Useful way to compensate for intrinsic birefringence caused by calcium fluoride in optical lithography systems

Calcium fluoride is widely used in optical lithography lenses and causes retardation that cannot be ignored.However, few studies have been conducted to compensate for the retardation caused by calcium fluoride in optical lithography systems. In this Letter, a new index based on orientation Zernike polynomials is established to describe the value of retardation. Then, a method of retardation compensation is described. The method is implemented by clocking calcium fluoride lens elements, and the optimal rotation angles are calculated using a population-based stochastic optimization algorithm. Finally, an example is provided to validate the method.     

基于DDE接口技术的计算机辅助装调方法

根据波前拟合和计算机辅助装调的算法原理,基于动态数据交换（DDE）接口技术和泽尼克多项式拟合技术,采用阻尼最小二乘法,运用MATLAB和ZEMAX联合计算实现了计算机辅助装调。利用DDE接口在ZEMAX和MATLAB之间进行通讯,并用MATLAB编写了能够生成各阶泽尼克圆多项式和环多项式的基底矩阵函数的可视化的计算机辅助装调程序,实现了程序的通用性及易用性。使用该程序对大遮拦比光学系统进行模拟装调,验证了程序的正确性。为了确认泽尼克圆多项式在环域上的相关性对计算机辅助装调结果的影响,分别采用泽尼克圆多项式和环多项式进行模拟装调。模拟结果表明：计算机辅助装调使用这两种泽尼克多项式均可行。     

由重构矩阵存储精度引起的波前重构误差的研究

 采用Southwell区域法波前重构模型对泽尼克多项式的前几项进行了波前重构的数值模拟, 研究了由重构矩阵存储精度原因而引起的波前重构误差。结果表明,对于前六项,以8Bit的数据精度来存储重构矩阵就能保证波前重构误差不超过1.0%,而对于具有更高阶的泽尼克多项式,比8Bit更高的数据精度不会使误差减小。对一个测量所得的波前进行了研究,所得结果和上述结论相符合。     

Analytic design of multiple-axis, multifocal diffractive lenses

In this Letter, we introduce an analytic procedure for designing diffractive lenses using the combination of wavefronts aberrated by Zernike polynomials. We show how to design amplitude-only, phase-only, continuous, and binary lenses providing equivalent results. As an example we apply it to the design of a multiple-axis, multifocal lens. The number of foci and their positions can be easily controlled. Theoretical predictions have been experimentally confirmed. The main advantage of this procedure is that, because it is simple and intuitive, it can be used successfully for the design of complex lenses.     

Nd:YAG棒中的热光效应计算

激光通过工作物质时所产生的热透镜以及应力双折射两种效应是高功率二极管泵浦激光器(DPL)的设计和优化过程中必需考虑的因素.讨论了采用有限差分光束传播法计算热透镜效应及结合材料的弹光系数矩阵计算应力双折射效应的一般方法,并具体针对长6 cm、均匀泵浦的NdYAG棒,计算了这两种效应对光束的影响.结果表明,由于热透镜的聚焦效应,棒中心的光场振幅在通过棒后增大了6%,棒中心与边缘产生了20个波长的相位差;同时,应力引起的双折射造成了约30%的功率损失.     

波面重构中非圆域Zernike正交基底构造方法

针对非圆域波面拟合中Zernike多项式失去正交特性、拟合系数交叉耦合的问题,提出非圆域Zernike正交基底函数构造方法。以圆Zernike为基底,采用Gram-Schimdt正交组构造方法,线性表出单位正交基底。通过构造不同遮光比环形光阑下的正交基底与环Zernike多项式进行比较,验证了此方法的正确性。然后采用圆Zernike多项式和构造的新基底对矩形光阑下的波面进行了拟合,从拟合残余误差、各项基底系数的稳定性、传递矩阵的条件数等分析,结果表明针对特定的非圆域构造的新基底可靠性和抗扰动能力优于圆Zernike多项式。此方法不需要具体求出基底的解析表达式,不同非圆域仅是正交化系数矩阵发生改变,为非圆域正交基底构造提供了一种新途径。     

