Effective parameters of metamaterials: a rigoroushomogenization theory via Whitney interpolation

Igor Tsukerman

Department of Electrical and Computer Engineering, The University of Akron,Akron, Ohio 44325-3904, USA (igor@uakron.edu)

Received November 17, 2010; accepted December 16, 2010;posted January 10, 2011 (Doc. ID 138379); published February 28, 2011

A rigorous homogenization theory of metamaterials—artificial periodic structures judiciously designed to controlthe propagation of electromagnetic (EM) waves—is developed. The theory is an amalgamation of two concepts:Smith and Pendry’s physical insight into field averaging and the mathematical framework of Whitney-like inter-polation. All coarse-grained fields are unambiguously defined and satisfyMaxwell’s equations exactly. Fields withtangential and normal continuity across boundaries are associated with two different kinds of interpolation, whichreveals the physical and mathematical origin of “artificial magnetism.” The new approach is illustrated with sev-eral examples and agrees well with the established results (e.g., theMaxwell–Garnett formula and the zero cell-sizelimit) within the range of applicability of the latter. The sources of approximation error and the respective suitableerror indicators are clearly identified, along with systematic routes for improving the accuracy further. The pro-posed methodology should be applicable in areas beyond metamaterials and EM waves (e.g., in acoustics andelasticity). © 2011 Optical Society of America

OCIS codes: 050.2065, 050.5298, 160.3918, 260.2110, 350.4238, 050.1755.

1. INTRODUCTIONElectromagnetic (EM) and optical metamaterials are periodicstructures with features smaller than the vacuum wave-length, judiciously designed to control the propagation ofwaves. Typically, resonance elements—variations of split-ringresonators—are included to produce intriguing effects, suchas backward waves and negative refraction, cloaking, andslow light in “electromagnetically induced transparency”(see, e.g., [1–7] and references therein).

To gain insight into the behavior of such artificial structuresand to be able to design useful devices, one needs to approx-imate a given metamaterial by an effective medium with a di-electric permittivity εeff and a magnetic permeability μeff , or, inmore general cases where magnetoelectric coupling may ex-ist, with a 6 × 6 parameter matrix. A variety of approacheshave been explored in the literature (e.g., [8–14]), with notableaccomplishments cited earlier. However, most studies aresemiheuristic, and there is a clear need for a consistent andrigorous theory—rigorous in the sense that the “macroscopic”(coarse-grained) fields E, H, B, and D are unambiguously andprecisely defined in terms of the “microscopic” (rapidly vary-ing) fields e, b, and d, giving rise to equally well-defined effec-tive parameters.

The main objective of this paper is to put forward such atheory. The methodology advocated here is an amalgamationof two very different lines of thinking: one relatively new anddriven primarily by physical insight, the other well establishedand mathematically rigorous. The physical insight is due toSmith and Pendry [13], who prescribed different averagingprocedures for the microscopic h and b fields (and similarlyfor the e and d pair). This is justified in [13] and other publica-tions by the analogy with staggered grid approximations infinite difference (FD) methods, but it is puzzling why physics

should be subordinate to numerical methods and not the otherway around.

The second root of the proposed methodology is the math-ematical framework developed by Whitney [15] and advancedin numerical analysis by Nedelec, Bossavit, and Kotiuga[16–20]. It should be emphasized, however, that this Whitney–Nedelec–Bossavit–Kotiuga (WNBK) framework is used herenot for computational purposes, but to define, analytically,the coarse-grained fields.

The end result of combining WNBK interpolation withSmith and Pendry’s insight is a mathematically and physicallyconsistent model that is rigorous, general (e.g., applicable tomagnetoelectric coupling), and yet simple enough to be prac-tical. The theory is supported by analytical and numerical casestudies and is consistent with the existing theories and results[e.g., with the Maxwell–Garnett (MG) mixing formula and withthe zero cell-size limit] within the ranges of applicability of thelatter. Approximations that have to be made, and the respec-tive sources of error as well as several routes for furtheraccuracy improvement, are clearly identified.

2. FORMULATION AND SOME PITFALLSConsider a periodic structure composed of materials that areassumed to be (i) intrinsically nonmagnetic (which is true atsufficiently high frequencies [21,22]) and (ii) described by alinear local constitutive relation d ¼ εe. For simplicity, we as-sume a cubic lattice with cells of size a. The microscopic fieldsb, e, and d satisfy Maxwell’s equations ∇ × e ¼ iωc−1b; ∇ ×b ¼ −iωc−1d in the frequency domain and with the expð−iωtÞphasor convention; c is the speed of light in a vacuum. Thetask is to define the respective coarse-grained fields E, B,D, an additional field H, and the constitutive relationships be-tween them, so that the standard Maxwell’s equations andboundary conditions hold on the coarse level. It is instructive

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to review some averaging procedures that appear to bequite natural and yet upon closer inspection turn out tobe flawed. A physically valid alternative is introduced inSections 3 and 4.

A. Passing to the Limit of the Zero Cell SizeA distinguishing feature of the homogenization problem formetamaterials is that the cell size a, while smaller than thevacuum wavelength λ0 at a given frequency ω, is not vanish-ingly small. The typical range in practice is a ∼ 0:1–0:3λ0.Therefore, classical homogenization procedures valid fora → 0 (e.g., Fourier [23] or two-scale analysis [24]) have lim-ited applicability here. Independent physical [22] and mathe-matical [25] arguments show that the finite cell size is aprincipal limitation rather than just a constraint of fabrication.If the cell size is reduced relative to the vacuum wavelength,the nontrivial physical effects, such as “artificial magnetism,”ultimately disappear, provided that the intrinsic dielectric per-mittivity ε of the materials remains bounded. On physicalgrounds [25], this can be explained by the operating pointon the normalized Bloch band diagram falling on the “uninter-esting” acoustic branch.

B. MollifiersThe classical approach to defining the macroscopic fieldsis via convolution with a smooth mollifier function (e.g.,Gaussian-like) [26]. The mollifier must operate on an inter-mediate scale, much coarser than the cell size but still muchfiner than the wavelength in the material. For natural materi-als, this requirement is fulfilled because the cell size is on theorder of molecular dimensions. In contrast, for metamaterials,with a ∼ 0:1–0:3λ0, no intermediate scale is available for themollifier.

