Signal-reference correlation for modal structure retrieval in disordered optical fields

We experimentally demonstrate a signal-reference modal correlation method for retrieving structured optical modal signatures after propagation through a scattering medium. Hermite-Gaussian and Laguerre-Gaussian modes transmitted through the medium produce randomized speckle patterns in direct intensity measurements, obscuring their spatial structure. By computing the spatial cross-correlation between each signal speckle and a single fixed Gaussian reference, distinct modal morphology emerges in the central correlation peak. Cartesian parity symmetry is recovered for Hermite-Gaussian modes, while charge-dependent radial scaling is resolved for Laguerre-Gaussian modes. The approach requires no transmission-matrix calibration, wavefront shaping, or machine-learning-assisted reconstruction, providing a compact and experimentally accessible framework for structured-light discrimination in complex scattering environments.