亚波长衍射微透镜的严格矢量分析
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摘要
本论文围绕亚波长衍射微透镜的严格矢量分析与设计展开。由于亚波长衍射微透镜的特征尺寸小于入射波长，因此不能采用标量衍射理论来分析，又由于它是有限非周期的结构，严格的耦合波方法对它也不适用。本论文把电磁场的数值计算方法作为分析亚波长衍射微透镜的严格的矢量衍射理论，给出了亚波长衍射微透镜的设计方法，对具体的设计实例做了分析。
第一部分为理论分析方法。论文中将旋转体时域有限差分法(BOR FDFD)作为分析具有轴对称结构的亚波长衍射微透镜的理论模型。论文中将相关的光学知识融入到BOR FDTD算法中，给出了它所包含的主要子算法的内容及公式。作为BOR FDTD中的一个最重要的子算法，首次提出了非分劈场量的完全匹配层吸收边界条件(PML ABCs)，大大简化了计算过程。改进了平面波谱传播算法(VPWS)，推导出的公式可以由BORFDTD计算空间内的近场输出数据直接计算出远场的电磁场分布。给出了应用BOR FDTD与VPWS算法分析亚波长衍射微透镜的分析过程和程序流程图。成功地完成了整套BOR FDTD与VPWS算法程序的编制与调试，在使用Matlab语言编程过程中通过矩阵赋值代替循环赋值，极大的提高了计算速度。
第二部分为衍射微透镜的设计。本论文给出了多台阶衍射微透镜和二台阶亚波长衍射微透镜的设计方法、面形公式、设计实例。重点说明了二台阶亚波长衍射微透镜的两种设计方法，并对两种方法作了比较，得出了亚波长结构的脉冲宽度调制法优于连续位相的线性进似法的结论。讨论了深度误差和宽度误差对衍射微透镜的影响，发现相对误差不大时制作误差对衍射微透镜的影响不大。
第三部分是对衍射微透镜光学特性的分析。首先研究了设计参数变化对衍射微透镜的影响，给出了设计焦距和透镜材料折射率变化对衍射微透镜衍射效率、爱里斑半径等的影响，并对分析结果作了定性讨论。然后研究了亚波长衍射微透镜的色散特性，表明亚波长衍射微透镜的焦距随着波长变短而增加，但是波长与焦距之间并非简单的完全反比例关系，而是随着波长的减小焦距增加得更快，同时指出标量理论已不能准确分析亚波长衍射微透镜的色散特性。
This dissertation is about the rigorous vector analysis and design of subwavelength diffractive microlenes. Because the feature size of subwavelength diffractive microlens is smaller than the incident wavelength, scalar diffractive theory cannot be used to the analysis. It also precludes rigorous vector coupled-wave theory for its finite and aperiod structure. In this dissertation an electromagnetic computational method is employed as rigorous vector diffractive theory for the analysis of subwavelength diffractive microlens. The design methods of subwavelength diffractive microlens are presented. The design examples are analyzed by the theory in the dissertation.
In the first section, analysis methods are developed. In the dissertation body-of-revolution finite-difference time-domain method (BOR FDTD) is employed as the theory mode for the analysis of axially symmetric subwavelength diffractive microlens. Relative optical theory is considered in BOR FDTD algorithm. The major sub-algorithms in BOR FDTD are explained, the formulas in each sub-algorithm are given. As one important sub-algorithm of BOR FDTD, Perfect matched layer absorbing boundary conditions (PML ABCs) algorithm with split field components is presented for the first time, which greatly simplified computational process. The vector-based plane-wave spectrum method (VPWS) is improved, which can directly obtain the far field diffractive pattern depending on the output near field data in the BOR FDTD meshes by use of the deduced formulas in this dissertation. The above two algorithms of BOR FDTD and VPWS are used to the analysis of subwavelength diffractive microlenses, the analysis process and program flow
chart are shown. The editing and debugging of the whole program of BOR FDTD and VPWS algorithms by Matlab code are finished successfully, in the program I used matrix to put values for variables instead of using cycle, which significantly increase the computing speed.
The second section is to design diffractive microlenses. In the dissertation multilevel microlenses and binary subwavelength diffractive microlenses are designed, design methods and configuring formulas and design examples are illustrated. Two kinds method of design binary subwavelength diffractive microlenses are illustrated with emphasis. I obtained the result that the method of subwavelength pulse-width modulation is better than the method of linear approximation for continuous phase piece in designing binary subwavelength diffractive microlenses by comparing them. The effects of depth error and width error on diffractive microlenses are discussed. I find that there are little influences on diffractive microlenses when the relative fabrication error is small.
The third section is to analyze the optical characteristic of diffractive microlenses. The influence of changing design parameters on diffractive microlens is investigated. The effects of changing design focal length and microlens material refractive index on diffractive efficiency and airy diffraction disc radius are presented. The numerical results are discussed by qualitative analysis. The dispersion property of subwavelength diffractive microlens is investigated. Numerical result illustrates that focal length becomes longer when incident wavelength becomes shorter, however the relationship between focal length and incident wavelength isn't absolute inverse ratio, it is that the increasing of focal length is faster than the decreasing of incident wavelength. At the same time the numerical result also show that scalar theory can't be used in the rigorous analysis of the dispersion property of subwavelength diffractive microlens.
