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Physical Review A

Doubly resonant stimulated Brillouin scattering in a microdroplet

P. T. Leung, K. Young

Published Jul 1, 1991• 16 citations

Abstract

Stimulated Brillouin scattering in a micrometer-sized liquid droplet is treated theoretically using recently developed techniques, with emphasis on the doubly resonant case, in which the frequencies of both the laser field and the scattered Brillouin field fall within a morphology-dependent resonance, leading to input and output enhancement. An important difference from a bulk medium is the replacement of the selection rules based on momentum conservation by those based on angular momentum conservation, with the latter most conveniently expressed through the spherical wave modes. Special attention is paid to the eigenvalue equation defining the gains and Brillouin shifts, in particular to the difference between small and large droplets. For large droplets, and for given angular momenta of the laser field and the scattered field, there is only one allowed shift with a positive gain. Numerical results for gain and threshold are shown for a droplet of size parameter about 30, and a simple approximate analytic formula for the gain is presented.

DOI: 10.1103/physreva.44.593

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Stimulated Brillouin scattering in a micrometer-sized liquid droplet is treated theoretically using recently developed techniques, with emphasis on the doubly resonant case, in which the frequencies of both the laser field and the scattered Brillouin field fall within a morphology-dependent resonance, leading to input and output enhancement. An important difference from a bulk medium is the replacement of the selection rules based on momentum conservation by those based on angular momentum conservation, with the latter most conveniently expressed through the spherical wave modes. Special attention is paid to the eigenvalue equation defining the gains and Brillouin shifts, in particular to the difference between small and large droplets. For large droplets, and for given angular momenta of the laser field and the scattered field, there is only one allowed shift with a positive gain. Numerical results for gain and threshold are shown for a droplet of size parameter about 30, and a simple approximate analytic formula for the gain is presented.

Continuous-wave stimulated Raman scattering was observed in 11-13-microm-diameter benzene and toluene microdroplets at pump intensities as low as 8 and 24 kW/cm(2), respectively. Low thresholds were achieved by exploiting simultaneous pump and Stokes wave resonance in the droplets and Raman gains that were cavity QED enhanced ~50 times with respect to bulk liquid values. Based on a photon-state conservation argument, the cavity gain enhancement factor may be approximated by the ratio of the spectral spacing between resonant modes of the same order to that of the homogeneous Raman linewidth. This relation appears to be consistent with the relative experimental behavior of benzene, ethanol, and toluene.

Nonlinear optical processes in spherical microdroplets whose studies have been pioneered by Chang and co-workers are intriguing from the point of view of applications because of the extremely low thresholds they exhibit for the generation of stimulated output, but also from the theoretical point of view, because they require a synthesis between the methods of nonlinear optics and Mie scattering theory. The basis for such a synthesis has been given in the semiclassical theory of Kurizki and Nitzan, which implies that the nonlinearly amplified component of the scattered field is spatially orthogonal ('out of phase') with the linearly scattered field component (the ordinary Mie solution). This approach allows the calculation of amplification coefficients in various nonlinear processes, as a function of the Mie resonant denominators and spatial overlap of the input and output spherical waves. Recently, this approach has been extended to quantized solutions for parametric amplification and oscillation via four-wave mixing in microdroplets. The very low thresholds for oscillation at Mie resonances are predicted to correspond to strong squeezing, i.e., suppression of the photocurrent noise associated with the detection of the output waves below the noise level associated with an ideal laser. Even well below threshold, the spatially orthogonal output waves allow for strong squeezing at periodically recurring distances from the droplet.

Significant cavity QED enhancement have been observed of both spontaneous and stimulated emission in 10- to 14-micrometers diameter liquid microdroplets containing fluorophors. Direct emission lifetime reduction was observed as well as greatly enhanced lasing gain. Use of droplets provides an excellent test of cavity QED theory for spherical geometries. The mode number and order of the spherical cavity resonances responsible for the observed emission peaks were identified and their Q values calculated from Lorenz-Mie theory. An experimental procedure, based on equating stimulated gain to cavity losses, is used to estimate the magnitude of the enhancement. Spherical cavity Qs of only 2600 were found to lead to cavity enhancements of over 100 in the emission cross-section of Rhodamine 6G in agreement with theory.

A variety of stimulated nonlinear optical processes are observed under relatively low intensity cw excitation in droplet spherical microcavities. These include stimulated Raman and Rayleigh-wing scattering as well as four-wave parametric oscillation. The low thresholds for these stimulated processes are accounted for by all waves being in resonance with high-

Q

droplet modes and by the existence of a significant cavity QED enhancement (> 100 times) in the nonlinear gain coefficients.