Exact control of parity-time symmetry in periodically modulated nonlinear optical couplers

We propose a mechanism for the realization of exact control of parity-time $\left(\mathcal{PT}\right)$ symmetry by using a periodically modulated nonlinear optical coupler with balanced gain and loss. It is shown that for certain appropriately chosen values of the modulation parameters, we can construct a family of exact analytical solutions for the two-mode equations describing the dynamics of such nonlinear couplers. These exact solutions give explicit examples that allow us to precisely manipulate the system from nonlinearity-induced symmetry breaking to $\mathcal{PT}$ symmetry, thus providing an analytical approach to all-optical signal control in nonlinear $\mathcal{PT}$-symmetric structures.

Figure 1

Schematic drawing of a Kerr nonlinear $\mathcal{PT}$-symmetric coupler. The periodic change of color along the $z$ axis denotes the periodic modulation of the linear refractive index. One waveguide experiences gain while the other loss.

Figure 2

Beam dynamics of the coupled-mode equation (1) for two different initial conditions: (a), (b) ${\psi }_1\left(0\right)=1$, ${\psi }_2\left(0\right)=0$ and (c), (d) ${\psi }_1\left(0\right)=0$, ${\psi }_2\left(0\right)=1$. (a) and (c) are for the no-modulation case; (b) and (d) are for the case with modulation. Shown here are the numerical results of intensity $I_1={\left|{\psi }_1\right|}^2$ in the first waveguide (blue dashed line), $I_2={\left|{\psi }_2\right|}^2$ in the second waveguide (black dashed-dotted line), and the total intensity $I=I_1+I_2$ (red solid line). The open circles denote the exact analytical results of the total intensity. The system parameters are chosen as $v=1$, $\gamma =0.6$, $\chi =1.2$, and the modulation parameters for (b) and (d) are given by Eqs. (24) and (28), respectively.

Figure 3

Numerical solution of the coupled equation (1) describing the nonlinear $\mathcal{PT}$-symmetric system with the initial condition ${\psi }_1\left(0\right)={\psi }_2\left(0\right)=1/\sqrt{2}$. (a) and (c) are for the case with no modulation; (b) and (d) are for the case with modulation (31). (a), (b) Dependencies of intensity on propagation distance $z$ in the first (blue dashed line) and second (black dashed-dotted line) waveguides; the solid line shows the total intensity. (c) and (d) give visualization of the intensity evolution along the propagation direction. In all plots, $v=1$, $\gamma =0.6$, the nonlinearity strength $\chi =1.2$, and the modulation parameters are given by Eq. (31).

Figure 4

(a), (b) Quasienergies $\lambda$ of the linearized equation (38) corresponding to two different exact solutions. (a) and (b) are for the quasienergies $\lambda$ associated with two exact solutions shown in Figs. 2 and 2, respectively. (c), (d) The evolutions of the fidelity between the exact solution and the numerical solution. Here, (c) represents the fidelity between the exact solution shown in Fig. 2 and its corresponding numerical solution with initial random noise. (d) denotes the fidelity between the exact solution shown in Fig. 2 and the numerical one with initial random noise.