Search

We theoretically propose a multifunctional photonic differentiation (DIFF) scheme based on phase demodulation using two cascaded linear filters. The photonic DIFF has a diversity of output forms, such as the 1st order intensity DIFF, the 1st order field DIFF and its inversion, and the 2nd-order field DIFF, depending on the relative shift between the optical carrier and the filter's resonant notches. As a proof, we also experimentally demonstrate the DIFF diversity using a phase modulator and two delay interferometers (DIs). The calculated average deviation is less than 7% for all DIFF waveforms. Our schemes show the advantages of flexible DIFF functions and forms, which may have different optical applications. For example, high order field differentiators can be used to generate complex temporal waveforms, and intensity differentiators are useful for the ultra-wideband pulse generation. [ABSTRACT FROM AUTHOR]

When simulating sky images, one often takes a galaxy image FðxÞ defined by a set of pixelized samples and an interpolation kernel, and then wants to produce a new sampled image representing this galaxy as it would appear with a different point-spread function, a rotation, shearing, or magnification, and/or a different pixel scale. These operations are sometimes only possible, or most efficiently executed, as resamplings of the Fourier transform ~ FðuÞ of the image onto a u-space grid that differs from the one produced by a discrete Fourier transform (DFT) of the samples. In some applications, it is essential that the resampled image be accurate to better than one part in 103, so in this paper, we first use standard Fourier techniques to show that Fourier-domain interpolation with a wrapped sinc function yields the exact value of ~ FðuÞ in terms of the input samples and kernel. This operation scales with image dimension as N4 and can be prohibitively slow, so we next investigate the errors accrued from approximating the sinc function with a compact kernel. We show that these approximations produce a multiplicative error plus a pair of ghost images (in each dimension) in the simulated image. Standard Lanczos or cubic interpolators, when applied in Fourier domain, produce unacceptable artifacts. We find that errors less than one part in 103 can be obtained by (1) fourfold zero-padding of the original image before executing the x u DFT, followed by (2) resampling to the desired u-grid using a six-point, piecewise-quintic interpolant that we design expressly to minimize the ghosts, then (3) executing the DFT back to x-domain. [ABSTRACT FROM AUTHOR]

An analytical model of acoustic field excited by a pulsed-laser line source on a coated cylinder is presented. Surface wave dispersive behaviours for a cylinder with a slow coating are analysed and compared with that of a bare cylinder. Based on this analysis, the laser-generated transient response of the perturbed Rayleigh wave and the higher modes of steel cylinder with a zinc coating are calculated from the model using residue theory and FFT technique. The theoretical result from the superposed waveform of the perturbed Rayleigh wave and higher modes agree well with the waveform obtained in experiment. The results show that the model and numerical method provide a useful technique for quantitatively characterizing coating parameters of coated cylinder by the laser generated surface waves. [ABSTRACT FROM AUTHOR]

This paper is a further work of the authors' paper published previously (Liao T H and Gao Q 2005 Chin. Phys. Lett. 22 2316). The amplitudes of fractional Fourier transform of Cantor sets are analysed from the viewpoint of multifractal by wavelet transform maxima method (WTMM). An integral operation is carried out before the application of WTMM, such that the function obtained can be considered as the perturbed devil staircase. Also, wavelets with large number of vanishing moments are used, which makes the complete singularity spectrum more accessible. The validity of multifractal formalism is guaranteed by restricting parameter q to a proper range, so that the phenomenon of multifractal phase transition can be explained reasonably. Particularly, the method of determining the range of parameter q in the above paper is developed to be more operational and rigorous. [ABSTRACT FROM AUTHOR]

This paper extends the definition of fractional Fourier transform (FRFT) proposed by Namias V by using other orthonormal bases for L2 (R) instead of Hermite-Gaussian functions. The new orthonormal basis is gained indirectly from multiresolution analysis and orthonormal wavelets. The so defined FRFT is called wavelets-fractional Fourier transform. [ABSTRACT FROM AUTHOR]

We develop fast discrete Fourier transforms (and their adjoints) from a square in space to a disk in the Fourier domain. Since our new transforms are not unitary, we develop a fast inversion algorithm and derive corresponding estimates that allow us to avoid iterative methods typically used for inversion. We consider the eigenfunctions of the corresponding band-limiting and space-limiting operator to describe spaces on which these new transforms can be inverted and made useful. In the process, we construct polar grids which provide quadratures and interpolation with controlled accuracy for functions band-limited within a disk. For rapid computation of the involved trigonometric sums we use the unequally spaced fast Fourier transform, thus yielding fast algorithms for all new transforms. We also introduce polar grids motivated by linearized scattering problems which are obtained by discretizing a family of circles. These circles are generated by using a single circle passing through the origin and rotating this circle with the origin as a pivot. For such grids, we provide a fast algorithm for interpolation to a near optimal grid in the disk, yielding an accurate adjoint transform and inversion algorithm. [ABSTRACT FROM AUTHOR]

