Input-output relations for multiport ring cavities

Multiport ring cavities represent the natural generalization of two-port Fabry-Perot interferometers [1] to several modes of the radiation field. They find application in advanced interferometry, division multiplexing and optical cross connect. Recently, a three-port fiber ring laser was suggested and demonstrated to improve sensing resolution [2], whereas multiport optical circulators have been used for interconnecting single-fiber bidirectional ring networks [3]. In addition, a three-port reflection grating was demonstrated [4] and the corresponding input-output relations have been derived [5]. From a more fundamental perspective, multiport couplers, either multiport beam splitters or ring cavities, are crucial devices to generate and engineer multiphoton entangled states [6]. In fact, the cavity response is linear in the input modes for both kind of devices, with ring cavities offering the additional feature of a high nonlinearity with respect to the internal phase-shifts of the cavity. The use of ring cavities, supplemented by nonlinear media, has been also suggested to realize nondemolitive measurement and photon filtering [7]. In this letter, fully quantum input-output relations for a generic n-port ring cavity are obtained by modeling the ring as a cascade of n interlinked, suitably matched, beam splitters. In this way, the cavity response to an impinging beam, as well as the use of the cavity in interferometry, can be evaluated as a function of the beam-splitter reflectivities, the internal phase-shifts, and the number of ports. Let us first illustrate the results in details for the case of a three-port ring cavity, which has been schematically depicted in Fig. 1. We assume that the three beam splitters used to build the cavity have the same transmissivity τ . We also assume that losses at the beam splitters are negligible. The reflectivity of each coupler is thus given by ρ = 1 − τ . The input-output relation for the three beam splitters are given by