On the Issue of Possibility of Low-Threshold Nonlinear Excitation of Oblique Langmuir Waves in Plasma Filament Using Ordinary Polarized Microwaves

Abstract

The results are presented from the theoretical analysis of the low-threshold parametric decay instability of the pumping microwave beam with the ordinary polarization and the frequency in the electron cyclotron resonance frequency range. This instability develops in the cylindrical plasma filament, which results in the excitation of two oblique Langmuir waves.

APPENDIX

Let us derive the expression for the nonlinear plasma susceptibility, which describes the nonlinear interaction of the OL waves in the presence of the ordinary wave. We use the Cartesian coordinate system and assume that the plasma is homogeneous. In this case, the electron density is the sum of the unperturbed component and the first and second order perturbations at frequencies of the daughter waves:

$${{n}_{1}} = n_{1}^{{(1)}}\left( {{{\varphi }_{1}}} \right) + n_{1}^{{(2)}}\left( {E_{z}^{*},{{\varphi }_{2}}} \right)$$

$$\begin{gathered} \, \propto \exp \left( {i{{q}_{{1x}}}x + i{{q}_{{1y}}}y + i{{q}_{z}}z - i{{\omega }_{1}}t} \right), \\ {{n}_{2}} = n_{2}^{{(1)}}\left( {{{\varphi }_{2}}} \right) + n_{2}^{{(2)}}\left( {{{E}_{z}},{{\varphi }_{1}}} \right) \\ \end{gathered} $$

(А.1)

$$\, \propto \exp \left( {i{{q}_{{2x}}}x + i{{q}_{{1y}}}y + i{{q}_{z}}z + i{{\omega }_{2}}t} \right).$$

In the electric fields of the high-frequency oscillations, the corresponding electron velocities also have the linear and non-linear components:

$$\begin{gathered} {{{\mathbf{u}}}_{0}} = u_{{0z}}^{{(1)}}\left( {{{E}_{z}}} \right){{{\mathbf{e}}}_{z}} \propto \exp \left( {i{{k}_{x}}x + i{{k}_{y}}y - i{{\omega }_{0}}t} \right), \\ {{{\mathbf{u}}}_{1}} = {\mathbf{u}}_{1}^{{(1)}}\left( {{{\varphi }_{1}}} \right) + {\mathbf{u}}_{1}^{{(2)}}\left( {E_{z}^{*},{{\varphi }_{2}}} \right) \\ \end{gathered} $$

$$\, \propto \exp \left( {i{{q}_{{1x}}}x + i{{q}_{{1y}}}y + i{{q}_{z}}z - i{{\omega }_{1}}t} \right),$$

(A.2)

$$\begin{gathered} {{{\mathbf{u}}}_{2}} = {\mathbf{u}}_{2}^{{(1)}}\left( {{{\varphi }_{2}}} \right) + {\mathbf{u}}_{2}^{{(2)}}\left( {{{E}_{z}},{{\varphi }_{1}}} \right) \\ \, \propto \exp \left( {i{{q}_{{2x}}}x + i{{q}_{{2y}}}y + i{{q}_{z}}z + i{{\omega }_{2}}t} \right). \\ \end{gathered} $$

The linear components of the electron oscillation velocities in the electric fields of three microwaves can be found by means of solving the moment equations written in the following form:

$$u_{{0z}}^{{(1)}} = - i\frac{{{\text{|}}e{\text{|}}}}{{{{m}_{e}}{{\omega }_{0}}}}{{E}_{z}},$$

$$\begin{gathered} u_{{1,2x}}^{{(1)}} = \frac{{{\text{|}}e{\text{|}}}}{{{{m}_{e}}}}\frac{{ \mp {{q}_{{1,2x}}}{{\omega }_{1}} - i{{q}_{{1,2y}}}{{\omega }_{{ce}}}}}{{{{\Delta }_{{1,2}}}}}{{\varphi }_{{1,2}}}, \\ u_{{1,2y}}^{{(1)}} = \frac{{{\text{|}}e{\text{|}}}}{{{{m}_{e}}}}\frac{{ \mp {{q}_{{1,2y}}}{{\omega }_{1}} + i{{q}_{{1,2x}}}{{\omega }_{{ce}}}}}{{{{\Delta }_{{1,2}}}}}{{\varphi }_{{1,2}}}, \\ \end{gathered} $$

