2. GENERAL FORMALISM

We now analyze the behavior of an electromagnetic wave that propagates in a homogeneous and magnetized plasma with initial density n ₀. A constant magnetic field B₀ in the y-direction is considered in the plasma channel. Here, the pump laser pulse is assumed to be a short pulse and have a high frequency, and since the duration of the short pulse is less than the plasma period, we ignore filamentation and self-focusing effects, hence is valid when the laser spot size is much bigger than c/ω _p. We consider an electromagnetic wave propagates perpendicularly to B₀. For an extraordinary wave (defined also as X-mode) the electric field is perpendicular to B₀, and therefore the electric field induces an electron velocity in the x–z-plane. As a result, the force v × B₀ introduces another velocity term in this plane. As a wave propagates through a region in which ω_p and ω _c are changing, it may encounter cutoffs and resonances. A wave generally reflected at a cutoff and absorbed at a resonance. There are two cutoffs that occur when the index of refraction goes to a zero value. The cutoffs of the extraordinary wave are found by setting k ₀ equal to zero in the dispersion relation of the incident electromagnetic wave. The cutoff frequencies are ${{\rm \omega} _1}={\textstyle{1 \over 2}}[ {{\rm \omega} _{\rm c}}+{\lpar 4{\rm \omega} _{\rm p}^2+{\rm \omega} _{\rm c}^2 \rpar ^{1/2}}] $ and ${{\rm \omega} _2}={\textstyle{1 \over 2}}[ - {{\rm \omega} _{\rm c}}+{\lpar 4{\rm \omega} _{\rm p}^2+{\rm \omega} _{\rm c}^2 \rpar ^{1/2}}] $, which correspond to plasmas with ω _p < ω ₀ and ω _p > ω ₀, respectively. Since we consider the interaction of a high-frequency laser pulse with an underdense plasma, in which ω _p≪ω ₀ and ω _c≪ω ₀, the cutoff points therefore do not appear in our investigation.

Consider an extraordinary laser pump wave propagating in a homogeneous underdense plasma embedded in a constant external magnetic field ${{\bf B}_0}\hat y$ (see Fig. 2). The electric vector of the radiation field propagating along the z-direction is represented by

$${\bf E}\lpar z\comma \; t\rpar ={E_0}\lpar \hat x+i{{\rm \alpha} _0}\hat z\rpar \, {e^{i\lpar {{\rm k}_0}z - {{\rm \omega} _0}t\rpar }}$$

where E ₀, k ₀, and ω ₀ are amplitude, wave number, and frequency of the radiation field, respectively. Similarly, the vector potential of the radiation field is defined as ${\bf A}\lpar z\comma \; t\rpar ={A_0}\lpar - i\hat x+{{\rm \alpha} _0}\hat z\rpar \, {e^{i\lpar {k_0}z - {{\rm \omega} _0}t\rpar }}$. Parameter α ₀, which is attributable to the extraordinary property of the incident light wave in the presence of ion motion, is given by

$${{\rm \alpha} _0}=\displaystyle{{\lpar {{\rm \omega} _{\rm c}}/{{\rm \omega} _0}\rpar \lpar {\rm \omega} _{\rm p}^2 /\lpar {\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 \rpar \rpar - \lpar {\Omega _{\rm c}}/{{\rm \omega} _0}\rpar \lpar \Omega _{\rm p}^2 /\lpar {\rm \omega} _0^2 - \Omega _{\rm c}^2 \rpar \rpar } \over {1 - \lpar {\rm \omega} _{\rm p}^2 /\lpar {\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 \rpar \rpar - \lpar \Omega _{\rm p}^2 /\lpar {\rm \omega} _0^2 - \Omega _{\rm c}^2 \rpar \rpar }}\comma \;$$

