Complex-conjugate Pole-residue Research Articles

Accurate and Numerically Stable FDTD Modeling of Human Skin Tissues in THz Band

We propose finite-difference time-domain (FDTD) modeling suitable for healthy skin, basal cell carcinoma, dysplastic pigmentary nevus, and non-dysplastic pigmentary nevus in the frequency range of 0.25 THz to 1.05 THz. Toward this purpose, we utilize the complex-conjugate pole-residue (CCPR) dispersion model, because it is very simple to extract the accurate CCPR coefficients using the powerful vector fitting tool. In the FDTD method, it is of great importance to check the numerical stability conditions. If the coefficients extracted through the vector fitting tool do not satisfy the numerical stability conditions, the particle swarm optimization (PSO) algorithm is employed to obtain the accurate and numerically stable coefficients. Numerical examples are provided to validate our proposed FDTD modeling.

Modeling of dispersive media in ADI-FDTD method with complex–conjugate pole residue pairs

This work presents an alternating-direction implicit (ADI) finite-difference time-domain (FDTD) scheme for the study of structures that involve materials with arbitrary frequency dispersion. The material dispersion is fitted to the complex–conjugate pole-residue (CCPR) terms model, and a novel, to the best of our knowledge, numerical formulation is presented based on auxiliary differential equations and two-step ADI methodology. Additionally, the proposed technique is combined with the concept of the perfectly matched layer, and a new implicit scheme is introduced for the termination of media with CCPR dispersion in the ADI-FDTD framework. The ADI-FDTD formulation is compared with the explicit FDTD scheme for several benchmark two-dimensional problems in terms of accuracy and efficiency. The suggested algorithm is proven to be robust and capable of simulating applications in different frequency regions, spanning from microwaves to optical frequencies. It can provide a powerful tool for the analysis of nanostructures involving both strongly dispersive and nanosized materials, such as plasmonic metasurfaces, antennas, core–shell nanoparticle systems, light-trapping plasmonic solar cells, surface-enhanced Raman spectroscopy substrates, or nanodevices based on epsilon-near-zero materials.

Numerical Stability of Modified Lorentz FDTD Unified From Various Dispersion Models.

The finite-difference time-domain (FDTD) method has been widely used to analyze electromagnetic wave propagation in complex dispersive media. Until now, there are many reported dispersion models including Debye, Drude, Lorentz, complex-conjugate pole-residue (CCPR), quadratic complex rational function (QCRF), and modified Lorentz (mLor). The mLor FDTD is promising since the mLor dispersion model can simply unify other dispersion models. To fully utilize the unified mLor FDTD method, it is of great importance to investigate its numerical stability in the aspects of the original dispersion model parameters. In this work, the numerical stability of the mLor FDTD formulation unified from the aforementioned dispersion models is comprehensively studied. It is found out that the numerical stability conditions of the original model-based FDTD method are equivalent to its unified mLor FDTD counterparts. However, when unifying the mLor FDTD formulation for the QCRF model, a proper Courant number should be used. Otherwise, its unified mLor FDTD simulation may suffer from numerical instability, different from other dispersion models. Numerical examples are performed to validate our investigations.

Time-Domain Studies of General Dispersive Anisotropic Media by the Complex-Conjugate Pole–Residue Pairs Model

A novel finite-difference time-domain formulation for the modeling of general anisotropic dispersive media is introduced in this work. The method accounts for fully anisotropic electric or magnetic materials with all elements of the permittivity and permeability tensors being non-zero. In addition, each element shows an arbitrary frequency dispersion described by the complex-conjugate pole–residue pairs model. The efficiency of the technique is demonstrated in benchmark numerical examples involving electromagnetic wave propagation through magnetized plasma, nematic liquid crystals and ferrites.

Numerical Stability and Accuracy of CCPR-FDTD for Dispersive Medi

The complex-conjugate pole-residue (CCPR) model has been popularly adopted because CCPR-finite-difference time domain (FDTD) can reduce the memory requirement with the help of complex conjugate property of auxiliary variables. To fully utilize CCPR-FDTD, it is of great necessity to investigate its numerical stability since the FDTD method is conditionally stable. Nonetheless, the numerical stability conditions of CCPR-FDTD have not been studied because its derivation is not straightforward. In this communication, the numerical stability conditions of CCPR-FDTD are systematically derived by combining the von Neumann method with Routh–Hurwitz criterion. It is found that the numerical stability conditions of CCPR-FDTD are the same as those of the modified Lorentz-FDTD with bilinear transform. Moreover, the numerical accuracy of CCPR-FDTD is studied, and numerical examples are employed to validate this work.