C. Simple Cell AveragingSimple cell averaging of the fields is, for metamaterials, inade-quate. To understand why, consider, qualitatively, the behav-ior of a tangential component of the microscopic b field in thedirection normal to a material–air interface (Fig. 1). The pre-cise field distribution is unimportant; one may have in mind,for example, a single Bloch wave moving away from the sur-face, although later it will be essential to consider a super-position of Bloch waves.

Clearly, the average field B0 ¼ hbi over the cell is not, ingeneral, equal to the field bð0þÞ inside the material immedi-ately at the boundary, yet it is bð0þÞ that couples with the fieldin the air: bð0þÞ ¼ bð0−Þ, where bð0−Þ is the field in the airimmediately at the boundary. (Intrinsically nonmagneticmaterials are assumed throughout.) The classical boundary

condition is recovered if an auxiliary H field is introducedin such a way that Hð0þÞ ¼ bð0þÞ; magnetization then is4πm ¼ b −H (i.e., it is the difference between the cell-averaged field b and its value at the boundary that ultimatelydefines magnetization and the permeability).

This picture, albeit simplified (in particular, it ignores acomplicated surface wave at the interface [11,27]), does serveas a useful starting point for a proper physical definition offield averages and effective material parameters.

Since the cell-averaged field is, for a nonvanishing cellsize, not generally equal to its boundary value, the use ofsuch fields in a model would result in nonphysical equivalentelectric/magnetic currents on the surface (jumps of a tangen-tial component of H or E), nonphysical electric/magnetic sur-face charges (jumps of the normal component of D or B),and incorrect reflection/transmission conditions at the bound-ary. It is true that, as a “zero-order” approximation, the cell-averaged field is approximately equal to its pointwise value;however, equating them means ignoring the variation of thefields over the cell and, hence, throwing away the very phys-ical effects that are being investigated.

D. Magnetic Dipole Moment per Unit VolumeThis textbook definition of magnetization turns out to beflawed as well. Simovski and Tretyakov [12] give a counter-example for a system of two small particles, but, in fact, theirargument is general. Suppose that a large volume of a low-lossmetamaterial has been in some way homogenized and is nowrepresented, to an acceptable level of approximation, by effec-tive parameters μeff , εeff . Consider then a standing EM wave inthis material (as produced, e.g., in a cavity or by reflection offa mirror). At any E node, the coarse-grained electric field iszero, and if in addition the lattice cell possesses mirror sym-metry in the direction of the wave, the microscopic field iszero, too. This implies zero polarization/conduction currentsat the node, hence zero magnetic dipole moment and μeff ¼ 1.Since an E node can be easily arranged to occur at any givenlocation, μeff must be equal to 1 everywhere if “magnetic di-pole moment per unit volume” is used to define magnetization.

E. Bulk ParametersIt is known that, even for a homogeneous isotropic infinitemedium, the pair of parameters ε and μ are not defined un-iquely. Indeed, the total microscopic current j can be splitup—in principle, fairly arbitrarily—into the “electric” and“magnetic” parts, j ¼ ∂tpþ c∇ ×m [27–29]. The h field is thendefined accordingly, as b − 4πm, giving rise to the respectivevalue of μeff that depends on the choice of m. A more general“Serdyukov–Fedorov” transformation leaves Maxwell’s equa-tions invariant, but changes the values of the materialparameters [28,30]: d0 ¼ dþ∇ ×Q, h0 ¼ hþ c−1∂tQ, b0 ¼ bþ∇ × F, and e0 ¼ eþ c−1∂tF, where Q and F are arbitrary fields(with a valid curl). It is possible [28,31] to set μ≡ 1.

Thus, even for a homogeneous infinite medium it is only theproduct εμ that is unambiguously defined, with its direct phys-ical relation to phase velocity vp ¼ c=ðεμÞ12. The situationchanges thoroughly when a material interface (for simplicity,with air) is considered. Classical boundary conditions for thetangential continuity of the H and E fields fix the ratio of thematerial parameters via the intrinsic impedance η ¼ ðμ=εÞ12,which, taken together with their product, identifies these

Fig. 1. (Color online) Sketch of the fields in the direction normal tothe interface. The cell-averaged b field (dashed line) may differ fromits boundary value. For simplicity, factors 4π (present in the Gaussiansystem) and μ0 (in the SI system) are not shown.

578 J. Opt. Soc. Am. B / Vol. 28, No. 3 / March 2011 Igor Tsukerman

two parameters separately and uniquely. It is clear then thatany complete and rigorous definition of the effective EM pa-rameters of metamaterials must account for boundary effects.

3. COARSE-GRAINED FIELDSA. Guiding PrinciplesConsider a periodic structure composed of materials that areassumed to (i) be intrinsically nonmagnetic (which is true atsufficiently high frequencies [21,22]) and (ii) satisfy a linearlocal constitutive relation d ¼ εe. For simplicity, we assumea cubic lattice with cells of size a.

Maxwell’s equations for the microscopic fields are, in thefrequency domain and with the expð−iωtÞ phasor convention,

∇ × e ¼ iωc−1b; ∇ × b ¼ −iωc−1d:

The coarse-grained fields B, H, E, and D must be defined insuch a way that the boundary conditions are honored. Fromthe mathematical perspective, these fields must lie in their re-spective functional spaces

E;H ∈ Hðcurl;ΩÞ; B;D ∈ Hðdiv;ΩÞ; ð1Þ

where Ω is a domain of interest that, for mathematical simpli-city, is assumed finite. Ωwill henceforth be dropped to shortenthe notation. Rigorous definitions of these functional spacesare available in the mathematical literature (e.g., [32]). Fromthe physical perspective, constraints (1) mean that the E andH fields have a valid curl as a regular function (not just as aSchwartz distribution), while B and D have a valid divergence.This implies, most importantly, tangential continuity of E andH and normal continuity of B and D across material and cellinterfaces. The fields in HðcurlÞ are also said to be curl con-forming, and those in HðdivÞ to be div conforming.

In principle, any choice of a curl-conforming H field pro-duces the respective “magnetization” 4πm≡ b −H and leavesMaxwell’s equations intact. However, most of such choiceswill result in technically valid but completely impracticaland arbitrary constitutive laws, with the “material parameters”depending more on the choice of H than on the material itself.

