Simple algebraic recursion formulas for calculation of propagation characteristics of graded-index waveguides

Using the perturbation method with the aid of the hypervirial theorem and Hellmann–Feynman theorem in quantum mechanics, simple algebraic recursion formulas for calculating propagation constants of graded-index planar optical waveguides in successive orders of approximation are derived. Algebraic formulas for calculation of the modal field functions are also presented. Numerical results for typical graded-index waveguides (Gaussian, exponential, complementary error function and symmetric Epstein profiles) demonstrate that the present method can be applied to an accurate calculation of the propagation characteristics to an arbitrary order of perturbation approximation.