Fourier-Domain Electromagnetic Wave Theory for LayeredMetamaterials of Finite Extent

Kenneth J. Chau,1, ∗ Mohammed H. Al Shakhs,2 and Peter Ott2

1School of Engineering, The University of British Columbia, Kelowna, British Columbia, Canada2Heilbronn University, Heilbronn, Germany

compiled: September 22, 2019

Stratified media consisting of metal and dielectric layers have been presented on the premise that they aremetamaterials with electromagnetic properties describable by plane-wave parameters obtained by the process ofhomogenization. The validity of this assumption rests upon consistency between derived plane-wave parametersand complete wave solutions to Maxwell’s equations, a comparison most easily performed if the wave solutionis expressed in the Fourier domain. Here, we analytically develop the general Fourier-domain electromagneticwave solution in a lossy, layered medium of finite extent, condensing the solution into a compact productof three terms: one governed by reflections from the medium boundaries, a second associated with complex-valued Floquet-Bloch modes due to layer periodicity, and a third that is dependent on layer thickness andcomposition. Decomposition of the wave solution into simpler parts enables the underlying mechanics of thesolution to be understood by inspection and provides insight into its dependence on physical parameters farbeyond that which can be inferred by plane-wave parameters. Numerical examples are presented in whichthe factorized, Fourier-domain wave solution is used to examine the electromagnetic fields and power flow invarious metal-dielectric layered structures and to check the accuracy of plane-wave parameters derived fromhomogenization methods including effective medium theory, scattering parameter retrieval, and Floquet-Blochanalysis.

1. Introduction

Recent contributions to the long tradition of inquiry intothe electromagnetic properties of planar layered struc-tures [1–10] have been sparked by the novel conceptual-ization of these structures - particularly those composedof sub-wavelength-thick layers of metal - as metama-terials. The metamaterial concept is used to describea structure with sub-wavelength scale heterogeneity interms of plane-wave parameters such as refractive indexand impedance. Planar layered structures, which pos-sess heterogeneity along just a single direction, are thesimplest metamaterial form and provide an experimen-tally feasible template for metamaterial devices operat-ing at visible frequencies and beyond due to the availabil-ity of thin film deposition techniques with layer thicknesscontrol on sub-nanometer scales. Exciting applicationsfor layered metamaterials include flat lens imaging [11–14] and analog computation [15].

Classification of a heterogeneous structure as a meta-material begins by seeking an analog homogeneousstructure possessing a plane-wave solution that mim-ics the more intricate wave solution corresponding theoriginal structure. This process, known as homoge-nization, yields familiar plane-wave parameters to ap-proximate wave behavior in the heterogeneous struc-ture. Each homogenization technique invokes a unique

∗ Corresponding author: kenneth.chau@ubc.ca

set of assumptions, which are not always justified, toarrive at its plane-wave parameters. Effective mediumtheory can be used to define an effective permittivitytensor through volumetric averaging of the local permit-tivity values [2, 16, 17], which, for a layered structure,simplifies to a thickness-weighted average of the layerpermittivity values. Although effective medium theoryis intuitive, it relies upon the electrostatic approxima-tion which neglects time-derivative terms in Maxwell’sequations. The scattering parameter method [18–21] isbased on equating the reflection and transmission co-efficients of a heterogeneous structure to those of anequivalent homogeneous structure. Drawbacks includenon-uniqueness [22, 23] and the absence of correlationto the fields inside the structure. It is possible to deriveeffective constitutive parameters by averaging local per-mittivity values weighted by the fields [24] or energy den-sities [25] inside a structure, although this method alsosuffers from non-uniqueness. If a structure is periodic,a unique set of homogeneous parameters can be definedusing the Floquet-Bloch theorem [26, 27]. The waveinside a structure is decomposed into a set of Floquet-Bloch modes kFB [28–31] and if one mode carries dom-inant power, it is assumed to approximate the entirewave and its plane-wave parameters are conferred to thestructure [32–38]. The Floquet-Bloch modes of a peri-odic layered structure can be found by imposing trans-lational invariance of the wave over a period within amultiplicative exponential factor [7], a procedure thatimplicitly assumes infinite extent and no loss. When the

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medium is lossy, the Floquet-Bloch modes are complex-valued [39–41], but no longer discrete [32]. When themedium is finite, translational invariance is altogetherlost due to reflections from the end facets of the mediumand a dependence on the excitation conditions in the ex-ternal bounding media [29].

The validity of the metamaterial approximation de-pends upon consistency between plane-wave parametersderived by homogenization and complete wave solutionsto Maxwell’s equations. Direct comparison between thetwo can be challenging because plane-wave parametersare most naturally interpreted in the spatial-frequencydomain, whereas wave solutions from numerical solversand analytical techniques (such as those based on matrixformalism [3, 7, 9, 10, 42]) are commonly communicatedin the spatial domain. The obvious resolution is to ob-tain Fourier-domain wave solutions, but the most com-mon solution techniques based on Fourier expansion of asingle field component (known as the plane-wave expan-sion method [43, 44]) or Fourier expansion of the elec-tric and magnetic field components into a series of planewaves [30, 31, 38] are limited to lossless, non-dispersive,and infinitely periodic media not applicable to finite-sized, metal-dielectric metamaterials. Recently, Fouriertransformation of the numerically-computed wave solu-tion in a finite-sized, lossless periodic layered mediumhas revealed a correlation between the spatial frequencycontent of the wave and the Floquet-Bloch modes cor-responding to an unbounded version of the periodicmedium [22]. This correlation, however, has not beenmade mathematically explicit and has yet to be estab-lished for the case of loss, a condition vital for analysisof metamaterials which frequently incorporate metals.

