K.P理論在光子晶體上的應用

Abstract

利用K.P理論推導光子在光子晶體之現象，發現光在周期性的介電物質中會造成光具有慣性質量，而光被侷限在缺陷中就好像粒子被侷限在位能井裡。在一維無限小的缺陷中，光能被侷限在裡面；而在二維和三維中缺陷中，卻需要足夠大的缺限陷才能侷限光。當要計算光侷限在缺陷中的性質時，我們可以從實驗或其他數值模擬得到等效慣性質量，在將這些係數代入雙能帶的 理論可以準確的計算出光在一維或二維異質結構的光子晶體中侷陷能量。當其他的數值解中只能解出結果卻很難去分出析， 理論提供了一個解析的來方法來解馬克斯威爾方程式，而且讓我們很容易解釋和預測一些光在光子晶體中的現象。

Using the K.P theory into photonic crystals (PCs), the photon excites an inertial mass form the periodic background. The phenomenon that the light trapped in the defect is similar to the particle trapped in the potential. Any arbitrary small defect will bind a state in 1-D defect, but a finite disorder is needed to bind a state in 2-D or 3-D defects. When we calculate the properties of the defects, the effective inertial mass can be gotten from the experiments or the simulation results from numerical methods of bulk PCs. Introducing the parameter into two-band model of K.P theory in the heterosturcture of PCs, the envelop function and the bound state energy can be solving quite correctly no matter in 1-D PCs or 2-D PCs. The K.P method provides an analytic method to solve the Maxwell’s equation in PCs, and by the method, we can predict and explain the trapping phenomenon of the PCs with defect whereas other numerical simulation methods just can tell us just a simulation result.