Mode Coupling and Scattering Parameters in a Parity-Time Symmetric Resonant Integrated Coupler

We investigate theoretically, in the frame of coupled-mode theory, a parity-time symmetric directional coupler inserted into a symmetric resonant cavity, all ports being terminated by four identical lossless reflectors. The scattering parameters are expressed under analytic closed forms in several equivalent bases, underlining the fact that PT Symmetry can be thought of as mode coupling between the old Eigenmodes of the passive structure. We also derive the oscillation threshold of such a complex cavity, and comment upon this apparent paradox: as a matter of fact, a laser effect is not necessarily expected from a device with more losses than gain.

I. Introduction

Since the rediscovery of non-Hermitic Hamiltonian operators [1]–[3] and their transposition to optical systems [4], [5], Parity-Time Symmetric (PTS) photonic structures have attracted a lot of attention, both theoretically and practically, either in directional couplers, distributed feedback structures or coupled resonators [6]–[8].