Scattering matrix analysis of 3×3 fiber couplers

A general scattering matrix (S-matrix) theory lossy reciprocal 3×3 couplers is developed. It predicts the possible values of the phases of the complex S-matrix coefficient if the magnitude of these coefficients is known. Depending on symmetry, and in particular the loss of the coupler, allowed ranges of the phase angles are given. As an example, an optimized coupler for a coherent homodyne transmission system with optoelectric 90° hybrid is investigated. The results show a strong tradeoff between the desired ideal behavior and the fabrication tolerances