The goal of this work is to analytically derive theFourier-domain electromagnetic wave solution for a gen-eral layered configuration that accommodates periodicor aperiodic layer arrangements under the realistic con-straints of loss and finite extent. The wave solution isdistilled into a compact, factorized form consisting of aterm due to reflections from the medium boundaries, an-other associated with the complex-valued Floquet-Blochmodes of the unit cell, and a third that is dependenton the layer thickness and composition. In contrast toplane-wave parameters obtained by homogenization, thefactorized, Fourier-domain wave solution is fully consis-tent with Maxwell’s equations and provides physical in-sight into the effects of loss, finite extent, and period-icity. We apply the wave solution to analyze the elec-tromagnetic fields and power flow in several representa-tive metal-dielectric layered metamaterials and to val-idate plane-wave parameters obtained by homogeniza-tion techniques such as effective medium theory, scat-tering parameter retrieval, and Floquet-Bloch analysis.Of the three, Floquet-Bloch analysis provides the clos-est approximation to the complete wave solution, an ob-servation that is consistent with previous reports [22]and can now be explained by the explicit appearance ofFloquet-Bloch modes in the factorized, Fourier-domain

wave solution.A Fourier-domain wave theory contributes to the res-

olution of the following research questions pertinent tolayered implementations of metamaterials: How accu-rately do plane-wave solutions derived by homogeniza-tion approximate the complete wave solution? How dothe conditions of material loss and finite extent, bothof which are important for the analysis of any practicallayered metamaterial device, affect the validity of themetamaterial concept? Is there a systematic method tounderstand the relationship between the internal electro-magnetic fields and power flow in a layered structure andthe physical parameters of the structure? Finally, arethere better ways to describe the electromagnetic prop-erties of layered metamaterials than through approxi-mate plane-wave solutions provided by homogenization?

2. Generalized Description of a Layered Medium

We consider a generalized one-dimensionally periodicmedium (Figure 1) immersed in free space and com-posed of M repeated unit cells, each consisting of Jlayers, yielding a total of MJ layers. The unit cellsare referenced by the integer m = 0, ...,M − 1 and thelayers within any unit cell are referenced by the inte-ger j = 1, ..., J . Each layer in the medium is uniquelylabeled by the integer ` = mJ + j = 1, ...,MJ and,for sequential consistency, the free-space half-spaces tothe left and right of the medium are labeled ` = 0 and` = MJ + 1, respectively. Layer ` has a thickness ofd` and its linear electromagnetic properties are gener-ally specified by a complex-valued relative permittiv-ity ε` (the underline denotes a complex variable) anda complex-valued relative permeability µ

`, resulting in a

complex-valued refractive index [45]

n` = sgn(<[ε`]|µ`|+ <[µ`]|ε`|)

√ε`µ`, (1)

which is permitted to have a real part that is posi-tive (right-handed) or negative (left-handed). The total

thickness of the layered medium is L =∑MJ`=1 d`. Due to

periodic repetition of the unit cell, the quantities d`, ε`,µ`, and n`, corresponding to layer j of an arbitrary unit

cell m, can be equivalently denoted dj , εj , µj , and njcorresponding to layer j of unit cell m = 0. The plane ofthe layers is aligned parallel to the xy plane and we de-note the location of the plane between layer ` and `+1 asz`, setting the position of the plane of the first interfacez0 = 0 without loss of generality. An electromagneticplane wave is incident onto the medium from the half-space z < 0 inclined at an angle θ in the xz plane. Dueto the independence of this configuration with respectto the y coordinate, any solution can be expressed as alinear combination of solutions obtained by assuming ei-ther transverse-electric (TE) polarization (electric field~E aligned to the y axis) or transverse-magnetic (TM)

polarization (magnetic field ~H aligned to the y axis).Here, we treat the case of TM polarization, noting that

the transformations ~E → −~H, ~H → ~E, and ε� µ yield

3

complimentary equations for TE polarization [9].

3. Wave Solution in the Spatial Domain

We derive a representation of the electromagnetic wavesolution in the finite one-dimensional periodic layeredstructure for the case in which one of the end facetsis subject to plane-wave illumination. An incident

TM-polarized electromagnetic wave is given by ~H =yH0e

i(kx,0x+kz,0z), where H0 is the amplitude, kx,0 andkz,0 are the real-valued wave-vector components along

the x- and z-axes, respectively, and k0 =√kx,0 + kz,0 is

the free-space wave vector. The wave is time-harmonicwhere an e−iωt dependence is assumed but suppressed.Invoking field continuity across the interfaces, the mag-netic field in an arbitrary layer ` can be written as a sumof two counter-propagating waves using matrix formal-ism

~H`(x, z) = H`(x, z)y

= eikx,0x(

eikz,`(z−z`)

e−ikz,`(z−z`)

)T (A`B`

)y,

(2)

where T denotes the transpose operator, A` and B` arethe wave coefficients, and kz,` is the wave-vector com-ponent in layer ` along the z-axis (note that kz,` corre-sponding to layer j of an arbitrary unit cell m can beequivalently denoted kz,j corresponding to layer j of unitcell m = 0). The wave-vector component kz,` is related

to the layer refractive index by

kz,` = n`

√k20 −

(kx,0n`

)2

, (3)

where, according to Eqn. 1, n` can have a real partthat is either positive or negative, describing a right- orleft-handed medium, respectively. If a medium is right-handed, A` describes waves with phase velocities alongthe +z direction and B` describes waves with phasevelocities along the −z direction. If the layer is left-handed, A` describes waves with phase velocities alongthe −z direction and B` describes waves with phase ve-locities along the +z direction.

