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Equivalent Circuit (EC) FDTD Methodfor Dispersive Materials: Derivation, StabilityCriteria and Application Examples

A. Rennings, A. Lauer, C. Caloz and I. Wolff

1 Introduction

Although many computational methods are currently available to analyze electro-magnetic problems of high complexity, further progress is necessary to unify thedifferent approaches and to improve the robustness of some algorithms. This chap-ter aims at reaching these two goals. It presents an equivalent circuit (EC) finitedifference time domain (FDTD) method which both represents a complete circuitalformulation of conventional FDTD [1] based on the electric and magnetic fields Yeescheme and provides guaranteed stability criteria for dispersive media.

A circuital formulation of FDTD offers several benefits. Since its update valuesare voltages and currents (as opposed to electric and magnetic fields), it facilitatesthe integration of electronic components, such as for instance diodes and transistors,into the simulation. In addition, it provides increased computational speed thanks toa reduced number of multiplications in the update equations [2, 3]. Finally, it maybe easily generalized to arbitrary dispersive media via simple manipulations of themesh immittances.

While no convenient explicit guaranteed stability criterion for dispersive mediais available in the framework of the conventional FDTD technique, the EC FDTDprovides such a criterion. This is a distinct advantage for both emerging metamate-rials [4], which are inherently dispersive, and for various natural materials [5], suchas metals at the optical frequencies, ferrimagnetic and ferromagnetic materials andbiological tissues, which exhibit various types of more or less complex dispersionresponses.

A. RenningsIMST GmbH, D-47475 Kamp-Lintfort, Germany, e-mail: [email protected]

A. LauerIMST GmbH, D-47475 Kamp-Lintfort, Germany

C. CalozEcole Polytechnique, Montreal, H3T 1J4, Quebec, Canada, e-mail: [email protected]

I. WolffIMST GmbH, D-47475 Kamp-Lintfort, Germany

P. Russer, U. Siart (eds.), Time Domain Methods in Electrodynamics, 211c© Springer-Verlag Berlin Heidelberg 2008

212 A. Rennings et al.

The chapter is organized as follows. Section 2 introduces a compact and conve-nient notation for the method, derives the EC FDTD method first for the simplestcase of non-dispersive materials and then extends it for the case of dispersive ma-terials, including the cases of Drude, Lorentz and Debye media. Section 3 developswithin the framework of the EC FDTD a discretized and circuital version of thePoynting theorem both for the time-continuous and time-discrete cases. Upon thisfoundation and following a Lyapunov energy approach, Sect. 4 establishes a guaran-teed stability criterium, which may be particularized to arbitrary types of dispersionresponses. Application examples of metamaterial and natural material dispersivestructures and devices are next presented in Sect. 5. Finally, Sect. 6 outlines themain results.

2 Derivation of the Method

2.1 Notation of the Discrete Calculus

Here a compact and convenient notation for the EC FDTD method is introduced.Additionally to conventional mathematical notations here all operators includingmatrices are denoted by upright characters (like S[·]), and vectorial/tensorial objectsare denoted by capital bold characters (vector field E or matrix A).

2.1.1 Discrete Fields

Each of the six continuous scalar fields of the form Fconti(x, t), x ∈ R3, t ∈ R of the

electromagnetic field are represented by six discrete fields of the form Fndisc(X), X ∈

Z3, n ∈ Z, where X is the three-dimensional spatial index vector and n is the time

index. The index vector X is usually not given [Fndisc and not Fn

disc(X)], since it isassumed that all discrete fields depend on X. Whereas the time index n is alwaysgiven for the time discrete case (conventional technique with upper index).

2.1.2 Modulo-Three Notation and Variable Indices

For the sake of simpler and compacter notation, component and spatial directionindices are numbered in a modulo-three sense, e.g. F0 ≡ Fx, F1 ≡ Fy, F2 ≡ Fz andF3 ≡ F0, F4 ≡ F2, i.e., the effective index is always element of the set {0, 1, 2}.

In 3D figures the direction indices d ∈ {0, 1, 2} are explicitly given (as absolutenumbers), whereas in 2D figures (projections of a 3D arrangement along the Carte-sian axes) variable indices (d, d + 1, d + 2) are used. By inserting {0, 1, 2} for done gets the projection onto a desired plane. The variable indices are also utilized tospecify vectorial equations in a scalar manner. Again, by inserting {0, 1, 2} for d the

EC FDTD Method for Dispersive Materials 213

specific component is obtained. The fact that d is element of the set {0, 1, 2} is notrepeated in each equation with variable indices, it is part of the general assumptionsoutlined in this section.

2.1.3 Discrete Spatial-Shift and Index-Shift Operators

In finite difference schemes the application of spatial operators onto discrete fieldsis usually specified by using index changes together with appropriate weightingfactors. Quite often the resulting equations are cumbersome. In order to circumventthis, we introduce shift operators Sd and its inverse S−1

d . Together with the null 0[·]and the identity element I[·] the family of shift operators generate an operator spaceO, which in algebraic terms is called a commutative ring. The shift operators aredefined via their action onto the discrete fields. It is

SdF(XXX) = F(XXX +EEEd),{

S−1d +Sd+1

}F(XXX) = F(XXX −EEEd)+F(XXX +EEEd+1), (1)

with the index unit vectors E0 = (1, 0, 0)T, E1 = (0, 1, 0)T and E2 = (0, 0, 1)T.In Eq. (1) the summation in O is also shown. If a summation of several operators isapplied to a field, braces {. . .} are used around the operator sum as in Eq. (1). Theapplication of Sd to a field that does not depend on Xd yields the unchanged field(identity operation). This rule is important for the application of shift-operators tofields that depend only on one component of the index vector, like the dimensionsof the mesh cells. For instance Δd is only a function of Xd . Thus expressions likeSdΔd can be simplified to SΔd .

The following example should clarify the multiplication in O:

α Sd S−1d+2F(XXX) = S−1

d+2 Sd αF(XXX) = α F(XXX +EEEd −EEEd+2) (2)

Due to inhomogeneous material distributions the simple multiplication with aconstant factor in O needs to be extended to a multiplicative operator M(X) thatdepends on X. In this case the commutativity of the multiplication in O is lost,since e.g.

Sd [M(XXX)F(XXX)] = M(XXX +EEEd)F(XXX +EEEd) = M(XXX)SdF(XXX). (3)

If an operator is applied to more than the field directly behind it, brackets [. . .]are used, for instance Sd [M(X)Fd(X)+Fd+2(X)].

The index-shift operator R and its inverse R−1 are defined by

RFd = Fd+1, R−1Fd = Fd−1. (4)

This operator is used in Sect. 3 where a discrete Poynting theorem will be de-rived. R is used for re-formulations and simplification of terms that are summed upover all three directions d ∈ {0, 1, 2}. The overall sum does not change if theseoperators are applied due to the modulo-3 sense of the indices.


2.2 Basic EC FDTD for Non-Dispersive Materials

The basic EC FDTD scheme is derived from an approximation of Maxwell’s equa-tions in integral form, including electric loss σe, magnetic loss σm, and a currentdensity source Jsrc (soft excitation),

∮

∂AHHH ·d�� = +

ddt

∫∫

AεEEE ·dAAA+

∫∫

Aσe EEE ·dAAA−

∫∫

AJJJsrc ·dAAA, (5)

∮

∂AEEE ·d�� = − d

dt

∫∫

AμHHH ·dAAA−

∫∫

Aσm HHH ·dAAA, (6)

where all material constitutive parameters (ε,μ ,σe,σm) are considered non-dispersive, i.e. independent of frequency (but possibly inhomogeneous).

These space-time continuous equations are spatially discretized using the well-known Yee cell [1], which is depicted in Fig. 1. Within the Yee cell, the electric field issampled at the edges of the main mesh subcell while the magnetic field is sampled atthe edges of the dual mesh subcell. The field samples within the Yee cell are locatedby the index vector X = (X0, X1, X2)T, according to the notations of Sect. 2.1.