where Ω_c is the ion cyclotron frequency and Ω_p is the ion oscillation frequency in the plasma (Krall & Trivelpicec, Reference Krall and Trivelpicec1973). The laser beam polarization is assumed to remain unaltered while propagating through a transversely magnetized plasma. Following the same procedure from our previous work (Paknezhad, Reference Paknezhad2013c) and replacing α ₀ with ${{\rm \beta} _0}\lpar =\lpar {{\rm \omega} _{\rm c}}/{{\rm \omega} _0}\rpar \lpar {\rm \omega} _{\rm p}^2 /\lpar {\rm \omega} _0^2 - {\rm \omega} _{{\rm UH}}^2 \rpar \rpar \rpar $ in all computations, the x-component of the nonlinear wave equation of the extraordinary wave in magnetized plasma turns out to be

(1)$$\left({\displaystyle{{{\partial ^2}} \over {\partial {t^2}}} - {c^2}\displaystyle{{{\partial ^2}} \over {\partial {z^2}}}} \right){A_x}=\displaystyle{{2{\rm \pi} in{e^2}} \over m}{{\rm \mu} _0}{A_0}\, {e^{i\lpar {k_0}z - {{\rm \omega} _0}t\rpar }}\comma \;$$

where

$$\eqalign{& {{\rm \mu} _0}\lpar {{\rm \omega} _0}\rpar = \left({\displaystyle{{{\rm \omega} _0^2 + {{\rm \alpha} _0}{{\rm \omega} _0}{{\rm \omega} _{\rm c}}} \over {{\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 }}} \right)\cr & \qquad \qquad + a_0^2 \left({\displaystyle{{{c^2}k_0^2 {{\rm \omega} _0}{M_1}} \over {4{{\lpar {\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 \rpar }^4}\lpar 4{\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 \rpar }} - \displaystyle{3 \over 8}\displaystyle{{{{\rm \omega} _0}{M_2}} \over {{{\lpar {\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 \rpar }^4}}}} \right)}$$

with

$$\eqalign{{M_1}& =2{\rm \alpha} _0^2 {\rm \omega} _0^7+{{\rm \alpha} _0}\lpar 6{\rm \alpha} _0^2 - 25\rpar {\rm \omega} _0^4 {\rm \omega} _{\rm c}^3+\lpar 26{\rm \alpha} _0^2 - 5\rpar {\rm \omega} _0^3 {\rm \omega} _{\rm c}^4 \cr & \quad - \lpar 30{\rm \alpha} _0^2+5\rpar {\rm \omega} _0^5 {\rm \omega} _{\rm c}^2 +27{{\rm \alpha} _0}{\rm \omega} _0^2 {\rm \omega} _{\rm c}^5 - {{\rm \alpha} _0}\lpar 6{\rm \alpha} _0^2 - 5\rpar {\rm \omega} _0^6 {{\rm \omega} _{\rm c}}\cr & \quad +10{{\rm \omega} _0}{\rm \omega} _{\rm c}^6+{{\rm \alpha} _0}{\rm \omega} _{\rm c}^7 }$$

and

$$\eqalign{& {M_2} = \lpar 3{\rm \alpha} _0^2 + 1\rpar \lpar {\rm \omega} _0^7 + 6{\rm \omega} _0^5 {\rm \omega} _{\rm c}^2 + {\rm \omega} _0^3 {\rm \omega} _c^4 \rpar \cr & \quad \quad + 4{{\rm \alpha} _0}\lpar {\rm \alpha} _0^2 + 3\rpar \lpar {\rm \omega} _0^6 {{\rm \omega} _{\rm c}} + {\rm \omega} _0^4 {\rm \omega} _{\rm c}^3 \rpar .}$$

Fig. 2.

The electric field components (E _x, E _z), the magnetic field B, and the wave vector k ₀ of the extraordinary pump laser wave in a transversely magnetized plasma. The electric field is elliptically polarized in a plane perpendicular to the external magnetic field B₀.