Numerical differentiation is commonly used by a number of science students and researchers for data analysis. The differentiation of vectors of data points representing discrete samples of some underlying signal can be implemented in a computer using the central differencing scheme or the fast Fourier transform (FFT)-based approach. We point out that a simple extension of the continuous Fourier transform derivative identity to the discrete case, however, gives rise to results that are inconsistent with the central differencing scheme. In particular, for functions with step-like discontinuities, the numerical derivative computed with the FFT-based approach is corrupted by ringing artifacts. We describe the idea of a modified wave number for FFT-based numerical differentiation which leads to results that are consistent with the central differencing scheme. Modified wave number identities for numerical computation of both first and second order derivatives are described. The pitfalls of using a simplistic extension of the derivative identity for continuous Fourier transform to the discrete case and the methodology to avoid them are illustrated with numerical examples. [ABSTRACT FROM AUTHOR]

In isotropic homogeneous media, far fields of time-harmonic electromagnetic waves radiated by compactly supported volume currents, and elastic waves radiated by compactly supported body force densities can be modelled in very similar fashions. Both are projected restricted Fourier transforms of vector-valued source terms. In this work we generalize two types of uncertainty principles recently developed for far fields of scalar-valued time-harmonic waves in Griesmaier and Sylvester (2017 SIAM J. Appl. Math. 77 154–80) to this vector-valued setting. These uncertainty principles yield stability criteria and algorithms for splitting far fields radiated by collections of well-separated sources into the far fields radiated by individual source components, and for the restoration of missing data segments. We discuss proper regularization strategies for these inverse problems, provide stability estimates based on the new uncertainty principles, and comment on reconstruction schemes. A numerical example illustrates our theoretical findings. [ABSTRACT FROM AUTHOR]

We are concerned with time-dependent inverse source problems in elastodynamics. The source term is supposed to be the product of a spatial function and a temporal function with compact support. We present frequency-domain and time-domain approaches to show uniqueness in determining the spatial function from wave fields on a large sphere over a finite time interval. The stability estimate of the temporal function from the data of one receiver and the uniqueness result using partial boundary data are proved. Our arguments rely heavily on the use of the Fourier transform, which motivates inversion schemes that can be easily implemented. A Landweber iterative algorithm for recovering the spatial function and a non-iterative inversion scheme based on the uniqueness proof for recovering the temporal function are proposed. Numerical examples are demonstrated in both two and three dimensions. [ABSTRACT FROM AUTHOR]

In the present work, we develop a low-cost and simple experiment to visualise Fourier’s synthesis using a short, soft, and light plastic coiled spring oscillating in a horizontal plane, and a basic camera (120 fps). It is shown that the spring obeys a linear wave differential equation, as gravitational influence is neglected. A nonlinear criterion is evaluated to determine if magnitudes of the parameters in the initial conditions satisfy the linear wave equation. Our setup promotes some desirable characteristics that make Fourier’s synthesis experiments feasible, visual, and enlightening: (i) it requires few, common, and cheap resources, and the experiment can be carried out even in a high-school laboratory; (ii) since the spring’s tension is small (∼1 N, on average), the frequencies of normal modes are low (close to 2 Hz), and therefore, it is possible to record the oscillations just with the camera and extract a considerable number of position and time data in just one cycle; (iii) when the video is loaded in the Tracker free software, it can be reproduced in slow motion. Since the frequencies involved are low, an interesting and instructive temporal sequence of images of the spring displaying the typical trapezoidal shape appears clearly; (iv) the tools associated with the Tracker software tools can yield the relevant oscillation parameters, such as the damping constant, amplitudes, frequencies, and phases; and (v) it is possible to carry out superposition of a snapshot of the spring in Tracker at any time, and to draw the related Fourier synthesis graphs. The visual match between the shape of the spring and the theoretical graph is remarkable, and can be enhanced by adding the damping term. [ABSTRACT FROM AUTHOR]

The linear canonical transforms (LCTs) are a parameterised group of linear integral transforms. The LCTs encompass a number of well-known transformations as special cases, including the Fourier transform, fractional Fourier transform, and the Fresnel integral. They relate the scalar wave fields at the input and output of systems composed of thin lenses and free space, along with other quadratic phase systems. In this paper, we perform a systematic search of all algorithms based on up to five stages of magnification, chirp multiplication and Fourier transforms. Based on that search, we propose a novel algorithm, for which we present numerical results. We compare the sampling requirements of three algorithms. Finally, we discuss some issues surrounding the composition of discrete LCTs. [ABSTRACT FROM AUTHOR]

