# Temporal Evolution of the Light Emitted by a Thin, Laser-ionized Plasma
Source

Valentina Lee  Center for Integrated Plasma Studies, Department of Physics,
University of Colorado Boulder, Boulder, Colorado 80309, USA  Robert Ariniello
SLAC National Accelerator Laboratory, 2575 Sand Hill Rd, Menlo Park, CA 94025,
USA  Christopher Doss  Center for Integrated Plasma Studies, Department of
Physics, University of Colorado Boulder, Boulder, Colorado 80309, USA  Kathryn
Wolfinger  RadiaSoft LLC, 1790 38th St Suite 306, Boulder, CO 80301, USA
Peter Stoltz  Zap Energy, 5901 23rd Dr W Everett, WA 98203, USA  Claire Hansel
Center for Integrated Plasma Studies, Department of Physics, University of
Colorado Boulder, Boulder, Colorado 80309, USA  Spencer Gessner  SLAC National
Accelerator Laboratory, 2575 Sand Hill Rd, Menlo Park, CA 94025, USA  John
Cary  Center for Integrated Plasma Studies, Department of Physics, University
of Colorado Boulder, Boulder, Colorado 80309, USA  Tech-X, 5621 Arapahoe
Avenue Suite A Boulder, CO 80303, USA  Michael Litos  Center for Integrated
Plasma Studies, Department of Physics, University of Colorado Boulder,
Boulder, Colorado 80309, USA

###### Abstract

We present an experimental and simulation-based investigation of the temporal
evolution of light emission from a thin, laser-ionized Helium plasma source.
We demonstrate an analytic model to calculate the approximate scaling of the
time-integrated, on-axis light emission with the initial plasma density and
temperature, supported by the experiment, which enhances the understanding of
plasma light measurement for plasma wakefield accelerator (PWFA) plasma
sources. Our model simulates the plasma density and temperature using a split-
step Fourier code and a particle-in-cell (PIC) code. A fluid simulation is
then used to model the plasma and neutral density, and the electron
temperature as a function of time and position. We then show the numerical
results of the space-and-time-resolved light emission and that collisional
excitation is the dominant source of light emission. We validate our model by
measuring the light emitted by a laser-ionized plasma using a novel
statistical method capable of resolving the nanosecond-scale temporal dynamics
of the plasma light using a cost-effective camera with microsecond-scale
timing jitter. This method is ideal for deployment in the high radiation
environment of a particle accelerator that precludes the use of expensive
nanosecond-gated cameras. Our results show that our models can effectively
simulate the dynamics of a thin, laser-ionized plasma source and this work is
useful to understand the plasma light measurement, which plays an important
role in the PWFA.

††preprint: AIP/123-QED

##  I Introduction

Plasma-based accelerators have demonstrated accelerating gradients two to
three orders of magnitude larger than conventional radio-frequency
accelerators, making them an enticing alternative for future high-energy
particle accelerator applications Blumenfeld  _et al._ (2007); Litos  _et al._
(2014, 2016). In a beam-driven plasma wakefield accelerator (PWFA), an
electron drive beam generates a wake as it propagates through a plasma. A
second electron beam follows the drive beam at a distance on the order of the
plasma skin depth. The strong longitudinal electric field of the plasma wake
accelerates this “witness” beam. As the performance of a PWFA depends strongly
on the plasma density profile Xu  _et al._ (2016); Floettmann (2014);
Ariniello  _et al._ (2019, 2022), control and understanding of the plasma
source are essential to interpret the outcome of experiments and optimize the
properties of the plasma source.

A typical PWFA plasma source is a long, narrow filament, 10’s to 100’s of
micrometers in diameter and 10’s of centimeters in length, with a core density
of $10^{16-18}\ {\rm cm^{-3}}$. One technique to form a suitable plasma is to
laser-ionize a gas using an optic with a long depth of focus, such as an
axicon lens or diffractive optic Honkanen and Turunen (1998); Ouadghiri-
Idrissi  _et al._ (2016); Ariniello (2022); Lee  _et al._ (2019). In such a
plasma source, an $\mathcal{O}(10\,{\rm TW})$, ultrashort laser pulse is
focused into a volume of gas, such as Hydrogen, Helium, or Lithium, ionizing a
thin filament prior to the arrival of the electron beams. Characterization of
the plasma source is often performed in the absence of the electron beams. One
common technique is to look at the light emitted by the plasma.

Multiple mechanisms with varying time scales contribute to the plasma light
emission process of these thin laser-ionized plasma sources. The plasma is
locally formed on the time scale of the ionizing laser pulse duration, which
is around $100\ {\rm fs}$. The plasma electrons thermalize within $10\ {\rm
ps}$ of formation. Then, the plasma expands quickly outward within a few
nanoseconds while the neutral gas diffuses inward, followed by a slower
diffusion and thermalization phase between the plasma and the neutral gas that
lasts 10’s to 100’s of nanoseconds. During this time, plasma electrons collide
with neutral atoms, exciting them, while also colliding with plasma ions and
recombining. Both of these mechanisms contribute to plasma light emission. The
latter will also lead to the eventual neutralization of the plasma over a few
hundred microseconds time scale Newton and Sexton (1968).

The dynamics of plasma expansion have been studied through simulations and
experiments Shalloo  _et al._ (2018); Mil (2016); Capitelli  _et al._ (2004),
and the electron thermalization time scale has been demonstrated in Ref. Zhang
_et al._ , 2020a, b. Experimental and theoretical studies of plasma
neutralization through recombination are presented in Ref. Baravian  _et al._
, 1972; Bates, Kingston, and Mcwhirter, 1962. However, there remains a gap in
understanding the light emission process in thin, laser-ionized plasmas, such
as PWFA plasma filaments. The amount of light emitted depends on the electron
temperature and the number densities of the plasma and neutral gas.
Determining the relative dominance of excitation versus recombination is a
complex question influenced by the plasma’s geometry. In this work, we show an
analytical and computational model of the plasma light emission process of a
thin laser-ionized Helium plasma and confirmed by the experiment.

Our measurements of plasma light emission are taken using an inexpensive
camera. These cameras are often used to verify plasma formation in PWFA
experiments. For example, many of the experiments at the Facility for Advanced
Accelerator Experimental Tests-II (FACET-II) at SLAC National Accelerator
Laboratory rely on cameras to observe the laser-ionized plasma source Joshi
_et al._ (2018); Deng  _et al._ (2019); Adli  _et al._ (2016); Claveria  _et
al._ (2022); Amorim and Vafaei-Najafabadi (2019); Corde  _et al._ (2016); Doss
_et al._ (2019). Cost-effective cameras, such as Gigabit Ethernet (GigE)
machine vision cameras, are typically used in PWFA experiments because the
high radiation environment in the accelerator housing leads to rapid cycles of
camera failure and replacement. Unfortunately, these cameras have a large
trigger timing jitter ($\sim 10\ {\rm\mu s}$) and long exposure time ($\sim
10\ {\rm\mu s}$) compared to the short time scales (10’s-100’s ${\rm ns}$) of
plasma light emission, making images from these kinds of cameras integrate
over various dynamics described previously. In order to verify our model using
these cameras, we demonstrate a novel technique to measure the time-resolved
light signal using these low-cost cameras.

Our three step light emission model: plasma formation, plasma expansion, and
plasma light emission is presented in section II. In section III, we introduce
our experimental setup where a laser-ionized Helium plasma is viewed by a GigE
machine vision camera. In section IV, we present a novel technique for
studying plasma light emission with 1 ns resolution using a low-cost GigE
camera. In section V, we demonstrate that the experimental data confirms our
Helium plasma light emission model. Furthermore, we show the broad
applicability of this diagnostic tool in PWFA experiments by demonstrating
that a simple theoretical model can describe the intensity of the plasma light
in a time-integrated image, permitting estimation of the initial plasma
parameters with a single image of the time-integrated plasma light.

##  II Modeling

This section outlines a workflow for modeling the laser-ionized plasma
formation, expansion, and light emission. We demonstrate the entire workflow
using a single set of parameters (laser pulse duration: $\tau$= 50 fs,
focusing optic: $1^{\circ}$ axicon lens, laser energy: $E$= 160.33 mJ, gas
pressure: $P$= 118 mbar) and compare the time-resolved simulated plasma light
emission pattern with experimental data. We also conducted multiple plasma
formation simulations for a range of parameters ($E$= 140-175 mJ and $P$=
20-80 mbar). The simulated electron temperatures were used to calculate plasma
light emission, which is compared to experimental measurements in Section V.

###  II.1 Plasma Formation

In a laser-ionized PWFA plasma source, ionization occurs through field
ionization, typically in the tunneling regime where the electric field from
the laser distorts the atomic potential allowing an electron to tunnel out of
the atomic barrier. This process is well described by the ADK model Keldysh
_et al._ (1965). Particle-in-cell (PIC) simulations are able to capture this
process, however, simulating the ionization process over a 5 ns window (axicon
focus of 1.5 m) while resolving the laser period would require an impractical
amount of computational resources. As an alternative, we use an in-house code
to simulate the 3D plasma profile produced by the laser. The ADK model is used
to calculate the ionization rate; while a the split-step Fourier (SSF)
algorithm Stoffa  _et al._ (1990) takes into account refraction of the rear of
the laser pulse due to the presence of plasma ionized by the front of the
laser pulse. Figure 1 shows an example of the simulated laser intensity and
plasma density profiles formed by an axicon lens by our SSF code. This code,
however, cannot provide information about the initial plasma electron
temperature. To accommodate this, we use a PIC code, VSim Nieter and Cary
(2004) provided by Tech-X, to simulate ionization of a typical localized
region along the laser ionization path, importing the local laser pulse
profile retrieved from the SSF code. We then extract the plasma electron
temperature from the resulting kinetic energy distribution of the electrons in
the PIC simulation.

