Spatial Fourier Transformation Research Articles

A novel vibration barrier for low-frequency surface waves in the ground

As for certain engineering scenarios, such as open-air quarries and underground pipeline areas, vibration barriers are in urgent need. However, their design and construction face various challenges, especially for attenuating surface waves with low frequencies. Inspired by the topography of hills and periodic phononic crystal structures, a novel vibration barrier consisting of periodic soil strips is proposed. Firstly, utilizing the finite element method and the theory of periodic structures, the dispersion band diagram is calculated for the periodic soil strips. The corresponding band gap mechanisms are discussed by the displacement field of modes in the lower boundary of band gaps. Besides, a simplified model and a theoretical expression are proposed to analyze the minimum frequency of the band gaps. Through frequency domain analysis, the influence of the damping of soil strips on isolation effects is examined, and the dissipated energy is also quantified using a spatial Fast Fourier Transform. Finally, a comparison of the isolation effect between soil strip barriers and typical trench barriers is conducted. The results show that periodic soil strips can exhibit several band gaps, within which surface waves (SWs) can be effectively isolated. Soil strip barriers exhibit similar or even better vibration isolation effects compared to trench barriers, especially for isolating low-frequency vibrations. This study enhances our understanding of surface wave mitigation using periodic barriers and provides fresh insights for designing seismic metamaterials.

Fourier-transform-only method for random phase shifting interferometry

An accurate and computationally simple phase shifting interferometry (PSI) method is developed to reconstruct phase maps without a priori knowledge of the phase shift. Previous methods developed for random PSI either do not address general sources of error or require complex iterative processes and increased computational time. Here we demonstrate a novel method that is able to extract the phase using only Fourier transform (FT). With spatial FT analysis, randomly phase-shifted data is reordered to allow performing temporal FT on the intensity, which is a function of the phase shift. Since the entire process, including order analysis and phase calculation, is based only on Fourier analysis, it is rapid, easy to implement, and addresses general sources of error. The method exhibits high performance in experiments containing random phase shifts. Moreover, simulations incorporating common experimental error sources such as random intensity noise, intensity harmonics, and phase shift errors demonstrate that the proposed method performs as good as or better than the state-of-the-art phase reconstruction techniques in terms of accuracy and time.

3-D Microwave Field Synthesis via Projected Spatial Fourier Transform

3-D Microwave Field Synthesis via Projected Spatial Fourier Transform

Tailoring Group Delay Dispersion of Spatial Phaser using Anisotropic Metasurface and Polarized Incident Waves

AbstractMetasurface‐based spatial phaser, exhibiting dispersive group delay responses, is able to perform real‐time analogue signal processing on the temporal spectrum of spatial electromagnetic waves, e.g., Fourier transformation, pulse chirping, pulse spread, and compression. The functionality of a spatial phaser is closely related to its group delay dispersion slope. For instance, for an up‐chirped incident pulse, the positive‐slope‐dispersion phaser spreads the pulse whereas the negative‐slope‐dispersion one compresses the pulse. Most reported spatial phasers exhibit fixed dispersion slopes and hence perform single functionality only, which cannot meet the demand of multifunctional systems in communication and sensing applications. In this paper, a new way is proposed to control and tune the dispersion slope of spatial phasers using polarization of incident waves, which is simpler and more efficient than conventional approaches based on tunable circuits. To verify the proposed method, a first‐order spatial phaser is designed and experimentally demonstrated within 9.4–10.6 GHz. It is able to generate negative‐slope, positive‐slope, and zero‐slope group delay dispersions, in case of x‐, y‐, and ±45°‐polarized waves. Such a reconfigurable phaser may find wide applications in integrated communication and sensing.

An azimuthal mode detection method based on angular deviation and its application in compressor

A deep understanding of the generation of rotating pressure fluctuations is a prerequisite for designing compressors with higher efficiency, lower noise, and a broader operating range. Azimuth modal analysis is an effective method to analyze the characteristics and sources of rotating pressure fluctuations. This study proposes a method for azimuthal mode detection based on equally spaced sensors with uncertain angular deviations, combining the spatial Fourier transform (SFT) method with the compressed sensing (CS) technique. The result of numerical experiments indicates that the angular deviation of the sensors causes an error in mode reconstruction, especially when the mode order is high. Two optimization schemes are developed to obtain the angular deviation. The effectiveness of the optimization schemes was analyzed through numerical experiments. Finally, the effectiveness of the proposed method was verified by a compressor mode test. The general procedure of this method for a fan/compressor test is given.