Equation (1) gives the nonlinear dispersion relation of the extraordinary pump laser wave in the magnetized plasma:

(2)$$\displaystyle{{{c^2}k_0^2 } \over {{\rm \omega} _0^2 }}=1 - {{\rm \mu} _0}\displaystyle{{{\rm \omega} _{\rm p}^2 } \over {{\rm \omega} _0^2 }}.$$

By neglecting the second and fourth terms in coefficient μ ₀ and then replacing it in the dispersion relation (2) and after some algebra, we recover the linear dispersion relation of the extraordinary electromagnetic wave in a non-relativistic magnetized plasma, which is known as (Krall & Trivelpicec, Reference Krall and Trivelpicec1973)

$$\eqalign{& \displaystyle{{{c^2}k_0^2 } \over {{\rm \omega} _0^2 }} = 1 - \displaystyle{{{\rm \omega} _{\rm p}^2 } \over {{\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 }} - \displaystyle{{\Omega _{\rm p}^2 } \over {{\rm \omega} _0^2 - \Omega _{\rm c}^2 }} \cr & \qquad \quad - \displaystyle{{{{\lpar \lpar {{\rm \omega} _{\rm c}}/{{\rm \omega} _0}\rpar \lpar {\rm \omega} _{\rm p}^2 /\lpar {\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 \rpar \rpar - \lpar {\Omega _{\rm c}}/{{\rm \omega} _0}\rpar \lpar \Omega _{\rm p}^2 /\lpar {\rm \omega} _0^2 - \Omega _{\rm c}^2 \rpar \rpar \rpar }^2}} \over {1 - \lpar {\rm \omega} _{\rm p}^2 /\lpar {\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 \rpar \rpar - \lpar \Omega _{\rm p}^2 /\lpar {\rm \omega} _0^2 - \Omega _{\rm c}^2 \rpar \rpar }}.}$$

3. THIRD-HARMONIC BRILLOUIN INSTABILITY ANALISIS

As the Brillouin instability switches into the third-harmonic scattering process, the density perturbation at (2ω ₀ ± ω, k ± 2k ₀) couples with the electron velocity at (ω ₀, k ₀) to generate a third-harmonic nonlinear current density at (3ω ₀ ± ω, k ± 3k ₀). Therefore, the wave equation for the third-harmonic scattered waves becomes

(3)$$\left({\displaystyle{{{\partial ^2}} \over {\partial {t^2}}} - {c^2}\displaystyle{{{\partial ^2}} \over {\partial {z^2}}}} \right){A_{x\lpar 3{{\rm \omega} _0} \pm {\rm \omega} \rpar }}=4{\rm \pi} {J_{x\lpar 3{{\rm \omega} _0} \pm {\rm \omega} \rpar }}\comma \;$$

where ${J_{x\lpar 3{{\rm \omega} _0} \pm {\rm \omega} \rpar }}$ is the third-harmonic nonlinear current density associated with the third-harmonic Brillouin scattering and is determined using the equation of continuity for density perturbation at (2ω ₀ ± ω), which has the form

(4)$$\eqalign{& {J_{x\lpar 3{{\rm \omega} _0} \pm {\rm \omega} \rpar }} = \displaystyle{i \over 2}ec\left({{n_0}{a_{3{\rm s}}}{{\rm \mu} _{3{\rm s}}} + \displaystyle{1 \over 2}{n_{\rm \omega} }{{\rm \sigma} _0}{{\rm \mu} _0}{a_0}} \right){e^{i[\lpar {\rm k} \pm 3{{\rm k}_0}\rpar z - \lpar 3{{\rm \omega} _0} \pm {\rm \omega} \rpar t] }}}$$

with

$${{{\rm \sigma} _0}=- {{{{c^2}a_0^2\, k_0^2 [\lpar {\rm \alpha} _0^2+4\rpar {\rm \omega} _0^2 {\rm \omega} _{\rm c}^2+2\lpar {\rm \alpha} _0^2 - 1\rpar {\rm \omega} _0^4+2{{\rm \alpha} _0}{\rm \omega} _0^3 {{\rm \omega} _{\rm c}}+4{{\rm \alpha} _0}{{\rm \omega} _0}{\rm \omega} _{\rm c}^3+{\rm \omega} _{\rm c}^4] } \over {4{{\lpar {\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 \rpar }^2}\lpar 4{\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 \rpar }}}}.$$

