PHYSICAL REVIEWER A VOLUME 41, NUMBER 9 1 MAY 1990 Spontaneous Brillouin scattering in a microdroplet S. C. Ching, * P. T. Leung, and K. Young Department of Physics, The Chinese University of Hong Kong, Hong Kong (Received 20 November 1989) Spontaneous Brillouin scattering in a micrometer-sized liquid droplet is analyzed from first prin- ciples, using the spherical-wave normal-mode basis. Instead of the conservation of linear momen- tum, this interaction is governed by a selection rule due to the conservation of angular momentum. The Brillouin spectrum is then calculated, both for observation at a given angle and for the sum over all angles, and compared with scattering in a bulk medium. Special attention is paid to the case where the incident and scattered radiation fall on an electromagnetic resonance of the droplet. The analysis lays the foundation for formulating stimulated Brillouin scattering in the same system. I. INTRODUCTION Intense hypersonic waves can be generated coherently in a bulk solid or liquid through stimulated Brillouin scattering (SBS). ' The selection rules due to the conser- vation of energy and momentum determine the Brillouin shift to be b, co =+2to(nc, /c) sin(8/2), where co is the incident frequency, c, is the velocity of second, c/n is the velocity of light in a medium with re- fractive index n, and 8 is the scattering angle. SBS has also been observed in finite and microscopic media such as liquid cells, optical fibers, and, most re- cently, individual micrometer-sized droplets. The pur- pose of this paper is to formulate the simpler case of spontaneous Brillouin scattering (BS) in a microdroplet, in order to clarify concepts necessary for understanding SBS in this geometry. SBS and BS in finite media differ from the analogous phenomena. in bulk media in a num- ber of ways. First of all, the electromagnetic (em) energy may be confined (as in a fiber) or concentrated (as in a droplet, with its surface acting as a lens and as a partial refiector). Secondly, optical feedback (by external mir- rors in a liquid cell or by total internal reflection at the surface of a droplet) may amplify both the incident and the scattered radiation, enhancing the SBS intensity and lowering its threshold. An equivalent description is that the density of states p(to) is redistributed, with many states squeezed into narrow resonances or quasimodes, and transition rates are enhanced when the incident or scattered frequency falls on a resonance. ' Since the redistribution preserves the total number of states, first- order processes such as fiuorescence, going like p(to), are not afFected on average. However, higher-order processes are signi6cantly enhanced; for example, energy transfer frorp a donor molecule to an acceptor molecule, as a second-order process, goes as p(to), which is substantial- ly increased even in the average sense when p(to) is changed from a smooth distribution to a sum of reso- nance. ' For stimulated processes, including SBS, stimulated Raman scattering, " and lasing, ' the effect of bto =2to(nc, /c)-41. 3 ns ' (0.22 cm ') . (2) the resonances is even more pronounced. In microscopic systems such as microdroplets, the acoustic modes are also redistributed and may appear discrete rather than continuous, allowing further enhancement under suitable conditions. Thus acoustic disturbances of relatively large amplitude can be generated, possibly leading to shape dis- tortions and even shattering of the droplet. ' More importantly, the selection rules in BS and SBS, which determine which em and acoustic modes are cou- pled and amplified, are fundamentally altered in a micro- scopic system, since without translational invariance, momentum is no longer conserved. For example, in a fiber there is only conservation of longitudinal momen- tum, while in a droplet, angular momentum rather than linear momentum is conserved. Thus a ray picture, with rectilinearly propagating em and acoustic modes, would in principle be inappropriate, and the relation (1) fails. The violation of momentum conservation is A/hr-fi/a, where hr is of the order of the droplet radius a. So for a large droplet, momentum conservation and the ray pic- ture must be recovered approximately. The physical consequences of the selection rules are therefore quite different for small and large droplets. In this context, "large" means a »A, , the optical wavelength, or the size parameter x = 2m a/1, » 1. Although simple accounts of SBS have been pro- posed, ' a thorough understanding of the selection rules and the resonant enhancement (i. e. , optical and acoustical feedback), as well as the interplay between the two, is necessary before a satisfactory theory can be formulated. This paper addresses precisely these problems, in the much simpler context of spontaneous BS. To be definite, we take the liquid to be water (refractive index n =1. 33, velocity of sound c, =1. 483X10 ms ') and the incident wavelength in vacuum to be A, -0.6 pm. The droplet size will be taken in the range 0. 5 pm (a (5 pm or 5 +x + 50. The actual experiment used much larger droplets (a -45 pm, x -500), but we limit the calculation to smaller sizes because the amount of computation rises rapidly as x and x-20-30 is al- ready large enough to exhibit asymptotic behavior. With these parameters, the maximum Brillouin shift is 41 5026 1990 The American Physical Society