Xuan Wang

Kevin Ehrhardt

Clémence Tallet

Marc Warenghem

Alexandre Baron

Ashod Aradian

Mortem Kildemo

Virginie Ponsinet
email: ponsinet@crpp-bordeaux.cnrs.fr

Morten Kildemo

Hyperbolic-by-design self-assembled metamaterial based on block copolymers lamellar phases

Keywords: plasmonics, self-assembly, hyperbolic media, block copolymers, localized surface plasmon resonance, effective dielectric permittivity, lamellar phase

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

The breakthroughs and innovations in the information and communication technologies rely, to a large extent, on improved performances of the devices, and their constitutive materials. Optical metamaterials lie among these envisioned breakthroughs in the near future: they are artificial materials structurally designed with the purpose of producing extraordinary optical properties. Their potential in revolutionizing the optical technologies has been largely recognized [START_REF] Pendry | Controlling electromagnetic fields[END_REF][START_REF] Engheta | Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials[END_REF][START_REF] Wang | Shaping the radiation pattern with mu and epsilon-near-zero metamaterials[END_REF] . While precise control of both the dielectric permittivity () and the magnetic permeability () of a 3D artificial material, in the visible or infra-red ranges, would open access to transformation optics and negative index materials, leading to the capacity of engineering the propagation of light in devices in an unprecedented manner [START_REF] Engheta | Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials[END_REF] , these goals are still far from reach. On the other hand, by combining nanostructuration and anisotropy, it is possible to engineer novel and non-natural dispersion relations, in order to control original propagation properties. This is why a recent interest has focused on a special case of uniaxial anisotropic metamaterials, called hyperbolic materials 4 , 5 , 6 , 7 , which exhibit strongly anisotropic dispersion relations in the visible or infra-red wavelength range. These materials are considered amongst the most promising metamaterials, because of their ability to provide a multi-functional platform [START_REF] Ferrari | Hyperbolic metamaterials and their applications[END_REF] to reach different meta-properties.

For example, hyperbolic nanowire and lamellar stacked metamaterials have been studied theoretically and experimentally [START_REF] Ferrari | Hyperbolic metamaterials and their applications[END_REF][START_REF] Poddubny | Hyperbolic metamaterials[END_REF][START_REF] Wood | Directed subwavelength imaging using a layered metal-dielectric system[END_REF][START_REF] Shukovsky | Physical nature of volume plasmon polaritons in hyperbolic metamaterials[END_REF] . Such structures have displayed a variety of promising properties, generating a surge in the activity on the topic over the recent years: negative refraction [START_REF] Yao | Optical Negative Refraction in Bulk Metamaterials of Nanowires[END_REF] , super-resolution [START_REF] Zhang | Superlenses to overcome the diffraction limit[END_REF] , subwavelength modes [START_REF] Narimanov | Beyond diffraction[END_REF] , perfect multi-band absorption [START_REF] Sreekanth | A multiband perfect absorber based on hyperbolic metamaterials[END_REF] , optical topological transition [START_REF] Krishnamoorthy | Topological Transitions in Metamaterials[END_REF] , epsilon-near-zero light propagation [START_REF] Kurilkina | Features of hyperbolic metamaterials with extremal optical characteristics[END_REF] , spontaneous emission and Purcell effect enhancement [START_REF] Cortes | Quantum nanophotonics using hyperbolic metamaterials[END_REF][START_REF] Jacob | Engineering photonic density of states using metamaterials[END_REF][START_REF] Sreekanth | Large spontaneous emission rate enhancement in grating coupled hyperbolic metamaterials[END_REF][START_REF] Jacob | Broadband Purcell effect: radiative decay engineering with metamaterials[END_REF][START_REF] Lu | Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials[END_REF] , thermal emission engineering [START_REF] Biehs | Hyperbolic Metamaterials as an Analog of a Blackbody in the Near Field[END_REF] including super-Planckian regimes [START_REF] Guo | Broadband super-Planckian thermal emission from hyperbolic metamaterials[END_REF] , or biosensing [START_REF] Kabashin | Plasmonic nanorod metamaterials for biosensing[END_REF][START_REF] Sreekanth | Extreme sensitivity biosensing platform based on hyperbolic metamaterials[END_REF] .

One very attractive property is the so-called "super-resolution" (i.e., sub-diffraction imaging), because the required structures are relatively less demanding than other metamaterials, and because superresolution could be profitable to several technological fields. Optical microscopy is indeed an essential tool in many fields such as microelectronics, biology and medicine. It is, however, hindered by the intrinsic diffraction limitation as it cannot obtain a better resolution than half the wavelength of light. This is because the finer detail-carrying, high wavevector components of the field emitted from the object are evanescent, decay exponentially in a medium with positive permittivity and permeability, and cannot contribute to the subwavelength image formation. Near-field scanning techniques [START_REF] Dürig | Near-field optical scanning microscopy[END_REF] , and fluorescence-based imaging methods [START_REF] Klar | Fluorescence microscopy with diffraction resolution barrier broken by stimulated emission[END_REF] have put forward ways to circumvent this limitation and have brought optical microscopy into the nanoworld, with the fascinating and revolutionary goal to visualize the pathways of individual molecules inside living cells [START_REF] Boettiger | Super-resolution imaging reveals distinct chromatin folding for different epigenetic states[END_REF] . Similar intrinsic resolution limitations also hinder optical lithography, one of the most important tools equipping the semiconductor industry, ubiquitous in our societies.

In hyperbolic materials, two components of the dielectric permittivity tensor  have opposite signs, as if the material behaved like a metal ( i < 0) along at least one direction and like a dielectric ( j > 0) along at least another. Let us consider a uniaxial material with principal axes (x,y,z) and permittivity tensor

           z     0 0 0 0 0 0 // //
with  //   // ' + i  // " and  z   z ' + i  z " Equation 1where the parallel (//) symbol denotes the (x,y)-plane. The isofrequency dispersion relation in this material is [START_REF] Ferrari | Hyperbolic metamaterials and their applications[END_REF][START_REF] Mackay | Toward optical sensing with hyperbolic metamaterials[END_REF]

   0 ) ( 2 // 2 // 2 2 2 // 2 2                  o o z z z y x o o y x k k k k k , Equation 2
where the two terms describe the behavior of waves of different polarizations: polarization in the (x,y) plane for the first term and polarization in a plane containing the z direction for the second.

The latter can be written

2 // 2 2 2 o z z y x k k k k      , with 2 2    o o o k  Equation 3
It exhibits a hyperboloid branch when the product 0 ' '

//  z  
, contrary to dielectrics, for which this relation defines a closed ellipsoidal shape (spherical if isotropic). This peculiar shape allows notably for the propagation of large magnitude wavevectors, carrying details finer than half the wavelength, otherwise corresponding to evanescent non-propagative waves in a usual dielectric. Hyperbolic material fabrication usually relies on metal deposition within top-down produced uniaxial nanostructures, either lamellar or cylindrical, and their permittivity tensor is usually inferred from basic effective medium models. The spectral tunability of the effective medium properties of the material is limited by the large spectral dispersion of the optical properties of the metal, which generally follows the Drude model.

One of the main bottlenecks in the research on metamaterials and hyperbolic materials in particular remains the lack of robust, simple and large scale fabrication technologies which permit tailoring final optical properties at chosen frequency ranges (including visible frequencies), and setting the technological basis for the development of devices: the field is open to innovative fabrication proposals. The powerful 'top-down' techniques such as nanolithography have been successful in manufacturing nanostructured surfaces and in evidencing meta-material properties, such as hyperlensing [START_REF] Pendry | Negative Refraction Makes a Perfect Lens[END_REF] and cloaking 1 , at wavelengths larger or close to the visible domain [START_REF] Soukoulis | Past achievements and future challenges in the development of threedimensional photonic metamaterials[END_REF] and even at ultraviolet frequencies [START_REF] Xu | All-angle negative refraction and active flat lensing of ultraviolet light[END_REF] . The use of chemistry and self-assembly of metallic nanoparticles, acting as plasmonic resonators, into dense ordered structures, was anticipated as a promising 'bottom-up' alternative 32 , 33 , especially in order to produce large-scale, 3D and tunable metamaterials. The versatility of chemical synthesis and the diverse mechanisms of self-assembly can be combined to produce materials with both nanostructuration and anisotropy, presenting novel propagation properties in the visible or infrared ranges.

