Finite-difference Time-domain Approximation Research Articles

Optimized 3D Finite-Difference-Time-Domain Algorithm to Model the Plasmonic Properties of Metal Nanoparticles with Near-Unity Accuracy

The finite difference time domain (FDTD) method is a grid-based, robust, and straightforward method to model the optical properties of metal nanoparticles (MNPs). Modelling accuracy and optical properties can be enhanced by increasing FDTD grid resolution; however, the resolution of the grid size is limited by the memory and computational requirements. In this paper, a 3D optimized FDTD (OFDTD) was designed and developed, which introduced new FDTD approximation terms based on the physical events occurring during the plasmonic oscillations in MNP. The proposed method not only required ~52% less memory than conventional FDTD, but also reduced the calculation requirements by ~9%. The 3D OFDTD method was used to model and obtain the extinction spectrum, localized surface plasmon resonance (LSPR) frequency, and the electric field enhancement factor (EF) for spherical silver nanoparticles (Ag NPs). The model’s predicted results were compared with traditional FDTD as well as experimental results to validate the model. The OFDTD results were found to be in excellent agreement with the experimental results. The EF accuracy was improved by 74% with respect to FDTD simulation, which helped reaching a near-unity OFDTD accuracy of ~99%. The λLSPR discrepancy reduced from 20 nm to 3 nm. The EF peak position discrepancy improved from ±5.5 nm to only ±0.5 nm.

Elastic full-waveform inversion based on GPU accelerated temporal fourth-order finite-difference approximation

Elastic full-waveform inversion (FWI) estimates the subsurface velocity model by inverting multi-component seismic data. Most of the current elastic FWI solves the elastic wave equation by a temporal second-order finite-difference time-domain (FDTD) method. However, the traditional temporal second-order FDTD introduces large temporal dispersion in the case of a large time-stepping size. To suppress the temporal dispersion, a relatively small time step should be used, which increases the computational cost of FWI. We introduce a temporal fourth-order and spatial arbitrary even-order FDTD method for elastic wave modeling and inversion. The temporal fourth-order scheme can simulate a highly accurate wavefield, even when a large time step is used for temporal extrapolation. Compared with the traditional temporal second-order FDTD scheme, the temporal fourth-order FDTD enables a larger time step for its more relaxed stability condition and smaller dispersion. As a result, FWI with temporal fourth-order FDTD scheme can achieve high accuracy inversion without reducing much computational efficiency. We run our elastic FWI codes on a Graphic Processing Unit (GPU) computing platform, and to accelerate the GPU computing, we propose to use the GPU shared memory to reduce the data access delay. By transferring the data frequently accessed for the FDTD approximations, from the global memory to the shared memory, we realize the FDTD calculation in a more efficient way. Forward modeling and FWI examples are carried out to confirm the effectiveness of our elastic FWI.

Direct computation of acoustic scattering and difference frequency generation

A simulation of the scattered field at the difference-frequency resulting from the nonlinear interaction of two beams at differing frequencies incident on a fluid scatterer is performed. The total wavefield response is modeled using a finite-difference time-domain approximation and a perfectly matched layer to estimate the spatial scattering features of the medium. A pulsed ultrasonic transducer beam is modeled and its interaction with a fluid scatterer with spatially varying compressibility contrast is visualized. The second-order pressure field is determined as a response to the difference-frequency source field constructed from the first order-response of the scatterer to the two incident beams as well as a spatial distribution of the nonlinear parameter. The performance and accuracy of the numerical simulation is examined.

Improving the Accuracy of FDTD Approximations to Tangential Components of the Coupled Electric and Magnetic Fields at a Material Interface

The finite-difference time-domain (FDTD) method is one of the most popular tools used for modeling important electromagnetic wave propagation and scattering problems. Of the many variations on the basic formulation, conventional orthogonal schemes remain prevalent because of their high level of accuracy and relative ease of implementation. Inaccuracies do persist, however, and in this paper a simple, computationally efficient remedy to an error that occurs when simulating waves that impinge on a material interface is identified. The detail of when and how the error occurs is illustrated using a simplistic interface scenario, with an analytic electric field being used to drive a single FDTD magnetic field update step. The resulting FDTD approximations are compared to the analytic solution to reveal the extent of the error. The proposed modification is then introduced and its remedial effect shown by repeating the above comparison using the modified FDTD equations. The broader effects of the proposed modification are demonstrated using a practical, 3D, full-wave FDTD forward modeling tool, taking scenario examples from the application area of ground penetrating radar (GPR). It is concluded that the overall accuracy of conventional, orthogonal FDTD schemes is improved, without unduly increasing their computational burden or algorithmic complexity.

Numerical and experimental study of microwave-excited microplasma and micronozzle flow for a microplasma thruster