Here, n _ω is the low-frequency density fluctuation associated with the electrostatic ion acoustic upper-hybrid wave arising in the stimulated Brillouin scattering, and a _3s and μ_3s, which are defined by replacing ω₀ with (3ω₀ ± ω) in the definition of parameter μ₀, are the normalized amplitude and nonlinear coefficient of the scattered wave, respectively. Hence, the linearized nonlinear wave equation governing the propagation of scattered Stokes and anti-Stokes waves with amplitude A _3± associated with the Brillouin scattering in magnetized plasma is obtained as

(5)$$\lpar {\rm \omega} _{3 \pm }^2 - {c^2}k_{3 \pm }^2 - {{\rm \mu} _{3{\rm s}}}{\rm \omega} _{\rm p}^2 \rpar {A_{3 \pm }}\lpar {{\rm \omega} _{3 \pm }}\comma \; {k_{3 \pm }}\rpar =\displaystyle{{2{\rm \pi} {e^2}} \over m}{{\rm \mu} _0}{{\rm \sigma} _0}{A_0}{n_{\rm \omega} }\lpar k\comma \; {\rm \omega} \rpar \comma \; $$

where k _3± = k ± 3k ₀ and ω _3± = 3ω ₀ ± ω are the wave number and frequency of the scattered Stokes/anti-Stokes waves, respectively. Combining the equation of electron's motion with the continuity equation in the direction of laser wave gives the second-coupled equation associated with the Brillouin scattering of X-mode laser wave in a magnetized plasma

(6)$$\displaystyle{e \over m}\left({1+\displaystyle{{{\rm \omega} _{\rm c}^2 } \over {{\rm \omega} _{\rm p}^2 }}} \right)\displaystyle{{\partial {{\rm \phi} _{\rm s}}} \over {\partial z}}=\displaystyle{{{e^2}} \over {{m^2}{c^2}}}\displaystyle{\partial \over {\partial z}}[{{\rm \mu} _{3{\rm s}}}{{\bf A}_{3{\rm s}}} \cdot {{\bf A}_{3x}}+{{\rm \chi} _{3{\rm s}}}{{\bf A}_{3{\rm s}}} \cdot {{\bf A}_{3z}}] +\displaystyle{{v_{\rm T}^2 } \over {{n_0}}}\displaystyle{{\partial {n_{\rm \omega} }} \over {\partial z}}.$$

The second term on the right-hand side of Eq. (6) is due to the thermal pressure of plasma electrons, where ${v_{\rm T}}\, \lpar =\sqrt {{T_{\rm e}}/m} \rpar $ is the electron thermal velocity. In this equation, ϕ_s is the electrostatic potential of plasma fluctuations, and A₃, A_3s are the vector potentials of the third-harmonic incident and scattered waves, respectively. Also, μ_3s is defined by replacing (3ω₀ ± ω) with ω₀ in the definition of parameter μ₀, and parameter χ _3s [which is determined by replacing (3ω₀ ± ω) with ω₀ in the definition of parameter χ ₀] is given by

$$\eqalign{& {{\rm \chi} _0} = \displaystyle{{{{\rm \alpha} _0}{\rm \omega} _0^2 + {{\rm \omega} _0}{{\rm \omega} _{\rm c}}} \over {{\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 }} + \displaystyle{{a_0^2 } \over {{{\lpar {\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 \rpar }^4}}} \cr & \qquad \times \left[\displaystyle{{{c^2}k_0^2 } \over {4\lpar 4{\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 \rpar }}\lpar {M_1}{{\rm \omega} _{\rm c}} + {M_3}\lpar {\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 \rpar \rpar \right. \cr & \qquad \left. - \displaystyle{3 \over 8}\lpar {M_2}{{\rm \omega} _{\rm c}} + {M_4}\lpar {\rm \omega} _0^2 - {\rm \omega} _{\rm c}^2 \rpar \rpar \right]}$$

