General coupled-mode analysis of a geometrically symmetric waveguide array with nonuniform gain and loss

The exceptional point(EP)is one of the typical properties of parity–time-symmetric systems,arising from modes coupling with identical resonant frequencies or propagation constants in optics.Here we show that in addition to two different modes coupling,a nonuniform distribution of gain and loss leads to an offset from the original propagation constants,including both real and imaginary parts,resulting in the absence of EP.These behaviors are examined by the general coupled-mode theory from the first principle of the Maxwell equations,which yields results that are more accurate than those from the classical coupled-mode theory.Numerical verification via the finite element method is provided.In the end,we present an approach to achieve lossless propagation in a geometrically symmetric waveguide array.