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Department of Electrical and Computer Engineering, The University of Akron,. Akron, Ohio 44325-3904, USA ([email protected]). Received November 17, 2010; ...
Igor Tsukerman

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

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Effective parameters of metamaterials: a rigorous homogenization theory via Whitney interpolation Igor Tsukerman Department of Electrical and Computer Engineering, The University of Akron, Akron, Ohio 44325-3904, USA ([email protected]) 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 control the 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 interpolation. All coarse-grained fields are unambiguously defined and satisfy Maxwell’s equations exactly. Fields with tangential and normal continuity across boundaries are associated with two different kinds of interpolation, which reveals the physical and mathematical origin of “artificial magnetism.” The new approach is illustrated with several examples and agrees well with the established results (e.g., the Maxwell–Garnett formula and the zero cell-size limit) within the range of applicability of the latter. The sources of approximation error and the respective suitable error indicators are clearly identified, along with systematic routes for improving the accuracy further. The proposed methodology should be applicable in areas beyond metamaterials and EM waves (e.g., in acoustics and elasticity). © 2011 Optical Society of America OCIS codes: 050.2065, 050.5298, 160.3918, 260.2110, 350.4238, 050.1755.

1. INTRODUCTION Electromagnetic (EM) and optical metamaterials are periodic structures with features smaller than the vacuum wavelength, judiciously designed to control the propagation of waves. Typically, resonance elements—variations of split-ring resonators—are included to produce intriguing effects, such as backward waves and negative refraction, cloaking, and slow light in “electromagnetically induced transparency” (see, e.g., [1–7] and references therein). To gain insight into the behavior of such artificial structures and to be able to design useful devices, one needs to approximate a given metamaterial by an effective medium with a dielectric permittivity εeff and a magnetic permeability μeff , or, in more general cases where magnetoelectric coupling may exist, with a 6 × 6 parameter matrix. A variety of approaches have been explored in the literature (e.g., [8–14]), with notable accomplishments cited earlier. However, most studies are semiheuristic, and there is a clear need for a consistent and rigorous theory—rigorous in the sense that the “macroscopic” (coarse-grained) fields E, H, B, and D are unambiguously and precisely defined in terms of the “microscopic” (rapidly varying) fields e, b, and d, giving rise to equally well-defined effective parameters. The main objective of this paper is to put forward such a theory. The methodology advocated here is an amalgamation of two very different lines of thinking: one relatively new and driven primarily by physical insight, the other well established and mathematically rigorous. The physical insight is due to Smith and Pendry [13], who prescribed different averaging procedures for the microscopic h and b fields (and similarly for the e and d pair). This is justified in [13] and other publications by the analogy with staggered grid approximations in finite difference (FD) methods, but it is puzzling why physics 0740-3224/11/030577-10$15.00/0

should be subordinate to numerical methods and not the other way around. The second root of the proposed methodology is the mathematical framework developed by Whitney [15] and advanced in numerical analysis by Nedelec, Bossavit, and Kotiuga [16–20]. It should be emphasized, however, that this Whitney– Nedelec–Bossavit–Kotiuga (WNBK) framework is used here not for computational purposes, but to define, analytically, the coarse-grained fields. The end result of combining WNBK interpolation with Smith and Pendry’s insight is a mathematically and physically consistent model that is rigorous, general (e.g., applicable to magnetoelectric coupling), and yet simple enough to be practical. The theory is supported by analytical and numerical case studies and is consistent with the existing theories and results [e.g., with the Maxwell–Garnett (MG) mixing formula and with the zero cell-size limit] within the ranges of applicability of the latter. Approximations that have to be made, and the respective sources of error as well as several routes for further accuracy improvement, are clearly identified.

2. FORMULATION AND SOME PITFALLS Consider a periodic structure composed of materials that are assumed to be (i) intrinsically nonmagnetic (which is true at sufficiently high frequencies [21,22]) and (ii) described by a linear local constitutive relation d ¼ εe. For simplicity, we assume a cubic lattice with cells of size a. The microscopic fields b, e, and d satisfy Maxwell’s equations ∇ × e ¼ iωc−1 b; ∇ × b ¼ −iωc−1 d in the frequency domain and with the expð−iωtÞ phasor convention; c is the speed of light in a vacuum. The task is to define the respective coarse-grained fields E, B, D, an additional field H, and the constitutive relationships between them, so that the standard Maxwell’s equations and boundary conditions hold on the coarse level. It is instructive © 2011 Optical Society of America

