We investigate parametric processes in magnetized 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 present a self-contained derivation of a “parametric” dispersion relation for magnetized three-wave interactions, meaning the pump wave is included in the equilibrium, similar to the unmagnetized work of Drake et al., Phys. Fluids 17, 778 (1974). For this, we use a multi-species plasma fluid model and Maxwell's equations. The application of an external B field causes right- and left-polarized light waves to propagate with differing phase velocities. This leads to Faraday rotation of the polarization, which can be significant in HED conditions. Phase-matching and linear wave dispersion relations show that Raman and Brillouin scattering have modified spectra due to the background B field, though this effect is usually small in systems of current practical interest. We study a scattering process we call stimulated whistler scattering, where a light wave decays to an electromagnetic whistler wave () and a Langmuir wave. This only occurs in the presence of an external B field, which is required for the whistler wave to exist.
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, Bz 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 . 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 , 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.
II. PARAMETRIC DISPERSION RELATIONS FOR MAGNETIZED 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 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).
A. Governing equations
The subscript s will be used to denote species, with mass ms and charge qs = Zse (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 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, fs (particles per , where denotes velocity, and we have integrated over x and y) can be written in a separable form: . allows for transverse electromagnetic waves and is normed such that . Fs is the 1D distribution (particles per ). Standard manipulations lead to the following 1D Vlasov–Maxwell system:
Beq > 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 and in terms of scalar and vector potentials, and : and . We choose the Weyl gauge in which and . Faraday's law is then automatic, and the remaining Maxwell's equations become
We arrive at fluid equations by taking moments of the equation for Fs, for 0, 1, and 2
with pressure and heat flux . 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 Ks is a constant
Common choices for linearized dynamics are isothermal and adiabatic , which follows from setting Qs = 0 in the pressure equation. Let us recap the complete fluid-Maxwell system, with the substitutions (units of speed), , and 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 Beq in the absence of the parametric coupling since we chose .
B. Linearization: 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 ), 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, nseq and a perturbation induced by the electrostatic wave, . We assume that no background flows exist in the plasma (vseq = 0), no external electric fields are imposed upon it (aeq = 0), and quasi-neutrality holds (). We write
where , and 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)
where . The 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 scalar and s + 1 vector equations for scalar , and and s + 1 vector and 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.
C. 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)
Subscript f denotes the Fourier amplitude, phase , 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 kj and ωj can be complex. It is assumed that the pump amplitude is fixed (no damping or pump depletion); hence, k0 and ω0 are real, and . We choose our definitions of , 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[ 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[. Especially with magnetized waves, the discussion of circular polarization for Re[ can become confusing.
1. Plasma waves in Fourier space
respectively. Repeating for Eq. (31) gives
where the parametric coupling terms are contained in (units of frequency × speed), and
where Res2 denotes terms which are resonant with mode 2. Using Eq. (37) to substitute for
Rearranging for ,
Substituting this result into Eq. (36), we obtain
2. EM waves in Fourier space
Writing Eq. (32) in terms of Fourier modes, we obtain
Let denote Zy and , 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 and eliminating gives
, and . , and similarly for . 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 in terms of , a2, and wave 0 quantities. We will finally arrive at a 5 × 5 system for , 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.
D. Left–right co-ordinate system
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 for the right- and left-polarized basis vectors, respectively. We define the dot product such that . Thus, dot products do not commute: . This normalization ensures . 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 , where we explicitly indicate Fourier amplitudes with subscript f
Note that . As an explicit example, for an R wave with real and vfL = 0, . At fixed z, rotates clockwise as time increases when looking toward , which is opposite to . We, therefore, follow the convention used by Stix,32 in which circular polarization is defined relative to and not .
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 and
where the subscripts i, j are the wave indices.
