ㄧ維光子晶體結構之數值分析

Numerical Analysis of One-Dimensional Photonic Bandgap Structures

陳又菁(You-Jing Chen)

指導教授 ： 楊士禮

國立清華大學/電機資訊學院/光電工程研究所/碩士(2008年)

摘要

本論文主要探討電磁波在一維光子晶體結構中的數值分析方法。我們採用Floquet-Bloch（FB） Wave Approximation假設在週期性結構內的電場形式，FB Wave在結構內滿足Maxwell’s equation在數學上表示為一二階非齊次的耦合微分方程，我們結合第一階微擾法（first-order perturbation method）對Mawell’s equation進行求解。在求解過程中，我們不使用傳統的大量數值迭代，將推導出可描述在結構內的電場收斂解析解，進而分析電場輻射能量與不同結構幾何的關係。
此外在透過多值的色散方程式求解結構內允許的模態（複數零根）時，我們使用保形映射轉換色散方程式為單值函數，而後以APM（argument principle method）尋找所有的導模（guided mode）或溢漏模（leaky mode）解，APM不需透過迭代運算可有效提升運算上的效率且能成功求解出所有零根。

關鍵字

一維光子晶體 ； 週期性波導 ； 微擾法 ； 布拉赫波 ； 幅角原理法 ； 色散方程式

並列摘要

In this thesis, we concentrate on the perturbation method as the numerical model in the analysis of electromagnetic fields one-dimensional photonic band gap（PBG） structures. Approximating that the electric fields is in the form of scalar Floquet-Bloch waves, the Maxwell’s equation in the corrugated region is a second-order inhomogeneous coupled differential equation and can be analytically solved by the first-order perturbation method that involves usually massive numerical iterative calculations. We suggest an approximated analytic solution for the electric field in the corrugated region instead of the numerical iterative results. After that, the radiated power with various geometrical parameters are presented.
Also, we study the implementation of mode determination inside one-dimensional PBG. The allowed modes, including guided modes and leaky modes, are obtained by finding zeros of the complex multi-valued dispersion equation. With conformal mapping, the four-valued dispersion equation is transformed into a single-valued equation with another new variable. We solve this single-valued equation by applying argument principle method（APM）. APM is a rigorous mathematical technique based on the complex number theory but not merely numerical iterative process. It is capable of finding all the zeros of any analytic function in the complex plane. With APM algorithms, the roots-finding of dispersion equation with PBG structure is more effective and accurate.

並列關鍵字

one-dimensional photonic crystals ； periodic waveguide ； perturbation method ； Floquet-Bloch Wave ； Argument principle method ； dispersion equation

參考文獻

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