Spectral Accuracy Research Articles

Inverse Differential Quadrature Based Model for Static Behaviour of Variable Stiffness Curved Composite Beams

Recent advancements in fibre placement technologies have expanded the potential applications of variable stiffness curved composite beams in industries such as aerospace, automotive, and naval engineering. Accurate solution techniques for examining these beams, especially those composed of advanced composite materials, are indispensable. In view of this demand, this study proposes a new high-order computational tool that combines the higher-order accuracy of the emerging inverse differential quadrature method (iDQM) and the simple kinematics of the Timoshenko beam theory for efficient and accurate prediction of the static behaviour of both constant stiffness and variable stiffness curved beam structures. This novel application of iDQM to curved beam analysis is leveraged upon its excellent potential to mitigate differentiation-induced errors by using the so-called indirect approximation strategy. Simple procedures for implementing different orders of iDQM models are presented to analyse curved beam problems, and are independently benchmarked against closed-form Navier's solutions, as well as numerical solutions obtained through the differential quadrature method (DQM) and finite element method (FEM), demonstrating excellent spectral accuracy. Furthermore, the iDQM scheme offers outstanding potential in recovering transverse shear stress, achieving superior accuracy over lower-order FEM in approximating higher-order derivatives. Remarkably, iDQM predictions for variable stiffness curved beams exhibit satisfactory agreement with the Strong Unified Formulation, achieving over 98% computational efficiency. Finally, convergence analysis of iDQM solutions reveal up to three orders of improved accuracy and faster convergence rates compared to the DQM, constituting a new benchmark for curved beam analysis.

A Spectrally Accurate Step-by-Step Method for the Numerical Solution of Fractional Differential Equations

In this paper we consider the numerical solution of fractional differential equations. In particular, we study a step-by-step procedure, defined over a graded mesh, which is based on a truncated expansion of the vector field along the orthonormal Jacobi polynomial basis. Under mild hypotheses, the proposed procedure is capable of getting spectral accuracy. A few numerical examples are reported to confirm the theoretical findings.

Consideration of the influence of aberrations in the operation diffraction prism spectral device

Spectral analysis is one of the most important methods of physico-technical research for solving many scientific and practical problems. The key parameter of a spectral instrument is its resolving power. Instruments based on diffraction gratings face a fundamental limitation: their resolving power is limited to operation in the first two orders of diffraction. The limitation is due to the fact that the intensity of diffraction maxima drops sharply with increasing diffraction order. As a result, such devices are unable to distinguish closely spaced spectral lines. Increasing the resolving power of spectral priors is an important and urgent task for spectroscopy. The aim of the research was to develop an optical scheme of a spectral instrument based on a high-resolution prism for the detection of nearby spectral lines with minimal influence of aberrations. The optical scheme of the diffraction prism spectral instrument was developed. The possibility of detecting nearby spectral lines is shown. Also a comprehensive study of the influence of aberrations and the development of methods of their minimisation to achieve the optimum quality of the spectral instrument and accuracy of measurements is carried out. The obtained results confirm the possibility of increasing the resolving power of the spectral instrument for the detection of closely spaced spectral lines. Taking into account the influence of aberrations in the diffraction prism spectral instrument will improve the quality of its work and measurement accuracy. The obtained results open new possibilities for studying complex spectra with high detail.

Accelerating X-ray photoemission spectroscopy measurements using Bayesian super-resolution

This study applies Bayesian super-resolution to X-ray photoelectron spectroscopy (XPS), achieving up to a 20-fold reduction in measurement time while preserving data quality. Traditional XPS, crucial for surface analysis, typically requires extensive measurement durations. Our methodology significantly accelerates the process, as demonstrated with glass and Polytetrafluoroethylene samples, where we reduced measurement times by up to 1/20th without compromising spectral accuracy. This approach decreases noise levels and retains spectral integrity, offering a highly efficient solution for XPS. This innovation is particularly valuable in material science, enabling rapid, reliable surface analysis.

