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Physical Review Letters

Novel Analytic Solutions to General Four-Wave-Mixing Problems in a Raman Medium

Jay R. Ackerhalt

Published Apr 6, 1981• 33 citations
Publisher

Abstract

A new approach to the problem of steady-state plane-wave propagation in a Raman-active medium has lead to general analytical solutions. Any arbitrary number of fields can interact via both Raman and four-wave processes, but all fields must satisfy the phase-matching condition. The solutions will be illustrated with a problem of current interest; multiple first Stokes generation from a multiple-mode pump field.

DOI
10.1103/physrevlett.46.922

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A new approach to the problem of steady-state plane-wave propagation in a Raman-active medium has lead to general analytical solutions. Any arbitrary number of fields can interact via both Raman and four-wave processes, but all fields must satisfy the phase-matching condition. The solutions will be illustrated with a problem of current interest; multiple first Stokes generation from a multiple-mode pump field.
The equations for plane-wave steady-state propagation of n fields interacting via four wave processes in a Raman medium have been known since at least 1962. Complete analytic solutions have only been found for the process of harmonic generation. However, solutions have been obtained in the gain regime assuming zero pump depletion. Assuming all fields satisfy the phase matching condition we have found general analytic solutions to these equations. The solutions are complete in that they describe both pump deletion and saturation. Some example problems will be studied.
A general theory of transient stimulated Raman scattering and the phase-matching diagrams are developed based on an interaction Hamiltonian which includes all the possible competing interaction channels. In a Raman system pumped by multiple crossing beams with multiple Stokes and anti-Stokes components, these channels are defined by the interactions among fields at different frequencies and/or k vectors of the pump pairs and Stokes and anti-Stokes fields. An ensemble-averaged equation of motion for the active phonon mode is derived within the Markovian approximation in which the phonon-phonon coupling (pressure effect) is reduced to an energy relaxation and a phase fluctuation. A complete set of transient coupled equations including phase modulation, Stark shift, medium (anharmonic) saturation, and all the possible four-wave-mixing channels is presented. In the transient regime, it is found that the effects of the Stark shift on the Raman efficiency are mainly governed by the input pump energy in contrast to the steady-state theory where the pump intensity plays the main role. Both analytic and numerical methods are developed in describing the nonlinear effects due to the energy feedback mechanism in the strong pump-depletion regime. Twelve possible four-wave-mixing channels are analyzed for nonlinear phase diagrams characterized by the angles among pump pairs, and Stokes and anti-Stokes fields. Finally, numerical examples are shown for the effects of the coherence lengths of high-order Stokes fields and pump pairs on the generation of low-order Stokes fields.
A theoretical analysis is made of the simultaneous generation of Stokes and anti-Stokes waves initiated by two-photon additional illumination. This is done allowing for Raman and parametric processes, saturation effects, and also for a possible difference between the Raman polarizabilities for the Stokes and anti-Stokes waves. The conditions for the establishment of various asymptotic regimes are determined and general expressions are derived for the ultimate conversion efficiencies, assuming that pump and additional illumination are fed into a medium (inverted or uninverted) under exact resonance and phase-matching conditions. The spatial evolution of the amplitudes of the interacting waves is investigated by an accurate analytic solution of the problem in an important particular case and using solutions in the approximation of a constant pump wave or constant additional illumination.
Analytic and numerical solutions are obtained for the equations describing the propagation of several pulses of different frequencies through a Raman medium. The transient case is treated, and the analysis includes Stokes, pump, and anti-Stokes frequency components. The effect of phase mismatch \ensuremath{\Delta}k between these waves is investigated. An analytic solution is found that describes the phenomenon of transient Stokes--anti-Stokes gain suppression in the limit \ensuremath{\Delta}k=0. For nonzero values of \ensuremath{\Delta}k, the behavior of the transient gain coefficient is found to follow a scaling rule that is the analog of a similar result known from the steady-state limit. Calculations are performed to identify conditions that enhance conversion to anti-Stokes radiation. It is found that there is an optimum strength of the Stokes seed, and that proper phase matching of the initial waves is essential.