Auxiliary Differential Equation Finite-difference Time-domain Research Articles

Newton-ADE FDTD Method for Time-Varying Plasma

A novel finite-difference time-domain (FDTD) method is proposed in this article for electromagnetic (EM) wave propagation in time-varying plasma. It is formulated based on the fundamental Newton’s equation of motion, which governs the relationships among the time-varying electron density, current density and electric field. Utilizing the auxiliary differential equation (ADE) FDTD scheme, the method is aptly referred to as the Newton-ADE FDTD method for time-varying plasma. Traditionally, the previous ADE FDTD methods directly apply the time-varying electron density for the plasma frequency in the update equations of current density and electric field. This is inadequate and incorrect in general time-varying plasma conditions. The formulation of the Newton-ADE FDTD method is provided and compared to that of the traditional ADE FDTD method. It is found that the key difference lies in the new compact term being the time-derivative of logarithm of electron density. Two discretization schemes for this term are provided. The Newton-ADE FDTD method is validated based on the matrix exponential method. Novel stability and convergence analyses are provided for the proposed method in time-varying dispersive media. These analyses show that our proposed method is stable and achieves second-order temporal accuracy. Numerical results are presented for various time-varying plasma conditions. It is demonstrated that when the absolute value for time-derivative of logarithm of electron density is large, e.g., greater than the collision frequency, the Newton-ADE FDTD method can provide correct results, while the traditional ADE FDTD method yields significant differences.

Investigating the stability of an explicit ADE-FDTD scheme for modeling graphene: Avoiding erroneous conclusions

In recent years, an explicit auxiliary differential equation finite difference time domain (ADE-FDTD) scheme has been frequently used in graphene simulations. In this respect, a differencing scheme in which the magnetic field and the current density are collocated in time was developed and it has been recently stated that this explicit ADE-FDTD implementation is “conditionally stable with the maximum time-step size bounded by the common Courant–Friedrichs–Lewy (CFL) limit”. In this communication, the stability of this explicit ADE-FDTD scheme is revisited and it is shown that the conventional CFL constraint is not retained, and a time-step less than the CFL limit must be used in all domain grids to insure the stability in the whole system. Although this time-step stringent can be of little effect for some graphene applications, and in order to avoid drawing erroneous conclusions, it is shown that when the given ADE-FDTD scheme is used for simulating graphene structures that include noble and other plasmonic materials the derived time-step stringent can be of significant effect. To retain the CFL stability constraint, alternative formulation based on the Runge–Kutta (RK) time-differencing scheme is also presented in this communication. Contrary to the other existing CFL-stable ADE-FDTD approaches, the presented RK-FDTD method is equally applicable to both magnetized and unmagnetized graphene structures without incurring additional computational cost. Furthermore, the RK-FDTD scheme is extended for modeling the complex-frequency-shifted perfectly matched layer (CFS-PML) mesh truncating technique. Finally, the stability and accuracy of the explicit ADE/RK-FDTD schemes are demonstrated through numerical examples.

An Area-Averaged Conformal Technique for the Electromagnetic Simulations of Debye Medium

A novel area-averaged dielectric conformal technique combined with the finite-difference time-domain (FDTD) method has been investigated in accuracy and efficiency for modeling curved dispersive medium objects in this article. The proposed method utilizes an effective average dielectric constant, which is derived by area averages of different dielectric regions in irregular cells cut by curved surfaces, to modify the magnetic field updating equations in auxiliary differential equation FDTD method. The accuracy and efficiency of the method are verified by comparing the radar cross section results produced by the method of moments and the FDTD method with very fine meshes, respectively. Numerical results show that good agreements can be achieved by using coarse mesh size (12 cells per wavelength). Further, the proposed method is devoted to analyzing the bio-electromagnetic effects of a dispersive human body in a vehicle platform to show its good ability in solving such complex problems.

Detection and Localization of Breast Cancer Using UWB Microwave Technology and CNN-LSTM Framework

Ultrawideband (UWB) microwave detection technology, which is low cost and harmless, has been intensively studied and developed for breast cancer detection. In this study, a composite end-to-end framework that consists of convolutional neural network (CNN) and long-short-term memory (LSTM) is proposed, which can realize the tasks of detecting and quadrant locating the breast tumor simultaneously without any complicated microwave imaging processing. In order to verify the proposed network, three datasets are constructed. First, the microwave signals are solved by the auxiliary differential equation-finite-difference time-domain (ADE-FDTD) solver and the magnetic resonance imaging (MRI)-derived breast models with varied breast densities. Then, Simulation datasets 1 and 2 with varied dielectric properties are constructed. Furthermore, an experimental dataset is constructed through an experimental system consisting of the UWB pulse generation module, UWB antennas, the realistic breast phantom, and sampling oscilloscope. With the proposed network, the overall prediction accuracies of the three datasets reach 99.56%, 98.94%, and 89.50%. These promising results demonstrate the effectiveness and accuracy of the proposed deep learning framework for microwave breast cancer detection.

