J Sci Comput (2016) 68:438–463DOI 10.1007/s10915-015-0144-y

Developing a Time-Domain Finite Element Method forthe Lorentz Metamaterial Model and Applications

Wei Yang1 · Yunqing Huang1 · Jichun Li2

Received: 20 October 2015 / Revised: 17 November 2015 / Accepted: 19 November 2015 /Published online: 28 November 2015© Springer Science+Business Media New York 2015

Abstract In this paper, we propose a new time-domain finite element method for solvingthe time dependent Maxwell’s equations coupled with the Lorentz metamaterial model. TheLorentz metamaterial Maxwell’s equations are much more complicated than the standardMaxwell’s equations in free space. Our fully discrete scheme uses edge elements to approxi-mate the unknowns in space, and uses the leap-frog scheme in time discretization. Numericalstability and the optimal error estimate in the L2 norm are proved for our proposed scheme.Extensive numerical results are presented to confirm the theoretical analysis and applicationsof our scheme to model many interesting phenomena happened when wave propagates in theLorentz metamaterials. Examples include the convergence effect happened in the concavelenses formed by the negative refraction index metamatrials, and total reflection and totaltransmission observed in the zero index metamaterials.

Keywords Maxwell’s equations · Finite element method · Lorentz metamaterial model ·Edge elements · Total transmission · Total reflection

Mathematics Subject Classification 78M10 · 65N30 · 65F10 · 78-08

This work was supported by NSFC Projects 11401506 and 11271310, NSFC Key Project 91430213, HunanEducation Department Project (15B236), and NSF Grant DMS-1416742.

B Jichun Lijichun.li@unlv.edu

Wei Yangyangweixtu@126.com

Yunqing Huanghuangyq@xtu.edu.cn

1 Hunan Key Laboratory for Computation and Simulation in Science and Engineering, XiangtanUniversity, Xiangtan, China

2 Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154-4020,USA

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J Sci Comput (2016) 68:438–463 439

1 Introduction

Since 2000, there has been a growing interest in studying metamaterials and its potentialapplications in many areas such as construction of perfect lens, design of cloaking devices,and sub-wavelength imaging. Numerical simulation plays a very important role in meta-material design and its exotic properties in many devices. Currently most simulations arecarried out by commercial packages such as COMSOL and the simple finite difference timedomain (FDTD) method, which have their limitations. For example, COMSOL is excellentin modeling Maxwell’s equations in the frequency domain, but is inefficient in solving time-dependent Maxwell’s equations in three-dimensional space. The FDTD method is simpleand easy to implement, but it lacks flexibility in dealing with complex geometrical domains.Readers can find more details on metamaterials and its numerical simulations in some recentmonographs (e.g., [11,18,27,30] and references therein).

Metamaterials exhibit interesting properties (such as negative reflection index, backwardwave propagation, reversed Doppler effect, reversed Cherenkov radiation) when both thepermittivity and permeability become negative in some common frequencies. One of themost popular models used to describe metamaterials is the so-called Lorentz model, whosepermittivity and permeability are described by:

ε(ω) = ε0

(1 + ω2

pe

jγeω + ω2e0 − ω2

), μ(ω) = μ0

(1 + ω2

pm

jγmω + ω2m0 − ω2

),

where ωpe and ωpm are respectively the electric and magnetic plasma frequencies, γe and γmare respectively the electric and magnetic damping frequencies,ωe0 andωm0 are respectivelythe electric and magnetic resonance frequencies, j = √−1 is the imaginary unit, and ω is ageneral wave frequency. Assuming a time-harmonic variation of exp( jωt), and incorporatingthe Lorentz model into the general Maxwell’s equations, we can obtain the time domainMaxwell’s equations in Lorentz metamaterials (details see [25]):

ε0∂E∂t

= ∇ × H − J, (1.1)

μ0∂H∂t

= −∇ × E − K , (1.2)

∂ J∂t

+ γe J + ω2e0

∫ t

0Jds = ε0ω

2peE, (1.3)

∂K∂t

+ γmK + ω2m0

∫ t

0Kds = μ0ω

2pmH . (1.4)

Tomake the problemwell-posed, we assume that (1.1)–(1.4) satisfy the perfect conducting(PEC) boundary condition

n × E = 0, (1.5)

where n is the unit outward normal to the boundary ∂� of �, which is assumed to be abounded Lipschitz domain of Rd , d = 2, 3.

Over the last three decades, many excellent results have been published on developmentand analysis of finite element methods (FEMs) for solving the Maxwell’s equations in freespace (e.g., papers [1,3–5,7–10,15,16,19–21,32,33,35,40], books [12,27,31] and referencestherein). Analysis and application of FEMs for Maxwell’s equations in general dispersivemedia (e.g., [24,29,36,37]) and in metamaterials [6,13,28,39] have been active in recentyears. Though there are many engineering papers dealing with the Lorentz metamaterials

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440 J Sci Comput (2016) 68:438–463

by the FDTD methods, to our best knowledge, there exist only a few publications on finite-element time-domain (FETD) methods devoted to the Lorentz model. In [2], Banks et alcarried out the dispersion analysis for Debye and Lorentz media. The well-posedness of(1.1)–(1.5) was first investigated by Li [25] and two finite element methods were proposedfor solving this model written in different forms as well.

The main goal of this paper is to propose a new leap-frog finite element method forsolving the Lorentz metamaterial Maxwell’s equations, and apply it to simulate many inter-esting phenomena happened when wave propagates in the Lorentz metamaterials. Examplesinclude wave propagation in convex and concave lenses made with metamaterials, total trans-missions and total reflections in zero index metamaterials, and unidirectional transmissionphenomenon.

The rest of the paper is organized as follows. In Sect. 2, we develop a leap-frog FETDscheme for the Lorentz metamaterial model. In Sect. 3, we prove the numerical stability andoptimal error estimate for our scheme. In Sect. 4, we first present a numerical result justifyingthe optimal convergence rate, then we show many interesting simulations obtained with theLorentz metamaterial model. Finally, we conclude our paper in Sect. 5.

2 The FETD Scheme

Without loss of generality, here we assume that� is a bounded Lipschitz domain ofR2. Notethat the scheme presented below and its theoretical analysis are all the same (except for thefinite element spaces) for both 2D and 3D problems. For 2D models, readers just need to becareful about the different usages of the curl operators.

To design our time-domain finite element method, we partition the physical domain �

by a family of regular meshes Th with maximum mesh size h. To easily accommodate thesimulations in a complicated domain�, we use a hybrid mesh with mixed types of elements:triangles in both the metamaterial region and the free space region; rectangles in the perfectlymatched layer (PML) region.

Due to the low regularity of the solution of our problem, which often involves differentmaterials in difference regions, here we just consider the lowest-order Raviart–Thomas–Nédélec finite element spacesUh and Vh given as follows: for a rectangular element e ∈ Th ,we choose

Uh = {ψh ∈ L2(�) : ψh |e ∈ Q0,0, ∀e ∈ Th

},

V h = {φh ∈ H(curl;�) : φh |e ∈ Q0,1 × Q1,0, ∀e ∈ Th

},

where Qi, j denotes the space of polynomials whose degrees are less than or equal to i andj in variables x and y, respectively; while for a triangular element, we choose

Uh = {ψh ∈ L2(�) : ψh |e is a constant, ∀e ∈ Th

},

V h = {φh ∈ H(curl;�) : φh |e ∈ span{λi∇λ j − λ j∇λi }, i, j = 1, 2, 3, ∀e ∈ Th

},

where λi denotes the standard linear basis function at vertex i of element e. To impose thePEC boundary condition (1.5), we introduce the space

V 0h = {φh ∈ V h, n × φh = 0, on ∂�}.

