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Research ArticleNewmark-FDTD Formulation for Modified Lorentz DispersiveMedium and Its Equivalence to AuxiliaryDifferential Equation-FDTD with Bilinear Transformation

Hongjin Choi,1 Jeahoon Cho,1 Yong Bae Park,2 and Kyung-Young Jung 1

1Department of Electronics and Computer Engineering, Hanyang University, Seoul 04763, Republic of Korea2Department of Electrical and Computer Engineering, Ajou University, Suwon 16499, Republic of Korea

Correspondence should be addressed to Kyung-Young Jung; [email protected]

Received 5 March 2019; Accepted 3 June 2019; Published 20 June 2019

Academic Editor: Rodolfo Araneo

Copyright © 2019 Hongjin Choi et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The finite-difference time-domain (FDTD) method has been popularly utilized to analyze the electromagnetic (EM) wavepropagation 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-FDTDmethod was recently proposed for the EM analysis of dispersive media and it was shown that the proposedNewmark-FDTDmethod can give higher stability and better accuracy compared to the conventional auxiliary differential equation-(ADE-) FDTDmethod. In this work, we extend theNewmark-FDTDmethod tomodified Lorentzmedium, which can simply unifyaforementioned dispersion models. Moreover, it is found that the ADE-FDTD formulation based on the bilinear transformationis exactly the same as the Newmark-FDTD formulation which can have higher stability and better accuracy compared to theconventional ADE-FDTD. Numerical stability, numerical permittivity, and numerical examples are employed to validate our work.

1. Introduction

The finite-difference time-domain (FDTD) method has beenwidely utilized to analyze various electromagnetic wave (EM)problems owing to its simplicity, robustness, and accuracy[1–3]. It is of great importance to choose well-fitted disper-sion model to obtain accurate results in dispersive media.A variety of dispersion models were proposed such asDebye, Drude, Lorentz, quadratic complex rational function(QCRF), complex-conjugate pole-residue, and critical pointmodels [4–14]. Note that the modified Lorentz dispersionmodel can unify the aforementioned dispersion models andhas more degree of freedom than Debye, Drude, and Lorentzmodels [15, 16].

Recently, the Newmark time-stepping algorithm wasapplied to the dispersive FDTD modeling of Debye, Drude,Lorentz, and QCRF dispersion models [17, 18]. The authorsshowed that the Newmark-FDTD method can lead to sim-ple arithmetic implementation, higher stability, and betteraccuracy compared to the conventional auxiliary differential

equation- (ADE-) FDTD method [19]. In this work, theNewmark-FDTD method is successfully extended to themodified Lorentz dispersion model. It is found that theformulation of the ADE-FDTDmethod based on the bilineartransformation (BT) [20–22] leads to the same formulationof the Newmark-FDTD method. Moreover, the ADE-FDTDformulations with BT for Debye, Drude, Lorentz, and QCRFdispersion models are also considered and it will be shownthat the resulting FDTD formulations are equivalent to theNewmark-FDTD counterparts.

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

HindawiInternational Journal of Antennas and PropagationVolume 2019, Article ID 4173017, 7 pageshttps://doi.org/10.1155/2019/4173017

https://orcid.org/0000-0002-7960-3650https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/4173017

2 International Journal of Antennas and Propagation

2. Formulations

Before proceeding with the Newmark-FDTD method, itis worth reviewing the Newmark time-stepping algorithmbriefly [23, 24]. Let us consider the second-order differentialequation of the form

𝑀𝑥 + 𝐶𝑥 + 𝐾𝑥 = 𝑓, (1)where 𝑥 denotes the displacement [24].The Taylor expansionof the velocity (𝑥) about (𝑛 + 1) Δ 𝑡 is

𝑥𝑛+1 = 𝑥𝑛 + Δ 𝑡𝑥𝑛 + Δ2𝑡

2! 𝑥𝑛 + . . . , (2)

where 𝑥𝑛 denotes the value of 𝑥 at 𝑛Δ 𝑡. For sufficientlysmooth functions, the above expansion can be truncated as