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to review some averaging procedures that appear to be quite natural and yet upon closer inspection turn out to be flawed. A physically valid alternative is introduced in Sections 3 and 4. A. Passing to the Limit of the Zero Cell Size A distinguishing feature of the homogenization problem for metamaterials is that the cell size a, while smaller than the vacuum wavelength λ0 at a given frequency ω, is not vanishingly small. The typical range in practice is a ∼ 0:1–0:3λ0 . Therefore, classical homogenization procedures valid for a → 0 (e.g., Fourier [23] or two-scale analysis [24]) have limited applicability here. Independent physical [22] and mathematical [25] arguments show that the finite cell size is a principal 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 permittivity ε of the materials remains bounded. On physical grounds [25], this can be explained by the operating point on the normalized Bloch band diagram falling on the “uninteresting” acoustic branch. B. Mollifiers The classical approach to defining the macroscopic fields is via convolution with a smooth mollifier function (e.g., Gaussian-like) [26]. The mollifier must operate on an intermediate scale, much coarser than the cell size but still much finer than the wavelength in the material. For natural materials, this requirement is fulfilled because the cell size is on the order of molecular dimensions. In contrast, for metamaterials, with a ∼ 0:1–0:3λ0 , no intermediate scale is available for the mollifier. C. Simple Cell Averaging Simple cell averaging of the fields is, for metamaterials, inadequate. To understand why, consider, qualitatively, the behavior of a tangential component of the microscopic b field in the direction normal to a material–air interface (Fig. 1). The precise field distribution is unimportant; one may have in mind, for example, a single Bloch wave moving away from the surface, although later it will be essential to consider a superposition of Bloch waves. Clearly, the average field B0 ¼ hbi over the cell is not, in general, equal to the field bð0þÞ inside the material immediately at the boundary, yet it is bð0þÞ that couples with the field in the air: bð0þÞ ¼ bð0−Þ, where bð0−Þ is the field in the air immediately at the boundary. (Intrinsically nonmagnetic materials are assumed throughout.) The classical boundary

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

Igor Tsukerman

condition is recovered if an auxiliary H field is introduced in such a way that Hð0þÞ ¼ bð0þÞ; magnetization then is 4πm ¼ b − H (i.e., it is the difference between the cellaveraged field b and its value at the boundary that ultimately defines magnetization and the permeability). This picture, albeit simplified (in particular, it ignores a complicated surface wave at the interface [11,27]), does serve as a useful starting point for a proper physical definition of field averages and effective material parameters. Since the cell-averaged field is, for a nonvanishing cell size, not generally equal to its boundary value, the use of such fields in a model would result in nonphysical equivalent electric/magnetic currents on the surface (jumps of a tangential component of H or E), nonphysical electric/magnetic surface charges (jumps of the normal component of D or B), and incorrect reflection/transmission conditions at the boundary. It is true that, as a “zero-order” approximation, the cellaveraged field is approximately equal to its pointwise value; however, equating them means ignoring the variation of the fields over the cell and, hence, throwing away the very physical effects that are being investigated. D. Magnetic Dipole Moment per Unit Volume This textbook definition of magnetization turns out to be flawed as well. Simovski and Tretyakov [12] give a counterexample for a system of two small particles, but, in fact, their argument is general. Suppose that a large volume of a low-loss metamaterial has been in some way homogenized and is now represented, to an acceptable level of approximation, by effective parameters μeff , εeff . Consider then a standing EM wave in this material (as produced, e.g., in a cavity or by reflection off a mirror). At any E node, the coarse-grained electric field is zero, and if in addition the lattice cell possesses mirror symmetry in the direction of the wave, the microscopic field is zero, too. This implies zero polarization/conduction currents at the node, hence zero magnetic dipole moment and μeff ¼ 1. Since an E node can be easily arranged to occur at any given location, μeff must be equal to 1 everywhere if “magnetic dipole moment per unit volume” is used to define magnetization. E. Bulk Parameters It is known that, even for a homogeneous isotropic infinite medium, the pair of parameters ε and μ are not defined uniquely. Indeed, the total microscopic current j can be split up—in principle, fairly arbitrarily—into the “electric” and “magnetic” parts, j ¼ ∂t p þ c∇ × m [27–29]. The h field is then defined accordingly, as b − 4πm, giving rise to the respective value of μeff that depends on the choice of m. A more general “Serdyukov–Fedorov” transformation leaves Maxwell’s equations invariant, but changes the values of the material parameters [28,30]: d0 ¼ d þ ∇ × Q, h0 ¼ h þ c−1 ∂t Q, b0 ¼ bþ ∇ × F, and e0 ¼ e þ c−1 ∂t F, 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 the product εμ that is unambiguously defined, with its direct phys1 ical relation to phase velocity vp ¼ c=ðεμÞ2 . The situation changes thoroughly when a material interface (for simplicity, with air) is considered. Classical boundary conditions for the tangential continuity of the H and E fields fix the ratio of the 1 material parameters via the intrinsic impedance η ¼ ðμ=εÞ2 , which, taken together with their product, identifies these

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two parameters separately and uniquely. It is clear then that any complete and rigorous definition of the effective EM parameters of metamaterials must account for boundary effects.