E. EM waves in left–right coordinates
respectively, where . The definitions of , and are analogous to that of . We now have uncoupled equations for which is the advantage of using rotating coordinates. This is unlike the original x and y coordinates, which are coupled due to the force. For the pump wave, we have these equations with subscript and set the RHS to 0. Thus,
Rearranging Eq. (55) to obtain an expression for
Substituting this into Eq. (56) and moving parametric coupling terms to the right-hand side, we obtain
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 , the usual unmagnetized result. To explain the notation, gives the linear dispersion relation for the scattered upshifted R wave, and is the parametric coupling coefficient for that wave and wave 2 (the plasma wave). Please see the parametric dispersion relation equation (67).
F. Plasma waves in left–right coordinates
Writing the term in Eq. (50) in terms of right and left circularly polarized waves, we obtain
Equation (50) can now be written in a more condensed form
We now have a plasma–wave relation involving just as.
G. Parametric dispersion relation
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 or . Taking for definiteness, waves and decouple from the dispersion relation, leaving the following dispersion matrix:
Setting the determinant to 0 gives
then gives the three linear dispersion relations for the upshifted L, downshifted L, and plasma waves: , or . 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 . Let a1 be either or in Eq. (67) to obtain the free-wave dispersion relation
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 , we have , which gives
For e/s waves, with , we have and
Note that the background B field has no effect at all on the e/s waves, for our geometry of .
A. Waves in an unmagnetized plasma
By setting , 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
and the ion acoustic wave (IAW) for
with . We must retain finite Te for an IAW to exist.
B. Waves with magnetic field
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 , so we can drop and set . 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: . In this case, Eq. (76) becomes (removing one ω = 0 root)
We assume , which is typical in the HED regime. For light waves, we consider Eq. (77) for . For k = 0, we find
For all k, we write ω as plus a correction
1. Whistler wave
We can also solve Eq. (77) for the whistler wave, which has . We call this full set of roots for ω the whistler though some authors only use this term for the small k domain and “electron cyclotron wave” when ω is near ωce. We derive expressions for this wave by considering two limits: first, for (but still large enough that we can neglect ion motion, discussed below), we obtain
We restrict interest to waves, which for the whistler requires the R wave (σ = 1)
Second, for , we obtain
For ω near ωce, the whistler group velocity 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.
The full numerical solutions of the dispersion relations for the whistler wave and the right- and left-polarized light waves are shown in Fig. 2(a). Note that here and throughout the rest of the paper, λDe is used to normalize k, as is customary for stimulated scattering. For large , the whistler wave tends to , shown in Fig. 2(a) as a dashed black line.
2. Ion cyclotron wave
We now consider the ICW which requires the retention of terms involving ion motion. As with the whistler wave, we consider two regimes. For , we seek solutions with , which gives
where the Alfvén velocity, . This solution applies for both values of σ, 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 , where we obtain two solutions with ω independent of k: for σ = 1 (the right-polarized whistler), and for (the left-polarized ICW). Including the next correction term for the ICW gives
Figure 2(a) is re-plotted in Fig. 2(b) for to show the IAW and ICW clearly. The ICW tends to , denoted by a dashed black line. The numerical and approximate analytic solutions to the ICW dispersion relation are shown in Fig. 2(b) in blue and dark blue, respectively. As can be seen from Eq. (83), at low , the ICW approaches the Alfvén frequency, which is represented by a dashed cyan line in Fig. 2(b). For large values of , the ICW frequency tends to ωci, marked by a dashed black line. The parameters used to plot the dispersion relations shown in Figs. 2(a) and 2(b) are given in Table I. A plasma comprising helium ions and electrons was considered.
