Imane Demnati

David Grossin

Olivier Marsan

Ghislaine Bertrand

Gérard Collonges

Christèle Combes

Maria Parco

Inigo Braceras

Joel Alexis

Yannick Balcaen

Christian Rey
email: christian.rey@ensiacet.fr

Comparison of Physical-chemical and Mechanical Properties of Chlorapatite and Hydroxyapatite Plasma Sprayed Coatings

Keywords: Chlorapatite, FTIR, hydroxyapatite, low energy plasma spraying, mechanical testing, Raman, XRD

Chlorapatite can be considered a potential biomaterial for orthopaedic applications. Its use as plasma-sprayed coating could be of interest considering its thermal properties and particularly its ability to melt without decomposition unlike hydroxyapatite. Chlorapatite (ClA) was synthesized by a high-temperature ion exchange reaction starting from commercial stoichiometric hydroxyapatites (HA). The ClA powder showed similar characteristics as the original industrial HA powder, and was obtained in the monoclinic form. The HA and ClA powders were plasma-sprayed using a low-energy plasma spraying system with identical processing parameters. The coatings were characterized by physicalchemical methods, i.e. X-ray diffraction (XRD), Fourier transform infrared spectroscopy (FTIR) and Raman spectroscopy, including distribution mapping of the main phases detected such as amorphous calcium phosphate (ACP), oxyapatite (OA), and HA or ClA. The unexpected formation of oxyapatite in ClA coatings was assigned to a side reaction with contaminating oxygenated species (O 2 , H 2 O). ClA coatings exhibited characteristics different from HA, showing a lower content of oxyapatite and amorphous phase. Although their adhesion strength was found to be lower than that of HA coatings, their application could be an interesting alternative, offering, in particular, a larger range of spraying conditions without formation of massive impurities.

INTRODUCTION

Orthopaedic implants, especially those intended to restore the function of load-bearing joints (hip, knee, shoulder, elbow, ankle, finger...) of aged or injured human beings, should possess several attributes to meet the high level of mechanical stresses, wear, and fatigue in the course of normal activity. Although they may differ depending on the type of joints, these attributes include mechanical properties (elasticity, yield stress, ductility, toughness, and wear resistance) as well as biological properties (biocorrosion resistance, biocompatibility, good and longterm osseointegration). Metallic materials are most commonly used for load bearing implants. Stainless steel, cobalt alloys and titanium alloys are accordingly commercially available for bone substitutes, artificial joints and dental implants [START_REF] Brunski | Biomaterials Science: An Introduction to Materials in Medicine[END_REF][START_REF] Niinomi | Metallic biomaterials[END_REF]. Durability and functionality of the implant are governed mainly by the properties of the bulk material, whereas the biological response depends, among others, on the surface characteristics of the implantable device. Indeed, a crucial step for the implant stability is the formation of the effective interface between the bone and the implanted material. Several surface characteristics favour this osseointegration: mechanical interlocking related to surface roughness and physical characteristics, and direct chemical interactions involving bonds between the implant surface and bone tissue constituents. To promote the osseointegration of metallic implants, morphological (3D texturing, surface roughness and porosity) and chemical (ion implantation, grafting, and coating) modifications of the implant surface have been proposed [START_REF] Wang | Functional coatings or films for hard-tissue applications[END_REF][START_REF] Sun | Material fundamentals and clinical performance of plasma-sprayed hydroxyapatite coatings: A review[END_REF]. Since the mid-1980s, plasma sprayed hydroxyapatite (HA) (Ca 10 (PO 4 ) 6 (OH) 2 ) coatings appear as the most developed and successful treatments of metallic, especially titanium-based, implants for a fast integration to bone tissues [START_REF] Heimann | Thermal spraying of biomaterials[END_REF][START_REF] Paital | Calcium phosphate coatings for bioimplant applications: Materials, performance factors, and methodologies[END_REF][START_REF] Surmenev | A review of plasma-assisted methods for calcium phosphate-based coatings fabrication[END_REF][START_REF] Surmenev | Significance of calcium phosphate coatings for the enhancement of new bone osteogenesis -A review[END_REF][START_REF] Berndt | Thermal spraying for bioceramic applications[END_REF][START_REF] De Groot | Hydroxyapatite as coating for implants[END_REF][START_REF] De Groot | Plasma sprayed coatings of hydroxyapatite[END_REF][START_REF] Thomas | Hydroxyapatite coatings[END_REF][START_REF] Soballe | Hydroxyapatite coating enhances fixation of porous coated implants[END_REF]. Indeed, atmospheric plasma spraying (APS) is the most widely used commercial technique due to its process simplicity, low cost, and bulk production [START_REF] Pawlowski | The science and engineering of thermal spray coating[END_REF]. During plasma spray, the material to be deposited (feedstock), typically as powder, is introduced into the core of a plasma plume emanating from a d.c. torch. The powder is fully or partially melted, and accelerated towards the work piece, upon which it impacts and cools down to form a coating by overlapping of splats. Depending on the plasma-forming gases, electrical characteristics, feedstock attributes, and kinematics conditions, the thermal input transferred to the in-flight particles and coating, together with the cooling rate, can result in severe powder decomposition and in lack of crystallinity. Starting from HA feedstock it is noteworthy that plasma spray can induce the formation of amorphous calcium phosphate (ACP), tricalcium phosphate ( or -TCP), tetracalcium phosphate (TTCP), calcium oxide (CaO) and dehydroxylated products (oxyhydroxyapatite or oxyapatite) [START_REF] Heimann | Thermal spraying of biomaterials[END_REF][START_REF] Surmenev | A review of plasma-assisted methods for calcium phosphate-based coatings fabrication[END_REF][START_REF] Overgaard | Calcium phosphate coatings for fixation of bone implants[END_REF][START_REF] Cizek | Role of in-flight temperature and velocity of powder particles on plasma sprayed hydroxyapatite coating characteristics[END_REF][START_REF] Gross | Thermal processing of hydroxyapatite for coating production[END_REF][START_REF] Ellies | Crystallographic changes in calcium phosphates during plasma-spraying[END_REF][START_REF] Wang | Characteristics and osteoconductivity of three different plasmasprayed hydroxyapatite coatings[END_REF][START_REF] Mcpherson | Structural characterization of plasma-sprayed hydroxylapatite coatings[END_REF][START_REF] Yang | The influence of plasma spraying parameters on the characteristics of hydroxyapatite coatings: A quantitative study[END_REF]. The decomposition sequence occurs along the powder trajectory in the plasma jet that is determined mainly by the particles' size, density and their dwell-time in conjunction with many parameters, for example the viscosity and density of the plasma. The degree of crystallinity of plasma sprayed HA coatings is usually between 65 and 85 vol. % (in some operating conditions it may reach 93 vol. %) [START_REF] Surmenev | A review of plasma-assisted methods for calcium phosphate-based coatings fabrication[END_REF][START_REF] Overgaard | Calcium phosphate coatings for fixation of bone implants[END_REF][START_REF] Dalton | In vivo mechanical and histological characteristics of HA-coated implants vary with vendor[END_REF][START_REF] Roy | Induction plasma sprayed nano hydroxyapatite coatings on titanium for orthopaedic and dental implants[END_REF][START_REF] Khor | Process-phase-properties relationship in radio frequency (RF) plasma synthesized hydroxyapatite (HA)[END_REF][START_REF] Gross | Amorphous phase formation in plasma-sprayed hydroxyapatite coatings[END_REF][START_REF] Legeros | Variations in composition and crystallinity of hydroxyapatite (Ha) preparations[END_REF]. Crystalline HA is seen as a crucial factor for long-term coating stability in vivo. Although the presence of relatively soluble non-apatitic calcium phosphate salts such as ACP, TCP, and TTCP has been found to accelerate the bone attachment, these phases are related to long-term non-uniform coating degradations that adversely affect the prosthetic stability of the implants [START_REF] Sun | Material fundamentals and clinical performance of plasma-sprayed hydroxyapatite coatings: A review[END_REF][START_REF] De Bruijn | Structural arrangements at the interface between plasma sprayed calcium phosphates and bone[END_REF][START_REF] Klein | Biodegradation behavior of various calcium phosphate materials in bone tissue[END_REF][START_REF] Xue | Effect of hydroxyapatite coatings crystallinity on dissolution and osseointegration in vivo[END_REF][START_REF] Overgaard | The influence of crystallinity of the hydroxyapatite coating on the fixation of implants[END_REF][START_REF] Bonfante | Physicochemical characterization and in vivo evaluation of amorphous and partially crystalline calcium phosphate coatings fabricated on Ti-6Al-4V implants by the plasma spray method[END_REF][START_REF] Wang | Early bone apposition in vivo on plasma-sprayed and electrochemically deposited hydroxyapatite coatings on titanium alloy[END_REF][START_REF] Bidar | Long-term results of the ABG-1 hydroxyapatite coated total hip arthroplasty: Analysis of 111 cases with a minimum follow-up of 10 years[END_REF][START_REF] Macdonald | Physicochemical study of plasma-sprayed hydroxyapatite-coated implants in humans[END_REF]. To limit the amount of easily soluble CaP phases, the steep temperature gradients during plasma spraying have to be reduced by increasing the substrate temperature and controlling the cooling rate and/or by limiting the thermal transfer. One method to control the degree of melting of the powder could be achieved by micro plasma spray (MPS, also called MIPS) or low-energy plasma spray (LEPS) techniques [START_REF] Junker | Bone-supportive behavior of micro-plasma-sprayed CaP-coated implants: mechanical and histological outcome in the goat[END_REF][START_REF] Junker | Bone reaction adjacent to micro-plasma-sprayed CaP-coated oral implants subjected to occlusal load, an experimental study in the dog. Part I: short-term results[END_REF][START_REF] Junker | Loaded microplasma-sprayed CaP-coated implants in vivo[END_REF][START_REF] Dey | Characterization of microplasma sprayed hydroxyapatite coating[END_REF][START_REF] Dey | Development of hydroxyapatite coating by microplasma spraying[END_REF][START_REF] Dey | In vitro dissolution, microstructural and mechanical characterizations of microplasma-sprayed hydroxyapatite coating[END_REF][START_REF] Dey | Physico-chemical and mechanical characterization of bioactive ceramic coating[END_REF][START_REF] Garcia-Alonso | Low-energy plasma spray (LEPS) deposition of hydroxyapatite/poly-ecaprolactone biocomposite coatings[END_REF]. Due to the low heat input of these plasma torches, overheating of the powder particles as well as excessive local overheating of the coating can be avoided. LEPS also allows the production of highly porous (around 20 vol. %)-thick coatings (about 250 µm) without accumulation of residual stresses (tensile stresses are detrimental because they promote cracks and accelerate dissolution) and without negatively affecting the mechanical properties (fracture toughness of about 0.6 MPa.m ½ ). In a previous work, we have implemented such an innovative low energy plasma spray system that has resulted in producing thin dense HA coatings with a good degree of crystallinity (around 70%) and improved purity, compared to conventional high-energy APS coatings, as demonstrated by Raman studies [START_REF] Demnati | Hydroxyapatite coating on titanium by a low energy plasma spraying mini-gun[END_REF]. Another strategy to overcome the decomposition of HA and thus limit the presence of soluble CaP phases is to use calcium apatites with higher thermal stability such as fluorapatite (FA) or chlorapatite (ClA) [START_REF] Tonsuaadu | A review on the thermal stability of calcium apatites[END_REF][START_REF] Shepherd | Substituted hydroxyapatites for bone repair[END_REF][START_REF] Lindahl | Incorporation of active ions into calcium phosphate coatings, their release behavior and mechanism[END_REF][START_REF] Overgaard | Resorption of hydroxyapatite and fluorapatite ceramic coatings on weight-bearing implants: A quantitative and morphological study in dogs[END_REF][START_REF] Demnati | A comparative physico-chemical study of chlorapatite and hydroxyapatite: from powders to plasma sprayed thin coatings[END_REF]. Indeed, it has already been demonstrated that both materials lead to coatings with superior adhesive strength to the substrate as well as without significant thermal degradation. Fluorapatite and fluorhydroxyapatite (FHA) as raw material or coatings have been extensively studied [START_REF] Shepherd | Substituted hydroxyapatites for bone repair[END_REF][START_REF] Lindahl | Incorporation of active ions into calcium phosphate coatings, their release behavior and mechanism[END_REF][START_REF] Overgaard | Resorption of hydroxyapatite and fluorapatite ceramic coatings on weight-bearing implants: A quantitative and morphological study in dogs[END_REF]. In vitro tests have demonstrated the ability of such material to favour the osteoblast attachment, to support the cytocompatibility, to exhibit an antibacterial effect, and to stimulate cell proliferation and differentiation. In weight-bearing model (in vivo), no statistical difference in loss of ceramic coating was found between HA and FA coatings and durable implant fixation was always concluded.

