Complex-conjugate Pole-residue Pairs Research Articles

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.

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.

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.

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.

Generalized Finite-Difference Time-Domain Method Utilizing Auxiliary Differential Equations for the Full-Vectorial Analysis of Photonic Crystal Fibers

We present the generalized finite-difference time-domain full-vectorial method by reformulating the time-dependent Maxwell's curl equations with electric flux density and magnetic field intensity, with auxiliary differential equations using complex-conjugate pole-residue pairs. The model is generic and robust to treat general frequency-dependent material and nonlinear material. The Sellmeier equation is implicitly incorporated as a special case of the general formulation to account for material dispersion of fused silica. The results are in good agreement with the results from the multipole method. Kerr nonlinearity is also incorporated in the model and demonstrated. Nonlinear solutions are provided for a one ring photonic crystal fiber as an example.

A generalized 2D FDTD model for photonic crystal fibers with frequency dependent media

A generalized model based on the two-dimensional finite-difference time-domain (2D FDTD) method for photonic crystal fibers (PCF) with frequency dependent media is presented. The Maxwell’s curl equations are formulated using flux density and the magnetic field. Auxiliary differential equations are used with complex-conjugate pole-residue pairs which incorporate the material dispersion of the dispersive media. The model is demonstrated to be a unified approach for arbitrary dispersive materials; therefore, it definitely reduces implementation cost when dealing with different frequency-dependent materials.

Model dispersive media in finite-difference time-domain method with complex-conjugate pole-residue pairs

In this letter, we show that both Debye poles and Lorentz pole pairs are special cases of complex-conjugate pole-residue pairs, and the general form of such pairs is in fact far more efficient than the commonly used Debye poles and Lorentz pole pairs for modeling real dispersive media with the finite-difference time-domain method. We first derive an alternative formulation of the auxiliary differential equation method for arbitrary dispersive media based on general complex-conjugate pole-residue pairs. We then numerically demonstrate the efficiency of using these pairs in modeling dispersive media.