Universal linear optics by programmable multimode interference

We introduce a constructive algorithm for universal linear electromagnetic transformations between the $N$ input and $N$ output modes of a dielectric slab. The approach uses out-of-plane phase modulation programmed down to $N^2$ degrees of freedom. The total area of these modulators equals that of the entire slab: our scheme satisfies the minimum area constraint for programmable linear optical transformations. We also present error correction schemes that enable high-fidelity unitary transformations at large $N$. This ``programmable multimode interferometer’’ (ProMMI) thus translates the algorithmic simplicity of Mach-Zehnder meshes into a holographically programmed slab, yielding DoF-limited compactness and error tolerance while eliminating the dominant sidewall-related optical losses and directional-coupler-related patterning challenges.