The Yee discretization details of the proposed EC FDTD formulation are shownin Fig. 2. The magnetic field samples are located normally at the center of the facesof the main mesh subcells and are symmetrically surrounded by four adjacent tan-gential electric field samples. In contrast, in the general case of a non-uniform mesh,the electric field samples are not located symmetrically with respect to the surround-ing magnetic field samples.

E0 E2

S0E1

H1S1

–1H2

H2

H0S1E0

E1

S2–1H1

Fig. 1 Yee cell, which consists of a main mesh subcell (parallelepipedic structure in the figure)and a dual mesh subcell (staggered parallelepipedic structure, of which only one face is shownin the figure). The Yee cell includes six unshifted field samples. Additionally four positively ornegatively shifted field samples are shown, which are required for the approximation of Eqs. (5)and (6)


ed ed+1

ed+2

Ed

Ed+2

Ed+1

Hd

Hd+2

Hd+1

Sd+1Hd+2

Δd+2

Ad

Sd+1Ed+2

Δd+1

Δd+2

Δd+1

Ad

Sd+2Hd+1

Sd+2Ed+1

−1

−1

Fig. 2 Details of the Yee cell (Fig. 1) discretization. Used components and corresponding subcelldimensions for the spatial discretization of Eqs. (5) and (6) leading to Eqs. (8) and (9)

Here in this scheme the electric and also the magnetic material parameters areapproximated by main mesh “blocks” with piecewise constant constitutive parame-ters1, which allows a natural meshing of inhomogeneous materials, since a changein the material parameters is mostly due to an object boundary. In the case of a non-uniform mesh, the dimensions of the mesh cells vary across space, and the edgelength Δd is only a function of the component Xd of the index vector X. Accordingto the notations of Sect. 2.1, the dimensions of the dual mesh subcell, Δ d , are relatedto the dimensions of the main mesh subcell, Δd , by

Δ d =12

(S−1Δd +Δd

)=

12

{S−1 +1

}Δd , ∀ d ∈ {0,1,2}. (7)

Evaluating Maxwell’s Eqs. (5) and (6) on all faces of the main and dual mesh sub-cells, as represented in Fig. 2, yields the (still time-continuous) spatially discretizedgeneric equations∮

∂Ad

HHH ·d�� ≈{

1−S−1d+1

}Hd+2 Δ d+2 −

{1−S−1

d+2

}Hd+1 Δ d+1 = (8)

+(

εd,effddt

Ed +σ ed,eff Ed − Jsrc

d

)Δ d+1Δ d+2 ≈

∫∫

Ad

. . . ·dAAA,

∮

∂Ad

EEE ·d�� ≈ {Sd+1 −1}Ed+2 Δd+2 −{Sd+2 −1}Ed+1 Δd+1 = (9)

−(

μd,effddt

Hd +σmd,eff Hd

)Δd+1Δd+2 ≈

∫∫

Ad

. . . ·dAAA,

1 Usually the magnetic material parameters are approximated by dual mesh blocks with piecewiseconstant constitutive parameters.


where the four effective constitutive parameters εd,eff, μd,eff, σ ed,eff, σm

d,eff will bedetermined now.

The effective electric parameters are approximated by averaging the fluxes∫∫Ad{ε,σe}E ·dA on each face Ad (Fig. 2) of the dual mesh subcells, leading to

εd,eff.=

14

{1+S−1

d+1 +S−1d+2 +S−1

d+1S−1d+2

}[ε Δd+1 Δd+2]

14

{1+S−1

d+1 +S−1d+2 +S−1

d+1S−1d+2

}[Δd+1 Δd+2]

, (10)

for εd,eff, as illustrated in Fig. 3(a), and σ ed,eff is evaluated in the same manner.

The magnetic parameters are constant in each Ad–transversal direction (whereAd is defined by the directions d + 1 and d + 2, Fig. 2, right) of the main meshsubcells, and therefore no useful averaging quantity may defined from the fluxes∫∫

Ad{μ ,σm}H · dA. In contrast, two different magnetic materials may exist in the

longitudinal direction (d), and therefore the effective magnetic parameters maybe approximated by averaging the magneto-motive force2 HdΔ d as illustrated inFig. 3(b)

ed+2

ed+2ed+1

Ed

ε∂Ad

S –1

(a) (b)

Δd+2

S−1Δd+1

S–1μ

ed

ed+1ed

Δd+1

S−1Δd+2

S –1 S

–1d

Hd,–

Hd,+

ΔdHd

Δd+2

μ

S−1Δd

d+1ε

S –1d+2εd+1 d+2ε

Fig. 3 Averaging process for the approximation of the effective constitutive parameters. (a) Localvirtual capacitance filled with 4 different materials to define εd,eff. (b) Local virtual inductancefilled with 2 different materials to define μd,eff

2 The magnetic flux density Bd is continuous here.


HdΔ d = Hd,+12

Δd +Hd,−12

S−1d Δd = Bd

12

{1+S−1

d

}[Δd

μ

].= Bd

Δ d

μd,eff, (11)

from which

μd,eff.=

Δ d

12

{1+S−1

d

}[Δdμ

] =12

{1+S−1

d

}Δd

12

{1+S−1

d

}[Δdμ

] , (12)

for μd,eff, and σmd,eff is evaluated in the same manner.

Inserting the so-defined effective constitutive parameters into Eqs. (8) and (9)yields the complete space-discretized Maxwell’s equations

{1−S−1

d+1

}Hd+2 Δ d+2 −

{1−S−1

d+2

}Hd+1 Δ d+1 = −Δ d+1Δ d+2 Jsrc

d

+14

{1+S−1

d+1 +S−1d+2 +S−1

d+1S−1d+2

}[ε Δd+1 Δd+2]

1Δd

ddt

EdΔd (13)

+14

{1+S−1

d+1 +S−1d+2 +S−1

d+1S−1d+2

}[σe Δd+1 Δd+2]

1Δd

EdΔd ,

{Sd+1 −1}Ed+2 Δd+2 −{Sd+2 −1}Ed+1 Δd+1 = (14)

− Δd+1Δd+2

12

{1+S−1

d

}[Δdμ

] ddt

HdΔ d+Δd+1Δd+2

12

{1+S−1

d

}[ Δdσm

]HdΔ d ,

where the terms EdΔd.= vd may be interpreted as the electric voltages along the

edges of the main mesh subcells and the terms HdΔ d.= id may be interpreted as the

“magnetic voltages” along the edges of the dual mesh subcells.3

This circuit analogy and inspection of Eqs. (13) and (14) naturally lead tothe definition of the equivalent lumped elements

Cd =14

{1+S−1

d+1 +S−1d+2 +S−1

d+1S−1d+2

}[ε Δd+1 Δd+2]

1Δd

, (15)

Gd =14

{1+S−1

d+1 +S−1d+2 +S−1

d+1S−1d+2

}[σe Δd+1 Δd+2]

1Δd

, (16)

which are interpreted as the local capacitance and conductance, and

Ld =Δd+1Δd+2

12

{1+S−1

d

}[Δdμ

] , Rd =Δd+1Δd+2

12

{1+S−1

d

}[ Δdσm

] , (17)

which are interpreted as the local inductance and resistance. To complete the circuitanalogy, a lumped current source isrc

d is defined

3 Alternatively the term HdΔ d.= id can be interpreted as corresponding loop current in the face Ad

of the main mesh cell (Fig. 4), which is related to the orthogonal magnetic field Hd component bythe right-hand rule.


isrcd =

14

{1+S−1

d+1 +S−1d+2 +S−1

d+1S−1d+2

}[Δd+1 Δd+2]Jsrc

d . (18)

With these local lumped elements Ampere’s (5) and Faraday’s (6) laws take thevery compact form

{1−S−1

d+1

}id+2 −

{1−S−1

d+2

}id+1 + isrc

d =+Cdddt

vd +Gd vd , (19)

{Sd+1 −1}vd+2 −{Sd+2 −1}vd+1 =−Ldddt

id −Rd id . (20)

For each d ∈ {0, 1, 2} Eq. (19) represents the Kirchhoff’s current law for acurrent node and Eq. (20) represents the Kirchhoff’s voltage law for a voltage loop.This “circuitization” of the Yee FDTD discretization of space may be representedby the equivalent circuit cell shown in Fig. 4.

