二维光子准晶的带隙及成像特性研究
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摘要
光子准晶，又称为准周期光子晶体，是一种特殊结构的光子晶体，具有长程取向序、结构自相似性及旋转对称性，但不具有平移对称性。在光子带隙与聚焦及成像方面，光子准晶比传统周期光子晶体更具优势。在过去二十六年中，虽周期光子晶体的理论、实验及应用已被大量研究，但光子准晶的研究则相对薄弱。本论文开展了二维光子准晶带隙、折射及成像特性的理论研究，以期望能为二维光子准晶相关物理特性的实验及应用研究提供指导。本论文的研究内容主要体现在以下几个方面。
采用平面波展开法研究了十重光子准晶基本结构单元中不同晶胞结构分别在不同散射子半径、不同基质相对介电常数且两种构建情况(即空气柱排布于介质中，介质柱排布于空气中)下的光子带隙。把二维光子准晶中的“类布拉格散射效应”类比于物理学中的“弹性碰撞”，建立了与散射子半径相关的光子带隙影响因素定性关系；结合电磁场理论，该关系又转化为与基质相对介电常数相关的光子带隙影响因素定性关系。对比了十重光子准晶与二维三角晶格光子晶体、二维正方晶格光子晶体分别在给定基质相对介电常数且不同散射子半径、给定散射子半径且不同基质相对介电常数，且均在两种构建情况下的光子带隙，并对比了锗基二维光子晶体与硅基二维光子晶体的光子带隙。得到了二维光子准晶更易产生光子带隙及完全带隙，锗基二维光子晶体更易产生带隙的结论。
基于十重光子准晶，设计了二维光子准晶楔形棱镜模型。为解决最大光强出射光束的出射位置偏离入射光束与楔形棱镜斜边交点且以往单一半圆形轨道探测装置不能准确探测出射光束折射角的问题，提出了可用于探测二维光子准晶楔形棱镜折射特性的双直线轨道探测方法。根据该探测方法，且采用时域有限差分法计算并分析了给定波长两种偏振模式且分别在不同入射位置、不同光束宽度下对应的折射角、等效折射率及出射位置偏移量。得到了二维光子准晶楔形棱镜折射特性存在非规律性的结论，并指出了其根本原因在于棱镜中散射子排布的非均匀性导致的类布拉格散射效应的非规律性。
基于十重光子准晶，设计了二维光子准晶平板透镜模型。确立了二维光子准晶平板透镜的能带结构与聚焦波长的关系。从波长、物距、散射子半径、基质折射率、透镜厚度及透镜宽度等方面，采用时域有限差分法分析了介质柱型二维光子准晶平板透镜的聚焦及成像特性，即像强、像质及物像关系等分别随各参量的变化关系。得到了二维光子准晶平板透镜成像的物像关系，以及聚焦及成像特性随波长的变化关系。发现了二维光子准晶平板透镜对给定点光源TM偏振模式的分光双聚焦效应及环形光子局域现象。建立了二维光子准晶平板透镜可发生非稳定聚焦现象的临界状态宽度分析模型及稳定聚焦的聚焦模型。发现了不同透镜厚度具有不同非稳定聚焦的临界状态宽度范围及稳定聚焦的宽度极限。提出了透镜最小尺寸与点光源波长满足的近似关系式。指出了二维光子准晶平板透镜对点光源能够发生聚焦及成像的根本原因及物理机制。设计了三散射子二维光子准晶平板透镜并分析了其聚焦成像特性与散射子半径的关系。
Photonic quasicrystals (PQCs), also known as quasi-periodic photonic crystals(QPCs), are a special class of photonic crystals (PCs). These have properties thatinclude long-range orientational order, structural self-similarity, and rotationalsymmetry, but no translational symmetry. With regard to their photonic band gaps(PBG), focusing, and imaging, PQCs have many advantages over traditionalperiodic photonic crystals (PPCs). There have been a number of studies on thetheoretical, experimental, and applied research on PPCs in the past26years, butrarely on PQCs. For this dissertation, theoretical research on PBG, refraction, andimaging for2D PQCs has been conducted to provide guidance in experimental andapplied research on the relevant physical characteristics. The main research contentsof this dissertation are as follows.
Using the plane wave expansion (PWE) method, a study was conducted on thePBGs of different unit cells in the basic structural unit of a decagonal PQC withdifferent scatterer radii and different substrate relative permittivity values, and twoconstruction cases (viz., air cylinders arranged in a dielectric medium, array ofdielectric cylinders in air). Assuming Bragg-like scattering effect in the2D PQC asthe elastic collision in physics, a qualitative relationship of PBG impact factors wasestablished showing a dependence on the scatterer radius. Based on theelectromagnetic theory, this relationship has been transformed into a qualitativerelationship for the PBG impact factors in terms of the relative permittivity of thesubstrate. The different PBGs of the decagonal PQC and2D triangular-lattice PCand2D square-lattice PC with given substrate relative permittivity and differentscatterer radii have been analyzed. A similar analysis was performed with fixedscatterer radius and different substrate relative permittivity as well as for the twoconstruction cases. For comparison, the analysis was extended to the different PBGsof the germanium-based2D PCs and the silicon-based2D PCs. The conclusionsobtained were that the2D PQC generates a PBG and complete band gap more easilythan the2D PC, and a germanium-based2D PC is more likely to generate a bandgap than a silicon-based2D PC.
