Guillaume Herlem
email: guillaume.herlem@univ-fcomte.fr

Olivier Segut

Alexandros Antoniou

Christine Achilleos

Didier Dupont

Virginie Blondeau-Patissier

Tijani Gharbi

Electrodeposition and characterization of silane thin films from 3-(aminopropyl)triethoxysilane

Keywords: Electrodeposition, 3-(aminopropyl)triethoxysilane, Scanning tunneling microscopy, Ellipsometry, X-ray photoelectron spectroscopy, Surface plasmon resonance

3-(aminopropyl)triethoxysilane based films have been electrodeposi t e dd i r e c t l yo np o l y c r y s t a l l i n eg o l da n dg o l d( 1 1 1 ) e l e c t r o d e si n aqueous 3-(aminopropyl)triethoxysilane based electrolyte and in tetrahydrofurane based electrolyte. These films were characterized by means of IR-ATR and X-ray photoelectron spectroscopies. The film morphology was investigated by scanning tunneling microscopy while the film growth was observed by ellipsometry measurements. The vibrationnal and X-ray photoelectron analysis suggest that the chemical composition of the electrodeposited films either in liquid tetrahydrofurane or in liquid 3-(aminopropyl)triethoxysilane is identical. The resulting coating thickness is different for the same biasing time in the two liquid media. The gold surface is coated irreversibly by an amino terminated film of great interest for sensor applications which was used as the functionalized part of a surface plasmon resonance biosensor to monitor α-lactalbumin graft.

Introduction

As an alternative to sol-gel method, there were some reports of the electrochemical reduction of silanes yielding linear Si backbone polymers [1,2]. Recently, the electrodeposition of trimethoxysilane (TMOS) on negative biased conducting electrode surfaces to form thin silane films was also reported [3].The deposition process of TMOS in aqueous solution is induced by the electrochemical reduction of oxygen to hydroxyl ions which acts as the catalyst for the hydrolysis and condensation of TMOS on the surface electrode.

A few years ago, we have discovered a new way to electrosynthesize thin linear polypropylenimine films on gold electrodes by anodic oxidation of pure 1,3-diaminopropane [4][5][6]. Our aim is to diversify the way to graft amine based molecule on conducting surfaces and if possible using a molecule which plays at the same time the part of solvent and monomer (to electropolymerize). This is the reason why we used 3-(aminopropyl)triethoxysilane (3-APTES) monomer, a liquid in standard conditions, which contains both amino and silano groups. In this connection, we transposed the work made with TMOS with 3-APTES so that the gold electrode presents amino terminated groups on its surface and makes it possible to graft biological species.

Effectively, two recent papers reported the electrodeposition of 3-APTES on gold or on recordable gold-type compact disks but first modified with a monolayer of alkanedithiol [7,8]. Here we used the electrochemical generation of the condensation catalyst [9,10] and applied it to 3-APTES.

Experimental

All products were from Sigma-Aldrich (France) and ultra pure water (Milli-Q, Millipore) was used when needed. Bovine α-lactalbumin was solubilized at 2 mg/mL in distilled water and stored at -20 °C. A standard three-electrode electrochemical cell setup was used and all potentials are quoted vs. a silver wire reference electrode (SRE).

Several gold substrates have been used in this study as working electrodes. 5 MHz polished gold coated AT cut quartz (Maxtek, USA) was used (in shear mode) and mounted on a probe for electrochemical quartz crystal microbalance (EQCM) experiments (1.37 cm 2 surface area). For scanning tunnel microscopy (STM) imaging, gold coated mica surface (Molecular Imaging, USA) with 1500 of Au (111) was used. 50 nm thickness gold on glass (10 × 12 mm 2 surface area) was used for surface plasmon resonance (SPR) measurements (Au Kit, Biacore).

Functional group analyses were carried out by means of the Omnisampler IR-ATR module plugged to the FT-IR spectrometer (Nexus 470; ThermoElectron, France).

Ellipsometric measurements were performed via an UVISEL (Horiba, France) phase-modulated spectroscopic ellipsometer. The incident angle was set at 70°and the light spot was 1 mm in diameter. The measurements were performed in the spectral range 250-850 nm, with sampling steps of 5 nm.

A Picoscan system (Molecular Imaging, USA) equipped with a 10 × 10 mm 2 scanner (1 nA/V sensitivity) was used for STM measurements using Pt-Ir tips prepared by electrochemical etching [11].

X-ray photoelectron spectroscopy (XPS) measurements were conducted on a ThermoElectron VG spectrometer equipped with a monochromatized Mg Kα X-ray source (1486.6 eV photons) using pass energies of 150 and 20 eV for survey and detailed scans, respectively. C1s peak of binding energy (284.6 eV) was used as the reference.

