Caustics in a meridional plane produced by plano-convex conic lenses

We study the formation of caustic surfaces formed in both convex-plano and plano-convex conic lenses by considering a plane wave incident on the lens along the optical axis. By using the caustic formulas and a paraxial approximation, we derive analytic expressions to evaluate the spherical aberration to the third order, and a formula to reduce this aberration is provided. Furthermore, we apply the formulas to evaluate the circle of least confusion for a positive lens as a function of all parameters involved in the process of refraction through the conic lenses.     

Properties of caustics produced by a positive lens: meridional rays

We study the formation of the caustic surfaces formed in both convex-plane and plano-convex spherical lenses by considering a plane wave incident on the lens along the optical axis. Using the caustic formulas and a paraxial approximation we derive analytical expressions to evaluate the spherical aberration to third order. Furthermore, we apply the formulas to evaluate the circle of least confusion for a positive lens.     

Designing reflectors to generate a line-shaped directivity diagram

A novel method of designing a mirror surface to generate a directivity diagram represented as a vector function of one argument is presented. A general relationship for the mirror surface for an arbitrary illuminating beam wavefront is derived as an envelope of a parametric family of surfaces. Each surface in the family transforms the input beam into a beam with plane wavefront of desired direction. For the spherical illuminating beam the mirror surface is given as the envelope of the family of rotational paraboloids. The envelope is represented as a family of curves given by the intersections of paraboloids with circular cones of rays from the point source.     

自由电子激光中的等效折射率

推导出适用于三维自由电子激光中的电子束的等效折射率，并说明高斯平面波分布的激光场是电子束等效折射率为平方分布率即类透镜介质的结果，对指数增长区的复数增益的三次方程作了改进，而且，还选用典型的自由电子激光的参数进行了数值计算，并对结果作了简要的分析。     

Measurement of parameters of simple lenses using digital holographic interferometry and a synthetic reference wave

The use of digital holographic intrerferometry in the testing of simple thin lenses is explored. Focal length, radius of curvature, and refractive index are the lens parameters that can be determined using this method. The digital holograms using the lens under test are recorded at various positions of the test lens using off-axis geometry. This is combined with a digitally computed plane wavefront to determine the curvature of the light beam emerging from the test lens. Focal length is the position of the test lens where a single fringe results. The radius of curvature of the test lens is also determined similarly using a long focal length lens to concentrate a collimated beam onto the test lens. The nonuniformities on the lens surface could also be found by using this method. The implementation of the method is shown by using computer simulations in the case of biconvex lenses. The method can be utilized to measure the parameters of plano-convex and concave lenses also.     

Examination of the effect of the fibrous structure of a lens on the optical characteristics of the human eye: a computer-simulated model

We introduce a model of the human eye for which we take into consideration the laminated nature of lens fibers. The thickness of each lamina is 5.6 μm; thus the lens comprises 300 eccentric lenses of minute dimensions. The index gradient of the lens is such that the index of refraction increases exponentially from the lens core to its peripheral zone. A vector ray-tracing technique is employed to study the optical haracteristics of the system. Both paraxial and marginal rays are simulated, and the angles of incidence vary from 0° to ±20°. Special attention is given to the meridional caustic surfaces as well as the wave-front distortion of the refracted rays. A quasi-Newton optimization technique is employed to obtain the best parameters for the system. A computer modeling program, written in FORTRAN 77, is used to simulate a ray's refraction through the multisurfaces of the eye. The results show full agreement with previous data and that the cornea is responsible for eliminating possible spherical aberration of the system.     

Electromagnetic field for a tightly focused beam incident upon ordinary and layered plane surfaces

A solution procedure is developed for the determination of the electromagnetic field that results from the interaction of a tightly focused beam with a plane surface with and without a layer. The effects of angle of incidence, relative index of refraction, polarization, layer thickness, and incident beam profile on the resulting electromagnetic field distribution are demonstrated.     

On the nature of the so-called subsurface longitudinal wave and/or the surface longitudinal “creeping” wave