As argued below, construction of the coarse-grained fieldsvia the WNBK interpolation has particular mathematical andphysical elegance, which leads to practical advantages in thecomputation of fields in periodic structures.

B. Background: WNBK InterpolationFor a rigorous definition of the coarse-grained H field, weshall need an interpolatory structure referred to in the litera-ture as the “Whitney complex” [16,17]; however, acknowledg-ment of the seminal contributions of Nedelec, Bossavit, andKotiuga [16,18,19] is quite appropriate and long overdue. Still,the subscripts of quantities related to the Whitney complexwill for brevity be just “W” rather than WNBK.

WNBK complexes form a basis of modern finite element(FE) methods with edge and facet elements. However, our ob-jective here is not to develop a numerical procedure; rather,the mathematical structure that has served so well in numer-ical analysis is borrowed and applied to fields in metamater-ial cells.

The original Whitney forms [15] are rooted very deeply indifferential geometry, and the interested reader can find a

complete mathematical exposition in the literature cited ear-lier. Here, we need a small subset of this theory where theusual framework of vector calculus is sufficient. AlthoughWhitney’s construction was developed for simplices [tetrahe-dra in three dimensions (3D)], the key ideas are also applic-able to a cubic reference cell ½−1;þ1�3 that can be linearlytransformed into an arbitrary rectangular parallelepiped or,if necessary, to any hexahedron [33]. For simplicity, this paperdeals only with cubic lattice cells.

We shall need two interpolation procedures: (i) given thecirculations ½q�α ≡

Rα q · dl of a field q over the 12 edges of

the cell (α ¼ 1; 2;…; 12), extend that field into the volumeof the cell; and (ii) given the fluxes ½½q��β ≡

Rβ q · dS of a field

q over the six faces of the cell (β ¼ 1; 2;…; 6), extend that fieldinto the volume of the cell. Single brackets denote line inte-grals over an edge; double brackets denote surface integralsover the faces. Typically, item (i) will apply to the e and hfields and item (ii) will apply to the d and b fields.

Consider an edge α along a ξ direction, where ξ is one of thecoordinates x, y, z, and let η and τ be the other two coordi-nates, with ðξ; η; τÞ being a cyclic permutation of ðx; y; zÞ and,hence, a right-handed system. Edge α is then formally definedas −1 ≤ ξ ≤ 1; η ¼ ηα; τ ¼ τα; ηα; τα ¼ �1. Associated with thisedge is a vectorial interpolating function wα ¼ ξð1þ ηαηÞð1þ τατÞ=8, where the hat symbol denotes the unit vectorin a given direction. For illustration, a two-dimensional(2D) analog of this vector function is shown in Fig. 2. In3D, there are 12 interpolating functions of this kind—oneper edge—in the cell. It is straightforward to verify that theedge circulations of these functions have the Kronecker-deltaproperty: ½wα�α0 ¼ δαα0 . This guarantees that the interpolatingfunctions are linearly independent over the cell and span a12-dimensional space of vectors that can all be representedby interpolation from the edges into the volume of the cell:

q ¼X12α¼1

½q�αwα: ð2Þ

We shall call this 12-dimensional spaceWcurl, whereW hon-ors Whitney and “curl” indicates fields whose curl is a regularfunction rather than a general distribution. This implies, inphysical terms, the absence of equivalent surface currents andthe tangential continuity of the fields involved. Any adjacent

Fig. 2. (Color online) 2D analog of the vectorial interpolation func-tion wα (in this case, associated with the central vertical edge sharedby two adjacent cells). Tangential continuity of this function is evidentfrom the arrow plot; its circulation is equal to 1 over the central edgeand to zero over all other edges.

Igor Tsukerman Vol. 28, No. 3 / March 2011 / J. Opt. Soc. Am. B 579

lattice cells sharing a common edge will also share, by con-struction of interpolation (2), the field circulation overthat edge.

The curls of wα are not linearly independent but rather, ascan be demonstrated, lie in the six-dimensional (6D) spaceWdiv spanned by functions v1−6 ¼ fxð1� xÞ=8, yð1� yÞ=8,zð1� zÞ=8g. A 2D analog of a typical function v is shown inFig. 3. The v functions satisfy the Kronecker-delta propertywith respect to the face fluxes: ½½vβ��β0 ¼ δββ0 . Hence, onecan consider vector interpolation from fluxes on the cell facesinto the volume of the cell, conceptually quite similar to edgeinterpolation (2): q ¼ P

6β¼1½½q��βvβ. The 6D space spanned in a

lattice cell by the v functions will be denoted with Wdiv, re-flecting the easily verifiable fact that the normal componentof these functions is continuous across the common face oftwo adjacent cells. Importantly, as already mentioned, thecurls of functions from Wcurl lie in Wdiv, or symbolically interms of the functional spaces,

∇ ×Wcurl ∈ Wdiv: ð3Þ

To summarize, the following properties are critical for ourconstruction of the coarse-grained fields. (1) Twelve functionswα (one per cell edge) interpolate the field from its edge cir-culations into the cell. The resulting field is tangentially con-tinuous across all cell boundaries. The w functions span the12-dimensional functional space Wcurl. (2) Six functions vβ(one per cell face) interpolate the field from its face fluxesinto the cell. The resulting field has normal continuity acrossall cell faces. The v functions span a 6D functional spaceWdiv.(3) Any vector field in Wcurl is uniquely defined by its 12 edgecirculations. (4) Any vector field inWdiv is uniquely defined byits six face fluxes. (5) ∇ ×Wcurl ∈ Wdiv (see also [34]).

