JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 20, NO. 8, AUGUST 2002 1627 Eigenmode Analysis of Optical Waveguides by a Yee-Mesh-Based Imaginary-Distance Propagation Method for an Arbitrary Dielectric Interface Takashi Ando, Student Member, IEEE, Hiroki Nakayama, Satoshi Numata, Junji Yamauchi, Member, IEEE, and Hisamatsu Nakano, Fellow, IEEE Abstract—A modified finite-difference (FD) formula for an arbi- trary dielectric interface is applied to an imaginary-distance prop- agation method based on Yee’s mesh. To confirm the validity of the modified method, we first analyze the eigenmode of a step-index circular fiber. A reduction in a discretization error is demonstrated in the evaluation of the field profile. Calculations of the normalized propagation constant show that the convergence rate of the modi- fied FD formula is faster than that of traditional techniques and is comparable to that of a body-of-revolution technique. As a further application, we analyze the eigenmodes of sloped-side rib guides. These data agree well with previously published data. Index Terms—Dielectric waveguides, electromagnetic fields, fi- nite-difference methods, Maxwell equations, numerical analysis, optical waveguides. I. INTRODUCTION T HE KNOWLEDGE of the eigenmode in an optical wave- guide is of fundamental importance in the design of pho- tonic integrated circuits. To date, many methods have been pro- posed for this important issue [1]–[4]. One of them is an imagi- nary-distance propagation method [2] based on Yee’s mesh [4], [5]. The use of Yee’s mesh has the advantage that the obtained eigenmode fields can directly be utilized for the finite-difference time-domain (FDTD) analysis [6]. In the FDTD method, the FD equations are conventionally obtained by discretizing Maxwell’s equations with the use of a uniform orthogonal Yee mesh [7]. The space derivatives of Maxwell’s equations are approximated by central-difference operators. It is well known that the conventional FD formula based on the central-difference operators induces a discretiza- tion error when a staircase approximation is introduced to describe an arbitrary dielectric interface. To reduce the discretization error, researchers have devel- oped some techniques [7]–[12]. One is the contour-path tech- nique [7], [8], in which space cells local to a dielectric inter- face are deformed to conform with the interface position and the FD equations for the fields adjacent to the interface are de- rived on the basis of Faraday’s and Ampere’s laws. Unfortu- nately, the derived FD equations cannot naturally be incorpo- Manuscript received October 23, 2001; revised March 7, 2002. The authors are with the Faculty of Engineering, Hosei University, Koganei, 184-8584 Tokyo, Japan. Digital Object Identifier 10.1109/JLT.2002.800360 rated into the imaginary-distance scheme. Another is the ef- fective-dielectric-constant technique (EDCT) [7], [9]–[12], in which permittivities along the stepped edges of Yee’s mesh are estimated by various averaging procedures. In contrast to the contour-path technique, the EDCT can be incorporated into the imaginary-distance scheme without difficulties [5]. Although the EDCT appears to improve the overall accuracy [9]–[13], the use of the conventional FD formula results in slow convergence as a function of transverse mesh size. This is be- cause the conventional FD formula assumes the continuity of the field at a dielectric interface. In practice, the field and its derivative are often discontinuous at the interface. To exactly evaluate the field discontinuity, we have to take into account the boundary conditions. Recently, a modified FD formula, which is constructed by combining the boundary conditions and one-sided difference operators, has been proposed for the FDTD algorithm [14]–[17]. It was demonstrated that the numerical errors caused by the tra- ditional techniques are reduced in the analysis of scattering by a grating coupler [14] and by a dielectric cylinder [15]–[17]. However, no attempt has been made for the eigenmode analysis of an optical waveguide except for our preliminary report [18]. In this paper, we apply the modified FD formula [16], [17] to the imaginary-distance propagation method based on Yee’s mesh and show the effectiveness of the modified method in the eigenmode analysis of an optical waveguide with an arbitrary dielectric interface. Furthermore, a body-of-revolution (BOR) [7] imaginary-distance propagation method is newly derived for a comparative study. After the derivation of the modified and BOR methods, we first analyze a step-index circular fiber, since the exact field pro- file and propagation constant are available. Calculations of the field profiles show that the discretization error is substantially reduced by virtue of the modified FD formula. In the evaluation of the normalized propagation constant against transverse mesh size, the modified FD formula also achieves faster convergence than the staircase approximation and the typical EDCT [12]. It is worth mentioning that the results of the modified method al- most coincide with those of the BOR method. We next analyze the eigenmodes of sloped-side rib guides [19]. The evaluated effective indexes are in agreement with pre- viously published data, including calculated and experimental results. 0733-8724/02$17.00 © 2002 IEEE Authorized licensed use limited to: HOSEI UNIVERSITY KOGANEI LIBRARY. Downloaded on July 27, 2009 at 05:24 from IEEE Xplore. Restrictions apply.