WITHDRAWN: Layer homogenization of a 2D periodic array of scatterers

+Model METMAT 118 1–6

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Metamaterials xxx (2013) xxx–xxx

Layer homogenization of a 2D periodic array of scatterers

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Didier Felbacq ∗

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University of Montpellier 2 and Institut Universitaire de France, Laboratory Charles Coulomb Unité Mixte de Recherche du Centre National de la Recherche Scientifique, 5221 Montpellier, France

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Received 8 November 2012; received in revised form 30 April 2013; accepted 1 May 2013

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Abstract

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The homogenization of a metamaterial made of a collection of scatterers periodically disposed is studied from an asymptotic theory and an optimization algorithm. Detailed numerical results are given for resonant scatterers and the spatial dispersion is investigated. © 2013 Published by Elsevier B.V.

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Keywords: Homogenization theory; Metamaterials; Photonic crystals; Scattering theory

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1. Introduction

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Giving a general definition of what homogenization is difficult, because of the various meanings attached to it. The physics at stake is the situation when a wave illuminates a complicated object, generally consisting of a periodic set of scatterers (but the periodicity assumption can be relaxed [1–3]), contained in some domain  and gives rise to a diffracted field Us . Loosely speaking, the homogenization problem consists in identifying homogeneous constitutive relations such that the same domain  inside which these constitutive relations hold, leads to a diffracted field Uhs such that Us and Uhs are close to each other, in some meaning to be specified. This can be done reasonably only if the wavelength λ is larger than twice the period d. This identifies a small parameter η = d/λ. From a mathematical point of view, the definition that would be considered the best one is the following: consider a partial differential operator Pη with oscillating coefficients aη (x). Consider a solution Uη of the

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equation: Pη (Uη ) = f, where f is some convenient source term. Then the goal of homogenization theory is to find a convenient topology in which Uη converges to a function U0 satisfying an equation: P0 (U0 ) = f. The operator P0 is called the homogenized operator [1,4–8]. This definition is quite clear and at the end, it leads to results such as: “when η tends to 0, Uη tends to U0 in some specific meaning”. The point of being able to specify a convergence is very interesting, in that it gives a clear meaning to the question “how close are Us and Uhs ?”. Also, it allows an extension of the results to higher order terms, as a function of the parameter η. Sometimes, it is not directly the field Us that is matched, but instead the Bloch spectrum. However, care should be taken because of the following result: Proposition 1. Given any isofrequency dispersion curves given implicitly in the form F(kx , ky ) = 0, there exists a spatially and temporally dispersive permittivity ε(k, ω) reproducing these curves. Proof. Write: k2 = k2 + F(kx , ky ) and define: ε(k, ω) = (k2 + F (kx , ky )/k02 ), where k0 = ω/c. 

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Please cite this article in press as: D. Felbacq. Layer homogenization of a 2D periodic array of scatterers, Metamaterials (2013), http://dx.doi.org/10.1016/j.metmat.2013.05.001

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φ(M) = Hp (k0 |r|)eipθ . For the infinite set of scatterers, this gives a diffracted field that reads as:  U s (M) = bpn φp (M − Mn ) (1) n

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p

the field diffracted by the nth scatterer is obtained by writing that it is the response to the incident field Ui and the field difffracted by the other scatterers: ⎞ ⎛  (2) (bpn )p = S ⎝Ei (Pn ) + Tnm (blm )l ⎠

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Fig. 1. Sketch of the structure under study. It is made of a stack of gratings, each consisting of periodically disposed scatterers.

If the incident field is pseudo-periodic, Ui (Mn ) = eikd Ui (Mn−1 ), then it holds:

i.e.

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U i (M0 ) (3) S −1 −  where = m =/ 0 eikm T0m is a quasi-periodic lattice sum. There are many efficient ways of computing this series [12]. Once these coefficients are known, the scattered field can be written as a Rayleigh series:  U s (x, y) = Uns ei(αn x+βn |y|) , (4) (bp0 )p =

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This shows that given any dispersion curves, it is always possible to reproduce it by using a spatially dispersive permittivity. However, nothing can be said on the complete electromagnetic field inside the medium: the Bloch diagram only accounts for the plane wave part of the field. Plus, a permittivity depending on a k vector only makes sense for an infinite medium. Dealing with a finite (physically reasonable) medium requires the introduction of ad hoc additional boundary conditions that seem unnatural in a consistent formulation. Despite the above, video meliora, proboque deteriora sequor, the approach adopted in this work relies on a Green function approach [10,11]. In the following, the structure considered as a model problem is a periodic set of 2D point scatterers (cf. Fig. (1)) electromagnetically characterized by their scattering matrix. The structure is illuminated by an incident monochromatic plane wave (time dependence of e−iωt ), in s-polarization (electric field parallel to the z axis).

