M Alaya

R Benabderrahmane Zaghouani

S Khamlich

J.-L Lazzari

W Dimassi

Rabia Benabderrahmane
email: rabia.benabderrahmane@gmail.com

Enhancement of physical properties of stain-etched porous silicon by integration of WO 3 nanoparticles

Keywords: Porous silicon, Passivation, Tungsten oxide, Photoluminescence, Surface recombination velocity, Minority carrier lifetime

In this work, we report on the passivation of porous silicon (PS) by tungsten trioxide (WO 3 ) nanostructured thin films deposited via dip-coating of PS in tungsten hexachloride and water/ethanol solution. Structural analysis by Fourier transform infrared spectroscopy showed a partial disappearance of SiH x and Si-O-Si peaks after WO 3 thin film deposition on the PS due to the replacement of H atoms by WO 3 . Additionally, PS/WO 3 sample revealed weakly intense peaks which can be ascribed to O-W-O and W=O bridging bonds. Morphological analysis by atomic force spectroscopy showed the coverage of the PS surface by a nanostructured thin film of dense WO 3 nanoparticles with a size ranging from 40 to 60 nm. The photoluminescence signal intensity of PS/WO 3 samples presents a continuous increase as a function of the immersion time in the precursor solution. This is attributed to a better decoration of the PS surface by WO 3 nanoplatelets and a decrease of surface recombination velocity (200 x 10 2 cm/s for untreated silicon, 137 x 10 2 cm/s for PS and 99 x 10 2 cm/s for PS/WO 3 samples). Hence, a ~ 50 % improvement in the effective lifetime of minority carriers is obtained.

Moreover, the deposited WO 3 contributed not only to the passivation of PS, but acted as an antireflection layer which enhanced the optical properties of PS by reducing the reflection of light. These obtained results confirmed the benefits of using WO 3 thin film as passivation/antireflection coating for possible solar cell application.

A C C E P T E D

M A N U S C R I P T 2

Introduction

Although silicon being the dominant and mature material in the manufacturing of photovoltaic modules, there is a permanent quest for high-volume, high-throughput and cost-effectiveness coating technologies with multilayered materials having different functionalities. Indeed, transparency, conductivity, impermeability, photocatalytic [START_REF] Zhao | Photocatalytic oxidation for indoor air purification: a literature review[END_REF], electrochromic, photochromic, passivation and antireflection properties are among those that could be combined in such multifunctional materials. In addition to the progress of coating technologies, the extension of nanotechnologies and the discovery of new nanostructuration processes have generated much interest. The nanostructuration of silicon have paved the way for different applications in microelectronic, photonic, photocatalysis [START_REF] Rong | Preparation of hierarchical micro/nanostructures Bi 2 S 3 -WO 3 composites for enhanced photocatalytic performance[END_REF][START_REF] Gondal | Visible light photocatalytic degradation of herbicide(Atrazine) using surface plasmon resonance induced in mesoporous Ag-WO 3 /SBA-15 composite[END_REF][START_REF] Zhang | PdO loaded WO3 COMPOSITE WITH Na2W4O13 flake: A 2-D heterostructure composite material[END_REF], gaz sensors [START_REF] Guérin | Modeling of the conduction in a WO3 thin film as ozone sensor[END_REF][START_REF] Li | Hydrogen peroxide sensing using Cu2O nanocubes decorated by Ag-Au alloy nanoparticles[END_REF] and photovoltaic devices [START_REF] Rajabi | Current improvement of porous silicon photovoltaic devices by using double layer porous silicon structure: applicable in porous silicon solar cells[END_REF]. Recent studies demonstrated that the quantum confinement of carriers in nanostructured silicon contributed to the efficiency enhancement when integrated in solar cells [START_REF] Gourbilleau | Silicon nanostructures for solar cell applications[END_REF][START_REF] Conibeer | Silicon nanostructures for third generation photovoltaic solar cells[END_REF]. Even if the silicon emits weak infrared photoluminescence due to its indirect band gap, the finding of a high visible photoluminescence from electrochemical etched silicon (porous silicon) [START_REF] Gelloz | Conduction and luminescent properties of wet porous silicon[END_REF][START_REF] Elhouichet | Laser induced degradation of photoluminescence intensity of porous silicon[END_REF] was well exploited by the photovoltaic community [START_REF] Schirone | Chemically etched porous silicon as an anti-reflection coating for high efficiency solar cells[END_REF]. Although largely debated, this discovery was attributed to quantum size crystalline structures [START_REF] Gelloz | Conduction and luminescent properties of wet porous silicon[END_REF]. In addition to its photoluminescence, porous silicon (PS)