For error analysis, we need to use the standard L2 projection Phv ∈ Uh :

(Phv − v, φh) = 0, ∀ φh ∈ Uh .

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J Sci Comput (2016) 68:438–463 441

It is known that (e.g., [26]):

‖v − Phv‖0 ≤ Chα‖v‖α, ∀ v ∈ Hα(�), (2.1)

where ‖ · ‖0 denotes the standard L2 norm, and ‖ · ‖α denotes the standard norm in theSobolev space Hα(�). We also need to use the so-called Nédélec interpolant uI ∈ V h forany u ∈ Hα(curl;�), 1

2 < α ≤ 1. It is known that the interpolant uI is well defined, andsatisfies the interpolation estimate [31]:

‖u− uI ‖0 + ‖∇ × (u− uI )‖0 ≤ Chα‖u‖α,curl , ∀ u ∈ Hα(curl;�),1

2< α ≤ 1, (2.2)

where the Sobolev space Hα(curl;�) is defined as [31]: for d = 2, 3,

Hα(curl;�) ={v ∈ (Hα(�))d : curlv ∈ (Hα(�))d

},

and is equipped with the norm

‖v‖α,curl = (‖v‖2α + ‖curlv‖2α)1/2

.

To define a fully-discrete scheme, we divide the time interval I = [0, T ] into N uniformsubintervals Ik = [tk−1, tk] by points tk = kτ, k = 0, . . . , N , where τ = T

N .Now we can construct a leap-frog type finite element scheme for solving the modeling

Eqs. (1.1)–(1.4): Given initial approximations E0h, K

0h, H

12h , J

12h , for any k ≥ 0, find Ek+1

h ∈V 0

h, Kk+1h ∈ Uh, H

k+ 32

h ∈ Uh, Jk+ 3

2h ∈ V h such that

ε0

(Ek+1h − Ek

h

τ, φh

)=

(H

k+ 12

h ,∇ × φh

)−

(Jk+ 1

2h , φh

), ∀ φh ∈ V 0

h, (2.3)

μ0

⎛⎝ H

k+ 32

h − Hk+ 1

2h

τ, ϕh

⎞⎠ = −

(∇ × Ek+1

h , ϕh

)−

(K k+1

h , ϕh

), ∀ ϕh ∈ Uh,

(2.4)

Jk+ 3

2h − J

k+ 12

h

τ+ γe

2

(Jk+ 3

2h + J

k+ 12

h

)+ ω2

e0Pk+1h = ε0ω

2peE

k+1h , (2.5)

K k+1h − K k

h

τ+ γm

2

(K k+1

h + K kh

)+ ω2

m0Qk+ 1

2h = μ0ω

2pmH

k+ 12

h , (2.6)

where Pk+1h and Q

k+ 12

h are updated by the following formulas:

P0h = 0, Pk+1

h = Pkh + τ J

k+ 12

h , (2.7)

Q− 1

2h = 0, Q

k+ 12

h = Qk− 1

2h + τ K k

h . (2.8)

This explicit scheme is very easy to implement. At each time step k, we perform as follows:

1. Solve (2.3) for Ek+1h ; update Pk+1

h with (2.7); update Qk+ 3

2h with (2.8). Note that these

three steps can be done in parallel.

2. Solve (2.5) for Jk+ 3

2h ; solve (2.6) for K k+1

h . Note that these two steps can be done inparallel.

3. Solve (2.4) for Hk+ 3

2h .

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442 J Sci Comput (2016) 68:438–463

4. Then go back to step 1 and repeat the above process for the next time step.

To make the implementation more efficient, we pre-calculate and store the inverses ofthose two mass matrices of (2.3) and (2.4).

3 Analysis of the FETD Scheme

In this section, we carry out the stability and error estimate analysis for our explicit FETDscheme (2.3)–(2.8).

Lemma 3.1 For any k ≥ 1, we have

(1)∥∥∥Pk+1

h

∥∥∥20

≤ (k + 1)τ 2k+1∑i=1

∥∥∥∥J i− 12

h

∥∥∥∥2

0,

(2)

∥∥∥∥Qk+ 12

h

∥∥∥∥2

0≤ (k + 1)τ 2

k∑i=0

∥∥K ih

∥∥20.

Proof By the definition of Pk+1h and P0

h=0, we obtain∥∥∥Pk+1h

∥∥∥0

=∥∥∥∥Pk

h + τ Jk+ 1

2h

∥∥∥∥0

≤∥∥∥Pk

h

∥∥∥0+ τ

∥∥∥∥Jk+ 12

h

∥∥∥∥0

≤ τ

(∥∥∥∥Jk+ 12

h

∥∥∥∥0+

∥∥∥∥Jk− 12

h

∥∥∥∥0+ · · · +

∥∥∥∥J 12h

∣∣∣∣0

)= τ

k+1∑i=1

∥∥∥∥J i− 12

h

∥∥∥∥0. (3.1)

Using the Cauchy–Schwarz inequality to (3.1), we immediately have

∥∥∥Pk+1h

∥∥∥20

≤ (k + 1)τ 2k+1∑i=1

∥∥∥∥J i− 12

h

∥∥∥∥2

0,

which completes the proof of (1).By the same argument, we can prove (2).

Theorem 3.2 Let Cv = 1/√

ε0μ0 be the wave speed in vacuum, and Cinv > 0 be theconstant in the standard inverse estimate

‖∇ × vh‖0 ≤ Cinvh−1‖vh‖0, ∀vh ∈ V h .

Then under the time step constraint

τ < min

{h

CinvCv

,1

ωpe,

1

ωpm,

1

2ω2e0

,1

2ω2m0

}, (3.2)

for any 1 ≤ n ≤ N we have

ε0∥∥En

h

∥∥20 + μ0

∥∥∥∥Hn+ 12

h

∥∥∥∥2

0+ 1

ε0ω2pe

∥∥∥∥Jn+ 12

h

∥∥∥∥2

0+ 1

μ0ω2pm

∥∥K nh

∥∥20

≤ C

(ε0

∥∥E0h

∥∥20 + μ0

∥∥∥∥H 12h

∥∥∥∥2

0+ 1

ε0ω2pe

∥∥∥∥J 12h

∥∥∥∥2

0+ 1

μ0ω2pm

∥∥K 0h

∥∥20

),

where the positive constant C depends on the physical parameters ε0, μ0, T , Cv and Cinv .

