# Magnetized Laser-Plasma Interactions in High-Energy-Density Systems:
Parallel Propagation

E. E. Los  Physics Department, Imperial College London, South Kensington
Campus, London SW7 2AZ  D. J. Strozzi  Lawrence Livermore National Laboratory,
Livermore, CA, 94551, USA

###### Abstract

We investigate parametric processes in magnetised plasmas, driven by a large-
amplitude pump light wave. Our focus is on laser-plasma interactions relevant
to high-energy-density (HED) systems, such as the National Ignition Facility
and the Sandia MagLIF concept. We derive dispersion relations for three-wave
interactions in a multi-species plasma using Maxwell’s equations, the warm-
fluid momentum equation and the continuity equation. The application of an
external B field causes right and left polarised light waves to propagate with
differing phase velocities. This leads to Faraday rotation of the
polarisation, which can be significant in HED conditions. Raman and Brilllouin
scattering have modified spectra due to the background B field, though this
effect is usually small in systems of current practical interest. We identify
a scattering process we call stimulated whistler scattering, where a light
wave decays to an electromagnetic whistler wave ($\omega\lesssim\omega_{ce}$)
and a Langmuir wave. This only occurs in the presence of an external B field,
which is required for the whistler wave to exist. We compute the scattered
wavelengths for Raman, Brillouin, and whistler scattering.

††preprint: AIP/123-QED

##  I Introduction

Imposing a magnetic field on HED systems is a topic of much current interest.
This has several motivations, including reduced electron thermal conduction to
create hotter systems (such as for x-ray sources [1]), laboratory astrophysics
[2], and magnetised inertial confinement fusion (ICF) schemes. If successful,
they could provide efficient, low-cost alternatives to unmagnetised, laser-
driven ICF. In the most successful case, the Sandia MagLIF concept [3, 4], an
external axial magnetic field is used to magnetise the deuterium-tritium (DT)
gas contained within a cylindrical conducting liner. A pulsed-power machine
then discharges a high current through the liner, generating a Lorentz force
which causes the liner to implode. The DT fuel is pre-heated by a laser as the
implosion alone is not sufficient to heat the fuel to the ignition
temperature. The magnetic field is confined within the liner and hence obeys
the MHD frozen in law, which states

$B_{z}\pi r^{2}=c$ (1)

where $r$ is the radius of the cylindrical liner, $B_{z}$ is the axial
magnetic field and $c$ is a constant. Over the course of the implosion, the
magnetic field strength perpendicular to the direction of compression
increases as $1/r^{2}$. Thus, following the implosion, the magnetic field
traps fusion alpha particles and thermal electrons, insulating the target and
aiding ignition.

The MagLIF scheme, as well as magnetised laser-driven ICF [5, 6], motivate us
to consider magnetised laser-plasma interactions (LPI), specifically
parametric scattering processes [7]. Parametric coupling involves the decay of
a large-amplitude or “pump” wave into two or more daughter waves. We focus on
those involving one electromagnetic (e/m) and one electrostatic (e/s) daughter
wave. In order for parametric coupling to occur, the following frequency and
wave-vector matching conditions must be met:

$\omega_{0}=\omega_{1}+\omega_{2}$ (2) $\vec{k}_{0}=\vec{k}_{1}+\vec{k}_{2}$
(3)

where the subscripts 0, 1 and 2 denote the pump, scattered and plasma waves,
respectively. Equations 2 and 3 are required by energy and momentum
conservation, respectively. Parametric processes can give rise to resonant
modes which grow exponentially in the plasma and remove energy from the target
[8]. Additionally, light waves which are back-scattered through the optics of
the experiment can cause significant damage and even be re-amplified[9, 10,
11]. Finally, electron plasma waves can generate superthermal or “hot”
electrons which can pre-heat the fuel, thereby increasing the work required to
compress it [12].

Laser-driven parametric processes have been extensively researched in
unmagnetised plasmas. However, the advent of experiments such as MagLIF and
the possibility of magnetised experiments on the National Ignition Facility
(NIF) [13, 14, 15] necessitate re-examining the impact of a magnetic field on
them, which is usually neglected. This is not unexplored territory. For
instance, prior work studied how an external axial B field affects Raman
backscattering in a hot, inhomogeneous plasma [16], and the decay of
circularly polarised EM waves in cold, homogeneous plasma [17]. Recently,
excellent theoretical work on a warm-fluid model for magnetized LPI has been
done by Shi [18]. Winjum et al.[19] have studied SRS in a magnetized plasma
with a particle-in-cell code in conditions relevant to indirect-drive ICF.
This work focuses on how the B field affects large-amplitude Langmuir waves,
which can nonlinearly trap resonant electrons and modify the Landau damping.
Our work ignores nonlinearity and damping, both of which are important in real
systems.

Besides modifying existing processes, a background B field gives rise to new
waves, one of which is an electromagnetic "whistler" wave which has
$\omega\leq\omega_{ce}$, the electron cyclotron frequency. Thus, a plethora of
new parametric processes involving this wave can occur, including one which we
call “stimulated whistler scattering”, in which the pump light wave decays to
an electrostatic Langmuir wave and a whistler wave. Parametric processes
involving whistlers have been known for some time. For instance, a collection
of new instabilities (mostly involving whistler waves) which include purely
growing, modulational and beat-wave instabilities in hot, inhomogeneous
plasmas has been explored by Forslund et al. [20]. The decay of a high-
frequency whistler wave into a Bernstein wave and a low-frequency whistler
wave in hot, inhomogeneous plasmas have also been investigated [21].
Additionally, parametric decays involving three whistler waves in cold,
homogeneous plasmas have been studied [22]. In magnetized fusion, parametric
interactions of large-amplitude RF waves launched by external antennas, for
plasma heating and current drive, have been explored since the 1970s. [23]

This paper aims to present the theory of magnetized LPI in a self-contained
way, for a simple enough situation where that is feasible. Namely, we consider
all wavevectors parallel to the background B field, and use warm-fluid theory
with multiple ion species. The results are mostly special cases of prior ones,
especially by Shi, but we hope the reader benefits from a relatively simple
presentation. We obtain a parametric dispersion relation, meaning one where
the pump light wave is included in the equilibrium, in the spirit of Drake et
al. [24]. We then study the “kinematics” of magnetized three-wave
interactions, based on phase-matching, in HED-relevant conditions (e.g. for
NIF and MagLIF). We consider magnetized modifications to stimulated Raman and
Brillouin scattering, as well as stimulated whistler scattering.

The rest of the paper is organized as follows. In section 2, we use the warm-
fluid equations to derive the parametric dispersion relation. These are then
linearized and decomposed in Fourier modes. Only resonant terms satisfying
phase-matching are retained. In section 3, the resulting free-wave dispersion
relations in a magnetised and unmagnetised plasma are discussed, along with
the Faraday rotation of light-wave polarization. Section 4 studies the impact
of the external magnetic field on stimulated Raman and Brillouin scattering in
typical HED plasmas. Stimulated whistler scattering is also explored. Section
5 concludes and discusses future prospects.

##  II Parametric Dispersion Relations for Magnetised Plasma Waves

This section develops a parametric dispersion relation, meaning one where the
pump is included in the equilibrium. This approach is in the spirit of the
paper by Drake [24] for kinetic, unmagnetised plasma waves, and also for
magnetised waves [25]. Subsequent kinetic work was done which extended the
Drake approach to include a background B field [26, 27]. While our approach
does not contain new results compared to the latter, we believe it is useful
to work through the details explicitly - especially in a form familiar to the
unmagnetised LPI community. The upshot of the lengthy math is Eq. 67, which
the reader should understand in physical terms before delving into the details
of this section. Our goal is expressions for the amplitude-independent $D$’s
(which give linear dispersion relations) and $\Delta$’s (which give parametric
coupling).

Figure 1: Geometry of the experimental setup considered throughout the paper.
The pump frequency, $\omega_{0}$, is set by the laser. An external magnetic
field, $B_{eq}\hat{z}$, is imposed parallel to the propagation direction of
the laser, $\hat{k}_{0}$. The laser is incident from vacuum on a plasma with
density, $n_{e}$, which varies with z. The wave vector is therefore also z
dependent.

###  II.1 Governing Equations

The subscript s will be used to denote species, with mass $m_{s}$ and charge
$q_{s}=Z_{s}e$ ($e>0$ the positron charge). The subscript j will denote the
wave or mode. We start with the 3D, non-relativistic Vlasov-Maxwell system
with no collisions, and assume spatial variation only in the z direction.
Hence, all vectors directed along $\hat{z}$ are longitudinal, and all vectors
which lie in the x-y plane are transverse. An experimental configuration for
which these assumptions hold is shown in figure 1. We further assume that the
distribution function for species $s$, $f_{s}$ (particles per $dz*d^{3}w$,
where $\vec{w}$ denotes velocity, and we have integrated over $x$ and $y$) can
be written in a separable form:
$f_{s}(t,z,\vec{w})=f_{s\perp}(t,z,\vec{w}_{\perp})F_{s}(t,z,w_{z})$.
$f_{s\perp}$ allows for transverse electromagnetic waves, and is normed such
that $\int f_{s\perp}d^{2}w_{\perp}=1$. $F_{s}$ is the 1D distribution
(particles per $dz*dw_{z}$). Standard manipulations lead to the following 1D
Vlasov-Maxwell system:

$\partial_{t}F_{s}+w_{z}\partial_{z}F_{s}=-\frac{q_{s}}{m_{s}}(E_{z}+(\vec{v}_{s}\times\vec{B})_{z})\partial_{w_{z}}F_{s}$
(4)
$(\partial_{t}+v_{sz}\partial_{z})\vec{v}_{s\perp}=\frac{q_{s}}{m_{s}}(\vec{E}_{\perp}+(\vec{v}_{s}\times\vec{B})_{\perp})$
(5) $\displaystyle n_{s}=\int F_{s}\mbox{d}w_{z},\quad\vec{v}_{s\perp}=\int
f_{s\perp}\vec{w}_{\perp}\mbox{d}^{2}w_{\perp},$ (6) $\displaystyle
v_{sz}=n_{s}^{-1}\int F_{s}w_{z}\mbox{d}w_{z}$ $B_{z}=B_{eq}=\mbox{const}$ (7)
$\partial_{t}\vec{B}_{\perp}=\partial_{z}(E_{y},-E_{x})$ (8)
$\partial_{t}\vec{E}_{\perp}=c^{2}\partial_{z}(-B_{y},B_{x})-\frac{e}{\epsilon_{0}}\sum_{s}Z_{s}n_{s}\vec{v}_{s\perp}$
(9) $\partial_{t}E_{z}=-\frac{e}{\epsilon_{0}}\sum_{s}Z_{s}n_{s}v_{sz}$ (10)

$B_{eq}>0$ and the subscript eq indicates a non-zero, zeroth order background
term. Poisson’s equation is not listed since the inclusion of Ampere’s law and
charge continuity render it redundant. It is possible to satisfy Maxwell’s
equations (equations 8, 9, and 10) by writing $\vec{E}$ and $\vec{B}$ in terms
of scalar and vector potentials, $\phi$ and $\vec{A}$:
$\vec{E}=-\vec{\nabla}\phi-\frac{\partial\vec{A}}{\partial t}$ and
$\vec{B}=\vec{\nabla}\times\vec{A}+B_{eq}\hat{z}$. We choose the Weyl gauge,
in which $\phi=0$ and $\vec{A}=\vec{A}_{\perp}+A_{z}\vec{z}$. Faraday’s law is
then automatic, and the remaining Maxwell’s equations become:

$\partial_{t}^{2}A_{z}=\frac{e}{\epsilon_{0}}\sum_{s}Z_{s}n_{s}v_{sz}$ (11)
$(\partial_{t}^{2}-c^{2}\partial_{z}^{2})\vec{A}_{\perp}=\frac{e}{\epsilon_{0}}\sum_{s}Z_{s}n_{s}\vec{v}_{s\perp}$
(12)