Newton-ADE-FDTD Method for Oblique Incident Magnetized Time-varying Plasma

In this paper, the Newtonian equation of motion describing the movement of electrons when electromagnetic waves propagate in a magnetized plasma is combined with the traditional auxiliary differential equation finite difference time domain (ADE-FDTD) method. The FDTD iterative formulas of transverse magnetic (TM) wave and transverse electric (TE) wave of the electromagnetic wave obliquely incident on the magnetized time-varying plasma plate are derived. The biggest difference between this method and the ordinary ADE-FDTD algorithm is the addition of the logarithmic derivative of the time-varying plasma electron density to calculate the current density, which is called the Newton-ADE-FDTD method. Through Example 1, the reflection coefficient of electromagnetic wave incident on the magnetization time-varying plasma plate was calculated, and the correctness of the improved algorithm was verified. At the same time, the Newton-ADE-FDTD algorithm is used to calculate the reflection coefficient of electromagnetic waves incident on the magnetized plasma-dielectric photonic crystal. The results show that different incident angles have a greater impact on the reflection coefficients of left-handed circularly polarized wave (LCP) and right-handed circularly polarizedwave (RCP).

Modeling Dispersive Media in Finite-Difference Time-Domain Method for Radiative Cooling Applications

Radiative cooling offers a new strategy to address building cooling problems in an environmentally friendly way, compared with traditional cooling methods that require high-power consumption. The goal of radiative cooling is to maximize absorption at the mid-infrared (MIR) spectrum (8– <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$13~\mu m$ </tex-math></inline-formula> ). Thus, it is important to characterize the dispersive relationship of the radiative cooling materials (TiO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , Si <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> , SiC, SiO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) in the MIR regime. In this work, we use the multi-pole Lorentz dispersion equation to model frequency-dependent dielectric constants of radiative cooling materials, which requires particle swarm optimization to solve multidimension optimization problems. This modeling can be directly applicable to the auxiliary differential equation (ADE)-based finite-difference time-domain (FDTD) method for simulating emissivity of radiative cooling materials. Further, we limit the number of Lorentz poles to five, which minimizes the computational burden in ADE–FDTD. Then, we validate our FDTD results against an analytic prediction calculated by the transfer matrix method. Finally, we demonstrate a highly efficient radiative cooling structure based on a simple SiO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> grating. The proposed material modeling can accelerate the nano/microstructural design of the radiative cooling strategies.

Modified Lorentz Model-Based ADE-WLP-FDTD Method Simulation on THz Wave Propagation Properties of Tumor Tissue

An auxiliary differential equation (ADE) finite-difference time-domain (FDTD) method based on the weighted Laguerre polynomials (WLP) algorithm is developed to study wave propagation in biological tissue in the Terahertz (THz) frequency band. The relative permittivity of dispersive tissue can be expressed by the double Debye model or Cole–Cole model, which can be transformed into the modified Lorentz model by using the Padé approximation technique. The modified Lorentz model is numerically solved by means of the ADE-WLP-FDTD method. Numerical results validate the accuracy and efficiency of the proposed method. The effects caused by the slide and medium parameters on THz waves propagation properties of biological tissue are investigated. This letter enriches the high efficient ADE-WLP-FDTD method for terahertz imaging and tumor diagnosis to detect medium parameters of biological tissue.

An ANN-based surrogate model for wave propagation in uncertain media

ABSTRACT The parameter values of complex media can be uncertain in reality and uncertainty quantification (UQ) in the simulation of wave propagation is essential. In this paper, an efficient surrogate model based on the artificial neural network (ANN) is developed to imitate the concerned ground penetrating radar (GPR) calculation. A two-dimensional GPR system is simulated with the auxiliary differential equation (ADE) finite-difference time-domain (FDTD), and the soil is considered as a nonmagnetic medium with the frequency-dependent dielectric permittivity represented by a two-term Debye model. A solid metallic object and a piece of dry granite are buried in the dispersive soil whose parameter values are considered uncertain. The framework of the surrogate model is presented for GPR system modeling. Different popular activation functions are applied to the proposed surrogate model and their results are analyzed. Compared with other activation functions, the activation function of exponential linear unit has a better performance in the training process. Moreover, the dropout method is used to reduce overfitting to a large extent. At last, the well trained surrogate model is applied for UQ to GPR calculation instead of running one thousand full-wave GPR simulations, and the results show its accuracy and efficiency.