SPR experiments were conducted on a Biacore 3000 (Biacore International AB, Uppsala, Sweden) using 3-APTES electrodeposited on bare gold sensor chip. After electrodeposition, the sensor chip was inserted in the Biacore device and a continuous flow of 10 mM Hepes, 0.15 M NaCl, 3 mM EDTA, 0.005% surfactant P20, pH = 7.4 (HBS-EP), over the sensor surface at 5 μL/min, was maintained. All the following experiments were done at 25 °C. First, the surface was rinsed with water and activated by injecting 300 μL of glutaraldehyde 1%. Then, after washing with water, 300 μL α-lactalbumin at 2 mg/mL were injected on the surface.

Results and discussion

Electrochemical modifications

Initially we wished to use pure 3-APTES as solvent for electrolyte synthesis. But due to its low dielectric constant, it has never been regarded as a solvent of interest. Concerning salts, we had to avoid those containing alkali cations M + which could give the deposition of insoluble hydroxides when water is reduced (M + salts yield insoluble MOH which coats and deeply modifies the interface). We used N(C 4 H 9 ) 4 PF 6 which is slightly soluble (notice that LiPF 6 , LiBF 4 , LiSO 3 CF 3 , LiN(SO 2 CF 3 ) 2 are insoluble). When the 3-APTES based electrolyte is saturated by this salt (1 mM NBu 4 PF 6 ), its conductivity is low (κ = 0.03 μS/cm at 20 °C), but it allows current-potential curves.

EQCM experiments coupled to cyclic voltammetry (CV) method were performed in 3-APTES charged with 10 -3 MN (C 4 H 9 ) 4 PF 6 in a wide potential window from 0 to -4 V. No mass variation and no faradic peak nor gas bubbling on the electrode surfaces (working and counter electrodes) has been observed. Then, in order to have similar conditions of the works already made on alkylsilanes [1][2][3][4][5][6][7][8] and evocated previously, we used 3-APTES charged with 10 -3 M N(C 4 H 9 ) 4 PF 6 plus freshly added water (10 -3 M). The conductivity increased up to κ =1μS/cm at 20 °C. The resulting electrochemical measurements in the range [0 V; -4 V] are gathered in the same Fig. 1A. The upper part concerns CV scans while the lower part is the gravimetric part. Both voltammograms and mass variation are plotted vs. potential. Although no reduction wave was observed even after 10 scans, there was an increase of current at the end of each cycle. In the same time, the mass deposition on gold electrode surface increased up to 2 mg cm -2 at the end of the 10th scan, according to the Lewis and Lu relationship implemented in the EQCM [12]. This corresponds to a frequency change of 115 Hz which is in excellent agreement with the 5 MHz quartz crystal AT cut sensitivity of 56.6 Hz.cm 2 mg -1 . Compared to anhydrous 3-APTES based electrolyte (charged only with N(C 4 H 9 ) 4 PF 6 ), there is a great influence of water traces: when the solvent is water free no deposition takes place, while when water is added, the deposition happens. Therefore, water presence is a key factor for the electrodeposition of 3-APTES on the metal surface. Water reduction yields hydroxyl ion formation on the surface electrode allowing then the hydrolysis and condensation of the 3-APTES.

In a second step, we wanted to examine the possibility to obtain well defined faradic peaks to elucidate the involved reactions by classical tests. So, we investigated further the electrodeposition of 3-APTES 10 -2 M (which corresponds to a 3-APTES:THF molar ratio of 1:10 6 ) in tetrahydrofurane (THF) because of its very negative cathodic limit, and a good solubility of siloxane and ammonium salt [START_REF]Organic Electrochemistry: An Introduction and Guide[END_REF]. Although N(C 4 H 9 ) 4 PF 6 is very soluble, we kept its concentration to 1 mM plus 1 mM H 2 O(pureTHF+1mM N(C 4 H 9 ) 4 PF 6 : κ =7.8 μS/cm ; pure THF + 1 mM N(C 4 H 9 ) 4 PF 6 + 1m MH 2 O: κ =9.5 μS/cm at 20 °C) for working in the same conditions as 3-APTES charged with 10 -3 MN (C 4 H 9 ) 4 PF 6 ). In addition: the resulting electrolytes are oxygen free. Consequently, only reduction of water takes place in the range [0 V; -2V ]a s shown Fig. 1B. We observed on the lower part of this figure that the mass deposition occurs below -0.7 V. The electrogenerated hydroxide ions are only due to the water decomposition, acting as the catalyst for the hydrolysis and condensation of 3-APTES. Below -1 V there is a slope change of the gravimetric curve and the mass deposition rate is optimum between -0.7 Vand -1V .As there is no need to bias down to -4 V as it was the case Fig. 1A, potential scans were limited in the potential range [-0.5 V; -2V] in THF (with the same scan rate). At the end of the 6th scan, the mass deposition is more important than in pure 3-APTES electrolyte, reaching 4.7 mg/cm 2 . Clearly, 3-APTES has not to be concentrated in THF because the mass deposition is twice in THF based electrolyte than in pure 3-APTES one and water concentration has to be in the same range. Moreover, the deposition rate depends on the current which flows, consequently it depends on the conductivity. This can be related to the conductivity of THF charged with 1 mM N(C 4 H 9 ) 4 PF 6 and 1 mM H 2 O which is ten time superior of that found in 3-APTES charged with 1 mM N(C 4 H 9 ) 4 PF 6 and 1 mM H 2 O.