𝑥𝑛+1 = 𝑥𝑛 + Δ 𝑡𝑥𝑞 , (3)where 𝑛 ≤ 𝑞 ≤ 𝑛 + 1. Using the linear interpolation, (3) canbe expressed as

𝑥𝑛+1 = 𝑥𝑛 + (1 − 𝛾)�𝑡𝑥𝑛 + 𝛾�𝑡𝑥𝑛+1, (4)where 0 ≤ 𝛾 ≤ 1. Similarly, the Taylor expansion of thedisplacement about (𝑛 + 1)Δ 𝑡 can be written as

𝑥𝑛+1 = 𝑥𝑛 + �𝑡𝑥𝑛 + �2𝑡

2! 𝑥𝑟 , (5)

where 𝑛 ≤ 𝑟 ≤ 𝑛 + 1. Employing the linear interpolation tothe above equation, we have

𝑥𝑛+1 = 𝑥𝑛 + �𝑡𝑥𝑛 + (1 − 2𝛽) �2𝑡

2! 𝑥𝑛 + 2𝛽�

2𝑡

2! 𝑥𝑛+1, (6)

where 0 ≤ 2𝛽 ≤ 1. When the governing equation (1) isexpressed at time step 𝑛 + 1, 𝑛, and 𝑛 − 1, the followingequations can be obtained:

𝑀𝑥𝑛+1 + 𝐶𝑥𝑛+1 + 𝐾𝑥𝑛+1 = 𝑓𝑛+1, (7)𝑀𝑥𝑛 + 𝐶𝑥𝑛 + 𝐾𝑥𝑛 = 𝑓𝑛, (8)

𝑀𝑥𝑛−1 + 𝐶𝑥𝑛−1 + 𝐾𝑥𝑛−1 = 𝑓𝑛−1. (9)Similar procedure to (4) and (6) is applied to the value of 𝑛Δ 𝑡:

𝑥𝑛 = 𝑥𝑛−1 + (1 − 𝛾)�𝑡𝑥𝑛−1 + 𝛾�𝑡𝑥𝑛 , (10)

𝑥𝑛 = 𝑥𝑛−1 + �𝑡𝑥𝑛−1 + (1 − 2𝛽) �2𝑡

2! 𝑥𝑛−1 + 2𝛽�

2𝑡

2! 𝑥𝑛 . (11)

There are three nonderivative terms (𝑥𝑛−1, 𝑥𝑛, 𝑥𝑛+1) andsix derivative terms (𝑥𝑛−1, 𝑥𝑛, 𝑥𝑛+1, 𝑥𝑛−1, 𝑥𝑛 , 𝑥𝑛+1) in sevenequations (see (4) and (6)–(11)). After simple mathematical

manipulation, we can finally obtain the update equationwithout the derivative terms:

[𝐾𝛽�2𝑡 + 𝐶𝛾�𝑡 +𝑀]𝑥𝑛+1+ [𝐾 (0.5 − 2𝛽 + 𝛾)�2𝑡 − 𝐶 (2𝛾 − 1)�𝑡 − 2𝑀]𝑥𝑛+ [𝐾 (0.5 + 𝛽 − 𝛾)�2𝑡 + 𝐶 (𝛾 − 1)�𝑡 +𝑀]𝑥𝑛−1

= [𝛽�2𝑡 ] 𝑓𝑛+1 + [(0.5 − 2𝛽 + 𝛾) Δ2𝑡] 𝑓𝑛+ [(0.5 + 𝛽 − 𝛾)�2𝑡 ] 𝑓𝑛−1.