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J Sci Comput (2016) 68:438–463 443

Proof Choosing φh = τ(Ek+1h + Ek

h) and ψh = τ(Hk+ 3

2h + H

k+ 12

h ) in (2.3) and (2.4),

respectively, multiplying the Eqs. (2.5) and (2.6) by τε0ω2

pe(J

k+ 32

h + Jk+ 1

2h ) and τ

μ0ω2pm

(K k+1h +

K kh) and integrating over �, respectively, then summing up the resultants, we obtain

ε0

(∥∥∥Ek+1h

∥∥∥20−

∥∥∥Ekh

∥∥∥20

)+ μ0

(∥∥∥∥Hk+ 32

h

∥∥∥∥2

0−

∥∥∥∥Hk+ 12

h

∥∥∥∥2

0

)

+ 1

ε0ω2pe

(∥∥∥∥Jk+ 32

h

∥∥∥∥2

0−

∥∥∥∥Jk+ 12

h

∥∥∥∥2

0

)+ 1

μ0ω2pm

(∥∥∥K k+1h

∥∥∥20−

∥∥∥K kh

∥∥∥20

)

≤ τ

[(H

k+ 12

h ,∇ ×(Ek+1h + Ek

h

))−

(∇ × Ek+1

h , Hk+ 3

2h + H

k+ 12

h

)]

+τ

[(Ek+1h , J

k+ 32

h + Jk+ 1

2h

)−

(Jk+ 1

2h , Ek+1

h + Ekh

)]

−τ

[(K k+1

h , Hk+ 3

2h + H

k+ 12

h

)−

(H

k+ 12

h , K k+1h + K k

h

)]

− τω2e0

ε0ω2pe

(Pk+1h , J

k+ 32

h + Jk+ 1

2h

)− τω2

m0

μ0ω2pm

(Q

k+ 12

h , K k+1h + K k

h

). (3.3)

Summing up (3.3) for k from 0 to n − 1, we obtain

ε0

(∥∥Enh

∥∥20 − ∥∥E0

h

∥∥20

)+ μ0

(∥∥∥∥Hn+ 12

h

∥∥∥∥2

0−

∥∥∥∥H 12h

∥∥∥∥2

0

)

+ 1

ε0ω2pe

(∥∥∥∥Jn+ 12

h

∥∥∥∥2

0−

∥∥∥∥J 12h

∥∥∥∥2

0

)+ 1

μ0ω2pm

(∥∥K nh

∥∥20 − ∥∥K 0

h

∥∥20

)

≤[τ

(H

12h ,∇ × E0

h

)− τ

(∇ × En

h, Hn+ 1

2h

)]

+[τ

(Enh, J

n+ 12

h

)− τ

(E0h, J

12h

)]+

[τ

(K 0

h, H12h

)− τ

(K n

h, Hn+ 1

2h

)]

− τω2e0

ε0ω2pe

n−1∑k=0

(Pk+1h , J

k+ 32

h + Jk+ 1

2h

)− τω2

m0

μ0ω2pm

n−1∑k=0

(Q

k+ 12

h , K k+1h + K k

h

)

=5∑

k=1

Erri . (3.4)

Below we will estimate all Erri terms. First, using the inverse estimate and the Cauchy–Schwarz inequality, we have

Err1 = τ

(H

12h ,∇ × E0

h

)− τ

(∇ × En

h, Hn+ 1

2h

)

≤ τCvCinvh−1

2

(ε0

∥∥E0h

∥∥20 + μ0

∥∥∥∥H 12h

∥∥∥∥2

0

)

+τCvCinvh−1

2

(ε0

∥∥Enh

∥∥20 + μ0

∥∥∥∥Hn+ 12

h

∥∥∥∥2

0

).

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444 J Sci Comput (2016) 68:438–463

Similarly, we can obtain

Err2 = τ

(Enh, J

n+ 12

h

)− τ

(E0h, J

12h

)

≤ τwpe

2

(ε0

∥∥Enh

∥∥20 + 1

ε0ω2pe

∥∥∥∥Jn+ 12

h

∥∥∥∥2

0

)+ τwpe

2

(ε0

∥∥E0h

∥∥20 + 1

ε0ω2pe

∥∥∥∥J 12h

∥∥∥∥2

0

),

and

Err3 = τ

(K 0

h, H12h

)− τ

(K n

h, Hn+ 1

2h

)

≤ τwpm

2

(1

μ0ω2pm

∥∥K nh

∥∥20 + μ0

∥∥∥∥Hn+ 12

h

∥∥∥∥2

0

)+ τwpm

2

(1

μ0ω2pm

∥∥K 0h

∥∥20 + μ0

∥∥∥∥H 12h

∥∥∥∥2

0

).

Using Lemma 3.1 and the Cauchy–Schwarz inequality, we have

Err4 = − τω2e0

ε0ω2pe

n−1∑k=0

(Pk+1h , J

k+ 32

h + Jk+ 1

2h

)

≤ τω2e0

2ε0ω2pe

n−1∑k=0

(∥∥∥Pk+1h

∥∥∥20+

∥∥∥∥Jk+ 32

h + Jk+ 1

2h

∥∥∥∥2

0

)

≤ τω2e0

2ε0ω2pe

n−1∑k=0

((k + 1)τ 2

k+1∑i=1

∥∥∥∥J i− 12

h

∥∥∥∥2

0

)+ 2τω2

e0

ε0ω2pe

n−1∑k=0

∥∥∥∥Jk+ 12

h

∥∥∥∥2

0+ τω2

e0

ε0ω2pe

∥∥∥∥Jn+ 12

h

∥∥∥∥2

0

≤ τ 3ω2e0

2ε0ω2pe

n∑i=1

∥∥∥∥J i− 12

h

∥∥∥∥2

0

n−1∑k=0

(k + 1) + 2τω2e0

ε0ω2pe

n−1∑k=0

∥∥∥∥Jk+ 12

h

∥∥∥∥2

0+ τω2

e0

ε0ω2pe

∥∥∥∥Jn+ 12

h

∥∥∥∥2

0

≤ τω2e0T

2

2ε0ω2pe

n∑i=1

∥∥∥∥J i− 12

h

∥∥∥∥2

0+ 2τω2

e0

ε0ω2pe

n−1∑k=0

∥∥∥∥Jk+ 12

h

∥∥∥∥2

0+ τω2

e0

ε0ω2pe

∥∥∥∥Jn+ 12

h

∥∥∥∥2

0.

By the same argument, we can obtain

Err5 = − τω2m0

μ0ω2pm

n−1∑k=0

(Q

k+ 12

h , K kh + K k+1

h

)

≤ τω2m0T

2

2μ0ω2pm

n−1∑k=0

∥∥∥K kh

∥∥∥20+ 2τω2

m0

μ0ω2pm

n−1∑k=0

∥∥∥K kh

∥∥∥20+ τω2

m0

μ0ω2pm

∥∥K nh

∥∥20 .

The proof is completed by substituting the above estimates into (3.4) with the choice ofthe time step τ satisfying the CFL constraint (3.2), and then using the Gronwall inequality.

Nowweprovide an optimal error estimate for our FETDscheme (2.3)–(2.8). The followingestimates are needed.

