Spectral Singularities of Complex Scattering Potentials and Infinite Reflection and Transmission Coefficients at Real Energies

Spectral singularities are spectral points that spoil the completeness of the eigenfunctions of certain non-Hermitian Hamiltonian operators. We identify spectral singularities of complex scattering potentials with the real energies at which the reflection and transmission coefficients tend to infinity, i.e., they correspond to resonances having a zero width. We show that a waveguide modeled using such a potential operates like a resonator at the frequencies of spectral singularities. As a concrete example, we explore the spectral singularities of an imaginary $\mathcal{P}\mathcal{T}$-symmetric barrier potential and demonstrate the above resonance phenomenon for a certain electromagnetic waveguide.

Figure 1

Cross section of a waveguide with gain ($+$) and loss ($-$) regions in the $x\mathrm{\text{−}}z$ plane. Arrows labeled by $I$, $R$, and $T$ represent the incident, reflected, and transmitted waves.

Figure 2

Graphs of ${log}_{10}(|T^{r,l}{|}^2)$ (solid blue curves), ${log}_{10}(|R^l{|}^2)$ (dotted red curves), and ${log}_{10}(|R^r{|}^2)$ (dashed green curves) as a function of $\omega /{\omega }_{0,1}$, for $m=1$, $n=0$, $\hslash {\omega }_{0,1}=5\mathrm{eV}$, $\hslash {\omega }_p=0.2\mathrm{eV}$, $\hslash \delta =1.25\mathrm{eV}$. For the figure on the left (right) $\alpha =1004.17\mathrm{nm}$, $\beta =62.0464\mathrm{nm}$ ($\alpha =1004\mathrm{nm}$, $\beta =62\mathrm{nm}$).