Self-Similar Modes of Coherent Diffusion

Self-similar solutions of the coherent diffusion equation are derived and measured. The set of real similarity solutions is generalized by the introduction of a nonuniform phase, based on the elegant Gaussian modes of optical diffraction. In a light-storage experiment, the complex solutions are imprinted on a gas of diffusing atoms, and the self-similar evolution of both their amplitude and phase pattern is demonstrated. An algebraic decay depending on the mode order is measured. Notably, as opposed to the regular diffusion spreading, a subset of the solutions exhibits a self-similar contraction.

Figure 1

The effect of diffusion on Gaussian beams is an effective expansion of the waist, even if it occurs away from the waist plane. Therefore far enough from the waist, the diffusion contracts the transverse shape. Right (color) map: Phase gradients of the field envelope (black lines are equal phase contours; white lines are the beam outline). From the viewpoint of the microscopic atomic motion, the contraction far from the waist occurs due to destructive interference of atoms diffusing through the rapidly oscillating phase pattern (red colored).

Figure 2

(a) Self-similar diffusion of the complex field of atomic coherence in a light-storage experiment. Top to bottom: the basic Gaussian mode, Laguerre-Gaussian modes of radial order $p=0$ and orbital orders $m=1$, 2, and a Hermite-Gaussian mode of Cartesian orders (0, 1). All images are $1.6\times 1.6\mathrm{mm}$. (b) Images after storage, diffracted by a binary grating mask with a fork dislocation, for confirming the conservation of phase. The retrieved Gaussian mode (top) yields the two $m=\pm 1$ vortex modes, while the retrieved vortex $m=+1$ (bottom) produces an $m=0$ and an $m=+2$ modes.

Figure 3

(a) Linear growth of $w^2$ with respect to the time duration of diffusion. The line is $w^2-w_0^2=4D\tau$, with $D$ and $w_0$ fit parameters ($w_0=0.4–0.55\mathrm{mm}$, varying for the different modes, and $D=10.8{\mathrm{cm}}^2/\mathrm{s}$). (b) Self-similarity: cross sections (Cartesian or radially weighted) at different times are congruent when the coordinate is scaled with $s(\tau {)}^{-1/2}$. The solid lines are the exact elegant HG and LG modes.

Figure 4

Decay of the total power in the retrieved images $P(\tau )=\int dxdy|\psi (\tau ){|}^2$. In the inset, $P^{*}$ compensates for the algebraic decay [$s(\tau )$ known from Fig. 3] and collapses onto a single straight line in the semilog scale, yielding the homogenous decay rate $2\gamma =(68\mu \mathrm{s}{)}^{-1}$. In the main graph, the dashed line is $e^{-2\gamma \tau }$ and the solid lines are $e^{-2\gamma \tau }s(\tau {)}^{-2(N+1)}$, demonstrating the faster decay of the higher-order modes. The difference between the ${\mathrm{LG}}_{0,1}$ and the ${\mathrm{HG}}_{0,1}$, both having $N=1$, is due to slightly different initial waist radii ($w_0$).

Figure 5

The self-similar contraction and subsequent expansion of the ${\mathrm{LG}}_{0,1}$ mode upon diffusion. Here, we refer to the original beam as the initial condition (left, taken off resonantly), since much contraction occurs already during the slow-light propagation. The original mode has a radius of $700\mu \mathrm{m}$ and a phase curvature of $(250\mathrm{mm}{)}^{-1}$. The minimal radius, of $340\mu \mathrm{m}$, is obtained after slowing for $5\mu \mathrm{s}$ plus storage for $7\mu \mathrm{s}$. The black line is calculated from $C(\tau {)}^2$ with no fit parameters. In these conditions, the power decays substantially faster than in the expansion experiments, and substantial noise is already apparent after $15\mu \mathrm{s}$ of storage.