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.     

Nonseparable two-dimensional fractional fourier transform

Previous generalizations of the fractional Fourier transform to two dimensions assumed separable kernels. We present a nonseparable definition for the two-dimensional fractional Fourier transform that includes the separable definition as a special case. Its digital and optical implementations are presented. The usefulness of the nonseparable transform is justified with an image-restoration example.     

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.     

True beam smoothing in the fractional fourier transform domain

Abstract

The design of a diffractive optical element (DOE) for true beam smoothing in the fractional Fourier transform domain is described. Based on the Fresnel integrals, the intensity distribution on the output plane is calculated accurately and the discretization error of the spherical phase factor is avoided. The ‘fine design' of the DOE for true beam smoothing is completed with the sampling interval chosen as half of the traditional sampling interval. Simulation results show that the intensity at any point on the output plane fully meets the required demands, not just those sampling points used in the optimization.     

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.     

Recursive algorithm for phase retrieval in the fractional fourier transform domain

We first discuss the discrete fractional Fourier transform and present some essential properties. We then propose a recursive algorithm to implement phase retrieval from two intensities in the fractional Fourier transform domain. This approach can significantly simplify computational manipulations and does not need an initial phase estimate compared with conventional iterative algorithms. Simulation results show that this approach can successfully recover the phase from two intensities.     

Generalized joint fractional fourier transform correlators: a compact approach

Fractional correlation was introduced recently. We generalize the architecture of a joint (Fourier) transform correlator (JTC) to achieve the joint fractional (Fourier) transform correlator (JFrTC) such that fractional correlation can be obtained. Here the Fourier transform in the JTC is replaced by the fractional Fourier transform, and four different JFrTC architectures can be implemented. The mathematical derivations for these JFrTC architectures are given, together with the simulation verifications. The JFrTC can provide a correlation signal similar to a delta function but with a small discrimination ratio, such that it is insensitive to additive noise. In a conventional JTC the distance between the two desired correlation signals at the output plane is fixed and depends on the distance between the input and the reference signals. However, with a given fractional order and an additional phase mask the separation distance between the two correlation signals at the output plane of a JFrTC can be larger or smaller than that of a JTC. This property is useful for the applications of real-time target tracking. Unlike in a previous approach [Appl. Opt. 36, 7402 (1997)], we need only two fractional Fourier transformations instead of three to achieve fractional correlation.     

A simple optical fourier transform system for the analysis of high frequency speckle patterns

A novel optical Fourier transform system is described that simplifies the experimental setup for examining speckle patterns in strain analysis.     

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.     

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.     

基于分数阶Fourier变换的机械变速过程信号分析

研究了旋转机械的变速测量过程，通过引入分数阶傅立叶变换（FRFT），推导证明了分数阶傅立叶变换对扫频信号具有良好的时频聚焦性，因此应用分数阶傅立叶变换来分析旋转机械的变速过程的测量信号，可以有效地解决机械转速不恒定对信号分析的影响，提高信号分析、状态监测以及故障诊断的准确性。     

Anamorphic fractional Fourier transform: optical implementation and applications

An additional degree of freedom is introduced to fractional-Fourier-transform systems by use of anamorphic optics. A different fractional Fourier order along the orthogonal principal directions is performed. Alaboratory experimental system shows preliminary results that demonstrate the proposed theory. Applications such as anamorphic fractional correlation and multiplexing in fractional domains are briefly suggested.     

Separation of overlapping linear frequency modulated (LFM) signals using the fractional fourier transform

Linear frequency modulated (LFM) excitation combined with pulse compression provides an increase in SNR at the receiver. LFM signals are of longer duration than pulsed signals of the same bandwidth; consequently, in many practical situations, maintaining temporal separation between echoes is not possible. Where analysis is performed on individual LFM signals, a separation technique is required. Time windowing is unable to separate signals overlapping in time. Frequency domain filtering is unable to separate signals with overlapping spectra. This paper describes a method to separate time-overlapping LFM signals through the application of the fractional Fourier transform (FrFT), a transform operating in both time and frequency domains. A short introduction to the FrFT and its operation and calculation are presented. The proposed signal separation method is illustrated by application to a simulated ultrasound signal, created by the summation of multiple time-overlapping LFM signals and the component signals recovered with ±0.6% spectral error. The results of an experimental investigation are presented in which the proposed separation method is applied to time-overlapping LFM signals created by the transmission of a LFM signal through a stainless steel plate and water-filled pipe.     

