Geometrical aberrations of self-imaged line gratings

We analyze the properties of a self-imaging system from the point of view of aberration theory. We examine analytically and numerically the geometrical aberrations that are observed in the self-image of a parallel-line grating. We first derive the raytracing equations for determining the optical path of a self-imaging ray with the order of diffraction l. We then obtain the third- and fifth-order contributions to ray aberrations which arise from the difference between the optical paths of a self-imaging ray and of an actual ray. We show that the overall ray aberrations are entirely undercorrected. The ray aberrations approach zero as the ratio of the grating constant to the wavelength of light becomes large enough. In a case of unit magnification, no curvatures are observed in the self-imaged lines. If the magnification is bigger than unity, the light rays passing through the point in a positive or negative domain of the aperture variable contribute to the formation of the curved images. The image evaluation technique discussed here can be useful in the various applications related to the self-image formation of a parallel-line grating and it can also provide insight into the self-imaging of other periodic objects.     

Aberrations of self-imaged patterns with two-dimensional periodicity

We develop a geometrical theory of aberration in the self-imaged patterns with two-dimensional periodicity. The patterns are considered to be a crossed-line grating or a periodic array of finite apertures. We first derive the raytracing equations for determining the optical path of a self-imaging ray. We then find the third- and fifth-order contributions to the wavefront aberration which arise from the difference between the optical paths of a self-imaging ray and of an actual ray. We also derive the expression of the ray aberration from the wavefront aberration. The ray aberration is classified into five distinct types by analogy with the cases in a refracting lens system. We show that the overall ray aberration is entirely undercorrected and the aberrated image patch is decentered from an ideal image point in the direction parallel to the direction tangent vector of a chief ray. The image evaluation technique discussed here will be useful in various applications related to self-image formation of two-dimensionally periodic patterns.     

First-order aberration of a misfocused self-imaging system

We analyze the characteristics of a self-imaging system that appear across a misfocused image plane. We approach this problem from the point of view of aberration theory. First, we derive the aberration functions (corresponding to several self-imaging rays of different orders) which are linear in a small shift of focus, and then we examine their roles in the amplitude spectra of misfocused self-images. We show that the aberration of the mth-order self-imaging ray is responsible for a lateral shift of the image amplitude component of frequency m/Mp, where M is the magnification of the system and p is the fundamental period of an object grating. We also analyze the role of wavefront aberrations in the image irradiance as an observable quantity. We then show that the theoretical estimation of irradiance of the aberrated image is in good agreement with the experiment and the focus-shift aberration can exert a severe effect on the irradiance spectrum of image in a complicated way.     

面对称光学系统的初级波像差理论研究

非旋转轴对称光学系统波像差理论的建立有利于理解旋转 轴对称光学系统的装调误差和离轴三反射光学系统等非旋转轴对称光学系统的选型设计. 本文利用旋转轴对称球面光学系统的全口径初级波像差理论推导了子孔径面对称光学系统的初级波像差分布公式, 证明了面对称光学系统中的节点像差理论, 即除球差外的所有初级像差的零值节点偏离视场中心, 而不再是视场的旋转对称函数; 并首次阐述了多零值节点初级非对称像差产生的根源和变化特性. 该理论可以有效指导非对称光学系统初始结构的选择和优化设计过程. 关键词： 面对称 像差 几何光学     

Aberrations in fractional-Talbot images of periodic binary gratings

We discuss the ray-optical aberrations which appear in the fractional-Talbot image of a periodic binary grating with coherent illumination. First we examine the complex amplitude of an aberration-free imaging field at a fractional-Talbot plane. We then trace the path of a diffracted ray of specific order which contributes to the fractional-Talbot imaging. Next we formulate the focus-shift and third-order aberrations which arise from a focusing error and a fourth-order approximation of the path length, respectively. We then evaluate the amplitude and phase of an aberrated imaging field that are represented in terms of aberration functions. When the grating period decreases to approach the optical wavelength, the aberrations of lower-order rays are shown to be more influential on the fractional-Talbot imaging field than those of higher-order rays. The theory of aberration discussed here could be very useful in evaluating the fractional-Talbot image of a periodic binary grating.     

