Stimulated Raman scattering of intense laser pulses in air J. R. Pen ˜ ano, P. Sprangle, P. Serafim,* B. Hafizi, ² and A. Ting Plasma Physics Division, Naval Research Laboratory, Washington, DC 20375, USA ~Received 8 May 2003; published 24 November 2003! Stimulated rotational Raman scattering ~SRRS! is known to be one of the processes limiting the propagation of high-power laser beams in the atmosphere. In this paper, SRRS, Kerr nonlinearity effects, and group velocity dispersion of short laser pulses and pulse trains are analyzed and simulated. Fully time-dependent, three-dimensional, nonlinear propagation equations describing the Raman interaction, optical Kerr nonlinearity due to bound electrons, and group velocity dispersion are presented and discussed. The effective time- dependent nonlinear refractive index containing both Kerr and Raman processes is derived. Linear stability analysis is used to obtain growth rates and phase matching conditions for the SRRS, modulational, and filamentation instabilities. Numerical solutions of the propagation equations in three dimensions show the detailed evolution of the Raman scattering instability for various pulse formats. The dependence of the growth rate of SRRS on pulse duration is examined and under certain conditions it is shown that short ( ;psec) laser pulses are stable to the SRRS instability. The interaction of pulses in a train through the Raman polarization field is also illustrated. DOI: 10.1103/PhysRevE.68.056502 PACS number~s!: 41.60.Cr, 42.65.Dr, 42.68.Mj, 42.65.Re I. INTRODUCTION Recent advances in laser technology have generated tre- mendous opportunities for applications which require the propagation of high-intensity, short laser pulses through the atmosphere. For example, free electron lasers ~FELs! have the potential for both high peak power and higher average power than existing systems, along with a flexible pulse for- mat @1#. FELs capable of delivering megawatts of average power are in the foreseeable future. The pulse train of a MW-class FEL driven by a radio frequency ~RF! linac will likely be characterized by individual pulses with durations of ;1 psec, peak powers in the GW range, separated by ;1 nsec. These short, high intensity laser pulses can un- dergo unique interactions with the atmosphere in which both linear and nonlinear processes play a central role @2#. As a result of the high intensities, bound electron anharmonicity ~optical Kerr effect! and stimulated rotational Raman scatter- ing ~SRRS! can affect laser beam propagation. Because of the short duration of these pulses, group velocity dispersion ~GVD! can also significantly affect propagation. SRRS of a laser pulse in air is a quantum mechanical process involving the excitation of the rotational states of the molecular constituents of air by the laser pulse. It can be characterized as an instability that scatters laser energy into multiple Stokes and anti-Stokes frequency bands @3# which, because of the dispersive properties of air, can propagate at different velocities and at large angles with respect to the initial laser pulse, causing a severe distortion of the laser envelope @4#. Theoretically, SRRS can be understood as a three-level interaction @3,5# through the energy level dia- grams shown in Fig. 1. The molecular scatterer is assumed to have two rotational eigenstates, 1 ~the ground state! and 2, with corresponding energies W 1 and W 2 , and an excited state, e.g., an electronic or translational state, with energy W 3 @W 2 2W 1 . In this paper we consider the nonresonant scattering process in which the central laser frequency v 0 V 31 , V 32 , where V nm 5V n 2V m , and V n is the fre- quency associated with state n . It is also assumed that V 31 , V 32 @v 0 @v R , where v R [V 21 is defined as the rotational frequency. In this situation, state 3 is not populated and the laser excites a virtual state which can decay to produce the Stokes and anti-Stokes radiation. The generation of Stokes radiation consists of a transition from state ~1! to a virtual state followed by a transition from the virtual state to state ~2!. In the process, a photon with frequency v 2 5v 0 2v R is emitted. The generation of anti-Stokes radiation consists of a transition from state ~2! to a virtual state followed by a tran- sition from the virtual state to state ~1!, thereby emitting a photon at frequency v 1 5v 0 1v R . Since the population of state ~2! is much smaller than that of state ~1! in thermal equilibrium, the anti-Stokes lines are generally much weaker than the Stokes lines @3#. Stimulated Raman scattering of laser pulses propagating through air had been studied extensively in the 1980s for longer ( ;nsec) laser pulses @6–10#. For altitudes below 100 *Present address: Northeastern University, Department of Electri- cal Engineering, Boston, MA 02115. ² Present address: Icarus Research, Bethesda, MD. FIG. 1. Energy level diagram of Stokes and anti-Stokes line generation from a three-level model of stimulated rotational Raman scattering ~SRRS!. PHYSICAL REVIEW E 68, 056502 ~2003! 1063-651X/2003/68~5!/056502~16!/$20.00 ©2003 The American Physical Society 68 056502-1