High-order FDTD methods for transverse electromagneticsystems in dispersive inhomogeneous media

Shan ZhaoDepartment of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487, USA (szhao@bama.ua.edu)

Received June 8, 2011; accepted July 18, 2011;posted July 22, 2011 (Doc. ID 148966); published August 15, 2011

This Letter introduces a novel finite-difference time-domain (FDTD) formulation for solving transverse electromag-netic systems in dispersive media. Based on the auxiliary differential equation approach, the Debye dispersionmod-el is coupled with Maxwell’s equations to derive a supplementary ordinary differential equation for describing theregularity changes in electromagnetic fields at the dispersive interface. The resulting time-dependent jump condi-tions are rigorously enforced in the FDTD discretization by means of the matched interface and boundary scheme.High-order convergences are numerically achieved for the first time in the literature in the FDTD simulations ofdispersive inhomogeneous media. © 2011 Optical Society of AmericaOCIS codes: 000.4430, 160.2710, 260.2030.

Recently, much attention has been devoted to the devel-opment of efficient time-domain methods for solvingMaxwell’s equations in dispersive media, including theFDTD methods [1,2], finite element time-domain meth-ods [3], discontinuous Galerkin time-domain methods[4–6], and so on. The success of these methods lies inthe accurate implementation of the frequency-dependentmaterial constitutive relations in the time domain. How-ever, great challenges exist in all dispersive simulationson how to restore the accuracy reduction at the disper-sive interface, where the wave solution loses its regular-ity in a complicated way.To account for the nonsmoothness of wave solution at

the dispersive interface, several dispersive FDTD algo-rithms [7–9] have been constructed by averaging the per-mittivity in the vicinity of the interface so that a smootheffective permittivity is produced for wave simulation.Consequently, the numerical convergence can be accel-erated. On the other hand, it is well known in the com-putational electromagnetics literature [10] that a moreaccurate way to treat the material interface is to incorpo-rate the physical jump conditions into the numerical dis-cretization in a proper manner. The resulting interfaceschemes can deliver higher-order convergences irrespec-tive of the presence of material interfaces [10–12]. How-ever, to the author’s knowledge, no rigorous interfacescheme has ever been formulated for treating the disper-sive interface. This is because, as to be shown in this Let-ter, the jump conditions at the dispersive interface couldbe time dependent—a difficulty that has not been en-countered in the previous interface studies [10–12].It is of great interest to develop higher-order interface

methods for the dispersive interface. To this end, wepresent in this Letter a novel FDTD formulation for one-dimensional (1D) Maxwell’s equations. Consider anx-directed, z-polarized transverse electromagnetic sys-tem in dispersive inhomogeneous media

∂Dz

∂t¼ ∂Hy

∂x;

∂Hy

∂t¼ 1

μ∂Ez

∂x; ð1Þ

where Ez and Hy are, respectively, electric and magneticfield components, and Dz is the electric displacementcomponent. For dispersive media, the constitutive rela-

tion is better prescribed in the frequency domain, i.e.,Dz ¼ ϵðωÞEz, in which the time-harmonic componentsare obtained via the Fourier transforms of the time-varying components.

For simplicity, we focus ourselves on the single-orderDebye dispersion model

ϵðωÞ ¼ ϵ0�ϵ∞ þ ϵs − ϵ∞

1þ jωγ

�; ð2Þ

where ϵ0, ϵs, and ϵ∞ are, respectively, permittivities offree space, at static frequency, and at high frequency lim-it. Here ω is the angular frequency and γ is the relaxationtime constant. The auxiliary differential equation ap-proach [1] is used to represent the constitutive Eq. (2)in time domain

γ ∂Dz

∂tþ Dz ¼ ϵ0ϵ∞γ

∂Ez

∂tþ ϵ0ϵsEz: ð3Þ

Equations (1) and (3) form a closed system. The classicaldispersive FDTDmethod [1] can be formulated by discre-tizing such a system.

The proposed FDTD formulation is driven by theconsideration of the dispersive interface. Consider a1D domain x ∈ ½a; b� with μ ¼ μ0 throughout. Assumean interface location at x ¼ d and the left subdomain½a; dÞ being vacuum with ϵ ¼ ϵ0, while the right subdo-main ½d; b� is a dispersive medium. Define function limitsand jump of function uðxÞ at x ¼ d to be uþ≔ limx→dþ u,u−≔ limx→d− u, and ½u�≔uþ − u−. To unify the notation,we assume Eq. (2) in vacuum as well with ϵ−s ¼ϵ−∞ ¼ 1. Then γ− is a free parameter so that we canassume γ ¼ γ− ¼ γþ being a constant throughout ½a; b�.