In particular, diblock copolymers [START_REF] Bates | Block Copolymers-Designer Soft Materials[END_REF][START_REF] Hamley | The physics of block copolymers[END_REF] have long been known to self-assemble into nanomaterials with strong structural anisotropy. They are macromolecules made of two molecular chains of distinct chemical nature linked together, called the blocks, and present solid state spontaneous organizations with long-range order and tunable characteristic sizes, ranging typically from a few nanometers to a few hundred nanometers. The morphology is selected mostly in relation with the volume fractions of the blocks f A and f B =1f A , due to interfacial curvature effects [START_REF] Hamley | The physics of block copolymers[END_REF][START_REF] Bates | Block Copolymer Thermodynamics: Theory and Experiment[END_REF] , among the choice of body-centered cubic array of spherical cores, bicontinuous gyroid, hexagonally packed cylinders and lamellae. It is thus largely possible to design a desired material by choosing the chemical nature and the length of the blocks in order to obtain a given morphology, a given characteristic size, as well as a given chemical functionality. Using such ordered block copolymer phases for optical applications is limited by the usually low dielectric contrast between the blocks: the refractive index of almost all easily accessible polymers is in the range 1.48 -1.61. One way of improving the contrast, among others [START_REF] Fink | Block Copolymers as Photonic Bandgap Materials[END_REF][START_REF] Salvatore | Tunable 3D Extended Self-Assembled Gold Metamaterials with Enhanced Light Transmission[END_REF] , or of conferring specific optical properties, is to disperse adequate nanoparticles within the ordered polymer phase in such a way that the particles get ordered by the matrix. This is what is reported in this work, in which we show that nanocomposites based on metal nanoparticles embedded in a dielectric host, can present a strongly anisotropic structure, as well as spectrally selective optical anisotropy, as a consequence of the spectral tunability of the plasmon resonance. Indeed, we propose in this article a novel fabrication technique for lamellar metal-dielectric nanocomposites using self-assembled block copolymer nanostructures, and we demonstrate the relevance of such bottom-up methodology for the fabrication of hyperbolic metamaterials. We first describe the preparation and structural study of thin films of self-assembled nanocomposites of block copolymers and gold nanoparticles prepared by wet chemistry. We then present the study of the relation between the structure of the composites, and in particular the density of the gold nanoparticles, and the anisotropy of their effective dielectric permittivity tensor, which is extracted from variableangle spectroscopic ellipsometry. We finally discuss the hyperbolic nature of the self-assembled material.

I. Nanostructure design and fabrication

As a first requirement, a structural uniaxial symmetry appears highly desirable, in order to obtain a hyperbolic regime of optical anisotropy. Diblock copolymer thin film engineering [START_REF] Ponsinet | Nanopatterns Produced by Directed Self-Assembly in Block Copolymer Thin Films[END_REF][START_REF] Lazzari | Methods for the alignment and the large-scale ordering of block copolymer morphologies[END_REF] offers a genuine technological platform for the generation of nanostructured thin films with different morphologies of the domains. Two specific thin film morphologies are particularly attractive here, which present a high degree of order and a uniaxial symmetry: the lamellar phase in parallel alignment, which we study in the following, and the phase of hexagonally-packed cylinders in perpendicular orientation, which has been considered for applications including lithography masks [START_REF] Kennemur | Sub-5 nm domains in ordered poly(cyclohexylethylene)-block-poly(methyl methacrylate) block polymers for lithography[END_REF][START_REF] Tang | Thin film morphology of block copolymer blends with tunable supramolecular interactions for lithographic applications[END_REF][START_REF] Park | Block copolymer lithography: Periodic arrays of similar to 1011 holes in 1 square centimeter[END_REF] and enhanced patterned LEDs [START_REF] Alam | Variety of nanopatterns on different substrates using PS-b-PMMA and their applications[END_REF] . A second requirement, for the light propagation properties to be described by a homogenized dispersion relation, is that the internal structure of the material presents a subwavelength characteristic size:

d o << .
This can be easily achieved with self-assembled block copolymer structures, with an appropriate choice of the total chain length.

We have studied several poly(styrene)-b-poly(2-vinyl pyridine) (PS-P2VP) copolymers, of different block lengths. A small-angle X-ray scattering study of thick samples of these copolymers (see Supplementary Information, Figure S1) shows the signatures of phases of either stacked lamellae or hexagonally-packed cylinders, depending on the ratio between the block lengths. The characteristic structural sizes, as given by the center-to-center distance between P2VP domains, range from 37 to 84 nm. As expected [START_REF] Hamley | The physics of block copolymers[END_REF] , a lamellar nanostructure is obtained when both blocks have similar volume fraction f A ~ f B ~ 0.5. Finally, we choose to work with the lamellar PS-P2VP copolymer with both blocks of molar mass 25 kg/mol, which spontaneously develops a multilayered structurally uniaxial thin film of alternate poly(styrene) and poly(2-vinyl pyridine) layers with a period of 37 nm in the bulk. We use a process combining spin-coating of a relatively dilute polymer solution on a selective surface (silicon wafer having a higher affinity for P2VP rather than PS) and thermal annealing, which produces a flat and homogeneous film with a parallel alignment of the organized structures, throughout the thickness of the film between 200 and 700 nm. This ordered structure spans an area as large as the spin-coating process can produce (we use 15x15 mm² wafers).

The third requirement to obtain a hyperbolic medium, which is not provided by the simple copolymer self-assembly, is to induce dielectric permittivity of opposite sign depending on the probed direction.

Thus, the combination of a metallic response ( < 0) and a dielectric response ( > 0) is necessary.

Reported examples of multilayer hyperbolic materials have been fabricated by alternate deposition of metal and dielectric materials by electron-beam or sputter deposition in vacuum. A self-assembled block copolymer lamellar phase equivalent requires a pseudo-metallic nature for one of the layer types, which we obtain by doping one of the polymer phase with metallic nanoparticles, thus producing alternate stacks of pure polymer layers (polystyrene, referred to as PS in the following), and composite layers of gold nanoparticles and poly(2-vinyl pyridine) (referred to as NC for Nano-Composite layers) (see Figures 1 and4).

The optical properties of nanocomposites of gold particles and polymer with no order, equivalent to the NC layers of the lamellar stacks, have been studied previously, using spectroscopic ellipsometry.

We showed that such nanocomposite films present a plasmon resonance due to the metallic nanoparticles [START_REF] Vieaud | Effective Medium Description of Plasmonic Couplings in Disordered Polymer and Gold Nanoparticle Composites[END_REF] and that the resonance amplitude increases as the gold volume fraction in the nanocomposite increases. As the resonance deepens, the system will ultimately reach < 0 in some frequency range: in fact, the Maxwell Garnett Effective Medium Approximation (MG-EMA), although not truthfully applicable for such high-fractions, suggests this can roughly occur for gold volume fractions beyond 25%. In order to transform one of the block copolymer domain into such a pseudo-metallic nanocomposites, we can choose among several routes [START_REF] Maxit | Facile formulation of high density well-ordered nanoparticle-copolymer nanocomposites[END_REF][START_REF] Chiu | Control of Nanoparticle Location in Block Copolymers[END_REF][START_REF] Mendoza | Block Copolymers with Gold Nanoparticles: Correlation between Structural Characteristics and Mechanical Properties[END_REF] for selective nanoparticle incorporation. Here, we developed an impregnation methodology, adapted from the literature [START_REF] Yun | Electrically Anisotropic Thin Films Consisting of Polymeric and Metallic Nanolayers from Self-Assembled Lamellae of Diblock Copolymers[END_REF][START_REF] Sohn | Fabrication of the multilayered nanostructure of alternating polymers and gold nanoparticles with thin films of self-assembling diblock copolymers[END_REF] , in which a previously ordered and aligned sturdy copolymer nanostructure can be selectively swollen with a precursor solution. More precisely, the film-bearing wafer is dipped into an Au salt (HAuCl 4 ) solution and then into a reducing agent (NaBH 4 ) solution. Due to the strong insolubility of PS in polar solvents, on the one hand, and the strong affinity of P2VP to Au, on the other hand, Au nanoparticles form selectively within the P2VP layers, thus producing a structure of alternating pure PS and (NC) (Au nanoparticles:P2VP nanocomposite) layers. One of the advantages of this infiltration procedure is that we can slowly increase, in a controlled way, the volume fraction of nanoparticles in the (NC) layers, by repeating the double dipping process N times. Each cycle increases the purple color of the film, related to the localized surface plasmon resonance of the nucleated gold nanoparticles. Structural and optical studies of the film were performed at different steps along the fabrication process, so that the same film was investigated for different values of N, which we will refer to in the following as "kinetic" measurements.