FDTD Modeling for the Accurate Electromagnetic Wave Analysis of Graphene

We develop a finite-difference time-domain (FDTD) method suitable for the electromagnetic (EM) analysis of graphene. In this work, we employ the modified Lorentz model for dispersion modeling, the two-dimensional (2-D) sheet model for geometrical modeling, and the complex-frequency-shifted (CFS)-perfectly matched layer (PML) for the absorbing boundary condition. In specific, the accurate complex-conjugate pole-residue (CCPR) dispersion model is first adapted for the electrical modeling of graphene by using the robust vector fitting. Next, the CCPR parameters are converted to the modified Lorentz parameters and then the modified Lorentz-based dispersive FDTD formulation is used to enhance the computational efficiency. In FDTD cell modeling, the 2-D sheet cells are allocated for graphene rather than the conventional FDTD cell-based modeling. Finally, CFS-PML are employed for terminating the computational domain to avoid the late-time instability. The presented FDTD approach is validated in numerical examples for graphene-based parallel plate waveguides.

Efficient and stable generalized auxiliary differential equation FDTD implementation of graphene dispersion

PurposeThe purpose of this paper is to present efficient and stable generalized auxiliary differential equation finite difference time domain (G-ADE-FDTD) implementation of graphene dispersion.Design/methodology/approachA generalized dispersive model is used for describing the graphene’s intraband and interband conductivities in the terahertz and infrared frequencies. In addition, the von Neumann method combined with the Routh-Hurwitz criterion are used for studying the stability of the given implementation.FindingsThe presented G-ADE-FDTD implementation allows modeling graphene’s dispersion using the minimal number of additional auxiliary variables, which will reduce both the CPU time and memory storage requirements. In addition, the stability of the implementation retains the standard non-dispersive Courant–Friedrichs–Lewy (CFL) constraint.Practical implicationsThe given implementation is conveniently applicable for most commonly used dispersive models, such as Debye, Lorentz, complex-conjugate pole residue, etc.Originality/valueThe presented G-ADE-FDTD implementation not only unifies the implementation of both graphene’s intraband and interband conductivities, with the minimal computational requirements but also retains the standard non-dispersive CFL time step stability constraint.

Newmark-FDTD Formulation for Modified Lorentz Dispersive Medium and Its Equivalence to Auxiliary Differential Equation-FDTD with Bilinear Transformation

The finite-difference time-domain (FDTD) method has been popularly utilized to analyze the electromagnetic (EM) wave propagation in dispersive media. Various dispersion models were introduced to consider the frequency-dependent permittivity, including Debye, Drude, Lorentz, quadratic complex rational function, complex-conjugate pole-residue, and critical point models. The Newmark-FDTD method was recently proposed for the EM analysis of dispersive media and it was shown that the proposed Newmark-FDTD method can give higher stability and better accuracy compared to the conventional auxiliary differential equation- (ADE-) FDTD method. In this work, we extend the Newmark-FDTD method to modified Lorentz medium, which can simply unify aforementioned dispersion models. Moreover, it is found that the ADE-FDTD formulation based on the bilinear transformation is exactly the same as the Newmark-FDTD formulation which can have higher stability and better accuracy compared to the conventional ADE-FDTD. Numerical stability, numerical permittivity, and numerical examples are employed to validate our work.

Accurate and Efficient Finite-Difference Time-Domain Simulation Compared With CCPR Model for Complex Dispersive Media

Recently, it has received a great deal of attention to analyze the electromagnetic wave problems in dispersive media by using the finite-difference time-domain (FDTD) method. Accordingly, it is of great importance to employ a proper dispersion model which can fit the frequency-dependent permittivity of a medium considered. The reported dispersion models include Debye, Drude, Lorentz, modified Lorentz, quadratic complex rational function, complex-conjugate pole-residue (CCPR) models. The CCPR dispersion model has advantage over other dispersion models in the fact that accurate CCPR dispersion parameters can be simply extracted by using the powerful and robust vector fitting tool which has been widely used in the circuit theory. However, the arithmetic operation of CCPR-based FDTD implementation is involved with complex-valued numbers and thus its numerical computation is not efficient. In this work, we propose an accurate and efficient FDTD simulation for complex dispersive media. In specific, an accurate CCPR dispersion model is simply obtained using the vector fitting tool and then the CCPR dispersion model is converted to the modified Lorentz dispersion model which leads to the arithmetic operation of only real-valued numbers in its FDTD implementation. Numerical examples are used to illustrate the accuracy and efficiency of our dispersive FDTD simulation.

