Wavelet-fractional Fourier transforms

This paper extends the definition of fractional Fourier transform （FRFT） proposed by Namias V by using other orthonormal bases for L^2（R） instead of Hermite-Gaussian functions. The new orthonormal basis is gained indirectly from multiresolution analysis and orthonormal wavelets. The so defined FRFT is called wavelets-fractional Fourier transform.     

High order generalized permutational fractional Fourier transforms

We generalize the definition of the fractional Fourier transform (FRFT) by extending the new definition proposed by Shih. The generalized FRFT, called the high order generalized permutational fractional Fourier transform (HGPFRFT), is a generalized permutational transform. It is shown to have arbitrary natural number M periodic eigenvalues not only with respect to the order of Hermite－Gaussian functions but also to the order of the transform. This HGPFRFT will be reduced to the original FRFT proposed by Namias and Liu's generalized FRFT and Shih's FRFT at the three limits with M=+∞, M=4k(k is a natural number), and M=4, respectively. Therefore the HGPFRFT introduces a comprehensive approach to Shih's FRFT and the original definition. Some important properties of HGPFRFT are discussed. Lastly the results of computer simulations and symbolic representations of the transform are provided.     

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.     

Spectrum measurement in the fractional Fourier domain

Fractional Fourier transform (FRFT) plays an important role in many fields of optics and signal processing. This paper considers the problem of real-time measurement of the spectrum of a signal in the FRFT domain. In this paper, we propose two approaches for real-time measurement of the FRFT of a signal based on modulation and bandpass filtering systems. The relation is established between the linear frequency modulation (LFM or chirp) spectrum and the FRFT of its envelope. In addition, two applications for spectrum measurement are presented in the FRFT domain. The LFM signal can be bandlimited in the Fourier transform (FT) domain through spectrum measurement associated with bandpass filtering method. The results can also be useful in the problems related to swept-frequency filter for measurement in the FRFT domain.     

基于柱坐标系下的空心高斯光束的分数傅里叶变换

推导出分数傅里叶变换基于柱坐标系下的解析表达式,应用此解析表达式对空心高斯光束进行分析,得到其分数傅里叶变换的解析表达式。讨论其在分数傅里叶变换平面的光强分布与各种光束参数之间的变换关系,并进行数值计算,从而得出基于柱坐标系下的空心高斯光束的分数傅里叶变换的传输性质。所得结果为分析和计算这种光束的分数傅里叶变换提供了方便,同时对此类光束的传输控制提供了基础理论支持。     

Signal separation using linear canonical and fractional Fourier transforms

Signal separation is an important operation in many areas such as communications, geophysics, and optics including optical signal processing. In this article, the issue of signal separation based on the linear canonical transform (LCT) and the fractional Fourier transform (FRFT) is considered. An iterative algorithm for signal separation in the 2-ray ground reflected model using the FRFT is also presented along with the simulation results.     

超洛伦兹-高斯光束SLG11模的分数傅里叶变换特性

 引入了一簇互相正交的超洛伦兹-高斯光束以描述半导体激光器所产生的大角度高阶模远场分布。将分数傅里叶变换应用于超洛伦兹-高斯光束SLG­₁₁模的传输特性的研究中。利用傅里叶变换的卷积原理,导出了SLG₁₁模经分数傅里叶变换系统后场分布的解析表达式。根据所得到的公式进行了数值计算,系统分析了分数傅里叶变换阶数和光束各参数对SLG₁₁模在分数傅里叶变换面上光强分布的影响。结果显示：SLG₁₁模在分数傅里叶变换面上的归一化强度分布随分数傅里叶变换的阶数呈周期性变化,周期为2;随着光束参数的增大,SLG₁₁模在分数傅里叶变换面上的光斑尺寸增大。     

强非局域非线性介质中的互诱导分数傅里叶变换

在强非局域非线性介质中,用强抽运光可以诱导另一弱信号光实现互诱导分数傅里叶变换效应.信号光的分数傅里叶变换阶数与传输距离和抽运光功率有关,当传输距离不变时与抽运光功率的开方成正比.互诱导分数傅里叶变换是实现光控光的又一方法,其特性有助于研发新型的分数傅里叶变换器件,并在光信息处理、光学成像等多个领域有潜在的应用. 关键词： 分数傅里叶变换 非局域介质 互诱导 光孤子     

分数域滤波的分数傅里叶变换光学成像性能分析

将分数域滤波同光学成像相结合,提出了一种基于Lohmann Ⅰ型的带有滤波孔径的两级联分数傅里叶变换光学成像系统.根据分数傅里叶变换和菲涅耳衍射之间的关系以及分数傅里叶变换的分数阶可加性,结合分数域滤波分析了分数傅里叶变换光学成像的基本理论.以点扩散函数和调制传递函数作为评价成像质量的准则,详细分析了不同分数阶光学系统...     

An optical waveform pre-distortion method based on time domain fractional Fourier transformation

An optical waveform pre-distortion method is proposed in reducing the temporal broadening of pulse based on the optical implementation of time domain fractional Fourier transform (FRFT). Moreover, a new analysis of the dispersion and self-phase modulation effects is investigated. The simulation results show that prechirp coefficient and proper fractional orders of FRFT can be beneficial in mitigating pulse broadening.     

