t=0$, it would have to exceed an energy level of
$1/($6300\text{\,}{\mathrm{J}}^{-1}$*$175\text{\,}\mathrm{J}$)\approx$1\text{\times}{10}^{-6}$$
of the main pulse as well as matching additional path length from a scattering
surface. For typical high power laser systems, prepulses can be suppressed
well below this level. Photons scattered from optics likely remain the
strongest contribution to the background and must be reduced by using surfaces
with (much) higher optical quality. The static-scattering improvement will
need to be determined by future experiments.

Most critical is the aspect that a realistic quantum vacuum experiment will
require at least two beams and the few signal photons will not emerge in
$None$, therefore the collection optic will point more towards one of the
focusing optics. In this case we expect a rapid, yet to be determined, growth
of the static scattering background contribution. Shortening the gating time
of the detector from ns to ps or even fs therefore seems necessary to
discriminate these photons.

##  4 Conclusion

We have investigated scattering from $175\text{\,}\mathrm{mJ}$ laser pulses
tightly focused in the residual gas environment of an interaction chamber.
Scaling to 1000 times larger pulse energy reveals that the residual gas
density in the focus is likely not the limiting factor, in particular due to
the ponderomotive cavitation. Significant improvement of surface quality of
involved optics and reduction of temporal gate times from nanoseconds to
femtoseconds are key to expedient experiments. Ideally, those technical
improvements can yield a discrimination of static scatter by 5-6 orders of
magnitude. Finally, an ideal setup should avoid all direct light paths from
single scattering events into the collection optics of the signal chain.

##  5 Acknowledgements

The authors wish to thank all technicians and colleagues at the JETi laser for
their support during the experimental campaign. This work has been funded by
the Deutsche Forschungsgemeinschaft (DFG) under Grant No. 416702141 within the
Research Unit FOR2783/1.