Surface morphology and thickness of the samples

After rinsing and drying the modified gold surfaces at ambient air, the deposited films give a slight and milky aspect to the gold surface. The topography of the 3-APTES thin film was examined Fig. 2 by STM after one and three CV scans of the electrodeposition process in THF as described previously. Fig. 2A shows a typical STM image of freshly annealed Au(111) substrate at air and illustrates t h ep r e s e n c eo fa t o m i c a l l yflat Au(111) terraces over hundreds of nanometers. Fig. 2B shows an image at air of the previous gold substrate, biased between -0.5 and -2V during one cycle at 20 mV/s. 3-APTES nanoparticle growths (white spot with a diameter of about 5-10 nm) took place with high coverage rate and were covalently bound on gold via oxo bridges from hydroxyl groups that were electrogenerated on the whole electrode surface. At the end of the three CV scans (Fig. 2C), the gold substrate is uniformly passivated. The condensation of the hydrolyzed 3-APTES nanoparticles with the maximum degree of condensation happens on the surface in THF compared to 3-APTES based electrolyte. In fact, it is very difficult to image the 3-APTES film at this stage and impossible beyond three CV cycles because it becomes insulating. Moreover, one month after the surface modification, the coating does not crackle and flake off. Several reasons can explain this observation, i.e., the chemical bonding and the thickness of this film on gold that we will examine in the next three paragraphs.

The film thickness on the gold surface was measured by ellipsometric technique at air in a clean room. Ψ and ▵ values were then fitted using, as a model, a layer of amorphous silica on 3000 Å-thick gold deposited on quartz crystal. The resulting thickness values (Fig. 2D), reported as a function of the CV cycle follow the same evolution as the mass deposition observed with EQCM measurements. On the other hand, there is a net difference between the mass deposition in THF and in 3-APTES for the same CV potential range [-0.5; -2V ] . Indeed, the thickness vs. the CV cycles in THF is best fitted with a sigmoid curve (square point plot), whereas in pure 3-APTES a linear regression matches very well the experimental data (star point plot). Moreover, at the end of ten CV cycles the film thickness still increases either in THF or in 3-APTES but to a lesser extent than for the first cycles. The use of THF during the cathodic polarization process affects greatly the mass deposition compared to 3-APTES based electrolyte.

Spectroscopic characterizations

The chemical characterization was performed on the electrochemically modified gold coated quartz crystal in THF based electrolyte by IR-ATR spectroscopy (Fig. 3). The spectrum of pure 3-APTES (dot line) shows typical absorption bands at 3374 and 3282 cm -1 (ν N-H for-NH 2 ) also observed on silanized glass too [START_REF] Beyer | [END_REF]. The spectrum of the electromodified gold surface (solid line), quite different from the previous one enables us to detect-NH 2 groups despite the noisy band at about 1600 cm -1 .I ti s possible to guess even the very weak but broad bands of the -NH 2 stretching modes in the frequency range between 3200 and 3400 cm -1 (arrows on the Fig. 3) for electrodeposited 3-APTES. The strong doublet at 1104 and 1084 cm -1 as well as the stronger band at 1022 cm -1 give evidence of the Si-OCH 2 CH 3 presence. Between 1000 and 900 cm -1 , shoulders at 972 and 930 cm -1 are in favor of the Si-O-metal bonding (Si-O-Au stretching) and can be related to the good adhesion of the film on the gold surface [15].

