傳統上，要將平行光聚焦於一個小光點，需要使用一個凸透鏡。近年來關於光子晶體的研究提供了另一種途徑：選擇光子晶體的負折射頻帶為工作頻帶，則以此光子晶體做成的凹透鏡就可以將光聚焦。已有大量的論文研究過光子晶體的負折射特性，此外也已經有幾篇關於光子晶體凹透鏡的研究被發表。這類透鏡做為一類奈米光學元件，具有短焦距以及能進行次波長成像的優點。然而這類透鏡的設計與實作還是顯得複雜，因此本論文提出空氣透鏡的概念以取代並簡化相關設計，並探討其成像特性。
　　本論文首先探討二維光子晶體奈米平凹透鏡的負折射行為與成像特性，接著將二維光子晶體以一維光子晶體取代，最後提出空氣透鏡的概念，以期不必使用光子晶體的複雜結構，亦能達到於奈米尺度之環境中有效將入射光聚焦的目標。
　　我們在介電質背景中置入一空氣平凹透鏡，其中之凹面為圓柱面，並選取背景介電係數與透鏡介電係數相差甚大的參數進行模擬。當光從背景介質傳播進入凹透鏡形狀的凹洞中(由光密介質進入光疏介質)再回到背景介質中，光線會產生與玻璃凸透鏡一樣的匯聚行為，達到聚焦的效果。為了找出較好的聚焦特性，我們對空氣平凹透鏡做非球面的處理，將凹面以幾種圓錐曲面取代，並比較在各種圓錐曲面下，以時域有限差分法（以 Maxwell 方程組為基礎的波動光學）與光線追跡法（幾何光學）進行聚焦位置的預測及成像點大小的分析比較。
　　在我們的分析中，以橫橢圓2的成像品質最佳(幾何光學預測與波動光學現象吻合度最高)。作為一個應用的例子，我們將空氣透鏡與光子晶體波導組合在一起，形成光學耦合器，並展示其有效將光強集中導入波導的結果。我們相信，適當設計的奈米級空氣透鏡元件將能在未來的奈米光子學中獲得普遍應用。

　Traditionally, to focus a beam of light into a tiny spot, we need a convex lens. Recently, alternative approach based on the researches of the propagating waves in photonic crystals (PhC) has been proposed: by choosing a negative-refraction (NR) photonic pass-band as operating band, a plano-concave lens of PhC makes a beam of incident light converge. Up to now, a huge amount of research papers discussing the features of NR in PhC together with a few papers studying the focusing ability of concave lenses of PhC have been published. As a kind of nano-optical component, a concave lens of PhC has the advantage of shortening the focal length and focusing the incident light beam into a spot of sub-wavelength size. However, the design and fabrication of a PhC lens are still too complicated. We therefore propose in this thesis the new idea of ‘air lens’ to simplify the previous designs. In addition, we investigate thoroughly the focusing characteristics of this device.
　In the thesis, we begin with the investigation of the NR phenomenon and imaging characteristics of plano-concave nanolens in a 2D PhC. We then replace the 2D PhC structure with a 1D PhC and explore if the same work can still be done. Finally, the idea of air lens is proposed. We hope to avoid such a complicated structure like PhC but achieve the same goal of making light converged effectively in the nanoscale environment.
　We first create a plano-concave air lens in a dielectric medium, whose concave surface is cylindrical. As one of the simulation parameters, a large dielectric constant of the background medium is assumed in order to focus the incident light effectively. When light propagates from a denser medium into a less dense one and back to the denser medium, the concave air lens converges light like the convex glass lens does in air. To find better converging characteristics, we then make the concave surface non-cylindrical. We replace the cylindrical surface with several conic surfaces. Moreover, we use FDTD (the wave optics based on Maxwell equations) method as well as the ray-tracing (geometrical optics) method to simulate and predict the locations of the focal points. In addition, we compare the spot sizes and focus locations for different surfaces.
　In our analysis, the second type of oblate elliptical surface has the best imaging performance, in which the geometrical prediction matches the wave phenomenon very well. As an example of application, we assemble an optical coupler of concave air lens and a photonic-crystal waveguide (PCW) and show how efficiently the light beam can be coupled into the PCW. We believe that appropriately designed air nanolens will become a commonly used component in the nanophotonics in the future.

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