Finite-difference Frequency-domain Research Articles

Double Resonance Characteristic of Photonic Crystal Structure with Defect Metal Layer

The optical response of the photonic crystal structure has been analyzed using the Finite Difference Frequency Domain (FDFD) method. Simulation using this method aims to analyze the characteristics of light transmission when interacting with related structure. The results of the study of photonic crystal structures with two defects show that there are two transmittance bands called double resonances at two specific wavelengths. Changes in the resonance peak with respect to the variations of refractive index in the second defect layer resulted in a linear change with significant sensitivity. For photonic crystals with two defects, the first defect is a metal layer and the second defect is material sensing, which produces double resonance with a sensitivity of 7,825 and 3,2675, while for photonic crystals with the first defect, a dielectric layer, and the second defect is material sensing, produces single resonance with a sensitivity of 2,022. The choice of the first defect is that the metal layer is quite important, in addition to producing greater sensitivity, it also produces a field enhancement between the dielectric and the metal layer which is called surface plasmon resonance (SPR). The sensitivity value can also be set by determining the thickness of the second defect layer. These results can be used as a standard for fabrication and developed for sensor applications.

Seismic wave modeling of fluid-saturated fractured porous rock: including fluid pressure diffusion effects of discretely distributed large-scale fractures

Abstract. The scattered seismic waves of fractured porous rock are strongly affected by the wave-induced fluid pressure diffusion effects between the compliant fractures and the stiffer embedding background. To include these poroelastic effects in seismic modeling, we develop a numerical scheme for discretely distributed large-scale fractures embedded in fluid-saturated porous rock. Using Coates and Schoenberg's local-effective-medium theory and Barbosa's dynamic linear slip model characterized by complex-valued and frequency-dependent fracture compliances, we derive the effective viscoelastic compliances in each spatial discretized cell by superimposing the compliances of the background and the fractures. The effective governing equations for fractured porous rocks are viscoelastic anisotropic and numerically solved by the mixed-grid-stencil frequency-domain finite-difference method. The main advantage of our proposed modeling scheme over poroelastic modeling schemes is that the fractured domain can be modeled using a viscoelastic solid, while the rest of the domain can be modeled using an elastic solid. We have tested the modeling scheme in a single fracture model, a fractured model, and a modified Marmousi model. The good consistency between the scattered waves off a single horizontal fracture calculated using our proposed scheme and the poroelastic modeling validates that our modeling scheme can properly capture the fluid pressure diffusion (FPD) effects. In the case of a set of aligned fractures, the scattered waves from the top and bottom of the fractured reservoir are strongly influenced by the FPD effects, and the reflected waves from the underlying formation can retain the relevant attenuation and dispersion information. The proposed numerical modeling scheme can also be used to improve migration quality and the estimation of fracture mechanical characteristics in inversion.

Accelerating finite-difference frequency-domain simulations of inverse designed structures in nanophotonics using deep learning

The inverse design of nanophotonic devices is becoming increasingly relevant for the development of complex photonic integrated circuits. Electromagnetic first-order simulations contribute to the overwhelming computational cost of the optimization routines in established inverse design algorithms, requiring more efficient methods for enabling improved and more complex design process flows. Here we present such a method to predict the electromagnetic field distribution for pixel-discrete planar inverse designed structures using deep learning. Our model is able to infer accurate predictions used to initialize a conventional finite-difference frequency-domain algorithm and thus lowers the average time required for simulating the electromagnetic response of nanophotonic device layouts by up to 53% in iterative design process flows. We demonstrate the applicability of our deep learning method for the inverse design of photonic integrated powersplitters and mode converters, and we highlight the possibility of exploiting previous learning results in subsequent design tasks of novel functionalities via fine-tuning reduced data sets, thus improving computational speed further.