Using scanning electron microscopy (SEM), we fully characterize the structure of the studied multistack films. As can be seen on an example for N=5 (Figure 1), SEM observation clearly evidenced the contrast in electron density between adjacent layers in the multilayer film, which can unambiguously be attributed to the selective gold loading of the P2VP layer. More precisely, the film is composed of 9 bilayers (PS)+(NC) of thickness 14 + 14 = 28 nm, with an additional (NC) layer of 6 nm along the substrate surface and an additional (PS) layer of 6 nm along the air interface 51 . As demonstrated by the SEM images, the films are structurally uniaxial and homogeneous and we can define their dielectric permittivity tensor with the ordinary (parallel to the substrate,  ord =   ) and extraordinary (normal to the substrate,  extraord =  z ) components. We can follow the structure of a sample as the amount of gold is increased in the film, by repeating the cycle of salt impregnation and reduction within the lamellar matrix N times. Figure 2 presents the evolution of the structure as followed by SEM. We can thus check that the lamellar structure is preserved and unchanged for all values of N for which the film was imaged. The contrast of the images cannot be quantitatively compared to confirm the increase of the gold volume fraction. It is noteworthy that the thickness of the NC layers does not show any detectable increase upon gold loading with increasing N. This can be understood as due to a reorganization of the polymer chains conformation, as when such films are swollen with solvent, as well as a possible degradation of the P2VP segments involved in the chemical reduction of the gold salt. Beyond N=20, the SEM micrographs show the evidence of an uncontrolled gold deposit on the upper surface of the film. In order to gain more precise information on the shape and nature of the gold objects introduced within the lamellar stack, which SEM images do not provide due to lack of resolution, we performed complementary studies by transmission electron microscopy (TEM) as well as SAXS on dispersions obtained by full dissolution of the samples in a good solvent of the diblock copolymer at the end of the fabrication process. Both techniques show that the NC layers are composed of individual gold nanoparticles (NPs) of relatively homogeneous size (diameter D ~ 7 ± 2 nm). Figure 3 shows an example of the results obtained by TEM in the case of N = 25. We can therefore infer that such 7nmgold NPs are dispersed with no specific order and surrounded by a P2VP matrix, in the NC layers. The data were analyzed using both the DeltaPsi2 software from Horiba Scientific and the Complete Ease Software from JA Woolam Company. SE data measured on the bare silicon substrate were analyzed using the Si and SiO 2 tabulated dielectric functions and yielded a thickness value (2.0 nm) for the native silica layer on the surface, which was fixed in the further analysis. SE of the PVP-PS diblock copolymer film, before gold introduction, gave the initial thickness, and demonstrated a finite amount of depolarization quantified by , which could be well modeled as a thickness non-uniformity. The latter non-uniformity was estimated to be close to 8 % and kept fixed through-out the fits. One of the possible causes for this thickness non-uniformity is the presence of thickness defects observed on block copolymer thin films when the total thickness of the film is not exactly commensurate with the lamellar period: T = n d o or T = (n+1/2)d o , where n, a small integer, is the number of complete bilayers 52,39,53,54 .

II.2 Ellipsometric modelling.

In order to extract the effective optical properties of the studied films from the SE data, two different models were built (see Figure 4). Due to the small sizes with respect to wavelength of both the gold NPs D ~ 0.02 and the copolymer nanostructure lamellae d o ~ 0.05 at = 500 nm, an effective medium (EM) description is appropriate, for all or part of the multilayer stack. The models are built using the information on the layer thicknesses, appropriate interface layers and number of bi-layer repetitions inferred from the SEM images and schematized in Figure 5.a. The N-dependent dielectric functions of the polymer-gold nanocomposite effective media NC and EM2 were "kinetically" analyzed, meaning in consistency with a progressive evolution with increasing N, in order to refine the initial guesses. The stack structure can be schematized as shown on Figure 5.a. In the Model 1 (Figure 5.b), a perfect periodic superlattice is considered to determine the optical properties of the individual layers of the stack. Although this multilayer ellipsometric model does not provide a direct access to the anisotropic effective permittivity of the material, it is used as an intermediate step, from which an effective uniaxial dielectric function will be derived in a second step, as will be explained below. As seen on When the three mentioned conditions are fulfilled, the optical properties of the uniaxial effective medium  // (  // '() + i  // "() and  z (  z '() + i  z "() can be written [START_REF] Kang | Validity of effective medium theory for metal-dielectric lamellar gratings[END_REF][START_REF] Belov | Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime[END_REF][START_REF] Centeno | Effective properties of superstructured hyperbolic metamaterials: How to beat the diffraction limit at large focal distance[END_REF] : Equation 4Equation 5

As was explained in §I., the condition of deeply subwavelength thicknesses was one of our requirements for the design of a hyperbolic medium. With d NC ~ d PS < 0.04 on the explored wavelength range, these conditions appear valid even at the resonance, where we expect that n NC can be large [START_REF] Vieaud | Effective Medium Description of Plasmonic Couplings in Disordered Polymer and Gold Nanoparticle Composites[END_REF][START_REF] Baron | Bulk optical metamaterials assembled via microfluidic evaporation[END_REF] . Moreover, the number of repetitions is in our case m = 9, which we regard as sufficient, in particular in view of the absorption in the layers [START_REF] Rytov | Electromagnetic Properties of a Finely Stratified Medium Soviet[END_REF] . Finally, Model 1 first allows us to extract  NC from the experimental VASE data, from which we then calculate  // and  z , using Equations 4 and 5.

The Model 2 (Figure 5.c) is a direct uniaxial effective medium approach (compared to the indirect approach using the fitted lamellar stack) in which we use the super-lattice results as initial guesses. It combines the surrounding half layers, i.e. the bottom d NC /2 and top layer d PS /2 into the effective uniaxial layer, in order to account for the whole self-assembled diblock copolymer film. The uniaxial permittivity ( // ,  z ) of the medium EM2 is determined by fitting the Model 2 to the SE data using the BSPLINE function available in the Complete Ease software and using the results of Model 1 as an initial guess. This fitting procedure allows to extract  // () and  z () for the different values of N.

II.3 Results

Let us now present and discuss the optical parameters that were extracted from the ellipsometric study for the fabricated nanostructures.

A/ Nanocomposite (NC) layers

We start by extracting the optical properties of the individual NC layers which make up the lamellar stack. The NC dielectric functions are accessible via Model 1 (Figure 5.b.). The uncontrolled gold deposit layer cannot be represented by a simple effective medium with a few parameters only and was analyzed using a free BSPLINE. Using the kinetic SE data, the fit converged to a top layer dominated by a resonance near a photon energy of 1 eV ( = 1240 nm), which is typical of a partially connected network of Au particles on the surface. The thickness of this Au pollution layer was found to increase from 4 nm at N = 5 to 20 nm at N = 25, which is in agreement with the SEM observations. In fact, for N ≤ 15, the results show very little variation when removing the top layer from the model (d SAu = 0), confirming its presence is negligible.

Finally, the resulting dielectric function  NC () = NC '() + i  NC "() for the NC layers in the superlattice, from N = 5 to N = 30, are shown on Figure 6. The optical properties of the Au-loaded polymer NC layer are dominated by a resonance at  = 580 nm, close to that expected [START_REF] Vieaud | Effective Medium Description of Plasmonic Couplings in Disordered Polymer and Gold Nanoparticle Composites[END_REF] for the plasmon resonance of the gold NPs present in the NC layers, with an amplitude increasing with the number of loading cycles N, as expected. In particular, the NC medium presents a pseudo-metallic behavior beyond N = 15 with  NC ' < 0 on a large wavelength range, of width 80 nm for N = 25 and 50 nm for N = 20. This "pseudo-metal" behavior being resonant in nature is associated with a significant level of losses as can also be seen on Figure 6.