光刻机投影物镜的像差原位检测新技术

提出了一种新的光刻机投影物镜像差原位检测(AMF)技术。详细分析了该技术利用特殊测试标记检测投影物镜球差、像散、彗差的基本原理,论述了该技术利用对准位置坐标计算像差引起的成像位置偏移量的方法。实验结果表明AMF技术可实现球差、彗差、像散等像差参量的精确测量。AMF技术考虑了光刻胶等工艺因素对像差引起的成像位置偏移量的影响,有效避免了目前基于硅片曝光方式的彗差原位检测技术对离焦量、像面倾斜等像质参量限制的依赖。     

紫外高分辨率像散校正型罗兰全息光栅设计

为了降低紫外高分辨率罗兰光栅的像散对光谱像高展宽的影响,提出了在罗兰圆上使用球面波非对称曝光设计思路。推导完全校正离焦和子午彗差的表达式,讨论了多种罗兰光栅记录结构的局限性,优选适合校正紫外高分辨率罗兰光栅的优化方法。通过全息光栅像面展宽表达式, 指出像散和弧矢彗差是影响光谱像高的主要因素,并分配了两者的优化权重。利用这种优化思路,设计了工作波段110～200 nm紫外高分辨率罗兰光栅,同时对比分析了传统光栅的像差系数和像高变化规律、像面结构和光谱分辨率。结果表明,和传统罗兰光栅分辨率处于同一数量级的情况下,所设计光栅光谱像高由25 mm降低到1.5 mm,谱面能量集中度有显著的提高。     

A Variation of Schwarzschild Telescope: Golden Section Solution with Two Concentric Spheres and Its Extension to Finite Distance Solutions

Imaging by two concentric spheres is studied by geometrical method. As the system is symmetric with respect to the common center of curvature, peripheral aberrations, namely coma, astigmatism, and distortion, do not appear. The imaging surface is a sphere that shares the center of curvature with two mirrors. As we solve the zero spherical aberration condition for imaging of an object at infinite distance, the golden section number appears as the ratio of the distance between two mirrors and the radius of the primary mirror. For imaging of an object at finite distance, we have to increase the radius of the second mirror. This system also has no peripheral aberrations. Making the object surface spherical, we can obtain a flat image surface. As the spherical aberration is zero in the original system, spherical aberration, coma, and astigmatism remain zero when the object surface is curved.     

畴腐蚀掺镁铌酸锂可调阵列光分束器的研究

研究了不同畴腐蚀深度的掺镁铌酸锂二维六角可调阵列光分束器的分数Talbot效应.对不同Talbot分数β和不同畴腐蚀深度的阵列光分束器Talbot衍射像进行了数值模拟理论研究.模拟结果表明,Talbot分数β可以改变Talbot衍射像的周期及结构分布,而畴腐蚀深度可有效调制衍射像的光强分布.在理论研究的基础上,设计并制备了具有不同畴腐蚀深度的掺镁铌酸锂二维六角阵列光分束器,对其在不同Talbot分数β条件下的分数Talbot效应进行了通光实验研究,实现了畴腐蚀阵列光分束器对近场Talbot衍射光强分布的调制,实验结果与理论研究结果一致.     

Third-order aberration analysis of tilted-pupil optical systems

The aberration field of an optical system with a tilted pupil is explored through expanding the vector expressions of the third-order wavefront aberrations. First, the vector forms of the wavefront aberrations are modified to obtain the aberration expressions with the pupil tilted; full field displays of coma and astigmatism in this situation are given. Then, the third-order aberration formulas with the pupil decentered and tilted simultaneously are derived and discussed. Finally, an example is taken to certify the validity of aberration distribution properties.     

