Fast numerical algorithm for the linear canonical transform

The linear canonical transform (LCT) describes the effect of any quadratic phase system (QPS) on an input optical wave field. Special cases of the LCT include the fractional Fourier transform (FRT), the Fourier transform (FT), and the Fresnel transform (FST) describing free-space propagation. Currently there are numerous efficient algorithms used (for purposes of numerical simulation in the area of optical signal processing) to calculate the discrete FT, FRT, and FST. All of these algorithms are based on the use of the fast Fourier transform (FFT). In this paper we develop theory for the discrete linear canonical transform (DLCT), which is to the LCT what the discrete Fourier transform (DFT) is to the FT. We then derive the fast linear canonical transform (FLCT), an N log N algorithm for its numerical implementation by an approach similar to that used in deriving the FFT from the DFT. Our algorithm is significantly different from the FFT, is based purely on the properties of the LCT, and can be used for FFT, FRT, and FST calculations and, in the most general case, for the rapid calculation of the effect of any QPS.     

Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane

An analytical and concise formula is derived for the fractional Fourier transform (FRT) of partially coherent beams that is based on the tensorial propagation formula of the cross-spectral density of partially coherent twisted anisotropic Gaussian-Schell-model (GSM) beams. The corresponding tensor ABCD law performing the FRT is obtained. The connections between the FRT formula and the generalized diffraction integral formulas for partially coherent beams passing through aligned optical systems and misaligned optical systems are discussed. With use of the derived formula, the transformation and spectrum properties of partially coherent GSM beams in the FRT plane are studied in detail. The results show that the fractional order of the FRT has strong effects on the transformation properties and the spectrum properties of partially coherent GSM beams. Our method provides a simple and convenient way to study the FRT of twisted anisotropic GSM beams.     

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.     

Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam

The truncated fractional Fourier transform (FRT) is applied to a partially coherent Gaussian Schell-model (GSM) beam. The analytical propagation formula for a partially coherent GSM beam propagating through a truncated FRT optical system is derived by using a tensor method. Furthermore, we report the experimental observation of the truncated FRT for a partially coherent GSM beam. The experimental results are consistent with the theoretical results. Our results show that initial source coherence, fractional order, and aperture width (i.e., truncation parameter) have strong influences on the intensity and coherence properties of the partially coherent beam in the FRT plane. When the aperture width is large, both the intensity and the spectral degree of coherence in the FRT plane are of Gaussian distribution. As the aperture width decreases, the diffraction pattern gradually appears in the FRT plane, and the spectral degree of coherence becomes of non-Gaussian distribution. As the coherence of the initial GSM beam decreases, the diffraction pattern for the case of small aperture widths gradually disappears.     

Coincidence fractional Fourier transform implemented with partially coherent light radiation

We introduce the coincidence fractional Fourier transform (FRT) implemented with incoherent and partially coherent light radiation. Optical systems for implementing the coincidence FRT are designed. The results show that the visibility and quality of the coincidence FRT of an object are closely related to the light source's transverse size, coherence, and spectral width. As an example, we numerically study the coincidence FRT of a single slit.     

BCD optical system and the fractional fourier transform

Abstract

A general and systematic analysis about the relationship between ABCD optical systems and the fractional Fourier transform (FRT) is provided. It is shown that the FRT can be implemented with an ABCD system but usually different scaling factors for the input and output functions must be used. The requirement for the property of direct additivity of the FRT order is derived for a cascade system; and the method of finding the final order of the FRT for a general cascade ABCD system by using the similarity theorem is discussed. As an application example of the results, an approach to observation of the FRT of continuously variable orders with a scale invariant input is demonstrated.     

Sensitivity of the fractional Fourier transform to parameters and its application in optical measurement

The fractional Fourier transform (FRT) is becoming important in optics and can be used as a new tool to analyze many optical problems. However, we point out that the FRT might be much more sensitive to parameters than the conventional Fourier transform. This sensitivity leads to higher requirements on the optical implementation. On the other hand, high parametric sensitivity can be used in optical diffraction measurements. We give the first proposal, to our knowledge, of the FRT's applications in optical measurement.     