3. COARSE-GRAINED FIELDS A. Guiding Principles Consider a periodic structure composed of materials that are assumed to (i) be intrinsically nonmagnetic (which is true at sufficiently high frequencies [21,22]) and (ii) satisfy a linear local constitutive relation d ¼ εe. For simplicity, we assume a cubic lattice with cells of size a. Maxwell’s equations for the microscopic fields are, in the frequency domain and with the expð−iωtÞ phasor convention, ∇ × e ¼ iωc−1 b;

∇ × b ¼ −iωc−1 d:

The coarse-grained fields B, H, E, and D must be defined in such a way that the boundary conditions are honored. From the mathematical perspective, these fields must lie in their respective functional spaces E; H ∈ Hðcurl; ΩÞ;

B; D ∈ Hðdiv; ΩÞ;

ð1Þ

where Ω is a domain of interest that, for mathematical simplicity, is assumed finite. Ω will henceforth be dropped to shorten the notation. Rigorous definitions of these functional spaces are available in the mathematical literature (e.g., [32]). From the physical perspective, constraints (1) mean that the E and H fields have a valid curl as a regular function (not just as a Schwartz distribution), while B and D have a valid divergence. This implies, most importantly, tangential continuity of E and H and normal continuity of B and D across material and cell interfaces. The fields in HðcurlÞ are also said to be curl conforming, and those in HðdivÞ to be div conforming. In principle, any choice of a curl-conforming H field produces the respective “magnetization” 4πm ≡ b − H and leaves Maxwell’s equations intact. However, most of such choices will result in technically valid but completely impractical and 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 fields via the WNBK interpolation has particular mathematical and physical elegance, which leads to practical advantages in the computation of fields in periodic structures. B. Background: WNBK Interpolation For a rigorous definition of the coarse-grained H field, we shall need an interpolatory structure referred to in the literature as the “Whitney complex” [16,17]; however, acknowledgment of the seminal contributions of Nedelec, Bossavit, and Kotiuga [16,18,19] is quite appropriate and long overdue. Still, the subscripts of quantities related to the Whitney complex will 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 objective here is not to develop a numerical procedure; rather, the mathematical structure that has served so well in numerical analysis is borrowed and applied to fields in metamaterial cells. The original Whitney forms [15] are rooted very deeply in differential geometry, and the interested reader can find a

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complete mathematical exposition in the literature cited earlier. Here, we need a small subset of this theory where the usual framework of vector calculus is sufficient. Although Whitney’s construction was developed for simplices [tetrahedra in three dimensions (3D)], the key ideas are also applicable to a cubic reference cell ½−1; þ13 that can be linearly transformed into an arbitrary rectangular parallelepiped or, if necessary, to any hexahedron [33]. For simplicity, this paper deals only with cubic lattice cells. We shall need two R interpolation procedures: (i) given the circulations ½qα ≡ α q · dl of a field q over the 12 edges of the cell (α ¼ 1; 2; …; 12), extend that fieldRinto the volume of the cell; and (ii) given the fluxes ½½qβ ≡ β q · dS of a field q over the six faces of the cell (β ¼ 1; 2; …; 6), extend that field into the volume of the cell. Single brackets denote line integrals over an edge; double brackets denote surface integrals over the faces. Typically, item (i) will apply to the e and h fields and item (ii) will apply to the d and b fields. Consider an edge α along a ξ direction, where ξ is one of the coordinates x, y, z, and let η and τ be the other two coordinates, with ðξ; η; τÞ being a cyclic permutation of ðx; y; zÞ and, hence, a right-handed system. Edge α is then formally defined as −1 ≤ ξ ≤ 1; η ¼ ηα ; τ ¼ τα ; ηα ; τα ¼ 1. Associated with this edge is a vectorial interpolating function wα ¼ ^ξð1 þ ηα ηÞ ð1 þ τα τÞ=8, where the hat symbol denotes the unit vector in a given direction. For illustration, a two-dimensional (2D) analog of this vector function is shown in Fig. 2. In 3D, there are 12 interpolating functions of this kind—one per edge—in the cell. It is straightforward to verify that the edge circulations of these functions have the Kronecker-delta property: ½wα α0 ¼ δαα0 . This guarantees that the interpolating functions are linearly independent over the cell and span a 12-dimensional space of vectors that can all be represented by interpolation from the edges into the volume of the cell: q¼

12 X ½qα wα :

ð2Þ

α¼1

We shall call this 12-dimensional space W curl , where W honors Whitney and “curl” indicates fields whose curl is a regular function rather than a general distribution. This implies, in physical terms, the absence of equivalent surface currents and the tangential continuity of the fields involved. Any adjacent

Fig. 2. (Color online) 2D analog of the vectorial interpolation function wα (in this case, associated with the central vertical edge shared by two adjacent cells). Tangential continuity of this function is evident from the arrow plot; its circulation is equal to 1 over the central edge and to zero over all other edges.