Analysis and optimization of electromagnetic scattering characteristics of two-dimensional metal airfoil covered with metamaterial

Airfoil is one of the strong scattering sources of aircraft, so it is of great significance to study the stealth characteristics of airfoil for the reduction of the radar cross section of the whole aircraft. As is known, the airfoil of aircraft is the main component that generates lift. Shaping the airfoil will reduce the aerodynamic performance of the aircraft. However, the emergence of metamaterials allows aircraft designers to achieve a good balance between stealth and aerodynamics. Considering the high cost of stealth test, numerical simulation is proposed to guide the design of metamaterial stealth. Then, the bistatic radar cross section (RCS) of airfoil covered with metamaterial is researched in this paper. We employ the 2-D auxiliary differential equation finite-difference time-domain (ADE-FDTD) to simulate the airfoil’s stealth characteristics. The two-dimensional situation can be regarded as a special approximate when electromagnetic waves are perpendicularly incident on the three-dimensional infinitely long airfoil. The tuning parameters including coating thickness, Drude frequency and collision frequency are considered on the airfoil’s bistatic RCS. The calculation program verified its correctness by simulating the perfect lens characteristics of electromagnetic waves propagating in the metamaterial plate. The numerical experiments show that the meta-material Drude frequency and collision frequency mechanism are more complicated than the coating thickness. The dual-station RCS increases at some angles and decreases at other angles than metal airfoils. But by choosing appropriate metamaterial parameters, airfoils with excellent stealth effect can be obtained.

Improved direct integration auxiliary differential equation FDTD scheme for modeling graphene drude dispersion

In this communication, an improved direct-integration auxiliary differential equation (IM-DI-ADE) scheme is introduced for incorporating graphene's Drude dispersion into the finite difference time domain (FDTD) simulations in the GHz and THz frequency ranges. Stability analysis is performed and it is shown that the time-step stability limit of the presented IM-DI-ADE scheme coincides with the conventional non-dispersive FDTD Courant–Friedrichs–Lewy (CFL) limit and removes the additional stability stringent of the classical DI-ADE counterpart. In addition, the presented scheme does not increase the memory storage overhead. Numerical dispersion analysis is also addressed and it is shown that the presented scheme provides high accuracy. The formulation is validated by numerical tests that investigate the tunneling of electromagnetic wave through a graphene sheet and the existence of the surface plasmon polaritons (SPPs) waves created at the interface between the graphene sheet and a dielectric material.

BCC-Grid versus SC-Grid in the modeling of a sheet of graphene as a surface boundary condition in the context of ADE-FDTD

Here, we model a thin layer of graphene located above a metal surface by means of a surface boundary condition in two different stencils, namely, a simple cubic grid (SC-Grid) and a body centered cubic grid (BCC-Grid). We extend the methodology described in the literature by taking into account the interband contribution of the graphene’s conductivity in addition to the intraband contribution. The mathematical description of the presented developments is explained. Besides this, the metal in contact with the graphene sheet is considered as a dispersive medium; therefore, we deal with the problem by using the auxiliary differential equation finite-difference time-domain (ADE-FDTD) method. In this context, we compared the two stencils and demonstrated that BCC-Grid does not present discontinuities in the normal components of the electric and magnetic fields located on the graphene surface; in this respect, BCC-Grid is also more appropriate than the traditional Yee’s cell for these type of applications.

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.

Uncertainty Analysis in Dispersive and Lossy Media for Ground-Penetrating Radar Modeling

The uncertainty analysis (UA) in simulation results is essential when the acceptability of the results is considered. This letter introduces an improved intrusive generalized polynomial chaos expansion (gPCE) method for UA in ground-penetrating radar modeling. The uncertainty in simulation results induced by the uncertain parameters of two-dimensional dispersive and lossy media is quantified with the auxiliary differential equation finite-difference time-domain method combined with gPCE. The results from the new method are compared against the Monte Carlo method. The new method shows its considerable advantage in the computational expense and speed.

Frequency-dependent radiation properties of negative permittivity metamaterial reflector antenna

This paper reports a novel type of microwave reflector antenna that uses the frequency-dependent properties of an epsilon negative (ENG) metamaterial medium. The frequency-dependent characteristics of this artificial plasma medium is analyzed using the dispersive auxiliary differential equation finite difference time domain (ADE-FDTD) method by employing a Gaussian pulse and the results are verified experimentally using an artificial plasma medium fabricated by an array of thin conducting wires. The radiation pattern of plain and corner plasma reflector antennas modeled using Drude equations are obtained and are compared with a plasma reflector and conventional metallic reflector antennas. The results obtained for the plasma reflector antenna is experimentally verified using a 90° corner reflector antenna fabricated using wire medium. This new class of plasma antennas show marked variations in radiation pattern for frequencies above and below plasma frequency, which may find potential use in various frequency selective applications.

