Nonstandard Finite-difference Time-domain Method Research Articles

Path Integral Representation Model to Extend the Analytical Capability of the Nonstandard Finite-difference Time-domain Method

The nonstandard finite-difference time-domain (NS-FDTD) method is a powerful tool for solving Maxwell’s equations in their differential form on orthogonal grids. Nonetheless, to precisely treat arbitrarily shaped objects, very fine lattices should be employed, which often lead to unduly computational requirements. Evidently, such an issue hinders the applicability of the technique in realistic problems. For its alleviation, a new path integral (PI) representation model, equivalent to the NS-FDTD concept, is introduced. The proposed model uses a pair of basic and complementary path integrals for the H-nodes. To guarantee the same accuracy and stability as the NS-FDTD method, the two path integrals are combined via optimization parameters, derived from the corresponding NS-FDTD formulae. Since in the PI model, E-field computations on the complementary path are not necessary, the complexity is greatly reduced. Numerical results from various real-world problems prove that the proposed method improves notably the efficiency of the NS-FDTD scheme, even on coarse orthogonal meshes.

A Nonstandard Path Integral Model for Curved Surface Analysis

The nonstandard finite-difference time-domain (NS-FDTD) method is implemented in the differential form on orthogonal grids, hence the benefit of opting for very fine resolutions in order to accurately treat curved surfaces in real-world applications, which indisputably increases the overall computational burden. In particular, these issues can hinder the electromagnetic design of structures with electrically-large size, such as aircrafts. To alleviate this shortcoming, a nonstandard path integral (PI) model for the NS-FDTD method is proposed in this paper, based on the fact that the PI form of Maxwell’s equations is fairly more suitable to treat objects with smooth surfaces than the differential form. The proposed concept uses a pair of basic and complementary path integrals for H-node calculations. Moreover, to attain the desired accuracy level, compared to the NS-FDTD method on square grids, the two path integrals are combined via a set of optimization parameters, determined from the dispersion equation of the PI formula. Through the latter, numerical simulations verify that the new PI model has almost the same modeling precision as the NS-FDTD technique. The featured methodology is applied to several realistic curved structures, which promptly substantiates that the combined use of the featured PI scheme greatly improves the NS-FDTD competences in the case of arbitrarily-shaped objects, modeled by means of coarse orthogonal grids.

Simulation of light scattering by two nano-sized circular cylinders using NS-FDTD method and interference analysis

Scattering problem by one cylinder whose radius is comparable to incident wavelength can be solved by Mie theorem. Due to complicated multiple scattering, scattering characteristics by two adjacent cylinders cannot be easily expounded. In this paper, high-accuracy non-standard finite-difference time domain (NS-FDTD) method is utilized to simulate electromagnetic field scattered by two nano-sized circular cylinders. Scattering intensities calculated by the NS-FDTD method demonstrate distinct polarization-dependence feature which can be explained by interference analyses including first-order and second-order scattering. The interference analyses indicate that the two-cylinder scattering problem in the transverse magnetic mode can be approximately solved by one-cylinder Mie scattering solution superposed with a first-order interference term, whereas both first-order and second-order interference should be considered into the superposition term in the transverse electric mode.

Non-standard finite-difference time-domain method for solving the Schrödinger equation

In this paper, an improvement of the finite-difference time-domain (FDTD) method using a non-standard finite-difference scheme for solving the Schrödinger equation is presented. The standard numerical scheme for a second derivative in the spatial domain is replaced by a non-standard numerical scheme. In order to apply the non-standard FDTD (NSFDTD), first, the estimates of eigenenergies of a system are needed and computed by the standard FDTD method. These first eigenenergies are then used by the NSFDTD method to obtain improved eigenenergies. The NSFDTD method can be employed iteratively using the resulting eigenenergies to obtain more accurate results. In this paper, the NSFDTD method is validated using infinite square well, harmonic oscillator and Morse potentials. Significant improvements are found when using the NSFDTD method.