Figure 1:  Simulated formation of an axicon plasma using a split-step Fourier
code, with an initial laser pulse energy of 175 mJ in a Helium gas pressure of
28 mbar, corresponding to a fully ionized plasma of $n_{e}=\ 6.9\times
10^{17}\ {\rm cm^{-3}}$. The axicon lens ($\alpha=1^{\circ}$) is positioned at
$z=0$. (a) illustrates the laser intensity at the middle of the pulse on the
$y=0$ plane. (b) shows the plasma density on the $y=0$ plane. Because of the
long Bessel focus from the axicon lens, the full ionization regime extends to
up to 1 meter.

For all PIC simulations in this work, 1 particle per cell is used with a time
step of $dt=3.25\times 10^{-17}\ {\rm s}$ and a spatial increment of
$dx=7\times 10^{-8}\ {\rm m}$. The transverse size of the simulation box is
approximately 14 times the full width at half maximum (FWHM) of the laser’s
spatial profile, and the temporal dimension is roughly 4 times the FWHM
duration of the laser pulse. Modeling ionization through a combination of the
SSF and PIC simulations allows us to acquire the plasma density and plasma
electron temperature with reasonable computational resources. Figure 2 shows a
histogram of the electron kinetic energy distribution as a function of radius.
It also demonstrates that the intensity of the ionization laser profoundly
influences the azimuthally averaged electron energy. As we will show, our
experimental observations confirm that the method described above is a
reliable and relatively inexpensive way to model a laser-ionized PWFA plasma
source.

Figure 2:  The plot illustrates the plasma electron energy histogram as a
function of the radius, ionized by a $175\ {\rm mJ}$ pulse in Helium gas with
a density of $6.9\times 10^{17}\ {\rm cm^{-3}}$, simulated by a PIC
simulation. The azimuthally averaged radial electron energy is represented by
the red curve and the ionizing laser intensity is represented by the green
curve. The azimuthally averaged radial electron energy follows the intensity
of the ionization laser.

###  II.2 Plasma Expansion

The plasma electrons thermalize on the order of the Spitzer electron self-
collision time, given by $\tau\approx(1.40/(8\pi
r_{e}^{2}c^{4}n_{e}\ln\Lambda))(3k_{B}T_{e}/m_{e})^{3/2}$ where $r_{e}$ is the
classical electron radius, $c$ is the speed of light, $n_{e}$ is the plasma
electron density, $\ln\Lambda$ is the Coulomb logarithm, $k_{B}$ is the
Boltzmann constant, $T_{e}$ is the average electron temperature, and $m_{e}$
is the mass of electron Spitzer (1962). For example, for a plasma with
$k_{B}T_{e}=10\,{\rm eV}$ and $n_{e}=1\times 10^{17}\,{\rm cm^{-3}}$,
$\tau\approx 9\,{\rm ps}$. This fast thermalization process has been measured
in Ref. Zhang  _et al._ , 2020a, b. The thermalized electron temperature of
the plasma is $k_{B}T_{e}=2/3<E_{k}>$, where $<E_{k}>$ is the average electron
kinetic energy immediately after ionization. The expansion of a laser-ionized
plasma source can be modeled using fluid simulations, as demonstrated in Ref.
Shalloo  _et al._ , 2018. Using the plasma density profile and the plasma
electron temperature acquired from previous steps, we model the plasma
expansion using Tech-X’s fluid simulation software, USim. USim is an Eulerian
computational fluid dynamics code optimized for plasma fluid simulations by
solving the Magnetohydrodynamic (MHD) equations Brio and Wu (1988). We
simulate the first $3\,{\rm ns}$ of the expansion using a two-temperature,
single-fluid MHD diffusion code, where the ion temperature is $0.025\,{\rm
eV}$ ($300\,{\rm K}$) and the average electron temperature is taken from the
PIC simulation, $<T_{e}>\sim 13\,{\rm eV}$. The simulated expansion is shown
in figure 3(a). For the subsequent expansion from $3\,{\rm ns}$ until
$t=200\,{\rm ns}$, plasma-neutral collisions are accounted for using a two-
fluid model that calculates the mass diffusion, energy exchange, and
temperature exchange between two species (plasma and neutral) Meier and
Shumlak (2012); Chaplin and Bellan (2015). The subsequent expansion is shown
in Fig. 3(b). It has been tested that the results of the second simulation
period are relatively insensitive to the exact choice of the transition time,
so long as the change in density is relatively small. As shown by the red
curve (density lineout) in Fig. 3(a), the expansion (decay) reaches $1/e^{2}$
before $3\,{\rm ns}$. Simulating the expansion in two steps simplifies the
complexity of the model and allows for a higher temporal resolution during the
initial rapid expansion.

Figure 3:  Simulated plasma expansion using fluid codes. (a) shows the plasma
expansion on the $y=0$ plane within the first three nanoseconds simulated by a
single-fluid MHD model. (b) shows the plasma expansion on the $y=0$ plane from
the 3 ns to 200 ns, including plasma-neutral collision effects. In both
figures, the red and green curves show the plasma density ($n_{e}$) and plasma
temperature ($T_{e}$) at $x=0$, respectively. Note that the first 5 ns
(numerical thermalization time) of the green curve ($T_{e}$) in (b) is
excluded from the plot.

###  II.3 Plasma Light Emission

Light emission from plasma can be generated by two primary processes:
electron-neutral collisional excitation and plasma electron-ion recombination.
Both processes are intricate and require Monte Carlo atomic electron tracking
models Vahedi and Surendra (1995) to simulate exactly, which is both
theoretically and computationally demanding. To simplify the problem, we focus
on the most probable transition process that emits visible/NIR light in either
process, since that is what Complementary Metal-oxide Semiconductor (CMOS)
GigE camera can detect.

In the collisional excitation model, the electrons have a significantly higher
temperature ($T_{e}\sim 15\,{\rm eV}$) compared to the ions and neutrals
($T_{i}\approx T_{n}\sim\,{\rm 0.025eV}$). Therefore, electron-neutral
collisions dominate while ion-neutral and neutral-neutral collisions are
negligible. We also assume that all neutral atoms are in their ground state
before the collision. In a Helium atom, $3d-2p$ de-excitation has the
strongest persistent lines. The dipole-allowed transitions from the ground
state occur from the $1s$ ($l=0$) state to the $p$ ($l=1$) states. Of these,
excitation to the $4p$ state has the largest cross-section among allowed
excited states that can generate a visible/NIR photon from a subsequent
$3d-2p$ de-excitation. Therefore, we choose this $1s-4p-3d-2p$ excitation/de-
excitation process to calculate the lower bound of the visible photon emission
rate via a single excitation path. Appendix A presents a more detailed
explanation of this model. To calculate the local excitation rate per unit
volume, we start with the collision rate for one electron with a population of
neutral particlesTheodosiou (1987); Boffard, Lin, and DeJoseph (2004);
Ralchenko  _et al._ (2008),

$\displaystyle\upsilon_{en}=n_{n}\sigma(v)_{1s-4p}\ v,$ (1)

where $n_{n}$ is the local neutral density, $\sigma(v)_{1s-4p}$ is the
electron-impact excitation cross-section from the 1s state to the 4p state,
and $v$ is the relative velocity between the electron and the neutrals, which
is approximated as the electron velocity ($v_{e}>>v_{n}$). The probability
density function for a given speed ($v=|\vec{v}|$) for a Maxwellian population
of electrons is

$\displaystyle f(v)=\left(\frac{m_{e}}{2\pi k_{B}T_{e}}\right)^{3/2}\ 4\pi
v^{2}\exp\left({-\frac{m_{e}v^{2}}{2k_{B}T_{e}}}\right),$ (2)

where $\int_{0}^{\infty}f(v)dv=1$. The average collision rate per electron is

$\displaystyle<\upsilon_{en}>=\frac{\int_{0}^{\infty}f(v)n_{n}\sigma(v)vdv}{\int_{0}^{\infty}f(v)dv}$
$\displaystyle=n_{n}4\pi\left(\frac{m_{e}}{2\pi
k_{B}T_{e}}\right)^{3/2}\int_{0}^{\infty}\exp\left({-\frac{m_{e}v^{2}}{2k_{B}T_{e}}}\right)\sigma(v)v^{3}dv.$
(3)

This average collision rate can be written in terms of the kinetic energy,
$K$:

$\displaystyle<\upsilon_{en}>=$ $\displaystyle\frac{8\pi
n_{n}}{m_{e}^{2}}\left(\frac{m_{e}}{2\pi k_{B}T_{e}}\right)^{3/2}$ (4)
$\displaystyle\int_{0}^{\infty}\exp\left({-\frac{K}{k_{B}T_{e}}}\right)\sigma(K)K\
dK.$

Thus, the excitation rate per unit volume at location $\vec{r}$ and time $t$,
for a population of electrons colliding with a population of neutrals is
$n_{e}<\nu_{en}>$, which depends on $\vec{r}$ ant $t$ through $n_{e}$,
$n_{n}$, and $T_{e}$.

After the collisional excitation, some excited electrons in the $4p$ state
spontaneously de-excite to the $3d$ state, and then to the $2p$ state,
yielding a detectable photon emission. The decay transition probability from
state $i$ to state $j$ is

$P_{ij}=\frac{A_{ij}}{\sum_{k}A_{ik}},$ (5)

where $k$ indicates all dipole-allowed final transition states from initial
state $i$, and $A$ is the transition rate, here taken from Ref. Theodosiou,
1987.