3D numerical simulation of frequency-domain elastic wave equation with a compact second-order scheme

Finite-difference frequency-domain (FDFD) modeling has gained popularity in recent years. However, its main drawback lies in the need to solve large sparse linear systems, which is computationally expensive, especially for 3D elastic wave simulations. The existing elastic wave optimal schemes improve the simulation accuracy at the cost of using large-sized stencils or high-order finite-difference (FD) operators, and most of them only consider equal grid spacing and solid media. To address these issues, we have proposed a compact second-order scheme for simulating 3D elastic wave propagation. Starting from the second-order FD discretization of the first-order velocity-stress expression, the compact discrete scheme of the second-order elastic wave equation is obtained by using a parsimonious staggered-grid strategy to eliminate the stress variables. In this way, the scheme preserves the ability to handle high-contrast velocity and liquid-solid media, and has a more compact and sparse impedance matrix than existing schemes, which helps save computational costs. We also use an antilumped mass strategy to further improve the modeling accuracy. Dispersion analysis indicates that our compact second-order scheme has better accuracy than the classic second-order scheme, and even better accuracy than the average derivative 27-point scheme at large Poisson's ratio. Compared with the existing 3D elastic wave schemes, the compact second-order scheme has advantages in computational cost for the same grid. Finally, we employ these schemes to implement 3D numerical simulations, validating the effectiveness of our proposed scheme.

Wavefield simulation of acoustic VTI wave equation based on adaptive-coefficient finite-difference frequency-domain method

Abstract Many simulation methods have been developed for P-waves in vertically transversely isotropic (VTI) media. These methods are based on the acoustic approximation. The finite-difference frequency-domain (FDFD) method stands out for its ability to simulate multi-shot or narrowband seismic data. It has no temporal dispersion, facilitates attenuation modelling, and enables parallelization. The optimal FDFD method is commonly used to simulate the acoustic VTI wave equation, but it applies the same FDFD coefficients for different frequencies and model velocities, which cannot fully minimize the numerical dispersion error. To enhance its accuracy and effectiveness, we develop an adaptive-coefficient FDFD method specifically for the acoustic VTI wave equation. The FDFD coefficients depend on two factors: the number of wavelengths in each grid and the Thomsen parameters. The dispersion analysis reveals that the proposed FDFD method can achieve a reduction in the necessary number of grid points from 4 to 2.5 compared to the optimal 9-point average derivative method (ADM), while maintaining a maximum dispersion error of 1%. From three numerical examples, the developed FDFD method can obtain more accurate wavefield results than the ADM optimal FDFD method, while taking comparable computational time and memory.

Adaptive-Coefficient Finite Difference Frequency Domain Method for Solving Time-Fractional Cattaneo Equation with Absorbing Boundary Condition

The time-fractional Cattaneo (TFC) equation is a practical tool for simulating anomalous dynamics in physical diffusive processes. The existing numerical solutions to the TFC equation generally deal with the Dirichlet boundary conditions. In this paper, we incorporate the absorbing boundary condition as a complex-frequency-shifted (CFS) perfectly matched layer (PML) into the TFC equation. Then, we develop an adaptive-coefficient (AC) finite-difference frequency-domain (FDFD) method for solving the TFC with CFS PML. The corresponding analytical solution for homogeneous TFC equation with a point source is proposed for validation. The effectiveness of the developed AC FDFD method is verified by the numerical examples of four typical TFC models, including the different orders of time-fractional derivatives for both the homogeneous model and the layered model. The numerical examples show that the developed AC FDFD method is more accurate than the traditional second-order FDFD method for solving the TFC equation with the CFS PML absorbing boundary condition, while requiring similar computational costs.