Aberrations of a slightly inclined Gaussian beam in a symmetric optical system

We discuss the Seidel-type aberrations which degrade the quality of the Gaussian beam, while the beam axis is slightly inclined to the axis of an optical system. First we examine the evolution of a complex-source-point spherical wave through multiple spherical boundaries. Then we find a ray-optical solution for the diffracted beam that can be represented as a function of Seidel-type aberrations. We also formulate the characteristic parameters of the aberrated beam. If the beam axis is placed at the yz plane, the x component of beam quality factor is degraded by both spherical aberration and field curvature, and the difference in the x and y components of beam quality factor is caused by linear coma, linear astigmatism, and distortion. The fourth order theory of propagation of the Gaussian beam presented here may be applied to determine the parameters of a slightly inclined Gaussian in a symmetric optical system.     

Aberrations of plane-symmetric multi-element optical systems

Based on a recently developed aberration theory of plane-symmetric grating systems, its aberration coefficients are firstly extended to be applicable to the plane-symmetric refractive optical surface. To justify the linear approximation of aperture ray in a multi-element system, the ray aberrations of double-element systems with the third-order accuracy of aperture ray are derived and expressed as the form of wave aberrations. The numerical comparisons are then made between the ray tracing, the aberration expressions with the linear and with third-order accuracy of the aperture ray. Moreover, the modifications of the wave and ray aberrations are discussed in the paper when the position of the principal ray changes.     

Design of Plane-Symmetric Low <Emphasis Type="Italic">F</Emphasis>-number Reflective System Based on Perturbation Aberration Model

All-reflective optical systems have theoretically no chromatic aberration and are suitable for wide spectral application, but co-axial reflective systems are difficult to design as low F-number optical systems. In this paper, the aberration of plane-symmetric optical systems with low F-number is analyzed based on the wavefront derived from optical path length. Up to third-order aberrations are classified into three categories by their characteristics. The reduction method of the dominant aberration, astigmatism-like aberration, is proposed. In the method, the Gauss image plane is modeled by means of a simplified mirror surface with third-order approximation. Because the system is plane-symmetric, symmetric cross section and asymmetric cross section are modelled differently. The design example of a three-mirror system with F/2 is shown.     

Ray aberrations of two-dimensional oblique lattices on the self-image plane

We extend the geometrical theory of aberration for a self-imaging system to the case of two-dimensional oblique lattices. In our approach, the fundamental translation vectors of the lattice are not restricted in both length and orientation. Evaluating the disturbance of light through the oblique lattice under coherent illumination, we find the conditions of constraint which limit the self-imaging of the oblique lattice. Various types of oblique lattices are shown to obey the self-imaging conditions. We derive the equations to trace the optical paths of self-imaging rays and then analyze the ray aberrations which arise from the difference between the optical paths of a self-imaging ray and of the corresponding actual ray. The ray aberrations are shown to disappear when the periods of the lattice are large compared with the wavelength of light. We find that the ray aberrations carried by self-imaged oblique lattices are totally undercorrected and the aberrated image patches are displaced along the direction tangent vector of a chief ray.     

Diffraction patterns of off-axis gaussian beams in the presence of third-order spherical aberration in the optical system

The intensity distribution of diffraction patterns for off-axis gaussian beams at the image plane of an optical system with third-order spherical aberration is calculated by using the concept of the beam aberration function previously introduced by the present authors. The results show how the peak intensity of the diffraction patterns diminishes and how the transversel focal shift occurs as a function of the incident position of the beams at the pupil plane and their beam size.     

