6×6 matrix formalism of optical elements for modeling and analyzing 3D optical systems

Although conventional 2×2 ray matrices, i.e. ABCD matrices, provide a convenient means of obtaining initial estimates of the performance of an optical system during the early stages of the design process, they are suitable only for optical systems with the axisymmetric property. Accordingly, this study utilizes the 6×6 matrix formalism, which was proposed by our group, for optical boundary surfaces to develop a new approach for modeling and analyzing 3D optical systems comprising multiple lenses and/or mirrors and clarifying many of the system’s basic properties, e.g. the effective focal length, primary aberration and cardinal points, and so forth. To reduce the complexity of the modeling process, general matrix formalisms of sub-systems in a 3D optical system are presented. The validity of the proposed approach is evaluated by modeling and analyzing a simple 3D optical system and comparing the results with those obtained from the skew ray-tracing computer program and commercial optical design software. The results confirm that the proposed methodology provides a convenient means of obtaining initial insights into a variety of 3D optical systems with non-coplanar axes.     

General method for determining the first order gradients of skew rays of optical systems with non-coplanar optical axes

Current commercial software for the analysis and design of optical systems uses finite difference (FD) approximation methodology to estimate the gradient matrix of a ray with respect to system variables. However, FD estimates are intrinsically inaccurate and are subject to gross error when the denominator is excessively small relative to the numerator. This paper avoids these problems and determines the gradient matrix of the exit ray traveling along an optical system with a non-coplanar axis. To achieve this, the gradient matrix of the rays reflected/refracted by flat or spherical boundary surfaces are first determined by directly differentiating the skew-ray tracing equations. By introducing a Jacobian matrix, which represents the partial derivatives specifying the rates of changes between boundary variables and element variables, one can obtain the gradient matrix of the exit ray of an element with respect to its independent variables. This methodology will be useful in the analysis of rays and in design of optical systems with non-coplanar axis. A right-angle prism is used as illustrative example to validate its applications.     

同轴柱面光学系统的象差

本文首先由柱面系统中描写光线所须的参数和对称性,确定了同轴柱面系统的独立象差数是八个,而对无限远物体而言则只有六个。而后由矢量代数方法证明空间光线在柱面系统中的折射光路与发生折射率改变后的子午截面内光路全同,由此得出柱面系统有别于球面系统的二种象差与色差相当。从而导出了校正柱面象差的条件并对柱面望远镜象差作了估值。结果表明,校正柱象差一般而言是很困难的,另一方面虽不加以校正影响也不算严重。最后讨论了柱面望远镜的高斯光学,结果表明,望远镜使一方向余弦按定比例缩小,而另一方向余弦不变时,它所成出的直线的象是弯曲的。     

复杂棱镜的实际光路追迹

本文解决了在光学软件中对各种复杂棱镜进行实际光路追迹的问题,给出了对其中各种偏心、倾斜工作面通用的坐标变换公式,并着重介绍了对屋脊棱镜、角反射镜等具有特殊几何光路的棱镜的光路计算方法。     

A novel ray refraction matrix for curved interface

A problem in fundamental 2 × 2 ray matrix for refraction at curved interface proposed by A. E. Siegman was found out and a novel one was derived in this paper. An experiment is introduced in detail to verify the reasonability of the novel ray matrix. Using the novel 2 × 2 ray matrix, augmented 5 × 5 ray matrix of refraction at misaligned curved interface between media of different refractive indices was deduced. With the refraction matrix, it is easy to characterize the effect of an astigmatic thick lens. The augmented ray matrix approach was applied to model and estimate the performance of an optical alignment system. Utilizing these matrices, one can readily design and evaluate optical systems, where contain astigmatic elements such as tilted spherical or cylindrical lenses, mirrors and so on. These results are also useful for cavity design, alignment, ray tracing and beam position control in 3D optical systems.     

