平板光子晶體帶隙結構與缺陷態之研究

On the band structures and defect states of planar photonic crystals

林志昌(Jhih-Chang Lin)

指導教授 ： 薛文証

國立臺灣大學/工學院/工程科學及海洋工程學研究所/博士(2007年)

https://doi.org/10.6342/NTU.2007.10463

摘要

光子晶體帶隙材料在光電領域上的應用日趨重要，其中具缺陷態光子晶體在雷射、濾波器及調變器等光電元件上的應用越來越受到重視，而研究重點之一為分析其帶隙結構及缺陷態。在帶隙結構分佈方面，一般傳統分析的方法是去計算布洛赫相位的餘弦值，進而描繪出帶隙結構的分佈，然而利用此法在求解某些帶隙結構時，會產生數值計算上不穩定的缺點。所以本篇論文提出一種新的模型去表示多層結構中每一層介面中電磁場的關係，並提出以帶隙邊緣方程式取代傳統的色散方程式，因此可以更容易的分析出多層光子晶體正確的帶隙結構。

另外關於缺陷態的研究，使用傳統分析方法時，除了會有數值不穩定的現象外，在計算缺陷態時可能會產生多餘的解，必須使用布洛赫之指數函數加以檢查及去除，這不只會增加分析的困難，也可能造成不正確的結果。所以本論文提出一個新的方法分析缺陷態，本方法的優點除了分析時可避免數值發散的情形發生，更可直接求得正確的結果，不會有多餘或不正確的缺陷態產生。

關鍵字

光子晶體 ； 缺陷態 ； 帶隙結構 ； 平面波導 ； 週期結構

並列摘要

Photonic bandgap materials are becoming more and more important in optoelectronic applications. Recently, the study of the defect states of photonic crystals with microcavity has received the attention on the applications of lasers, filters, and modulators. Band structures and defect modes are two main characteristics of the photonic crystals. According to the analysis of band structures, traditional methods are usually used to determine the band structures by the cosine of Bloch phase. However, it may have the numerical instability problem about solving band structures of some photonic crystals. As a result, a novel model to express the relationships among the tangential fields at any layer interface of multi-layer waveguides is presented in this study. The bandedge equations instead of the traditional dispersion equations are derived in order to determine band structures accurately and more easily.

Based on the analysis of defect states, it may cause the numerical instability by using traditional methods. In addition, it will result spurious solutions from solving defect states of photonic crystals and eliminate the spurious solutions by checking the exponential function of the Bloch phase. Therefore, this study presents a new method to analyze the defect states of photonic crystals. Adopting this method, the problem of numerical instability and spurious solutions can be avoided in this analysis.

並列關鍵字

photonic crystal ； band structure ； defect state ； planar waveguide ； periodic structure

參考文獻

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