The resulting dielectric functions are consistent with what can be expected for a disordered composite of spherical inclusions within a homogeneous matrix. In fact, such composite can be described, at least at small gold volume fraction by the Maxwell-Garnett effective medium function [START_REF] Garnett | Colours in metal glasses and in metallic films[END_REF] , written:

  m Au m Au m Au m m MG f f                2 3
Equation 6

where  MG is the effective permittivity of the gold NP-polymer composite,  m is the polymer matrix permittivity,  Au is the NP gold permittivity, and f is the gold volume fraction in the nanocomposite NC. Figure 7 shows how the extracted values of  NC compare with those of  MG , calculated using For small N ≤ 5 (small gold fraction), the agreement is very good on all the red part of the spectrum, for wavelength above the resonance value at ~580 nm, and satisfying at lower wavelength (Figure 7). When pushing the Maxwell-Garnett effective medium approximation (MG-EMA) to higher N, a degraded agreement is naturally expected: it remains nevertheless reasonably good especially above 580 nm. These partial agreements provide rough estimates of the loading concentration in gold: we The analysis of the SE data through direct inversion as a uniaxial effective medium (Model 2) leads to a similar or better goodness-of-fit, compared to Model 1. Figure 9 shows the comparison between the components  // '() extracted for N = 15, N = 20 and N = 25 using Models 1 and 2. Models 1 and 2 were found to give highly similar results, within the experimental accuracy, in particular for  // , although  z shows a little more variation between the models. The agreement between the extractions of the two models substantiates the main result of this experimental study showing that the dielectric functions  // and  z have opposite signs in a finite wavelength range. As is shown on the Figure 8, the dielectric functions  // and  z both present a resonance at the wavelength  =580 nm (or alternatively a photon energy of 2.1 eV), close to the plasmon resonance of the gold nanoparticles. However, the resonance amplitudes of the two components significantly differ.

In the case of  z , this amplitude is limited and varies only little with the value of N, in direct relation with the amount of gold nanoparticles in the nanostructure. By contrast, the resonance of  // is stronger and its amplitude significantly increases with N, reaching a regime of negative values beyond N=20.

As a result, strong artificial anisotropies with different characteristics are found across the spectrum.

Finally note that attempting to combine the top Au deposit layer (of thickness d SAu ) into the effective uniaxial medium smears its properties and does not produce negative  // , in contrast with the analyses based on the two used models. Therefore, future studies will aim at lifting off the uncontrolled detrimental gold deposit top layer. In the current analysis, it is regarded as an experimental artifact whose analysis needs to be isolated from the analysis of the targeted nanostructure below it.

C/ Hyperbolic dispersion relation

We consider three regions of strong anisotropy denoted A:  z ' >  // ' > 0 ( = 481 nm), B:  // ' < 0 <  z ' ( = 539 nm) and C:  // ' >  z ' > 0 ( = 670 nm) indicated on Figure 10, which displays the values of  // and  z for N = 25 (corresponding to a MG-extracted f = 31%). The iso-frequency dispersion diagrams of the constitutive material can be produced using the experimental dielectric functions and are also , shown for reference. The plot labeled D, shown for reference, is a fictional hyperbolic case without losses: at  = 539 nm,  // ' and  z ' are set to the values for case B, whereas  // " and  z " are set to zero.

Cases A and C present typical elliptical dispersion diagrams, when both  // ' and  z ' are positive: one can observe, for small values of k x  (k x  < k x *), two symmetrical elliptical branches where the real part of k z is significantly larger than the imaginary part, typical of an attenuated propagation. For larger k x  (k x  > k x *), hyperbolic branches are observed, which are not usually present in ideal elliptical cases, and which are characterized by a predominantly evanescent behavior (imaginary part of k z much larger than the real part). In the case B when  // ' < 0 <  z ', we find that the dispersion diagram present hyperbolic branches for k x  > k x *, as well as elliptical curves at small k x .

These diagrams are related to the strong optical of the lamellar stack, obtained as a consequence of the resonant nature of the NC layers. The significant losses (Re(k z ) ≈ Im(k z )) associated with the hyperbolic propagation are another consequence of this resonant nature.

To understand the significance of these dispersion curves, it is useful to recall the usual description of and with imaginary parts set to zero:  // " =  z " = 0 (lossless materials). We represent the dispersion relation (Equation 3) in the k y = 0 plane. In the elliptical ideal case (Fig. 11.a), the propagative modes (Re(k z ) ≠ 0) are restricted to small wavenumbers (k x  < k x *), while all possible large-k x  modes are forbidden [START_REF] Repän | Dark-field hyperlens: Super-resolution imaging of weakly scattering objects[END_REF] and only purely evanescent (Re(k z ) = 0). In the hyperbolic ideal case (Fig. 11.d), we observe the opposite situation, with forbidden, purely evanescent modes (Re(k z ) = 0) for small k x  and purely propagative modes (Re(k z ) ≠ 0, Im(k z ) = 0) for large k x , the latter potentially providing the super-resolution property. The hyperbolic branches for k x  > k x * present the shape of a one-fold hyperboloid, with a forbidden gap in k x , as expected for a "metallic" [START_REF] Ishii | Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium[END_REF] or type II 4 hyperbolic medium, when the permittivity tensor  has two negative and one positive components.

To depart from these idealized situations, let us now introduce a small amount of losses in the materials, see Figure 11.b and 11.e (here we arbitrarily took  // " = 4  z " = 0.8): the dispersion curves become hybrid rather than purely elliptical or hyperbolic and present finite Re(k z ) and Im(k z ) at all values of k x , allowing previously forbidden modes to propagate weakly. Purely evanescent waves for the hyperbolic medium at small k x  and the elliptical medium at large k x  become weakly propagative with Re(k z ) ≠ 0 and Im(k z ) >> Re(k z ). The large-k x  branches of the hyperbolic medium (Fig. 11.f) present the same one-fold hyperboloid shape as that of the no-loss case (Fig. 11.e). Interestingly, when the imaginary parts  // " and  z " are further increased to simulate more losses (Fig. 11.(c-d) and (g-h)), the modification of the propagative and evanescent modes continues to occur, so much so that over most of the whole k x range, more or less strongly attenuated, propagative modes are observable.

In particular, we note that the intermediate hyperbolic cases (as represented in Figure 11.f and 11.g) still present predominantly propagative modes at large k x , which can support super-resolution properties: these are the cases corresponding to our experimental system in the B situation in Figure 10. Finally, these results demonstrate the capacity of our bottom-up self-assembled multilamellar stack to respond as a hyperbolic effective medium in a given region of the visible spectrum (520 <  < 560 nm). In the hyperbolic region, the dispersion relation allows propagative modes for large values of k x , potentially providing super-resolution. These hyperbolic modes are however significantly attenuated due to the lossy nature of the media, in which the hyperbolic property is obtained from the resonant nature of the permittivity. These detrimental losses need now to be minimized by tuning the nature, size, size dispersity and organization of the nanoparticles. Furthermore, the same self-assembly methodology for lamellar stacks fabrication could be used with other nanoparticles with a resonance of better quality factor, or with a lamellar system including gain species (fluorophores or quantum dots) which could amplify the plasmons and compensate the losses.

In any case, we have demonstrated for the first time the possibility of using a self-assembly methodology for the fabrication of bulk hyperbolic material, which opens new fabrication routes for metamaterials aiming at super-resolution lenses.

Figure 1 .

 1 Figure 1. Backscattering scanning electron microscopy side-view image (SEM) of a 370 nm-thick film of alternating layers of poly(styrene) (PS, appearing black) and Au nanoparticles:P 2 VP nanocomposite (NC, appearing white). The lower and upper black domains of the micrograph are the substrate and the air, respectively. The trihedral indicates the ordinary ((x,y) or //) and extraordinary (z) directions.

Figure 2 .

 2 Figure 2. Backscattering scanning electron microscopy side-view image (SEM) of the 265 nm-thick film of alternating layers of pure polymer (PS, appearing black) and of Au nanoparticles:P2VP nanocomposite (NC, appearing white), for a number of cycles of gold impregnation and reduction of N=5 (a), 10 (b), 20 (c), 30 (d).

Figure 3 .

