Free-space Fresnel diffraction for the approximation of fractional Fourier transform

Fractional Fourier transformation of an object can be approximated by the object's free-space Fresnel diffraction pattern under some restricted conditions and plane wave illumination according to Hua's method. A better approximation is achieved under least-squared conditions developed in this paper. Simulation results verify that our theoretical development works for any fractional order a compared with the previous approach.     

一种改良的啁啾变换算法及其应用

快速啁啾算法引入两次快速傅里叶变换（FFT）及一个解析高斯核,计算复杂度低于卷积算法.通过对啁啾算法实现过程进行的改进,避免了该算法在实现过程中存在的一些问题,比如输出窗口小、信号丢失、计算复杂度稍大等缺点. 把算法用于简单的可求得解析解的系统并与之做比较. 对高斯函数,最大误差通常在10^-15数量级,而对矩形函数,由于受FFT算法计算精度的影响,误差在10^-3数量级,但这并不影响算法的性能. 最后把算法用于一种典型的标量衍射系统及分数傅里叶变换的计算,获得了很好的结果. 关键词： 快速啁啾算法 啁啾Z变换 菲涅耳变换 分数傅里叶变换     

Relation between Fresnel transform of input light field and the two-parameter Radon transform of Wigner function of the field

This paper proves a new theorem on the relationship between optical field Wigner function's two-parameter Radon transform and optical Fresnel transform of the field, i.e., when an input field ψ( x') propagates through an optical [ D( -B) ( -C) A] system, the energy density of the output field is equal to the Radon transform of the Wigner function of the input field, where the Radon transform parameters are D,B. It prove this theorem in both spatial-domain and frequency-domain, in the latter case the Radon transform parameters are A,C.     

光束分数傅里叶变换的Wigner分布函数分析方法

利用Wigner分布函数(WDF)方法，对光束的分数傅里叶变换特性进行了研究.以厄米 高斯(H G)光束为例，导出了H G光束在分数傅里叶变换面上光强分布的解析公式和H G光束在分数傅里叶变换面上束宽的解析计算公式.通过数值计算研究了H G光束光强随分数傅里叶变换阶数变化的规律.研究表明：选取适当的分数傅里叶变换阶数p，在x，y方向可以得到相等束宽的对称光强分布. 关键词： Wigner分布函数 厄米 高斯(H G)光束 分数傅里叶变换     

菲涅耳衍射和分数傅里叶变换

介绍了衍射孔的菲涅耳衍射和分数傅里叶变换的对应关系，使得可以用分数傅里叶变换来描述光电原始光场经过菲涅耳衍射区一直到无穷远处夫琅禾费衍射区的自由空间标量衍射传播全过程。     

规范光学分数傅里叶变换下的光学传递函数

基于规范光学分数傅里叶变换,从光学传递函数概念出发,根据菲涅耳衍射、夫琅和费衍射定义,应用线性系统理论,分别给出了菲涅耳与夫琅和费衍射系统在规范分数傅里叶变换下的光学传递函数数学表达式。证明了其具有分数傅里叶变换的级联性,指出常规傅里叶变换下的光学传递函数为分数传递函数的特例。所得结果对光学分数傅里叶变换在信息处理、像质评价等方面的应用具有实用价值。     

用Wigner分布函数方法研究平顶多高斯光束的分数傅里叶变换

本文用Wigner分布函数方法分析了平顶多高斯光束的分数傅里叶变换。导出了分数傅里叶变换面上光的强度、束宽、远场发散角、M2因子和K参数的解析表达式。通过数值模拟研究了分数傅里叶变换阶数对平顶多高斯光束传输性质的影响。     

激光光路追踪Wigner分布函数方法

提出了一种考虑衍射效应的激光几何光路追踪方法。引入由激光标量场定义的Wigner激光能量相空间分布函数,并给出该函数满足的刘维尔运动方程。Wigner分布函数用来描述经过空间任一点沿任一方向传输的激光光线上的能量分配。激光能量沿由波包色散关系定义的光线轨迹保持不变（真空中）或者衰减（等离子体中）。与传统几何光路追踪方法相比,该方法从理论上给出了激光光线初始携带能量份额的计算方法,并且将激光标量场的相位信息自然地包含在Wigner分布函数的定义里。算例表明,该方法与解析模型及广泛使用的菲涅耳衍射积分方法结果一致。     

Bivariate averaging and the Wigner distribution function

In order to gain insight into the nature of the Wigner and related distribution functions, bivariate averaging functions of real unbounded variables with absolutely continuous marginals that are ordinary probabilities are considered. Accordingly variables are chosen to be phase space variables that are respectively eigenvalues of position and momentum operators. The impact of the condition that the marginals are squared magnitudes of amplitudes that are Fourier transforms of one another is emphasized by the delay of the introduction of this Fourier transform condition until after the form for a bivariate distribution with the given marginals is obtained. When the respective amplitudes are fourier transforms of one another, special cases of the bivariate averaging function correspond to generalized Wigner functions characterized by a parameterα. Such anα-Wigner function can be used as the basis of a consistent averaging procedure if an appropriate corresponding representation for underlying operators to be averaged is specified. Properties of theα-Wigner functions are summarized.     