C. Construction of the Coarse-Grained FieldsWhile the naive averaging of the E and H fields over the cellmay break the tangential continuity of these fields across theboundaries, WNBK interpolation provides a mathematicallyand physically valid alternative. Let us define the coarse-grained fields as WNBK interpolants of the actual edge circu-lations via the w functions, in accordance with Eq. (2). Withineach lattice cell,

E≡X12α¼1

½e�αwα ≡Wcurlð½e�1−12Þ ð4Þ

with a completely similar expression for the H field. ≡ indi-cates that this is the definition of E and H as well as of theWNBK curl-interpolation operator Wcurl. Similarly, the Band D fields are defined as Wdiv interpolants from the actualface fluxes into the cells via the v functions

B ¼X6β¼1

½½b��βvβ ≡Wdivð½½b��1−6Þ ð5Þ

and a completely analogous expression for the D field.We now decompose the microscopic fields e, b, and d into

coarse-grained parts E, B, and D defined above and rapidlyvarying remainders e∼, b∼, and d∼:

e ¼ Eþ e∼; d ¼ Dþ d∼; b ¼ Bþ b∼: ð6Þ

Importantly, b has an alternative decomposition where H istaken as a basis:

b ¼ Hþ 4πm∼: ð7Þ

With these splittings, Maxwell’s equations become

∇ × ðEþ e∼Þ ¼ iωc−1ðBþ b∼Þ; ð8Þ

∇ × ðHþ 4πm∼Þ ¼ −iωc−1ðDþ d∼Þ: ð9Þ

At this point, the role of the WNBK interpolation becomes ap-parent: the scales separate. Indeed, E is, by construction, inWcurl, and, therefore, ∇ × E is in Wdiv and so is, by construc-tion, B. In that sense, the capital-letter terms in Eq. (8) are fullycompatible. Furthermore, E—again by definition—has thesame edge circulations as the microscopic field e; hence,for any face of any cell,

Zface

ð∇ × EÞ · dS ¼Zface edges

E · dl ¼ iωc−1Zface

b · dS;

where the Stokes theorem and the microscopic Maxwell’sequations were used. However, the B field has the same facefluxes as b by construction and, since these face fluxes definethe field in Wdiv uniquely, we have

∇ × E ¼ iωc−1B: ð10Þ

Analogously,

∇ ×H ¼ −iωc−1D: ð11Þ

Thus, the coarse level has separated out, and, remarkably, theWNBK fields satisfy the Maxwell’s equations as well as theproper continuity conditions exactly. The underlying reasonfor that is the compatibility of the curl and div interpolations,i.e., condition (3).

For the rapidly changing components, straightforwardalgebra yields, from Eq. (8) and (9),

∇ × e∼ ¼ iωc−1b∼; 4π∇ ×m∼ ¼ −iωc−1d∼ ð12Þ

with the “constitutive relationships”

Fig. 3. (Color online) 2D analog of the vectorial interpolation func-tion vβ (in this case, associated with the central vertical edge). Normalcontinuity of this function is evident from the arrow plot; its flux isequal to 1 over the central edge and zero over all other edges.

580 J. Opt. Soc. Am. B / Vol. 28, No. 3 / March 2011 Igor Tsukerman

d∼ ¼ εe∼ þ ðεE − DÞ; 4πm∼ ¼ b∼ þ ðB −HÞ: ð13Þ

All edge circulations of e∼,m∼ and all face fluxes of b∼, d∼ arezero. If/when the coarse-grained fields have been found fromtheir Maxwell’s equations, one may convert the two equationsfor the rapid fields into a single equation for e∼:

∇ ×∇ × e∼ − ðω=cÞ2εe∼ ¼ ðω=cÞ2ðεE − DÞ − iω=c∇ × ðB −HÞ:ð14Þ

This equation can be solved for each cell, with the Dirichlet-type zero-circulation boundary conditions. In principle, thisfast-field correction will increase the accuracy of the overallsolution. However, this paper remains focused on the coarsefields and the corresponding effective parameters.

4. MATERIAL PARAMETERSA. Procedure for the Constitutive MatrixWe are now in a position to define effective material para-meters, i.e., the linear relationships between D, B and E, H.Given a metamaterial lattice, let us construct coarse-grainedcurl-conforming fields E, H using the lattice cell edges as a“scaffolding”: the field is interpolated into the cells from theedge circulations of the respective microscopic field (Fig. 4).Thus, the edge circulations of the microscopic and the coarse-grained curl-conforming fields are the same. Similar consid-erations apply to div-conforming fields, but with interpolationfrom the faces.

Next, let the EM field be approximated as a linear combina-tion of some basis waves (modes) ψα:

Ψeh ¼Xαcαψeh

α ; Ψdb ¼Xαcαψdb

α :

In the most general case, Ψ and all ψα are six-component vec-tor comprising both microscopic fields; e.g., Ψeh ≡ fΨe;Ψhg,etc. However, in the absence of magnetoelectric coupling, it is

natural to consider each pair of fields ðe;dÞ and ðh;bÞ sepa-rately, and then all ψs have three components rather thansix. A common example of such basis waves is Bloch modesin periodic media.

To each basis wave ψα, there correspond the WNBKcurl interpolants EαðrÞ ¼ Wcurlð½ψe

α�1−12Þ and HαðrÞ ¼Wcurlð½ψb

α�1−12Þ. Similarly, there are the WNBK div interpolantsDαðrÞ ¼ Wdivð½½dα��1−6Þ and BαðrÞ ¼ Wdivð½½bα��1−6Þ.

It is possible to find a 6 × 6 constitutive matrix ζ that mini-mizes the L2 norm of the discrepancy ψDBðrÞ − ζðrÞψEHðrÞ; away to do so will be described in a future publication. Here, asimpler procedure is adopted. For all basis waves α at anygiven point r in space, we seek a linear relation

ψDBα ðrÞ ¼ ζðrÞψEH

α ðrÞ;

where ζ characterizes, in general, anisotropic material behav-ior with (if the off-diagonal blocks are nonzero) magnetoelec-tric coupling. Similar to ψdb, the six-component vectors ψDB

comprise both fields, but coarse grained; the same appliesto ψEH . In matrix form, the above equations are

ζðrÞΨEHðrÞ ¼ ΨDBðrÞ; ð15Þ

where each column of the matricesΨDB andΨEH contains therespective basis function. (Illustrative examples in Section 5may help to clarify these notions and notation.)

If exactly six basis functions are chosen, one obtains theconstitutive matrix by straightforward matrix inversion; ifthe number of functions is more than six, the pseudoinverse[35] is appropriate:

ζðrÞ ¼ ΨDBðrÞðΨEHÞþðrÞ: ð16Þ

A notable feature of this construction is that the coarse-grained fields corresponding to a particular basis wave satisfythe Maxwell’s equations with this material parameter ζðrÞ ex-actly. The approximate parameters for each cell are thenfound simply by cell averaging ζðrÞ.