电磁复合聚焦—偏转球面阴极透镜的相对论象差理论

本文应用变分原理研究了电磁复合聚焦-偏转球面阴极透镜的相对论象差理论。在考虑阴极面逸出电子具有一定的初能量和初角度分布,物场和象场弯曲以及阴极面上磁场和横向电场不为零的情况下,导出了任意理想象面上的一级近轴横向象差和包括色球差在内的全部三级几何横向象差,以及各种特殊类型象差系数的明显表达式。本文导出的象差公式不仅普遍适用于宽束和细束阴极透镜,而且普遍适用于相对论或非相对论,阴极和屏为球面或平面时的各种情况。本文以复数描写轨迹,用矩阵表示象差,形式简洁,适合于计算机计算。 关键词：     

Effects of low order atmosphere-turbulence aberrations on the vortex carrying Gaussian beam

The normalized intensity distributions at the focal plane of a vortex carrying Gaussian beam propagating in weak/middle turbulent atmosphere channel with z-tilt aberration, defocus aberration, astigmatism aberration or total turbulent aberration are discussed by numerical calculation. Our results show that the effect of z-tilt aberration on the intensity distribution of optical vortex beam is main effect of total turbulent aberration. In weak turbulent region, the effects of defocus and astigmatism aberration on the intensity distribution of a vortex carrying Gaussian beam can be ignored. In middle turbulent region, the effect of z-tilt aberration is still the most significant, but the effects of defocus and astigmatism aberrations, specially the effect of astigmatism on the center dark core of the intensity distribution at focal plane, can no longer be disregarded. Our results also show that for three low order aberrations, the beam with values of the topological charge, the beam have larger beam-radius and undergo smaller effects of z-tilt turbulent aberration on the doughnut distribution. For defocus aberration, the radius of the center dull of beam intensity increases with the values of topological charges increasing. And for astigmatism aberration, the beam with odd number topological charges, the center dull speck of beam intensity becomes to the bright speck. But for even number topological charges, the center dull speck of beam intensity maintains the dull one.     

Effect of primary aberrations on transverse displacement of laser speckle patterns

The effect of primary aberrations on the transverse displacement of laser speckle patterns in the real image plane due to transverse displacement of an object is studied in the approximation of Fourier optics. Primary aberrations cause: (i) a complementary displacement of the speckle pattern; and (ii) its decorrelation. The main features of complementary displacements are as follows: (a) the overall complementary displacement is the sum of complementary displacements caused by each one of the primary aberrations; (b) the complementary displacement caused by the spherical aberration is uniform in the image plane; (c) spherical aberration and curvature of field cause complementary displacements that are parallel to the object's displacement; (d) the complementary displacement caused by distortion is always zero on the optical axis; (e) the complementary displacement caused by distortion is free of decorrelation; (f) complementary displacements are non-proportional to the object's displacement; and (g) complementary displacements depend on the position of the illuminating source with the exception of the complementary displacement caused by distortion.     

泽尼克多项式校正全息阵列光镊像差的实验研究

像差会影响光镊对粒子的捕获效果. 全息阵列光镊中, 像差不仅来自光学元件, 由特定算法设计的光阱相位片也会在光路中引入像差. 本文通过液晶空间光调制器加载泽尼克多项式相位图, 对全息阵列光镊中由光栅透镜组型算法引起的像差进行校正. 结果显示: 利用三阶泽尼克多项式可有效消除光路中由光栅透镜组型算法引 起的慧差, 使得捕获2 μm聚苯乙烯小球的阵列光阱刚度提高了约40%; 对比不同项的像差校正结果发现, 全息阵列光镊中由算法引起的慧差 与光学元件引起的像差一样, 也会对阵列光阱的捕获效果产生较大影响; 同时根据一阶像差校正结果可得光栅透镜 组型算法对于一阶泽尼克像差具有鲁棒性. 实验结果表明, 对全息阵列光镊中由 算法引起的像差进行校正, 对于提高光阱的捕获效果和深化对算法特性的认识都具有重要意义.     

Differential algebraic method for arbitrary-order chromatic aberration analysis of electrostatic lenses

In this paper, differential algebra is applied to calculate arbitrary high-order chromatic aberrations of electrostatic electron lenses. Expressions of differential algebraic form of high-order combined chromatic aberration coefficients are obtained and arbitrary-order chromatic aberrations can be calculated numerically. As an example, a typical Schiske's electrostatic lens has been studied. All the first- to third-order chromatic aberration coefficients of the lens have been calculated, and the pattern of the first-order chromatic aberration has been given as well.