# Reconstruction and interpretation of photon Doppler velocimetry spectrum for
ejecta particles from shock-loaded sample in vacuum

Xiao-Feng Shi  Institute of Applied Physical and Computational Mathematics,
Beijing 100094, China  Dong-Jun Ma<EMAIL_ADDRESS>Institute of
Applied Physical and Computational Mathematics, Beijing 100094, China  Song-
lin Dang  Jiangxi University of Applied Science, Nanchang 330103, China  Zong-
Qiang Ma  Institute of Applied Physical and Computational Mathematics, Beijing
100094, China  Hai-Quan Sun  Institute of Applied Physical and Computational
Mathematics, Beijing 100094, China  An-Min He  Institute of Applied Physical
and Computational Mathematics, Beijing 100094, China  Pei Wang
<EMAIL_ADDRESS>Institute of Applied Physical and Computational
Mathematics, Beijing 100094, China  Center for Applied Physics and Technology,
Peking University, Beijing 100871, China

###### Abstract

The photon Doppler velocimetry (PDV) spectrum is investigated in an attempt to
reveal the particle parameters of ejecta from shock-loaded samples in a
vacuum. A GPU-accelerated Monte-Carlo algorithm, which considers the multiple-
scattering effects of light, is applied to reconstruct the light field of the
ejecta and simulate the corresponding PDV spectrum. The influence of the
velocity profile, total area mass, and particle size of the ejecta on the
simulated spectra is discussed qualitatively. To facilitate a quantitative
discussion, a novel theoretical optical model is proposed in which the single-
scattering assumption is applied. With this model, the relationships between
the particle parameters of ejecta and the peak information of the PDV spectrum
are derived, enabling direct extraction of the particle parameters from the
PDV spectrum. The values of the ejecta parameters estimated from the
experimental spectrum are in good agreement with those measured by a
piezoelectric probe.

##  I Introduction

The strong shock wave released from the metal–vacuum/gas interface may eject a
great number of metal particles.Sollier and Lescoute (2020); Monfared  _et
al._ (2014); Asay, Mix, and Perry (1976); Speight, Harper, and Smeeton (1989);
Ogorodnikov  _et al._ (1998) Most of these particles are of micrometer-scale
in size. This phenomenon of ejecta, or microjetting, was first observed by
Kormer et al. in a plane impact experiment in the 1950s.Ogorodnikov  _et al._
(1998) And the earliest available technical report on ejecta is from research
by the Atomic Weapons Research Establishment, Aldermaston(UK).Bistow and Hyde
(1969) The physics of ejecta are understood as a special limiting case of
impulse driven Richtmyer–Meshkov.Richtmyer (1960); Meshkov (1969) In recent
decades, extensive investigations on particle ejection have been performed
because of its important role in many scientific and engineering fields,
including explosion damage,Yeager  _et al._ (2017) pyrotechnics,Held (1996)
and inertial confinement fusion.Tokheim  _et al._ (1999); Masters  _et al._
(2016) Many experimental approaches have attempted to measure the ejection
production, such as the Asay foil,Asay, Mix, and Perry (1976); Asay (1978)
foam recovery,He  _et al._ (2014) piezoelectric probes,Speight, Harper, and
Smeeton (1989); Vogan  _et al._ (2005) Fraunhofer holography,Sorenson  _et
al._ (2002, 2017) X-ray/proton radiography,Monfared  _et al._ (2014);
Hammerberg  _et al._ (2017) Mie scattering,Monfared  _et al._ (2015);
Hammerberg  _et al._ (2017), and photon Doppler velocimetry (PDV).La-Lone  _et
al._ (2015); Ogorodnikov  _et al._ (2017); Andriyash  _et al._ (2018);
Franzkowiak  _et al._ (2018a); Sun  _et al._ (2016); Fedorov, Gnutov, and
Yagovkin (2018); Kondrat’Ev, Andriyash, and Kuratov (2020) The main quantities
of interest are the particles’ velocity, diameter, and total area mass. Most
approaches can only measure some of these ejecta parameters. To reveal the
full particle field of ejecta, multiple measurement approaches must be
equipped. However, in real-world conditions, these approaches are hard to
apply simultaneously because of limitations on the experimental space or
configuration. Recently, PDV has attracted considerable attentionAndriyash
_et al._ (2018); Franzkowiak  _et al._ (2018a); Kondrat’Ev, Andriyash, and
Kuratov (2020) owing to its ability to recover the total area mass and the
distributions of particle velocity and diameter at the same time. In addition,
the light path of PDV is rather concise and its application is convenient. In
some complex experimental configurations, PDV may be the only approach that
can measure the ejecta particles.

A standard PDV setup is shown in Fig. 1. The photodetector records a mixture
of reference and backscattering light. The reference wave is in the carrier
frequency, and the backscattering wave from ejecta particles has a shifted
frequency due to the Doppler effect. The interference of the two light waves
in the photodetector leads to temporal beats of light intensity. The beat
signal consists of a large number of harmonics with different amplitudes and
phases. The heterodyne signal may change according to variations in the
particles’ position and velocity. A discrete Fourier transform is applied to
sweep the beats over time, giving a two-dimensional spectrogram on the
“frequency/velocity–time” plane. In the spectrogram, the brightness of each
point represents the corresponding spectral amplitude. The spectrogram is
composed of the integral of all particles’ scattering effects. Hence,
interpreting the spectrogram in detail remains a challenging task.

Figure 1:  Standard PDV setup. (1) Laser; (2) Reference light; (3) Incident
light; (4) Ejecta; (5) Metal plate; (6) Shock; (7) Detonation; (8)
Backscattering light; (9) Optical circulator; (10) Photodetector; (11)
Photoelectric signal; (12) PDV spectrogram; (13) Instantaneous spectrum.

There have been several studies on the interpretation of the PDV spectrum.
ButtlerButtler  _et al._ (2009, 2012) used the spectrogram boundaries to
determine the velocities of the spike and bubble of Richtmyer–Meshkov
instability in loaded metal surface. The evolution of the PDV spectrogram in a
gas environment was discussed by Sun et al.,Sun  _et al._ (2016) and the upper
boundary of the spectrogram was used to obtain the particle size by
considering aerodynamic deceleration effects. Fedorov et al.Fedorov, Gnutov,
and Yagovkin (2018) discussed the influence of different particle sizes on the
spectrogram boundary in further detail. Recently, Franzkowiak et
al.Franzkowiak  _et al._ (2018a) and Andriyash et al.Andriyash  _et al._
(2018); Kondrat’Ev, Andriyash, and Kuratov (2020) reconstructed the light
field of ejecta and obtained the simulated PDV spectrum using single- and
multiple-scattering theory, respectively. They varied the particles’
parameters and fitted the simulated PDV spectrum to the experimental data. In
this way, the particle velocity profile, diameter, and total area mass were
recovered. Andriyash et al. considered the aerodynamic deceleration effects in
a gas environment, whereas Franzkowiak et al. only discussed the case of a
vacuum.

Franzkowaik et al.Franzkowiak  _et al._ (2018a) and Andriyash et al.Andriyash
_et al._ (2018); Kondrat’Ev, Andriyash, and Kuratov (2020) proposed similar
approaches for recovering the ejecta parameters from the PDV spectrum through
reconstruction and then fitting. However, some assumptions were introduced in
the reconstruction of the light field. Franzkowaik et al. assumed that only
backscattering light was present, while Andriyash et al. set the light
scattering direction to be uniform and random in space. These assumptions
affect the accuracy of the spectrum reconstruction, and thus influence the
recovery of the ejecta parameters. The fitting model is another factor that
affects the interpretation of the PDV spectrum. Different convergence criteria
may produce different results. The quantitative relationship between the
ejecta parameters and the PDV spectrum remains unclear. Hence, it is difficult
to obtain definite ejecta parameters from the PDV spectrum. These issues
provide the motivation for the present work.

In this study, we improve the reconstruction method of the ejecta light field,
and propose a novel model for extracting the ejecta parameters directly from
the PDV spectrum. Mie theory, which gives a rigorous mathematical solution to
Maxwell’s equations, is applied to calculate the light scattering effects, and
a Monte-Carlo (MC) algorithm is used to describe the light transport process
realistically. This reconstruction method provides a high-fidelity simulation
for the PDV spectrum. The procedure is discussed in detail in Section II. The
influence of the ejecta parameters on the PDV parameters is then explored
through MC simulations in Section III.A. In Section III.B, we propose an
optical model that reveals the relationships between the PDV spectrum
characteristics and the ejecta parameters. With this model, the ejecta
parameters can be extracted directly from the PDV spectrum, instead of fitting
to experimental data. In Section III.C, the estimated ejecta parameters from
an experimental PDV spectrum are verified against those measured by a
piezoelectric probe. Finally, the conclusions to this study are presented in
Section IV.

##  II Reconstruction of PDV spectrum

###  II.1 Theoretical background

The photodetector records reference and backscattering light waves. The
scattering process of incident light is illustrated in Fig. 2. The scattering
light from the ejecta is governed by the superposition of waves propagating in
the ejection particles along different light paths $i$:

${E_{bs}}\left(t\right)=\sum\limits_{i}{{E_{i}}\left(t\right)}$ (1)

where $E_{bs}$ and $E_{i}$ are the electric vectors of total and partial
scattering waves, respectively.

The light intensity measured by the detector can be represented as:

$\begin{array}[]{l}I\left(t\right)={\left({{E_{r}}\left(t\right)+{E_{bs}}\left(t\right)}\right)^{2}}={\left({{E_{r}}\left(t\right)+\sum\limits_{i}{{E_{i}}\left(t\right)}}\right)^{2}}\\\
=E_{r}^{2}\left(t\right)+\sum\limits_{i}{E_{i}^{2}\left(t\right)}+2\sum\limits_{i}{{E_{r}}\left(t\right){E_{i}}\left(t\right)+}\sum\limits_{i\neq
j}{{E_{i}}\left(t\right){E_{j}}\left(t\right)}\end{array}$ (2)

where $E_{r}^{2}$ and $E_{i}^{2}$ denote the intensity of the reference and
scattering light signals, respectively. The third term represents the
heterodyne beats between the reference and scattering light, and the last term
represents the heterodyne beats between the different scattering light paths.
The Fourier transform of $I(t)$ is determined by the relation:

$\begin{array}[]{l}I\left(\omega\right)=\int{dt\exp\left({i\omega
t}\right)}I\left(t\right)\\\ \approx\int{dt\exp\left({i\omega
t}\right)}\left({2\sum\limits_{i}{{E_{r}}\left(t\right){E_{i}}\left(t\right)}}\right)\\\
=2{\left|{{E_{r}}}\right|}\cdot{\left.{\sum\limits_{i}{\left|{{E_{i}}}\right|}}\right|_{\omega={\omega_{i}}-{\omega_{r}}}}\end{array}$
(3)

where $\left|{{E}}\right|$ is the amplitude of the light wave field. In Eq.
(3), only the third term appearing in Eq. (2) remains. This is because the
frequencies of the first two terms are too high to be measured by the detector
and the value of the last term is much smaller than that of the third term.
$\omega_{r}$ is the carrier frequency, and $\omega_{i}$ is the Doppler-shifted
frequency, which corresponds to a sequence of scattering events along path
$i$:

${\omega_{i}}=\frac{{{\omega_{r}}}}{c}\sum\limits_{k}{\left({{\bf{n}}_{k,i}^{s}-{{\bf{n}}_{k,i}}}\right){{\bf{v}}_{k,i}}}$
(4)

where $c$ is the speed of light, ${\bf{n}}_{k,i}$ and ${\bf{n}}_{k,i}^{s}$ are
the directions of wave propagation before and after scattering by particle
$k$, and ${\bf{v}}_{k,i}$ is the velocity of this particle.

