Balanced gain-and-loss optical waveguides: Exact solutions for guided modes in susy-QM

The construction of exactly solvable refractive indices allowing guided TE modes in optical waveguides is investigated within the formalism of Darboux–Crum transformations. We apply the finite-difference algorithm for higher-order supersymmetric quantum mechanics to obtain complex-valued refractive indices admitting all-real eigenvalues in their point spectrum. The new refractive indices are such that their imaginary part gives zero if it is integrated over the entire domain of definition. This property, called condition of zero total area, ensures the conservation of optical power so the refractive index shows balanced gain and loss. Consequently, the complex-valued refractive indices reported in this work include but are not limited to the parity-time invariant case.