acts as an antireflection layer with a surface reflectance attaining less than 10% in the visible spectrum due to the light-trapping into its nano-pores. Porous silicon can be elaborated by different methods with a formation mechanism leading to H-terminated surface which is unstable in ambient conditions [START_REF] Schirone | Chemically etched porous silicon as an anti-reflection coating for high efficiency solar cells[END_REF]. Particularly, the appearance of dangling bonds on PS surface when subjected to an atmospheric oxidation leading to a deterioration of PS electronic and optical properties, specially, the minority carrier lifetime and the photoluminescence signal. In silicon-based solar cells, to obtain the maximum of current, photo-generated carriers diffusing to metallic contacts should have a high diffusion length and then a high lifetime. Many researchers have demonstrated the necessity of a porous silicon passivation step in order to achieve PS stability before its integration in final devices [START_REF] Jeske | Electrodeposition of metals into porous silicon[END_REF]. Several materials used as PS passivation layer are reported in the literature. Ben Rabha et al. [START_REF] Ben Rabha | Monocrystalline silicon surface passivation by Al 2 O 3 /porous silicon combined treatment[END_REF] reported porous silicon passivation by means of Al improve solar cells efficiency such as silicon carbide [START_REF] Barbouche | New process of silicon carbide purification intended for silicon passivation[END_REF] and alumina [START_REF] Rahman | Passivation of all-angle black surfaces for silicon solar cells[END_REF].

To improve the PS properties, this work covers the use of tungsten oxide (WO 3 ) as passivation layer. This oxide is known as a semi-conductor with a band gap around 3 eV and good chemical stability in aqueous solutions.

This binary material has attracted considerable attention for its outstanding electrochromic, photochromic and photocatalytic properties [START_REF] Blackman | Atmospheric Pressure Chemical Vapor Deposition of Crystalline Monoclinic WO 3 and WO 3-x Thin Films from Reaction of WCl 6 with O-Containing Solvents and Their Photochromic and Electrochromic Properties[END_REF][START_REF] Liao | Electrochromic properties of nanocomposite WO 3 films[END_REF][START_REF] Gomez-Solis | Facile synthesis of m-WO 3 powders via precipitation in ethanol solution and evaluation of their photocatalytic activities[END_REF]. There have been tremendous reports on the use of metal-oxide-semi-conductor structures as gas sensors. In this context, the majority of the applications adopted WO 3 as sensing material due to its high sensitivity and stability. Most of the recent publications studied the properties of WO 3 under different

A C C E P T E D M A N U S C R I P T 3
gases such as NO 2 [START_REF] Wang | Room temperature NO2 gas sensing of Au-loaded tungsten oxide nanowires/porous silicon hybrid structure[END_REF]. WO 3 thin films could be synthesized by different physical and chemical methods such as: chemical vapor deposition [START_REF] Blackman | Atmospheric Pressure Chemical Vapor Deposition of Crystalline Monoclinic WO 3 and WO 3-x Thin Films from Reaction of WCl 6 with O-Containing Solvents and Their Photochromic and Electrochromic Properties[END_REF][START_REF] Gordona | Atmospheric pressure chemical vapor deposition of electrochromic tungsten oxide films[END_REF], spray pyrolysis [START_REF] Li | Electrochromic films produced by ultrasonic spray deposition of tungsten oxide nanoparticles[END_REF], sol gel [START_REF] Sharma | FTIR and absorption edge studies on tungsten oxide based precursor materials synthesized by sol-gel technique[END_REF], electrodeposition [START_REF] More | Electrodeposition of nano-granular tungsten oxide thin films for smart window application[END_REF][START_REF] Regragui | Preparation and characterization of pyrolytic spray deposited electrochromic tungsten trioxide films[END_REF] and atomic layer deposition (ALD) [START_REF] Malm | Atomic layer deposition of WO 3 thin films using W(CO) 6 and O 3 precursors[END_REF]. Its properties are motivating and encouraging to use it in photovoltaic devices. This work highlights an original study with a focus on the improvement of porous silicon electronic and optical properties by passivation with WO 3 thin layers deposited by a dip-coating method. A specific emphasis was put on the improvement of the PS photoluminescence signal in addition to the increase of minority carrier lifetime by depositing WO 3 thin layers.

Experimental techniques

To synthesis porous silicon, solar grade boron-doped single crystalline silicon (100) substrates (Czochralski)

with a resistivity of ~1-3 Ω.cm and a thickness of 200 µm were used. Before PS elaboration, silicon substrates were cleaned with boiling acetone for 10 min, followed by immersion in ethanol for 5 min and rinsed after that in deionized water in order to eliminate organic greases. Finally, the substrates were etched in 10 % hydrofluoric acid for 1 min to eliminate native oxide, rinsed in deionized water and dried with nitrogen flux. Porous silicon was formed by "stain etching" in a solution of hydrofluoric acid (40 wt %), nitric acid (65 wt %) and deionized water (HF: HNO 3: H 2 O) = (1:2:3) in volumetric ratio for 2 min. Immediately, after formation of PS layer, the substrates were dip-coated in a solution containing tungsten hexachloride (WCl 6 ) prepared previously in order to deposit WO 3 layers [START_REF] Badilescu | Study of sol-gel prepared nanostructured WO 3 thin films and composites for electrochromic applications[END_REF]. WCl 6 was mixed with ethanol (50 %) and water (50 %) under stirring at ambient temperature for 3 hours, leading to a blue solution. The latter solution was stored in air for 7 days until the blue color disappears and then it is ready to be used.

