Complex signal recovery from multiple fractional Fourier-transform intensities

The problem of recovering a complex signal from the magnitudes of any number of its fractional Fourier transforms at any set of fractional orders is addressed. This problem corresponds to the problem of phase retrieval from the transverse intensity profiles of an optical field at arbitrary locations in an optical system involving arbitrary concatenations of lenses and sections of free space. The dependence of the results on the number of orders, their spread, and the noise is investigated. Generally, increasing the number of orders improves the results, but with diminishing return beyond a certain point. Selecting the measurement planes such that their fractional orders are well separated or spread as much as possible also leads to better results.     

Optical fractional Fourier transforms with complex orders

A definition of real and complex domains for the order of fractional Fourier transforms is introduced in association with diffraction either in free space or through real and complex lenses. The geometrical and the optical conditions that lead to real and complex orders of the fractional Fourier transforms and their relevance to cascading optical systems are discussed.     

Eigenfunctions of the offset Fourier,fractional Fourier,and linear canonical transforms

The offset Fourier transform (offset FT), offset fractional Fourier transform (offset FRFT), and offset linear canonical transform (offset LCT) are the space-shifted and frequency-modulated versions of the original transforms. They are more general and flexible than the original ones. We derive the eigenfunctions and the eigenvalues of the offset FT, FRFT, and LCT. We can use their eigenfunctions to analyze the self-imaging phenomena of the optical system with free spaces and the media with the transfer function exp[j(h2x2 + h1x + h0)] (such as lenses and shifted lenses). Their eigenfunctions are also useful for resonance phenomena analysis, fractal theory development, and phase retrieval.     

Chirp filtering in the fractional Fourier domain

In the Wigner domain of a one-dimensional function, a certain chirp term represents a rotated line delta function. On the other hand, a fractional Fourier transform (FRT) can be associated with a rotation of the Wigner-distribution function by an angle connected with the FRT order. Thus with the FRT tool a chirp and a delta function can be transformed one into the other. Taking the chirp as additive noise, the FRT is used for filtering the line delta function in the appropriate fractional Fourier domain. Experimental filtering results for a Gaussian input function, which is modulated by an additive chirp noise, are shown. Excellent agreement between experiments and computer simulations is achieved.     

Simplified fractional Fourier transforms

The fractional Fourier transform (FRFT) has been used for many years, and it is useful in many applications. Most applications of the FRFT are based on the design of fractional filters (such as removal of chirp noise and the fractional Hilbert transform) or on fractional correlation (such as scaled space-variant pattern recognition). In this study we introduce several types of simplified fractional Fourier transform (SFRFT). Such transforms are all special cases of a linear canonical transform (an affine Fourier transform or an ABCD transform). They have the same capabilities as the original FRFT for design of fractional filters or for fractional correlation. But they are simpler than the original FRFT in terms of digital computation, optical implementation, implementation of gradient-index media, and implementation of radar systems. Our goal is to search for the simplest transform that has the same capabilities as the original FRFT. Thus we discuss not only the formulas and properties of the SFRFT's but also their implementation. Although these SFRFT's usually have no additivity properties, they are useful for the practical applications. They have great potential for replacing the original FRFT's in many applications.     

Multiple fractional fourier transform hologram by use of holographic lens

Abstract

A new method is presented to record multiple fractional Fourier transform holograms by use of a holographic lens. With holographic lenses, several fractional Fourier transform holograms of different objects can be recorded in a simple way, and images of these recorded objects can be reconstructed in different positions and directions in three-dimensional space. In this paper, the theory and characteristics of the hologram recorded using the holographic lens are analysed. Using this method, a multiple fractional Fourier transform hologram was fabricated, with satisfactory results.     

Lie algebraic structure and generalized Poisson conservation law for fractional generalized Hamiltonian systems

Based on definition of Riemann–Liouville fractional derivatives, we find that the fractional generalized Hamiltonian system possesses consistent algebraic structure and Lie algebraic structure, and the generalized Poisson conservation law of the fractional generalized Hamiltonian system is investigated. As special cases of this paper, the conditions under which a fractional generalized Hamiltonian system can be reduced to a generalized Hamiltonian system and a classical Hamiltonian system are given, and the Lie algebraic structure and Poisson conservation law of these special dynamical systems are obtained, respectively. Then, by using the method and results of this paper, we construct three kinds of new fractional dynamical model, i.e. a fractional Henon–Heiles model, a fractional Euler–Poinsot model of a rigid body and a fractional Volterra model of the three species groups, and their Poisson conserved quantities are obtained.     