Fractional Fourier transform of off-axial elliptical cosh-Gaussian beams

Cosh-Gaussian beams are of practical interest because they are more efficient at extracting energy from conventional laser amplifiers, and they have some important applications because their profiles can closely resemble the flat-top field distribution by choosing suitable beam parameters of cosh parts. The fractional Fourier transform (FRFT) is applied to treat the propagation of off-axial elliptical cosh-Gaussian beams (EChGBs). By the use of vector integration, the analytical expression for an off-axial EChGB passing through a FRFT system is derived in terms of tensor method. Our formula provides a convenient way for studying the FRFT of off-axial EChGBs; some numerical simulations are also given.     

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     

一种基于角谱理论的光学FRFT的数值模拟算法

以角谱传播理论和分数傅里叶变换(FRFT)的单透镜和双透镜光学实现单元为理论基础,提出了一种基于实际光学结构的光学FRFT的数值模拟算法,推导了该算法中采样间隔的选取方法,并利用Matlab语言编写了相应的处理程序,将该程序对方孔分数傅里叶变换的计算结果与实际实验结果相比较,结果表明该算法是实际光学结构在理想情况下的模拟,因此,该数值计算在一定程度上可以代替柱面镜对光场实现分数傅里叶变换,也可以为实际实验提供很好的参考和指导。     

基于FRFT域分形特征的压制干扰存在性检测（英）

基于分数阶傅里叶变换（FRFT）域的分形特征研究压制干扰的存在性检测问题。首先,分析了典型的三种压制干扰的分形特征,说明了典型压制干扰信号具有分形特性,并采用盒维数和信息维数定量描述它们的分形特征。然后,发现了压制干扰和高斯白噪声在FRFT域具有不同的分形特征,进而提出了一种检测压制干扰存在性的方法。最后,仿真验证了该方法的有效性和优越性。     

光学图像加密与多参数加权类分数傅里叶变换

为了提高图像加密的安全性, 提出了一种多参数加权类分数傅里叶变换。此类多参数加权类分数傅里叶变换是C.C.Shih提出的四项加权类分数傅里叶变换的一种扩展, 除了分数阶数, 还有四个在四项加权系数之中的自由参数, 称其为向量参数。同时给出此多参数加权类分数傅里叶变换的离散形式, 并把这种算法应用到光学图像加密中。此算法在应用一次二维分数傅里叶变换可以有十个密键：一类为阶数参数; 另一类为向量参数, 因此这种加密算法在增加了安全性的同时, 加密过程的复杂度降低。数值仿真验证了此算法的有效性和可靠性。     

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。     

Application of fractional Fourier transform on spatial filtering

Starting from the relationship between the Wigner transformation and the fractional Fourier transform (FRFT), we apply the FRFT to the spatial filtering. In the process of filtering, the noise can be simulated as the chirp functions or their linear combination, which can be filtered at the corresponding fractional planes, and also the actual positions can be exactly obtained. Some numerical simulations prove its possibility.     

分数傅里叶变换面上余弦-高斯光束的变换特性

 利用Wigner分布函数的方法，研究了余弦-高斯光束的分数傅里叶变换特性。导出了余弦-高斯光束在分数傅里叶变换面上光强分布和束宽的解析计算公式，并对此进行了数值模拟计算。研究表明：分数傅里叶变换阶数对余弦-高斯光束的光强分布有明显影响，余弦-高斯光束的轴上光强随分数傅里叶变换阶数呈周期性变化，束宽随分数傅里叶变换阶数也呈周期性变化，周期为2；对给定调制参数的余弦-高斯光束，通过适当选取分数傅里叶变化阶数可以获得平顶的光强分布。     

Novel uncertainty relations associated with fractional Fourier transform

In this paper the relations between two spreads, between two group delays, and between one spread and one group delay in fractional Fourier transform (FRFT) domains, are presented and three theorems on the uncertainty principle in FRFT domains are also developed. Theorem 1 gives the bounds of two spreads in two FRFT domains. Theorem 2 shows the uncertainty relation between two group delays in two FRFT domains. Theorem 3 presents the crossed uncertainty relation between one group delay and one spread in two FRFT domains. The novelty of their results lies in connecting the products of different physical measures and giving their physical interpretations. The existing uncertainty principle in the FRFT domain is only a special case of theorem 1, and the conventional uncertainty principle in time-frequency domains is a special case of their results. Therefore, three theorems develop the relations of two spreads in time-frequency domains into the relations between two spreads, between two group delays, and between one spread and one group delay in FRFT domains.     

超洛伦兹-高斯光束SLG11模的分数傅里叶变换特性

引入了一簇互相正交的超洛伦兹-高斯光束以描述半导体激光器所产生的大角度高阶模远场分布。将分数傅里叶变换应用于超洛伦兹-高斯光束SLG11模的传输特性的研究中。利用傅里叶变换的卷积原理,导出了SLG11模经分数傅里叶变换系统后场分布的解析表达式。根据所得到的公式进行了数值计算,系统分析了分数傅里叶变换阶数和光束各参数对SLG11模在分数傅里叶变换面上光强分布的影响。结果显示：SLG11模在分数傅里叶变换面上的归一化强度分布随分数傅里叶变换的阶数呈周期性变化,周期为2;随着光束参数的增大,SLG11模在分数傅里叶变换面上的光斑尺寸增大。