New classes of non-parity-time-symmetric optical potentials with all-real spectra and exceptional-point-free phase transition

TL;DR

This paper introduces new classes of non-parity-time-symmetric optical potentials with all-real spectra and phase transitions that occur without exceptional points, expanding understanding of complex potentials in optical waveguides.

Contribution

The authors construct novel non-PT-symmetric complex potentials with conjugate eigenvalue symmetry, demonstrating all-real spectra and unique phase transition behaviors without exceptional points.

Findings

Spectra of these potentials are often all-real due to eigenvalue symmetry.

Phase transitions can occur via bifurcation from interior continuous eigenvalues.

Unidirectional propagation behaviors are demonstrated in diffraction patterns.

Abstract

Paraxial linear propagation of light in an optical waveguide with material gain and loss is governed by a Schr\"odinger equation with a complex potential. Properties of parity-time-symmetric complex potentials have been heavily studied before. In this article, new classes of non-parity-time-symmetric complex potentials featuring conjugate-pair eigenvalue symmetry in its spectrum are constructed by operator symmetry methods. Due to this eigenvalue symmetry, it is shown that the spectrum of these complex potentials is often all-real. Under parameter tuning in these potentials, phase transition can also occur, where pairs of complex eigenvalues appear in the spectrum. A peculiar feature of the phase transition here is that, the complex eigenvalues may bifurcate out from an interior continuous eigenvalue inside the continuous spectrum, in which case a phase transition takes place without going through an exceptional point. In one spatial dimension, this class of non-parity-time-symmetric complex potentials is of the form $V(x)=h'(x)-h^2(x)$, where $h(x)$ is an arbitrary parity-time-symmetric complex function. These potentials in two spatial dimensions are also derived. Diffraction patterns in these complex potentials are further examined, and unidirectional propagation behaviors are demonstrated.