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lattice cells sharing a common edge will also share, by construction of interpolation (2), the field circulation over that edge. The curls of wα are not linearly independent but rather, as can be demonstrated, lie in the six-dimensional (6D) space ^ð1  yÞ=8, W div spanned by functions v1−6 ¼ f^ xð1  xÞ=8, y ^zð1  zÞ=8g. A 2D analog of a typical function v is shown in Fig. 3. The v functions satisfy the Kronecker-delta property with respect to the face fluxes: ½½vβ β0 ¼ δββ0 . Hence, one can consider vector interpolation from fluxes on the cell faces into the volume of theP cell, conceptually quite similar to edge interpolation (2): q ¼ 6β¼1 ½½qβ vβ . The 6D space spanned in a lattice cell by the v functions will be denoted with W div , reflecting the easily verifiable fact that the normal component of these functions is continuous across the common face of two adjacent cells. Importantly, as already mentioned, the curls of functions from W curl lie in W div , or symbolically in terms of the functional spaces, ∇ × W curl ∈ W div :

C. Construction of the Coarse-Grained Fields While the naive averaging of the E and H fields over the cell may break the tangential continuity of these fields across the boundaries, WNBK interpolation provides a mathematically and physically valid alternative. Let us define the coarsegrained fields as WNBK interpolants of the actual edge circulations via the w functions, in accordance with Eq. (2). Within each lattice cell, 12 X ½eα wα ≡ W curl ð½e1−12 Þ



6 X ½½bβ vβ ≡ W div ð½½b1−6 Þ

ð5Þ

β¼1

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 rapidly varying remainders e∼ , b∼ , and d∼ : e ¼ E þ e∼ ;

d ¼ D þ d∼ ;

b ¼ B þ b∼ :

ð6Þ

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

ð3Þ

To summarize, the following properties are critical for our construction of the coarse-grained fields. (1) Twelve functions wα (one per cell edge) interpolate the field from its edge circulations into the cell. The resulting field is tangentially continuous across all cell boundaries. The w functions span the 12-dimensional functional space W curl . (2) Six functions vβ (one per cell face) interpolate the field from its face fluxes into the cell. The resulting field has normal continuity across all cell faces. The v functions span a 6D functional space W div . (3) Any vector field in W curl is uniquely defined by its 12 edge circulations. (4) Any vector field in W div is uniquely defined by its six face fluxes. (5) ∇ × W curl ∈ W div (see also [34]).

E≡

with a completely similar expression for the H field. ≡ indicates that this is the definition of E and H as well as of the WNBK curl-interpolation operator W curl . Similarly, the B and D fields are defined as W div interpolants from the actual face fluxes into the cells via the v functions

b ¼ H þ 4πm∼ :

ð7Þ

With these splittings, Maxwell’s equations become ∇ × ðE þ e∼ Þ ¼ iωc−1 ðB þ b∼ Þ;

ð8Þ

∇ × ðH þ 4πm∼ Þ ¼ −iωc−1 ðD þ d∼ Þ:

ð9Þ

At this point, the role of the WNBK interpolation becomes apparent: the scales separate. Indeed, E is, by construction, in W curl , and, therefore, ∇ × E is in W div and so is, by construction, B. In that sense, the capital-letter terms in Eq. (8) are fully compatible. Furthermore, E—again by definition—has the same edge circulations as the microscopic field e; hence, for any face of any cell, Z

Z

E · dl ¼ iωc−1

ð∇ × EÞ · dS ¼ face

face edges

Z b · dS; face

where the Stokes theorem and the microscopic Maxwell’s equations were used. However, the B field has the same face fluxes as b by construction and, since these face fluxes define the field in W div uniquely, we have

ð4Þ

α¼1

∇ × E ¼ iωc−1 B:

ð10Þ

∇ × H ¼ −iωc−1 D:

ð11Þ

Analogously,

Thus, the coarse level has separated out, and, remarkably, the WNBK fields satisfy the Maxwell’s equations as well as the proper continuity conditions exactly. The underlying reason for that is the compatibility of the curl and div interpolations, i.e., condition (3). For the rapidly changing components, straightforward algebra yields, from Eq. (8) and (9), Fig. 3. (Color online) 2D analog of the vectorial interpolation function vβ (in this case, associated with the central vertical edge). Normal continuity of this function is evident from the arrow plot; its flux is equal to 1 over the central edge and zero over all other edges.

