Relation between transmission and energy stored in random media with gain

Abstract

Received 16 January 2010; revised manuscript received 5 July 2010; published 13 September 2010 In this work, we investigate a possibility of using the ratio between optical transmission, T, and energy stored inside the system, E, as a quantitative measure of the enhanced mesoscopic corrections to diffusive transport of light through a random medium with gain. We obtain an expression for T / E as a function of amplification strength in the diffusive approximation and show that it does not a have tendency to diverge when the threshold for random lasing is approached, as both T and E do. Furthermore, we find that a change in T / E signifies a change in the electric field distribution inside the random medium. In the localization regime, we also investigate the correlations between transmission and energy stored in the medium with and without amplification. Our results suggest that T / E is a promising parameter which can help characterize the nature of wave transport in random medium with gain. In this work, we investigate the properties of the ratio between transmission T and the energy inside a random me- dium E, with the goal of formulating a criterion of AL which would be applicable in the presence of gain. Both parameters should be experimentally accessible in planar systems, e.g., perforated dielectric films, in which the spatial field distribu- tion inside the medium can be obtained via near-field scan- ning optical microscopy. In Sec. II, to motivate our choice of the parameter in the form T / E, we consider random medium in regime of diffu- sive transport. First we note that when taken separately, both transmission and the energy inside the system E exhibit an expected tendency to increase with an increase in the gain strength and to diverge when threshold for random lasing is approached no saturation mechanism is assumed. However, when taken in a form of a ratio, the divergence is eliminated and the tendency to increase is greatly reduced. Second, we show that in passive systems the ratio can be related to the spatially dependent diffusion constant Dz. The latter con- cept has been recently invoked 17,18 in the self-consistent theory of AL to extend the applicability of diffusion approxi- mation into the localized regime. The connection between T / E and Dz demonstrates that the former can, indeed, be used as a quantitative measure of the contribution of local- ization interference phenomena in transport through disor- dered systems. In Sec. II D, we obtain an expression for T / E from the solution of the diffusion equation in a random medium us- ing the slab geometry with gain. This establishes a baseline—a decrease in the parameter T / E below the diffu- sion prediction obtained in this section can be used to quan- tify the extent to which the gain promotes the localization effects. In order to further assess the usefulness of T / E as a local- ization criterion, in Sec. III we also consider random medium in the localized regime where we investigate the correlations between the transmission and stored energy in the same dis- order configuration. We show that the ratio strongly depends on the spatial location of localization center. This introduces an additional geometrical source of fluctuation which is not present in the transmission coefficient alone. Interestingly,