S0L0

L1

C2 C1

C0

L2S1C0

v0 v0

v1

S1v0

S0v1S2C0

i1

i1 i2

i2

S−11 i2 S−1

2 i1

iC0

vL2

Fig. 4 3D equivalent circuit cell for a non-dispersive material with (shunt) capacitances located atthe edges of the main mesh cells (e-nodes) and (series) inductances (represented by the rectanglein the loop) located at the faces of the main mesh cells. The 1:1-transformers are neccessary toensure a well defined loop current with a one-to-one correspondance to the orthogonal magneticfield component. The direction of the voltages vd is positive in the parallel Cartesian directioned and the one for the loop currents id is positive with respect to the corresponding magneticfield component (right-hand rule). The values iC0 = C0

ddt v0 and vL2 = L2

ddt i2 are used as an

abbreviation within the figure


The final step in the derivation of the EC FDTD scheme consists in time-discretize by finite difference the previous equations in order to set up the updateequations. Defining a time step Δ t and using the traditional leap-frog scheme, thevoltage values are sampled at time multiples of Δ t while the current values are sam-pled between these instances, i.e. at odd multiples of Δ t/2. This leads to

{1−S−1

d+1

}in− 1

2d+2 −

{1−S−1

d+2

}in− 1

2d+1 = +Cd

vnd − vn−1

d

Δ t+Gd

vnd + vn−1

d

2− i

n− 12

src,d ,

(21)

{Sd+1 −1}vnd+2 −{Sd+2 −1}vn

d+1 = −Ldin+ 1

2d − i

n− 12

d

Δ t−Rd

in+ 1

2d + i

n− 12

d

2, (22)

and solving these equations for the latest voltages vnd and currents i

n+ 12

d yield thefinal EC FDTD update equations for a non-dispersive material region of the compu-tational domain

vnd =

1− Δ tGd2Cd

1+ Δ tGd2Cd

vn−1d +

Δ tCd

{1−S−1

d+1

}in− 1

2d+2 −

{1−S−1

d+2

}in− 1

2d+1 + i

n− 12

src,d

1+ Δ tGd2Cd

, (23)

in+ 1

2d =

1− Δ tRd2Ld

1+ Δ tRd2Ld

in− 1

2d − Δ t

Ld

{Sd+1 −1}vnd+2 −{Sd+2 −1}vn

d+1

1+ Δ tRd2Ld

. (24)

2.3 Extended Method for General Dispersive Materials

In Sect. 2.2 the basic EC FDTD scheme for non-dispersive materials is derived. Thishas led to the emergence of the local impedance Z(ω) = jωL ∝ jωμ constituted ofthe inductance L (with resistance R for loss) and local admittance Y (ω)= jωC ∝ jωεconstituted of a capacitance C (with conductance G for loss), which are character-istic of non-dispersive media in the transmission line framework and correspond tothe material non-dispersive constitutive parameters.

However, many problems in electrodynamics involve dispersive media, includ-ing metamaterials and various natural materials, for which application examplesare presented in Sects. 5.1 and 5.2, respectively. In order to handle such problems,we generalize now the EC FDTD scheme to an extended EC FDTD scheme fordispersive materials. This extended scheme fundamentally consists in adding tothe equivalent impedance Z(ω) and admittance Y (ω) the appropriate lumped el-ements to account for the specific dispersions considered, μ(ω) = Z(ω)/(jωΔm)and ε(ω) = Y (ω)/(jωΔe) [the lengths Δm and Δe are specified later]. This trans-formation will require additional update equations. The present section specificallyfocuses on double-Drude dispersion, which is the dispersion encountered in the so-called composite right/left-handed (CRLH) metamaterials [4]. Dispersions of theLorentz and Debye types are also discussed.


The relation between the generalized-dispersive admittance and the correspond-ing dispersive function ε(ω) = ε∞ + Δε ′(ω) − jσe/ω − jε ′′(ω) [where ε∞

.=ε(ω → ∞)] reads

εd,eff(ω) =Yd(ω)

jωΔd

14

{1+S−1

d+1 +S−1d+2 +S−1

d+1S−1d+2

}[Δd+1 Δd+2]

=Yd(ω)jωΔe,d

, (25)

where Δe,d represents the formfactor of an artificial subcell with cross section Ad ofthe dual mesh and the height Δd of the main mesh and has the dimension of a length.Similarly, the dispersive function μ(ω) = μ∞ +Δ μ ′(ω)− jσm/ω − jμ ′′(ω) [whereμ∞

.= μ(ω → ∞)] can be determined from its corresponding generalized-dispersiveimpedance via

μd,eff(ω) =Zd(ω)

jω

12

{1+S−1

d

}Δd

Δd+1Δd+2=

Zd(ω)jωΔm,d

, (26)

where Δm,d represents the formfactor of the artificial cell with cross section Ad ofthe main mesh subcell and height Δ d of the dual mesh subcell.

2.3.1 Drude Dispersion

Let us now particularize these generalized dispersive relations to the important caseof Drude materials, which include for instance noble metals at optical frequencies(ε-Drude) or CRLH metamaterials (both ε and μ-Drude). In this case, the relevantadmittance Yd includes an inductance LL,d in parallel with the capacitance Cd asrepresented in Fig. 5(a) and the relevant impedance Zd includes a capacitance CL,d

in series with the inductance Ld , [4], as represented in Fig. 5(b). In terms of losses,we consider the most general situation, i.e., in series to each inductance there isa resistance, and parallel each capacitor a conductance consideres the losses. Thegeneral dispersive parameters in Eqs. (25) and (26) become then

εext-Drude(ω) =Yext-Drude(ω)

jωΔe=

CΔe

⎧⎨⎩1−

1LLC

ω(

ω − j RLLL

)⎫⎬⎭− j

GωΔe

= ε0εr,∞

{1−

ω2pe

ω (ω − jΩ)

}− j

σe

ω, (27)

μext-Drude(ω) =Zext-Drude(ω)

jωΔm=

LΔm

⎧⎨⎩1−

1LCL

ω(

ω − j GLCL

)⎫⎬⎭− j

RωΔm

= μ0μr,∞

{1−

ω2pm

ω (ω − jΩ)

}− j

σm

ω. (28)


iL,dvd

LL,d

RL,d

Cd Gd

(a)

idvL,d

Ld Rd

CL,d

GL,d

(b)

Fig. 5 Extended equivalent circuit models for Drude dispersive and lossy media. (a) AdmittanceYext-Drude(ω) located at E-field nodes and related to the dispersive εext-Drude(ω). (b) ImpedanceZext-Drude(ω) located at H-field nodes and related to the dispersive μext-Drude(ω)

Due to the presence of the additional Drude lumped elements, the time-continuousAmpere’s and Faraday’s circuit law system of Eqs. (19) and (20) must be extendedto the system

{1−S−1

d+1

}id+2 −

{1−S−1

d+2

}id+1 + isrc,d = +Cd

ddt

vd +Gd vd + iL,d , (29)

vd = +LL,dddt

iL,d +RL,d iL,d , (30)

{Sd+1 −1}vd+2 −{Sd+2 −1}vd+1 = −Ldddt

id −Rd id − vL,d , (31)

id = +CL,dddt

vL,d +GL,d vL,d , (32)

where the two original equations include one more term and which has been in-creased by additional equations, as seen from the models of Fig. 5.