Assuming a linear electromagnetic response, the elec-tric field in layer ` can then be obtained from theAmpere-Maxwell law and written in matrix formalismas

~E`(x, z) = Ex,`x+ Ez,`z

=eikx,0x

ε`ε0ω

(kz,`e

ikz,`(z−z`)

−kz,`e−ikz,`(z−z`)

)T (A`B`

)x

+eikx,0x

ε`ε0ω

(−kx,0eikz,`(z−z`)

−kx,0e−ikz,`(z−z`))T (

A`B`

)z,

(4)

where ε0 is the free-space permittivity. For complete-ness, we also express the time-averaged Poynting vectorin layer ` as

〈~S`(x, z)〉 =1

2<(~E` × ~H

∗` )

=1

2<[kx,0ε`ε0ω

(|A`|2 − |B`|2 +A`B

∗`e

2ikz,`z −A∗`B`e−2ikz,`z)]x

+1

2<[kz,`ε`ε0ω

(|A`|2 − |B`|2 +A`B

∗`e

2ikz,`z −A∗`B`e−2ikz,`z)]z,

(5)

where, in the limit of no loss, |A`|2 describes forwardpower flow along the +z direction and |B`|2 describesbackwards power flow along the −z direction. If thelayer is lossless and also right-handed, both A` andB` describe forward-propagating waves having parallelphase velocity and power flow. If the layer is lossless andalso left-handed, both A` and B` describe backwards-propagating waves having anti-parallel phase velocityand power flow.

The wave solution can now be solved by relating thewave coefficients A` and B` across the MJ + 1 bound-aries. The wave coefficients in an arbitrary layer ` canbe related to the coefficients in an adjacent layer ` + 1

by (A`+1

B`+1

)= T `P `

(A`B`

), (6)

where the propagation matrix P ` corresponding to layer` is given by

P ` =

(eikz,`d` 0

0 e−ikz,`d`

), (7)

and the transmission matrix T ` corresponding to theinterface between layer ` and `+ 1 is given by

T ` =1

2

(1 + p

`1− p

`1− p

`1 + p

`

), (8)

4

Fig. 1. Geometry under consideration consisting of a one-dimensional periodic layered medium immersed in free space andcomposed of M repeated unit cells, each consisting of J layers. The medium is excited from one half-space by an incident planeinclined at an arbitrary angle θ in the xz plane.

with p`

= (ε`+1kz,`)/(ε`kz,`+1). Assuming uni-directional wave excitation from the left half-space, thewave coefficients in layer 1 are related to the coefficientsin the left half-space by(

A1

B1

)= T 0

(A0

B0

)= T 0

(1r

), (9)

where r is the complex-valued reflection coefficient andthe incident wave amplitude has been assumed to beunity. The wave coefficients in layer MJ are related tothe coefficients in the right half-space by(

AMJ+1

BMJ+1

)=

(t0

)= TMJ

(AMJ

BMJ

), (10)

where t is the complex-valued transmission coefficient.Relation of the wave across the MJ+1 boundaries yields2MJ + 2 linear equations, which is sufficient to solve forthe 2MJ+2 unknowns (r and t, in addition to the 2MJwave coefficients in the MJ layers).

Upon solving for the unknown quantities, the fields~H` and ~E` in each layer ` are completely specified andwe can succinctly express the total field distributions inthe spatial domain as

~H(x, z) = H(x, z)y =

MJ∑`=1

rect

(z − zc,`d`

)H`(x, z)y

(11)and

~E(x, z) = Ex(x, z)x+ Ez(x, z)z

=

MJ∑`=1

rect

(z − zc,`d`

)[Ex,`(x, z)x+ Ez,`(x, z)z],

(12)

where zc,` is the location of the center of layer ` and therect function is defined as

rect

(z − zc,`d`

)=

{1 zc,` − d`/2 ≤ z ≤ zc,` + d`/2

0 otherwise.

The solutions given by Eqns. 11 and 12 offer valid rep-resentations of the fields as piece-wise functions subdi-vided into spatial intervals corresponding to the layerregions. Although this form is amenable to numericalroutines for solving sets of linear equations, there areat least two disadvantages to using piece-wise spatial-domain representations of the fields to study layeredsystems. First, compartmentalization of the wave so-lution into the individual layers does not afford phys-ical insight into the collective behavior of the solutionacross repeated sets of layers. Second, representationof the solution in the spatial domain does not produceimmediate connections to homogenization parameters,which are generally represented in the spatial-frequencydomain. In the next section, we apply Fourier transfor-mation to the piece-wise wave solution and demonstratethe utility of this approach in decomposing the wave so-lution into meaningful terms with direct connection toFloquet-Bloch theory.

4. Factorized Fourier-Domain Wave SolutionWe re-express the general wave solution given by Eqn. 11in the spatial frequency domain by

H(κx, κz) =

∫ zMJ

0

∫ ∞−∞

H(x, z)e−iκxxe−iκzzdxdz,

(13)where κx and κz are the spatial frequency variablesalong the respective x and z directions. Substitution

5

of Eqn. 11 into Eqn. 13 and development of the inte-grand using well-known Fourier relations and theoremsyields

H(κx, κz) = (2π)2δ(κx − kx,0)

MJ∑`=1

d` sinc

(κzd`2π

)

e−iκzzc,` ∗(e−ikz,`z`−1δ(κz − kz,`)eikz,`z`−1δ(κz + kz,`)

)T(A`B`

),

(14)

where δ is the delta dirac function. We now re-write thesingle summation in Eqn. 14 as a nested double sum-mation over the number of layers in a unit cell and thenumber of unit cells by making the variable substitu-tions d` = dj and kz,` = kz,j and the index substitution` = mJ + j, resulting in

H(κx, κz) = (2π)2δ(κx − kx,0)

M−1∑m=0

J∑j=1

dj sinc

(κzdj2π

)

e−iκzzc,mJ+j ∗(e−ikz,jzmJ+j−1δ(κz − kz,j)eikz,jzmJ+j−1δ(κz + kz,j)

)T(AmJ+jBmJ+j

).

