Spectral Singularity in confined PT symmetric optical potential

TL;DR

This paper analytically investigates scattering in a confined PT symmetric optical potential, revealing normal and anomalous regimes, spectral singularities, and unidirectional invisibility depending on parameters, with implications for optical device design.

Contribution

It provides new analytical results on scattering behavior, spectral singularities, and unidirectional invisibility in a confined PT symmetric optical potential, extending understanding of non-Hermitian optics.

Findings

Normal scattering for V_0 ≤ 0.5 with Hermitian mapping

Infinite spectral singularities for V_0 > 0.5

Unidirectional invisibility at V_0=0.5 for L=2nπ

Abstract

We present an analytical study for the scattering amplitudes (Reflection $|R|$ and Transmission $|T|$), of the periodic ${\cal{PT}}$ symmetric optical potential $ V(x) = \displaystyle W_0 \left( \cos ^2 x + i V_0 \sin 2x \right) $ confined within the region $0 \leq x \leq L$, embedded in a homogeneous medium having uniform potential $W_0$. The confining length $L$ is considered to be some integral multiple of the period $ \pi $. We give some new and interesting results. Scattering is observed to be normal ($|T| ^2 \leq 1, \ |R|^2 \leq 1$) for $V_0 \leq 0.5 $, when the above potential can be mapped to a Hermitian potential by a similarity transformation. Beyond this point ($ V_0 > 0.5 $) scattering is found to be anomalous ($|T| ^2, \ |R|^2 $ not necessarily $ \leq 1 $). Additionally, in this parameter regime of $V_0$, one observes infinite number of spectral singularities $E_{SS}$ at different values of $V_0$. Furthermore, for $L= 2 n \pi$, the transition point $V_0 = 0.5$ shows unidirectional invisibility with zero reflection when the beam is incident from the absorptive side ($Im [V(x)] < 0$) but finite reflection when the beam is incident from the emissive side ($Im [V(x)] > 0$), transmission being identically unity in both cases. Finally, the scattering coefficients $|R|^2$ and $|T|^2 $ always obey the generalized unitarity relation : $ ||T|^2 - 1| = \sqrt{|R_R|^2 |R_L|^2}$, where subscripts $R$ and $L$ stand for right and left incidence respectively.