Angular Memory Effect of Transmission Eigenchannels

The optical memory effect has emerged as a powerful tool for imaging through multiple-scattering media; however, the finite angular range of the memory effect limits the field of view. Here, we demonstrate experimentally that selective coupling of incident light into a high-transmission channel increases the angular memory-effect range. This enhancement is attributed to the robustness of the high-transmission channels against perturbations such as sample tilt or wave front tilt. Our work shows that the high-transmission channels provide an enhanced field of view for memory-effect-based imaging through diffusive media.

Figure 1

Angular memory effect, the experimental setup, and data. (a) Sketch of the angular memory effect for a diffusive slab. An incident beam generates a transmitted speckle pattern in far field (green). When the input wave front is tilted by a small angle $\theta$, the output wave front is tilted by the same angle $\theta$, leading to a lateral shift of far-field speckle pattern (red). (b) Simplified schematic of the experimental setup. The laser beam is modulated by a phase-only SLM, imaged onto the pupil of a microscope objective by a pair of lenses, and directed onto a ZnO nanoparticle film. The transmitted (reflected) light is measured by a CCD camera CCD1 (CCD2) in the far field. NA and $P$ stand for numerical aperture and linear polarizer, respectively. (c) Experimentally measured intensity correlation function $C(\theta )$ of the transmitted speckle patterns as a function of the normalized tilt angle $\theta /{\theta }_0^{(r)}$, for a high-transmission channel (with $T/\overline{T}=2.29$, blue dashed line), a low-transmission channel (with $T/\overline{T}=0.58$, red dot-dashed line), and a random incident wave front (black solid line). ${\theta }_0^{(r)}$ denotes the width of $C(\theta )$ for the random wave fronts where $C({\theta }_0^{(r)})=C(0)/2$, and its value is about 1°.

Figure 2

Numerical and theoretical results. (a) Angular correlation width ${\theta }_0^{(n)}$ of transmission eigenchannels vs their transmittance ${\tau }_n$. Each point represents an average over 10 disorder realizations. The horizontal black line denotes the angular correlation width for random incident wave fronts ${\theta }_0^{(r)}$. (b) Transverse spread $\mathrm{\Delta }D$ of high ($\overline{\tau }<{\tau }_n<1$, blue line) and low (${10}^{-4}<{\tau }_n<\overline{\tau }$, red line) transmission eigenchannels vs tilt angle $\theta$ of their incident wave fronts. (c) Sketch of the transmission matrix of a tilted sample $t_{\theta }=R_{\theta }^{†}t{R}_{\theta }$, where $R_{\theta }$ is the tilting matrix and $t$ the field transmission matrix without tilting. (d) Intensity correlation coefficient $C_n$ of transmission eigenchannels at the tilt angle $\theta ={\theta }_0^{(r)}\approx 0.8°$, obtained from numerical simulation (red solid line), and the prediction of the phenomenological model, Eq. (1), with ${\sigma }^2$ as the only fitting parameter (black dashed line). The number of channels is $N=3239$. The diffusive slabs have thickness $k_{0}L=100$, width $k_{0}W=6000$, transport mean free path $k{l}_t=4.6$, average refractive index $n_0=1.5$, where $k=n_0{k}_0$, $k_0=2\pi /\lambda$, and $\lambda$ is the vacuum wavelength.

Figure 3

Comparison between the experimental and the theoretical results. Correlation coefficient $C_n$ in (a) [$C_n^{(R)}$ in (b)] of transmitted (reflected) eigenchannel intensity patterns for $\theta =0$ and $\theta ={\theta }_0^{(r)}$ as a function of their normalized transmittance $T/\overline{T}$. ${\theta }_0^{(r)}$ is the width at half maximum of the angular correlation function in transmission $C(\theta )$ for (a) and in reflection $C^{(R)}(\theta )$ for (b) for random incident wave fronts. Blue solid line and black dots: numerical simulation results in the case of complete channel control (CCC) and incomplete channel control (ICC), respectively. Green open squares (red open circles): experimental data for ten highest (lowest) transmission channels. The simulation data represent an average over 50 disorder realizations. The simulation parameters are slab thickness $L=10\mu \mathrm{m}$, width $W=508\mu \mathrm{m}$, $l_t=1\mu \mathrm{m}$, $n_0=1.4$, background refractive index $n_1=1.0$ (in front of the slab), and $n_2=1.5$ (at the back).