The final update equations representing the Drude counterparts of Eqs. (23)and (24) are finally obtained as

vnd =

1− Δ tGd

2Cd

1+Δ tGd

2Cd

vn−1d +

Δ tCd

{1−S−1

d+1

}in− 1

2d+2 −

{1−S−1

d+2

}in− 1

2d+1 − i

n− 12

L,d + in− 1

2src,d

1+Δ tGd

2Cd

,

(33)

in+ 1

2L,d =

1− Δ tRL,d

2LL,d

1+Δ tRL,d

2LL,d

in− 1

2L,d +

Δ tLL,d

vnd

1+Δ tRL,d

2LL,d

, (34)

in+ 1

2d =

1− Δ tRd

2Ld

1+Δ tRd

2Ld

in− 1

2d − Δ t

Ld

{Sd+1 −1}vnd+2 −{Sd+2 −1}vn

d+1 + vnL,d

1+Δ tRd

2Ld

, (35)


vnL,d =

1− Δ tGL,d

2CL,d

1+Δ tGL,d

2CL,d

vn−1L,d +

Δ tCL,d

in− 1

2d

1+Δ tGL,d

2CL,d

. (36)

2.3.2 Other Dispersive Media: Lorentz and Debye

Lorentz-type (one pole and one zero in ε or μ) dispersive materials may be an-alyzed by similar extended EC FDTD models [6]. Such Lorentz media are oftenencountered in natural materials (e.g. ferrimagnetic and ferromagnetic materials)and in metamaterials (e.g. split ring resonator structures). The circuit model for thistype of material is shown in Fig. 6(a), and the extended EC FDTD scheme equa-tions may be obtained following a procedure formally analogous to that for theDrude case.

As a final important example of dispersive media which can be analyzed by theextended EC FDTD scheme, let us consider the Debye-type dispersive case [typ-ically characterized by the parameters (ε∞,σe,Δεn,ωn = 1/τn)], which are verycommon in biological systems. A Debye material typically exhibits a permittivityfunction of the form

εN-Debye(ω) =YN-Debye(ω)

jωΔe=

1Δe

{C∞ − j

G0

ω+

N

∑n=1

ΔCn

1+ jωΔCnRn

}

= ε∞ − jσe

ω+

N

∑n=1

Δεn

1+ jω/ωn, (37)

which may include N poles and whose circuit model is shown in Fig. 6(b), whileits permeability is usually constant and equal to μ0 (non-magnetic medium). Again,the extended EC FDTD scheme equations may be obtained following a procedureformally analogous to that for the Drude case.

vd

CP GP

LL RL

CR GR

(a)

C∞ G0

ΔC1

R1

ΔCn

Rn

vd

(b)

Fig. 6 Extended equivalent circuit admittance models for a Lorentz and a Debye dispersive andlossy media. (a) Extended lossy Lorentz model. (b) N-pole Debye circuit


3 Poynting Theorem for EC FDTD

This section presents the Poynting theorem for the EC FDTD scheme. It will showthat the space-discrete Poynting theorem has exactly the same form as the space-continuous Poynting theorem in the time-continuous case but a slightly differentform (with additional terms) in the time-discrete case. These results will be utilizedin Sect. 4 to derive stability criteria. The procedure for the derivation of the space-discrete Poynting theorem is somewhat analogous to that for the space-continuouscase. For generality, the derivations will be made for the double-Drude case. Theycan be easily particularized to the case of non-dispersive media or applied to othertypes of dispersive media.

3.1 Time-Continuous Case

In order to evaluate a Poynting quantity (E×H, or scalar EdHd+1), one needs toconsider perpendicular components of the E and H fields. Since the (Drude) currentEqs. (31) and (32) correspond to a H-field sample which is parallel to the E-fieldsample corresponding to the associated voltage Eqs. (29) and (30) (see Fig. 1, withHd ‖ Ed), it is then necessary to consider, for one of the two fields, a different andperpendicular (i.e. rotated) component. Let us for instance keep the Ed sample un-changed and right-handedly rotate the Hd sample to select the appropriate perpen-dicular component, where the current and voltages considered are those orthogonalto each other; this is achieved by rotating or index-shifting (d → d + 1) Eqs. (31)and (32) by the operator R (Sect. 2.1).

The first step of the procedure consists then in multiplying Eqs. (29), (30), index-shifted (31) and index-shifted (32) with vd , iL,d ,(−id+1) and vL,d+1, respectively,and summing for each index-direction d ∈ {0, 1, 2} over all the cells (X ∈ Ω) ofthe computational domain to determine its total power. This yields

∑XXX∈Ω

2

∑d=0

(vd{

1−S−1d+1

}id+2 − vd

{1−S−1

d+2

}id+1 + vd iL,d

−id+1 {Sd+2 −1}vd + id+1 {Sd −1}vd+2 + vL,d+1id+1

)(38)

= ∑XXX∈Ω

2

∑d=0

(Cd vd

ddt

vd +Gd v2d + vd iL,d − vd isrc,d +LL,d iL,d

ddt

iL,d +RL,d i2L,d +

Ld+1id+1ddt

id+1 +Rd+1i2d+1 + vL,d+1id+1 +CL,d+1vL,d+1ddt

vL,d+1 +GL,d+1v2L,d+1

).

The different terms of this relation may be rearranged as


∑XXX∈Ω

2

∑d=0

Psrc,d = ∑XXX∈Ω

2

∑d=0

vdisrc,d

= ∑XXX∈Ω

2

∑d=0

(Gd v2

d +RL,d i2L,d +Rd+1 i2d+1 +GL,d+1 v2L,d+1

+ddt

[12

Cdv2d +

12

LL,di2L,d +12

Ld+1i2d+1 +12

CL,d+1v2L,d+1

]

− vdid+2 + vdS−1d+1id+2 + vdid+1 − vdS−1

d+2id+1

+Sd+2vdid+1 − vdid+1 −Sdvd+2id+1 +R−1 [vdid+2]

)(39)

where the following different total powers of the 3D EC mesh within the compu-tational domain are identified: (i) left-hand side: power delivered by the sources(∫

Ω E · Jsrc dv in the field-continuous case); (ii) right-hand side – first row: powerdissipated; (iii) right-hand side – second row: power deriving from the net reactiveenergy stored in the inductances and capacitances ( d

dt [Wm +We]); right-hand side -third and fourth row: power radiated within the volume (

∫Ω div(E×H) dv in the

field-continuous case), where 4 out of the 8 products involving the terms vdid+1 andvdid+2 cancel out4.

Furthermore, applying spatial shifting (operator S and its inverse S−1), it is foundthat the radiated power terms inside the volume Ω cancel out so that only the termsat the surface ∂Ω of this volume contribute to the radiated power, leading to the sub-stitution: ∑ X∈Ω → ∑ X∈∂Ω . This fact constitutes the discretized version of Gausstheorem (

∫Ω div(E×H) dv =

∫∂Ω (E×H) ·dA). The previous relation may then be

written in the form

∑XXX∈Ω

2

∑d=0

Psrc,d = ∑XXX∈Ω

2

∑d=0

(PGd +PRL,d +PRd+1 +PGL,d+1

+ddt

[WCd +WLL,d +WLd+1 +WCL,d+1

])

+2

∑d=0

{∑

Xd+1

∑Xd+2

(Sdvd+1id+2

∣∣Xd=N+,d

−Sdvd+2id+1∣∣Xd=N+,d

+ vd+2S−1d id+1

∣∣Xd=N−,d

− vd+1S−1d id+2

∣∣Xd=N−,d

)}, (40)

which is the EC FDTD discretized time-continuous version of the Poynting theorem,stating that the power delivered by the sources is equal to the sum of the power

4 To indicate the cross-cancellation of the 1st and 8th term, the 8th has been index-shifted by theoperator R−1.


transmitted through the surface5 ∂Ω , the power lost to heat in the volume Ω andthe net reactive power stored in the volume Ω .

3.2 Time Discrete Case

Let us now derive the time-discrete counterpart of EC FDTD discretized Poyntingtheorem. This will lead to considerations of stability in the next section.