## References

  * [1] Werner Heisenberg and Heinrich Euler  “Folgerungen Aus Der Diracschen Theorie Des Positrons”  In _Zeitschrift für Physik_ 98.11-12, 1936, pp. 714–732
  * [2] F. Moulin and D. Bernard  “Four-Wave Interaction in Gas and Vacuum: Definition of a Third-Order Nonlinear Effective Susceptibility in Vacuum: $X$vacuum(3)”  In _Optics Communications_ 164.1-3, 1999, pp. 137–144  DOI: 10.1016/S0030-4018(99)00169-8
  * [3] F. Moulin, De Bernard and François Amiranoff  “Photon-Photon Elastic Scattering in the Visible Domain”  In _Zeitschrift für Physik C: Particles and Fields_ 72.4, 1996, pp. 607–611
  * [4] D. Bernard et al.  “Search for Stimulated Photon-Photon Scattering in Vacuum”  In _The European Physical Journal D-Atomic, Molecular, Optical and Plasma Physics_ 10.1, 2000, pp. 141–145
  * [5] Federico Della Valle et al.  “The PVLAS Experiment: Measuring Vacuum Magnetic Birefringence and Dichroism with a Birefringent Fabry–Perot Cavity”  In _The European Physical Journal C_ 76.1, 2016  DOI: 10.1140/epjc/s10052-015-3869-8
  * [6] Agathe Cadène et al.  “Vacuum Magnetic Linear Birefringence Using Pulsed Fields: Status of the BMV Experiment”  In _The European Physical Journal D_ 68.1, 2014, pp. 16  DOI: 10.1140/epjd/e2013-40725-9
  * [7] Xing Fan et al.  “The OVAL Experiment: A New Experiment to Measure Vacuum Magnetic Birefringence Using High Repetition Pulsed Magnets”  In _The European Physical Journal D_ 71.11, 2017, pp. 308  DOI: 10.1140/epjd/e2017-80290-7
  * [8] Thomas Heinzl et al.  “On the Observation of Vacuum Birefringence”  In _Optics Communications_ 267.2, 2006, pp. 318–321  DOI: 10.1016/j.optcom.2006.06.053
  * [9] Felix Karbstein and Chantal Sundqvist  “Probing Vacuum Birefringence Using X-Ray Free Electron and Optical High-Intensity Lasers”  In _Physical Review D_ 94.1, 2016  DOI: 10.1103/PhysRevD.94.013004
  * [10] Felix Karbstein et al.  “Vacuum Birefringence at X-Ray Free-Electron Lasers”  In _New Journal of Physics_ 23.9, 2021, pp. 095001  DOI: 10.1088/1367-2630/ac1df4
  * [11] David Enterria and Gustavo G. Silveira  “Observing Light-by-Light Scattering at the Large Hadron Collider”  In _Physical Review Letters_ 111.8, 2013, pp. 080405  DOI: 10.1103/PhysRevLett.111.080405
  * [12] G. Aad et al.  “Observation of Light-by-Light Scattering in Ultraperipheral Pb + Pb Collisions with the ATLAS Detector”  In _Physical Review Letters_ 123.5, 2019, pp. 052001  DOI: 10.1103/PhysRevLett.123.052001
  * [13] A.M. Sirunyan et al.  “Evidence for Light-by-Light Scattering and Searches for Axion-like Particles in Ultraperipheral PbPb Collisions at s NN = 5.02 TeV”  In _Physics Letters B_ 797, 2019, pp. 134826  DOI: 10.1016/j.physletb.2019.134826
  * [14] Holger Gies, Felix Karbstein and Christian Kohlfürst  “All-Optical Signatures of Strong-Field QED in the Vacuum Emission Picture”  In _Physical Review D_ 97.3, 2018  DOI: 10.1103/PhysRevD.97.036022
  * [15] E. Lundström et al.  “Using High-Power Lasers for Detection of Elastic Photon-Photon Scattering”  In _Physical Review Letters_ 96.8, 2006  DOI: 10.1103/PhysRevLett.96.083602
  * [16] Ben King, Antonino Di Piazza and Christoph H. Keitel  “A Matterless Double Slit”  In _Nature Photonics_ 4.2, 2010, pp. 92–94  DOI: 10.1038/nphoton.2009.261
  * [17] Daniele Tommasini and Humberto Michinel  “Light by Light Diffraction in Vacuum”  In _Physical Review A_ 82.1, 2010, pp. 011803  DOI: 10.1103/PhysRevA.82.011803
  * [18] Felix Karbstein and Elena A. Mosman  “Enhancing Quantum Vacuum Signatures with Tailored Laser Beams”  In _Physical Review D_ 101.11, 2020, pp. 113002  DOI: 10.1103/PhysRevD.101.113002
  * [19] W T Hill and L Roso  “Probing the Quantum Vacuum with Petawatt Lasers”  In _Journal of Physics: Conference Series_ 869, 2017, pp. 012015  DOI: 10.1088/1742-6596/869/1/012015
  * [20] Alexander Blinne et al.  “All-Optical Signatures of Quantum Vacuum Nonlinearities in Generic Laser Fields”  In _Physical Review D_ 99.1, 2019  DOI: 10.1103/PhysRevD.99.016006
  * [21] Felix Karbstein, Alexander Blinne, Holger Gies and Matt Zepf  “Boosting Quantum Vacuum Signatures by Coherent Harmonic Focusing”  In _Physical Review Letters_ 123.9, 2019  DOI: 10.1103/PhysRevLett.123.091802
  * [22] Leonhard Klar  “Detectable Optical Signatures of QED Vacuum Nonlinearities Using High-Intensity Laser Fields”  In _Particles_ 3.1, 2020, pp. 223–233  DOI: 10.3390/particles3010018
  * [23] Elena A. Mosman and Felix Karbstein  “Vacuum Birefringence and Diffraction at an X-Ray Free-Electron Laser: From Analytical Estimates to Optimal Parameters”  In _Physical Review D_ 104.1, 2021  DOI: 10.1103/PhysRevD.104.013006
  * [24] Felix Karbstein and Rashid Shaisultanov  “Stimulated Photon Emission from the Vacuum”  In _Physical Review D_ 91.11, 2015  DOI: 10.1103/PhysRevD.91.113002
  * [25] John C. Stover  “Optical Scattering: Measurement and Analysis”  Bellingham, Washington USA: SPIE Press, 2012
  * [26] Britton Chang, Paul R. Bolton and David N. Fittinghoff  “Closed-Form Solutions for the Production of Ions in the Collisionless Ionization of Gases by Intense Lasers”  In _Physical Review A_ 47.5, 1993, pp. 4193–4203  DOI: 10.1103/PhysRevA.47.4193
  * [27] Paul Gibbon  “Short Pulse Laser Interactions with Matter: An Introduction”  PUBLISHED BY IMPERIAL COLLEGE PRESS AND DISTRIBUTED BY WORLD SCIENTIFIC PUBLISHING CO., 2005  DOI: 10.1142/p116
  * [28] I… Turcu et al.  “High Field Physics and QED Experiments at ELI-NP”  In _Rom. Rep. Phys._ 68, 2016, pp. S145