(12)

Now, let us derive the Newmark-FDTD method for themodified Lorentz dispersion model. The modified Lorentzmodel [15, 16] can be expressed as 𝜀𝑟(𝜔) = 𝜀𝑟,∞ +𝜒(𝜔), where

𝜒 (𝜔) = 𝑎0 + 𝑎1 (𝑗𝜔)𝑏0 + 𝑏1 (𝑗𝜔) + 𝑏2 (𝑗𝜔)2 (13)

and 𝜀𝑟,∞ is a relative permittivity at the infinite frequency.Note that the update equation of magnetic field can beimplemented by using standard central difference scheme(CDS) to Faraday’s law [1]:

∇ × E = −𝜇0 𝜕H𝜕𝑡 . (14)

Ampere’s law can be written as

∇ ×H = 𝜀∞ 𝜕E𝜕𝑡 +𝜕P𝜕𝑡 , (15)

where 𝜀∞ = 𝜀0𝜀𝑟,∞, 𝜀0 is the permittivity in the free space,and P(𝜔) = 𝜀0𝜒(𝜔)E(𝜔) is the constitutive relation in thefrequency domain. Inverse Fourier transform (IFT) is appliedto the constitutive relation:

𝑏0P (𝑡) + 𝑏1 𝜕P (𝑡)𝜕𝑡 + 𝑏2𝜕2P (𝑡)𝜕𝑡2

= 𝑎0𝜀0E (𝑡) + 𝑎1𝜀0 𝜕E (𝑡)𝜕𝑡 = W (𝑡) ,(16)

where a temporary variable W is introduced to apply theNewmark time-stepping algorithm.

Therefore, we have two sets of differential equations:

𝑏0P (𝑡) + 𝑏1 𝜕P (𝑡)𝜕𝑡 + 𝑏2𝜕2P (𝑡)𝜕𝑡2 = W (𝑡) , (17)

𝑎0𝜀0E (𝑡) + 𝑎1𝜀0 𝜕E (𝑡)𝜕𝑡 = W (𝑡) . (18)

By applying the Newmark time-stepping algorithm to theabove equations, the solution in terms of P and E can beobtained as follows:

𝐶𝑎P𝑛+1 + 𝐶𝑏P𝑛 + 𝐶𝑐P𝑛−1 = 𝐶𝑑E𝑛+1 + 𝐶𝑒E𝑛 + 𝐶𝑓E𝑛−1, (19)

International Journal of Antennas and Propagation 3

where

𝐶𝑎= [𝑏0𝛽�2𝑡 + 𝑏1𝛾�𝑡 + 𝑏2] ,𝐶𝑏= [𝑏0 (0.5 − 2𝛽 + 𝛾)�2𝑡 − 𝑏1 (2𝛾 − 1)�𝑡 − 2𝑏2] ,𝐶𝑐= [𝑏0 (0.5 + 𝛽 − 𝛾)�2𝑡 + 𝑏1 (𝛾 − 1)�𝑡 + 𝑏2] ,

𝐶𝑑=𝜀0 [𝑎0𝛽�2𝑡 + 𝑎1𝛾�𝑡] ,𝐶𝑒=𝜀0 [𝑎0 (0.5 − 2𝛽 + 𝛾)�2𝑡 − 𝑎1 (2𝛾 − 1)�𝑡] ,𝐶𝑓=𝜀0 [𝑎0 (0.5 + 𝛽 − 𝛾)�2𝑡 + 𝑎1 (𝛾 − 1)�𝑡] .

(20)

Therefore, the E field can be updated by inserting P𝑛+1 of (19)into the CDS version of Ampere’s law (15):

(𝐶𝑎𝜀∞ + 𝐶𝑑𝐶𝑎�𝑡 ) E𝑛+1=(𝐶𝑎𝜀∞ − 𝐶𝑒𝐶𝑎�𝑡 )E

𝑛 − 𝐶𝑓𝐶𝑎�𝑡E𝑛−1

+ ∇ ×H𝑛+1/2 + 𝐶𝑏 + 𝐶𝑎𝐶𝑎�𝑡 P𝑛 + 𝐶𝑐𝐶𝑎�𝑡P

𝑛−1.(21)

In what follows, we choose 𝛽=0.25 and 𝛾=0.5 the same asin [17, 18] because this Newmark-FDTD can have higherstability and better accuracy compared to the conventionalADE-FDTDmethod [18].

Next, let us derive the conventional ADE-FDTDmethod.Toward this purpose, CDS is applied to (16):

𝑏0P𝑛 + 𝑏1P𝑛+1 − P𝑛−12�𝑡 + 𝑏2

P𝑛+1 − 2P𝑛 + P𝑛−1�2𝑡

= 𝑎0𝜀0E𝑛 + 𝑎1𝜀0 E𝑛+1 − E𝑛−12�𝑡 .