The elastic wave field of certain angle beam probes used for nondestructive testing of solid materials, like steel, has been shown to exhibit a so-called subsurface longitudinal wave, i.e., a wavefront traveling with the pressure wave speed having a beam angle of approximate 74° in steel. In addition, this wavefront is supposed to be connected to the stress-free surface via a headwave, which radiates a nearly plane wave with shear velocity into the bulk material under an angle of 33°, approximately, and giving rise to a strongly attenuated longitudinal creeping wave on the surface. In the present paper we utilize a new numerical scheme for the computation of elastodynamic wave fields in nearly arbitrary environments, called elastodynamic finite integration technique (EFIT), to predict the above-mentioned wave features quantitatively. Furthermore, we employ several analytical and analytical-numerical integration procedures to evaluate the angle beam probe plane wave spectrum in terms of an inverse spatial Fourier transform. This gives rise to a theoretical interpretation of the physical origin of the numerically computed EFIT wavefronts. Essential results are as follows: the particular wavefronts of angle beam probes, as referred to in this paper, are exclusively associated with afinite aperture radiating into an elastic half-space; they cannot be explained in terms ofsingle homogeneous and inhomogeneousplane waves. The subsurface longitudinal wave emerges from the superposition of the edge pressure waves of the transducer, resulting in a propagation with pressure wave speed, but, in the near-field, where it is often employed, it is not longitudinally polarized. On the surface, and very close to it, the superposition of the subsurface longitudinal wave and the head waves associated with the probe edges gives rise to a strongly attenuated wavefront exhibiting longitudinal as well as transverse particle displacement components, but neither a surface wave nor a creeping wave is really involved. The bulk shear wavefront is not an appendix of the head wave but the geometric optical shear wave radiation pattern of the finite probe.This article is dedicated to Professor Dr. Paul Höller on the occasion of his 65th birthday.     

Caustics caused by refraction in the interface between an isotropic medium and a uniaxial crystal

In general, a caustic by refraction at an arbitrary surface is commonly known as a diacaustic. We study the formation of the diacaustic in a plane interface between an isotropic medium and a uniaxial crystal, for both ordinary and extraordinary rays, when the crystal axis is perpendicular to the plane of incidence and when it lies in the plane of incidence. For the latter case two special positions of the crystal axis with respect to the normal to the refracting surface for the extraordinary rays are treated.     

用于毫米波焦面阵成像系统的扩展半球介质透镜

毫米波成像是近年来毫米波领域的一个研究热点,而焦面阵成像因其具有实时成像的优点更加受到重视。中分析可用为面阵成像的扩展半球介质透镜。这种焦面阵成像结构将集成天线阵贴附在透镜背面接收透镜聚焦的功率,消除了集成天线工作在毫米波频段时存在的表面波对天线性的影响,具有尺寸紧凑、损耗小的特点。采用Stratton-Chu公式和射线追迹分析了电磁波入射到扩展半球透镜上时在其背面的场分布,即透镜的焦区场分布,以获得透镜用于焦面阵成像时的性能。为验证分析方法的正确性,对平面波垂直入射和会聚高斯束入射两种情形进行了实验验证,实验结果与理论分析吻合较好。该透镜天线还可用来消除常规集成毫米波系统中抛物面天线与集成前端之间的过渡,以降低损耗,改善系统性能,也可用于与准光系统的连接或耦合。该结果将对上述应用提供理论指导。     

Zonal wavefront sensor with reduced number of rows in the detector array

In this paper, we describe a zonal wavefront sensor in which the photodetector array can have a smaller number of rows. The test wavefront is incident on a two-dimensional array of diffraction gratings followed by a single focusing lens. The periodicity and the orientation of the grating rulings of each grating can be chosen such that the +1 order beam from the gratings forms an array of focal spots in the detector plane. We show that by using a square array of zones, it is possible to generate an array of +1 order focal spots having a smaller number of rows, thus reducing the height of the required detector array. The phase profile of the test wavefront can be estimated by measuring the displacements of the +1 order focal spots for the test wavefront relative to the +1 order focal spots for a plane reference wavefront. The narrower width of the photodetector array can offer several advantages, such as a faster frame rate of the wavefront sensor, a reduced amount of cross talk between the nearby detector zones, and a decrease in the maximum thermal noise. We also present experimental results of a proof-of-concept experimental arrangement using the proposed wavefront sensing scheme.     

Reflection and refraction of plane waves at interface between two piezoelectric media

This paper analyses reflection and refraction of plane waves at a perfect interface between two anisotropic piezoelectric media. The equations of elastic waves, quasi-static electric field, and constitutive relationships for the piezoelectric media are derived. A solution based on the inhomogeneous wave theory is developed to address the inconsistency between the numbers of independent wave modes in the media and the numbers of interfacial boundary conditions to obtain accurate reflection and refraction coefficients in case of strong piezoelectric media, where all the elastic and electric continuity conditions across the interface are satisfied simultaneously. The study shows that there exist independent and zero energy wave modes satisfying the general Snell’s law and propagating along the interface for any incident wave angle. These waves can be treated as pseudo surface waves. It is further found that all the reflection/refraction waves including the pseudo surface waves obey the energy conservation law at the interface boundary. In addition, the analysis also reveals that the reflection and refraction elastic waves can turn into pseudo surface waves at some critical incident angles.     