Figure 2: Multiple scattering of light waves in ejection particles.

To reconstruct the PDV spectrogram [Eq. (3)], the key is to obtain
$\left|{{E_{i}}}\right|$ and $\omega_{i}$, i.e., the detailed scattering
process in the particles. For multiple-particle systems, the scattering field
can be described by the transport equation:Ishimaru (1978); Reguigui  _et al._
(1997); Binzoni  _et al._ (2016)

$\begin{split}&\left({\bf{n}\frac{\partial}{{\partial
r}}+\sigma_{s}+\kappa}\right)I\left({{\bf r,n},t}\right)\\\
&=\int{\left\langle{\sigma_{s}p\left({\bf
n,n^{\prime}}\right)\exp\left({i{k_{0}}\left({\bf n-n^{\prime}}\right){\bf
v}t}\right)}\right\rangle I\left({{\bf r,n^{\prime}},t}\right)d\bf
n^{\prime}}\end{split}$ (5)

where $I$ is the light intensity in the field, which depends on both the
detection position $\bf r$ and the direction $\bf n$. $\sigma_{s}$ and
$\kappa$ are the coefficients of scattering and absorption, respectively.
$p\left({\bf n,n^{\prime}}\right)$ is the scattering phase function. The
right-hand side of this equation represents the contributions of scattering
light from other positions.

At the boundaries of the ejection, the light intensity has the form:

$\left\\{{\begin{array}[]{*{20}{l}}{I\left({{\bf{r}}=0,{\bf{n}}={{\bf{n}}_{\bf{0}}},t}\right)={I_{0}}}\\\
{I\left({{\bf{r}}=0,{\bf{n}}=-{{\bf{n}}_{\bf{0}}},t}\right)={I_{bs}}={{(\sum\limits_{i}{\left|{{E_{i}}}\right|}})^{2}}}\end{array}}\right.$
(6)

where $\bf n_{0}$ is the direction of incident light, which is usually
perpendicular to the free surface. $I_{0}$ and $I_{bs}$ are the intensities of
incident light and backscattering light, respectively, which correspond to the
input and output of the transport equation.

###  II.2 Monte-Carlo algorithm

Andriyash et al.Andriyash  _et al._ (2018); Kondrat’Ev, Andriyash, and Kuratov
(2020) used the discrete ordinate method to solve the transport equation [Eq.
(5)]. In this paper, a more convenient and accurate method of the MC algorithm
is applied to calculate the scattering effects.

In the MC algorithm, the incident light is assumed to be a great number of
photons. When passing through random granular media, only part of the photons
can penetrate. The proportion of permeable photons is approximated by
Beer–Lambert’s law:Bohren and Huffman (2004)

$p_{r}=\exp\left({-\tau L}\right)$ (7)

where $L$ is the thickness of the medium and $\tau$ denotes the inverse
extinction length, given by:

$\tau=N{K_{ext}}\bar{A}=\frac{{\sum\limits_{i}{\frac{\pi}{4}{d_{i}}^{2}\cdot{K_{ext}}\left({{d_{i}}}\right)}}}{{{A_{s}}L}}$
(8)

where $N$ is the number of particles per unit volume, $K_{ext}$ is the light
extinction coefficient (determined by the particle diameter, light wavelength,
and metal relative refraction index), $\bar{A}$ is the mean cross-section area
of the particles, $d_{i}$ is the diameter of particle $i$, and $A_{s}$ is the
light exposure area.

For particles in the light exposure area, the total mass has the form:

${m_{0}}{A_{s}}=\sum\limits_{i}{\frac{1}{6}\pi{d_{i}^{3}}{\rho_{0}}}$ (9)

where $m_{0}$ is the total area mass of ejection and $\rho_{0}$ is the density
of the particle material.

Combining Eqs. (8) and (9), we can rewrite Eq. (7) as:

${p_{r}}=\exp\left({-\frac{{3{m_{0}}\sum\limits_{i}{d_{i}^{2}{K_{ext}}\left({{d_{i}}}\right)}}}{{2{\rho_{0}}\sum\limits_{i}{d_{i}^{3}}}}}\right)$
(10)

The photons staying in the medium are scattered or absorbed by particles. The
probabilities of scattering and absorption are calculated by the formula:

$\left\\{\begin{array}[]{l}{p_{s}}={K_{sca}}/{K_{ext}}\\\
{p_{a}}={K_{abs}}/{K_{ext}}\end{array}\right.$ (11)

where the scattering coefficient $K_{sca}$ and the absorption coefficient
$K_{abs}$ are calculated by Mie theoryMie (1908) as:

$\left\\{\begin{array}[]{l}{K_{ext}}=\frac{2}{{{\alpha^{2}}}}\sum\limits_{n=1}^{\infty}{\left({2n+1}\right){\mathop{\rm
Re}\nolimits}\left({{a_{n}}+{b_{n}}}\right)}\\\
{K_{sca}}=\frac{2}{{{\alpha^{2}}}}\sum\limits_{n=1}^{\infty}{\left({2n+1}\right)\left({{{\left|{{a_{n}}}\right|}^{2}}+{{\left|{{b_{n}}}\right|}^{2}}}\right)}\\\
{K_{abs}}={K_{ext}}-{K_{sca}}\end{array}\right.$ (12)

where $\alpha$ is a dimensionless particle diameter parameter, $\alpha=\pi
d/\lambda$, $\lambda$ is the light wavelength, and $a_{n}$, $b_{n}$ are Mie
coefficients. It is clear that $p_{s}+p_{a}=1$.

If the photon is absorbed by particles, it will completely disappear and be
converted into the particle’s internal energy. If the photon is scattered, its
propagation direction and frequency will change, as shown in Fig. 3. The phase
function of the scattering angle $\theta$ is calculated by the formula:

$p(\theta)=\frac{{{\lambda^{2}}}}{{2\pi{K_{sca}}}}({\left|{{S_{1}}(\theta)}\right|^{2}}+{\left|{{S_{2}}(\theta)}\right|^{2}})$
(13)

where $S_{1}$ and $S_{2}$ denote the scattering intensity components in the
perpendicular and parallel directions:

$\left\\{\begin{array}[]{l}{S_{1}}\left(\theta\right)=\sum\limits_{n=1}^{\infty}{\frac{{2n+1}}{{n\left({n+1}\right)}}\left({{a_{n}}\frac{{d{P_{n}}\left({\cos\theta}\right)}}{{d\cos\theta}}+{b_{n}}\frac{{dP_{n}^{(1)}\left({\cos\theta}\right)}}{{d\theta}}}\right)}\\\
{S_{2}}\left(\theta\right)=\sum\limits_{n=1}^{\infty}{\frac{{2n+1}}{{n\left({n+1}\right)}}\left({{a_{n}}\frac{{dP_{n}^{(1)}\left({\cos\theta}\right)}}{{d\theta}}+{b_{n}}\frac{{d{P_{n}}\left({\cos\theta}\right)}}{{d\cos\theta}}}\right)}\end{array}\right.$
(14)

where $P_{n}$ and $P_{n}^{(1)}$ are Legendre and first-order associative
Legendre functions, respectively.

The incident light is assumed to be non-polarized, so the azimuth angle
$\varphi$ after scattering obeys the uniform random distribution:

$p(\varphi)=\frac{1}{2\pi}$ (15)

Figure 3: Light scattering on a particle.

After scattering, the scattering angle and azimuth angle are added to the
original light direction. The new direction cosine
$\hat{u}=[\hat{u}_{x},\hat{u}_{y},\hat{u}_{z}]$ has the form:

$\left\\{\begin{array}[]{l}{{\hat{u}}_{x}}=\frac{1}{{\sqrt{1-u_{z}^{2}}}}\sin\left(\theta\right)\left[{{u_{x}}{u_{y}}\cos\left(\varphi\right)-{u_{y}}\sin\left(\varphi\right)}\right]+{u_{x}}\cos\left(\theta\right)\\\
{{\hat{u}}_{y}}=\frac{1}{{\sqrt{1-u_{z}^{2}}}}\sin\left(\theta\right)\left[{{u_{x}}{u_{z}}\cos\left(\varphi\right)-{u_{x}}\sin\left(\varphi\right)}\right]+{u_{y}}\cos\left(\theta\right)\\\
{{\hat{u}}_{z}}=-\sin\left(\theta\right)\cos(\varphi)\sqrt{1-u_{z}^{2}}+{u_{z}}\cos\left(\theta\right)\end{array}\right.$
(16)

where $u=[u_{x},u_{y},u_{z}]$ is the original direction cosine. When
$\left|{{u_{z}}}\right|\approx 1$, the direction cosine $\hat{u}$ is
calculated by the formula:

$\left\\{\begin{array}[]{l}{{\hat{u}}_{x}}=\sin\left(\theta\right)\cos\left(\varphi\right)\\\
{{\hat{u}}_{y}}=\sin\left(\theta\right)\sin\left(\varphi\right)\\\
{{\hat{u}}_{z}}=\cos\left(\theta\right){u_{z}}/\left|{{u_{z}}}\right|\end{array}\right.$
(17)

After scattering, the frequency of the photon will have changed. The new
frequency of scattering light has the form:

$\hat{\omega}=\omega\left({1+\frac{{{\bf{\hat{v}}}-{\bf{v}}}}{c}{\bf{n}}}\right)$
(18)

where $\bf{v}$ is the velocity of the last particle that the photon left,
$\bf{\hat{v}}$ is the current particle velocity, $\bf n$ is the photon
direction, and $\omega$ is the photon frequency before scattering.

After passing through the entire ejection layer, few photons reach the free
surface. An ideal diffuse reflection is assumed for these photons. After
reflection, the space angles $\theta$ and $\varphi$ are uniformly random in
$[\pi/2,\pi]$ and $[0,2\pi]$, respectively.

Because the ejection is blocked by the free surface, eventually all the
photons are either absorbed by particles or backscattered out from the top of
the ejection. The frequency shifts of these “out” photons are summarized as
the theoretical PDV spectrogram:

$I\left(\omega\right)\propto\sum\limits_{i}{\left|{{E_{i}}}\right|}=\sum\limits_{i}{\sqrt{{I_{i}}}}={\left.{{n_{out}}}\right|_{\omega=\hat{\omega}-{\omega_{r}}}}\cdot\sqrt{\frac{{{I_{0}}}}{{{n_{0}}}}}$
(19)

where $n_{0}$ is the number of initial photons and $n_{out}$ is the
distribution of out photons in terms of their frequency.

The detailed steps of the calculation procedure are as follows:

(1) First, the initial conditions of the photons and particles are set, such
as the number and frequency of photons, and the diameter, velocity, and
position of the particles. The photons start at the top of the ejection and
then move towards the free surface.

