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Physical Review A

Theory of traveling-wave electronic Raman lasers — Stokes-anti-Stokes coupling in the steady state

S. Y. Yuen, Benjamin Lax, Peter A. Wolff

Published Jul 1, 1974• 7 citations

Abstract

A detailed theory of traveling-wave electronic Raman lasers is constructed, which applies in particular to the spin-flip Raman lasers of Patel and others. Ten equations are set up describing explicitly the behavior of the Stokes field, the anti-Stokes field, the input laser field, and the electronic system. Solutions of these equations are obtained in the sinusoidal steady-state limit. Depletion of the input laser field is accounted for in an approximate but novel fashion in the form of photon-number conservation. We are able to express the steady-state output power in terms of the system parameters and the boundary pump excitation alone. Analytic formulas are given for the linear gain, the threshold pump power, and the Stokes as well as the anti-Stokes power output. They describe, among other things, the saturation and fluorescence behavior of the electronic Raman laser. Various intuitive interpretations are provided for the analytical results. Our detailed predictions are in good agreement with experiments.

DOI: 10.1103/physreva.10.416

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A detailed theory of traveling-wave electronic Raman lasers is constructed, which applies in particular to the spin-flip Raman lasers of Patel and others. Ten equations are set up describing explicitly the behavior of the Stokes field, the anti-Stokes field, the input laser field, and the electronic system. Solutions of these equations are obtained in the sinusoidal steady-state limit. Depletion of the input laser field is accounted for in an approximate but novel fashion in the form of photon-number conservation. We are able to express the steady-state output power in terms of the system parameters and the boundary pump excitation alone. Analytic formulas are given for the linear gain, the threshold pump power, and the Stokes as well as the anti-Stokes power output. They describe, among other things, the saturation and fluorescence behavior of the electronic Raman laser. Various intuitive interpretations are provided for the analytical results. Our detailed predictions are in good agreement with experiments.

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.

I study the coherent interaction of a three-level atom in the lambda configuration with pump, Stoke, and anti-Stokes laser fields. The pump laser field drives transitions from the two non-degenerate lower levels to the excited state which is highly detuned for either transition. This excited state is also coupled to each of the lower states by Stokes or anti-Stokes transitions so that this system may represent a model of four-wave mixing. Assuming large detunings, the excited state is adiabatically eliminated from obtaining an effective two-level system with Raman coupling between the levels. Optical Bloch equations are obtained for this effective two-level system, and phenomenological damping terms are added for collisional dephasing of the Raman transitions. The equations are solved for the steady-state as well as by the method of adiabatic following. From these solutions we obtain the polarization.

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.

The theory of stimulated Raman scattering has been treated at various levels of sophistication from plane-wave steady-state models through 3-D transient quantum-electrodynamic calculations.1 Many physical effects, such as pump depletion, dispersion, level degeneracy, ac Stark shift, and parametric coupling of Stokes and anti-Stokes light, can be important in experimental setups. We describe a recently developed computer code which treats these effects in detail. The code is based on integration of the coupled Maxwell-Bloch equations of quantum optics. We discuss numerical predictions of the code relating to these effects and present applications to experimental investigations such as Raman beam cleanup and transient amplification.2