Lemma 3.3 Let P(t) = ∫ t0 J(s)ds, Pk+1 = P(tk+1), Jk+

12 = J(tk+ 1

2), Pk+1

h and Qk+ 1

2h

be defined by (2.7) and (2.8), respectively. Then for any J, K ∈ H2(0, T ; (L2(�))3), wehave

(i)

∣∣∣∣∫ tk+1

tk

(J(s) − Jk+

12

)ds

∣∣∣∣2

≤ τ 5

96

∫ tk+1

tk|J t t (s)|2 ds,

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J Sci Comput (2016) 68:438–463 445

(ii)

∣∣∣∣∣∣ 1τ∫ t

k+ 32

tk+ 1

2

J(s)ds − Jk+ 32 + Jk+ 1

2

2

∣∣∣∣∣∣2

≤ τ 3

120

∫ tk+ 3

2

tk+ 1

2

|J t t (s)|2 ds,

(iii)

∣∣∣∣∣∣ 1τ∫ t

k+ 32

tk+ 1

2

P(s)ds − Pkh

∣∣∣∣∣∣2

≤ τ 3

96

∫ tk+ 3

2

tk+ 1

2

|J t (s)|2 ds,

(iv)

∣∣∣∣ 1τ∫ tk+1

tkQ(s)ds − Q

k+ 12

h

∣∣∣∣2

≤ τ 3

96

∫ tk+1

tk|K t (s)|2 ds,

(v)∣∣Pn − Pn

h

∣∣2 ≤ T48τ

4∫ tn

0|J t t (s)|2 ds + 2T τ

n−1∑k=0

∣∣∣∣Jk+ 12 − J

k+ 12

h

∣∣∣∣2

,

(vi)

∣∣∣∣Qn+ 12 − Q

n+ 12

h

∣∣∣∣2

0≤ T

48τ 4

∫ tn+ 1

2

0|K t t (s)|2 ds + 2T τ

n∑k=0

∣∣∣K k − K kh

∣∣∣2.Proof (i) Using the integral identity

J(s) − Jk+12 =

(s − tk+ 1

2

)J t

(tk+ 1

2

)+

∫ s

tk+ 1

2

(s − r)J t t (r)dr,

we have

∣∣∣∣∫ tk+1

tk

(J(s) − Jk+

12

)ds

∣∣∣∣2

=∣∣∣∣∣∣∫ tk+1

tk

⎛⎝∫ s

tk+ 1

2

(s − r)J t t (r)dr

⎞⎠ ds

∣∣∣∣∣∣2

≤(∫ tk+1

tk12ds

) ⎛⎜⎝∫ tk+1

tk

∣∣∣∣∣∣∫ s

tk+ 1

2

(s − r)J t t (r)dr

∣∣∣∣∣∣2

ds

⎞⎟⎠

≤ τ

∫ tk+1

tk

⎛⎝

∣∣∣∣∣∣∫ s

tk+ 1

2

(s − r)2dr

∣∣∣∣∣∣⎞⎠

⎛⎝

∣∣∣∣∣∣∫ s

tk+ 1

2

|J t t (r)|2dr∣∣∣∣∣∣⎞⎠ ds

≤ τ

∫ tk+1

tk

⎛⎝

∣∣∣∣∣∣∫ s

tk+ 1

2

(s − r)2dr

∣∣∣∣∣∣⎞⎠ ds

(∫ tk+1

tk|J t t (r)|2dr

)= τ 5

96

∫ tk+1

tk|J t t (r)|2ds.

(ii) The proof is all the same as (i) except we use the following identity

Jk+ 32 + Jk+ 1

2

2− 1

τ

∫ tk+ 3

2

tk+ 1

2

J(s)ds = 1

2τ

∫ tk+ 3

2

tk+ 1

2

(s − tk+ 1

2

) (tk+ 3

2− s

)J t t (s)ds.

(iii) Using the following identity

P(s) = P(tk) + (s − tk)J(tk) +∫ tk

s(r − s)J t (r)dr,

and following similar procedures to (1), we can obtain the estimate.(iv) The proof is all the same as (iii).(v) By the definition of Pk+1, we have

Pk+1 =∫ tk+1

0J(s)ds =

∫ tk

0J(s)ds +

∫ tk+1

tkJ(s)ds = Pk +

∫ tk+1

tkJ(s)ds. (3.5)

123

446 J Sci Comput (2016) 68:438–463

Subtracting (3.5) from (2.7), we have

Pk+1 − Pk+1h = Pk − Pk

h +∫ tk+1

tkJ(s)ds − τ J

k+ 12

h

= Pk − Pkh +

∫ tk+1

tk

(J(s) − Jk+

12

)ds + τ

(Jk+

12 − J

k+ 12

h

),

which leads to∣∣∣Pk+1 − Pk+1h

∣∣∣ ≤∣∣∣Pk − Pk

h

∣∣∣ +∣∣∣∣∫ tk+1

tk

(J(s) − Jk+

12

)ds

∣∣∣∣ + τ

∣∣∣∣(Jk+

12 − J

k+ 12

h

)∣∣∣∣ ,(3.6)

Summing both sides of (3.6) over k = 0, 1, 2, . . . , n − 1, and using P0 = P0h = 0, we have

∣∣Pn − Pnh

∣∣ ≤n−1∑k=0

∣∣∣∣∫ tk+1

tk

(J(s) − Jk+

12

)ds

∣∣∣∣ + τ

n−1∑k=0

∣∣∣∣(Jk+

12 − J

k+ 12

h

)∣∣∣∣ , (3.7)

Squaring (3.7) and using the Cauchy–Schwarz inequality and (i), we obtain

∣∣Pn − Pnh

∣∣2 ≤ nτ 5

48

n∑k=0

∣∣∣∣∫ tk+1

tk|J t t (s)|2ds

∣∣∣∣ + 2τ 2nn∑

k=0

∣∣∣∣Jk+ 12 − J

k+ 12

h

∣∣∣∣2

≤ T

48τ 4

∫ tn+1

0|J t t (s)|2ds + 2T τ

n∑k=0

∣∣∣∣Jk+ 12 − J

k+ 12

h

∣∣∣∣2

,

where we used the fact nτ ≤ T for any 1 ≤ n ≤ N .(vi) Similar to (v), it is easy to complete the proof.

Theorem 3.4 Let (En, Hn) and (Enh, H

nh) be the analytic and finite element solutions an

time t = tn , respectively. Under the CFL time step constraint

τ < min

{h

CinvCv

,ε0

5 + 2ε0,

1

1.5 + γe + ε0ω2pe + ω2

e0

,1

0.5 + γm + μ0ω2pm + ω2

m0

},

(3.8)and the regularity assumptions

E,∇ × E, J ∈ L∞(0, T ; Hα(curl,�)),

H, K ∈ L∞(0, T ; (Hα(�))3),

Et , J t ∈ L2(0, T ; Hα(curl,�)),

Et t , J t t , Ktt ,∇ × Et t ,∇ × H t t ∈ L2(0, T ; L2(�)),

for any 1 ≤ n ≤ N we have

max1≤n≤N

(∥∥En − Enh

∥∥20 +

∥∥∥∥Hn+ 12 − H

n+ 12

h

∥∥∥∥2

0+

∥∥∥∥Jn+ 12 − J

n+ 12

h

∥∥∥∥2

0+ ∥∥K n − K n

h

∥∥20

)

≤ C

((τ 2 + hα)2 + ∥∥E0 − E0

∥∥20 +

∥∥∥∥H 12 − H

12h

∥∥∥∥2

0+

∥∥∥∥J 12 − J

12h

∥∥∥∥2

0+ ∥∥K 0 − K 0

h

∥∥20

),

where the positive constant C is independent of h and τ .

123

J Sci Comput (2016) 68:438–463 447

Proof Denote

ξ kh = EkI − Ek

h, ξ̂k+ 1

2h = J

k+ 12

I − Jk+ 1

2h , η

k+ 12

h = PhHk+ 12 − H

k+ 12

h , η̂kh = PhK k − K kh .