基于泽尼克多项式进行面形误差拟合的频域分析

获得泽尼克多项式的频谱信息是正确利用该多项式进行误差拟合的关键。推导出了泽尼克多项式的傅里叶变换公式,在频域中分析了不同阶数该多项式的径向频谱信息和幅角频谱信息,得到了有限项泽尼克多项式能够有效表达面形误差的最大径向空间频率和角频率。基于频域分析理论,利用泽尼克多项式对不同口径局部误差进行了拟合,并利用齐戈（Zygo）干涉仪对带有不同面形误差的光学元件进行了试验分析。结果表明,当误差的径向空间频率或角频率超出泽尼克多项式所能表达的频谱范围时,拟合误差迅速变大。     

Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials,Logan–Shepp ridge polynomials,Chebyshev–Fourier Series,cylindrical Robert functions,Bessel–Fourier expansions,square-to-disk conformal mapping and radial basis functions

We compare seven different strategies for computing spectrally-accurate approximations or differential equation solutions in a disk. Separation of variables for the Laplace operator yields an analytic solution as a Fourier–Bessel series, but this usually converges at an algebraic (sub-spectral) rate. The cylindrical Robert functions converge geometrically but are horribly ill-conditioned. The Zernike and Logan–Shepp polynomials span the same space, that of Cartesian polynomials of a given total degree, but the former allows partial factorization whereas the latter basis facilitates an efficient algorithm for solving the Poisson equation. The Zernike polynomials were independently rediscovered several times as the product of one-sided Jacobi polynomials in radius with a Fourier series in θ. Generically, the Zernike basis requires only half as many degrees of freedom to represent a complicated function on the disk as does a Chebyshev–Fourier basis, but the latter has the great advantage of being summed and interpolated entirely by the Fast Fourier Transform instead of the slower matrix multiplication transforms needed in radius by the Zernike basis. Conformally mapping a square to the disk and employing a bivariate Chebyshev expansion on the square is spectrally accurate, but clustering of grid points near the four singularities of the mapping makes this method less efficient than the rest, meritorious only as a quick-and-dirty way to adapt a solver-for-the-square to the disk. Radial basis functions can match the best other spectral methods in accuracy, but require slow non-tensor interpolation and summation methods. There is no single “best” basis for the disk, but we have laid out the merits and flaws of each spectral option.     

头盔显示器自由曲面光学组件的优化设计

根据设计要求计算了非共轴球面光学系统的初始结构,选定泽尼克多项式为自由曲面面型,利用正交泽尼克多项式与像差的关系,调整曲面参数校正系统的像差,运用CODE V进行系统优化。通过设计实例对比了直接优化和分步优化方法,由对比结果可知,分步优化更适用于自由曲面光学组件的设计。仿真结果表明:系统在整个视场范围内,点列斑直径小于27 m,网格畸变的最大值为-7.15%,30 lp/mm处的MTF大于0.197。     

用Zernike多项式实现光机分析的技术方法

由于光学软件不能直接利用有限元分析的结果，而Zernike多项式的各项与光学像差有对应关系，因此常用Zernike多项式作为光机接口。针对目前常用轴向位移作为拟合量描述拟合面形的不足，给出了几种常用的表面位移校正方法并说明了其优缺点。用具体实例比较各校正位移，并对其进行Zernike多项式拟合，从拟合系数的差异可以看出,曲率比较大的表面必须采用校正位移进行拟合。最后指出：在不知道初始表面方程的情况下，轴向和法向校正位移均采用从初始表面出发的方法，如果已知初始表面方程，则轴向校正位移采用从变形表面出发的方法，法向校正位移仍采用从初始表面点出发进行计算。     

Modal Description of Wavefront Aberration in Non-circle Apertures

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自适应光学系统的模式法数值模拟

建立了利用模式法笃自适应光学系统进行数值模拟的理论模型，编制了计算程序，并与激光大气传输计算程序衔接起来，进行了大量数值模拟计算。首次发现：存在泽尼特多项式展开的最佳项数。大于一定项数的展开式的效果迅速变坏，竖排和斜排经特面式展开有类似的结果。文献中认为可以采用的１５项经特多项式展开的效果不好，最佳项数随着横向风速的增加而减小，在风速较大时最佳项数下的模范地结果稍好于直接斜率控制法的结果。     

Zernike多项式拟合干涉面方法研究

本文就专著和文献中关于Zernike多项式拟合干涉波面几乎都建设采用Gram-Schimdt正交化方法,而不采用比较简单的传统经典的最小二乘法问题进行了深入研究,从理论和实践上严格地证明了两种方法的等价性。实践中发现,用最小二乘法求解Zernike多项式拟合系数的速度比用Gram-Schimdt正交化方法提高了三倍之多。由于在精密光测技术中,Zernike多项式已被广泛采用,因此,“等价性”的证明具有重要意义,并对于其它类似问题也有着普遍的参考价值。     

Gram-Schmidt orthogonalization of the Zernike polynomials on apertures of arbitrary shape

An orthonormal hexagonal Zernike basis set is generated from circular Zernike polynomials apodized by a hexagonal mask by use of the Gram-Schmidt orthogonalization technique. Results for the first 15 hexagonal Zernike polynomials are shown. The Gram-Schmidt orthogonalization technique presented can be extended to both apertures of arbitrary shape and other basis functions.     

