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We extend the generalize conservation law of light propagating in aone-dimensional $\cal PT$-symmetric system, i.e., $|T-1|=\sqrt{R_LR_R}$ for thetransmittance $T$ and the reflectance $R_{L,R}$ from the left and right, to amultimode waveguide with either $\cal PT$ or $\cal RT$ symmetry, in whichhigher dimensional investigations are necessary. These conservation laws existnot only in a matrix form for the transmission and reflection matrices; theyalso exist in a scalar form for real-valued quantities by defining generalizedtransmittance and reflectance. We then discuss, for the first time, how amultimode $\cal PT$-symmetric waveguide can be used to observe spontaneoussymmetry breaking of the scattering matrix, which typically requires tuning thenon-hermiticity of the system (i.e. the strength of gain and loss). Here theadvantage of using a multimode waveguide is the elimination of tuning anysystem parameters: the transverse mode order $m$ plays the role of the symmetrybreaking parameter, and one observes the symmetry breaking by simply performingscattering experiment in each waveguide channel at a single frequency and fixedstrength of gain and loss.

physical review. a, atomic, molecular, and optical physics/physical review, a, atomic, molecular, and optical physics

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