∇ × e∼ ¼ iωc−1 b∼ ;

4π∇ × m∼ ¼ −iωc−1 d∼

with the “constitutive relationships”

ð12Þ

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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∼ are zero. If/when the coarse-grained fields have been found from their Maxwell’s equations, one may convert the two equations for the rapid fields into a single equation for e∼ : ∇ × ∇ × e∼ − ðω=cÞ2 εe∼ ¼ ðω=cÞ2 ðεE − DÞ − iω=c∇ × ðB − HÞ: ð14Þ This equation can be solved for each cell, with the Dirichlettype zero-circulation boundary conditions. In principle, this fast-field correction will increase the accuracy of the overall solution. However, this paper remains focused on the coarse fields and the corresponding effective parameters.

4. MATERIAL PARAMETERS A. Procedure for the Constitutive Matrix We are now in a position to define effective material parameters, i.e., the linear relationships between D, B and E, H. Given a metamaterial lattice, let us construct coarse-grained curl-conforming fields E, H using the lattice cell edges as a “scaffolding”: the field is interpolated into the cells from the edge circulations of the respective microscopic field (Fig. 4). Thus, the edge circulations of the microscopic and the coarsegrained curl-conforming fields are the same. Similar considerations apply to div-conforming fields, but with interpolation from the faces. Next, let the EM field be approximated as a linear combination of some basis waves (modes) ψ α : Ψeh ¼

X cα ψ eh α ; α

Ψdb ¼

natural to consider each pair of fields ðe; dÞ and ðh; bÞ separately, and then all ψs have three components rather than six. A common example of such basis waves is Bloch modes in periodic media. To each basis wave ψ α , there correspond the WNBK curl interpolants Eα ðrÞ ¼ W curl ð½ψ eα 1−12 Þ and Hα ðrÞ ¼ W curl ð½ψ bα 1−12 Þ. Similarly, there are the WNBK div interpolants Dα ðrÞ ¼ W div ð½½dα 1−6 Þ and Bα ðrÞ ¼ W div ð½½bα 1−6 Þ. It is possible to find a 6 × 6 constitutive matrix ζ that minimizes the L2 norm of the discrepancy ψ DB ðrÞ − ζðrÞψ EH ðrÞ; a way to do so will be described in a future publication. Here, a simpler procedure is adopted. For all basis waves α at any given point r in space, we seek a linear relation EH ψ DB α ðrÞ ¼ ζðrÞψ α ðrÞ;

where ζ characterizes, in general, anisotropic material behavior with (if the off-diagonal blocks are nonzero) magnetoelectric coupling. Similar to ψ db , the six-component vectors ψ DB comprise both fields, but coarse grained; the same applies to ψ EH . In matrix form, the above equations are ζðrÞΨEH ðrÞ ¼ ΨDB ðrÞ;

ζðrÞ ¼ ΨDB ðrÞðΨEH Þþ ðrÞ:

α

ð16Þ

A notable feature of this construction is that the coarsegrained fields corresponding to a particular basis wave satisfy the Maxwell’s equations with this material parameter ζðrÞ exactly. The approximate parameters for each cell are then found simply by cell averaging ζðrÞ. One may wonder why such cell averaging could not be done in the very beginning, for the original microscopic Maxwell’s equations, in which case one would get the trivial value of unity by averaging the intrinsic permeability μ ¼ 1. First, as already discussed, the cell-averaged fields violate the boundary conditions and, therefore, are inadequate for metamaterials. Further, in the case of the coarse-grained fields, splitting the material matrix into its cell average plus a fluctuating component, ζ ¼ hζi þ ζ ∼ , makes more sense. Indeed, consider the Maxwell’s equations (in the 6D form) for the coarse-grained fields:  ∇×

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

ð15Þ

where each column of the matrices ΨDB and ΨEH contains the respective basis function. (Illustrative examples in Section 5 may help to clarify these notions and notation.) If exactly six basis functions are chosen, one obtains the constitutive matrix by straightforward matrix inversion; if the number of functions is more than six, the pseudoinverse [35] is appropriate:

X cα ψ db α :

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

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E H

 ¼

iω c



B −D

 ¼

iω c



0 −I 3

   I3 E ζðrÞ ; 0 H

ð17Þ

where I 3 is the 3 × 3 unit matrix. The approximate replacement of ζðrÞ with hζi introduces a perturbation that is proportional to ΨDB − hζiΨEH and that our construction of hζi is intended to minimize. It would be a mistake to manipulate the original microscopic equations in a similar manner: the respective “perturbation” would not be small due to the strong variation of the intrinsic permittivity over the lattice cell. This strong variation is at the heart of the resonance effects [22,25].