We arrive at fluid equations by taking moments $\int w_{z}^{p}\mbox{d}w_{z}$
of the equation for $F_{s}$, for $p=$ 0, 1, and 2:

$\partial_{t}n_{s}+\partial_{z}(n_{s}v_{sz})=0$ (13)
$\partial_{t}(n_{s}v_{sz})+\partial_{z}\left(n_{s}v^{2}_{sz}+\frac{P_{s}}{m_{s}}\right)=\frac{q_{s}n_{s}}{m_{s}}(E_{z}+(\vec{v}_{s}\times\vec{B})_{z})$
(14)
$(\partial_{t}+v_{sz}\partial_{z})P_{s}=-3P_{s}\partial_{z}v_{sz}-2\partial_{z}Q_{s}$
(15)

with pressure $P_{s}\equiv m_{s}\int F_{s}(w_{z}-v_{sz})^{2}\mbox{d}w_{z}$ and
heat flux $Q_{s}\equiv(m_{s}/2)\int F_{s}(w_{z}-v_{sz})^{3}\mbox{d}w_{z}$.
Note that the pressure is the $zz$ component of the 3D pressure tensor, _not_
the scalar, isotropic pressure. We can close the fluid-moment system by
replacing the pressure equation with a polytrope equation of state, where
$K_{s}$ is a constant:

$P_{s}=n_{s}T_{s}=K_{s}n_{s}^{\gamma_{s}}$ (16)
$\partial_{z}P_{s}=K_{s}\gamma_{s}n_{s}^{\gamma_{s}-1}\partial_{z}n_{s}=\gamma_{s}T_{s}\partial_{z}n_{s}$
(17)

Common choices for linearised dynamics are isothermal $(\gamma_{s}=1)$ and
adiabatic $(\gamma_{s}=3)$, which follows from setting $Q_{s}=0$ in the
pressure equation. Let us recap the complete fluid-Maxwell system, with the
substitutions $\vec{a}=\frac{e}{m_{e}}\vec{A}$ (units of speed),
$\omega_{ps}^{2}=\frac{q_{s}^{2}n_{seq}}{\epsilon_{0}m_{s}}$,
$\omega_{cs}=|\frac{q_{s}}{m_{s}}B_{eq}|$, $\mu_{s}=\frac{m_{s}}{m_{e}Z_{s}}$
and $s_{s}=-1,1$ for electrons and ions respectively:

$\partial_{t}^{2}a_{z}-\sum_{s}\omega_{ps}^{2}\mu_{s}\frac{n_{s}}{n_{seq}}v_{sz}=0$
(18)
$(\partial_{t}^{2}-c^{2}\partial_{z}^{2})\vec{a}_{\perp}=\sum_{s}\omega_{ps}^{2}\mu_{s}\frac{n_{s}}{n_{seq}}\vec{v}_{s\perp}$
(19)
$\displaystyle\partial_{t}v_{sz}+\mu_{s}^{-1}\partial_{t}a_{z}+v_{sz}\partial_{z}v_{sz}+\gamma_{s}\frac{T_{s}}{m_{s}n_{s}}\partial_{z}n_{s}=\mu_{s}^{-1}\vec{v}_{s\perp}\cdot\partial_{z}\vec{a}_{\perp}$
(20)
$\displaystyle\partial_{t}\vec{v}_{s\perp}+\mu_{s}^{-1}\partial_{t}\vec{a}_{\perp}-s_{s}\omega_{cs}\vec{v}_{s\perp}\times\hat{z}=-\mu_{s}^{-1}v_{sz}\partial_{z}\vec{a}_{\perp}-v_{sz}\partial_{z}\vec{v}_{s\perp}$
(21) $\partial_{t}n_{s}+\partial_{z}(n_{s}v_{sz})=0$ (22)

Terms that can give rise to parametric couplings of interest have been moved
to the RHS. These involve at least one e/m wave, which will become the pump,
and one e/m or e/s wave, which will become one of the daughters. All other
terms have been moved to the LHS, namely those that are purely linear or
contain 2nd-order terms not of interest. It is clear that the longitudinal
dynamics are unaffected by $B_{eq}$ in the absence of parametric coupling,
since we chose $\vec{k}||B_{eq}\hat{z}$.

###  II.2 Linearisation: Physical Space

We consider parametric processes involving the decay of a fixed, finite-
amplitude, electromagnetic pump to an electromagnetic and an electrostatic
daughter wave, denoted by subscripts 0, 1 and 2, respectively. The daughter
waves are assumed to be much lower in amplitude than the pump. We write the
velocity and vector potential pertaining to each wave as an infinite sum of
terms of increasing order in amplitude. We neglect all terms of second-order
or higher in the pump amplitude (such as the ponderomotive term, which scales
as $a_{0}^{2}$), retaining only terms which are strictly linear in wave
amplitudes or involve the product of one pump and one daughter amplitude. The
plasma density is approximated by the sum of a static, uniform equilibrium
term, $n_{seq}$ and a perturbation induced by the electrostatic wave,
$n_{s2}$. We assume that no background flows exist in the plasma
($v_{seq}=0$), no external electric fields are imposed upon it ($a_{eq}=0$),
and quasi-neutrality holds ( $\sum_{s}q_{s}n_{seq}=0$). We write

$\vec{a}_{\perp}=\vec{a}_{0\perp}+\vec{a}_{1\perp}$ (23) $a_{z}=a_{2}$ (24)
$\vec{v}_{s\perp}=\vec{v}_{s0\perp}+\vec{v}_{s1\perp}$ (25) $v_{sz}=v_{s2}$
(26) $n_{sz}=n_{seq}+n_{s2}$ (27)

where $\vec{a}_{j}$, $\vec{v}_{j}$ and $n_{s2}$ are functions of $t,z$. Since
we are only interested in second order terms which give rise to parametric
coupling, we can linearise equation 17:

$\frac{\partial_{z}P_{s}}{n_{s}}=\gamma_{s}\frac{T_{seq}}{n_{seq}}\partial_{z}n_{s2}$
(28)

Substituting these results and equations 23-27 into equations 18-22 and
keeping only coupling terms of interest, we obtain, for waves 1 and 2:

$\partial_{t}^{2}a_{2}-\sum_{s}\omega_{ps}^{2}\mu_{s}v_{s2}=0$ (29)
$(\partial_{t}^{2}-c^{2}\partial_{z}^{2})\vec{a}_{1}-\sum_{s}\omega_{ps}^{2}\mu_{s}\vec{v}_{s1}=\sum_{s}\omega_{ps}^{2}\mu_{s}\frac{n_{s2}}{n_{seq}}\vec{v}_{s0}$
(30)
$\displaystyle\partial_{t}v_{s2}+\mu_{s}^{-1}\partial_{t}a_{2}+\gamma_{s}\frac{v_{Ts}^{2}}{n_{seq}}\partial_{z}n_{s2}$
(31)
$\displaystyle=\mu_{s}^{-1}\left(\vec{v}_{s0}\cdot\partial_{z}\vec{a}_{1}+\vec{v}_{s1}\cdot\partial_{z}\vec{a}_{0}\right)$
$\displaystyle\partial_{t}\vec{v}_{s1}+\mu_{s}^{-1}\partial_{t}\vec{a}_{1}-s_{s}\omega_{cs}\vec{v}_{s1}\times\hat{z}=-v_{s2}\partial_{z}(\vec{v}_{s0}+\mu_{s}^{-1}\vec{a}_{0})$
(32) $\partial_{t}n_{s2}+n_{seq}\partial_{z}v_{s2}=0$ (33)

where $v_{Ts}^{2}=\frac{T_{eqs}}{m_{s}}$. The $-v_{sz}\partial_{z}v_{sz}$ term
in equation 20 has been neglected because it is second order in the daughter
wave amplitude. Wave 0 satisfies the same eqs. as wave 1 (i.e. Eqs. 30 and 32)
without the coupling terms (RHS = 0). For the daughter waves 1 and 2, we now
have $2s+1$ scalar and $s+1$ vector equations for $2s+1$ scalar
$(n_{s2},v_{s2}$ and $a_{2})$ and $s+1$ vector $(\vec{v}_{s1}$ and
$\vec{a}_{1})$ unknowns, with all vectors in the 2D transverse $(xy)$ plane.
Our plan is to move to Fourier space, retain only linear and parametric-
coupling terms, and arrive at a closed system just involving the $a$’s.

###  II.3 Fourier Decompositions

If the variable X is used to represent the electric field, electron density or
wave velocity, then X can be written as a Fourier decomposition, in which j
denotes the wave (0,1,2):

$X_{j}(t,\vec{r})=\frac{1}{2}X_{fj}e^{i\psi_{j}}+cc$ (34)

Subscript f denotes the Fourier amplitude, phase
$\psi_{j}=(\vec{k_{j}}\cdot\vec{r}-\omega_{j}t)\equiv k_{j}z-\omega_{j}t$ and
cc is an abbreviation of complex conjugate. Since all successive amplitudes
will be Fourier amplitudes, the subscript f will henceforth be neglected. Wave
1 can be written in terms of two e/m waves, with either an up-shifted or a
down-shifted frequency vs. wave 0, denoted by subscripts + and - respectively.
The phase matching conditions are hence

$\psi_{-}=\psi_{0}-\psi_{2}^{*}\qquad\psi_{+}=\psi_{0}+\psi_{2}$ (35)

Growth due to parametric coupling means the daughter-wave $k_{j}$ and
$\omega_{j}$ can be complex. It is assumed that the pump amplitude is fixed
(no damping or pump depletion), hence $k_{0}$ and $\omega_{0}$ are real, and
$\psi_{0}^{*}=\psi_{0}$. We choose our definitions of $\psi_{\pm}$ so they and
$\psi_{2}$ have the same imaginary part, i.e. the same parametric growth rate.
We also choose all frequencies to have a positive real part: the companion
field for Re[$\omega]<0$ follows from the condition that the physical field is
real. Although one can mix positive and negative frequency waves, we find the
analysis simpler with all Re[$\omega]>0$. Especially with magnetized waves,
the discussion of circular polarization for Re[$\omega]<0$ can become
confusing.