Method of reconstructing wavefront aberrations by use of Zernike polynomials in radial shearing interferometers

A novel method of reconstructing wavefront aberrations by use of Zernike polynomials for radial shearing interferometers is discussed. This method uses matrix formalism to calculate the Zernike coefficients of a wavefront under test and shows the validity of reconstructing an arbitrary wavefront aberration from an interferogram taken by a radial shearing interferometer. We also propose a new interferometer setup to determine the shape and the direction (concave or convex) of wavefront aberration in a single measurement.     

Orthonormal polynomials in analysis of single-mode fiber coupling

Zernike polynomials have been widely used for wave-front analysis because of their orthogonality over a uniform circular pupil. However, the pupil is not uniform but weighted by the backpropagated fiber mode in analyzing fiber coupling efficiency. Zernike polynomials are not appropriate for a weighted pupil due to their lack of orthogonality over such pupil. We emphasize the advantages of using orthonormal polynomials in fiber coupling systems. The orthonormal polynomials over weighted pupil are derived by matrix approach. The effects of primary aberrations are investigated based on the orthonormal polynomials. The accuracy of the Strehl ratio approximation for primary aberrations is evaluated.     

利用泽尼克系数求取衍射光栅的分辨本领

光栅分辨本领检测设备的焦距通常达几米甚至十几米,采用直接测量法难度大、成本高,利用衍射波前间接求取光栅分辨本领是解决该问题的有效途径之一。在光栅光谱成像傅里叶变换理论基础上,建立了利用泽尼克多项式拟合系数求解衍射光栅分辨本领的归一化模型,揭示了光栅衍射波前与分辨本领的求取关系,提出了依据泽尼克多项式拟合系数求取衍射光栅分辨本领的新方法。根据该方法实测了一块衍射光栅的分辨本领,并与直接测量法进行对比测试。结果表明该方法误差小于4.42%,降低了分辨本领的测试难度,是衍射光栅分辨本领求取的有效手段,应用于ZYGO干涉仪等仪器中,通过简单的波前测试即可得到定量的衍射光栅分辨本领指标。     

Zernike aberration coefficients transformed to and from Fourier series coefficients for wavefront representation

The set of Fourier series is discussed following some discussion of Zernike polynomials. Fourier transforms of Zernike polynomials are derived that allow for relating Fourier series expansion coefficients to Zernike polynomial expansion coefficients. With iterative Fourier reconstruction, Zernike representations of wavefront aberrations can easily be obtained from wavefront derivative measurements.     

采用密度泛函理论与分子动力学对聚甲基丙烯酸甲酯双折射性的理论计算

双折射性是各种光学材料的重要性能之一,具有高双折射率的光学材料在诸多研究及工业领域的应用越来越广泛.然而,作为常用的光学薄膜及光波导材料之一的聚合物材料的双折射性通常却很弱,只能通过实验对其双折射率进行大致的表征,缺乏对其双折射率的系统性理论计算,从而限制了提高聚合物双折射性的研究.本文建立了从聚合物的单体分子结构到多分子链的系统性的双折射率理论计算方法,并借助此方法研究了导致聚合物弱双折射性的限制因素.以聚甲基丙烯酸甲酯(PMMA)为研究对象,运用密度泛函理论研究了其本征双折射率,这里的本征双折射率是指分子链完全取向时其单体单元的双折射率.计算结果表明其本征双折射率高达0.0738左右,并且通过计算给出了PMMA单体单元的平均双折射率色散曲线.采用分子动力学方法研究了该聚合物(包含20个分子链)的材料双折射率.理论计算结果表明,尽管该聚合物具有较大的本征双折射率,但是由于聚合物中分子链取向度极低,聚合物材料最终表现出来的双折射率只有0.00052.本文建立的研究方法及研究结果为研究增强聚合物材料双折射性提供了理论依据.