Diffraction transfer function and its calculation of classic diffraction formula

In scalar diffraction theory, Kirchhoff formula, Rayleigh–Sommerfeld formula, and angular spectrum transmission formula are classic formulas that express diffraction correctly. But as complex calculation, Fresnel diffraction integral, which is the paraxial approximate solution to these formulas, is used widely. This paper presents diffraction transfer function corresponding to each classic diffraction formula, gives discussion on methods when using fast Fourier transform to calculate these formulas, derives conditions each formula must meet when it is calculated correctly based on sampling theorem, and finally real examples are finished to certify the results.     

A general analytic procedure for calculating two-centre integrals involving the one-electron dipolar coupling operator and Slater atomic orbitals

Analytic expressions are derived for the hybrid-type two-centre matrix elements of the one-electron dipolar coupling operator between Slater atomic orbitals. The method used is of wide applicability. The matrix elements of interest are written in terms of a generalized form of overlap integral. The overlap integrals are evaluated using the convolution theorem/contour integration technique. Computational results are discussed and times given.     

显微数字全息中物光波前重建方法研究和比较

根据全息理论和线性系统理论,采用离轴无透镜傅里叶变换全息记录光路,对利用菲涅耳近似法、基于瑞利—索末菲衍射积分的卷积法以及角谱理论方法数值重建全息图进行了比较研究,并做了计算机模拟验证.结果表明：菲涅耳近似法和角谱方法重建像质比较好,且菲涅耳方法重建速度快;在记录距离极小的情况下,尽管记录距离不满足通常的菲涅耳近似条件,菲涅耳近似公式仍然成立;自由空间脉冲响应的快速傅里叶变换的性质与距离有关,由卷积方法得到的再现像只在某一特定距离下比较理想;对于极小物场、大孔径显微数字全息来说,菲涅耳近似重建方法是较为有效的方法.     

New convolution theorem for the linear canonical transform and its translation invariance property

The linear canonical transform (LCT), which is a generalization of the Fourier transform (FT), has many applications in several areas, including signal processing and optics. Many properties for this transform are already known, but an extension of convolution theorem of FT is still not having a widely accepted closed form expression. In the literature of recent past different authors have tried to formulate convolution theorem for LCT, but none have received acclamation because their definition do not generalize very nicely the classical result for the FT. Moreover, those definitions exhibit only partial invariance properties which prevent their actual use in many applications of signal processing. The purpose of this paper is to introduce a new convolution structure for the LCT that preserves the translation invariance property. Indeed, an effective translation invariance is obtained by slightly modifying the former definitions and by introducing linear canonical translation operators.     

Multichannel sampling expansion in the linear canonical transform domain and its application to superresolution

The linear canonical transform (LCT) describes the effect of first-order quadratic phase optical system on a wave field. In this paper, we address the problem of signal reconstruction from multichannel samples in the LCT domain based on a new convolution theorem. Firstly, a new convolution structure is proposed for the LCT, which states that a modified ordinary convolution in the time domain is equivalent to a simple multiplication operation for LCT and Fourier transform (FT). Moreover, it is expressible by a one dimensional integral and easy to implement in the designing of filters. The convolution theorem in FT domain is shown to be a special case of our achieved results. Then, a practical multichannel sampling expansion for band limited signal with the LCT is introduced. This sampling expansion which is constructed by the new convolution structure can reduce the effect of spectral leakage and is easy to implement. Last, the potential application of the multichannel sampling is presented to show the advantage of the theory. Especially, the application of multichannel sampling in the context of the image superresolution is also discussed. The simulation results of superresolution are also presented.     