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n

 where Uns = l bl0 φˆ l (k + 2πn/d) ( ˆ· denotes the Fourier transform). The entire 2D periodic structure can now be dealt with by considering the transfer operator: T(Es )(x, y) = Es (x, y + d) [14,15]. Our point here is to study the low-frequency regime where the scattering matrix S reduces to a scalar s0 . In that case, one obtains (taking Ei (P0 ) = 1): b00 =

1 s0−1

− 0

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where the series 0 can be written:  0 = eikmd H0 (k0 |m|d)

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m= / 0

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2. An asymptotic result The medium is assumed infinite in the x direction and it is made out of a stack of basic layers made of an infinite number of scatterers periodically disposed at points x = nd. First we consider the transmission spectrum of one basic layer. It is obtained by means of a multiple scattering approach [10,11]. For the convenience of the reader and for later calculations, the theory is briefly recalled. Each scatterer “n” at position Mn is characterized by a scattering matrix S. When it is illuby an incident field Ui , it gives rise to a field minated  s U (M) = p bp φp (M − Mn ) where (bp )p = SUi (Mn ) and

 4π 2 2i 2i γ + ln + π π k0 d dβ0  2 1 1 d + + − d βn β−n iπ|n| n>0



where β0 = k02 − k2 and βn = k02 − (k + n2π/d). An asymptotic analysis of this series [13] allows to write the following expansion: = −1 −

Lemma 1.  2 1 1 d + − = O[(k0 d)2 ] d βn β−n iπ|n|

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n>0

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Because the only relevant coefficient is b00 , the series in (4) simplifies to the following form:  U s (x, y) = Uns ei(αn x+βn |y|) , (7) n

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where Uns = 2b00 /dβ0 . This series splits into a propagative part and an evanescent part. Because, we are interested in the low frequency behavior, we expect the evanescent field to be irrelevant. We then obtain the following representation for the scattered field: Proposition 2. Above the grating, the propagative part of the electric field reads as: U+ (y) = e−iβ0 y + r(k0 , β0 )eiβ0 y and below it reads U− (y) = t(k0 , β0 )e−iβ0 y where r(k0 , β0 ) = 2b00 /β0 d and t = 1 + r The expression of the reflection coefficient takes a particular form, because of the conservation of energy: Lemma 2. There exists a function χ(k0 , β0 ) such that: −1 . 1 + iχ(k0 , β0 )

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r(k0 , β0 ) =

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The function χ is real for real β0 .

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Proof. Energy conservation implies that |r|2 + |1 + r|2 = 1 and therefore R(r)/|r|2 = −1, the result follows by defining χ = I (r)/|r|2 .  3. Single layer homogenization

Fig. 2. The homogenization of the sandwich structure. The horizontal black line represents the grating.

the transfer matrix of this basic sandwich structure (cf. Fig. (2)) being: T = T(h/2)Tg T(h/2), where: ⎞   ⎛ β0 h/2 β0 h sin cos ⎟ 2 β0 h ⎜ ⎜ ⎟ T =⎜ (9)   ⎟ 2 ⎝ β0 h β0 h ⎠ −β0 sin cos 2 2 A direct calculation shows that:

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Proposition 4. One grating layer can be approximated by the following 2 × 2 transfer matrix:

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⎛ ⎜ T =⎝

cos (β0 h) +

sin(β0 h) 2 L β0

cos(β0 h/2)2 − β0 sin(β0 h) L

sin(β0 h/2)2 L β02

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sin(β0 h) β0

sin(β0 h) cos(β0 h) + 2 L β0

⎞ ⎟ ⎠

Proposition 3. For the propagative part of the field, the grating is equivalent to an infinitely thin slab whose transfer matrix is: ⎞ ⎛ 1 0 ⎠ (8) Tg = ⎝ 1 1 L

Once the basic layer is characterized, the transfer matrix for a stack of “n” gratings is described by Tn . This effective transfer matrix can be used directly to compute the transmission spectrum or the Bloch diagram of the structure.

where L = χ/2β0 .

3.1. Numerical investigations

Proof. The electric field is continuous. The matrix Tg is obtained by computing the jump of the normal derivative of the field: ∂y U+ (0) − ∂y U− (0) = 2iβ0 r. Using Proposition 1, we get U(0) = 1 + r = iχ/1 + iχ = − iχ r. Hence 2iβ0 r = − (2β0 /χ)U(0). 

The transfer matrix is known to be an unstable object for numerical purposes. The preferred quantity is the scattering matrix [14,15]. The asymptotic behavior is performed numerically by searching for a permittivity εeff (k0 , β0 ) such that the scattering matrix of a homogeneous slab with this permittivity fits that of the sandwich structure. Parameter extraction from the S-matrix is a rather old subject [16]. Its application to metamaterials

A layer of the structure can thus be described by this slab surrounded by homogeneous layers of height h/2,

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Reflection spectrum

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Fig. 3. Domain of the parameter space for which the scattering matrix of a dielectric rod can be limited to the term s0 . The region in white corresponds to a ratio |s1 /s0 | less than 1/10.