# DiCOVA-Net: Diagnosing COVID-19 using Acoustics based on Deep Residual
Network for the DiCOVA Challenge 2021

###### Abstract

In this paper, we propose a deep residual network-based method, namely the
DiCOVA-Net, to identify COVID-19 infected patients based on the acoustic
recording of their coughs. Since there are far more healthy people than
infected patients, this classification problem faces the challenge of
imbalanced data. To improve the model’s ability to recognize minority class
(the infected patients), we introduce data augmentation and cost-sensitive
methods into our model. Besides, considering the particularity of this task,
we deploy some fine-tuning techniques to adjust the pre-training ResNet50.
Furthermore, to improve the model’s generalizability, we use ensemble learning
to integrate prediction results from multiple base classifiers generated using
different random seeds. To evaluate the proposed DiCOVA-Net’s performance, we
conducted experiments with the DiCOVA challenge dataset. The results show that
our method has achieved 85.43% in AUC, among the top of all competing teams.

Index Terms: COVID-19 diagnosis, imbalanced data, data augmentation, focal
loss, transfer learning

##  1 Introduction

Since the COVID-19 outbreak in 2020, countries around the world have been
shrouded in the haze of the epidemic. Until now, people in many countries were
at risk of infection. In the process of combating the epidemic, countries are
actively working to avoid large-scale infection in their countries. Countries
such as China, the United States and the United Kingdom have all developed
vaccines against COVID-19 and called on people to get vaccinated. But the
vaccine’s protection period is limited, and many people are reluctant to get
vaccinated. Therefore, effective testing is still needed to detect COVID-19
infected persons.

In the early days of the outbreak, the primary tests for COVID-19 were X-rays
and computed tomography (CT). When medical images of the lungs show symptoms
such as shadows, the patient is more likely to be infected with COVID-19 [1].
At the beginning of the epidemic, this detection method achieved relatively
good results, and played an important role in the further spread of the
epidemic. With the development of research on COVID-19, swab test is
considered to be a reliable method for detecting COVID-19 infection. At
present, most countries use throat swab or nose swab test. In addition, some
studies have shown that saliva may also be an effective test method [2].

In addition to the testing methods for infected persons mentioned above, other
testing methods are also being studied. Numerous studies have shown that
COVID-19 mainly damages lung tissue and affect the respiratory system [3].
Therefore, it is hypothesized that COVID-19 may lead to significant changes in
the acoustic characteristics of the infected patients. Inspired by this, in
this paper, we will explore a non-invasive method for detecting COVID-19
infected persons, which uses cough audio data to construct a classification
model.

In recent years, machine learning and deep learning have made remarkable
achievements in many fields such as image processing [4], speech recognition
[5] and text analysis [6]. In the wake of the COVID-19 outbreak, researchers
are also trying to apply artificial intelligence technology to the fight
against the epidemic. For example, Tuli et al. [7] designed a model based on
cloud computing and machine learning to predict the growth and trend of
COVID-19 pandemic. Yan et al. [8] developed an XGBoost machine learning-based
model that could predict the mortality rates of patients more than 10 days.
Ardakani et al. [9] applied the deep learning technique to manage COVID-19 in
routine clinical practice using CT images. Inspired by these studies, this
paper aims to identify COVID-19 infected patients from the audio recordings of
their coughing.

In order to enhance the recognition ability, we equipped our model, namely the
the DiCOVA-Net, with various data augmentation, transfer learning, cost-
sensitive learning and ensemble learning techniques. To overcome the
limitation in the robustness of the model, we leverage a novel randomness
method to further improve the model’s robustness and the ensemble results
outperform the previous methods.

The remaining of the paper is organized as follows. Section 2 reviews related
work. Section 3 describes the proposed method in detail. Section 4 presents
the experiment and results. In the last section, this study is summarized.

##  2 Related Work

In essence, the task of identifying COVID-19 infected people using acoustic
data in this paper falls under the category of speech recognition. In the
early stage of speech recognition research, scholars used the acoustic model
developed based on Gaussian Mixture Models (GMMs), and achieved certain
recognition effect. However, with the recent advancement of the deep learning
methods, scholars found that the deep neural networks (DNNs) performs much
better than GMMs [10]. A spectrogram is a visual representation of the
spectrum of frequencies of a signal as it varies with time which helps deep
learning model to model the audio data. [11].