Brillouin microscopic imaging of ex-vivo porcine eye using VIPA-CMOS-based spectrometer

The research on non-invasive ophthalmic imaging has become a hot topic in diagnosing ophthalmic diseases, especially assessment of corneal biomechanics has been an unmet clinical need in ophthalmology for many years. The Brillouin microscopy technique based on spontaneous Brillouin scattering has emerged as a unique elastography method due to its advantages of non-contact, label-free, and high-resolution biomechanical imaging of biological tissues. Here, we developed a three-dimensional (3D) optical microscope system for ex-vivo porcine cornea Brillouin imaging. To do this, we designed and experimentally demonstrated a setup with double-stage VIPA etalons and CMOS camera that integrate to a fully parallel dispersive Brillouin spectrometer. The designed spectrometer exhibits a smaller dispersion rate to enhance the spectral sampling rate and downsizes the size of the double-stage-based system. Using the optical microscope system, we measured and mapped the Brillouin frequency shifts of the cornea and crystalline lens in ex-vivo porcine eyes at micron-scale spatial resolution, and a spectral accuracy of < 50 MHz can be achieved without interpolating the spectra. This work is essential for future applications of Brillouin microscopy in clinical diagnosis and treatment of ophthalmic diseases.

An exact in time Fourier pseudospectral method with multiple conservation laws for three-dimensional Maxwell’s equations

Maxwell’s equations describe the propagation of electromagnetic waves and are therefore fundamental to understanding many problems encountered in the study of antennas and electromagnetics. The aim of this paper is to propose and analyse an efficient fully discrete scheme for solving three-dimensional Maxwell’s equations. This is accomplished by combining Fourier pseudospectral methods in space and exact formulation in time. Fast computation is efficiently implemented in the scheme by using the matrix diagonalisation method and fast Fourier transform algorithm which are well known in scientific computations. An optimal error estimate which is not encumbered by the CFL condition is established and the resulting scheme is proved to be of spectral accuracy in space and exact in time. Furthermore, the scheme is shown to have multiple conservation laws including discrete energy, helicity, momentum, symplecticity, and divergence-free field conservations. All the theoretical results of the accuracy and conservations are numerically illustrated by two numerical tests.

A space-time Galerkin Müntz spectral method for the time fractional Fokker–Planck equation

In this paper, we propose a space-time Galerkin spectral method for the time fractional Fokker–Planck equation. This approach is based on combining temporal Müntz Jacobi polynomials spectral method with spatial Legendre polynomials spectral method. Based on the well-posedness and regularity for the re-scaled problem of a linear model problem which reflects the main difficulty for solving the equivalent equation (i.e. the time fractional convection-diffusion equation): the singularity of the solution in time, we explain in detail why we use the Müntz polynomials to approximate in time. The well-posedness and stability of the discrete scheme as well as its continuous problem are established. Moreover, the error estimation of the space-time approach is derived. We find that the proposed method can attain spectral accuracy regardless of whether the solution of the original equation is smooth or non-smooth. Numerical experiments substantiate the theoretical results.

A dual-camera hyperspectral laparoscopic imaging system.

Minimally invasive surgery (MIS) has expanded broadly in the field of abdominal and pelvic surgery. However, there are still prevalent issues surrounding intracorporeal surgery, such as iatrogenic injury, anastomotic leakage, or the presence of positive tumor margins after resection. Current approaches to address these issues and advance laparoscopic imaging techniques often involve fluorescence imaging agents, such as indocyanine green (ICG), to improve visualization, but these have drawbacks. Hyperspectral imaging (HSI) is an emerging optical imaging modality that takes advantage of spectral characteristics of different tissues. Various applications include tissue classification and digital pathology. In this study, we developed a dual-camera system for high-speed hyperspectral imaging. This includes the development of a custom application interface and corresponding hardware setup. Characterization of the system was performed, including spectral accuracy and spatial resolution, showing little sacrifice in speed for the approximate doubling of the covered spectral range, with our system acquiring 29 spectral images from 460-850 nm. Reference color tiles with various reflectance profiles were imaged and a RMSE of 3.56 ± 1.36% was achieved. Sub-millimeter resolution was shown at 7 cm working distance for both hyperspectral cameras. Finally, we image ex vivo tissues, including porcine stomach, liver, intestine, and kidney with our system and use a high-resolution, radiometrically calibrated spectrometer for comparison and evaluation of spectral fidelity. The dual-camera hyperspectral laparoscopic imaging system can have immediate applications in various surgeries.

A statistical approach to system suitability testing for mass spectrometry imaging.