(15)

Carrying out the convolution operation in Eqn. 15 andusing the relation zc,mJ+j − zmJ+j−1 = dj/2 yields

H(κx, κz) = (2π)2δ(κx − kx,0)

M−1∑m=0

J∑j=1

dje−iκzzc,mJ+j

(eikz,jdj/2 sinc[(κz − kz,j)dj/2π]

e−ikz,jdj/2 sinc[(κz + kz,j)dj/2π]

)T(AmJ+jBmJ+j

).

(16)

The unit cell summation in Eqn. 16 can be simplifiedusing the relationship

zc,mJ+j = mD + zj−1 + dj/2, (17)

where D =∑Jj=1 dj is the thickness of the unit cell and

zj−1 is the position of the interface between layer j − 1and j within unit cell m = 0. Substitution of Eqn. 17

into Eqn. 16 gives

H(κx, κz) = (2π)2δ(κx − kx,0)

J∑j=1

dje−iκzzj−1

(e−i(κz−kz,j)dj/2 sinc[(κz − kz,j)dj/2π]

e−i(κz+kz,j)dj/2 sinc[(κz + kz,j)dj/2π]

)TM−1∑m=0

e−iκzmD(AmJ+jBmJ+j

).

(18)

Equation 18 expresses the wave solution in terms of ar-bitrary wave coefficients AmJ+j and BmJ+j distributedthroughout the medium. The solution can be furthersimplified in terms of the wave coefficients in just thefirst unit cell by using the matrix relationship betweenwave coefficients in different layers. The wave coeffi-cients in layer ` are related to the coefficients in an arbi-trary layer `+ s (where the integer s ≤MJ − `) withinthe layered medium by(

A`+sB`+s

)= W `+s,`

(A`B`

), (19)

where the transfer matrix W `+s,` is determined from thetransmission and propagation matrices by

W `+s,` =

`+s−1∏q=`

T qP q. (20)

The coefficients AmJ+j and BmJ+j corresponding tolayer j within an arbitrary unit cell m can be relatedto the coefficients Aj and Bj corresponding to layer jwithin unit cell m = 0 by(

AmJ+jBmJ+j

)= Umj

(AjBj

), (21)

where U j is the unit cell transfer matrix from layer j toj + J and can be expressed as

U j = W j+J,j . (22)

We can now further simplify Eqn. 18 to

H(κx, κz) = (2π)2δ(κx − kx,0)

J∑j=1

dje−iκzzj−1

(e−i(κz−kz,j)dj/2 sinc[(κz − kz,j)dj/2π]

e−i(κz+kz,j)dj/2 sinc[(κz + kz,j)dj/2π]

)T(M−1∑m=0

(e−iκzDU j

)m)( AjBj

).

(23)

Note that the matrix element in the layer summationterm in Eqn. 23 contains an upper (nominally labeled

6

“forward”) component associated with a sinc functioncentered at +kz,j and a lower (nominally labeled “back-ward”) component associated with a sinc function cen-tered at −kz,j . The unit cell transfer matrix U j , whichis referenced from the layer j, can be related to the unitcell transfer matrix referenced from layer 1 using therelation

Umj = W j,1 Um1 W−1j,1 . (24)

Substitution of Eqn. 24 into Eqn. 23 yields

H(κx, κz) = (2π)2δ(κx − kx,0)

J∑j=1

dje−iκzzj−1

(e−i(κz−kz,j)dj/2 sinc[(κz − kz,j)dj/2π]

e−i(κz+kz,j)dj/2 sinc[(κz + kz,j)dj/2π]

)T

W j,1

(M−1∑m=0

(e−iκzDU1

)m)W−1j,1 ·

(AjBj

),

(25)

and application of the transfer matrix W−1j,1 to the wavecoefficients in layer j results in

H(κx, κz) = (2π)2δ(κx − kx,0)

J∑j=1

dje−iκzzj−1

(e−i(κz−kz,j)dj/2 sinc[(κz − kz,j)dj/2π]

e−i(κz+kz,j)dj/2 sinc[(κz + kz,j)dj/2π]

)T

W j,1

(M−1∑m=0

(e−iκzDU1

)m)( A1

B1

).

(26)

Applying an eigen-decomposition to the unit cell trans-fer matrix, U1 = Q λ Q−1, where λ is the diago-nal eigenvalue matrix whose diagonal elements are thecorresponding eigenvalues λ1,2, and using the identity

Um1 = Q λm Q−1, Eqn. 26 can be re-written as

H(κx, κz) = (2π)2δ(κx − kx,0)

J∑j=1

dje−iκzzj−1

(e−i(κz−kz,j)dj/2 sinc[(κz − kz,j)dj/2π]

e−i(κz+kz,j)dj/2 sinc[(κz + kz,j)dj/2π]

)T

W j,1Q

(M−1∑m=0

(e−iκzDλ

)m)Q−1T 0

(1r

).

(27)

At this point, the wave solution has been condensed intotwo manageable summation terms: one summed over thenumber of layers in the unit cell and another summedover the number of unit cells describing the overall peri-odicity of the structure. Additional insight can be gained

concerning the latter term. Because the determinant ofthe unit cell transfer matrix is unity, the diagonal el-ements of the eigenvalue matrix λ are inverses of eachother, λ2 = 1/λ1. We therefore introduce the Floquet-Bloch mode, kFB , which is related to the eigenvalue λ1of the unit cell matrix by

kFB =−i ln(λ1)

D, (28)

and can be re-written as

λ1 = eikFBD, (29)

corresponding to a solution that is nominally labeled the“forward” Floquet-Bloch solution. Similarly, the eigen-value λ2 is related to kFB by

λ2 = e−ikFBD, (30)

corresponding to a solution that is nominally labeledthe “backward” Floquet-Bloch solution. We note thatin typical applications of Floquet-Bloch analysis, thebackward Floquet-Bloch solution is discarded a priori,a procedure which we will later show is valid for infiniteperiodic systems but invalid for finite periodic systems.