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.

Stability-Improved ADE-FDTD Method for Wideband Modeling of Graphene Structures

A stability-improved auxiliary differential equation finite-difference time-domain (FDTD) method is developed for broadband modeling graphene-based devices in this letter. By introducing an electric polarization term characterized with an auxiliary equation, the graphene sheet is incorporated into the FDTD method without any decrease in the Courant–Friedrich–Levy number. The stability condition is analyzed and numerical experiments are carried out to validate the proposed method with analytical results. It is further applied to investigate some properties of metal–graphene split-ring resonator.

Study on stealth characteristics of metamaterials based on simulated annealing algorithm

Auxiliary differential equation finite-difference time domain (ADE-FDTD) expressions of drude dispersive medium model are applied to simulate stealth characteristics of metamaterials coating. Then, simulated annealing algorithm is used to optimize electromagnetic parameters considered include metamaterials coating, collision frequency drude frequency and coating thickness. The results show that monostatic RCS of metal cylinder covering with metamaterial coating can reduce 6.3dB and bistatic RCS of metal cylinder covering with metamaterial coating can shrink 0.1dB less than before. Simulated annealing algorithm can play an important effect on stealth characteristics of metamaterials.

Hybrid sub-gridding ADE–FDTD method of modeling periodic metallic nanoparticle arrays**Project supported by the National Natural Science Foundation of China (Grant Nos. 61471105 and 61331007).

In this paper, a modified sub-gridding scheme that hybridizes the conventional finite-difference time-domain (FDTD) method and the unconditionally stable locally one-dimensional (LOD) FDTD is developed for analyzing the periodic metallic nanoparticle arrays. The dispersion of the metal, caused by the evanescent wave propagating along the metal-dielectric interface, is expressed by the Drude model and solved with a generalized auxiliary differential equation (ADE) technique. In the sub-gridding scheme, the ADE–FDTD is applied to the global coarse grids while the ADE–LOD–FDTD is applied to the local fine grids. The time step sizes in the fine-grid region and coarse-grid region can be synchronized, and thus obviating the temporal interpolation of the fields in the time-marching process. Numerical examples about extraordinary optical transmission through the periodic metallic nanoparticle array are provided to show the accuracy and efficiency of the proposed method.

Backward Scattering Characteristics of a Reentry Vehicle Enveloped by a Hypersonic Flow Field

The hypersonic flow field around a reentry vehicle has a significant influence on the ground-vehicle communication as well as on the detection and recognition of the reentry vehicle. Backward scattering characteristics of a reentry vehicle enveloped by a hypersonic flow field are analyzed using a high-order auxiliary differential equation finite difference time-domain (ADE-FDTD) method in this paper. Flow field parameters, including electron density, neutral particle density, and temperature, are obtained by solving the Navier-Stokes (NS) equations numerically. According to the flow field parameters, distributions of the plasma frequency and the collision frequency are then derived. Based on a validity of the physical model and the high-order ADE-FDTD method, backward radar cross sections (RCSs) of a perfect electrical conductor (PEC) sphere enveloped by a hypersonic flow field under different Mach numbers, heights, and incident angles of the electromagnetic (EM) wave are then investigated. Numerical results show that the incident angle of the EM wave exerts noticeable effects on the backward RCS, which is due to an inhomogeneous distribution of the plasma. The flight height and Mach number have significant influences on the distribution of the plasma that they play an important role in the variation of the RCSs. The results presented in this paper provide useful reference data for practical tests in ballistic range or in the high-frequency plasma wind tunnels, where a sphere target is usually used due to its simple shape.

Localized Plasmonic Resonances of Prolate Nanoparticles in a Symmetric Environment: Experimental Verification of the Accuracy of Numerical and Analytical Models

We study the evolution of the surface-plasmon resonances of individual ion-beam-shaped prolate gold nanoparticles embedded in a dielectric SiO 2 environment by electron-energy-loss spectroscopy mapping in a scanning transmission electron microscope. The controlled symmetric dielectric environment obtained through the ion-beam-shaping method allows a direct quantitative comparison with numerical results obtained through simulations (auxiliary differential-equation finite-difference time-domain and boundary-element method) and with theoretical results obtained through analytical models (quasistatic model for prolate nanoellipsoids and waveguide model for infinite one-dimensional plasmonic waveguides), with which our experimental results are in very good agreement. We confirm the accuracy of state-of-the-art numerical tools and analytical theories that establish ion-beam shaping as a very promising method to design metal-dielectric nanocomposites with well-predicted optical properties, and with many possible applications in surface-enhanced Raman spectroscopy and second-harmonic generation, as well as in conventional applications of metamaterials like negative refraction, superimaging, and invisibility cloaking.