部分相干同步辐射照射下光栅分数塔尔博特效应

基于高斯谢尔光束的相干模式分解理论和波动光学,建立了部分相干同步辐射硬X射线光束通过光学器件的传播模型。模拟了微聚焦X射线光束照射下光栅的分数塔尔博特效应,得到了聚焦光束的光强分布和相干特性变化,并分析了散焦光束入射下的光栅自成像。然后通过模拟准直光束入射下的光栅塔尔博特效应,得到了不同传播距离处自成像条纹的变化情况,分析了影响光栅自成像条纹形状的因素。通过光栅衍射条纹测量同步辐射相干度,发现对矩形相位光栅来说,应该对衍射图样进行傅里叶分解,求出各级傅里叶系数随传播距离的变化曲线,从而得到入射光束相干特性。     

Fractional Fourier transform for partially coherent beam in spatial-frequency domain

By using Fourier transform and the tensor analysis method, the fractional Fourier transform (FRT) in the spatial-frequency domain for partially coherent beams is derived. Based on the FRT in the spatial-frequency domain, an analytical transform formula is derived for a partially coherent twisted anisotropic Gaussian－Schell model (GSM) beam passing through the FRT system. The connections between the FRT formula and the generalized diffraction integral formulae for partially coherent beams through an aligned optical system and a misaligned optical system in the spatial-frequency domain are discussed, separately. By using the derived formula, the intensity distribution of partially coherent twisted anisotropic GSM beams in the FRT plane are studied in detail. The formula derived provide a convenient tool for analysing and calculating the FRTs of the partially coherent beams in spatial-frequency domain.     

X-ray coherent diffraction interpreted through the fractional Fourier transform

We propose to use the fractional Fourier transform to deal with diffraction of coherent X-ray beams from the Fresnel to the Fraunhofer regime. We will illustrate the benefits of the approach compared to the Fresnel wave propagation theory from situations commonly encountered in diffraction experiments: the successive diffraction of two objects and coherent diffraction of a periodic modulation as a charge density wave, containing or not a phase shift.     

分数傅里叶变换和失调光学系统的衍射积分公式

空域中光束通过失调的ABCD光学系统的衍射积分公式在满足一定条件下,可实现分数傅里叶变换.同时,对角谱通过失调的ABCD光学系统变换的频域衍射积分公式也可实现分数傅里叶变换.最后,以单透镜失调光学系统为例说明其应用.     

Second-order fractional Fourier transform with incoherent radiation

Based on the coherent optical theory, we extend the fractional Fourier transform of first-order correlation to a fractional Fourier transform of second-order correlation. An optical system for implementing a second-order fractional Fourier transform was designed. As a numerical example, we investigate the second-order fractional Fourier transform for a single slit.     

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。     

利用分数傅里叶变换设计衍射光学元件

讨论了分数傅里叶变换与菲涅尔衍射的关系,提出球面衍射过程就是一种具有尺度因子的分数傅里叶变换,并应用分数傅里叶变换进行衍射光学元件的设计。     

Femtosecond digital in-line holography with the fractional Fourier transform: application to phase-contrast metrology

The possibility to analyze transparent phase objects using femtosecond digital in-line holography is demonstrated. We show that the diffraction patterns produced by the diffraction of 20-fs pulses by pure phase objects can be advantageously processed by using the fractional Fourier transform. The optimal fractional orders lead to the longitudinal location of the phase object while the analysis of the reconstructed patterns leads to its diameter and to the value of the phase shift. Simulations and experimental results are in good concordance. This work shows new important applications of these reconstruction techniques in the domain of digital phase-contrast metrology. We further show that the use of femtosecond pulses results in a significant reduction of coherent noise.     

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.     

部分相干分数域滤波改善光刻分辨率新方法

基于部分相干成像和分数傅里叶变换,提出在投影光刻系统中,利用分数域滤波改善光刻图形质量的新方法.理论和模拟分析表明:在曝光成像系统中的适当位置加入分数域滤波器,能增强滤波操作的灵活性和效果,可更有效地改善光刻图形质量,为提高曝光系统的分辨力提供一种新的有效选择.     