Let’s now more closely examine the unit cell summa-tion term in Eqn. 27. Writing the eigenvalue matrix interms of the Floquet-Bloch mode, the unit cell summa-tion can be expanded to yield

M−1∑m=0

(e−iκzDλ

)m=

M−1∑m=0

(e−i(κz−kFB)D 0

0 e−i(κz+kFB)D

)m

=

(1−e−i(κz−kFB)MD

1−e−i(κz−kFB)D 0

0 1−e−i(κz+kFB)MD

1−e−i(κz+kFB)D

).

(31)

The diagonal elements in Eqn. 31 can be re-written as

1− e−i(κz±kFB)MD

1− e−i(κz±kFB)D=

e−i(κz±kFB)M−12 DM

sin[M(κz ± kFB)D/2]

M sin[(κz ± kFB)D/2],

(32)

which contains an expression that is related to the peri-odic Dirichlet function

sin[M(κz ± kFB)D/2]

M sin[(κz ± kFB)D/2]=

∆ 2πD

[κz ±<(kFB)] ∗ sinc

(M [κz ± i=(kFB)]D/2

2π

),

(33)

where ∆ 2πD

[κz −<(kFB)] is the Dirac comb given by

∆ 2πD

[κz −<(kFB)] =

∞∑N=−∞

δ [κz − 2πN/D −<(kFB)] .

(34)

7

Equation 31 can now be written in the insightful form

1− e−i(κz±kFB)MD

1− e−i(κz±kFB)D=e−i(κz±kFB)M−1

2 D×

M∆ 2πD

[κz ±<(kFB)]∗

sinc

(M [κz ± i=(kFB)]D/2

2π

).

(35)

From Eqn. 35, the basic effects of periodicity on the wavesolution can be understood. The Dirac comb indicates

the presence of spatial frequency harmonics spaced by2π/D with a principle harmonic located at the real partof kFB . The Dirac comb, however, is convolved with asinc function that widens the harmonics, either throughthe effect of a finite number of unit cells M or the in-fluence of material losses through the imaginary partof kFB . It is noted that this result has some similari-ties with description of far-field diffraction from a finitegrating, which is not too surprising given the commonreliance of Fourier transformation along periodic struc-tures [17].

Putting all the pieces of the magnetic field solutiontogether yields

H(κx, κz) = (2π)2δ(κx − kx,0)

J∑j=1

dje−iκzzj−1

(e−i(κz−kz,j)dj/2 sinc[(κz − kz,j)dj/2π]

e−i(κz+kz,j)dj/2 sinc[(κz + kz,j)dj/2π]

)TW j,1Q︸ ︷︷ ︸

Lj(e−i(κz−kFB)M−1

2 DMsin[M(κz−kFB)D/2]M sin[(κz−kFB)D/2] 0

0 e−i(κz+kFB)M−12 DM

sin[M(κz+kFB)D/2]M sin[(κz+kFB)D/2]

)︸ ︷︷ ︸

FB

Q−1T 0

(1r

)︸ ︷︷ ︸

C

,

(36)

which can be compactly written as

H(κx, κz) = (2π)2δ(κx − kx,0)

J∑j=1

Lj

FB C, (37)

consisting of a product of three factors including a layer-

dependent term Lj determined by the composition andgeometry of the jth layer of the unit cell, a Floquet-Bloch-dependent term FB determined by the periodic-ity of the unit cell, and a coefficient term C dependent,in part, on reflection from the first facet of the medium.The corresponding x- and z-components of the electricfield solution can be similarly expressed as

Ex(κx, κz) = (2π)2δ(κx − kx,0)

J∑j=1

djεjε0ω

e−iκzzj−1

(kz,je

−i(κz−kz,j)dj/2 sinc[(κz − kz,j)dj/2π]

−kz,je−i(κz+kz,j)dj/2 sinc[(κz + kz,j)dj/2π]

)TW j,1Q FB C

(38)

and

Ez(κx, κz) = (2π)2δ(κx − kx,0)

J∑j=1

djεjε0ω

e−iκzzj−1

(kx,0e

−i(κz−kz,j)dj/2 sinc[(κz − kz,j)dj/2π]

−kx,0e−i(κz+kz,j)dj/2 sinc[(κz + kz,j)dj/2π]

)TW j,1Q FB C,

(39)

respectively. Given the vector spectral magnetic field ~H(κx, κz) = H(κx, κz)y and the vector spectral electric

8

field ~E(κx, κz) = Ex(κx, κz)x+ Ez(κx, κz)z, it is possi-ble to define the spectral time-averaged Poynting vector

〈~S(κx, κz)〉 =1

2<[~E(κx, κz)× ~H

∗(κx, κz)

]. (40)

This is similar to the spectral Poynting vector proposedin Ref. [30, 31] and used in Ref. [38] to analyze energypropagation of discrete Floquet-Bloch modes in infinite,lossless dielectric photonic crystals, except now extendedto accommodate a continuous range of Fourier field com-ponents in a finite, lossy periodic system. It should alsobe noted that the spectral time-averaged Poynting vec-tor defined in Eqn. 40 is not equivalent to the Fouriertransform of the spatial time-averaged Poynting vector,which would involve the convolution of the spectral elec-

tric and magnetic fields. It does, however, enable thespatial frequency κz present in the electric and magneticfields to be envisioned as an electromagnetic plane wavehaving a well-defined time-averaged direction of powerflow, which is useful, for instance, for the characteri-zation of the wave in terms of forward- and backward-propagating wave components.

