Year: 2015
Type: article
Abstract: We outline a general perturbative method of evaluating scattering features of finite-range complex potentials and use it to examine complex perturbations of a rectangular barrier potential. In optics, these correspond to modulated refractive index profiles of the form $\mathfrak{n}(x)={n}_{0}+f(x)$, where ${n}_{0}$ is real, $f(x)$ is complex valued, and $|f(x)|\ensuremath{\ll}1\ensuremath{\le}{n}_{0}$. We give a comprehensive description of the phenomenon of unidirectional invisibility for such media, proving five general theorems on its realization in $\mathcal{P}\mathcal{T}$-symmetric and non-$\mathcal{P}\mathcal{T}$-symmetric material. In particular, we establish the impossibility of unidirectional invisibility for $\mathcal{P}\mathcal{T}$-symmetric samples whose refractive index has a constant real part and show how a simple scaling transformation of a unidirectionally invisible $\mathcal{P}\mathcal{T}$-symmetric index profile with ${n}_{0}=1$ may be used to generate a hierarchy of unidirectionally invisible $\mathcal{P}\mathcal{T}$-symmetric index profiles with ${n}_{0}>1$. The results pertaining to unidirectional invisibility for ${n}_{0}>1$ open the way for the experimental studies of this phenomenon in a variety of active materials. As an application of our general results, we show that a medium with $\mathfrak{n}(x)={n}_{0}+\ensuremath{\zeta}{e}^{iKx}$, $\ensuremath{\zeta}$ and $K$ real, and $|\ensuremath{\zeta}|\ensuremath{\ll}1$ can support unidirectional invisibility only for ${n}_{0}=1$. We then construct unidirectionally invisible index profiles of the form $\mathfrak{n}(x)={n}_{0}+{\ensuremath{\sum}}_{\ensuremath{\ell}}{\mathfrak{z}}_{\ensuremath{\ell}}{e}^{i{K}_{\ensuremath{\ell}}x}$ with ${\mathfrak{z}}_{\ensuremath{\ell}}$ complex, ${K}_{\ensuremath{\ell}}$ real, $|{\mathfrak{z}}_{\ensuremath{\ell}}|\ensuremath{\ll}1$, and ${n}_{0}>1$.
Source: Physical Review A
Author: Ali Mostafazadeh
Institution: Koç University
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W1912410022