Integrating the Lorentz model Eqs. (1.1) and (1.4) from tk to tk+1, and (1.2) and (1.3) from

tk+ 12to tk+ 3

2, and multiplying the respective resultants by

ξ k+1h +ξ kh

τ,ηk+ 3

2h +η

k+ 12

hτ

,ξ̂k+ 3

2h +ξ̂

k+ 12

hτ

,

η̂k+1h +η̂kh

τ, and then subtracting (2.3)–(2.6), we have

ε0

(∥∥∥ξ k+1h

∥∥∥20−

∥∥∥ξ kh

∥∥∥20

)− τ

(ηk+ 1

2h ,∇ ×

(ξ k+1h + ξ kh

))

= ε0τ(δτ

(Ek+1

I − Ek+1)

, ξ k+1h + ξ kh

)+

(τ J

k+ 12

h −∫ tk+1

tkJ(s)ds, ξ k+1

h + ξ kh

),

−(

τ PhHk+ 12 −

∫ tk+1

tkH(s)ds,∇ ×

(ξ k+1h + ξ kh

)),

μ0

(∥∥∥∥ηk+ 3

2h

∥∥∥∥2

0−

∥∥∥∥ηk+ 1

2h

∥∥∥∥2

0

)+ τ

(∇ × ξ k+1

h , ηk+ 3

2h + η

k+ 32

h

)

= μ0τ

(δτ

(PhHk+ 3

2 − Hk+ 12

), η

k+ 32

h + ηk+ 1

2h

)

+(

τ K k+1h −

∫ tk+ 3

2

tk+ 12

K (s)ds, ηk+ 3

2h + η

k+ 12

h

)

−(

τ∇ ×(Ek+1

I −∫ t

k+ 32

tk+ 12

E(s)ds

), η

k+ 32

h + ηk+ 1

2h

),

(∥∥∥∥ξ̂k+ 3

2h

∥∥∥∥2

0−

∥∥∥∥ξ̂k+ 1

2h

∥∥∥∥2

0

)≤ τ

(δτ

(Jk+ 3

2I − Jk+

32

), ξ̂

k+ 32

h + ξ̂k+ 1

2h

)

+ γe

⎛⎝τ

2

(Jk+ 3

2I + J

k+ 12

I

)−

∫ tk+ 3

2

tk+ 1

2

J(s)ds, ξ̂k+ 3

2h + ξ̂

k+ 12

h

⎞⎠

+ ε0ω2pe

⎛⎝∫ t

k+ 32

tk+ 1

2

E(s)ds − τ Ek+1h , ξ̂

k+ 32

h + ξ̂k+ 1

2h

⎞⎠

+ω2e0

(τ Pk+1

h −∫ t

k+ 32

tk+ 12

(∫ t

0J(s)ds

)dt, ξ̂

k+ 32

h + ξ̂k+ 1

2h

),

and (∥∥∥η̂k+1h

∥∥∥20−

∥∥∥η̂kh

∥∥∥20

)≤ τ

(δτ

(PhK k+1 − K k+1

), η̂k+1

h + η̂kh

)

+ γm

(τ

2

(PhK k+1 + PhK k

)−

∫ tk+1

tkK (s)ds, η̂k+1

h + η̂kh

)

+μ0ω2pm

(∫ tk+1

tkH(s)ds − τ H

k+ 12

h , η̂k+1h + η̂kh

)

123

448 J Sci Comput (2016) 68:438–463

+ω2m0

(τ Q

k+ 12

h −∫ tk+1

tk

(∫ t

0K (s)ds

)dt, η̂k+1

h + η̂kh

).

Summing up the above four identities from k = 0 to k = n − 1 and following our earlierwork [26], we can obtain

ε0

(∥∥ξnh

∥∥20 − ∥∥ξ0h

∥∥20

)+ μ0

(∥∥∥∥ηn+ 1

2h

∥∥∥∥2

0−

∥∥∥∥η12h

∥∥∥∥2

0

)+

∥∥∥∥ξ̂n+ 1

2h

∥∥∥∥2

0−

∥∥∥∥ξ̂12h

∥∥∥∥2

0+ ∥∥η̂nh

∥∥20 − ∥∥η̂0h

∥∥20

≤20∑i=1

Erri .

(3.9)

Below we need to estimate each error term Erri . Using the inequality ab ≤ δa2 + 14δ b

2,we have

Err1 = τε0

n−1∑k=0

(δτ

(Ek+1

I − Ek+1)

, ξ k+1h + ξ kh

)

≤ 2τε0δ1∥∥ξnh

∥∥20 + 4τε0δ1

n−1∑k=0

∥∥∥ξ kh

∥∥∥20+ Cε0h2α

4δ1‖Et‖2L2(0,T ;Hα(curl,�))

,

Err2 =n−1∑k=0

(τ Hk+ 1

2 −∫ tk+1

tkH(s)ds,∇ ×

(ξ k+1h + ξ kh

))

≤ 2τδ2∥∥ξnh

∥∥20 + 4τδ2

n−1∑k=0

∥∥∥ξ kh

∥∥∥20+ τ 4

384δ2‖∇ × H t t‖2L2(0,T ;L2(�))

,

Err3 = τ

n−1∑k=0

(Jk+ 1

2I − Jk+

12 + Jk+

12 − 1

τ

∫ tk+1

tkJ(s)ds, ξ k+1

h + ξ kh

)

≤ 2τδ3∥∥ξnh

∥∥20 + 4τδ3

n−1∑k=0

∥∥∥ξ kh

∥∥∥20+ τ 4

192δ3‖J t t‖2L2(0,T ;L2(�))

+Ch2αT

2δ3‖J‖2L∞(0,T ;Hα(curl,�)) .

By the projection property, we have

Err4 =n−1∑k=0

μ0τ

(δτ

(PhHk+ 3

2 − Hk+ 32

), η

k+ 32

h + ηk+ 1

2h

)= 0,

Similarly, we have

Err5 = τ

n−1∑k=0

(∇ × Ek+1

I − 1

τ

∫ tk+ 3

2

tk+ 12

∇ × E(s)ds, ηk+ 3

2h + η

k+ 12

h

)

≤ 2τδ5

∥∥∥∥ηn+ 1

2h

∥∥∥∥2

0+ 4τδ5

n−1∑k=0

∥∥∥∥ηk+ 1

2h

∥∥∥∥2

0

+Ch2αT

2δ5‖∇ × E‖2L∞(0,T ;Hα(�)) + τ 4

192δ5‖∇ × Et t‖2L2(0,T ;L2(�))

,

123

J Sci Comput (2016) 68:438–463 449

Err6 =n−1∑k=0

(τ PhK k+ 1

2 −∫ t

k+ 32

tk+ 12

K (s)ds, ηk+ 3

2h + η

k+ 12

h

)

≤ 2τδ6

∥∥∥∥ηn+ 1

2h

∥∥∥∥2

0+ 4τδ6

n−1∑k=0

∥∥∥∥ηk+ 1

2h

∥∥∥∥2

0+ τ 4

384δ6‖K t t‖2L2(0,T ;L2(�))

,

Err7 = τ

n−1∑k=0

(δτ

(Jk+ 3

2I − Jk+

32

), ξ̂

k+ 32

h + ξ̂k+ 1

2h

)

≤ 2τδ7

∥∥∥∥ξ̂n+ 1

2h

∥∥∥∥2

0+ 4τδ7

n−1∑k=0

∥∥∥∥ξ̂k+ 1

2h

∥∥∥∥2

0+ Ch2α

4δ7‖J t‖2L2(0,T ;Hα(curl;�))

