Mesoscopic Theory of Wavefront Shaping to Focus Waves inside Disordered Media

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 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. It appears that the energy density is formally equal to intensity speckle described by the so-called C1, C2, C3 and even C0 correlations in mesoscopic transport theory, yet the density also obeys a diffusion equation. The C1 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 C2 speckle is incorporated. The role of C0 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.