Self-similar solutions are associated with the long-time asymptotic behavior of many dynamic processes. In many real-life situations, similarity solutions predominate, with the most familiar being the Gaussian solution of the diffusion (heat) equation, and the family of Gaussian solutions of optical diffraction in free space. The latter are only partially self-similar throughout their propagation -- their shape is preserved up to scaling and normalization, while their phase pattern curves or straightens. Here we explore, theoretically and experimentally, the self-similarity properties of the modes of the complex diffusion equation. We expand a previously known set of similarity solutions of the diffusion equation [1] by introducing a curved phase surface, utilizing the elegant Gaussian modes of optical diffraction and employing the correspondence between diffusion and diffraction. Using the technique of light storage in a gas of diffusing atoms [2,3], we imprint a complex initial condition and follow its dynamics. The self-similarity of both the amplitude and the phase pattern is demonstrated, and the geometrical decay, associated with the modal order, is measured. A fascinating result in the context of dissipative processes such as diffusion is the self-similar contraction appearing in a special subset of these solutions.

[1] R. C. Kloosterziel, Jr. Eng. Math. 24, 213-236 (1990).

[2] R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, N. Davidson, Phys. Rev. Lett. 98, 203601 (2007).

[3] M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, N. Davidson, Phys. Rev. Lett. 100, 223601 (2008).