In order to study the effect of WO 3 layers deposition on PS properties, the WCl 6 solution concentration and immersion times (t imm ) were varied. First, the morphological features of porous silicon before and after WO 3 treatment were characterized by a digital instrument, Nanoscope 3100 Atomic Force Microscopy (AFM) operating in tapping mode. WO 3 deposition was also studied by a Fourier Transform Infrared (FTIR) Nicolet MAGNA-IR 560 Spectrometer. The optical properties of the prepared samples based on the surface reflectivity were analyzed using PerkinElmer Lambda 950 UV/VIS. Photoluminescence (PL) analyses were performed at ambient temperature with a 447 nm excitation laser. Finally, the minority carrier's lifetime was measured by the "Sinton WCT-120 lifetime tester" in the quasi-steady-state mode. The measurement was based on the recombination dynamics of optically generated carriers. The excess carrier density was evaluated via inductively measured photoconductance signal. The minority carrier lifetime was determined by measuring simultaneously the photoconductance variation in the sample and the generation rate. The carrier lifetime was then presented as a function of the excess carrier density.

Results and discussion

Structural and morphological Studies of the deposited WO 3 on PS

FTIR analysis was performed on freshly prepared PS and PS/WO 3 samples. The FTIR absorption spectra are presented in Fig. 1 (a). The principal recorded vibration bands observed for the two spectra are SiH x stretching mode at 2112 cm -1 , Si-O-Si stretching mode at 1092 cm -1 and SiH 2 scissors mode at 907 cm -1 . Absorption bands

A C C E P T E D M A N U S C R I P T

4 at 628 cm -1 and 672.7 cm -1 corresponding to Si-Si stretching mode and SiH n (n=1and n=2) wagging mode were also observed. Moreover, the appearance of a peak at 861 cm -1 was attributed to the O-SiH x wagging mode. We clearly noticed a reduction of SiH x and Si-O-Si peaks intensity for PS/WO 3 sample, suggesting the replacement of H atoms by WO 3 entities [START_REF] Haddadi | Improvement of solar cells performances by surface passivation using porous silicon chemically treated with LiBr solution[END_REF]. In order to determine precisely the different W-O bands in the spectral region (400 cm -1 -1000 cm -1 ), we have zoomed at different zones on the FTIR spectra of PS and PS/WO 3 samples. We observed appearance and disappearance of some peaks in the FTIR spectrum after a successful deposition of WO 3 nanostructured thin film. Fig. 1 (b) and Fig. 1 (c) show the amplification of zone 2 in PS and PS/WO 3 samples, respectively. After deconvolution of the large peak around 906 cm -1 , we obtained new peaks for PS/WO 3 sample at 927.7 cm -1 and 898.9 cm -1 corresponding to W═O and W-O-W bridging modes respectively [32,[START_REF] Deepa | Electrodeposited tungsten oxide films: annealing effects on structure and electrochromic performance[END_REF]. In the PS FTIR spectrum, peaks situated at 962.5 cm -1 and 934.9 cm -1 disappeared. After amplification of zone 3 and deconvolution (Fig. 1 (d) and Fig. 1 (e)), a band at 782.9 cm -1 attributed to W-O intra -W bridging stretch mode was easily distinguished [34], other peak situated at 802 cm -1 was clearly disappeared.

The surface topography of the untreated PS was investigated by AFM, and the 2D micro-image was depicted in Fig. 2(a). After immersion in a WCl 6 solution, porous silicon surface was totally covered with nanostructured features of WO 3 (Fig. 2(b)). Quasi-spherical WO 3 nanoparticles with a 60 nm in average size were clearly observed when the immersion time of 360 s in WCL 6 solution was used; this was confirmed by the top-view SEM image of treated PS sample as shown in Fig. 2(c). region of the solar spectrum, an efficient photovoltaic device could be fabricated [START_REF] Atyaoui | Improvement of photoluminescence and electrical properties of porous silicon layer treated with lanthanum[END_REF]. As can be seen in Fig. 3 (a), WO 3 deposition leads to a reflectance decrease as regard to PS sample, specifically between 250 nm and 550 nm of the spectral region.

