Magnetized laser–plasma interactions in high-energy-density systems: Parallel propagation

Imposing a magnetic field on high-energy-density (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 sources¹), laboratory astrophysics,² and magnetized inertial confinement fusion (ICF) schemes. If successful, they could provide efficient, low-cost alternatives to unmagnetized, laser-driven ICF. In the most successful case, the Sandia MagLIF concept,^3,4 an external axial magnetic field, is used to magnetize 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 in the absence of diffusion or other flux loss obeys flux conservation, which states

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 $1/r2$⁠. 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 magnetized laser-driven ICF,^5,6 and magnetized parametric laser amplification⁷ motivate us to consider magnetized laser–plasma interactions (LPI), specifically parametric scattering processes.⁸ The parametric coupling involves the decay of a large-amplitude or “pump” wave into two or more daughter waves. We focus on the decay of an electromagnetic (e/m) pump wave to one e/m and one electrostatic (e/s) daughter wave. In an unmagnetized plasma, this is limited to stimulated Brillouin (SBS) and Raman (SRS) scattering. In order for parametric coupling to occur, the following frequency and wave-vector matching conditions must be met:

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. In the context of ICF, parametric processes can give rise to resonant modes which grow exponentially in the plasma and remove energy from the target.⁹ Additionally, light backscattered through the optics of the experiment can cause significant damage and even be re-amplified.^10–12 Finally, electron plasma waves can generate superthermal or “hot” electrons which can pre-heat the fuel, thereby increasing the work required to compress it.¹³ 

By contrast, plasma-based laser amplification schemes utilize parametric processes to transfer energy from a long-duration, high-amplitude laser pump to a short, low-amplitude seed pulse. While both Raman and Brillouin amplification have been realized experimentally with some success,^14,15 Raman amplification is restricted to densities less than a quarter of the critical density and requires the seed pulse frequency to be significantly downshifted compared to the pump. Brillouin amplification can occur up to the critical density and negates the frequency downshift but has a lower growth rate.¹⁶ Recently, a magnetized amplification scheme has been proposed, where the external magnetic field is neither parallel nor perpendicular to the pump wavevector. In this scheme, low-frequency magnetohydrodynamic waves mediate the energy transfer between the seed and pump. The advantages of this approach compared to unmagnetized Raman amplification are twofold: the pump and seed are closer in frequency, and the growth rate is higher. This method is additionally suitable for short pulse amplification and compression.⁷ 

Laser-driven parametric processes have been extensively researched in unmagnetized plasmas. However, the advent of experiments such as MagLIF, the possibility of magnetized experiments on the National Ignition Facility (NIF),^17–19 and proposed magnetized parametric laser amplification schemes⁷ necessitate a re-examining of 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,²⁰ and the decay of circularly polarized electromagnetic waves in cold, homogeneous plasma.²¹ Recently, excellent theoretical work on a warm-fluid model for magnetized LPI has been done by Shi.²² Winjum et al.²³ 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 $ω≤ωce$⁠, the electron cyclotron frequency. Thus, a plethora of new parametric processes involving this wave can occur, including one which we call “stimulated whistler scattering” (SWS), 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.²⁴ The decay of a high-frequency whistler wave into a Bernstein wave and a low-frequency whistler wave in hot, inhomogeneous plasmas has also been investigated.²⁵ Additionally, parametric decays involving three whistler waves in cold, homogeneous plasmas have been studied.²⁶ 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.²⁷ 

This paper has two main objectives. First, to present the theory of magnetized LPI in a didactic and 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. We work with left and right circularly polarized waves, a natural choice for this magnetic field configuration, which allows for Faraday rotation. We obtain the uncoupled, linear waves and a parametric dispersion relation [Eq. (67)], meaning one where the pump light wave is included in the equilibrium, in the spirit of Drake et al.²⁸ (unmagnetized) and Manheimer and Ott²⁹ (magnetized—but details lacking). This allows the calculation of growth rates and the inclusion of strong coupling, where the parametric coupling significantly alters the daughter waves from their linear, uncoupled dispersion relations. It also includes both frequency up- and down-shifted scattered light waves. This paper is, to our knowledge, the only published, detailed derivation of such a magnetized parametric dispersion relation.

