Perturbative Unidirectional Invisibility

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 $n(x)=n_0+f(x)$, where $n_0$ is real, $f(x)$ is complex-valued, and $|f(x)|\ll 1\le n_0$. 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 unidirectional invisibility for $n_0>1$ open up the way for the experimental studies of this phenomenon in a variety of active material. As an application of our general results, we show that a medium with $n(x)=n_0+\zeta e^{iKx}$, $\zeta$ and $K$ real, and $|\zeta |\ll 1$ can support unidirectional invisibility only for $n_0=1$. We then construct unidirectionally invisible index profiles of the form $n(x)=n_0+{\sum }_{\ell }{z}_{\ell }{e}^{i{K}_{\ell }x}$, with $z_{\ell }$ complex, $K_{\ell }$ real, $|z_{\ell }|\ll 1$, and $n_0>1$.

15 pages, 3 figures

References in corpus (6)

• Invisibility and PT-symmetry
• The transfer matrix: a geometrical perspective
• Spectral singularities in PT-symmetric periodic finite-gap systems
• Lasing Threshold Condition for Oblique TE and TM Modes, Spectral Singularities, and Coherent Perfect Absorption
• Spectral Singularities and CPA-Laser Action in a Weakly Nonlinear PT-Symmetric Bilayer Slab
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Cited by in corpus (12)

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• Unidirectional reflectionlessness and invisibility in the TE and TM modes of a PT-symmetric slab system
• Transfer Matrix Formulation of Scattering Theory in Two and Three Dimensions
• Unidirectional Invisibility and PT-Symmetry with Graphene
• Perfect Broadband Invisibility in Isotropic Media with Gain and Loss
• Reflectionless and invisible potentials in photonic lattices
• Unidirectional Invisibility and Nonreciprocal Transmission in Two and Three Dimensions
• Non-Hermitian time-dependent perturbation theory: asymmetric transitions and transitionless interactions
• Broadband and Wide-Angle Invisibility with PT-Symmetric 2D-Weyl Semimetal
• Dynamical Theory of Scattering, Exact Unidirectional Invisibility, and Truncated $\mathfrak{z}{e}^{-2i{k}_{0}x}$ potential
• Low-Frequency Scattering of TE and TM Waves by an Inhomogeneous Medium with Planar Symmetry
• Refractive index profiles for a $\mathcal{P}\mathcal{T}$-symmetric optical structure