In the practice of medical field, the problem of imbalanced data is common.
Through analysis, we found that the recognition of COVID-19 infected people
also has the problem of imbalanced data. The existence of imbalanced data will
cause problems in the training of the model, that is, the model will pay more
attention to the recognition of majority samples, which is not the desired
result of medical staff or managers. To deal with the imbalance, researchers
have come up with a variety of approaches. For instance, Jiang et al. [12]
used the data augmentation technique to add noise to the data, where some
promising results have been achieved. Moreover, in addition to processing the
data to deal with the problem of imbalanced data, the loss function of the
model can also be modified in a cost-sensitive approach to achieve the purpose
of modifying the model. For example, Lin et al. [13] proposed a new loss
function called Focal Loss in the dense object detection task as an extension
of the conventional Cross-Entropy (CE) loss function. Experiments verified
that the Focal Loss function does help to improve the detection ability of
minority samples for imbalanced data.

In the field of image processing and recognition, the popular deep learning
models include AlexNet [14], VGG [15] and ResNet [16], etc. Among these
various deep learning models, ResNet is one of the most widely used and the
most prominent ones. Thus, our proposed model is customized from the ResNet50
network. With the development of deep learning technology, scholars have found
that it is very inconvenient to retrain the whole deep neural network every
time they encounter a new task, which requires a lot of time and computing
resources. Therefore, the researchers came up with a technique called “pre-
training” [17]. An already trained deep learning model can be loaded through
pre-training, and then the network weight can be partially adjusted (fine-
tuning) for specific tasks in the domain, so as to effectively solve new
problems. The ”pre-train then tune” paradigm is also the core idea of transfer
learning [18].

Another very popular technique in machine learning is ensemble learning. The
purpose of ensemble learning is to enhance the generalization ability and
prediction accuracy of the entire model by combining the outputs of multiple
base learners. Some studies have found that combining the outputs of multiple
deep neural networks through ensemble learning method is helpful to improve
the accuracy of prediction [19, 20]. To generate the base model for ensemble
learning, the classical methods mainly focus on changing the training data
distribution or changing the model’s structure. In this paper, we explore the
function of randomness on the model’s diversity.

In this paper, we propose a model based on deep residual network for COVID-19
infection recognition, which integrates the technologies of imbalanced data
processing, transfer learning and ensemble learning.

##  3 The Proposed DiCOVA-Net Method

The proposed DiCOVA-Net method is illustrated in Figure 1. First of all, the
input acoustic data needs to be transformed into spectrogram data. After that,
the data is divided into multiple data subsets for the purpose of cross
validation, and Gaussian noise is added to the minority samples to form new
minority samples on each subset. The new data set obtained through data
augmentation is fed into the ResNet50 pre-training model for fine-tuning,
where focal loss is used as the loss function. Finally, ensemble learning
methods are used to combine the outputs of the various models which are
generated by different random seeds when forming the final prediction results.

Figure 1: The proposed DiCOVA-Net method.

###  3.1 Gussian Noise-based Data Augmentation

Data augmentation is often necessary to enhance the diversity of the training
datasets. In additon, data augmentation can also reduce the domain mismatch
between the enrolled and test set data [12]. The data augmentation methods
commonly used include flip, rotation, scale, crop, translation, etc., while
what we adopt is to add Gaussian noise. Gaussian noise has a mean of zero and
a standard deviation of one and can be generated as needed using a
pseudorandom number generator. For our proposed model, Gaussian noise was
added to the data of minority samples (namely COVID-19 infected persons) to
generate some new synthetic minority samples, which is very beneficial for
DNNs and helps to reduce the incidence of overfitting, because DNNs have a
large number of parameters.

###  3.2 Deep Residual Network with Transfer Learning

Residual neural network [16] is a classical method which achieve the best
performance in the ILSVRC 2015 classification task. Residual neural network is
a new convolutional neural network structure. With the success of the VGG
network, the depth of the neural network has attracted more and more
attention, but it is accompanied by the problem of gradient disappearance that
is difficult to solve. The residual neural network uses the residual block to
solve this problem very well. The residual block can be calculated as Eq.(1):

$y_{l}=h(x_{l})+F(x_{l},W_{l})$ (1)

where $y_{l}$ represents the label of sample $l$, and $h(x_{l})$ is the
prediction that neural network output at the step $l$. The $F(x_{l},W_{l})$ is
the residual result of the step $l$.

