Compactifying linear optical unitaries using multiport beamsplitters

Abstract

We show that any $N$-dimensional unitary matrix can be realized using a finite sequence of concatenated identical fixed multiport beamsplitters (MBSs) and phase shifters (PSs). Our construction is based on a Lie group theorem applied to existing decompositions. Using the Bell-Walmsley-Clements framework, we prove that any $N$-dimensional unitary requires $N+2$ phase masks, $N-1$ fixed MBSs, and $N-1$ BSs. Our scheme requires only $\mathcal{O}(N)$ fixed, identical components (MBSs and BSs) compared to the $\mathcal{O}(N^2)$ fixed BSs required by conventional schemes (e.g., Clements), all while keeping the same number of PSs. Experimentally, these MBS can be realized as a monolithic element via femtosecond laser writing, offering superior performance through reduced insertion losses. As an application, we present a reconfigurable linear optical circuit that implements a three-dimensional unitary emerging in the unambiguous discrimination of two nonorthogonal qubit states.