彩色数字全息的非插值波面重建算法研究

彩色数字全息及多波长照明的数字全息检测研究中,避免插值误差的可变放大率波面重建是一个重要的研究内容.由于Fresnel衍射积分可以表示成Fourier变换及卷积两种形式,对应地存在两种波面重建算法：其一,将重建距离分为两段的衍射“接力”算法;其二,用球面波为重建波的卷积算法.文中对这两种算法进行理论分析及实验研究,讨论让重建计算满足取样定理的条件.结果表明,卷积算法较容易满足取样定理,能够获得较好的重建物光场. 关键词： 彩色数字全息 波面重建 衍射计算     

彩色数字全息的非插值波面重建算法研究

彩色数字全息及多波长照明的数字全息检测研究中,避免插值误差的可变放大率波面重建是一个重要的研究内容.由于Fresnel衍射积分可以表示成Fourier变换及卷积两种形式,对应地存在两种波面重建算法：其一,将重建距离分为两段的衍射“接力”算法;其二,用球面波为重建波的卷积算法.文中对这两种算法进行理论分析及实验研究,讨论让重建计算满足取样定理的条件.结果表明,卷积算法较容易满足取样定理,能够获得较好的重建物光场.     

Rubinowicz transform of the MTPO surface integrals

The surface integral of the modified theory of physical optics is reduced to a line integral by using the Rubinowicz transform for the incident scattered fields by an arbitrary aperture in a black surface. The integral theorem of Kirchhoff is applied to the scattering geometry and the diffracted fields are expressed in terms of a line integral along the contour of the diffracting edge.     

Papoulis-like generalized sampling expansions in fractional Fourier domains and their application to superresolution

We present a generalized convolution theorem in the fractional Fourier domains that preserves the convolution theorem of the conventional Fourier transform. The Papoulis-like generalized sampling expansions in the fractional Fourier domains using this generalized convolution theorem are also derived and it is shown that the classical generalized Papoulis sampling expansion is a special case of it. Its application in the context of the image superresolution is also discussed.     

New twistorial integral formulas for massless free fields of arbitrary spin

A manifestly scaling-invariant version of the Kirchoff-D'Adhemar-Penrose field integrals is presented. The invariant integral expressions for the spinning massless free fields are directly transcribed into the framework of twistor theory. It is then shown that the resulting twistorial field integrals can be thought of as being equivalent to the universal Penrose contour integral formulas for these fields.     

An efficient method for computing backscattering from Born objects of arbitrary shape

A method is presented for efficiently computing the propagating pressure field backscattered by an arbitrarily shaped, weakly scattering, three-dimensional object. This is accomplished by drawing upon a previously reported relationship between the boundary condition on a two-dimensional radiating aperture and the pressure propagating along an axis normal to the aperture, and the fundamental theorem of diffraction tomography, which relates the Fourier transform of an object function to its scattered pressure field. Together, these two results are used to derive an integral formula that expresses the pressure field backscattered from an object as a one-dimensional Fourier transform of its scattering amplitude. This formula is then utilized to compute the backscattered pressure field from a uniform fluid sphere in the first Born approximation; the results of which are compared to the rigorous partial wave expansion.     

Closed form solutions for the acoustical impulse response over a masslike or an absorbing plane

The transient sound field caused by a Dirac delta impulse function above an infinite locally reacting plane can be calculated by applying the inverse Fourier transform of the corresponding half-space Green's function in frequency domain. As a starting point, the representation given by Ochmann [J. Acoust. Soc. Am. 116(6), 3304-3311 (2004)] is used, which consists of discrete and continuous superposition of point sources. For a locally reacting plane with masslike character and also with pure absorbing behavior, it is possible to express the resulting impulse response in closed form. Such a result is surprising, because corresponding formulations in the frequency domain are not available yet. Hence, the first main result is the closed form solution Eq. (22) for an impulse response over an infinite plane with a pure imaginary impedance. The second main result is the closed form solution Eq. (53) for an impulse response over an infinite plane with a pure real impedance. As a particular application of both main results, a convolution technique is used for deriving formulas for point sources with a general time dependency. For special signals like an exponentially decaying time signal or a triangular shaped impulse, the resulting sound field can be presented in terms of elementary functions.     

Vertical component of the extremely low-frequency electric field excited by the grounded horizontal dipole

A representation of the vertical component of the electric field emitted by the grounded linear dipole is obtained in the form of an integral of an oscillating function and a method of its numerical integration is described. For the particular case in which the medium under the antenna is homogeneous and the field is calculated on the interface, the function obtained is transformed with the help of the Fock integrals to the form convenient for deriving simple approximate formulas and for analyzing the field behavior. Calculations performed by the expressions obtained corroborate the theoretical results.     