(2) The step sizes of all photons are set to the same and equal to one
thousandth of the height of the ejecta.

(3) In one iteration, all photons take one step in the direction of their
propagation. Some photons may penetrate the current ejection layer, and the
proportion $p_{r}$ is determined by Eq. (10). For each photon, a random number
is generated in (0, 1). If the random number is less then $p_{r}$, the
corresponding photon travels over the ejection layer boundary successfully.
Otherwise, the corresponding photon is absorbed or scattered by particles in
the ejection layer.

(4) For the photons that remain in the ejection layer, we use Eq. (11) to
determine whether they are scattered or absorbed. If the photon is scattered,
the direction change is calculated by Eqs. (13)–(17). Because the phase
function of the scattering angle is very complex, an acceptance–rejection
method is applied. The new frequency of the scattered photons is determined by
Eq. (18).

(5) Overall, if the photon travels across the ejection layer boundary, its
position is updated; if the photon is scattered, its direction, frequency, and
position are updated; if the photon is absorbed, it is labeled as such and
removed from subsequent calculations.

(6) After updating the state of the photons, we check which of them have
reached the free surface or left through the top of the ejection. For all
photons that have reached the free surface, the ideal diffuse reflection is
applied. If any photons have left the ejection, they are labeled accordingly
and removed from subsequent calculations.

(7) Steps (3)–(6) are repeated until all photons have been absorbed or have
left the ejection. The frequency shifts of outgoing photons are summarized as
the spectrogram.

###  II.3 Particle models

The MC algorithm indicates that the PDV spectrum is related to the particle
size $d$, velocity $v$, position $z$, and number $N$ (i.e., total area mass
$m_{0}$). This algorithm can be applied in cases where these parameters are
completely random. In real situations, however, the particles of the ejecta
usually satisfy certain distributions in terms of velocity and
diameter.Monfared  _et al._ (2014, 2015); Durand and Soulard (2015, 2012);
Schauer  _et al._ (2017); Sorenson  _et al._ (2002, 2017); He  _et al._ (2017)
For the sake of discussion, these assumptions are applied in this paper.
Previous studiesMonfared  _et al._ (2014); Durand and Soulard (2015) indicate
that the initial velocities of particles in the ejecta can be approximated by
an exponential law:

$f\left(v\right)=\frac{{m(v)}}{{{m_{0}}}}=\frac{\beta}{{{v_{fs}}}}\exp\left[{-\beta\left({\frac{v}{{{v_{fs}}}}-1}\right)}\right]$
(20)

where $v_{fs}$ is the velocity of the free surface and $\beta$ is the velocity
distribution coefficient. Under this exponential law, most of the particles
are located in the low-velocity region, which is near the free surface.
$\beta$ determines the non-uniformity of this distribution.

In this paper, we only consider the ejecta in a vacuum environment. After
being ejected, the particles retain an almost constant velocity and the ejecta
expands in a self-similar manner over time. The particle position $z$ is only
related to its initial velocity $v$ and ejection time $t_{e}$, $z=vt_{e}$. The
corresponding PDV spectrum exhibits slight changes over time.Franzkowiak  _et
al._ (2018a); Bell  _et al._ (2017)

The particle size distribution is assumed to obey a log-normal law:Schauer
_et al._ (2017); Sorenson  _et al._ (2017)

$n\left(d\right)=\frac{1}{{\sqrt{2\pi}\sigma
d}}\exp\left({-\frac{{{{\ln}^{2}}\left({d/{d_{m}}}\right)}}{{2{\sigma^{2}}}}}\right)$
(21)

where $\sigma$ is the width of the distribution and $d_{m}$ is the median
diameter. These parameters depend on the roughness of the metal surface,
shock-induced breakout pressure, and surrounding gas properties. When $\sigma$
= 0, the function becomes a Dirac equation and all of the particles have the
same diameter. Obviously, this is the ideal situation. The particle
distribution can also be described by a power law,Durand and Soulard (2012);
Sorenson  _et al._ (2002); He  _et al._ (2017) but this description may be
invalid in the range of small particle sizes (less than 10 $\rm\mu
m$).Sorenson  _et al._ (2017) In this paper, the particle size is assumed to
be independent of its velocity.

With these assumptions, the determining factors of the PDV spectrum change to
the velocity profile coefficient $\beta$, total area mass $m_{0}$, median
diameter $d_{m}$, and size distribution width $\sigma$. The aim of this paper
is to discuss the influence of these parameters on the PDV spectrum, and to
explore how they can be extracted from the PDV spectrum most accurately.

###  II.4 Convergence and comparison

The accuracy of the MC algorithm mainly depends on the initial number of
photons. Theoretical PDV spectra with $10^{4}$, $10^{5}$, $10^{6}$, and
$10^{7}$ initial photons are shown in Fig. 4. The calculation assumes a vacuum
environment and the particles distribution assumptions are applied. In this
case, the PDV spectra have a single peak. As the initial number of photons
increases, the spectrum curves tend to be smooth. The difference between the
spectra with $10^{6}$ and $10^{7}$ initial photons is very slight. Thus,
$10^{7}$ initial photons are applied in the following calculations.

Figure 4: Theoretical PDV spectra with different initial numbers of photons.
The calculation assumes the ejection of Sn particles in a vacuum environment.
The particle velocities obey an exponential distribution $(\beta=10)$ and the
particle sizes follow a log-normal distribution $(d_{m}=1.5\;\rm{\mu
m},\sigma=0.5)$. The total area mass is $20\;\rm{mg/cm^{2}}$. $\omega_{fs}$ is
the Doppler frequency shift corresponding to the free surface velocity,
$\omega_{fs}=2\omega_{r}\cdot v_{fs}/c$. The probing wavelength is
$\lambda=1550\;\rm{nm}$.

The high initial number of photons leads to considerable computational cost.
For the case of $10^{7}$ photons, a single-core CPU requires approximately 3 h
to determine the spectrum. GPUs can be applied to accelerate the calculation.
Although the frequency of GPU processors is much lower than that of CPUs, GPUs
contain hundreds or thousands of stream processors that can work
simultaneously. The acceleration ratio of a GPU compared to a CPU is shown in
Fig. 5. The Intel Xeon W-2102 CPU (frequency 2.9 GHz) and two GPUs (Quadro
P600 and Nvidia GTX960) are applied. As the initial number of photons
increases, the acceleration ratio of the GPUs is enhanced. For the case of
$10^{7}$ photons, the acceleration ratio reaches a factor of 8 for the Quadro
P600 and a factor of 20 for the Nvidia GTX960. Because there are many judgment
events in the procedure, and the GPUs have few logical units, it is difficult
to improve the acceleration ratio with these GPUs. Thus, the Nvidia GTX960,
which requires approximately 500 s to compute each case, is used in the
following calculations.

Figure 5: Acceleration ratio of GPU calculation compared with CPU for
different initial numbers of photons. The CPU is an Intel Xeon W-2102 and its
basic frequency is 2.9 GHz. The Quadro P600 GPU has 384 stream processors; the
clock speed of each processor is about 1.3 GHz. The Nvidia GTX960 GPU has 1024
stream processors; the clock speed of each processor is about 1.1 GHz.

The PDV spectra simulated by the present procedure are compared with those
reported by Andriyash et al. and Franzkowiak et al. in Fig. 6. We use the
equivalent ejecta area mass and particle size instead of the transport optical
thickness used by Andriyash et al. With the uniform scattering assumption, our
simulation (Case 2) is almost the same as that of Andriyash et al. (Case 3),
which validates the adequacy of the present numerical method. However, when
the Mie scattering theory is applied, there is a remarkable difference between
the present procedure (Case 1) and the results of Andriyash et al. (Case 3)
and Franzkowiak et al. (Case 4). The difference with Andriyash et al. is
mainly in the low-velocity part. This is because the change in the scattering
phase function has a great influence on the multiple scattering, which is the
main form of scattering in the low-velocity dense part. The difference with
Franzkowiak et al. is in the location of the spectrum peak. Franzkowiak et al.
applied the single-scattering theory and assumed that all of the light
scattered backward. This implies that the optical thickness is overestimated,
and so little light would reach the deep region of the ejecta. Thus, the
spectrum moves toward high velocities. These differences in spectra indicate
that the scattering assumption may introduce some reconstruction inaccuracy
that cannot be neglected.

Figure 6: Comparison of PDV spectra calculated by different reconstruction
methods. Case 1: MC + Mie scattering theory (proposed procedure); Case 2: MC +
uniform scattering assumption; Case 3: Discrete coordinates + uniform
scattering assumption (Andriyash et al.); Case 4: Single scattering theory
(Franzkowiak et al.) . The data for Case 3 were extracted directly from the
paper of Andriyash et al. The calculations were carried out for a transport
scattering thickness of $\tau_{tr}=10$, which corresponds to
$m_{0}=10.7\;{\rm{mg/cm^{2}}},d_{m}=1.5\;\rm{\mu m}$, and $\sigma=0.5$. The
material is Sn and the ejection velocity profile has $\beta=8$.

##  III Interpretation of PDV spectrum

###  III.1 Influence of ejecta parameters

The results of numerical calculations that demonstrate the sensitivity of the
PDV spectrum to changes in the ejecta parameters ($\beta$, $m_{0}$, $d_{m}$,
and $\sigma$) are presented in Figs. 7–10. The PDV spectra were simulated
using the MC algorithm described in the previous section for Sn particles in a
vacuum environment. The initial ejecta parameters were set to $\beta=10$,
$m_{0}=20\;\rm{mg/cm^{2}}$, $d_{m}=1.5\;\rm{\mu m}$, and $\sigma=0.5$. In each
figure, one of the parameters changes and the others remain constant.

The simulated PDV spectra with different values of the velocity coefficient
$\beta$ are shown in Fig. 7. With an increase in $\beta$, the spectrum peak
moves towards the low velocities and its magnitude decreases. Furthermore, the
spectrum shape becomes sharper and the high-velocity part of the spectrum
becomes invisible. The coefficient $\beta$ determines the distribution of
particles in the ejecta. With larger $\beta$, fewer particles are located at
the top of the ejecta and the incident light can penetrate deeper. This
results in the movement of the spectrum peak and a decrease in the
observability of high-velocity particles.

Figure 7: Simulated PDV spectra with different velocity coefficients. The
calculation was carried out for Sn particles in a vacuum environment. The
total area mass was $20\;\rm mg/cm^{2}$. The log-normal distribution
$(d_{m}=1.5\;{\rm\mu m},\sigma=0.5)$ was applied to the particle sizes.

The simulated PDV spectra with different values of the total area mass $m_{0}$
are shown in Fig. 8. The changes in the spectra can be divided into two
sections. When $m_{0}\geq 10\;\rm mg/cm^{2}$, the spectrum displays a single
peak. With a decrease in $m_{0}$, this peak moves to the left, and its
magnitude and slope exhibit slight changes. When $m_{0}\leq 5\;\rm mg/cm^{2}$,
a new peak appears around the free surface, and the spectrum exhibits a
double-peak shape. A smaller area mass produces a more remarkable new peak.
For $m_{0}=2\;\rm mg/cm^{2}$, the original peak disappears and the spectrum
again exhibits a single peak. The double-peak spectrum has been observed in
previous experimentsLa-Lone  _et al._ (2015); Andriyash  _et al._ (2020) and
simulations,Kondrat’Ev, Andriyash, and Kuratov (2020); Franzkowiak  _et al._
(2018b) and is the result of the direct exposure of incident light at the free
surface.