Total-Field/Scattered-Field Separation Based on $H$ -field Correction for the Nonstandard Finite-Difference Time-Domain

The nonstandard finite-difference time-domain (NS-FDTD) methodology is a very competent numerical tool for the radar cross section (RCS) analysis of scattering objects with complicated geometries and electrically large sizes, due to its high accuracy at fixed frequencies. During the RCS calculation via the typical FDTD scheme, the total-field/scattered-field (TF/SF) approach is generally used to separate the SF from TF simulation region. However, existing TF/SF separation algorithms for the NS-FDTD method are rather laborious for practical arrangements. Therefore, we develop a new TF/SF separation technique, solely based on H-field correction, and apply it in both cubic- and rectangular-cell computational domains, which discretize demanding structures. In all scenarios, the proposed algorithm is found to be superior over previously derived procedures, while, concurrently, remaining much simpler in its overall NS-FDTD implementation.

Wide-Angle Elimination of TF/SF-Generated Spurious Waves in the Nonstandard FDTD Technique

In the radar-cross section (RCS) analysis of electrically large objects via the nonstandard finite-difference time-domain method, spurious reflection or scattering waves arise from the total-field/scattered-field (TF/SF) separation surface. To suppress such waves, a novel TF/SF separation scheme with a multilayered structure for the suitable assumed-known incident-field calculations, is proposed. The scheme, which uses two adjustable parameters, is numerically verified and found to attain a drastic reduction of the spurious scattered field over a wide-angle regime, unlike previous TF/SF approaches, which is necessary for accurate RCS studies.

ADI Scheme of a Nonstandard FDTD Method and Its Numerical Properties

An alternating direction implicit (ADI) scheme for a nonstandard (NS) finite-difference time-domain (FDTD) method is proposed. The NS FDTD method is derived by analyzing the dispersion relation. Two different groups of update coefficients are used to update the electric field and magnetic field, respectively. The dispersion property of the NS method is similar with that of the standard higher order method, though it is only second order accurate in the space domain. The total number of nonzero update coefficients for this method is fewer than that of the latter. This small stencil property can be used to reduce computational burden in implicit formulation of this method. The one-step leapfrog ADI scheme is adopted to examine this idea. The detailed update equations of this ADI scheme are derived. The dispersion and stability properties are also analyzed and compared with the corresponding standard one. The results show that the computational efficiency of the proposed method is higher than that of the standard one.

A Stability Improvement Technique Using PML Condition for the Three-Dimensional Nonuniform Mesh Nonstandard FDTD Method

We propose a technique to improve the stability of three-dimensional (3D) nonuniform mesh nonstandard FDTD (NS-FDTD) calculations. To mitigate the numerical instability arising from the connection of different-sized meshes, our method combines the usual interpolation fields and perfectly matched layer (PML) condition at mesh interfaces. We examine a connection for a mesh size ratio of 3:1, and show numerically that our technique improves the stability of 3D NS-FDTD calculation on nonuniform meshes. Our technique is also applied to analyze the fields around a periodic slot panel.

Rendering Morpho butterflies based on high accuracy nano-optical simulation

A rendering method, based on high accuracy nano-optical simulations, is developed and applied to render the iridescent colors of the Morpho butterfly. The wings of the male display blue structural colors and backscatterings. In order to capture the Morpho butterfly features, subwavelength interactions on the scales must be rigorously taken into account. We calculate optical interference in the subwavelength scale structure using the high accuracy nonstandard finite-difference time-domain method, and validate the results by comparing with experimental measurements. Using our simulation results, realistic Morpho butterfly images are rendered.

