# Focusing inside Disordered Media with the Generalized Wigner-Smith Operator

## TL;DR

This paper presents a wavefront shaping method using a generalized Wigner-Smith operator to focus inside disordered media, leveraging the scattering matrix and its derivative, enabling targeted focusing without optimization.

## Contribution

It introduces a novel wavefront shaping protocol based on a generalized Wigner-Smith operator, applicable to disordered media, demonstrated experimentally in the microwave regime.

## Key findings

- Eigenstates sorted by focusing strength
- Effective focusing without optimization or phase-conjugation
- Potential applications in optical imaging

## Abstract

We introduce a wavefront shaping protocol for focusing inside disordered media based on a generalization of the established Wigner-Smith time-delay operator. The key ingredient for our approach is the scattering (or transmission) matrix of the medium and its derivative with respect to the position of the target one aims to focus on. A specifc experimental realization in the microwave regime is presented showing that the eigenstates of a corresponding operator are sorted by their focusing strength - ranging from strongly focusing on the designated target to completely bypassing it. Our protocol works without optimization or phase-conjugation and we expect it to be particularly attractive for optical imaging in disordered media.