Fig. 4 shows the XPS low resolution survey spectrum of the electrodeposited 3-APTES on gold before etching the surface. The N 1s peaks at about 400 eV correspond to the amino groups on the surface of the thin film and tend to vanish with etching time. This is a direct indication of 3-APTES grafting on gold surface via silano groups. In addition amino groups are present on the film surface. An enlargement towards 400 eV shows two distinct peaks (Insert) after baseline correction and peak deconvolution. One at 399.7 eV is typical of -NH 2 while the other at higher BE (403 eV) is due to -NH 3 + [16]. That can be put in relation with the results obtained in IR spectroscopy. The -NH 2 groups belonging to the electrodeposited 3-APTES film have a protonated form explaining why there are two weak absorption bands beyond 3200 cm -1 compared to primary amine group of the pure 3-APTES. These two stretching vibrations have band intensities decreased to the profit of the bending mode (at about 1600 cm -1 ). It appears that the-NH 2 (and -NH 3 + ) groups are engaged in intermolecular interactions and acid-base bonds respectively what reduces these stretching mode intensities.

Biosensor application

The interest to obtain a surface covered with amino groups is to be able to graft in a covalent way thereafter biological molecules in mild conditions. Consequently, we followed the α-lactalbumin graft on the 3-APTES film electrodeposited on ab a r eg o l dc h i p( c o r r e s p o n d i n gt o2C Vc y c l e sd e p o s i t i o n from the Fig. 2D) by means of the SPR shift (Fig. 5). After rinsing with distilled water ( q u o t e d1o nt h eg r a p h ) ,1 % glutaraldehyde was injected on the surface (arrow A) during 1400s (quoted 2). The sensor chip was rinsed three times with water (quoted 1) and α-lactalbumin (2 mg/mL) was injected on the 3-APTES surface (arrow B) during 1700s (quoted 3). The injection of α-lactalbumin is then stopped (arrow C) and the difference in resonance units before and after αlactalbumin injection corresponded to the amount of protein covalently attached to the 3-APTES surface (quoted 4). This result confirms that -NH 2 groups on the top of the 3-APTES thin film are available for covalent binding of proteins. Furthermore, the electrodeposited 3-APTES thin film on gold surface for SPR experiments allows graft and detection of macromolecules such as α-lactalbumin.

Conclusion

We have directly modified gold surfaces electrochemically with 3-APTES when biased negatively below -0.7 V/SRE. The resulting siloxane based film is not only grafted covalently to gold metal via oxo bridges but has also several tens of nanometers of thickness on gold surface suggesting a multilayer coating. Electrochemical studies of 3-APTES based electrolytes showed that gold surface modification is irreversible and mass deposition is larger in THF than in 3-APTES based electrolyte. More, depending on the electrode bias time, an accurate 3-APTES film thickness can be obtained making our finding attractive for biosensor applications.

Fig. 1 .

 1 Fig. 1. CVon polycrystalline gold (1.37 cm 2 surface area), (A) of 3-APTES containing 1 mM of N(C 4 H 9 ) 4 PF 6 plus water (10 -3 M) in the range [0 V, -4 V], (B) of 3-APTES (10 -2 M) plus water (10 -3 M)inTHFintherange[0V ,-2 V] at 20 mV/S and the simultaneous mass deposition as a function of the potential applied to a 5 MHz gold coated AT cut quartz crystal.

Fig. 2 .

 2 Fig. 2. STM pictures of: (A) freshly annealed Au(111) on mica (10 × 12 mm 2 surface area), (B) electrodeposited 3-APTES in THF between -0.5 and -2 V during one scan and during 3 scans (C) at 20 mV/s. 3-APTES layer thickness as a function of the number of cyclic voltammetry scans (D) in either 3-APTES or THF based electrolyte by ellipsometry method.

Fig. 3 .

 3 Fig. 3. IR-ATR spectra of pure 3-APTES (dot line) solution and electrodeposited 3-APTES on gold plated quartz crystal (solid line).

Fig. 4 .

 4 Fig. 4. XPS spectrum survey before surface etching and (inset) the N 1s peaks around 400 eV.

Acknowledgements

This work was financially supported by a Grant-in-Aid for Scientific Research (RNTS 2003 No. 03 B 294) on allergy biochip from the Ministry of National Education and Research (MNESR).


Didier Felbacq
email: dfelbacq@univ-montp2.fr

Guy Bouchitté

Brahim Guizal

Antoine Moreau

Two-scale approach to the homogenization of membrane photonic crystals

Keywords: homogenization, photonic crystals, metamaterials, effective properties

Wave propagation and diffraction in a membrane photonic crystal with finite height were studied in the case where the free-space wavelength is large with respect to the period of the structure. The photonic crystals studied are made of materials with anisotropic permittivity and permeability. Use of the concept of two-scale convergence allowed the photonic crystals to be homogenized.

