Observing the fractional Fourier transform by free-space Fresnel diffraction

The fractional Fourier transform of an object can be observed in the free-space Fresnel diffraction pattern of the object.     

Double-lens extended fractional Fourier transform

A double-lens extended fractional Fourier transform (EFRT) is proposed. In the double-lens setup, arbitrary-order EFRTs including real orders and complex orders can be carried out. To verify that the single-lens EFRT and the double-lens EFRT are equivalent, numerical simulations and optical experiments for the real-order EFRT and the complex-order EFRT are performed. Their results indicate that the double-lens EFRT is consistent with the single-lens EFRT. Thus the double-lens setup can be regarded as another basic optical configuration. Compared with the single-lens EFRT, the double-lens EFRT has some advantages.     

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.     

Free-space diffraction and the fractional fourier transform

Abstract

An explicit expression is derived for free-space propagation in terms of the fractional Fourier transform. The expression contains two arbitrary parameters, which can be, for example, an original phase curvature together with a scaling parameter or, alternatively, the order of fractional transformation. Appropriate choice of these parameters is discussed. This approach is then applied to the free-space propagation of Gaussian beams.     

Nonlinear effects of phase blurring on Fourier transform holograms

Liquid-crystal light valves can have intensity-dependent resolution. We find for a nematic liquid-crystal light valve that this effect is well modeled as a phase that has been blurred by a linear space-invariant filter. The phase point-spread function is measured and is used in simulations to demonstrate that it introduces intermodulation products to the diffraction patterns of computer-generated Fourier transform holograms. Also, the influence of phase blurring on a pseudorandom-encoding algorithm is evaluated in closed form. This analysis applied to a spot array generator design indicates that nonlinear effects are negligible only if the diameter of the point-spread function is a small fraction of the pixel spacing.     

Discrete fractional Fourier transform as a fast algorithm for evaluating the diffraction pattern of pulsed radiation

A technique is proposed for computing the field radiated from a rectangular aperture. This technique, based on the discrete fractional Fourier transform, avoids the complexities of computing the diffraction pattern by the direct evaluation of the Fresnel integral. The advocated approach provides a fast and accurate computational tool, especially in the case of evaluating pulsed fields radiated through two-dimensional screens of complex amplitude. A detailed numerical study that demonstrates the efficacy of this approach is carried out.     

Image-scaling problem in the optical fractional Fourier transform

The significance of scale factors and cascade of optical fractional Fourier transform is emphasized. Exact and cascadable fractional Fourier transforms in practical applications mandate that the image scale be the reciprocal of the scale of the input plane controlled by the optical setup. The practical setup of the optical fractional Fourier transform must be without any quadratic phase term at the spectrum plane.     

Optical correlation based on the fractional Fourier transform

Some properties of optical correlation based on the fractional Fourier transform are analyzed. For a particular set of fractional orders, a filter is obtained that becomes insensitive to scale variations of the object. An optical configuration is also proposed to carry out the fractional correlation in a flexible way, and some experimental results are shown.     

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.     

Geometry and dynamics in the fractional discrete Fourier transform

The N x N Fourier matrix is one distinguished element within the group U(N) of all N x N unitary matrices. It has the geometric property of being a fourth root of unity and is close to the dynamics of harmonic oscillators. The dynamical correspondence is exact only in the N-->infinity contraction limit for the integral Fourier transform and its fractional powers. In the finite-N case, several options have been considered in the literature. We compare their fidelity in reproducing the classical harmonic motion of discrete coherent states.     

Incoherent fractional Fourier transform and its optical implementation

The fractional Fourier transform is redefined for working with incoherent light. As a real transformation, the incoherent fractional Fourier transform overcomes coherent system disadvantages such as the speckle effect and the need for incoherent-coherent conversion. It also might have some applications for digital image and signal processing owing to its decreased computing complexity. An incoherent optical implementation of the new transform based on the shearing interferometer is suggested. Laboratory experimental results are given.     

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.     

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.     

Wigner distribution moments in fractional Fourier transform systems

It is shown how all global Wigner distribution moments of arbitrary order in the output plane of a (generally anamorphic) two-dimensional fractional Fourier transform system can be expressed in terms of the moments in the input plane. Since Wigner distribution moments are identical to derivatives of the ambiguity function at the origin, a similar relation holds for these derivatives. The general input-output relationship is then broken down into a number of rotation-type input-output relationships between certain combinations of moments. It is shown how the Wigner distribution moments (or ambiguity function derivatives) can be measured as intensity moments in the output planes of a set of appropriate fractional Fourier transform systems and thus be derived from the corresponding fractional power spectra. The minimum number of (anamorphic) fractional power spectra that are needed for the determination of these moments is derived. As an important by-product we get a number of moment combinations that are invariant under (anamorphic) fractional Fourier transformation.     

Reconstruction of complex-valued electron density with x-ray in-line holograms

A reconstruction algorithm is proposed to analyze the complex-valued electron-density distribution in an object with x-ray in-line holograms. The real and imaginary parts of the electron density correspond to local variations of x-ray phase shift and absorption, respectively. In the algorithm, the least-squares error of the holograms is iteratively minimized under sign constraints on the real and imaginary parts. The constraints, which are derived from the physical conditions in the interaction between x rays and materials, facilitate robust reconstruction. The reconstruction was applied to holograms of a biological specimen that caused both phase shift and absorption.     

散斑分数傅里叶联合变换相关测量研究

提出数字散斑联合变换分数相关测量方法,利用分数相关可以锐化相关峰的作用,在数字散斑联合变换相关运算中用分数傅里叶变换代替傅里叶变换,提高测量精度。通过对散斑图像进行相位调制,有效地解决了分数傅里叶变换的移变性带来的谱移问题。编程模拟和对拉伸试件位移场测量的结果表明,只要选择合适的分数傅里叶变换级次和相位调制函数,可以使相关峰的半宽度从4～5pixel锐化到仅1pixel,得到优于傅里叶变换相关的理想输出。     

The relation between light diffraction and the fractional fourier transform

Abstract

The relation between the diffraction integral and the fractional Fourier transform is analysed in detail from the geometrical and energy points of view. It is shown that the optical matrices associated with the two transformations in the geometrical approximation, although very different, are consistent with the simple relation between the respective integral transforms. Moreover, the meaning of the complex degree of fractionality of the fractional Fourier transform is found to be related to the energy variation of the beam. Its influence on the phase-space representation of the optical beam is shown to be different compared with the diffraction through an energy variant system.     

Affine cryptosystem of double-random-phase encryption based on the fractional Fourier transform

An affine mapping mathematical expression of the double-random-phase encryption technique has been deduced utilizing the matrix form of discrete fractional Fourier transforms. This expression clearly describes the encryption laws of the double-random-phase encoding techniques based on both the fractional Fourier transform and the ordinary Fourier transform. The encryption process may be regarded as a substantial optical realization of the affine cryptosystem. It has been illustrated that the encryption process converts the original image into a white Gaussian noise with a zero-mean value. Also, the decryption process converts the data deviations of the encrypted image into white Gaussian noises, regardless of the type of data deviations. These noises superimpose on the decrypted image and degrade the signal-to-noise ratio. Numerical simulations have been implemented for the different types of noises introduced into the encrypted image, such as the white noise with uniform distribution probability, the white noise with Gaussian distribution probability, colored noise, and the partial occlusion of the encrypted image.