Stall Warning of Axial Compressor Using Spatial FFT and Combined Analysis of Multiple Statistical Parameters

In order to study an effective algorithm for the pre-stall warning of axial compressor, a pre-stall warning algorithm based on Spatial Fast Fourier Transform (Spatial FFT) and combined analysis of multiple statistical parameters is proposed to solve the problems of insufficient accuracy of stall judgment and short warning time margin in view of the existing pre-stall warning research. Taking the dynamic pressure signal at the tip of the first stage stator of a multi-stage axial compressor as the research object, the circumferential multi-channel signals which are reconstructed from the single-channel sensor signal by using the signal translation method is decomposed to obtain the multi-order modal amplitudes through the spatial FFT. Analyzing the 1st-order modal amplitude coefficient by using a variety of statistical parameter methods, several conclusions were obtained. Compared with the one-dimensional analysis methods, the spatial FFT analysis method has better warning time margin; Compared with the two statistical parameters of kurtosis and autocorrelation coefficient, the parameter analysis method combining autocovariance and autocorrelation function is less computationally intensive, has a larger difference between stable and stall conditions, lower misjudgment rate, better early warning effect, and can achieve a warning time margin of 0.04~0.137s.

Wave transmission and vibration response in periodically stiffened plates using a free wave approach.

There are various structures constructed with periodically stiffened thin plates. Vibration prediction of such structures is not easy compared to the structures comprised of uniform plates only due to the mathematical complexity stemmed from the periodic nature. This study provides the analytic method to predict the wave transmission at junctions connecting two semi-infinite periodic structures and the response of a finite periodic structure to an external harmonic point force. The same theoretical framework is employed for dealing with both phenomena. First, free wave solutions are obtained by solving the governing equation for the bending motion of a periodically stiffened, infinite plate using the spatial Fourier Transform and the Floquet's theorem. Then, the free wave solutions are linearly superposed, and the linear coefficients are calculated by applying the appropriate boundary conditions. Numerical simulation is conducted. In dealing with the periodic finite structure, the result is compared with that by the finite element analysis. It is revealed that the periodic nature of the structures affects both the energy transmission and the vibration response of the periodically stiffened plates.

Analog signal processing through space-time digital metasurfaces

Abstract In the quest to realize analog signal processing using subwavelength metasurfaces, in this paper, we present the first demonstration of programmable time-modulated metasurface processors based on the key properties of spatial Fourier transformation. Exploiting space-time coding strategy enables local, independent, and real-time engineering of not only amplitude but also phase profile of the contributing reflective digital meta-atoms at both central and harmonic frequencies. Several illustrative examples are demonstrated to show that the proposed multifunctional calculus metasurface is capable of implementing a large class of useful mathematical operators, including 1st- and 2nd-order spatial differentiation, 1st-order spatial integration, and integro-differential equation solving accompanied by frequency conversions. Unlike the recent proposals based on the Green’s function (GF) method, the designed time-modulated signal processor effectively operates for input signals containing wide spatial frequency bandwidths with an acceptable gain level. Proof-of-principle simulations are also reported to demonstrate the successful realization of image processing functions like edge detection. This time-varying wave-based computing system can set the direction for future developments of programmable metasurfaces with highly promising applications in ultrafast equation solving, real-time and continuous signal processing, and imaging.

Grating Spectrometry and Spatial Heterodyne Fourier Transform Spectrometry: Comparative Noise Analysis for Raman Measurements

The desire for portable Raman spectrometers is continuously driving the development of novel spectrometer architectures where miniaturisation can be achieved without the penalty of a poorer detection performance. Spatial heterodyne spectrometers are emerging as potential candidates for challenging the dominance of traditional grating spectrometers, thanks to their larger etendue and greater potential for miniaturisation. This paper provides a generic analytical model for estimating and comparing the detection performance of Raman spectrometers based on grating spectrometer and spatial heterodyne spectrometer designs by deriving the analytical expressions for the performance estimator (signal-to-noise ratio, SNR) for both types of spectrometers. The analysis shows that, depending on the spectral characteristics of the Raman light and on the values of some instrument-specific parameters, the ratio of the SNR estimates for the two spectrometers () can vary as much as by two orders of magnitude. Limit cases of these equations are presented for a subset of spectral regimes which are of practical importance in real-life applications of Raman spectroscopy. In particular, under the experimental conditions where the background signal is comparable or larger than the target Raman line and shot noise is the dominant noise contribution, the value of is, to a first order of approximation, dependent solely on the relative values of each spectrometer’s etendue and on the number of row pixels in the detector array. For typical values of the key instrument-specific parameters (e.g., etendue, number of pixels, spectral bandwidth), the analysis shows that spatial heterodyne spectrometer-based Raman spectrometers have the potential to compete with compact grating spectrometer designs for delivering in a much smaller footprint (10–30 times) levels of detection performance that are approximately only five to ten times poorer.