 3 Figure 3. Transmission electron microscopy (TEM) image obtained with a grid on which was deposited a drop of the dispersion obtained by full dissolution of a film, at the step N = 20 of the fabrication process, in a good solvent of the diblock copolymer. The image evidences gold nanoparticles of mean diameter 7 nm. Note that this observation technique being destructive, it was performed on different samples than the ones studied by SEM and VASE.

Figure 4 .

 4 Figure 4. Evolution of the measured ellipsometric angles  and  as a function of the photon energy for an angle of incidence of  o =65° and different values of N between 0 and 25. The spectroscopic features are damped as N increases, due to the increasing absorption of the film.

Figure 5 .

 5 Figure 5. The models are built based on the detailed film nanostructure obtained from the cross-section SEM images, as schematized in (a). The ellipsometric models used in Models 1 and 2 are represented in (b) and (c), respectively. PS stands for pure polystyrene and NC for the nanocomposite composed of gold nanoparticles and poly(2-vinylpyridine), of varying composition when N increases. The thickness of the native silica layer covering the silicon wafer was determined by VASE on a bare wafer. The "kinetic" SE data are analyzed so as to extract the N-independent dimensions d NC , d PS , d EM2 , and the N-dependent parameters, which are the thickness d SAu and dielectric function of the upper gold layer, and the dielectric functions of the polymer-gold nanocomposite effective media NC, and EM2.

Figure 5 .

 5 Figure 5.b, the thickness of the individual layers are d PS , d NC , where the bottom and top half-layer

Equation 6

 6 and a dielectric function for gold modified from the Johnson & Christy tabulated data 62 in order to take into account finite size effects (see reference 45). All values of  m and  Au used in the  MG calculation are displayed in the Supplementary Information, Figure S2.

Figure 6 .

 6 Figure 6. Real (upper plot) and imaginary (lower plot) parts of the dielectric function  NC of the NC layers in the lamellar stack, as extracted from the SE study for different values of N between 5 and 25. The dotted line is the dielectric function used for the PS layer, with Im( PS ) = 0. The resonance amplitude varies as N increases, due to the increasing introduction of plasmonic NPs.

Figure 7 .

 7 Figure 7. Comparison between the real (upper plot) and imaginary (lower plot) parts of the dielectric function  NC of the NC layers in the lamellar stack, as extracted from the SE study (solid lines), and the values calculated as a Maxwell-Garnett effective medium  MG (dashed lines) for different volume fractions in gold: f = 7% (yellow line), 18% (orange line), 31% (red line).

  find N=5 to correspond to a MG-extracted value of f = 7%, and N = 10 and N = 25 to approximately f = 18% and f = 31%, respectively.B/ Uniaxial effective mediumFollowing the second step of Model 1, the dielectric constants of the uniaxial effective medium  // (  // '() + i  // "() and  z (  z '() + i  z "() are first determined using the optical properties of the fitted Au loaded polymer layers (NC) and the PS layer, through Equations 4 and 5. The resulting effective optical properties are shown in Figure8.

Figure 8 .

 8 Figure 8. Real (upper plots) and imaginary (lower plots) parts of the components  // (left) and  z (right) of the lamellar nanoplasmonic thin films, as computer using Equations 4 and 5 from the Model 1 SE extractions for different values of N between 5 and 25 (MGextracted f between 7 and 31%,). The resonance amplitude varies as N increases, due to the increasing volume fraction of introduced plasmonic nanoparticles.

Figure 9 .

 9 Figure 9. Comparison between the real part of the ordinary components in the cases of the multilayer (Model 1, continuous lines) and uniaxial model (Model 2, dashed lines) for N=15 (a), N=20 (b) and N=25 (c).

shown on Figure 10 .

 10 For each frequency ω considered (or equivalently, excitation wavelength), two wavevector ranges can be distinguished k x  < k x * and k x  > k x *, with ' k o ²= o  o ².

Figure 10 .

 10 Figure 10. Left: Parallel (red continuous line) and perpendicular (green dotted line) components of the uniaxial dielectric function of the lamellar nanoplasmonic stack for N = 25 (f Au ~ 30% in the NC layers). The vertical blue dashed lines indicate the three conditions (A:  = 481 nm; B:  = 539 nm; C:  = 670 nm) chosen for the calculation of the dispersion relations, are displayed in the right part. Right: Dispersion relation at k y = 0, showing the real (blue lines) and imaginary (orange lines) parts of the k z component of the wave-vectors in the material as a function of the k x component. Thick continuous and dashed lines correspond to opposite solutions of Equation 3. The thin black continuous circle corresponds to the wavevectors propagating in air 2 2 2 o z x k k k  

  elliptical and hyperbolic media. The ideal cases are displayed in the Figures 11.a and 11.d, where two fictitious materials are considered with the real parts of the permittivity tensor set to the experimentally determined values in cases C ( // ' = 1.77,  z ' = 3.85) and B ( // ' = -1.42,  z ' = 5.42);

Figure 11 .

 11 Figure 11. Dispersion relations at k y = 0, showing the real (blue line) and imaginary (orange line) parts of the k z component of the wave-vectors in the material as a function of the k x component. Continuous and dashed lines correspond to opposite solutions of Equation 3. The plots (a)-(d) (resp. (e)-(h)) correspond to fictional elliptical (resp. hyperbolic) cases with  // ' and  z ' set to the values for case C (resp. B), and  // " =4  z " = 0 (a) and (e);  // " =4  z " = 0.8 (b) and (f);  // " =4  z " = 4 (c) and (g); and  // " =4  z " = 8 (d) and (h).

ACKNOWLEDGMENTS

We thank I. Ly and C. Picard for help with the SEM imaging, and P. Barois, E. Garcia-Caurel, B. Gallas, and M. Stchakovsky for useful discussions and comments. This work was supported by the LabEx AMADEus (ANR-10-LABX-42) in the framework of IdEx Bordeaux (ANR-10-IDEX-03-02), France. It has benefited from an Aurora Hubert Curien Partnership, funded by the Norges Forskningsraad (NFR) and by the French ministères des Affaires étrangères et du Développement international (MAEDI) and de l'Education nationale l'Enseignement supérieur et de la Recherche (MENESR).


Zeng-Hui Wang

Yan-Jun Zhao

Yang Liu

Dong-Yan Huang

Application of Growth Curve in Agricultural Scientific Research

Keywords: Logistic curve, Parameter estimation, division method

This paper introduces the application of logistic curve in agricultural science, and gives a division method of parameter estimation of logistic curve. Because the logistic curve contains three parameters, it is not possible to use least square method to estimate the parameters directly. In this paper, the parameter C is estimated by the three (four) point method at first. Then the two steps method is used to estimate the parameters b and a respectively. Through a practical example with the new method, the example shows that the precision of this method is relatively high..

Introduction

The logistic growth curve is the most famous curve among the building-up curve, which was originally developed in the study of population growth, and later P.F.Ver hulst , a mathematician in Belgium has summed up it as a mathematical model.

The parameter t represents time, the parameter y is the increase in the t times.

The characteristic of this kind of curve is that when the value of the t is small, the value of the y grows slowly. As time goes on, y changes into blocks, when the parameter t gets the critical value, and the increase of the value of Y begins to slow down again. Finally, the value of Y is close to a certain value.According to this characteristic, logistic growth curve is widely applied in life science research.For example, the logistic curve is used to describe the growth process of plant height , leaf area, the growth of the height and weight of the young in the study of juveniles. Especially, the accumulation of starch, fat and protein in food crops is consistent with the logistic curve. Therefore, it is very meaningful to study the parameter estimation method of logistic curve.

The ordinary least square method and the Marquardr method can estimate the parameters in the logistic growth curve, but the accuracy of ordinary least squares method is low, and the Marquardr method is an iterative method for technicians, for the calculation is more complex. A new approach in logistic growth curve of parameter estimation is given in this paper.

