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Journal of the Optical Society of America B

Raman-resonant four-wave mixing and energy transfer

Bradley L. Bobbs, Charles Warner

Published Feb 1, 1990• 26 citations
Publisher

Abstract

A common assumption that Raman-resonant four-wave mixing does not transfer energy between the light and the Raman medium is shown to be incorrect. The derivation of the correct energy-transfer picture is compared with those for stimulated Raman scattering and for nonresonant mixing. The physical mechanism for Raman-gain suppression due to Stokes/anti-Stokes coupling under phase-matched conditions is clarified by this understanding of the energy transfer. An experimental demonstration of the nature of the energy transfer is proposed.

DOI
10.1364/josab.7.000234

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A common assumption that Raman-resonant four-wave mixing does not transfer energy between the light and the Raman medium is shown to be incorrect. The derivation of the correct energy-transfer picture is compared with those for stimulated Raman scattering and for nonresonant mixing. The physical mechanism for Raman-gain suppression due to Stokes/anti-Stokes coupling under phase-matched conditions is clarified by this understanding of the energy transfer. An experimental demonstration of the nature of the energy transfer is proposed.
When nonlinear four-wave mixing (FWM) is resonant with a Raman transition, it causes coupling between lightwaves separated by two Raman frequency shifts. This coupling is manifested in effects such as Raman gain suppression, higher-order Stokes generation, and coherent Raman spectroscopies (e.g., CARS). An understanding of these effects, however, has been impeded by a widespread misconception that Raman-resonant FWM does not transfer energy between the lightwaves and the medium. The equations for the photon densities at each light frequency show that, in actuality, two medium excitations are created or destroyed for each photon produced by FWM. This point may be further clarified by consideration of the driving nonlinear polarization and by comparisons with broadband Raman conversion and nonresonant FWM. The Raman gain suppression effect may be understood qualitatively in terms of destructive interference between FWM and the usual stimulated Raman process.
We present a detailed derivation of a new, rotationally invariant formalism for the theory of stimulated Raman scattering. The formalism is applied to Raman transitions of well-defined rotational symmetry, e.g., rotational Raman (S) transitions, yielding the explicit dependence of gain on light polarization, phase mismatch, and frequency offset. For a linearly polarized laser, the Stokes fields parallel and perpendicular to the laser field are decoupled. Far from Stokes--anti-Stokes phase matching, their gain coefficients are in the ratio of 4:3; at phase matching, both are suppressed and have zero exponential gain. For a circularly polarized laser, the circularly polarized Stokes fields circulating in the same and opposite senses as the laser field are decoupled. The gain of the opposite-sense field, which is decoupled from the anti-Stokes field, is independent of phase mismatch. Far from phase matching, the gain of the same-sense Stokes field is (1/6 that of the opposite-sense field; at phase matching, the same-sense field has zero exponential gain. An unpolarized laser is shown to have the lowest Raman gain; far from phase matching, its Stokes fields decouple into two incoherent, oppositely circularly polarized fields, but at phase matching all Stokes polarizations have zero exponential gain. The gradual transition away from phase matching is also treated explicitly, and it is shown that when the gain is suppressed the maximum Stokes growth occurs when the Stokes-laser beat is a fraction of a Raman linewidth off resonance.
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 system for phase conjugation of Stokes radiation by a four-wave interaction involving intracavity stimulated Raman scattering (STRS) in a Raman-active medium with a nonresonant nonlinearity. This method can be used in construction of high-power sources of highly directional Stokes radiation on the basis of round-trip STRS amplifiers coupled to a phase-conjugating element. It is shown that an STRS converter utilizing organic liquids and subjected to laser pumping of less than 100 MW/cm2 intensity inside a resonator can ensure a highly efficient phase conjugation with a peak reflection coefficient in excess of unity. Depending on the ratio of the Raman and nonresonant susceptibilities of a medium, there is an optimal value of the coefficient representing the degree of conversion of laser pump radiation into STRS radiation, which ensures the maximum phase conjugation efficiency.