Three-dimensional diffraction of a thin metallic cylinder illuminated in conical incidence: application to diameter estimation

We present a model to determine the far-field diffraction pattern of a metallic cylinder of infinite length when it is illuminated in oblique incidence. This model is based on the Helmholtz-Kirchhoff integral using the Beckmann conditions for reflection. It considers the three-dimensional nature of the diffracting object as well as the material of which the cylinder is made. This model shows that the diffraction orders are placed in a cone of light. The amplitude at the far field can be divided into three terms: the first term accounts for Babinet's principle, that is, the contribution of the cylinder projection; the second term accounts for the three dimensionality of the cylinder; and the third term accounts for the material of which the cylinder is made. This model is applied to the diameter estimation of the cylinder. Since the amplitude of the Babinet contribution is much larger than the light reflected by the surface, the cylinder diameter can be obtained in a simple way. With this approximation, the locations of the diffraction minima do not vary when the cylinder is inclined. On the other hand, when the reflected light is considered the location of the minima and, hence, the estimation of the diameter, varies. Also, a modification of the diffraction minima is produced by the material of which the cylinder is made. Experimental results are also obtained that corroborate the theoretical approach.     

Generalized Jinc functions and their application to focusing and diffraction of circular apertures

A family of generalized Jinc functions is defined and analyzed. The zero-order one is just the traditional Jinc function. In terms of these functions, series-form expressions are presented for the Fresnel diffraction of a circular aperture illuminated by converging spherical waves or plane waves. The leading term is nothing but the Airy formula for the Fraunhofer diffraction of circular apertures, and those high-order terms are directly related to those high-order Jinc functions. The truncation error of the retained terms is also analytically investigated. We show that, for the illumination of a converging spherical wave, the first 19 terms are sufficient for describing the three-dimensional field distribution in the whole focal region.     

Tolerance on defocus precisely locates the far field (exactly where is that far field anyway?)

The Fraunhofer criterion defines the location of the boundary between the Fresnel and the Fraunhofer diffraction regions and thus determines the location of that region commonly referred to as the far field. The Fraunhofer criterion is usually given as an axial distance much greater than some amount relative to the maximum dimension of the aperture. By recognizing that Fresnel diffraction patterns are merely defocused Fraunhofer diffraction patterns, we show that the Fraunhofer criterion can be written precisely in terms of an allowable tolerance on defocus. This new criterion provides insight that is useful to optical designers and engineers who routinely deal with such tolerances.     

Vector diffraction analysis of optical disk readout

The optical disk readout signals from ROM disks are presented by use of a rigorous three-dimensional vector diffraction method. The optical disk is modeled as a crossed metal grating without restriction on the form of the information marks, and the permittivity of the metal is taken into account. The diffracted field from the disk is obtained by means of decomposing the focused incident beam into a spectrum of plane waves and then calculating the diffracted plane waves for each respective incident component. The readout signal is obtained by integration of the energy-flux density of the diffracted field according to the detection scheme of the optical disk system. A typical digital versatile disk (DVD) system is applied with this theory, and the result is far from that of scalar diffraction theory.     

Fraunhofer Diffraction by Koch Fractals

Abstract

Analytical expressions are derived for the complex amplitude in the Fraunhofer diffraction field of an arbitrary Koch fractal with a finite range of self-similarity. Results of the numerical evaluation for the intensity distribution of Fraunhofer diffraction patterns are compared with those obtained experimentally. It is shown that the diffraction pattern of the Koch fractal can be divided into two areas, a central fractal area and a periodic area, and that the former is surrounded by the latter. The existence of the periodic area is a consequence of the finite inner cut-off of the self-similarity of the object fractal. On the other hand, the outer cut-off gives rise to a small core area at the centre of the diffraction pattern.     

Fraunhofer diffraction of a slit aperture between a knife-edge and a metal cylinder

The Fraunhofer diffraction pattern of a slit aperture formed between a reference knife-edge and a metal-cylinder surface is different from that of an ideal slit aperture. This pattern should include reflected light coming from both the front and rear sides of a cylinder surface. To investigate the influence of light reflected from the cylinder surface, we discuss the theoretical consideration based on the simple model of the reflected light on the surface. The experimental setup is designed and constructed to measure the actual diffraction pattern produced by the slit between the knife-edge and the cylinder surface. As a result it is obvious that the reflection of diffracted light on the rear side is dominant in both the simulation and the experiment.     

On the Fraunhofer (Far Field) Diffraction Patterns of Opaque and Transparent Objects with Coherent Background

The diffracting objects considered are opaque and semi-transparent objects of known cross section which are situated in an otherwise transparent area and illuminated with a collimated quasi-monochromatic beam of light. The plane in which the resultant diffraction pattern is considered is at a sufficient distance from the object for the approximation of Fraunhofer (far field) diffraction to be made whilst still remaining in the near field of the surrounding transparent area. The resultant intensity distribution is investigated both theoretically and experimentally for objects which present both circular and rectangular cross sections to the beam.     

