以時域有限差分法分析二維光子晶體聚光現象

Abstract

過去在光子晶體的研究中，大部分都是探討其具有光子頻隙的特性並依此製造出波導、共振腔、光纖等等，在應用上較少提到傳導帶的部分。而在光子晶體的傳導帶中，我們可以觀測到異常折射的現象，甚至有類似負折射現象出現。2000年J.B. Pendry[6]提出利用負折射平板可製造出完美透鏡，使得很多人開始對負折射物質產生興趣。一部分人積極尋找左手物質的存在，另一部分人則研究光子晶體中的等效負折射。本文透過平面波展開法求出二維光子晶體的頻帶圖(即色散曲面)，並以等頻率曲線近似為圓形時且等效為負折射的二維三角晶格平板，利用時域有限差分法模擬其聚光現象，分析以光子晶體作為負折射透鏡對點光源的解析度及聚光強度。而模擬結果指出，將光子晶體平板視為均勻的負折射是不明智的，因為等頻率曲線所預測的為無邊界的光子晶體。我們確實可以利用光子晶體平板實現負折射聚光，但卻無法超越繞射極限，並不能作為完美透鏡。

We investigate the focusing effect in 2D hexagonal photonic crystal (PhC) slab with an effective negative refraction. The effective negative refraction of PhC does not guarantee the existence of left-handed meta-material, so amplification of the evanescent modes is not observed. In this thesis, 2D PhC equi-frequency contour is used to calculate the effective refraction index. We present the point source image properties in 2D PhC slab with uniform effective negative refraction index for various angles by employing the finite-difference time-domain (FDTD) technique. It is shown that the diffraction limit is not overcome by the PhC lens and large angle incident waves are mostly total reflected. The focusing properties of the photonic crystal slab are mainly due to the negative refraction effect, rather than the self-collimation effect.