In the residual neural network, there are two hypotheses: (1) $h(x)$ is the
direct mapping. (2) $F(x,W)$ is the direct mapping. Then, the loss $\epsilon$
gradient between the beginning layer and layer $l$ can calculated as Eq.(2):

$\frac{\partial{\epsilon}}{x_{0}}=\frac{\partial{\epsilon}}{x_{l}}(1+\alpha)$
(2)

where $\alpha$ can be presented as Eg.(3):

$\alpha=\frac{\partial{F}}{x_{0}}$ (3)

where $F$ can be presented as Eg.(4):

$F=\sum_{i=0}^{l-1}{f(x_{i},W_{i})}$ (4)

Throughout the training process, $\alpha$ cannot always be $-1$, which means
that the problem of gradient vanishing will not occur in the residual network

Transfer learning [23] is innovated to reduce the human annotation cost by
employing the model which is trained on annotated source domain data to
predict on the unannotated target domain data. And transfer learning is proved
to be efficient in the few data classification [24]. We follow this insight,
firstly we pre-train the ResNet50 on the ImageNet [25] which has 12 subtrees
with 5247 synsets and 3.2 million images in total. Then we fine-tune the pre-
training neural network structure on the DiCOVA dataset. With the help of pre-
training, we find the model will reach the best point in only few epoches.

###  3.3 Focal Loss

In the model of machine learning and deep learning, the loss function is used
to measure the degree of difference between the predicted value and the real
value. If the model prediction is not accurate, the value of the loss function
will be larger. In general, the better the loss function is designed, the
better the performance of the model will be. The loss function plays the role
of ”supervisor” in DNNs, which guides the model training to move towards the
direction of reducing the loss function, so as to find the network parameter
combination that minimizes the loss function. At present, cross entropy (CE)
is often used as the loss function when DNNs are trained, which is calculated
as Eq.(5). Compared with the classification error rate and the mean square
error (MSE), CE can better capture the differences between different models.
Moreover, CE has the property of convex function, so it can find the global
optimal value when taking the derivative.

$L_{CE}=-\sum_{i=1}^{m}{y_{i}\cdot{log(p_{i})}}$ (5)

where $y_{i}$ represents the label of sample $i$, and $p_{i}$ is the
probability that sample $i$ is predicted to be a positive class.

Cross entropy can achieve good results when the number of samples in each
class is not very different, but it is no longer effective when the data is
unbalanced. Aiming at the problem of unbalanced data, Lin et al. [13] proposed
a new loss function called focal loss which is calculated as Eq.(6).

$L_{FL}=-\sum_{i=1}^{m}{\alpha_{i}(1-p_{i})^{\gamma}{log(p_{i})}}$ (6)

where $\gamma$ $(\gamma\geq 0)$ is the focusing parameter which is used to
adjust the weight of difficult samples and easy samples, and the
$(1-p_{i})^{\gamma}$ is called the modulating factor. In addition,
$\alpha_{i}$ is used to adjust the weight of positive and negative samples.

From Eq.(6), we can get two important properties of the focal loss: (1) When
the value of $p_{i}\rightarrow 1$, $(1-p_{i})^{\gamma}\rightarrow 0$,
indicating that the prediction of the model is accurate and the sample is an
easy sample, so the contribution of the sample to the loss is small. (2) The
focusing parameter $\gamma$ smoothly adjusts the rate at which easy examples
are down-weighted. When $\gamma=0$, focal loss equivalent to CE, and as
$\gamma$ increases, the weight of easy samples will be further reduced.
Therefore, using Eq.(6) as the loss function of the model in this paper, on
the one hand, the problem of imbalanced data can be considered, and on the
other hand, the difficulty of sample discrimination can be taken into account.

###  3.4 Ensemble Learning

An ensemble framework aggregates the predictions of multiple base models to
get better prediction. Formally, suppose that we have $K$ base models with
predictions $\bm{y}_{t+1}=[y_{t+1}^{1},\cdots,y_{t+1}^{K}]$, ensemble is an
aggregation function $y_{t+1}=g(\bm{y}_{t+1};\bm{\Phi})$, which has various
implementations such as voting, averaging, and stacking [26]. We used the
averaging method to ensemble different the same model trained on different
distributed feature space.