Figure 8: Simulated PDV spectra with different total area mass. The area mass
unit is $\rm{mg/cm^{2}}$. The calculation was carried out for Sn particles in
a vacuum environment. The velocity coefficient $\beta=10$ and the size
coefficients $d_{m}=1.5\;\rm{\mu m},\sigma=0.5$.

There are two parameters that determine the particle size distribution—the
median diameter $d_{m}$ and the distribution width $\sigma$. Their influence
on the PDV spectrum is illustrated in Figs. 9 and 10, respectively. $d_{m}$
and $\sigma$ exhibit similar effects: as $d_{m}$ or $\sigma$ increases, the
original peak of the spectrum moves towards the low velocities and the peak
value decreases. A new peak then appears in the position of the free surface
and the original peak gradually attenuates. This change in the form of the
spectrum peak is similar to that for the area mass.

Figure 9: Simulated PDV spectra with different particle median diameters. The
diameter unit is $\rm{\mu m}$. The calculation was carried out for Sn
particles in a vacuum environment. The total area mass was
$20\;\rm{mg/cm^{2}}$. The velocity coefficient $\beta=10\;\rm{mg/cm^{2}}$ and
the size coefficient $\sigma=0.5$. Figure 10: Simulated PDV spectra with
different particle size coefficients $\sigma$. The calculation was carried out
for Sn particles in a vacuum environment. The total area mass was
$20\;\rm{mg/cm^{2}}$. The velocity coefficient $\beta=10$ and the median
diameter $d_{m}=1.5\;\rm\mu m$.

###  III.2 Theoretical optical model

The simulations described above using the MC algorithm provide a qualitative
understanding of the influence of the ejecta parameters on the PDV spectrum.
However, how to solve the reverse problem, i.e., extracting the ejecta
parameters from the PDV spectrum, remains unclear. To obtain the quantitative
relationships between the ejecta parameters and the characteristics of the PDV
spectrum, we introduce the single-scattering theory. In this theory, the light
is assumed to be scattered only once, and the scattering direction is always
backward. With this assumption, the light direction is always parallel to the
motion of the particles. The frequency shift of the light is proportional to
the particle velocity, $\omega=2\omega_{0}\cdot v/c$. The PDV spectrum can be
expressed in terms of velocity, $I(\tilde{v})$.

The above calculations have shown that the single scattering leads to an
overestimation of the optical thickness. Here, we assume that the extinction
process of particles only includes the backscattering and absorption effects,
and the forward scattering is ignored. With this assumption, the PDV spectrum
is calculated by the formula:

$\begin{split}I\left(\omega\right)&=I\left({\tilde{v}}\right)\propto\sum\limits_{i}^{\tilde{v}<v<\tilde{v}+dv}{\sqrt{{I_{back,i}}}}\\\
&=\sum\limits_{i}^{\tilde{v}<v<\tilde{v}+dv}{\sqrt{\frac{{{I_{0}}}}{{{A_{s}}}}{A_{back,i}}\left(v\right)\cdot\exp\left({-2\int_{\tilde{v}}^{\infty}{\tau\left(v\right)dv}}\right)}}\\\
&\propto\sum\limits_{i}^{\tilde{v}<v<\tilde{v}+dv}{\sqrt{{A_{back,i}}\left(v\right)}}\cdot\sqrt{\exp\left({-2\int_{\tilde{v}}^{\infty}{\tau\left(v\right)dv}}\right)}\end{split}$
(22)

where $\tilde{v}$ is the velocity corresponding to the frequency shift,
$\tilde{v}=2\omega c/\omega_{r}$, and $A_{back}$ is the backscattering cross-
section area of particles with velocity $\tilde{v}$:

$\begin{split}&\sum\limits_{i}^{\tilde{v}<v<\tilde{v}+dv}{\sqrt{{A_{back,i}}\left(v\right)}}=\sum\limits_{i}^{\tilde{v}<v<\tilde{v}+dv}{\sqrt{\frac{1}{4}\pi
d_{i}^{2}{K_{back}}}}\\\
&=N\left({\tilde{v}}\right)\frac{1}{2}\bar{d}\sqrt{\pi{K_{back}}}=\frac{{m\left({\tilde{v}}\right){A_{s}}}}{{\frac{1}{6}\pi\overline{{d^{3}}}{\rho_{0}}}}\frac{1}{2}\bar{d}\sqrt{\pi{K_{back}}}\\\
&=\frac{{3{A_{s}}\sqrt{{K_{back}}}}}{{\sqrt{\pi}{\rho_{0}}}}\frac{{{m_{0}}}}{{\overline{{d^{3}}}/\overline{d}}}f(\tilde{v})\end{split}$
(23)

where $\overline{d}$ is the average particle diameter and $K_{back}$ is the
backscattering coefficient, which is calculated by Mie theory.

The exponent in Eq. (22) denotes the extinction effects, where the coefficient
2 signifies the back and forth of light in the ejection process. The integral
represents the contributions from the extinction of particles above the layer
of velocity $\tilde{v}$:

$\begin{split}\int_{\tilde{v}}^{\infty}{\tau\left(v\right)}dv&=\frac{{\sum\limits_{i}^{v>\tilde{v}}{\frac{\pi}{4}d_{i}^{2}{K^{*}_{ext}}}}}{{{A_{s}}}}=\frac{{\int_{\tilde{v}}^{\infty}{N\left(v\right)}dv\cdot\frac{\pi}{4}\overline{{d^{2}}}{K^{*}_{ext}}}}{{{A_{s}}}}\\\
&=\frac{{3{K^{*}_{ext}}}}{{2{\rho_{0}}}}\frac{{\int_{\tilde{v}}^{\infty}{m\left(v\right)}dv}}{{\overline{{d^{3}}}/\overline{{d^{2}}}}}\end{split}$
(24)

where the equivalent extinction coefficient $K^{*}_{ext}$ only considers the
backscattering and absorption effects:

$K^{*}_{ext}=K_{back}+K_{abs}=g_{back}K_{ext}$ (25)

where the coefficient $g_{back}$ is 0.5–0.7 for particle diameters of 1–10
$\rm\mu m$. When $g_{back}$ = 1, the present model reduces to that of
Franzkowiak et al.

Because the velocity profile is exponential,
${\int_{\tilde{v}}^{\infty}{m\left(v\right)}dv}=m_{0}f(\tilde{v})v_{fs}/\beta$.
Equation (25) has the form:

$\int_{\tilde{v}}^{\infty}{\tau\left(v\right)}dv=\frac{{3{K^{*}_{ext}}}}{{2{\rho_{0}}}}\frac{{{m_{0}}}}{{\overline{{d^{3}}}/\overline{{d^{2}}}}}\frac{v_{fs}}{{{\beta}}}f(v)$
(26)

When $\tilde{v}$ is equal to the free surface velocity, the integral
represents the amount of light that is able to reach the free surface. The
optical thickness of the ejecta is defined as:

${\tau_{0}}=\int_{{v_{fs}}}^{\infty}{\tau\left(v\right)}dv=\frac{{3{K^{*}_{ext}}}}{{2{\rho_{0}}}}\frac{{{m_{0}}}}{{\overline{{d^{3}}}/\overline{{d^{2}}}}}$
(27)

Combining Eqs. (23), (26), and (27), we can write Eq. (22) as:

$I\left({\tilde{v}}\right)\propto\frac{{\sqrt{{K_{back}}}}}{{K_{ext}^{*}}}\cdot\frac{{\overline{d}}}{{\overline{{d^{2}}}}}\cdot{\tau_{0}}f\left({\tilde{v}}\right)\exp\left({-\frac{{{v_{fs}}}}{\beta}{\tau_{0}}f\left({\tilde{v}}\right)}\right)$
(28)

This equation provides the theoretical form for determining the PDV spectrum
from the ejecta parameters in a vacuum environment. In this formula, the PDV
spectrum is proportional to the product of the velocity profile and the
extinction term. These two parts are illustrated in Fig. 11. As the velocity
decreases, the corresponding ejecta position moves closer to the free surface,
and the incident light becomes weaker because of particle extinction. However,
the particles become dense deeper within the ejecta, and this enlarges the
cross-section area of scattering. With the contribution of these two parts,
the PDV spectrum exhibits a single peak shape.

Figure 11: Extinction ratio and particle distribution with respect to
velocity.

The simulated PDV spectra of the present model are compared with those of the
MC algorithm in Fig. 12. With the correction of the backscattering
coefficient, there is a good agreement between the results, both in the main
peak position and the curve shape. However, in the case of a small ejecta mass
($m_{0}=5\;\rm mg/cm^{2}$), the present model cannot simulate the peak around
the free surface. Although the spectrum is nonzero at the position of the free
surface in this model, neglecting the multiple-scattering results prevents the
second peak from appearing.Kondrat’Ev, Andriyash, and Kuratov (2020);
Franzkowiak  _et al._ (2018b) In this paper, we mainly consider the
information supplied by the original peak of the PDV spectrum. Thus, this
defect has only a very slight influence on the accuracy of the present model.

Figure 12: Simulated PDV spectrum by MC algorithm and single-scattering (SS)
model. The area mass unit is $\rm mg/cm^{2}$. The particle settings are
$\beta=10$, $d_{m}=1.5\;\rm\mu m$, $\sigma=0.5$. The backscattering
coefficient is $g_{back}=0.67$.

We now analyze the spectrum function [Eq. (28)]. First, we take its
derivative:

$\begin{split}I^{\prime}\left({\tilde{v}}\right)&\propto{\tau_{0}}f^{\prime}\left({\tilde{v}}\right)\exp\left({-\frac{{{v_{fs}}}}{\beta}{\tau_{0}}f\left({\tilde{v}}\right)}\right)\left({1-\frac{{{v_{fs}}}}{\beta}{\tau_{0}}f\left({\tilde{v}}\right)}\right)\\\
&=-\frac{\beta}{{{v_{fs}}}}I\left({\tilde{v}}\right)\left({1-\frac{{{v_{fs}}}}{\beta}{\tau_{0}}f\left({\tilde{v}}\right)}\right)\end{split}$
(29)

where $f^{\prime}(\tilde{v})=-\beta/v_{fs}f(\tilde{v})$.