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B. Recipe Let the vectorial dimension of the problem—i.e., the total number of vector field components—be N. Most generally, N ¼ 6 (three components of E and three of H), but if only one field is involved, then N ¼ 3, and if that field has only one component, then N ¼ 1, etc. It is convenient to summarize the procedure described earlier as a “recipe” for finding the effective parameters: (1) Choose a set of M ≥ N basis waves (modes) ψ α (e.g., Bloch waves) that provide a good approximation of the fields within a cell. (2) Find the curl- and div-WNBK interpolants of each basis wave. (This step requires the computation of face fluxes and edge circulations of the respective fields in the wave.) (3) Assemble these WNBK interpolants into the ΨDB and ΨEH matrices of Eq. (15). (4) Find the coordinate-dependent parameter matrix ζðrÞ from Eq. (16). (5) The cell average of this matrix gives the final result: an N × N (6 × 6 in the most general case) matrix of effective parameters. As already noted, instead of steps (4) and (5), one may want to minimize the discrepancy ∥ψ DB ðrÞ − ζðrÞψ EH ðrÞ∥. C. Errors and Error Indicators Three sources of error can be distinguished in the proposed homogenization procedure. 1. In-the-basis error. If the number of basis waves is strictly greater than the vectorial dimension of the problem (M > N), then Eq. (15) does not generally have an exact solution and is solved in the least-squares sense, as the pseudoinverse in Eq. (16) indicates. From Eqs. (15) and (16), the discrepancy between the fields is (with the dependence on r implied) ΨDB − ζΨEH ¼ ΨDB ðI − ðΨEH Þþ ΨEH Þ, where I is the 6 × 6 identity matrix. Therefore, the ratio ∥I − ðΨEH Þþ ΨEH ∥=∥ðΨEH Þþ ΨEH ∥ is a suitable indicator of the relative in-the-basis error. If M ¼ N (for example, six basis waves in a generic problem of electrodynamics), then the in-the-basis error vanishes. It also vanishes in the longwavelength limit. 2. Out-of-the-basis error. Any field can be represented as a linear combination of the basis waves, plus a residual field. For a good basis set, this residual term is small. Any expansion of the basis carries a trade-off: the residual field and the outof-the-basis error will decrease, but the in-the-basis error may increase. 3. Parameter averaging. As an intermediate step, the proposed homogenization procedure yields a coordinatedependent parameter matrix ζðrÞ that is, in the end, averaged over the lattice cell. This error was discussed earlier. The limitations of the effective-medium approximation now become apparent as well. If a sufficient number of modes with substantially different characteristics exist in a metamaterial (e.g., in cases of strong anisotropy), then the in-the-basis error will be high. This is not a limitation of the specific procedure advocated in this paper, but a reflection of the fact that the behavior of fields in the material is, in such a case, too rich to be adequately described by a single effective-parameter tensor. On a positive side, several specific ways of improving the approximation accuracy can be identified. Obviously, no accuracy gain is completely free. (i) The cell problem for the rapidly varying fields can be solved, once the coarse-grained

Igor Tsukerman

fields have been found from the effective-parameter model. (ii) If the relative weights of different modes in a particular model can be roughly evaluated a priori, then Eq. (15) could be biased toward these modes; some columns of the Ψ matrices (i.e., some basis functions) could even be eliminated. The downside is that the material parameters are no longer a property of the material alone, but become partly problem dependent. (iii) Instead of reducing the number of columns in the Ψ matrices, one can increase the number of rows. This requires new degrees of freedom in addition to the six components of the fields, e.g., higher-order moments of the fields over edges and faces. This idea was put forward by Rodin [36] in a different context. Physically, this is a manifestation and an acknowledgment of the nonlocality of the problem. The material parameter matrix ζ also becomes expanded and might be called a “lattice cell response matrix.” (iv) The size and composition of the basis set can be optimized for common classes of problems. (v) The last step of the proposed procedure—the cell averaging of matrix ζ—could be omitted. The coarse-grained fields are then evaluated accurately, but the numerical solution may be computationally expensive, as the material parameter varies within the cell. There is a spectrum of practical compromises where ζ would be approximated within a cell not as a constant, but to some higher order. (vi) One may envision adaptive procedures whereby the basis waves are updated after the problem has been solved, and then new material parameters are derived from the new basis set. Again, the downside is that, in addition to the increased cost, the material parameters become partly problem dependent. In connection with the last item, it is clear that the basis set should, in general, reflect the symmetry and reciprocity properties of the problem. In particular, the basis should as a rule include pairs of waves traveling in the opposite directions. D. Causality and Passivity Physical considerations indicate that the proposed procedure should be expected to produce causal and passive effective media, at least if M ¼ N. Indeed, suppose that the opposite is true, e.g., ε00eff < 0 (with anisotropy neglected for simplicity), violating passivity. The effective parameters, however, apply by construction to all basis waves exactly; hence, ε00eff < 0 would imply power generation in the actual physical modes in the actual passive metamaterial, which is impossible. The above argument is, strictly speaking, valid only prior to the cell averaging of the coordinate-dependent material parameter matrix. Another caveat to this argument is that causality and passivity apply to the total fields that are the sums of coarse-grained and rapid components. If the rapid components and energies associated with them are significant, then the coarse-grained fields and the related material parameters may, in principle, violate passivity without breaking physical laws. In such cases, however, the validity of homogenization would be questionable, and further analysis is desirable in the future.