The paper's second goal is to study magnetized LPI in HED-relevant conditions (e.g., for NIF and MagLIF). We do this via the “kinematics” of magnetized three-wave interactions, based on phase-matching of linear, uncoupled waves. This approach treats the parametric coupling as small and is, thus, a special case of prior work, especially the weakly coupled, warm-fluid model developed by Shi.²² We quantify the effect of the imposed magnetic field on SRS and SBS spectra, which is small for fields that have been achieved on existing facilities. Analytic approximate expressions for the shifts in scattered wavelength due to the B field are also given. We then consider SWS, which, to the best of our knowledge, is the first such explicit analysis in the HED and LPI contexts. We believe the HED and LPI communities will find this self-contained theory and simple application to present experiments useful.

The rest of the paper is organized as follows. In Sec. II, 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 Sec. III, the resulting free-wave dispersion relations in a magnetized and unmagnetized plasma are discussed, along with the Faraday rotation of light-wave polarization. Section IV 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 V concludes and discusses future prospects.

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 et al.²⁸ for kinetic, unmagnetized plasma waves and also for magnetized waves.²⁹ Subsequent kinetic work was done which extended the Drake approach to include a background B field.^30,31 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 unmagnetized 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 Ds (which give linear dispersion relations) and Δs (which give parametric coupling).

The subscript s will be used to denote species, with mass m_s and charge q_s = Z_se (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 $ẑ$ 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 Fig. 1. We further assume that the distribution function for species s, f_s (particles per $dz×d3w$⁠, where $w→$ denotes velocity, and we have integrated over x and y) can be written in a separable form: $fs(t,z,w→)=fs⊥(t,z,w→⊥)Fs(t,z,wz)$⁠. $fs⊥$ allows for transverse electromagnetic waves and is normed such that $∫fs⊥d2w⊥=1$⁠. F_s is the 1D distribution (particles per $dz×dwz$⁠). Standard manipulations lead to the following 1D Vlasov–Maxwell system:

B_eq > 0 and the subscript eq indicates a nonzero, zeroth order background term. Poisson's equation is not listed since the inclusion of Ampère's law and charge continuity render it redundant. It is possible to satisfy Maxwell's equations [Eqs. (8)–(10)] by writing $E→$ and $B→$ in terms of scalar and vector potentials, $ϕ$ and $A→$⁠: $E→=−∇→ϕ−∂A→∂t$ and $B→=∇→×A→+Beqẑ$⁠. We choose the Weyl gauge in which $ϕ=0$ and $A→=A→⊥+Azz→$⁠. Faraday's law is then automatic, and the remaining Maxwell's equations become

We arrive at fluid equations by taking moments $∫wzpdwz$ of the equation for F_s, for $p=$ 0, 1, and 2

with pressure $Ps≡ms∫Fs(wz−vsz)2dwz$ and heat flux $Qs≡(ms/2)×∫Fs(wz−vsz)3dwz$⁠. 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

Common choices for linearized dynamics are isothermal $(γs=1)$ and adiabatic $(γs=3)$⁠, which follows from setting Q_s = 0 in the pressure equation. Let us recap the complete fluid-Maxwell system, with the substitutions $a→=emeA→$ (units of speed), $ωps2=qs2nseqε0ms, ωcs=|qsmsBeq|,μs=msmeZs$⁠, and $ss=−1,1$ for electrons and ions, respectively,

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 second-order terms not of interest. It is clear that the longitudinal dynamics are unaffected by B_eq in the absence of the parametric coupling since we chose $k→||Beqẑ$⁠.

