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Soviet Journal of Quantum Electronics

Theory of parametric and Raman interaction in an incoherent pumping field

Guerman A. Pasmanik, Gennadii I Freidman

Published Mar 31, 1974• 7 citations
Publisher

Abstract

An approximate theory of parametric amplification and stimulated Raman scattering is developed for a pumping field which is incoherent in terms of the transverse coordinates and time. Expressions are obtained for the increments of the average field and the conditions of validity of the obtained expressions are determined. It is shown, in particular, that if the longitudinal length of the correlation between the pump radiation and the signal wave is small in comparison with the length of correlation of this radiation with the idler wave, then the employed approximation describes with sufficient accuracy the transition from the case of δ-correlated pumping (incoherent interaction) to the case when the gain achieved in a distance equal to the correlation length of the pump and idler waves exceeds unity (quasicoherent interaction). The intensity increment is then close to double the increment of the average field or the signal wave. In addition, for the indicated ratio of the correlation lengths, it is relatively easy to calculate the correlation function at a distance from the point of entry into a crystal, which is short but sufficient for an appreciable gain. Analogous conclusions hold also for stimulated Raman scattering in a pumping field with a spectral width exceeding the spontaneous-scattering linewidth.

DOI
10.1070/qe1974v004n03abeh006730

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An approximate theory of parametric amplification and stimulated Raman scattering is developed for a pumping field which is incoherent in terms of the transverse coordinates and time. Expressions are obtained for the increments of the average field and the conditions of validity of the obtained expressions are determined. It is shown, in particular, that if the longitudinal length of the correlation between the pump radiation and the signal wave is small in comparison with the length of correlation of this radiation with the idler wave, then the employed approximation describes with sufficient accuracy the transition from the case of δ-correlated pumping (incoherent interaction) to the case when the gain achieved in a distance equal to the correlation length of the pump and idler waves exceeds unity (quasicoherent interaction). The intensity increment is then close to double the increment of the average field or the signal wave. In addition, for the indicated ratio of the correlation lengths, it is relatively easy to calculate the correlation function at a distance from the point of entry into a crystal, which is short but sufficient for an appreciable gain. Analogous conclusions hold also for stimulated Raman scattering in a pumping field with a spectral width exceeding the spontaneous-scattering linewidth.
Stimulated Raman scattering of a randomly modulated pump is investigated by the method of successive approximations. This involves expanding solutions in terms of small parameters, which are ratios of the correlation scales of random effects to other characteristic dynamic scales of the problem. Systems of closed equations are obtained for the moments of the amplitudes of the Stokes and pump waves and of the molecular vibrations. These describe the dynamics of the process allowing for changes in the pump intensity and statistics due to a three-wave interaction. By analyzing equations in higher-order approximations, it is possible to establish the conditions of validity of the first (Markov) and second approximations. In particular, it is found that these are valid for pump intensities JL both above and below the critical value Jcr near which the gain begins to increase rapidly and reproduction of the pump spectrum by the Stokes wave is initiated. Solutions are obtained for average intensities of the Stokes wave and molecular vibrations in the first approximation in a constant pump field. It is established that, for JLJcr, the Stokes wave undergoes rapid nonsteady-state amplification which is associated with an increase in the amplitude of the molecular vibrations. The results of the calculations show good agreement with known experimental data.
A theoretical and experimental investigation was made of the amplification of a Stokes signal in a multimode pump field when the ratio of the spectral widths of the interacting fields was varied. In the pump intensity range I p < I cr ( I cr was a critical intensity) the gain increment was considerably less than the static value and it was governed entirely by the spectral distributions of the pump and Stokes signal. In the range I p > I cr the gain increment tended to the static value. An anomalous scatter of the Raman amplifier gain was observed experimentally for Δ ν s ≈Δ ν p ≫Δ ν l (Δ ν s ,Δ ν p , and Δ ν l were the line widths of the Stokes signal, pump, and spontaneous stimulated Raman scattering). A theory developed for the purpose indicated that the scatter was observed because the phase relationships between the interacting waves at the input into the medium varied from one measurement to another.
A three-dimensional quantum theory of stimulated Raman scattering is presented which takes into account a general transverse shape of the pump laser. The transverse spatial autocorrelation function and the intensity of the Stokes pulse are derived analytically, by use of the Maxwell-Bloch equations in low-signal-gain conditions and introduction of a many-spatial-modes approximation. We find that the spatial and temporal coherence are coupled. The transverse coherence depends on the pump-laser shape, and the coherence area increases with increasing gain. Time scales for the buildup of stimulated intensity and correlations for the various transverse parts of the Stokes pulse are obtained.
A new method developed by the authors for the analysis of stimulated scattering of spatially noncoherent radiation is used in a comparative study of the concurrent and opposed interactions of a Stokes field with pump radiation. If there is no correlation between the Stokes and pump waves, the interaction is independent of the relative direction of propagation of the waves. However, if the Stokes and pump waves are correlated, the interaction efficiency depends strongly on the relative directions of the waves. Exact expressions are obtained for the structure of the wave fields and gain increments in the backscattering case and approximate expressions are obtained for the forward scattering. The physical reason for the asymmetry in the amplification is the reproduction by the Stokes signal spectrum of the pump spectrum in the forward scattering case and inversion of this spectrum in the backscattering case. The same reason accounts for a deterioration of the quality of wavefront reversal in the backscattering case compared with the forward scattering.
We present a statistical theory of forward and backward Raman amplification and spontaneous generation in dispersive media pumped by a phase-diffusion field of arbitrary bandwidth. It is shown that, unless the Stokes input is correlated with the pump, the Stokes output and the pump are statistically orthogonal, even if the Stokes field builds up spontaneously. The growth of the average Stokes field amplitude is not affected by dispersion. A general expression is given for the average Stokes intensity which is valid both above and below the critical pump intensity for overcoming the decorrelating effects of group-velocity dispersion. If the length of the medium is much greater than the coherence length associated with the dephasing of the two waves, then the average intensity of stimulated Raman scattering (SRS) depends on the pump bandwidth. For relatively short medium lengths the average intensity of spontaneously initiated SRS is essentially independent of the bandwidth of a phase-diffusion pump field.