The transition probability from the $4p$ state to the $3d$ state is
$P_{4p-3d}\approx 1.2\times 10^{-3}$, and from the $3d$ to the $2p$ state is
$P_{3d-2p}\approx 1$. The visible photon emission rate is thus

$\displaystyle\frac{dn_{\gamma
exc}(\vec{r},t)}{dt}=n_{e}(\vec{r},t)<\upsilon_{en}(\vec{r,t})>P_{4p-3d}\
P_{3d-2p}.$

We then proceed to work out an analytic prediction of the peak photon emission
$(x=0,y=0)$ scaling as a function of the initial plasma density and
temperature with a few assumptions explained in Appendix B:

$\displaystyle\Gamma=$
$\displaystyle\int_{0}^{\infty}n_{e}(x=0,y=0)<\upsilon_{en}(x=0,y=0)>dt$
$\displaystyle=$ $\displaystyle C\ n_{0}^{2}\sqrt{Rk_{B}T_{0}}\
\exp\left({-\frac{K_{th}}{Rk_{B}T_{0}}}\right),$ (6)

where $C$ is scaling constant left as a free parameter to fit to the data,
$n_{0}$ and $T_{0}$ are the initial electron density and temperature at
$x=0,y=0$, respectively, $K_{th}$ is the free electron kinetic energy
threshold for exciting a ground-state atom to the $4p$ state, and $R=0.09$ is
the empirically acquired temperature decay ratio; that is, the ratio between
the initial temperature and the final temperature after the initial fast
expansion as shown in Fig. 3 (a). This value was obtained from our simulation
and may vary in other circumstances, though it agreed well with the
experimental data over a relatively wide range of initial plasma parameters.
The detailed derivation of 6 can be found in Appendix B.

The other possible light emission process, recombination, can occur through
three different primary modes: three-body recombination, radiative
recombination, and dielectronic recombination Hahn (1997). Radiative
recombination dominates the production of visible photons Drawin and Emard
(1975), wherein a single free electron recombines with a singly ionized ion
into one of its high Rydberg states, yielding a highly-exited Helium atom. The
rate of electron-ion radiative recombination can be approximated as follows
Huba (1998):

$\alpha_{r}=2.7\times 10^{-13}Z^{2}\ T_{e}[eV]^{-1/2}\ \ [{\rm cm^{3}/s}],$
(7)

where $Z=1$ is the charge state of He+. The recombination rate Chen  _et al._
(1984) is, therefore,

$\frac{dn_{\gamma{\rm rec}}}{dt}=\alpha_{r}n_{e}^{2},$ (8)

assuming quasi-neutrality.

Because it is more energetically favorable, we assume that most of the
recombined electrons start in a highly-excited state and then eventually de-
excite to the ground state in a radiative process. To estimate an upper bound
of the detectable photon emission rate from a single recombination event, we
look at the strongest visible/NIR line in Helium atoms, the $3d-2p$
transition. We then assume that electrons in high Rydberg states eventually go
through the dominant transition process, $4f-3d-2p$ Theodosiou (1987); Ma,
Howard, and Thapar (2011). The details of the model are discussed in Appendix
A. The de-excitation transition probability for $4f$ states to $3d$ states is
$P_{4f-3d}\approx 1$. The local visible photon emission rate from electron-ion
recombination is thus

$\frac{dn_{\gamma rec}(\vec{r},t)}{dt}=\alpha_{r}n_{e}(\vec{r},t)^{2}\
P_{4f-3d}P_{3d-2p}.$ (9)

Note that 9 is predicated on the assumption that all recombination events lead
to the emission of a detectable photon, which is not strictly true, but does
provide an upper-bound estimation of the recombination photon emission rate.

The collisional excitation and recombination rates per unit volume are orders
of magnitude less than the electron density, so their effects on the expansion
of the plasma are negligible. This allows us to model the time-resolved plasma
expansion first and then numerically estimate the visible photon emission
density from excitation and recombination using Eq. II.3 and Eq. 9 at each
moment in time, $t$, and at every position, $\vec{r}$. The cross-section in
Eq. 1 is numerically interpolated from Ref. Stone, Kim, and Desclaux, 2002.
Our simulation indicates that the visible/NIR photon emission rate from the
collisional excitation is three orders of magnitude greater than that from
recombination. Recall that in our estimation, the photon emission rate arising
from collisional excitation is a lower bound, while that from plasma
recombination is an upper bound. Thus, even with the simplifications
introduced in our model, the assertion that collisional excitation principally
dominates the light emission process remains valid. Figures 8 (a) illustrates
the time-resolved simulated photon emission pattern from an expanding Helium
plasma.

##  III Experimental Setup

Figure 4:  Experimental setup. 50 fs laser pulses generated by a Ti:Sapphire
laser system enter Chamber A through the rightmost window in this figure,
bouncing off mirrors M1, M2, M3, M4, and the deformable mirror (DM). The
leakage light from M5 exits the vacuum chamber and splits into three paths to
the near-field camera (NF), far-field camera (FF), and the Shack-Hartmann
wavefront sensor (WF) to monitor the laser position, pointing, and wavefront.
(L1, L2: lenses; MO: microscope objective.) The main laser beam reflects off
mirrors M5 and M6, passing through a $1^{\circ}$ axicon lens (AX), which
produces a Bessel focus within the vacuum pipe connecting Chamber A and
Chamber B. The Bessel focus is centered at the six-way cross (CR), as denoted
by the yellow region indicating plasma formation. The laser beam is terminated
at the beam dump (BD), while the leaked light from mirror M7 is used for
imaging the Bessel focus by a single lens (L5) to a camera (BC). On the
diagnostic side, a commercial macro lens lens (L4) is used to image the plasma
light onto a camera (AG). The vacuum pressure is measured shot-to-shot using a
pressure gauge (PG) on the figure’s far left. (a) shows an example of the raw
plasma light image taken by camera AG.

The plasma channel is ionized by a $50\,{\rm fs}$, $450\,{\rm mJ}$ pulse from
a terawatt-class Ti:Sapphire laser system. The wavefront of the pulse is
optimized with a deformable mirror before being sent through an axicon lens
($\alpha=1^{\circ}$) that generates a non-diffracting Bessel-beam focus over a
length of $\sim 150\,{\rm cm}$. The experimental vacuum chamber is filled with
Helium gas, creating the plasma source within. The vacuum system consists of
two vacuum chambers connected with vacuum pipes and a six-way vacuum cross as
shown in Fig. 4. The plasma is formed inside the vacuum pipe and viewed
through an optical window. The gas pressure is monitored by a vacuum gauge and
recorded on a shot-by-shot basis. The laser intensity profile is recorded
upstream and downstream of the axicon lens. The former is used as an input
parameter for the simulation, and the latter is used to monitor the Bessel
focus profile.

The diagnostic system views the plasma light at 1 m downstream of the axicon.
A commercial camera lens (AT-X M100 PRO D Macro from Tokina, $f=100\ {\rm
mm}$) is installed immediately outside the vacuum window. A $6$-OD notch
filter (central wavelength= $785\ {\rm nm}$, FWHM= $33\ {\rm nm}$) filters out
most of the scattered laser light that appears as a background to the plasma
light. The imaged plasma light is recorded with a CMOS GigE camera chip (Sony
IMX265), forming the primary source of data collected in this experiment.

The camera is externally triggered by a trigger pulse train from a Signal
Delay Generator (SDG), which is synchronized with the Ti:Sapphire laser
system. The timing of the laser system is precise to the sub-picosecond scale;
the SDG is precise to the 10’s of picoseconds scale. However, the CMOS GigE
camera has a significant internal timing jitter between the arrival of the
trigger signal to the initiation of the exposure time, on the scale of $10\
{\rm\mu s}$. The shortest integration time for this type of camera is on the
order of $10\ {\rm\mu s}$, whereas the dynamic timescale of light emission
from the plasma is 10’s to 100’s of nanoseconds. Therefore, this type of
camera always captures a time-integrated signal of the plasma light that is
much longer than the dynamic timescale.

Despite the limitations of CMOS GigE cameras, their cost-effectiveness makes
them a preferred choice in high-radiation environments. For example, in the
FACET-II accelerator tunnel at SLAC National Accelerator Laboratory,
approximately 10 GigE cameras are employed solely to monitor the laser and
plasma source. During operations, FACET has reported the loss of more than 20
cameras per year due to high radiation exposure. Therefore, cost-efficient
CMOS cameras (tyipcally a few hundred dollars each) are far more suitable than
would be a fast (nanosecond) gated camera, which can cost tens of thousands of
dollars. In Section IV, we introduce an innovative analysis method to extract
time-resolved plasma light emission data from time-integrated measurements
using inexpensive GigE cameras.

##  IV Analysis Methodology

###  IV.1 Statistical Analysis

As explained in the previous section, CMOS GigE cameras are preferable in
high-radiation environments but are conventionally unsuitable for high-timing-
precision measurements. In this section, we discuss an inventive statistical
analysis to extract time-resolved light emission from multi-shot time-
integrated data.