with

$$\eqalign{& {M_3} = {{\rm \alpha} _0}\lpar 3{\rm \alpha} _0^2 - 25\rpar {\rm \omega} _0^4 {\rm \omega} _{\rm c}^2 + \lpar 14{\rm \alpha} _0^2 - 9\rpar {\rm \omega} _0^3 {\rm \omega} _{\rm c}^3 - \lpar 26{\rm \alpha} _0^2 + 1\rpar {\rm \omega} _0^5 {{\rm \omega} _{\rm c}} \cr & \quad \quad - {{\rm \alpha} _0}\lpar 6{\rm \alpha} _0^2 + 3\rpar {\rm \omega} _0^6 + 15{\rm \alpha} _0^2 {\rm \omega} _{\rm c}^4 + 7{{\rm \omega} _0}{\rm \omega} _{\rm c}^5 }$$

and

$$\eqalign{& {M_4} = {{\rm \alpha} _0}\lpar {\rm \alpha} _0^2 + 3\rpar {\rm \omega} _0^6 + \lpar 3{\rm \alpha} _0^2 + 1\rpar \lpar {\rm \omega} _0^3 {\rm \omega} _{\rm c}^3 + 3{\rm \omega} _0^5 {{\rm \omega} _{\rm c}}\rpar \cr & \quad \quad + {{\rm \alpha} _0}\lpar 3{\rm \alpha} _0^2 + 6{{\rm \alpha} _0} + 3\rpar {\rm \omega} _0^4 {\rm \omega} _{\rm c}^2 .}$$

Now, combining the equations of ion motion and continuity gives (Paknezhad, Reference Paknezhad2013c)

(7)$$\displaystyle{{{{\partial} ^2}{n_{\rm \omega} }} \over {{\partial} {t^2}}}=\displaystyle{{{n_0}Ze} \over M}\displaystyle{{{{\partial} ^2}{{\rm \phi} _{\rm s}}} \over {{\partial} {z^2}}} - \Omega _{\rm c}^2 {n_{\rm \omega} }$$

where M and Z are the ion rest mass and the ion charge state respectively and Ω_c (=e B₀/Mc) is the ion cyclotron frequency, and we assumed that both electron and ion fluids have the same density in the plasma. By substituting ϕ_s from Eq. (6) into Eq. (7), we finally obtain a nonlinear equation for the low-frequency density fluctuation in plasma:

(8)$$\eqalign{&\left[{\displaystyle{{{{\partial} ^2}} \over {{\partial} {t^2}}}+\Omega _{\rm c}^2 - c_{\rm p}^2 \displaystyle{{{\rm \omega} _{\rm p}^2 } \over {{\rm \omega} _{{\rm UH}}^2 }}\displaystyle{\partial {^2} \over {{\partial} {z^2}}}} \right]{n_{\rm \omega} } \cr & \quad=\displaystyle{{{\rm \omega} _{\rm p}^2 } \over {{\rm \omega} _{{\rm UH}}^2 }}\displaystyle{{{n_0}Z{e^2}} \over {mM{c^2}}}\displaystyle{{{\partial ^2}} \over {\partial {z^2}}}[{{\rm \mu} _{3{\rm s}}}{{\bf A}_{3{\rm s}}} \cdot {{\bf A}_{3x}}+{{\rm \chi} _{3{\rm s}}}{{\bf A}_{3{\rm s}}} \cdot {{\bf A}_{3z}}] \comma \; }$$