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 $a02$⁠), 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, $ns2$⁠. 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 (⁠$∑sqsnseq=0$⁠). We write

where $a→j, v→j$⁠, and $ns2$ are functions of t, z. Since we are only interested in second-order terms which give rise to the parametric coupling, we can linearize equation (17)

Substituting these results and Eqs. (23)–(27) into Eqs. (18)–(22) and keeping only coupling terms of interest, we obtain, for waves 1 and 2,

where $vTs2=Teqsms$⁠. The $−vsz∂zvsz$ term in Eq. (20) has been neglected because it is second order in the daughter wave amplitude. Wave 0 satisfies the same equations 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 $(ns2,vs2$⁠, and $a2)$ and s + 1 vector $(v→s1$ and $a→1)$ 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 as.

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)

Subscript f denotes the Fourier amplitude, phase $ψj=(kj→·r→−ωjt)≡kjz−ωjt$⁠, and c.c. 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,

Growth due to the parametric coupling means the daughter-wave k_j and ω_j can be complex. It is assumed that the pump amplitude is fixed (no damping or pump depletion); hence, k₀ and ω₀ are real, and $ψ0*=ψ0$⁠. We choose our definitions of $ψ±$⁠, so they and ψ₂ 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[$ω]<0$ 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[$ω]>0$⁠. Especially with magnetized waves, the discussion of circular polarization for Re[$ω]<0$ can become confusing.

We shall eliminate $ns2$ and $v→s2$ in favor of the as. Substituting Eq. (34) into Eqs. (29) and (33), we obtain

and

respectively. Repeating for Eq. (31) gives

where the parametric coupling terms are contained in $PCs2$ (units of frequency × speed), and

where Res₂ denotes terms which are resonant with mode 2. Using Eq. (37) to substitute for $ns2$

Rearranging for $vs2$⁠,

Substituting this result into Eq. (36), we obtain

Writing Eq. (32) in terms of Fourier modes, we obtain

Let $Zy+,Zy−$ denote Z_y and $Zy*$⁠, 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

Finally, Eq. (30), once written in terms of Fourier modes, becomes

Keeping terms resonant with $ψ±$ and eliminating $ns2$ gives

Using Eq. (41) to eliminate $vs2$ from Eqs. (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

$Ks±=k0ω2±2Ps±2μsω±ωps2, βs±=ssωcsω±$⁠, and $Ps±=ωps2ω2±2−γsk2±2vTs2$⁠. $ω2+=ω2,ω2−=ω2*$⁠, and similarly for $k2±$⁠. The equations for wave 0 are equivalent to those for the ± waves, neglecting second-order terms.

At this point, the remaining task is to solve for $v→s±$ in terms of $a→±$⁠, a₂, and wave 0 quantities. We will finally arrive at a 5 × 5 system for $a→+, a→−*$⁠, and a₂, which includes both the linear waves and parametric coupling to wave 0. For magnetized waves, this is most easily done in a rotating coordinate system, where R and L circularly polarized waves are the linear light waves.

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

In condensed notation,

where $σ=+1,−1$ for the right- and left-polarized basis vectors, respectively. We define the dot product such that $a→·b→=∑iaibi*$⁠. Thus, dot products do not commute: $a→·b→=(b→·a→)*$⁠. This normalization ensures $σ̂·σ̂=1$⁠. Using this convention, any vector can be re-written in terms of right- and left-polarized unit vectors and amplitudes. Consider, for example, the physical velocity vector $v→⊥$⁠, where we explicitly indicate Fourier amplitudes with subscript f

Note that $v→·σ̂=2−1/2eiψ(vx−iσvy)=eiψvfσ+c.c$⁠. As an explicit example, for an R wave with $vfR=V$ real and v_fL = 0, $v→⊥=21/2V(cos ψ,−sin ψ)$⁠. At fixed z, $v→⊥$ rotates clockwise as time increases when looking toward $−ẑ$⁠, which is opposite to $B→eq$⁠. We, therefore, follow the convention used by Stix,³² in which circular polarization is defined relative to $B→eq$ and not $k→$⁠.

We use the result given in the last line of Eq. (53) to produce the definition of a dot product of two vectors in Fourier space in this coordinate system. Consider the vectors $v→$ and $a→$

where the subscripts i, j are the wave indices.