(A.3)

$$u_{{1,2z}}^{{(1)}} = \mp {{q}_{z}}\frac{{{\text{|}}e{\text{|}}}}{{{{m}_{e}}{{\omega }_{{1,2}}}}}{{\varphi }_{{1,2}}},$$

where ${{\Delta }_{{1,2}}} = \omega _{{1,2}}^{2} - \omega _{{ce}}^{2}$. Then, using the continuity equation, we obtain the expressions for the linear perturbations of the background density at frequencies of the interacting waves:

$$\begin{gathered} n_{0}^{{(1)}} = 0, \\ n_{{1,2}}^{{(1)}} = \pm \frac{{{{q}_{{1,2x}}}u_{{1,2x}}^{{(1)}} + {{q}_{{1,2y}}}u_{{1,2y}}^{{(1)}} + {{q}_{z}}u_{{1,2z}}^{{(1)}}}}{{{{\omega }_{{1,2}}}}}. \\ \end{gathered} $$

(А.4)

The second-order charge density at a frequency of the first OL wave is determined by the following expression:

$$\begin{gathered} \rho _{2}^{{(2)}} \\ = \frac{{{\text{|}}e{\text{|}}}}{{{{\omega }_{2}}}}\left( {\bar {n}\left( {{{q}_{{2x}}}u_{{2x}}^{{(2)}} + {{q}_{{2y}}}u_{{2y}}^{{(2)}} + {{q}_{z}}u_{{2z}}^{{(2)}}} \right) + n_{1}^{{(1)}}{{q}_{z}}u_{{0z}}^{{(1)*}}} \right). \\ \end{gathered} $$

(A.5)

where the second-order correction to the components of the electron oscillation velocity at a frequency of the first OL wave is the solution of the moment equations

$$\begin{gathered} \left( {\begin{array}{*{20}{c}} {u_{{2x}}^{{(2)}}} \\ {u_{{2y}}^{{(2)}}} \\ {u_{{2z}}^{{(2)}}} \end{array}} \right) = - \left( {\begin{array}{*{20}{c}} {\frac{1}{{{{\Delta }_{2}}}}}&0&0 \\ 0&{\frac{1}{{{{\Delta }_{2}}}}}&0 \\ 0&0&{\frac{1}{{\omega _{2}^{2}}}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{\omega }_{2}}}&{ - i{{\omega }_{{ce}}}}&0 \\ {i{{\omega }_{{ce}}}}&{{{\omega }_{2}}}&0 \\ 0&0&{{{\omega }_{2}}} \end{array}} \right) \\ \times \left( {\begin{array}{*{20}{c}} {{{q}_{z}}u_{{0z}}^{{(1)*}}u_{{1x}}^{{(1)}} + i\frac{{{\text{|}}e{\text{|}}}}{{{{m}_{e}}c}}u_{{1z}}^{{(1)}}H_{y}^{*}} \\ {{{q}_{z}}u_{{0z}}^{{(1)*}}u_{{1y}}^{{(1)}} - i\frac{{{\text{|}}e{\text{|}}}}{{{{m}_{e}}c}}u_{{1z}}^{{(1)}}H_{x}^{*}} \\ {{{q}_{z}}u_{{0z}}^{{(1)*}}u_{{1z}}^{{(1)}}\, + \,{{k}_{x}}u_{{1x}}^{{(1)}}u_{{0z}}^{{(1)*}}\, - \,i\frac{{{\text{|}}e{\text{|}}}}{{{{m}_{e}}c}}(u_{{1x}}^{{(1)}}H_{y}^{*}\, - \,u_{{1y}}^{{(1)}}H_{x}^{*})} \end{array}} \right){\kern 1pt} , \\ \end{gathered} $$