where c _s = (ZT _e/M)^1/2 is the ion acoustic velocity. Here, ${{\bf A}_{3x}}={A_0}\, {e^{i[ 3\lpar {k_0}z - {{\rm \omega} _0}t\rpar ]}}$ and ${{\bf A}_{3z}}=i{{\rm \alpha} _{3{{\rm \omega} _0}}}{A_0}\, {e^{i[3\lpar {k_0}z - {{\rm \omega} _0}t\rpar }}$ are the third-harmonic vector potential components, and ${A_{3{\rm s}x}}=$${A_{3 \pm }}{e^{i[\lpar k \pm 3{k_0}\rpar z - \lpar 3{{\rm \omega} _0} \pm {\rm \omega} \rpar t\rpar]}}$ and ${{\bf A}_{3{\rm s}z}}=i{{\rm \alpha} _{3 \pm }}{A_{3 \pm }}\, {e^{i[ \lpar k \pm 3{k_0}\rpar z - \lpar 3{{\rm \omega} _0} \pm {\rm \omega} \rpar t\rpar]}}$ are the vector potential components of the scattered Stokes and anti-Stokes waves associated with the Brillouin scattering, where ${{\rm \alpha} _{3{{\rm \omega} _0}}}$ and α _3± are respectively determined by replacing 3ω ₀ and (3ω ₀ ± ω) with ω ₀ in the definition of α ₀. Fourier transform of Eq. (8), and then combining with Eq. (5) gives the dispersion relation of the exited ion acoustic upper-hybrid wave

(9)$$\eqalign{& {{\rm \omega} ^2} - \Omega _{\rm c}^2 - {k^2}c_{\rm s}^2 \displaystyle{{{\rm \omega} _{\rm p}^2 } \over {{\rm \omega} _{{\rm UH}}^2 }} \cr &= \displaystyle{{{c^2}{k^2}\Omega _{\rm p}^2 a_0^2 {{\rm \mu} _0}{{\rm \sigma} _0}} \over 4}\displaystyle{{{\rm \omega} _{\rm p}^2 } \over {{\rm \omega} _{{\rm UH}}^2 }} \left[\displaystyle{{\lpar {{\rm \mu} _{3 + }} + {{\rm \chi} _{3 + }}{{\rm \alpha} _{3{{\rm \omega} _0}}}{{\rm \alpha} _{3 + }}\rpar } \over {{D_{3 + }}}} \right. \cr & \left. \;\quad+ \displaystyle{{\lpar {{\rm \mu} _{3 - }} + {{\rm \chi} _{3 - }}{{\rm \alpha} _{3{{\rm \omega} _0}}}{{\rm \alpha} _{3 - }}\rpar } \over {{D_{3 - }}}} \right].}$$

Here, ${\Omega _{\rm p}}\; \lpar ={{\rm \omega} _{\rm p}}\sqrt {Zm/M} \rpar $ is the plasma ion frequency, and ${D_{3 \pm }}\; [\!\! ={\rm \omega} _{3 \pm }^2 - {c^2}k_{3 \pm }^2 - {{\rm \mu} _{3 \pm }}{\rm \omega} _p^2 ]$ is the dispersion coefficient of the third-harmonic scattered waves (Stokes/anti-Stokes) in the Brillouin scattering process.

For Brillouin instability, maximum growth rate occurs in backward scattering in which both the scattered Stocks light wave and the exited ion acoustic upper-hybrid wave are resonant modes (ω = Ω, D ₃₋ = 0), that is, when

(10)$${\lpar 3{{\rm \omega} _0} - \Omega \rpar ^2} - {c^2}{\lpar k - 3{k_0}\rpar ^2} - {{\rm \mu} _{3 - }}{\rm \omega} _{\rm p}^2=0\comma \; $$

where $\Omega={\lpar \Omega _{\rm c}^2+\lpar {\rm \omega} _{\rm p}^2 /{\rm \omega} _{{\rm UH}}^2 \rpar {k^2}c_{\rm s}^2 \rpar ^{1/2}}$. The frequency-matching condition for Stokes wave is ω ₃₋ = 3ω ₀ − Ω and the wave number k is determined from Eq. (10). Assuming Ω_c≪ω ₀, we obtain the normalized wave number of the ion acoustic upper-hybrid wave in a transversely magnetized plasma

