Mesoscopic Theory of Wavefront Shaping to Focus Waves inside Disordered Media

TL;DR

This paper develops a mesoscopic theoretical framework for wavefront shaping to focus waves inside disordered media, deriving energy density distributions and linking them to mesoscopic correlations and diffusion, with applications to optimized transmission.

Contribution

It introduces a comprehensive mesoscopic theory connecting wavefront shaping, energy density, and mesoscopic correlations, including the roles of C1, C2, and C0, in focusing inside disordered media.

Findings

Energy density near focus is described by mesoscopic correlations.

C1 correlations explain focusing but do not add energy.

Incorporating C2 correlations introduces an effective energy source.

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.