The mathematical notion of a spectral singularity admits a physical interpretation as a zero-width resonance. It finds an optical realization as a certain type of lasing effect that occurs at the threshold gain. We explore spectral singularities of a complex spherical barrier potential and study their realization as transverse spherical electromagnetic waves emitted by a gain medium with a spherical geometry. In particular, for a typical dye laser material, we obtain a lower bound on the size of the gain medium for the occurence of this kind of spectral singularities.An interesting by-product of the recent study of complex potentials supporting a real spectrum [1,2] is the discovery of the optical realizations [3,4,5,6] of the mathematical notion of a spectral singularity [7]. For a complex scattering potential defined on the real line, these are certain points of the continuous spectrum of the corresponding non-Hermitian Schrödinger operator at which the reflection and transmission coefficients diverge [3]. As a result, they correspond to scattering states that behave exactly like resonances. This observation has also found applications in condensed matter systems [8] and triggered further study of the subject [9].The study of the optical realizations and applications of spectral singularities has so far been confined to effectively one-dimensional models involving infinitely long waveguides [3,4] or infinite planar slab gain media [5,6]. In the present article we examine spectral singularities of a threedimensional complex spherical barrier potential that admits a physical realization involving an optical gain medium with a spherical geometry. *
We study spectral singularities and their application in determining the threshold gain coefficient g (E/M ) for oblique transverse electric/magnetic (TE/TM) modes of an infinite planar slab of homogenous optically active material. We show that g (E) is a monotonically decreasing function of the incidence angle θ (measured with respect to the normal direction to the slab), while g (M ) has a single maximum, θc, where it takes an extremely large value. We identify θc with the Brewster's angle and show that g (E) and g (M ) coincide for θ = 0 (normal incidence), tend to zero as θ → 90 • , and satisfy g (E) < g (M ) for 0 < θ < 90 • . We therefore conclude that lasing and coherent perfect absorption are always more difficult to achieve for the oblique TM waves and that they are virtually impossible for the TM waves with θ ≈ θc. We also give a detailed description of the behavior of the energy density and the Poynting vector for spectrally singular oblique TE and TM waves. This provides an explicit demonstration of the parity-invariance of these waves and shows that the energy density of a spectrally singular TM wave with θ > θc is smaller inside the gain region than outside it. The converse is true for the TM waves with θ < θc and all TE waves.
Complex scattering potentials can admit scattering states that behave exactly like a zero-width resonance. Their energy is what mathematicians call a spectral singularity. This phenomenon admits optical realizations in the form of lasing at the threshold gain, and its time-reversal is responsible for antilasing. We study spectral singularities and whispering gallery modes (WGMs) of a cylindrical gain medium. In particular, we introduce a new class of WGMs that support a spectral singularity and, as a result, have a divergent quality factor. These singular gallery modes (SGMs) are excited only if the system has a positive gain coefficient, but typically the required gain is extremely small. More importantly given any amount of gain, there are SGMs requiring smaller gain than this amount. This means that, in principle, the system lacks a lasing threshold. Furthermore, the abundance of these modes allows for configurations where a particular value of the gain coefficient yields an effective excitation of two distant SGMs. This induces lasing at two different wavelengths.
An optical spectral singularity is a zero-width resonance that corresponds to lasing at threshold gain. Its time-reversal causes coherent perfect absorption of light and forms the theoretical basis of antilasing. In this article, we explore optical spectral singularities of a two-layer spherical medium. In particular, we examine the cases that a gain medium is coated by a thin layer of high-refractive index glass and a spherical glass covered by a layer of gain material. In the former case, the coating reduces the minimum radius required for exciting spectral singularities and gives rise to the formation of clusters of spectral singularities separated by wide spectral gaps. In the latter case, the coating leads to a doubling of the number of spectral singularities.
Unidirectional invisibility of a PT -symmetric optical system is of great interest, but challenging as well since it is infeasible to fulfill it through wide optical frequency ranges in all angular directions. Accordingly we study reflectionless and invisible patterns in the TE and TM modes of an optical slab system consisting of adjacent or separated pair of balanced gain and loss layers with a gap. We provide a comprehensive study of one of the simplest experimentally accessible examples of a unidirectionally reflectionless and invisible PT -symmetric optical slab system. We obtain the physically optimal conditions for the realization of these phenomena. We derive analytic expressions, and show that only certain gain amounts restricted to take values between certain minimum and maximum values give rise to uni/bi-directionally invisible configurations. The size of gap decides the measure of reflectionlessness and invisibility parameters, especially on gain value and incident angle.
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