WO 3 nanostructured thin film influence on the PS physical properties

Surface reflectivity

Minority carrier lifetime

As mentioned above, the mechanism of porous silicon formation leads to a discontinuity in silicon crystal structure, which directly results in large density of dangling bonds at its surface giving rise to a high density of recombination-active defect levels. The impact of these interface defects on electrical porous silicon properties depends on their density, their capture cross section and their energy levels. The surface recombination is directly quantified by the surface recombination velocity S as given in the (Eq.1) [START_REF] Rein | Lifetime Spectroscopy[END_REF]:

n U S s   (Eq.1)
Where U s is the surface recombination rate and n  is the excess carrier density. The determination of S permits the quality evaluation of the surface passivation, which reduces surface recombination effect by optimizing the interface properties. In this work, porous silicon is passivated by a thin layer of tungsten oxide. FTIR results

A C C E P T E D M A N U S C R I P T

5 revealed the presence of W atoms. These atoms are filling the dangling bonds and changing the interface which result in a change in the defect properties. The reduction of the interface trap density and the capture cross section results in the reduction of the surface recombination rate, and so to the decrease of the surface recombination velocity. Consequently, with WO 3 deposition, the increasing of the effective minority carrier lifetime τ eff in the porous silicon samples is expected. The minority carrier lifetime is measured using the WCT-120 photoconductance lifetime tester in the quasi-steady-state mode (QSSPC). The τ eff takes into account the impact of the different bulk recombination channels and also the surface recombination. These different mechanisms may occur simultaneously in the same sample. The τ eff is provided by (Eq.2):

surface bulk eff    1 1 1   (Eq.2)
In the surface minority carrier lifetime term, the back and front surface contributions are considered:

surface back surface front surface        1 1 1 (3)
In this work, back and front surfaces are identically passivated (S 1 =S 2 = S). τ eff is then expressed by:

w S bulk eff 2 1 1     (4)
Where w is the sample thickness.

In order to optimize the deposition parameters, we have varied WO 3 solution concentration and measured τ eff for different immersion times. We prepared three different concentrations: C 1 =0.086 mol/L, C 2 = 0.035 mol/L and C 3 = 0.018 mol/L. In Fig. 4, the evolution of the minority carrier lifetime as a function of excess carrier density ∆n for the concentrations C 1 , C 2 and C 3 with a deposition time of 30 s is reported. Porous silicon sample, freshly prepared, has an H-terminated surface. Consequently, minority carrier lifetime is greater in PS sample than in silicon substrate. Comparing PS and PS/WO 3 samples for the different concentrations, we observed that almost the same τ eff for C 1 and C 2 was obtained. This behavior can be explained by the increase of WCl 6 solution viscosity at relatively high concentrations. WO 3 may not diffuse into the PS structure forming a glass layer and doesn't passivate PS surface defects. So, we have decreased the solution concentration and as we can observe for C 3 τ eff is increased. In Fig. 5 (a), we notice an enhancement in the effective carrier lifetime as the immersion time increases for the concentration C 3 . This is attributed to defects passivation and so to the decrease of the surface recombination velocity S. This velocity can be obtained via the following equation:

) 1 1 ( 2 bulk eff w S     (5)
To calculate bulk lifetime ( bulk  ), the surface recombination must be reduced until we can consider that the measured effective lifetime reflects the bulk recombination contribution. This is possible by passivating the surfaces. In this work, a bare silicon sample was chemically passivated using Iodine-Ethanol solution [START_REF] Aouida | Light induced enhancement of minority carrier lifetime of chemically passivated crystalline silicon[END_REF].

Silicon sample was covered on both sides with Iodine-Ethanol solution, placed in a transparent bag and measured with the Sinton WCT-120 lifetime tester giving a bulk minority carrier lifetime of the order of 19 µs.

A C C E P T E D M A N U S C R I P T 6

The passivation mechanism was explained in details previously [START_REF] Aouida | Light induced enhancement of minority carrier lifetime of chemically passivated crystalline silicon[END_REF]. Fig. 5 (b) depicts the surface recombination velocity S for silicon, PS and PS/WO 3 samples. It can be noted that a significant reduction of S from 137.5 x10 2 cm/s for porous silicon sample to 94.4x10 2 cm/s for PS/WO 3 sample with an immersion time of 360 s. These results confirmed the effectiveness of the use of tungsten oxide as a passivation layer. They have to be compared with rear-surface passivation of low-resistivity p-type crystalline silicon of passivated emitter and rear cell (PERC) solar cells using an Atomic-Layer-Deposited 130 nm-thick Al 2 O 3 single layer, resulting in a rear surface recombination velocity as low as 90 cm/s and device efficiency as high as 20 % [START_REF] Schmidt | Surface passivation of high-efficiency silicon solar cells by atomic-layer-deposited Al 2 O 3[END_REF].

