## Abstract

This paper reviews and extends two classes of algorithms for the design of planar couplers with any unitary transfer matrix as design goals. Such couplers find use in optical sensing for fading free interferometry, coherent optical network demodulation, and also for quantum state preparation in quantum optical experiments and technology. The two classes are (1) “atomic coupler algorithms” decomposing a unitary transfer matrix into a planar network of $2\times 2$ couplers, and (2) “Lie theoretic algorithms” concatenating unit cell devices with variable phase delay sets that form canonical coordinates for neighborhoods in the Lie group $U(N)$, so that the concatenations realize any transfer matrix in $U(N)$. As well as review, this paper gives (1) a Lie theoretic proof existence proof showing that both classes of algorithms work and (2) direct proofs of the efficacy of the “atomic coupler” algorithms. The Lie theoretic proof strengthens former results. $5\times 5$ couplers designed by both methods are compared by Monte Carlo analysis, which would seem to imply atomic rather than Lie theoretic methods yield designs more resilient to manufacturing imperfections.

© 2015 Optical Society of America

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