Thermoacoustics and combustion instability analysis for multi-burner combustors

This paper presents a low-order thermoacoustic model for multi-burner combustors and proposes a control theoretic approach to linear combustion instability analysis in frequency domain, taking acoustic and flame interaction effects among burners into account under a non-zero mean flow condition. Our thermoacoustic model can deal with can-annular and annular combustors in a unified way and is given as a multi-input multi-output transfer function matrix from heat fluctuation input vector to velocity fluctuation output vector. We also propose a flame transfer function matrix description where off-diagonal components describe the effects of a local acoustic field of one burner location on the flame response at another burner location. The discrete rotational symmetry of burner arrangement in a combustor makes it possible to diagonalize both the aforementioned thermoacoustic and flame transfer function matrices analytically from a spatial discrete-time Fourier transformation. The decoupled thermoacoustic transfer function matrix provides a characterization of acoustic resonances, particularly the longitudinal-azimuthal coupled modes unique to multi-burner combustors. In addition, from the Bloch wave interpretation, we find general patterns in the mode shape of a multi-burner combustor and its relations to the mode shape of a single-burner combustor. Furthermore, from a diagonalized closed loop transfer function matrix, a simple combustion instability condition for a multi-burner combustor is developed. This stability condition shows how a single-burner combustor and a multi-burner combustor are different quantitatively, when it comes to combustion instability. Numerical examples are presented to validate our thermoacoustic model against three-dimensional FEM results, and to illustrate our developments and findings.

Meta-optics for spatial optical analog computing

AbstractRapidly growing demands for high-performance computing, powerful data processing, and big data necessitate the advent of novel optical devices to perform demanding computing processes effectively. Due to its unprecedented growth in the past two decades, the field of meta-optics offers a viable solution for spatially, spectrally, and/or even temporally sculpting amplitude, phase, polarization, and/or dispersion of optical wavefronts. In this review, we discuss state-of-the-art developments, as well as emerging trends, in computational metastructures as disruptive platforms for spatial optical analog computation. Two fundamental approaches based on general concepts of spatial Fourier transformation and Green’s function (GF) are discussed in detail. Moreover, numerical investigations and experimental demonstrations of computational optical surfaces and metastructures for solving a diverse set of mathematical problems (e.g., integrodifferentiation and convolution equations) necessary for on-demand information processing (e.g., edge detection) are reviewed. Finally, we explore the current challenges and the potential resolutions in computational meta-optics followed by our perspective on future research directions and possible developments in this promising area.

Analysis in reciprocal space of the band-pass filter effect in uniform and non-uniform grating couplers

A detailed study on the origin of the bandwidth reduction in the Coupling- Efficiency (CE) spectrum for non-uniform grating couplers (nu-GCs) due to a band-pass filter effect is reported. This effect is deeply investigated through Finite Difference Time Domain simulations (FDTD) in order to look at the spatial Fourier Transform (sFT) of the nu-GC’s emission, which gives the energies and the wavevectors sustained by the overall structure. We show: how to evaluate the bandwidth from the sFT and how the band-pass filter effect arises from the sFT affecting the CE spectrum giving a physical insight on the bandwidth’s reduction. Moreover, we also point out how the concept of photonic crystal bandstructure can be applied for such nu-GCs and used as a powerful tool to describe the dispersion of the light inside these non periodic structures. Finally, both the sFT and the bandstructure approaches are shown to give the same outcomes in terms of the bandwidth evaluation and to be in agreement with the CE spectra’s behaviours. The entire analysis is proposed also for uniform-GC in order to directly compare the two different types of structures.

