Fig. 1. (a) Illustration of a cylindrical diffusive-optical core-shell neutral inclusion with radii ${R_1}$ and ${R_2}$ in an otherwise homogeneously disordered surrounding. The piecewise constant light diffusivities ${D_0},$ ${D_1}$ , and ${D_2}$ are indicated. (b) Comparison of the (large) invisibility-cloak sample used in our previous work [13,23] (using a hollow core rather than a solid core) and the (small) neutral-inclusion sample used in the present work (see white arrow). (c) Photographs of the samples discussed in the present work: reference sample (left), obstacle sample (middle), and core-shell neutral-inclusion sample (right). The width of these cuboid samples is ${L_x} = 15\;{\rm{mm}}$ , their height is ${L_y} = 8\;{\rm{mm}}$ , and their thickness is ${L_z} = 3\;{\rm{mm}}$ . Furthermore, we have ${R_1} = 0.8\;{\rm{mm}}$ and ${R_2} = 1.2\;{\rm{mm}}$ .

Fig. 2. Photographs of the transmitted light of the reference, obstacle, and neutral-inclusion samples used in this work [cf. Fig. 1(c)] under illumination with white light from the other side of the sample. The top row corresponds to homogeneous illumination of the sample front surface and the bottom row to point-like illumination. The yellow curves superimposed onto these photographs exhibit intensity cuts through these photographs, averaged over the range indicated by the dashed gray horizontal lines. For clarity, these cuts are normalized to the peak brightness value of the reference sample (a) for (a)–(c) and to that of (d) for (d)–(f), respectively.

Fig. 3. Scheme of the optical transmission-matrix experiment. (a) Overview over the entire interferometer containing a sample arm (red) and reference arm (blue). The focal lengths of the lenses are L1: 75 mm, L2: 50 mm, L3: ${-}9\;{\rm{mm}}$ , L4: 200 mm, L5: 75 mm, and L6: 100 mm. The distance between lenses of a pair is adjusted such that a collimated beam results. A1 is a 50 µm pinhole, M1–M7 are fixed mirrors, M8 is a tunable micro-mirror. BS1 and BS2 are 50%/50% beam splitters and GF1 is a variable gray filter. MO1 and MO2 are microscope objective lenses and LP is a linear polarizer. (b) Magnified view onto the sample arm and the combination of sample and reference arms on the CCD camera. The reference beam and the optical axis intentionally include an angle $\gamma = 2.5^\circ$ , which is exaggerated in the drawing for clarity. This angle leads to a tilt in both the horizontal and vertical directions on the CCD camera chip [cf. dashed white circle in Fig. 4(b)]. The MEMS mirror allows us to vary the direction of the incident light while keeping the position of the illuminated spot on the front side of the sample fixed. We vary the angle by ${\pm}23^\circ$ in both directions with a spacing between adjacent directions of 0.47° in both directions. This combination leads to 7,845 different and independent illumination configurations. This number defines the number of columns of the transmission matrix. The number of rows of the transmission matrix results from the choices discussed in Fig. 4 and is ${({361})^2} = 130{,}321$ .

Fig. 4. (a) Example camera raw image, a speckle pattern, for one selected incident direction of light (cf. Fig. 3). Light from the sample arm and reference arm (cf. Fig. 3) interferes on the camera pixels. The inset highlights the resulting interference stripes due to the angle $\gamma \ne 0$ and the fact that the speckle pattern is spatially resolved. The side length of one pixel on the camera chip corresponds to 0.1 µm in the sample plane [cf. SP in Fig. 3(b)]. (b) Modulus of the Fourier transform of (a) on a logarithmic false-color scale. The contribution arising from the interference of the sample and reference arms is highlighted by the dashed white circle. All pixels outside of this white circle are set to zero, and the center of the white circle is chosen as the new coordinate origin (0,0) in momentum space. Inverse Fourier transformation of these data leads to panels (c) and (d). (c) Modulus and (d) phase. These two panels each contain ${({361})^2} = 130{,}321$ pixels, which defines the numbers of rows of the transmission matrix (cf. caption of Fig. 3).

Fig. 5. (a) Modulus of the $i$ th singular value, ${\sigma _i} = | {{\sigma _{{ii}}}} |$ , versus index $i = 1 \ldots 7{,}845$ for the case of no sample (black), reference sample (red), and core-shell neutral-inclusion sample (blue). (b) Same, but normalized to the singular value distribution of the homogeneously disordered reference sample.

Fig. 6. (a) Same as Fig. 5(b), but for seven different positions on the core-shell neutral-inclusion sample (see inset). (b) Same as Fig. 5(b) but for three different positions (see inset) on the same reference sample. For each case, we show the average of five individual measurements by the full curve and the error bar (plus/minus one standard deviation) by the gray area. Note that the differences between the curves in (a) are much larger than those in (b). This observation means that the transmission-matrix experiments can reveal the invisible core-shell neutral-inclusion sample.