|Quantity .||Value .|
|Quantity .||Value .|
C. Faraday rotation
Three unique waves exist in an unmagnetized 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 polarizations). If the electromagnetic wave is linearly polarized, it can be written as the sum of two circularly polarized waves of equal amplitude and opposite handedness (R and L waves). If an external B field, is applied, the R and L waves experience different indices of refraction and propagate with differing phase velocities. Consequently, the overall polarization 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 Eq. (71),
Two first-order Taylor expansions of Eq. (85), assuming and yield
Consider a linearly polarized plane electromagnetic wave. We can write the physical electric field as the sum of the electric fields of two circularly polarized waves with opposite handedness
Writing this in Cartesian co-ordinates,
Assuming ε is real,
At a fixed z, always lies along the same line in the xy plane, with its exact position varying in time. As z varies, the angle this line makes with respect to the x axis increases at the rate
The final formula is in practical units. We have introduced the critical density , which is the usual definition for the unmagnetized plasma. When discussing LPI, ncrit is for the pump wave ω0. Significant Faraday rotation is, thus, possible in current ICF platforms with modest B fields. For instance, with and Beq = 10 T, we obtain mm. This could be used to diagnose ne (a common technique when feasible) and could affect LPI processes such as crossed-beam energy transfer.33–35
IV. IMPACT OF EXTERNAL B FIELD ON THE PARAMETRIC COUPLING
We apply the above theory to magnetized LPI in HED relevant conditions, all for . We consider how the imposed field modifies SRS and SBS as well as SWS which only occurs in a background field. Recall , and we choose . k1 and k2 can have either sign. Let sign(ki) 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 , and “backward scatter” to the opposite case . To satisfy k matching, we cannot have both and . For SRS and SBS, c2 must equal +1, but for SWS, 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 ω1 since this is what escapes the plasma and is measured experimentally. As discussed in Sec. III C, causes the R and L waves to propagate with different phase velocities. Therefore, a laser or other external source that imposes a linearly polarized light wave of frequency ω0 couples to an R and L wave in a magnetized plasma. For stimulated scattering, we are mostly interested in down-shifted scattered waves for which , which have the same polarization as the pump: an R or L pump couples to a down-shifted R or L scattered wave, respectively; hence, which we, sometimes, denote as σ. 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 II summarizes the processes we study.
|Process .||Pump e/m wave .||Scattered e/m wave .||Plasma wave .||Geometries (c1, c2) .||ω1 range .||range .|
|SRS||R,L||R,L||EPW||(1,1) (−1, 1)|
|SBS||R,L||R,L||IAW||(1,1) (−1, 1)||<1|
|SWS||R||R-whistler||EPW||(1,−1) (−1, 1)||for Te = 0|
|Process .||Pump e/m wave .||Scattered e/m wave .||Plasma wave .||Geometries (c1, c2) .||ω1 range .||range .|
|SRS||R,L||R,L||EPW||(1,1) (−1, 1)|
|SBS||R,L||R,L||IAW||(1,1) (−1, 1)||<1|
|SWS||R||R-whistler||EPW||(1,−1) (−1, 1)||for Te = 0|
In order to derive a dispersion relation for ω1 in terms of known inputs, we begin with the identity k2 = k2. We use k matching to write on the left side, and the plasma–wave dispersion relation of interest to rewrite the right side in terms of ω2. We then use the e/m dispersion relation to write k1 in terms of ω1 and use ω matching to write . For SRS and SWS, this yields . The same method is applied for SBS, where k2 is written in terms of ω2 using the simple IAW dispersion relation, , for an approximate analysis (the numerical roots use the full e/s dispersion relation). That is, . The resulting dispersion relations can be summarized as follows:
where Y is either RW, for SRS and SWS, or B, for SBS. For SRS and SWS, , where . For SBS, , with . This is usually very small, with a typical magnitude. , where X denotes any angular frequency subscript in Eq. (92). The frequency of scattered light which satisfies phase matching is given by the roots of Eq. (92), which can be found by plotting MY vs . This is illustrated for SRS and SWS in Fig. 3, and for SBS in Fig. 4, for the parameters given in Table I and .
