Optical-reciprocity-induced symmetry in photonic heterostructures and its manifestation in scattering $\mathcal{PT}$-symmetry breaking

The scattering matrix $S$ obeys the symmetry property $\mathcal{PT}S\mathcal{PT}=S^{-1}$ in a parity-time $\left(\mathcal{PT}\right)$ symmetric system and the unitary relation $S^{†}S=1$ in the absence of gain and loss. Here we report a different symmetry relation of $S$ in a one-dimensional heterostructure, which is given by the amplitude ratio of the incident waves in the scattering eigenstates. It originates from the optical reciprocity and holds independently of the presence of gain and loss in the system. Guided by this symmetry relation, we probe the remnant of the spontaneous symmetry breaking of a $\mathcal{PT}$-symmetric $S$ matrix, when the system does not have exact $\mathcal{PT}$ symmetry due to unbalanced gain and loss and even in the absence of gain. We show that the additional symmetry relation provides clear evidence of a quasitransition, even when all previously found signatures of the $\mathcal{PT}$-symmetry breaking of $S$ are completely erased.

Figure 1

Exceptional points in a two-layer heterostructure (black dots) with weak unbalanced gain and loss. Loss is fixed at $\text{Im}\left[n_1\right]=0.05$ in the left half and the gain is reduced from $\text{Im}\left[n_2\right]=-0.05$ to $-0.04$ in the gain half. $\text{Re}\left[n\right]=3$ in the system and its length is $L=23\mu m$. A false color plot of the product $\text{Re}[\mathcal{G}-i]\text{Im}[\mathcal{G}-i]$ is also shown. Nearly vertical and wavy diagonal lines show the zeros of $\text{Re}[\mathcal{G}-i]$ and $\text{Im}[\mathcal{G}-i]$, respectively. Their intersections show the locations of the exceptional points. $\mathcal{G}=-i$ does not hold in this region. Inset: Schematic of scattering from a two-layer heterostructure.

Figure 2

Contrast of the scattering behaviors when gain and loss are balanced [(a), (c), (e): $\text{Im}\left[n_2\right]=-\text{Im}\left[n_1\right]=-0.05]$ or weakly unbalanced [(b), (d), (f): $\text{Im}\left[n_1\right]=0.05$ and $\text{Im}\left[n_2\right]=-0.04]$. $\text{Re}\left[n\right]=3$ and $L=23\mu m$ as in Fig. 1. Shadowed areas in (a), (c), and (e) indicate the broken symmetry phase. The dashed line in (f) marks the wavelength of the closest exceptional point at $\{\text{Im}\left[n_2\right]=-0.0401,\lambda =1326\text{nm}\}$ shown in Fig. 1. Note that we have chosen $\text{Im}\left[n_2\right]$ to be slightly different from the value of this EP intentionally, since the exact value of the latter is unknown in an actual setup before we conduct the experiment.

Figure 3

Amplitude ratios $\nu$ in the scattering eigenstates of the $S$ matrix with strongly unbalanced gain and loss (a) and loss only (c). $\text{Im}\left[n_2\right]=-0.0168$ in (a) and 0.04 in (c), and $\text{Im}\left[n_1\right]=0.05$ is fixed. Dashed lines in (a) and (c) mark the wavelength of the closest exceptional point at $\{\text{Im}\left[n_2\right]=-0.0170,\lambda =1440\text{nm}\}$ and $\{\text{Im}\left[n_2\right]=0.038,\lambda =1440\text{nm}\}$, respectively [see (b) and (d)]. The system length is chosen to be $L=36\mu \mathrm{m}$ and the other parameters are the same as in Figs. 1. (b) and (d) Similar to Fig. 1 for (a) and (c), showing the EPs near the quasitransition of $|{\nu }_{\pm }|$. In (d) $\mathcal{G}-i$ is replaced by $\mathcal{G}+i$.

Figure 4

Amplitude ratios $\nu$ in the scattering eigenstates of the $S$ matrix in a four-layer heterostructure. $\text{Re}\left[n\right]=3$ and $L=72\mu \mathrm{m}$, and the four layers have equal length. (a) $\text{Im}\left[n\right]=0.05,0.01,-0.01,-0.05$ for a $\mathcal{PT}$-symmetric heterostructure [37]. (b) $\text{Im}\left[n\right]=0.05,0.01,0.005,0.025$ for a loss-only waveguide.