含有各向異性材質之平面波導與表面電漿子結構之導波模態研究

Studies of Guided Modes on Planar-Waveguide and Surface-Plasmon Structures Involving Anisotropic Materials

劉暄浩(Hsuan-Hao Liu)

指導教授 ： 張宏鈞

國立臺灣大學/電機資訊學院/光電工程學研究所/博士(2014年)

https://doi.org/10.6342/NTU.2014.01250

摘要

本研究根據有限元法建立一套平面光波導模態解析法，得以求解由各向異性材料構成的波導之導波和洩漏波模態。所考慮的各向異性材料可具有任意的介電常數張量，例如單軸材料之光軸可指向任意方向。本研究特別著重複數有效折射率的正確數值求解，並與可取得之解析解比較，尤其著重決定模態洩漏量的有效折射率之虛部部分。當光軸僅在平行於波導界面之平面上改變方向時，本研究以相較於三十年之前所發表的解析方程式更有系統的方式分別推導出導波和洩漏波模態的解析特徵方程式，並展示所得到的複數有效折射率解析值與有限元素數值結果極為吻合。
有限元素的推導是根據三個電場分量或三個磁場分量，採用二次元的節點基底，獲得二次特徵值方程式，然後採用shift-and-invert Arnoldi方法求解。本研究發現，在求解洩漏模態時，完全匹配層的厚度和其理論反射係數應適當選擇才能獲得良好的有限元素解的結果。
此外本研究亦探討存在於金屬材料與單軸各向異性介電材料之接面上的表面波或是表面電漿子模態，當光軸落於接面之平面時，由求解所推得的特徵方程式，發現洩漏表面波的存在。金屬材料可假設為具有負數介電常數的無損材料或是介電常數具有虛部的實際金屬材料(如銀) ，本研究比較兩者之間在洩漏模態解得的洩漏損耗特性的差異。數值結果顯示，上述情況的解析解與以本研究所建立的有限元素法獲得的結果相當吻合。 最後，解析解與有限元素分析進一步推展於具有兩個接面結構上的表面電漿子模態的研究，此兩個接面結構由單軸各向異性介電材料、金屬薄膜與各向同性介電基座所組成。

關鍵字

各向異性光學材料 ； 雙折性 ； 晶體光學 ； 電磁光學 ； 導波 ； 洩漏模態 ； 液晶 ； 平面波導 ； 表面電漿子 ； 表面波

並列摘要

A planar optical waveguide mode solver is established based on a finite-element (FE) formulation for determining the guided and leaky modes that exist on waveguides made of anisotropic materials with arbitrary permittivity tensor, for example, with arbitrary optic-axis orientation in the uniaxially anisotropic material case. Correct numerical determination of the complex effective index, especially its imaginary part which gives the modal leakage, is particularly emphasized referring to available analytical solutions. For the situation when the optic axis changes its direction only in the plane parallel to the waveguide interface planes, analytical characteristic equations for solving purely guided and leaky modes are separately derived in a more systematic manner compared with prior analytical formulae reported more than three decades ago, with the obtained complex effective indices agreeing excellently with FE solutions. It is found that in the FE analysis of leaky modes, the thickness of the perfectly matched layer (PML) and the PML theoretical reflection coefficient should be properly chosen. The FE formulation is based on either the three electric-field components or the three magnetic-field components using quadratic nodal bases, resulting in a quadratic eigenvalue equation which is then solved by the shift-and-invert Arnoldi method.

Furthermore, by solving analytically derived characteristic equations, leaky surface waves or leaky surface plasmon polariton (SPP) modes are found to exist at an interface between a metal material and a uniaxially anisotropic dielectric material with its optic axis falling in the plane of the interface. Both the assumed lossless metal material with negative relative permittivity and the real metal such as silver having an imaginary part in its permittivity are considered to reveal the difference in the solved leakage loss behavior of the leaky modes. Analytical solutions are also confirmed by the FE eigenmode analysis. The analytical and FE analysis are finally extended to SPP modes on two-interface structures composed of a uniaxially anisotropic dielectric material, a metal thin film, and an isotropic dielectric substrate.

並列關鍵字

Anisotropic optical materials ； birefringence ； crystal optics ； electromagnetic optics ； guided waves ； leaky modes ； liquid crystals ； planar waveguides ； surface plasmons ； surface waves

參考文獻

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