(22)

This equation can also be written in the same form as (19),where the coefficients are expressed as

𝐶𝑎= [0.5𝑏1�𝑡 + 𝑏2] ,𝐶𝑏= [𝑏0�2𝑡 − 2𝑏2] ,𝐶𝑐= [−0.5𝑏1�𝑡 + 𝑏2] ,𝐶𝑑=𝜀0 [0.5𝑎1�𝑡] ,𝐶𝑒=𝜀0 [𝑎0�2𝑡 ] ,

𝐶𝑓=𝜀0 [−0.5𝑎1�𝑡] .

(23)

Now, let us consider the ADE-FDTD formulation based onthe BT.TheBT is basically an approximate of 𝑗𝜔 ≈ (2/�𝑡)((1−𝑍−1)/(1 + 𝑍−1)) [20–22]. When the BT is applied to theconstitutive relation, (16) can be expressed as

𝑏0P𝑛+1 + 2P𝑛 + P𝑛−1

4 + 𝑏1P𝑛+1 − P𝑛−1

2�𝑡

+ 𝑏2P𝑛+1 − 2P𝑛 + P𝑛−1

�2𝑡

= 𝑎0𝜀0E𝑛+1 + 2E𝑛 + E𝑛−1

4 + 𝑎1𝜀0E𝑛+1 − E𝑛−1

2�𝑡 .(24)

The above equation can also be expressed as the form of (19),where the coefficients are

𝐶𝑎= [0.25𝑏0�2𝑡 + 0.5𝑏1�𝑡 + 𝑏2] ,𝐶𝑏= [0.5𝑏0�2𝑡 − 2𝑏2] ,

𝐶𝑐= [0.25𝑏0�2𝑡 − 0.5𝑏1�𝑡 + 𝑏2] ,𝐶𝑑=𝜀0 [0.25𝑎0�2𝑡 + 0.5𝑎1�𝑡] ,

𝐶𝑒=𝜀0 [0.5𝑎0�2𝑡 ] ,𝐶𝑓=𝜀0 [0.25𝑎0�2𝑡 − 0.5𝑎1�𝑡] .

(25)

Please note that the above coefficients are exactly the same as(20) with 𝛽 = 0.25 and 𝛾 = 0.5.

It is worth comparing the numerical stability of theNewmark-FDTD formulation, the conventional ADE-FDTDformulation, and the ADE-FDTD formulation based on theBT. The numerical stability conditions for FDTD formula-tions can be obtained by using von Neumann method [4, 25–27]. Z transform is applied to the wave equation and theconstitutive relation. For the two Z-transformed equations,the determinant must be zero to get nonzero solutions. To bestable, all the roots of the polynomial must be inside or onthe unit circle. Instead of finding the roots of the polynomial,let us transform 𝑍 = (𝑟 + 1)/(𝑟 − 1). By transformingthe Z-domain into the r-domain, roots inside the unit circlein the Z-domain map onto the left half-plane in the r-domain. Next, Routh-Hurwitz (R-H) criterion can be used toderive the numerical stability conditions [4]. To be stable, allcomponents of the first column on the Routh table must beequal to or larger than zero. Following the above procedure,the numerical stability conditions of the conventional ADE-FDTD formulation are as follows:

𝑏0 ≥ 0,𝑏1 ≥ 0,

Q�2𝑡 + 𝑏1𝜀𝑟,∞]2 (4𝑏2 − 𝑏0�2𝑡 ) ≥ 0,Q (𝑎1 + 𝑏1𝜀𝑟,∞)�2𝑡 + 4𝑎1𝑏1𝑏2𝜀𝑟,∞]2 ≥ 0,(4𝑏2 − 𝑏0�2𝑡 ) (1 − ]2) 𝜀𝑟,∞ − 𝑎0�2𝑡 ≥ 0,