Based on the decagonal PQC, a2D PQC wedge-shaped prism model has beendesigned. A problem arose requiring solution in that the exit position for themaximum-power outgoing light beam is offset from the intersection of the incidentbeam and the prism slope. Moreover, the real refraction angle of the outgoing beamcan not be detected accurately using previous detection equipment with a singlesemicircular track. Our solution uses a double-straight-line-track detection method that can be used to detect the refractive characteristics of the2D PQC wedge-shapedprism. According to this method, the refraction angle, the effective refractive index(ERI), and the offset at the exit position can be calculated and analyzed using thefinite-difference time-domain (FDTD) method for a given wavelength with twopolarization modes, at different incident positions in the prism, and with differentincident beam widths. A conclusion drawn is that the refraction characteristic of the2D PQC wedge-shaped prism is irregular. The fundamental reason, which is pointedout, is the non-uniformity of the arrangement of the scatterers in the prism whichinduce irregularities in the Bragg-like scattering effect.
On the basis of the decagonal PQC, a2D PQC flat lens model has beendesigned. The relationship between the band structure and the focusing wavelengthhas been developed. The focusing and imaging characteristics of thedielectric-cylinder-type2D PQC flat lens have been analyzed using the FDTDmethod. This included the dependence of image power and quality and therelationship between object distance and image distance, on parameters such aswavelength, object distance, scatterer radius, substrate refractive index, thickness,and width of lens. The relationship between object distance and image distance andthe wavelength dependence of the focusing and imaging characteristics for the2DPQC flat lens have been obtained. A spectroscopic double-focusing effect and a ringphoton localization phenomenon have been found in the2D PQC flat lens for agiven point light source with TM polarization mode. A critical-state width analysismodel of the2D PQC flat lens performing unstable focusing and a focus model ofthe2D PQC flat lens performing stable focusing have been proposed. Different lensthicknesses have different critical-state width ranges for unstable focusing and widthlimits for stable focusing. An approximate relationship between the minimum lenssize and wavelength of the point light source is proposed. The fundamental reasonand the physical mechanism underlying focusing and imaging of the2D PQC flatlens for a point light source have been pointed out. The2D PQC flat lens with threescatterers has been designed and the relationship between its focusing and imagingcharacteristics and scatterer radius has been analyzed.
采用平面波展开法研究了十重光子准晶基本结构单元中不同晶胞结构分别在不同散射子半径、不同基质相对介电常数且两种构建情况(即空气柱排布于介质中，介质柱排布于空气中)下的光子带隙。把二维光子准晶中的“类布拉格散射效应”类比于物理学中的“弹性碰撞”，建立了与散射子半径相关的光子带隙影响因素定性关系；结合电磁场理论，该关系又转化为与基质相对介电常数相关的光子带隙影响因素定性关系。对比了十重光子准晶与二维三角晶格光子晶体、二维正方晶格光子晶体分别在给定基质相对介电常数且不同散射子半径、给定散射子半径且不同基质相对介电常数，且均在两种构建情况下的光子带隙，并对比了锗基二维光子晶体与硅基二维光子晶体的光子带隙。得到了二维光子准晶更易产生光子带隙及完全带隙，锗基二维光子晶体更易产生带隙的结论。
基于十重光子准晶，设计了二维光子准晶楔形棱镜模型。为解决最大光强出射光束的出射位置偏离入射光束与楔形棱镜斜边交点且以往单一半圆形轨道探测装置不能准确探测出射光束折射角的问题，提出了可用于探测二维光子准晶楔形棱镜折射特性的双直线轨道探测方法。根据该探测方法，且采用时域有限差分法计算并分析了给定波长两种偏振模式且分别在不同入射位置、不同光束宽度下对应的折射角、等效折射率及出射位置偏移量。得到了二维光子准晶楔形棱镜折射特性存在非规律性的结论，并指出了其根本原因在于棱镜中散射子排布的非均匀性导致的类布拉格散射效应的非规律性。
基于十重光子准晶，设计了二维光子准晶平板透镜模型。确立了二维光子准晶平板透镜的能带结构与聚焦波长的关系。从波长、物距、散射子半径、基质折射率、透镜厚度及透镜宽度等方面，采用时域有限差分法分析了介质柱型二维光子准晶平板透镜的聚焦及成像特性，即像强、像质及物像关系等分别随各参量的变化关系。得到了二维光子准晶平板透镜成像的物像关系，以及聚焦及成像特性随波长的变化关系。发现了二维光子准晶平板透镜对给定点光源TM偏振模式的分光双聚焦效应及环形光子局域现象。建立了二维光子准晶平板透镜可发生非稳定聚焦现象的临界状态宽度分析模型及稳定聚焦的聚焦模型。发现了不同透镜厚度具有不同非稳定聚焦的临界状态宽度范围及稳定聚焦的宽度极限。提出了透镜最小尺寸与点光源波长满足的近似关系式。指出了二维光子准晶平板透镜对点光源能够发生聚焦及成像的根本原因及物理机制。设计了三散射子二维光子准晶平板透镜并分析了其聚焦成像特性与散射子半径的关系。
Photonic quasicrystals (PQCs), also known as quasi-periodic photonic crystals(QPCs), are a special class of photonic crystals (PCs). These have properties thatinclude long-range orientational order, structural self-similarity, and rotationalsymmetry, but no translational symmetry. With regard to their photonic band gaps(PBG), focusing, and imaging, PQCs have many advantages over traditionalperiodic photonic crystals (PPCs). There have been a number of studies on thetheoretical, experimental, and applied research on PPCs in the past26years, butrarely on PQCs. For this dissertation, theoretical research on PBG, refraction, andimaging for2D PQCs has been conducted to provide guidance in experimental andapplied research on the relevant physical characteristics. The main research contentsof this dissertation are as follows.