We have the ordinary jump conditions [11] for Ez

½Ez� ¼ 0;

�∂Ez

∂x

�¼ 0; ð4Þ

which can be simply handled by the previous interfaceschemes [10–12]. For Hy, we have continuity condition½Hy� ¼ 0. However, the first-order jump condition forHy is physically unknown. We overcome this difficultyby solving a supplementary ordinary differential equation

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0146-9592/11/163245-03$15.00/0 © 2011 Optical Society of America

(ODE). Assume ½Dz� ¼ ψðtÞ for some unknown functionψðtÞ. We have ½∂Hy=∂x� ¼ ½∂Dz=∂t� ¼ ψ 0ðtÞ. By takingjumps for (3), we obtain an ODE

γψ 0ðtÞ þ ψðtÞ ¼ ϵ0γ½ϵ∞ _Ez� þ ϵ0½ϵsEz�; ð5Þwhere _Ez ¼ ∂Ez=∂t. To facilitate the solution of the ODE(5), we propose to utilize _Ez instead of Dz to form aclosed Maxwell system:

∂ _Ez

∂t¼ −

ϵsϵ∞γ

_Ez þ1

ϵ0μ0ϵ∞∂2Ez

∂x2þ 1ϵ0ϵ∞γ

∂Hy

∂x; ð6Þ

∂Ez

∂t¼ _Ez;

∂Hy

∂t¼ 1

μ0∂Ez

∂x: ð7Þ

In the proposed higher-order FDTD methods, the ODE(5) and Maxwell’s Eqs. (6) and (7) will be time integratedvia the classical fourth-order Runge–Kutta method [11].We note that other standard explicit or implicit timestepping methods may also be used for time integration.In the present study, at each time step tk ¼ kΔt, theright-hand side of (5) can be calculated as gðtkÞ≔ϵ0γðϵþ∞ − ϵ−∞Þ _E−

z ðtkÞ þ ϵ0ðϵþs − ϵ−s ÞE−zðtkÞ. Here E−

zðtkÞ and_E−z ðtkÞ are approximated via one-sided extrapolation

based on several function values of EzðtkÞ and _EzðtkÞfrom the left of the interface. With gðtkÞ, one can computeψðtkÞ and ψ 0ðtkÞ at each time step tk.With ψ 0ðtÞ being accurately estimated, a matched inter-

face and boundary (MIB) method [11,12] is utilized toimpose jump conditions at the dispersive interface, i.e.,

½Hy� ¼ 0;

�∂Hy

∂x

�¼ ψ 0ðtÞ; ð8Þ

for Hy and (4) for Ez. In particular, a uniform staggeredgrid system is used for Ez and Hy on the domain ½a; b�.The standard ð2MÞth order central FDTD approximation[11] is employed for spatial discretization, with M beingthe half stencil bandwidth. In the MIB algorithm, whenthe FDTD stencil refers to function values from the otherside of the interface, fictitious values instead of originalfunction values will be supplied. In general, to support að2MÞth order FDTD approximation, M fictitious nodesfrom the positive side and M ones from the negative sideof the interface are required for Ez. So does Hy.Since the jump conditions (4) and (8) are decoupled,

we consider only the MIB scheme for (8) here. In the firststep of the MIB scheme [11,12], H−

y and ∂H−y=∂x are dis-

cretized by using L original Hy values from the negativeside and one fictitious value of Hy from the positive side.With the similar discretization for Hþ

y and ∂Hþy =∂x, (8)

becomes two algebraic equations. Solving them, one at-tains two fictitious values for Hy. The MIB scheme itera-tively enforces the jump condition (8) [11,12] so that 2Mfictitious values can be obtained afterM iterations. Thesefictitious values of Hy will depend on 2L function valuesof Hy and the nonhomogeneous value ψ 0ðtÞ. The jumpcondition (4) for Ez can be similarly treated.We validate the proposed MIB time-domain (MIBTD)

methods by considering an air–water interface. We

choose a domain with a ¼ 0mm, b ¼ 30mm, andd ¼ 7:5πmm. Similar to [2], we assume ϵs ¼ 81,ϵ∞ ¼ 1:8, and γ ¼ 9:4 ps for the water. An incident pulseof the form EðtÞ ¼ expð−ðxþ 15 − c0tÞ2=ð2τ2ÞÞ is em-ployed as the boundary condition at x ¼ a. Here τ ¼2:5mm and c0 is the speed of light in air. The boundarycondition at x ¼ b is assumed to be perfectly conductingfor simplicity. A high-order boundary closure schemepresented in [13] is employed to treat such a boundary.

We first test the second-order MIBTD or MIBTD2 forcalculating the reflection coefficient of the air–water in-terface. For the MIBTD2, we choose M ¼ 1 and L ¼ 2.The number of nodes for Ez is chosen as N ¼ 801 andthe time increment isΔt ¼ 0:025 ps for Nt ¼ 10000 steps.Numerical values of Ez are tracked at one location nearthe left boundary and another to the left of the interface.Through a proper truncation, this yields two time his-tories for incident and reflected pulse, respectively.The reflection coefficient can then be calculated as theratio of the Fourier transform of the reflected wave overthat of the incident wave. The reflection coefficient ofthe MIBTD2 is plotted against the analytical one [2] inFig. 1. A good agreement over various frequencies isclearly seen.

