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.     

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.     

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.     

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.     

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.     

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.     

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 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.     

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.     

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.     

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.     

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.     

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.     

Encryption of digital hologram based on phase-shifting interferometry and virtual optics

A new information encryption system is presented, based on phase-shifting interferometry and virtual optics. Three-step phase-shifting interferometry is used to record a digital hologram of the input data and a virtual optical system based on the scaled optical fractional Fourier transform is used for encryption of the recorded digital hologram. In the virtual optical system, the digital hologram to be encrypted is fractional Fourier transformed two times, and a random phase mask is placed at the output plane of the first fractional Fourier transform. Both the encryption and decryption processes are performed digitally. The encrypted data and the keys for decryption can be stored and transmitted in a conventional communication channel. Numerical simulations are presented to verify validity and efficiency.     

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.     

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．     

Collins formulae in both space and frequency domains for ABCD optical systems with small deformations

Abstract

Generalized Collins formulae for arbitrary imperfect ABCD optical systems with small deformations (misalignments and/or deviations from ideal optical operations) are obtained in both the space domain and the frequency domain. These formulae can provide a unified way to analyse the performance of a practical optical system in both domains, including the ideal as its special case. Particularly, it shows that a reciprocally symmetrical ABCD system with small deformations can implement so-called almost-fractional Fourier transformation (FRT) simultaneously in both domains with the same order. Some other applications in practice are also discussed to verify the effectiveness of our proposed method.     

Algorithm study of Collins formula and inverse Collins formula

In the study of a diffraction field of a light wave passing through a symmetrical paraxial optical system, the Collins formula and its inverse are convenient for calculation. The algorithm study of the Collins formula demonstrates that both a single fast Fourier transform algorithm and a double fast Fourier transform algorithm can be used in diffraction calculation. But, whichever algorithm is adopted, only by meeting some specific conditions can the amplitude and phase distributions of a diffracted wave be calculated correctly. Based on the Nyquist sampling theorem, the indispensable conditions to calculate a diffraction field accurately are presented.     

Paraxial speckle-based metrology systems with an aperture

Digital speckle photography can be used in the analysis of surface motion in combination with an optical linear canonical transform (LCT). Previously [D. P. Kelly et al. Appl. Opt.44, 2720 (2005)] it has been shown that optical fractional Fourier transforms (OFRTs) can be used to vary the range and sensitivity of speckle-based metrology systems, allowing the measurement of both the magnitude and direction of tilting (rotation) and translation motion simultaneously, provided that the motion is captured in two separate OFRT domains. This requires two bulk optical systems. We extend the OFRT analysis to more general LCT systems with a single limiting aperture. The effect of a limiting aperture in LCT systems is examined in more detail by deriving a generalized Yamaguchi correlation factor. We demonstrate the benefits of using an LCT approach to metrology design. Using this technique, we show that by varying the curvature of the illuminating field, we can effectively change the output domain. From a practical perspective this means that estimation of the motion of a target can be achieved by using one bulk optical system and different illuminating conditions. Experimental results are provided to support our theoretical analysis.     

Application of the fractional Fourier transform to the design of LCOS based optical interconnects and fiber switches

It is shown that reflective liquid crystal on silicon (LCOS) spatial light modulator (SLM) based interconnects or fiber switches that use defocus to reduce crosstalk can be evaluated and optimized using a fractional Fourier transform if certain optical symmetry conditions are met. Theoretically the maximum allowable linear hologram phase error compared to a Fourier switch is increased by a factor of six before the target crosstalk for telecom applications of -40 dB is exceeded. A Gerchberg-Saxton algorithm incorporating a fractional Fourier transform modified for use with a reflective LCOS SLM is used to optimize multi-casting holograms in a prototype telecom switch. Experiments are in close agreement to predicted performance.