,

Err8 = τ

n−1∑k=0

γe

⎛⎝τ

2

(Jk+ 3

2I + J

k+ 12

I

)−

∫ tk+ 3

2

tk+ 1

2

J(s)ds, ξ̂k+ 3

2h + ξ̂

k+ 12

h

⎞⎠

≤ 2τγeδ8

∥∥∥∥ξ̂n+ 1

2h

∥∥∥∥2

0+ 4τγeδ8

n−1∑k=0

∥∥∥∥ξ̂k+ 1

2h

∥∥∥∥2

0

+ Ch2αγeT

2δ8‖J‖2L∞(0,T ;Hα(curl;�)) + τ 4γe

240δ8‖J t t‖2L2(0,T ;L2(�))

,

Err9 = ε0ω2pe

n−1∑k=0

⎛⎝∫ t

k+ 32

tk+ 1

2

E(s)ds − τ Ek+1I , ξ̂

k+ 32

h + ξ̂k+ 1

2h

⎞⎠

+ 2τδ9ε0ω2pe

∥∥∥∥ξ̂n+ 1

2h

∥∥∥∥2

0+ 4τε0ω

2peδ9

n−1∑k=0

∥∥∥∥ξ̂k+ 1

2h

∥∥∥∥2

0

+ Ch2αT

2δ9‖E‖2L∞(0,T ;Hα(curl;�)) + τ 4ε0ω

2pe

192δ9‖Et t‖2L2(0,T ;L2(�))

.

Using the Cauchy–Schwartz inequality and Lemma 3.3, we have

Err10/ω2e0 =

n−1∑k=0

(τ Pk+1

h −∫ t

k+ 32

tk+ 12

(∫ t

0J(s)ds

)dt, ξ̂

k+ 32

h + ξ̂k+ 1

2h

)

= τ

n−1∑k=0

(Pk+1h − Pk+1 + Pk+1 − 1

τ

∫ tk+ 3

2

tk+ 12

(∫ t

0J(s)ds

)dt, ξ̂

k+ 32

h + ξ̂k+ 1

2h

)

≤ 2τδ10

∥∥∥∥ξ̂n+ 1

2h

∥∥∥∥2

0+ 4τδ10

n−1∑k=0

∥∥∥∥ξ̂k+ 1

2h

∥∥∥∥2

0+ τ

2δ10

n−1∑k=0

∥∥∥Pk+1h − Pk+1

∥∥∥20

+ τ

2δ10

n−1∑k=0

∥∥∥∥∥Pk+1 − 1

τ

∫ tk+ 3

2

tk+ 12

(∫ t

0J(s)ds

)dt

∥∥∥∥∥2

0

≤ 2τδ10

∥∥∥∥ξ̂n+ 1

2h

∥∥∥∥2

0+ 4τδ10

n−1∑k=0

∥∥∥∥ξ̂k+ 1

2h

∥∥∥∥2

0+ τ 5T

96δ10

n−1∑k=0

∫ tk+1

0‖Jtt‖20

+τ 2T

δ10

n−1∑k=0

k∑i=0

∣∣∣∣J i+ 12 − J

i+ 12

h

∥∥∥∥2

0+ τ 4

192δ10

n−1∑k=0

∫ tk+ 3

2

tk+ 1

2

|J t (s)|2ds

123

450 J Sci Comput (2016) 68:438–463

≤ 2τδ10

∥∥∥∥ξ̂n+ 1

2h

∥∥∥∥2

0+ 4τδ10

n−1∑k=0

∥∥∥∥ξ̂k+ 1

2h

∥∥∥∥2

0+ τ 4T 2

96δ10‖J t t‖2L2(0,T ;L2(�))

+ 2τT 2

δ10

n−1∑k=0

∥∥∥ξ̂ k+12

∥∥∥20+ τ 4

192δ10‖J t‖2L2(0,T ;L2(�))

+ CT 3h2α

δ10‖J‖2L∞(0,T ;Hα(curl;�)) .

By the projection property, we see that

Err11 = τ

n−1∑k=0

(δτ

(PhK k+1 − K k+1

), η̂k+1

h + η̂kh

)= 0.

By the similar arguments, we can prove that

Err12 = γm

n−1∑k=0

(τ

2

(PhK k+1 − PhK k

)−

∫ tk+1

tkK (s)ds, η̂k+1

h + η̂kh

)

≤ 2τγmδ12∣∣η̂nh∥∥20 + 4τγmδ12

n−1∑k=0

∥∥∥η̂kh

∥∥∥20+ τ 4γm

480δ12‖K t t‖2L2(0,T ;L2(�))

,

Err13 = τμ0ω2pm

n−1∑k=0

(∫ tk+1

tkH(s)ds − τ H

k+ 12

h , η̂k+1h + η̂kh

)

≤ 2τμ0ω2pmδ13

∥∥η̂nh

∥∥20 + 4τμ0ω

2pmδ13

n−1∑k=0

∥∥∥η̂kh

∥∥∥20

+τ 4μ0ω2pm

480δ13‖H t t‖2L2(0,T ;L2(�))

,

Err14/ωω2e0

=n−1∑k=0

(τ Q

k+ 12

h −∫ tk+1

tk

(∫ t

0K (s)ds

)dt, η̂k+1

h + η̂kh

)

= τ

n−1∑k=0

(Q

k+ 12

h − Qk+ 12 + Qk+ 1

2 − 1

τ

∫ tk+1

tk

(∫ t

0K (s)ds

)dt, η̂k+1

h + η̂kh

)

≤ 2τδ14∥∥η̂nh

∥∥20 + 4τδ14

n−1∑k=0

∥∥∥η̂kh

∥∥∥20+ τ

2δ14

n−1∑k=0

∥∥∥∥Qk+ 12

h − Qk+ 12

∥∥∥∥2

0

+ τ

2δ14

n−1∑k=0

∥∥∥∥Qk+ 12 − 1

τ

∫ tk+1

tk

(∫ t

0K (s)ds

)dt

∥∥∥∥2

0

≤ 2τδ14∥∥η̂nh

∥∥20 + 4τδ14

n−1∑k=0

∥∥∥η̂kh

∥∥∥20+ τ 4T 2

96δ14‖K t t‖2L2(0,T ;L2(�))

+2τT 2

δ14

n−1∑k=0

∥∥∥η̂k∥∥∥20+ τ 4

192δ14‖K t‖L2(0,T ;L2(�))

+CT 3h2α

δ14‖K‖L∞(0,T ;(Hα(�))3) ,

123

J Sci Comput (2016) 68:438–463 451

Err15 = τ

(ξ̂n+ 1

2h , ξnh

)≤ τδ15

∥∥∥∥ξ̂n+ 1

2h

∥∥∥∥2

0+ τ

4δ15

∥∥ξnh

∥∥20 ,

Err16 = −τ

(ξ̂

12h , ξ0h

)≤ τδ16

∥∥∥∥ξ̂12h

∥∥∥∥2

0+ τ

4δ16

∥∥ξ0h

∥∥20 ,

Err17 = −τ

(η̂nh , η

n+ 12

h

)≤ τδ17

∥∥η̂nh

∥∥20 + τ

4δ17

∥∥∥∥ηn+ 1

2h

∥∥∥∥2

0,

Err18 = τ

(η̂0h, η

12h

)≤ τδ18

∥∥η̂0h

∥∥20 + τ

4δ18

∥∥∥∥η12h

∥∥∥∥2

0,

Err19 = τ

(∇ × ξ0h , η

12h

)≤ ε0δ19

∥∥ξ0h

∥∥20 + (CinvCvτ )2

h21

4δ19

∥∥∥∥η12h

∥∥∥∥2

0,

Err20 = −τ

(∇ × ξnh , η

n+ 12

h

)≤ ε0δ20

∥∥ξnh

∥∥20 + (CinvCvτ )2

h2μ0

4δ20

∥∥∥∥ηn+ 1

2h

∥∥∥∥2

0.