Photoluminescence

Fig. 6 depicts the photoluminescence spectra of porous silicon before and after treatment with WO 3 during different immersion times (30, 60, 300 and 360 s) for the concentration C 3 . The measurement was performed in the wavelength range of 480-750 nm at room temperature. We observed an enhancement of the PL intensity of all PS/WO 3 samples. The maximum intensity was obtained for an immersion time of 360 s. It is clearly seen that coating Ps with WO 3 could protect PS from its PL intensity degradation, which may be related to the coating effect of WO 3 on PS surface chemistry. The pumping source for PL measurement was the 447 nm laser which is not the excitation wavelength of WO 3 , so the PL amelioration observed is attributed to porous silicon properties changes by WO 3 deposition. This amelioration is due to the passivation of PS layer and the reduction of the surface recombination velocity. The principal peak of PS sample is blue shifted in the range of (2.11 -2.14 eV) after WO 3 treatment. We also observed the presence of a large peak around 2.4 eV. This behavior is due to WO 3 incorporation effect in the porous silicon matrix which may modify the surface electronic structure by creating new radiative centers [START_REF] Bouzourâa | Optical study of annealed cobalt-porous silicon nanocomposites[END_REF]. However, for longer immersion durations, the quantity of WO 3 deposited on the porous layer promotes the non-radiative energy transfer by creating excitation energy traps.

Conclusion

In this work, we have reported a detailed study on the deposition effect of WO 3 thin film on a porous silicon substrate by a simple and effective dip-coating technique. FTIR results demonstrated the deposition of WO 3 layer. AFM confirmed the total coverage of porous silicon surface by WO 3 nanostructured thin film. PS/WO 3 samples exhibits a low surface reflectivity up to about 5 %. A ~ 50 % enhancement of the minority carrier lifetime by decreasing the surface recombination velocity was achieved when depositing WO 3 with an optimum 0.018 mol/L concentration of tungsten hexachloride in water/ethanol solution, which leads to an increase in the photoluminescence intensity in comparison with untreated PS layer. Obtained results confirmed that tungsten oxide is a good passivation layer to improve the porous silicon electronic an optical properties, and thus encouraging its integration in many applications such as solar cells and gas detectors. Despite these interesting results, a further amelioration of the deposited films properties is in demand. In particular, the reduction of the

Fig. 3 (

 3 Fig. 3 (a) shows the surface reflectivity of PS and PS/WO 3 samples at different immersion times (30 s, 90 s, and 120 s) in a WCl 6 solution of 0.086 mol/L concentration. PS sample presents a reflectivity of about 5 % in the visible spectral region. This low reflectivity compared to the bare silicon (Fig.3 (b)) is attributed to light trapping in the porous surface of PS. This interesting property of PS must be conserved when performing any chemical or physical treatment of the PS surface. Having a low reflectivity, in particular around the UV-VIS

7 A 10 Figure captions Fig. 1 :Fig. 2 :

 710captions12 Figure captions

Fig. 3 :

 3 Fig. 3: (a) Reflectivity spectra of PS and PS/WO 3 samples with different immersion times (b) Reflectivity spectra of PS sample and untreated silicon substrate.

Fig. 4 :

 4 Fig. 4: Effective minority carrier lifetime as a function of excess carrier density of untreated silicon, PS and PS/WO 3 samples for different concentration C 1 =0.086 mol/L, C 2 =0.035 mol/L and C 3 =0.018 mol/L for an immersion time of 30 s.

Fig. 5 :Fig. 6 : 14 Figure 2 16 Figure 3 18 Figure 5

 56142163185 Fig. 5: (a) Effective minority carrier lifetime as a function of excess carrier density of untreated silicon, PS and PS/WO 3 samples for different immersion times and a concentration C 3 =0.018 mol/L (b) Effective Surface recombination velocity S eff as a function of immersion time

Acknowledgements

The authors wish to thank Alain Ranguis of the near-field microscopy service and Vasile Heresanu from the xrays service for measurements done at the Centre Interdisciplinaire de Nanoscience de Marseille (CINaM).

A C C E P T E D M A N U S C R I P T 20

Highlights:

 Deposition of Tungsten trioxide on porous silicon surface.  Enhancement of carrier lifetime and surface recombination velocity.  Increase of the photoluminescence of elaborated samples.


A Pichard

J A Alcántara-Núñez

R Alvès-Condé

M Dubois

R Frigot

P Jardin

P Lecomte

J Y Pacquet

M G Saint-Laurent

METHOD FOR EFFICIENCY AND TIME RESPONSE MEASUREMENT ON DIVERSE TARGET ION SOURCES WITH STABLE ALKALI

Developments of new setups for radioactive ion beam production by the isotope-separator-on-line (ISOL) method are underway at GANIL in the frame of the SPIRAL (Système de Production d'Ions Radioactifs Accélérés en Ligne) and SPIRAL-II projects. The measurement of total efficiency and time behaviour of these new target/ion-source systems (TISSs) is a crucial step for these devices which aims to produce short-lived isotopes with high intensity. The overall atom-to-ion transformation efficiency depends on several processes: diffusion of the atoms out of the production target, effusion towards the ion source and ionization. The efficiency can be extracted using the ratio between the emerging yield and implanted flux in the TISS. Several methods have already been developed to achieve these measurements: the use of stable or radioactive beams, gas injection, or the introduction of solid material into the TISS. This paper focuses primarily on a method that uses stable alkali. A pulsed/CW alkali ion gun has been built and will be used to optimise diverse TISSs: NanoNaKE [1] a compact multi-charged alkali ion production system for SPIRAL-I; SISTHE a surface ionisation source prototype to study the alkali ion source for SPIRAL-II; and one TISS for SPIRAL-II.

