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 ${\cal{PT}}$ 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.
 Publication:

Journal of Mathematical Physics
 Pub Date:
 November 2013
 DOI:
 10.1063/1.4829675
 arXiv:
 arXiv:1307.1844
 Bibcode:
 2013JMP....54k2106S
 Keywords:

 03.65.Nk;
 02.30.Uu;
 03.65.Ge;
 Scattering theory;
 Integral transforms;
 Solutions of wave equations: bound states;
 Quantum Physics;
 Mathematical Physics;
 Physics  Optics
 EPrint:
 13 pages, 7 figures