Reconfigurable Linear Optical Transformations in a Single Integrated Multimode Waveguide

While conventional wavefront shaping relies on free-space systems comprising bulk optical components, compact and fully integrated approaches are comparatively unexplored. Here, we introduce a concept based on a single multimode waveguide with distributed thermo-optic perturbations for controlling multimode propagation.

We develop a differentiable physical model and numerically demonstrate shaping the outgoing wavefront and programmable linear optical transformations, using gradient-based optimization. We achieve diffraction-limited focusing with 20 spatial modes and show that the theoretical focusing limit can be retained while maintaining more than $99\%$ of the input optical power. We extend this approach to complete input--output transformations and implement Sylvester--Hadamard matrix operations with dimensions up to $32\times32$ with correlations exceeding $99\%$.