###  3.5 Randomness

We proposed a novel and robust method to achieve high performance in this
challenging dataset in this work. Randomness is ubiquitous in deep learning
which adds uncertainty to deep learning. However, the researchers have
explored randomness in the weight initial for deep learning models which helps
to generate realized images without training [21]. And other researchers
further explored the randomness in layers chosen and models chosen for the
ensemble methods [22]. This method achieves state-of-art results on different
domain datasets. However, all the previous methods have not explored the
randomness in the input data. With our knowledge, we can find that neural
network pays more attention to the data entered at the beginning. So the
different order of input batches will make the model start evolving in
different places. Therefore, we find the different order of input batches in
this challenge data will surely improve the final performance through the
experiment. With the infinite search space is expensive, we suggested trying
four or five different random seeds and ensemble results together. The novel
method can be summarized in $\bm{f}_{t+1}=[f_{t+1}^{1},\cdots,f_{t+1}^{K}]$,
the $[f_{t+1}^{1}=W(x^{r}),r\in random(300,2000)$. The
$[f_{t+1}^{1},\cdots,f_{t+1}^{K}]$ are all based on the same model but with
different data input order. The experimental result shows our method improves
the model’s accuracy and robustness.

##  4 Experiments and Results

###  4.1 Datasets

We evaluated our proposed method in the DiCOVA 2021 challenge [27]. This
challenge provides a dataset of sound recordings collected from COVID-19
infected and non-COVID-19 individuals for a two-class classification. This
data set contains a total of 1040 samples, including 965 non-COVID-19 samples
and 75 COVID-19 infected samples, with an imbalanced ratio of 13:1, which
proves that this is an extremely imbalanced dataset. The average duration of
recordings across subjects is 4.72 (standard error (S.E.) $\pm$ 0.07) sec. In
addition, the organizers of this challenge adopt a blind test set containing
233 samples, so the results obtained in this paper on the test set are given
by the organizers of the challenge.

###  4.2 Experimental Settings

To be objective, the challenge organizers have already split the dataset into
train set and validation set. The train set contains 822 data which includes
772 non-COVID-19 and 50 COVID-19. The validation set contains 218 data which
includes 193 non-COVID-19 and 25 COVID-19. The organizers also generate five-
fold data set which helps participants to gain a more generalized and diverse
model.

We use the librosa111https://github.com/librosa/librosa to transform the audio
data into mel-spectrogram. In the guassian augmentation progress, we use the
skimage222https://github.com/scikit-image/scikit-image and set the random seed
between 0 and 5. We list other parameters in the Table 1.

In the ResNet50, we use the focal loss as the loss function and use the adam
as the optimization method. In model selection, we use the area under the
curve (AUC) as an evaluation indicator in each folds.

In the randomness experiment, we only change the random seeds at the beginning
of the pipeline.

Table 1: Main parameters Parameter name | Settings
---|---
Sampling Rate | 2048
Audio length | 4 sec
FFT window | 2048
Frame shift | 512
Epoch | 20
Batch size | 16
Learning rate | 0.0002
Focal loss $\alpha$ | 0.25
Focal loss $\gamma$ | 2
Random seeds (ensemble) | 1001,500,1500,2000

###  4.3 Results and Discussions

For this challenge, the organizers present the performance of three baseline
methods: Random Forest (RF), Multi-layer Perceptron (MLP), and Logistic
Regression (LR). These three methods could not receive audio data, so the
organizers use the mel-frequency cepstral coefficients (MFCC) and the delta
and delta-delta coefficients methods to extract features. The performance of
the three baseline methods which is provided by organizer is shown in Table 2
[27]. According to the results in Table 2, LR has the worst performance,
because compared with RF and MLP, LR has the worst nonlinear fitting ability.
In addition, the difference between RF’s performance on the validation set and
the test set is larger than that of MLP, so MLP is more robust.

Table 2: Baseline performance Methods | Validation AUC | Test AUC
---|---|---
RF | 70.63 | 67.59
MLP | 68.81 | 69.91
LR | 66.97 | 61.97

The experiment result of proposed method is shown in Table 3. We compare the
performance of fine-tuning model “Original” with two different loss functions
and two different data augmentation methods. The two different loss functions
respectively “cross entropy” (CE) loss and “focal loss” (FL). The two
different data augmentation methods are “simple duplication” (Dul) and
“Gaussian noise” (Gua). The “Ensemble” is the performance by integrating four
models with different random seeds.

