光子晶體平板中的耦合共振光波導以及指向性耦合器研究

Abstract

由光子晶體平板製成的耦合共振光波導以及指向性耦合器為一種能夠結合光減速、光通訊以及光學非線性效應等應用的結構。藉由廣義的緊密束縛理論所推導出來的解析方程式，我們可以用來分析此種結構的電磁場模態分布以及色散曲線。在此論文中主要針對兩種不同型態:介電柱以及空氣洞所形成的結構來加以分析，指向性耦合器基於波導之間的相對位置又可分為對立型以及交替型兩種。利用調整所設計結構中缺陷的大小，我們可以由數值模擬的方式來得到電場分布以及特徵頻率的變化，進而了解方程式中耦合係數的改變，因此色散曲線的變動趨勢亦可被預測。藉著比較平面波展開法的模擬結果，我們得知緊密束縛理論所推導出來的方程式可以成功的運用在上述缺陷調變所引起的現象，除了可以用少量的參數來控制此種結構的色散行為外，亦可用來進行調變的規則設計。

We present an analytical way to study the directional couplers (DCs) based on coupled resonant optical waveguides (CROWs) in the photonic crystal slab (PCS). It holds potential for combining the applications of slow-light propagation, nonlinear optical processes and optical signal coupling in integrated photonic circuits. From the analytical equations derived by the extended tight-binding theory (TBT), we can obtain the dispersion relations and the electromagnetic (EM) mode distribution of a single PCS-CROW and the PCS-DCs. In the dielectric-rod structures, we find that the dispersion curves of the opposite-type PCS-DCs never cross and the frequency difference of them remains constant. Additionally, the dispersion relation of the alternating-type PCS-DCs with larger defects possess a crossing point, which will shift to the smaller wavevector and the higher frequency by increasing the defect radius. At this crossing point, the energy in one waveguide will never transfer into the other one and is also called the decoupling point. On the other hand, in the air-hole structures, we know that the dispersion curves of both the opposite-type and alternating-type PCS-DCs have a decoupling point nearly fixed at a certain wavevector. Moreover, as increasing the wavevector, the frequency difference between the curves of the opposite-type PCS-DCs increases, and that of the alternating-type PCS-DCs increases and then decreases. In conclusion, the dielectric-rod structure can be used to form the demultiplexers, and the air-hole structures can be used to create the beam splitters. All of these theoretical analyses from the TBT agree well with the numerical ones using the plane-wave expansion method, and give the design rules for these kind of structures.