# Self Similar Modes of Coherent Diffusion

O. Firstenberg,^{A} D. Yankelev,^{A} P. London,^{A} M. Shuker,^{A} R. Pugatch,^{B} and N. Davidson^{B}

A. Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel ( This email address is being protected from spambots. You need JavaScript enabled to view it. )

B. Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel

When a resonant light pulse propagates slowly through a vapor medium of hot diffusing atoms, the electromagnetic field exhibits diffusion, in addition to the regular free-space diffraction [1]. The extent of the diffusion depends on the degree of atomic excitation, which is maximal when storage of light is performed. For the latter, the evolution of the atomic coherence is described by the *coherent* diffusion equation where ψ is a complex scalar.

It is well known that the shape of the "standard" Gaussian modes (such as HG or LG) evolves self-similarly under free-space diffraction, that is, they maintain their transverse intensity distribution up to scaling and normalization. Here, we demonstrate that the so called "elegant" Gaussian modes [2], which are not self-similar under diffraction, are in fact the analogous self-similar modes of diffusion

We experimentally demonstrate the self-similar behaviour of these modes [3]. The square of the waist radius of the modes expands linearly with respect to the storage time, while the intensely profile remains the same up to scaling and normalization (Fig. 1a and 1b). In addition, a fascinating phenomenon occurs when the light is stored far from the waist. In such scenario, the diffusion during storage actually causes the beam cross-section to self-similarly contract (Fig. 1d). This can be attributed to the fact that the diffusion effectively expands the beam at its waist plane, where the phase if flat, even if it is performed far from it (Fig. 1c).

**Figure 1: (a) The linear growth of the area of a Gaussian mode due to diffusion, independent of the mode order. (b) The self-similar profiles of several elegant Gaussian modes under diffusion. (c) Illustration of the effect of diffusion on a Gaussian beam. The increase of the waist radius causes the beam to contract far from the waist plane. (d) Experimental demonstration of self-similar contraction under coherent diffusion.**

1. O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, *Nature Phys.*** 5**, 665-668 (2009).

2. A. E. Siegman, *Lasers* (University Science Books, Sausalito, California, 1986).

3. O. Firstenberg, P. London, D. Yankelev, R. Pugatch, M. Shuker, N. Davidson, *Phys. Rev. Lett.* **105**, 183602 (2010).