INTRODUCTION

Photonic crystals, i.e. dielectric or metallic artificial periodic structures, are generally thought of as strongly scattering devices, displaying photonic band gaps. However, their actual electromagnetic behavior when the free-space wavelength is large with respect to the period is also interesting, because it can produce strongly anisotropic behaviors, plasmon frequencies, or even negative index materials [START_REF] Pendry | Beyond metamaterials[END_REF]. The study of the properties of these structures in this asymptotic regime comes under the theory of homogenization [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF][START_REF] Felbacq | Homogenization of a set of parallel fibres[END_REF][START_REF] Felbacq | Theory of mesoscopic magnetism in photonic crystals[END_REF]. A lot of results are by now well established both for 2D and 3D structures (even for non periodic structures [START_REF] Peng | Experimental observation of left-handed behavior in an array of standard dielectric resonators[END_REF][START_REF] Felbacq | Localization of light by a set of parallel cylinder[END_REF][START_REF] Felbacq | Negative refraction in periodic and random photonic crystals[END_REF]).

In this paper, we consider a photonic crystal made of a collection of parallel finite-size fibers embedded in a matrix. This covers the case of structures made out of a layer of bulk materials in which holes are made periodically (membrane photonic crystal) but also the case of structures made out of nanopillars (pillar photonic crystal [START_REF] Poborchii | Si pillar photonic crystal slab with linear defects: transmittance and waveguide properties[END_REF][START_REF] Benisty | Photonic crystals in twodimensions based on semiconductors: fabrication, physics and technology[END_REF]). The fibers are not supposed to be invariant in the direction of their axis (for instance they can be cone-shaped, see fig. 1). Our point is to derive the effective permittivity and permeability tensors of this structure when the ratio between the period of the structure and the free-space wavelength of the incident field is very small. We had already derived rigorous results for infinitely long cylindrical fibers [START_REF] Felbacq | Homogenization of a set of parallel fibres[END_REF], for which explicit formulas can be derived in some cases [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF][START_REF] Erikson | Homogenization and Effective Moduli of Materials and Media[END_REF][START_REF] Mcphedran | Homogenization of composites: dynamic and static theories[END_REF][START_REF] Mcphedran | Effective dielectric constant of arrays of elliptical cylinders[END_REF]. Here we shall get homogenized permeability and permittivity tensors with a dependence along the axis of the fibers. Let us note that our results hold for dispersive and lossy materials (it applies to imperfect metals as well as to good insulators).

TWO-SCALE APPROACH TO HOMOGENIZATION

Description of the structure and methodology

We consider a 2D photonic crystal such as that in fig. 1

. It is constructed from a basic cell Y pictured in fig. 2 ( Y = Y × (-L, L), where Y =( -1/2, 1/2) 2 ). A contraction ratio η is applied to obtain a contracted cell in the horizontal directions ( Y η = η 2 Y × (-L, L) ).
In the units of the free-space wavelength, the period of the lattice is thus η. The cells are contained in a cylindrical domain Ω ( Ω=O×(-L, L)) (cf. fig. 1). The domain Ω is thus periodically filled with contracted cells. The space coordinates are denoted by x =(x 1 ,x 2 ,x 3 ) and we also Fig. 1. Schematic of the photonic crystal. denote x ⊥ =( x 1 ,x 2 ). The coordinates in Y are denoted by y =( y 1 ,y 2 ). We consider time harmonic fields, the time dependence is chosen to be exp (-iωt). For a given monochromatic incident field E i , H i , we denote by (E η , H η ) the total electromagnetic field.

Our aim is to pass to the limit η → 0 and determine the limit of the couple (E η , H η ). In our methodology, we get at the limit a true electromagnetic scattering problem, for a given free-space wavelength λ and a bounded obstacle Ω characterized by some permittivity and permeability tensors. This situation is quite different from other homogenization techniques, making use of periodization conditions, in which the frequency tends to zero, thus not leading to a diffraction problem but rather to an electrostatic one. Such an approach can sometimes give useful explicit formulas but generally leads to complicated formulations [10-12, 20, 21]. Moreover, it does not handle the boundary effects which in some cases may lead to some miscomprehensions [START_REF] Felbacq | Anomalous homogeneous behaviour of metallic photonic crystals[END_REF]. The relative permittivity tensor ε η (x) and relative permeability tensor µ η (x) are described by:

ε η (x)=1,µ η (x)=1 for x ∈ R 3 \Ω ε η (x)=ε 0 x ⊥ η ,x 3 ,µ η (x)=µ 0 x ⊥ η ,x 3 for x ⊥ ∈ Ω (1)
where at fixed x 3 , the applications y → ε 0 (y,x 3 )= ε 0 ij (y,x 3 ) and y → µ 0 (y,x 3 )= µ 0 ij (y,x 3 ) are Y -periodic 3 × 3 matrix functions. The total electromagnetic field

(E η , H η ) satisfies curl E η = iωµ 0 µ η H η curl H η = -iωε 0 ε η E η (2)
and (E s,η , H s,η )= E η -E i , H η -H i satisfies Silver-Müller radiation conditions:

lim x →+∞ x Z 0 H s,η × x x -E s,η =0 (3)
where Z 0 is the impedance of vacuum.

