Absence of Exceptional Points in Square Waveguide Arrays with Apparently Balanced Gain and Loss.

Zhenzhen Liu,Qiang Zhang,Jun-Jun Xiao,Xiangli Liu,Y Yao

doi:10.1038/srep22711

Zhenzhen Liu, Qiang Zhang + Show 3 more

Open Access

https://doi.org/10.1038/srep22711

Copy DOI

Abstract

The concept of parity-time (PT) symmetry in the field of optics has been intensively explored. This study shows the absence of exceptional points in a three-dimensional system composed of a square waveguide array with diagonally-balanced gain/loss distribution. More specifically, we show that an array of four coupled waveguides supports eight fundamental propagation supermodes, four of which are singlet, and the other two pairs are double degenerated. It is found that the singlet states follow the routine PT phase transition; however, the double-degenerated modes never coalesce as the gain/loss-to-coupling strength level varies, showing no actual PT symmetry-derived behavior. This is evident in the phase rigidity which does not approach zero. The absence of exceptional points is ascribed to the coupling of non-symmetric supermodes formed in the diagonal waveguide pairs. Our results suggest comprehensive interplay between the mode pattern symmetry, the lattice symmetry, and the PT-symmetry, which should be carefully considered in PT-phenomena design in waveguide arrays.

Highlights

The equivalence of the Schrödinger equation in quantum mechanics with the paraxial wave equation of the approximated Maxwell’s equations[11], leads to the application of the concept of PT symmetry toward many optical systems
Combining the spatial coupled mode theory (SCMT)[21,25,26] and finite element method (FEM)[27], we analytically and numerically depict the process and the underlying mechanism, for the absence of the EP that emerges for four particular bands
For the configuration that gain is set in cylinder A and equal amount of loss is assumed in cylinder B, the system can be mathematically represented by the SCMT: i k0 d dz

Summary

Introduction

The equivalence of the Schrödinger equation in quantum mechanics with the paraxial wave equation of the approximated Maxwell’s equations[11], leads to the application of the concept of PT symmetry toward many optical systems. More significantly important is that the realization of PT is generally associated to the coupling and hybridized modes, abbreviated as “H-mode” in nanophotonic systems. It is these H-modes whose modal index should meet the requirement nH1(x) = nH⁎2 (−x), where H1 and H2 represent the elementary modes that couple to each other and form a supermode over the whole system, that really matters in terms of PT-symmetry breaking. There are many studies about 2D and quasi-2D waveguide PT-symmetric systems[22,23] In these structures, the modal effective index is determined by the material index where the guiding mode is concentrated and highly localized. Combining the spatial coupled mode theory (SCMT)[21,25,26] and finite element method (FEM)[27], we analytically and numerically depict the process and the underlying mechanism, for the absence of the EP that emerges for four particular bands

Methods

Results

Conclusion