Bragg solitons in nonlinear $\mathcal{PT}$-symmetric periodic potentials

It is shown that slow Bragg soliton solutions are possible in nonlinear complex parity-time ($\mathcal{PT}$) symmetric periodic structures. Analysis indicates that the $\mathcal{PT}$-symmetric component of the periodic optical refractive index can modify the grating band structure and hence the effective coupling between the forward and backward waves. Starting from a classical modified massive Thirring model, solitary wave solutions are obtained in closed form. The basic properties of these slow solitary waves and their dependence on their respective $\mathcal{PT}$-symmetric gain-loss profile are then explored via numerical simulations.

Figure 1

Band structure of a $\mathcal{PT}$-symmetric periodic grating (linear case) for different ratios of $g/\kappa$; (a) 0, (b) $0.8$, (c) 1, (d) $1.2$

Figure 2

Propagation dynamics of a solitary wave solution in a $\mathcal{PT}$-symmetric Bragg structure; intensity evolution for both the forward (left) and backward waves (right) during propagation.

Figure 3

Propagation dynamics of a Gaussian wave packet when injected only in the forward direction when the $\mathcal{PT}$ grating is operated below the $\mathcal{PT}$-symmetry breaking threshold. (a),(b) Forward and backward components, respectively, and (c) the associated energy as a function of normalized time.

Figure 4

The same as Fig. 3 when the $\mathcal{PT}$ grating is operated at the $\mathcal{PT}$-symmetry breaking threshold. (a),(b) Forward and backward components, respectively, and (c) the associated energy as a function of normalized time.