The objective of the current study is to compare chlorapatite and hydroxyapatite coatings obtained by a LEPS gun. Until today only a few authors have worked on the substitution of chloride ion in hydroxyapatite, and no commercial product is available. Hence, our first aim was to develop a simple and fast reaction synthesis transposable at an industrial scale. A method is first described to prepare easily chlorapatite from commercial HA preserving similar powders characteristics. The main physical-chemical and mechanical properties of the coatings obtained with a low energy plasma spraying process are compared using standardized procedures and specific techniques such as Raman microscopy mapping of the main mineral phases found in the coating and three points mechanical bending with stiffener.

POWDERS PREPARATION AND CHARACTERISTICS

The hydroxyapatite used in this study was a commercial product (Teknimed, L'Union, France). The XRD analyses, according to ISO standard (ISO 13779-3: 2008), did not detect any crystalline impurity (Fig. 1) and the FTIR spectrum (Fig. 2) was the same as that of stoichiometric hydroxyapatite.

Regarding the synthesis of chlorapatite, several methods have been proposed [START_REF] Hoekstra | The chemistry and luminescence of antimonycontaining calcium chlorapatite[END_REF][START_REF] Garcia-Tunon | Synthesis and characterization of large chlorapatite single-crystals with controlled morphology and surface roughness[END_REF]. Precipitation methods in aqueous media do not lead generally to stoichiometric chlorapatite and the apatite obtained contains always hydroxide ions. Several dry synthesis methods giving stoichiometric ClA with a high purity have been proposed. The reaction of CaCl 2 on Ca 3 (PO 4 ) 2 (800-1100°C) [START_REF] Garcia-Tunon | Synthesis and characterization of large chlorapatite single-crystals with controlled morphology and surface roughness[END_REF] results in ClA with a high yield, especially when using an excess of CaCl 2 , which can be removed by washing. Other methods can be used, especially Cl-OH ion exchange reactions on HA (800-1100°C), which offers the advantage of using an existing commercial product. Several chlorinating agents can be used for this exchange reaction, such as CaCl 2 :

CaCl 2 + Ca 10 (PO 4 ) 6 (OH) 2 → Ca 10 (PO 4 ) 6 Cl 2 + CaO + H 2 O [START_REF] Brunski | Biomaterials Science: An Introduction to Materials in Medicine[END_REF] This reaction leads to the formation of CaO, which is difficult to remove. Other chlorinating agents could be used and more interestingly, chlorinating gases, such as HCl, which would not produce any residues except water. This reaction has been studied by Hoekstra [START_REF] Hoekstra | The chemistry and luminescence of antimonycontaining calcium chlorapatite[END_REF] at about 1050°C and has been shown to lead to formation of stoichiometric ClA:

Ca 10 (PO 4 ) 6 (OH) 2 + HCl ↔ Ca 10 (PO 4 ) 6 Cl 2 + H 2 O ( 2
)
This reaction is supposed not to alter the powder characteristics (e.g. grain size, shape). One of the difficulties in such a reaction is to handle hydrogen chloride gas, and, to overcome this difficulty, we choose as a precursor of HCl, NH [START_REF] Sun | Material fundamentals and clinical performance of plasma-sprayed hydroxyapatite coatings: A review[END_REF] Cl which decomposes at about 340°C under atmospheric pressure, into NH 3 and HCl. In such a synthesis we should keep in mind, that non-stoichiometry has been reported to occur in ClA heated at high temperature, which has been assigned to the release of CaCl 2 [START_REF] Prener | Nonstoichiometry in calcium chlorapatite[END_REF]

Previous attempts to use the reaction of HA with NH 4 Cl by A. Hoekstra [START_REF] Hoekstra | The chemistry and luminescence of antimonycontaining calcium chlorapatite[END_REF] have probably failed to produce stoichiometric ClA because the treatment temperature (1180°C) was too high. The second point to consider is the fast hydrolysis of ClA into HA when traces of water molecules are present. Thus, we shall use a temperature as low as possible, resulting in a reasonable reaction rate and enabling to preserve a dry atmosphere.

The ion exchange reaction was performed in a tubular oven under a flux of dry nitrogen enriched with sublimated NH 4 Cl in large excess placed in a zone of the tubular oven at about 400°C [START_REF] Demnati | Developpement et characterisation de revêtements bioactifs d'apatite obtenus par projection plasma à basse énergie : application aux implants biomédicaux[END_REF]. The unreacted HCl in the gas flow recombines with NH 3 and regenerates NH 4 Cl, which crystallizes at the outlet of the oven and could possibly be recycled. The reaction was fast at 950°C and a few grams of HA were transformed into ClA in about 15 minutes. However, for production treatments of the order of several hundreds of grams on partly sintered HA grains of industrial HA the treatment time was two hours [START_REF] Demnati | Developpement et characterisation de revêtements bioactifs d'apatite obtenus par projection plasma à basse énergie : application aux implants biomédicaux[END_REF].

The XRD analyses (Fig. 1) of the samples obtained indicated the formation of chlorapatite in its monoclinic form, which is the stable low temperature isomorph. The monoclinic ClA has been shown to correspond to an order among the Cl -ions of the chlorapatite [START_REF] Mackie | Monoclinic structure of synthetic Ca5(PO4)3Cl, chlorapatite[END_REF]. The tiny peaks of the superstructure related to this ordering can be distinctly seen on the XRD diagram Fig. [START_REF] Brunski | Biomaterials Science: An Introduction to Materials in Medicine[END_REF]. The transition temperature to the hexagonal structure of ClA occurred as expected at about 200°C. The unit cell dimensions (Table 5) appeared consistent with that of a stoichiometric chlorapatite. Theoretically stoichiometric HA should also exist as a monoclinic apatite at room temperature. However, most industrial HA crystallize in the hexagonal systems. This anomaly is often attributed to the existence of defects and impurities, which stabilizes the hexagonal form of HA. On the other hand, chlorinating treatments have been shown to remove several impurities present in the apatite structure as volatile chlorides and this reaction has been proposed for the extraction of transition elements, rare earths and uranides from apatitic ores [START_REF] Montel | Progrès dans le domaine de la chimie des composés phosphorés solides à structure d'apatite. Application à la biologie et au traitement des minerais[END_REF]. Although we did not determine the effect of the chlorination treatment on the level of impurities, this possibility could be an interesting additional advantage of the chlorination treatment.

FTIR data (Fig. 2) confirm the removal of the OH -ions, the stretching (3570 cm -1 ) and librational (633 cm -1 ) lines of which cannot be seen on the spectra after chlorination. The lines characteristic of OH-Cl interactions in solid-solutions of HA and ClA, at 3495, 3510 cm -1 [START_REF] Maiti | Incorporation of chlorine into hydroxyapatite[END_REF] cannot be observed either, suggesting a complete replacement of OH -ions by Cl - ions in the apatite structure. The morphology of the HA and ClA powders are shown in Fig. [START_REF] Wang | Functional coatings or films for hard-tissue applications[END_REF]. Their main characteristics (Ca/P, particle size, specific surface area, flowability) are reported in Table 1.

The grains in both samples show irregular shapes related to the preparation of HA granules by grinding of sintered particles and subsequent sieving. The surface irregularities can be seen on ClA grains, which were not apparent on HA ones. Granulometry (Malvern Matersizer 2000) revealed a similar grain distribution for both HA and ClA powders. The grain sizes extended from 0.7 to 140 µm with a main distribution peak between 50 and 80 µm and smaller ones between 0.7-3µm and 3-20 µm [START_REF] Demnati | Developpement et characterisation de revêtements bioactifs d'apatite obtenus par projection plasma à basse énergie : application aux implants biomédicaux[END_REF]. On chlorination, the median grain size d 0.5 has decreased, presumably due to fragmentation of the initial HA grain during the chlorination process. The flowability was determined, according to the recommendations of the European Pharmacopoeia, by measuring the flow time of 100 g of powder through a funnel with specified dimensions [START_REF] Demnati | Developpement et characterisation de revêtements bioactifs d'apatite obtenus par projection plasma à basse énergie : application aux implants biomédicaux[END_REF]. The flowability of the powders appeared significantly different and this observation might be possibly related to a change in surface characteristics as observed by SEM, although this remains difficult to elucidate.

In addition, Rietveld analyses of XRD data reveal that the chlorination treatment induced, unexpectedly, a decrease in the average size of the ClA crystals compared to the industrial HA crystals in the starting powder (Tables 4 and5).

Comparison of the phase diagrams, CaO-P 2 O 5 for HA and CaCl 2 -Ca 3 (PO 4 ) 2 for ClA [START_REF] Garcia-Tunon | Synthesis and characterization of large chlorapatite single-crystals with controlled morphology and surface roughness[END_REF][START_REF] Elliott | Structure and chemistry of the apatites and other calcium orthophosphates[END_REF] indicate a decomposition of HA on heating, whereby ClA melts without decomposition. This property is a major argument in favour of the use of ClA for producing plasma sprayed coatings: the formation of foreign phases is expected to be limited, except for the amorphous phase resulting of the quenching of a melted phase.

