Physics of Spectral Singularities

TL;DR

Spectral singularities are special points in complex scattering potentials that have significant implications in optics and laser physics, linking mathematical properties to physical phenomena like laser thresholds.

Contribution

This paper provides a comprehensive review of spectral singularities, clarifies their physical meaning, and explores their applications in optics and laser physics, including new insights into laser threshold conditions.

Findings

Spectral singularities correspond to laser threshold conditions.

Optical realizations of spectral singularities demonstrate their physical relevance.

Nonlinear spectral singularities relate to laser output intensity dependence.

Abstract

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 ${\cal PT}$-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.