A general calculation of the 3-D disk of least confusion using skew ray tracing

A general and analytical method is proposed to determine the 3-D contour of the disk of least confusion (DLC) produced by an optical system for a point light source located at any arbitrary position in the object space. By using the skew ray tracing method, the sensitivity method, and the Jacobian matrix, we can analytically obtain the caustic and the marginal rays. Consequently, the DLC can be determined by the intersection of the obtained caustic and marginal rays. In addition, this proposed method covers common cases with special light sources, e.g. mutually parallel incident rays parallel to the optical axis, or a point light source located on the optical axis. This proposed methodology will be useful in the analysis of rays and in the design of optical systems with a general point light source. A general case is used as illustrative example to validate its applications.     

Measurement of Focusing Properties of Axially Asymmetric Lenses Using Moire Technique

Most lenses composed of spherical or aspherical surfaces have symmetrical focusing properties about an optical axis. On the other hand, unconventional optical systems such as cylindrical lenses, astigmatic or progressive power glasses have axially asymmetrical focusing properties. In this study, the method of measuring focusing properties by the moire technique is extended to asymmetrical lenses. Some unconventional lenses were experimentally tested, and the focusing power in the interesting region and azimuthal direction of the aperture was evaluated.     

失调多腔镜型环形谐振腔共轭光轴位置精度分析

为了详细分析由多个反射镜构成的谐振腔内的闭合光轴(共轭光轴)的特性以及存在条件, 首先从光束传输坐标变换的角度, 分析了由多个平面镜所构成的谐振腔的共轭光轴存在条件, 得出了奇数平面镜谐振腔仅当平面镜间具有高精度的相同垂直度时才存在闭合光轴, 而偶数平面镜谐振腔总是存在闭合光轴的结论, 并给出了腔共轭轴随腔镜方向失调而产生的角度变化关系. 然后从矩阵光学的角度分析包含球面镜的谐振腔的共轭光轴在腔镜间传输时的方向变换问题, 具体分析了由两个平面镜和一个球面镜构成的三角形环形谐振腔的闭合共轭光轴存在的问题. 结果表明, 当不同的镜子出现角度偏差时, 腔内仍然存在闭合的共轭光轴, 并给出了对应的腔轴变化以及新谐振面的位置及方向. 由此表明由于球面镜的加入降低了谐振腔共轭轴存在性对平面镜间高精度平行度的要求, 为基于高品质光学无源腔的光谱测量技术的高精度装调工作提供了理论指导.     

基于光线光学的非线性自聚焦现象的仿真分析

采用光线光学方法对非线性自聚焦现象进行仿真, 能够从宏观上直观地体现强激光的传输过程, 同时避免采用近轴近似、自相似近似等. 本文采用在光传输路径上垂直于光轴切片的方法, 将光的非线性传输转化为切片上的光对折射率的调制作用和切片间的线性传输. 在切片端面上统计光强后对量化误差进行了抑制, 而线性传输过程采用了亚当斯法求解光线方程从而解决了龙格库塔法等不能用于非线性光传输仿真的问题. 仿真结果显示, 强激光自聚焦在轴上有多个焦点, 且第一个焦点的位置随光功率的增大而更靠近入射位置; 由于追迹的是实际光线, 故可以得到近轴区以外区域自聚焦及成丝(环)的情况, 这对于强激光系统安全是有重要意义的. 利用已有的同样基于光线追迹方法的光学设计、仿真软件, 可以把非线性自聚焦介质和线性介质结合起来, 仿真光在实际强激光系统中的传输. 关键词： 实际光线追迹 非线性自聚焦 光传输仿真     

Veselago’s lens consisting of left-handed materials with arbitrary index of refraction

We study Veselago’s lens with arbitrary index of refraction and characteristic impedance. Using a full wave optics calculation, we show that this lens can be considered as an imaging system and we derive the appropriate lens formula. The lens with arbitrary index and impedance retains some of the properties of the matched lens, such as the invariance of its optical axis, three-dimensional imaging and easy manufacturing, but it loses the property of sub-wavelength resolution. We also show that identical results can be obtained for the impedance matched lens in the framework of paraxial geometrical optics, from which it can be inferred that optical systems containing such a lens can be studied and designed using traditional ray-tracing tools.