Table 3: Different methods comparison results Methods | Validation AUC | Test AUC
---|---|---
CE_Original | 69.77 | 75.27
CE_Dul | 71.15 | 72.92
CE_Gua | 71.57 | 75.59
FL_Original | 74.17 | 69.78
FL_Dul | 69.92 | 75.10
FL_Gua | 73.58 | 83.59
Ensemble | 76.29 | 85.43

In Table 3, the performance of the fine-tuning model based on ResNet50 is
better than that of the traditional machine learning methods such as RF, MLP
and LR. In terms of the loss function, CE performs better than FL on the test
set, and overfitting occurs when FL is used for training on the original
training set. However, after using “Dul” or “Gua” for data augmentation, FL
performed better than CE on the test set. This proves that a combination of
multiple imbalanced data processing methods will achieve better results. In
addition, it can also be seen that using Gaussian noise to process imbalanced
data is better than directly expanding minority samples. Finally, using
randomness to train multiple models for integration can further improve the
prediction performance of the entire model.

##  5 Conclusions

In order to use acoustics data to identify patients infected with COVID-19
more accurately, we propose a deep learning method that incorporates multiple
image processing techniques. First, we transform the acoustics data into
spectrogram data, which can better suit the deep learning model. After that,
Gaussian noise-based data augmentation and focal loss are introduced to solve
the problem of imbalanced data. Based on the pre-training model of ResNet50,
we combine the fine-tuning technology in transfer learning to adjust the
weight of the deep neural network to make it more suitable for the
identification of COVID-19 infected persons. In addition, in order to make the
model we designed more robust, we use ensemble learning to build multiple deep
learning models. When training these models, we adopt an advanced data
extraction method with randomness and uncertainty to build sample subsets. Our
experimental results show that the proposed method can effectively identify
the infected with COVID-19 and is superior to other state-of-the-art methods.

##  6 Acknowledgements

We are very grateful to the organizers of the DiCOVA 2021 Challenge for their
efforts in providing the participants with data and a platform for the
competition. And, this research is supported by the China Scholarship Council
(No. 202006060162).