Diagnostic ultrasound tooth imaging using fractional Fourier transform

An ultrasound contact imaging method is proposed to measure the enamel thickness in the human tooth. A delay-line transducer with a working frequency of 15 MHz is chosen to achieve a minimum resolvable distance of 400 μm in human enamel. To confirm the contact between the tooth and the transducer, a verification technique based on the phase shift upon reflection is used. Because of the high attenuation in human teeth, linear frequency-modulated chirp excitation and pulse compression are exploited to increase the penetration depth and improve the SNR. Preliminary measurements show that the enamel-dentin boundary creates numerous internal reflections, which cause the applied chirp signals to interfere arbitrarily. In this work, the fractional Fourier transform (FrFT) is employed for the first time in dental imaging to separate chirp signals overlapping in both time and frequency domains. The overlapped chirps are compressed using the FrFT and matched filter techniques. Micro-computed tomography is used for validation of the ultrasound measurements for both techniques. For a human molar, the thickness of the enamel layer is measured with an average error of 5.5% after compressing with the FrFT and 13.4% after compressing with the matched filter based on the average speed of sound in human teeth.     

Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier transform

We show that the fractional Fourier transform is a suitable mechanism with which to analyze the diffraction patterns produced by a one-dimensional object because its intensity distribution is partially described by a linear chirp function. The three-dimensional location and the diameter of a fiber can be determined, provided that the optimal fractional order is selected. The effect of compaction of the intensity distribution in the fractional Fourier domain is discussed. A few experimental results are presented.     

Optical aperture synthesis imaging with fractional Fourier-domain filtering

The fractional Fourier transform, which is a generalization of the classical Fourier transform, is introduced into an optical aperture synthesis (OAS) system by which imaging of an astronomical object can be achieved. We introduce fractional Fourier optical imaging and fractional Fourier-domain filtering (FFDF), and then present the schematic diagram of an OAS imaging system with FFDF. The modulation transfer function of an OAS system with FFDF is compared with that of an OAS system in the same condition. The result indicates that the OAS system with FFDF has larger practical cutoff frequency when the fill factor is smaller. Furthermore, the quality of imaging and restoration also demonstrates this conclusion.     

用于分数傅里叶域滤波位相元件设计新算法

基于部分相干成像和分数傅里叶变换理论,建立成像系统的分数傅里叶域滤波模型,提出了可快速设计分数域位相型滤波片的 FrPI 算法。此算法将逆向设计预设初位相与分数傅里叶迭代技术相结合,针对成像系统中不同的设计要求可快速设计出所需的分数域滤波片结构。模拟结果表明,对数值孔径有限的部分相干成像系统,使用新算法设计的分数域滤波片,可有效补偿由于高频损失带来的图像失真,从而使局部线宽偏差和面积偏差分别由滤波前的 36.548nm 和 39.09%下降到了 15.234nm 和 7.72%,明显减少了空间像畸变。     

Planar Diffractive Lenses: Fundamentals,Functionalities, and Applications

Traditional objective lenses in modern microscopy, based on the refraction of light, are restricted by the Rayleigh diffraction limit. The existing methods to overcome this limit can be categorized into near‐field (e.g., scanning near‐field optical microscopy, superlens, microsphere lens) and far‐field (e.g., stimulated emission depletion microscopy, photoactivated localization microscopy, stochastic optical reconstruction microscopy) approaches. However, they either operate in the challenging near‐field mode or there is the need to label samples in biology. Recently, through manipulation of the diffraction of light with binary masks or gradient metasurfaces, some miniaturized and planar lenses have been reported with intriguing functionalities such as ultrahigh numerical aperture, large depth of focus, and subdiffraction‐limit focusing in far‐field, which provides a viable solution for the label‐free superresolution imaging. Here, the recent advances in planar diffractive lenses (PDLs) are reviewed from a united theoretical account on diffraction‐based focusing optics, and the underlying physics of nanofocusing via constructive or destructive interference is revealed. Various approaches of realizing PDLs are introduced in terms of their unique performances and interpreted by using optical aberration theory. Furthermore, a detailed tutorial about applying these planar lenses in nanoimaging is provided, followed by an outlook regarding future development toward practical applications.     

自适应分数阶算子的图像增强

为了突出图像纹理细节的同时保留平滑区域,节省确定分数阶微分阶次的时间,提出了一种改进的自适应分数阶微分算子。首先将经典Tiansi模板分解为四个不同方向,分别与待处理像素点进行卷积,达到增强图像纹理细节的效果;其次针对Tiansi算子需通过多次实验确定最佳微分阶次的现状,结合图像的局部特征信息,构建了具有自适应能力的分数阶阶次模型,能够获得比原图像更丰富的细节信息。对多组不同场景图像的实验结果表明：构建的自适应分数阶微分算子有效地增强了图像的纹理细节,自适应分数阶微分算子的主观视觉效果和客观评价指标均优于原图像,其客观评价指标中的平均梯度、信息熵、对比度相比原图像平均提高190.3%、8.1%、18.3%;平均梯度、对比度相比Tiansi算子处理后的图像平均提高45.0%、9.6%。     

Optical illustration of a varied fractional Fourier-transform order and the Radon-Wigner display

Based on an all-optical system, a display of a fractional Fourier transform with many fractional orders is proposed. Because digital image-processing terminology is used, this display is known as the Radon-Wigner transform. It enables new aspects for signal analysis that are related to time- and spatial-frequency analyses. The given approach for producing this display starts with a one-dimensional input signal although the output signal contains two dimensions. The optical setup for obtaining the fractional Fourier transform was adapted to include only fixed free-space propagation distances and variable lenses. With a set of two multifacet composite holograms, the Radon-Wigner display has been demonstrated experimentally.     