基于单次傅里叶变换的分段衍射算法

针对单次傅里叶变换算法(S-FFT算法)受到采样定理的约束,衍射面画幅尺寸和有效内容像素数无法灵活控制,很容易出现衍射面画幅尺寸大小与衍射距离不匹配的情况,本文提出了一种分段衍射算法。首先,在采样数、光的波长、初始衍射面大小确定的情况下,利用拆分的衍射距离比控制最终衍射面画幅尺寸。然后,对单次衍射计算结果与分段衍射计算结果进行了图像相似度对比。实验表明,分段衍射算法可在画面强度分布不变的情况下,提高有效像素数目,数据量增加了2~3个数量级。此外,文章分析了造成误差的一个主要原因来自有效数据分辨率提高后,细节分布与低分辨率像素值之间的差别。在图像细节较丰富时,其差别较大。因此这种差别应视为优于直接计算的一种结果。本算法能够获得更加清晰的图像细节,灵活调整衍射面画幅尺寸,使得S-FFT算法在大衍射距离问题计算中能发挥其算法优势。     

分数傅里叶变换啁啾滤波

针对常规傅里叶变换所不能解决的啁啾噪声滤除问题,利用Wigner分布函数分析分数傅里叶变换的空域和频域特性,提出在分数傅里叶变换域进行啁啾滤波的方法。并将该方法应用到图像处理中,对分数傅里叶变换滤除一维和二维图像的啁啾噪声进行了计算机仿真,获得了满意的效果,结果表明该方法在图像处理中的有效性。     

Angular criterion to distinguish between Fraunhofer and Fresnel diffraction

The distinction between Fresnel and Fraunhofer diffraction is an essential condition for the accurate analysis of diffracting structures. In this paper we propose a criterion based on the angle subtended by the first minimum of the diffraction pattern from the centre of the diffracting aperture. The determination of the minimum of the diffraction pattern is the crucial point to assure the accuracy of the criterion. Therefore, the applicability of adequate thresholds for detection is discussed. The criterion is also generalized by expressing it in terms of the number of Fresnel zones delimited by the aperture. Simulations are reported to illustrate the feasibility of the criterion.     

激光光路追踪Wigner分布函数方法

提出了一种考虑衍射效应的激光几何光路追踪方法。引入由激光标量场定义的Wigner激光能量相空间分布函数,并给出该函数满足的刘维尔运动方程。Wigner分布函数用来描述经过空间任一点沿任一方向传输的激光光线上的能量分配。激光能量沿由波包色散关系定义的光线轨迹保持不变（真空中）或者衰减（等离子体中）。与传统几何光路追踪方法相比,该方法从理论上给出了激光光线初始携带能量份额的计算方法,并且将激光标量场的相位信息自然地包含在Wigner分布函数的定义里。算例表明,该方法与解析模型及广泛使用的菲涅耳衍射积分方法结果一致。     

Wigner Transforms and Fractional Fourier Transforms of the First Order Optical Systems

Based on the Collins formula, the relationship between the coordinate transform matrix (WCTM) of the Wigner distribution function (WDF) and the ray transfer matrix (RTM) of an arbitrary first-order optical system has been derived. By using this relation and the definition of fractional Fourier transform (FRT) in terms of WDF rotation, it is concluded that an arbitrary first-order optical system can be generally decomposed into a thin lens and a FRT sub-system whose order is not unique and depends on two concrete decomposing operations on the system. And when the system is reciprocally symmetric, a FRT can be implemented by it. In addition, the composition, that is also the decomposition condition of the complicated FRT optical system by cascading a series of FRT subsystems has also been derived by using the operations of RTM.     

Propagation and transformation of Bessel beam through the fractional Fourier transform system with two hard-edged apertures

The propagation properties of Bessel beam is a meaningful research. In this paper, based on the expanding the hard-edged circular aperture as a finite sum of complex Gaussian functions and the scalar diffraction theory, an approximate analytical solution for Bessel beam propagating through a fractional Fourier transform system is derived in the cylindrical coordinates. Then, the detailed numerical calculation for Bessel beam is presented. The simulation also shows that the beam parameter and the order of fractional Fourier transform result in the change of field distribution, including location, intensity and width.     

关于菲涅耳衍射图样相似性的分析与讨论

与夫琅禾费衍射不同,菲涅耳衍射图样对于给定波长,当衍射距离变化或输入图像几何缩放时一般都不具有相似性.但我们的分析表明,若波长、衍射距离和输入图像几何线度诸参数满足一定关系,或对于某些特殊类型孔径,菲涅耳衍射图样有可能彼此相似.本文导出了在上述诸参数发生变化时菲涅耳衍射图样保持相似性的普适条件,以及此时衍射图样大小与诸参数的定量关系,并对包括圆孔衍射和直边衍射等情况都得到了一系列有价值的推论.     

FFT-based convolution algorithm for fast and precise numerical evaluating diffracted field by photon sieve

FFT-based convolution is proposed to numerical solve Fresnel–Kirchhoff integral in Fresnel regime carefully and in a very shorter time in comparison to direct solving convolution. To show its capability, the algorithm was implemented to evaluate amplitude of a diffracted plane wave at the focal plane of photon sieves with different focal lengths. The calculated amplitudes are completely the same calculated via convolving operation but has advantageous of taking very very shorter time. The calculation was also repeated using single-FFT algorithm that produce same result for all ranges either below or upper the sampling criteria and different results in comparison to the other two methods.     

圆孔的菲涅耳衍射

得到了圆孔菲涅耳衍射光强分布的表达式,此式由贝塞尔函数的数列(或Lommel函数)给出,因此很容易利用普及的数学软件算出结果.并且利用CCD照相机量度了衍射光强分布,与运算的结果作了比较,也与Burch在1985报导的结果作了比较.