####  II.3.1 Plasma Waves in Fourier Space

We shall eliminate $n_{s2}$ and $\vec{v}_{s2}$ in favour of the $a$’s.
Substituting equation 34 into equations 29 and 33, we obtain:

$a_{2}+\frac{1}{\omega_{2}^{2}}\sum_{s}\omega_{ps}^{2}\mu_{s}v_{s2}=0$ (36)

and

$n_{s2}=n_{seq}\frac{k_{2}}{\omega_{2}}v_{s2}$ (37)

respectively. Repeating for equation 31 gives:

$\displaystyle\left(-\frac{\omega_{2}}{2}v_{s2}-\frac{\mu_{s}^{-1}}{2}\omega_{2}a_{2}+\frac{\gamma_{s}v^{2}_{Ts}}{2n_{eqs}}k_{2}n_{s2}\right)+cc$
(38) $\displaystyle=\frac{\mu_{s}^{-1}}{4}PC_{s2}+cc$

where the parametric coupling terms are contained in $PC_{s2}$ (units of
frequency*speed), and

$\displaystyle PC_{s2}$
$\displaystyle=-ie^{-i\psi_{2}}Res_{2}[(\vec{v}_{s0}e^{i\psi_{0}})\cdot(ik_{\pm}\vec{a}_{\pm}e^{i\psi_{\pm}})$
(39)
$\displaystyle+(\vec{v}_{s0}e^{i\psi_{0}})\cdot(-ik_{\pm}^{*}\vec{a}_{\pm}^{*}e^{-i\psi_{\pm}^{*}})+(\vec{v}_{s\pm}e^{i\psi_{\pm}})\cdot(ik_{0}\vec{a}_{0}e^{i\psi_{0}})$
$\displaystyle+(\vec{v}_{s\pm}e^{i\psi_{\pm}})\cdot(-ik_{0}^{*}\vec{a}_{0}^{*}e^{-i\psi_{0}^{*}})+cc$

where $Res_{2}$ denotes terms which are resonant with mode 2. Using equation
37 to substitute for $n_{s2}$:

$\displaystyle-\omega_{2}(v_{s2}+\mu_{s}^{-1}a_{2})+\gamma_{s}\frac{k_{2}^{2}v_{Ts}^{2}}{\omega_{2}}v_{s2}=\frac{\mu_{s}^{-1}}{2}PC_{s2}$
(40)

Rearranging for $v_{s2}$:

$\displaystyle
v_{s2}=-\frac{\omega_{2}P_{s}}{\mu_{s}\omega_{ps}^{2}}(\omega_{2}a_{2}+PC_{s2})$
(41)
$P_{s}=\frac{\omega_{ps}^{2}}{\omega_{2}^{2}-\gamma_{s}k_{2}^{2}v_{Ts}^{2}}$
(42)

Substituting this result into equation 36, we obtain:

$\displaystyle\left(1-\sum_{s}P_{s}\right)a_{2}=\frac{1}{2\omega_{2}}\sum_{s}P_{s}PC_{s2}$
(43)

####  II.3.2 EM Waves in Fourier Space

Writing equation 32 in terms of Fourier modes, we obtain:

$\displaystyle\frac{1}{2}\sum_{+,-}\left(-i\omega_{\pm}\vec{v}_{s\pm}-i\mu_{s}^{-1}\omega_{\pm}\vec{a}_{\pm}-s_{s}\omega_{cs}\vec{v}_{s\pm}\times\hat{z}\right)e^{i\psi_{\pm}}+cc=$
(44)
$\displaystyle-\frac{1}{4}[ik_{0}v_{s2}\vec{v}_{s0}e^{i\psi_{+}}+ik_{0}v_{s2}^{*}\vec{v}_{s0}e^{i\psi_{-}}$
$\displaystyle+\mu_{s}^{-1}(ik_{0}v_{s2}\vec{a}_{0}e^{i\psi_{+}}+ik_{0}\vec{a}_{0}v_{s2}^{*}e^{i\psi_{-}})]+cc$

Let $Z_{y+},Z_{y-}$ denote $Z_{y}$ and $Z_{y}^{*}$, respectively, where $Z$
denotes an amplitude, frequency or wavelength, and $y$ denotes a subscript
containing the mode and plasma species (if applicable) of Z. This allows us to
write generic equations for the + and - waves. Selecting only resonant terms
we obtain:

$\displaystyle\omega_{\pm}(\vec{v}_{s\pm}+\mu_{s}^{-1}\vec{a}_{\pm})-is_{s}\omega_{cs}\vec{v}_{s\pm}\times\hat{z}=\frac{k_{0}}{2}v_{s2\pm}(\vec{v}_{s0}+\mu_{s}^{-1}\vec{a}_{0})$
(45)

Finally, equation 30, once written in terms of Fourier modes, becomes:

$\displaystyle\frac{1}{2}\sum_{+,-}\left((-\omega_{\pm}^{2}+c^{2}k_{\pm}^{2})\vec{a}_{\pm}-\sum_{s}\omega_{ps}^{2}\mu_{s}\vec{v}_{\pm}\right)e^{i\psi_{\pm}}+cc=$
(46)
$\displaystyle\sum_{s}\omega_{ps}^{2}\frac{\mu_{s}}{4n_{seq}}(\vec{v}_{s0}n_{s2}e^{i\psi_{+}}+\vec{v}_{s0}n_{s2}^{*}e^{i\psi_{-}})+cc$

Keeping terms resonant with $\psi_{\pm}$ and eliminating $n_{s2}$ gives

$(-\omega_{\pm}^{2}+k_{\pm}^{2}c^{2})\vec{a}_{\pm}-\sum_{s}\omega_{ps}^{2}\mu_{s}\vec{v}_{s\pm}=\frac{1}{2}\frac{k_{2\pm}}{\omega_{2\pm}}\sum_{s}\omega_{ps}^{2}\mu_{s}v_{s2\pm}\vec{v}_{s0}$
(47)

Using equation 41 to eliminate $v_{s2}$ from equations 45 and 47, keeping only
terms up to second order we are left with the following equations, where we
restate the plasma-wave equation for convenience:

$\displaystyle\vec{v}_{s\pm}+\mu_{s}^{-1}\vec{a}_{\pm}-i\beta_{s\pm}\vec{v}_{s\pm}\times\hat{z}=-K_{s\pm}a_{2\pm}(\vec{v}_{s0}+\mu_{s}^{-1}\vec{a}_{0})$
(48)
$(-\omega_{\pm}^{2}+k_{\pm}^{2}c^{2})\vec{a}_{\pm}-\sum_{s}\omega_{ps}^{2}\mu_{s}\vec{v}_{s\pm}=-\frac{k_{2\pm}\omega_{2\pm}}{2}\sum_{s}P_{s\pm}a_{2\pm}\vec{v}_{s0}$
(49)
$\displaystyle(1-\sum_{s}P_{s})a_{2}=\frac{1}{2\omega_{2}}\sum_{s}P_{s}PC_{s2}$
(50)

$K_{s\pm}=\frac{k_{0}\omega_{2\pm}^{2}P_{s\pm}}{2\mu_{s}\omega_{\pm}\omega_{ps}^{2}}$,
$\beta_{s\pm}=s_{s}\frac{\omega_{cs}}{\omega_{\pm}}$ and
$P_{s\pm}=\frac{\omega_{ps}^{2}}{\omega_{2\pm}^{2}-\gamma_{s}k_{2\pm}^{2}v_{Ts}^{2}}$.
$\omega_{2+}=\omega_{2}$, $\omega_{2-}=\omega_{2}^{*}$, and similarly for
$k_{2\pm}$. The equations for wave 0 are equivalent to those for the $\pm$
waves, neglecting second order terms.

At this point, the remaining task is to solve for $\vec{v}_{s\pm}$ in terms of
$\vec{a}_{\pm}$, $a_{2}$, and wave 0 quantities. We will finally arrive at a
$5\times 5$ system for $\vec{a}_{+}$, $\vec{a}_{-}^{*}$, and $a_{2}$, which
includes both the linear waves and parametric coupling to wave 0. For
magnetised waves, this is most easily done in a rotating coordinate system,
where $R$ and $L$ circularly-polarised waves are the linear light waves.

###  II.4 Left-Right Co-ordinate System

It is convenient when dealing with Fourier amplitudes to formulate vectors in
terms of right and left polarised co-ordinates, which are defined in terms of
cartesian co-ordinates as follows:

$\displaystyle\hat{R}=\frac{1}{\sqrt{2}}(\hat{x}+i\hat{y})$ (51)
$\displaystyle\hat{L}=\frac{1}{\sqrt{2}}(\hat{x}-i\hat{y})$

In condensed notation,

$\hat{\sigma}=\frac{1}{\sqrt{2}}(\hat{x}+i\sigma\hat{y})$ (52)

where $\sigma=+1,-1$ for the right and left-polarised basis vectors,
respectively. We define the dot product such that
$\vec{a}\cdot\vec{b}=\sum_{i}a_{i}b_{i}^{*}$. Thus, dot products do not
commute: $\vec{a}\cdot\vec{b}=(\vec{b}\cdot\vec{a})^{*}$. This normalisation
ensures $\hat{\sigma}\cdot\hat{\sigma}=1$. Using this convention, any vector
can be re-written in terms of right and left polarised unit vectors and
amplitudes. Consider, for example, the physical velocity vector
$\vec{v}_{\perp}$, where we explicitly indicate Fourier amplitudes with
subscript $f$:

$\displaystyle\vec{v}_{\perp}$
$\displaystyle=(\hat{x}v_{fx}+\hat{y}v_{fy})e^{i\psi}+cc$ (53)
$\displaystyle=\frac{1}{\sqrt{2}}((\hat{R}+\hat{L})v_{fx}+i(\hat{L}-\hat{R})v_{fy})e^{i\psi}+cc$
$\displaystyle=\frac{1}{\sqrt{2}}(\hat{L}(v_{fx}+iv_{fy})+\hat{R}(v_{fx}-iv_{fy}))e^{i\psi}+cc$
$\displaystyle=\frac{1}{\sqrt{2}}(v_{fL}\hat{L}+v_{fR}\hat{R})e^{i\psi}+cc$
$\displaystyle=e^{i\psi}\sum_{\sigma}v_{f\sigma}\hat{\sigma}+cc$

Note that $\vec{v}\cdot\hat{\sigma}=2^{-1/2}e^{i\psi}(v_{x}-i\sigma
v_{y})=e^{i\psi}v_{f\sigma}+cc$. As an explicit example, for an R wave with
$v_{fR}=V$ real and $v_{fL}=0$,
$\vec{v}_{\perp}=2^{1/2}V(\cos\psi,-\sin\psi)$. At fixed $z$,
$\vec{v}_{\perp}$ rotates clockwise as time increases when looking toward
$-\hat{z}$, which is opposite to $\vec{B}_{eq}$. We therefore follow the
convention used by Stix [28], in which circular polarization is defined
relative to $\vec{B}_{eq}$ and not $\vec{k}$.

We use the result given in the last line of equation 53 to produce the
definition of a dot product of two vectors in Fourier space in this coordinate
system. Consider the vectors $\vec{v}$ and $\vec{a}$:

$\displaystyle\vec{v}.\vec{a}=e^{i(\psi_{i}-\psi_{j}^{*})}(v_{fRi}a_{fRj}^{*}+v_{fLi}a_{fLj}^{*})+cc$
(54)

where the subscripts $i,j$ are the wave indices.

###  II.5 EM Waves in Left-Right Coordinates

Taking $\hat{\sigma}\cdot$ (equations 48 and 49), we obtain

$\displaystyle(1+\sigma\beta_{s\pm})v_{s\pm\sigma}+\mu_{s}^{-1}a_{\pm\sigma}=-K_{s\pm}\left(\mu_{s}^{-1}a_{0\sigma}+v_{s0\sigma}\right)a_{2\pm}$
(55)
$(\omega_{\pm}^{2}-k_{\pm}^{2}c^{2})a_{\pm\sigma}+\sum_{s}\omega_{ps}^{2}\mu_{s}v_{s\pm\sigma}=\frac{k_{2\pm}\omega_{2\pm}}{2}\sum_{s}P_{s\pm}a_{2\pm}v_{s0\sigma}$
(56)

respectively, where $a_{\pm\sigma}\equiv\vec{a}_{\pm}\cdot\hat{\sigma}$. The
definitions of $v_{s\pm\sigma}$, $v_{s0\sigma}$ and $a_{0\sigma}$ are
analogous to that of $a_{\pm\sigma}$. We now have uncoupled equations for
$(a_{\pm\sigma},v_{s\pm\sigma})$ which is the advantage of using rotating
coordinates. This is unlike the original x and y coordinates, which are
coupled due to the $\vec{v}\times\vec{B}$ force. For the pump wave, we have
these equations with subscript $\pm\rightarrow 0$ and set the RHS to 0. Thus

$v_{s0\sigma}=-\frac{1}{\mu_{s}(1+\sigma\beta_{s0})}a_{0\sigma}$ (57)

Rearranging equation 55 to obtain an expression for $v_{s\pm\sigma}$

$(1+\sigma\beta_{s\pm})v_{s\pm\sigma}=-\mu_{s}^{-1}a_{\pm\sigma}-\frac{\sigma
K_{s\pm}\beta_{s0}}{\mu_{s}(1+\sigma\beta_{s0})}a_{0\pm\sigma}a_{2\pm}$ (58)

Substituting this into equation 56, and moving parametric coupling terms to
the right-hand side, we obtain:

$D_{\pm\sigma}a_{\pm\sigma}=-\Delta_{\pm\sigma 2}a_{0\sigma}a_{2\pm}$ (59)

where

$\displaystyle D_{\pm\sigma}$
$\displaystyle=\omega_{\pm}^{2}-k_{\pm}^{2}c^{2}-\sum_{s}\frac{\omega_{ps}^{2}}{1+\sigma\beta_{s\pm}}$
(60) $\displaystyle\Delta_{\pm\sigma 2}$
$\displaystyle=\frac{\omega_{2\pm}}{2}\sum_{s}\frac{P_{s\pm}}{\mu_{s}}\frac{1}{1+\sigma\beta_{s0}}\left(k_{2\pm}-k_{0}\frac{\omega_{2\pm}}{\omega_{\pm}}\frac{\sigma\beta_{s0}}{1+\sigma\beta_{s\pm}}\right)$

This has the desired form, where wave amplitudes are written only in terms of
$a$’s, not $v$’s. For no B field, all $\beta$’s are zero, and the parametric
coupling coefficient $\Delta_{\pm\sigma 2}\propto k_{2\pm}$, the usual
unmagnetised result. To explain the notation, $D_{+R}$ gives the linear
dispersion relation for the scattered upshifted R wave, and $\Delta_{+R2}$ is
the parametric coupling coefficient for that wave and wave 2 (the plasma
wave). Please see the parametric dispersion relation Eq. 67 below.

