Photon Delocalization Transition in Dimensional Crossover in Layered Media

We report a crossover in optical propagation in random layered media from localization towards diffusion as the interaction of the wave with the sample is transformed from one to three dimensional due to nonuniformity in the layer thickness. The crossover occurs at the point that the lateral spread of the wave equals the transverse coherence length in the transmitted speckle pattern.

Figure 1

(a) Example of a fringe pattern generated by a single cover slip. (b) Typical speckle pattern for two cover slips and intervening air gap. (c) and (d) Speckle patterns generated by samples with 20 and 80 cover slips, respectively. The local transmission coefficient indicated by the color bar may exceed unity due to the interference of waves with wave vector components in the layer plane.

Figure 2

(a) Semilogarithmic plot of measurements (red squares) and 1D simulations (black dots) of $⟨T⟩$ versus number of glass cover slips, $L$. Simulations of $⟨\ln T⟩$ are shown in the inset. (b) and (c) Schematic of interference between partial waves following two trajectories, $\alpha$ and $\beta$, which pass through the same layers an equal number of times, in samples with parallel and nonparallel layers, respectively.

Figure 3

(a) Schematic of the average transverse spread, $\sigma$, and the speckle size, $d$, along the $x$ direction at the output plane for an incident plane wave. (b) Measurement of ${\sigma }_x$ and $d_x$ versus $L$. Their crossing at $L\approx 35$ is consistent with the departures of $⟨T(L)⟩$ from results of 1D simulations beginning at $L=35$ [seen in Fig. 2a].

Figure 4

Comparison of $⟨T(L)⟩$ for samples with different additional wedge angles. Thin shims with different thicknesses are inserted between layers to introduce average wedge angles of 0.073°, 0.218°, and 0.364°, respectively, into the air gaps. The side of the cover slip in which the shims are placed is alternated so that the average angle of the cover slips is not changed. The total transmission could only be measured up to thicknesses at which the beam spread does not approach the edges of the cover slips.