The first step is quantifying the camera-trigger-jitter distribution, which is
device-dependent. Physically, a trigger pulse is sent at time $t_{\rm SDG}$ to
the GigE camera. After processing in the camera chip, the exposure time begins
at $t_{\rm start}$. This delay between $t_{\rm SDG}$ to $t_{\rm start}$ varies
shot-to-shot; we call this the camera-trigger-jitter, which follows a
distribution $J(t)$. The precision of the laser arrival time at the target
plasma location ($\mathcal{O}(10\,{\rm ps})$) is orders of magnitude smaller
than the camera-trigger-jitter ($\mathcal{O}(10\,{\rm\mu s})$). As a result,
the laser arrival time can serve as a fiducial signal to investigate $J(t)$.
The diagnostic camera is first synchronized to the laser arrival-time, so that
the laser pulse always arrives within the duration of the camera exposure time
and the laser scattered light is captured by the camera in every shot. This
starting time is defined as $t=0$ in Fig. 5. The trigger signal is then
advanced/delayed away from $t=0$. Because the exposure starting-time (and
ending-time) jitters, the laser pulse will sometimes fall outside the exposure
window as the delay time is scanned, resulting in images without laser signal.
Eventually, the trigger advance/delay is great enough that none of the images
taken capture the laser. The observed probability of a laser pulse falling
inside the camera exposure window as a function of advance/delay time, $j(t)$,
is plotted in Fig. 5. Note that the falling edge in this figure corresponds to
a relative delay of the SDG trigger, meaning that the laser arrives ahead of
the camera exposure’s starting-time. This implies that the falling edge of
$j(t)$ is the cumulative distribution function of the camera-trigger-jitter
distribution, $J(t)$,

$j(t)=\int_{-\infty}^{t}J(t)dt.$ (10)

The cumulative distribution function of a normal distribution is

$j(t)=\frac{1}{2}\left(1+\text{erf}\left(\frac{t-\mu_{J}}{\sigma_{J}\sqrt{2}}\right)\right),$
(11)

where $\mu_{J}$ is the mean of $J(t)$, $\sigma_{J}$ is the deviation of of
$J(t)$, and erf is the error function. The rising and falling edge data
presented in Fig. 5 is fitted with Eq. 11 and the camera-trigger-jitter
distribution, $J(t)$, is thus determined from Eq. 10, assuming normal
distribution and $\sigma_{J}$ is given by the fitting, and plotted in Fig. 6

Figure 5:  Measurement of the probability of a laser pulse appearing within
the camera’s exposure time at various relative delays, measured with 30, 40,
50, and $1000\,{\rm\mu s}$ exposure settings. Jitter of the camera’s exposure
start time and end time cause the falling and rising ramps to be continuous
functions instead of step functions. Repeatable measurements of the falling
edge at the various exposure settings (30, 40, 50, and $1000\,{\rm\mu s}$)
confirm that altering the exposure duration does not affect the start-time
jitter. The magenta and cyan curves show Eq. 11 fitted to the rising and
falling ramps, respectively.

The observable plasma light lasts for an extended duration of time $dt_{\rm
glow}$. Most of the images captured by the camera either include both the
plasma light and background light from the laser pulse, or neither. (Note:
Though the notch filter removes much of the scattered laser light, some always
remain visible on the camera.) For certain trigger delay times, however, it is
possible for the camera to capture a portion of the plasma light while the
prompt laser light falls outside of the camera’s exposure window. This happens
when the exposure begins after the laser arrives but before the plasma light
ends. The longer the plasma light decay time, $dt_{\rm glow}$, the more these
“lucky shots” accumulate.

Figure 6:  A conceptual diagram illustrating the relationship between camera
jitter, laser arrival time, and plasma light time scale. The blue curve
represents the camera starting time jitter distribution measured and
calculated by Eq. 10. The red dashed line indicates the laser arrival time.
Following the laser arrival, if a plasma is formed, the plasma light continues
to glow for an extended duration, $dt_{glow}$, until the intensity is
undetectable. This is represented as the purple area shown in the figure,
though the scale of $dt_{glow}$ is greatly exaggerated for illustration. When
the camera exposure starts at time $I$ (dashed green line), the captured image
contains both laser and plasma light. When it starts at time $III$ (dot-dashed
green line), no light is captured in the image. However, when it starts at a
propitious timing, e.g., at time $II$ (solid green line), the camera exposure
begins after the arrival of the laser but before the plasma light dissipates,
resulting in the capture of only the plasma light without the laser light.

The purple area under the curve in Fig. 6 represents these “lucky shots”
(shots with plasma light but without the laser signal). The ratio of “lucky
shots” to all recorded shots $R$ is expressed as

$\displaystyle R\ =\int_{t_{\rm laser}}^{t_{\rm laser}+dt_{\rm glow}}J(t-\
t_{\rm SDG})\,dt$ (12)

where $J(t-\ t_{\rm SDG})$ is the camera-trigger-jitter distribution centered
about a given SDG trigger time $t_{\rm SDG}$, and $t_{laser}\equiv
t_{SDG_{0}}$, where $t_{SDG_{0}}$ is the timing $t=0$ at Fig. 5. Accordingly,
we solve for $dt_{\rm glow}$, giving

$\displaystyle dt_{glow}|_{t_{SDG}}=\left.\frac{-J(t_{\rm
laser})+\sqrt{J(t_{\rm laser})^{2}-2\ I\ J^{\prime}(t_{\rm
laser})}}{J^{\prime}(t_{\rm laser})}\right|_{t_{SDG}},$ (13)

where $J^{\prime}$ denotes the time derivative of $J$, and
$I=R{\int_{-\infty}^{\infty}J(t-\ t_{\rm SDG})\,dt}$. Using Eq. 13, we can
measure the characteristic plasma emission time scale without a nanosecond
gated camera, so long as we record a statistically sufficient number of images
with our GigE camera.

###  IV.2 Data Processing

In this subsection, we explain how the raw experimental data is processed
before performing the statistical analysis described in the previous
subsection. We collected data at various SDG delays across a $120\,{\rm\mu s}$
interval with $10\,{\rm\mu s}$ increments. For each SDG delay, we set the
camera exposure time to $30$, $40$, $50$, and $1000\,{\rm\mu s}$. We captured
$300$ shots for each combination of SDG delay and exposure time, resulting in
a total of $12,000$ data shots.

Scattered laser light fills the vacuum chamber during the experiment,
resulting in plasma light emission images with high background noise at the
laser frequency. The background noise level is hundreds of times higher for
shots that include prompt laser light than for those that do not, making it
easy to identify shots where the laser arrival time fell within the camera
exposure window.

Due to the high and fluctuating noise level, distinguishing the plasma signal
from the background noise is somewhat challenging. A dynamic statistical
threshold is employed to determine whether any particular image includes a
signal from plasma emission light. A region of interest where the plasma light
may appear is identified as the “plasma region” and the rest of the image is
tagged as the “background region”. We work out the $99.7\%$ error bound for a
population mean (EBM), that is, the probability of the average of a random
sample falling below $\overline{x}_{{\rm max},99.7}$ is $99.7\%$ based on

$\overline{x}_{{\rm max},99.7}=\overline{x}+z_{99.7}\frac{\sigma}{\sqrt{n}},$
(14)

where $z_{99.7}=2.778$ is the $Z$ score for $99.7\%$ probability,
$\overline{x}$ is the average light count value of the “background region”,
$\sigma$ is the standard deviation of the “background region”, and $n$ is the
sample size of the “plasma region”. When the average value of the “plasma
region” exceeds the threshold defined in Eq. 14, it is considered a positive
signal of plasma light emission. This dynamic statistical threshold
effectively adapts to the varying background intensities and is more efficient
than a fixed intensity threshold.

By Eq. 13, we determined that the decay time scale of the detectable signal
for our laser-ionized Helium plasma light emission is $194\pm 14\,{\rm ns}$.
We identified $204$ shots that exhibited plasma light emission without laser
light based on Eq. 14. Each image is rotated to ensure the plasma light
filament appears completely horizontal. Since the plasma filament is
homogeneous over the majority of its length, we took the projection of these
images along the laser propagation axis and plotted them as a waterfall plot
in Fig 7(a), sorted by total intensity of the region of interest. Each column
in the waterfall plot represents one plasma-light-only shot. Because the
camera exposure window starts at $t_{\rm start}$, which is after the arrival
of the laser pulse, $t_{\rm laser}$, the plasma light in each shot is
integrated from $t_{\rm start}$ to the end of the detectable light emission,
$t_{\rm endEmission}$. Since the decay time scale of the plasma light is
orders of magnitude shorter than the characteristic camera jitter time scale,
the $204$ plasma-light-only shots can be treated as being roughly uniformly
distributed in time within the $194\,{\rm ns}$ plasma light emission window,
yielding an effective time resolution of approximately $1\,{\rm ns}$.

Figure 7:  Each column in (a) represents a single image (shot) containing
plasma light but without a laser signal. The images have been summed over the
laser/plasma axial dimension. The columns are sorted and aligned based on
their total intensity within the region of interest. Each image integrates
from $t=t_{start}$ to $t_{EndEmission}$. In the shot with the highest
intensity (leftmost column), $t_{start}\sim t_{laser}$; whereas in the
rightmost column, $t_{start}\sim t_{EndEmission}$. (b) shows the result after
a Gaussian (low-pass) filter is applied to (a) and each column is subtracted
from the next column, yielding time-resolved (spatially integrated) photon
emission profile with the temporal resolution $\sim 1$ ns.

Every column in Fig. 7 corresponds to an integrated plasma light signal from
$t_{\rm start}$ to $t_{\rm endEmission}$, (where $t_{\rm start}$ shifts $\sim
1\,{\rm ns}$ further away from $t_{\rm laser}$ from one column to the next),
thus subtracting each column from its neighboring column yields a time-
resolved plasma light emission plot with a resolution of $\sim 1\,{\rm ns}$.
We apply a Gaussian noise filter to the waterfall plot in Fig. 7 (a), then
subtract one column from the next to yield the time-resolved plasma light
emission pattern shown in Fig. 7 (b). The results of this analysis demonstrate
the ability to achieve $\mathcal{O}(1\,{\rm ns})$ time-resolved imagery of the
plasma light using a cost-effective GigE CMOS camera.

Finally, The camera records the light emission function projected along the
camera axis, the radial emission function is retrieved with Abel inversion
Hickstein  _et al._ (2019), as shown in Fig. 8 (b).

##  V Results and Discussion

We compare the results of our simulation work and our experimental data
analysis in Fig. 8, and observe good agreement overall. Figure 8 (a) and (b)
show the time evolution of the simulated and experimentally observed plasma
light emission along the $x$ axis ($y=0$) at an arbitrary $z$ location within
the homogeneous middle region of the plasma filament, respectively. The
simulations used to generate Fig. 8 (a) are described in section II, and the
data analysis methods lead to Fig. 8 (b) is described in Section IV.2. Figure
8 (a) and (b) are normalized for comparison.