By choosing δi = 12 and the time step τ satisfying the constraint (3.8) so that all right

hand side terms can be controlled by the corresponding terms on the left hand side of (3.9),we complete the proof.

4 Numerical Tests

In this section, we provide extensive numerical results to demonstrate the effectiveness ofour FETD method.

Example 1: Test of convergence rates Here we present an example to rigorously checkthe optimal convergence rates proved in Theorem 3.4. In this test, we choose the physicaldomain � = [0, 1]2, the time interval I = [0, 1], and all the physical parameters being one(i.e., ε0 = μ0 = γe = γm = ωpe = ωpm = ωe0 = ωm0 = 1) such that the exact solutions

E(x, y, t) = exp(−t) (− cos(πx) sin(πy), sin(πx) cos(πy))T ,

Hz(x, y, t) = exp(−t) cos(πx) cos(πy),

J(x, y, t) = t exp(−t) (− cos(πx) sin(πy), sin(πx) cos(πy))T ,

K (x, y, t) = t exp(−t) cos(πx) cos(πy)

satisfy (1.1)–(1.4) with source terms

f 1 = (t − 1 − π) exp(−t) (− cos(πx) sin(πy), sin(πx) cos(πy))T ,

f 2 = (2π + t − 1) exp(−t) cos(πx) cos(πy),

f 3 = (1 − exp(−t) − t exp(−t)) (− cos(πx) sin(πy), sin(πx) cos(πy))T ,

f 4 = (1 − exp(−t) − t exp(−t)) cos(πx) cos(πy)

imposed to the right hand sides of modeling equations (1.1)–(1.4), respectively.We implemented the leap-frog scheme (2.3)–(2.6) with added source terms to solve this

example with a fixed time step τ = 0.001 on various uniform rectangular meshes withh = 1/nx , where nx and ny = nx denote the number of divisions in the x and y directions,respectively. The obtained errors in the L2 norm at t = 1 are presented in Tables 1 and 2,which clearly show the convergence rates O(h) for ‖E − Eh‖0, ‖∇ × (E − Eh)‖0, ‖Hz −Hz,h‖0, ‖J− Jh‖0, ‖K−Kh‖0 as our theoretical analysis proved in Theorem 3.4. To confirm

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Table 1 The errors of E with τ = 0.001 obtained on uniform meshes

Meshes ‖E − Eh‖0 Rate ‖∇ × (E − Eh)‖0 Rate

h = 1/4 0.058772180 – 0.364193457 –

h = 1/8 0.029500380 0.9943 0.184376613 0.9820

h = 1/16 0.014747456 1.0002 0.092526147 0.9947

h = 1/32 0.007372667 1.0002 0.046306578 0.9986

h = 1/64 0.003686177 1.0000 0.023158764 0.9996

Table 2 The errors of Hz , J and K with τ = 0.001 obtained on uniform meshes

Meshes ‖Hz − Hz,h‖0 Rate ‖J − Jh‖0 Rate ‖K − Kh‖0 Rate

h = 1/4 0.064256947 – 0.061456021 – 0.057923796 –

h = 1/8 0.030240759 1.0873 0.029842468 1.0421 0.029351835 0.9807

h = 1/16 0.014836254 1.0273 0.014790130 1.0127 0.014727170 0.9949

h = 1/32 0.007381056 1.0072 0.007377995 1.0033 0.007370072 0.9987

h = 1/64 0.003685845 1.0018 0.003686842 1.0008 0.003685849 0.9996

Table 3 The errors of E, Hz and J with τ =√h

40 obtained on uniform meshes

Time steps ‖E − Eh‖0 Rate ‖Hz − Hz,h‖0 Rate ‖J − Jh‖0 Rate

τ = 180 0.0588571 – 0.0614534 – 0.0639118 –

τ = 1160 0.0147484 1.9966 0.0147900 2.0548 0.0147975 2.1107

τ = 1320 0.0036861 2.0003 0.0036868 2.0041 0.0036819 2.0068

τ = 1640 9.2153e−004 2.0000 9.2154e−004 2.0002 9.2083e−004 1.9994

the temporal convergence rate, we solve this example with a fixed τ =√h

40 by varying themesh size h. Obtained numerical results given in Table 3 show the convergence rate O(τ 2),which justifies our theoretical analysis.

Example 2: Simulation of convex and concave lenses made with negative refractionindex metamaterials In this example, we simulate how wave propagates in both convexand concave lenses made with negative refraction index metamaterials. To avoid the wavereflection at the physical boundaries, we use the perfectly matched layer (PML) around thephysical domain. Here, we use the same fully-discrete finite element scheme developed inour previous work [22] to solve the PML equations. In our simulation, the center frequencyis fixed at f = 6GHz , the time step is chosen as τ = 10−12s, the incidence source wavevaries in space as e−y2/(10h)2 (imposed as the Hz component) and in time as given in ourprevious work [22], and a PML with 12 cells in each direction around the physical domainis used.

For the convex lens simulation, the physical domain is chosen to be [−0.3, 0.3] hboxm ×[−0.225, 0.225] hboxm, and the convex lens (sketched by the black line in Fig. 1) is embed-ded inside. For the simulation, we use a finite element mesh with 74,752 and 9168 triangularand rectangular elements, respectively, which leads to a total number of Degrees of Freedom

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Fig.1

ElectricfieldsEh,y

atvarioustim

estepsforthe

simulationof

thenegativ

erefractio

nconvex

lens.Top

left1200

steps.Topmiddle1600

steps.Toprigh

t2000steps.Bottom

left2400

steps.Bottommiddle2800

steps.Bottomrigh

t40

00steps

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454 J Sci Comput (2016) 68:438–463

(DoFs) for E as 130,838. The physical parameters are chosen as

ωpe = ωpm = √2ω, ω = 2π f, ωe0 = ωm0 = ω√

5, γe = γm = 10−5ωpe,

which lead to the negative refraction index at the center frequency,

n(ω) = si

√ε(ω)μ(ω)

ε0μ0= 1 + ω2

pe

jγeω + ω2e0 − ω2

≈ −1.5,

where si ={1, in double positive material−1, in double negative material

is the chiral sign function.

Figure 1 shows the calculated Ey component of the electric fields at different time steps.From Fig. 1, we clearly see that the wave reverses its propagation direction when the waveenters into the convex lens, and the wave gets refocused as the wave exits the lens. Thisinteresting phenomenon is typical for wave propagation in negative refraction index meta-materials [27]. To appreciate the exotic property of the negative index metaterial convex lens,we solve the same problem with the same convex lens but made with a positive refractionindex material of n = 1.5. The calculated Ey fields at various time steps are presented inFig. 2, which shows that the convex lens with positive refraction can concentrate the wave.