Derivation of the microscopic equations

A short account of the two-scale expansion

In order to describe this problem, we will rely on a two-scale expansion of the fields. That way, the physical problem is described by two variables: a macroscopic one x and a microscopic one y representing the rapid variations of the material at the scale of one basic cell, measured by η. By noticing that there are no rapid variations in the vertical direction x 3 , the microscopic variable is set to be: y = x ⊥ /η. Differential operators with respect to variable y are denoted with a subscript y. The fields are periodic with respect to that microscopic variable (this corresponds to the neighborhood of the center of the first Brillouin zone). The limit problem obtained by letting η tend to 0, will then depend on the macroscopic, physical, variable x but also on the microscopic, hidden, variable y. The total limit fields will read E 0 (x, y) and H 0 (x, y) and the observable, physical, fields will be given by the mean value over the hidden variable y:

E(x)=|Y | -1 Y E 0 (x, y)dy and H(x)=|Y | -1
Y H 0 (x, y)dy, where |Y | is the measure area of Y . In order to lighten the notations, we denote by brackets the averaging over Y , hence H(x)= H 0 and E(x)= E 0 .

The main mathematical tool that we use is a mathematically rigorous version of the multiscale expansion, widely used in various areas of physics. More precisely, for a vector field F η in L 2 (Ω) 3 , we say, by definition, that F η two-scale converges towards F 0 if for every sufficiently regular function φ (x, y), Y -periodic with respect to y,wehave:

Ω E η (x) • φ (x, x ⊥ /ε) dx → Ω×Y E 0 (x, y) • φ(x, y)dxdy, (4)
as η tends to 0. The limit field F 0 is square integrable with respect to both variables x and y and is Yperiodic in the y variable (this is the definition of the space L 2 Ω; L 2 per (Y ) 3 ). A complete analysis of this new mathematical tool can be found in [START_REF] Allaire | Homogenization and 2-scale convergence[END_REF]. We make the physically reasonable assumption that the electromagnetic energy remains bounded when η tends to 0, which is equivalent to assume that (E η , H η ) are both locally square integrable. Then it is known [START_REF] Allaire | Homogenization and 2-scale convergence[END_REF] that (E η , H η ) two-scale converges towards limit fields E 0 , H 0 . This physical assumption could be justified mathematically, however it seems quite obvious, from the point of view of physics, that the limit fields exist. The point is then to give the system of equations that is satisfied by these fields and to derive the effective permittivity and permeability tensors.

The equations at the microscopic scale

First of all, we have to determine the set of equations that are microscopically satisfied, that is, satisfied with respect to the hidden variable y, for that will give the constitutive relations of the homogenized medium. Multiplying Maxwell-Faraday equation by a regular test function φ x, x ⊥ η , and integrating over Ω, we obtain:

Ω E η (x) • curl x (φ)+ 1 η curl y (φ) dx = iωµ 0 Ω µ η (x) H η (x) φ (x, x ⊥ /η) dx. (5)
Multiplying by η and letting η tend to 0, we get using (4):

Ω×Y E 0 (x, y) • curl y (φ) dxdy =0. (6)
This is equivalent to:

Ω×Y curl y E 0 (x, y) • φ (x, y) dxdy =0 (7)
which is nothing else but the variational form for: curl y E 0 =0. In a very similar way, but using now Maxwell-Ampere equation, we get curl y H 0 =0 . On the other hand, since ε η E η is divergence free, we have, for every test function φ(x, y),

Ω ε η (x) E η (x)• ∇ x φ + 1 η ∇ y φ dx =0.
Multiplying by η and letting η tend to 0, we get:

Ω×Y ε 0 (y,x 3 ) E 0 (x, y) •∇ y φdxdy =0, (8)
which can be written as (notice that the div y operator acts only on the transverse components):

div y ε 0 E 0 =0. (9)
Similarly, since the magnetic field is divergence free, we derive:

div y µ 0 H 0 =0. (10)
Summing up, we have the following microscopic equations, holding inside the basic cell Y :

div y µ 0 H 0 =0 curl y H 0 =0 , div y ε 0 E 0 =0 curl y E 0 =0 (11)