INTRODUCTION

All particle acceleration originates in ion sources. Due to the constant increase of their intensities various important nuclear physics research projects have been successfully completed. Most ion beams produced are stable. However to achieve optimum results the ion source must provide a high-intensity, high-charged state. For radioactive ion beam (RIB) production by ISOL method, the source feed material is limited since it is produced on-line by nuclear reactions occurring in a target. Also, the species of interest are often short-lived compared to the time spent in the target/ion-source system (TISS). Therefore, the ion source must have a high efficiency and a short delay time.

The total efficiency of TISSs depends on numerous parameters such as target design (conditioned by the energy and intensity of the primary beam), diffusion out of the target material, effusion toward the ion source and ionization. To delineate how to improve the total efficiency of a TISS we need to separate and characterize each of these processes.

In this paper, one method suggested to measure efficiency and time response of TISSs with stable alkali is presented and compared to other existing methods. The method will be applied to the SPIRAL-I and SPIRAL-II TISS development at GANIL.

EFFICIENCY MEASUREMENT METHODS

The total efficiency of a TISS is simply given by the ratio of the current leaving the source to the flux entering the system. Diverse methods for measurement of efficiency and time response are now presented. Several parameters of these methods will be qualitatively compared, such as the sensitivity, the perturbation induced and the chemical form of the species ionised.

Gas Injection

This method, easy to set up, and is mostly used with electron-cyclotron-resonance ion sources (ECRISs). The gas is injected into the ECRIS through a calibrated leak and then ionized. Time response measurements can be achieved using a pulsed valve [START_REF] Vondrasek | [END_REF]3].

Condensable Effusive Jet

External ovens heating a sample [4] allow evaporation and introduction of solids into an ECRIS or surface ion source (SIS), much like a gas. The injected flux can be found beforehand by calibrating the flux, using a piezoelectric quartz for example, or afterwards by mass measurement of the evaporated sample: neither method is very sensitive (of the order of mA).

Sputtering and Plasma Heating

Direct ion sputtering or plasma heating can be used to introduce solid materials into an ECRIS.

For ion sputtering [5], a negatively biased solid sample is placed just outside of the ECRIS plasma. Positive ions from the support gas are accelerated toward the sample, impact upon it and sputter solid material into the plasma.

A crucible or wire can be positioned at the plasma boundary, such that the sample is heated and evaporates by the plasma itself [6].

Laser Ablation

A laser is used to evaporate a small amount of material into the plasma of an ECRIS [7]. Laser pulses are sent onto the sample with peak energy densities about 10 7 W/cm². The solid atoms ablates from the samples and recoil into the plasma before being ionised.

Stable Ion Beam as Primary Beam

Implantation of stable isotopes ion beams in the target can be used to measure the release characteristics of a TISS. The radioactive ions produced by nuclear reactions or the ions of the implanted beam can be detected ___________________________________________ *Contact: pichard@ganil.fr

MOPC154

Proceedings of EPAC08, Genoa, Italy The second row presents the quantity measured at the entrance and exit of the source: F for Flux, I for Intensity, M for Mass. In the third row the measured processes for each method are listed: P for Production, D for Diffusion, E for Effusion and I for Ionisation. The last row describes the perturbations induced by the efficiency measurement on the production process downstream of the TISS. In both cases, time response measurements are achieved switching the primary beam on and off. For radioactive ions, the evolution of the decay rate of a particular radionuclide is monitored either in the target or after ionisation [8]. The time response is directly bound to the efficiency measured in CW mode, owing to the life-time of the element.

Radioactive Ion Beam as Primary Beam

This method [9] is quite similar to the ion beam method but the ions of the primary beam are radioactive. They are implanted in the target and the decay rate of the same isotope is detected downstream from the TISS after diffusion, effusion, ionisation and mass separation. The beam intensities of the primary beam can be much lower than with the previous method, limiting local modification of target heating.

