Mesoscopic theory of optical wavefront shaping to focus deep inside opaque media

¹ Université Grenoble Alpes, Centre National de la Recherche Scientifique (CNRS), Laboratoire de Physique et de Modélisation des Milieux Condensées (LPMMC), 38000 Grenoble, France
² Complex Photonic Systems (COPS), MESA+ Institute, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
³ Institut Langevin, ESPCI Paris, PSL University, CNRS, 1 Rue Jussieu, 75005 Paris, France

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Published online: 22 September 2025

Abstract

We describe the theory of optical wavefront shaping, where waves are focused to a predefined focal point deep inside a disordered three-dimensional (3D) opaque medium (e.g., tissue, paint, foam) by shaping the wavefield sent by N external array elements. Our work is motivated by recent studies where wavefront shaping (controlled interference of scattered waves) coexists with classical diffusion for which wave interference appears to be irrelevant. We derive the energy density both near the focus and anywhere in the medium, averaged over realizations after optimization. We find that the average energy density including focusing is described by the C1, C2, C3 and C0 intensity correlations known from mesoscopic transport theory. Remarkably, the background energy density obeys a diffusion equation wherein C2-correlations create an energy source inside. Thus, a classical property (diffusion) coexists with interference. We discuss several energy density profiles proposed in literature, that are associated with optimized transmission by a slab using wavefront shaping. Our results are relevant for applications where the internal energy density in opaque media is crucial, such as in white-light illumination, projection optics, semiconductor metrology, (bio)sensing, and photovoltaics.