Fund project: national key research and development project funded by National Science and Technology department (project number: 2016YFD0700300)

Author profile: Wang Zeng hui (1956 -), male, research direction: statistics and optimization method;

Corresponding author: Yan-Jun Zhao (1979 -), male, research direction: partial differential equation and its application, the application of time series analysis and data mining, E-mail:yjzchsnenu@163.com

2

The general estimation approach in logistic growth curve

Estimation of the parameter C

Assuming that some actual measured data as

) , i i y t ( ) , , 2 , 1 ( n i  =
.Make the value of i t is spaced equally, we have

h i t t i ) 1 ( 1 - + = , ) , , 2 , 1 ( n i  = .
If n is odd, select the initial point, midpoint and terminal point. In this case, a formula as follows:

n n n n n n y y y y y y y y y c 1 2 2 1 2 1 1 1 2 2 1 2 ) ( ˆ-- + = + + + (2)
If n is even number,select the

1 1 2 2 1 2 2 1 1 2 2 1 ) ( ) ( 1 2 2 1 1 1 2 2 1 2 2 1 ) ( ) ( + + + - + - + n n n n n n n n n y y y y y y y y y y y y (3)

Estimation of parameter a and b

When the parameter C is estimated by using the above formula, there are only two parameters to be estimated in the model 1.The least squares method can be used to solve it.

First of all, the type (1) needs to be became the reciprocal value, and then take the natural logarithm. We use the following notation:

y y c - = ln Y , a A ln = , b B - =
The above formula can be transformed into a linear model.

Bt A Y + = (4)          - - = - - = - = ∑ ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = = ∧ - ∧ ∧ n i i n i i i n i n i i i n i i n i n i i i i t n t Y t n Y t t n t Y t n Y t B t B Y A 1 2 _ 2 1 _ _ 1 1 2 2 1 1 1 _ ) ( 1
) )( ( 1 (5) Among, ∑ = = n i i Y n Y 1 _ 1 , ∑ = = n i i t n t 1 _ 1 ∧ = ∧ A e a , ∧ ∧ - = B b .
The estimation of the parameter A and the parameter B is calculated by the formula (5),we have

∧ = ∧ A e a , ∧ ∧ - = B b
,The three parameters of the model (1) are estimated.

A new approach in logistic growth curve of parameter estimation

Let's say that the logistic growth curve is the following formula bt ae

c y - + = 1 0 , 0 , 0 > > > c b a
(The estimated parameters)

In the case of the estimation of the pending parameters in model(1), we take three steps.

Estimation of parameter c

The estimation of parameter c is still used in the three points or four points method, as the formula [START_REF] Hong | New method of logistic model parameter estimation and application[END_REF] or (3).

Estimation of parameter b

A set of measured data is ( )

n i y i i , , 2 , 1 , 0 , , t  =
. The formula for the parameter b is derived. Take the inverse of the (1) and transform it into the following model:

i bt i ae Y - = ˆ, 1 1 ˆ-- -= i bt i ae Y , n i , , 3 , 2  = , i i i y y c Y - = , n i , , 2 , 1  = .
Eliminate the parameter a in the (6) model,therefore

( ) 1 1 1 ˆˆ- - - - - - - = = i i i i t t b bt bt i i e ae ae Y Y (7)
The ( 7) formula only has one parameter b which needs to be estimated. Now we use the least square method to estimate the parameter b.

Get the natural log for the(7) formula ( )

1 1 ˆln - - - - = i i i i t t b Y Y (8) Thus we set 1 1 ln Y ln ln - - - = = i i i i i Y Y Y x , ) ( ln ln ln 1 1 1 - ∧ - ∧ ∧ - ∧ ∧ - - = - = = i i i i i i t t b Y Y Y Y x , 1 - - = ∆ i i i t t t , n i , , 3 , 2  =
.So the formula (8) becomes the following model.

i i t b x ∆ - = ˆ
By the least squares principle we have

( ) ∑ ∑ = = = ∆ + = - = n i i n i i i i t b x x x b Q 1 2 2 2 min ) ( ) ( (9)
Use the principle of the extremum of the function ) . ln(ln ˆ [START_REF] Ai-Hua | Research on Logistic model[END_REF]

( ) 0

Estimation of parameter a

Exponential model ( 6) of the parameters is given by [START_REF] Ai-Hua | Research on Logistic model[END_REF], so only the parameter in the model ( 6) can be estimated, here we still using the least squares estimation, we note

( ) (

)

∑ ∑ = = - - = - = n i n i bt i i i i ae Y Y Y a Q 1 1 2 2
) ( [START_REF] Xu Liang | Application of animal growth curve model to analyze the growth and development of larva of different tussah silkworm varieties[END_REF] Use the principle of the extremum of the function. [START_REF]Genetic parameters of growth curve parameters in male and female chickens[END_REF] 5 Case analysis

Model application

We study the pattern of the weight of the chicken.The silky fowl come from Xichuan county in Henan province, is one of the excellent local breeds in Henan province, it has the high protein content, amino acid, a variety of vitamins and trace elements as well, while the cholesterol content is low, people enjoy it at the table.In order to get profit, it is meaningful to understand its growth rule. Table 1 shows the growth observation data of the silky fowl. Take t = 0, due to a total of nine data, namely, n = 9, therefore the three point method can be used to estimate the parameter c in the logistic growth curve.

Get the dates for 30.58, 434. The data in table 1 is evaluated in the formula [START_REF] Ai-Hua | Research on Logistic model[END_REF],the parameter b can be estimated, [START_REF]Genetic parameters of growth curve parameters in male and female chickens[END_REF] we conclude the correlation index, thus it can be seen that the fitting precision of the curve is relatively high.

Conclusion

The logistic model ( 14) was applied to study the weight growth rule of the silky fowl in Zhejiang province.

Take the derivative of the formula (14), we have .It is shown that the weight gain of the silky fowl in Zhejiang province at the time of 1 t is faster than the time of 2 t .

(2)Take the first derivative with respect to the formula (15) again, set '' 0 y = , we can find the inflection point of the curve, and the inflection point corresponds with the time point is 11.5. We know the speed of the second derivative reaction of chicken weight growth.

When 0 y > , it indicates that the chicken's weight gain is in the fast growth period. When '' 0 y < , it indicates that the weight gain of chickens enters the deceleration period. When 12 t < (yearling), the weight is in the fast growth period. When the time of 12 t > (yearling), although the weight is still growing, but it has been in the growing period with low speed.

  So we have a estimated model about parameter b

.

  The parameter a should satisfy the following model gains the minimum.

4 2 R

 2 we get the estimated formula of parameter a, The logistic growth curve is fitted to the optimum test.In general, the equation of logistic growth curve regression equation is measured by the correlation index ( )and the residual sum of the sum of the squares of residuals( Se ), which the definitions are as follows,

  So we have the logistic growth curve model of the pattern of growth of the silky fowl.

)

Table 1 t

 1 Now we use the logistic growth curve to fit the pattern of the growth of the chicken.

	(yearling)	1994	2	4	6	8	10	12	14	16
	y i	(weight)	30.58	82.45	172.28	312.27	434.90	589.70	759.25	921.93	1026.93
	y i ˆ(weight)	33	64.31	121.66	220.1	370.29	563.6	761	957.2	1037.7
							(weight: g)

  The accuracy of the model(14)needs to be thought.We count the fitting value for the first time.Counted fitting values are listed in table 1, from the data in table 1 in the type

	y	=	1	+	34	3451 4661 . 0 t e 6398 . 1181 . -		14 (
	( ˆ= -3451 . 0 6398 . 34 1 4661 . 1181 + = i e i t y i	, 1	) 9 ,..., 2

  The value of first derivative in the time of t is the reflection of the growing speed of the silky fowl in

				y	'	( t	)	=	. 14123	476 e	0 -	.	3451 t	1 (	+	34	.	6398 e	3451 . 0 t -	)	2	(15)
	(1)Zhejiang province.
	If 1 2 t t < , we have ' y t 1 ( )	<	' y t ( ) 2	.It is shown that the weight gain of the silky fowl in Zhejiang province
	at the time of 2 t is faster than the time of 1 t .If 1 2 t t < , we have ' y t 1 ( )	>	' y t ( ) 2


Loutfi Nuaymi
email: loutfi.nuaymi@enst-bretagne.fr

Association of Power Control and Data Rate Selection in WCDMA

Ahstrrrct-In this paper, we study the interaction between power control and data rate selection in a wireless CDMA network. Specifically, third-generation cellular WCDMA system is considered. The criterion is the average of the sum of data rates in a variable rates services environment. Simulation results show that the association (or the integration) of power control and data rate selection allows a given network to have better achievable performance. Interesting elements of the proposed algorithm are highlighted and useful comments given.