Quasi-diffraction-free beams.

A diffraction-free beam is obtained by the superposing of plane waves whose wave vectors make an angle with the propagation axis. These plane waves are realized with point sources that are distributed uniformly around a circle and an infinitely large aperture lens. After the field passes through the lens it has nondiffracting properties and is described by the zero-order Bessel function. Relaxing these conditions makes the beam diffraction free within only a limited region. The beam generated from such a geometry is referred to as a quasi-diffraction-free beam. The effects of the width of the annular source on the beam spread are discussed and compared with those for a Gaussian beam. Approximate expressions for quasi-diffraction-free beams are also obtained.     

Design of lenses to project the image of a pupil in optical testing interferometers

When an optical surface or lens in an interferometer (Twyman-Green or Fizeau interferometer) is tested, the wave front at the pupil of the element being tested does not have the same shape as at the observation plane, because this shape changes along its propagation trajectory if the wave front is not flat or spherical. An imaging lens must then be used, as reported many times in the literature, to project the image of the pupil of the system being tested over the observation plane. This lens is especially necessary if the deviation of the wave front from sphericity is large, as in the case of testing paraboloidal or hyperboloidal surfaces. We show that the wave front at both positions does not need to have the same shape. The only condition is that the interferograms at both places be identical, which is a different condition. This leads to some considerations that should be taken into account in the optical design of such lenses.     

Synthesis of an inhomogeneous dielectric axisymmetric lens

The problem of determining the index of refraction as a function of the distance from the axis of symmetry of an inhomogeneous dielectric lens is considered for the input and output wave fronts. An analytical method of obtaining a solution for a layered lens is proposed, which is based on the geometric optics approximation and the assumption that the index of refraction within each layer is constant. It is demonstrated that this method is applicable to both layered and gradient lenses (provided that the number of layers is sufficiently large).     

声子晶体平板组合成像特性分析

用多重散射方法分析了声子晶体单平板和多平板的成像特性。发现：在特定频率范围内,引入等效负折射率后,单板负折射成像可用反Snell定律描述;多层声子晶体平板成像服从依次成像规律,可类比传统几何光学中共轴球面系统的成像,引入实物实像、虚物虚像等概念;平板透镜不改变出射波的出射方向,仅使出射位置产生平移;多平板透镜的累加成像与等厚度的单一平板透镜成像具有相同的效果。平板透镜的这些成像特点,使其在成像质量、加工方法、系统组合等方面比球面透镜更具优势。     

The point-characteristic function, wavefronts, and caustic of a spherical wave refracted by an arbitrary smooth surface

The aim of this paper is to obtain expressions for the k-function, the wavefront train, and the caustic associated with the light rays refracted by an arbitrary smooth surface after being emitted by a point light source located at an arbitrary position in a three-dimensional homogeneous optical medium. The general results are applied to a parabolic refracting surface. For this case, we find that when the point light source is off the optical axis, the caustic locally has singularities of the hyperbolic umbilic type, while the refracted wavefront, at the caustic region, locally has singularities of the cusp ridge and swallowtail types.     

Aberration effects in two-beam laser interferometers

In displacement measurements by two-beam interferometers, the wavefront curvature of a laser beam causes a systematic increase of the fringe period. This increase depends on beam collimation: It is null for a plane wave and proportional to the squared divergence of the beam. With interfering beams not perfectly recombined, an additional fringe-period error is caused, with the effect of counteracting and also of compensating for and prevailing over the usual error. We describe this hitherto unsuspected effect and give a correction equation.     

Measurement of longitudinal displacement using lateral shearing cyclic path optical configuration setup and phase shifting interferometry

We present a technique for the measurement of longitudinal displacement using a lateral shearing cyclic path optical configuration (CPOC) setup and phase shifting interferometry. In the technique, a plane mirror mounted on a linear translation stage, placed slightly away from the focal plane of a lens, introduces a longitudinal focal shift to the incident focusing beam. The resulting spherical wavefront emerging from the lens is sheared into two orthogonally polarized beams using the CPOC setup. By applying polarization phase shifting interferometry (PPSI), the longitudinal focal shift of the beam focus is calculated by determining the slope of the optical path difference variation between the sheared beams. Similarly, the additional focal shift introduced due to longitudinal translation of the mirror, by an unknown amount, is determined using PPSI. Half of the difference between the two longitudinal focal shifts measured gives the longitudinal displacement of the mirror. The technique can be used for an extended range of distance measurement. The novelty of the technique is the introduction of CPOC for the distance measurement. The advantages of the technique compared to other related methods are discussed.