For the concave lens simulation, the physical domain is chosen to be [−0.3, 0.3]m ×[−0.14, 0.14]m, and the concave lens (sketched by the black line in Fig. 4) is embeddedinside. The finite element mesh used for the simulation contains 42496 and 7344 triangularand rectangular elements, respectively, which leads to the total number of DoFs for E as78742. The physical parameters are chosen as

ωpe = ωpm = ω, ωe0 = ωm0 = ω√5/3

, γe = γm = 10−5ωpe,

which also lead to a negative refraction index

n(ω) = si

√ε(ω)μ(ω)

ε0μ0= 1 + ω2

pe

jγeω + ω2e0 − ω2

≈ −1.5.

The calculated Ey components at various time steps are plotted in Fig. 3, which showsclearly the refocusing property and negative refraction phenomenon. As a comparison, inFig. 4 we also present the calculated Ey components for the concave lens with n = 1.5.From these simulations, we can see that the concave lens with negative refraction has aconvergence effect, while the concave lens with positive refraction has a divergence effect.

Example 3: Simulation of total transmission and total reflection Recently, it is foundthat total transmission and total reflection can be realized by embedding proper dielectricdefects in zero index metamaterials (ZIMs). Such fantastic properties have been realizedwith defects of normal dielectric materials of different shapes, such as cylindrical defects[17,34], rectangular defects [38], and triangular defects [23]. All the simulation results inthose papers are obtained by the commercial package COMSOL with frequency-domainmethods, hence researchers only have the distribution of the electromagnetic fields in thesteady state. However, it would be very interesting to see how wave changes at differenttime as the wave propagates through ZIM regions. To our best knowledge, this is the firsttime-domain simulation of wave propagation in ZIMs.

First, we use our FETD method (2.3)–(2.7) to model the phenomenon of total transmis-sion and total reflection realized by embedding rectangular dielectric defects in ZIM. Thecenter frequency is fixed at f = 15GHz , the time step is chosen as τ = 3 · 10−13 s,

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Fig.2

ElectricfieldsEh,y

atvariou

stim

estepsforthe

simulationof

thepo

sitiv

erefractio

nconvex

lens.Top

left1200

steps.Topmiddle1600

steps.Toprigh

t2000steps.Bottom

left2400

steps.Bottommiddle2800

steps.Bottomrigh

t40

00steps

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456 J Sci Comput (2016) 68:438–463

Fig.3

ElectricfieldsEh,y

atvarioustim

estepsforthesimulationof

thenegativ

erefractio

nconcavelens.T

opleft1200

steps.Topmiddle1600

steps.Toprigh

t2000

steps.

Bottomleft2400

steps.Bottommiddle2800

steps.Bottomrigh

t60

00steps

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J Sci Comput (2016) 68:438–463 457

Fig.4

Magnetic

fieldsEh,y

atvariou

stim

estepsforthesimulationof

cloaking

byZIM

.Top

left1200

steps.Topmiddle1600

steps.Toprigh

t2000

steps.Bottom

left24

00steps.Bottommiddle2800

steps.Bottomrigh

t60

00steps

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458 J Sci Comput (2016) 68:438–463

P

LM

PEC

L

P

PEC

ZIM

defect

ε0,μ

0

M

ε0,μ

0

P

LM

PEC

L

P

PEC

ε0,μ

0

M

ε0,μ

0

ZIM

ZIM

ZIM

(b)(a)

Fig. 5 The schematic description of the waveguide structure with ZIM: a for total transmission and totalreflection simulation; b for the unidirectional propagation simulation

the incident wave is generated by a plane wave source Hz = sin(ωt) imposed at linex = −0.048m located inside the physical domain [−0.06, 0.06]m × [−0.032, 0.032]m.The ZIM region is [−0.03, 0.03]m × [−0.032, 0.032]m, and contains a rectangular defectregion [−0.01, 0.0]m × [−0.01, 0.01]m. The simulation setup is shown in Fig. 5a, and weuse a uniform rectangular mesh with the mesh size h = 0.001m for this simulation. Thephysical parameters are chosen as

ωpe = ωpm = ω, ωe0 = ωm0 = 0, γe = γm = 10−5ωpe, (4.1)

which lead to the zero refractive index

n(ω) = si

√ε(ω)μ(ω)

ε0μ0= 1 + ω2

pe

jγeω + ω2e0 − ω2

≈ 0.

From [38], we know that the total transmission and reflection phenomena can be obtainedby setting the permittivity of the rectangular defect ε = 4.25 and ε = 1.25, respectively. Thecalculated magnetic fields Hz at various time steps are plotted in Figs. 6 and 7, respectively,which show clearly how wave propagates in the ZIM and defect regions. The magnetic fieldchanges with time in the ZIM region, and it becomes constant after the wave passes thedefects. Our time domain results at times long enough are similar to those obtained by thefrequency domain methods [38].

Example 4: Simulation of unidirectional transmission High efficiency unidirectionaltransmission can be achieved by properly arranging prisms made of ZIMs [14]. Here we useour FETD method to simulate this unidirectional transmission phenomenon with the setupshown in Fig. 5b. The physical domain is chosen to be [−0.5, 0.5]m × [−0.2, 0.2]m, andthe ZIM region is made of three triangles. The center frequency is fixed at f = 10GHz ,the time step is chosen as τ = 10−12 s, and the incident wave is generated by a plane wavesource Hz = sin(ωt) imposed at line x = −0.5m. For this simulation, we used a hybridmesh, which contains 247,838 totals edges, 159,744 triangular elements and 3840 rectangularelements. The physical parameters are chosen as (4.1). The calculated magnetic fields Hz

at various time steps are plotted in Fig. 8, which clearly show the efficiency unidirectionaltransmission phenomenon: the transmitted wave is mainly along two directions from theinterfaces of ZIMs, and the magnetic field is also a constant in the triangular ZIM region. Ourtime domain results at a time long enough are also similar to those obtained by the frequencydomain methods [14].

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Fig.6

Magnetic

fieldsHzatvariou

stim

estepsforthesimulationof

cloaking

byZIM

.Top

left600steps.Topmiddle800steps.Toprigh

t1200

steps.Bottomleft1800

steps.

Bottommiddle9000

steps.Bottomrigh

t10

,000

steps

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460 J Sci Comput (2016) 68:438–463

Fig.7

Magnetic

fieldsHzatvarioustim

estepsforthesimulationof

totalreflectio

nby

ZIM

.Top

left600steps.Topmiddle800steps.Toprigh

t1200

steps.Bottomleft18

00steps.Bottommiddle9000

steps.Bottomrigh

t10

,000

steps

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J Sci Comput (2016) 68:438–463 461

Fig.8

Magnetic

fieldsHzatvariou

stim

estepsforthesimulationof

unidirectio

naltransmission

.Top

left1800

steps.Topmiddle2000

steps.Toprigh

t2500

steps.Bottomleft

3000

steps.Bottommiddle3500

steps.Bottomrigh

t40

00steps

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5 Conclusions

In this paper, we proposed an explicit FETD method for solving the Maxwell’s equations inLorentz metamaterials. We first established the stability and optimal error estimate of ourmethod. Extensive numerical results are presented to confirm our theoretical analysis anddemonstrate the effectiveness of our FETD method. Many interesting phenomena (such aswave propagation in both convex and concave lenses made with negative refraction indexmetamaterials, total transmissions and total reflections with buried defects in ZIMs, and theunidirectional transmission) are successfully simulated by the FETDmethod for the first time.Finally, we like to point out that superconvergence O(h2) at element centers can be provedfor the lowest rectangular edge element for the Lorentz model by following our previouswork (see [27, Ch.5]).

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