(26)

where Q = 𝑎0𝑏1 − 𝑎1𝑏0, ]2 = (𝑐∞�𝑡)2∑𝛼=𝑥,𝑦,𝑧(sin2(𝑘𝛼�𝛼/2)/�2𝛼), 𝑐∞ = 1/√𝜇0𝜀∞, and 𝑘𝛼 are the numerical wavenumbersin the 𝛼 direction.Thenumerical stability conditions of ADE-FDTD formulation based on the BT are the same as those ofthe Newmark-FDTD formulation, since both methods have


the same formulation. Their numerical stability conditionsare

𝑏0 ≥ 0,𝑏1 ≥ 0,

Q�2𝑡 + 4𝑏1𝑏2𝜀𝑟,∞]2 ≥ 0,Q [𝑎1 + 𝑏1𝜀𝑟,∞ (1 − ]2)]�2𝑡 + 4𝑎1𝑏1𝑏2𝜀𝑟,∞]2 ≥ 0,

𝑏2 (1 − ]2) ≥ 0.

(27)

Next, the numerical permittivity can be derived insertingthe numerical solution P0𝑒𝑗𝜔𝑛�𝑡 and E0𝑒𝑗𝜔𝑛�𝑡 into the con-stitutive relation [24, 28]. The numerical permittivity of theADE-FDTD formulation can be obtained as follows:

𝜀𝑟 (𝜔) = 𝜀𝑟,∞ + 𝑎0 + 𝑎1 (𝑗�̃�)𝑏0 + 𝑏1 (𝑗�̃�) + 𝑏2 (𝑗�̃�)2, (28)

where �̃� = 2tan(𝜔�𝑡/2)/�𝑡, 𝑎0 = 𝑎0/cos2(𝜔�𝑡/2), and𝑏0 = 𝑏0/cos2(𝜔�𝑡/2).The numerical permittivity of the ADE-FDTD formulation based on the BT is the same as that of theNewmark-FDTD formulation:

𝜀𝑟 (𝜔) = 𝜀𝑟,∞ + 𝑎0 + 𝑎1 (𝑗�̃�)𝑏0 + 𝑏1 (𝑗�̃�) + 𝑏2 (𝑗�̃�)2. (29)

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

𝜀𝐷𝑒𝑏𝑦𝑒𝑟 (𝜔) = 𝜀𝑟,∞ + �𝜀1 + 𝑗𝜔𝜏 ,

𝜀𝐷𝑟𝑢𝑑𝑒𝑟 (𝜔) = 𝜀𝑟,∞ + 𝜔20

𝑗𝜔𝛾 + (𝑗𝜔)2 ,

𝜀𝐿𝑜𝑟𝑒𝑛𝑡𝑧𝑟 (𝜔) = 𝜀𝑟,∞ + �𝜀 ⋅ 𝜔20

𝜔20 + 2𝑗𝜔𝛿 + (𝑗𝜔)2,

𝜀𝑄𝐶𝑅𝐹𝑟 (𝜔) = 𝐴0 + 𝐴1 (𝑗𝜔) + 𝐴2 (𝑗𝜔)2

𝐵0 + 𝐵1 (𝑗𝜔) + 𝐵2 (𝑗𝜔)2.

(30)

ForDebye,Drude, and Lorentz dispersionmodels, 1/𝜒(𝜔) canbe written as 𝑀(j𝜔)2 + 𝐶(j𝜔) + 𝐾, where M, C, and 𝐾 areconstants for each model. By applying the Newmark-FDTD,one can obtain

𝑆P𝑛+1 = 𝑤1P𝑛 + 𝑤2P𝑛−1 + 𝑢0E𝑛+1 + 𝑢1E𝑛 + 𝑢2E𝑛−1, (31)where

𝑆 = 𝑀 + 0.5𝐶�𝑡 + 0.25𝐾�2𝑡 ,𝑤1 = 2𝑀 − 0.5𝐾�2𝑡 ,𝑤2 = −𝑀 + 0.5𝐶�𝑡 + 0.25𝐾�2𝑡 ,𝑢0 = 0.25𝜀0�2𝑡 ,𝑢1 = 0.5𝜀0�2𝑡 ,𝑢2 = 0.25𝜀0�2𝑡 .