Multiplying the time-discretized versions of Eqs. (29), (30), (31) and (32) with

vn− 1

2d , inL,d ,

(−ind+1

)and v

n− 12

L,d+1, respectively, yields

∑XXX∈Ω

2

∑d=0

(v

n− 12

d

{1−S−1

d+1

}in− 1

2d+2 − v

n− 12

d

{1−S−1

d+2

}in− 1

2d+1 + vn

d inL,d

− ind+1 {Sd+2 −1}vn

d + ind+1 {Sd −1}vn

d+2 + vn− 1

2L,d+1i

n− 12

d+1

)

= ∑XXX∈Ω

2

∑d=0

(Cd v

n− 12

d

vnd − vn−1

d

Δ t+Gd v

n− 12

d

vnd + vn−1

d

2+ v

n− 12

d in− 1

2L,d − v

n− 12

d in− 1

2src,d

+LL,d inL,d

in+ 1

2L,d − i

n− 12

L,d

Δ t+RL,d in

L,d

in+ 1

2L,d + i

n− 12

L,d

2

+Ld+1 ind+1

in+ 1

2d+1 − i

n− 12

d+1

Δ t+Rd+1 in

d+1

in+ 1

2d+1 + i

n− 12

d+1

2+ vn

L,d+1ind+1

+CL,d+1 vn− 1

2L,d+1

vnL,d+1 − vn−1

L,d+1

Δ t+GL,d+1 v

n− 12

L,d+1

vnL,d+1 + vn−1

L,d+1

2

). (41)

Next partly replacing the time samples vn− 1

2d , in

L,d , ind+1 and v

n− 12

L,d+1 by their aver-

ages 12 (vn

d + vn−1d ), 1

2

(in+ 1

2L,d + i

n− 12

L,d

), 1

2

(in+ 1

2d+1 + i

n− 12

d+1

)and 1

2

(vn

L,d+1 + vn−1L,d+1

),

respectively, transforms Eq. (41) into the re-arranged relation

∑XXX∈Ω

2

∑d=0

Pn− 1

2src,d = ∑

XXX∈Ω

2

∑d=0

vn− 1

2d i

n− 12

src,d = ∑XXX∈Ω

2

∑d=0

12

(vn

d + vn−1d

)in− 1

2src,d

= ∑XXX∈Ω

2

∑d=0

(Gd

(v

n− 12

d

)2

+RL,d(inL,d

)2+Rd+1(ind+1

)2+GL,d+1

(v

n− 12

L,d+1

)2

+12

Cd

(vn

d

)2 − (vn−1d

)2Δ t

+12

LL,d

(in+ 1

2L,d

)2

−(

in− 1

2L,d

)2

Δ t

5 The indices values N−,d and N+,d in the last summation of Eq. (40) are the lower and upperbounds of the cuboidal domain Ω in the direction d. The inner two sums over Xd+1 and Xd+2 aredue to the summation over the plane boundaries of the volume Ω .


+12

Ld+1

(in+ 1

2d+1

)2

−(

in− 1

2d+1

)2

Δ t+

12

CL,d+1

(vn

L,d+1

)2−(

vn−1L,d+1

)2

Δ t

+12

[− vn

d in− 1

2d+2 − vn−1

d in− 1

2d+2 + vn

d S−1d+1i

n− 12

d+2 + vn−1d S−1

d+1in− 1

2d+2

+ vnd i

n− 12

d+1 + vn−1d i

n− 12

d+1 − vnd S−1

d+2in− 1

2d+1 − vn−1

d S−1d+2i

n− 12

d+1

+Sd+2vnd i

n+ 12

d+1 +Sd+2vnd i

n− 12

d+1 − vnd i

n+ 12

d+1 − vnd i

n− 12

d+1

−Sdvnd+2i

n+ 12

d+1 −Sdvnd+2i

n− 12

d+1 + vnd+2i

n+ 12

d+1 +R−1[

vnd i

n− 12

d+2

]

− vnd i

n+ 12

L,d − vnd i

n− 12

L,d − vnL,d+1i

n− 12

d+1 − vn−1L,d+1i

n− 12

d+1

+ vnd i

n− 12

L,d + vn−1d i

n− 12

L,d + vnL,d+1i

n+ 12

d+1 + vnL,d+1i

n− 12

d+1

]), (42)

which includes 16 non-dispersive and 8 dispersive vi products, over which 4 non-dispersive and 4 dispersive cancel out6. As for the time-continuous case, applyingspatial shifting (operator S and its inverse S−1), it is found that some of the radiatedpower terms inside the volume Ω cancel out and only 4 surface terms remain. Theprevious relation may then be written in the form

∑XXX∈Ω

2

∑d=0

Pn− 1

2src,d = ∑

XXX∈Ω

2

∑d=0

(P

n− 12

Gd+Pn

RL,d+Pn

Rd+1+P

n− 12

GL,d+1+

W nCd

−W n−1Cd

Δ t

+W

n+ 12

LL,d−W

n− 12

LL,d

Δ t+

Wn+ 1

2Ld+1

−Wn− 1

2Ld+1

Δ t+

W nCL,d+1

−W n−1CL,d+1

Δ t

+12

[− vn

d

{1−S−1

d+2

}in+ 1

2d+1 + vn−1

d

{1−S−1

d+2

}in− 1

2d+1

+ vnd

{1−S−1

d+1

}in+ 1

2d+2 − vn−1

d

{1−S−1

d+1

}in− 1

2d+2

− vnd i

n+ 12

L,d + vn−1d i

n− 12

L,d + vnL,d+1i

n+ 12

d+1 − vn−1L,d+1i

n− 12

d+1

])

+2

∑d=0

{∑

Xd+1

∑Xd+2

(Sdvn

d+1ind+2

∣∣Xd=N+,d

−Sdvnd+2in

d+1

∣∣Xd=N+,d

+ vn− 1

2d+2 S−1

d in− 1

2d+1

∣∣Xd=N−,d

− vn− 1

2d+1 S−1

d in− 1

2d+2

∣∣Xd=N−,d

)}, (43)

6 To indicate the cross-cancellation of the 1st and 16th term, the 16th has been index-shifted by theoperator R−1.


which is the time-discrete EC FDTD version of the Poynting theorem7. In con-trast to its time-continuous counterpart [Eq. (40)], this relation does not simplifyto a form exactly identical to the space-continuous Poynting theorem: 12 vi “time-discretization-compensatory” product terms (8 for the non-dispersive case and 4additional ones for the double Drude material), due to the two different time stepsn− 1

2 and n involved, remain present in the power expressions.

4 Stability Criteria Based on a Lyapunov Energy

Time-discretized Maxwell’s equations are subject to instability when the time stepΔ t is larger then the computational cell size Δ divided by the speed of light c,and multiplied by a factor depending on the dimension of the computational do-main. Whereas in the case of non-dispersive media stability is ensured by follow-ing the simple Courant–Friedrichs–Lewy criterion [Δ t < Δ/(

√3c)] [1], no explicit

guaranteed-stability criterion is available for dispersive media in FDTD schemes.The EC FDTD scheme provides such a criterium, which will be derived now apply-ing a Lyapunov energy approach to the time-discrete Poynting theorem [Eq. (43)].Here the volume Ω in the discrete Poynting theorem must be the whole computa-tional domain in order to include the overall energy in the 3D EC network.

In order to simplify the derivation, we assume that the computational domain istruncated either by PECs, by PMCs or by lossy material for absorption (with PECor PMC walls behind)8, so that the last sum in Eq. (43) vanishes.

The Lyapunov energy stability criterium states that a system is stable if its en-ergy: (1) is a positive definite scalar function, and (2) is constant or monotonicallydecreasing after all sources have been switched off [7].

