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arXiv:1303.4874 · Physical Review A

Nonlinear spectral singularities of a complex barrier potential and the lasing threshold condition

Ali Mostafazadeh

Published Jun 24, 2013• 38 citations
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Abstract

A spectral singularity is a mathematical notion with an intriguing physical realization in terms of certain zero-width resonances. In optics it manifests as lasing at the threshold gain. We explore the recently developed nonlinear generalization of spectral singularities for a complex barrier potential and study the associated resonance effect for an infinite planar slab of gain medium. In particular, for a Kerr nonlinearity, we show that the first-order perturbative equation that determines the nonlinear spectral singularities provides an explicit expression for the intensity of the emitted waves from the slab. This is a mathematical derivation of the known linear relationship between the output intensity of the slab and its gain coefficient. We also discuss the implications of our results for the time-reversed system which acts as a coherent perfect absorber.

DOI
10.1103/physreva.87.063838

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A spectral singularity is a mathematical notion with an intriguing physical realization in terms of certain zero-width resonances. In optics it manifests as lasing at the threshold gain. We explore the recently developed nonlinear generalization of spectral singularities for a complex barrier potential and study the associated resonance effect for an infinite planar slab of gain medium. In particular, for a Kerr nonlinearity, we show that the first-order perturbative equation that determines the nonlinear spectral singularities provides an explicit expression for the intensity of the emitted waves from the slab. This is a mathematical derivation of the known linear relationship between the output intensity of the slab and its gain coefficient. We also discuss the implications of our results for the time-reversed system which acts as a coherent perfect absorber.
We develop a perturbative method of computing spectral singularities of a Schr\"odinger operator defined by a general complex potential that vanishes outside a closed interval. These can be realized as zero-width resonances in optical gain media and correspond to a lasing effect that occurs at the threshold gain. Their time-reversed copies yield coherent perfect absorption of light that is also known as antilasing. We use our general results to establish the exactness of the nnnth-order perturbation theory for an arbitrary complex potential consisting of nnn delta functions, obtain an exact expression for the transfer matrix of these potentials, and examine spectral singularities of complex barrier potentials of arbitrary shape. In the context of optical spectral singularities, these correspond to inhomogeneous gain media.
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 PT-symmetric barrier potential and demonstrate the above resonance phenomenon for a certain electromagnetic waveguide.
We propose a scheme for a coherent perfect absorber (CPA) and a laser in an atomic medium with gain and loss, obeying simultaneously a spectral singularity and a time-reversed spectral singularity, both occurring at different wavevectors (or frequencies). We term this system a CPA- and -laser and investigate its features allowing to obtain asymmetric lasing and absorbing properties, switches, etc. We show that the CPA-and-laser can be obtained by modifying characteristics of a CPA-laser of an initial -symmetric configuration, provided there are at least three tunable parameters. The physical mechanism of emergence of a CPA-and-laser in this way, is based on splitting of a self-dual spectral singularity of the -symmetric CPA-laser into the spectral singularity and time-reversed spectral singularity. After the discussion of a particular example of a bilayer consisting of one active and one passive slabs, we suggest a realistic physical system for implementing a CPA-and-laser. It consists of two adjacent atomic cells filled with isotopes of Λ-type three-level rubidium atoms interacting with probe and control fields, allowing for the required number of tunable parameters.
Recent works [Chong et al., Phys. Rev. Lett. 105, 053901 (2010); Wan et al., Science 331, 889 (2011)] have shown that the time-reversed process of lasing at threshold realizes a coherent perfect absorber (CPA). In a CPA, a lossy medium in an optical cavity with a specific degree of dissipation, equal in modulus to the gain of the lasing medium, can perfectly absorb coherent optical waves at discrete frequencies that are the time-reversed counterpart of the lasing modes. Here the concepts of time reversal of lasing and CPA are extended for optical radiation emitted by a laser operated in an arbitrary regime, i.e., for transient, chaotic, or periodic coherent optical fields. We prove that any electromagnetic signal E(t)E(t)E(t) generated by a laser system SSS operated in an arbitrary regime can be perfectly absorbed by a CPA device ${S}^{\ensuremath{'}}$ which is simply realized by placing inside SSS a broadband linear absorber (attenuator) of appropriate transmittance. As examples, we discuss CPA devices that perfectly absorb a chaotic laser signal and a frequency-modulated optical wave.
We investigate a two-parametric family of one-dimensional non-Hermitian complex potentials with parity-time (PT\mathcal{PT}PT) symmetry. We find that there exist two distinct types of phase transitions, from an unbroken phase (characterized by a real spectrum) to a broken phase (where the spectrum becomes complex). The first type involves the emergence of a pair of complex eigenvalues bifurcating from the continuous spectrum. The second type is associated with the collision of such pairs at the bottom of the continuous spectrum. The first transition type is closely related to spectral singularities (SSs), at which point the transmission and reflection coefficients are divergent simultaneously. The second is associated with the emergence of bound states. In particular, under specific parameter conditions, we construct an exact bound state solution. By systematically exploring the parameter space, we establish a universal relationship governing the number of SSs in these potentials. These findings provide a fundamental theoretical framework for manipulating wave scattering in non-Hermitian systems, offering promising implications for designing advanced optical and quantum devices.
Existence of a spectral singularity (SS) in the spectrum of a Schr\"odinger operator with a non-Hermitian potential requires exact matching of parameters of the potential. We provide a necessary and sufficient condition for a potential to have a SS at a given wavelength. It is shown that potentials with SSs at prescribed wavelengths can be obtained by a simple and effective procedure. In particular, the developed approach allows one to obtain potentials with several SSs and with SSs of the second order, as well as potentials obeying a given symmetry, say, PT\mathcal{PT}PT-symmetric potentials. We illustrate all these opportunities with examples. We also describe splitting of a second-order SS under change of the potential parameters, and discuss possibilities of experimental observation of SSs of different orders.