(11)$$\eqalign{\displaystyle{{k}\over{k_o}} & =3 \left[\displaystyle{{1 - \lpar {{\rm \omega} _{\rm p}}/{{\rm \omega} _{{\rm UH}}}\rpar \lpar {c_{\rm s}}/c\rpar {{\lpar 1 - {{\rm \mu} _0}\lpar {\rm \omega} _{\rm p}^2 /{\rm \omega} _0^2 \rpar \rpar }^{ - 1/2}}} \over {1 - \lpar {\rm \omega} _{\rm p}^2 /{\rm \omega} _{{\rm UH}}^2 \rpar \lpar c_{\rm s}^2 /{c^2}\rpar }} \right. \cr & \left.\quad + \displaystyle{{\matrix{[ {\lpar 1 - \lpar {{\rm \omega} _{\rm p}}/{{\rm \omega} _{{\rm UH}}}\rpar \lpar {c_{\rm s}}/c\rpar {{\lpar 1 - {{\rm \mu} _0}\lpar {\rm \omega} _{\rm p}^2 /{\rm \omega} _0^2 \rpar \rpar }^{ - 1/2}}\rpar }^2 \cr + \; \lpar 1 - \lpar {\rm \omega} _{\rm p}^2 /{\rm \omega} _{{\rm UH}}^2 \rpar \lpar c_{\rm s}^2 /{c^2}\rpar \rpar {{\lpar \lpar \lpar 1 + \lpar \Omega _{\rm c}^2 /9{\rm \omega} _0^2 \rpar \rpar}} \cr \lpar {{\rm \mu} _{3 - }} + {{\rm \chi} _{3 - }} {{\rm \alpha} _{3 - }} \rpar \lpar {\rm \omega} _{\rm p}^2 /9{\rm \omega} _0^2 \rpar \rpar /\lpar \lpar 1 - {{\rm \mu} _0}\lpar {\rm \omega} _{\rm p}^2 /{\rm \omega} _0^2 \rpar \rpar - 1\rpar ] ^{1/2} \cr} } \over {1 - \lpar {\rm \omega} _{\rm p}^2 /{\rm \omega} _{{\rm UH}}^2 \rpar \lpar c_{\rm s}^2 /{c^2}\rpar }} \right]}.$$

By ignoring the nonlinear terms, Eq. (11) reduces to

(12)$$\displaystyle{{{k_{\rm p}}} \over {{k_0}}} \simeq 6\, \left({1 - \displaystyle{{{{\rm \omega} _{\rm p}}} \over {{{\rm \omega} _{{\rm UH}}}}}\displaystyle{{{c_{\rm s}}} \over c}} \right){\left({1 - \displaystyle{{{\rm \omega} _{\rm p}^2 } \over {{\rm \omega} _{{\rm UH}}^2 }}\displaystyle{{c_{\rm s}^2 } \over {{c^2}}}} \right)^{ - 1}}\comma \; $$

which is three times the normalized linear wave number of ion acoustic upper-hybrid wave in fundamental Brillouin backscattering in a magnetized plasma (Paknezhad, Reference Paknezhad2013a). Note that for Brillouin backscattering, only the downshifted light wave needs to be retained (Kruer, Reference Kruer1998; Paknezhad, Reference Paknezhad2012; Reference Paknezhad2013a), and assuming the weakly coupled regime (Γ≪kc _s), we substitute ω = Ω + iΓ_NBBS in Eq. (9) to find