One may wonder why such cell averaging could not bedone in the very beginning, for the original microscopic Max-well’s equations, in which case one would get the trivial valueof unity by averaging the intrinsic permeability μ ¼ 1. First, asalready discussed, the cell-averaged fields violate the bound-ary conditions and, therefore, are inadequate for metamater-ials. Further, in the case of the coarse-grained fields, splittingthe material matrix into its cell average plus a fluctuating com-ponent, ζ ¼ hζi þ ζ∼, makes more sense. Indeed, consider theMaxwell’s equations (in the 6D form) for the coarse-grainedfields:

∇ ×

�EH

�¼ iω

c

�B−D

�¼ iω

c

�0 I3−I3 0

�ζðrÞ

�EH

�; ð17Þ

where I3 is the 3 × 3 unit matrix. The approximate replace-ment of ζðrÞ with hζi introduces a perturbation that is propor-tional to ΨDB − hζiΨEH and that our construction of hζi isintended to minimize. It would be a mistake to manipulatethe original microscopic equations in a similar manner: therespective “perturbation”would not be small due to the strongvariation of the intrinsic permittivity over the lattice cell. Thisstrong variation is at the heart of the resonance effects [22,25].

Fig. 4. (Color online) Lattice (with arbitrary inclusions) serves as a“scaffolding” for the construction of coarse-grained fields. The curl-conforming fields ðE;HÞ are interpolated into the cells from the edgecirculations, while the div-conforming fields ðB;DÞ are interpolatedfrom the face fluxes. Only E and B are shown.

Igor Tsukerman Vol. 28, No. 3 / March 2011 / J. Opt. Soc. Am. B 581

B. RecipeLet the vectorial dimension of the problem—i.e., the totalnumber of vector field components—be N . Most generally,N ¼ 6 (three components of E and three of H), but if onlyone field is involved, then N ¼ 3, and if that field has onlyone component, then N ¼ 1, etc. It is convenient to summar-ize the procedure described earlier as a “recipe” for findingthe effective parameters: (1) Choose a set of M ≥ N basiswaves (modes) ψα (e.g., Bloch waves) that provide a good ap-proximation of the fields within a cell. (2) Find the curl- anddiv-WNBK interpolants of each basis wave. (This step requiresthe computation of face fluxes and edge circulations of therespective fields in the wave.) (3) Assemble these WNBKinterpolants into the ΨDB and ΨEH matrices of Eq. (15).(4) Find the coordinate-dependent parameter matrix ζðrÞ fromEq. (16). (5) The cell average of this matrix gives the finalresult: an N × N (6 × 6 in the most general case) matrix of ef-fective parameters.

As already noted, instead of steps (4) and (5), one may wantto minimize the discrepancy ∥ψDBðrÞ − ζðrÞψEHðrÞ∥.

C. Errors and Error IndicatorsThree sources of error can be distinguished in the proposedhomogenization procedure.

1. In-the-basis error. If the number of basis waves isstrictly greater than the vectorial dimension of the problem(M > N), then Eq. (15) does not generally have an exact solu-tion and is solved in the least-squares sense, as the pseudo-inverse in Eq. (16) indicates. From Eqs. (15) and (16), thediscrepancy between the fields is (with the dependence onr implied) ΨDB − ζΨEH ¼ ΨDBðI − ðΨEHÞþΨEHÞ, where Iis the 6 × 6 identity matrix. Therefore, the ratio ∥I −ðΨEHÞþΨEH∥=∥ðΨEHÞþΨEH∥ is a suitable indicator of therelative in-the-basis error. If M ¼ N (for example, six basiswaves in a generic problem of electrodynamics), then thein-the-basis error vanishes. It also vanishes in the long-wavelength limit.

2. Out-of-the-basis error. Any field can be represented as alinear combination of the basis waves, plus a residual field.For a good basis set, this residual term is small. Any expansionof the basis carries a trade-off: the residual field and the out-of-the-basis error will decrease, but the in-the-basis error mayincrease.

3. Parameter averaging. As an intermediate step, theproposed homogenization procedure yields a coordinate-dependent parameter matrix ζðrÞ that is, in the end, averagedover the lattice cell. This error was discussed earlier.

The limitations of the effective-medium approximation nowbecome apparent as well. If a sufficient number of modes withsubstantially different characteristics exist in a metamaterial(e.g., in cases of strong anisotropy), then the in-the-basis errorwill be high. This is not a limitation of the specific procedureadvocated in this paper, but a reflection of the fact that thebehavior of fields in the material is, in such a case, too richto be adequately described by a single effective-parametertensor.

On a positive side, several specific ways of improving theapproximation accuracy can be identified. Obviously, no ac-curacy gain is completely free. (i) The cell problem for therapidly varying fields can be solved, once the coarse-grained

fields have been found from the effective-parameter model.(ii) If the relative weights of different modes in a particularmodel can be roughly evaluated a priori, then Eq. (15) couldbe biased toward these modes; some columns of the Ψ ma-trices (i.e., some basis functions) could even be eliminated.The downside is that the material parameters are no longera property of the material alone, but become partly problemdependent. (iii) Instead of reducing the number of columns inthe Ψ matrices, one can increase the number of rows. Thisrequires new degrees of freedom in addition to the six com-ponents of the fields, e.g., higher-order moments of the fieldsover edges and faces. This idea was put forward by Rodin [36]in a different context. Physically, this is a manifestation and anacknowledgment of the nonlocality of the problem. The ma-terial parameter matrix ζ also becomes expanded and mightbe called a “lattice cell response matrix.” (iv) The size andcomposition of the basis set can be optimized for commonclasses of problems. (v) The last step of the proposedprocedure—the cell averaging of matrix ζ—could be omitted.The coarse-grained fields are then evaluated accurately, butthe numerical solution may be computationally expensive,as the material parameter varies within the cell. There is aspectrum of practical compromises where ζ would be ap-proximated within a cell not as a constant, but to some higherorder. (vi) One may envision adaptive procedures wherebythe basis waves are updated after the problem has beensolved, and then new material parameters are derived fromthe new basis set. Again, the downside is that, in additionto the increased cost, the material parameters become partlyproblem dependent.

In connection with the last item, it is clear that the basis setshould, in general, reflect the symmetry and reciprocity prop-erties of the problem. In particular, the basis should as a ruleinclude pairs of waves traveling in the opposite directions.