Derivation of the homogenized parameters

The systems in [START_REF] Mcphedran | Homogenization of composites: dynamic and static theories[END_REF] are respectively of electrostatic and magnetostatic types. This means that, with respect to the hidden variable y, the electric field and magnetic field satisfy the static Maxwell system. This property is true only at that scale and not at the macroscopic scale. However, it is these static equations that determine the effective permittivity and permeability. Indeed let us analyze this system starting with the electric field. From the curl relation, we get ∇ y E 0 3 =0 , and so E 0 3 (x, y) ≡ E 3 (x). Besides, the basic cell having the geometry of a flat torus, we get the existence of a regular periodic function w E (y) such that:

E 0 ⊥ = E ⊥ + ∇ y w E . (12)
The function w E is the electrostatic potential associated with the microscopic electrostatic problem. Inserting (9) in equation ( 12) and projecting on both horizontal axis, we obtain:

div y ε 0 ⊥ (e i + ∇ y w E,i ) =0,i∈{1, 2} (13)
where ε 0 ⊥ denotes the 2 × 2 matrix extracted from ε 0 by removing the last line and last column. By linearity, denoting E ⊥ =( E 1 ,E 2 ), we derive that the potential w E is given by w E = E 1 w E,1 + E 2 w E,2 , where w E,i are the periodic solutions of [START_REF] Felbacq | Anomalous homogeneous behaviour of metallic photonic crystals[END_REF]. We stress that these potentials depend upon y but also on x 3 . In fact, we get a family of homogenization problems parametrized by the vertical coordinate. By [START_REF] Mcphedran | Effective dielectric constant of arrays of elliptical cylinders[END_REF] we obtain:

E 0 (x, y)=E (y,x 3 ) E (x) (14)
where:

E (y,x 3 )= ⎛ ⎝ 1+∂ y1 w E,1 ∂ y1 w E,2 0 ∂ y2 w E,1 1+∂ y2 w E,2 0 00 1 ⎞ ⎠ ( 15
)
The magnetic field H 0 can be handled in the same way since it satisfies exactly the same kind of equations as H 0 (see [START_REF] Mcphedran | Homogenization of composites: dynamic and static theories[END_REF]). In particular, we may represent its tranversal component in the form:

H 0 ⊥ = H ⊥ + ∇ ⊥ w H
, where w H is a periodic magnetic potential (the possibility of this representation is due to the curl-free condition which means that no microscopic current is present). Analogously as in [START_REF] Allaire | Homogenization and 2-scale convergence[END_REF][START_REF] Guenneau | Homogenization of 3D finite chiral photonic crystals[END_REF], we find:

H 0 (x, y)=M (y,x 3 ) H (x) (16)
where

M (y)= ⎛ ⎝ 1+∂ y1 w H,1 ∂ y1 w H,2 0 ∂ y2 w H,1 1+∂ y2 w H,2 0 00 1 ⎞ ⎠ (17)
where:

div y µ 0 ⊥ (e i + ∇ y w H,i ) =0,i∈{1, 2} (18)
and µ 0 ⊥ denotes the 2×2 matrix extracted from µ 0 by removing the last line and last column. Of course, the same remark as in the case of electric potentials holds: the functions w H,i depend on the vertical coordinate x 3 . The above results show that, at the microscopic scale, the limit fields (E 0 , H 0 ) are completely determined by the physical fields (E, H). Now that the microscopic behavior is precised, we are able to determine the macroscopic system satisfied by (E, H).T o that aim, let us choose a regular test function φ (x) independent of variable y. From Maxwell equations we get

Ω H η (x) .curl (φ) dx = -iωε 0 Ω ε η (x) E η (x) φ (x) dx Ω E η (x) • curl (φ) dx = iωµ 0 Ω µ η (x) H η (x) φ (x) dx (19)
passing to the limit η → 0, we get:

Ω×Y H 0 (x, y) • curl (φ) dxdy = -iωε 0 Ω×Y ε 0 (y,x 3 ) E 0 (x, y) φ (x) dxdy Ω×Y E 0 (x, y) • curl (φ) dxdy = iωµ 0 Ω×Y µ 0 (y,x 3 ) H 0 (x, y) φ (x) dxdy (20)
Recalling that E 0 = E and that H 0 = H, we get:

curl E = iωµ 0 µ 0 H 0 curl H = -iωε 0 ε 0 E 0 (21)
which, taking into account [START_REF] Allaire | Homogenization and 2-scale convergence[END_REF][START_REF] Bouchitté | Homogenization of a wire photonic crystal: the case of small volume fraction[END_REF], brings to the limit system:

curl E = iωµ 0 µ 0 M (x 3 ) H curl H = -iωε 0 ε 0 E (x 3 ) E (22)