5. VERIFICATION A. Empty Cell Although this case is seemingly trivial, the effective parameters of empty cells produced by alternative methods in the literature often contain spurious Bloch-like factors that are

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then discarded by fiat. In contrast, direct calculation that follows the methodology of this paper produces μeff ¼ 1, εeff ¼ 1 exactly, without any spurious factors and, of course, with no magnetoelectric coupling, both in 2D and 3D. This is valid as long as the cell size does not exceed one half of the wavelength. B. Example: One-Component Static Fields This is another simple consistency check for the proposed methodology. Let a static field (e.g., electrostatic) have only one component (say, z) that must be independent of z due to the zero-divergence condition. Then, the lattice cells become effectively 2D and the div-conforming WNBK interpolant for d reduces just to a constant D0 whose flux through the cell is equal to the flux of d. The curl-conforming WNBK interpolant W curl ðeÞ would generally be a bilinear function of x and y, but since the e field is constant in the xy plane, this interpolant also reduces to a constant E0 ¼ E. Thus, the dielectric permittivity is D0 ¼ E0

εeff ≡

R

d · dS ¼ S −1 SE 0

Z εdS

cell

cell

exactly as should be expected. C. Example: the MG Formula The next test is consistency with the classical MG mixing formula for a two-component medium in a static field. (We do not deal with radiative corrections to the polarizability and the MG in this paper.) The MG expression for the effective permittivity is, in 2D, εMG;2D ¼ ð1 þ f χÞ=ð1 − f χÞ; where χ is the polarizability of inclusions in a host medium and f is the fill factor. In particular, for cylindrical inclusions in a nonpolarizable host, χ ¼ ðεcyl − 1Þ=ðεcyl þ 1Þ, f ¼ πr 2cyl =a2 . The proposed methodology specializes to this case as follows. First, one has to define a set of basis fields; naturally, for the static problem this is more easily done in terms of the potential rather than the field. At zero frequency, the Bloch wavenumber is also zero; hence, the Bloch conditions for the field are periodic, but the potential may have an offset corresponding to the line integral of the field across the cell. Thus, the first basis function ψ 1 is defined by   a a ¼ ; ψ1 x ¼  2 2

∇ · ε∇ψ 1 ¼ 0;

583

where ε is that of the host if cell boundaries do not cut through the inclusions. Because the fluxes and circulations of the fields over the pairs of opposite edges are in this case equal, the WNBK interpolant is simply constant and the effective parameters are obtained immediately, without the intermediate stage of computing and averaging their pointwise values. Formally, Eq. (15) reduces to 

εxx εyx

εxy εyy



1 0 0 1



 ¼

½½d1  0

0 ½½d2 

 ð18Þ

(cell size a normalized to unity for simplicity). The identity matrix is written out explicitly to point out its origin: its first column represents the WNBK curl-conforming interpolant of e1 ; the second column represents that of e2 . The rightmost matrix contains the div-conforming WNBK interpolants of d1 and d2 . The system of equations is trivial in this case; since the fluxes ½½d1  and ½½d2  are equal, the permittivity is a scalar quantity equal to these fluxes. For numerical illustration, let us consider a cylindrical inclusion with a varying radius in a nonpolarizable host. The commercial FE package Comsol is used to compute the basis function and its edge flux ½½d1  (¼ ½½d2 ). (There are ∼30; 000 triangular elements in a typical FE simulation; fourth-order numerical 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 and εcyl ¼ 10) show that the agreement between the new method and MG is excellent even for fairly large radii of the inclusions. While such an agreement may be surprising at first glance, MG does, in fact, retain high accuracy beyond its “theoretical” range of applicability (see, e.g., a summary in [37]). D. Example: Bloch Bands and Wave Refraction Let us now consider wave propagation through a photonic crystal (PhC) slab—an array of cylindrical rods with no defects. For consistency with previous work, the geometric

ψ 1 ðy þ aÞ

¼ ψ 1 ðyÞ: The potential difference across the cell corresponds to a unit uniform field applied in the −x direction. The second basis function is completely analogous and corresponds to a field in the −y direction. Once the basis functions have been found, the next steps are to compute the circulations and fluxes, and then the WNBK interpolants, of the basis functions. For ψ 1 , the circulation of the respective field e1 ¼ −∇ψ 1 over each “horizontal” edge y ¼ a=2 is, by construction, equal to a and is zero over the two “vertical” edges. The fluxes of d1 ¼ εe1 are zero over the horizontal edges; for the vertical ones, Z ½½d1  ¼

y¼a=2

Z d1x dy ¼ −

y¼a=2

ε∂x ψ 1 dy;

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.