'

I. Introduction

One of the major innovations of third-generation cellular networks is the presence of different communication services including different data rates, which can have relatively high values. Data rates can reach 2 or more Mbis. The multiple access system selected for third-generation cellular systems is Code-Division Multiple Access (CDMA). For wireless CDMA systcms. capacity limitations are mainly due to interference ([ 11). Radio resource management allows lower interference and thus higher capacity. In this paper, we consider the application of an algorithm conibining power control and data rate selection for WCDMA.

For WCDMA ( [START_REF] Holnia | WCDMA ,for UMTS[END_REF]), the European and Japanese common proposal for third-generation 1MT-2000, power control is defined in [START_REF]Partnership Project; Technical Specification Group Radio Access Networks[END_REF] for Frequency-Division Duplex (FDD) system and [START_REF]Partnership Project; Technical Specification Group Radio Access Networks[END_REF] for Time-Division Duplex (TDD) system.

I n a multimedia environment such as in WCDMA, different services with different constraints are transmitted simultaneously. In this paper, we try to optimise resource allocation (selection of transmitted power and rate) in a wireless network where each communication has different possible data rates, including zero (no-transmission). This is, for example, the case of Internet communications. The criterion of optimisation is the instantaneous sum of data rates.

Evidently this is a restrictive hypothesis since in real networks some services will have a mandatory minimal data rate in addition to transmission delay constraints; this is, for example, the casc of voice transmission. Yet, the computation of the maximal sum of data rates can be useful for a general wireless multimedia CDMA system.

Considering a given user and rate distribution, the existence of a transmitted power vector such that all communications verify their quality constraint has been discussed in [5].

Integrated power control and rate selection for the uplink of a cellular CDMA network with two classes of users, typically voice and data users, is studied in [6]. A transmission scheduling scheme for delay-tolerant users allowing relatively higher throughput and lower intercell interference, while respecting different quality requirements, is proposed. The same topic for downlink non real-time services is considered in 1171. Scheduling is such that one user transmits at a time in each cell. A distributed power control algorithm allows intercell interference management. Other references about the association of power control and data rate can be found in [START_REF] Berggren | Joint Power Control and lntracell Scheduling of DS-CDMA Nonreal Time Data[END_REF].

In this paper, we study uplink variable data rate services. A simple approach is used in order to show interesting results of capacity increase when power control and data rate are associated. We start by reviewing some previously-proposed studies on that topic. The system model considered is described and some details about power control and data rates in WCDMA given. Then we propose some interesting results concerning the total throughput of a CDMA cell. A discussion of the results and conclusions are given in the last section.

Although the implementation topic is not directly addressed in our paper, the results and the comparisons obtained may bc used for performance evaluation.

System model

The uplink (mobile-to-station) of a CDMA 3G system is considered. In an FDD-CDMA system, an uplink physical channel is defined by its code and its frequency. The methods proposed in this paper could be applied to downlink andor to TDD systems after some slight adaptations. Power control is only applied at Layer 1 (inner closed-loop PC). Circuitswitched systems are considered. Packet transmission for packet-switched services is assumed to be taken into account at layer 3 or higher layers. A fairly simple simulation rvlodel is used. This allows an illustration of the results of this work. A maximal transmitted power limitation is considered. The simulation model consists of M mobiles that transmit information sharing a common radio channel (uplink of a cellular CDMA system). Each mobile should communicate with one of the B base stations (BS). Interference between the reverse links of a CDMA network is illustrated in Fig. 1.

The algorithms considered are applied for a fixed set of mobiles, their application being supposed fast enough to consider the mobile set and geographical positions all fixed (snapshot). The speed of convergence of the iterative algorithms is verified in the following.

The index of the base to which mobile nz is assigned is a,,, .

A mobile is connected to the best-received power level base station (sometimes called the best-server), which is not always the closest one due to the fadings. Mobile nz transmitted power is p,,, and the link gain from mobile IIZ to base b is noted gh,,,, so matrix G = (ya,,,) is of size ( B x M). Mobile m received signalto-interference ratio (SIR) on base 0, noted yJrr,h , is then given by: where nh is the receiver noise, also called thermal noise, at base 0. The received SIR of mobile m on its assigned BS is noted y,,,, so we can write: y,,, = y,,,,,,,,, . A mobile is in outage state when its SIR falls below its SIR target. In a CDMA system, the received BER is a direct function of the energy-per-bit over total noise spectral density ratio, noted E,,INo. Thus, E,,INo i s considered as a good quality indicator. It is related to the received SIR, noted y, by the following formula:

where R is the data (or information) bit rate and W the total frequency bandwidth (5 MHz for WCDMA). The value Eh/& i s sometimes called the despread SIR. The spreading factor (SF), or VIR, is then an important parameter for the decision of the quality threshold.

For a data rate I-" and for a required threshold for ,!?,/No, given by:

(3)

Communications have different possible data rates and Eh/No targets. Thus, different SIR targets are considered. In this work, we consider that each data communication has D possible data rates :

where rl is equal to zero (corresponding to a no-transmission state).

Paging or synchronization channels are not included in the simulation. Performance of the algorithm is evaluated by the average sum of transmitted rates. More details about the simulation are given below in the following sections.

Power control and data rate selection in WCDMA system

Power control in WCDMA is a closed-loop power control (PC) which is a combination of the outer and inner closed loop controls. The inner (also called fast) closed loop PC adjusts the transmitted power in order to keep the received SIR equal to a given target. This SIR target is fixed according to the received Bit Error Rate (BER) by the higher layer outer-loop PC in order to achieve a given BLock Error Rate (BLER) target. Ensuring that the lowest possible SIR target is used results in greater network capacity. The BER target is a function of the communication service. In this paper, we use fixed values for SIR thresholds (see below). The results should still hold for varying values of SIR thresholds (e.g., by the outer-loop).

The inner closed-loop PC measures the received quality, represented by received SIR and then sends commands to the transmitter (i.e., the mobile in the case of the uplink) for the power update. The receiver estimates the received power of the connection to be power controlled. Simultaneously, the receiver estimates the received interference. The obtained SIR estimate, noted SIR,,,, is then used by the receiver to generate PC commands.

The frame duration in WCDMA System is 10 ms and contains 15 time slots. The transmitted power is fixed for a given time slot. Each slot then has one PC information bit, which gives a PC update rate of 1500 b/s. Each PC command increases or decreases the transmitted power level by a fixed value of around 1 dB, according to Algorithm 1 or 2 of [SI. In this work, the power update step size is always equal to 1 dB.

We studied Adaptive-Step Power Control for WCDMA in [START_REF] Nuaymi | A Power Control Algorithm for 3G WCDMA System[END_REF].

Fast PC is made by the OS1 model (PHYsical) Layer I .

Outer-loop PC is realized by Layer 3 RRC (Radio Resource Control). The outer loop PC update frequency is 10-100 Hz ( [START_REF] Holnia | WCDMA ,for UMTS[END_REF]). RRC can also make the data rate selection.

Data rates and SIR thresholds

In WCDMA System, different data rates can be used ([SI).

In our simulation, we considered the uplink data rates of the DPCH (Dedicated Physical Channel), rewritten in Table 1. The threshold of E/,/No corresponding to a given BER is theoretically independent of the data rate. In WCDMA system, / the energy of the pilot bits is more important for high data rates than for smaller rates. This leads to easier demodulation, and then smaller Eh/& thresholds, for higher data rates than for smaller data rates. For a given BER, the value of the E,,/No threshold is a function of many factors: modulation, coding, average mobile speed, radio channel details (multipath, etc), receiver type, antenna type and diversity and other operating environment parameters. In our simulation we ~ised the values of Table 1 for E,INo thresholds. According to what is said above for WCDMA, these values decrease when the data rates increase. Formula ( 3) is then used for the computation of the SIR threshold for the different data rates (and thus the different spreading factors). Results of this paper still hold for different values of EhINo thresholds, and even for time-varying Eh/No thresholds, e.g., as a result of outer-loop PC.

The association of data rate selection and power control has many advantages: one of these is the possibility of choosing the highest data rate that has an achievable SIR threshold. This choice can be updated according to the network parameters (and. specifically, saturation) for communication services where it is possible. Table 1 . WCDMA data rates. The thresholds for &/No used in our simulation and corresponding signal-to-interference ratios are also given.

