Inverse theory of wavefront shaping in nonlinear scattering media

Scattering fundamentally limits the propagation of light in complex media, yet controlling it is essential for transformative advances in imaging, sensing, and optical communication. While decades of research have established powerful methods for linear wavefront shaping, the control of nonlinear scattering remains dominated by feedback-based optimization and neural networks - approaches that lack interpretability and theoretical bounds. Here, we establish the analytic inverse theory of nonlinear wavefront shaping under open-geometry scattering conditions with circular complex Gaussian statistics. By formulating an explicit scattering tensor model, we reveal how the optimal input field emerges from the dominant eigenchannel of the tensor's spectral diagonalization. This framework directly leads to a closed-form enhancement bound for second-harmonic generation. We experimentally confirm the theory by shaping wavefronts to realize single-point focusing, multi-point focusing, and global second-harmonic signal enhancement in nonlinear scattering media. By bridging nonlinear optics and inverse wavefront control, this work transforms nonlinear wavefront shaping from an optimization-driven practice into a principled, interpretable, and prediction-capable discipline.