Solitons in -symmetric ladders of optical waveguides

Abstract

We consider a ${ \mathcal P }{ \mathcal T }$ -symmetric ladder-shaped optical array consisting of a chain of waveguides with gain coupled to a parallel chain of waveguides with loss. All waveguides have the focusing Kerr nonlinearity. The array supports two co-existing solitons, an in-phase and an antiphase one, and each of these can be centred either on a lattice site or midway between two neighbouring sites. We show that both bond-centred (i.e. intersite) solitons are unstable regardless of their amplitudes and parameters of the chain. The site-centred in-phase soliton is stable when its amplitude lies below a threshold that depends on the coupling and gain–loss coefficient. The threshold is lowest when the gain-to-gain and loss-to-loss coupling constant in each chain is close to the interchain gain-to-loss coupling coefficient. The antiphase site-centred soliton in the strongly-coupled chain or in a chain close to the ${ \mathcal P }{ \mathcal T }$ -symmetry breaking point, is stable when its amplitude lies above a critical value and unstable otherwise. The instability growth rate of solitons with small amplitude is exponentially small in this parameter regime; hence the small-amplitude solitons, though unstable, have exponentially long lifetimes. On the other hand, the antiphase soliton in the weakly or moderately coupled chain and away from the ${ \mathcal P }{ \mathcal T }$ -symmetry breaking point, is unstable when its amplitude falls in one or two finite bands. All amplitudes outside those bands are stable.

Keywords

• optical waveguides
• gain and loss
• parity-time symmetry
• discrete nonlinear Schroedinger equation
• solitons
• stability