Finite-difference frequency-domain method with QR-decomposition-based complex-valued adaptive coefficients for 3D diffusive viscous wave modelling

Abstract The diffusive viscous (DV) model is a useful tool for interpreting low-frequency seismic attenuation and the influence of fluid saturation on frequency-dependent reflections. Among present methods for the numerical solution of the corresponding DV wave equation, the finite-difference frequency-domain (FDFD) method with complex-valued adaptive coefficients (CVAC) has the advantage of efficiently suppressing both numerical dispersion and numerical attenuation. In this research, the FDFD method with CVAC is first generalized to a 3D DV equation. In addition, the current calculation of CVAC involves the numerical integration of propagation angles, conjugate gradient (CG) iterative optimization, and the sequential selection of initial values, which is difficult and inefficient for implementation. An improved method is developed for calculating CVAC, in which a complex-valued least-squares problem is constructed by substituting the 3D complex-valued plane-wave solutions into the FDFD scheme. The QR-decomposition method is used to efficiently solve the least-squares problem. Numerical dispersion and attenuation analyses reveal that the FDFD method with CVAC requires ∼2.5 spatial points in a wavelength within a dispersion deviation of 1% and an attenuation deviation of 10% for the 3D DV equation. An analytic solution for 3D DV wave equation in homogeneous media is proposed to verify the effectiveness of the proposed method. Numerical examples also demonstrate that the FDFD method with CVAC can obtain accurate wavefield modelling results for 3D DV models with a limited number of spatial points in a wavelength, and the FDFD method with QR-based CVAC requires less computational time than the FDFD method with CG-based CVAC.

Complex-valued adaptive-coefficient finite-difference frequency-domain method for wavefield modeling based on the diffusive-viscous wave equation

The diffusive-viscous wave (DVW) equation is an effective model for analyzing seismic low-frequency anomalies and attenuation in porous media. To effectively simulate DVW wavefields, the finite-difference or finite-element method in the time domain is favored, but the time-domain approach proves less efficient with multiple shots or a few frequency components. The finite-difference frequency-domain (FDFD) method featuring optimal or adaptive coefficients is favored in seismic simulations due to its high efficiency. Initially, we develop a real-valued adaptive-coefficient (RVAC) FDFD method for the DVW equation, which ignores the numerical attenuation error and is a generalization of the acoustic adaptive-coefficient FDFD method. To reduce the numerical attenuation error of the RVAC FDFD method, we introduce a complex-valued adaptive-coefficient (CVAC) FDFD method for the DVW equation. The CVAC FDFD method is constructed by incorporating correction terms into the conventional second-order FDFD method. The adaptive coefficients are related to the spatial sampling ratio, number of spatial grid points per wavelength, and diffusive and viscous attenuation coefficients in the DVW equation. Numerical dispersion and attenuation analysis confirm that, with a maximum dispersion error of 1% and a maximum attenuation error of 10%, the CVAC FDFD method only necessitates 2.5 spatial grid points per wavelength. Compared with the RVAC FDFD method, the CVAC FDFD method exhibits enhanced capability in suppressing the numerical attenuation during anelastic wavefield modeling. To validate the accuracy of our method, we develop an analytical solution for the DVW equation in a homogeneous medium. Three numerical examples substantiate the high accuracy of the CVAC FDFD method when using a small number of spatial grid points per wavelength, and this method demands computational time and computer memory similar to those required by the conventional second-order FDFD method. A fluid-saturated model featuring various layer thicknesses is used to characterize the propagation characteristics of DVW.

Wideband close perfect meta surface absorber for solar photovoltaic and optical spectrum applications