Fractional Fourier transform for partially coherent Gaussian-Schell model beams

The fractional Fourier transform (FRT) is applied to partially coherent twisted anisotropic Gaussian-Schell model (GSM) beams based directly on the cross-spectral density. An analytical and concise formula is derived for the cross-spectral density of partially coherent twisted anisotropic GSM beams passing through a FRT system in terms of the tensor method. The corresponding tensor ABCD law for performing a FRT is obtained. The connection between the FRT formula and the generalized Collins formula for partially coherent beams is discussed. The formulas derived provide a powerful tool for analyzing and calculating the FRTs of partially coherent beams.     

失调单球面折射系统的衍射积分和分数傅里叶变换

根据球面的几何特征，分析了单球面折射光学系统折射面倾斜和偏心时的失调特征；由失调元件矩阵和共轴元件矩阵的依次相乘，得到了完备的描述失调单球面折射光学系统的矩阵；从系统衍射积分与失调矩阵元的关系导出了该系统的失调衍射积分，并利用单球面折射光学系统的分数傅里叶变换结构参数得到了系统失调时的分数傅里叶变换结果.这些结果可用于数值计算和模拟该系统失调时对输入复振幅的衍射和分数傅里叶变换.     

Coherence vortex properties of partially coherent flat-topped vortex beams

The analytical propagation expression of partially coherent flat-topped vortex beams through a paraxial optical ABCD system is derived, and it then is used to investigate the coherence vortex properties of partially coherent flat-topped vortex beams in the Fourier transform and fractional Fourier transform systems. It is shown that in the Fourier transform system the coherence vortex depends on the flat-topped beam order N, spatial coherence parameter α and position (x ₁,y ₁) of the reference point, whereas in the fractional Fourier transform system the flat-topped beam order N does not affect the spectral degree of coherence. Furthermore, in both transform systems, depending on the choice of the reference point, the zero value point of the spectral degree of coherence may be present or absent. In particular, if x ₁=y ₁=0 is selected, the phase at the zero value point of the spectral degree of coherence may be determinate; thus the coherence vortex does not exist.     

利用分数傅里叶变换相关实现散斑相关测量

提出了一种在匹配滤波相关中利用分数傅里叶变换代替傅里叶变换实现散斑相关测量的方法.利用分数傅里叶变换的指数性质,采用把移变后的散斑图像先进行相位调制,再进行分数傅里叶变换的方法,有效地解决了分数傅里叶变换的移变性带来的谱移问题.编程模拟和拉伸试件位移场测量的结果表明,只要选择合适的分数傅里叶变换级次和相位调制函数,可以得到优于傅里叶变换相关的理想输出.     

失调分数傅里叶变换及其光学作用

利用失调光学系统的衍射积分公式,分析了失调的Lohmann分数傅里叶变换光学系统和准非对称非均匀介质的衍射特性,得到了两者具有相同的失调衍射积分表达式,表明两者具有等效性,从中提出了失调分数傅里叶变换新概念. 关键词： 分数傅里叶变换 失调光学系统 衍射积分 梯度折射率介质     

Experimental observation of coincidence fractional Fourier transform with entanglement photon pairs

Experiment of coincidence fractional Fourier transform with spontaneous parametric down-conversion photon pairs is described in this work. Results agree well with theoretical calculations with Gaussian beams. Visibilities of the results are much greater than those obtained with partially coherent beam.     

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.     

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。     

Partially coherent beam and its applications

The recent progresses on the propagation, generation and application of partially coherent beam (PCB) are reviewed in this paper. A new tensor ABCD law for treating the propagation of partially coherent Gaussian-Schell model (GSM) beams through a paraxial optical system is derived. The focusing, spectral shift and fractional Fourier transform of the GSM beam are investigated by using the tensor method. The generation of PCB with special optical resonator is studied both theoretically and experimentally. Furthermore, the ghost imaging and superluminal propagation of the PCB is discussed. The results show clearly that the coherence of light have strong influences on the ghost imaging and superluminal propagation.