## References

  * [1] X. Xie, Z. Zhong, W. Zhao, C. Zheng, F. Wang, and J. Liu, “Chest CT for Typical Coronavirus Disease 2019 (COVID-19) Pneumonia: Relationship to Negative RT-PCR Testing,” Radiology, vol. 296, no. 2, pp. E41–E45, 2020.
  * [2] A. K. J. Teo, Y. Choudhury, I. B. Tan, C. Y. Cher, S. H. Chew, Z. Y. Wan, L. T. E. Cheng, L. L. E. Oon, M. H. Tan, K. S. Chan and L. Y. Hsu, “Saliva is more sensitive than nasopharyngeal or nasal swabs for diagnosis of asymptomatic and mild COVID-19 infection,” Scientific Reports, vol. 11, no. 1, pp. 1–8, 2021.
  * [3] D. L. Grieco, F. Bongiovanni, L. Chen, L. S. Menga, S. L. Cutuli, G. Pintaudi, S. Carelli, T. Michi, F. Torrini, G. Lombardi, G. M. Anzellotti, G. D. Pascale, A. Urbani, M. G. Bocci, E. S. Tanzarella, G. Bello, A. M. Dell’anna, S. M. Maggiore, L. Brochard and M. Antonelli, “Respiratory physiology of COVID-19-induced respiratory failure compared to ARDS of other etiologies,” Critical Care, vol. 24, no. 1, pp. 1–11, 2020.
  * [4] A. O. de Carvalho Filho, A. C. Silva, A. C. de Paiva, R. A. Nunes, and M. Gattass, “Classification of patterns of benignity and malignancy based on CT using topology-based phylogenetic diversity index and convolutional neural network,” Pattern Recognition, vol. 81, pp. 200–212, 2018.
  * [5] Y. Wu, H. Mao, and Z. Yi, “Audio classification using attention-augmented convolutional neural network,” Knowledge-Based Systems, vol. 161, pp. 90–100, 2018.
  * [6] S. Sun, C. Luo, and J. Chen, “A review of natural language processing techniques for opinion mining systems,” Information Fusion, vol. 36, pp. 10–25, 2017.
  * [7] S. Tuli, S. Tuli, R. Tuli, and S. S. Gill, “Predicting the Growth and Trend of COVID-19 Pandemic using Machine Learning and Cloud Computing,” Internet of Things, p. 100222, 2020.
  * [8] L. Yan, H.-T. Zhang, J. Goncalves, Y. Xiao, M. Wang, Y. Guo, C. Sun, X. Tang, L. Jing, M. Zhang, X. Huang, Y. Xiao, H. Cao, Y. Chen, T. Ren, F. Wang, Y. Xiao, S. Huang, X. Tan, N. Huang, B. Jiao, C. Cheng, Y. Zhang, A. Luo, L. Mombaerts, J. Jin, Z. Cao, S. Li, H. Xu and Y. Yuan, “An interpretable mortality prediction model for COVID-19 patients,” Nature Machine Intelligence, vol. 2, 2020.
  * [9] A. A. Ardakani, A. R. Kanafi, U. R. Acharya, N. Khadem, and A. Mohammadi, “Application of deep learning technique to manage COVID-19 in routine clinical practice using CT images: Results of 10 convolutional neural networks,” Computers in Biology and Medicine, vol. 121, p. 103795, 2020.
  * [10] L. Deng, G. Hinton, and B. Kingsbury, “New types of deep neural network learning for speech recognition and related applications: An overview,” in 2013 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2013, pp. 8599–8603.
  * [11] J. Sharma, O. Granmo, and M. Goodwin, “Environment Sound Classification using Multiple Feature Channels and Attention based Deep Convolutional Neural Network,” in INTERPEECH 2020, 2020, pp. 1186–1190.
  * [12] T. Jiang, M. Zhao, L. Li, and Q. Hong, “The XMUSPEECH system for short-duration speaker verification challenge 2020,” in INTERSPEECH 2020, 2020, pp. 736–740.
  * [13] T. Y. Lin, P. Goyal, R. Girshick, K. He, and P. Dollár, “Focal loss for dense object detection,” in Proceedings of the IEEE International Conference on Computer Vision (ICCV), 2017, pp. 2980–2988.
  * [14] A. Krizhevsky, I. Sutskever, and G. E. Hinton, “ImageNet Classification with Deep Convolutional Neural Networks,” in Advances in neural information processing systems (NIPS), 2012, pp. 1097–1105.
  * [15] K. Simonyan and A. Zisserman, “Very deep convolutional networks for large-scale image recognition,” in 3rd International Conference on Learning Representations (ICLR), 2015, pp. 1–14.
  * [16] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016, pp. 770–778.
  * [17] D. Hendrycks, L. Kimin, and M. Mazeika, “Using Pre-Training Can Improve Model Robustness and Uncertainty,” in Proceedings of the 36th International Conference on Machine Learning (ICML), 2019, pp. 2712–2721.
  * [18] Y. Guo, H. Shi, A. Kumary, K. Grauman, T. Rosing, and R. Feris, “Spottune: Transfer learning through adaptive fine-tuning,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2019, pp. 4805–4814.
  * [19] T. Pang, K. Xu, C. Du, N. Chen, and J. Zhu, “Improving adversarial robustness via promoting ensemble diversity,” in Proceedings of the 36th International Conference on Machine Learning (ICML), 2019, pp. 4970–4979.
  * [20] X. Yang, Z. Zeng, T. Sing G., L. Wang, V. Chandrasekar, and S. C. H. HOI, “Deep learning for practical image recognition: Case study on kaggle competitions,” in Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2018, pp. 923–931.
  * [21] K. He, Y. Wang, J. Hopcroft, “A Powerful Generative Model Using Random Weights for the Deep Image Representation”, Proceedings of the 30th International Conference on Neural Information Processing Systems (NIPS), 2016, pp. 631-639.
  * [22] K. Kowsari, M. Heidarysafa, D. E. Brown, K. J. Meimandi, L. E. Barnes, “RMDL: Random Multimodel Deep Learning for Classification”,  Proceedings of the 2nd International Conference on Information System and Data Mining , 2018, pp. 19-28.
  * [23] SJ. Pan, Q. Yang, “A survey on transfer learning”,  IEEE Transactions on knowledge and data engineering , vol. 22, no. 10, pp. 1345-1359, 2009.
  * [24] Q. Sun, Y. Liu, T. Chua, B. Schiele, “Meta-Transfer Learning for Few-Shot Learning”,  Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) , 2019, pp. 1345-1359.
  * [25] J. Deng, W. Dong, R. Socher, L. Li, K. Li, F. Li, “ImageNet: A large-scale hierarchical image database”, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2009, pp. 248-255.
  * [26] Z. Zhou, “Ensemble methods: foundations and algorithms”,  Chapman and Hall/CRC press, 2012.
  * [27] A. Muguli, L. Pinto, N. R, N. Sharma, P. Krishnan, P, K, Ghosh, R. Kumar, S. Ramoji, S. Bhat, S. R. Chetupalli, S. Ganapathy, V. Nanda, “DiCOVA Challenge: Dataset, task, and baseline system for COVID-19 diagnosis using acoustics”,  arXiv preprint ArXiv: 2103.09148, 2021.