Electronic parametric instabilities of an ultrarelativistic laser pulse in a plasma

Electronic parametric instabilities of an ultrarelativistic circularly polarized laser pulse propagating in underdense plasmas are studied by numerically solving the dispersion relation which includes the effect of the radiation reaction force in laser-driven plasma dynamics. Emphasis is placed on studying the different modes in the laser-plasma system and identifying the absolute and convective nature of the unstable modes in a parameter map spanned by the normalized laser vector potential and the plasma density. Implications for the ultraintense laser-plasma experiments are pointed out.

Figure 1

Growth rates of the Raman [solid blue (solid dark gray) line], and modulational [dotted red (dotted light gray) line] instabilities with the normalized wave vector, $k/k_0$. The dot-dashed blue (dot-dashed dark gray) and the dashed red (dashed dark gray) lines represent the real parts of the corresponding modes, respectively. In panel (a), the dashed cyan line (dashed light gray) represents the laser mode. This laser mode gets unstable and its real part represented by the dashed cyan line (upper dashed light gray) and imaginary part shown by the solid cyan line (lower solid light gray) are depicted in panels (b) and (c). The color legends in the brackets are for the black and white printing of the figures. Parameters are (a) $a_0=0.25,n=0.01n_c$; (b) $a_0=80,n=2.5n_c$; (c) $a_0=250,n=4n_c$, where $n_c=m_e{\omega }_0^2/4\pi e^2$ is the nonrelativistic critical density for the laser propagation.

Figure 2

The resonance of the anti-Stokes $[D_{+}$, dashed red (dashed gray) line] and Stokes $[D_{-}$, solid blue (solid dark gray) line] modes with (a) and without (b) the RR force in a plasma.

Figure 3

Parameter map for the distinguishable unstable branches with (a) and without (b) radiation reaction force. The colorbar denotes the number of distinguishable unstable branches. Dark blue color (dark gray shading) in the lower right corner denotes the relativistically opaque regions of plasma density.

Figure 4

Parameter map for the maximum normalized growth rate ($\mathrm{\Gamma }/{\omega }_0$) of the unstable branches with respect to the normalised laser vector potential $a_0$ and the plasma density $n/n_c$. The colorbar is on ${\log }_{10}$ scale. White region in the lower right corner denotes the relativistically opaque regions of plasma density.

Figure 5

Parameter map for the normalized maximum wave vector ($k/k_0$) of any unstable perturbation with (a) without (b) the RR force. The colorbar is on ${\log }_{10}$ scale. White region in lower right corner denotes the relativistically opaque regions of plasma density.

Figure 6

Solution of dispersion relation Eq. (3) in a complex $k$ plane with (a) and without (b) radiation reaction force. Panels (1), (2), (3), and (4) are for $\Im \left(\omega \right)=0.4,0.3,0.2,$ and 0.1, respectively, in each case. The other parameters are $a_0=250,n=10n_c$. Both real and imaginary parts of the wave vector $k$ are normalized with $k_0$.

Figure 7

Solutions of equation, $D-\partial D/\partial k=0$, together with $\partial \omega /\partial k=0$ condition in a complex $k$ plane with (a) and without (b) radiation reaction force. Panels (1), (2), (3), and (4) are for $\Im \left(\omega \right)=2.0,1.8,0.5$, and 0.05, respectively, in each case. The other parameters are $a_0=250,n=10n_c$. Both real and imaginary parts of the wave vector $k$ are normalized with $k_0$.

Figure 8

Magnified view of panel (4) of Fig. 7 with (a) and without (b) radiation reaction force. It can be seen that one branch of the $k$-solution has crossed the $\Re \left(k\right)$-axis and is in the upper-half plane, signifying the absolute instability as seen in Fig. 6. Other parameters are same as in Fig. 7. Both real and imaginary parts of the wave vector $k$ are normalized with $k_0$.

Figure 9

Parameter map showing the regions of absolute and convective instabilities with (a) and without (b) the inclusion of the radiation reaction force. Green (light gray shading) and yellow (light shading) colors represent the absolute and convective regions, respectively. Blue color (dark gray shading) denotes the relativistically opaque regions of plasma density.