一維與二維光子晶體之分析與模擬

Abstract

當兩種不同介電常數的材質在空間中成週期性的排列，則某些特定波長的光波或電磁波會被排斥，亦即在該頻率電磁波不能存在其間。材料的結構具備這種光子能隙的性質者，稱為光子晶體。 本論文從一個平面波展開法的數學通式出發，探討光子晶體的TE及TM的波動行為。在一維光子晶體的分析，實際的情形是有限週期的結構及材質具有耗損(σ≠0)性質，因為偏離平面波展開法所能處理之範圍，改採傳輸矩陣法，其結果和平面波展開法在零損耗的的結果大致符合。從一維光子晶體的分析嘗試設計工作波長為1500nm的光濾波器的設計，其通帶的寬度約為130nm；以矽及砷化鎵為例，光晶的空間週期長度約為1μm，所設計的濾波器允許入射角的範圍約為15°，在此角度範圍內的光子能隙相當完整。同樣使用平面波展開法應用到二維光子晶體的計算，其結果和已知文獻之結果大致符合。本論文主要以建立光子晶體之模擬方法為主，數學的計算包括MathCAD及C語言的程式編寫，但尚未進行光子晶體的實驗驗證。

Materials with periodic dielectric structure have the property to suppress or allow the propagation of the electromagnetic waves in them for only some specific wavelengths. It means that light with specific wavelength can not subsist in such a material structure. Materials with this property is named as photonic crystal. This study aims at the development of simulation technique to investigate the photonic crystal or photonic bandgap structure. We use MathCAD and C++ -Language in calculations. We formulate the photonic crystal equations mainly with the Plane Wave Method Expansion Method. This method is efficient in investigating the behavior of transverse-electric modes and transverse–magnetic modes of electromagnetic waves in photonic crystal. However, for the analysis of 1-dimensional (1-D) photonic crystal where the material is lossy (σ≠0) and the dielectric distribution has finite periodicity, we adopt the Transfer Method for the reason that the Plane Wave Expansion Method is not appropriate in dealing with those problems. The solutions for the lossless material obtained by the Transfer Method agree basically with that of the Plane Wave Method Expansion Method. From the analysis for 1-D photonic crystal, we also propose the implementing structures of an optical filter which has a center wavelength 1500nm with a bandwidth of nearly 130nm. Taking Si and GaAs as base materials, according to the simulations, the proposed 1-D structure has period around 1μm in space. This filter possesses still wide frequency bandgap for an oblique incidence of light at angle within ±30°. We also apply the Plane Wave Expansion Method to deal with the 2- dimensional (2-D) photonic crystal. Since at this stage of study no experimental work is conducted to verify our simulations, the numerical results of 1-D and 2-D simulations can only be compared with the published data found in the literature. The fair agreement of our results with that of the literature would suggest that the so far self-developed simulation technique is acceptable, although it needs surely further work of improvement.