利用緊束縛理論研究光子晶體波導之耦合行為與多工分波器設計

Abstract

利用固態物理中的能帶緊束縛(Tight-binding)理論，可以正確地描述一個光子晶體波導的傳輸行為。利用此理論得到的光子晶體波導的色散關係方程式，可以進一步去描述兩條耦合光子晶體波導的傳輸行為和色散關係方程，並正確地計算其耦合長度(coupling length)，進而用以設計光通訊元件。 當兩個相同的光子晶體波導彼此靠得很近時，其能帶便會由於波導之間的耦合效應，而分裂為偶對稱與奇對稱的本徵模。由於耦合光子晶體波導與普通耦合光波導不同，除了橫向之耦合效應外，並具有縱向(傳播方向)之耦合，導致此兩種模態會發生能帶交叉的現象。因此，我們可以利用緊束縛理論所推導出來的色散關係方程式找到正確非耦合頻率(decoupled frequency)。利用此耦合波導的特殊特性，我們用“時域有限差分法(FDTD)＂之數值模擬，完成了多工分波器(WDM)元件的設計。本論文中，我們可以將三道不同波長的光分開，且均達到光通訊的標準。

By using tight-binding theory of solid-state physics, we can analytically describe the dispersion relation of the propagation in a photonic crystal waveguide (PCW). In turn, we can derive the dispersion curves of two coupled identical PCWs . Due to not only the transverse coupling as the conventional coupled waveguides but also the longitudinal coupling of two coupled identical PCWs. “Band-crossing” may occur at which the PCWs will not couple with each other (or decoupled) when the coupled PCWs are placed close enough to each other. By employing the tight-binding theory to this problem, we can accurately determine the decoupling frequency as well as calculate the coupling length for every frequency. We have designed a wavelength division multiplexer which can route three wavelengths into different channels with the power ratio of all outputs reach 20 dB, the specification of optical communication.