Characteristics of the Boundary Model in the 2-D NS-FDTD Method

This paper investigates the representation of the dielectric boundary and the absorbing boundary conditions (ABC) that are major concerns in the nonstandard FDTD (NS-FDTD) method. The effective permittivity model of the dielectric boundary is examined and it is found that its accuracy decreases when the boundary coincides with a node. To overcome this difficulty, we propose an alternative, highly accurate boundary model based on an analytic boundary condition. The accuracy of Berenger's split-field perfectly matched layer (PML) as an ABC is also demonstrated for the NS-FDTD algorithm.

Origin of Retroreflection from a Wing of theMorphoButterfly

We have performed detailed angular and spectroscopic measurements on a typical species of Morpho butterflies, Morpho rhetenor , and have found that the butterfly wing shows characteristic retroreflection in a blue region, when light is incident around 10–30° inclined from the normal to the scales on the wing. Various angular and spectroscopic measurements such as detector and sample rotations, and θ–2θ scan are tested, and apart from an inclusive method of θ–φ scan, the sample rotation method is found to be the most appropriate to characterize its optical properties. In order to investigate the origin of the retroreflection, non-standard finite-difference time-domain method is applied and alternate shelf structure is found to be responsible for it. Its optical characteristics are more intuitively understood by two simple models, destructive interference between two units of one-sided shelf structure and two inversely inclined multilayers, although such models are not complete in a strict sense. The presence of retroreflection in this type of animal will be extremely beneficial to show their colors clearly without specular reflection that has no color selectivity.

3차원 ID-FDTD 알고리즘의 Stability Condition과 광대역 특성 분석

Abstract The stability condition and wideband characteristics of 3D ID-FDTD algorithm which has low dispersion error with isotropic dispersion are presented in this paper. 3D ID-FDTD method was proposed to improve the defect of the Yee FDTD such as the anisotropy and large dispersion error. The published paper calculated the stability condition of 3D ID-FDTD algorithm by using numerical method, however, it is thought that the examples were not sufficient to verify the stability condition. Thus, in this paper, various simulations are included in order to hold reliability under the con-ditions that the plane wave propagation is assumed with a single frequency and a wideband frequency. Also, the 3D ID-FDTD algorithm is compared to those that have the similar FDTD algorithm with ID-FDTD such as Forgy's method and non-standard FDTD method in a wideband. Finally, the radar cross section(RCS) for the large sphere with high dielectric constant is calculated. Key words : FDTD, Isotropic Dispersion, Rcs, Stability Condition

Coefficients of Finite Difference Operator for Rectangular Cell NS-FDTD Method

The nonstandard finite difference time domain (NS-FDTD) method is a dispersion error reduction technique for the FDTD method. High accuracy is achieved by the optimal selection of the coefficients used for the finite difference (FD) operators in the NS-FDTD method. However, the previously published method of obtaining the coefficients for rectangular cells, using a full numerical technique, takes an enormous amount of computing time. To solve this difficulty, we propose a semi-analytical method to obtain the optimal coefficients with a far higher computational efficiency. It is shown, by numerical tests, that the optimal coefficients need six significant digits for accurate NS-FDTD solutions. Consequently our new method can reduce the computing time to obtain the optimal coefficients from an estimated 42 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> minutes by the previous method to about 7 seconds for three dimensional rectangular cells.

A high resolution nonstandard FDTD method for the TM mode of Maxwell’s equations

In this work, a high resolution nonstandard finite-difference time-domain method for the transverse magnetic (TM) mode of 2D-Maxwell’s equations is derived. Stability and convergence analysis and numerical studies are presented for the proposed method. The accuracy of this numerical scheme is validated by simulating a light scattering cross-section by a perfectly-conducting circular cylinder.

Nonstandard FDTD Method for Multifrequency Analysis

The nonstandard finite difference time domain (NS-FDTD) method is a new high-accuracy FDTD method, but until now, it was applicable only at a single frequency. In this paper, we extend the NS-FDTD method to handle multiple frequencies in one computer run. We apply our new method to analyze the scattering analysis of a large cavity.