(А.6)

where ${{H}_{x}} = {{k}_{y}}\frac{c}{{{{\omega }_{0}}}}{{E}_{z}}$ and ${{H}_{y}} = - {{k}_{x}}\frac{c}{{{{\omega }_{0}}}}{{E}_{z}}$. Substituting expressions (А.3), (A.4), and (A.6) into expression (A.5), we obtain the following expressions:

$$\rho _{2}^{{(2)}} = - \frac{{{{{\hat {\chi }}}_{{21}}}\left( {{{\varphi }_{1}}E_{z}^{*}} \right)}}{{4\pi H}},$$

(А.7)

$$\begin{gathered} {{{\hat {\chi }}}_{{21}}} = i\frac{{\omega _{{pe}}^{2}\left| {{{\omega }_{{ce}}}} \right|{{q}_{z}}c}}{{{{\omega }_{0}}\omega _{1}^{2}\omega _{2}^{2}{{\Delta }_{1}}{{\Delta }_{2}}}}\left( {q_{z}^{2}\left( {{{\omega }_{1}} - {{\omega }_{2}}} \right){{\Delta }_{1}}{{\Delta }_{2}}\mathop {}\limits_{} } \right. \\ \, + {{\omega }_{1}}\left( {{{k}_{y}}\left( {{{q}_{{2y}}}\omega _{2}^{2}{{\Delta }_{1}} + {{q}_{{1y}}}\omega _{1}^{2}{{\Delta }_{2}}} \right.} \right. \\ \left. {\mathop - \limits_{} \;i\left| {{{\omega }_{{ce}}}} \right|\left( {{{\omega }_{2}}{{\Delta }_{1}}{{q}_{{2x}}} + {{\omega }_{1}}{{\Delta }_{2}}{{q}_{{1x}}}} \right)} \right) \\ \end{gathered} $$

$$\begin{gathered} + \;{{\omega }_{2}}\left( {\left( {i\left| {{{\omega }_{{ce}}}} \right|\left( {{{q}_{{1y}}}{{\Delta }_{2}} - {{\omega }_{1}}\left( {{{\omega }_{1}} - {{\omega }_{2}}} \right){{q}_{{2y}}}} \right)\mathop {}\limits_{} } \right.} \right. \\ \, - \left. {{{\omega }_{1}}{{\Delta }_{1}}{{q}_{{1x}}} - {{\omega }_{1}}\left( {{{\omega }_{1}}{{\omega }_{2}} - {{{\left| {{{\omega }_{{ce}}}} \right|}}^{2}}} \right){{q}_{{2x}}}} \right){{q}_{{1x}}} \\ \end{gathered} $$

(А.8)

$$\begin{gathered} \, + {{q}_{{1y}}}\left( {\left| {{{\omega }_{{ce}}}} \right|\left( {{{\omega }_{1}}\left( {{{\omega }_{1}} - {{\omega }_{2}}} \right){{q}_{{2x}}} - i{{\Delta }_{2}}{{q}_{{1x}}}} \right)} \right. \\ \, - \left. {{{\omega }_{1}}{{q}_{{2y}}}\left( {{{\omega }_{1}}{{\omega }_{2}} - {{{\left| {{{\omega }_{{ce}}}} \right|}}^{2}}} \right) - {{q}_{{1y}}}{{\omega }_{1}}{{\Delta }_{2}}} \right) \\ \left. {\left. {\left. {\mathop + \limits_{_{{}}} {{\Delta }_{1}}\left( {{{\omega }_{2}}{{q}_{{2x}}} + i{{q}_{{2y}}}\left| {{{\omega }_{{ce}}}} \right|} \right){{k}_{x}}} \right)} \right)} \right). \\ \end{gathered} $$

It can be demonstrated that the second-order charge density at a frequency of the second OL wave obeys the following equation:

$$\rho _{1}^{{(2)}} = \rho _{2}^{{(2)*}}.$$

(А.9)