Plasma and aerodynamic features have been investigated for a microplasma thruster of electrothermal type using azimuthally symmetric microwave-excited microplasmas. The thruster developed consisted of a microplasma source 1.5 mm in diameter, 10 mm long with a rod antenna on axis, and a converging-diverging micronozzle 1 mm long with a throat 0.2 mm in diameter. The feed or propellant gas employed was Ar at pressures of 10–50 kPa with flow rates of 10–70 SCCM (SCCM denotes standard cubic centimeter per minute at STP) and the surface wave-excited plasmas were established by 4.0 GHz microwaves at powers of ≤6 W. Numerical analysis was made for the plasma and flow properties by developing a self-consistent, two-dimensional model, where a two-temperature fluid model was applied to the entire region through the microplasma source to the micronozzle (or through subsonic to supersonic); in the former, an electromagnetic model based on the finite difference time-domain approximation was also employed for analysis of microwaves interacting with plasmas. In experiments, optical emission spectroscopy was employed with a small amount of additive gases of H2 and N2, to measure the plasma electron density and gas temperature in the microplasma source around the top of the microwave antenna, just upstream of the micronozzle inlet; in practice, the numerical analysis exhibited a maximum thereabout for the microwave power density absorbed, plasma density, and gas temperature. The Stark broadening of H Balmer line and the vibronic spectrum of N2 second positive band indicated that the electron density was in the range of (3–12)×1019 m−3 and the gas or rotational temperature was in the range of 700–1000 K. The thrust performance was also measured by using a microthrust stand with a combination of target and pendulum methods, giving a thrust in the range of 0.2–1.4 mN, a specific impulse in the range of 50–80 s, and a thrust efficiency in the range of 2%–12%. These experimental results were consistent with those of numerical analysis, depending on microwave power and gas flow rate.

A simplified Lax‐Wendroff correction for staggered‐grid FDTD modeling of electromagnetic wave propagation in frequency‐dependent media

The Lax‐Wendroff correction is an elegant method for increasing the accuracy and computational efficiency of finite‐difference time‐domain (FDTD) solutions of hyperbolic problems. However, the conventional approach leads to implicit solutions for staggered‐grid FDTD approximations of Maxwell’s equations with frequency‐dependent constitutive parameters. To overcome this problem, we propose an approximation that only retains the purely acoustic, i.e., lossless, terms of the Lax‐Wendroff correction. This modified Lax‐Wendroff correction is applied to an O(2, 4) accurate staggered‐grid FDTD approximation of Maxwell’s equations in the radar frequency range (≈10 MHz–10 GHz). The resulting pseudo-O(4, 4) scheme is explicit and computationally efficient and exhibits all the major numerical characteristics of an O(4, 4) accurate FDTD scheme, even for strongly attenuating and dispersive media. The numerical properties of our approach are constrained by classical numerical dispersion and von Neumann‐Routh stability analyses, verified by comparisons with pertinent 1-D analytical solutions and illustrated through 2-D simulations in a variety of surficial materials. Compared to the O(2, 4) scheme, the pseudo-O(4, 4) scheme requires 64% fewer grid points in two dimensions and 78% in three dimensions to achieve the same level of numerical accuracy, which results in large savings in core memory.

Analytical solution to the FDTD scalar‐wave equation

A general analytical solution to the finite-difference time-domain (FDTD) scalar-wave equation is derived in terms of its initial conditions using Fourier analysis. The fundamental sources of FDTD modeling error, namely, phase and sampling errors, are demonstrated by comparing this expression with a corresponding analytical solution to the continuous scalar-wave equation. A detailed analysis of the phase error is given which proves mathematically why increasing the time step will decrease error rather than increase it. This explanation, in turn, is used to conclude that the FDTD approximation will, in general, suffer from less numerical dispersion than a frequency-domain finite-difference approximation of the same order. In addition, it is shown that sources can exist implicitly in the FDTD scalar-wave formulation, even though no such sources are explicitly defined. By accounting for these implicit sources, the equivalence between the FDTD scalar-wave equation and the Yee algorithm is demonstrated. ©1999 John Wiley & Sons, Inc. Microwave Opt Technol Lett 20: 240–245, 1999.

Numerical analysis of surface wave excitation in a planar-type nonmagnetized plasma processing device

Two-dimensional numerical simulations based on the finite-difference time-domain approximation to Maxwell's equations coupled with nonmagnetized cold electron plasma equations are used to study surface wave propagation in a four-layer planar type surface wave plasma processing structure (Komachi et al., 1994) under the assumption that the plasma is overdense. Simulations are used to verify the existence of surface waves along the plasma-dielectric window interface as well as to investigate the excitation mechanism of surface waves. In addition, simulations are used to study the effect of the air gap on the electric field distribution at the plasma-dielectric window interface. It is found that metal edges located along the processing chamber wall sufficiently close to the interface are needed to excite surface waves along the interface. It is also found that the air gap layer helps achieve a more uniform electric field distribution over the processing chamber.

Accurate and efficient FDTD modeling of ground‐penetrating radar antenna radiation

We present a 3‐D finite‐difference time domain (FDTD) solution of Maxwell's equations for modeling the near‐field radiation of dipole antennas in ground‐penetrating radar surveys. The antenna is represented as a conducting wire with a narrow gap in the middle. The electromagnetic field in the immediate vicinity of the wire is evaluated by numerical solution of an integral representation of Maxwell's equations. Integration contours and surfaces are defined by the finite‐difference cells surrounding the wire. Everywhere else the electromagnetic wavefield is calculated using conventional FDTD approximations of Maxwell's equations. The accuracy of the algorithm has been tested by comparing the simulated near‐field radiation of a dipole antenna between two dielectric media with corresponding experimental observations.

A simple feed model that reduces time steps needed for FDTD antenna and microstrip calculations

The finite-difference time-domain (FDTD) method is being widely applied to antenna and microstrip calculations. One aspect of this application is accurately and efficiently modeling antenna and microstrip feeds within the constraints of the FDTD approximations. Several relatively straightforward approaches have been suggested, including gap and frill feeds. More complicated approaches, which involve including the coaxial feed cable in the FDTD calculation space, have also been suggested. A related aspect is the desirability of reducing the number of time steps required for the FDTD calculations to converge, especially for transient excitation. We illustrate that for many geometries a simple gap model with an internal source resistance provides accurate results while greatly reducing the number of time steps required for convergence.