(32)

300 1000 1500 2000 2500 3000Frequency [MHz]

Newmark-FDTDADE-FDTD with BTADE-FDTD

Relat

ive E

rror

of Ｌ(

)

10−3

10−4

10−5

10−6

Figure 1: Relative error of the numerical permittivity.

Note that this update equation is the same as the ADE-FDTD formulation based on the BT. For QCRF dispersionmodel, the constitutive relation between electric flux densityD and electric field E is simpler. The update equation forthe Newmark-FDTDmethod is the same as the ADE-FDTDformulation based on the BT and it can be expressed as

[0.25𝐵0�2𝑡 + 0.5𝐵1�𝑡 + 𝐵2]D𝑛+1+ [0.5𝐵0�2𝑡 − 2𝐵2]D𝑛+ [0.25𝐵0�2𝑡 − 0.5𝐵1�𝑡 + 𝐵2]D𝑛−1

= [0.25𝐴0𝜀0�2𝑡 + 0.5𝐴1𝜀0�𝑡 + 𝐴2𝜀0]E𝑛+1+ [0.5𝐴0𝜀0�2𝑡 − 2𝐴2𝜀0]E𝑛+ [0.25𝐴0𝜀0�2𝑡 − 0.5𝐴1𝜀0�𝑡 + 𝐴2𝜀0]E𝑛−1.

(33)

All things considered, the Newmark-FDTD formulation with𝛽 = 0.25 and 𝛾 = 0.5 is the same as the ADE-FDTDformulation based on the BT.

3. Results and Discussion

In this section, numerical examples are used to validateour study. First, we consider human blood from 300MHzto 3GHz. The modified Lorentz parameters are extractedby using the particle swarm optimization [29, 30] for thecomplex relative permittivity data [31]. They are 𝑎0 = 6.9379 ⋅1021, 𝑎1 = 1.5057 ⋅ 1012, 𝑏0 = 6.1637 ⋅ 1018, 𝑏1 = 4.5425 ⋅1010, 𝑏2 = 1, and 𝜀𝑟,∞ = 31.1662. The space step size �𝑧 =1𝑚𝑚 and time step size �𝑡 = 𝐶𝑛�𝑧/𝑐∞, where 𝐶𝑛 is aclassic Courant number [1] and 𝐶𝑛 = 0.99 is used in thiswork. The relative errors of the numerical permittivity of thethree FDTD formulations are shown inFigure 1.The accuracyof the ADE-FDTD with BT is the same as the Newmark-FDTD and they have higher accuracy than the ADE-FDTD.This result can be explained from the fact that the numerical


0 5 10 15 20 25 30 35 40Time [ns]

−1

0

1

(a)

0 5 10 15 20 25 30 35 40Time [ns]

−1

0

1

(b)

Figure 2: 1D FDTDsimulations for different sets ofmodified Lorentz parameters. (a) All FDTD formulations are stable. (b)Only ADE-FDTDis unstable. Legends are the same as Figure 1.

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

(a)

0.5

1

1.5

30

210

60

240

90

270

120

300

150

330

180 0

(b)

Figure 3: Root locus of the stability polynomial in the Z-domain for Figure 1. Legends: Black circle: Newmark-FDTD/ADE-FDTDwith BT;blue-gray square: ADE-FDTD.

coefficients 𝑎0 and 𝑏0 are involved in the ADE-FDTDmethodin addition to the numerical angular frequency �̃�, differentlyfromboth theNewmark-FDTD and the ADE-FDTDwith BT(see (28)-(29)).