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was studied in [17] (see also [18] for a more comprehensive bibliography). In the present work, we do not apply inversion formulas to the transfer matrix (10) but rather use an optimisation algorithm on the cost function (11). The use of the cost function allows to quantify directly to what extend there is a pertinent effective behavior (for instance one or several minima) especially when the ratio d/λ is not very small: while the inversion formulas always gives an answer (possibly inaccurate), the optimisation algorithm might not converge, indicating that the problem is not well-posed. It can also be noted that the optimisation could be performed over more than one parameter, for instance the permeability, the width of the layer, the losses (if added in the original model). Specifically, the cost function is:



reff

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Fig. 4. The reflection spectrum for the homogenized structure (dashed) and the sandwich structure (continuous).

ratio |S1 /S0 | as a function of λ/a and ε. The white region corresponds to |S1 /S0 | < 1/10. The reflection spectra for a plane wave in normal incidence and for both the sandwich structure and the homogeneous slab are given in Fig. (4). It can be shown that the homogenization works very well for λ/d > 15. The resulting homogenized permittivity, depending on both the frequency and the horizontal Bloch vector, is given in Fig.(5). The averaged value is εlw = 7.28318 ± 10−5 . The value obtained numerically for λ/d = 1000 is: εeff = 7.2832. The homogenized permittivity was calculated for two angles of incidence 0 and π/4: it can be seen that the effects of spatial

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where rg and tg are the reflection and transmission coefficients of the sandwich structure and reff , teff that of the homogeneous slab. First the scatterer in the basic cell is a dielectric rod with permittivity εr = 9 and radius a/d = 1/2. In that case, the low-frequency behavior is well-known: the sandwich structure is equivalent to a homogeneous slab with permittivity: εlw = 1 + (εr − 1)π(a/d)2 . This serves to test the proposed approach and also to evaluate the spatial dispersion effect in the medium. It is first necessary to determine the region of parameters where each scatterer can be represented by one scattering coefficient only. In Fig. (3), we have plotted the

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Fig. 5. The effective permittivity obtained by using the cost function (11). The continuous curve corresponds to β0 = 0 and the dashed √ one to β0 = k0 2/2. The horizontal line corresponds to the averaged permittivity εlw .

Please cite this article in press as: D. Felbacq. Layer homogenization of a 2D periodic array of scatterers, Metamaterials (2013), http://dx.doi.org/10.1016/j.metmat.2013.05.001

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Fig. 6. The effective permittivity obtained by using the cost function (11) for a grating of resonant scatterers with scattering coefficient given by (12).

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Fig. 7. The effective permittivity obtained by using the cost function (11). 0.1 0.095

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dispersion are quite small. The quasi-static rule represents correctly the slab only for wavelengths that are very large with respect to the period of the lattice, also the medium can be very well represented by a frequency dependent permittivity at much shorter wavelengths. Second, we choose a resonant scatterer with a scattering coefficient of the form ω − ωz s0 = g(ω) , ω − ωp

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where g(ω) is regular and satisfy g → 0 as ω → 0. The sandwich structure air-grating-air can be replaced by a slab with permittivity εeff (ω, β0 ) obtained from the optimisation procedure described above. The reflection spectrum is given in Fig. (6), the homogenization procedure works very well for λ/d > 12. The corresponding homogenized permittivity is given in Fig.(7). The spatial dispersion effects are negligible in that situation. Interestingly, it seems that resonant structures can be homogenized at smaller ratio λ/d than non resonant ones. Moreover the homogenization regime is obtained above the resonant frequency of the system. In Fig. (8), we give the Bloch diagram of the structure (continuous line) and the Bloch diagram of the homogenized structure (black crosses). An excellent fit is seen, below and above the first gap. This shows that a gap can be opened due to internal resonances (a fact already encountered in [20,21]) and that a homogenized description is possible on the second band of the system.

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Fig. 8. Bloch diagram for the periodic array of resonant scatterers (continuous line) and for the homogenized structure (black crosses).

4. Conclusion We have described an approach to the homogenization of a two dimensional dielectric metamaterial by means of an asymptotic approach consisting in replacing a grating of rods by an impedance condition. This led to an explicit form of the transfer matrix of one basic layer. Then we used an optimisation algorithm to derive an effective permittivity. It was demonstrated that resonant scatterers admit homogenization above the first gap, i.e. in a regime very different from the quasistatic homogenization. This asymptotic study could be straightforwardly extended to deal with the new emerging field of quantum metamaterials [22], as for periodic atomic lattices [23,24]. Moreover, the transfer matrix

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formalism presented here could be used to study out-ofequilibrium properties with the formalism developped in [25]. Work is now in progress to study the quantum thermalization mecanism for a quantum system close to such a periodic array [26,27].

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