Nonstandard Finite-difference Time-domain Research Articles

Accurate Nonstandard Path Integral Models for Arbitrary Dielectric Boundaries in 2-D NS-FDTD Domains.

An efficient path integral (PI) model for the accurate analysis of curved dielectric structures on coarse grids via the two-dimensional nonstandard finite-difference time-domain (NS-FDTD) technique is introduced in this paper. In contrast to previous PI implementations of the perfectly electric conductor case, which accommodates orthogonal cells in the vicinity of curved surfaces, the novel PI model employs the occupation ratio of dielectrics in the necessary cells, providing thus a straightforward and instructive means to treat an assortment of practical applications. For its verification, the reflection from a flat plate and the scattering from a cylinder using the PI model are investigated. Results indicate that the featured methodology can enable the reliable and precise modeling of arbitrarily shaped dielectrics in the NS-FDTD algorithm on coarse grids.

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.

Partial RCS Evaluation Method for Low-Observable Aircraft Design using PEC and PMC Boundaries

For the design of low-observable aircrafts, more commonly termed stealth, it is necessary to analyze the influence of each airframe part for radar cross section (RCS) reduction. To achieve this aim, a partial RCS evaluation method is developed via the nonstandard finite-difference time-domain (NS-FDTD) technique. The new scheme handles the scattering waves caused by body parts, such as the intake and cockpit of the aircraft, by masking the perfectly electric conductor (PEC) and perfectly magnetic conductor (PMC) boundaries. Compared to the use of the perfectly matched layer (PML), the proposed algorithm is straightforward to apply to the overall RCS evaluation of various aircraft configurations with electrically large size.

Adaptable Nonstandard FDTD Schemes for the Precise Evaluation of Electrostatic Fields

The reduction of the total computational overhead in the design of complex geometries, such as modern aircrafts, is a very challenging problem, particularly when electrostatic fields (ESF) for lightning protection, are considered. To this aim, an efficient ESF evaluation scheme, based on the nonstandard finite-difference time-domain (NS-FDTD) method, is proposed. Combining the total-field/scattered-field (TF/SF) concept with a distinct sine-wave form, the novel technique cancels the accumulative errors caused by the static field component. Numerical results reveal that the featured method enables the use of high-frequency discretization models to ESF problems, with notable accuracy and seriously decreased design costs.

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.

MITHRA 1.0: A full-wave simulation tool for free electron lasers

Free Electron Lasers (FELs) are a solution for providing intense, coherent and bright radiation in the hard X-ray regime. Due to the low wall-plug efficiency of FEL facilities, it is crucial and additionally very useful to develop complete and accurate simulation tools for better optimizing a FEL interaction. The highly sophisticated dynamics involved in a FEL process was the main obstacle hindering the development of general simulation tools for this problem. We present a numerical algorithm based on finite difference time domain/Particle in cell (FDTD/PIC) in a Lorentz boosted coordinate system which is able to fulfill a full-wave simulation of a FEL process. The developed software offers a suitable tool for the analysis of FEL interactions without considering any of the usual approximations. A coordinate transformation to bunch rest frame makes the very different length scales of bunch size, optical wavelengths and the undulator period transform to values with the same order. Consequently, FDTD/PIC simulations in conjunction with efficient parallelization techniques make the full-wave simulation feasible using the available computational resources. Several examples of free electron lasers are analyzed using the developed software, the results are benchmarked based on standard FEL codes and discussed in detail. Program summaryProgram Title: MITHRAProgram Files doi:http://dx.doi.org/10.17632/9f5k4zbtkg.1Licensing provisions: GNU General Public License 3Programming language: C++Nature of problem: Full-wave simulation of the free electron laser radiation is accomplished by the code. MITHRA transforms the particle positions and momenta to the bunch rest frame using the Lorentz transformation. Electrons entering the undulator start radiating due to the induced wiggling motion. The back-effect of the radiation on the bunch results in the modulation of the electron position, which in turn generates a coherent radiation. This process as the main principle behind the operation of free electron lasers is simulated using Maxwell equations, electron motion equations and relativity principles.Solution method: Non-standard Finite Difference Time Domain (NSFDTD) combined with Particle-in-Cell (PIC) is implemented in the Lorentz-boosted framework to calculate the FEL radiation. Parallelization is done using both multi-threading (open-MP) and message passing interface (MPI) to maximize the computation efficiency.

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.