D. Causality and PassivityPhysical considerations indicate that the proposed procedureshould be expected to produce causal and passive effectivemedia, at least if M ¼ N . Indeed, suppose that the oppositeis true, e.g., ε00eff < 0 (with anisotropy neglected for simplicity),violating passivity. The effective parameters, however, applyby construction to all basis waves exactly; hence, ε00eff < 0would imply power generation in the actual physical modesin the actual passive metamaterial, which is impossible.

The above argument is, strictly speaking, valid only prior tothe cell averaging of the coordinate-dependent material pa-rameter matrix. Another caveat to this argument is that caus-ality and passivity apply to the total fields that are the sums ofcoarse-grained and rapid components. If the rapid compo-nents and energies associated with them are significant, thenthe coarse-grained fields and the related material parametersmay, in principle, violate passivity without breaking physicallaws. In such cases, however, the validity of homogenizationwould be questionable, and further analysis is desirable in thefuture.

5. VERIFICATIONA. Empty CellAlthough this case is seemingly trivial, the effective param-eters of empty cells produced by alternative methods in theliterature often contain spurious Bloch-like factors that are

582 J. Opt. Soc. Am. B / Vol. 28, No. 3 / March 2011 Igor Tsukerman

then discarded by fiat. In contrast, direct calculation that fol-lows the methodology of this paper produces μeff ¼ 1, εeff ¼ 1exactly, without any spurious factors and, of course, with nomagnetoelectric coupling, both in 2D and 3D. This is valid aslong as the cell size does not exceed one half of the wave-length.

B. Example: One-Component Static FieldsThis is another simple consistency check for the proposedmethodology. Let a static field (e.g., electrostatic) have onlyone component (say, z) that must be independent of z due tothe zero-divergence condition. Then, the lattice cells becomeeffectively 2D and the div-conforming WNBK interpolant for dreduces just to a constant D0 whose flux through the cell isequal to the flux of d. The curl-conforming WNBK interpolantWcurlðeÞ would generally be a bilinear function of x and y, butsince the e field is constant in the xy plane, this interpolantalso reduces to a constant E0 ¼ E. Thus, the dielectric permit-tivity is

εeff ≡D0

E0¼

Rcell d · dSSE0

¼ S−1

Zcell

εdS

exactly as should be expected.

C. Example: the MG FormulaThe next test is consistency with the classical MG mixing for-mula for a two-component medium in a static field. (We do notdeal with radiative corrections to the polarizability and theMG in this paper.) The MG expression for the effective permit-tivity is, in 2D, εMG;2D ¼ ð1þ f χÞ=ð1 − f χÞ; where χ is the po-larizability of inclusions in a host medium and f is the fillfactor. In particular, for cylindrical inclusions in a non-polarizable host, χ ¼ ðεcyl − 1Þ=ðεcyl þ 1Þ, f ¼ πr2cyl=a2.

The proposed methodology specializes to this case as fol-lows. First, one has to define a set of basis fields; naturally, forthe static problem this is more easily done in terms of the po-tential rather than the field. At zero frequency, the Blochwavenumber is also zero; hence, the Bloch conditions forthe field are periodic, but the potential may have an offset cor-responding to the line integral of the field across the cell.Thus, the first basis function ψ1 is defined by

∇ · ε∇ψ1 ¼ 0; ψ1

�x ¼ � a

2

�¼ �a

2; ψ1ðyþ aÞ

¼ ψ1ðyÞ:

The potential difference across the cell corresponds to a unituniform field applied in the −x direction. The second basisfunction is completely analogous and corresponds to a fieldin the −y direction. Once the basis functions have been found,the next steps are to compute the circulations and fluxes, andthen the WNBK interpolants, of the basis functions. For ψ1,the circulation of the respective field e1 ¼ −∇ψ1 over each“horizontal” edge y ¼ �a=2 is, by construction, equal to a andis zero over the two “vertical” edges. The fluxes of d1 ¼ εe1 arezero over the horizontal edges; for the vertical ones,

½½d1�� ¼Zy¼�a=2

d1xdy ¼ −

Zy¼�a=2

ε∂xψ1dy;

where ε is that of the host if cell boundaries do not cut throughthe inclusions.

Because the fluxes and circulations of the fields over thepairs of opposite edges are in this case equal, the WNBK in-terpolant is simply constant and the effective parametersare obtained immediately, without the intermediate stage ofcomputing and averaging their pointwise values. Formally,Eq. (15) reduces to

�εxx εxyεyx εyy

��1 00 1

�¼

� ½½d1�� 00 ½½d2��

�ð18Þ

(cell size a normalized to unity for simplicity). The identitymatrix is written out explicitly to point out its origin: its firstcolumn represents the WNBK curl-conforming interpolant ofe1; the second column represents that of e2. The rightmost ma-trix contains the div-conforming WNBK interpolants of d1 andd2. The system of equations is trivial in this case; since thefluxes ½½d1�� and ½½d2�� are equal, the permittivity is a scalarquantity equal to these fluxes.

For numerical illustration, let us consider a cylindrical in-clusion with a varying radius in a nonpolarizable host. Thecommercial FE package Comsol is used to compute the basisfunction and its edge flux ½½d1�� (¼ ½½d2��). (There are ∼30; 000triangular elements in a typical FE simulation; fourth-ordernumerical quadratures are used to compute the edge flux.)The plots of εeff versus the radius of the cylinder (Fig. 5)for two different values of the permittivity (εcyl ¼ 5 andεcyl ¼ 10) show that the agreement between the new methodandMG is excellent even for fairly large radii of the inclusions.While such an agreement may be surprising at first glance, MGdoes, in fact, retain high accuracy beyond its “theoretical”range of applicability (see, e.g., a summary in [37]).

D. Example: Bloch Bands and Wave RefractionLet us now consider wave propagation through a photoniccrystal (PhC) slab—an array of cylindrical rods with no de-fects. For consistency with previous work, the geometric

Fig. 5. (Color online) Effective ε for a 2D periodic array of cylinders.Curves, the proposed procedure (with nine cylindrical harmonics);markers, the MG formula.

Igor Tsukerman Vol. 28, No. 3 / March 2011 / J. Opt. Soc. Am. B 583

and physical parameters of the crystal are chosen to be thesame as in [38] and are taken from [39]. Namely, the radiusof the rod is rcyl ¼ 0:33a and its dielectric permittivity isεcyl ¼ 9:61. The p mode (H mode, with the one-componentmagnetic field along the rods) is considered because this isa more interesting case for homogenization.