Diffraction of an e.m. Wave by an Absorbing Circular Cylinder

Some expressions for the e.m. field diffracted by an absorbing circular cylinder of large radius in the Fresnel region are developed. Until recently results were available only for the far field of ‘good conductors’. The derived equations agree with Young's theory of diffraction since the diffracted field may be expressed as two cylindrical waves which seem to arise from the two ‘edges’ of the cylinder. Numerical calculations are also presented and compared with previous experimental results.     

Collimating cylindrical diffractive lenses: rigorous electromagnetic analysis and scalar approximation

Practical collimating diffractive cylindrical lenses of 2, 4, 8, and 16 discrete levels are analyzed with a sequential application of the two-region formulation of the rigorous electromagnetic boundary-element method (BEM). A Gaussian beam of TE or TM polarization is incident upon the finite-thickness lens. F/4, F/2, and F/1.4 lenses are analyzed and near-field electric-field patterns are presented. The near-field wave-front quality is quantified by its mean-square deviation from a planar wave front. This deviation is found to be less than 0.05 free-space wavelengths. The far-field intensity patterns are determined and compared with the ones predicted by the approximate Fraunhofer scalar diffraction analysis. The diffraction efficiencies determined with the rigorous BEM are found to be generally lower than those obtained with the scalar approximation. For comparison, the performance characteristics of the corresponding continuous Fresnel (continuous profile within a zone but discontinuous at zone boundaries) and continuous refractive lenses are determined by the use of both the BEM and the scalar approximation. The diffraction efficiency of the continuous Fresnel lens is found to be similar to that of the 16-level diffractive lens but less than that of the continuous refractive lens. It is shown that the validity of the scalar approximation deteriorates as the lens f-number decreases.     

Fraunhofer Diffraction at Apertures in the Form of Regular Polygons. I.

In this first contribution the symmetry properties of the Fraunhofer diffraction phenomena at the apertures possessing a symmetry are discussed. These properties are used for the derivation of the wave function describing Fraunhofer diffraction at apertures of the form of regular polygons. Besides the general formula expressions are derived giving the wave function in the directions where the diffraction patterns show some particular properties.     

基于实验的圆柱体流固耦合升力谱模型研究

基于均匀流场的弹性圆柱体涡激振动实验,提出了一个考虑流固耦合的涡激升力谱模型。该升力模型采用了Webull 分布模型,以流速、约化速度、圆柱体直径和圆柱体固有频率为模型参数,通过引入圆柱体固有频率来考虑流固耦合的影响,避免了时域模型中直接计算圆柱体的顺流向振动响应问题。与实验结果比较表明,该模型可用于均匀流场的涡激振动预测。     

Application of the two-dimensional fractional-order Fourier transformation to particle field digital holography

We demonstrate that the fractional-order Fourier transformation is a suitable method to analyze the diffraction patterns of particle field holograms. This method permits reconstruction of in-line digital holograms beyond the Fraunhofer condition (d2/lambdaz approximately/= 10). We show that the diameter of spherical particles is measured with good accuracy. Simulation and experimental results are presented.     

J. H. Kulick, J. M. Jarem, R. G. Lindquist, S. T. Kowel, and M. W. Friends are with the Department of Electrical and Computer Engineering, University of Alabama in Huntsville, Huntsville, Alabama 35899

The development and modeling of a liquid-crystal phase grating for real-time diffractive three-dimensional displays are discussed. The system being developed, which is called the ICVision system, utilizes a number of ideas that will result in a rugged, low-power three-dimensional display offering both vertical and horizontal parallax and eventually full color. Fringing fields created between interdigitated electrodes formed on top of VLSI die will induce a diffraction pattern in a thin layer of liquid crystal that will cover the die. A detailed electrostatic and diffraction analysis of liquid-crystal phase-grating regions that will make up the final display is given here. The electrostatic analysis is developed by use of the method of moments. The diffraction analysis is developed by use of rigorous coupled-wave diffraction theory. The numerical results obtrained from the mathematical model are compared with experimental diffraction results from preliminary LCD cells that have been assembled as prototype ICVision devices.     

Issues in Optical Diffraction Theory

This paper focuses on unresolved or poorly documented issues pertaining to Fresnel’s scalar diffraction theory and its modifications. In Sec. 2 it is pointed out that all thermal sources used in practice are finite in size and errors can result from insufficient coherence of the optical field. A quarter-wave criterion is applied to show how such errors can be avoided by placing the source at a large distance from the aperture plane, and it is found that in many cases it may be necessary to use collimated light as on the source side of a Fraunhofer experiment. If these precautions are not taken the theory of partial coherence may have to be used for the computations.In Sec. 3 it is recalled that for near-zone computations the Kirchhoff or Rayleigh-Sommerfeld integrals are applicable, but fail to correctly describe the energy flux across the aperture plane because they are not continuously differentiable with respect to the assumed geometrical field on the source side. This is remedied by formulating an improved theory in which the field on either side of a semi-reflecting screen is expressed as the superposition of mutually incoherent components which propagate in the opposite directions of the incident and reflected light.These components are defined as linear combinations of the Rayleigh-Sommerfeld integrals, so that they are rigorous solutions of the wave equation as well as continuously differentiable in the aperture plane. Algorithms for using the new theory for computing the diffraction patterns of circular apertures and slits at arbitrary distances z from either side of the aperture (down to z = ± 0.0003 λ) are presented, and numerical examples of the results are given. These results show that the incident geometrical field is modulated by diffraction before it reaches the aperture plane while the reflected field is spilled into the dark space. At distances from the aperture which are large compared to the wavelength λ these field expressions are reduced to the usual ones specified by Fresnel’s theory. In the specific case of a diffracting half plane the numerical results obtained were practically the same as those given by Sommerfeld’s rigorous theory.The modified theory developed in this paper is based on the explicit assumption that the scalar theory of light cannot explain plolarization effects. This premise is justified in Sec. 4, where it is shown that previous attempts to do so have produced dubious results.     