To fully demonstrate the potential of higher-orderMIBTD methods, we consider next a rigorous conver-gence analysis. The parameters for higher-order MIBTDare chosen as: ðM;LÞ ¼ ð2; 4Þ and (3,6), respectively, forthe MIBTD4 and MIBTD6. We select the stop time to bet ¼ 140 ps, which is short enough such that both reflectedand transmitted pulses have not reached the boundaries,see Fig. 2. We choose the number of time steps to beNt ¼10000 in all computations. The corresponding Δt ¼0:014 ps is small enough so that the approximation errorsare guaranteed to be mainly produced by the spatial dis-cretizations. To benchmark the numerical results, wegenerate a reference solution by employing the MIBTD6with a very dense grid N ¼ 12801. Convergence analysiscan then be conducted by considering mesh refinementsusing nodes that are also covered in the reference solu-tion. We note that since the Hy nodes are staggered tothe Ez nodes, the present convergence analysis is notapplicable to Hy.

The maximum errors in Ez of the MIBTD methods aredepicted in Fig. 3. For a comparison, the results of theclassical dispersive FDTD [1] are also shown. In all cases,the numerical errors based on different mesh size N areplotted as dashed lines. A linear least-squares fitting [12]

0 10 20 30 40 50 60 700.5

0.6

0.7

0.8

0.9

1

Frequency (GHz)

Mag

nitu

de o

f Ref

lect

ion

Coe

f.

Analytical solutionMIBTD2

Fig. 1. (Color online) Reflection coefficient at an air–waterinterface.

3246 OPTICS LETTERS / Vol. 36, No. 16 / August 15, 2011

is then conducted in the log-log scale. The fitted conver-gence lines are shown as solid lines in Fig. 3. Moreover,the fitted slope essentially represents the numerical con-vergence rate r of the scheme. The conventional FDTDalgorithm attains r ¼ 1:18, because it degrades to the firstorder of accuracy due to the loss of solution regularitiesat the dispersive interface [11]. In particular, it is clearfrom Fig. 2 that Ez is C1 continuous, but Hy is only C0

continuous. Higher-order convergence can be achievedonly when special interface treatments, such as the pro-posed MIB schemes, are incorporated in the discretiza-tion. In the present study, the numerical order r of theMIBTD2, MIBTD4, and MIBTD6 methods is found tobe 2.12, 4.32, and 5.74, respectively, which confirmsthe theoretical orders of two, four, and six.We finally present an efficiency study. Based on the

samemesh sizeN andNt, the MIBTDmethods need moreexecution time than the FDTD scheme. However, the

MIBTD methods can be more efficient, if the same accu-racy level is required to be achieved by the FDTDscheme. For example, if one specifies the maximum errorbeing at most 5:0 × 10−5, the MIBTD2, MIBTD4, andMIBTD6 can accomplish this goal by choosing N to be1601, 801, and 401, respectively. By using a fixedNt ¼ 10000, the execution time for MIBTD2, MIBTD4,and MIBTD6, is 3.99, 2.21, and 1:25 s, respectively. Never-theless, a very large N ¼ 102401 has to be used in theFDTD for the desired accuracy. Moreover, due to timestability constraint, a very large Nt ¼ 160000 has to beused for time integration. Such an FDTD simulation costs243:71 s. Thus, the MIBTD6 scheme is about 195 timesmore efficient that the FDTD method.

In summary, we have proposed a novel interface for-mulation for treating Debye dispersive interface in 1D.The time-dependent jump conditions are coupled withMaxwell’s equations via a supplementary ODE. Higher-order convergences are numerically achieved for the firsttime in the literature in the FDTD simulation of disper-sive inhomogeneous media. The dispersive interface for-mulation for higher dimensions is currently under ourinvestigation.

This work is supported in part by the National ScienceFoundation (NSF) grantDMS-1016579 and by aUniversityof Alabama Research Grants Committee Award.

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1690 (2009).

0 15 interface 30−0.8

−0.4

00.2

x (mm)

Ez

0 15 interface 30

−3

−2

−1

0x 10

−3

x (mm)

Hy

Fig. 2. (Color online) MIBTD6 solution with N ¼ 401 att ¼ 140 ps. Top, Ez; bottom, Hy.

101 201 401 801 1601

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

N

Max

Err

or

FDTD, r=1.18MIBTD2, r=2.12MIBTD4, r=4.32MIBTD6, r=5.74

Fig. 3. (Color online) Numerical convergence tests of Ez.

August 15, 2011 / Vol. 36, No. 16 / OPTICS LETTERS 3247