Speckle from phase-ordering systems

The statistical properties of coherent radiation scattered from phase-ordering materials are studied in detail using large-scale computer simulations and analytic arguments. Specifically, we consider a two-dimensional model with a nonconserved, scalar order parameter (model A), quenched through an order-disorder transition into the two-phase regime. For such systems it is well established that the standard scaling hypothesis applies, consequently, the average scattering intensity at wave vector $\mathbf{k}$ and time $\tau$ is proportional to a scaling function which depends only on a rescaled time, $t\sim |\mathbf{k}{|}^2\tau$. We find that the simulated intensities are exponentially distributed, and the time-dependent average is well approximated using a scaling function due to Ohta, Jasnow, and Kawasaki. Considering fluctuations around the average behavior, we find that the covariance of the scattering intensity for a single wave vector at two different times is proportional to a scaling function with natural variables $\delta t=|t_1-t_2|$ and $\overline{t}{=(t}_1{+t}_2)/2\mathrm{}$. In the asymptotic large-$\overline{t}$ limit this scaling function depends only on $z=\delta t/{\overline{t}}^{1/2}$. For small values of $z$, the scaling function is quadratic, corresponding to highly persistent behavior of the intensity fluctuations. We empirically establish that the intensity covariance (for $\mathbf{k}\ne 0$) equals the square of the spatial Fourier transform of the two-time, two-point correlation function of the order parameter. This connection allows sensitive testing, either experimental or numerical, of existing theories for two-time correlations in systems undergoing order-disorder phase transitions. Comparison between theoretical scaling functions and our numerical results requires no adjustable parameters.