Instability and dynamics of two nonlinearly coupled laser beams in a two-temperature electron plasma

We consider nonlinear interactions between two colliding laser beams in an electron plasma, accounting for the relativistic electron mass increase in the laser fields and radiation pressure driven electron-acoustic (EA) perturbations that are supported by hot and cold electrons. By using the hydrodynamic and Maxwell equations, we obtain the relevant equations for nonlinearly coupled laser beams and EA perturbations. The coupled equations are then Fourier analyzed to obtain a nonlinear dispersion relation. The latter is numerically solved to show the existence of new classes of the parametric instabilities in the presence of two colliding laser beams in a two-electron plasma. The dynamics of nonlinearly coupled laser beams in our electron plasma is also investigated. The results should be useful in understanding the nonlinear propagation characteristics of multiple electromagnetic beams in laser-produced plasmas as well as in space plasmas.

Figure 1

(Color online) The growth rate $\gamma ∕{\omega }_{p0}$ as a function of the wave numbers $K_y$ and $K_z$, for a single laser beam ${\mathbf{A}}_1$ propagating in the positive $y$ direction with the wave number $(k_{1y},k_{1z})=(0.8,0){\omega }_{p0}∕c$, having the amplitude $e\mid {\mathbf{A}}_{10}\mid ∕m{c}^2=0.1$. We used $n_{c0}=0.7n_0$, $n_{h0}=0.3n_0$, and $V_{Th}=0.05c$.

Figure 2

(Color online) The growth rate $\gamma ∕{\omega }_{p0}$ as a function of the normalized wave numbers $K_{y}c∕{\omega }_{p0}$ and $K_{z}c∕{\omega }_{p0}$, for two coupled laser beams ${\mathbf{A}}_1$ and ${\mathbf{A}}_2$ propagating in the positive $y$ and $z$ directions, respectively, with wave numbers $(k_{1y},k_{1z})=(0.8,0){\omega }_{p0}∕c$ and $(k_{1y},k_{1z})=(0,0.8){\omega }_{p0}∕c$, respectively. Each beam has the amplitude $e\mid {\mathbf{A}}_{10}\mid ∕m{c}^2=e\mid {\mathbf{A}}_{20}\mid ∕m{c}^2=0.1$. We used $n_{c0}=0.7n_0$, $n_{h0}=0.3n_0$, and $V_{Th}=0.05c$.

Figure 3

(Color online) The growth rate $\gamma ∕{\omega }_{p0}$ as a function of the normalized wave numbers $K_{y}c∕{\omega }_{p0}$ and $K_{z}c∕{\omega }_{p0}$, for two counter-propagating laser beams ${\mathbf{A}}_1$ and ${\mathbf{A}}_2$ propagating in the positive and negative $y$ directions, respectively, with wave numbers $(k_{1y},k_{1z})=(0.8,0){\omega }_{p0}∕c$ and $(k_{1y},k_{1z})=(-0.8,0){\omega }_{p0}∕c$, respectively. Each beam has the amplitude $e\mid {\mathbf{A}}_{10}\mid ∕m{c}^2=e\mid {\mathbf{A}}_{20}\mid ∕m{c}^2=0.1$. We used $n_{c0}=0.7n_0$, $n_{h0}=0.3n_0$, and $V_{Th}=0.05c$.

Figure 4

(Color online) The dynamics of two coupled laser beams ${\mathbf{A}}_1$ and ${\mathbf{A}}_2$ that initially are propagating in the positive $y$ and $z$ directions, respectively, with wave numbers $(k_{1y},k_{1z})=(0.8,0){\omega }_{p0}∕c$ and $(k_{1y},k_{1z})=(0,0.8){\omega }_{p0}∕c$, respectively. Each beam has initially the amplitude $e\mid {\mathbf{A}}_{10}\mid ∕m{c}^2=e\mid {\mathbf{A}}_{20}\mid ∕m{c}^2=0.1$. (Same parameters as in Fig. 2.) The coupled beams decay into localized waves that are propagating obliquely towards larger $y$ and $z$.