Mesoscopic Theory of Wavefront Shaping to Focus Waves inside Disordered Media

TL;DR

The paper develops a mesoscopic framework to describe how wavefront shaping focuses waves to a point inside a disordered 3D medium and how the resulting energy density is composed of a diffraction-limited peak governed by $C_1$ correlations and a background driven by long-range $C_2$ correlations. Critically, $C_2$ not only raises background energy away from the focus but also creates a genuine internal energy source at the focal point, enabling enhanced (non-Ohmic) transmission through a slab. By extending to slab geometries and analyzing both focusing and optimized transmission, the work shows that focusing is fundamentally about interference at the focus, while optimized transmission requires a reconstructed internal source that is constrained by the angular extent of the wavefront array. The results emphasize that the energy-density profile near focus mimics the first diffusion eigenmode, and that discriminating between transmission-optimized models rests on the internal source density rather than the energy density alone.

Abstract

We describe the {theory of focusing waves} to a predefined spatial point {inside} a disordered {three-dimensional medium} by the external shaping of {$N$} different field sources outside the medium, {also known as wavefront shaping}. We {derive} the energy density of the wave field {both} near the focal point and anywhere else inside the medium, {averaged over realizations\textit{ after} focusing}. {To this end, we conceive of a point source at the focal point that emits waves to a detector array that - by time reversal - emits the desired shaped fields. }%endcolor {It appears that the energy} density is formally equal to intensity speckle described by {the so-called} $C_1$, $C_2$, $C_3$ and even $C_0$ {correlations} in mesoscopic transport theory, {yet the density also obeys a diffusion equation}. The $C_1$ {correlations} describes the focusing in the random medium very well, but do not generate a new source of energy that {is conceived} at the focal point. A source emerges {only} when the $C_2$ speckle is incorporated. The role of $C_0$ speckle, describing fluctuations in the {local density of optical states (LDOS)} is also investigated, {but hardly plays a role in the focusing. } Finally, we use the {concept of an energy source inside the medium} to model the {well-known} optimized transmission by a slab using wavefront shaping.

Figures (5)

• Figure 1: Schematic of wavefront shaping to focus waves to a point $S$ inside a disordered 3D half-space on the right. The medium has a skin layer with a depth equal to about one mean free path $\ell$. A wave packet from a virtual point source at $S$ is sent towards the detector array with $N$ detectors at positions $\{ \mathbf{x}_i\}$ in the "far field" of the medium, meaning here that $a \gg z_S$. The optimal shaping is established by time-reversal of the entire wave packet. It necessarily refocuses on a diffraction-limited spot $R \approx S$ with a background energy that is calculated in this work for all positions $R$. The path of the wave from source to array is separated into 3 parts: a) release of energy by a virtual source at focus $S$ to a nearby scatterer $2$, b) radiative transport towards a position $1$ in the skin layer with $T$-matrix $T_\omega(2,1)$, and c) ballistic propagation from skin layer towards array. The same is done for the time-reversed signal, here equal to the complex conjugate and shown as a dashed line.
• Figure 2: Hikami diagrams for $C_2$ correlations in the average $\langle T(2,1) T^*(3,4) T^*(2',1')T(3',4') \rangle$ to focalization. Dashed lines denote complex conjugates. The Hikami vertex $H$ exchanges momenta at position $\mathbf{H}$ in the medium, over which is to be integrated.
• Figure 3: wavefront shaping and the energy density inside a slab. Left: if we optimize the total transmission, it is unity ($T = 1$), whereas total reflection vanishes ($R = 0$) and the energy density $\rho(z)$ varies wildly. Center: if we apply time-reversal, the incident pattern (from left panel) is incident from the back surface without reflection ($R = 0$) and perfectly transmitted ($T = 1$) to the left entrance surface. The energy density $\rho(z)$ has the same pattern as it is not affected by time reversal. Right: if we apply a mirror operation, the incident wavefront from the center image is incident from the left, enters perfectly with ($R = 0$) and transmits perfectly ($T = 1$) to the right. The energy density $\rho(z)$ is mirrored with respect to left and center since the sample structure has been mirrored. Averaging over configurations yields a mirror symmetric energy density, hence $\rho_M (z) = \rho_M(L-z)$.
• Figure 4: Normalized energy densities (top) and normalized sources (bottom) for the six different models that are specified in Table \ref{['table:6_different_models']}.
• Figure 5: $C_0$ contributions to the energy density at position $R$ caused by a scatterer (in red) close to the focus locus $S$. Dashed lines denote complex conjugate wave fields. Both actually generate a contribution to the peak, but only the left diagram suffers from decorrelation between the $N$ channels. Complex conjugates of these diagrams exist but are not shown.