Determination of longitudinal and transverse shifts of three-dimensional Gaussian beams reflected at a grating near an anomaly

Spatial displacements of three-dimensional Gaussian beams diffracted at a reflection grating are studied theoretically and numerically. The complex diffraction coefficients (amplitudes and phases) of the grating diffracted plane waves calculated by a rigorous method for conical diffraction are the basis for this investigation. The classical analytical formula for the longitudinal shift depending on the gradient of the reflection phase is generalized to the simultaneous analytical treatment of the longitudinal Goos—Hänchen like shift as well as a transverse shift. A second method uses the full integration on the whole spectrum of plane waves of the diffracted beam.     

An integral equation method for the diffraction of oblique waves by an infinite cylinder

A hybrid integral equation method is formulated to study the diffraction of oblique waves by an infinite cylinder. The water depth and the geometry of the floating cylinder are assumed to be uniform in the y-direction (one of the horizontal axes). Numerical discretization and integrations are performed in the vertical plane. Analytical solutions are used in far fields such that radiation boundary conditions are satisfied. Numerical results are obtained for the case of wave scattering by a floating rectangular cylinder in a constant water depth. The accuracy and efficiency of present method are compared with those obtained by other numerical techniques.     

The effect of acoustic diffraction and acoustic attenuation on the dynamic range of acoustooptic Bragg cells

A three-dimensional model for the propagation of finite acoustic waves in nonlinear media is developed. This model implicitly includes the effects of acoustic attenuation and divergence due to diffraction. The generation of intermodulation products in the case of a two-tone input signal is numerically analyzed. It is found that acoustic diffraction can have a significant effect on the dynamic range of a Bragg cell if the acoustic field extends well into the Fraunhofer region. Inclusion of the effect of diffraction in the model predicts a dynamic range that can be considerably larger than the value obtained by using the infinite plane wave assumption. It is shown that acoustic attenuation significantly reduces the level of the acoustic intermodulation products relative to the level of the fundamental modes. This also increases the dynamic range. The influence of these effects on design considerations for Bragg cells is discussed.     

Speckle noise arising from coherently illuminated polydisperse and continuous size distributions of discs

Abstract

An existing model of the speckle noise affecting images in far-field in-line holographic imaging is extended to distributions of disc-shaped particles which are no longer of uniform diameter. In this model the signal-to-noise ratio in replayed images from ideal in-line holograms in the Fraunhofer regime is limited by the speckle field arising from the overlapping Airy patterns. The original study was motivated by holographic particle image velocimetry and assumed that all the particles were of uniform diameter. We extend this model to cover polydisperse, power-law, exponential and Gaussian distribution functions and include an explicit limit on our maximum particle diameter to ensure that the hologram is recorded in the far-field of all the particles.     

单根和双分裂导线风噪声的数值模拟研究

针对单根和双分裂导线风噪声采用混合计算方法进行数值模拟。基于导线的绞线结构进行建模,获得单导线风场和声场的分布特征,并与同直径的光滑圆柱结果进行对比,并针对水平和垂直布置双分裂导线风场和声场进行研究。研究表明:导线的阻力系数和总声压级均小于光滑圆柱,而斯特劳哈尔数大于光滑圆柱;在卓越频率段导线和光滑圆柱的风噪声呈现了明显的8字形指向性,而在远离卓越频率段呈现圆形指向,说明导线和圆柱风噪声是一种偶极子声源;导线和光滑圆柱风噪声的总声压级总体呈圆形、略呈8字形分布;垂直布置双分裂圆柱因相互之间无干扰,总声压级为两根单导线风噪声声压级之和;垂直布置的双分裂导线风噪声略小于水平布置双分裂导线风噪声。     

超大型结构物受波浪力作用的数值模拟

为了计算波浪效应下超大型结构物所受的波浪力,该文运用σ坐标变换来追踪随时间变化的自由液面,添加了浸没边界法(IBM)来处理不规则结构物表面,建立了基于大涡模拟(LES)方法的三维数值模型。为验证该数值模型的正确定,运用该模型来模拟规则波作用下大直径垂直圆柱的绕射问题,圆柱周围的波浪爬高数值解与解析解吻合良好。运用该三维数值模型,模拟了波浪与南京长江三桥南塔钢套箱相互作用的问题,数值计算出的结构物所受的波浪力与试验数据吻合良好,表明该数值模型能够很好地模拟工程中三维波浪与超大型结构物相互作用的问题。