In the remainder of this paper, we consider that all the communications can at any instant have one of the data rates of Table I , Rate 0 (i.e., no transmission) is also possible. This could be the case for Internet-style communication. Obviously, in real networks, some services will have a fixed rate or a mandatory minimal data rate i n addition to transmission delay constraint: this is, for example, the case of voice transmission. However, the computation of the maximal sum of data rates, with the hypothesis we consider, can be useful for a general wireless multimedia CDMA system.

IV. Power control and rate selection

Power control is .a part of the radio resource control in a CDMA system. Association of the power control and the base station selection is studied in [l 01.

The association of the processes of power control and data rate selection can be made by choosing simultaneously and according to a given criterion the transmitted power and the data rate for each transmitter (mobile or base station). These choices can be made by a centralized or a distributed algorithm.

The Selective Power Control (SPC) Algorithm proposed in [5] is a generalization of the power control algorithm known as Distributed Power Control (DPC) of [ I I ] and [12]. The SPC tries to maximize the sum of data rates with minimal energy consumption. It is an iterative distributed algorithm.

This iterative algorithm is applied for a fixed set of mobiles, its application being assumed fast enough to consider the mobile set and geographical positions all fixed (snapshot). The transmitted power of mobile i at instant v is noted pI'"', where the time index is v. Its received SIR is noted y,(v). The maximum (resp. minimum) transmitted power is PMtu (resp. PILIIII). Each transmitter data rate and transmitted power is computed, iteratively, as follows. Starting from an arbitrary initial transmitted power vector P'O', the next iteration transmitted power is determined by: where ( E ) is the indicator function of the event E. The data rate of mobile i, at convergence (or at any instant v ) is noted rj,fc/eL--j(v), where index-i(v) is given by: One can note that Eq. ( 4) considers that mobile i may be the only one to update its power and data rate at a given step; the SPC is a completely distributed algorithm. Data rate 0 (no transmission) is selected if no transmitted power can be used.

In this paper, we consider the application of the SPC in the environment of WCDMA. The data rates of Section 111 are used. The SPC is a quality-based power control, i.e., power control update information is based on estimated received quality. In WCDMA System, one information bit is allocated for the update of the transmitted power, at each power control step. Combined power control and rate selection in WCDMA could be achieved in the RRC layer in WCDMA radio protocol architecture.

V. Simulation results

This section contains the simulation results. Numerical hypothesis of the simulated system model for the mobile network are now given. The sum of data rates per cell is evaluated in simulation and compared to the same parameter in the Stepwise Removal Algorithm (SRA) of [l 11.

The SRA applies for a fixed data rate system, in our case 240 kb/s, which is equivalent to an SIR threshold q, = 0.15 (see Table 1). The SRA is distributed and iterative. The power update formula is: Communications are removed until all these remaining achieve their SIR. The data rates (all equal in the case of SRA) of the remaining conlniunications are added in order to have the total throughput.

A fairly simple simulation model is used in order to illustrate the conclusions of this paper. A CDMA system with 25 base stations ( 5x5 ) is considered. The base stations are placed on a rectangular grid. The side of the average square cell is 200m. Mobiles are uniformly distributed. The link gain between niobilej and base i is modeled as gij = c,, 1 djj where coefficient c, corresponds to obstacles (shadow fading) and d, is the mobile-to-base distance. Coefficient c,, is a zero-mean log-normal random variable whose standard deviation is o=6dB. Average values are evaluated by Monte Carlo simulations for 500 independent configurations (mobiles di stributjons).

Class 3 mobile stations are considered. The maximum transmitted power is 24 dBni (0.25 Watt) and the minimum transmitted power is -50 dBm as specified in [13]. The receiver thermal noise for the base stations is -103 dBni. A receiver thermal noise is considered in order to have absolute values for transmitted powers computed by the PC algorithms (see Formula (I)). The initial transmitted powers of the algorithms have a random value between the minimum value and IOYn of the (real) value of the maximum transmitted power. A perfect estimation of the received SIR is assumed.

The admission control is not addressed in this work, i.e., an 'admit-all' policy is considered. The two algorithms considered can converge to data rate zero when needed, which is a sort of admission control. The association of the proposed algorithm with an admission control policy is an interesting topic of research. This would, for example, allow the presence of niinimuni data rate services such as voice services.

We consider a simulation with fixed-position mobiles (snapshot configuration). The convergence of the algorithms is assumed to be fast.enough with regard to the variation of the radio channel to use this condition. After thirty iterations, the values computed by the iterative algorithms used are almost definitive (see below). Given that one iteration corresponds to one time slot ( I O nis 115) and that the coherence time of the radio channel is of the order of 100 ms, we can assume the snapshot hypothesis.

According to ( 3) and ( 4), data rate zero would lead to a transmitted power equal to zero also, which would not allow a fiirther change of this transmitted power (see update formula ( 4)). The use of a minimal (non-null) transmitted power, as noted above, eliminates this problem. Figure 2 shows the sum of the data rates in a cell as a function of the average number of mobiles per cell for the SRA. The values of the data rates after 100 iterations are used for the computation of the average sum. At this point, they ai-e sufficiently convergent. Figure 3 shows the same average values for the SPC Algorithm.

We notice in Figure 2 that, in the case of SRA, the average sum of data rates per cell is the same when the number of mobiles per cell changes. This means that (for the high data rates considered) only some mobiles are allowed to transmit while the others are silent. In our case, only 3 or 4 mobiles. in each cell, can transmit at 240 kbis. Fig. 3. Sum of the data rates in a cell as a function of the average number of mobiles per cell for the SPC Algorithm: the association of data rate selection and power control leads to a clearly greater sun of data rates per cell.

Figures 2 and3 show that the association of data rate selection and power control leads to a clearly greater sum of data rates per cell. In addition, this latter value is no longer fixed whcn the number of mobiles increases. Having different possible data rates allows greater flexibility. The SPC works in a way such that smaller data rates are used, when needed, in addition to large values which are used whenever it is possible. This allows a higher sum of data rates per cell, a sum that increases when the number of mobiles per cell increases.

The sum of the data rates is plotted for one configuration in Fig. 4 for the SPC Algorithm. The total number of niobiles is 325 (or 1 3 per cell). Convergence is almost complete after 10 iterations. For a relatively greater number of mobiles (20 or 25 mobiles per cell), the number of iterations for convergence remains smaller than 30. We verify that the snapshot hypothesis can be assumed with a good approximation.

VI. Conclusions

In this paper, we have studied the association of power control and data rate selection in a cellulai-CDMA network.

This was done for a specific environment, where different data rates. including data rate zero, can be used, which is a restrictive hypothesis. This is, for example, the case for Intelnet comniunications. The coniputation of the maximal sum of data rates can be useful for any general wireless multimedia CDM A system The criterion of optimisation is the instantaneous sum of data ratcs. We notice that this sum has practically a fixed value for a fixed data rate environment, when the number of demanding (but not necessarily served) mobiles increases. An algorithm with different possible data rates allows greater flexibility. The association of power control and data rate selection works in a way such that smaller data rates are used, when needed, in addition to large values which are used whenever it is possible. With UMTS parameters, the convergence o f the iterative algorithm is fast enough to consider that the radio channel is non variable during one data rates values convergence time.

The association of the algorithm studied with an admission control policy is an interesting topic of research. This would, for example, allow the presence of minimum data rate services such as voice transmission. Results of this work can also be used for scheduling policy studies

Fig. 1 :

 1 Fig. 1 : Reverse link of a CDMA cellular network. AI1 mobiles use the same Frequency Bandwidth W.

The 1

 1 4m IEEE 2003 International Symposium on PersonalJndoor and Mobile Radio Communication Proceedings (Kbitds) [ (dB) [ (real values)

The 1

 1 4Ih IEEE 2003 International Symposium on Persona1,lndoor and Mobile Radio Communication

Fig. 2 .

 2 Fig.2. Sum of the data rates in a cell as a function of the average number of mobiles per cell for the SRA (Stepwise Removal Algorithm).

Fig. 4 .

 4 Fig.4. Sum of the data rates per cell as a function of the number of' itei-ations. plotted for the SPC Algorithm. The total number of mobiles in the simulated area is 325.

Acknowledgement: I would like to thank Ziad El Murr for his help in some o f the computer simulations of this work. I also thank Janet Orinrod for her helpful comments.