Analysis of rib waveguides by a fourth‐order accurate finite‐difference beam‐propagation method

Abstract

SUMMARYWhen a step index waveguide is analyzed by the finite-difference method, the treatment of the discontinuous refractive index profile is a problem. In this paper, while the boundary condition of the field is taken into account, the improved finite-difference equation in which the truncation error is reduced to the fourth order is combined with the alternate direction implicit solution method, formulating a three-dimensional beam propagation method with high computational efficiency. First, by means of the imaginary axis method, an eigenmode analysis of the rib waveguide is carried out so that the effectiveness and validity of the method are proven. Next, a beam propagation analysis is carried out for the rib waveguide taper. By taking into account the discontinuity location of the refractive index, the quantization error is found to be reduced.

Semivector Finite-Difference Formula for the Analysis of a Step-Index Waveguide With a Tilted Interface

“…Since the relations among , , and are given by and , can be written as (2) Note that the superscripts and , respectively, means the normal and tangential components. The coefficients, , , and are defined as follows: where (6) (7) (8) in which [8], where is the free-space wavenumber. and are appropriately chosen according to Table I. As expected, for , (2) reduces to the expression of the standard modified FD formula whose accuracy is second-order for [2], [8].…”

“…The coefficients, , , and are defined as follows: where (6) (7) (8) in which [8], where is the free-space wavenumber. and are appropriately chosen according to Table I. As expected, for , (2) reduces to the expression of the standard modified FD formula whose accuracy is second-order for [2], [8]. Note, however, that the accuracy of the modified FD formula becomes first-order for .…”

Fourth‐order accurate sub‐sampling for finite‐difference analysis of surface plasmon metallic waveguides

“…For example, the early work of Stern [5] and Vassallo [6] was extended by Yamauchi et al [7] to provide a FD scheme with O͑h 2 ͒ truncation error, regardless of the placement of material interfaces within the mesh. Subsequently, Chiou et al [8] [9,10]. However, these fourth order accurate FD formulas have yet to be used to investigate the characteristics of SPP waveguides.…”

“…First, in comparison with dielectric waveguides, SPP metallic waveguides are more sensitive to the accuracy of the FD scheme due to the substantial difference in the refractive indices of the metal and dielectric materials. To accurately model the discontinuous fields of such step index profiles, researchers have proposed various modified FD formulations [5][6][7][8][9][10]. For example, the early work of Stern [5] and Vassallo [6] was extended by Yamauchi et al [7] to provide a FD scheme with O͑h 2 ͒ truncation error, regardless of the placement of material interfaces within the mesh.…”

“…The second consideration is that conventional FD techniques (including the higher accurate techniques proposed by Chiou et al [8] and Yamauchi et al [10]) are not capable of dealing with multiple material interfaces occurring between sampling points. This problem is particularly acute when simulating SPP waveguides where an extremely thin metal film, whose thickness is usually less than a few hundred nanometers, is loaded on a dielectric substrate, usually several micrometers in thickness.…”