Analysis of Source Influence on Guided Wave Excitation in Cylindrical Structures Using Spatial Fourier Transform Method

Guided wave transducers, such as electromagnetic acoustic transducers and piezoelectric transducers, generate multimode waves at a given excitation frequency in a cylindrical structure, making it difficult to detect flaws in such structures. To accurately identify the flaws, the transducers must be well designed to suppress the nonaxisymmetric modes. Instead of using the normal mode expansion (NME) method, a spatial Fourier transform (SFT) method is proposed to analyze source influence on the guided wave excitation in a cylindrical structure. A two-dimensional SFT is performed on the spatial distribution function of the surface loading applied to the cylindrical structure. The spatial distribution function is represented in a cylindrical coordinate system. The circumferential-direction SFT is carried out from the angular coordinate to the circumferential orders of the guided wave modes. The axial-direction SFT is carried out from the axial coordinate to the wavenumbers of the guided wave modes. The results of the two-dimensional SFT represent guided wave excitation capabilities for different circumferential orders and wavenumbers. The specific surface loading conditions on the outer surface of a pipe are analyzed to predict source influence on the guided wave excitation. The results are consistent with those obtained using the NME method. Experiments corresponding to the specific surface loading conditions are carried out on a stainless steel pipe. The results confirm the effectiveness of the SFT method.

Investigation of Chaotic Image Encryption in Spatial and FrFT Domains for Cybersecurity Applications

The need for cybersecurity increases to protect the exchange of information for improving the data privacy. This paper presents an investigation of the encryption efficiency of the chaotic-based image block ciphering in the spatial and Fractional Fourier Transform (FrFT) domains. The main aim of this investigation is to examine the efficiency of different chaotic maps, while considering the parameters of the FrFT as additional keys for encryption and achieving reliable cybersecurity for robust image communication. In this paper, Cat, Baker, and Logistic map confusion approaches are applied in the spatial and FrFT domains to study and analyze the cybersecurity and ciphering efficiency of chaos-based image cryptosystems. The confusion features of the chaotic maps in spatial and FrFT domains are investigated using information entropy, differential analysis, histograms, visual observation, attack analysis, impact of noise, and encryption quality tests. Simulation results prove that the chaotic-based image encryption in the FrFT domain increases the efficiency of the confusion process and achieves a high nonlinear relation between the plainimage and the cipherimage in a symmetric ciphering approach. Moreover, the results demonstrate that the Cat-FrFT scheme is more susceptible to channel noise attacks than the Baker-FrFT and the Logistic-FrFT schemes. Hence, they can be implemented efficiently in the scenarios of noisy channels due to their high robustness to channel noise.

Analysis of eigenmodes in a swirling jet and flame: 3D PIV and PLIF study

The existence of spiral structures was directly confirmed experimentally on the basis of 3D Particle Image Velocimetry (PIV) and planar laser-induced fluorescence (PLIF) measurements both in a strongly swirling non-reacting jet and in a fuel-lean premixed methane-air flame. Flows were characterized by the presence of vortex breakdown and precession of the vortex core. The differences in magnitude and spatial distribution of flow eigenmodes with distance from the nozzle are analyzed by using Proper Orthogonal Decomposition (POD) based on Spatial Fourier Transform over azimuthal coordinate. The analysis of flow eigenmodes reveal that for a high-swirl isothermal jet, the vortex core co-existed with the pair of counter-rotating helical vortices, which were located in outer shear layer and inside the recirculation zone. This double-vortex helical structure was also detected in a swirling flame, but its magnitude was suppressed compared with isothermal flow. It was determined that the change in the shape of the chemical reaction area was associated with two types of large-scale coherent structures: nearly axisymmetric mode m = 0 of the flame front deformation, presumably due to the effect of buoyancy forces on the combustion products, and the quasi-solid rotation of the asymmetric global mode |m| = 1 in the form of double-vortex structure due to precession of the swirling flow. It was shown that the energy of the axisymmetric mode m = 0 in a reacting jet increased downstream by almost 10 %, unlike other modes, whose energy was only diminished.

Retrospective motion compensation for edited MR spectroscopic imaging

Edited magnetic resonance spectroscopic imaging (MRSI) is capable of mapping the distribution of low concentration metabolites such as gamma-aminobutyric acid (GABA) or and glutathione (GSH), but is prone to subtraction artifacts due to head motion or other instabilities. In this study, a retrospective motion compensation algorithm for edited MRSI is proposed. The algorithm identifies movement-affected signals by comparing residual water and lipid peaks between different transients recorded at the same point in k-space, and either phase corrects, replaces or removes affected spectra prior to spatial Fourier transformation. The method was tested on macromolecule-unsuppressed GABA-edited spin-echo MR spectroscopic imaging data acquired from 8 healthy adults scanned at 3T. Relative to non-motion compensated data sets, the motion compensated data had significantly less subtraction artifacts across subjects. The residual choline (Cho) peak in the spectrum (which is well resolved from as a different chemical shift from GABA and is completely absent in a spectrum without subtraction artifact) was used as a metric of motion artifact severity. The normalized Cho area was 5.14 times lower with motion compensation than without motion compensation. A ‘removal-only’ version of the technique is also shown to be promising in removing motion-corrupted artifacts in a GSH-edited MRSI acquisition acquired in 1 healthy subject. This study introduces a motion compensation technique and demonstrates that retrospective compensation in k-space is possible and significantly reduces the amount of subtraction artifacts in the resulting edited spectra.