When $I^{\prime}(\tilde{v})=0$, the solution provides the position of the
spectrum peak:

$1-\frac{{{v_{fs}}}}{\beta}{\tau_{0}}f\left({{{\tilde{v}}_{peak}}}\right)=0$
(30) ${{\tilde{v}}_{peak}}/{v_{fs}}=1+\ln\left({{\tau_{0}}}\right)/\beta$ (31)

The peak value of the PDV spectrum is:

$I\left({{{\tilde{v}}_{peak}}}\right)\propto\frac{{\sqrt{{K_{back}}}}}{{K_{ext}^{*}}}\cdot\frac{{\bar{d}}}{{\overline{{d^{2}}}}}\cdot\frac{\beta}{{{v_{fs}}}}{e^{-1}}\propto\frac{{\beta}}{{\overline{{d^{2}}}/\overline{{d}}}}$
(32)

Finally, the relative curvature at the spectrum peak is given by the second
derivative of the spectrum function:

$\begin{split}I^{\prime\prime}\left({\tilde{v}}\right)/I\left({{{\tilde{v}}_{peak}}}\right)=\frac{{{v_{fs}}}}{\beta}e{\left({{\tau_{0}}f\left({\tilde{v}}\right)\exp\left({-\frac{{{v_{fs}}}}{\beta}{\tau_{0}}f\left({\tilde{v}}\right)}\right)}\right)^{\prime\prime}}\end{split}$
(33)

$I^{\prime\prime}\left({{{\tilde{v}}_{peak}}}\right)/I\left({{{\tilde{v}}_{peak}}}\right)=-{\left({\frac{\beta}{{{v_{fs}}}}}\right)^{2}}$
(34)

In summary, the relationships between the ejecta parameters and the
characteristics of the PDV spectrum have the form:

$\left\\{\begin{aligned}
&{{\tilde{v}}_{peak}}/{v_{fs}}=1+\ln\left({{\tau_{0}}}\right)/\beta\\\
&I\left({{{\tilde{v}}_{peak}}/{v_{fs}}}\right)\propto\frac{\beta}{{\overline{{d^{2}}}/\overline{d}}}\\\
&I^{\prime\prime}\left({{{\tilde{v}}_{peak}}/{v_{fs}}}\right)/I\left({{{\tilde{v}}_{peak}}/{v_{fs}}}\right)=-{\beta^{2}}\end{aligned}\right.$
(35)

where the independent variable is normalized by the velocity of the free
surface, $v_{fs}$.

The above relationships allow some of the ejecta parameters to be determined.
First, the velocity profile coefficient $\beta$ is given by the relative
curvature at the spectrum peak. In addition, the surface mean diameter
$d^{*}=\overline{d}^{2}/\overline{d}=d_{m}e^{0.5\sigma^{2}}$ can be derived
from the value of the spectrum peak. Finally, the optical thickness of the
ejecta $\tau_{0}$ is obtained from the position of the spectrum peak, where
$\tau_{0}$ is related to the ejecta area mass $m_{0}$ and the Sauter mean
diameter $d_{s}=\overline{d^{3}}/\overline{d^{2}}=d_{m}e^{2.5\sigma^{2}}$. If
$d_{s}$ is assumed to be approximately $d^{*}$, the area mass $m_{0}$ can be
determined.

Figures 13–15 compare these theoretical relationships with the MC simulation
results. Two optical thickness and a large range of velocity profiles are
considered. It can be observed that the theoretical relationships largely
conform to the MC simulations in these cases, which verifies the present model
to some extent.

Figure 13: Peak positions of PDV spectrum calculated by MC algorithm and
theoretical formula. Two optical thickness are considered, where the square
points denote $\tau_{0}=23.4$ and the circular points denote $\tau_{0}=8.6$.
The corresponding ejecta parameters are $m_{0}=20\;{\rm
mg/cm^{2}},d_{m}=1.5\;\mu{\rm m},\sigma=0.5$ and $m_{0}=10\;{\rm
mg/cm^{2}},d_{m}=2.0\;\mu{\rm m},\sigma=0.5$, respectively.  Figure 14: Peak
values of PDV spectrum calculated by MC algorithm and theoretical formula. Two
optical thickness are considered, where the square points denote
$\tau_{0}=23.4$ and the circular points denote $\tau_{0}=8.6$. The
corresponding ejecta parameters are $m_{0}=20\;{\rm
mg/cm^{2}},d_{m}=1.5\;\mu{\rm m},\sigma=0.5$ and $m_{0}=10\;{\rm
mg/cm^{2}},d_{m}=2.0\;\mu{\rm m},\sigma=0.5$, respectively.  Figure 15:
Relative curvature of PDV spectrum at the peak calculated by MC algorithm and
theoretical formula. Two optical thickness are considered, where the square
points denote $\tau_{0}=23.4$ and the circular points denote $\tau_{0}=8.6$.
The corresponding ejecta parameters are $m_{0}=20\;{\rm
mg/cm^{2}},d_{m}=1.5\;\mu{\rm m},\sigma=0.5$ and $m_{0}=10\;{\rm
mg/cm^{2}},d_{m}=2.0\;\mu{\rm m},\sigma=0.5$, respectively.

###  III.3 Experimental verification

In the above relationships, the peak value of the spectrum is difficult to use
in the analysis of PDV experiments. In the experiments, the PDV spectrum is
scaled by the reference light intensity, probe reception, photoelectric
conversion efficiency, and circuit amplification factor, among other factors.
Additional PDV experiments are required to calibrate this scaled factor. For a
single PDV vacuum experiment, only the velocity profile $\beta$ and optical
thickness $\tau_{0}$ of the ejecta can be extracted. If there is an additional
particle granularity measurement, the ejecta area mass $m_{0}$ can also be
determined.

A set of ejecta PDV experiments performed by Franzkowiak et al.Franzkowiak
_et al._ (2018a) was used to verify the present theoretical model. The
experiment was carried out in a vacuum environment using Sn material with the
surface machined into $60\times 8\;\rm\mu m$ grooves. The shock-induced
breakout pressure was $P_{SB}=28\;\rm GPa$. The velocity of the free surface
was found to be approximately $2013\;\rm m/s$. We extracted the PDV spectrum
from the experimental spectrogram over the period $0.2-0.8\;\rm\mu s$, as
shown in Fig. 16(a). The PDV data were then averaged and smoothed using the
low-pass filtering of the fast Fourier transform. We converted the spectrum
units [dBm] to volts and then took the second derivative to give the smoothed
PDV spectrum shown in Fig. 16(b). The peak of this spectrum is located at
$\tilde{v}/v_{fs}=1.32$ and the corresponding relative curvature is
approximately $-110$. Combined with Eq. (35), this suggests a velocity profile
coefficient of $\beta=10.5$ and an optical thickness of $\tau_{0}=28.79$.

Schauer et al.Schauer  _et al._ (2017) conducted a Mie-scattering experiment
with similar conditions, where the surface roughness was $50\times 8\;\mu m$
and the breakout pressure was about $30\;\rm GPa$. The particle size
distribution was measured to be $d_{m}=0.6\;\mu{\rm{m}},\sigma=0.5$. Using
this data, the total area mass was determined to be
$m_{0}=7.5\;{\rm{mg/cm^{2}}}$.

In their PDV experiment, Franzkowiak et al. simultaneously measured the area
mass with respect to velocity using a piezoelectric probe. The PDV spectrum
and area mass given by our estimations and their experiments are compared in
Figs. 17 and 18, respectively. These two results are in good agreement, which
verifies the present theoretical model.

Figure 16: (a) PDV spectrogram extracted from ejecta experiment of Franzkowiak
et al. Franzkowiak  _et al._ (2018a) and (b) second derivative of the smooth
data. Figure 17: Comparison of the experimental and simulated PDV spectra.
Figure 18: Comparison of the area mass and velocity profile between the
piezoelectric probe measurement Franzkowiak  _et al._ (2018a) and our
estimation.

##  IV Conclusion

This paper has discussed the PDV spectrum of ejecta particles from shock-
loaded samples in a vacuum. A GPU-accelerated MC algorithm that rebuilds the
PDV spectrum for the ejecta particles has been proposed, and Mie theory was
applied to describe the scattering process. Compared with the reconstruction
methods of Andriyash et al. and Franzkowiak et al., a reasonable scattering
model is the key to simulating the PDV spectrum accurately. The simulations
using the MC algorithm indicate that the particle velocity profile, particle
size, and ejecta area mass have a significant influence on the shape and
values of the PDV spectrum. As the velocity profile coefficient or particle
size increases, the spectrum peak moves to lower velocities. However, this
change in the spectrum peak is reversed for the total area mass. In addition,
for small values of the optical thickness (few ejecta mass or large particle
size), a new spectrum peak appears near the free surface and the original peak
gradually decreases or even disappears. For a quantitative analysis, a
corrected single-scattering model was proposed for deriving the relationships
between the ejecta parameters and the characteristics of the PDV spectrum. It
was found that the relative curvature of the spectrum peak is equal to the
square of the velocity profile coefficient $\beta$. The peak value of the
spectrum is proportional to the ratio of $\beta$ to the particle size $d$, and
the peak position of the spectrum is related to $\beta$ and the total
extinction coefficient $\tau_{0}$, where $\tau_{0}$ is calculated from the
total area mass $m_{0}$ and the particle size $d$. Thus, the ejecta parameters
$\beta$, $d$, and $m_{0}$ can be resolved using information about the spectrum
peak. However, the spectrum is scaled by multiple experimental parameters, and
the relationship with the particle size is difficult to determine. For a
single PDV spectrum, only the velocity profile and optical thickness can be
determined. Finally, the theoretical interpretation was found to be in good
agreement with the MC simulations and PDV experiments of ejecta in a vacuum
environment.

The present theoretical model does not consider the multiple scattering near
the free surface. When the optical thickness is sufficiently small and the
original peak disappears, the present model may be invalid. How to determine
the particle size in the PDV experiment is another unsolved issue. In a gas
environment, the particles slow down because of aerodynamic deceleration, and
this introduces a series of changes to the PDV spectrum over time. The
particle deceleration is related to the particle size. In future work, the PDV
spectrum in a gas environment will be discussed in an attempt to recover more
comprehensive quantities of the ejecta.

###### Acknowledgements.

This work was supported by a joint fund from the National Natural Science
Foundation of China (Grant Nos. 11902043, 11772065) and the Science Challenge
Project (Grant No. TZ2016001).

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# Quantitative evaluation of laser-induced fluorescence in magnetized plasma
accounting for disalignment effect

Roman Bergert, Slobodan Mitic, Markus H. Thoma  Institute of Experimental
Physics I, Justus-Liebig-University Giessen Heinrich-Buff-Ring 16, D-35392
Giessen, Germany<EMAIL_ADDRESS>
###### Abstract

Quantitative evaluation of tunable diode laser induced fluorescence (TDLIF)
measurements in magnetized plasma take into account Zeeman splitting of
energetic levels and intra-multiplet mixing defining the density distribution
(alignment) of excited $\rm 2p_{8}$ multiplet is discussed in this paper.
TDLIF measurements were used to evaluate light-transport properties in a
strongly magnetized optically thick argon plasma under different pressure
conditions. Therefore, a coupled system of rate balance equations were
constructed to describe laser pumping of individual magnetic sub-levels of
$\rm 2p_{8}$ state through frequency separated sub-transitions originating
from $\rm 1s_{4}$ magnetic sub-levels.

The density distribution of $\rm 2p_{8}$ multiplet was described by balancing
laser pumping with losses including radiative decay, transfer of excitation
between the neighboring multiplets driven by neutral collisions and quenching
due to electron and neutral collisions. Resulting $\rm 2p_{8}$ magnetic sub-
level densities were then used to model polarization dependent fluorescence,
consider self-absorption, which could be directly compared with measured
polarization resolved TDLIF measurements. This enables to obtain unique
solutions for the $\rm 1s_{4}$ and $\rm 1s_{5}$ magnetic sub-level densities
which were in good agreement with the densities obtained by laser absorption
measurements. It is shown that LIF measurements in magnetized plasma
conditions have strong pressure dependence that should be corrected consider
effective disalignment rate. The presented measurement method and model can
help further understanding and improve description of optical emission of
argon in magnetized conditions.