Evaluation of Some Lattice SU(2) Group Integrals

A procedure for deriving a set of formulas of SU(2) group integrals in the lattice gauge theory by extensively using the properties of modified Bessel functions is presented. These formulas are extremely useful in the higher-order calculations of the cumulant expansion.     

A convolution integral for Fourier transforms on the groupSL(2,C)

The Fourier transform of a product of two functions onSL(2,C) is expressed as a convolution integral of the Fourier transforms of its factors. With the help of this convolution integral we present the Fourier transform of a polynomially bounded function as a finite linear combination of analytic delta functionals applied to a continuous function on the real line in an improper sense.     

失调光学系统衍射的分数傅里叶变换表述

本文对一般的失调多元件光学系统衍射积分公式进行了整理分析,在空域和频域中分别引进缩放变量,讨论得到了满足一定条件下失调多元件光学系统衍射的分数傅里叶变换表述.     

New formula for calculating coulomb energies in the liquid-drop model

A new transformation of double volume integrals into double surface integrals is presented. A simple regular method for deriving integrands in a surface integral is proposed. This method is used to calculate the Coulomb energy of a nucleus within the model of a liquid drop with a sharp boundary. Numerical results obtained on the basis of the new formula are compared with those calculated by one of the formulas employed previously.     

Fractional Fourier transform of Lorentz beams

This paper introduces Lorentz beams to describe certain laser sources that produce highly divergent fields. The fractional Fourier transform (FRFT) is applied to treat the propagation of Lorentz beams. Based on the definition of convolution and the convolution theorem of the Fourier transform, an analytical expression for a Lorentz beam passing through a FRFT system has been derived. By using the derived formula, the properties of a Lorentz beam in the FRFT plane are illustrated numerically.     

Quadratic algebras for three-dimensional superintegrable systems

The three-dimensional superintegrable systems with quadratic integrals of motion have five functionally independent integrals, one among them is the Hamiltonian. Kalnins, Kress, and Miller have proved that in the case of nondegenerate potentials with quadratic integrals of motion there is a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral implies that the integrals of motion form a ternary parafermionic-like quadratic Poisson algebra with five generators. In this contribution we investigate the structure of this algebra. We show that in all the nondegenerate cases there is at least one subalgebra of three integrals having a Poisson quadratic algebra structure, which is similar to the two-dimensional case.     

Calculating the forced response of two-dimensional homogeneous media using the wave and finite element method

The forced response of two-dimensional, infinite, homogenous media subjected to time harmonic loading is treated. The approach starts with the wave and the finite element (WFE) method where a small segment of a homogeneous medium is modelled using commercial or in-house finite element (FE) packages. The approach is equally applicable to periodic structures with a periodic cell being modelled. This relatively small model is then used, along with periodicity conditions, to formulate an eigenvalue problem whose solution yields the wave characteristics of the whole medium. The eigenvalue problem involves the excitation frequency and the wavenumbers (or propagation constants) in the two directions. The wave characteristics of the medium are then used to obtain the response of the medium to a convected harmonic pressure (CHP). Since the Fourier transform of a general two-dimensional excitation is a linear combination of CHPs, the response to a general excitation is a linear combination of the responses to CHPs. Thus, the response of a two-dimensional medium to a general excitation can be obtained by evaluating an inverse Fourier transform. This is a double integral, one of which is evaluated analytically using contour integration and the residue theorem. The other integral can be evaluated numerically. Hence, the approach presented herein enables the response of an infinite two-dimensional or periodic medium to an arbitrary load to be computed via (a) modelling a small segment of the medium using standard FE methods and post-processing its model to obtain the wave characteristics, (b) formulating the Fourier transform of the response to a general loading, and (c) computing the inverse of the Fourier transform semi-analytically via contour integration and the residue theorem, followed by a numerical integration to find the response at any point in the medium. Numerical examples are presented to illustrate the approach.