發展平行有限差分時域法： 研究蜂巢狀光子晶體負折射等新穎特性

Abstract

左手材料具有克服光學繞射極限的超級透鏡的效應。近年來，光子晶體能夠模擬全角度左手材料的一些性質。其中美國麻省理工學院的光子晶體能帶程式被用來計算二維蜂巢狀結構光子晶體的能帶結構。另一方面，我們利用有限時域差分法計算蜂巢狀結構光子晶體內外在不同時刻的電場分佈。但是，蜂巢狀結構光子晶體包含了細微的結構。這導致以序列的有限時域差分法計算模擬此種問題會花費大量的時間和計算資源。本論文的目的是發展平行的有限時域差分法以有效率的計算包含細微結構光子晶體內外的電磁場分佈。同時，我們將重新評估以光子晶體做為全角度左手材料的條件。模擬結果顯示：蜂巢狀結構光子晶體的負折射的現象和入射光的入射方向有著非常重要的關係。更進一步，我們提出：非偶合模態隨著入射光的不同入射方向而扮演著不同的角色，某些情形下，它幫助負折射的現象發生；某些情形下，它讓負折射無法發生。這結果顯示：雖然我們選擇的負折射頻率在動量空間的等頻率線是各向同性的，它並不保證可以產生全角度的負折射現象。所以當我們考慮負折射光子晶體時，不可以忽略光子晶體的對稱性。雖然本研究是考慮二維的蜂巢狀結構光子晶體，但是我們的研究結果也可以應用於三維負折射光子晶體和三維的超級透鏡。另外一方面，我們發展的平行有限時域差分法不只可以運用於光子晶體，它也可以增加其它電磁學問題的計算效率以模擬更複雜和更大結構的電磁問題。

Left-handed materials have superlensing effects that enable them to surmount the optical diffraction limit. A photonic crystal is able to mimic some properties of all-angle left-landed materials. The MIT Photonic-Bands program is employed to calculate the band structure of walled honeycomb photonic crystals. Furthermore, the finite-difference time-domain (FDTD) method is used to calculate the electromagnetc field distribution inside and outside of walled honeycomb photonic crystals. However, the walled honeycomb photonic crystals include fine structures. It takes time and computational resource to simulate this kind of problem set in executing sequential FDTD program. The objective of this thesis is to develop parallel FDTD program to simulate the propagation of electromagnetic waves in the photonic crystals with the fine structures efficiently; meanwhile, the all-angle negative refraction criteria of photonic crystals are evaluated. The simulation results indicate that the all-angle negative refraction phenomena of the honeycomb photonic crystals are correlated with the orientation of the photonic crystals, even if the equal frequency contour is in the shape of a perfect circle. Moreover, the role of the uncoupled modes varies based on their orientation to the all-angle negative refraction photonic crystals, in one case assisting negative refraction and in the other case preventing it. The results suggest that symmetric properties should not be ignored when considering the negative refraction of photonic crystals. Although we studied on the two-dimensional walled honeycomb photonic crystals, it would be also useful to design the three-dimensional negative refraction devices of photonic crystals. On the other hand, the development of the parallel FDTD program should progress the existing FDTD method and simulate more complicated and larger structure efficiently. While, the photonic crystals are the simulation problem of parallel FDTD program in the thesis, the method could be applied to other electromagnetic problems.