(13)$$\eqalign{& 2i{\Gamma _{{\rm NBBS}}}\Omega \, [\!\!-\!\! 2i{\Gamma _{{\rm NBBS}}}\lpar {{\rm \omega} _0} - \Omega \rpar] \cr & \quad= \displaystyle{{{\rm \omega} _{\rm p}^2 } \over {{\rm \omega} _{{\rm UH}}^2 }} \displaystyle{{{c^2}{k^2}a_0^2 \Omega _{\rm p}^2 {{\rm \mu} _0}{{\rm \sigma} _0}\lpar {{\rm \mu} _{3 - }} + {{\rm \chi} _{3 - }}{{\rm \alpha} _{3{{\rm \omega} _0}}}{{\rm \alpha} _{3 - }}\rpar } \over 4}.}$$

The real part of Eq. (13) gives the nonlinear growth rate of 3HBBS instability in the backscattering of an extraordinary laser wave in a weakly coupled magnetized plasma, where Γ_NBBS≪ω _UH and $a_0^2 \ll 1$,

(14)$${\Gamma _{3{\rm HBBS}}}=\displaystyle{{ck{a_0}{\Omega _{\rm p}}} \over 4}\displaystyle{{{{\rm \omega} _{\rm p}}} \over {{{\rm \omega} _{{\rm UH}}}}}\sqrt {\displaystyle{{{{\rm \mu} _0}{{\rm \sigma} _0}\lpar {{\rm \mu} _{3 - }} - {{\rm \chi} _{3 - }}{{\rm \alpha} _{3 - }}{{\rm \alpha} _{3{{\rm \omega} _0}}}\rpar } \over {\Omega \, \lpar 3{{\rm \omega} _0} - \Omega \rpar }}} .$$

In Eq. (6), parameter σ₀ arises as a result of the effect of 3HBBS of laser in a mildly relativistic magnetized plasma. Now, we investigate the variation of the growth rate of 3HBBS instability on the basis of Eq. (14).

In Figure 3, the variation of normalized growth rate of 3HBBS instability Γ_3HBBS/ω₀ has been plotted against the normalized wave number k/k ₀ for a ₀ = 0.02, and two values of ω_c/ω₀ = 0 and ω_c/ω₀ = 0.1. It is demonstrated that for the given values, the starting point of k/k ₀ for 3HBBS instability in the unmagnetized plasma (ω_c/ω₀ = 0) is around 5.8824, whereas it is 6 for the magnetized plasma (ω_c/ω₀ = 0.1). For both cases, as the normalized wave number decreases (by increasing plasma density), the growth rate increases and then saturates for a low value of k/k ₀ (high plasma densities).

Fig. 3.

Variation of the normalized growth rate of 3HBBS instability as a function of normalized wave number k/k ₀ in unmagnetized (ω_c/ω₀ = 0) and magnetized (ω_c/ω₀ = 0.1) plasmas for a ₀ = 0.02, c _s/c = 0.02.

Figure 4 describes the variation of normalized growth rate of 3HBBS instability Γ_3HBBS/ω₀ with normalized plasma frequency ω_p/ω₀ for ω_c/ω₀ = 0.05, and two values of a ₀ = 0.02 and a ₀ = 0.03. It is observed that an increase in the plasma frequency (plasma density) leads to an increase in the growth rate. The growth rate also increases with laser amplitude.

Fig. 4.

Variation of normalized growth rate of 3HBBS instability as a function of normalized plasma frequency for ω_c/ω₀ = 0.05, and c _s/c = 0.02.

Figure 5 depicts the variation of normalized growth rate of 3HBBS instability as a function of normalized cyclotron frequency ω_c/ω₀ at a ₀ = 0.02 and ω_p/ω₀ = 0.1. Similar to the third-harmonic Raman backscattering instability (Paknezhad, Reference Paknezhad2013c), the growth rate decreases by increasing the external magnetic field. However, in the case of third-harmonic Raman backscattering, the growth rate has a further decrease because of the absence of ion's motion.

Fig. 5.

Variation of normalized growth rate of 3HBBS instability as a function of normalized electron cyclotron frequency for a ₀ = 0.02, c _s/c = 0.02, and ω_p/ω₀ = 0.1.