Locally parity-time-symmetric and globally parity-symmetric systems

Abstract

We introduce a class of systems holding parity-time ($\mathcal{PT}$) symmetry locally, whereas being globally $\mathcal{P}$ symmetric. The potential, $U=U(|r|)$, fulfills $\mathcal{PT}$ symmetry with respect to periodically distributed points ${r}_{0}:U(|{r}_{0}+r|)={U}^{*}(|{r}_{0}\ensuremath{-}r|)$ being ${r}_{0}\ensuremath{\ne}0$. We show that such systems hold unusual properties arising from the merging of the two different symmetries, leading to a strong field localization and enhancement at the double-symmetry center, $r=0$, when the coupling of outward to inward propagating waves is favored. We explore such general potentials in one and two dimensions, which could have actual realizations combining gain-loss and index modulations in nanophotonic structures. In particular, we show how to render a broad aperture vertical-cavity surface-emitting laser into a bright and narrow beam source, as a direct application.