Keywords: magnetized plasma diagnostics, plasma jet, dielectric barrier
discharge (DBD), low-pressure plasma, laser-induced absorption spectroscopy
(LAS), magnetized plasma, magnetic sub-level population, argon plasma, laser-
induced fluorescence (LIF)

##  1 Introduction

Influence of an external magnetic field $B$ changes the plasma behavior by
influencing movement of a charged particles by the Lorentz force and changing
energetic structures of a plasma components by the Zeeman effect [1] which
overall influence the plasma properties. The presence of an external magnetic
field in plasma induces anisotropy in motion of a charged particles by
reducing cross-field diffusion, while also inducing strong anisotropy in
plasma optical emission spectra. Total emitted line intensity is therefore
artificial construct of two polarized components orientated parallel
($\pi$-component) or perpendicular ($\sigma\pm$-component) to the magnetic
field vector. Due to such complex effects of magnetic field on plasma
components, standard (non-magnetized) description of plasma emission spectra
and light transport properties are not suitable. The complexity of optical
properties mainly comes from the splitting of energy levels. The total angular
momentum quantum number $J$ of an energetic level will result in $2J+1$
magnetic sub-levels , each sub- level is described by the magnetic quantum
number $m$, in the presence of a magnetic field compared to the one single
level without a magnetic field. This changes the emission into a system of
sub-transitions symmetrically redistributed around the initial unshifted line
center. As a result, each spectral transition will have to be rewritten as a
sum of optically allowed transitions between corresponding upper and lower
magnetic sub-levels [2]. The degeneracy of the levels vanishes under higher
external magnetic fields so that an individual description of transitions
between magnetic sub-levels is necessary [3]. In optically thick plasma,
density distribution of $\rm 1s$ sub-states should be taken into account for
correct description of the self-absorption effect and thus the light transport
[4].

Description of polarized plasma spectroscopy has been introduced by Fujimoto
and colleagues [3, 5, 6, 7] who presented a population-alignment collisional-
radiative model (PACRM) based on collisional and radiative interactions
between the multiplets and intra-multiplet transitions. Such approach has been
demonstrated on the experiments where polarization of plasma optical emission
spectroscopy (OES) was induced by highly anisotropic electron energy
distribution function (electron beam), mostly in helium or hydrogen, while
only limited number of reports considered argon. Also such description is
highly non-trivial accounting for less available collisional cross-sections
between different multiplets (ground, 1s and 2p states). The possibilities of
PACRM for helium has been demonstrated in articles deducing highly interesting
plasma properties, such as an anisotropy in the electron energy distribution
function based on the modelling of the polarities of a plasma emission lines
[8, 9].

The main goal of this work is to evaluate optical emission properties in
magnetized argon plasma accounting for Zeeman splitting of radiating ($\rm
2p_{8}$) and $\rm 1s$ stats, density distribution between magnetic sub-levels
(alignment) and polarization of plasma emission. Therefore, laser absorption
spectroscopy (LAS) and laser induced fluorescence (LIF) were used as
diagnostic tools. Description of LAS and evaluation of argon $\rm 1s_{4}$ and
$\rm 1s_{5}$ sub-level state densities has been introduced in a recent article
[4]. Based on the sub-level densities the self-absorption (SA) for different
sub-transitions was discussed. Self-absorption was included in the
interpretation of linear polarized and circular polarized TDLIF at 842 nm and
801 nm. Optical sub-transitions between $\rm 1s_{4}$ and $\rm 2p_{8}$ sub-
states were used to pump different magnetic sub-level of $\rm 2p_{8}$.

A model describing the polarization dependent fluorescence was developed
including self-absorption and sub-level mixing (disalignment) due to neutral
collisions which strongly influence the ratio between the two polarities of
the fluorescence spectral lines. Such effect changes the description of 2p
state densities by introducing disalignment as additional production and
losses mechanism competing with radiative decay. The correct analytical
description will result in quite strong coupling between involved states
densities through disalignment and self-absortion so that LIF measurements can
be uniquely quantitatively solved resulting in estimations of the 1s magnetic
sub-levels densities and disalignment constant. The disalignment constant has
been estimated in [10] where in the presence of weak magnetic field, time
dependent polarized LIF measurements has been modeled. However, in most of
such experiments, broad and powerful lasers were used for inducing excitation
followed by modeling of polarized emission. Another interesting aspect of
disalignment constant is the temperature dependence which can cause quite
opposite effect depending on the quenching atoms [10, 11]. In the reported
case, krypton atoms reduced disaligment with temperature increase while for
helium, neon and argon disalignment constant was increased. Such findings are
illustrating the complexity of the effect and quite interesting non-linear
response of different argon 2p levels. Although such unique optical properties
for magnetized plasma conditions has been investigated since 1960’s using
highly complicated PACRM formalism, strong and clearly visible effect of
neutral gas density on plasma spectra and its practical applications has been
poorly conveyed to the plasma spectroscopy audience. Therefore, we present a
model for the quantitative evaluation of LIF measurements using tunable diode
laser, carried out on an compact dielectric barrier discharge low pressure
plasma in the presence of an external magnetic field of 0.3 T.

##  2 Methods

Correct interpretation of optical transitions (absorption and induced
fluorescence) in magnetized plasma conditions is based on line splitting
formalism due to Zeeman effect (resulting in polarization) and its further
implications in light transport properties under optically thick conditions.
Direct detection of individual induced fluorescence profiles were done with a
spectrometer, where a slow laser scan over a broad frequency range was used
for pumping frequency shifted sub-transitions. Additionally, laser absorption
spectroscopy of the $\rm 1s_{4}$ state was used as a reference for the sub-
level densities evaluated from TDLIF model.

Under the influence of the magnetic field, each energetic level will split
into the 2J+1 multiplet producing system of transitions described by the
transition rules. In a strong magnetic field, sub-transitions will be shifted
in frequency producing absorption structure symmetric around unshifted line
center. The observed energetic structure and system of transitions of 801 and
842 nm in argon are represented by a Kastler diagram as shown in 1. The
allowed electric dipole transitions originating from $\rm 2p_{8}$ to $\rm
1s_{4}$ and $\rm 1s_{5}$ are shown by their polarization. The $\rm 2p_{8}$ and
$\rm 1s_{5}$ state splits into 5 sub-levels. Hence, 12 different electric
dipole transitions are formed (4 $\sigma+$, 4 $\pi$ and 4 $\sigma-$), which is
indicated with red arrows between the sub-levels of $\rm 2p_{8}$ and $\rm
1s_{5}$. Nine sub-transitions can be found between $\rm 2p_{8}$ and $\rm
1s_{4}$ (3 $\sigma+$, 3 $\pi$ and 3 $\sigma-$), which is indicated with blue
arrows in figure 1.

Figure 1: Kastler diagram indicating possible optical transitions under the
influence of an external magnetic field. ($\pi$ solid lines, $\sigma-$ dashed
lines and $\sigma+$ dotted lines) Additionally, the disalignment effect with
the disalignment constant $k_{\rm dis}$ between the adjacent sub-levels of
$\rm 2p_{8}$ is indicated with solid black arrows.

###  2.1 Tunable diode laser absorption spectroscopy

The description of LAS, polarization dependent sub-level transitions and
resulting self-absorption has been described in our previous work [4] and it
will be used as a basis for LIF interpretation. Therefore, just a short
overview will be given on LAS and description of magnetic sub-transitions
rates since this has a direct impact on optical properties like self-
absorption and light transport. Based on the reconstructed absorption profile
of individual sub-transition $\kappa(\nu)$ the magnetic sub-level density
$n_{m}{{}_{j}}$ of targeted 1s states can be calculated according to:

$n_{m}{{}_{j}}=\frac{8\pi}{c^{2}}\frac{\nu_{m_{i},m_{j}}^{2}}{A_{J_{i}J_{j},m_{i}m_{j}}}\kappa(\nu)$
(1)

where $c$ describes the vacuum speed of light, $\nu_{m_{i},m_{j}}$ is a
central wavelength of the (shifted) transition dependent on the external
magnetic field strength, and the Einstein coefficient of transition
$A_{J_{i},J_{j},m_{i},m_{j}}$ between upper $\rm 2p_{8}$ $m_{i}$ and lower
$\rm 1s_{4}$ or $\rm 1s_{5}$ magnetic sub-level states $m_{j}$. The expression
for the Einstein coefficients describing transitions between upper and lower
magnetic sub-levels can be written as [12, 13]:

$\begin{split}A_{J_{i},J_{j},m_{i},m_{j}}=\left(2J_{i}+1\right)A_{ij}\times
W\\\ \ W=\left|\left(\begin{matrix}J_{j}&q&J_{i}\\\ -m_{j}&\Delta
m_{j,i}&m_{i}\end{matrix}\right)\right|^{2}\end{split}$ (2)

where W represents the Wigner 3-j coefficient to the power of two with the
multi-polarity of the transition $q$ equivalent to 1 for electrical dipole
transitions, and difference in the magnetic quantum number of the transition
$\Delta m_{j,i}=m_{j}-m_{i}$. $A_{ij}$ describes the Einstein coefficient
(life time) for spontaneous decay of the observed emission in unmagnetized
case.

###  2.2 Tunable diode laser induced fluorescence in magnetized plasma

The laser induced fluorescence measurements were based on the same laser
excitation scheme consecutively pumping different $\rm 2p_{8}$ sub-levels
through 842 nm sub-transitions separated by frequency. The efficiency of laser
pumping is therefore proportional to the laser intensity, density of the
targeted $\rm 1s_{4}$ $m_{j}$-multiplet and Einstein coefficient of sub-
transition assuming a linear regime for laser pumping. Each transition
interacts only with corresponding polarity from the un-polarized pumping laser
so that laser intensities for different polarization should scale as
$I_{\pi}=2I_{\sigma}$.

Pumping of an individual $\rm 2p_{8}$ sub-level would lead to fluorescence of
the light emission with polarization determined by the ratio of Einstein
coefficients of $\pi$ and $\sigma$ component originating from the pumped level
and corrected for self-absorption.

However under the observed conditions (magnetically separated transitions) it
is evident that the polarization of the induced fluorescence is strongly
effected by an additional process leading to appearance of an additional
intensities, changing the expected polarization of the fluorescence, which in
some cases could not be described by the transition rules. The observed
deviations from the predicted polarization are described by inter-multiplet
state density mixing driven by collision with neutrals. Such effect would try
to equalize the density distribution within multiplets and destroy the
alignment. The disalignment effect needs to be considered, along other
mechanism, as an additional production and loss mechanism in the rate balance
equation describing $\rm 2p_{8}$ sub-state densities.

