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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.

The finite-difference time-domain (FDTD) method has been widely utilized to analyze various electromagnetic wave (EM) problems owing to its simplicity, robustness, and accuracy [

Recently, the Newmark time-stepping algorithm was applied to the dispersive FDTD modeling of Debye, Drude, Lorentz, and QCRF dispersion models [

The remainder of this paper is organized as follows. The Newmark time-stepping algorithm is reviewed and then the Newmark-FDTD formulation is derived for modified Lorentz medium. Numerical stability and numerical permittivity of the Newmark-FDTD formulation are discussed and the equivalence of the Newmark-FDTD formulation to the ADE-FDTD formulation based on the BT is also addressed. In the next section, numerical examples involving homogenous one-dimensional (1D) structure and three-dimensional (3D) plasmonic nanosphere are used to validate our work.

Before proceeding with the Newmark-FDTD method, it is worth reviewing the Newmark time-stepping algorithm briefly [

Now, let us derive the Newmark-FDTD method for the modified Lorentz dispersion model. The modified Lorentz model [

Therefore, we have two sets of differential equations:

Next, let us derive the conventional ADE-FDTD method. Toward this purpose, CDS is applied to (

It is worth comparing the numerical stability of the Newmark-FDTD formulation, the conventional ADE-FDTD formulation, and the ADE-FDTD formulation based on the BT. The numerical stability conditions for FDTD formulations can be obtained by using von Neumann method [

Next, the numerical permittivity can be derived inserting the numerical solution

We also consider Debye, Drude, Lorentz, and QCRF dispersion models. The dispersion models are as follows:

In this section, numerical examples are used to validate our study. First, we consider human blood from 300 MHz to 3 GHz. The modified Lorentz parameters are extracted by using the particle swarm optimization [

Relative error of the numerical permittivity.

Next, actual FDTD simulations are performed. A sinewave with the frequency of 300 MHz is excited in 1D homogenous modified Lorentz medium. The 10-layer perfectly matched layer (PML) [

1D FDTD simulations for different sets of modified Lorentz parameters. (a) All FDTD formulations are stable. (b) Only ADE-FDTD is unstable. Legends are the same as Figure

Root locus of the stability polynomial in the Z-domain for Figure

As a final example, a 3D Ag nanosphere with the radius of 40 nm is considered. In this case, the parameters in [

3D FDTD simulations for an Ag nanosphere with the radius of 40 nm. (a) Spectral response. (b) Relative error.

In this work, the Newmark-FDTD method is applied to modified Lorentz dispersion model which can systematically unify various existing dispersion models. It is found that the Newmark-FDTD is equivalent to the ADE-FDTD with the BT in terms of update formulation, numerical stability, and numerical accuracy. In addition, it is figured out that both methods can yield better accuracy and higher stability against the conventional ADE-FDTD method. Moreover, Debye, Drude, Lorentz, and QCRF dispersion models are extended to the Newmark-FDTD method and its equivalence to the BT-based ADE-FDTD counterpart is also observed. Numerical permittivity and the 1D blood example are used to illustrate that both the Newmark-FDTD method and the ADE-FDTD method based on the BT are better than the conventional ADE-FDTD method in terms of numerical accuracy and numerical stability. A further numerical example involving 3D plasmonic nanosphere is presented to demonstrate that both the Newmark-FDTD simulation and the BT-based ADE-FDTD simulation are in good agreement with the Mie solution and they are superior to the conventional ADE-FDTD method.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This research was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2017R1D1A1B03034537) and in part by Ministry of Culture, Sports and Tourism (MCST) and Korea Creative Content Agency (KOCCA) in the Convergence Tourism Service Research and Business Development Program (no. SF0718106).