  Programmable photonic integrated circuits represent an emerging technology
that amalgamates photonics and electronics, paving the way for light-based
information processing at high speeds and low power consumption. Programmable
photonics provides a flexible platform that can be reconfigured to perform
multiple tasks, thereby holding great promise for revolutionizing future
optical networks and quantum computing systems. Over the past decade, there has
been constant progress in developing several different architectures for
realizing programmable photonic circuits that allow for realizing arbitrary
discrete unitary operations with light. Here, we systematically investigate a
general family of photonic circuits for realizing arbitrary unitaries based on
a simple architecture that interlaces a fixed intervening layer with
programmable phase shifter layers. We introduce a criterion for the intervening
operator that guarantees the universality of this architecture for representing
arbitrary $N \times N$ unitary operators with $N+1$ phase layers. We explore
this criterion for different photonic components, including photonic waveguide
lattices and meshes of directional couplers, which allows the identification of
several families of photonic components that can serve as the intervening
layers in the interlacing architecture. Our findings pave the way for
efficiently designing and realizing novel families of programmable photonic
integrated circuits for multipurpose analog information processing.
