Raman scattering of intense, short laser pulses in modulated plasmas

We examine the exponentiation of the Raman forward scattering instability in modulated plasma channels computationally and analytically. An evolution equation for the complex phases of the Raman scattered waves treating the spatial localization and discrete nature of the channel modes is derived. Simulations with WAKE [P. Mora and T. M. Antonsen Jr., Phys. Plasmas 4, 217 (1997)] verify the theory in the linear growth regime and provide insight into the nonlinear stage of the instability when cascading and pump depletion play a role. We find that the exponentiation in modulated channels depends on two factors: the increase in coupling due to the increased plasma wavenumber in the high-density regions of the channel and a decreased coupling due to the reduced longitudinal spatial coherence. For the parameters considered, simulations show that the finite extent of the pump pulse is more significant in determining the exponentiation than phase mixing due to the transverse variation of the channel. Both the theory and simulations confirm that modulated channels allow for the stable guiding of longer pulses than nonmodulated channels.

Figure 1

A comparison of the exponentiation in transversely uniform plasmas in the axially uniform case, the black (upper) line, and the modulated case, the red (lower) line), as a function of propagation distance. The pump amplitude is ${\mathrm{\alpha ̂}}_0=0.25$, and the scattered eigennumber is $n=1$.

Figure 2

A comparison of our theoretical exponentiations with the simulation results in an axially uniform channel for a pump amplitude of ${\mathrm{\alpha ̂}}_0=0.14$. (a) $n=0$ scattered eigenmode; (b) $n=1$ scattered eigenmode. The black (oscillating) lines are the simulation results, the red (solid) lines are the uncoupled sideband theory results, and the red dashed is line the coupled sideband theory result.

Figure 3

A comparison of our theoretical exponentiations with the simulation results in a modulated channel for a pump amplitude of ${\mathrm{\alpha ̂}}_0=0.21$. (a) $n=0$ scattered eigenmode; (b) $n=1$ scattered eigenmode. The black oscillating lines are the simulation results, and the red lines are the theory results.

Figure 4

A comparison of the growth in the axially uniform and modulated plasma channels as a function of propagation distance. (a) Initial pump amplitude is ${\mathrm{\alpha ̂}}_0=0.28$; (b) amplitude is ${\mathrm{\alpha ̂}}_0=0.42$. The dashed red line is the theory, the black solid (upper) line is the axially uniform exponentiation, and the blue solid (lower) line is the modulated exponentiation.

Figure 5

Normalized pump amplitude as a function of time for three different initial amplitudes. (a) Axially uniform channel; (b) modulated channel. The black (topmost) line is for a pump amplitude ${\mathrm{\alpha ̂}}_0=0.14$, the blue (middle) line is for ${\mathrm{\alpha ̂}}_0=0.28$, and the red (lowest) line is for ${\mathrm{\alpha ̂}}_0=0.42$.

Figure 6

Longitudinal pulse shape at two different times during the propagation. (a) Uniform channel; (b) modulated channel. The black solid line is the pulse shape at $0R_T$, and the dashed red line is at $2R_T$.

Figure 7

Log scale of the electromagnetic spectrum as a function of time for the $n=1$ eigenmode. Figure (a) is the axially uniform channel and (b) the modulated channel. The right streak in each curve is the seed level of the eigenmode at the pump central wavenumber. The left streak is the first Stokes sideband resulting from the scattering instability.

Figure 8

Laser pulse shape for the pump wave, black dashed line (top), and scattered waves of the $n=1$ spatial harmonic in the axially uniform, blue solid line (middle), and modulated, red solid line (lower), channels evaluated at the $k_z$ for maximum growth on a log scale as a function of $\xi$. (a) Pump amplitude ${\mathrm{\alpha ̂}}_0=0.28$; (b) pump amplitude ${\mathrm{\alpha ̂}}_0=0.35$.

Figure 9

Laser pulse shape at three different times in a modulated channel (top) and an axially uniform channel bottom for an initial amplitude of ${\mathrm{\alpha ̂}}_0=0.35$.

Figure 10

Log scale of the electromagnetic spectrum as a function of time for the $n=0$ eigenmode and a pump amplitude of ${\mathrm{\alpha ̂}}_0=0.28$. (a) Channel is axially uniform; (b) channel is modulated.

Figure 11

A comparison of an axial sinusoidal and triangular density profile for the $n=1$ eigenmode and a pump amplitude of ${\mathrm{\alpha ̂}}_0=0.35$. (a) Maximum exponentiation is plotted for the initial redshifted sideband as a function of propagation distance. The black line (left) and the red line (right) correspond to the sine and triangle profiles, respectively. (b) and (c) Spectra of the scattered eigenmode as a function of time in the sinusoidal and triangular channels.

Figure 12

(a) Instability growth rates in the strong transverse phase-mixing regime as a function of plasma wavenumber in the axially uniform channel, red line (left) and modulated channel, black line (right). Cutoff in the growth rate is due to the resonance radius being imaginary. (b) Maximum growth rate as a function of modulation amplitude.