Characteristics of Evanescent Waves in the Nonstandard FDTD Method

This paper investigates the characteristics of evanescent waves in the generalized nonstandard finite-difference time-domain (NS-FDTD) method and derives the dispersion equation and stability condition for two-dimensional space. The validity of the dispersion equation is confirmed by comparison with results of numerical simulations. The NS-FDTD method is applied to the analysis of an optical switching device using a photon tunneling effect. It is shown that the NS-FDTD method has highly accurate characteristics not only for propagating waves, but also for evanescent waves compared with the standard FDTD method

Overlap Algorithm for the Nonstandard FDTD Method Using Nonuniform Mesh

In this paper, the overlap algorithm is applied to the nonstandard finite-difference time-domain (NS-FDTD) method using a nonuniform mesh in a 2-D space. The characteristics of the overlap algorithm are numerically examined and compared with a no overlap algorithm. The results show that the reflection rate from the interface of the two meshes using the overlap algorithm is less than -80 dB, which is smaller than that of the NS-FDTD method without the overlap. Consequently, it is shown that the overlap algorithm is more suitable for the NS-FDTD method when using a nonuniform mesh. The overlap algorithm is successfully applied to the analysis of a dielectric flat panel and a corrugated surface

Colour characterization of a Morpho butterfly wing-scale using a high accuracy nonstandard finite-difference time-domain method

In certain species of moths and butterflies iridescent colours arise from subwavelength diffractive structures. The optical properties of such a structure depend strongly on wavelength, incidence angle and state of polarization of illuminating radiation and on the viewing angle. Such structures can be analyzed only by solving Maxwell's equations, but since analytical solutions exist for only a few simple, highly symmetric structures numerical methods must be employed. We investigated the optical properties of butterfly wings in two dimensions by simulating a scale structure using a high accuracy version of nonstandard finite-difference time-domain algorithm. The simulated structure is a computer-generated model of a certain quasi-periodic arrangement of tree-like structures observed in the transmission electron micrograph (TEM) image of a transverse cross-section of a single scale from Morpho butterfly wings. We assumed that the structure is made of a slightly lossy dielectric material. We checked the accuracy and validity of our approach, by computing scattered field intensities due to an infinite cylinder and compared the results with analytical calculations using Mie theory. Next we deduced the wavelength dependence of a real refractive index and an absorption coefficient for the ground scales on the wings of Morpho sulkowskyi butterfly by computing the reflectivity and transmissivity spectrum of a scale at normal incidence, and comparing with experimental measurements. Finally, we calculated the tristimulus values and corresponding colour coordinates for various viewing directions from the scale's far-field reflectivity and transmissivity spectra to characterize its colour rendering abilities.

Calculation of Diffraction Characteristics of Sub wavelength Conducting Gratings Using a High Accuracy Nonstandard Finite-Difference Time-Domain Method

Gratings with subwavelength groove depth and period are frequently used in optics for various purposes. The polarization dependent diffraction characteristics of subwavelength (high frequency) gratings can only be calculated by solving Maxwell’s equations of electromagnetism. In this paper, we calculate the classical diffraction characteristics of subwavelength conducting gratings numerically, using a new high accuracy version of nonstandard finite-difference time-domain (NS-FDTD) algorithm. For the purpose of analysis, we employ a gold sinusoidal grating with light incident at a large angle. We have compared high accuracy NS-FDTD simulation results with those obtained from standard finite-difference time-domain (S-FDTD), and the finite element method (FEM) simulations.

New optimization parameters for the nonstandard FDTD method

In this paper, new simpler optimization parameters for the nonstandard FDTD (NS-FDTD) method are proposed by redefining the spatial-difference operator. Consequently, the number of optimization parameters required in the method is reduced to one from two in 2D space, and three from nine in 3D space, respectively. In addition, it is shown that the NS-FDTD method using new optimization parameters has more accurate characteristics than the NS-FDTD method using conventional parameters. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 47: 161–163, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21112