High-Accuracy Finite-Difference Equations for Dielectric Waveguide Analysis II:

TL;DR: In this paper, the authors present a finite-difference equation for waveguide eigenmode solution valid for a dielectric corner, which is based on an expansion of the field components as powers of the radius at the corner.

Abstract: We present a discussion of the behavior of the electric and magnetic fields satisfying the two-dimensional Helmholtz equation for waveguides in the vicinity of a dielectric corner. Although certain components of the electric field have long been known to be infinite at the corner, it is shown that all components of the magnetic field are finite, and that finite-difference equations may be derived for these fields that satisfy correct boundary conditions at the corner. These finite-difference equations have been combined with those derived in the previous paper to form a full-vector waveguide solution algorithm of unprecedented accuracy. This algorithm is employed to provide highly accurate solutions for the fundamental modes of a previously studied standard rib waveguide structure. Indeed, with second-order accurate code it may not be worth the trouble to change the way corners are handled. However, for those intent on more accurate simulations, correct treatment of the corner points is essential. The history of the behavior of fields near a dielectric or metal wedge dates to the early work of Bouwcamp and Meixner (2), (3), who pointed out some of the peculiarities of this problem, including field components that diverge weakly at the corner as a small negative power of the distance from the corner (ra- dius). Meixner further published a solution formalism (3) based on an expansion of the field components as powers of the ra- dius. This expansion was later shown to be incorrect by An- derson and Solodukhov (4), and although papers continued to appear, almost ten years elapsed before a correct expansion was published. Makarov and Osipov (5) explained the reason for the problems with Meixner's series and correctly identified the missing terms as containing various powers of the logarithm of the radius. However, the latter authors did not treat a "gen- eral" corner, but considered only a single field component, and left their expansion in a very general form quite unsuitable for the derivation of finite-difference equations. This entire body of work, and in fact the existence of the problem itself, re- mained largely unknown to the optics community until 1992, when Sudbo (6) described the importance of the problem in the modeling of optical waveguide structures. Apparently unaware of Makarov and Osipov's work, he correctly pointed out the in- herent difficulties of including corner effects in any numerical algorithm for modeling waveguides. Since that time only a few authors have attempted to include corner effects in their mod- eling work (7), (8) and no one has attempted a thorough deriva- tion. This paper derives a finite-difference equation for waveguide eigenmode solution valid for a dielectric corner. This equation is ostensibly first-order accurate, and when combined with highly accurate difference equations in the interior regions and at simple planar boundaries (1), has resulted in a highly accurate full-vector waveguide eigenmode solver whose trun- cation error ranges from second to third order. This departure from the expected third-order behavior apparently results from the presence of infinite derivatives at the corner and will be discussed in more depth below. The present work may be viewed as an extension of the work of Makarov and Osipov (5), in that the standard solutions of the Helmholtz equation in a uniform region (i.e., Bessel functions and sines and cosines) are augmented by terms containing logarithms that appear similar to their solutions. However, here we consider the full vector