Reflection characterization of nano-sized dielectric structure in Morpho butterfly wings

Morpho butterflies living in Central and South America are well-known for their structural-colored blue wings. The blue coloring originates from the interaction of light with nano-sized dielectric structures that are equipped on the external surface of scales covering over their wings. The high-accuracy nonstandard finite-difference time domain (NS-FDTD) method is used to investigate the reflection characterization from the nanostructures. In the NS-FDTD calculation, a computational model is built to mimic the actual tree-like multilayered structures wherever possible using the hyperbolic tangent functions. It is generally known that both multilayer interference and diffraction grating phenomena can occur when light enters the nano-sized multilayered structure. To answer the question that which phenomenon is mainly responsible for the blue coloring, the NS-FDTD calculation is performed under various incidence angles at wavelengths from 360 to 500 nm. The calculated results at one incident wavelength under different incidence angles are visualized in a two-dimensional mapping image, where horizontal and vertical axes are incidence and reflection angles, respectively. The images demonstrate a remarkable transition from a ring-like pattern at shorter wavelengths to a retro-reflection pattern at longer wavelengths. To clarify the origin of the pattern transition, the model is separated into several simpler parts and compared their mapping images with the theoretical diffraction calculations. It can be concluded that the blue coloring at longer wavelengths is mainly caused by the cooperation of multilayer interference and retro-reflection while the effect of diffraction grating is predominant at shorter wavelengths.

Constructing nonstandard and high order FDTD schemes in cylindrical coordinates using spectral domain and modified equation methodologies – 1-D case

One of the major drawbacks of the original Yee’s algorithm is “staircasing.” In an attempt to overcome this flaw, especially when trying to solve electromagnetic problems involving cylindrical geometries, an FDTD scheme was formulated in cylindrical coordinates. The vast majority of works published describe an algorithm with a (2, 2) order of accuracy. In this work, we use a recently introduced methodology, that of constructing finite difference schemes in the spectral domain, combined with the well-established one known as the “modified equation” to analyze existing cylindrical FDTD schemes and synthesize new nonstandard and high-order ones.

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.

An Enhanced Total-Field/Scattered-Field Scheme for the 3-D Nonstandard Finite-Difference Time-Domain Method

The nonstandard finite-difference time-domain (NS-FDTD) method exhibits high accuracy at fixed frequencies, and, thus, proves to be a powerful means of the radar cross section analysis of scattering objects with complex shapes and materials. To this aim, significant enhancement can be pursued in its combination with the FDTD total-field/scattered-field (TF/SF) concept. However, the principal implementation characteristics of the latter in the NS-FDTD technique have yet to be elaborately studied, especially for electrically large domains or structures. Hence, in this paper, a new advanced TF/SF scheme for the 3-D NS-FDTD algorithm is developed and fully validated. Numerical results reveal that the proposed method, considering the features of the NS-FDTD operators, is far more efficient and versatile than the original FDTD one.

A 4-D Subgrid Scheme for the NS-FDTD Technique Using the CNS-FDTD Algorithm With the Shepard Method and a Gaussian Smoothing Filter

A precise subgrid technique for the nonstandard finite-difference time-domain (NS-FDTD) method in 4-D (i.e., 3-D space and 1-D time) is proposed in this paper. The novel algorithm is efficiently blended with the Shepard scheme and a Gaussian smoothing filter to minimize the error in the interpolated values used for the spatial connection process. Moreover, the required time interpolation is performed via the complex (C)NS-FDTD approach. A key advantage of the proposed formulation is its structural simplicity, due to the prior interpolation concepts and the absence of any nonphysical convention, which enables its straightforward application to a variety of realistic problems. The numerical results validate the benefits of the method by means of different subgrid simulation scenarios.

A Novel Efficient Nonstandard High-Order Finite-Difference Time-Domain Method Based on Dispersion Relation Analysis

A novel approach to get new high-order nonstandard finite-difference time-domain methods is proposed. By using different coefficients in electric field and magnetic field update equations, schemes up to order (2, 8) are derived. These new methods maintain comparable accuracy to the conventional high-order finite-difference time-domain counterparts while a reduction in computational cost is achieved. The dispersion and anisotropic behaviors of these new methods are investigated. Some numerical examples are studied to examine the accuracy and efficiency of the new method.

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.

Precise and rapid distance measurements by scatterometry

We found that the distances between isolated scatterers with similar columnar shapes could be measured by taking a single Fourier transform of their diffraction intensity. If the scatterers have different shapes, the distances between similar shapes can be selected from the distances between all the shapes. The distance from a specific scatterer can be measured with a resolution of 0.8 wavelengths and a precision of 0.01 wavelengths. This technique has the potential to be used in a novel optical memory that has a memory density as high as that of holographic memory, while can be fabricated by simple transfer molding. We used rigorous coupled-wave analysis to calculate the diffraction intensity. Some of the results were verified by nonstandard finite-difference time-domain simulations and experiments.