Three-wave scattering in magnetized plasmas: From cold fluid to quantized Lagrangian

Large amplitude waves in magnetized plasmas, generated either by external pumps or internal instabilities, can scatter via three-wave interactions. While three-wave scattering is well known in collimated geometry, what happens when waves propagate at angles with one another in magnetized plasmas remains largely unknown, mainly due to the analytical difficulty of this problem. In this paper, we overcome this analytical difficulty and find a convenient formula for three-wave coupling coefficient in cold, uniform, magnetized, and collisionless plasmas in the most general geometry. This is achieved by systematically solving the fluid-Maxwell model to second order using a multiscale perturbative expansion. The general formula for the coupling coefficient becomes transparent when we reformulate it as the scattering matrix element of a quantized Lagrangian. Using the quantized Lagrangian, it is possible to bypass the perturbative solution and directly obtain the nonlinear coupling coefficient from the linear response of the plasma. To illustrate how to evaluate the cold coupling coefficient, we give a set of examples where the participating waves are either quasitransverse or quasilongitudinal. In these examples, we determine the angular dependence of three-wave scattering, and demonstrate that backscattering is not necessarily the strongest scattering channel in magnetized plasmas, in contrast to what happens in unmagnetized plasmas. Our approach gives a more complete picture, beyond the simple collimated geometry, of how injected waves can decay in magnetic confinement devices, as well as how lasers can be scattered in magnetized plasma targets.

Figure 1

The most general geometry of three-wave scattering in a uniform plasma with a constant magnetic field. The three wave vectors ${\mathbf{k}}_1={\mathbf{k}}_2+{\mathbf{k}}_3$ are in the same plane, and are at angles $\theta$ with respect to the magnetic field.

Figure 2

Scattering of a parallel pump laser in uniform hydrogen plasmas. The pump laser has frequency ${\omega }_1/{\omega }_p=10$, and the scattered laser propagates at angle ${\theta }_2$ with respect to ${\mathbf{k}}_1\parallel {\mathbf{B}}_0$. The laser can scatter from the upper resonance (red), the lower resonance (orange), and the bottom resonance (blue). When the plasma is overdense, e.g., $|{\mathrm{\Omega }}_e|/{\omega }_p=0.8$ (a), (c), the upper resonance is of Langmuir type, while the lower and bottom resonances are cyclotronlike; when the plasma is underdense, e.g., $|{\mathrm{\Omega }}_e|/{\omega }_p=1.2$ (b), (d), the lower resonance is of Langmuir type, while the upper and bottom resonances are cyclotronlike. For Langmuir-type resonance, the frequency shift (c), (d) $\mathrm{\Delta }\omega \to {\omega }_p$, and the normalized growth rate (a), (b) is monotonously increasing; while for cyclotronlike resonances, $\mathrm{\Delta }\omega \to |{\mathrm{\Omega }}_e|,{\mathrm{\Omega }}_i$, and the normalized growth rate $|{\mathcal{M}}_T|$ peaks at intermediate ${\theta }_2$, while becoming zero for exact backscattering. See text for how $|{\mathcal{M}}_T|$ scales with plasma parameters.

Figure 3

Normalized growth rate $|{\mathcal{M}}_T|$ for scattering of a perpendicular pump laser (${\mathbf{k}}_1\perp {\mathbf{B}}_0$) in a uniform hydrogen plasma with ${\omega }_1/{\omega }_p=10$ and $|{\mathrm{\Omega }}_e|/{\omega }_p=0.8$. In spherical coordinate, the scattered laser propagates at polar angle ${\theta }_2$ with respect to ${\mathbf{B}}_0$, and azimuthal angle ${\varphi }_2$ measured from ${\mathbf{k}}_1$. The laser can scatter from the upper resonance (a), in which case backscattering is the strongest scattering mode. Alternatively, the laser can scatter off the lower resonance (b). In this case, one maximum of $|{\mathcal{M}}_T|$ is attained for backscattering, and another maximum is attained when the scattered laser propagates almost perpendicular to the incident laser along the magnetic field. Finally, the laser can scatter off the bottom resonance (c). In this case, exact backscattering is suppressed while nearly backward scattering is strong.

