Semiclassical theory of speckle correlations

Coherent wave propagation in random media results in a characteristic speckle pattern, with spatial intensity correlations with short-range and long-range behavior. Here, we show how the speckle correlation function can be obtained from a ray picture for two representative geometries, namely a chaotic cavity and a random waveguide. Our calculation allows us to study the crossover between a “ray limit” and a “wave limit,” in which the Ehrenfest time ${\tau }_{\mathrm{E}}$ is larger or smaller than the typical transmission time ${\tau }_{\mathrm{D}}$, respectively. Remarkably, long-range speckle correlations persist in the ray limit ${\tau }_{\mathrm{E}}\gg {\tau }_{\mathrm{D}}$.

Figure 1

Schematic picture of a trajectory $\alpha$ connecting the source at $\mathbf{r}$ and the detector at $\mathbf{R}$.

Figure 2

Schematic picture of a diagonal trajectory contribution, $\alpha =\beta$, to the mean intensity $\langle I_{\omega }(Y,y)\rangle$.

Figure 3

Trajectory constellation contributing to the $C_1$ contribution of the speckle correlation function.

Figure 4

Trajectory constellation contributing to $C_2$. The top panel shows the trajectory configurations for which correlations remain short-ranged as a function of the source position difference $\Delta y$; the bottom panel refers to short-ranged correlation as a function of the detector position difference $\Delta Y$. In both panels, the trajectories undergo a small-angle encounter, indicated by the gray area. Phase-space points at the beginning and end of the encounter are denoted ${\Omega }_B$ and ${\Omega }_C$, respectively.

Figure 5

The function $A_2\left(0\right)$ vs ${\tau }_{\mathrm{E}}/{\tau }_{\mathrm{D}}$ for the case of a quasi-one-dimensional random waveguide.

Figure 6

Three different trajectory constellations contribution to $C_3$. All three constellations contain two small-angle encounters. The bright gray areas indicate single encounters, the dark gray areas indicate overlapping encounters. The constellations (b1), (b2), (c1), and (c2) involve a periodic reference trajectory $\gamma$, shown as a thin line. The figure shows only the “minimal” version of the constellations (b1), (b2), (c1), and (c2), in which the trajectory $\alpha$ winds once around $\gamma$, and ${\alpha }^{\prime }$ does not wind around $\gamma$ at all. Other contributions, in which $\alpha$ and ${\alpha }^{\prime }$ wind $n$ and $n-1$ times around $\gamma$, respectively, with $n>1$, are not shown in the figure, as well as constellations in which $\alpha$ and ${\alpha }^{\prime }$ wind $n-1$ and $n$ times around $\gamma$, respectively, with $n\ge 1$.

Figure 7

The rescaled correlation function $A_3=C_3{g}^2$ at equal frequencies, $\Delta \omega =0$, for the random waveguide, as a function of the Ehrenfest time ${\tau }_{\mathrm{E}}$, together with its three contributions $A_{3,a}$, $A_{3,b}$, and $A_{3,c}$.