ALKALI ION GUN

A pulsed alkali ion gun has been built to measure efficiency and time response of diverse TISS: NanoNaKE, a compact 1 + /N + alkali production system for SPIRAL-I; SISTHE a surface ionisation source to study the alkali ion source for SPIRAL-II; and the target/ionsource prototype for SPIRAL-II. This method is related to the ion implantation presented previously but is much simpler to implement. In fact, most of ion beams used arise from particle accelerators with very costly and limited beam time. This compact device (8 cm long, diameter 15 cm) can provide mono-isotopic continuous or pulsed alkali ion beam for hundreds of hours with a high intensity (15µA). The chosen beam energy is lower than 1 keV to implant the ions on the surface of the target, restricting the efficiency measurement to the diffusion and ionisation. The low error on the current measurement and the possibility to measure the current injected in and extracted from the TISS on line make this method interesting.

Technical Description

Our device is composed of three mains parts (part a, b and c of fig 3): a stable alkali ion source, an extraction lens and a focusing lens. A Faraday cup is added to measure the extracted beam current and its transport efficiency. The alkali ion source is provided by Heat Wave Labs [START_REF]Specifications of the stable alkali ion sources[END_REF]. Their operational principle relies on thermionic emission of positive ions from a porous tungsten layer doped with alkali heated by an isolated molybdenum heater wire. Temperatures of about 1100°C are reached.

Characterisation

The alkali ion gun has been characterised with a potassium source. The first tests have shown the reliability of the device. The ion gun delivers currents of about 15 µA for an extraction potential of 400 V and 9 W on the heater wire. At 8 cm from the emission point, the beam can be focused to a diameter less than 6 mm. The current emitted is measured with an error rate about 10%. A pulsed beam of 1ms has been obtained for time response measurement.

NanoNaKE

The compact multi-charged alkali ion source NanoNaKE developed at GANIL [START_REF] Eléon | Proceedings of the XV e International Conference EMIS[END_REF] is composed of MonoNaKE, a SIS for radioactive singly-charged ions production, associated with the multi-charged ECRIS Nanogan III currently used on SPIRAL-I. On-line tests carried out in April 2007 shown evidence for 1 + to N + 6mm conversion but with an efficiency lower than expected. Since performance of the 1 + ionization source MononaKE has already been measured on-line, it has been replaced with the ion gun to measure the transport and the 1 + N + conversion efficiency of the system restricted to the 1 + beam line associated to Nanogan III.

SISTHE

SISTHE is a source dedicated to the study of the surface ionisation process. It will allow us to acquire fundamental knowledge to develop a future surface ion source for the SPIRAL-II project. Temperature measurements of the carbon ionizer tube have been carried out during June 2007. The next step is to measure the ionisation efficiency and the effect of the pollution over a period of up to three months for different alkali elements, using the ion gun.

The beam delivered by the ion gun is stopped on a carbon sheet placed into a cavity upstream of the ioniser tube. As the energy of the ions (1 keV) is low, they are implanted on the surface (a range of 2 nm for sodium in C). Heating this cavity leads to the release in atomic or ionic form of the implanted ions outside the surface and the cavity. Assuming that:

• ions or atoms collide at least once with the ionizer, and • the probability to become an ion after a contact with the ionizer is the same for ions and atoms, the state of the atom (neutral or ionized) at the entrance of the ionizer does not affect the state of the atoms at the exit of the source and thus our measurement. The proportion of atoms exiting from the cavity toward the ionizer tube is easy to calculate through the conduction ratio or by a Monte-Carlo simulation (code RIBO [START_REF]Specifications of the stable alkali ion sources[END_REF]). Time response measurements are achieved with the pulsed beam of the ion gun. The volume of the cavity has been kept as small as possible to limit the contribution of the cavity to the time response.

SPIRAL 2 Prototype

With the results from SISTHE, a SIS prototype will be developed for SPIRAL-II. This source will be linked to the production target through a transfer tube. The beam from the ion gun will be injected into the surface of the target at the opposite end of the transfer tube. In this case, measurement of effusion efficiency into the target cavity toward the ion source will be possible, and also of the ionisation efficiency and the time response.

Figure 1 :

 1 Figure 1: Photo of the stable alkali ion source used in the ion gun.

Figure 2 :

 2 Figure 2: Photo of the ion gun assembled on NanonaKE for transport and conversion measurement: a-ion gun, btransfert line, c-Nanogan III multi-charged ECRIS.First results have already been obtained on the effect of the residual gas on the transport of the beam in the 1 + beam line, and also evidence for losses induced by effects not observable during on-line tests (Penning and RF discharges, secondary electron emission, unexpected influence of the extraction voltage on the transport efficiency). These results have led to a new version of the 1 + beam line, with an improved pumping speed. More detailed results will be published later.

Figure 3 :

 3 Figure 3: Technical design of the ion gun on SISTHE for efficiency and time response measurements: a-stable alkali ion source, b-extraction lens, c-focusing lens, doven, e-cavity, f-ioniser tube, g-extraction lens.