Reply to the Comment on the Fractional Fourier Transform in Optical Propagation Problems

The application of the fractional Fourier transform (FRT) to optical propagation problems is re-examined as a reply to the recent comment by S. Abe and J. T. Sheridan. It is shown that their criticism to our previous consideration of Fresnel diffraction in the context of the FRT is not appropriate.     

Fractional Fourier transform for partially coherent off-axis Gaussian Schell-model beam

The fractional Fourier transform (FRT) is applied to a partially coherent off-axis Gaussian Schell-model (GSM) beam, and an analytical formula is derived for the FRT of a partially coherent off-axis GSM beam. The corresponding tensor ABCD law for performing the FRT of a partially coherent off-axis GSM beam is also obtained. As an application example, the FRT of a partially coherent linear laser array that is expanded as a sum of off-axis GSM beams is studied. The derived formulas are used to provide numerical examples. The formulas provide a convenient way to analyze and calculate the FRT of a partially coherent off-axis GSM beam.     

White-light optical implementation of the fractional fourier transform with adjustable order control

An optical implementation of the fractional Fourier transform (FRT) with broadband illumination is proposed by use of a single imaging element, namely, a blazed diffractive lens. The setup displays an achromatized version of the FRT of order P of any two-dimensional input function. This fractional order can be tuned continuously by shifting of the input along the optical axis. Our compact and flexible configuration is tested with a chirplike input signal, and the good experimental results obtained support the theory.     

Scaled fractional Fourier transform and its optical implementation

The scaled fractional Fourier transform is suggested and is implemented optically by one lens for different values of phi and output scale. In addition, physically it relates the FRT with the general lens transform-the optical diffraction between two asymmetrically positioned planes before and after a lens.     

遗传算法在分数阶Fourier变换域极值优化中的应用

分数阶Fourier变换采用线性调频基,因此,线性调频（Linear Frequency Modulation,LFM）信号在分数阶Fourier域平面能够聚焦,并形成峰值。为了克服传统步进式搜索法在LFM信号峰值搜索中效率低下的缺点,将遗传算法引入到分数阶Fourier变换极值搜索中。仿真结果表明,该方法优于传统的步进式搜索法。     

Chromatic compensation of broadband light diffraction: ABCD-matrix approach

Compensation of chromatic dispersion for the optical implementation of mathematical transformations has proved to be an important tool in the design of new optical methods for full-color signal processing. A novel approach for designing dispersion-compensated, broadband optical transformers, both Fourier and Fresnel, based on the collimated Fresnel number is introduced. In a second stage, the above framework is fully exploited to achieve the optical implementation of the fractional Fourier transform (FRT) of any diffracting screen with broadband illumination. Moreover, we demonstrate that the amount of shift variance of the dispersion-compensated FRT can be tuned continuously from the spatial domain, which is totally space variant, to the spectral domain, which is totally space invariant, with the chromatic correction remaining unaltered.     

An enhanced Fourier transform spectrometer with a search algorithm

Interferometric instruments have the following serious weak points: (1) the necessity of doing a Fourier transform that involves a vast amount of calculation; (2) the lack of knowledge of suitable measuring conditions until the Fourier transform is finished; and (3) the spectral resolution of the conventional Fourier-based techniques is significantly affected by the sampling rate, data length, and noise in signal processing. In this paper, an enhanced spectrometer is proposed using the modified forward-backward linear prediction method (MFBLP) with a search algorithm. To document the advantage of the method presented, a computer simulation for multiple-wavenumber estimation is investigated. The MFBLP method is truly superior to the fast Fourier transform (FFT) method. In general, the spectral resolution using the FFT method is proportional to the data length. In this paper, however, it is shown that excellent results can also be obtained from only 60 sample points using the FFT method. Moreover, from experimental results, we also conclude that the sampling rate must be consistent with the condition 632p·t·D<3164, where f_p , represents the value of the pulse generator frequency in Hz, t the observation time, and D the decimation factor     

Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics

We report the experimental observation of the fractional Fourier transform (FRT) for a partially coherent optical beam with Gaussian statistics [i.e., partially coherent Gaussian Schell-model (GSM) beam]. The intensity distribution (or beam width) and the modulus of the square of the spectral degree of coherence (or coherence width) of a partially coherent GSM beam in the FRT plane are measured, and the experimental results are analyzed and agree well with the theoretical results. The FRT optical system provides a convenient way to control the properties, e.g., the intensity distribution, beam width, spectral degree of coherence, and coherence width, of a partially coherent beam.     

基于FFT的舍入不确定度评估

舍入不确定度是快速傅里叶算法不确定度评估中的一个重要来源。将表示成向量矩阵的FFT算法分解为稀疏矩阵,以此确定信号传递流图,从而得出舍入不确定度在每一级的传递形式。再利用GUM中的B类方法对其进行评定,评定时假设舍入测量不确定度为均匀分布,通过对FFT算法中基-2算法的舍入不确定度进行评定,最终经过算法传递后得到其舍入不确定度的结果,在此基础上建立了FFT舍入测量不确定度的通用评估方法。     

免疫算法在分数阶Fourier变换域极值优化中的应用

利用线性调频(Linear Frequency Modulation,LFM)信号在分数阶Fourier域上的聚焦性,通过搜索可实现LFM信号的检测和参数估计。通常采用步进式搜索法,效率低下。为了克服该缺点,通过对分数阶Fourier域优化问题的研究,将免疫算法引入到分数阶Fourier变换极值搜索中。仿真结果表明:该方法优于传统的步进式搜索法。     

ON THE USE OF FFT ALGORITHM FOR THE CIRCUMFERENTIAL CO-ORDINATE IN BOUNDARY ELEMENT FORMULATION OF AXISYMMETRIC PROBLEMS

A new numerical method is proposed for the boundary element analysis of axisymmetric bodies. The method is based on complex Fourier series expansion of boundary quantities in circumferential direction, which reduces the boundary element equation to an integral equation in (r–z) plane involving the Fourier coefficients of boundary quantities, where r and z are the co-ordinates of the (r, θ, z) cylindrical co-ordinate system. The kernels appearing in these integral equations can be computed effectively by discrete Fourier transform formulas together with the fast Fourier transform (FFT) algorithm, and the integral equations in (r–z) plane can be solved by Gaussian quadrature, which establishes the Fourier coefficients associated with boundary quantities. The Fourier transform solution can then be inverted into (r, θ, z) space by using again discrete Fourier transform formulas together with FFT algorithm. In the study, first we present the formulation of the proposed method which is outlined above. Then, the method is assessed by using three sample problems. A good agreement is observed in the comparisons of the predictions of the method with those available in the literature. It is further found that the proposed method provides considerable saving in computer time compared to existing methods of literature. © 1997 by John Wiley & Sons, Ltd.     

Comment on ‘The Fractional Fourier Transform in Optical Propagation Problems’

Although their mathematical forms apparently resemble each other, the diffraction integral and fractional Fourier transformation (FRT) have completely different physical meanings. We point out that an interpretation of the FRT given recently in a paper by Alieva et al. is not physically appropriate. We then show how those integral transformations can be treated in a unified way within the framework of the special affine Fourier transformation. Finally the multidimensional FRT presented in the above paper is further generalized to allow n independent fractional degrees.     

Coincidence subwavelength fractional Fourier transform

The coincidence subwavelength fractional Fourier transforms (FRTs) with entangled photon pairs and incoherent light radiation are introduced as an extension of the recently introduced coincidence FRT. Optical systems for implementing the coincidence subwavelength FRTs are designed. The width of the coincidence subwavelength FRT pattern is two times narrower than the width of the coincidence FRT. The coincidence subwavelength FRT with partially coherent light radiation is also studied numerically. Differences between the coincidence subwavelength FRT with entangled photon pairs and that with incoherent light radiation are discussed.