The leading research challenge is to develop a meta-surface absorber design for optimal performance across ultraviolet, visible, and near-infrared wavelengths. In ongoing research, a meta-surface absorber is being introduced, designed to achieve near-perfect absorption of 99% across the 250–2500 nm UV–Vis-NIR spectrum for both Transverse electric (TE) and Transverse magnetic (TM) polarized waves. The aimed design features a symmetrical resonator pattern, resulting in a polarization-insensitive absorption response. Additionally, the absorption response demonstrates robustness when altering the geometric parameters of the meta-surface unit cell. Precise simulation and modelling of the proposed meta-surface absorber are achieved by Employing a Finite Difference Frequency Domain (FDFD) solver, coupled with the Finite Integration Technique (FIT) and a tetrahedral mesh component. Verification of absorption characteristics is conducted through three frameworks: impedance matching, interference, and an equivalent circuit model. The suggested meta-structure design exhibits a uniform absorption response, as determined through considerations of impedance matching, interference theory, and the equivalent circuit theory. The PCR value is significantly lower than recent studies, reinforcing that the proposed meta-surface primarily serves as an absorber, not a polarization converter. The absorption response of the suggested meta-structure design remains consistent across the range of electromagnetic wave polarization angles from 0 to 90 for both TE and TM waves. Furthermore, the absorption response of the proposed design demonstrates stability up to an incident angle of 70 for electromagnetic waves. The distribution of electric and magnetic fields, along with surface current density on the proposed design, validates the existence of localized, propagating, and cavity surface plasmon resonance. These collectively create anti-parallel current flow, leading to the absorption of incident EM waves. Furthermore, the suggested design achieves a solar energy thermal conversion efficiency of 99% across the operational spectrum at a working temperature of 573 K, substantiating its potential for use as a solar absorber. Thus, this study presents a meta-structure unit cell that addresses the challenge with near-perfect absorption, polarization independence, and angle robustness. Applications include photodetection, thermal mapping, solar harvesting, and photo-trapping.

Design Simulation of Circular Photonic Crystal Structures for the Development of Biomedical Sensor Based on Finite Difference Method

Optical analysis of a circular photonic crystal structure consisting of a metal-dielectric layer and a defect layer sandwiched between two dielectric layers has been carried out using the Finite Difference Frequency Domain (FDFD) method and the Finite Difference Time Domain (FDTD) method. The FDTD method can describe the propagation of electromagnetic waves on the structure at any time, while the FDFD method can describe the characteristics of the wave interaction with the structure in a steady state. We observed strong resonances at certain wavelengths when using three different dielectric materials: , , and . The simulation results show a change in the resonance peak in the sensing material in the form of normal blood and blood-containing glucose in the defect layer for concentration: . The use of three different dielectric materials produces different sensitivity values. The highest sensitivity value when using material that produces a sensitivity value of at the refractive index intervals . The simulation results can be used to build biomedical sensors that measure blood sugar levels.

A 17-point composite average-derivative optimal method for high-accuracy frequency-domain scalar wave modeling

The frequency-domain finite-difference (FD) technique, especially the average-derivative method (ADM), is widely utilized in wavefield modeling. However, conventional low-order FD techniques are insufficient for high-accuracy demands. Applying high-order FD operator and optimizing their coefficients are effective measures for improving the accuracy of seismic modeling. However, these approaches suffer from the limitations of constant density media and approximant perfectly matched layer (PML). To address this issue, we have developed a composite-ADM 17-point scheme for a scalar-wave equation. This novel scheme is based on the composition of ADM discretizations with different grid intervals, which successfully removes the above limitations and achieves high accuracy. To suppress the numerical dispersion, we take the L∞ norm to construct the objective function, which we minimize using the Grey Wolf Optimizer (GWO) and Particle Swarm Optimization (PSO) algorithms. Numerical analysis indicates that the composite-ADM greatly reduces phase velocity error compared with the traditional-ADM, requiring only 2.44 grid points per wavelength within a 1% phase velocity error. Using numerical examples in homogeneous and complex models, we demonstrate the accuracy and versatility of the composite-ADM 17-point scheme in variable density media with exact PML. Furthermore, we extend this optimal scheme to the case of 2D viscous and 3D scalar-wave equations, making it a more practical and versatile solution.

