Gaussian-to-Le ´vy statistical crossover in a random amplifying medium: Monte Carlo simulation Divya Sharma * , Hema Ramachandran, N. Kumar Raman Research Institute, Sadashivanagar, Bangalore 560 080, India Received 12 October 2006; received in revised form 5 December 2006; accepted 5 December 2006 Abstract We report Monte Carlo simulations of photons diffusing through a random amplifying medium and study the statistics of emitted intensity fluctuations. We observe a crossover from the Gaussian to the Le ´vy statistics corresponding to divergence of the variance of intensity fluctuations, as we go to large gain and strong scattering, consistent with our earlier analytical and experimental works. In the limit of very high gain, our simulations suggest a ‘‘Le ´vy microscope effect’’, where the emitted mean intensity is expected to diverge, and the sum of intensities is dominated by the typical largest event. Ó 2006 Elsevier B.V. All rights reserved. PACS: 42.55.Zz; 05.10.Ln; 05.40.a; 05.40.Fb Keywords: Random amplifying medium; RAM; Random walk; Le ´vy statistics 1. Introduction A Random Amplifying Medium (RAM) combines mul- tiple scattering of light and photon multiplication (Fig. 1) so as to give light sources with interesting features that have attracted much attention in recent years [1–9]. Because of the synergy between multiple scattering and amplification, these sources can emit light that is broad- band (spectral width 40 nm), or narrowband (4 nm), or even laser-like spiked (<1 nm) emission. Further, the coherent effects such as the weak [10,11] or the strong local- ization [8,9,12] of photons within the medium may also be reinforced by the gain narrowing. These random lasing sys- tems have also been studied for their possible applications. Lasing in a RAM was experimentally first demonstrated by Lawandy et al. in an optically pumped laser dye solution (Rhodamine 640 perchlorate in methanol) in which point- like scatterers (TiO 2 microspheres of diameter <1 lm) were randomly suspended [1]. A collapse of the linewidth of emission was observed above a threshold pump power, and was interpreted as the onset of lasing when the gain due to the enhanced path lengths within the active medium, brought about by multiple scattering, exceeded the inten- sity loss from the system. Thus, the ‘‘mirror-less lasing’’ is aided by the deliberate introduction of scattering centres within the amplifying bulk that provide a non-resonant dis- tributed positive feedback. Numerous studies of RAMs over the last decade have addressed two mechanisms for the random lasing involv- ing, (a) incoherent, classical diffusion of the photons giving increased path lengths within the amplifying medium [1–7], and, (b) coherent wave-mechanical localization of light [8,9]. These studies range from theoretical investigations to experiments to numerical simulations. A striking feature of the emission from a RAM is its non-selfaveraging nature at high gain and strong scattering, wherein the emission intensity fluctuates randomly over the ensemble of different microscopic realizations of the disorder in macroscopically identical bulk samples. Normally, these statistical fluctua- tions in the emission intensity exhibit Gaussian statistics, 0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.12.009 * Corresponding author. Tel.: +91 80 2361 0122; fax: +91 80 2361 0492. E-mail addresses: divya@rri.res.in (D. Sharma), hema@rri.res.in (H. Ramachandran), nkumar@rri.res.in (N. Kumar). www.elsevier.com/locate/optcom Optics Communications 273 (2007) 1–7