Parity-Time Symmetry in Laterally Coupled Bragg Waveguides

We investigate theoretically, in the frame of Coupled-Mode Theory (CMT), a system made of two laterally coupled, perfectly phase-matched, index-coupled Bragg waveguides, the modal gain of the amplifying one compensating exactly the modal losses of the other. Reverting to the supermode eigenbasis of the underlying "naked" (non-periodic) Parity-Time Symmetric (PTS) directional coupler, we derive analytically the scattering parameters of the whole structure. We also consider a system made of two purely gain-coupled Bragg waveguides, either in phase or in anti-phase. In each case, we comment upon the gain requirement for achieving a laser oscillation.