The comparison of the solubility of HA and ClA, which could determine possible differences in bioresorption of the coatings, is also interesting. However, published data on ClA solubility are scarce and provide incoherent values. Thus ClA has been reported to be less soluble than HA [START_REF] Narasaraju | Determination of solubility products of hydroxyapatite, chlorapatite, and their solid solutions[END_REF], although thermodynamic data strongly suggest that it should be more soluble than HA with a solubility product pK sp (ClA) = 108 at 25°C (pK sp (HA) = 116.8 for Ca 10 (PO 4 ) 6 (OH) 2 ) [START_REF] Valyashko | Stability of fluorapatite, chlorapatite, and hydroxyapatite in aqueous solutions at different temperatures[END_REF]. However, considering the concentration of Cl -ions in body fluids of more than 10 5 times higher than that of OH -in serum at physiological pH, the effect of this difference would be minor. It should be noticed that the incorporation of chloride ions in biological apatites, especially enamel, has been confirmed [START_REF] Dykes | The occurrence of chloride ions in the apatite lattice of Holly Springs hydroxyapatite and dental enamel[END_REF].

COATINGS DEPOSITION

Thermal Spraying

The powders (HA and ClA) were sprayed in air on surgical grade (grade 4) Ti disks (12 mm in diameter and 3 mm in thickness), using a low energy plasma spray (LEPS) device [START_REF] Demnati | Hydroxyapatite coating on titanium by a low energy plasma spraying mini-gun[END_REF]. This system was an experimental low energy plasma spray system designed and developed by Inasmet-Tecnalia (Spain) from commercial Sulzer Metco torch (Switzerland). Ti substrates were previously degreased and then sandblasted with corundum F150 using a pressure of 2 bars to produce a surface roughness (Ra) of about 1 µm. Coupons were separately sprayed with chlorapatite or hydroxyapatite under the same thermal spraying conditions (Table 2). For practical reasons, the gun operated with standard electrodes. The entire gun was cooled with water and a glass fibre-reinforced epoxy housing separated the electrical poles. A more detailed description of the LEPS system has been provided elsewhere [START_REF] Barykin | Thermal spraying method and device[END_REF]. Coatings with various thicknesses were obtained simply by changing the number of torch runs (1, 6, 12 and 20 runs) for both HA and ClA powders. We can see on Table 3 that the coating thickness depended not only on the number of runs, but also on the nature of the apatite: whatever the number of spray runs, the HA-coatings were always thicker than the ClAcoatings. This observation can be related to the decrease of the median grain size associated with the chlorination treatment, and possibly to a certain additional fragility of the grains, although other causes could be envisioned such as the viscosity of the melted phase for example, which is supposed to be totally liquid in the case of ClA whereas it may contain solid CaO in the case of HA.

PHYSICAL-CHEMICAL CHARACTERISTICS OF THE COATINGS Structural and Chemical Composition of the Coatings

Surface and bulk composition of HA and ClA coatings were determined. The coating surface was analysed directly on the as-sprayed coatings. For the bulk characterizations, the coatings were scrapped and ground as indicated in the ISO standard (13779-3: 2008). Investigations on coating microstructure were also performed on sample cross-sections prepared as follows: the coupons were cut with a diamond disc (Struers) and then the half-coupon was embedded into an epoxy resin under vacuum. The embedded cross-sections were polished using a silicon carbide paper (1000 and 1200 grade SiC paper). Further polishing was conducted using 2400 then 4000 grade SiC paper to obtain a flat surface suitable for analysis.

X-ray Diffraction Analysis of the Coatings

Standards have been published concerning the coating's chemical composition (Ca/P ratio), phase composition, and crystallinity (ISO 13779-2 and 3:2008). These standards are based essentially on X-ray diffraction (XRD) analyses.

X-ray diffraction studies on both powders and coatings were carried out in order to identify the composition of the coating (obtained after 12 spray runs) and determine its crystallinity, according to the ISO 13779-3: 2008 standard. A Seifert XRD-3000 diffractometer (Germany) with a CuK  radiation over a 2 range of 20-65° was used. Patterns were collected at a step increment of 0.02°, with a collection time of 24 seconds at each step. According to the standard, the HA-coatings crystallinity (Cr) was defined as the ratio between the sum of integrated intensities of ten diffraction lines (A hkl ) of the coating diagram, and of the same ten lines of a fully crystalline HA reference powder (A 0 hkl ): Cr = (A hkl )/ (A 0 hkl ). The same approach was used to determine the ClA coating crystallinity. However, the (102) line used in crystallinity determination of HA is weak in ClA and it has a minor contribution in the crystallinity index; thus it was replaced by (201) line. Both coatings diagrams can be indexed in the hexagonal system indicating that a change of the structure of sprayed ClA powder has occurred. As we will see later, this structural modification is related to the alteration of the ClA crystals in the coatings and the formation of oxyapatite (OA) as well.

For both coatings, we noticed an increase of the full width at half maximum (FWHM) value of the diffraction lines compared to that of the initial powders (Figs. 4 and5), which is generally assigned to two main phenomena: the decrease of the crystallite size, and the increase of the microstrain. In order to evaluate these contributions, Rietveld refinement analyses were performed using MAUD software. The crystallographic parameters determined for the HA and ClA initial powders and for the 12-run HA-and ClA-coatings are reported in Table 4 andTable 5, respectively. For HA-and ClA-coatings, we can notice both a significant increase in micro-strains and a decrease in crystallite size. However, these observations and the related calculations neglect an important cause of line broadening associated with the formation of heterogeneous solid solutions of oxyhydroxyapatite and oxy-chlorapatite, which would result in broadened diffraction lines, corresponding to the sum of unresolved diffraction peaks of individual crystalline domains considered as homogeneous. For the OClA this effect would result in a higher broadening of diffraction lines than for OHA, due to larger differences in unit-cell dimensions between OA and ClA [START_REF] Barykin | Thermal spraying method and device[END_REF]. The average unit-cell dimensions for ClA coatings can be compared to that of the powder considering that b=2a in the monoclinic ClA powder. The formation of solid solutions (OHA or OClA) can be related to a clear decrease of the a unit-cell dimensions noticed on the coatings. This effect appears The calculated crystallinity ratio, determined according to the standard, was equal to 68% for HA-coating and to 98% for ClA-coating. The ClA-coating showed a much higher crystallinity ratio than HA-coating, although it was not determined in the same way. This observation should not be only related to the amount of amorphous phase formed in the coatings. Other causes can produce crystallinity variations such as fast recrystallization of small apatite crystals from the melt or the amorphous phase, or the formation of solid solutions with OA.

The Ca/P ratio of HA-and ClA-coatings obtained after 12 spray runs was determined according to the ISO 13779-3: 2008 standard on a scraped and ground coating, which had been heated at 1000°C during 15 h. The XRD patterns showed only apatite crystals for both HA-and ClA-coatings, indicating that plasma spraying using the LEPS system did not appear to alter the stoichiometry of the apatite.

Vibrational Spectroscopy Analysis of the Coatings.

Several additional characteristics of the apatite coatings are more difficult to determine using X-ray diffraction analyses: the main phase proportions and their distribution within the coating, and, more importantly, the amount of amorphous phase, which is not specifically considered in the ISO standard, although it could determine the coating behaviour in vitro and in vivo. A more accurate characterization of HA and ClA powders and the related plasma sprayed apatite coatings can be proposed, using vibrational spectroscopies [START_REF] Trombe | Some features of the incorporation of oxygen in different oxidation states in the apatite lattice[END_REF], Fourier transform infrared spectroscopy and Raman spectroscopy, with a special focus on Raman micro-spectroscopy imaging on HA-and ClAcoatings.

FTIR Spectroscopy Characterization

Both initial HA and ClA powders and scraped HA-and ClA-coatings were analysed by transmission Fourier transform infrared (FTIR) spectroscopy (Nicolet 5700 spectrometer, ThermoElectron) using KBr pellets. We examined especially the 1100-950 cm -1 ( 1 PO 4 and  3 PO 4 ) and 600-450 cm -1 ( 4 PO 4 ) domains, corresponding to the different phosphate vibration bands. In these domains, FTIR spectra of both HA-and ClA-coatings Figs. (6 and 7) showed broader and less resolved bands than that of the initial powders. This phenomenon can be related to a decrease in crystallinity. The enhanced relative intensity of bands observed in the  2 PO 4 domain (434 and 471 cm -1 ), in the spectrum of the HA coating, can be correlated to the splitting of the  1 PO 4 band and the decrease in the relative intensity of the OH bands at 3572 and 630 cm -1 (Fig. 6) which have been shown to correspond to the formation of oxyapatite [START_REF] Trombe | Some features of the incorporation of oxygen in different oxidation states in the apatite lattice[END_REF]. Although less visible than in HA, the formation of oxyapatite is also apparent on FTIR spectra of ClA coatings (Fig. 7) which exhibit lines with a higher intensity in the  2 PO 4 domain than the starting powder.

Fig. (6). FTIR spectra of HA powder and coating.

Raman Spectroscopy Characterization

All phases detected in a plasma-sprayed coating of HA can theoretically be distinguished by Raman spectroscopy, especially the most important ones, in some instances with better sensitivity than according to XRD analyses [START_REF] Demnati | Developpement et characterisation de revêtements bioactifs d'apatite obtenus par projection plasma à basse énergie : application aux implants biomédicaux[END_REF]. In addition, Raman scattering imaging allows local investigation of the coating's heterogeneity [START_REF] Demnati | Developpement et characterisation de revêtements bioactifs d'apatite obtenus par projection plasma à basse énergie : application aux implants biomédicaux[END_REF][START_REF] Rey | Characterization of calcium phosphates using vibrational spectroscopies[END_REF]. The  1 PO 4 domain, with strong thin lines, seems well adapted for imaging [START_REF] Paital | Calcium phosphate coatings for bioimplant applications: Materials, performance factors, and methodologies[END_REF][START_REF] Demnati | Plasma sprayed apatite coatings: review of physical-chemical aspects and their biological consequences[END_REF]. The amorphous calcium phosphate (ACP) especially is characterized by a broad Raman line around 950 cm -1 and all crystalline phases show lines distinct from that of apatite at 961 cm -1 . The xyapatite (OA), rather difficult to distinguish by other methods, exhibits very specific Raman scattering lines at about 951 cm -1 and 966 cm -1 [START_REF] Trombe | Contribution à l'étude de la décomposition et de la réactivité de certaines apatites hydroxylées, carbonatées ou fluorées alcalino-terreuses[END_REF]. The only phase that cannot be detected in this domain is CaO.

Fig. (7). FTIR spectra of ClA powder and coating.