The special case of a 1D grating

Let us specialize the above results to the case of a one dimensional grating (fig. 3) that is, the pillars are invariant in the x 2 and x 3 directions (the basic cell Y is depicted in fig. 3). We assume also that the pillars are made on a non-magnetic material and that the relative permittivity tensor is given by: . The homogenized relative permittivity tensor is given by:

ε 0 (y 1 )= ⎛ ⎝ ε 1 (y 1 )0 0 0 ε 2 (y 1 )0 00 ε 3 (y 1 ) ⎞ ⎠ (
ε 0 E = ⎛ ⎜ ⎝ 1 ε1 -1 00 0 ε 2 0 00 ε 3 ⎞ ⎟ ⎠ (24)
We retrieve a well-established result concerning the homogenization of 1D photonic crystals [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF].

DISCUSSION

The homogenized permeability and permittivity tensors are respectively µ 0 M and ε 0 E . Both are matrix functions of the coordinate x 3 . In case of cylindrical fibers (i.e. invariant along their axis), we would find that the effective tensors coincide with that obtained in the polarized cases [START_REF] Bensoussan | Asymptotic Analysis for Periodic Structures[END_REF][START_REF] Felbacq | Homogenization of a set of parallel fibres[END_REF][START_REF] Guenneau | Homogenization of 3D finite chiral photonic crystals[END_REF]. In fact, because the permittivity and permeability tensors are not renormalized with respect to η, the homogenization process is purely local, that is, the effective constitutive relations are local ones. However, we emphasize that the locality is lost if we change the scale of the permittivity or permeability coefficients in the obstacles. In particular, the results obtained in the case of infinite conductivities in the polarized case [START_REF] Felbacq | Homogenization of a set of parallel fibres[END_REF] cannot be transposed to the case of fibers with finite length, due to the emergence of surprising non local effects which are studied in [START_REF] Bouchitté | Homogenization of a wire photonic crystal: the case of small volume fraction[END_REF][START_REF] Felbacq | Left-handed media and homogenization of photonic crystals[END_REF][START_REF] Zolla | Bloch vector dependance of the plasma frequency in metallic photonic crystals[END_REF]. In this context, another problem is interesting: it is the study of the situation where the height L of the pillar is small with respect to the free-space wavelength. An ansymptotic study can be performed in that case, leading to a simplified formulation of the diffraction problem [START_REF] Ammari | Scattering of electromagnetic waves by thin dielectric planar structures[END_REF].

We also note that an approach relying on explicit calculations, for instance using Blochwaves theory or Fourier-Bessel expansions, cannot work here, due the lack of an explicit representation of the fields in case of finite size fibers. The method that is used here proves to be particularly efficient, the results are obtained at the cost of very few calculations. Besides, it gives an interesting physical pictures of the homogenization process in terms of hidden variables: the homogenized parameters are obtained through clearly defined electrostatic and magnetostatic problems with respect to these variables. It should also be noted that the case of dielectric materials with losses is handled by our result. This result can be straightforwardly applied to the study of membrane photonic crystal in the low frequency range where phenomena of birefringence and dichroism are obtained [START_REF] Halevi | Photonic crystal optics and homogenization of 2D periodic composites[END_REF][START_REF] Halevi | Long-wavelength limit (homogenization) for two-dimensional photonic crystals[END_REF].

In our homogenization result, it is clear that the main numerical problem is the solving of the annex problems [START_REF] Felbacq | Anomalous homogeneous behaviour of metallic photonic crystals[END_REF][START_REF] Zolla | Bloch vector dependance of the plasma frequency in metallic photonic crystals[END_REF] for they give the effective matrices E and M. In certain simple cases, for instance that of circular isotropic non magnetic rods and a permittivity constant in each connected region, it is possible to find an explicit expression for the effective permittivity (it is in fact a very old problem). However, for more complicated geometries, there is a general numerical procedure based on fictitious sources, that allows to solve both annex problems at a low numerical cost. It should also be stressed that if the wavelength is not very large with respect to the period, the device may still be homogenizable, but then the evanescent fields play an important role [START_REF] Felbacq | Bloch modes dressed by evanescent waves and the generalized Goos-Hänchen effect in photonic crystals[END_REF] that can make the effective parameters be dependent upon the environnement of the device.

Fig. 2 .

 2 Fig. 2. Schematic of the basic cell.

23 )Fig. 3 .

 233 Fig. 3. Schematic of the 1D photonic crystal. The inset shows the basic cell Y .

Acknowledgments

This work was realized in the framework of the ANR project POEM PNANO 06-0030.