The dispersion relations given in Eq. (92) are plotted as a function of and for scattering geometries , in Figs. 5–7, respectively. The two dispersion relations, MRW and MB, have been overplotted. To distinguish between them, MRW has been cross-hatched, while MB has not. The color scale for M applies to both MRW and MB. The regions of Figs. 5–7 where M is not real are colored gray. The regions of the plot where serve only to illustrate the root-finding method employed: to ensure we have correctly identified roots, we check that has changed the 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 normalized plasma density and scattered EMW frequency for each of these processes. The contours which correspond to a given parametric process are appropriately labeled.
In Figs. 5 and 6, a sharp decrease can be seen in the frequency of SRS scattered light with increasing plasma density. Also in Figs. 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 unmagnetized plasma, SRS is only possible for . The region of parameter space in which SWS can occur is also restricted, as . Using the same method as for SRS, the following inequality is obtained for the normalized electron densities at which SWS can occur in a cold plasma
A. Stimulated Raman scattering: SRS
The dispersion relation for SRS is given by Eq. (92), where . For a cold plasma with Ve = 0, we find always, so . This is true with or without a background field Beq. Thus, any effect of Beq on is “doubly small,” in that it also relies on thermal effects. For no background field , we obtain the usual solutions, which for and are for (forward scatter), and for (backscatter).
Including a weak background field, we write where is the solution for : . We have , which gives with . All partials are evaluated at and . One can find a formula for α, but it is unilluminating. We quote the result in the limit that and
The full numerical solution of MRW [see Eq. (92)] is plotted in Figs. 8 and 9 for the plasma conditions given in Table I and the first row of Table III. The frequencies, wave vectors, and, if applicable, the polarizations of the e/m and e/s waves at which phase-matching conditions are met are illustrated by parallelograms. Specifically, Figs. 8 and 9 correspond to forward and back-SRS, respectively.
|Laser wavelength (μm) .||.||ne () .||Beq (T) .|
|Laser wavelength (μm) .||.||ne () .||Beq (T) .|
The shift in wavelength of SRS light due to the presence of the external magnetic field, , is given by
Substituting from Eq. (85), and treating temperature and magnetic field as small perturbations in as detailed above, we derive the following expression for to first order in Ωce and
in practical units. Under the conditions given in Table I, for and B = 100 T for SRS backscattered light from a left-polarized pump wave, the analytic approximation yields nm, compared to the full numerical solution, which gives nm. Typically, in NIF-type experiments, the wavelength of back-SRS light is in the range of 500–600 nm, with a spectral width of 5–10 nm 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 agrees reasonably closely with the full numerical computation of , which is plotted as a function of for Te = 2 keV, 4 keV, and , 0.15 in Figs. 10(a) and 10(b), for forward and back-SRS light, respectively. Similarly, the first order approximation and full numerical solution for is shown as a function of Te for forward and back SRS in Figs. 11(a) and 11(b), respectively, for and . The effect of electron density and temperature becomes particularly significant for forward and backward-SRS light from a right-polarized pump as , as in this limit, , respectively.
B. Stimulated Brillouin scattering: SBS
The phase-matching relation for SBS, MB = 0 is derived in Sec. IV, and given in Eq. (92). A phase-matching diagram is shown in Fig. 12 for the same conditions as Fig. 8. Exact forward SBS () is not considered since in our strictly 1D geometry it does not occur. MB = 0 has a spurious root for , which connects to near-forward scatter for small but nonzero angle between and . The SBS growth rate is zero for , so we discuss only backscatter (). For , the exact solution is
with . The approximate form for is typically quite accurate. The correction for a weak B field and to leading order in is
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 and Ωce. The scattered wavelength shift , evaluated at , is
In practical units,
This is an extremely small value for ICF conditions. For the parameters shown in Table III, with nm, , B = 100T, and a right-polarized pump, the analytic approximation gives pm, whereas the full numerical solution gives pm. The variation of with and Te is shown in Figs. 13 and 14, respectively.