Figure 8 (c) shows the photon emission density from Fig. 8 (a) and (b) at the
center location ($x=0,y=0$) as a function of time. We see good agreement,
including the approximate time of the second peak in brightness between 75 and
100 nanoseconds. There is a slight discrepancy in the first $10\ {\rm ns}$,
where the data shows a sharp rise from zero, and the simulation immediately
yields a high photon emission density. This is due to the fact that the model
we used in our simulation does not take into account the atomic decay lifetime
of the excited atoms, which leads to a slight delay between the formation of
the plasma and the onset of photon emission. The lifetime of the $4p$ state is
3.975 ns and the $3d$ state is 15.696 ns Theodosiou (1987), which
qualitatively matches the delayed peak observed at 13.5 ns.

Figure 8 (d) compares the full photon emission density profile along $x$ at a
few different times, $10$, $40$, and $80\ {\rm ns}$. We again see generally
good agreement, though the data does not seem to capture the $\sim
400\,{\rm\mu m}$-wide diffuse region of light at later times. This is likely
an artifact of our limited signal-to-noise ratio in the experimental data, and
the noise level can be observed in Fig 8 (c) as well after $\sim 100$ ns.

Figure 8: Light emission from the plasma as a function of radius and time from
(a) simulation and (b) measurement. Measurements were taken at $z=1$ m from
the axicon. (c) shows the temporal evolution at $r=0$ with simulation in red
and experimental data in blue. The shaded region is the statistical temporal
measurement error. (d) shows radial lineouts at 10, 40, and 80ns, solid lines
are experiment and dashed lines simulation.

Lastly, Figure 9 (a) and (b) show the experimental results of a laser energy
scan and a gas pressure scan, respectively. The data points of each plot
correspond to the peak photon emission observed in the time-integrated (i.e.
the entire plasma light lifetime falls within the camera integration time),
spatially resolved data (average of 100 shots). The lines correspond to an
analytic prediction of the temporally integrated peak photon emission,
$\Gamma$, using Eq. 6. Despite the complexity of the plasma light emission
process, the experimental data shows remarkable agreement with our simple
prediction of the peak photon density scaling as a function of the initial
plasma parameters, $n_{0}$ and $T_{0}$.

Figure 9:  Time-integrated peak photon emission as a function of (a) laser
energy and (b) gas pressure. Each data point represents the average result of
100 measurements. The solid lines show the calculated quantity from Eq. 6,
where $T_{e}$ is simulated using PIC simulations and $n_{0}$ is assumed to
correspond to a fully ionized gas at the given pressure. The purple dot-dashed
line in (a) shows the laser energy at which $99\%$ ionization occurs,
approximately $147\,{\rm mJ}$. The purple dot-dashed line in (b) shows the
curve corresponding to calculation at which $99\%$ ionization occurs, below
which the gas is not expected to be fully ionized. The analytic formula agrees
well with the data when the plasma is fully ionized.

##  VI Conclusions

We have demonstrated an experimental comprehension of the decay process of the
plasma light emitted from a thin, laser-ionized Helium plasma source,
supported by numerical simulations. We used a three-step model to model the
photon emission pattern, which showed that electron-neutral collisional
excitation dominates over electron-ion recombination. The photon emission
pattern was measured in the experiment from a $6.9\times 10^{17}\ {\rm
cm^{-3}}$, $\sim 30\rm{\mu m}$ wide plasma and agrees well with the
simulation. We constructed an analytic model to predict the scaling of the
time-integrated, on-axis light emission with the initial plasma density and
temperature according to Eq. 6 and we saw the expected scaling in the
experiment.

We also presented a novel statistical approach for measuring the temporal
evolution of plasma light with nanosecond resolution using a cost-effective
GigE CMOS machine vision camera. By leveraging the prompt scattered laser
light as a fiducial signal, we quantified the exposure timing jitter
distribution of the camera and measured the observable plasma light lifetime
at $194\pm 14\,{\rm ns}$. Furthermore, by sorting and applying image
processing techniques, we reconstructed the time-resolved plasma light
emission profile with nanosecond-level precision.

This statistical method is generalizable and offers new diagnostic
possibilities in PWFA experiments and related applications where cost-
effective diagnostics are a necessity. To prove effective, there must exist an
appropriate temporal fiducial signal (e.g. prompt laser light) and the the
detector’s timing jitter distribution must exhibit statistical repeatability.
In PWFA experiments, these two requirements are often met due to the presence
of a laser and an electron beam, which can serve as reliable temporal fiducial
signals. This technique allows commonly used sensors to achieve measurements
previously exclusive to expensive gated sensors (e.g. gated cameras or gated
spectrometers) on a statistical basis, giving inexpensive GigE CMOS cameras
access to nanosecond dynamics. For example, this technique could be applied to
Stark broadening of emission lines in the plasma glow Griem  _et al._ (1962)
detected by a spectrometer; a CMOS camera equipped with a wide-angle lens can
capture the formation of laser-ionized meter-scale plasma. Some potential
applications in PWFA include imagining the dissipation of the plasma wake Wan
_et al._ (2022), investigating the plasma heating mechanism by an electron
beam for laser-electron alignment purposes Scherkl  _et al._ (2022), and
measuring the recovery time of a plasma-wakefield accelerator D’Arcy  _et al._
(2022).

Finally, we come to two more significant conclusions regarding the work
presented here. First, our results enhance our confidence in the validity of
the plasma formation models. In particular, we have confirmed that the
relatively inexpensive SSF simulation is able to accurately predict the laser
ionization process, making it a valuable tool in the iterative plasma source
design process. Second, our advances in understanding of the readily
accessible plasma light emission have enhanced its utility as a diagnostic
tool. Numerous studies have shown that the electron temperature of the PWFA-
like plasma source impacts a wide range of applications Diederichs  _et al._
(2023); Ma  _et al._ (2020); Miao  _et al._ (2022), and the methods described
in our work can make it possible to design and diagnose experiments that aim
to study such effects with significantly greater confidence.

###### Acknowledgements.

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of High Energy Physics under Award No. DE-SC0017906
and the National Science Foundation under Grant Number PHY-2047083.

##  Appendix A Excitation and recombination model

Detectable visible/NIR photons are emitted by transitions in Helium between
the $n=3$ and $n=2$ energy levels. Among these transitions, the $3d-2p$
transition exhibits the highest rate and the most persistent atomic spectral
line of neutral Helium Theodosiou (1987). Consequently, our focus is on
processes that populate the $3d$ state. To populate the $3d$ state, electrons
can transition from $p$ or $f$ states. Although the $4f-3d$ transition has a
higher transition than the $4p-3d$ transition, populating the $f$ states
requires a higher excitation energy, which corresponds to a significantly
smaller collisional cross-section. Therefore, in a collisional excitation
scenario, the most probable transition from the ground state that will yield a
visible/NIR photon is the sequence $1s-4p-3d-2p$.

On the other hand, in plasma electron-ion recombination, it takes less energy
to capture a free electron into a highly excited state Robben, Kunkel, and
Talbot (1963). As a result, most recombination events result in highly excited
states, thereby easily populating the $4f$ state. Therefore, in a
recombination and de-excitation process, the most probable transition leading
to a visible/NIR photon is $4f-3d-2p$.

##  Appendix B Temporally integrated peak photon emission density

The analytic formalism for the temporally integrated peak photon emission is
derived by integrating $n_{e}<\nu_{en}>$ from $t=0$ to $t=\infty$ with a few
key approximations and assumptions. Recall $<\nu_{en}>$ is given by Eq. 4. We
note that the peak photon emission occurs at $\vec{r}\simeq 0$ and for
simplification, we drop the notation $\vec{r}\simeq 0$. To calculate the
integral in Eq. 4, we first plot the electron-impact excitation cross-section
from Ref. Stone, Kim, and Desclaux, 2002 in Fig. 10, and fit it with the
following function:

$\displaystyle\sigma(K)_{1s-4p}=A(K-\kappa-
K_{th})\exp({-\frac{K}{K_{w}}})+\psi,$ (15)

where $A=2.1\times 10^{-4}\ {\rm cm^{2}},\kappa=34.2\ {\rm eV},K_{w}=26\ {\rm
eV},\psi=7.5\times 10^{-3}\ {\rm cm^{2}}$ are the free parameters found by the
fit. The integral in Eq. 4 becomes

$\displaystyle I_{exc}\approx A\int_{K_{th}}^{\infty}$
$\displaystyle\exp({-\frac{K}{k_{B}T_{e}(t)}})$ (16)
$\displaystyle((K-\kappa)\exp({-\frac{K}{K_{w}}})+\psi)KdK.$

Because the electron temperature during the plasma glowing process (as shown
in Fig. 3 (b)) is orders of magnitude less than the initial temperature, we
make the following approximations, $K_{th}>>k_{B}T_{e}$, $C>>k_{B}T_{e}$, and
$B>>k_{B}T_{e}$. The integral now becomes

$\displaystyle I_{exc}\approx
Ak_{B}T_{e}(t)\exp({-\frac{K_{th}}{k_{B}T_{e}(t)}})(\psi
K_{th}-\sigma_{1s-4p}),$ (17)

where
$\sigma_{1s-4p}=\frac{4}{K_{w}^{2}}\exp{(\frac{K_{th}}{K_{w}})K_{th}^{4}}$.
Now, we plug the result of the integral in Eq. 17 back into Eq. 4,

$\displaystyle n_{e}(t)<\upsilon_{en}(t)>\approx$ $\displaystyle
A^{\prime}n_{e}(t)n_{n}(t)(k_{B}T_{e}(t))^{-3/2}$ (18) $\displaystyle
k_{B}T_{e}(t)\exp({-\frac{K_{th}}{k_{b}T_{e}(t)}}),$

where $A^{\prime}$ is a collective constant. To calculate the temporally
integrated photon emission, we integrate Eq. 18 from $t=0$ to $t=\infty$,

