Focusing inside Disordered Media with the Generalized Wigner-Smith Operator

We introduce a wave front shaping protocol for focusing inside disordered media based on a generalization of the established Wigner-Smith time-delay operator. The key ingredient for our approach is the scattering (or transmission) matrix of the medium and its derivative with respect to the position of the target one aims to focus on. A specific experimental realization in the microwave regime is presented showing that the eigenstates of a corresponding operator are sorted by their focusing strength—ranging from strongly focusing on the designated target to completely bypassing it. Our protocol works without optimization or phase conjugation and we expect it to be particularly attractive for optical imaging in disordered media.

Figure 1

Sketch of the experimental setup. The system consists of a rectangular aluminum waveguide of height $H=8\mathrm{mm}$, width $W=10\mathrm{cm}$, and total length $L=2.38\mathrm{m}$, see middle panel. The top plate (not shown) can be removed. The wave front is injected from the left using ten monopole antennas, see bottom panel. The scattering region displayed in the top panel has a length $L_s=60\mathrm{cm}$ and consists of 18 cylindrical Teflon scatterers (black cylinders, index of refraction $n=1.44$, radius 2.55 mm, height 8 mm). A (re-)movable brass scatterer of radius 8.825 mm and height 8 mm is located in the central part of the scattering region. The placement of scatterers in this sketch was randomly generated and matches the actual scatterer positions in the experiment. The distance between the injecting antenna array and the scanning antenna is $L_a=1.50\mathrm{m}$. The grained area around the movable scatterer indicates the region shown in Figs. 2 and 3.

Figure 2

Measured spatial distribution of the microwave intensity (see Supplemental Material [38]) inside the disordered scattering system realized in the experiment. The region shown is a zoom on the vicinity of the movable brass scatterer in the middle (as highlighted in the top panel of Fig. 1). In the top row this brass scatterer is included and in the bottom row it is removed. In both cases we display the eigenstates of $q_{y_b}$ with the largest eigenvalues $|{\vartheta }_n^{y_b}|=96.9$, 81.6, and 66.9 [a.u.] from left to right. We can clearly observe that the intensity distribution is enhanced in the region around the brass scatterer (top row), such that removing the scatterer changes the intensity pattern strongly (bottom row). In each of the plots (also in Fig. 3) the color scale has been adjusted to match the maximum intensity (shown in dark red).

Figure 3

Same as Fig. 2, but for the eigenstates of $q_{y_b}$ with the smallest eigenvalues $|{\vartheta }_n^{y_b}|=1.9$, 2.1, and 6.0 [a.u.] from left to right. The measured intensity pattern clearly avoids the brass scatterer in the middle (top row), such that removing the scatterer leaves the intensity pattern almost unchanged (bottom row).