C. 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 (). For a cold plasma, this imposes a minimum density of to satisfy frequency matching, as opposed to a maximum of for SRS. Forward () and backward () SWS are both kinematically allowed though forward SWS can only occur for a plasma wave propagating counter to the pump: . The phase-matching condition MRW for SWS, given in Eq. (92), is identical to that of SRS except that . Figures 15 and 16 show SWS phase-matching diagrams for the allowed geometries and for a range of and Te.
The relationship between and is shown in Figs. 17 and 18 for , 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 and the values of and at which saturation occurs vary with plasma density and temperature. Increasing Te decreases the at which the trend saturates, while increasing causes the observed trend to saturate at lower and . is plotted to indicate the magnitude of Landau damping, which is expected to significantly reduce SWS growth for . In the opposite limit, the SWS growth rate approaches zero as .
The wavelength of SWS scattered light is
For the bottom rows of Table IV, , and . For a pump wavelength of , we have B = 5000 T and 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 λ1 much longer.
|c1 .||c2 .||.||Te (keV) .||.||.||.||.|
|c1 .||c2 .||.||Te (keV) .||.||.||.||.|
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 Eq. (77). Rearranging for k, we obtain
We see that for , k is real and the wave can propagate. If the reverse is true, k is imaginary and the wave is evanescent. ωpe and ωce 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 unmagnetized one, . In this case, if ne 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, and , 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 Te, 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 Eq. (93). We see this in Fig. 16, where the lowest density shown, , corresponds to the highest pump frequency and a very high Langmuir-wave frequency, 2. This requires a large , 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 (for some power p) when is small and decrease with increasing Landau damping for large . This means there is an effective low-density cut-off, below which SWS is kinematically allowed but strongly damped. In the opposite limit, as ne approaches ncrit (such as in Figs. 16 and 17 and Table IV), k2 becomes small, and Landau damping is negligible; however, the growth rate of SWS also tends to 0. There is, thus, an intermediate range of ne in which the growth rate is optimal, and is moderate. The case where and Te = 2 keV shown in Figs. 16 and 15 and Table IV typifies this regime.
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 polarized e/m waves become non-degenerate and form the natural basis, as opposed to linearly polarized 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, which have no analogues in an unmagnetized plasma.
We found a parametric dispersion relation to first order in the parametric coupling, Eq. (67), analogous to the classic 1974 work of Drake et al.28 We then focused on the kinematics of phase matching for three-wave interactions. Since the right and left circular polarized 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 unmagnetized 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 Ang. for SRS, and 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 for new parametric processes, such as 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 1–100 μm for fields of 10 kT–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 (Te = 0), there is a minimum density for SWS to satisfy phase matching, namely, . Finite Te allows us to circumvent this limit, at the price of high Langmuir-wave , and thus, strong Landau damping. We expect an analysis of SWS growth rates, including Landau damping, to show maximum growth for moderate .
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.22 It is important to know when the two circularly polarized light waves generated by a single linearly polarized 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 modeling tools analogous to existing ones for unmagnetized 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 backscatter36 and crossed-beam energy transfer.35 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 unmagnetized limit. A suitable linear, kinetic, magnetized dielectric function will be one of the key enablers.
The dispersion relation presented in this work does not account for plasma inhomogeneities, which are highly significant for NIF and MAGLIF campaigns. To include these effects, an approach similar to the one utilized for DEPLETE36 could be employed. This treatment assumes that the length scale of the inhomogeneity is greater than the wavelength of the pump and scattered waves, which allows the scattered and plasma waves to be treated as collections of monoenergetic carrier waves with slowly varying amplitudes in time and space. Frequency-matching conditions, and, hence, the gain rate for SRS and SBS, vary with the plasma density, as do the refracted paths of the scattered light, which are computed using ray-tracing. The spectrum of scattered light is obtained by integrating the spatially varying gain rate over the inhomogeneous density profile along the paths of the refracted rays.
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 No. 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.
Conflict of Interest
The authors have no conflicts to disclose.
Data sharing is not applicable to this article as no new data were created or analyzed in this study.