$\displaystyle\int_{0}^{\infty}n_{e}(t)<\upsilon_{en}(t)>dt$
$\displaystyle\approx\int_{0}^{\infty}A^{\prime}n_{e}(t)n_{n}(t)(k_{B}T(t))^{-1/2}\exp({-\frac{K_{th}}{k_{B}T(t)}})dt.$
(19)

To continue the calculation, a few assumptions are made: $n_{n}(t)=n_{0}$,
$k_{B}T_{e}(t)\approx k_{B}T_{e0}e^{-t/\tau_{d}}$, $n_{e}(t)\approx
n_{0}e^{-t/\tau_{d}}$, where $\tau_{d}$ is the decay lifetime and $T_{e0}$ is
the plasma electron temperature before the decay. The first assumption is
justified because the neutral particles fill in the plasma column much faster
than the full expansion time scale, leading to a quasi-homogenous background
of neutral gas. The second and third assumptions are justified by the lineout
in Fig. 3(b). So Eq. 19 now becomes

$\displaystyle\int_{0}^{\infty}$ $\displaystyle
A^{\prime}n_{0}^{2}e^{-t/\tau_{d}}(k_{B}T_{e0}e^{-t/\tau_{d}})^{-1/2}\exp\left({-\frac{K_{th}}{k_{B}T_{e0}e^{-t/\tau_{d}}}}\right)dt$
$\displaystyle\approx$ $\displaystyle
A^{\prime}n_{0}^{2}(k_{B}T_{e0})^{-1/2}2\tau_{d}$ (20)
$\displaystyle\left(e^{-\frac{K_{th}}{k_{B}T_{e0}}}-\frac{K_{th}\sqrt{\pi}\text{Erfc}(\sqrt{\frac{K{th}}{k_{b}T_{e0}}})}{\sqrt{K_{th}k_{B}T_{e0}}}\right),$

where Erfc is the complementary error function, and when
$k_{B}T_{e0}<<K_{th}$, $\text{Erfc}(\sqrt{\frac{K{th}}{k_{b}T_{e0}}})\approx
0$. This simplifies Eq. 20 to

$\displaystyle\int_{0}^{\infty}n_{e}(t)<\upsilon_{en}(t)>dt$
$\displaystyle\approx$ $\displaystyle
A^{\prime}n_{0}^{2}(k_{B}T_{e0})^{-1/2}2\tau_{d}e^{-\frac{K_{th}}{k_{B}T_{e0}}},$
(21)

We empirically find from our simulations that the decay lifetime $\tau_{d}$ is
linearly proportional to the initial kinetic energy as shown in Fig. 11. So
Eq. 21 becomes

$\displaystyle\int_{0}^{\infty}n_{e}(t)<\upsilon_{en}(t)>dt\approx
2A^{\prime}n_{0}^{2}(k_{B}T_{e0})^{1/2}e^{-\frac{K_{th}}{k_{B}T_{e0}}}.$ (22)

Finally, we note that the observed plasma light arises almost entirely from
collisions occurring after the initial rapid plasma expansion, so we plug in
the temperature after the initial rapid expansion $T_{e0}=RT_{0}$ where we
empirically find $R=0.09$ from our simulations. Thus, we finally arrive at the
following expression of proportionality for the time-integrated peak light
emission $n_{\gamma,{\rm peak}}$:

$\displaystyle\Gamma\propto
n_{0}^{2}\sqrt{Rk_{B}T_{0}}\exp\left({-\frac{K_{th}}{Rk_{B}T_{0}}}\right).$
(23)

Figure 10:  The electron-impact excitation cross-section of the $1s$ state to
the $4p$ state, $\sigma_{1s-4p}$. The blue data points are taken from Ref.
Stone, Kim, and Desclaux, 2002 and the red curve corresponds to a fit using
Eq. 15. The green dashed line denotes the excitation threshold energy,
$K_{th}$. Figure 11:  The peak photon density decay half-life (at $x=0,y=0$)
for various initial average electron temperatures for a thin, laser ionized
PWFA plasma with density $n_{e}=8e17\ {\rm cm^{-3}}$.

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# Peek into the Future Camera-based Occupant Sensing in Configurable Cabins
for Autonomous Vehicles

Avinash Prabu1, Renran Tian2, Lingxi Li1, Jialiang Le3, Srinivasan
Sundararajan3, and Saeed Barbat3 The authors would thank Ford to support the
research work.1Avinash Prabu and Lingxi Li are with the Department of
Electrical & Computer Engineering, Indiana University Purdue University
Indianapolis, 723 West Michigan Street, Indianapolis, IN, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS>Tian is with the Department of
Computer Information Technology, Indiana University Purdue University
Indianapolis, 799 West Michigan Street, Indianapolis, IN, USA
<EMAIL_ADDRESS>Le, Srinivasan Sundararajan and Saeed Barbat are with
the Ford Motor company, Research and Innovation Center, Dearborn, MI, USA.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The development of fully autonomous vehicles (AVs) can potentially eliminate
drivers and introduce unprecedented seating design. However, highly flexible
seat configurations may lead to occupants’ unconventional poses and actions.
Understanding occupant behaviors and prioritize safety features become eye-
catching topics in the AV research frontier. Visual sensors have the
advantages of cost-efficiency and high-fidelity imaging and become more widely
applied for in-car sensing purposes. Occlusion is one big concern for this
type of system in crowded car cabins. It is important but largely unknown
about how a visual-sensing framework will look like to support 2-D and 3-D
human pose tracking towards highly configurable seats. As one of the first
studies to touch this topic, we peek into the future camera-based sensing
framework via a simulation experiment. Constructed representative car-cabin,
seat layouts, and occupant sizes, camera coverage from different angles and
positions is simulated and calculated. The comprehensive coverage data are
synthesized through an optimization process to determine the camera layout and
overall occupant coverage. The results show the needs and design of a
different number of cameras to fully or partially cover all the occupants with
changeable configurations of up to six seats.

##  I INTRODUCTION

The development of autonomous vehicles (AVs) has been rapidly growing, leading
to more machine control and lesser human interventions. This revolutionary
progress will eventually remove drivers from the cars and give occupants the
freedom to carry out more flexible activities. Besides the expected
improvement of mobility, efficiency, and comfort, this change also increases
the difficulty of efficiently triggering safety features and better protecting
occupants when crashes happen. Without driver interventions, AVs need to
prioritize safety features based on external conditions and occupant behaviors
via enhanced sensor systems [1]. Thus, ”researching occupant protection in
alternative vehicle designs” becomes a higher-focused research topic by the
National Science and Technology Council and the United States Department of
Transportation.

###  I-A Related Research

Towards traditional manual-driving vehicles, driver monitoring systems have
been frequently researched. Different impaired driver states like distraction
[2, 3], drowsiness [4], fatigue [5, 6], and mind wandering [7] are widely
investigated about their effects on driving safety. Research also focuses on
the corresponding detection algorithms and mitigation means. Although
physiological sensors like EEG, ECG, heart rate, and other measurements are
valid and reliable indicators for different driver states [7, 4, 8], camera-
based systems are more widely developed and implemented. Studies and practices
show that visual sensors are non-intrusive solutions and more convenient to be
integrated into vehicle-cabins [2, 9, 4, 10]. These driver state sensing and
mitigation systems are initially designed to reduce impaired driver status and
improve safety. However, with the advantage of vehicle automation levels,
driver state sensing also extends its functionalities and is used in more
complicated conditions.

As level 2 and level 3 autonomous vehicles are coming into reality, driver
monitoring systems become core components of the overall advanced driver
assist systems and the autonomous driving systems. AVs road test in California
has demonstrated the problems of automation disengagement and highlighted the
importance of the transition process from auto-control to manual-control [11,
12]. This control hand-over process has a list of human factors concerns,
including driver inattention and distraction, situational awareness, and over-
reliance on the technology [13]. These issues may result in a take-over
reaction time between 2 to 10 seconds, a prolonged take-over completion time
from 2 to 20 seconds [14], and even longer time for a driver to fully
stabilize the car [15]. Considering the various take-over scenarios in terms
of urgency, predictability, and criticality [16], it is important to monitor
the driver’s states before and during the take-over process to re-engage them
via optimized interface design [13], and adopt appropriate transition
strategies [17].

Although more attention starts to focus on rear-seat occupants’ safety, full-
cabin occupant sensing is much less studied than the driver state sensing
systems. Not surprisingly, rear-seat occupants have more differences in terms
of individual characteristics [18] when these seats are treated as safer for
young and older passengers and people in a more vulnerable status. However,
these rear-seat occupants are equally likely to be involved in car crashes
[18], and their protection becomes a more challenging task. Most of the full-
cabin or rear-seat occupant sensing studies implement seat-embedded sensors to
detect vital signs or occupancy of different seats, with one typical study
exampled in [19]. In recent years, researchers started to use seat-independent
sensors like Kinect [20] and Ultra Wide-Band Radar [21] to detect and track
all occupants, including those who sit in the back. However, most of these
studies are still in the initial investigation states and usually only focus
on traditional fixed cabin layouts.

###  I-B Research Objectives and Contributions

To our best knowledge, few studies discuss the sensing of all occupants in a
highly configurable vehicle cabin. To address this research gap, this study
focuses on camera-based full-cabin sensing given such a dynamic environment. A
car cabin was created on a simulation platform with multiple seat layouts.
Occupants/passengers were designed, and their key upper body points were
identified. Computer simulation was first conducted to obtain the camera
coverage of the key points across all occupants in different seat layouts from
different angles and positions. An algorithm was then developed to find the
combinations of camera locations and angles to optimize overall coverage
towards different goals. The contributions of the paper are itemized below:

  * •

Propose the concept of full-cabin occupant sensing with a highly configurable
seat layouts.

