Refractive index profiles for a $\mathcal{P}\mathcal{T}$-symmetric optical structure

By mapping the scalar Helmholtz equation (SHE) to the Schrodinger form we investigate the behaviour of $\mathcal{P}\mathcal{T}$ optical structure when the refractive index distribution $n$ admits variation in the longitudinal direction only. Interpreting the Schrodinger equation in terms of a superpotential we determine the supersymmetric partners for $n$. We also obtain new analytical solutions for the refractive index profiles and provide graphical illustrations for them.