Nonlinear harmonic wave manipulation in nonlinear scattering medium via scattering-matrix method

Scattering of waves, e.g., light, due to medium inhomogeneity is ubiquitous in physics and is considered detrimental for many applications. Wavefront shaping technology is a powerful tool to defeat scattering and focus light through inhomogeneous media, which is vital for optical imaging, communication, therapy, etc. Wavefront shaping based on the scattering matrix (SM) is extremely useful in handling dynamic processes in the linear regime. However, the implementation of such a method for controlling light in nonlinear media is still a challenge and has been unexplored until now. We report a method to determine the SM of nonlinear scattering media with second-order nonlinearity. We experimentally demonstrate its feasibility in wavefront control and realize focusing of nonlinear signals through strongly scattering quadratic media. Moreover, we show that statistical properties of this SM still follow the random matrix theory. The scattering-matrix approach of nonlinear scattering medium opens a path toward nonlinear signal recovery, nonlinear imaging, microscopic object tracking, and complex environment quantum information processing.