Scattering in $\cal PT$ and $\cal RT$ Symmetric Multimode Waveguides: Generalized Conservation Laws and Spontaneous Symmetry Breaking beyond One Dimension

TL;DR

This paper extends conservation laws for light scattering in one-dimensional $\ ext{PT}$-symmetric systems to multimode waveguides, revealing spontaneous symmetry breaking observable through mode analysis without parameter tuning.

Contribution

It introduces generalized conservation laws for multimode waveguides with $\ ext{PT}$ or $\ ext{RT}$ symmetry and demonstrates spontaneous symmetry breaking detection via mode analysis at fixed parameters.

Findings

Conservation laws extend to multimode systems in matrix and scalar forms.

Spontaneous symmetry breaking can be observed by mode analysis without tuning parameters.

Transverse mode order acts as a symmetry breaking parameter.

Abstract

We extend the generalize conservation law of light propagating in a one-dimensional $\cal PT$-symmetric system, i.e., $|T-1|=\sqrt{R_LR_R}$ for the transmittance $T$ and the reflectance $R_{L,R}$ from the left and right, to a multimode waveguide with either $\cal PT$ or $\cal RT$ symmetry, in which higher dimensional investigations are necessary. These conservation laws exist not only in a matrix form for the transmission and reflection matrices; they also exist in a scalar form for real-valued quantities by defining generalized transmittance and reflectance. We then discuss, for the first time, how a multimode $\cal PT$-symmetric waveguide can be used to observe spontaneous symmetry breaking of the scattering matrix, which typically requires tuning the non-hermiticity of the system (i.e. the strength of gain and loss). Here the advantage of using a multimode waveguide is the elimination of tuning any system parameters: the transverse mode order $m$ plays the role of the symmetry breaking parameter, and one observes the symmetry breaking by simply performing scattering experiment in each waveguide channel at a single frequency and fixed strength of gain and loss.