Defining as a generalized energy expression the Lyapunov function

Lnglobal = ∑

XXX∈Ω

2

∑d=0

(W n

Cd+W

n+ 12

LL,d+W

n+ 12

Ld+1+W n

CL,d+1(44)

+Δ t2

[− vn

d

{1−S−1

d+2

}in+1

2d+1+vn

d

{1−S−1

d+1

}in+ 1

2d+2 −vn

din+ 1

2L,d + vn

L,d+1in+ 1

2d+1

]),

the time-discrete Poynting theorem [Eq. (43)] may be reformulated as

Lnglobal −Ln−1

global

Δ t= ∑

XXX∈Ω

2

∑d=0

(P

n− 12

src,d −Pn− 1

2Gd

−PnRL,d

−PnRd+1

−Pn− 1

2GL,d+1

)≤ 0

for Pn− 1

2src,d = 0, (45)

7 The indices values N−,d and N+,d in the last summation of Eq. (43) are the lower and upperbounds of the cuboidal domain Ω in the direction d. The inner two sums over Xd+1 and Xd+2 aredue to the summation over the plane boundaries of the volume Ω .8 Special absorbing boundary conditions of traveling-wave or perfectly matched layer type are notconsidered.


which means that the global energy of the system remains constant or decreases intime after the source has been switched off, and the second Lyapunov condition isthus fulfilled.

The demonstration of the first Lyapunov condition, namely the positive def-initeness of the Lyapunov function, or equivalently the properness of Ln

global inEq. (44) [7], is more intricate. The sought stability criterion will be obtained interms of the maximum Δ t delimiting the range where Ln

global is positive-definite.Since the positive-definiteness of the global energy function Ln

global is difficult toestablish, we decompose it in terms of the local Lyapunov function Ln

local,d(X)

Lnglobal = ∑

XXX∈Ω

2

∑d=0

Lnlocal,d , (46)

which involves only local voltages and currents. Once the positive definiteness ofthis local function will have been shown, the positive definiteness of the global onewill follow by superposition.

For the non-dispersive case each summand of the global Lyapunov energy func-tion [Eq. (44)] includes the two orthogonal values vd and id+1 (via the energies WCd

and WLd+1 ) and additionally the adjacent currents9 id+2, and (the negatively shifted)S−1

d+2id+1 and S−1d+1id+2. This local configuration is also used for the Ln

local,d . But in-stead of just considering the energy of the the current id+1 as in Eq. (44), the energiesof all four surrounding currents are included. If this arrangement is summed up overall directions d and all X as defined in Eq. (46) the energies of the four surroundingcurrents would have been counted four times. Thus a weighting factor of 1/4 must beused for proper energy summation. The advantage of the mentioned re-arrangementof the energy terms is the fact that such a local Lyapunov function can be evaluatedby a quadratic form. The positive definiteness of Ln

local,d is equivalent to the positivedefiniteness of the associated quadratic form matrix A. This can be mostly done byapplying Sylvester’s criterion, which states that all of the leading principal minorsof the square matrix must have a positive determinant for the matrix to be positivedefinite.

Within the following subsections three important practical cases are consideredfor stability analysis: the non-dispersive material, the plasmonic ε-Drude case andfinally the double Drude metamaterial.

4.1 Stability Criterion for Non-Dispersive Materials

By using the advantageous computational configuration for Lnlocal,d(X) where the ad-

mittance Yd is at its center together with only one quarter (above-mentioned weight-ing factors) of the four adjacent and orthogonal impedances, the quadratic form

9 These terms are due to the remaining 4 vi products.


including only the non-dispersive reactive EC elements10 and the voltages/currentvector Θ5 reads11

2 Lnlocal,d = ΘT

5 ·A5 ·Θ5

= ΘT5 ·

⎛⎜⎜⎜⎜⎜⎜⎝

Cd −Δ t2 +Δ t

2 +Δ t2 −Δ t

2−Δ t

2Ld+1

4 0 0 0

+Δ t2 0

S−1d+2Ld+1

4 0 0+Δ t

2 0 0 Ld+24 0

−Δ t2 0 0 0

S−1d+1Ld+2

4

⎞⎟⎟⎟⎟⎟⎟⎠

·

⎛⎜⎜⎜⎜⎜⎜⎜⎝

vnd

in+ 1

2d+1

S−1d+2i

n+ 12

d+1

in+ 1

2d+2

S−1d+1i

n+ 12

d+2

⎞⎟⎟⎟⎟⎟⎟⎟⎠

(47)

Sylvester’s criterion demands that all of the leading principal minors of thesquare matrix must have a positive determinant in order to have a positive overallmatrix, which leads to several stability criteria. Since the determinant of the over-all matrix A5 is the most limiting one for the time step, we have for the maximalΔ tnon-disp

Δ tnon-disp <2

maxXXX∈Ω , d∈{0,1,2}

√1

Cd

(4

Ld+1+ 4

S−1d+2Ld+1

+ 4Ld+2

+ 4S−1

d+1Ld+2

) . (48)

For an uniform mesh the relation Δ tnon-disp =√

3/4 Δ tCFL ≈ 87%Δ tCFL holds,where Δ tCFL is the conventional time step based on the Courant-Friedrich-Levy(CFL) condition [1]. However, in non-uniform (graded) meshes, where the small-est cell is adjacent to a larger one (with at least 33% longer edges), the EC-basedcriterion yields a larger time step. This improvement depends on the size-ratioof directly adjacent cells. For extreme meshing situations an improvement factorΔ tnon-disp/Δ tCFL of 10 can easily be achieved. The ordinary CFL criteria has beenderived for an uniform mesh and a homogeneous material distribution, therefore itseems to be too conservative in highly graded meshes, since the smallest cell sizehas to be used in the formula [1].

4.2 Stability Criterion for Plasmonic (ε-Drude) Materials

For a plasmonic (ε-Drude) medium, which has been deployed in the optical couplerexample to be presented in Sect. 5.2, the admittances at the edges of the EC unit cell[Fig. 4] are extended by a LH inductance LL in parallel, and the quadratic form reads

10 All elements with an index L are set to zero.11 The index 5 should indicate the number of components of this vector.


ΘT6 ·A6 ·Θ6 =

ΘT6 ·

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

Cd −Δ t2 −Δ t

2 +Δ t2 +Δ t

2 −Δ t2

−Δ t2 LL,d 0 0 0 0

−Δ t2 0 Ld+1

4 0 0 0

+Δ t2 0 0

S−1d+2Ld+1

4 0 0+Δ t

2 0 0 0 Ld+24 0

−Δ t2 0 0 0 0

S−1d+1Ld+2

4

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

·

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

vnd

in+ 1

2L,d

in+ 1

2d+1

S−1d+2i

n+ 12

d+1

in+ 1

2d+2

S−1d+1i

n+ 12

d+2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (49)

which yields by Sylvester’s criterion (6×6-matrix determinant) the maximal possi-ble Δ t for an ε-Drude material

Δ tε-Drude <2

maxXXX∈Ω , d∈{0,1,2}

√1

Cd

(1

LL,d+ 4

Ld+1+ 4

S−1d+2Ld+1

+ 4Ld+2

+ 4S−1

d+1Ld+2

)

(50)

Inspection of Eq. (50) confirms that a smaller time step limit must be used com-pared to the non-dispersive case [Δ tε-Drude < Δ tnon-disp] due to the additional term1/LL,d in the denominator. This fact makes the novel criterion even more importantfor stable time domain simulation of plasmonic structures.