Both initial HA and ClA powders and the as-prepared HA-and ClA-coatings were analysed by Raman spectroscopy in the range of 100-1200 cm -1 and 3500-3600 cm -1 (Horiba Jobin-Yvon HR 800 spectrometer equipped with a helium-neon laser: = 632.8 nm). The quantitative evaluation of the components of the  1 PO 4 band was performed using LabSpec 5 software (Horiba). The  1 PO 4 band Raman spectra were decomposed according to the same protocol for all the samples in order to facilitate their comparison [START_REF] Demnati | Developpement et characterisation de revêtements bioactifs d'apatite obtenus par projection plasma à basse énergie : application aux implants biomédicaux[END_REF]: a baseline correction was performed on each spectrum. Peak positions and some curve-fitting parameters were recorded and used as initial input in the curve-fitting program. Iterations continued until the best fit was obtained. 8) and ( 9) together with those of the initial powders. The most intense Raman line,  1 PO 4 , appeared the most altered after spraying. The integrated intensity of the different contributing sub-bands has been determined to estimate quantitatively the changes induced by spraying. The resulting curve-fittings are presented in Figs. [START_REF] De Groot | Hydroxyapatite as coating for implants[END_REF] and [START_REF] De Groot | Plasma sprayed coatings of hydroxyapatite[END_REF] for HA-and ClA-coatings respectively. The detection of oxyapatite phase in both coatings confirmed the results obtained by FTIR spectroscopy; however, the proportion of OA appears always significantly less important in ClA than in HA coatings. The same observation is true for the amorphous phase. An accurate mapping of the different phases in apatite coatings can be obtained by micro-Raman imaging involving curve fitting in the  1 PO 4 domain, extraction of the characteristic line(s) associated with specific phases, and of their relative intensities [START_REF] Demnati | Developpement et characterisation de revêtements bioactifs d'apatite obtenus par projection plasma à basse énergie : application aux implants biomédicaux[END_REF]. It should be noticed that at this stage, without accurate standardization, and due to uncertainties in decomposition results. The data do not represent the real content of phases present in the coating but offer a base for visualizing the phase distribution in a first semi-quantitative approach. Figs. ( 12) and ( 13) show the main constituents distributions, i.e. HA or ClA, OA and ACP, in the cross section of HA and ClA coatings. The heterogeneity in the distribution of the constituents is immediately apparent. Domains with a high content in HA appear to be embedded in an amorphous phase acting as a binder. These observations are consistent with other descriptions of phenomenon occurring during plasma spraying [START_REF] Sun | Material fundamentals and clinical performance of plasma-sprayed hydroxyapatite coatings: A review[END_REF][START_REF] Heimann | Thermal spraying of biomaterials[END_REF][START_REF] Demnati | Plasma sprayed apatite coatings: review of physical-chemical aspects and their biological consequences[END_REF][START_REF] Berndt | A review of hydroxyapatite coatings manufactured by thermal spray" Chapter 9 in Advances in Calcium Phosphate Biomaterials[END_REF]. The nodules of HA, which appear as well delimited zones, are believed to correspond to the unmelt cores of the splats. In some cases smaller HA domains grouped in a zone may appear, corresponding possibly to the crash and dispersion into smaller particles of the solid core of splats impacting the substrate or to recrystallization of HA from the melted or amorphous phase. The OA rich domains are distributed more uniformly than the HA nodules and do not form well-defined zones with dominant concentrations. This distribution could result of a recrystallization process in the melted part of the splats, possibly associated with a direct formation of OA in the unmelt cores.

For ClA coatings the layered structure of the coating related to the splat accumulation is more apparent than in HA coatings, especially in the ACP mapping. The ClA domains appear elongated and rather diffuse, indicating a better spreading of the splats; they are surrounded by the amorphous phase. The OA seems more homogeneously distributed at the surface of ClA domains or within amorphous domains; in these domains, zones very poor in ClA and rich in OA can be clearly observed, suggesting that AO forms mainly in a recrystallization processes of the molten phase or the amorphous phase.

Therefore, Raman micro-spectroscopy can be used as a routine non-destructive tool to obtain rapid analysis of the composition of apatite coatings and even of different domains of a splat. This technique could be adapted for the chemical characterization of the coating directly on biomedical implants. Image analyses could then be used to describe the coating characteristics with accuracy.

Surface Roughness

The average roughness of the ClA and HA coatings was determined by optical interferometry (Zygo) as the average of four measurements. The surface arithmetic roughness of HA and ClA coatings (Ra) was determined as a function of the number of spray runs Fig. [START_REF] Pawlowski | The science and engineering of thermal spray coating[END_REF]. Ra increases strongly after the first run and reaches rapidly a constant value depending on the nature of the apatite: it is always higher in HA coatings than in ClA coatings. The increase of Ra is commonly observed and has been related to the structure of splats generally smaller at the fringe and larger in the centre [START_REF] Li | Thermal sprayed hydroxyapatite splats: nanostructures, pore formation mechanism and TEM characterization[END_REF]. These data indicate that HA splats are globally thicker than ClA ones, and seem consistent with the variations of the coating thickness according to the number of runs. The difference between HA and ClA coatings could be due, in part, to the alteration, already mentioned, of the particle size related to the chlorinating treatment.

The coating surfaces and cross-sections were studied by scanning electron microscopy (SEM). The coating crosssections were carbon coated, then SEM micrographs were recorded in the secondary electron mode using a LEO 435 VP microscope. The typical surface morphologies of the assprayed HA-and ClA-coatings (12 spray runs) are presented as supportive materials.

MECHANICAL PROPERTIES

Several methods can be used to determine the adherence of a coating to its substrate. Concerning calcium phosphate coatings, an adherence measurement test was standardized (ASTM C633-01, ISO 13779-4) and the results obtained according to this procedure will be first presented. Among other methods used, the three point bending with a stiffener has also been applied and will be discussed.

Fig. (12).

Raman spectroscopy mappings of HA, OA and ACP in hydroxyapatite coatings. Light blue oval: domain rich in HA, poor in ACP and with a little OA, tentatively associated with an unmelt core of a splat. White circle: domain rich in OA, poor in HA, with some ACP, attributed to the recrystallization of a molten domain or recrystallization of solid ACP phase.

Fig. (13).

Raman spectroscopy mappings of ClA, OA and ACP in chlorapatite coatings. Light blue oval: domain rich in ClA, poor in ACP and OA, tentatively associated with an unmelt core of a splat. White circle: domain rich in OA, poor in HA, within two ACP layers, attributed to the recrystallization of a molten domain or of solid ACP phase. The layered structure of the splats is clearly visible on ACP maps (the top surface is corrupted by the spectra of the embedding resin).

Fig. (14).

Surface roughness of the coatings vs. number of sprayruns (0: initial roughness of Ti plate). Test samples were made on 25 mm disks coated with HA or ClA (12 spray runs). These coated disks were glued to a dummy cylinder with the same diameter with FM1000 glue, (Cytec, NJ, USA) applied at 200°C for 90 to 100 minutes. The ultimate tensile strength of the assembly was determined on an Instron instrument, without pre-loading at a traction rate of 1mm/minute. The data obtained (mean of six measurements) are presented in Table 6. There was no significant difference between ClA and HA coatings. The failure appeared mainly cohesive for HA coupons (it occurred at the interface between the coating and the substrate), and it was mixed cohesive (at 80%) and adhesive for ClA coupons. The ultimate loads observed are in the lower range of those reported for HA coatings obtained by plasma spraying [START_REF] Berndt | A review of hydroxyapatite coatings manufactured by thermal spray" Chapter 9 in Advances in Calcium Phosphate Biomaterials[END_REF]. The main drawback of this basic standardized test is the diffusion of the glue in the porous coating and its limitation to rather thick coatings (above 0.38 mm). Other measurements were performed especially the three point bending with stiffener.

Three Point Bending Test with Stiffener

This test was developed particularly by A. Roche et al. [START_REF] Roche | A three-point flexure test configuration for improved sensitivity to metal/ adhesive interfacial phenomena[END_REF] and lead to the ISO 14679-1997 standard. It is designed to determine adhesive or cohesive properties of paints, varnish and others coatings. A stiffener epoxy resin (Araldite ® AW134B with HV997 hardener) is cured onto the coated substrate in accordance with the standard to generate a stress concentration at the interface substrate-deposit. Such loading is known to produce mainly shear stress at the substrate/coating interface and appears more consistent with the surgical use of coated prosthesis. The determination of the critical shear stress of the interface requires, without considerations to the actual geometry of the sample, to know the mechanical properties of all the materials implied in the elaboration of the samples: substrate, coating, and bulk stiffener Young's modulus, as well as the thickness and elastic properties of a possible interphase between the coating and the stiffener. Once these parameters are determined, critical shear stress and critical strain energy release rate of the substrate coating interface can be determined from experimentally measured critical load, thanks to analytical and/or numerical models [START_REF] Roche | The Role of the Residual Stresses of the Epoxy-Aluminum Interphase on the Interfacial Fracture Toughness[END_REF]. This complexity led us, as many authors [START_REF] Coulaud | Role des interfaces et interphases dans les assemblages collés[END_REF][START_REF] Aufray | Residual stresses and practical adhesion: effect of organo-metallic complex formation and crystallization[END_REF] to consider this practical adhesion test as a comparative test, considering critical load F max an indicator of adhesion.

The tests were performed on corundum-blasted titanium coupons plasma-sprayed at 12 spray runs. The stiffener was casted in a silicon mould, directly on coated samples. The mechanical test apparatus used in this experiment consists in an electromechanical Instron test machine fitted with a load cell capacity of 500 N. The crosshead speed is set to 0.5 mm. min -1 . A schematic illustration of the testing conditions is given as supportive material. Ten samples were successfully tested for ClA coatings, only eight for HA and two of them had the rupture of the stiffener itself. Load versus displacement curves are displayed on Fig. [START_REF] Overgaard | Calcium phosphate coatings for fixation of bone implants[END_REF]. To check the quality of the stiffener, we measured the stiffness of the sample. It was found to be 319 ± 16.9 N.mm -1 for HA coatings and 305 ± 9.83 N.mm -1 for CIA coatings. These preliminary results confirmed the good quality of the polymeric stiffener.

An abrupt decrease in load-displacement behaviour was observed beyond F max . The maximum load F max achieved before rupture of the substrate-coating interface, thanks to the geometrical effect generated by the stiffener, is the indicator of the practical adhesion. For HA, F max = 165 ± 30.7 N (18.6 %), and for ClA, F max = 128 ± 7.92 N (6.16 %). Coatings exhibited mixed modes of failure (cohesive and adhesive). By image analysis on Energy-dispersive X-ray spectroscopy (EDX) maps, adhesive failure was estimated to be closed to 65% for both HA and ClA deposits. (EDX maps are given as supportive material)

DISCUSSION

The data reveal some differences between HA and ClA coatings obtained in similar deposition conditions with powders showing slightly different characteristics: -ACP content is also lower in ClA coatings compared to HA ones.