Figure 4

Scattering of a perpendicular pump laser with ${\omega }_1/{\omega }_p=10$ in a uniform hydrogen plasma. Figure (a) can be obtained from Fig. 3 by taking a one dimensional cut along the unit sphere using the plane spanned by ${\mathbf{k}}_1\perp {\mathbf{B}}_0$. The scattered laser, propagating at angle ${\alpha }_2$ with respect to ${\mathbf{k}}_1$, can scatter from the upper resonance (red), the lower resonance (orange), and the bottom resonance (blue). Both the normalized growth rate $|{\mathcal{M}}_T|$ (a), (b) and the frequency shifts $\mathrm{\Delta }\omega$ (c), (d) behave qualitatively the same in overdense plasma, e.g., $|{\mathrm{\Omega }}_e|/{\omega }_p=0.8$ (a), (c), and in underdense plasma, e.g., $|{\mathrm{\Omega }}_e|/{\omega }_p=1.2$ (b), (d). As ${\alpha }_2$ increases from $0^{\circ }$ to ${180}^{\circ }, |{\mathcal{M}}_T|$ increases monotonously for scattering from the upper resonance. For scattering off the lower resonance, $|{\mathcal{M}}_T|$ hits zero near ${\alpha }_2\sim {176}^{\circ }$, where electron and ion contributions exactly cancel, and then increase to finite value at exact backscattering. In contrast, when the laser is scattered from the bottom resonance, $|{\mathcal{M}}_T|$ strongly peaks near ${\alpha }_2\sim {170}^{\circ }$, and becomes zero for exact backward scattering. See text for how $|{\mathcal{M}}_T|$ scales with plasma parameters.

Figure 5

Scattering of a parallel electrostatic pump wave in uniform hydrogen plasmas, when observed at angle ${\theta }_2$ with respect to ${\mathbf{k}}_1\parallel {\mathbf{B}}_0$. At each ${\theta }_2$, due to the degeneracy $2\leftrightarrow 3$, the wave vector has two possible values $k_2^{\pm }$, corresponding to ${\mathcal{M}}_L^{+}$ (blue, ${\theta }_3>{90}^{\circ }$) and ${\mathcal{M}}_L^{-}$ (red, ${\theta }_3<{90}^{\circ }$). The pump wave can be the Langmuir wave (a), (b); the electron cyclotron wave (c), (d); and the ion cyclotron wave (e). The normalized growth rate attains local extrema for symmetric scattering, where the two decay waves have the same frequency ${\omega }_r\left({\theta }_s\right)={\omega }_1/2$. In overdense plasma, e.g., $|{\mathrm{\Omega }}_e|/{\omega }_p=0.8$ (a), (c), $u\to l,l$ happens for ${\theta }_2<{\theta }_b^o$, where ${\omega }_l\left({\theta }_b^o\right)={\omega }_p-\left|{\mathrm{\Omega }}_e\right|$; $l\to l,l$ happens for ${\theta }_2>{\theta }_a^o$, where ${\omega }_l\left({\theta }_a^o\right)=\left|{\mathrm{\Omega }}_e\right|-{\omega }_{LH}$; and $l\to l_2,b_3$ happens for ${\theta }_2<{\theta }_i^o$, where ${\omega }_l\left({\theta }_i^o\right)=\left|{\mathrm{\Omega }}_e\right|-{\mathrm{\Omega }}_i$. In underdense plasma, e.g., $|{\mathrm{\Omega }}_e|/{\omega }_p=1.2$ (b), (d), $u\to l,l$ happens for ${\theta }_2<{\theta }_b^u$, where ${\omega }_l\left({\theta }_b^u\right)=\left|{\mathrm{\Omega }}_e\right|-{\omega }_p$; $l\to l,l$ happens for ${\theta }_2>{\theta }_a^u$, where ${\omega }_l\left({\theta }_a^u\right)={\omega }_p-{\omega }_{LH}$; and $l\to l_2,b_3$ happens for ${\theta }_2<{\theta }_i^u$, where ${\omega }_l\left({\theta }_i^u\right)={\omega }_p-{\mathrm{\Omega }}_i$. Regardless of plasma density (e), $b\to b,b$ can always happen, for which the growth rate peaks near ${\theta }_s\sim {88}^{\circ }$, where the decay is symmetrical. The gray lines indicate the symmetric angles and the asymptotic maxima obtained in the text.

