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Journal of the Optical Society of America B

Quantum theory of Stokes generation with a multimode laser

Michael G. Raymer, Lynn A. Westling

Published Sep 1, 1985• 19 citations
Publisher

Abstract

The quantum theory-of Stokes generation by stimulated Raman scattering of a multimode laser is developed and analyzed. For large laser-mode spacing, the generated Stokes intensity is found to be independent of the number of laser modes for both high and low gains. The result applies to both free-running and mode-locked lasers and is in contrast to that in the Raman amplifier, in which gain is found to depend on the number of modes. The Stokes light is generated with mode amplitudes and phases in perfect correlation with those of the laser. Also, in steady state the instantaneous Stokes intensity is found to follow exactly the rapid fluctuations of the laser intensity.

DOI
10.1364/josab.2.001417

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Physical Review A•Oct 1, 1981•358 citations
The quantum theory-of Stokes generation by stimulated Raman scattering of a multimode laser is developed and analyzed. For large laser-mode spacing, the generated Stokes intensity is found to be independent of the number of laser modes for both high and low gains. The result applies to both free-running and mode-locked lasers and is in contrast to that in the Raman amplifier, in which gain is found to depend on the number of modes. The Stokes light is generated with mode amplitudes and phases in perfect correlation with those of the laser. Also, in steady state the instantaneous Stokes intensity is found to follow exactly the rapid fluctuations of the laser intensity.
An experimental and theoretical study of the temporal correlations between the intensities of a broadband pump laser and Stokes light in stimulated Raman generation has been carried out. Direct measurements have been made of the pump laser and Stokes light intensity autocorrelation functions and of the intensity cross-correlation function between the two. Strong cross correlations are found. Nevertheless, the correlations are not perfect; that is, the Stokes intensity does not follow exactly the intensity variations of the laser. Transient effects and quantum fluctuations are shown, by theoretical modeling, to be responsible for the absence of perfect correlations.
The development of the Stokes spectrum and the dependence of Raman gain upon pump laser bandwidth in a narrow linewidth Raman medium are analyzed in terms of a closed set of multimode equations. The analysis predicts that the Stokes gain coefficient takes on its maximum value, which is independent of bandwidth, when the Stokes spectrum replicates the pump spectrum. It is also shown that in the absence of dispersion the spectrum of a weak Stokes wave is driven to duplicate the pump spectrum, and that for noise input the average power generated in the Stokes wave is independent of the pump bandwidth.
A quantum theory of stimulated Raman scattering is presented that takes into account three-dimensional propagation and collisional dephasing, allowing the study of the spatial and temporal coherence properties of the generated Stokes light. Maxwell-Bloch equations for the Stokes field operator and the collective atomic operators are solved analytically under low-signal-gain conditions, where the laser field and the atomic ground states remain undepleted. The intensity and the space-time autocorrelation function of the Stokes field are calculated. The Stokes field is expanded into a set of statistically independent ``coherence modes,'' which are determined explicitly for the case of a cylindrically shaped pumped volume. The Stokes pulse energy W is found to fluctuate from pulse to pulse. The probability distribution function for pulse energies P(W) is calculated for a range of Fresnel numbers of the excited volume and collisional dephasing rates. For small values of Fresnel number and dephasing rate, P(W) is a negative exponential distribution. For large values of either, P(W) narrows and approaches a Gaussian-shaped distribution. This occurs because many independent modes contribute to the Stokes emission, making it spatially and/or temporally incoherent.
The authors have extended the theory of stimulated Raman scattering to include the effects of laser bandwidths, in both the transient and steady-state regimes. The case of two interacting laser beams, a pump laser and a probe (Stokes) laser, is treated. Using the phase-diffusion model for laser bandwidth, the authors demonstrate that in the absence of dispersion, the forward Raman gain is essentially independent of the laser bandwidths in the high-gain limit, while in the low-gain limit the gain coefficient is inversely proportional to the sum of the bandwidths. It is further shown that when the pump-laser bandwidth is much larger than the linewidth of the Raman medium, the stimulated Stokes output assumes the same spectrum as the pump laser in the high-gain limit. A possible interpretation of these results is discussed assuming a "phase-locking" of the Stokes phase to the fluctuations in the pump-laser phase, due to the nonlinear interaction of the two beams through the Raman medium.
The quantum theory of forward Stokes generation by means of stimulated Raman scattering is extended to the case of a colored (finite-bandwidth) chaotic pump, which has both amplitude and phase fluctuations. In both transient and steady-state limits the mean Stokes intensity is found to be enhanced over that resulting from a coherent pump. In the limit that the chaotic-pump bandwidth becomes large, the mean Stokes intensity becomes identical with that resulting from a coherent pump of the same total power.
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.