Computational scaling in inverse photonic design through factorization caching

Inverse design coupled with adjoint optimization is a powerful method to design on-chip nanophotonic devices with multi-wavelength and multi-mode optical functionalities. Although only two simulations are required in each iteration of this optimization process, these simulations still make up the vast majority of the necessary computations and render the design of complex devices with large footprints computationally infeasible. Here, we introduce a multi-faceted factorization caching approach to drastically simplify the underlying computations in finite-difference frequency-domain (FDFD) simulations and significantly reduce the time required for device optimization. Specifically, we cache the numerical and symbolic factorizations for the solution of the corresponding system of linear equations in discretized FDFD simulations and re-use them throughout the device design process. As proof-of-concept demonstrations of the resulting computational advantage, we present simulation speedups reaching as high as 9.2× in the design of broadband wavelength and mode multiplexers compared to conventional FDFD methods. We also show that factorization caching scales well over a broad range of footprints independent of the device geometry, from as small as 16μm2 to over 7000μm2. Our results present significant enhancements in the computational efficiency of inverse photonic design and can greatly accelerate the use of machine-optimized devices in future photonic systems.

Design and performance evaluation of a novel metamaterial broadband THz filter for 6G applications

Terahertz (THz) radiation, which has applications in the imaging of objects, non-destructive testing, satellite communication, medical diagnostics, and biosensing, has generated a great deal of attention due to its remarkable properties. This paper proposes a novel broadband filter for THz applications. The main idea is to overcome the insertion loss and bandwidth issues by modeling a frequency-domain finite difference method and guided-mode resonance (GMR). The optimal design scheme of the wideband pass filter based on the circular resonant ring is discussed by comparing the transmission parameters under various parameters. This scheme overcomes the restriction of the narrow passband bandwidth of the prior THz filters and achieves approximately 3 dB bandwidth of 0.54 THz. The proposed THz filter paper also has the advantages of a straightforward structure, low processing costs, and ease of conformal with other structures, and it can be used for stealth fighters, new communication technology, and precise instruments. In addition, when compared to existing models, the suggested filter offers higher 3 dB BW operation, increased transmittance, low insertion loss, and stable performance at various oblique angles.

Finite-Difference Frequency-Domain Scheme for Sound Scattering by a Vortex with Perfectly Matched Layers

Finite-Difference Frequency-Domain Scheme for Sound Scattering by a Vortex with Perfectly Matched Layers

A simplified calculation for adaptive coefficients of finite-difference frequency-domain method

A simplified calculation for adaptive coefficients of finite-difference frequency-domain method

Multiscale model reduction of finite-difference frequency-domain wave modelling in acoustic media

SUMMARY Frequency-domain wave modelling can easily describe the visco-acoustic behaviour of wave propagation using frequency-dependent velocities. Conventional finite-difference (FD) modelling in the frequency domain is computationally prohibitive for solving the acoustic Helmholtz equation in complicated and large geological models. To reduce the computational cost of traditional FD Helmholtz solvers, we develop a multiscale FD frequency-domain method that uses multiscale basis functions to significantly reduce the dimension of system matrices associated with the Helmholtz equation. Due to the insufficient accuracy of the first-order multiscale basis functions in the case of strongly heterogeneous models, we introduce the multinode coarse-element scheme into the scalar Helmholtz equation, a scheme previously developed in the extended multiscale finite-element method for vector problems. This multinode scheme enables multiscale basis functions to capture accurate fine-scale medium property variations. We use one homogeneous model and two heterogeneous models to validate our multiscale method for accuracy and computational cost. Numerical results demonstrate that our new approach can significantly reduce the time and memory costs of acoustic wave modelling while maintaining accuracy, indicating the great potential of our multiscale method in large-scale modelling applications.

Analog of electromagnetically induced transparency based on asymmetric nickel-ferrite metamaterials in THz regime

The electromagnetically induced transparency-like characteristics of asymmetric nickel-ferrite metamaterials through near-field coupling are investigated in the terahertz regime. The nickel-ferrite metamaterial consists of a cut wire resonator on the front side and a disk resonator at the back of the quartz substrate. The analog of electromagnetically induced transparency is characterized by employing a finite difference frequency domain technique. By designing the nickel-ferrite metamaterial to overcome Snoek’s limitation of ferrite materials in the THz spectral region, electromagnetically induced transparency-like spectral features with a high quality factor and slow-light effect are achieved. These results may provide an important way to understand the coupling mechanism and ferrite material application in slow-light devices, THz sensors, and tunable switching in THz frequency bands.

