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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.
Linear optics underpins tests of fundamental quantum mechanics and computer science, as well as quantum technologies. Here we experimentally demonstrate the longstanding goal of a single reprogrammable optical circuit that is sufficient to implement
Reconfigurable quantum circuits are fundamental building blocks for the implementation of scalable quantum technologies. Their implementation has been pursued in linear optics through the engineering of sophisticated interferometers. While such optic
Interference-fit joints are typically adopted to produce permanent assemblies among mechanical parts. The resulting contact pressure is generally used for element fixing or to allow load transmission. Nevertheless, some special designs take advantage
We develop a universal approach enabling the study of any multimode quantum optical system evolving under a quadratic Hamiltonian. Our strategy generalizes the standard symplectic analysis and permits the treatment of multimode systems even in situat
On-chip optical interconnect has been widely accepted as a promising technology to realize future large-scale multiprocessors. Mode-division multiplexing (MDM) provides a new degree of freedom for optical interconnects to dramatically increase the li