Next, actual FDTD simulations are performed. Asinewave with the frequency of 300MHz is excited in1D homogenous modified Lorentz medium. The 10-layerperfectly matched layer (PML) [1] is used to truncate thecomputational domain of 10,000 FDTD cells. First, weconsider the aforementioned modified Lorentz parametersthat satisfy the numerical stability conditions of (26) and(27) for all possible values of ]. As expected, all FDTDsimulations are stable for 10,000 time steps in Figure 2(a). Asshown in Figure 3(a), all roots of the stability polynomial inthe Z-domain are inside or on the unit circle for all cases.The memory requirement and the central processing unit(CPU) time of all three FDTD formulations are the sameas 6.6 MB and 39.6 s, respectively, because the number ofauxiliary variables and the number of arithmetic operationsare completely the same for all three formulations. Now, only𝑏2 is changed to 0.8, where the stability conditions of (26)

are not satisfied. For the ADE-FDTD method, instability canbe found in Figure 2(b). It can be observed that some rootsof stability polynomial in the Z-domain are outside the unitcircle for the ADE-FDTD method but all roots of stabilitypolynomial in the Z-domain are inside or on the unit circlefor the others, as shown in Figure 3(b). As can be seen inthese FDTD simulations, both the Newmark-FDTD andthe ADE-FDTD with BT have the same accuracy and theyare better than the ADE-FDTD in terms of the numericalstability.

As a final example, a 3DAg nanosphere with the radius of40 nm is considered. In this case, the parameters in [32] areused, 𝑎0 = 1.9136 ⋅ 1032, 𝑎1 = 0, 𝑏0 = 0, 𝑏1 = 2.7362 ⋅ 1013, 𝑏2 =1, and 𝜀𝑟,∞ = 3.7. The x-polarized Gaussian-modulatedsinewave planewave is excited and the computational domainis terminated by a 10-layer complex frequency shifted- (CFS-)PML [33, 34]. The FDTD computational domain comprised260x260x260 cells and the total number of time marchingsteps is 40,000. To compare FDTD results with Mie theory[35], we calculate the magnitude of E𝑥 at the centre of thenanosphere normalized by the magnitude of the incident


300 400 500 600 700 800Frequency [THz]

0

0.5

1

1.5

2

2.5

Mie theoryNewmark-FDTDADE-FDTD with BTADE-FDTD

Nor

mal

ized

％Ｒ

(a)

300 400 500 600 700 800Frequency [THz]

Rela

tive E

rror

Newmark-FDTDADE-FDTD with BTADE-FDTD

10−1

10−2

10−3

10−4

10−5

10−6

(b)

Figure 4: 3D FDTD simulations for an Ag nanosphere with the radius of 40 nm. (a) Spectral response. (b) Relative error.

electric field. As shown in Figure 4, the Newmark-FDTD isthe same as the ADE-FDTD with BT and they have higheraccuracy than the ADE-FDTD. For all 3DFDTD simulations,the memory requirement is 4.5GB and the CPU time is 6hours.

4. Conclusions

In this work, the Newmark-FDTD method is applied tomodified Lorentz dispersion model which can systematicallyunify various existing dispersion models. It is found thatthe Newmark-FDTD is equivalent to the ADE-FDTD withthe BT in terms of update formulation, numerical stability,and numerical accuracy. In addition, it is figured out thatboth methods can yield better accuracy and higher stabilityagainst the conventional ADE-FDTD method. Moreover,Debye, Drude, Lorentz, and QCRF dispersion models areextended to the Newmark-FDTDmethod and its equivalenceto the BT-based ADE-FDTD counterpart is also observed.Numerical permittivity and the 1D blood example are usedto illustrate that both the Newmark-FDTD method andthe ADE-FDTD method based on the BT are better thanthe conventional ADE-FDTD method in terms of numer-ical accuracy and numerical stability. A further numericalexample involving 3D plasmonic nanosphere is presented todemonstrate that both the Newmark-FDTD simulation andthe BT-based ADE-FDTD simulation are in good agreementwith the Mie solution and they are superior to the conven-tional ADE-FDTD method.

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported in part by Basic ScienceResearch Program through the National Research Founda-tion of Korea (NRF) funded by the Ministry of Education(no. 2017R1D1A1B03034537) and in part by Ministry ofCulture, Sports and Tourism (MCST) and Korea CreativeContent Agency (KOCCA) in the Convergence TourismService Research and Business Development Program (no.SF0718106).

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Newmark-FDTD Formulation for Modified Lorentz Dispersive ... › journals › ijap › 2019 › 4173017.pdf · Before proceeding with the Newmark-FDTD method, it is worth reviewing