5. Asymptotic Approach to the Floquet-Bloch Solu-tion

In the case of an infinite repetition of unit cells (M →∞), the diagonal elements of Eqn. 31 (which describethe effect of periodicity on the wave solution) approachthe asymptotic limits

limM→∞

∣∣∣∣1− e−i(κz±kFB)MD

1− e−i(κz−kFB)D

∣∣∣∣ =

{M∆2π/D(κz ± kFB) for =(kFB) = 0

∆2π/D(κz ±<(kFB)) ∗ 1

D/2√κ2z+(=(kFB))2

for =(kFB) 6= 0 , (41)

where we have distinguished two possibilities corre-sponding to the case of either a lossless periodic sys-tem where =(kFB) = 0 or a lossy periodic system where=(kFB) 6= 0. In the case of an infinite system that islossless, the field is convolved with an ideal Dirac comband can be interpreted as a weighted linear superposi-tion of Floquet-Bloch modes, where the relative weightsof the modes are determined by the layer matrices ofthe unit cell. In the case of an infinite system that islossy, the field can be interpreted as a continuum of spa-tial frequency components, distributed among a comb ofpeaks spaced by 2π/D with widths determined by theimaginary part of the Floquet-Bloch mode and a prin-cipal peak located at a spatial frequency correspondingto the real part of the Floquet-Bloch mode.

In the case of a lossy periodic system, it has beenobserved that

limM→∞

Q−1T 0

(1r

)= limM→∞

C =

(c0

), (42)

where c is a positive constant. Since the components ofC serve to weight the forward and backward componentsof the Floquet-Bloch matrix, this limiting behavior indi-cates that an infinite, lossy periodic system sustains onlythe forward Floquet-Bloch modes having =(kFB) > 0.Although this has not been mathematically proven, it isin accordance with the principle of energy conservation,which do not allow for unbounded growth of backwardscomponents of the field.

6. Example: Metal-Dielectric Layered MetamaterialWe next apply the factorized representation of the wavesolution to analyze a metal-dielectric layered structureinspired by recently proposed designs of layered meta-materials [13, 14]. The layered system consists of a

bi-layer unit cell composed of a 30- nm-thick Ag layer(with a complex refractive index 0.076 + 1.605i interpo-lated from experimental data [46]) and a 30- nm-thickTiO2 layer (with a real refractive index of 2.80). Thelayers are excited by a normally incident TM-polarizedwave with a free-space wavelength λ0 = 365 nm. Asshown in Fig. 2, the spectral contents of the wave aredictated by two contributions: the components of thelayer matrix establish a broad spectral envelope andthe components of the Floquet-Bloch matrix define thesharper spectral features within that envelope. Thefinite extent of the envelop functions cancel out con-tributions from higher order Floquet-Bloch harmonics.The principal Floquet-Bloch mode for this system iskFB = 32.5 + 0.4i µm−1 (corresponding to a Floquet-Bloch refractive index nFB = 1.89 + 0.02i), resulting inforward and backward Floquet-Bloch spatial harmonicsthat are shifted about the spatial frequency axis. As thenumber of repetitions increase, the Floquet-Bloch spa-tial harmonics narrow, yielding a series of well-definedpeaks in the total field spectrum. Due to the lossesin the metallic layer, the harmonics have finite widtheven in the limit of an infinitely repeated system, under-scoring the limitation of the asymptotic Floquet-Blochsolution for describing the spectral composition of thewave in lossy periodic systems. The negative spatialfrequency components of the wave are characterized bya time-averaged spectral Poynting vector that is nega-tive, suggesting that these wave components are actu-ally forward-propagating waves (having parallel phaseand energy velocities) that have been simply reflectedin the system. This is counter to the widely heldbelief that the negative spatial frequency componentsof a Floquet-Bloch field decomposition are backward-propagating waves [28]. Both the forward and backward

9

Fig. 2. Decomposition of the wave solution in a metal-dielectric bi-layer system consisting of alternating layers of 30- nm-thickAg and 30- nm-thick TiO2, assuming a normally incident TM-polarized wave with a free-space wavelength of λ0 = 365 nm.a) The forward and backward components of the layer-dependent term L1 corresponding to the 30- nm-thick Ag layer. b)The forward and backward components of the layer-dependent term L2 corresponding to the 30- nm-thick TiO2 layer. c), d),and e) depict the Floquet-Bloch spectrum, magnetic field spectrum, and z-component of the time-averaged spectral Poyntingvector, respectively, for the case of 2 unit cells. f), g), and h) depict the Floquet-Bloch spectrum, magnetic field spectrum, andz-component of the time-averaged spectral Poynting vector, respectively, for the case of 10 unit cells. The horizontal gray linesin e) and h) correspond to zero values of the spectral Poynting vector.

Floquet-Bloch spatial harmonics make contributions tothe total field spectrum, although the forward harmon-ics are dominant, as seen in Fig. 3. In the limit of a largenumber of repetitions, the backward component of thecoefficient term gradually approaches zero, suppressingthe backward Floquet-Bloch mode which has a negativeimaginary part.

It should be noted that the Floquet-Bloch refractiveindex can take on values with negative real parts, par-ticularly in finely layered structures composed of metalat frequencies near the bulk plasma frequency of themetal [14, 47]. The physical significance of the sign of thereal part of the Floquet-Bloch refractive index, however,is not immediately clear. As seen here, the total fieldspectrum is generally dispersed in peaks spanning acrosspositive and negative spatial frequencies and a numberof factors conspire to determine the relative weights ofthese peaks. Moreover, in systems composed of right-handed materials, it has been consistently observed thatthe positive and negative spatial frequency componentsof the wave are forward-propagating waves.