###  II.6 Plasma Waves in Left-Right Coordinates

Writing the $PC_{s2}$ term in equation 50 in terms of right and left
circularly polarised waves, we obtain:

$\displaystyle
PC_{s2}=-k_{-}^{*}(v_{s0R}a_{-R}^{*}+v_{s0L}a_{-L}^{*})+k_{0}(v_{s-R}^{*}a_{0R}+v_{s-L}^{*}a_{0L})$
(61)
$\displaystyle+k_{+}(v_{s0R}^{*}a_{+R}+v_{s0L}^{*}a_{+L})-k_{0}(v_{s+R}a_{0R}^{*}+v_{s+L}a_{0L}^{*})$

Substituting for $\vec{v}_{s0}$ using equation 57, and $\vec{v}_{s\pm}$ using
equation 58

$\displaystyle-\mu_{s}PC_{s2}=a_{0R}a_{-R}^{*}\left(\frac{k_{0}}{1+\beta_{s-}^{*}}-\frac{k_{-}^{*}}{1+\beta_{s0}}\right)+$
(62) $\displaystyle
a_{0L}a_{-L}^{*}\left(\frac{k_{0}}{1-\beta_{s-}^{*}}-\frac{k_{-}^{*}}{1-\beta_{s0}}\right)$
$\displaystyle+a_{0R}^{*}a_{+R}\left(-\frac{k_{0}}{1+\beta_{s+}}+\frac{k_{+}}{1+\beta_{s0}}\right)$
$\displaystyle+a_{0L}^{*}a_{+L}\left(-\frac{k_{0}}{1-\beta_{s+}}+\frac{k_{+}}{1-\beta_{s0}}\right)$

Equation 50 can now be written in a more condensed form:

$D_{2}a_{2}=-\sum_{\sigma}\left(\Delta_{2+\sigma}a_{0\sigma}^{*}a_{+\sigma}+\Delta_{2-\sigma}a_{0\sigma}a_{-\sigma}^{*}\right)$
(63) $D_{2}=1-\sum_{s}P_{s}$ (64)
$\Delta_{2+\sigma}=\frac{1}{2\omega_{2}}\sum_{s}\frac{P_{s}}{\mu_{s}}\left(\frac{k_{+}}{1+\sigma\beta_{s0}}-\frac{k_{0}}{1+\sigma\beta_{s+}}\right)$
(65)
$\Delta_{2-\sigma}=\frac{1}{2\omega_{2}}\sum_{s}\frac{P_{s}}{\mu_{s}}\left(-\frac{k_{-}^{*}}{1+\sigma\beta_{s0}}+\frac{k_{0}}{1+\sigma\beta_{s-}^{*}}\right)$
(66)

We now have a plasma-wave relation involving just $a$’s.

###  II.7 Parametric Dispersion Relation

Equations 59 (really 4 equations: equation 59 and its complex conjugate for
$\sigma=R,L$) and 63 form a system of 5 linear equations, which can be
summarised in matrix form:

$\displaystyle\begin{bmatrix}D_{+R}&0&0&0&\Delta_{+R2}a_{0R}\\\
0&D_{-R}^{*}&0&0&\Delta_{-R2}^{*}a_{0R}^{*}\\\
0&0&D_{+L}&0&\Delta_{+L2}a_{0L}\\\
0&0&0&D_{-L}^{*}&\Delta_{-L2}^{*}a_{0L}^{*}\\\
\Delta_{2+R}a_{0R}^{*}&\Delta_{2-R}a_{0R}&\Delta_{2+L}a_{0L}^{*}&\Delta_{2-L}a_{0L}&D_{2}\\\
\end{bmatrix}\begin{bmatrix}a_{+R}\\\ a_{-R^{*}}\\\ a_{+L}\\\ a_{-L^{*}}\\\
a_{2}\\\ \end{bmatrix}$ (67) $\displaystyle=0$

The structure of this matrix matches our physical understanding of plasma-wave
dispersion relations: the diagonal terms are independent of $a$, and give rise
to linear waves. The off-diagonal terms are all proportional to $a_{0}$ and
represent parametric coupling between the two daughter waves, one e/m and the
e/s plasma wave. Nonzero solutions exist when the determinant is zero, which
gives the parametric dispersion relation including the pump light wave in the
equilibrium. This is analogous to Drake [24], but generalized to include a
background magnetic field, and specialized to our 1D geometry and fluid
instead of kinetic plasma-wave response.

The parametric dispersion relation couples a pump and scattered e/m wave of
the same R or L polarization. Consider the case where there is only one pump
wave: i.e. either $a_{0R}=0$ or $a_{0L}=0$. Taking $a_{0R}=0$ for
definiteness, waves $a_{-R}$ and $a_{-R}^{*}$ decouple from the dispersion
relation, leaving the following dispersion matrix:

$\begin{bmatrix}D_{+L}&0&\Delta_{+L2}a_{0L}\\\
0&D_{-L}^{*}&\Delta_{-L2}^{*}a_{0L}^{*}\\\
\Delta_{2+L}a_{0L}^{*}&\Delta_{2-L}a_{0L}&D_{2}\\\
\end{bmatrix}\begin{bmatrix}a_{+L}\\\ a_{-L}^{*}\\\ a_{2}\\\ \end{bmatrix}=0$
(68)

Setting the determinant to 0 gives

$D_{+L}D_{-L}^{*}D_{2}=|a_{0L}|^{2}(D_{+L}\Delta_{2-L}\Delta_{-L2}^{*}+D_{-L}^{*}\Delta_{2+L}\Delta_{+L2})$
(69)

$a_{0L}=0$ then gives the three linear dispersion relations for the upshifted
L, downshifted L, and plasma waves: $D_{+L}=0$, $D_{-L}=0$, or $D_{2}=0$.
$a_{0L}\neq 0$ couples the linear waves and gives parametric interaction.

##  III Impact of External B Field on Free Waves

This section considers the linear or free waves, with $a_{0}=0$. Let $a_{1}$
be either $a_{+}$ or $a_{-}$ in equation 67 to obtain the free-wave dispersion
relation:

$\begin{bmatrix}D_{1L}^{*}&0&0\\\ 0&D_{1R}^{*}&0\\\ 0&0&D_{2}\\\
\end{bmatrix}\begin{bmatrix}a_{1L}^{*}\\\ a_{1R}^{*}\\\ a_{2}\\\
\end{bmatrix}=0$ (70)

$\vec{a}\neq 0$ solutions exist if the determinant of this matrix equals 0.
This gives rise to the following dispersion relations, for a single ion
species. For the e/m waves, with $a_{2}=0$, we have $D_{1L}D_{1R}=0$, which
gives

$\omega_{1}^{2}=k_{1}^{2}c^{2}+\frac{\omega_{pe}^{2}}{1-\sigma\frac{\omega_{ce}}{\omega_{1}}}+\frac{\omega_{pi}^{2}}{1+\sigma\frac{\omega_{ci}}{\omega_{1}}}$
(71)

For e/s waves, with $a_{1L}=a_{1R}=0$, we have $D_{2}=0$ and

$\omega_{2}^{2}=\frac{\omega_{pe}^{2}}{1-\gamma_{e}\frac{k_{2}^{2}v_{Te}^{2}}{\omega_{2}^{2}}}+\frac{\omega_{pi}^{2}}{1-\gamma_{i}\frac{k_{2}^{2}v_{Ti}^{2}}{\omega_{2}^{2}}}$
(72)

Note that the background $B$ field has no effect at all on the e/s waves, for
our geometry of $\vec{k}||\vec{B}_{eq}$.

###  III.1 Waves in an unmagnetised Plasma

By setting $\omega_{ce}=0$, we recover the unmagnetised dispersion relation
for electromagnetic waves from equation 71:

$\omega_{1}^{2}=c^{2}k_{1}^{2}+\omega_{pe}^{2}+\omega_{pi}^{2}$ (73)

The ion contribution is usually negligible. Equation 72 gives the
electrostatic waves, with the conventional approximations, like neglecting
ions for electron plasma waves (EPWs), being highly accurate. Namely, we find
the EPW for $\gamma_{e}=3$:

$\omega_{2}^{2}=\omega_{pe}^{2}+3v_{Te}^{2}k_{2}^{2}$ (74)

and the ion acoustic wave (IAW) for $\gamma_{e}=1,\gamma_{i}=3$:

$\omega_{2}^{2}=\frac{Z_{i}T_{e}}{m_{i}}\left(\frac{1}{1+(k_{2}\lambda_{De})^{2}}+\frac{3T_{i}}{Z_{i}T_{e}}\right)k_{2}^{2}$
(75)

with $\lambda_{De}\equiv v_{Te}/\omega_{pe}$. We must retain finite $T_{e}$
for an IAW to exist.

###  III.2 Waves with Magnetic Field

The dispersion relation for free electromagnetic waves in a magnetised plasma
is given in equation 71. As is usual in LPI literature, we view this as giving
$\omega$ as a function of real $k$. This gives a 4th order polynomial for
$\omega$ with four real solutions, each of which corresponds to an e/m wave:

$\displaystyle\omega^{4}-\sigma(\omega_{ce}-\omega_{ci})\omega^{3}-(c^{2}k^{2}+\omega_{ce}\omega_{ci}+\omega_{pe}^{2}+\omega_{pi}^{2})\omega^{2}$
(76)
$\displaystyle+\sigma(\omega_{ce}-\omega_{ci})c^{2}k^{2}\omega+\omega_{ce}\omega_{ci}c^{2}k^{2}=0$

Note one can solve this trivially in closed form for $k$ given $\omega$. In
the following analysis, but not in the numerical solutions, we assume
$Z_{i}m_{e}/m_{i}\ll 1$, so we can drop $\omega_{pi}^{2}$ and set
$\omega_{ce}-\omega_{ci}\rightarrow\omega_{ce}$. In order of descending
frequency, these waves are: the right and left-polarised light waves, the
whistler wave and the ion cyclotron wave (ICW). In addition to these waves,
two electrostatic waves are obtained by solving equation 72: the EPW and the
IAW.