Wigner distribution and fractional Fourier transform for two-dimensional symmetric optical beams

A useful relationship between the fractional Fourier transform power spectra of a two-dimensional symmetric optical beam, on the one hand, and its Wigner distribution, on the other, is established. This relationship allows a significant simplification of the standard procedure for the reconstruction of the Wigner distribution from the field intensity distributions in the fractional Fourier domains. The Wigner distribution of a symmetric optical beam is analyzed, both in the coherent and in the partially coherent case.     

Nontoxic and chemically stable hollow optical fiber probe for fourier transform infrared spectroscopy

Remote spectroscopy systems based on hollow optical fiber probes are proposed and experimental results using a Fourier transform spectroscope are presented. A hollow optical-fiber probe with a silver and polymer inner coating is used to deliver incoherent light to a target and another separate hollow fiber is used to collect the reflected light. The reflectance spectra of teeth, skin, and oral mucosa were successfully measured with the probe even from surfaces with reflectances lower than 0.5%. The preliminary results obtained using attenuated total reflection spectroscopy are also presented. This remote infrared spectroscope is useful for endoscopic measurements inside the body because it is flexible, durable, nontoxic, and has the low transmission losses associated with hollow-fiber-based probes.     

Fractional fourier transform as a tool for analysing the diffraction of misaligned optical systems

Abstract

It is shown that fractional Fourier transform could be viewed as a tool for analysing the diffraction of misaligned optical systems, just as the fractional Fourier transforms are adapted to the mathematical expression of Fresnel diffraction. Some special cases are also discussed.     

Magnitude and direction of motion with speckle correlation and the optical fractional Fourier transform

The optical fractional Fourier transform (OFRT) in combination with speckle photography has previously been used to measure the magnitude of surface tilting and translation. Previous OFRT techniques used to determine motion have not been able to discern the direction of the tilt and translation. A simple new approach involving use of correlation is presented to overcome this limitation. Controlled variation of the minimum resolution and dynamical range of measurement is demonstrated. It is then experimentally confirmed that if a rigid body's motion is captured by two OFRT systems of different orders, the direction and magnitude of both the tilting and the in-plane translation motion of the body can be independently determined without a priori knowledge. The experimental results confirm the validity of previous theoretical predictions.     

Shah convolution fourier transform detection

A new convolution-detection method was developed which converts multiple-point (Shah function) detection, time-domain electropherograms into frequency-domain plots by means of a Fourier transformation, allowing the analytes' speeds to be viewed in terms of their "blinking" frequency; we have named this method Shah convolution Fourier transform detection, or SCOFT. This paper represents proof of principle of the detection concept. A micromachined glass stucture with a patterned layer of Cr on its top surface to form regularly spaced detection slits was used to perform capillary electrophoresis separations with 55-point, laser-induced fluorescence detection over 3.78 cm of the 6.6 cm separation channel. While this method can be easily integrated into a miniaturized total analysis system (μ-TAS), the principle is equally applicable to detection in full-sized analytical instrumentation. Single-component samples (fluorescein) migrating through the separation channel yielded a single peak in the frequency domain, and two-component samples (fluorescein and fluorescein isothiocyanate) yielded two resolved peaks, each at the expected frequency; harmonics were also observed. Advantages were seen in terms of isolation of the analyte peaks from interference such as baseline drift and line noise. Resolution is somewhat inferior to that seen in single-point detection, but it is thought that improved chip design and mathematical and instrument optimization will lead to performance superior to that of single-point detection.     

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.