One-Step Leapfrog ADI-FDTD Method Using the Complex-Conjugate Pole-Residue Pairs Dispersion Model

A novel one-step leapfrog alternating-direction-implicit finite-difference time-domain method for the modeling of dispersive media described by the complex-conjugate pole-residue pairs model is proposed. The efficiency of the technique is demonstrated in benchmark numerical examples.

An Extended Newmark-FDTD Method for Complex Dispersive Media

Based on polarizability in the form of a complex quadratic rational function, a novel finite-difference time-domain (FDTD) approach combined with the Newmark algorithm is presented for dealing with a complex dispersive medium. In this paper, the time-stepping equation of the polarization vector is derived by applying simultaneously the Newmark algorithm to the two sides of a second-order time-domain differential equation obtained from the relation between the polarization vector and electric field intensity in the frequency domain by the inverse Fourier transform. Then, its accuracy and stability are discussed from the two aspects of theoretical analysis and numerical computation. It is observed that this method possesses the advantages of high accuracy, high stability, and a wide application scope and can thus be applied to the treatment of many complex dispersion models, including the complex conjugate pole residue model, critical point model, modified Lorentz model, and complex quadratic rational function.

Marching-on-in-Degree Time-Domain Integral Equation Solver for Transient Electromagnetic Analysis of Graphene

The marching-on-in-degree (MOD) time-domain integral equation (TDIE) solver for the transient electromagnetic scattering of the graphene is presented in this paper. Graphene’s dispersive surface impedance is approximated using rational function expressions of complex conjugate pole-residue pairs with the vector fitting (VF) method. Enforcing the surface impedance boundary condition, TDIE is established and solved in the MOD scheme, where the temporal surface impedance is carefully convoluted with the current. Unconditionally stable transient solution in time domain can be ensured. Wide frequency band information is obtained after the Fourier transform of the time domain solution. Numerical results validate the proposed method.

Internal Network Boundary Condition Incorporated in TLM for Efficiently Modeling Thin Layer of Periodic Structures

It is a problem when modeling the thin layer of periodic structures with numerical method by directly discretizing the inner part, because the fine mesh applied to simulate the thin layer would result in great computational data. The internal network boundary condition (INBC) incorporated in the transmission line matrix (TLM) scheme is proposed in this paper, to avoid the directly discretization of the thin layer and achieve a high accuracy. The thin layer of periodic structure is regarded as an easily analyzed two-port network. Vector fitting approach is used to approximate the network parameters into a series of rational expressions with either real term or complex conjugate pole–residue pairs. The INBC equation with discrete-time form is derived and then incorporated in TLM scheme by introducing the approximate pole–residue pairs to the TLM update equation. Compared with the extremely fine-discretized TLM unit cell used to deal with the thin layer by the conventional TLM scheme, a much larger coarse TLM unit cell is accurate enough to be used to discretize the structure by the proposed method. Compared with the conventional TLM scheme, substantial savings in computational storage are achieved. A common frequency selective surface (FSS) structure is first analyzed for the validation of the method. Good agreement in scattering characteristics is observed in frequency domain between the proposed method and simulation by HFSS software. To further demonstrate the validity of the proposed method, an aperture-coupled resonator-based FSS is designed and fabricated. Good agreement among the S-parameters from the proposed method, simulation, and measurement is observed, verifying the accuracy of the proposed method.

SIBC incorporated in conformal FDTD to efficiently simulate the thin conductive layer of periodic structure

In this study, surface impedance boundary condition (SIBC) incorporated in conformal finite-difference time-domain (CFDTD) to efficiently simulate the thin conductive layer of periodic structure is proposed. The method is based on the vector fitting (VF) technique to approximate the frequency-dependent surface impedance of the thin conductive layer by a series of partial fractions in terms of real or complex conjugate pole-residue pairs. A discrete-time equation of SIBC is then derived and incorporated in CFDTD update equations by Laplace transform and time central difference methods. The proposed method takes the following advantages, (i) the time step is no longer bounded by the fine resolution of the thin layer; (ii) The complicated recursive convolution approach is avoided, and the approximation of the frequency-dependent surface impedance of the thin conductive layer can be easy to handle with the VF method. This method is numerically verified for the time- and frequency-dependent scattering characteristics of several periodic structures behaved under normal and oblique incident waves.