We investigate the structure-preserving numerical algorithm of the Degasperis—Procesi equation which can be transformed into a bi-Hamiltonian form using the discrete variational derivative method. Based on two different space discretization methods, the Fourier pseudospectral method and the wavelet collocation method, we develop two modified structure-preserving schemes under the periodic boundary condition. These proposed schemes are proved to preserve the Hamiltonian invariants theoretically and numerically. Meanwhile, the numerical results confirm that they can simulate the propagation of solitons effectively for a long time. [ABSTRACT FROM AUTHOR]

We propose a universal analytical method of studying the dynamics of a multi-anticrossing system subjected to driving by a single large-amplitude triangle pulse, within a time scale smaller than the dephasing time. Our approach can explain the main features of the Landau--Zener--Stuckelberg interference patterns recently observed in a tripartite system [Nature Communications 1 51 (2010)]. In particular, we focus on the effect of the size of the anticrossing on interference and compare the calculated interference patterns with numerical simulations. In addition, a Fourier transform of the patterns can extract the information about the energy level spectrum. [ABSTRACT FROM AUTHOR]

This paper studies quantitatively the generation of Lamb waves in thin bonded plates subjected to laser illumination, after considering the viscoelasticity of the adhesive layer. The displacements of such plates have been calculated in the frequency domain by using the finite element method, and the time domain response has been reconstructed by applying an inverse fast Fourier transform. Numerical results are presented showing the normal surface displacement for several configurations: a single aluminum plate, a three-layer bonded plate, and a two-layer plate. The characteristics of the laser-generated Lamb waves for each particular case have been investigated. In addition, the sensitivity of the transient responses to variations of material properties (elastic modulus, viscoelastic modulus, and thickness) of the adhesive layer has been studied in detail. [ABSTRACT FROM AUTHOR]

Fourier modal method incorporating staircase approximation is used to study tapered crossed subwavelength gratings in this paper. Three intuitive formulations of eigenvalue functions originating from the prototype are presented, and their convergences are compared through numerical calculation. One of them is found to be suitable in modeling the diffraction efficiency of the circular tapered crossed subwavelength gratings without high absorption, and staircase approximation is further proven valid for non-highly-absorptive tapered gratings. This approach is used to simulate the "moth-eye" antireflection surface on silicon, and the numerical result agrees well with the experimental one. [ABSTRACT FROM AUTHOR]

We consider a near-field inverse scattering problem for the wave equation: find the shape of a Dirichlet scattering object from time domain measurements of scattered waves. For this time-domain inverse problem, we propose a linear sampling method, a well-known technique for corresponding frequency domain inverse scattering problems. The problem setting and the algorithm incorporate two basic features. First, the data for the method consist of measurements of causal waves, that is, of waves that vanish before some moment in time. Second, the inversion algorithm directly works on the time-domain data without using a Fourier transformation. The first point is related to the applications we have in mind, which include for instance ground-penetrating radar imaging. The second feature allows us to naturally incorporate multiple (in fact, a continuum of) frequencies in the inversion algorithm. Consequently, it offers the potential of improving the quality of the reconstruction compared to frequency domain methods working with a single frequency. We demonstrate this potential by several numerical examples. [ABSTRACT FROM AUTHOR]

A novel three-dimensional wide-angle beam propagation method based on the split-step fast Fourier transform is developed. The formulation is based on the three-dimensional Helmholtz wave equation. Each propagation step is performed by utilizing both the FFT and split-step scheme. The solution of Helmholtz wave equation does not make the slowly varying envelope and one-way propagation approximations. To validate the efficiency and accuracy, numerical results for a propagation beam in a tilted step-index optical waveguide are compared with other beam propagation algorithms. [ABSTRACT FROM AUTHOR]

In this paper we consider the numerical solution of a phase retrieval problem for a compactly supported, linear spline f: {\bb R} \to {\bb C} with the Fourier transform {\skew5\hat{f}} , where values of |f| and |{\skew5\hat{f}}| at finitely many equispaced nodes are given. The unknown phases of complex spline coefficients fulfil a well-structured system of nonlinear equations. Thus the phase reconstruction leads to a nonlinear inverse problem, which is solved by a multilevel strategy and iterative Tikhonov regularization. The multilevel strategy concentrates the main effort of the solution of the phase retrieval problem in the coarse, less expensive levels and provides convenient initial guesses at the next finer level. On each level, the corresponding nonlinear system is solved by an iteratively regularized Gauss-Newton method. The multilevel strategy is motivated by convergence results of IRGN. This method is applicable to a wide range of examples as shown in several numerical tests for noiseless and noisy data. [ABSTRACT FROM AUTHOR]