Optical non-volatile correction of SERS wavelength using optical pumping

In this paper, we propose a reconfigurable two-dimensional photonic crystal (PC) made from a new ultralow loss phase change material for detecting and imaging applications of surface-enhanced Raman scattering (SERS). The proposed all-dielectric structure is composed of identical holes periodically distributed on Sb2S3 as a substrate, tuned by optical pumping. The proposed PC is investigated using full-wave simulation of the CST with the finite-difference frequency domain method over a wide bandwidth of optical wavelengths. In this study, compensation for the error of the Fano resonance wavelength due to the fabrication process is realized through tuning optical pumping applied to the configuration. Also, the numerical results show the designed PC supports two high-quality Fano resonance modes, leading to uniform and high field enhancement with a SERS enhancement factor of 1.23 × 1012, which is significant for the application of SERS enhancement.

Applicability of the Effective Index Method for the Simulation of X-Cut LiNbO3 Waveguides

Photonic integrated circuits (PIC) find applications in the fields of microwaves, telecoms and sensing. Generally, PICs are fabricated on a base of isotropic materials such as SOI, Si3N4, etc. However, for some applications, anisotropic substrates such as LiNbO3 are used. A thin film of LiNbO3 on an insulator (LNOI) is a promising material platform for complex high-speed PICs. The design and simulation of PICs on anisotropic materials should be performed using rigorous numerical methods based on Maxwell’s equations. These methods are characterized by long calculation times for one simulation iteration. Since a large number of simulation iterations are performed during the PIC design, simulation methods based on approximations should be used. The effective index method (EIM) is an approximation-based method and is widely applied for simulations of isotropic waveguides. In this study, the applicability of EIM for simulations of anisotropic waveguides is analyzed. The results obtained by EIM are compared with the calculation results of a rigorous finite-difference frequency-domain (FDFD) method for evaluation of the EIM’s applicability limits. In addition, radiation losses in waveguides with rough sidewalls are estimated using the Payne–Lacey model and EIM. The results demonstrate the applicability of EIM for the simulation of anisotropic LNOI-based waveguides with cross-section parameters specified in this paper.

A Fast Butterfly-Compressed Hadamard–Babich Integrator for High-Frequency Helmholtz Equations in Inhomogeneous Media with Arbitrary Sources

We present a butterfly-compressed representation of the Hadamard–Babich (HB) ansatz for the Green’s function of the high-frequency Helmholtz equation in smooth inhomogeneous media. For a computational domain discretized with discretization cells, the proposed algorithm first solves and tabulates the phase and HB coefficients via eikonal and transport equations with observation points and point sources located at the Chebyshev nodes using a set of much coarser computation grids, and then butterfly compresses the resulting HB interactions from all cell centers to each other. The overall CPU time and memory requirement scale as for any bounded two-dimensional (2D) domains with arbitrary excitation sources. A direct extension of this scheme to bounded 3D domains yields an CPU complexity, which can be further reduced to quasi-linear complexities with proposed remedies. The scheme can also efficiently handle scattering problems involving inclusions in inhomogeneous media. Although the current construction of our HB integrator does not accommodate caustics, the resulting HB integrator itself can be applied to certain sources, such as concave-shaped sources, to produce caustic effects. Compared to finite-difference frequency domain methods, the proposed HB integrator is free of numerical dispersion and requires fewer discretization points per wavelength. As a result, it can solve wave propagation problems well beyond the capability of existing solvers. Remarkably, the proposed scheme can accurately model wave propagation in 2D domains with 640 wavelengths per direction and in 3D domains with 54 wavelengths per direction on a state-of-the-art supercomputer at Lawrence Berkeley National Laboratory.