Let us consider the high-frequency e/m waves, the light and whistler waves,
where ion motion can be neglected: $\omega_{ci}\rightarrow 0$. In this case,
equation 76 becomes (removing one $\omega=0$ root)

$\omega^{3}-\sigma\omega_{ce}\omega^{2}-(c^{2}k^{2}+\omega_{pe}^{2})\omega+\sigma\omega_{ce}c^{2}k^{2}=0$
(77)

We assume $\omega_{pe}\gg\omega_{ce}$, which is typical in the HED regime. For
light waves, we consider equation 77 for $\omega\gg\omega_{ce}$. For $k=0$, we
find

$\omega(k=0)\approx\omega_{pe}+\frac{\sigma}{2}\omega_{ce}$ (78)

For all $k$ we write $\omega$ as
$\omega(B_{eq}=0)\equiv(c^{2}k^{2}+\omega_{pe}^{2})^{1/2}$ plus a correction:

$\omega\approx\omega(B_{eq}=0)+\frac{\sigma}{2}\frac{\omega_{pe}^{2}}{\omega(B_{eq}=0)^{2}}\omega_{ce}$
(79)

Whistler wave: We can also solve equation 77 for the whistler wave, which has
$\omega\leq\omega_{ce}$. We call this full set of roots for $\omega$ the
whistler, though some authors only use this term for the small $k$ domain and
use “electron cyclotron wave” when $\omega$ is near $\omega_{ce}$. We derive
expressions for this wave by considering two limits: first, for $k\rightarrow
0$ (but still large enough that we can neglect ion motion, discussed below),
we obtain:

$\omega\approx\sigma\frac{c^{2}k^{2}}{\omega^{2}_{pe}}\omega_{ce}$ (80)

We restrict interest to $\omega>0$ waves, which for the whistler requires the
R wave ($\sigma=1$):

$\omega\approx\frac{c^{2}k^{2}}{\omega^{2}_{pe}}\omega_{ce}\qquad\sigma=1$
(81)

Secondly, for $ck\gg\omega_{pe}$, we obtain:

$\omega\approx\omega_{ce}\left(1-\frac{\omega^{2}_{pe}}{c^{2}k^{2}}\right)\qquad\sigma=1$
(82)

For $\omega$ near $\omega_{ce}$, the whistler group velocity $d\omega/dk$
approaches zero. Since this is the relevant wave propagation speed for three-
wave interactions, such a localized whistler wavepacket would propagate very
slowly. This impacts how stimulated whistler scattering evolves, and how to
practically realize the process in experiments or simulations.

\begin{overpic}[width=227.62204pt,trim=14.22636pt 2.84544pt 8.5359pt
8.5359pt,clip]{epw_emw_b_5000_with_analytic.pdf}
\put(20.0,30.0){$\omega_{ce}$} \end{overpic} (a) (b)

Figure 2: Numerical solutions to the free-wave dispersion relations in a
magnetized plasma, for the conditions in Table 1. Red: right-polarised e/m,
blue: left-polarised e/m, purple: unmagnetised e/m, and black: electrostatic.
Top: high-frequency waves, in decreasing order: e/m light, electron plasma,
and whistler. The black dashed line lies at $\frac{\omega_{ce}}{\omega_{pe}}$.
Bottom low-frequency waves: electrostatic ion acoustic wave, right-polarised
whistler, and left-polarised ion cyclotron waves. Also plotted are the
analytic approximations to the ion cyclotron wave for $ck\gg\omega_{pe}$ (dark
blue) (equation 84), which tends to $\frac{\omega_{ci}}{\omega_{pe}}$ (dashed
black line), and $k\rightarrow 0$, which yields the Alfven frequency (dashed
cyan line), given in equation 83.

The full numerical solutions of the dispersion relations for the whistler wave
and the right and left-polarised light waves are shown in figure 2(a). Note
that here and throughout the rest of the paper, $\lambda_{De}$ is used to
normalise $k$, as is customary for stimulated scattering. For large
$k\lambda_{De}$, the whistler wave tends to $\omega=\omega_{ce}$, shown in
figure 2(a) as a dashed black line.

Ion cyclotron wave: We now consider the ion cyclotron wave (ICW) which
requires the retention of terms involving ion motion. As with the whistler
wave, we consider two regimes. For $k\rightarrow 0$, we seek solutions with
$\omega\propto k$, which gives

$\omega\approx v_{A}k\qquad\sigma=-1\textrm{ or }+1$ (83)

where the Alfven velocity,
$v_{A}=c\frac{\omega_{ci}}{\omega_{pi}}=B/(\rho\mu_{0})^{1/2}$. This solution
applies for both values of $\sigma$, meaning there is both an R wave (the
whistler, including ion motion), and an L wave (the ICW). To see which is
which, we need to take the opposite limit $ck\gg\omega_{pe}$, where we obtain
two solutions with $\omega$ independent of $k$: $\omega=\omega_{ce}$ for
$\sigma=1$ (the right-polarised whistler), and $\omega=\omega_{ci}$ for
$\sigma=-1$ (the left-polarised ICW). Including the next correction term for
the ICW gives

$\omega\approx\omega_{ci}\left(1-\frac{\omega^{2}_{pi}}{c^{2}k^{2}}\right)\qquad\sigma=1$
(84)

Figure 2(a) is re-plotted in figure 2(b) for $\omega\ll\omega_{pe}$ to show
the IAW and ICW clearly. The ICW tends to $\omega=\omega_{ci}$, denoted by a
dashed black line. The numerical and approximate analytic solutions to the ICW
dispersion relation are shown in figure 2(b) in blue and dark blue
respectively. As can be seen from equation 83, at low $k\lambda_{De}$ the ICW
approaches the Alfven frequency, which is represented by a dashed cyan line in
figure 2(b). For large values of $k\lambda_{De}$, the ICW frequency tends to
$\omega_{ci}$, marked by a dashed black line. The parameters used to plot the
dispersion relations shown in figures (2(a)-2(b)) are given in table 1. A
plasma comprised of helium ions and electrons was considered.

Quantity | Value
---|---
Z | 2
A | 4
$T_{e}$ | 2 keV
$T_{i}$ | 1 keV
$\frac{\omega_{ce}}{\omega_{pe}}$ | 0.423
Table 1: Parameters used to plot dispersion relations. Laser wavelength [$\mu$m] | $n_{e}/n_{crit}$ |  $n_{e}$ [$cm^{-3}$] |  $B_{eq}$ [T]
---|---|---|---
0.351 (NIF) | 0.15 | $1.36\times 10^{21}$ | 5000
0.351 | 0.01 | $9.05\times 10^{19}$ | 1290
10.6 (CO2) | 0.15 | $1.49\times 10^{18}$ | 166
10.6 | 0.01 | $9.92\times 10^{16}$ | 42.7
Table 2: Electron densities and magnetic field strengths which correspond to
the normalised parameters considered throughout this paper, for typical NIF
and CO2 laser wavelengths. $\frac{\omega_{ce}}{\omega_{pe}}=0.423$ in all
cases.

###  III.3 Faraday Rotation

Three unique waves exist in an unmagnetised plasma, of which two are
electrostatic (the electron plasma wave, (EPW) and the ion acoustic wave,
(IAW)) and one is electromagnetic (light wave, with two degenerate
polarisations). If the electromagnetic wave is linearly polarised, it can be
written as the sum of two circularly polarised waves of equal amplitude and
opposite handedness (R and L waves). If an external B field, $\vec{B_{eq}}$ is
applied, the R and L waves experience different indices of refraction and
propagate with differing phase velocities. Consequently, the overall
polarisation of the electromagnetic wave, found by summing the R and L waves,
rotates as the electromagnetic wave propagates through the plasma. This is the
well-known Faraday effect, which is briefly derived below.

An expression for the wavenumber of the electromagnetic wave can be obtained
by rearranging equation 71.

$k_{\sigma}=\frac{\omega}{c}\left(1-\frac{\omega_{pe}^{2}}{\omega^{2}(1-\sigma\frac{\omega_{ce}}{\omega})}\right)^{\frac{1}{2}}\
$ (85)

Two first-order Taylor expansions of equation 85, assuming
$\omega\gg\omega_{ce}$, and $\omega\gg\omega_{pe}$ yield:

$k_{\sigma}\approx K-\sigma\Delta K$ (86)

where

$K=\frac{\omega}{c}\left(1-\frac{\omega_{pe}^{2}}{2\omega^{2}}\right),\qquad\Delta
K=\frac{\omega_{pe}^{2}}{2\omega^{2}}\frac{\omega_{ce}}{c}$ (87)

Consider a linearly polarised plane electromagnetic wave. We can write the
physical electric field $\vec{E}=Re[\vec{E}_{F}]$ as the sum of the electric
fields of two circularly polarised waves with opposite handedness:

$\vec{E}_{F}=\epsilon(\hat{R}e^{i\psi_{R}}+\hat{L}e^{i\psi_{L}})\qquad\psi_{R,L}\equiv
k_{R,L}z-\omega t$ (88)

Writing this in Cartesian co-ordinates,

$\frac{2^{1/2}}{\epsilon}\vec{E}_{F}=\hat{x}(e^{i\psi_{L}}+e^{i\psi_{R}})+i\hat{y}(e^{i\psi_{L}}-e^{i\psi_{R}})$
(89)

Assuming $\epsilon$ is real,

$\displaystyle\vec{E}=E(\cos\phi,-\sin\phi)$ (90) $\displaystyle
E=|2^{1/2}\epsilon\cos[(1/2)(k_{L}+k_{R})z-\omega t]|$
$\displaystyle\phi=\frac{1}{2}(k_{L}-k_{R})z=\Delta Kz$

At a fixed $z$, $\vec{E}$ always lies along the same line in the $xy$ plane,
with its exact position varying in time. As $z$ varies, the angle $\phi$ this
line makes with respect to the $x$ axis increases at the rate

$\frac{\partial\phi}{\partial z}=\Delta
K=16.8\frac{n_{e}}{n_{crit}}B_{eq}\textrm{[T] [deg/mm]}$ (91)

The final formula is in practical units. We have introduced the critical
density $n_{crit}\equiv(\epsilon_{0}m_{e}/e^{2})\omega^{2}$, which is the
usual definition for unmagnetised plasma. When discussing LPI, $n_{crit}$ is
for the pump wave $\omega_{0}$. Significant Faraday rotation is thus possible
in current ICF platforms with modest B fields. For instance, with
$n_{e}/n_{crit}=0.1$ and $B_{eq}=10$ T, we obtain
$\partial_{z}\phi=16.8^{\circ}/$mm. This could be used to diagnose $n_{e}$ (a
common technique when feasible), and could affect LPI processes such as
crossed-beam energy transfer. [29, 30, 31]

##  IV Impact of external B field on Parametric Coupling

We apply the above theory to magnetized LPI in HED relevant conditions, all
for $\vec{k}||\vec{B_{eq}}||\hat{z}$. We consider how the imposed field
modifies stimulated Raman (SRS) and Brillouin (SBS) scattering, as well as
stimulated whistler scattering (SWS) which only occurs in a background field.
Recall $\vec{k}_{i}=k_{i}\hat{z}$ and we choose $k_{0}>0$. $k_{1}$ and $k_{2}$
can have either sign. Let $c_{i}=$sign($k_{i}$) for $i=1,2$. For all three
parametric processes we discuss, “forward scatter” refers to the case where
the scattered e/m wave propagates in the same direction as the pump
$(c_{1}=+1)$, and “backward scatter” to the opposite case $(c_{1}=-1)$. To
satisfy $k$ matching, we cannot have both $c_{1}=-1$ and $c_{2}=-1$. For SRS
and SBS, $c_{2}$ must equal +1, but for SWS $c_{2}=-1$ is possible.