Measured intensities were corrected by subtracting plasma emission consider
polarization dependency. Due to the optical selection rules there are no
allowed electric dipole $\pi$ transitions for $m_{i}=2$ to $m_{j}=2$ of the
$\rm 2p_{8}$ to $\rm 1s_{4}$ system and $m_{i}=0$ to $m_{j}=0$ of the $\rm
2p_{8}$ to $\rm 1s_{5}$ system. Nevertheless, the signals were very strong
when observing these systems especially when pumping $m_{i}=2$ and measuring
$\pi$ fluorescence of 842 nm. These intensities raise with higher pressure
indicating the role of level mixing by neutral atoms. Such effect would
transfer population from laser pumped level ($m_{j}=2$) to the adjacent
magnetic sub-levels resulting in polarization of light emission determined by
final density distribution achieved among magnetic sub-levels. The resulting
disalignment constants for different 2p states of argon are discussed in [11,
14, 15].

##  3 Experimental Setup

The experimental setup and discharge configuration is presented in figure 2
with indicated laser path (red arrow) and line of sight for LIF detection
(blue arrow). A cone shaped glass discharge chamber followed by a tube of 4 mm
inner diameter were used to create a compact plasma jet. Open end of the tube
was attached to 500 mm long expansion chamber with the vacuum system attached
at the end. Driven electrode made of aluminum tape was attached from the
outside of the cone base with an opening in the center to introduce a laser
beam in axial direction.

Figure 2: Schematic representation of the experimental setup.

A base pressure of 1.4$\times 10^{-1}$ Pa could be reached. The minimum
working pressure inside the tube was estimated to be around 46 Pa at 0.014
slpm gas flow and highest 431 Pa at 0.32 slpm. The working pressure inside the
tube was estimated based on the measured Doppler shift in the gas velocity
distribution function caused by the induced gas flow through the plasma jet
similar as in [16, 17]. The particle velocity, extracted from Doppler shift
measurements, can be correlated to the pressure inside the jet through the
conservation of volume flow through the jet accounting for the cross-section
of the tube and temperature of the gas. The Doppler shift was measured in
fluorescence mode using axial laser excitation, along the gas flow, and radial
fluorescence measurements at the point between the magnets. The fluorescence
signal was detected by photo-diode and recorded together with Fabry-Pérot
signal for scale and reference. The intensity of the shift was estimated by
comparing with the unshifted position determined by radial laser absorption
measurements.

Plasma was created by a bipolar 30 kHz sinusoidal high voltage
($V_{pp}\approx$ 4 kV) signal. The external magnetic field was introduced by
two cylindrical permanent magnets of 6 mm in diameter inducing a magnetic
field of 0.3 T. The field strength was confirmed by a gaussmeter. The magnets
were mounted perpendicular to the line of sight at the opposite sides at the
discharge tube.

Laser absorption measurements were done in radial direction crossing the tube
at position between the magnets, including common optical elements like argon
reference cell and a Fabry-Pérot interferometer to monitor laser scanning
range and quality. Absorption at 842.47 nm transition was scanned by TOPTICA
DLC 100 laser. A system of collimators and multimode $200\leavevmode\nobreak\
\mathrm{\mu m}$ optical fiber was used to manipulate the laser beams,
producing an unpolarized probing laser light. A band pass filters at 840 nm
with a full width at half-maximum of 10 nm were used in order to suppress the
rest of the plasma emission. The absorption measurements were done using
optics crossing the tube in radial direction in orientation perpendicular to
the magnetic field lines allowing to probe all polarization transitions.

For fluorescence measurements, laser light was introduced perpendicular to the
external magnetic field in axial direction of the discharge tube along the gas
flow. The 3 $\pi$ and 3 $\sigma+$ transitions of $1s_{4}$ at 842 nm were used
to pump the $2p_{8}$ sub-levels. Laser scan was set at low scanning frequency
allowing to measure simultaneously the induced fluorescence with a high
sampling rate (10 ms integration time) using $200\leavevmode\nobreak\
\mathrm{\mu m}$ optical fiber and a collimator connected to Ocean Optics
(USB2000+) spectrometer. The fluorescence was observed perpendicular to the
illumination laser using identical optics from the LAS measurements. A linear
polarization filter from Thorlabs GmbH with a total transmission of 40 % for
unpolarized light was used between the plasma and collection optics to isolate
desired polarization from the induced fluorescence. With linear polarization
filter oriented parallel to the magnetic field lines only $\pi$ transitions
could be transmitted while perpendicular orientation would transmit only
$\sigma$. Polarization filter was not completely isolating desired polarity so
that 3 % transmissions of the opposite polarity was measured which was further
taken into account for correct reconstruction of polarization dependent
intensity. Relative sensitivity for linear and circular polarized light was
evaluated to 1:0.7, which is usually dependent on an optics and detector
system. Such difference in sensitivity of the two polarities was found to be
between 0.581 to 0.67 in other studies [10, 18, 19].

##  4 Results and Discussion

Figure 3: Pressure dependency of measured polarized fluorescence at 842 nm for
four different pressures. The $A_{21}$ peak of the $\pi$ profile exhibits the
crucial intensity increase due to the disalignment effect. Numbers in the
lower part of the figure are indicating the upper $m_{i}$ and lower $m_{j}$
magnetic quantum number of each transition.

Fluorescence emission recorded at 842 nm branch for 4 different gas pressures
are presented in figure 3 colour-coded with thick lines for $\pi$ and dashed
for $\sigma$ component. The magnetic sub-level quantum number of the levels
involved in the transitions are indicated at the bottom of the figure. The
relative changes of the intensities with pressure are noticeable for each
polarization component. The intensity of not allowed $\pi$ component
originating at $A_{21}$ transition is clearly visible demonstrating the
influence of the disalignment on the light transport properties in magnetized
plasma.

Systematic changes in polarization of the fluorescence induced at $m_{j}=-1$
to $m_{i}=-1$ and $m_{j}=1$ to $m_{i}=2$ sub-transitions are better presented
in figure 4 with intensities scaled by the intensity of the respective
$m_{i}=-1$ to $m_{j}=-1$ $\pi$ peak at each pressure. From such representation
it is evident that $\pi$ component only slightly changes the shape with
pressure increase for all except $m_{i}=2$ to $m_{j}=1$ transition which
should be pure sigma by the optical selection rules for electrical dipole
transitions.

Figure 4: Systematic changes of measured 842 nm polarized fluorescence with
pressure scaled to the $A_{-1-1}$ $\pi$ component. Strong increase of not
optically allowed $A_{21}$ $\pi$ component is clearly increasing with pressure
as a result of more efficient excitation transfer between the sub-states.

Considering the mentioned Kastler diagram (figure 1) and possible transitions
it is evident that excitation of different $\rm 2p_{8}$ multiplet
($m_{i}=-2,-1,...,2$) will produce unique ratio of $\pi$ to $\sigma$
fluorescence components for both 842 nm and 801 nm branches. Furthermore,
optical pumping of $\rm 2p_{8}$ $m_{i}=\pm 2$ is only possible by $\sigma$
polarized component from $\rm 1s_{4}$ $m_{j}=\pm 1$ and according to the
optical selection rules for electrical dipole transitions only $\sigma$
polarized light at 842 nm branch should be emitted. However, from the
presented measurements the $\pi$ component detected by the excitation of
$m_{i}=2$ sub-level state is clearly visible and rather strong with systematic
increase with pressure. This is a solid evidence of the disalignment process
or intra-multiplet density mixing driven by collisions with the neutral atoms.
Such effect would transfer excitation from the laser pumped sub-level
($m_{i}=2$) to the neighboring magnetic sub-levels and vice-versa producing a
light emission with polarization determined by the final density distribution
achieved within the multiplet called alignment. Such effect is also
responsible for detection of not optically allowed $\pi$ component on 801 nm
branch when $\rm 2p_{8}$ $m_{i}=0$ state is optically pumped.

Measured fluorescence was first analyzed by identifying the positions and
width of the transitions from measured fluorescence structure, quite similar
to the LAS structure analyzed in our previous work [4]. Laser induced
fluorescence was modeled including collisional and radiative processes
contribution to final state density distribution. In order to model measured
fluorescence rate balance equation describing $\rm 2p_{8}$ sub-levels density
induced by laser pumping was constructed including radiative decay, self-
absorption, disalignment effect, stimulated emission and quenching effects.
With slow laser scan, steady state solution for each of $\rm 2p_{8}$ multiplet
can be described in following form:

$\begin{split}n_{{\rm 2p_{8}},m_{i}}&=\frac{I_{\pi/\sigma}B^{\rm
842}_{m_{j}m_{i}}n_{{\rm 1s_{4}},m_{j}}+Nk_{\rm dis}\sum n_{{\rm
2p_{8}},{m_{i}\pm 1}}}{\sum\left(A^{\rm
842}_{{m_{i}m_{j}}}\gamma_{m_{i},m_{j}}(n_{{\rm 1s_{4}},m_{j}})+A^{\rm
801}_{{m_{i}m_{j}}}\gamma_{m_{i},m_{j}}(n_{{\rm
1s_{5}},m_{j}})\right)+2Nk_{\rm dis}+Q}\\\ Q&=I_{\pi/\sigma}B^{\rm
842}_{m_{j}m_{i}}\gamma_{m_{i},m_{j}}(n_{{\rm
1s_{4}},m_{j}})+c_{e}+c_{q}\end{split}$ (3)

The $I_{\pi/\sigma}$ describes either the linear or circular polarized laser
intensity, $N$ is the number of the neutral atoms, $B_{m_{j},m_{i}}$ is the
Einstein coefficient for absorption between magnetic sub-levels. The term
under the sum in the denominator is describing the lifetime of the 2p sub-
level including all the radiative transitions corrected by corresponding self-
absorption factors $\gamma_{m_{i},m_{j}}$. The self-absorption coefficient is
dependent on the magnetic sub-level density $n_{1s,m_{j}}$ and the absorption
coefficient. If a sub-state is not directly pumped than the first factor in
numerator in equation 3 describing laser pumping should be neglected. The
laser stimulated emission for the 842 nm branch is taken into account only for
the pumped sub-states and is described by the term $I_{\pi/\sigma}B^{\rm
842}_{m_{j}m_{i}}\gamma_{m_{i},m_{j}}(n_{{\rm 1s_{4}},m_{j}})$ in the
denominator. Quenching of 2p states by collision with electrons $c_{e}$ [20]
and neutrals $c_{q}$ [21, 22] are also accounted as a loss mechanisms in the
rate balance equation. Electron densities of few $10^{17}\,\rm m^{-3}$ were
assumed to calculate the rate constant due to electron collision. The electron
density is assumed to be similar to un-magnetized conditions from our previous
work [16].

Rate balance equation accounts the laser pumping as a source of population
proportional to Einstein coefficient and targeted 1s sub-level density while
additional production due to disalignment is included as transfer of
excitation from neighboring sub-levels described with rate constant $k_{\rm
dis}$ in $\rm m^{3}/s$. The disalignment effect is accounted as a loss
mechanism, competing mostly with radiative decay, in depopulation of the
described state. The term $2Nk_{\rm dis}$ in the denominator of the equation
(3) losses the factor 2 in description of $m_{i}=\pm 2$ states since transfer
from only one neighboring 2p sub-level is possible.

The laser intensity in our experiments was measured to be 15 mW focused on the
2 mm diameter spot resulting in laser intensity of few hundreds $\rm W/m^{2}$
which according to [23] is well above laser saturation intensity. With
saturated laser intensity, the laser excitation is no longer dependent on the
laser intensity. But secondary emission start to play a role and is taken into
account.

