## Physics of Spectral Singularities

Spectral singularities are certain points of the continuous spectrum of generic complex scattering potentials. We review the recent developments leading to the discovery of their physical meaning, consequences, and generalizations. In particular, we give a simple definition of spectral singularities, provide a general introduction to spectral consequences of PT-symmetry (clarifying some of the controversies surrounding this subject), outline the main ideas and constructions used in the pseudo-Hermitian representation of quantum mechanics, and discuss how spectral singularities entered in the physics literature as obstructions to these constructions. We then review the transfer matrix formulation of scattering theory and the application of complex scattering potentials in optics. These allow us to elucidate the physical content of spectral singularities and describe their optical realizations. Finally, we survey some of the most important results obtained in the subject, drawing special attention to the remarkable fact that the condition of the existence of linear and nonlinear optical spectral singularities yield simple mathematical derivations of some of the basic results of laser physics, namely the laser threshold condition and the linear dependence of the laser output intensity on the gain coefficient.

http://arxiv.org/abs/1412.0454
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)

## Unidirectionally Invisible Potentials as Local Building Blocks of all Scattering Potentials

We give a complete solution of the problem of constructing a scattering potential v(x) that possesses scattering properties of one’s choice at an arbitrary prescribed wavenumber. Our solution involves expressing v(x) as the sum of at most six unidirectionally invisible finite-range potentials for which we give explicit formulas. Our results can be employed for designing optical potentials. We discuss its application in modeling threshold lasers, coherent perfect absorbers, and bidirectionally and unidirectionally reflectionless absorbers, amplifiers, and phase shifters.

http://arxiv.org/abs/1407.1760
Quantum Physics (quant-ph); Optics (physics.optics)

## Generalized Unitarity and Reciprocity Relations for PT-symmetric Scattering Potentials

We derive certain identities satisfied by the left/right-reflection and transmission amplitudes, $$R^{l/r}(k)$$ and $$T(k)$$, of general $${\cal PT}$$-symmetric scattering potentials. We use these identities to give a general proof of the relations, $$|T(-k)|=|T(k)|$$ and $$|R^r(-k)|=|R^l(k)|$$, conjectured in [Z. Ahmed, J. Phys. A 45 (2012) 032004], establish the generalized unitarity relation: $$R^{l/r}(k)R^{l/r}(-k)+|T(k)|^2=1$$, and show that it is a common property of both real and complex $${\cal PT}$$-symmetric potentials. The same holds for $$T(-k)=T(k)^*$$ and $$|R^r(-k)|=|R^l(k)|$$.

http://arxiv.org/abs/1405.4212
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)

## Spectral Singularities and CPA-Laser Action in a Weakly Nonlinear PT-Symmetric Bilayer Slab

We study optical spectral singularities of a weakly nonlinear PT-symmetric bilinear planar slab of optically active material. In particular, we derive the lasing threshold condition and calculate the laser output intensity. These reveal the following unexpected features of the system: 1. For the case that the real part of the refractive index η of the layers are equal to unity, the presence of the lossy layer decreases the threshold gain; 2. For the more commonly encountered situations when η−1 is much larger than the magnitude of the imaginary part of the refractive index, the threshold gain coefficient is a function of η that has a local minimum. The latter is in sharp contrast to the threshold gain coefficient of a homogeneous slab of gain material which is a decreasing function of η. We use these results to comment on the effect of nonlinearity on the prospects of using this system as a CPA-laser.

http://arxiv.org/abs/1404.1737
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)

## Adiabatic Approximation, Semiclassical Scattering, and Unidirectional Invisibility

The transfer matrix of a possibly complex and energy-dependent scattering potential can be identified with the S-matrix of a two-level time-dependent non-Hermitian Hamiltonian H(t). We show that the application of the adiabatic approximation to H(t) corresponds to the semiclassical description of the original scattering problem. In particular, the geometric part of the phase of the evolving eigenvectors of H(t) gives the pre-exponential factor of the WKB wave functions. We use these observations to give an explicit semiclassical expression for the transfer matrix. This allows for a detailed study of the semiclassical unidirectional reflectionlessness and invisibility. We examine concrete realizations of the latter in the realm of optics.

http://arxiv.org/abs/1401.4315
Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)

## A Dynamical Formulation of One-Dimensional Scattering Theory and Its Applications in Optics

We develop a dynamical formulation of one-dimensional scattering theory where the reflection and transmission amplitudes for a general, possibly complex and energy-dependent, scattering potential are given as solutions of a set of dynamical equations. By decoupling and partially integrating these equations, we reduce the scattering problem to a second order linear differential equation with universal initial conditions that is equivalent to an initial-value time-independent Schrodinger equation. We give explicit formulas for the reflection and transmission amplitudes in terms of the solution of either of these equations and employ them to outline an inverse-scattering method for constructing finite-range potentials with desirable scattering properties at any prescribed wavelength. In particular, we construct optical potentials displaying threshold lasing, anti-lasing, and unidirectional invisibility.

http://arxiv.org/abs/1310.0592
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Optics (physics.optics)

## Spectral Singularities Do Not Correspond to Bound States in the Continuum

We show that, contrary to a claim made in arXiv:1011.0645, the von Neumann-Winger bound states that lie in the continuum of the scattering states are fundamentally different from Naimark’s spectral singularities.

http://arxiv.org/abs/1207.2278

Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Quantum Physics (quant-ph)

## Invisibility and PT-symmetry

For a general complex scattering potential defined on a real line, we show that the equations governing invisibility of the potential are invariant under the combined action of parity and time-reversal (PT) transformation. We determine the PT-symmetric an well as non-PT-symmetric invisible configurations of an easily realizable exactly solvable model that consists of a two-layer planar slab consisting of optically active material. Our analysis shows that although PT-symmetry is neither necessary nor sufficient for the invisibility of a scattering potential, it plays an important role in the characterization of the invisible configurations. A byproduct of our investigation is the discovery of certain configurations of our model that are effectively reflectionless in a spectral range as wide as several hundred nanometers.

http://arxiv.org/abs/1206.0116
Mathematical Physics (math-ph); Optics (physics.optics); Quantum Physics (quant-ph)

## Self-dual Spectral Singularities and Coherent Perfect Absorbing Lasers without PT-symmetry

A PT-symmetric optically active medium that lases at the threshold gain also acts as a complete perfect absorber at the laser wavelength. This is because spectral singularities of PT-symmetric complex potentials are always accompanied by their time-reversal dual. We investigate the significance of PT-symmetry for the appearance of these self-dual spectral singularities. In particular, using a realistic optical system we show that self-dual spectral singularities can emerge also for non-PT-symmetric configurations. This signifies the existence of non-PT-symmetric CPA-lasers.

http://arxiv.org/abs/1205.4560
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th); Mathematical Physics (math-ph); Optics (physics.optics)