Mass spectrometry imaging (MSI) elevates the power of conventional mass spectrometry (MS) to multidimensional space, elucidating both chemical composition and localization. However, the field lacks any robust quality control (QC) and/or system suitability testing (SST) protocols to monitor inconsistencies during data acquisition, both of which are integral to ensure the validity of experimental results. To satisfy this demand in the community, we propose an adaptable QC/SST approach with five analyte options amendable to various ionization MSI platforms (e.g., desorption electrospray ionization, matrix-assisted laser desorption/ionization [MALDI], MALDI-2, and infrared matrix-assisted laser desorption electrospray ionization [IR-MALDESI]). A novel QC mix was sprayed across glass slides to collect QC/SST regions-of-interest (ROIs). Data were collected under optimal conditions and on a compromised instrument to construct and refine the principal component analysis (PCA) model in R. Metrics, including mass measurement accuracy and spectral accuracy, were evaluated, yielding an individual suitability score for each compound. The average of these scores is utilized to inform if troubleshooting is necessary. The PCA-based SST model was applied to data collected when the instrument was compromised. The resultant SST scores were used to determine a statistically significant threshold, which was defined as 0.93 for IR-MALDESI-MSI analyses. This minimizes the type-I error rate, where the QC/SST would report the platform to be in working condition when cleaning is actually necessary. Further, data scored after a partial cleaning demonstrate the importance of QC and frequent full instrument cleaning. This study is the starting point for addressing an important issue and will undergo future development to improve the efficiency of the protocol. Ultimately, this work is the first of its kind and proposes this approach as a proof of concept to develop and implement universal QC/SST protocols for a variety of MSI platforms.

Tensor-Based Sparse Representation for Hyperspectral Image Reconstruction Using RGB Inputs

Hyperspectral image (HSI) reconstruction from RGB input has drawn much attention recently and plays a crucial role in further vision tasks. However, current sparse coding algorithms often take each single pixel as the basic processing unit during the reconstruction process, which ignores the strong similarity and relation between adjacent pixels within an image or scene, leading to an inadequate learning of spectral and spatial features in the target hyperspectral domain. In this paper, a novel tensor-based sparse coding method is proposed to integrate both spectral and spatial information represented in tensor forms, which is capable of taking all the neighboring pixels into account during the spectral super-resolution (SSR) process without breaking the semantic structures, thus improving the accuracy of the final results. Specifically, the proposed method recovers the unknown HSI signals using sparse coding on the learned dictionary pairs. Firstly, the spatial information of pixels is used to constrain the sparse reconstruction process, which effectively improves the spectral reconstruction accuracy of pixels. In addition, the traditional two-dimensional dictionary learning is further extended to the tensor domain, by which the structure of inputs can be processed in a more flexible way, thus enhancing the spatial contextual relations. To this end, a rudimentary HSI estimation acquired in the sparse reconstruction stage is further enhanced by introducing the regression method, aiming to eliminate the spectral distortion to some extent. Abundant experiments are conducted on two public datasets, indicating the considerable availability of the proposed framework.

Single-pixel p-graded-n junction spectrometers.

Ultra-compact spectrometers are becoming increasingly popular for their promising applications in biomedical analysis, environmental monitoring, and food safety. In this work, we report a single-pixel-photodetector spectrometer with a spectral range from 480 nm to 820 nm, based on the AlGaAs/GaAs p-graded-n junction with a voltage-tunable optical response. To reconstruct the optical spectrum, we propose a tailored method called Neural Spectral Fields (NSF) that leverages the unique wavelength and bias-dependent responsivity matrix. Our spectrometer achieves a high spectral wavelength accuracy of up to 0.30 nm and a spectral resolution of up to 10 nm. Additionally, we demonstrate the high spectral imaging performance of the device. The compatibility of our demonstration with the standard III-V process greatly accelerates the commercialization of miniaturized spectrometers.

Fast Fourier-Chebyshev Approach to Real-Space Simulations of the Kubo Formula.

The Kubo formula is a cornerstone in our understanding of near-equilibrium transport phenomena. While conceptually elegant, the application of Kubo's linear-response theory to interesting problems is hindered by the need for algorithms that are accurate and scalable to large lattice sizes beyond one spatial dimension. Here, we propose a general framework to numerically study large systems, which combines the spectral accuracy of Chebyshev expansions with the efficiency of divide-and-conquer methods. We use the hybrid algorithm to calculate the two-terminal conductance and the bulk conductivity tensor of 2D lattice models with over 10^{7} sites. By efficiently sampling the microscopic information contained in billions of Chebyshev moments, the algorithm is able to accurately resolve the linear-response properties of complex systems in the presence of quenched disorder. Our results lay the groundwork for future studies of transport phenomena in previously inaccessible regimes.