7. Comparison with Parameters Derived from Ho-mogenization Techniques

The Fourier-domain wave solution can be directly com-pared to wave-vector diagrams derived from homoge-nization techniques such as effective medium theory,scattering parameter retrieval, and Floquet-Bloch anal-ysis (a more complete discussion on the application ofthese homogenization techniques to described layeredsystems is provided in the Appendix). We examine theelectromagnetic properties (again for TM polarizationat a free-space wavelength λ0 = 365 nm) of a finite pe-riodic layered system immersed in free space and con-sisting of two bi-layer unit cells, each composed of a 30-nm-thick Ag layer and a 30- nm-thick TiO2 layer. TheFourier-domain wave solution is presented as the mag-netic field spectrum mapped as a function of κz and|κx| < k0, and the wave-vector diagrams are presentedas complex-valued kz obtained by homogenization tech-niques mapped over |kx| < k0. As shown in Fig.4,homogenization yields plane-wave solutions describedby wave-vector diagrams consisting of either a singlediscrete branch (effective medium theory) or a fam-

10

Fig. 3. Forward and backward components of the coefficientterm |C|2 corresponding to a metal-dielectric bi-layer systemconsisting of alternating layers of 30- nm-thick Ag and 30-nm-thick TiO2, assuming a normally incident TM-polarizedwave with a free-space wavelength of λ0 = 365 nm.

ily of discrete branches (scattering parameter retrievaland Floquet-Bloch analysis), whereas the magnetic fieldspectrum consists of a series of broad bands. Effectivemedium theory and scattering parameter retrieval areparticularly poor at describing the spatial-frequency fea-tures of the wave solution; the former yields too few solu-tions and the latter yields too many solutions. Floquet-Bloch analysis gives the closest approximation of thewave solution, with the two most prominent bands ofthe wave solution fairly well aligned to two branches inthe wave vector diagram. This is not surprising giventhe explicit appearance of Floquet-Bloch modes in oneof the terms of the factorized Fourier-domain wave so-lution. However, the more complex features of the wavesolution - variations of the magnetic field amplitude bothwithin a band and between the bands and the presenceof narrower subsidiary bands with weaker amplitudes -are beyond the descriptive capabilities of wave-vectordiagrams obtained by Floquet-Bloch analysis. It shouldbe noted that scattering parameter retrieval applied toa single unit cell of a symmetric layered system withidentical bounding spaces produces the same solutionsas those obtained by Floquet-Bloch analysis due to theequivalence of the two procedures under these condi-tions [20].

8. Conclusion

The classification of a heterogeneous structure as a meta-material is based on approximating the complete wavesolution in the structure with plane-wave solutions ob-tained through the inexact method of homogenization.It is challenging to assess the consistency between plane-

wave solutions and complete wave solutions in generalterms, in part because wave solutions are typically ex-pressed as a function of space and obtained throughnumerical simulation valid only for a particular geom-etry. Thus, we have analytically developed a factorized,Fourier-domain electromagnetic wave solution generallyvalid for lossy layered media of finite extent. Comparedto standard analytical representations of the wave solu-tion in such systems as piece-wise function in the spa-tial domain, our representation offers insight into thecollective behavior of the wave solution across multiplelayers and explicitly reveals a connection to Floquet-Bloch modes associated with periodicity. The factorizedFourier-domain wave solution provides new perspectiveinto the relationship between the electromagnetic fieldsand power flow in a layered metamaterial and its phys-ical parameters. We have presented numerical exam-ples in which the wave solution is directly comparedto plane-wave solutions from homogenization techniquessuch as effective medium theory, scattering parameterretrieval, and Floquet-Bloch analysis. Of these three,Floquet-Bloch analysis provides the most direct link tothe complete wave solution, an observation consistentwith previous reports [22] and the explicit appearance ofFloquet-Bloch modes in one of the terms of the factor-ized wave solution. It is envisioned that this theory willhelp advance the science and technology of layered im-plementations of metamaterials, by providing completesolutions to Maxwell’s equations for these systems ina physically insightful and intuitive way (which can beapplied to study layered metamaterials of any genre)and a means to rigorously and directly validate currentapplications of plane-wave solutions critical to the verydefinition of a metamaterial.

9. AcknowledgmentsThis work was supported by Natural Sciences and Engi-neering Research Council of Canada (NSERC) DiscoveryGrant 366136. KJC thanks Thomas Johnson from TheUniversity of British Columbia and Henri Lezec fromthe National Institute of Standards and Technology forinvaluable discussions.

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Fig. 4. Comparison of wave-vector diagrams describing electromagnetic wave behavior in a periodic layered structure consistingof two bi-layer unit cells, each consisting of a 30-nm-thick layer of Ag and a 30-nm-thick layer of TiO2 for TM-polarization ata free-space wavelength λ0 = 365 nm. Diagrams of the real (solid line) and imaginary (dashed line) parts of the wave vectorderived from a) effective medium theory (green), b) scattering parameter retrieval (red), and c) Floquet-Bloch analysis (blue)are compared against d) the magnetic field spectrum obtained from the Fourier-domain wave solution.

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10. AppendixFor completeness, we detail here the application of effec-tive medium theory, scattering parameter retrieval, andFloquet-Bloch analysis to derive homogenization param-eters for planar layered structures.

10.A. Effective Medium Theory Applied to a Bi-layered StructureThere are several forms of effective medium theory thatcan be used to obtain an effective permittivity tensorof a layered structure. The simplest and most widelyused is the Maxwell-Garnett formulation, which can bederived for the case of a repeated planar bi-layer system(with layer thicknesses d1 and d2 and complex permittiv-ity values ε1 and ε2) by considering the average electricand displacement fields of an incident plane wave fordifferent propagation directions under the electrostaticapproximation. For plane-wave propagation perpendic-ular to the plane of the layers, continuity of the electricfield and volumetric averaging of the electric field overthe bi-layer yields an effective permittivity component

ε⊥ =ε1ε2

f1ε2 + f2ε1, (A.1)

where f1 = d1/(d1 + d2) and f2 = d2/(d1 + d2) are thefractions of total volume occupied by the layers. Forplane-wave propagation parallel to the plane of the lay-ers, continuity of the displacement field and volumet-ric averaging of the displacement field over the bi-layeryields an effective permittivity component

ε‖ = f1ε1 + f2ε2. (A.2)

13

Thus, the bi-layer system acts like a uniaxial crystal withan optical axis perpendicular to the plane of the layer.