The numerical simulation of the Bloch bands and wave pro-pagation through the PhC slab was performed with flexiblelocal approximation methods (FLAME) [38,40], a generalizedFD calculus that incorporates local analytical solutions intothe difference scheme—arguably, the most accurate numeri-cal method for this type of problem.

Figure 6 shows that the Bloch bands obtained with the ef-fective parameters are in excellent agreement with the accu-rate numerical simulation, except for a few data points at theband edge, where effective parameters cannot be expected toremain valid. Figure 7 displays the real parts of εeff and μeff as

a function of the Bloch wavenumber, in the ΓX direction.Among other features, a region of double-negative parameters(εeff < 0, μeff < 0) can be clearly identified for the normalizedfrequency approximately between 0.33 and 0.42. This agreesvery well with the band diagrams above and the results re-ported previously [39,38].

Numerical simulation (using FLAME) of EM waves in theactual PhC is compared with the analytical solution for ahomogeneous slab of the same thickness as the PhC as wellas effective material parameters εeff and μeff calculated as themethodology of this paper prescribes. In the numerical simu-lations, obviously, a finite array of cylindrical rods has to beused (in this case, 24 × 5) to limit the computational cost. Theresults reported below are at the normalized frequency of~ω≡ ωa=ð2πcÞ ¼ a=λ ¼ 0:24959, coinciding with one of thedata points in previous simulations [38]. A fragment of thecontour plot of ReðHÞ in Fig. 8 helps to visualize the wavefor the angle of incidence π=6.

The real part of the numerical and analytical magneticfields along the line perpendicular to the slab is plotted inFig. 9 for the angle of incidence π=6 (the plot for the imaginarypart is similar). The analytical and numerical results areshown to be in a good agreement, despite the approximate

Fig. 6. (Color online) ΓX Bloch bands obtained with the effectiveparameters (markers) versus accurate numerical simulation (solidcurves). εcyl ¼ 9:61, rcyl ¼ 0:33a.

Fig. 7. (Color online) Effective parameters (dashed curve, ε0eff ; solidcurve,μ0eff ) versus frequency in the ΓX direction. εcyl ¼ 9:61,rcyl ¼ 0:33a.

Fig. 8. (Color online) Contour plot of ReðHÞ. Angle of incidence π=6,εcyl ¼ 9:61, rcyl ¼ 0:33a.

Fig. 9. (Color online) ReðHÞ along the coordinate perpendicular tothe slab. The angle of incidence π=6. Other parameters are the same asin Figs. 7 and 8.

584 J. Opt. Soc. Am. B / Vol. 28, No. 3 / March 2011 Igor Tsukerman

nature of the effective parameters for the fairly large cell sizea ∼ λ0=4 and the high fill factor.

E. ConclusionThe proposed new methodology rigorously defines effectiveparameters of EM and optical metamaterials and explains,among other things, nontrivial magnetic effects arising in in-trinsically nonmagnetic media. The main underlying principleis that the coarse-grained E and H fields have to be curl con-forming (i.e., possessing a valid curl as a regular function,which, in particular, implies tangential continuity across ma-terial interfaces), while the B and D fields have to be div con-forming, with their normal components continuous acrossinterface boundaries. While some flexibility in the choice ofthese coarse-grained fields exists, an excellent frameworkfor their construction is provided by Whitney forms and theWNBK complex (Subsection 3.B). This construction ensuresnot only the proper continuity conditions for the respectivefields, but also the compatibility of the respective interpolants,so that, for example, the curl of E lies in the same approxima-tion space as B. As a result, remarkably, Maxwell’s equationsfor the coarse-grained fields are satisfied exactly.

Further, the EM field is approximated with a linear combi-nation of basis functions (modes), e.g., Bloch waves. Effectiveparameters are then devised to provide the most accurate lin-ear relation between the WNBK interpolants of these basisfunctions. In the limiting case of a vanishingly small cell size,this procedure yields the exact result; for a finite cell size, asmust be the case for all metamaterials of interest [22,25], theeffective parameters are an approximation. A number of waysto improve the accuracy are outlined in Subsection 4.C.

Proponents of the differential-geometric treatment of EMtheory have long argued that the H, E and B, D fields are ac-tually different physical entities, the first pair best character-ized via circulations (mathematically, as 1-forms) and thesecond one via fluxes (2-forms) [17,18,41]. The approach ad-vocated and verified in this paper buttresses this viewpoint.

There are several important issues to be addressed in futurework. First, the theory of this paper needs to be applied to 3Dstructures of practical interest. Second, physical considera-tions indicate that the theory should lead to causal and passiveeffective media (see Subsection 4.D and [11,12]), but a rigor-ous mathematical analysis or, in the absence of that, accumu-lated numerical evidence are highly desirable. Third, specificroutes for accuracy improvement, as outlined in this paper,need to be thoroughly investigated and tested.

Although the object of interest in this paper is artificial me-tamaterials, it is hoped that the new theory will also help tounderstand more deeply the nature of the fields in natural ma-terials, as rigorous definitions of such fields, especially of theH field, are nontrivial. The ideas and methodology of this pa-per are general and should find applications beyond electro-magnetism, such as acoustics and elasticity.

ACKNOWLEDGMENTSDiscussions with V. Markel and J. Schotland to a large extentinspired this work and are greatly appreciated. D. Golovatyhas offered invaluable insight and a number of helpfulsuggestions. Communication with A. Bossavit, C. T. Chan,Z. Zhang, Y. Wu, Y. Lai, G. Rodin, and L. Demkowicz is alsogratefully acknowledged. The author thanks C. T. Chan

(HKUST, Hong Kong), W. C. Chew and L. Jiang (HKU,Hong Kong), O. Bíró, C. Magele, K. Preis (TU Graz, Austria),S. Bozhevolnyi (Syddansk Universitet, Denmark), and G.Rodin (The University of Texas at Austin) for their hospitalityduring the period when this work was either contemplated orperformed.

REFERENCES1. A. Alù, F. Bilotti, N. Engheta L. Vegni, “Subwavelength, compact,

resonant patch antennas loaded with metamaterials,” IEEETrans. Antenn. Propag. 55, 13–25 (2007).

