Programmable linear optical interferometers are a core primitive in optical signal processing, quantum information processing, and photonic computing. Existing photonic-integrated implementations realize arbitrary
M-mode unitaries using Mach--Zehnder-interferometer meshes whose footprint and accumulated loss scale with
O(M2) optical components. Here we analyze and experimentally demonstrate a programmable architecture for implementing linear optical transformations directly on spatially tiled free-space {\em pixel modes} using multi-plane light conversion (MPLC). In this architecture,
M spatial modes arranged on a transverse lattice undergo a unitary transformation and are mapped to
M output modes of identical geometry through a sequence of programmable phase masks separated by free-space propagation segments. Numerical simulations show that arbitrary
M-mode unitaries can be compiled to a desired high fidelity using a number of phase planes that scales approximately linearly with
M. Using a spatial-light-modulator-based MPLC, we experimentally demonstrate programmable interferometers acting on up to
16 spatial pixel modes, including tunable beamsplitters, Hadamard unitaries, spatial permutations, and partial unitaries on select subsets of modes. These results establish MPLC-based pixel-mode interferometers as a promising architecture for programmable linear optics with applications in classical and quantum optical interconnects, photonic switching, and quantum information processing.
Abstract Linear optical operations are fundamental and significant for both quantum mechanics and classical technologies. We demonstrate a non-cascaded approach to perform arbitrary linear operations for N -dimensional phase-coherent spatial modes using meticulously designed phase gratings. As implemented on spatial light modulators (SLMs), the unitary transformation matrix has been realized with dimensionalities ranging from 7 to 24 and the corresponding fidelities are from 95.1% to 82.1%. For the non-unitary operators, a matrix is presented for the tomography of a 4-level quantum system with a fidelity of 94.9%. Thus, the linear operator has been successfully implemented with much higher dimensionality than that in previous reports. It should be mentioned that our approach is not limited to SLMs and can be easily applied on other devices. Thus, our proposal could provide another option to perform linear operations with a simple, fixed, error-tolerant and scalable scheme.
Photonic circuits are central to classical and quantum information processing. While integrated technologies dominate, free-space architectures are emerging as attractive alternatives, offering broad bandwidth and direct manipulation of optical modes without confinement in waveguides. A key challenge for scalability lies in circuit depth, as the number of layers manipulating the optical field typically grows with the system size. Here, we introduce a programmable free-space photonic platform that implements translation-invariant, high-dimensional unitary transformations using only three layers. Information is encoded in structured light modes defined by circular polarization and quantized transverse momenta, and processed with spatial light modulators interleaved with half-wave plates. We implement unitaries that are equivalent to quantum walks over up to 30 time steps, in one- and two-dimensional lattices, distributing a single input mode across more than 7,000 outputs, where conventional approaches would require tens or hundreds of layers. The platform supports diverse quantum walk dynamics, including disorder, synthetic gauge fields, and topological effects, previously explored only in separate experiments. Using coincidence detection with a time-tagging camera, we show compatibility with quantum optics protocols and provide examples of quantum walks of heralded single photons. These results contribute to establishing free-space optical processors as promising resources for high-dimensional quantum simulation and scalable optical information processing.