We do not consider growth rates, but focus instead on the “kinematics” of
three-wave interactions, through the phase-matching conditions among free
waves. We study the scattered e/m wave frequency $\omega_{1}$, since this is
what escapes the plasma and is measured experimentally. As discussed in
section III.3, $\vec{B_{eq}}$ causes the R and L waves to propagate with
different phase velocities. Therefore, a laser or other external source that
imposes a linearly-polarised light wave of frequency $\omega_{0}$ couples to
an R and an L wave in a magnetised plasma. For stimulated scattering, we are
mostly interested in down-shifted scattered waves for which
$\omega_{1}<\omega_{0}$, which have the same polarisation as the pump: an R or
L pump couples to a down-shifted R or L scattered wave, respectively, hence
$\sigma_{1}=\sigma_{0}$ which we sometimes denote as $\sigma$. We discuss SRS
and SBS, which can be driven by either an R or L pump, and SWS, which can only
be driven by an R pump (since the whistler wave is an R wave). Table 3
summarizes the processes we study.

Process | pump e/m wave | scattered e/m wave | plasma wave | geometries $(c_{1},c_{2})$ |  $\omega_{1}$ range |  $n_{e}/n_{crit}$ range
---|---|---|---|---|---|---
SRS | R,L | R,L | EPW | (1,1), (-1, 1) | $>\omega_{pe}$ | $<1/4$
SBS | R,L | R,L | IAW | (1,1), (-1, 1) | $\gtrsim\omega_{0}-\omega_{pi}$ | $<1$
SWS | R | R-whistler | EPW | (1,-1), (-1, 1) | $<\omega_{ce}$ |  $\gtrsim(1-\omega_{ce}/\omega_{0})^{2}$ for $T_{e}=0$
Table 3: Summary of parametric processes we study. L, R refer to left, right
polarised e/m waves.

In order to derive a dispersion relation for $\omega_{1}$ in terms of known
inputs, we begin with the identity $k_{2}=k_{2}$. We use $k$ matching to write
$k_{2}=k_{0}-k_{1}$ on the left side, and the plasma-wave dispersion relation
of interest to re-write the right side in terms of $\omega_{2}$. We then use
the e/m dispersion relation to write $k_{1}$ in terms of $\omega_{1}$, and use
$\omega$ matching to write $\omega_{2}=\omega_{0}-\omega_{1}$. For SRS and SWS
this yields
$k_{0}-k_{1}=(\omega_{2}^{2}-\omega_{pe}^{2})^{1/2}/v_{Te}3^{1/2}$. The same
method is applied for SBS, where $k_{2}$ is written in terms of $\omega_{2}$
using the simple IAW dispersion relation, $\omega_{2}=c_{s}|k_{2}|$, for an
approximate analysis (the numerical roots use the full e/s dispersion
relation). That is, $c_{s}^{2}=(Z_{i}T_{e}/m_{i})(1+3T_{i}/Z_{i}T_{e})$. The
resulting dispersion relations can be summarised as follows:

$\displaystyle
M_{Y}\equiv(1-\Omega_{pe}^{2}(1-\sigma_{0}\Omega_{ce})^{-1})^{1/2}$ (92)
$\displaystyle-
c_{1}\Omega_{1}(1-\Omega_{1}^{-2}\Omega_{pe}^{2}(1-\sigma_{1}\Omega_{ce}/\Omega_{1})^{-1})^{1/2}-P_{Y}=0$

where $Y$ is either RW, for SRS and SWS, or B, for SBS. For SRS and SWS
$P_{Y}=P_{RW}=c_{2}V_{e}^{-1}((1-\Omega_{1})^{2}-\Omega_{pe}^{2})^{1/2}$,
where $V_{e}\equiv v_{Te}3^{1/2}/c$. For SBS,
$P_{Y}=P_{B}=V_{s}^{-1}(1-\Omega_{1})$, with $V_{s}\equiv c_{s}/c$. This is
usually very small, with $10^{-3}$ a typical magnitude.
$\Omega_{X}\equiv\omega_{X}/\omega_{0}$, where $X$ denotes any angular
frequency subscript in equation LABEL:M_general. The frequency of scattered
light which satisfies phase matching is given by the roots of equation
LABEL:M_general, which can be found by plotting $M_{Y}$ vs. $\Omega_{1}$. This
is illustrated for SRS and SWS in figure 3, and for SBS in figure 4, for the
parameters given in table 1 and $n_{e}/n_{crit}=0.15$.

\begin{overpic}[width=241.84842pt,trim=11.38092pt 5.69046pt 9.95863pt
9.95863pt,clip]{MRW_c1=-1_c2=1_right_and_left_pol_pump.pdf}
\put(80.0,55.0){SRS} \put(25.0,50.0){SWS} \end{overpic} Figure 3: The
dispersion relation for SRS and SWS, $M_{RW}$ is plotted vs.
$\omega_{1}/\omega_{0}$. Its roots $M_{RW}=0$ are indicted by magenta points.
This is for backscatter ($c_{1}=-1,c_{2}=1$) and the parameters of Table 1
plus $n_{e}/n_{crit}=0.15$. SWS is possible for a right polarised pump (red),
but cannot occur when the pump is left polarised (blue). Figure 4: The
dispersion relation for SBS, $M_{B}$ is plotted vs. $\omega_{1}/\omega_{0}$,
for the same parameters as Fig. 3. Its roots $M_{B}=0$ are indicted by magenta
points. The roots of $M_{B}$ occur at similar, but not identical
$\omega_{1}/\omega_{0}$ for a left and right polarised pump.

The dispersion relations given in equation LABEL:M_general are plotted as a
function of $\omega_{1}/\omega_{0}$ and $n_{e}/n_{crit}$ for scattering
geometries $(c_{1},c_{2})=(-1,1),(1,1),(1,-1)$, in figures 5, 6 and 7,
respectively. The two dispersion relations, $M_{RW}$ and $M_{B}$ have been
overplotted. To distinguish between them, $M_{RW}$ has been cross-hatched,
whilst $M_{B}$ has not. The colour scale for M applies to both $M_{RW}$ and
$M_{B}$. The regions of figures 5, 6 and 7 where M is not real are coloured
gray. The regions of the plot where $M_{RW,B}\neq 0$ serve only to illustrate
the root-finding method employed: to ensure we have correctly identified
roots, we check that $M_{RW,B}$ has changed sign. The roots of M have been
computed numerically and are plotted as black contours. These contours
indicate whether SRS, SBS or SWS can occur for the geometry and plasma
conditions considered, and illustrate the relationship between the normalised
plasma density and scattered EMW frequency for each of these processes. The
contours which correspond to a given parametric process are appropriately
labelled.

In figures 5 and 6 a sharp decrease can be seen in the frequency of SRS
scattered light with increasing plasma density. Also in figures 5 and 7, the
frequency of SWS scattered light rises with electron density before reaching a
maximum, and falling. It is often useful to obtain limits in parameter space
beyond which phase matching cannot occur. For example, in an unmagnetised
plasma, SRS is only possible for $n_{e}/n_{crit}<0.25$. The region of
parameter space in which SWS can occur is also restricted, as
$\omega_{1}\leq\omega_{ce}$. Using the same method as for SRS, the following
inequality is obtained for the normalised electron densities at which SWS can
occur in a cold plasma:

$\frac{n_{e}}{n_{crit}}\geq(1-\omega_{ce}/\omega_{0})^{2}$ (93)

These three limits are shown in figures 5, 6 and 7 in cyan, magenta and
purple, respectively. Note that the contours for SRS and SWS always lie within
$n_{e}/n_{crit}<0.25$ and $\omega_{1}\leq\omega_{ce}$ respectively, as
expected. SWS does not respect Eq. 93, as discussed further below.

Figure 5: The dispersion relations for SWS and SRS $(M_{RW})$ and SBS
$(M_{B})$ vs. electron density and scattered light frequency. $T_{e}=4$ keV,
$T_{i}=2$ keV, $\omega_{ce}/\omega_{pe}=0.423$, and we consider backscatter
$(c_{1}=-1,c_{2}=1)$. $M_{RW}$ is distinguished by cross-hatching. The roots
of $M$ are plotted as black contours which have been labelled appropriately.
Three other curves have been plotted: $n_{e}/n_{crit}=0.25$, the maximum
density at which SRS occurs, $\omega_{1}=\omega_{ce}$, the maximum SWS
frequency, and $n_{e}/n_{crit}\geq(1-\omega_{ce}/\omega_{0})^{2}$, the minimum
density at which SWS can occur in a cold plasma. Note that $M_{RW}$ adheres to
only the first two of these approximate analytic
limits.\begin{overpic}[width=241.84842pt,trim=11.38092pt 11.38092pt 11.38092pt
8.5359pt,clip]{srs_w_sbs_w_sws_phase_space_warm_wce_wpe=0pt423_c1=-1_c2=1_reduced_M_range.png}
\put(18.0,60.0){SRS} \put(32.0,22.0){SWS} \put(20.0,80.0){SBS} \end{overpic}
Figure 6: As figure 5, but for forward scatter $(c_{1}=c_{2}=1)$. Only SRS can
occur for this geometry. While SBS is kinematically possible, the ion wave has
$k_{2},\omega_{2}=0$, and SBS has 0 growth rate. Thus, the solution plotted is
spurious. For this geometry, SWS is kinematically
disallowed.\begin{overpic}[width=241.84842pt,trim=11.38092pt 11.38092pt
11.38092pt
8.5359pt,clip]{srs_w_sbs_w_sws_phase_space_warm_wce_wpe=0pt423_c1=1_c2=1.png}
\put(18.0,64.0){SRS} \put(20.0,80.25){SBS} \end{overpic} Figure 7: As figure
5, but for $c_{1}=1$ and $c_{2}=-1$. For this geometry, phase matching is only
satisfied for SWS, and unphysical SBS as in figure 6. As in figure 5,
$M_{RW}=0$ is only satisfied for densities above the minimum normalised
electron density in a cold plasma,
$n_{e}/n_{crit}\geq(1-\omega_{ce}/\omega_{0})^{2}$, which is plotted in
purple.\begin{overpic}[width=241.84842pt,trim=11.38092pt 11.38092pt 11.38092pt
8.5359pt,clip]{srs_w_sbs_w_sws_phase_space_warm_wce_wpe=0pt423_c1=1_c2=-1.png}
\put(41.0,23.0){SWS} \put(20.0,80.25){SBS} \end{overpic}

###  IV.1 Stimulated Raman Scattering: SRS

The dispersion relation for SRS is given by equation LABEL:M_general, where
$c_{2}=1$. For a cold plasma with $V_{e}=0$, we find $\Omega_{2}=\Omega_{p}$
always, so $\Omega_{1}=1-\Omega_{p}$. This is true with or without a
background field $B_{eq}$. Thus, any effect of $B_{eq}$ on $\Omega_{1}$ is
“doubly small”, in that it also relies on thermal effects. For no background
field $\Omega_{ce}=0$, we obtain the usual solutions, which for $V_{e}\ll 1$
and $\Omega_{p}\ll 1$ are $\Omega_{1}\approx
1-\Omega_{p}-(\Omega_{p}/2)V_{e}^{2}$ for $c_{1}=1$ (forward scatter), and
$\Omega_{1}\approx 1-\Omega_{p}-(2/\Omega_{p})V_{e}^{2}$ for $c_{1}=-1$
(backscatter).