-Thickness of the coatings for the same number of spray runs is higher for HAp than for ClAp; -Surface roughness is higher in HA than in ClA -Mechanical properties appear also slightly different, showing lower strength in ClA than in HA coatings in a three point bending test with stiffener, although adherence tests according to the ISO standard 13779-4 did not revealed significant differences.

HA and ClA Powder Characteristics

The different physical characteristics of the HA and ClA coatings, for identical spray conditions, could be related to slight changes in the powder characteristics and specific differences in the apatites involved. The chlorination treatment was expected to preserve the crystal size and grain characteristics. Actually, the reverse treatment (the hydroxylation of ClA) on monocrystals allowed the preparation of HA monocrystals and its crystal structure determination [START_REF] Elliott | Conversion of single crystals of chlorapatite into single crystals of hydroxyapatite[END_REF], this ability of high temperature ion exchange reactions to preserve the crystalline structure having been recently confirmed [START_REF] Garcia-Tunon | Synthesis and optimization of the production of millimeter-sized hydroxyapatite single crystals by Cl -OHion exchange[END_REF]. This is not observed in our chlorination experiments and a significant decrease of the crystal size of the initial HA was observed during its transformation into ClA. The role of the change in unit-cell volume between HA and ClA could be involved, although it represents only 3.6 % of the total volume. It shall be noticed, however, that the change in unit cell dimensions between HA and ClA involves a decrease of the c dimension and an increase of the a dimension. These opposite variations in partly sintered grains could lead to a rupture of the initial crystals and eventually of the grains. The reduction of roughness in ClA coatings compared to HA ones, even for one spray run, indicates a better spreading of the splats, possibly due to a lesser amount of matter and/or higher content of liquid phase and/or a lower viscosity. The slightly lower melting point of ClA (1530°C) compared to that of HA (1570°C), according to phase diagrams, together with a smaller grain and crystal size, could facilitate its melting within sprayed grains, although other considerations on heat transfers in the plasma flame have also to be taken into account [START_REF] Dyshlovenko | Modelling of plasma particle interactions and coating growth for plasma spraying of hydroxyapatite[END_REF].

Oxyapatite Content

The strong differences between Raman spectra of HA and OA have been assigned to variations of the local environments of phosphate ions, i.e. OH -ions in HA and lattice vacancies or O 2-ions in OA. Involving a, correlative alterations of the vibrational energy level of the phosphate ion in the apatite lattice. The formation of OA in HA coatings is generally attributed to the dehydration of HA during heating. Actually, OA is not considered to exist as a separate phase but as a solid solution, OHA (oxyhydroxyapatite). Raman spectra do not allow to discriminate easily between solid solution of OHA and mixtures of HA and OA phases. However, Raman mapping reveals micro-domains containing both HA and OA environments, thus supporting the existence of solid solutions. In addition, very strong spatial variations are evidenced indicating a strong heterogeneity. The formation of OA environments appears as an intermediate stage before the decomposition of HA into TTCP and TCP takes place. A second process of formation of OA environments is due to the recrystallization of apatite in a melt practically deprived of OH -ions. The existence of domains very rich in OA and very poor in HA are consistent with this recrystallization process.

The formation of OA in ClA coatings is rather puzzling as there is no precursor of oxygenated species. Thus in ClA coatings the formation of OA involves necessarily a reaction with a contaminant. A reaction may be proposed with residual water molecules remaining in the ClA powder, which was stored and manipulated in air. In fact, this is the reverse reaction of the chlorination synthesis, which has been used to synthesize hydroxyapatite monocrystals:

2H 2 O +
although this analogue of Deacon's reaction [START_REF] Schmittinger | Chlorine" in Ullmann's Encyclopedia of Industrial Chemistry[END_REF] has never been reported in ClA. This reaction would occur preferably at the surface of splats during the cooling stage, when in contact with the air.

Other reactions might be possible between TiO 2 at the surface of the substrate and ClA. However, OA is distributed in the coating thickness and does not localize especially in close proximity to the titanium surface. OA rich domains are always observed in domains isolated from ClA rich ones, with no clear gradation, suggesting a preferential formation of OA in a recrystallization process from the molten phase, which would facilitate oxygen diffusion and reaction. The mode of formation of OA in ClA coatings, which implies reactions with contaminants, explains why these contain significantly less OA than the HA ones, where OA formation is a part of the thermal degradation.

Amorphous Phase Content

Concerning the formation of ACP, the Raman spectroscopic data appear more reliable than the crystallinity measurements done by XRD following standards. As already mentioned before, several causes may be behind the decrease in XRD crystallinity observed in the coatings, of which ACP formation is only one. On the contrary, the significant shift in the Raman  1 PO 4 of ACP, compared to that of apatite HA or ClA (-12 cm -1 ), appears as a distinctive characteristic of the amorphous phase. However the broadness of the amorphous PO 4 lines in Raman spectra and the superimposition with an OA line, do not allow a very accurate quantitative evaluation. The formation of ACP is assigned to the rapid cooling of the molten grain reaching the surface. However, a recrystallization process is supposed to either happen during the cooling of the splat or during an additional upper layer spray run, involving a local re-heating of an underlying amorphous phase. Thus, a difference in ACP content could be related to different ratios of molten phases and/or different rates of recrystallization. All these phenomena are related to heat transfer in the plasma or in the coating, which are difficult to evaluate and control, although simulations have been proposed [START_REF] Dyshlovenko | Modelling of plasma particle interactions and coating growth for plasma spraying of hydroxyapatite[END_REF]. Several spray parameters such as the angle of spraying; the spray distance and the plasma gas flow rate have been shown to determine the amorphous content of coatings [START_REF] Gross | Amorphous phase formation in plasma-sprayed hydroxyapatite coatings[END_REF]. However, these parameters were maintained constant in this study. A crucial parameter seems to be the particle size, which is related to strong amorphous phase variations [START_REF] Gross | Amorphous phase formation in plasma-sprayed hydroxyapatite coatings[END_REF]. The characteristics of ClA powders compared to HA ones (lower melting point, smaller grain size and crystal size) should favour the formation of coatings with a higher content in amorphous phase, which is not observed. Therefore, the recrystallization rate could be a determining factor in the ACP content in these experiments.

Different recrystallization processes may occur in coatings: crystal formation directly from the melt or in solid state, from the ACP phase. The recrystallization in the melt is expected to be a faster process, although it can be interrupted by the rapid cooling and solidification of the splat. Recrystallization from the melt could be easier in the case of ClA as it does not involve intermediate phases like those theoretically observed in HA recrystallization according to phase diagrams. The recrystallization from the amorphous solidified phase would then follow and may occur at a rather low temperature. In HA coatings for example, the recrystallization of ACP has been observed at about 700°C [START_REF] Kumar | Phase composition and heat of crystallisation of amorphous calcium phosphate in ultra-fine radio frequency suspension plasma sprayed hydroxyapatite powders[END_REF]. The low recrystallization temperature could also occur when a new hot layer is deposited on a cold one, due to the low thermal conductivity of apatites.

Mechanical Properties

The difference in mechanical properties between ClA and HA coatings is especially apparent in the three point bending test with stiffener, which seems more reliable than the standardized test. The adhesive strength of coatings has been correlated to the porosity of the coatings, the presence of cracks related to stresses due to differences in thermal coefficient expansion, and its composition. In wet media, rehydration of the coating, inducing cracks, has been identified as a crucial parameter in the degradation of mechanical properties. Chemical interactions between the coating and the oxide layer of the substrate have been suggested, but their role in the adhesion has not been clearly identified. Concerning the phase content, it appears that a high amount of ACP, as well as a high amount of crystalline phase, is not favourable [START_REF] Tsui | Plasma sprayed hdroxyapatite coatings on titanium substrates. Part 2: optimization of coating properties[END_REF]. The weaker strength of ClA coatings could thus be related to too high a crystallinity.

CONCLUSION

The main characteristics of ClA compared to HA coatings is a higher content in apatite and a lower content in amorphous calcium phosphate and oxyapatite. It is probable that slight modifications in the coating process would permit to limit even further the formation of OA in ClA coatings, especially a careful elimination of water from the ClA powder, although the role of OA in coatings is not known and this phase should not behave differently from other apatites. The expected advantage of ClA coatings would be a lesser degradation in vivo. A persisting problem remains the poor adhesion of the ClA coating on titanium substrates. In vivo experiments have been realized and the explants are currently examined. The first results show a direct attachment of bone to the coating surface, as it is usually found for this kind of coating. These results will be published in a forthcoming report.

Fig. ( 1 ).

 1 Fig. (1). Diffraction patterns of the HA and ClA powders used in this work. Superlattice lines on ClA pattern are shown by arrows.

Fig. ( 2 )

 2 Fig. (2). FTIR spectra of the HA and ClA powders used in this work. Apatitic phosphate absorption bands are indicated by dotted lines.

Fig. ( 3

 3 Fig. (3). SEM images of HA (left) and ClA (right) powders.

Fig. ( 4

 4 Fig. (4). X-ray diffraction patterns of HA powder and coating (12 spray runs).

Fig. ( 5 )

 5 Fig. (5). X-ray diffraction patterns of ClA powder and coating (12 spray runs).

Fig. ( 8 ).

 8 Fig. (8). Raman spectra of HA powder and coating.

Fig. ( 10

 10 Fig. (10). A-Gaussian-Lorentzian curve fitting of the ν 1 PO 4 Raman stretching domain of HA coatings (20 spray runs), with the LEPS plasma gun. (1) ACP: amorphous calcium phosphate; (2) and (4) OA: oxyapatite; (3) HA: hydroxyapatite.

Fig. ( 9 ).

 9 Fig. (9). Raman spectra of ClA powder and coating.

Fig. ( 15 )

 15 Fig.[START_REF] Overgaard | Calcium phosphate coatings for fixation of bone implants[END_REF]. three point bending tests loading curves for HA (left) and ClA (right) coatings.

-

  Formation of oxyapatite occurs in both coatings but ClA coatings exhibit much lower OA contents than HA.

