發展一能準確求解光波導方程之方法

Development of an accurate solver for optical waveguide equations

林耿正(Ken-Cheng Lin)

指導教授 ： 許文翰

國立臺灣大學/工學院/工程科學及海洋工程學研究所/碩士(2008年)

https://doi.org/10.6342/NTU.2008.01345

摘要

在本論文中，
吾人提出一種準確且有效率的有限差分精確解格式
(convection-diffusion-reaction (CDR) finite difference scheme)，
並將其運用於離散全向量與半向量之波導模態方程。
其中，
以一維精確解格式求解
光能僅侷限於一個方向的二維結構平面波導，
所採用的離散方式涉及三個空間的網格點；
以二維精確解格式求解
光能可允許在兩個方向的三維結構光波導，
所採用的離散方式涉及五個空間網格點。

在二維結構平面波導的部份，
吾人亦利用三點四階緊緻格式與三點六階緊緻格式求解，
以期在不增加計算資源的前提下，
以相同的離散格點點數得到更精確之解。

在本論文中的二維波導算例，
包含階變式折射率三層平板波導、階變式折射率多層平板波導(平面方向耦合器)及漸變式折射率平面波導，
而三維光波導結構的問題則探討了擴散型通道波導、矩形介電質波導及半導體肋型波導。
計算所得的結果經
詳盡地與先前許多研究者的結果比較後，
可以得知兩者所獲得的結果相當一致。

關鍵字

CDR精確解格式 ； 全向量 ； 半向量 ； 波導模態方程 ； 三點四階 ； 三點六階

並列摘要

We propose in this study a numerically accurate and computationally efficient
convection-diffusion-reaction (CDR) finite difference scheme
to discretize the full-vector and semi-vector optical waveguide equations.
The proposed schemes implemented in a stencil of three/ five nodal points
for solving the two/ three dimensional waveguide equations
are featured with the provision of locally analytic solutions.
We also use the three-point fourth-order compact scheme
and three-point sixth-order compact scheme
to solve the 2-D waveguide structures.
These two schemes can yield the accuracy order of fourth and sixth in a grid stencil
involving three points only.
The improved prediction accuracy
is not at the cost of an increasingly expensive matrix calculation and, thus, computational cost.

Computations of several 2-D and 3-D waveguide structures,
such as the three-layer,
multi-layer (planar directional coupler),
planar diffused,
diffused channel,
rectangular dielectric,
and semiconductor rib waveguides
have been carried out to test the accuracy and efficiency of the present method.
Good agreement between the current and other results justifies the
proper use of the proposed schemes.

並列關鍵字

convection-diffusion-reaction ； full-vector ； semi-vector ； waveguide equations ； three-point fourth-order ； three-point sixth-order

參考文獻

[1] M. J. Adams, An Introduction to Optical Waveguides, John Wiley & Sons, 1981.

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[4] A. Sharma and P. Bindal, Variational analysis of diffused planar and channel waveguides and directional couplers, J. Opt. Soc. Am. A, vol. 11, pp. 2244-2248, 1994.

[5] J. Chilwell and I. Hodgkinson, Thin-film field-transfer matrix theory of planar multiplayer waveguides and reflection from prism-loaded waveguides, J. Opt. Soc. Amer. A vol. 1, pp. 742-753, 1984

[6] M. S. Stern, Semivectorial polarized finite diRerence method for optical waveguides with arbitrary index profiles, IEE Proc. J., vol. 135, pp. 56-63, 1988.