Including a weak background field, we write
$\Omega_{1}\approx\Omega_{1U}+\delta\Omega_{1}$ where $\Omega_{1U}$ is the
solution for $\Omega_{ce}=0$: $M[\Omega_{1U},\Omega_{ce}=0]=0$. We have
$M[\Omega_{1U}+\delta\Omega_{1},\Omega_{ce}]\approx
M[\Omega_{1U},0]+\delta\Omega_{1}(\partial
M/\partial\Omega_{1})+\Omega_{ce}\partial M/\partial\Omega_{ce}=0$, which
gives $\delta\Omega_{1}\approx\alpha\Omega_{ce}$ with $\alpha=-(\partial
M/\partial\Omega_{ce})/(\partial M/\partial\Omega_{1})$. All partials are
evaluated at $\Omega_{1}=\Omega_{1U}$ and $\Omega_{ce}=0$. One can find a
formula for $\alpha$, but it is unilluminating. We quote the result in the
limit that $V_{e}\ll 1$ and $\Omega_{p}\ll 1$:

$\alpha\approx
c_{1}\left(2/\Omega_{p}^{2}+1/\Omega_{p}+2\right)^{\frac{1-c_{1}}{2}}\sigma_{0}V_{e}^{2}\Omega_{p}^{3}$
(94)

The full numerical solution of $M_{RW}$ (see equation LABEL:M_general) is
plotted in figures 8 and 9 for the plasma conditions given in table 1 and the
first row of table 2. The frequencies, wave vectors and, if applicable, the
polarisations of the e/m and e/s waves at which phase-matching conditions are
met are illustrated by parallelograms. Specifically, figures 8 and 9
correspond to forward and back-SRS, respectively.

Figure 8: Phase-matching parallelograms for forward-SRS light for plasma
conditions given in table 1, with $n_{e}/n_{crit}=0.15$. The right and left-
polarised e/m waves are plotted in red and blue, respectively, while the
unmagnetised e/m wave and the electrostatic EPW are shown in purple and black,
respectively. The phase-matching parallelograms are colour-coded according to
the polarisation of the pump wave. The pump frequency $\omega_{0}$ is fixed in
all cases, which gives slightly different $k_{0}$’s from the relevant
dispersion relations. The scattered e/m frequencies $\omega_{1}$ are nearly
but not exactly the same, though this is very hard to see visually. The pump
and scattered e/m waves have the same
handedness.\begin{overpic}[width=497.92322pt,trim=0.0pt 0.0pt 0.0pt
0.0pt,clip]{SRS_forward_scatter_R_L_and_0_ne_0point15.pdf}
\put(85.5,22.0){Pump} \put(56.0,27.0){Scattered} \put(58.0,15.0){EPW}
\end{overpic} Figure 9: Phase-matching parallelograms for backward-SRS light:
otherwise same as Fig. 8. \begin{overpic}[width=497.92322pt,trim=2.84544pt
0.0pt 0.0pt 0.0pt,clip]{SRS_backward_scatter_R_L_and_0_ne_0point15.pdf}
\put(80.0,41.0){Pump} \put(60.0,48.0){Scattered} \put(52.0,14.0){EPW}
\end{overpic}

The shift in wavelength of SRS light due to the presence of the external
magnetic field, $\Delta\lambda_{1}=\lambda_{1}-\lambda_{1}(\omega_{ce}=0)$, is
given by

$\frac{\Delta\lambda_{1}}{\lambda_{0}}=\omega_{0}\left(\frac{1}{\omega_{1}}-\frac{1}{\omega_{1}(\omega_{ce}=0)}\right)$
(95)

Substituting from equation 85, and treating temperature and magnetic field as
small perturbations in $\Omega_{1}$ as detailed above, we derive the following
expression for $\Delta\lambda_{1}$ to first order in $\Omega_{ce}$ and
$\Omega_{pe}^{2}$:

$\frac{\Delta\lambda_{1}}{\lambda_{0}}\approx-\frac{\delta\Omega_{1}}{\Omega_{1U}^{2}}$
(96)

or equivalently

$\displaystyle\Delta\lambda_{1}[nm]\approx-c_{1}\lambda^{2}_{0}[{\mu
m}^{2}]\frac{5.48\times
10^{-4}}{\Omega_{1U}^{2}}Te[keV]\left(\frac{n_{e}}{n_{crit}}\right)^{3/2}$
(97) $\displaystyle
B[T]\left(2\frac{n_{crit}}{n_{e}}+\sqrt{\frac{n_{crit}}{n_{e}}}+2\right)^{\frac{1-c_{1}}{2}}\sigma_{0}$

in practical units. Under the conditions given in table 1, for
$n_{e}/n_{crit}=0.15$ and $B=100$T for SRS backscattered light from a left-
polarised pump wave, the analytic approximation yields
$\Delta\lambda_{1}=-0.041$nm, compared to the full numerical solution, which
gives $\Delta\lambda_{1}=-0.046$nm. Typically, in NIF-type experiments, the
wavelength of back-SRS light is in the range 500-600nm, with a spectral width
of 5-10nm due to damping and gradients. Given that this is the case, detecting
sub-Angstrom shifts in this spectrum presents a significant challenge. This
first-order approximation of $\Delta\lambda_{1}$ agrees reasonably closely
with the full numerical computation of $\Delta\lambda_{1}$,which is plotted as
a function of $\omega_{ce}/\omega_{pe}$ for $T_{e}=2$keV, $4$keV and
$n_{e}/n_{crit}=0.05$, $0.15$ in figures 10(a) and 10(b), for forward and
back-SRS light, respectively. The effect of electron density and temperature
become particularly significant for forward and backward-SRS light from a
right-polarised pump as $\omega_{ce}\rightarrow\omega_{pe}$, as in this limit,
$\Delta\lambda_{1}\rightarrow\infty$, $-\infty$, respectively.

(a)

(b)

Figure 10: $\Delta\lambda_{1}$, the difference in wavelength of forward
(10(a)) and backward (10(b)) SRS light in a magnetised versus an unmagnetised
plasma, for $T_{e}=2.0$keV, $4.0$keV, $n_{e}/n_{crit}=0.05$, $0.15$ and
$\lambda_{0}=351$nm. For [forward, backward] SRS, $\Delta\lambda_{1}$ is
$[>0,<0]$ for a right-polarised pump and $[<0,>0]$ for a left-polarised pump.

(a)

(b)

Figure 11: $\Delta\lambda_{1}$ of forward (11(a)) and backward (11(b)) SRS
light, plotted for $\omega_{ce}/\omega_{pe}=0.1$,
$\frac{n_{e}}{n_{crit}}=0.05$ and $\lambda_{0}=351$nm. Full numerical
solutions are unbroken lines, first-order analytic approximations are dashed
lines.

###  IV.2 Stimulated Brillouin Scattering: SBS

The phase matching relation for SBS, $M_{B}=0$ is derived in section IV, and
given in equation LABEL:M_general. Exact forward SBS ($c_{1}=1$) is not
considered since in our strictly 1D geometry it does not occur. $M_{B}=0$ has
a spurious root for $k_{2}=\omega_{2}=0$, which connects to near-forward
scatter for small but nonzero angle between $\vec{k}_{0}$ and $\vec{k}_{1}$.
The SBS growth rate is zero for $k_{2}=0$, so we discuss only backscatter
($c_{1}=-1$, $c_{2}=1$). For $\Omega_{ce}=0$, the exact solution is

$\Omega_{1U}=\frac{1-2\eta_{0}V_{s}+V_{s}^{2}}{1-V_{s}^{2}}\approx
1-2\eta_{0}V_{s}$ (98)

with $\eta_{0}\equiv(1-\Omega_{pe}^{2})^{1/2}$. The approximate form for
$V_{s}\ll 1$ is typically quite accurate. The correction for a weak $B$ field
and to leading order in $V_{s}^{2}$ is

$\delta\Omega_{1}=\sigma_{0}\Omega_{pe}^{2}V_{s}\Omega_{ce}\left(1+V_{s}\right)$
(99)

For simplicity, we set the final factor to 1 below. As with SRS, the
correction is “doubly small” since it scales with the product of $V_{s}\propto
T_{e}^{1/2}$ and $\Omega_{ce}$. The scattered wavelength shift
$\delta\lambda_{1}\equiv\lambda_{1}-\lambda_{1}[\Omega_{ce}=0]$, evaluated at
$\Omega_{1U}=1$, is

$\frac{\delta\lambda_{1}}{\lambda_{0}}\approx-\sigma_{0}\Omega_{pe}^{2}V_{s}\Omega_{ce}\left(1+V_{s}\right)$
(100)

In practical units,

$\displaystyle\delta\lambda_{1}[Ang.]\approx-9.67\times
10^{-4}\sigma_{0}\frac{n_{e}}{n_{crit}}B[T]$ (101)
$\displaystyle\sqrt{\frac{Z_{i}T_{e}[keV]}{A_{i}}\left(1+\frac{3T_{i}[keV]}{Z_{i}T_{e}[keV]}\right)}\lambda_{0}^{2}[\mu
m^{2}]$

This is an extremely small value for ICF conditions. For the parameters shown
in table 2, with $\lambda_{0}=351$ nm, $n_{e}/n_{crit}=0.15$, $B=100$T and a
right-polarised pump, the analytic approximation gives
$\delta\lambda_{1}\approx-2.37$pm, whereas the full numerical solution gives
$\delta\lambda_{1}\approx-2.41$pm.

Figure 12: Phase-matching parallelograms for backward-SBS, otherwise same as
Fig.8. Electrostatic IAW shown in black.
\begin{overpic}[width=483.69684pt,trim=2.84544pt 0.0pt 0.0pt
0.0pt,clip]{SBS_backward_scatter_R_L_and_0_ne_0point15.pdf}
\put(80.0,40.0){Pump} \put(61.0,42.0){Scattered} \put(50.0,13.0){IAW}
\end{overpic}

[]

Figure 13: $\delta\lambda_{1}$, the difference in wavelength of backward-SBS
light in a magnetised versus an unmagnetised plasma, for three combinations of
electron temperatures and densities $T_{e}=2.0$keV, $4.0$keV, and
$n_{e}/n_{crit}=0.05$, $0.15$, where the ratio of electron and ion temperature
is kept constant: $T_{e}/T_{i}=2$. The laser wavelength, $\lambda_{0}=351$nm.
The full numerical solutions and their analytic counterparts are plotted as
unbroken and dashed lines, respectively. $\Delta\lambda_{1}$$[<0,>0]$ for a
right or left-polarised pump, respectively.

[]

Figure 14: $\Delta\lambda_{1}$ of backwards SBS light, plotted for
$\omega_{ce}/\omega_{pe}=0.423$, $\frac{n_{e}}{n_{crit}}=0.15$ and
$\lambda_{0}=351$nm. Full numerical solutions are unbroken lines, analytic
approximations as dashed lines.

###  IV.3 Stimulated Whistler Scattering: SWS

We now discuss SWS, which only occurs with a background magnetic field. It
resembles SRS, except the scattered e/m wave is a low-frequency whistler
($\omega_{1}<\omega_{ce}$). For a cold plasma, this imposes a minimum density
of $n_{e}/n_{crit}\geq(1-\omega_{ce}/\omega_{0})^{2}$ to satisfy frequency
matching, as opposed to a maximum of $n_{e}/n_{crit}<1/4$ for SRS. Forward
($c_{1}=+1,c_{2}=-1$) and backward ($c_{1}=-1,c_{2}=+1$) SWS are both
kinematically allowed, though forward SWS can only occur for a plasma wave
propagating counter to the pump: $c_{2}=-1$. The phase-matching condition
$M_{RW}$ for SWS, given in equation LABEL:M_general, is identical to that of
SRS except that $c_{2}=\pm 1$. Figures 17 and 15 show SWS phase matching
diagrams for the allowed geometries and for a range of $n_{e}/n_{crit}$,
$\omega_{ce}/\omega_{pe}$ and $T_{e}$.

The relationship between $\omega_{1}/\omega_{0}$, $k_{2}\lambda_{De}$ and
$\omega_{ce}/\omega_{pe}$ is shown in figures 18 and 16 for
$(c_{1},c_{2})=(-1,1),(1,-1)$, respectively, for a range of plasma densities
and temperatures. The frequency of the scattered EMW increases with increasing
magnetic field strength, before saturating. The rate of increase with
$\omega_{ce}/\omega_{pe}$, and the values of $\omega_{1}/\omega_{0}$ and
$\omega_{ce}/\omega_{pe}$ at which saturation occurs vary with plasma density
and temperature. Increasing $T_{e}$ decreases the $\omega_{1}/\omega_{0}$ at
which the trend saturates, while increasing $n_{e}/n_{crit}$ causes the
observed trend to saturate at lower $\omega_{1}/\omega_{0}$ and
$\omega_{ce}/\omega_{pe}$. $k_{2}\lambda_{De}$ is plotted to indicate the
magnitude of Landau damping, which is expected to significantly reduce SWS
growth for $k_{2}\lambda_{De}\gtrsim 0.5$. In the opposite limit, the SWS
growth rate approaches zero as $k_{2}\lambda_{De}\rightarrow 0$.