Figure 6

Normalized growth rate $|{\mathcal{M}}_L|$ when pumped by an upper-hybrid wave (${\mathbf{k}}_1\perp {\mathbf{B}}_0$) in a uniform hydrogen plasma with $|{\mathrm{\Omega }}_e|/{\omega }_p=1.2$. The growth rates are observed at polar angle ${\theta }_2$ with respect to ${\mathbf{B}}_0$ and azimuthal angle ${\varphi }_2$ with respect to ${\mathbf{k}}_1$. When ${\omega }_2$ is on the upper resonance (a), the $u\to u_2,l_3$ decay can happen for ${\theta }_2<{\theta }_u^a$, where ${\omega }_u\left({\theta }_u^a\right)={\omega }_{UH}-{\omega }_{LH}$. In this case, an important decay channel has ${\omega }_2\sim \left|{\mathrm{\Omega }}_e\right|$ propagating almost parallel to ${\mathbf{B}}_0$ in the backward direction, and ${\omega }_3\gg {\omega }_{LH}$ propagating almost perpendicular to ${\mathbf{B}}_0$ in the forward direction. This region corresponds to the $l_2,u_3$ region in (b), in which ${\omega }_2$ is on the lower resonance instead. The other decay mode is $u\to u_2,b_3$, which can happen in the narrow strip ${\theta }_2>{\theta }_u^b$ in (a), where ${\omega }_u\left({\theta }_u^b\right)={\omega }_{UH}-{\mathrm{\Omega }}_i$. Equivalently, exchanging the labels to $b_2,u_3$, this decay mode can happen in the colored region in (c), in which ${\omega }_2$ is on the bottom resonance instead. For this decay mode, the dominant decay channel has ${\omega }_2\sim {\omega }_{UH}$ propagating almost perpendicular to ${\mathbf{B}}_0$ in the forward direction, and ${\omega }_3\sim {\mathrm{\Omega }}_i$ propagating either in the forward or backward direction. The last decay mode is $u\to l_2,l_3$, which corresponds to the large colored region in (b). For this decay mode, the dominant decay channel is the symmetric decay, where ${\omega }_2\sim {\omega }_3\sim {\omega }_{UH}/2$ and both waves propagate at angles with ${\mathbf{B}}_0$ in the forward direction.

Figure 7

Linear wave dispersion relations (a) and polarization angles (b) in a cold electron-ion plasma with $m_i/m_e=10$ and $|{\mathrm{\Omega }}_e|/{\omega }_{pe}=1.2$, when $\theta ={45}^{\circ }$. Both the $n_{+}^2$ (red) and the $n_{-}^2$ (blue) solutions contain an electromagneticlike branch and electrostaticlike branches. The electromagneticlike branches asymptote to vacuum light wave $\omega \to ck$ when $k\to \infty$, where the waves become transverse ($\varphi \to {90}^{\circ }$, mod ${180}^{\circ }$). The electrostaticlike branches asymptote to resonances $\omega \to {\omega }_r$ as $k\to \infty$, where the waves become longitudinal ($\varphi \to 0^{\circ }$, mod ${180}^{\circ }$). The waves are in general elliptically polarized ($\psi \ne 0^{\circ }$, mod ${90}^{\circ }$), except at special angles.

Figure 8

Resonance frequencies in electron-ion plasma with $m_i/m_e=10$. In overdense plasma, e.g., $|{\mathrm{\Omega }}_e|/{\omega }_p=0.8$ (a), as $\theta$ increases from $0^{\circ }$ to ${90}^{\circ }$, the upper resonance (red) increases from ${\omega }_p$ to ${\omega }_{UH}$; the lower resonance (orange) decreases from $|{\mathrm{\Omega }}_e|$ to ${\omega }_{LH}$; and the bottom resonance (blue) decreases from ${\mathrm{\Omega }}_i$ to zero. In underdense plasma, e.g., $|{\mathrm{\Omega }}_e|/{\omega }_p=1.2$ (b), as $\theta$ increases from $0^{\circ }$ to ${90}^{\circ }$, the upper resonance (red) increases from $|{\mathrm{\Omega }}_e|$ to ${\omega }_{UH}$; the lower resonance (orange) decreases from ${\omega }_p$ to ${\omega }_{LH}$; and the bottom resonance (blue) decreases from ${\mathrm{\Omega }}_i$ to zero. This figure can be used to read out the frequency shift $\mathrm{\Delta }\omega$, once the scattering angle of the longitudinal wave is known.