  * •

Describe the possible designs of camera-based sensing framework for L3+
autonomous vehicles.

  * •

Provide a simulation platform to develop and evaluate camera-based occupant
sensing systems.

  * •

Create a novel optimization strategy to study camera locations and
orientations, given the constraints on the coverage requirements and camera
numbers.

(a) Side View

(b) Back View

Figure 1: Cabin Design

##  II Simulation Method

###  II-A Cabin Design

In order to set up a representative cabin, we surveyed a list of on-market
vehicles, including SUVs and mini-vans to acquire average cabin measurements.
The results of the survey is explained in Table I.

TABLE I: Cabin Measurements Axis | Parameter | Measurement (m)
---|---|---
Z | Cabin height | 1.41
| Occupant head to ceiling | 0.11
Y | Cabin length | 3.25
| Leg room row 1 | 0.11
| Leg room row 2 | 0.5
| Leg room row 3 | 0.35
X | Cabin width | 1.8
| Driver seat from side | 0.15
| Passenger seat from side | 0.15

The angle of the rear window is also averaging existing car cabins. The final
design of the cabin is shown in Fig. 1. The red line outlines the cabin inside
a Ford Transit van from two angles to illustrate the designed cabin sizes. The
cabin shape is not based on any existing car but represents a typical size of
current three-row vehicles. The cabin design sets up the simulation
boundaries. The same method presented in this study can be applied accordingly
by changing the cabin dimensions for any given car.

###  II-B Key Body Points

In the simulation environment, we focus on body key points for each seated
occupant. The simulation process will output the camera coverage of each of
these key points for each occupant in all interested seat layouts from
different positions and angles in the cabin. This research defines six body
locations, including nose, left shoulder, right shoulder, chest, right waist,
and left waist. The body locations are depicted in Fig. 2.

Figure 2: Key Body Points

###  II-C Simulation Space

In the simulation environment, we move the camera along a constructed frame in
the cabin. The paths along which the camera moves is shown as green lines in
Fig. (3), following a step size of 0.1m. At each step, the cameras are rotated
at various roll, pitch, and yaw angles. At all the locations and angles, the
coverage for each body key point is recorded. Three camera locations on the
windshield (illustrated as red dots) are also recorded.

Figure 3: Camera Locations

###  II-D Simulation Outputs

The results of the simulation are exported as a .csv file. The variables in
the dataset are _Position_Index, Seat_Index, Body_Area, Coverage Indicator,
Camera_X, Camera_Y, Camera_Z, Camera_Roll, Camera_Pitch, and Camera_Yaw_. The
_Position_Index_ is a unique ID to represent the position of the camera. The
_Seat_index_ represents the seat for the occupant in the vehicle. _Body_Area_
is the body key point, as explained in 2. The _Coverage Indicator/Luminance_
is a number showing coverage of the key point if the value is non-zero (in
this research, we use lighting sources to replace cameras, and use light
dominance on different body parts to estimate camera coverage as well as
distance). The _Camera_X, Camera_Y, Camera_Z, Camera_Roll, Camera_Pitch,
Camera_Yaw_ provides the pose of the camera at a particular position, based on
the vehicle coordinate system. An example of the data is provided in Fig. (4).

Figure 4: Example of Dataset

###  II-E Simulation Setup

The simulation aims to find the coverage towards each occupant in multiple
seat layouts from all possible angles and positions of the cameras in the
cabin. As an equivalent process, we decide to use light coverage to represent
camera coverage to use convenient off-the-shelf software. The idea behind this
is that in a 3D environment, if light can illustrate a body part in the cabin,
then a camera in the same position/angles should also capture the body part.
The light can be adjusted with the same angular ranges as the camera and can
be blocked by the same obstacles.

With that, a simulation environment is firstly constructed in a 3D virtual
environment, including a car CAD model and movable objects like seats,
occupants, and lighting sources. There are several main features of the
simulation environment:

  1. 1.

Any car CAD model can be imported into the simulation environment to reflect
the actual cabin dimensions.

  2. 2.

The movable objects can be arranged into different design layouts in the
cabin.

  3. 3.

The angles of the lights can be adjusted to represent the simulated camera’s
view angles. In this study, we set up the camera diagonal view range as 94
degrees.

  4. 4.

Multiple virtual sensors can be attached to the occupants at desired body
locations, which will report the strength of light coverage.

Based on sensor outputs, we can determine if certain body parts (where the
sensors are attached) are visible to the camera set up at the same
locations/angles as the lights. A program will automatically change the
position and orientation values for the light and enumerate all possible
angles and positions of interest. During this automated process, one light
will move along predetermined routes on the ceiling of the cabin and the
windshield window at a step of 10cm. At each light location, the light will
rotate Roll, Pitch, Yaw angles in steps of 10 degrees. At each location (XYZ
coordinates) and each direction (RPY angles) of the light, the script will
record all sensors’ readings attached to all occupants. These readings
directly measure the coverage of corresponding body parts from a camera in the
same pose.

This process can be completed for all the interested seat layouts separately,
and the results can then be integrated. As these light simulation data are
reflecting the coverage of cameras, an optimization algorithm (explained in
Section III) will select the optimal combinations of different camera
locations and angles to maximize the coverage of all occupants in all
simulated seat-layouts.

##  III Optimization Algorithm

The main goal of the optimization algorithm is to cover all the _Body_Area_ of
all the occupants with minimum number of camera. The optimization uses the
dataset from the simulation to find the best possible positions of the camera.
The simulation provides the intensity of light (_Luminance_) on each marker as
one of the primary variables. As explained earlier, if the light falls on the
marker (_Luminance_ $>$0), then the camera can capture the marker and if the
marker is not illuminated (_Luminance_ =0), then the camera cannot capture the
marker. With this idea in mind, the first step in the algorithm is to convert
the luminance measurements into binary values. This is explained in Algorithm
1.

Algorithm 1 Convert Luminance to Binary Luminance

$n\rightarrow$ total number of rows in the dataset

$BLuminance=[0]*len(Luminance)$

for $i=1:n$ do

if $Luminance(i)>0$ then

$BLuminance(i)=1$

end if

if $Luminance(i)=0$ then

$BLuminance(i)=0$

end if

end for

$\displaystyle Luminance$ $\displaystyle=[L1$ $\displaystyle L2$
$\displaystyle L3]$ (1) $\displaystyle L1$ $\displaystyle=[1.2$ $\displaystyle
5.6$ $\displaystyle 3.4$ $\displaystyle 9.3$ $\displaystyle 6.5$
$\displaystyle 0$ $\displaystyle 0$ $\displaystyle 0$ $\displaystyle 0$
$\displaystyle 0$ $\displaystyle 0$ $\displaystyle 0]$ $\displaystyle L2$
$\displaystyle=[0$ $\displaystyle 0$ $\displaystyle 0$ $\displaystyle 0$
$\displaystyle 0$ $\displaystyle 0$ $\displaystyle 0$ $\displaystyle 6.8$
$\displaystyle 4.3$ $\displaystyle 2.2$ $\displaystyle 8.7$ $\displaystyle
9.1]$ $\displaystyle L3$ $\displaystyle=[0$ $\displaystyle 0$ $\displaystyle
0$ $\displaystyle 0$ $\displaystyle 1.5$ $\displaystyle 4.9$ $\displaystyle
11.2$ $\displaystyle 3.5$ $\displaystyle 0$ $\displaystyle 0$ $\displaystyle
0$ $\displaystyle 0]$

For example, assuming that Equation. (1) provides the luminance of a two
occupant seat layout for three camera positions. After running through
Algorithm 1, the binary luminance is obtained as given in Equation (2).

$\displaystyle BLuminance$ $\displaystyle=[BL1$ $\displaystyle BL2$
$\displaystyle BL3]$ (2) $\displaystyle BL1$ $\displaystyle=[1$ $\displaystyle
1$ $\displaystyle 1$ $\displaystyle 1$ $\displaystyle 1$ $\displaystyle 0$
$\displaystyle 0$ $\displaystyle 0$ $\displaystyle 0$ $\displaystyle 0$
$\displaystyle 0$ $\displaystyle 0]$ $\displaystyle BL2$ $\displaystyle=[0$
$\displaystyle 0$ $\displaystyle 0$ $\displaystyle 0$ $\displaystyle 0$
$\displaystyle 0$ $\displaystyle 0$ $\displaystyle 1$ $\displaystyle 1$
$\displaystyle 1$ $\displaystyle 1$ $\displaystyle 1]$ $\displaystyle BL3$
$\displaystyle=[0$ $\displaystyle 0$ $\displaystyle 0$ $\displaystyle 0$
$\displaystyle 1$ $\displaystyle 1$ $\displaystyle 1$ $\displaystyle 1$
$\displaystyle 0$ $\displaystyle 0$ $\displaystyle 0$ $\displaystyle 0]$

The next step in the algorithm is to arrange the _Bluminance_ in matrix form.
This step helps in moving forward with formulating the optimization problem.
Each column of the matrix represents a particular position’s Bluminance.
Assuming there are $m$ total positions that were simulated and there are $p$
occupants in the scenario, the dimension of the matrix will be $6p\times m$.
As mentioned earlier, there are six markers for each occupant and each marker
has a luminance value, which equates to $6p$ _Bluminance_ values for each
position. The $q^{th}$ column of the _BMatrix_ corresponds to the $q^{th}$
position.

Algorithm 2 Arrange $BLuminance$ in matrix form

$m\rightarrow$ total number of positions in the dataset

$p\rightarrow$ number of occupants in the scenario

$BMatrix\rightarrow$ zero matrix of dimension $6p$ x $m$

$j=1$

for $i=1:m$ do

$BMatrix[:,i]=BLuminance[j:j+6p-1]$

$j=j+6p$

end for

Continuing with the example provided in Equation. (2), the BMatrix can be
obtained using Algorithm 2. The BMatrix is given in Equation. (3).

