Scattering in PT - and RT -symmetric multimode waveguides: Generalized conservation laws and spontaneous symmetry breaking beyond one dimension

We extend the generalized conservation law of light propagating in a one-dimensional PT -symmetric system, i.e., |T -1 | =√{R_LR_R } for the transmittance T and the reflectance R_{L ,R} from the left and right, to a multimode waveguide with either PT or 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 how a multimode 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 a scattering experiment in each waveguide channel at a single frequency and fixed strength of gain and loss.