10.B. Scattering Parameter Retrieval Applied to aFinite Layered StructureScattering parameter retrieval is a general techniquebased on the conceptual replacement of a finite het-erogeneous system with a hypothetical, finite homoge-neous system, where the parameters of the homogeneousmedium have been retrieved assuming reflection andtransmission coefficients identical to that of the hetero-geneous medium. Here, we detail its application to de-scribe the electromagnetic properties of the finite, lossy,periodic layered system generally depicted in Fig. 1. Thereflection and transmission coefficients of the mediumcan be related to each other using transfer matrix for-malism by (

0t

)= W

(r1

), (A.3)

where the total transfer matrix W can be expressed as

W = WMJ+1,0 =

TMJ+1T−1MJ (U1)

MT 0 =(

W 1,1 W 1,2

W 2,1 W 2,2

).

(A.4)

The reflection and transmission coefficients of themedium can be subsequently written as

r = −W 2,1

W 2,2

(A.5)

and

t = W 1,1 −W 2,1

W 2,2W 1,2

, (A.6)

respectively.We next conceptually replace the medium with a ficti-

tious, homogeneous, effective medium with an identicallength L and immersed in free space. The homogeneousmedium is characterized by an effective permittivity εe,an effective permeability µ

e, an effective refractive index

ne, and an effective impedance pe. The reflection coeffi-

cient, re, and transmission coefficient, te, of the effectivemedium are related to each other by(

0te

)= W e

(re1

), (A.7)

where the total transfer matrix of the effective mediumcan be expressed as

W e =(1 + p

e)2

4pe

×eikz,eL − (1−pe)2

(1+pe)2 e−ikz,e i2

1−pe

1+pe

sin(kz,eL)

−i2 1−pe

1+pe

sin(kz,eL) e−ikz,eL − (1−pe)2

(1+pe)2 e

ikz,eL

,

(A.8)

where kz,e is the z-component of the wave vector in theeffective medium and p

e= kz,e/(εekz,0). The transmis-

sion coefficient of the effective medium can be writtenas

te =4pe

(1 + pe)2e−ikz,eL − (1− p

e)2eikz,eL

=1

cos(kz,eL)− i(1 + p2e) sin(kz,eL)/(2p

e).

(A.9)

Similarly, the reflection coefficient re can be written as

re =(1− p2

e)(eikz,eL − e−ikz,eL

)(1 + p

e)2e−ikz,eL − (1− p

e)2eikz,eL

=i(1− p2

e) sin(kz,eL)

cos(kz,eL)/(2pe)− i(1 + p2

e) sin(kz,eL)/(2p

e).

(A.10)

Equating re = r and te = t, the parameters of the ef-fective medium yielding reflection and transmission coef-ficients identical to that of the layered medium are givenby

ne =± cos−1[(1− r2 + t2)/2t]±N2π

k0L

pe

= ∓

√t2 − (1 + r)2

t2 − (1− r)2, (A.11)

where N is an integer. There are two notable limita-tions associated with these parameters. First, there arean infinite number of possible solutions for the effectiverefractive index. Second, there exist two sets of solu-tions for p

eand ne, each having opposite sign (although

for a system with loss only one set satisfies the physicalconstraints that =(ne) ≥ 0 and <(p

e) ≥ 0). From the

effective parameters given in Eqn. A.11, the remainingeffective parameters can be obtained using the relations

εe = pene

µe

=n2eεe, (A.12)

for which there again exists two set of solutions, eachhaving opposite sign. Care must be taken if solutions forεe and µ

eare then used to calculate ne using Eqn.1 in the

main text, as only one set of solutions for εe and µe

yieldthe correct value of ne. It should also be pointed outthat if the bounding half-spaces are not identical or if theunit cell is not symmetric, scattering parameter retrievalyields different effective parameters in the forward andbackward direction.

10.C. Floquet-Bloch Analysis Applied to an InfinitePeriodic Layered StructureFloquet-Bloch analysis is commonly used to describe theproperties of periodic layered media. Rigorously appli-cable to only infinite periodic media, it is based on the

14

conceptual replacement of the heterogeneous unit cell ofa periodic system with a hypothetical homogeneous unitcell, where the refractive index of the homogeneous unitcell has been selected to have a transfer matrix traceidentical to that of the heterogeneous unit cell. To de-scribe its salient features, we first assume an infinitelyperiodic layered medium having a unit cell with thick-ness D, where the fields at the boundaries of the unitcell are related by the unit cell transfer matrix U . Wenext replace the unit cell with a fictitious, homogeneouseffective unit cell with an identical length D. The fieldsat the boundaries of the effective unit cell are related bythe effective unit cell transfer matrix

Ue =

(eikz,eD 0

0 e−ikz,eD

). (A.13)

Equating the traces of the transfer matrices for the unitcell and effective unit cell yields

tr(Ue) = tr(U)

2 cos(kz,eD) = U1,1 + U2,2. (A.14)

Isolating the effective wave vector to one side ofEqn. A.14 yields

kz,eD = ± cos−1(U1,1 + U2,2

2

)±N2π, (A.15)

where, notably, the effective wave vector is non-unique.Equation A.15 can be further simplified by re-writingthe trace of the unit cell matrix in terms of the Floquet-Bloch mode

tr(U) = λ1 + λ2

= eikFBD + e−ikFBD. (A.16)

Substituting Eqn. A.16 into Eqn. A.15 yields the com-pact relation

kz,e = kFB ± 2πN/D, (A.17)

which describes a infinitely large family of discreteFloquet-Bloch modes. It is interesting to note that incontrast to scattering parameter retrieval, which gen-erates two effective parameters based on two distinctequations (one for the transmission coefficient givenby Eqn. A.9 and another for the reflection coefficientgiven by Eqn. A.9), Floquet-Bloch mode analysis gener-ates just one effective parameter based on one equation(equating the traces of the transfer matrices given inEqn. A.14).