Rayleigh quotient solution of semivectorial field problems for optical waveguides with arbitrary index profiles

A new Rayleigh quotient, which was introduced and briefly considered in a previous paper, ibid., vol.135, p.56, 1988, is now examined in greater detail. When used in conjunction with polarised finite difference schemes established in earlier papers, it enables semivectorial E-field and H-field propagation modes of waveguide structures, which may contain any arbitrary distribution of horizontal and vertical dielectric discontinuities, to be determined without the need to solve matrix eigenvalue problems. The absence of a matrix enables the finite difference grid to incorporate substantially more points and smaller step lengths, thereby improving the accuracy of computed solutions. Results are presented for single rib waveguides and for rib waveguide directional couplers fabricated from III-V semiconductors.