Graph-based analysis of nonreciprocity in coupled-mode systems

In this workwe derive the general conditions for obtaining nonreciprocity inmulti-mode parametrically-coupled systems. The results can be applied to a broad variety of optical,microwave, and hybrid systems including recent electroand opto-mechanical devices. In deriving these results, we use a graph-basedmethodology to derive the scatteringmatrix. This approach naturally expresses the terms in the scattering coefficients as separate graphs corresponding to distinct coupling paths betweenmodes such that it is evident that nonreciprocity arises as a consequence ofmulti-path interference and dissipation in key ancillarymodes. These concepts facilitate the construction of newdevices inwhich several other characteristicsmight also be simultaneously optimized. As an example, we synthesize a novel three-mode unilateral amplifier design by use of graphs. Finally, we analyze the isolation generated in a commonparametricmulti-mode system, the dc-SQUID.