4.3 Stability Criterion for Double Drude Materials

For a double Drude (ε-Drude and μ-Drude) medium, the admittances at the edges ofthe EC unit cell are extended by a LH inductance LL in parallel and the impedancesin the faces of the EC unit cell [Fig. 4] are extended by a LH capacitance CL inseries. In this case the voltage/current vector reads

ΘT10

.=(

vnd , i

n+ 12

L,d , in+ 1

2d+1 ,S−1

d+2in+ 1

2d+1 , i

n+ 12

d+2 ,S−1d+1i

n+ 12

d+2 ,

vnL,d+1,S

−1d+2vn

L,d+1,vnL,d+2,S

−1d+1vn

L,d+2

), (51)

i.e. the equation |A10|(Δ t) > 0 must be solved for Δ t. In principle this might bepossible, but the result will be very cumbersome. Therefore the special case

1

ω2se,d

.= Ld+1 CL,d+1 (52)

= S−1d+2Ld+1 S−1

d+2CL,d+1 = Ld+2 CL,d+2 = S−1d+1Ld+2 S−1

d+1CL,d+2


is considered here instead, due to a simplified determination of the stability criterion.Solving the simplified determinant equation yields the maximal possible Δ t for adouble-Drude material

Δ tdouble-Drude <2

maxXXX∈Ω , d∈{0,1,2}√

ω2sh,d +ω2

se,d +ω2mix,d

<2

maxXXX∈Ω , d∈{0,1,2} ωdouble-Drude,d(53)

with

ω2double-Drude,d =

12

(ω2

sh,d +ω2se,d +ω2

mix,d

+√

ω4sh,d +ω4

se,d +ω4mix,d +2

(ω2

se,d ω2mix,d +ω2

sh,d ω2mix,d −ω2

sh,d ω2se,d

))

< ω2sh,d +ω2

se,d +ω2mix,d (54)

where

ω2sh,d =

1LL,d Cd

, (55)

ω2mix,d = 16 ω2

sh,d ω2se,d Ld CL,se,d with (56)

CL,se,d =14

(CL,d+1 +S−1

d+2CL,d+1 +CL,d+2 +S−1d+1CL,d+2

). (57)

The simplified denominator [ω2sh,d + ω2

se,d + ω2mix,d ] in equation (53) yields a

slightly lower limit for the time step, but is easier to compute. The term is an up-per estimate for the complicated ωdouble-Drude,d . Inspection of Eq. (53) reveals that asmaller time step limit must be used compared to the non-dispersive and also theε-Drude case [Δ tdouble-Drude < Δ tε-Drude < Δ tnon-disp], due to the additional termω2

se,d in the denominator of Eq. (53).

5 Simulation Examples

This section presents examples of dispersive material structures and devices com-puted by the EC FDTD method. The dispersive materials used are metamaterials inSect. 5.1 and natural materials in Sect. 5.2.

5.1 Applications based on Dispersive Metamaterials

A metamaterial is an artificial structure which is electromagnetically homogeneous(i.e. possesses a structural unit cell much smaller than the guided wavelength,


p � λg) and which exhibits unusual properties not readily found in nature, suchas for instance a negative refractive index (NRI) [4]. Metamaterials are generallydispersive media. This section presents a spatial-frequencial refractive filter andan open surface-plasmonic resonator. In addition, it discusses a few other possiblemetamaterial applications.

5.1.1 Spatial-Frequencial Refractive Filter

The spatial-frequencial refractive filter is shown in Fig. 7. It consists of a compositeright/left-handed (CRLH) [4] wedge placed in vacuum. This wedge is excited bya two-tone Gaussian (plane) wave with the lower frequency ( fl) in its left-handedrange and the higher frequency ( fh) in its right-handed range. As a consequence, thelower frequency part of the signal experiences negative refraction and is thereforerefracted toward the left, while the higher frequency part of the signal experiencespositive refraction and is therefore refracted toward the right. The structure pre-sented is specifically designed such that, with a 11.3◦ inclined-plane wedge, theoutput beams refract under the angles ±30◦ with respect to the direction of the inci-dent wave. The structure operates thus as a spatial (±θ) – frequencial ( fl/ fh) filter.Such a device may be used as quasi-optical diplexer.

–30°

(a)

+30°

(b)

Fig. 7 Spatial-frequencial refractive filter, consisting of a CRLH wedge excited by a two-tone sig-nal. (a) Negatively refracted lower-frequency beam, for which n(4.21GHz) =−1.63. (b) Positivelyrefracted higher frequency beam, for which n(12.56GHz) = +3.37

5.1.2 Open Surface-Plasmonic Resonator

The open surface-plasmonic resonator is shown in Fig. 8(a). It consists of fourNRI metamaterial (MTM) triangular regions with n = −1 interleaved with comple-


Fig. 8 Open surface plasmonic resonator. The metamaterial (MTM) regions are made of nega-tive refractive index (NRI) medium with n = −1 while the complementary regions are vacuum(n = +1). The structure is bounded in the xy-plane by 4 perfect matched layers (PMLs) for ab-sorption and in the z-direction by two perfect electric conductor (PEC) walls (bottom and top) forTMz operation. (a) Top view of the structure. (b) Ray-tracing description, for a source located atthe position of the black circle (3 rays at different angles are shown).

mentary vacuum (n = +1) regions, and this arrangement is bounded by perfectlymatched layer (PML) absorbing boundaries on the sides and by perfect electric(PEC) walls at the bottom and at the top to ensure TMz operation.

Figure 8(b) shows a naive ray-tracing description of the expected behavior ofthe resontator: due to the negative and iso-density refraction occurring at each in-terface, the electromagnetic wave radiated by a source within the structure shouldbe “trapped”, as indicated, despite the fact that no walls surround the structure toconfined the energy as in a conventional cavity resonator (hence the term “open”resonator). In addition, it is known that the evanescent components (correspondingto the near-field radiation contributions) of a wave incident at the interface betweena right-handed and a NRI medium excite surface plasmons along the interface [8],as suggested in Fig. 9(a). The time snapshots of Figs. 9(b), (c), (d), (e) and (f) illus-trate the refractive traping and surface plasmonic behavior of the resonator. Note thatonly a small amount of energy is radiated through the open boundaries. This typeof “energy traps” may lead to high-quality factor resonators, due to the absence oflossy metal walls, if purely dielectric NRI media are used.

5.1.3 Other Metamaterial Structures

Many other types of metamaterials structures may be easily modeled by the pro-posed EC FDTD method. CRLH metamaterials are double Drude (only one zero


(a) (b) (c)

(d) (e) (f)

Fig. 9 Open surface-plasmonic resonator. (a) Surface plasmon propagation, and distribution ofthe vertical component of the electric field [Ez(x,y)] at the resonance frequency fres at the phaseinstants (b) 0◦, (c) 45◦, (d) 90◦, (e) 135◦, (f) 180◦. Here, the excitation is located at the center ofthe structure

in both ε and μ) media [ε(ω)p = CR − 1/(ω2LL) and μ(ω)p = LR − 1/(ω2CL)].Other metamaterials are double Lorentz (one pole and one zero in both ε and μ) [6]or mixed Drude-Lorentz as the thin wire and split ring resonator structure whichwas the first experimentally verified NRI metamaterial [9] (Fig. 10), and more com-

p

(a)

v2

LP

LR RR

CL

GL

LL

RL

CR

length

(b)

H

E

b S

Fig. 10 Firstly reported practical left-handed metamaterial structure (a) of the mixed Drude-Lorentz short wire and split ring resonator kind [9] and its EC FDTD model (b). The series in-ductance LP and shunt capacitance CR represent the space wave propagating between the wire andring scatterers, the impedance anti-tank LR−CL models the split ring resonator (Lorentz) magneticdipolar effect and the admittance inductance LL models the wire (Drude) electric dipolar effect


plex possibilities exist. Corresponding dispersive EC FDTD models are straightfor-wardly available in the proposed EC FDTD scheme.

5.2 Applications Based on Natural Dispersive Materials

Many natural materials are dispersive. This includes for instance plasmas, ferromag-netic and ferroelectric materials, noble metals at optical frequencies and biologicaltissues. The EC FDTD method, by including various circuit-based dispersive (andalso lossy) models both for ε and for μ (Drude, Lorentz, Debye, and variationsthereof) can ideally analyze structures and devices made of such materials. Thissection presents the example of an optical coupled surface plasmon subwavelengthpower divider and discusses the applicability of the method to biological tissues.