:

  Ca 10 (PO 4 ) 6 Cl 2 → Ca 10-x (PO 4 ) 6 Cl 2-2x + xCaCl 2

Table 1 . Main characteristics of the powders Powders

 1

Ca/P Specific surface area (m 2 g -1 ) Granulometry d0.5 (µm) Flowability (s) HA 1.66 ± 0.02 1.44 ± 0.07 69.5 27.26 ± 3.44 ClA 1.67 ± 0.02 1.48 ± 0.15 56.5 4.75 ± 0.32

Table 2 . Torch characteristics and plasma spray conditions

 2

	Torch Type	Sulzer Metco; Switzerland
	Powder level (kW)	15 kW
	Current (A)	400
	Gas	Argon
	Spraying velocity (mm.s -1 )	200
	Plasma gas rate (Ar; L.mn -1 )	24
	Stand-off distance (mm)	40
	Carrier gas rate (Ar; L.mn -1 )	4
	Powder particle size (µm)	50-80
	Powder feeding rate (g.mn -1 )	8
	Number of spray runs	1; 6; 12; 20

Table 3 . Coating thickness as a function of the number of runs and of the type of apatite. Coating thickness (µm) Thickness per run (µm.run -1 )

 3

	Number of runs	HA-coating	ClA-coating	HA-coating	ClA-coating
	1	28 ± 4	16 ± 4	28 ± 4	16 ± 4
	6	85 ± 4	60 ± 4	14.2 ± 0.7	10.0 ± 0.7
	12	161 ± 4	120 ± 4	13.4 ± 0.3	10.0 ± 0.3
	20	274 ± 4	179 ± 4	13.7 ± 0.2	9.0 ± 0.2

Table 4 . Crystallographic parameters of HA powder and HA coating (12 spray runs). (The pattern obtained with Rietveld refinement of a HA coating is given as supportive material).

 4

	Space group: P63/m	HA-powder	HA-coating
	a (Å)	9.4157(2)	9.4107(2)
	c (Å)	6.8865(2)	6.8748(2)
	Crystallites size (µm)	0.4025(171)	0.2030(67)
	Microstrain (rms)	0.00058(2)	0.00099(2)
	Rexp (%)	5.47	6.00

Table 5 . Crystallographic parameters of ClA powder and ClA coating (12 spray runs).

 5

	Space group: P21/c	ClA-powder	ClA-coating
	a (Å)	9.652(1)	9.550(3)
	b (Å)	19.319(2)	-
	c (Å)	6.784(1)	6.818(2)
	 (°)	120.00(2)	-
	Crystallites size (µm)	0.1467(39)	0.0747(20)
	Microstrain (rms)	0.00106(3)	0.00135(3)
	Rexp (%)	6.30	5.85

Table 6 . Tensile strength of the coatings obtained at 12 spray runs

 6

	Coating	Tensile bond strength (MPa)
	HA	8.9 ± 1.5
	ClA	8.2 ± 2.2

  Ca 10 (PO 4 ) 6 Cl 2 ↔ Ca 10 (PO 4 ) 6 (OH) 2 + HCl ↔ Ca 10 (PO 4 ) 6 O +H 2 O + HCl (4) Such a reaction involves the formation of HA as an intermediate phase. Although there is no detection of OH - ions in the ClA coatings by Raman or FTIR spectroscopy, this reaction could occur in the plasma or on the surface of the coupon during cooling in air. A second possible reaction would be the direct oxidation of ClA with O 2 from air: O 2 + Ca 10 (PO 4 ) 6 Cl 2 ↔ Ca 10 (PO 4 ) 6 O + Cl 2

ACKNOWLEDGMENTS

This study was carried out under a MNT ERA-Net Project named NANOMED. The authors gratefully thank the Midi-Pyrénées region (MNT ERA Net Midi-Pyrénées Région, NANOMED2 project) and the Institute National Polytechnique de Toulouse (BQR INPT 2011, BIOREVE project) for supporting this research work, especially the financial support for research carried out in the CIRIMAT and the LGP laboratories (France), and the Basque government and Tratamientos Superficiales Iontech, S.A. for their financial and technical support under the IG-2007/0000381 grant for the development of the LEPS device and deposition of the coatings carried out in Inasmet-Tecnalia. The French industrial collaborators (TEKNIMED SA and 2PS SA) were financed by the OSEO programs.

SUPPLEMENTARY MATERIALS

Supplementary material is available on the publisher's web site along with the published article.

CONFLICT OF INTEREST

The authors confirm that this article content has no conflict of interest.


Excitation Retrieval of Microwave Linear Arrays from Phaseless Far Field Data Benjamin Fuchs, Member, IEEE, and Laurent Le Coq

Abstract-A methodology to recover the excitations of microwave linear arrays from the measurements of far field magnitude only is proposed. The approach combines tools from convex optimization (to solve the phase retrieval problem) and a simple measurement procedure (to mitigate the non uniqueness of the solution). Numerical simulations in various representative and realistic configurations of noise and measurement sampling are presented and discussed. They show that it is possible to perfectly retrieve the array excitations from only the knowledge of far field magnitude by simply solving a convex optimization problem. In addition, the proposed approach, that only calls for readily available routines, is stable with respect to noise since the reconstruction performances degrades gracefully as the signal to noise ratio decreases.

Index Terms-Antenna measurements, phaseless measurement, phase retrieval, convex optimization.

I. INTRODUCTION

M ANY applications, in particular in imaging and optics, seek to reconstruct an object from the measurements of a signal. However, in many experimental setups, the phase of this signal is often difficult to measure accurately. The problem is then to reconstruct the object from magnitude measurements only. This inverse problem, known as phase retrieval, is frequently encountered in applied physics and engineering including X-ray and crystallography imaging [START_REF] Harrison | Phase problem in crystallography[END_REF], optics [START_REF] Walther | The question of phase retrieval in optics[END_REF] and audio signal processing [START_REF] Griffin | Signal estimation from modified short-time Fourier transform[END_REF] to name just a few. A famous example that arises in a variety of fields is the one of recovering a signal from the magnitude of its Fourier transform. The phase retrieval problem is also of uppermost interest in microwave antenna measurements where measuring accurately the phase may be costly and difficult in particular when dealing with high frequencies. Many efficient phaseless measurement methods have thus been proposed for antenna metrology [START_REF] Bucci | Far-Field Pattern Determination from the Near-Field Amplitude on Two Surfaces[END_REF]- [START_REF] Migliore | Far-field antenna pattern estimation from near-field data using a low-cost amplitude-only measurement setup[END_REF], array diagnosis and imaging [START_REF] Morris | Phase retrieval in the radio holography of reflector antennas and radio telescopes[END_REF]- [START_REF] Bucci | Diagnosis of Array Faults from Far-Field Amplitude-Only Data[END_REF]. The advantages of such approaches are manifold. The probe positioning tolerances that is critical in near field measurement setups is relaxed. Moreover, magnitude-only measurements are by definition insensitive to potential instrument's phase drifts. Finally, phaseless measurements do not require the use of vector instruments whose cost is prohibitive as the frequency goes up.

Mathematically, the phase retrieval problem is notoriously very challenging. On the one hand, it requires the development of efficient and reliable reconstruction algorithms. A considerable amount of methods have been proposed in the literature. So far, most approaches were using alternating projection algorithms inspired by the seminal works of Gerchberg and Saxton [START_REF] Gerchberg | A practical algorithm for the determination of phase from image and diffraction plane pictures[END_REF] and Fienup [START_REF] Fienup | Phase retrieval algorithms: a comparison[END_REF]. These greedy techniques avoid an exhaustive search and they turn out to often work well in practice. However, these methods are not known to converge in general and they can stall in a local minima. Recently, phase retrieval problems have been treated using semidefinite relaxation and low-rank matrix recovery ideas [START_REF] Candes | Phase retrieval via matrix completion[END_REF]- [START_REF] Fogel | Phase Retrieval for Imaging Problems[END_REF]. To date, these convex approaches are the only ones providing guarantees on the recovery performances under some specific conditions (given in Section IV). On the other hand, the phase retrieval problem may be illposed since its solution is, in general, not unique. In particular, the 1-D phase retrieval problem is known to admit multiple solutions [START_REF] Bruck | On the ambiguity of the image reconstruction problem[END_REF]. Most existing methods seek to overcome this non uniqueness by imposing additional constraints on the signal to be retrieved such as its non-negativity or its sparsity. Recently, approaches assuming no prior information on the signal at all but using multiple structured measurements have been successfully proposed for optical setups [START_REF] Candes | Phase retrieval via matrix completion[END_REF], [START_REF] Candes | Phase Retrieval from masked Fourier transforms[END_REF].

In this paper, we focus our investigation on the phase retrieval for microwave 1-D (linear) arrays. Our objective is to retrieve the complex array element excitations from its far field magnitude measurements. An efficient approach combining convex programming (to solve the phase retrieval problem) and two measurement runs (to mitigate the non uniqueness of the solution) is proposed. To the authors' best knowledge, it is the first time that convex relaxation based approaches are exploited to address phase retrieval problems in microwave applications. The goal of this paper is to empirically show that under appropriate conditions (of sampling and noise level), it is possible to uniquely recover the excitations of 1-D array from phaseless far field data.

The paper is organized as follows. The phase retrieval problem is formulated for 1-D microwave array imaging in Section II. Two algorithms to efficiently retrieve the array excitations are described and compared in Section III. Their application to array imaging problem and specifically a simple procedure to mitigate the solution ambiguities is proposed in Section IV. The numerical performances of the proposed phase retrieval procedure, namely the influence of the oversampling and the robutness to noise, are assessed in Section V. The conclusion and perspectives of this work are finally discussed.

II. PROBLEM FORMULATION

The phase retrieval seeks to reconstruct N complex excitations x given only the magnitude of M linear measurements y. It can be formulated as follows: find x subject to |Ax| = y [START_REF] Harrison | Phase problem in crystallography[END_REF] where A ∈ C M×N is the sensing matrix, x ∈ C N and y ∈ R M . This problem is difficult to solve because the set of real or complex numbers with a given magnitude is non-convex. An exhaustive search is untractable since, even in the real case, there are 2 M possible assignments of sign to the M phaseless measurements. Moreover, the problem may be ill-posed. Depending on the properties of the matrix A, the mapping from the magnitude measurements y to the excitations x is or is not one-to-one and hence unique recovery is not possible in general.

The geometry of the investigated problem, i.e. the imaging of a linear array, is represented in Fig. 1. The magnitude of

x n θ 0 r n f n (θ)

Linear array excitations

Far field magnitude measurements y(θ m ) y(θ 1 ) y(θ M )

x N x 1 the measured far field in the direction θ m is related to the excitations x to be retrieved as follows:

y(θ m ) = N n=1 A m,n x n with A m,n = f n (θ m )e j 2π
λ rn sin θm where λ is the free space wavelength, r n and f n (θ m ) are the position and the far field pattern in the direction θ m of the n-th antenna respectively and the time convention e jωt is omitted for clarity. Let us denote with a m the vector of dimension N whose elements are those of the m-th row of A, it is the array steering vector in the direction θ m . The problem (1) can then be rewritten:

find x subject to |a m T x| = y m , for m = 1, ..., M (2)
with y m = y(θ m ) and where (.) T is the transpose operator. The radiated field is thus related to the excitations by a discrete Fourier transform in the case of isotropic sources, i.e. when f n (θ) = 1, ∀n. The problem is then ill-posed since many different sets of excitations have the same Fourier transform magnitude.

The ways (a) to solve the non-convex phase retrieval problem and (b) to mitigate the non-uniqueness (also called ambiguities) of the solution in order to uniquely recover the underlying excitations are addressed in Section III and IV respectively.