The system of equations can be build for each probed transition providing
laser-induced alignment of $\rm 2p_{8}$ state that can be further used to
model the intensity of the fluorescence. Solved sub-level densities were
further used to describe emission rate of both polarities induced by laser
pumping in a following form for 842 nm branch:

$\begin{split}I^{\rm 842nm}_{\pi}&=n_{2p_{8},0}A_{00}\gamma_{0,0}(n_{\rm
1s_{4},0})+n_{\rm 2p_{8},1}A_{11}\gamma_{1,1}(n_{\rm 1s_{4},1})+n_{\rm
2p_{8},-1}A_{-1-1}\gamma_{-1,-1}(n_{\rm 1s_{4},-1})\\\ \\\ I^{\rm
842nm}_{\sigma}&=n_{\rm 2p_{8},0}A_{01}\gamma_{0,1}(n_{\rm 1s_{4},1})+n_{\rm
2p_{8},1}A_{10}\gamma_{1,0}(n_{\rm 1s_{4},0})+n_{\rm
2p_{8},2}A_{21}\gamma_{2,1}(n_{\rm 1s_{4},1})\\\ &+n_{\rm
2p_{8},0}A_{0-1}\gamma_{0,-1}(n_{\rm 1s_{4},-1})+n_{\rm
2p_{8},-1}A_{-10}\gamma_{-1,0}(n_{\rm 1s_{4},0})\\\ &+n_{\rm
2p_{8},-2}A_{-2-1}\gamma_{-2,-1}(n_{\rm 1s_{4},-1})\end{split}$ (4)

and for 801 nm branch:

$\begin{split}I^{\rm 801nm}_{\pi}&=n_{\rm 2p_{8},1}A_{11}\gamma_{1,1}(n_{\rm
1s_{5},1})+n_{2p_{8},2}A_{22}\gamma_{2,2}(n_{\rm 1s_{5},2})\\\ &+n_{\rm
2p_{8},-1}A_{-1-1}\gamma_{-1,-1}(n_{\rm
1s_{5},-1})+n_{2p_{8},-2}A_{-2-2}\gamma_{-2,-2}(n_{\rm 1s_{5},-2})\\\ \\\
I^{\rm 801nm}_{\sigma}&=2n_{\rm 2p_{8},0}A_{01}\gamma_{0,1}(n_{\rm
1s_{5},1})+2n_{\rm 2p_{8},1}A_{10}\gamma_{1,0}(n_{\rm 1s_{5},0})\\\ &+2n_{\rm
2p_{8},2}A_{21}\gamma_{2,1}(n_{\rm 1s_{5},1})+2n_{\rm
2p_{8},1}A_{12}\gamma_{1,2}(n_{\rm 1s_{5},2})\\\ &+n_{\rm
2p_{8},0}A_{0-1}\gamma_{0,-1}(n_{\rm 1s_{5},-1})+n_{\rm
2p_{8},-1}A_{-10}\gamma_{-1,0}(n_{\rm 1s_{5},0})\\\ &+n_{\rm
2p_{8},-2}A_{-2-1}\gamma_{-2,-1}(n_{\rm 1s_{5},-1})+n_{\rm
2p_{8},-1}A_{-1-2}\gamma_{-1,-2}(n_{\rm 1s_{5},-2})\end{split}$ (5)

Similar sets of equations can be constructed for each pumped transition
resulting in $\pi$ and $\sigma$ radiation rates for both radiation branches
842 and 801 nm as observed in the measurements. With the proposed description,
it is evident that pumping of a same state through different transitions, as
$A_{11}$ and $A_{10}$, would lead to the identical polarization although with
different intensities determined mostly by the laser pumping rate through the
targeted 1s state densities ($\rm n_{1s4,1}$ and $\rm n_{1s4,0}$
respectively).

Equations constructed for each pumped transition
($A_{00},A_{11},A_{21},A_{10}$ and $A_{0-1}$) will effectively create 3
equations describing $\pi$ to $\sigma$ ratios at each radiation branch
resulting in 6 coupled equations with 6 variables ($n_{{\rm 1s_{4}},m_{j}=0}$,
$n_{{\rm 1s_{4}},m_{j}=1}$, $n_{{\rm 1s_{5}},m_{j}=0}$, $n_{{\rm
1s_{5}},m_{j}=1}$, $n_{{\rm 1s_{5}},m_{j}=2}$ and $k_{\rm dis}$) assuming
symmetry in the self-alignment of 1s states ($n_{-m_{j}}=n_{m_{j}}$). The
variables can be reduced to 3 ($n_{{\rm 1s_{4}},m_{j}=0}$, $n_{{\rm
1s_{5}},m_{j}=0}$ and $k_{\rm dis}$) because 1s sub-level densities belonging
to the same state have a constant relation to each other. This is confirmed
within this study and by the previous work [4] as also the symmetry by LAS.
The system of equations is mostly coupled through the self-absorption
coefficients. The system of equations was solved with 3 variables using least-
square method by minimizing the error between the measured and modeled rations
of polarities.

Figure 5: Modeled and measured fluorescence at 842 nm for two different
pressures and both polarization components. Figure 6: Modeled and measured
fluorescence at 801 nm for two different pressures and both polarization
components.

Solution of the system of equations will result in numerical evaluation of
$\rm 1s_{4}$ and $\rm 1s_{5}$ multiplet densities and effective disalignment
rate in form $c_{\rm dis}$=$Nk_{\rm dis}$. Measured fluorescence and modeled
intensities at two different pressures (red low and blue medium pressure) are
presented in figure 5 for 842 nm and figure 6 for 801 nm branch. The quality
of the fits is clearly visible from the presented example where besides the
reconstructed profiles, relative changes in the intensities between two
different pressures could also be reliably reproduced. Similar quality of the
fits is achieved for all measured pressure settings. In most cases the
deviation is not larger than few percent making this analysis highly
efficient. Here it should be emphasized that since only the ratios between the
two polarities for each branch were considered in evaluation, there is no need
for intensity calibration of the detection system for the two observed
wavelengths.

Reconstructed $\rm 1s_{4}$ and $\rm 1s_{5}$ multiplets densities are presented
in figures 7 and 8 respectively, for series of different pressures. The
estimated $\rm 1s_{4}$ densities were further compared with the results from
direct laser absorption measurements. The estimated densities by this two
independent methods are in close agreement accounting for the complexity of
the measurements and the evaluation. Alignment between the magnetic sub-levels
has been also successfully reconstructed for both states. Reconstructed $\rm
1s_{5}$ multiplet densities can be indirectly compared with the results from
the LAS measurements presented in [4] where similar densities and identical
negative alignment was observed under very similar conditions.

Figure 7: Density of $\rm 1s_{4}$ sub-levels measured by LAS and evaluated
from TDLIF under different pressures ($\propto$ disalignment rate $c_{dis}$).
Figure 8: Densities of $\rm 1s_{5}$ sub-levels evaluated from TDLIF
measurements under different pressures ($\propto$ disalignment rate
$c_{dis}$).

The pressure at the measurement point (between the magnets) evaluated for used
gas flow conditions was estimated to be in the range between 46 and 431 Pa.
The estimated pressures by LIF were in between the values measured at the
extraction side (lower pressure) and in gas flow line (higher pressure),
closer to the high pressure side. Based on the pressure estimations, the
disalignment rate coefficient $k_{dis}$ was estimated to be in the range from
1.26 to 1.41 $\times 10^{-9}\,\rm{\frac{cm^{3}}{s}}$ as presented in figure 9.

Figure 9: Estimation of disalignment rate coefficient at different measured
pressure.

The estimated values at different pressures are in a good agreement with the
reported value of 1.45 $\times 10^{-9}\,\rm{\frac{cm^{3}}{s}}$ in [15] which
is evaluated at higher temperature of 380 K and by a different method.
Evaluated temperature of 340 K by LAS in our conditions is lower which could
reduce the disalignment coefficient according to the temperature dependency
reported in [11]. The deviation in estimated rate coefficient at different
pressures could be the result of poor estimation of pressure inside the tube
based on measurements of relatively small shifts (of about 50 MHz) and robust
model for pressure estimation.

The disalignment rate coefficients for most of argon $\rm 2p$ states (in range
from 1 – 4 $\times 10^{7}\,\rm{s^{-1}}$) reported in [15] are in excellent
agreement with our estimated rate coefficients for $\rm 2p_{8}$ in range of 1
– 12 $\times 10^{7}\,\rm{s^{-1}}$ between 46 – 431 Pa, rendering disalignment
effect highly important for accurate description of the 2p state densities in
a magnetized plasma.

Figure 10: Normalized $\rm 2p_{8}$ sub-level densities solved for pumping $\rm
m_{j}=0$ to $\rm m_{i}=0$ transition. In this particular case $m_{i}=\pm 2,\pm
1$ is same so that $m_{i}=-2,-1$ are not shown.

The influence of pressure on sub-state densities in the presence of laser
pumping is illustrated in figure 10 where steady state solution for pumping of
$m_{j}=0$ state by $\rm A_{00}^{842\,nm}$ transition is shown. The pressure
dependent evolution of densities were calculated using constant and low values
of $\rm 1s_{4}$ and $\rm 1s_{5}$ sub-state densities normalized by the value
at $\rm c_{\rm dis}=0$. It is evident that increase of pressure would lead to
increase of the population of other not-pumped sub-levels effectively changing
polarization and overall intensity of the fluorescence signals. Since the
central ($A_{00}$) transition is pumped the resulting alignment will be
symmetric while pumping of other states $m_{j}=\pm 1,\pm 2$ would result in
strong asymmetric alignment.

The increase of the 2p sub-level densities which are not pumped is a result of
a more efficient mixing between the sub-levels until the point when densities
within the multiplet are equally distributed having no alignment.

##  5 Conclusion

A method to evaluate TDLIF measurements accounting for intra- and inter-
multiplet transitions has been proposed resulting in full quantitative
evaluation of 1s states multiplets involved in the LIF scheme and evaluation
of the disalignment coefficient for argon. The method was described for the
$\rm 2p_{8}$ to $\rm 1s_{4}$ and $\rm 1s_{5}$ schemes but can be easily
adopted for other 2p argon states. The evaluated densities were in good
agreement with the values measured by LAS for wide pressure range. The
disalignment rate constant evaluated within same analysis showed good
agreement compared with restricted literature values. We also showed that the
interpretation of LIF measurements in magnetized plasma conditions is strongly
pressure dependent through the disalignment interactions.

Proposed measurement method and analysis could further improve an evaluation
of a magnetized plasma conditions and provide information valuable for further
development of diagnostics methods based purely on optical emission
spectroscopy. With the systematic evaluation of disalignment for argon 2p
levels and quantitative LIF done on various pressures, the densities of all 1s
states would be well-known, providing basis for further development of an
appropriate line branching method for evaluation of electron densities and
temperature. These properties are making LIF measurements in magnetized plasma
as highly efficient tool for quantitative plasma diagnostics while providing
basis for further understanding and description of light transport properties.

This work is supported by the Deutsche Forschungsgemeinschaft (DFG).

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