Online Testing Method for the Fine Spectral Characteristics of Narrow-Band Interference Filters Based on a Narrow-Linewidth Tunable Laser.

The transmission spectrum of a narrow-band interference filter is crucial and highly influenced by factors such as the temperature and angle, thus requiring precise and online measurements. The traditional method of measuring the transmission spectrum of an interference filter involves the use of a spectrometer, but the accuracy of this method is limited. Moreover, placing a narrow-band interference filter inside a spectrometer hinders real-time online measurements. To address this issue, there is demand for high-precision online spectral testing methods. In response to this demand, we propose and experimentally validate a fine spectral characterization method for narrow-band interference filters. This method uses a narrow-linewidth tunable laser, achieving a spectral resolution in the MHz range for online testing. Two types of narrow-band interference filters were tested using the constructed laser spectroscopy experimental system, obtaining a transmission spectrum with a spectral resolution of 318 MHz. In comparison to spectrometer-based methods, our proposed method demonstrates higher spectral accuracy, enables online measurements, and provides more accurate measurements for special spectral interference filters. This approach has significant application value and promising development prospects.

Development of a Fourier‐expansion based differential quadrature method with lattice Boltzmann flux solvers: Application to incompressible isothermal and thermal flows

AbstractThis paper presents a high‐order Fourier‐expansion based differential quadrature method with isothermal and thermal lattice Boltzmann flux solvers (LBFS‐FDQ and TLBFS‐FDQ) for simulating incompressible flows. The numerical solution in the present method is approximated via trigonometric basis. Therefore, both periodic and non‐periodic boundary conditions can be handled straightforwardly without the special treatments as required by polynomial‐based differential quadrature methods. The incorporation of LBFS/TLBFS enables the present methods to efficiently simulated various types of flow problems on considerably coarse grids with spectral accuracy. The high‐order accuracy, efficiency and competitiveness of the proposed method are comprehensively demonstrated through a wide selection of isothermal and thermal flow benchmarks.

Error analysis of the exponential wave integrator sine pseudo-spectral method for the higher-order Boussinesq equation

AbstractIn this paper, we derive a new exponential wave integrator sine pseudo-spectral (EWI-SP) method for the higher-order Boussinesq equation involving the higher-order effects of dispersion. The method is fully-explicit and it has fourth order accuracy in time and spectral accuracy in space. We rigorously carry out error analysis and establish error bounds in the Sobolev spaces. The performance of the EWI-SP method is illustrated by examining the long-time evolution of the single solitary wave, single wave splitting, and head-on collision of solitary waves. Numerical experiments confirm the theoretical results.

Large dynamic range stellar radiation simulation optical system

For the current stellar spectral simulation can not realize the stellar color temperature information with large dynamic range simulation, this paper proposes a broad spectrum high-resolution subdivision and spatial beam zoning modulation combined with a large dynamic range of stellar radiation information simulation method, designed a kind of imaging and non-imaging stellar radiation information simulation optical system, using an optical system to achieve multi-color temperature spectrum and large dynamic range stellar simulation. The experimental results show that the designed system can simultaneously achieve the spectral simulation accuracy (single point evaluation) better than ± 7% in the range of spectral 450–1000 nm and color temperature 3000–11,000 K; on the premise of ensuring the spectral simulation accuracy, the magnitude simulation range reaches 0 to + 12 Mv, and the magnitude simulation accuracy is better than ± 0.05 Mv; Accurate simulation of stellar spectral information and energy large dynamic range tuning is realized, and the system is extended. The system function has been extended to realize the switching of broadband and narrowband modes, The half-peak width of the narrowband output beam is better than 4.1 nm, which extends the application of the spectral simulation technology and provides the theoretical and technical basis for the ground calibration of the development of the high-precision stellar radiation information ground simulation system.

Mollification of Fourier spectral methods with polynomial kernels

Many attempts have been made in the past to regain the spectral accuracy of the spectral methods, which is lost drastically due to the presence of discontinuity. In this article, an attempt has been made to show that mollification using Legendre and Chebyshev polynomial based kernels improves the convergence rate of the Fourier spectral method. Numerical illustrations are provided with examples involving one or more discontinuities and compared with the existing Dirichlet kernel mollifier. Dependence of the efficiency of the polynomial mollifiers on the parameter is analogous to that in the Dirichlet mollifier, which is detailed by analyzing the numerical solution. Further, they are extended to linear scalar conservation law problems.

