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.     

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.     

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.     

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.     

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.     

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.     

Characterization of chirped pulses with the fractional-order Fourier transformation.

We show that the fractional-order Fourier transformation (FRT) is a suitable method to describe chirped pulses submitted to group-velocity dispersion in a linear dispersive medium. Amplitudes exhibiting different chirp coefficients are easily separated with the FRT, although they are temporally superposed.     

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.     

Sheet hydroforming of woven FRT composites: non-orthogonal constitutive equation considering shear stiffness and undulation of woven structure

For the simulation of sheet hydroforming for the shaping of woven fabric reinforced thermo-plastic (FRT) composites, a non-orthogonal constitutive model was developed based on a homogenization method by considering the microstructures of composites including mechanical and structural properties of the fabric reinforcement. This model is modified to capture the wrinkling behavior due to the undulation geometry of the woven structure and shear stiffness at the crossover of the warp and weft yarns of woven FRT composites. The model was implemented in an explicit dynamic finite element code to analyze the forming behavior of woven FRT during the stamp thermo-hydroforming process. Wrinkling behavior was investigated based on the application of a counteracting fluid pressure and changes to the initial blank shape.     

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.     

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.     

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.     

Fractional-Fourier-transform calculation through the fast-Fourier-transform algorithm

A method for the calculation of the fractional Fourier transform (FRT) by means of the fast Fourier transform (FFT) algorithm is presented. The process involves mainly two FFT's in cascade; thus the process has the same complexity as this algorithm. The method is valid for fractional orders varying from -1 to 1. Scaling factors for the FRT and Fresnel diffraction when calculated through the FFT are discussed.     

血液成分无创光学检测中浮动基准理论的适用性

在浮动基准理论（FRT）应用于血糖无创光学检测研究成果的基础上,进一步研究其在其他血液成分（如胆红素）无创光学检测中的适用性．根据径向检测基准位置存在的条件,经蒙特卡洛模拟的结果表明胆红素测量中难以找到浮动基准位置;而通过研究胆红素与水的置换效应,发现在波长524nm处吸光度值与胆红素的浓度无关,将该波长作为基准波长,实际测量中可以用于去除背景噪声和环境干扰．综合FRT在血糖及胆红素两种不同血液成分中应用的研究结果表明：对于不同检测成分,在相应的检测波段,浮动基准位置和浮动基准波长有一定的特异性,从而进一步完善和扩展了FRT的应用领域．     

Kinematics for Chicane-beam electron coincidence experiments

The kinematics for coincidence electrodisintegration experiments are investigated. “Out-of-plane” measurements designed to completely determine the unpolarized cross section are analyzed by an alternate method that considers the situation where the primary electron beam is redirected through the plane formed by the target and the spectrometer. Specific cases, namely, pion production in a light nucleus, proton electroproduction in the giant resonance region and deutron electrodisintegration in the quasielastic region are examined.     

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.     

Detection of Cherenkov light emission in liquid argon

Detection of Cherenkov light emission in liquid argon has been obtained with an ICARUS prototype, during a dedicated test run at the Gran Sasso Laboratory external facility. Ionizing tracks from cosmic ray muons crossing the detector active volume have been collected in coincidence with visible light signals from a photo-multiplier (PMT) immersed in liquid argon. A 3D reconstruction of the tracks has been performed exploiting the ICARUS imaging capability. The angular distributions of the tracks triggered by the PMT signals show an evident directionality. By means of a detailed Monte Carlo simulation we show that the geometrical characteristics of the events are compatible with the hypothesis of Cherenkov light emission as the main source of the PMT signals.     

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.     

Measuring the absolute photodetection efficiency using photon number correlations

We present two methods for determining the absolute detection efficiency of photon-counting detectors directly from their singles rates under illumination from a nonclassical light source. One method is based on a continuous variable analog to coincidence counting in discrete photon experiments, but it does not actually rely on high detector time resolutions. The second method is based on difference detection, which is a typical detection scheme in continuous variable quantum optics experiments. Since no coincidence detection is required with either method, they are useful for detection efficiency measurements of photodetectors with detector time resolutions far too low to resolve coincidence events.     

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.