2. K. Buell, H. Mosallaei, and K. Sarabandi, “A substrate for smallpatch antennas providing tunable miniaturization factors,” IEEETrans. Microwave Theory Tech. 54, 135–146 (2006).

3. P. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A. Tretyakov,“On artificial magnetodielectric loading for improving the impe-dance bandwidth properties of microstrip antennas,” IEEETrans. Antenn. Propag. 54, 1654–1662 (2006).

4. N. Papasimakis, V. A. Fedotov, N. I. Zheludev, andS. L. Prosvirnin, “Metamaterial analog of electromagnetically in-duced transparency,” Phys. Rev. Lett. 101, 253903 (2008).

5. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry,A. F. Starr, and D. R. Smith, “Metamaterial electromagneticcloak at microwave frequencies,” Science 314, 977–980(2006).

6. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, andS. Schultz, “Composite medium with simultaneously negativepermeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187(2000).

7. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101,047401 (2008).

8. A. K. Sarychev and V. M. Shalaev, Electrodynamics of Metama-

terials (World Scientific, 2007).9. M. G. Silveirinha, “Metamaterial homogenization approach with

application to the characterization of microstructured compo-sites with negative parameters,” Phys. Rev. B 75, 115104 (2007).

10. C. Fietz and G. Shvets, “Current-driven metamaterial homogeni-zation,” Physica B 405, 2930–2934 (2010).

11. C. R. Simovski, “Material parameters of metamaterials (areview),” Opt. Spectrosc. 107, 726–753 (2009).

12. C. R. Simovski and S. A. Tretyakov, “On effective electromag-netic parameters of artificial nanostructured magnetic materi-als,” Photon. Nanostr. Fundam. Appl. 8, 254–263 (2010).

13. D. R. Smith and J. B. Pendry, “Homogenization of metamaterialsby field averaging,” J. Opt. Soc. Am. B 23, 391–403 (2006).

14. S. Tretyakov, Analytical Modeling in Applied Electromagnetics

(Artech House, 2003).15. H. Whitney, Geometric Integration Theory (Princeton

University, 1957).16. A. Bossavit, “Whitney forms: a class of finite elements for three-

dimensional computations in electromagnetism,” IEE Proc. A135, 493–500 (1988).

17. A. Bossavit, Computational Electromagnetism: Variational

Formulations, Complementarity, Edge Elements (Academic,1998).

18. P. R. Kotiuga, “Hodge decompositions and computational elec-tromagnetics,” PhD thesis (McGill University, 1985).

19. J.-C. Nédélec, “Mixed finite elements in R3,” Numer. Math. 35,315–341 (1980).

20. J.-C. Nédélec, “A new family of mixed finite elements in R3,”Numer. Math. 50, 57–81 (1986).

21. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous

Media (Pergamon, 1984).22. R. Merlin, “Metamaterials and the Landau–Lifshitz permeability

argument: large permittivity begets high-frequency magnetism,”Proc. Natl. Acad. Sci. USA 106, 1693–1698 (2009).

23. D. Sjöberg, C. Engstrom, G. Kristensson, D. J. N. Wall, andN. Wellander, “A Floquet–Bloch decomposition of Maxwell’sequations applied to homogenization,” Multiscale Mod. Simul.4, 149–171 (2005).

24. A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic

Methods in Periodic Media (North-Holland, 1978).

Igor Tsukerman Vol. 28, No. 3 / March 2011 / J. Opt. Soc. Am. B 585

25. I. Tsukerman, “Negative refraction and the minimum lattice cellsize,” J. Opt. Soc. Am. B 25, 927–936 (2008).

26. G. Russakoff, “A derivation of the macroscopic Maxwellequations,” Am. J. Phys. 38, 1188–1195 (1970).

27. V. A. Markel, University of Pennsylvania, Philadelphia,Pennsylvania (personal communication, 2010).

28. A. P. Vinogradov, “On the form of constitutive equations inelectrodynamics,” Phys. Usp. 45, 331–338 (2002).

29. V. A. Markel, “Correct definition of the Poynting vector in elec-trically and magnetically polarizable medium reveals that nega-tive refraction is impossible,” Opt. Express 16, 19152–19168(2008).

30. V. V. Bokut, A. N. Serdyukov, and F. I. Fedorov, “Form of con-stitutive equations in optically active crystals,” Opt. Spectrosc.37, 166–168 (1974).

31. V. M. Agranovich and V. L. Ginzburg, Crystal Optics with

Spatial Dispersion, and Excitons, 2nd ed. (Springer-Verlag, 1984).

32. P. Monk, Finite Element Methods for Maxwell’s Equations

(Clarendon, 2003).33. J. van Welij, “Calculation of eddy currents in terms of H on

hexahedra,” IEEE Trans. Magn. 21, 2239–2241 (1985).34. The following exactness property [16,17,42] is also fundamental

for Whitney complexes, but is not explicitly used in the paper.

Any divergence-free field in Wdiv is the curl of some fieldin Wcurl.

35. G. H. Golub and C. F. Van Loan, Matrix Computations (TheJohns Hopkins University, 1996).

36. G. J. Rodin, “Higher-order macroscopic measures,” J. Mech.Phys. Solids 55, 1103–1119 (2007).

37. A. Moroz, “Effective medium properties, mean-field description,homogenization, or homogenisation of photonic crystals,”http://www.wave‑scattering.com/pbgheadlines.html#Effective%20medium%20properties.

38. I. Tsukerman and F. Čajko, “Photonic band structure computa-tion using FLAME,” IEEE Trans. Magn. 44, 1382–1385(2008).

39. R. Gajic, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction andrightness in photonic crystals,” Opt. Express 13, 8596–8605(2005).

40. I. Tsukerman, Computational Methods for Nanoscale Applica-

tions: Particles, Plasmons and Waves (Springer, 2007).41. E. Tonti, “A mathematical model for physical theories,” Atti

Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. LII, 175–181;350–356 (1972).

42. L. Demkowicz, J. Kurtz, D. Pardo, M. Paszenski, W. Rachowicz,and A. Zdunek, Computing with hp-Adaptive Finite Elements

(Chapman & Hall/CRC, 2007), Vol. 2.

586 J. Opt. Soc. Am. B / Vol. 28, No. 3 / March 2011 Igor Tsukerman