The wavelength of SWS scattered light is

$\lambda_{1}[\mu m]=\frac{\omega_{ce}}{\omega_{1}}\frac{10709.7}{B[T]}$ (102)

For the bottom rows of table 4, $n_{e}/n_{crit}=0.15$,
$\omega_{ce}/\omega_{pe}=0.423$, and $\omega_{1}\approx\omega_{ce}$. For a
pump wavelength of $0.351\mu m$, we have $B=5000$T and $\lambda_{1}\approx
2.14\mu$m. This is in the near infrared, where detectors exist but are not
commonly fielded on ICF lasers. More realistic $B$ fields will be much lower,
and $\lambda_{1}$ much longer.

In order for SWS scattered light to be detected, it must first leave the
plasma and propagate to a detector. Given the long wavelength of SWS scattered
light, there is a possibility that changing plasma conditions experienced by
the wave as it propagates through the plasma may cause it to become
evanescent. Consider equation 77. Rearranging for k, we obtain:

$c^{2}k^{2}=\omega^{2}-\frac{\omega^{2}_{pe}}{1-\sigma\frac{\omega_{ce}}{\omega}}$
(103)

We see that for
$\omega^{2}>\frac{\omega^{2}_{pe}}{1-\sigma\frac{\omega_{ce}}{\omega}}$, $k$
is real and the wave can propagate. If the reverse is true, $k$ is imaginary
and the wave is evanescent. $\omega_{pe}$ and $\omega_{ce}$ vary in space, and
generally go to zero far from the target. If $B$ tends to zero too rapidly,
the dispersion relation tends to the unmagnetised one,
$c^{2}k^{2}=\omega^{2}-\omega^{2}_{pe}$. In this case, if $n_{e}$ exceeds the
critical density of the SWS scattered light wave, the wave will be reflected
and will not reach the detector. However, if the magnetic field strength
decreases slowly enough and/or the electron density decreases quickly enough,
the wave will escape the plasma. Then $\omega_{pe}=0$ and $ck=\omega$, that
is, it becomes a vacuum light wave and can propagate to the detector.

We now discuss the variation of SWS with plasma parameters. For finite
$T_{e}$, Langmuir-wave frequency increases, an effect comparable to an
increase in electron density. This enables SWS to occur at densities lower
than the minimum density in a cold plasma, given in eqn 93. We see this in
Fig. 15, where the lowest density shown, $n_{e}/n_{crit}=0.15$, corresponds to
the highest pump frequency and a very high Langmuir-wave frequency,
$\omega_{2}/\omega_{pe}>$2\. This requires a large $k_{2}\lambda_{De}>1$,
which entails considerable Landau damping and therefore a low SWS growth rate.
Although growth rates are beyond the scope of this paper, other work
establishes that they generally are $\propto k_{2}^{p}$ (for some power $p$)
when $k_{2}\lambda_{De}$ is small, and decrease with increasing Landau damping
for large $k_{2}\lambda_{De}$. This means there is an effective low-density
cut-off, below which SWS is kinematically allowed but strongly damped. In the
opposite limit, as $n_{e}$ approaches $n_{crit}$ (such as $n_{e}/n_{crit}=0.6$
in figures 15 and 18 and table 4), $k_{2}$ becomes small and Landau damping is
negligible, however the growth rate of SWS also tends to 0. There is thus an
intermediate range of $n_{e}$ in which the growth rate is optimal, and
$k_{2}\lambda_{De}$ is moderate. The case where $n_{e}/n_{crit}=0.4$ and
$T_{e}=2$ keV shown in the figures 15 and 17 and table 4 typifies this regime.

[]

Figure 15: Phase-matching parallelogram for forward SWS: $c_{1}=1,c_{2}=-1$,
where $\omega_{ce}/\omega_{pe}=0.423$ and $T_{i}=T_{e}/2$.

[]

Figure 16: Frequency (unbroken lines) of forward-SWS scattered light
$(c_{1}=1,c_{2}=-1)$, and Langmuir wave $k_{2}\lambda_{De}$ (dashed lines) for
various plasma densities, $n_{e}/n_{crit}=0.6,0.15$, and species temperatures,
$T_{e}=4,0.5$keV, $T_{i}=T_{e}/2$keV. $k_{2}\lambda_{De}$ is plotted to
indicate the strength of Landau damping.

[]

Figure 17: Phase-matching parallelogram for backward SWS ($c_{1}=-1,c_{2}=1$), for a range of electron densities and temperatures, where $\omega_{ce}/\omega_{pe}=0.423$ and $T_{i}=T_{e}/2$. $c_{1}$ | $c_{2}$ | $n_{e}/n_{crit}$ |  $T_{e}$ [keV] | $\omega_{1}/\omega_{0}$ | $\omega_{1}/\omega_{ce}$ | $k_{2}\lambda_{De}$ | $(1-\frac{\omega_{ce}}{\omega_{0}})^{2}$
---|---|---|---|---|---|---|---
-1 | 1 | 0.6 | 0.5 | 0.2212 | 0.6752 | 0.0592 | 0.452
1 | -1 | 0.6 | 0.5 | 0.224 | 0.6836 | 0.0336 | 0.452
-1 | 1 | 0.6 | 4 | 0.1995 | 0.609 | 0.1502 | 0.452
1 | -1 | 0.6 | 4 | 0.2163 | 0.6601 | 0.0883 | 0.452
-1 | 1 | 0.4 | 2 | 0.2557 | 0.9557 | 0.3582 | 0.5365
1 | -1 | 0.4 | 2 | 0.2615 | 0.9776 | 0.3479 | 0.5365
-1 | 1 | 0.15 | 4 | 0.1623 | 0.9904 | 1.1074 | 0.6992
1 | -1 | 0.15 | 4 | 0.1631 | 0.9955 | 1.106 | 0.6992
Table 4: Frequencies of stimulated whistler scattered light for several
$n_{e}/n_{crit}$ and $T_{e}$ (ion temperature, $T_{i}=T_{e}/2$), and their
corresponding values of the normalised EPW wavenumber. For all cases,
$\omega_{ce}/\omega_{pe}=0.423$. The rightmost column is the minimum
$n_{e}/n_{crit}$ for SWS to occur in a cold plasma.

[]

Figure 18: Frequency backward SWS light with $c_{2}=1$, for
$n_{e}/n_{crit}=0.6,0.15$ and $T_{e}=4,0.5$keV.

##  V Conclusion

We presented a warm-fluid theory for magnetized LPI, for the simple geometry
of all wavevectors parallel to a uniform, background field. The field affects
the electromagnetic linear waves in a plasma, though the electrostatic waves
are unaffected for our geometry. Specifically, the right and left circular
polarised e/m waves become non-degenerate, and form the natural basis, as
opposed to linearly polarised waves. This allows for Faraday rotation, which
could be significant on existing ICF laser facilities for magnetic fields
imposable with current technology. The field introduces two new e/m waves, the
ion cyclotron and whistler wave, with no analogues in an unmagnetised plasma.

We found a parametric dispersion relation to first order in parametric
coupling, Eq. 67, analogous to the classic 1974 work of Drake [24]. We then
focused on the kinematics of phase matching for three-wave interactions. Since
the right and left circular polarised light waves have different $k$ vectors
for the same frequency, the background field introduces a small shift in the
scattered SRS and SBS frequencies compared to the unmagnetised case. The sign
of the shift depends on the pump polarization and forward vs. backward
scatter. The shift’s magnitude increases with magnetic field, electron
temperature, and plasma density. The wavelength shifts are $\lesssim 1$ Ang.
for SRS, and $\lesssim 0.1$ Ang. for SBS, for plasma and magnetic field
conditions currently accessible on lasers like NIF. Such small shifts would be
extremely challenging to detect.

The new waves supported by the background B field also allow new parametric
processes, such as stimulated whistler scattering (SWS) which we studied in
detail. In this process, a light wave decays to a whistler wave and Langmuir
wave. This is analogous to Raman scattering, with the whistler replacing the
scattered light wave. We expect SWS scattered light to be infrared, with
wavelength 1 to 100 $\mu$m for fields of 10 kT to 100 T. The whistler
wavelength was found to decrease with increasing magnetic field strength, and
increase with increasing plasma density and temperature. In a cold plasma
($T_{e}=0$), there is a minimum density for SWS to satisfy phase matching,
namely $n_{e}/n_{crit}>(1-\omega_{ce}/\omega_{0})^{2}$. Finite $T_{e}$ allows
us to circumvent this limit, at the price of high Langmuir-wave
$k\lambda_{De}$ and thus strong Landau damping. We expect an analysis of SWS
growth rates, including Landau damping, to show maximum growth for moderate
$k\lambda_{De}$.

Much work remains to be done on magnetized LPI. This paper does not discuss
parametric growth rates, though they are contained in our parametric
dispersion relation (without damping or kinetics), and others have studied
them in the limit of weak coupling [18]. It is important to know when the two
circularly-polarised light waves generated by a single linearly-polarised
laser (incident from vacuum) should be treated as independent pumps, with half
the intensity of (and thus lower growth rates than) the original laser. This
likely occurs when the wavevector spread exceeds an effective bandwidth set by
damping, inhomogeneity, or parametric coupling

Two major limitations to our model are the restriction to wavevectors parallel
to the background field, and the lack of kinetic effects especially in the
plasma waves. Propagation at an angle to the B field opens up many rich
possibilities, including waves of mixed e/m and e/s character, and B field
effects on the e/s waves. In the case of perpendicular propagation, the e/s
waves become Bernstein waves. Adding kinetics is essential to understanding
parametric growth in many systems of practical interest, where collisionless
(Landau) damping is dominant. This also raises the so-called “Bernstein-Landau
paradox”, since Bernstein waves are naïvely undamped for any field strength.

If these issues can be resolved, we envisage magnetized LPI modelling tools
analogous to existing ones for unmagnetised LPI. This was one of the main
initial motivations for this work. For instance, linear kinetic coupling in
the convective steady state and strong damping limit has been a workhorse in
ICF for many years, such as for Raman and Brillouin backscatter [32] and
crossed-beam energy transfer [31]. A magnetized generalization of this needs
to handle propagation at arbitrary angles to the B field, as well as arbitrary
field strength. Among other things, it must correctly recover the unmagnetised
limit. A suitable linear, kinetic, magnetized dielectric function will be one
of the key enablers.

It is a pleasure to thank Y. Shi and B. I. Cohen for many fruitful
discussions. This work was performed under the auspices of the U.S. Department
of Energy by Lawrence Livermore National Laboratory under Contract DE-
AC52-07NA27344. This document was prepared as an account of work sponsored by
an agency of the United States government. Neither the United States
government nor Lawrence Livermore National Security, LLC, nor any of their
employees makes any warranty, expressed or implied, or assumes any legal
liability or responsibility for the accuracy, completeness, or usefulness of
any information, apparatus, product, or process disclosed, or represents that
its use would not infringe privately owned rights. Reference herein to any
specific commercial product, process, or service by trade name, trademark,
manufacturer, or otherwise does not necessarily constitute or imply its
endorsement, recommendation, or favoring by the United States government or
Lawrence Livermore National Security, LLC. The views and opinions of authors
expressed herein do not necessarily state or reflect those of the United
States government or Lawrence Livermore National Security, LLC, and shall not
be used for advertising or product endorsement purposes.

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