Conservation relations and anisotropic transmission resonances in one-dimensional<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric photonic heterostructures

Abstract

We analyze the optical properties of one-dimensional $\mathcal{PT}$-symmetric structures of arbitrary complexity. These structures violate normal unitarity (photon flux conservation) but are shown to satisfy generalized unitarity relations, which relate the elements of the scattering matrix and lead to a conservation relation in terms of the transmittance and (left and right) reflectances. One implication of this relation is that there exist anisotropic transmission resonances in $\mathcal{PT}$-symmetric systems, frequencies at which there is unit transmission and zero reflection, but only for waves incident from a single side. The spatial profile of these transmission resonances is symmetric, and they can occur even at $\mathcal{PT}$-symmetry-breaking points. The general conservation relations can be utilized as an experimental signature of the presence of $\mathcal{PT}$ symmetry and of $\mathcal{PT}$-symmetry-breaking transitions. The uniqueness of $\mathcal{PT}$-symmetry-breaking transitions of the scattering matrix is briefly discussed by comparing to the corresponding non-Hermitian Hamiltonians.