A Numerical Method Based on Operator Splitting Collocation Scheme for Nonlinear Schrödinger Equation

In this paper, a second-order operator splitting method combined with the barycentric Lagrange interpolation collocation method is proposed for the nonlinear Schrödinger equation. The equation is split into linear and nonlinear parts: the linear part is solved by the barycentric Lagrange interpolation collocation method in space combined with the Crank–Nicolson scheme in time; the nonlinear part is solved analytically due to the availability of a closed-form solution, which avoids solving the nonlinear algebraic equation. Moreover, the consistency of the fully discretized scheme for the linear subproblem and error estimates of the operator splitting scheme are provided. The proposed numerical scheme is of spectral accuracy in space and of second-order accuracy in time, which greatly improves the computational efficiency. Numerical experiments are presented to confirm the accuracy, mass and energy conservation of the proposed method.

Ultra-simplified diffraction-based computational spectrometer

Miniaturizing spectrometers for compact and cost-effective mobile platforms is a major challenge in current spectroscopy research, where conventional spectrometers are impractical due to their bulky footprint. Existing miniaturized designs primarily rely on precalibrated response functions of nanophotonic structures to encode spectral information captured in a snapshot by detector arrays. Accurate spectrum reconstruction is achieved through computational techniques, but this requires precise component design, high-precision fabrication, and calibration. We propose an ultra-simplified computational spectrometer that employs a one-to-broadband diffraction decomposition strategy facilitated by a numerical regularized transform that depends only on the spectrum of the diffracted radiation. The key feature of our design is the use of a simple, arbitrarily shaped pinhole as the partial disperser, eliminating the need for complex encoding designs and full spectrum calibration. Our spectrometer achieves a reconstructed spectral peak location accuracy of better than 1 nm over a 200 nm bandwidth and excellent resolution for peaks separated by 3 nm in a bimodal spectrum, all within a compact footprint of under half an inch. Notably, our approach also reveals a breakthrough in broadband coherent diffractive imaging without requiring any prior knowledge of the broadband illumination spectrum, assumptions of non-dispersive specimens, or correction for detector quantum efficiency.

Calibration method of relative spectral response function of indirect imaging spectrometer

In imaging spectrometers, area array detectors are usually used as photoelectric conversion devices, but the inconsistency of the spectral response among pixels can distort the collected target spectra. To improve the spectral radiometric accuracy of imaging spectrometers, calibrating and correcting the inconsistency of the spectral response among pixels is essential. The signal received by each pixel of area array detector of the indirect imaging spectrometer is usually the superposition of the target multi-spectral radiation signals or full-spectral radiation signals. Therefore, its relative spectral radiometric calibration requires measuring the spectral response of each pixel at different wavelengths on the array detector. Under the ideal conditions, the response values of each pixel in the area array detector are different, so the indirect imaging spectrometer cannot simply calibrate the relative spectral response (RSR) function between pixels by using the method of “monochromator + integrating sphere”. In this work, taking the interferometric imaging spectrometer for example, we analyze the influence of the inconsistency of the RSR among pixels on the target spectral radiation measurement accuracy, and propose a system-level RSR function measurement method for the indirect imaging spectrometer based on the Fourier transform modulation calibration source. In addition, we establish a mathematical model for calibrating the RSR function,and provide guidelines for selecting calibration system parameters such as light source, spectral resolution, and OPD sampling interval. The simulation results show that under the ideal noise-free condition, the 1% spectral response inconsistency among pixels results in a relative error of 1.02% to the recovered spectra. After RSR correction, the relative error of the recovered spectra of different rows decreases to 0.08%. Furthermore, in this work we simulate and analyse the influence of spectral signal-to-noise ratio on the calibration accuracy of the RSR function, and point out that increasing the brightness of the calibration light source, extending exposure time, and combining multi-frame interferograms can enhance RSR function calibration accuracy in practical applications. The research result can provide a theoretical basis for realizing the relative spectral radiometric calibration of indirect imaging spectrometer, which is of great significance in promoting quantitative spectral remote sensing.