5.2.1 Optical Coupled Surface Plasmon Subwavelength Power Divider

The optical power divider to be discussed here [10] is essentially constituted of athin metallic Ag film within SiO2 (εSiO2 = 2.09) and supporting two surface (in factinterface) surface plasmons coupling to each other. It is appropriate to first providean accurate description of the plasmonic response of Ag at optical frequencies.

It is well known that noble metals, such as silver (Ag), exhibit at optical frequen-cies a Drude-type permittivity response

εDrude(ω) = ε0

{1−

ω2p

ω (ω − jΩ)

}. (58)

This expression with the parameters ωp = 12.9 × 1015rad/s, Ω = 8.0 × 1013

rad/s was used for instance in [11] for the design of a plasmonic contra-directionalcoupler. Since this simple model fits experimental data [12] only in a very narrow-band region around 500 THz, we propose the following extended model

εext-Drude(ω) = ε0εr,∞

{1−

ω2p

ω (ω − jΩ)

}− j

σ0

ω, (59)

which includes the two additional parameters εr,∞ (asymptotic term) and σ0 (con-ductive term) to better fit experimental data in the frequency range of interest.The complex permittivity εr = ε ′r − jε ′′r obtained from this model for Ag withεr,∞ = 3.9, ωp = 7.0× 1015rad/s, Ω = 2.3× 1013rad/s and σ0 = 8.0× 103S/m,yielding the best fit to the measured values of [12] in the frequency range of450 THz–850 THz, is shown in Fig. 11.

The optical coupled surface plasmon subwavelength power divider is presentedin Fig. 12 along with its field distribution. It consists of a tapered metal (Ag) filmembedded in a dielectric (SiO2) material, with the coupling region in the center


Fig. 11 Measured complex permittivity εr = ε ′r − jε ′′r from [12] compared to the data obtained bythe simple Drude model given in Eq. (58) and an extended Drude function given in Eq. (59)

AgAg

SiO�2

SiO�2

x

y

z

λ0

dmax

dmininput (1)

isolated (2)

through (3)

coupled (4)

Fig. 12 Optical coupled surface plasmon subwavelength 3-dB/quadrature power divider (longitu-dinal section view) with intensity plot of the driven-mode TEz magnetic field Hz(x,y; f = 770THz)


and the input/output regions on either sides. The field distribution shown indicates3-dB/quadrature splitting of the incoming guided surface plasmon. The dimensionsare dmin = 15nm, dmax = 240nm, �narrow = 50nm and �taper = 375nm yielding anoverall length of the device, including the tapering sections, of 800 nm. The couplingregion of the device is therefore of ultra-compact subwavelength size, since the free-space wavelength λ0 at the operation frequency of 770 THz is 390 nm.

5.2.2 Ultra Wide-Band Exposure of Human Tissues

Humantissues illuminatedbyultrawide-bandelectromagneticfieldsgenerallyexhibita Debye-type response of the form given by Eq. (37), where the frequencies ωn aretypically determined by experimental and empirical techniques. Table 1 lists typical2-pole corresponding parameters for the brain, for muscles and for bones [13]. Thisbroadband model is accurate in the frequency range from 100 MHz up to 6 GHz.

Table 1 Parameters for a 2-pole Debye model for wide-band time-domain simulation for differenthuman tissues

Parameter Brain Muscles Bones

εr,∞ 20.20 21.02 5.48σ0 in S/m 0.355 0.699 0.104Δεr,1 30.89 12.08 3.19Δεr,2 24.44 34.20 6.20f1 in MHz 164.8 212.8 239.6f2 in GHz 9.086 11.737 6.707

Recently, specific emphasis has been put on ultra wide-band (UWB) imagingtechniques for medical diagnosis and biological detection, which are based, in con-trast to X-ray imaging, on low power and non-ionizing radiation while still provid-ing high resolution. These techniques require not only exact dispersive models butalso broadband analysis capability. Moreover, the complexity of the materials to in-vestigate is such that an explicit time domain technique with no matrix inversion isthe method of choice. In this context, time-domain methods such as the proposedEC FDTD technique may play a crucial role, since they allow almost unlimiteddescription of the geometrical and dispersive properties of these complex materials.

6 Conclusion

The EC FDTD method has been derived for the simplest case of non-dispersivematerials and for the case of dispersive materials, including Drude, Lorentz andDebye dispersions.

Within the framework of the EC FDTD a discretized and circuital version ofthe Poynting theorem both for the time-continuous and time-discrete cases was


established. In the former case the theorem has exactly the same form as the well-known one in the space-and-time-continuous world. Whereas the time-discrete the-orem with its additional power flow terms could be utilized to derive novel stabilitycriteria based on a Lyapunov energy function. With the help of this generalizedenergy function stability criteria have been derived for the non-dispersive, the ε-Drude-dispersive and the double-Drude-dispersive case. At least for these two dis-persive materials the maximal possible time iteration step was decreased comparedto a non-dispersive media due to additional reactances in the 3D equivalent circuit.

Finally several application examples of metamaterial and natural material dis-persive structures and devices have been presented. The diversity of the examplesindicates the versatility of the proposed EC FDTD method.

By incorporating the divergence information of the electrodynamic field, or inthe circuital world of the EC FDTD the conservation of charges, an even improvedstability criterion is expected. This will be investigated in the near future.

References

1. A. Taflove and S. Hagness. Computational Electrodynamics: The Finite-Difference Time-Domain Method, Third Edition, Artech House, 2005.

2. A. Rennings, S. Otto, A. Lauer, C. Caloz, and P. Waldow. “An extended equivalent circuitbased FDTD scheme for the efficient simulation of composite right/left-handed metamateri-als,” Proc. of the Euro. Microw. Assoc., vol. 2, no. 1, pp. 71–82, March 2006.

3. A. Rennings, S. Otto, C. Caloz, A. Lauer, W. Bilgic, and P. Waldow. “Composite right/left-handed extended equivalent circuit (CRLH-EEC) FDTD: stability, dispersion analysis withexamples,” Int. J. Numer. Modell., vol. 19, no. 2, pp. 141–172, March 2006.

4. C. Caloz and T. Itoh. Electromagnetic Metamaterials, Transmission Line Theory and Mi-crowave Applications, Wiley and IEEE Press, 2005.

5. E. M. Lifshitz, L. D. Landau, and L. P. Pitaevskii. Electrodynamics of Continuous Media:Volume 8, Second Edition, Butterworth-Heinemann, 1984.

6. A. Rennings, T. Liebig, C. Caloz, and I. Wolff. “Double Lorentz transmission line metamate-rials and their applications to triband devices,” IEEE MTT-S Int. Microwave Symp., Honolulu,USA, June 2007, pp. 1427–1430.

7. N. Rouche, P. Habets, and M. Laloy. Stability Theory by Liapunov’s Direct Method,Springer, 1977.

8. J. B. Pendry. “Negative refraction makes a perfect lens,” Phys. Rev. Lett., vol. 85, no. 18,pp. 3966–3969, Oct. 2000.

9. R. A. Shelby, D. R. Smith, and S. Schultz. “Experimental verification of a negative index ofrefraction,” Science, vol. 292, pp. 77–79, April 2001.

10. A. Rennings, J. Mosig, S. Gupta, C. Caloz, R. Kashyap, D. Erni, and P. Waldow, “Ultra-compact power splitter based on coupled surface plasmons,” International Symposium on Sig-nals, Systems and Electronics (ISSSE), Montreal, Canada, July–Aug. 2007, pp. 471–474.

11. Y. Wang, R. Islam, and G. V. Eleftheriades, “An ultra-short contra-directional coupler utili-zing surface plasmon-polaritons at optical frequencies,” Opt. Express, vol. 14, no. 16, pp.7279–7290, 2006.

12. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phy. Rev. B 6,pp. 4370–4379, 1972.

13. T. Wuren, T. Takai, M. Fujii, and I. Sakagami, “Effective 2-Debye-pole FDTD model ofelectromagnetic interaction between whole human body and UWB radiation,” Microw. Wirel.Compon. Lett., vol. 17, no. 7, pp. 483–485, 2007.

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