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Fig. 8. (Color online) Contour plot of ReðHÞ. Angle of incidence π=6, εcyl ¼ 9:61, r cyl ¼ 0:33a. Fig. 6. (Color online) ΓX Bloch bands obtained with the effective parameters (markers) versus accurate numerical simulation (solid curves). εcyl ¼ 9:61, r cyl ¼ 0:33a.

and physical parameters of the crystal are chosen to be the same as in [38] and are taken from [39]. Namely, the radius of the rod is r cyl ¼ 0:33a and its dielectric permittivity is εcyl ¼ 9:61. The p mode (H mode, with the one-component magnetic field along the rods) is considered because this is a more interesting case for homogenization. The numerical simulation of the Bloch bands and wave propagation through the PhC slab was performed with flexible local approximation methods (FLAME) [38,40], a generalized FD calculus that incorporates local analytical solutions into the difference scheme—arguably, the most accurate numerical method for this type of problem. Figure 6 shows that the Bloch bands obtained with the effective parameters are in excellent agreement with the accurate numerical simulation, except for a few data points at the band edge, where effective parameters cannot be expected to remain 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 normalized frequency approximately between 0.33 and 0.42. This agrees very well with the band diagrams above and the results reported previously [39,38]. Numerical simulation (using FLAME) of EM waves in the actual PhC is compared with the analytical solution for a homogeneous slab of the same thickness as the PhC as well as effective material parameters εeff and μeff calculated as the methodology of this paper prescribes. In the numerical simulations, obviously, a finite array of cylindrical rods has to be used (in this case, 24 × 5) to limit the computational cost. The results reported below are at the normalized frequency of ~ ≡ ωa=ð2πcÞ ¼ a=λ ¼ 0:24959, coinciding with one of the ω data points in previous simulations [38]. A fragment of the contour plot of ReðHÞ in Fig. 8 helps to visualize the wave for the angle of incidence π=6. The real part of the numerical and analytical magnetic fields along the line perpendicular to the slab is plotted in Fig. 9 for the angle of incidence π=6 (the plot for the imaginary part is similar). The analytical and numerical results are shown to be in a good agreement, despite the approximate

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

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

Igor Tsukerman

nature of the effective parameters for the fairly large cell size a ∼ λ0 =4 and the high fill factor. E. Conclusion The proposed new methodology rigorously defines effective parameters of EM and optical metamaterials and explains, among other things, nontrivial magnetic effects arising in intrinsically nonmagnetic media. The main underlying principle is that the coarse-grained E and H fields have to be curl conforming (i.e., possessing a valid curl as a regular function, which, in particular, implies tangential continuity across material interfaces), while the B and D fields have to be div conforming, with their normal components continuous across interface boundaries. While some flexibility in the choice of these coarse-grained fields exists, an excellent framework for their construction is provided by Whitney forms and the WNBK complex (Subsection 3.B). This construction ensures not only the proper continuity conditions for the respective fields, but also the compatibility of the respective interpolants, so that, for example, the curl of E lies in the same approximation space as B. As a result, remarkably, Maxwell’s equations for the coarse-grained fields are satisfied exactly. Further, the EM field is approximated with a linear combination of basis functions (modes), e.g., Bloch waves. Effective parameters are then devised to provide the most accurate linear relation between the WNBK interpolants of these basis functions. In the limiting case of a vanishingly small cell size, this procedure yields the exact result; for a finite cell size, as must be the case for all metamaterials of interest [22,25], the effective parameters are an approximation. A number of ways to improve the accuracy are outlined in Subsection 4.C. Proponents of the differential-geometric treatment of EM theory have long argued that the H, E and B, D fields are actually different physical entities, the first pair best characterized via circulations (mathematically, as 1-forms) and the second one via fluxes (2-forms) [17,18,41]. The approach advocated and verified in this paper buttresses this viewpoint. There are several important issues to be addressed in future work. First, the theory of this paper needs to be applied to 3D structures of practical interest. Second, physical considerations indicate that the theory should lead to causal and passive effective media (see Subsection 4.D and [11,12]), but a rigorous mathematical analysis or, in the absence of that, accumulated numerical evidence are highly desirable. Third, specific routes 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 metamaterials, it is hoped that the new theory will also help to understand more deeply the nature of the fields in natural materials, as rigorous definitions of such fields, especially of the H field, are nontrivial. The ideas and methodology of this paper are general and should find applications beyond electromagnetism, such as acoustics and elasticity.

ACKNOWLEDGMENTS Discussions with V. Markel and J. Schotland to a large extent inspired this work and are greatly appreciated. D. Golovaty has offered invaluable insight and a number of helpful suggestions. Communication with A. Bossavit, C. T. Chan, Z. Zhang, Y. Wu, Y. Lai, G. Rodin, and L. Demkowicz is also gratefully acknowledged. The author thanks C. T. Chan

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(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 hospitality during the period when this work was either contemplated or performed.

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