Chirped-pulse amplification system based on chirp reversal and near-field spatial reversal with common tiled grating pair as stretcher and compressor

Chirped-pulse amplification system based on chirp reversal in optical parametric chirped-pulse amplification is proposed and experimentally demonstrated. The operation of this system can be described as negative stretching-temporal chirp reversal-energy amplification-negative compression, in which the pulse is stretched and compressed with the same gratings. Stand-alone stretcher adopting lenses or concave mirrors with large aperture can be omitted. Simulations showed that this work mode can also increase the cut-off band-pass of the whole system and increase the output energy by 15-17%. In addition, the stability of a tiled-grating compressor can be improved with this work mode.     

应用分数傅立叶变换方法的水下目标特征提取

针对基于传统时频分析方法的水下目标特征提取方法的缺陷,引入了一种新的时频分析方法——分数阶傅立叶变换（FRFT）方法用于处理水下目标特征提取问题。仿真信号和实际试验数据的处理结果表明,当主动声纳的发射信号为线性调频信号时,分数阶傅立叶变换方法可以使目标的回波能量在分数阶域的对应区域聚集,使目标和混响的分数阶傅立叶变换在分数阶域上不仅呈现明显不同的特征,而且较其他时频分析方法具有抗混响的优点。     

Computational algorithms for FE formulations involving fractional operators

This paper considers the development of transient solution algorithms for finite element simulations of viscoelastic problems involving fractional integrodifferential operators. Specifically, numerical approximations are developed for the Grunwald-Liouville-Riemann formalism. This includes establishing formal error estimations. Based on the numerical representations of the fractional operators, implicit, explicit and predictor corrector type transient algorithms are derived for viscoelastic finite element simulations. To illustrate their computational properties, the results of several numerical benchmark experiments are presented. These emphasize the efficiency and stability of the various algorithms developed.Work partially supported by NASA Langley through Grant NAG-1-444     

A Wiener path integral technique for determining the stochastic response of nonlinear oscillators with fractional derivative elements: A constrained variational formulation with free boundaries

A technique based on the concept of Wiener path integral (WPI) is developed for determining approximately the joint response probability density function (PDF) of nonlinear oscillators endowed with fractional derivative elements. Specifically, first, the dependence of the state of the system on its history due to the fractional derivative terms is accounted for, alternatively, by augmenting the response vector and by considering additional auxiliary state variables and equations. In this regard, the original single-degree-of-freedom (SDOF) nonlinear system with fractional derivative terms is cast, equivalently, into a multi-degree-of-freedom (MDOF) nonlinear system involving integer-order derivatives only. From a mathematics perspective, the equations of motion referring to the latter can be interpreted as constrained. Second, to circumvent the challenge of increased dimensionality of the problem due to the augmentation of the response vector, a WPI formulation with mixed fixed/free boundary conditions is developed for determining directly any lower-dimensional joint PDF corresponding to a subset only of the response vector components. This can be construed as an approximation-free dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. Thus, the original SDOF oscillator joint PDF corresponding to the response displacement and velocity is determined efficiently, while circumventing the computationally challenging task of treating directly equations of motion involving fractional derivatives. Two illustrative numerical examples are considered for demonstrating the reliability of the developed technique. These pertain to a nonlinear Duffing and a nonlinear vibro-impact oscillators with fractional derivative elements subjected to combined stochastic and deterministic periodic loading. Note that alternative standard approximate techniques, such as statistical linearization, need to be significantly modified and extended to account for such cases of combined loading. Remarkably, it is shown herein that the WPI technique exhibits the additional advantage of treating such types of excitation in a straightforward manner without the need for any ad hoc modifications. Comparisons with pertinent Monte Carlo simulation data are included as well.     

Double-image encryption by iterative phase retrieval algorithm in fractional Fourier domain

A novel double-image encryption algorithm is proposed, which can simultaneously encrypt two images into a single one as the amplitude of a fractional Fourier transform with two different groups of fractional orders. The two original images can be retrieved independently by fractional Fourier transforms with two different groups of fractional orders, one public phase mask, and two different private phase masks. The proposed approach can enlarge the key space, achieve faster convergence in the iterative process, and avoid cross-talk between the two images in reconstruction. Numerical simulations are presented to verify its validity and efficiency.     

Quantisation of the free electromagnetic field implies quantisation of its angular momentum

It is shown that when the gauge-invariant Bohr–Rosenfeld commutators of the free electromagnetic field are applied to the expressions for the linear and angular momentum of the electromagnetic field interpreted as operators then, in the absence of electric and magnetic charge densities, these operators satisfy the canonical commutation relations for momentum and angular momentum. This confirms their validity as operators that can be used in quantum mechanical calculations of angular momentum.