Unidirectional nonlinear $\mathcal{PT}$-symmetric optical structures

We show that nonlinear optical structures involving a balanced gain-loss profile can act as unidirectional optical valves. This is made possible by exploiting the interplay between the fundamental symmetries of parity ($\mathcal{P}$) and time ($\mathcal{T}$), with optical nonlinear effects. This unidirectional dynamics is specifically demonstrated for the case of an integrable $\mathcal{PT}$-symmetric nonlinear system.

Figure 1

Beam propagation in two coupled linear waveguides. For the parameters of the simulation (we use normalized coupling units), the spontaneous $\mathcal{PT}$-breaking take place at ${\gamma }_{\mathcal{PT}}=1$. In all cases, left (right) panels correspond to an initial excitation at the left (right) channel. (a),(b) A passive system corresponding to $\gamma =0$. The propagation is reciprocal; (c),(d) $\gamma =0.4{\gamma }_{\mathcal{PT}}$ corresponding to the exact $\mathcal{PT}$ phase. A nonreciprocal beam propagation is evident. Although the dynamics is non-Hermitian, the evolution is “pseudounitary” and the total beam power remains bounded. (e),(f) For $\gamma =1.5{\gamma }_{\mathcal{PT}}$, the beam power grows exponentially (vertical scale is logarithmic) in both waveguides, while the beam propagation is again nonreciprocal with respect to the mirror axis of the two waveguides. Waveguides are color-coded, indicating balanced gain (red, left) and loss (green, right) regions. Gray waveguides indicate a passive ($\gamma =0$) system.

Figure 2

Beam propagation in two coupled nonlinear waveguides with nonlinearity strength $\chi$ and a complex $\mathcal{PT}$-symmetric refractive index profile. Waveguides are color coded, indicating balanced gain (red, left) and loss (green, right) regions ($\gamma =0.1$). Left columns correspond to an initial excitation at the gain waveguide port, while right columns correspond to an initial excitation at the lossy waveguide. (a),(b) The nonlinearity $\chi =1.9$ is below the critical value ${\chi }_d\approx 3.37$ while for (c) and (d) the nonlinearity strength $\chi =8$ is above.

Figure 3

(Main figure) A semilogarithmic plot of ${\chi }_d$ vs $\gamma$. For the numerical evaluation of ${\chi }_d$ we have integrated Eq. (5). (Inset) The efficiency factor $Q$ vs nonlinearity strength $\chi$, for a fixed gain (loss) parameter $\gamma =0.1$ and three different waveguide lengths $z=10$, 20, and $z=30$. For nonlinearity strength $\chi ={\chi }_d\approx 3.4$ the isolator reaches its optimal efficiency.

Figure 4

The numerically extracted first passage distance $z_{\mathrm{fpd}}$ versus the gain (loss) parameter $\gamma$. The initial conditions are chosen to be $S_0(0)=1$ and $S_3(0)=-1$. An inverse power law is observed. (Inset) The proportionality coefficient $f(\chi )$ is plotted versus the nonlinearity strength $\chi$ for $\chi >{\chi }_d$. The red line correspond to the best linear fit.

Figure 5

Dynamics of the rescaled Stokes variables $\mathcal{F}$ for $\chi =9$ and various gain (loss) parameters $\gamma$: dashed blue lines correspond to $\gamma =0.157$; pink lines ($\star$) correspond to $\gamma =0.15$; solid yellow lines correspond to $\gamma =0.12$; and red lines ($\hspace{1em}\hspace{1em}\hspace{1em}\blacksquare$) to $\gamma =0.174$. The green line ($•$) corresponds to the passive system $\gamma =0$ with critical nonlinearity $\chi =4$, where the motion of the trajectory is on the separatrix. The trajectory associated with $\gamma =0.157$ (see the red arrow in Fig. 4) is typical to the cases where $z_{\mathrm{fpd}}$ diverge and correspond to the closest one to the separatrix of the passive $\gamma =0$ system.