第一部分为理论分析方法。论文中将旋转体时域有限差分法(BOR FDFD)作为分析具有轴对称结构的亚波长衍射微透镜的理论模型。论文中将相关的光学知识融入到BOR FDTD算法中，给出了它所包含的主要子算法的内容及公式。作为BOR FDTD中的一个最重要的子算法，首次提出了非分劈场量的完全匹配层吸收边界条件(PML ABCs)，大大简化了计算过程。改进了平面波谱传播算法(VPWS)，推导出的公式可以由BORFDTD计算空间内的近场输出数据直接计算出远场的电磁场分布。给出了应用BOR FDTD与VPWS算法分析亚波长衍射微透镜的分析过程和程序流程图。成功地完成了整套BOR FDTD与VPWS算法程序的编制与调试，在使用Matlab语言编程过程中通过矩阵赋值代替循环赋值，极大的提高了计算速度。
第二部分为衍射微透镜的设计。本论文给出了多台阶衍射微透镜和二台阶亚波长衍射微透镜的设计方法、面形公式、设计实例。重点说明了二台阶亚波长衍射微透镜的两种设计方法，并对两种方法作了比较，得出了亚波长结构的脉冲宽度调制法优于连续位相的线性进似法的结论。讨论了深度误差和宽度误差对衍射微透镜的影响，发现相对误差不大时制作误差对衍射微透镜的影响不大。
第三部分是对衍射微透镜光学特性的分析。首先研究了设计参数变化对衍射微透镜的影响，给出了设计焦距和透镜材料折射率变化对衍射微透镜衍射效率、爱里斑半径等的影响，并对分析结果作了定性讨论。然后研究了亚波长衍射微透镜的色散特性，表明亚波长衍射微透镜的焦距随着波长变短而增加，但是波长与焦距之间并非简单的完全反比例关系，而是随着波长的减小焦距增加得更快，同时指出标量理论已不能准确分析亚波长衍射微透镜的色散特性。
This dissertation is about the rigorous vector analysis and design of subwavelength diffractive microlenes. Because the feature size of subwavelength diffractive microlens is smaller than the incident wavelength, scalar diffractive theory cannot be used to the analysis. It also precludes rigorous vector coupled-wave theory for its finite and aperiod structure. In this dissertation an electromagnetic computational method is employed as rigorous vector diffractive theory for the analysis of subwavelength diffractive microlens. The design methods of subwavelength diffractive microlens are presented. The design examples are analyzed by the theory in the dissertation.
In the first section, analysis methods are developed. In the dissertation body-of-revolution finite-difference time-domain method (BOR FDTD) is employed as the theory mode for the analysis of axially symmetric subwavelength diffractive microlens. Relative optical theory is considered in BOR FDTD algorithm. The major sub-algorithms in BOR FDTD are explained, the formulas in each sub-algorithm are given. As one important sub-algorithm of BOR FDTD, Perfect matched layer absorbing boundary conditions (PML ABCs) algorithm with split field components is presented for the first time, which greatly simplified computational process. The vector-based plane-wave spectrum method (VPWS) is improved, which can directly obtain the far field diffractive pattern depending on the output near field data in the BOR FDTD meshes by use of the deduced formulas in this dissertation. The above two algorithms of BOR FDTD and VPWS are used to the analysis of subwavelength diffractive microlenses, the analysis process and program flow
chart are shown. The editing and debugging of the whole program of BOR FDTD and VPWS algorithms by Matlab code are finished successfully, in the program I used matrix to put values for variables instead of using cycle, which significantly increase the computing speed.
The second section is to design diffractive microlenses. In the dissertation multilevel microlenses and binary subwavelength diffractive microlenses are designed, design methods and configuring formulas and design examples are illustrated. Two kinds method of design binary subwavelength diffractive microlenses are illustrated with emphasis. I obtained the result that the method of subwavelength pulse-width modulation is better than the method of linear approximation for continuous phase piece in designing binary subwavelength diffractive microlenses by comparing them. The effects of depth error and width error on diffractive microlenses are discussed. I find that there are little influences on diffractive microlenses when the relative fabrication error is small.
The third section is to analyze the optical characteristic of diffractive microlenses. The influence of changing design parameters on diffractive microlens is investigated. The effects of changing design focal length and microlens material refractive index on diffractive efficiency and airy diffraction disc radius are presented. The numerical results are discussed by qualitative analysis. The dispersion property of subwavelength diffractive microlens is investigated. Numerical result illustrates that focal length becomes longer when incident wavelength becomes shorter, however the relationship between focal length and incident wavelength isn't absolute inverse ratio, it is that the increasing of focal length is faster than the decreasing of incident wavelength. At the same time the numerical result also show that scalar theory can't be used in the rigorous analysis of the dispersion property of subwavelength diffractive microlens.
引文
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