Abstract

The beam-propagation method (BPM) is an accurate numerical procedure for describing the propagation of an arbitrary incoming optical field through a given optical system. The method can be based on either the full scalar-wave equation (Helmholtz equation) for the electric field, or on the Fresnel equation that is valid for paraxial field propagation. In this paper we adopt the more general, Helmholtz equation formalism. Although the numerical solution generated by BPM emphasizes the beam properties of the field, it contains implicitly all the information necessary for a complete description of the field in terms of modes. The Fourier transform of the field with respect to axial distance z at a single transverse point (x,y) yields the totality of all mode eigenvalues β_n, including those that correspond to leaky or decaying modes. It is also possible to determine from a propagating-beam solution the weights of the modes. We have done it with the aid of a correlation function formed by multiplying the conjugate of the complex field amplitude at z = 0, and the complex field amplitude for arbitrary z and integrating over the waveguide cross section. The numerical Fourier transform with respect to z of the complex correlation function, applied in conjunction with digital-filtering techniques, has given us values for both the mode weights and eigenvalues. Furthermore, once the eigenvalues have been determined, the mode eigenfunctions could be computed by evaluating the Fourier transform of the field for β = β_n.

© 1988 Optical Society of America