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.

https://doi.org/10.1103/PhysRevA.111.053507

B. A. van Tiggelen, A. Lagendijk, and W. L. Vos

2025

05/2025

Phys. Rev. A

111

053507: 1-14

10.1103/PhysRevA.111.053507

disordered materials, energy density control, focusing, multiple scattering media, time-reversal, Wavefront shaping