Systematic wave-equation finite difference time domain formulations for modeling electromagnetic wave-propagation in general linear and nonlinear dispersive materials

In this paper, systematic wave-equation finite difference time domain (WE-FDTD) formulations are presented for modeling electromagnetic wave-propagation in linear and nonlinear dispersive materials. In the proposed formulations, the complex conjugate pole residue (CCPR) pairs model is adopted in deriving a unified dispersive WE-FDTD algorithm that allows modeling different dispersive materials, such as Debye, Drude and Lorentz, in the same manner with the minimal additional auxiliary variables. Moreover, the proposed formulations are incorporated with the wave-equation perfectly matched layer (WE-PML) to construct a material independent mesh truncating technique that can be used for modeling general frequency-dependent open region problems. Several numerical examples involving linear and nonlinear dispersive materials are included to show the validity of the proposed formulations.

Unified Frequency-Dependent Split-Step FDTD Formulations Based on the Complex-Conjugate Pole-Residue Pairs Model

Unified and efficient frequency-dependent split-step finite difference time domain (SS-FDTD) formulations are presented. In the proposed formulations, the complex-conjugate pole-residue (CCPR) pairs model is adopted in the FDTD implementation of the domain's frequency-dependent relation with the minimal additional auxiliary variables. In addition, the formulations allow modeling different dispersive materials, such as Debye, Drude, and Lorentz, in the same manner. A numerical example is included to show the validity of the formulations.

A Method to Model Thin Conductive Layers in the Finite-Difference Time-Domain Method

This paper presents a new approach for modeling of electrically thin conductive shields in the finite-difference time-domain (FDTD) method. The method is based on representation of the relation between the fields at two faces of the shield as an impedance boundary network condition (INBC) in the frequency domain. The INBC includes frequency-dependent self and mutual impedances which are approximated by series of partial fractions in terms of real or complex conjugate pole-residue pairs. A discrete time-domain INBC at the shield is generated, which is then incorporated within the FDTD method. The primary advantages of the proposed approach are: 1) the convolution equations are not used in the formulation, 2) the approximation applied for discretizing the Maxwell equation has second-order accuracy in time and first-order of accuracy in space, and 3) the stability of the method is governed by the classical Courant Friedrichs Lewy stability condition. Numerical examples are presented to validate the new method and to demonstrate its efficiency and accuracy.

Wideband Modeling of Graphene Using the Finite-Difference Time-Domain Method

In this paper, we present a method to incorporate the intraband and interband terms of the surface conductivity of graphene into the finite-difference time-domain (FDTD) method. The method is based on approximating the surface resistivity of graphene by a series of partial fractions in terms of real or complex conjugate pole-residue pairs. Then, a discrete time-domain surface boundary condition at the graphene sheet is generated, which is then incorporated within the FDTD method using the infinitesimally thin sheet formulation. Numerical examples are presented to validate and demonstrate the capabilities and advantages of the proposed approach.

FDTD Modeling of Graphene Devices Using Complex Conjugate Dispersion Material Model

Graphene-based devices constitute a pioneering field of research for their extraordinary electromagnetic properties. The incorporation of appropriate models into numerical simulators is necessary in order to take advantage of these properties. In this work, we propose a method to incorporate graphene-sheet models into the FDTD method. The use of vector-fitting techniques expands the permittivity of graphene into a rational function series of complex conjugate pole-residue pairs, which is implemented into FDTD by an auxiliary differential equation formulation. Simple waveguiding problems validate our approach.

Modeling hemoglobin at optical frequency using the unconditionally stable fundamental ADI-FDTD method

This paper presents the modeling of hemoglobin at optical frequency (250 nm – 1000 nm) using the unconditionally stable fundamental alternating-direction-implicit finite-difference time-domain (FADI-FDTD) method. An accurate model based on complex conjugate pole-residue pairs is proposed to model the complex permittivity of hemoglobin at optical frequency. Two hemoglobin concentrations at 15 g/dL and 33 g/dL are considered. The model is then incorporated into the FADI-FDTD method for solving electromagnetic problems involving interaction of light with hemoglobin. The computation of transmission and reflection coefficients of a half space hemoglobin medium using the FADI-FDTD validates the accuracy of our model and method. The specific absorption rate (SAR) distribution of human capillary at optical frequency is also shown. While maintaining accuracy, the unconditionally stable FADI-FDTD method exhibits high efficiency in modeling hemoglobin.