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Physical Review A

Unified theory of spontaneous and stimulated rotational Raman scattering

Mark R. Hermann, David H. Chambers, Sham N. Dixit, Thomas J. Karr

Published Apr 1, 1991• 4 citations
Publisher

Abstract

A general analytic theory of spontaneous and stimulated rotational Raman scattering is presented. This analytic description incorporates arbitrary polarization, time dependence, phase matching, and parametric coupling of the pump, Stokes, and anti-Stokes fields and the degeneracy of the rotational states. The analytic expressions for the time-dependent Stokes and anti-Stokes fields provide valuable physical insight into the dynamics of spontaneous and stimulated rotational Raman scattering.

DOI
10.1103/physreva.43.4079

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A general analytic theory of spontaneous and stimulated rotational Raman scattering is presented. This analytic description incorporates arbitrary polarization, time dependence, phase matching, and parametric coupling of the pump, Stokes, and anti-Stokes fields and the degeneracy of the rotational states. The analytic expressions for the time-dependent Stokes and anti-Stokes fields provide valuable physical insight into the dynamics of spontaneous and stimulated rotational Raman scattering.
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 theory of stimulated Raman scattering (SRS) is presented which treats in a unified manner the buildup of SRS from spontaneous Raman scattering and the spatial propagation which leads to gain. Maxwell-Bloch equations describe the coupling between the Stokes field operator and the collective atomic operators, which are driven by a classical laser field under low-signal-gain conditions, where the atomic ground states and laser field remain undepleted. An analytic expression is derived for the forward-propagating Stokes field operator in a one-dimensional spatial approximation when the Fresnel number of the excited medium is unity. Steady-state and transient Stokes intensities are evaluated under low-gain (spontaneous) and high-gain (stimulated) conditions. The Stokes intensity is found to be exactly independent of a laser bandwidth arising solely from phase fluctuations. The power spectrum of steady-state SRS is evaluated, giving a gain-narrowed spectral profile when the laser is narrow band, and a spectrum identical to the laser spectrum when the laser is broad band. A strong analogy between SRS and two-level superfluorescence is found; in both cases the macroscopic, collective dipole moment is initiated by quantum fluctuations.
Transient stimulated rotational Raman scattering induced by a polarization-modulated femtosecond pulse whose polarization varies with time is reported. The peak intensity of the polarization-modulated pulse produced in this study is low compared to that of a pulse without polarization variation, and this leads to the low efficiency of self-phase-modulation and thus the high efficiency of stimulated Raman scattering compared to the case of a pump pulse without polarization modulation. The scattering process is unique in terms of the high peak ratio of the anti-Stokes to Stokes emissions compared to the ratio in the case of the scattering induced by linearly, circularly, and elliptically polarized pump pulses. Under optimum conditions, the ratio approaches 1, which is not realized in the general case of stimulated rotational Raman scattering.
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.