III. RESOLUTION METHODS

Two convex relaxation approaches are now described to approximate in order to efficiently solve the phase retrieval problem [START_REF] Walther | The question of phase retrieval in optics[END_REF].

A. PhaseLift

The first convex relaxation of the phase retrieval problem has been introduced by Candes et al. [START_REF] Candes | Phase retrieval via matrix completion[END_REF]- [START_REF] Candès | Solving quadratic equations via PhaseLift when there are about as many equations as unknowns[END_REF]. He observed that the non-convex measurements y on vectors x become linear measurements on matrices X = xx H where (.) H is the Hermitian conjugate. The measurements can be rewritten:

y 2 m = x H A m x = Tr(A m X)
where A m = a m a m H are hermitian matrices and X = xx H is a rank-one Hermitian matrix. The phase retrieval problem (2) becomes:

find X subject to Tr(A m X) = y 2 m , m = 1, ..., M X 0 rank(X) = 1 (3) that is equivalent to: minimize rank(X) subject to Tr(A m X) = y 2 m , m = 1, ..., M X 0 (4)
since there exists by definition a rank-one solution. The problem (4) is a combinatorially hard problem. However, for positive semidefinite matrices, i.e. in this case since X 0, the rank functional can be approximated by a convex surrogate, the trace norm as proposed in [START_REF] Fazel | Rank Minimization and Applications in System Theory[END_REF]. The problem (4) becomes:

minimize X Tr(X) subject to Tr(A m X) = y 2 m , m = 1, ..., M X 0 (5)
which is a semidefinite program that is then convex and efficiently solvable. The original vectorial phase retrieval problem is thus convexified by "lifting" it up to a matrix recovery problem hence the name PhaseLift introduced by [START_REF] Candes | Phase retrieval via matrix completion[END_REF], [START_REF] Candes | Phaselift : exact and stable signal recovery from magnitude measurements via convex programming[END_REF]. In practice, the measurements are contaminated by noise:

y m = |a m T x + n m |, for m = 1, • • • , M
where n m is a noise term. The equalities in [START_REF] Isernia | Radiation Pattern Evaluation from Near-field Intensities on Planes[END_REF] no longer hold in presence of noise. The following formulation has then been proposed in [START_REF] Candès | Solving quadratic equations via PhaseLift when there are about as many equations as unknowns[END_REF]:

minimize X M m=1 Tr(A m X) -y 2 m (6)
subject to X 0.

In words, solving [START_REF] Migliore | Far-field antenna pattern estimation from near-field data using a low-cost amplitude-only measurement setup[END_REF] amounts to find the positive semidefinite matrix X that best fits the observed data in an ℓ 1 sense. The solution of ( 6) will be from now on referred to as PL (for PhaseLift).

If the solution X of (6) happens to have rank one, then X = xx H and x is the optimal solution of the original phase retrieval problem [START_REF] Walther | The question of phase retrieval in optics[END_REF]. Otherwise, one extracts the best rank one approximation of X: X 1 = σ 1 u 1 u 1 H where σ 1 is the largest eigenvalue of X and u 1 is the associated eigenvector. The vector x = √ σ 1 u 1 is then an approximate solution of (2). For further information related to convex relaxations of quadratic problems, we refer the interested reader to the tutorial [START_REF] Luo | Semidefinite Relaxation of Quadratic Optimization Problems[END_REF].

B. PhaseCut

A different strategy has been proposed in [START_REF] Waldspurger | Phase Recovery, MaxCut and Complex Semidefinite Programming[END_REF], [START_REF] Fogel | Phase Retrieval for Imaging Problems[END_REF] to solve the phase retrieval problem by explicitly separating the amplitude and phase variables in order to only optimize the value of the phase variables. By factoring out the magnitude information, we can write Ax = diag(y)u where u ∈ C M is a phase vector satisfying |u m | = 1 for m = 1, ..., M . The phase retrieval problem can be expressed as follows:

minimize u∈C M ,|um|=1 x∈C N Ax -diag(y)u 2 2 ( 7
)
where we optimize over both variables u and x.

The minimization over x is a standard least square and can be solved explicitly: x = A † diag(y)u where A † is the pseudo inverse of A. The problem ( 7) is equivalent to:

minimize u∈C M ,|um|=1 AA † diag(y)u -diag(y)u 2 2 . (8)
The objective of ( 8) can be rewritten:

AA † diag(y)u -diag(y)u 2 2 = (AA † -I) diag(y)u 2 2 = u H diag(y T ) R diag(y)u
where R = (AA † -I) H (AA † -I) = (I -AA † ). The phase retrieval problem (1) becomes:

minimize u∈C M u H Ru (9) subject to |u m | = 1, m = 1, ..., M
where the Hermitian matrix R = diag(y)(I -AA † ) diag(y) is positive semidefinite. The problem ( 9) is equivalent to:

minimize U Tr(RU) subject to diag(U) = 1 U 0 rank(U) = 1. (10)
By dropping the non-convex rank constraint, we obtain the following convex relaxation:

minimize U Tr(RU) subject to diag(U) = 1 U 0 (11)
that is very similar to the MaxCut semidefinite program hence the name PhaseCut (denoted PC in the sequel).

As in the PL procedure, if the solution has a rank larger than one, the leading eigenvector of U is used to build an approximate solution û of [START_REF] Fienup | Phase retrieval algorithms: a comparison[END_REF] from which an approximate solution of ( 2) is derived: x = A † diag(y)û.

C. Discussion

By trading non-convex constraints into convex ones, we have transformed a vector problem into a matrix one. This "lifting" leads to a much larger representation of the state space which means a higher computational cost. It also implies that we must deal with a highly underdetermined problem since for the PL procedure, there are N 2 unknowns (instead of only N ) for M = O(N ) magnitude measurements. As explained in [START_REF] Demanet | Stable optimizationless recovery from phaseless linear measurements[END_REF], this apparent lack of data is compensated by the fact that we are only looking for rank one matrices. It is also worth noting that the computational complexity of the PC algorithm [START_REF] Fienup | Phase retrieval algorithms: a comparison[END_REF] is higher than the one of PL (6) since we are looking in the space of Hermitian matrices of dimension M for PC instead of only N for PL. Last but not least, both convex relaxations ( 6) and ( 11) can be solved optimally by readily available software such as CVX [START_REF]CVX: Matlab software for disciplined convex programming[END_REF].

IV. APPLICATION TO ARRAY IMAGING PROBLEMS

The problem of ambiguity in phase retrieval problems is first discussed, existing solutions to mitigate this non uniqueness are reviewed and a simple approach suitable to microwave antenna arrays is proposed. Then, the way to quantify the recovery performances of the proposed approach are detailed.

A. Non-uniqueness (Ambiguities) of the solution

In the general case, the solution to the phase retrieval problem (1) is not unique and the best we can strive for is to retrieve x n up to a phase α, a "mirror function", and an unknown shift δ. Indeed, if x n with n=1,...,N is a solution to [START_REF] Harrison | Phase problem in crystallography[END_REF], then e jα x n , x * N +1-n and x n at locations r n -δ are respectively also solution. Note that a * is the complex conjugate of a. Since we assume that the positions r n of the array elements are known, the last ambiguity (unknown shift) type does not exist. More specifically, for a linear array composed of N equispaced identical elements, there are up to 2 N -1 possible solutions as explained in [START_REF] Isernia | Shaped beam antenna synthesis problems: feasibility criteria and new strategies[END_REF].

To mitigate the ambiguities arising in phase retrieval problems, the oversampling in the Fourier domain (i.e. the increasing of the measurement points M ) has been shown to almost always work in the case of 2D Fourier transform [START_REF] Hayes | The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform[END_REF]. However, this technique offers no benefits for 1D recovery as discussed in [START_REF] Candes | Phase retrieval via matrix completion[END_REF], [START_REF] Waldspurger | Phase Recovery, MaxCut and Complex Semidefinite Programming[END_REF], [START_REF] Bruck | On the ambiguity of the image reconstruction problem[END_REF] and therefore cannot be used for our application. Another approach is to try to render the sensing matrix A as "random" as possible in order to make unique recovery possible. For that purpose, a measurement setup using of an electronically controlled reflectarray between the microwave antenna under test and the measurement probe has been proposed in [START_REF] Migliore | Phase-transition behavior in array diagnosis using sparse recovery techniques[END_REF]. The idea is to switch randomly the elements of the reflectarray to obtain a random sampling of the field radiated by the antenna under test. This solution, that is theoretically very appealing, seems in practice difficult and costly to implement. In optics, many techniques have been successfully applied to resolve the phase ambiguity problem. They consist in using multiple independent measurements to somehow gather different views of the object under test as reviewed in [START_REF] Candes | Phase retrieval via matrix completion[END_REF]. Thus, taking multiple diffraction patterns by adding a mask or modulating the light beam with an optical grating usually yields uniqueness. In that case, the sensing matrix can be seen as the product between the discrete Fourier transform matrix and a mask or filter that is a diagonal matrix with either random binary or Gaussian entries. Recently, an interesting framework [START_REF] Raz | Vectorial Phase Retrieval of 1-D Signals[END_REF] (called vectorial phase retrieval by the authors) has been proposed to efficiently solve the phase retrieval problem. It is shown that the unique reconstruction of two signals is possible by measuring their spectral intensity signals and their pairwise interferences.

The previously described techniques are difficult to implement in microwaves that is why we propose the following approach. As done in holography [START_REF] Osherovich | Phase Retrieval Combined With Digital Holography[END_REF], we assume that we have a reference antenna whose complex radiation pattern y ref (θ) (the vector y ref of dimension M after discretization) is known. We measure the magnitude of the far field radiated by: -the array under test y AUT = |A AUT x| and -the interference between the array under test and the reference antenna y AUT +ref = |A AUT x + αy ref | where α is the excitation (not necessarily known) of the reference antenna. This simple procedure allows to mitigate the ambiguities due to the "mirror function" and the excitations are then determined up a global phase. If in addition the phase of the excitation α is known (after a calibration for instance), then the solution x is unique. The matrix A ∈ C 2M×(N +1) and the measurement vector y in (1) becomes:

A = A AUT | 0 M A AUT | y ref and y = y AUT y AUT +ref (12)
where 0 M is a vector of zeros of dimension M .

B. Recovery Performances 1) Recovery Error:

The phase retrieval algorithms ( 6) and ( 11) compute an approximate solution x from y = |Ax|. Their performances are typically assessed by computing the relative "distance" between the exact solution x and the recovered one x. Special care must be taken since a solution may be unique up to a global phase (for complex values) / sign (for real values) (if the excitation of the reference antenna is not known). Thus, the error in excitation denoted ǫ(x, x) can be computed as:

min α∈[0,2π] e jα x -x 2 x 2 or xx H -xx H F xx H F (13)
where . F stands for the Frobenius norm. This second definition, easier to compute, will be used in Section V.