Collective modes of three-dimensional magnetic structures: A study of target skyrmions

• We explored the three-dimensional (3D) collective modes of target skyrmion under different external magnetic fields. • The three regimes captured by the spectrum calculations directly reflect the ground states of the target skyrmion. • All the collective modes were spatially resolved. • 3D analysis is necessary to understand the collective modes of 3D magnetic structures. Three-dimensional magnetic textures present their richness not only in variety of ground state structures, but also in their dazzling collective modes. As an example, the target skyrmion, a topological spin texture with a central skyrmion and an extra concentric helical ring, has been enabled by confining a chiral magnet in a nanodisk geometry. In this work we investigate the collective spin wave modes of the target skyrmion in the external field region of −400 mT to 400 mT. Through micromagnetic simulations we reproduced three major phases of the target skyrmion, and derived resonance modes at varying external magnetic field by the power spectral density study. All typical modes were resolved in terms of their dynamical snapshots in temporal domain and the corresponding spatial Fourier transformations. Some modes are shown to behave similarly in-plane but differently out-of-plane, and vice versa. This suggests, on a broader scope, that both responses will be necessary to fully characterize collective modes of three-dimensional spin textures.

Ring defects in n-type Czochralski-grown silicon: A high spatial resolution study using Fourier-transform infrared spectroscopy, micro-photoluminescence, and micro-Raman

We investigated ring defects induced by a two-step anneal in n-type Czochralski-grown silicon wafers using a combination of high spatial resolution Fourier Transform Infrared Spectroscopy (FTIR), micro-photoluminescence (PL) mapping, and micro-Raman mapping. Through FTIR measurements, we show the inhomogeneous loss in interstitial oxygen with a positive correlation with the inverse lifetime. Using high-resolution micro-PL mapping, we are able to distinguish individual recombination-active oxygen precipitates within the rings with a decreasing density from the center to the edge of the sample. The radial inhomogeneity of the oxygen precipitates is likely to be related to variations in the distribution of grown-in defects. We also demonstrate that micro-Raman mapping reveals the oxygen precipitates without the smearing effects of carrier diffusion that are present in micro-PL mapping.

Near-Field Interference Estimation Between Multilayer PCBs Using a Reciprocity-Based Filamentary-Element Coupling Model

In this paper, the near-field interference between two multilayer printed circuit boards (PCBs), which is resulted from the board-edge radiation and the board-edge coupling, is estimated by developing a reciprocity-based filamentary-element coupling (RBFEC) model. In the RBFEC model, the radiation of an $N$ -cavity multilayer PCB is modeled as the radiation from $N$ vertically parallel filamentary magnetic current sources. The near fields of a PCB are calculated based on the spatial Fourier transformation method. Then, by applying the reciprocity theorem to relate the radiated fields of the two PCBs that are on the same closed surface together, an analytical formula regarding the interference noise is obtained. The proposed model can be applied to handle the mutual-coupling problem between two multilayer PCBs that include vias and striplines. Because the coupling model is in the frequency domain, the dispersive behavior of the PCBs can be taken into account easily. The full-wave simulation based on the finite integration technique (FIT) and the experimental measurement are carried out to validate the proposed model. The results show that the proposed model has the same precision level as the FIT, but shows much higher computational efficiency than the FIT.

Dual Doppler Effect in Wedge-Type Photonic Crystals

The dual Doppler effect, in the simultaneous occurrence of both normal and inverse Doppler effect in one moving two-dimensional wedge-type photonic crystal, is proposed. An improved finite-different time-domain algorithm is used to verify this phenomenon. The spatial Fourier Transformation has been applied to complex electrical field data to reveal the mechanism. The harmonics with negative spatial frequencies show a lagging phase evolution, while those with positive spatial frequencies show a front phase evolution. Different wedge-type photonic crystals are designed to filter out the required harmonics based on the systematic study of spatial Fourier Transformation and wave vector diagram. Our work paves a new way for Doppler cooling of atomic gases, radar deception, invisibility cloaks, microstructure dual frequency interferometer and so on.