Optical Reciprocity Induced Symmetry of the Scattering Eigenstates in Non-$\mathcal{P}\mathcal{T}$-Symmetric Heterostructures

Abstract

The scattering matrix $S$ obeys the unitary relation $S^{†}S=1$ in a Hermitian system and the symmetry property $\mathcal{P}\mathcal{T}S\mathcal{P}\mathcal{T}=S^{-1}$ in a Parity-Time ($\mathcal{P}\mathcal{T}$) symmetric system. Here we report a different symmetry relation of the $S$ matrix 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 independent of the Hermiticity or $\mathcal{P}\mathcal{T}$ symmetry of the system. Using this symmetry relation, we probe a quasi-transition that is reminiscent of the spontaneous symmetry breaking of a $\mathcal{P}\mathcal{T}$-symmetric $S$ matrix, now with unbalanced gain and loss and even in the absence of gain. We show that the additional symmetry relation provides a clear evidence of an exceptional point, even when all other signatures of the $\mathcal{P}\mathcal{T}$ symmetry breaking are completely erased. We also discuss the existence of a final exceptional point in this correspondence, which is attributed to asymmetric reflections from the two sides of the heterostructure.

5 pages, 4 figures