Using the plane wave expansion (PWE) method, a study was conducted on thePBGs of different unit cells in the basic structural unit of a decagonal PQC withdifferent scatterer radii and different substrate relative permittivity values, and twoconstruction cases (viz., air cylinders arranged in a dielectric medium, array ofdielectric cylinders in air). Assuming Bragg-like scattering effect in the2D PQC asthe elastic collision in physics, a qualitative relationship of PBG impact factors wasestablished showing a dependence on the scatterer radius. Based on theelectromagnetic theory, this relationship has been transformed into a qualitativerelationship for the PBG impact factors in terms of the relative permittivity of thesubstrate. The different PBGs of the decagonal PQC and2D triangular-lattice PCand2D square-lattice PC with given substrate relative permittivity and differentscatterer radii have been analyzed. A similar analysis was performed with fixedscatterer radius and different substrate relative permittivity as well as for the twoconstruction cases. For comparison, the analysis was extended to the different PBGsof the germanium-based2D PCs and the silicon-based2D PCs. The conclusionsobtained were that the2D PQC generates a PBG and complete band gap more easilythan the2D PC, and a germanium-based2D PC is more likely to generate a bandgap than a silicon-based2D PC.
Based on the decagonal PQC, a2D PQC wedge-shaped prism model has beendesigned. A problem arose requiring solution in that the exit position for themaximum-power outgoing light beam is offset from the intersection of the incidentbeam and the prism slope. Moreover, the real refraction angle of the outgoing beamcan not be detected accurately using previous detection equipment with a singlesemicircular track. Our solution uses a double-straight-line-track detection method that can be used to detect the refractive characteristics of the2D PQC wedge-shapedprism. According to this method, the refraction angle, the effective refractive index(ERI), and the offset at the exit position can be calculated and analyzed using thefinite-difference time-domain (FDTD) method for a given wavelength with twopolarization modes, at different incident positions in the prism, and with differentincident beam widths. A conclusion drawn is that the refraction characteristic of the2D PQC wedge-shaped prism is irregular. The fundamental reason, which is pointedout, is the non-uniformity of the arrangement of the scatterers in the prism whichinduce irregularities in the Bragg-like scattering effect.
On the basis of the decagonal PQC, a2D PQC flat lens model has beendesigned. The relationship between the band structure and the focusing wavelengthhas been developed. The focusing and imaging characteristics of thedielectric-cylinder-type2D PQC flat lens have been analyzed using the FDTDmethod. This included the dependence of image power and quality and therelationship between object distance and image distance, on parameters such aswavelength, object distance, scatterer radius, substrate refractive index, thickness,and width of lens. The relationship between object distance and image distance andthe wavelength dependence of the focusing and imaging characteristics for the2DPQC flat lens have been obtained. A spectroscopic double-focusing effect and a ringphoton localization phenomenon have been found in the2D PQC flat lens for agiven point light source with TM polarization mode. A critical-state width analysismodel of the2D PQC flat lens performing unstable focusing and a focus model ofthe2D PQC flat lens performing stable focusing have been proposed. Different lensthicknesses have different critical-state width ranges for unstable focusing and widthlimits for stable focusing. An approximate relationship between the minimum lenssize and wavelength of the point light source is proposed. The fundamental reasonand the physical mechanism underlying focusing and imaging of the2D PQC flatlens for a point light source have been pointed out. The2D PQC flat lens with threescatterers has been designed and the relationship between its focusing and imagingcharacteristics and scatterer radius has been analyzed.
引文
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