The error over the measured field amplitudes is:

ǫ(|Ax|, |Ax|) = |Ax| -|Ax| 2 Ax 2 . ( 14
)
If a low excitation error ǫ(x, x) implies a low error over the measured field amplitudes ǫ(|Ax|, |Ax|), the opposite is not necessarily true. Indeed, when the phase retrieval problem does not have a unique solution, we may have a low ǫ(|Ax|, |Ax|) but ǫ(x, x) ≫ ǫ(|Ax|, |Ax|).

Although the quantities ( 13) and ( 14) are useful to assess and compare the efficiency of the algorithms PL and PC, they are not very speaking. In order to appreciate the results with more physical insights, we compute the mean values of the excitation amplitude ratio and phase difference (denoted µ m (x, x) and µ p (x, x) respectively):

µ m (x, x) = 1 N N i=1 δ mi with δ mi = |x i | |x i | ( 15
)
µ p (x, x) = 1 N N i=1 |δ pi | with δ pi = (∠x i -∆φ) -∠x i
where ∆φ the global phase shift between x and x is equal to ∆φ = 1 N N i=1 (∠x i -∠x i ). We derive from [START_REF] Waldspurger | Phase Recovery, MaxCut and Complex Semidefinite Programming[END_REF] the standard deviation in amplitude and phase (σ m (x, x) and σ p (x, x) respectively):

σ m (x, x) = 1 N N i=1 (δ mi -µ m (x, x)) 2 1/2 (16)
σ p (x, x) = 1 N N i=1 (δ pi -µ p (x, x)) 2 1/2
in order to measure the degree of confidence in the retrieved excitation magnitudes and phases.

2) Noise: The reconstruction of the excitations in presence of noise is crucial for practical applications. In our experiments, the magnitude measurement y m is polluted by a Gaussian white noise n m as follows: y m = |a m T x+n m |. The level of this noise is quantified by the Signal-to-Noise Ratio (SNR): SNR dB = 10 log 10 (P signal /P noise ) where

P signal = max m=1,••• ,M (|a m T x| 2 )
is the maximum measured power.

In order to estimate the SNR of a far field measurement in an anechoic chamber, both the full equipment system (noise floor of the receiver, transmitted power and losses due to the cables and rotary joints) and the reflectivity of the chamber itself must be taken into account. A SNR value of 60 dB, that corresponds to a reasonably good anechoic chamber, will be considered for the numerical applications of Section V.

3) Sampling (M/N ): Up to date, the convex approaches are the only ones to offer guarantees on the recovery performances for phase retrieval problems. It means that under appropriate conditions, the difficult problem ( 2) is equivalent to the convex program [START_REF] Migliore | Far-field antenna pattern estimation from near-field data using a low-cost amplitude-only measurement setup[END_REF] or [START_REF] Fienup | Phase retrieval algorithms: a comparison[END_REF] in the sense that they have the same unique solution. If the number M of equations |a m T x| = y m are sufficiently randomized and at least on the order of N , it has been demonstrated in [START_REF] Candès | Solving quadratic equations via PhaseLift when there are about as many equations as unknowns[END_REF] that the solution to the convex relaxation ( 6) is exact (i.e. it is the same as the one of ( 2)) with a probability approaching one exponentially, as M is increased. It is important to point out that these results have been established for sensing matrices A with Gaussian i.i.d. (independent and identically distributed) entries. In the next Section, we will numerically check if similar results hold in the case of 1-D microwave array imaging, i.e. when the sensing matrix is not Gaussian but of the form of a discrete Fourier Transform.

V. NUMERICAL RESULTS

The goal of this Section is to numerically assess the recovery performances of the phase retrieval algorithm ( 6) and [START_REF] Fienup | Phase retrieval algorithms: a comparison[END_REF]. In the retrieval procedure, the element positions and radiation patterns of the array under test, i.e. of the sensing matrix A, and the reference antenna y ref are assumed known. For each configuration, the simulation is repeated 100 times with different random excitations (whose magnitude and phase are chosen uniformly between [0, 1] and [0, 2π] respectively), different random measurement vectors (whose angle are chosen uniformly such that -1 ≤ sin θ ≤ 1) and a Gaussian white noise in order to get meaningful results. All presented results are average values over these 100 simulations.

A. Linear Array of Isotropic Sources 1) Settings:

We consider a linear array composed of N isotropic half wavelength spaced elements that are excited by a random amplitude and phase x n . A reference antenna (also isotropic) with an arbitrary excitation (fixed but chosen randomly in the simulations) is placed at an arbitrary distance (2 λ in the simulations) from the extremity of the array under test. It has been checked numerically that both the position and excitation value of this source with respect to the array under test do not impact the recovery performances.

2) Influence of the Sampling:

The influence of the ratio M/N (number of measurement points over number of excitations to retrieve) on the recovery performances is investigated. The error in excitations ǫ(x, x) is plotted as a function of the sampling for linear arrays of various elements N in Fig. 2. In absence of noise, there is a clear transition between 'bad' and 'good' recovery at M = 2N . In presence of a realistic noise (SNR=60 dB), a less sharp transition also exists around M = 2N . Generally speaking, for a given sampling M/N , the recovery performances are better for a small number of array elements N . To evaluate the reconstruction quality as a function of the sampling, let us plot in Fig. 3 the reconstruction errors in amplitude and phase for a linear array of N =10 elements in presence of noise (SNR=60 dB). We retrieve, as seen above, the significant improvement in recovery performances when M = 2N . Then, The oversampling helps to further improve up to a certain extent the recovery performances as shown in Fig. 3 for M > 2N . 3) Robustness to Noise: Let us choose a sampling M = 2N in order to see the influence of the noise on the excitation recovery. The influence of the SNR on the retrieved excitations and field amplitudes (ǫ(x, x) and ǫ(|Ax|, |Ax|) respectively) is plotted in Fig. 4. There is clearly a linear behavior between these errors and the SNR with a log-log scale. This graceful degradation shows that both convex approaches PL and PC provide a stable recovery in presence of noise. The reconstruction errors in excitation amplitude and phase are plotted in Fig. 5 as a function of the SNR. They show that above a given SNR (of 60 dB for M/N = 2), the reconstruction is very good in amplitude and fairly good in phase. In presence of important noise (SNR < 60 dB), the algorithm PC seems to yield a better reconstruction (especially in phase) than PL. This better performance may be explained by the decoupling between magnitude and phase reconstruction of the PC procedure (see Section III-B). 2) Influence of the Sampling: In absence of noise, a perfect recovery, i.e. an error in excitation amplitude and phase lower than 10 -5 dB and deg respectively and a very low standard deviation, is achieved as soon as M ≥ 2N . Generally speaking, better recovery performances are achieved in the case of realistic sources compared to isotropic ones. The ambiguity problem is indeed intrinsically reduced when the radiation patterns of each element are not all identical. The influence of the oversampling on the recovery performances in presence of noise (SNR=60 dB) are plotted in Fig. 7. An oversampling M > 6N does not bring any further improvement. 3) Robustness to Noise: The error in the retrieved excitations and field amplitudes increases linearly (in a log-log scale) with the noise. This robustness to the noise (not reported) is similar to the one of isotropic sources of Section V-A.3. The reconstruction errors in excitation magnitudes and phases are shown in Fig. 8 as a function of the SNR for an oversampling of M = 3N . In that case, a very good recovery is achieved for SNRs greater than 60 dB. The algorithm PC provides a slightly better recovery in case of low SNR as for the case of isotropic sources (see Section V-A.3).

µ m (x, x) [dB] σ m (x, x) [dB] µ p (x, x) [deg] σ p (x, x) [deg] SNR [dB] SNR [dB] SNR [dB]

VI. CONCLUSION

A methodology has been proposed to efficiently solve the phase retrieval problem arising in microwave linear array. The goal is to retrieve the complex array excitations from the measurements of the far field magnitude only. A procedure combining convex optimization and two measurement runs has been developed and numerically assessed in various representative and realistic conditions of noise and sampling. Promising results have been obtained: an excellent amplitude and phase excitation recovery is achieved for a sampling (number of measurement samples over excitation points) of above a factor two in presence of a realistic noise of SNR equal to 60 dB. This procedure is new in the field of microwave antenna measurements and seems to be promising to significantly reduce the cost of future microwave imaging systems.

This work calls for several appealing extensions. First, the application of the proposed approach to 2-D array phase retrieval should be straightforward all the more as the problem of ambiguities can be directly overcome by oversampling. The effort should then focus on reducing the computational complexity to solve the convex relaxations. Second, it would be of great interest to assess the efficiency of the proposed phase retrieval approach in the case of near field data. The proposed approach for far field data could be a priori used by only changing the sensing matrix. However, several crucial parameters such as the distance array under test -probe and the range of the probe measurements have to be carefully addressed. Finally, the phase retrieval method must be tested on real measurement data and we hope to report such experiment in a future publication.

Fig. 1 .

 1 Fig. 1. Geometry of the linear array imaging problem with the notations.

Fig. 2 .

 2 Fig. 2. Error in the retrieved excitation amplitudes as a function of the sampling with PhaseLift for linear array of various elements (N =5, 10, 20, 30, 40).

Fig. 3 .

 3 Fig. 3. Recovery performances with PhaseLift (solid line) and PhaseCut (dotted line) for a linear array of N =10 isotropic sources: influence of the sampling (M/N ) for measurements with a SNR of 60 dB.

Fig. 4 .

 4 Fig. 4. Error in the retrieved excitations and field amplitudes for a linear array of N =10 elements as a function of the SNR with PhaseLift (solid line) and PhaseCut (dotted line). The markers (circle for PL and square for PC) are the results without noise.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Recovery performances for a linear array of N =10 isotropic sources with a sampling M = 2N : influence of the SNR with PhaseLift (solid line) and PhaseCut (dotted line).

Fig. 7 .

 7 Fig. 7. Recovery performances for a linear array of five patches: influence of the sampling (M/N ) for measurements with a SNR of 60 dB with PhaseLift (solid line) and PhaseCut (dotted line).

  , x) [dB] σ m (x, x) [dB] µ p (x, x) [deg] σ p (x, x) [deg] SNR [dB] SNR [dB] SNR [dB] SNR [dB]

Fig. 8 .

 8 Fig. 8. Recovery performances for a linear array of five patches: influence of the SNR with PhaseLift (solid line) and PhaseCut (dotted line) for a sampling M = 3N .

Laurent Le Coq received the electronic engineering and radiocommunications degree and the french DEA degree (M.Sc.) in electronics in 1995 and the Ph. D. in 1999 from the National Institute of Applied Science (INSA), Rennes, France. In 1999, he joined IETR (Institute of Electronics and Telecommunications of Rennes), University of Rennes 1, as a research lab engineer, where he is responsible of the measurement technical facilities up to 110GHz, and of the prototyping service. His activities in antenna measurements and development of related procedures involved him in more than 30 research contracts of national or european interest. He is author and co-author of more than 35 journal papers and 35 papers in conference proceedings.