Taking $σ̂·$ [Eqs. (48) and (49)], we obtain

respectively, where $a±σ≡a→±·σ̂$⁠. The definitions of $vs±σ, vs0σ$⁠, and $a0σ$ are analogous to that of $a±σ$⁠. We now have uncoupled equations for $(a±σ,vs±σ)$ which is the advantage of using rotating coordinates. This is unlike the original x and y coordinates, which are coupled due to the $v→×B→$ force. For the pump wave, we have these equations with subscript $±→0$ and set the RHS to 0. Thus,

Rearranging Eq. (55) to obtain an expression for $vs±σ$

Substituting this into Eq. (56) and moving parametric coupling terms to the right-hand side, we obtain

where

This has the desired form, where wave amplitudes are written only in terms of as, not vs. For no B field, all βs are zero, and the parametric coupling coefficient $Δ±σ2∝k2±$⁠, the usual unmagnetized result. To explain the notation, $D+R$ gives the linear dispersion relation for the scattered upshifted R wave, and $Δ+R2$ is the parametric coupling coefficient for that wave and wave 2 (the plasma wave). Please see the parametric dispersion relation equation (67).

Writing the $PCs2$ term in Eq. (50) in terms of right and left circularly polarized waves, we obtain

Substituting for $v→s0$ using Eq. (57), and $v→s±$ using Eq. (58)

Equation (50) can now be written in a more condensed form

We now have a plasma–wave relation involving just as.

Equations (59) [really four equations: Eq. (59) and its complex conjugate for $σ=R,L$] and (63) form a system of five linear equations, which can be summarized in matrix form

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 a₀ and represent the parametric coupling between the e/m and e/s (plasma) daughter waves. 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 et al.,²⁸ but generalized to include a background magnetic field, and specialized to our 1D geometry and fluid instead of a kinetic plasma–wave response. It should also be a special case of the magnetized results in Manheimer and Ott,²⁹ which we find difficult to penetrate. One could also derive parametric growth rates from Eq. (67) and compare to those of Shi.²² We defer this to future work since we do not use growth rates in the subsequent application to HED conditions.

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 $a0R=0$ or $a0L=0$⁠. Taking $a0R=0$ for definiteness, waves $a−R$ and $a−R*$ decouple from the dispersion relation, leaving the following dispersion matrix:

Setting the determinant to 0 gives

$a0L=0$ then gives the three linear dispersion relations for the upshifted L, downshifted L, and plasma waves: $D+L=0, D−L=0$⁠, or $D2=0$⁠. $a0L≠0$ couples the linear waves and gives parametric interaction.

This section considers the linear or free waves, with $a0=0$⁠. Let a₁ be either $a+$ or $a−$ in Eq. (67) to obtain the free-wave dispersion relation

$a→≠0$ 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 $a2=0$⁠, we have $D1LD1R=0$⁠, which gives

For e/s waves, with $a1L=a1R=0$⁠, we have $D2=0$ and

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

By setting $ωce=0$⁠, we recover the unmagnetized dispersion relation for electromagnetic waves from Eq. (71)

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 $γe=3$

and the ion acoustic wave (IAW) for $γe=1,γi=3$

with $λDe≡vTe/ωpe$⁠. We must retain finite T_e for an IAW to exist.

The dispersion relation for free electromagnetic waves in a magnetized plasma is given in Eq. (71). As is usual in LPI literature, we view this as giving ω as a function of real k. This gives a fourth-order polynomial for ω with four real solutions, each of which corresponds to an e/m wave

Note one can solve this trivially in closed form for k given ω. In the following analysis, but not in the numerical solutions, we assume $Zime/mi≪1$⁠, so we can drop $ωpi2$ and set $ωce−ωci→ωce$⁠. In order of descending frequency, these waves are the right- and left-polarized light waves, the whistler wave, and the ion cyclotron wave (ICW). In addition to these waves, two electrostatic waves are obtained by solving Eq. (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: $ωci→0$⁠. In this case, Eq. (76) becomes (removing one ω = 0 root)