Mode pairs in $\mathcal{PT}$-symmetric multimode waveguides

We study mode properties in multimode optical waveguides with parity-time $\left(\mathcal{PT}\right)$ symmetry. We find that two guiding modes with successive orders $2m-1$ and $2m$ form a mode pair in the sense that the two components of the pair evolve into the same mode when the loss and gain coefficient increases to some critical values, and they experience $\mathcal{PT}$ symmetry breaking simultaneously. For waveguides that in their conservative limit support an odd number of guiding modes, a new mode with a proper order emerges upon the increase of the gain and loss level, so that it pairs with the already existing highest-order mode and then breaks their $\mathcal{PT}$ symmetry simultaneously. Depending on the specific realizations of $\mathcal{PT}$-symmetric potentials, higher-order mode pairs may experience symmetry breaking earlier or later than the lower-order mode pairs do.

Figure 1

Mode profiles for fundamental and dipole modes at different values of $\alpha$. The shaded region stands for the landscape of the Gaussian waveguide; $p=2,d=5$.

Figure 2

Dependence of the (a) real and (b) imaginary parts of the propagation constants of the first six modes on $\alpha$; $p=2,d=5$.

Figure 3

The real (solid lines) and imaginary (dashed lines) parts of the wave functions for the first (a, b) and the second (c, d) mode pairs, at ${\alpha }_1=0.02$ (green lines) and ${\alpha }_2=0.04$ (red lines); $p=2,d=5$.

Figure 4

Mode profiles for tripole and quadrupole modes at different values of $\alpha$; $p=2,d=5$.

Figure 5

Dependence of mode propagation constants on $\alpha$ for waveguides with (a, b) $p=2,d=0.8$ and (c, d) $p=2,d=2$. Insets in (a) and (c) show the mode profiles corresponding to the circles in the spectrum.

Figure 6

Propagation simulation of the dipole mode at (a) $\alpha =0.041(<{\alpha }_{\text{cr}}^{\left(1\right)})$, and of the mode bifurcating from the fundamental-dipole mode pair at (b) $\alpha =0.175(>{\alpha }_{\text{cr}}^{\left(1\right)})$. Propagation simulation of the quadrupole mode at (c) $\alpha =0.175(<{\alpha }_{\text{cr}}^{\left(2\right)})$, and of the mode bifurcating from the tripole-quadrupole mode pair at (d) $\alpha =0.22(>{\alpha }_{\text{cr}}^{\left(2\right)})$; $p=2,d=5$.

Figure 7

(a) Real $\left[\text{Re}\right(V\left)\right]$ and imaginary $\left[\text{Im}\right(V\left)\right]$ parts of the waveguide profile, with $\text{Re}\left(V\right)=pexp(x^2/d^2)$ for all $x$, while $\text{Im}\left(V\right)=ix/\left|x\right|\alpha$ for $x\in (-w,w)$, and 0 otherwise. (b) Dependence of ${\alpha }_{\text{cr}}$ of three mode pairs on $w$. The inset is the enlargement of the dashed-box portion. The dependence of mode propagation constants on $\alpha$ is shown for (c) $w=2.2$ and (d) $w=7.4$. In all the cases, $p=2,d=5$.