Spectral Singularity in confined PT symmetric optical potential

Abstract

We present an analytical study for the scattering amplitudes (Reflection $|R|$ and Transmission $|T|$), of the periodic $\mathcal{P}\mathcal{T}$ symmetric optical potential $V(x)=W_0\left({\cos }^{2}x+i{V}_0\sin 2x\right)$ confined within the region $0\le x\le L$, 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 ($|T{|}^2\le 1,|R{|}^2\le 1$) for $V_0\le 0.5$, 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 $\le 1$). 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 $L=2n\pi$, 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.

13 pages, 7 figures

References in corpus (5)

• PT-symmetry breaking and laser-absorber modes in optical scattering systems
• Unidirectional Nonlinear PT-symmetric Optical Structures
• Exponentially Fragile PT-Symmetry in Lattices with Localized Eigenmodes
• A note on the PT-invariant periodic potential V(x)=4 cos^2 x + 4 i V_0 sin 2x
• Spectral singularities in PT-symmetric periodic finite-gap systems

Cited by in corpus (7)

• Generalized Unitarity and Reciprocity Relations for PT-symmetric Scattering Potentials
• PT-symmetric spectral singularity and negative frequency resonance
• Perturbative Unidirectional Invisibility
• Interlace properties for the real and imaginary parts of the wave functions of complex-valued potentials with real spectrum
• Dynamical Theory of Scattering, Exact Unidirectional Invisibility, and Truncated $\mathfrak{z}{e}^{-2i{k}_{0}x}$ potential
• Adiabatic Series Expansion and Higher-Order Semiclassical Approximations in Scattering Theory
• Spectral singularities and threshold gain of a slab laser under illumination of a focused Gaussian beam