$BMatrix=\begin{bmatrix}1&0&0\\\ 1&0&0\\\ 1&0&0\\\ 1&0&0\\\ 1&0&1\\\ 0&0&1\\\
0&0&1\\\ 0&1&1\\\ 0&1&0\\\ 0&1&0\\\ 0&1&0\\\ 0&1&0\end{bmatrix}$ (3)

The optimization proceeds with the _BMatrix_ as one of the primary parameters.
The main goal of the optimization is to minimize the number of cameras and to
make sure all the markers are covered. The standard form of the optimization
problem is provided in Equation (4).

$\displaystyle\underset{x}{\text{min}}$ $\displaystyle c^{T}x$ (4) s.t
$\displaystyle Ax\geq b,\;$ $\displaystyle x=\\{0,1\\}$

In Equation (4), the one dimensional vector $c$ has all its elements as 1. The
dimension of $c$ is $m\times 1$. The vector $x$ is the solution to the
equation $Ax\geq b$, where $A$ is the $BMatrix$ and $b$ is a vector with all
the elements as 1, with dimension $6p\times 1$. The vector $b$ indicates that
all the _Body_Area_ needs to be covered. This ensures that the objective
function is the sum of all the elements of $x$. It can be tweaked to
accommodate other requirements also. For obvious reasons, the constraint on
elements of $x$ is that it can be only 0 or 1. Solving for $x$ using the above
optimization problem will provide the best position to place the cameras in
the cabin. The vector $x$ will have 1’s in the indices of the required
positions.

With the $BMatrix$ obtained in the example given in Equation. (3), the only
possible solution to the optimization problem is the column vector
$x=[111]^{T}$. The value $Ax$ is given in Equation. (5) and the value of
$c^{T}x$ is 3.

$\displaystyle Ax$ $\displaystyle=[1$ $\displaystyle 1$ $\displaystyle 1$
$\displaystyle 1$ $\displaystyle 2$ $\displaystyle 1$ $\displaystyle 1$
$\displaystyle 2$ (5) $\displaystyle 1$ $\displaystyle 1$ $\displaystyle 1$
$\displaystyle 1]$

The optimization problem can also be modified to accommodate various other
constraints. Assuming that the user can afford only two cameras in the cabin
and requires the algorithm to cover maximum possible markers. In this case,
the optimization algorithm can be set as given in Equation (6).

$\displaystyle\underset{x,b}{\text{max}}$ $\displaystyle\lambda^{T}b$ (6) s.t
$\displaystyle Ax=b,\;$ $\displaystyle c^{T}x=2,\;$ $\displaystyle
x=\\{0,1\\}\;$ $\displaystyle b=\\{0,1\\}$

In Equation (6), $\lambda$ is a one dimensional vector with all the elements
as 1 and the dimension is $6p\times 1$. $b$ is a one dimensional vector with
dimension $6p\times 1$, which indicates the coverage of markers. The objective
function ensures the sum of all the elements of $b$ is a global maximum across
all possible combinations of two cameras. The one dimensional vector $c$ has
all its elements as 1. The dimension of $c$ is $m\times 1$. The vector $x$ is
consistent with the explanation given for Equation (4), which stands for the
locations of the cameras needed. $c^{T}x=2$ ensures that only a total of two
cameras can be used. The constraints on $x$ and $b$ is to make sure that the
results don not contain cameras at the same location. Since this optimization
problem cannot be solved with linear programming, we use an iterative method
to find all the possible combinations that will provide the best coverage.

Again, continuing from the example given in Equation. (3), if the constraint
on number of cameras is 2, the best coverage possible with two cameras is
given in Equation. (7), and the best camera positions to obtain this coverage
is $x=[110]^{T}$.

$\displaystyle Ax=b$ $\displaystyle=[1$ $\displaystyle 1$ $\displaystyle 1$
$\displaystyle 1$ $\displaystyle 1$ $\displaystyle 0$ $\displaystyle 0$
$\displaystyle 1$ (7) $\displaystyle 1$ $\displaystyle 1$ $\displaystyle 1$
$\displaystyle 1]$

##  IV Optimization Result and Visualization

Three experiments are conducted using the developed simulation and
optimization methods to picture how a future camera-based full-cabin occupant
sensing frame may look like and perform in highly configurable cabins. Results
are reported and illustrated separately in this section.

###  IV-A Experiment 1: Coverage of all occupants for six seat layouts
independently

The simulation was performed on the six configurations given in Fig. 5. To
better demonstrate all possibilities, these configurations have two
(configurations 4 and 5), four (configurations 1 and 6), and six
(configurations 2 and 3) occupants correspondingly.

The optimization results show that configurations 2 and 3 require a minimum of
three cameras to cover all the markers. Configurations 1, 5 and 6 require two
cameras to cover all the occupants and configuration 4 requires just one
camera. The results of the optimized positions are shown in Fig. 5. In Fig. 5,
the red circles represent the camera positions.

Figure 5: Optimized Camera Positions

As explained earlier, the camera is substituted with a light source (94-degree
diagonal field of view). The light sources were positioned at the optimized
camera locations (see Fig. 5) in the cabin, and the resulting visualization is
given in Fig. 6. Here, the dark areas are the locations where the camera does
not see. For example, in Fig. 6(f), the legs of both the occupants are not
illuminated and thus look dark. This means that the cameras cannot see the
legs of the occupants, while the upper body of the occupants are well
illuminated, which infers that the cameras capture the whole upper body.
Likewise, concluding from all the sub-figures of Fig. 6, the cameras are able
to capture all the markers on the upper-body of all the occupants in the six
configurations, proving the concept of the experiment.

(a) Layout 1-1

(b) Layout 1-2

(c) Layout 2

(d) Layout 3-1

(e) Layout 3-2

(f) Layout 4

(g) Layout 5

(h) Layout 6-1

(i) Layout 6-2

Figure 6: Visualization of Coverage

###  IV-B Experiment 2: Coverage of occupants in a highly configurable cabin

In this experiment, we assume that a cabin with certain number of seats can be
reconfigured between different seat layouts. In Fig (5), configuration 1 can
be reconfigured to configuration 6 (Fig. (7)), configuration 2 can be
reconfigured to configuration 3 (Fig. (8)), and configuration 4 can be
reconfigured to configuration 5 (Fig. (9)). Then assuming that the cameras
will be fixed, the optimization goal is to find the camera numbers and
locations that maximize the occupant coverage towards the combinations of
configurations.

Figure 7: Optimized Camera Positions for Configurations 1 and 6  Figure 8:
Optimized Camera Positions for Configurations 2 and 3 Figure 9: Optimized
Camera Positions for Configurations 4 and 5

The results of the above mentioned optimized position are provided in Fig.
(7), Fig. (8), and Fig. (9), respectively. When there are four seats in the
cabin, two cameras are enough to cover all the four occupants in each of the
two configurations (as shown in Fig. (7). When there are six seats, four
cameras are needed to cover all the occupants’ upper bodies (as shown in Fig.
(8)) as opposed to three cameras needed to cover each configuration. This is
because the middle row in configuration 3 is facing rearward. If there are two
seats, two cameras are still needed to cover all the body parts in a highly
configurable cabin (as shown in Fig. (9)).

###  IV-C Experiment 3: Maximum coverage of occupants given constraints on the
number of cameras

Although the experiments above show that as the number of camera increases,
the cameras together can cover all the interested body parts from all
occupants in different configurations. However, the total number of cameras is
always limited in reality to reduce cost and system complexity. It is
important to see the trade-offs between camera numbers and coverage rates. In
this experiment, the optimization focuses on maximizing the coverage of
occupant body parts, given a certain number of cameras.

The first trial assumes that there are only three cameras available for the
six-seat highly configurable cabin (configuration 2 and 3). The optimization
towards this combination with three cameras show that out of 72 body parts in
total (across all occupants in all seats), 68 body parts can be covered with
three cameras. The visualization of the optimization can be observed in Fig.
10. Here, the uncovered points are the right shoulder and right hip of the
occupants marked in blue circle.

Figure 10: Optimized Camera Positions for configurations 2 and 3 with three
cameras

Another trial focuses on using only one camera for the highly configurable
cabin with two seats in total (configurations 4 and 5). The optimization
results in a maximum coverage of 21 out of the 24 body parts (across all
occupants in the two configurations). The visualization of the optimization is
illustrated on Fig. 11.

The experiment results suggest that with limited total number of cameras, the
body coverage is sacrificed. The trade-off between camera numbers and coverage
rates needs to be investigated based on the sensing needs and algorithm
development requirements.

Figure 11: Optimized Camera Positions for configurations 4 and 5 with three
cameras

##  V Conclusion

This paper peeks into the future camera-based sensing framework towards a
cabin with highly configurable seat layouts in L3+ autonomous cars. In
particular, a proposed simulation and optimization process can estimate the
number of cameras, their designs, and the coverage rates of occupant body
parts in all possible seats. The research pictures the camera design concept
for commonly-mentioned driver-less seat layouts, independently or combined,
based on representative cabin dimensions and occupant sizes.

We conducted three experiments using the proposed simulation and optimization
method. Towards seat layouts with two, four, or six seats in total, the result
shows that the sensing system needs one, two, or three cameras to cover all
the occupants’ upper bodies. When the cabin is highly configurable (meaning
the seat layout can change from one to another completely), the sensing system
may need more cameras at different poses and can still achieve full upper-body
coverage. Not surprisingly, additional investigation shows that reducing the
total number of cameras will sacrifice coverage rate to a certain degree. This
finding suggests future development to focus on the trade-off (between camera
numbers and coverage rates) towards specified sensing needs and system
development requirements.

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