Table 1 :

 1 Comparison of several efficiency-measurement methods.

	04 Hadron Accelerators
	436

Hadron Accelerators T01 Proton and Ion Sources

Proceedings of EPAC08, Genoa, Italy 04 Hadron Accelerators

T01 Proton and Ion Sources


Jean-Philippe Farrugia
email: jean-philippe.farrugia@univ-lyon1.fr

Christian Degrigny

Optimized visualization of Château de Germolles' wall paintings using mobile augmented reality with co-lighting

Keywords: Château de Germolles, mural decorations, augmented reality, appearance rendering, colighting

This paper presents the design of an augmented reality visualization of the wall paintings (late 14 th century) of the Château de Germolles. This visualization is done through a mobile application, does not require any external equipment, and considers the co-lighting: the decor can thus be visualized under several virtual ambient lightings, simulating different seasons, weather conditions and two times of the day as well as candle lighting. We present the methods and models that have enabled these simulations, as well as their results and limitations. We also briefly discuss the limitation of an approach based solely on physical plausibility in relation to the objective of a visit speech.

INTRODUCTION

The Château de Germolles, offered to Margaret of Flanders by her husband Philip the Bold, Duke of Burgundy and brother of King Charles V of France, is both the best-preserved residence of these Valois princes, and the one that still displays unique wall paintings from the end of the 14 th century. Rediscovered during World War II, they were partially conserved between 1989 and 1994. This intervention, although quite respectful, was not documented at the time. Thanks to archival documents and the in-depth analysis of the residual decorations [START_REF] Degrigny | Wall Paintings in the Château de Germolles: An Interdisciplinary Project for the Rediscovery of a Unique Fourteenth Century Decoration[END_REF]), the materials used and the painting technique could be better understood, especially for the metal decorations, which residues were partly masked during the conservation intervention. A partial derestoration would make it possible to better visualize them, without however appreciably improving the global rendering.

In discovering these painted decorations, the public visiting the Château de Germolles is far from imagining the delicacy of the original rendering, despite the information transmitted. Therefore, the possibilities of augmented reality have been tested, based on newly acquired scientific knowledge, in order to propose hypotheses of rendering, evaluated by the site's managers, who are also heritage professionals, and then readjusted (geometry, colour and luminosity) for better conformity. The developed application is adapted to the private nature of the site: it uses a light and nomadic support (tablet), is non-intrusive (no visible markers or locating devices) and easy to use. It is, moreover, entirely controlled by the person guiding the visitors: it is indeed an accompaniment to the speech (Degrigny et al. (2020)).

In order to be convincing, an augmented reality application must solve three problems: colocation, which is necessary to align real and virtual elements, co-occultation, which allows the virtual elements to be masked by the real ones (and vice versa), and co-lighting, which allows luminous interactions between virtual and real. In the framework of this project, the co-location is managed by environmental geometric markers materialized by the decor itself, and the co-occultation is not considered insofar as the virtual elements to be inserted are plane and directly visible without occultation. On the other hand, co-lighting required more advanced treatments which are the subject of this article.

In order to restore the original decor, it is necessary to enhance it with adequate models of materials and lighting. Three difficulties have to be reconciled: the limited computational capacities of the tablets, the complex models to obtain a satisfactory physical realism, and the adequacy with the visual expectations of those responsible for the monument. To satisfy these constraints, we have opted for the following solutions:

 A material model widely used in the industry, light and easily parameterizable while being physically plausible.

 An ambient lighting modelled with high dynamic range (HDR) environment maps.  A daytime local lighting modelled by an area light source, approximated by a set of point sources obtained with importance sampling.

 A local night lighting modelled by animating the photometric solid of a candle.

The whole was integrated with Unity software and deployed on an Apple iPad Air 2 tablet. After a state of the art around augmented reality and physical based rendering for heritage enhancement, this paper introduces the modelling of ambient lighting and the obtaining of HDR environment maps as well as the modelling of night lighting. The presentation of the results is accompanied by the appreciation by the public of the approach followed. Finally, we conclude by giving some perspectives for future development.

STATE OF THE ART

By modelling historical light sources [START_REF] Chalmers | Authentic illumination of archaeological site reconstructions In[END_REF], [START_REF] Gutierrez | Modelling light scattering for virtual heritage[END_REF], [START_REF] Rodrigues | Reconstruction and generation of virtual heritage sites[END_REF]) to illuminate virtual archaeological sites [START_REF] Devlin | Realistic visualisation of the Pompeii frescoes[END_REF], it has been shown that lighting and, more generally, the ambient lighting has a significant impact on the perception and understanding of a historical or archaeological scene. The ambition is to restore the historical visual sensation of the site. Although it is rather difficult to formally validate these results, they seem convincing from an archaeological and perceptual point of view (Goncalves et al. (2009)).

Other work by Callet focuses on light-matter interaction, especially colour, metallic effects and spectral simulation [START_REF] Cerise | Natural Lighting and Medieval Glass -Scientific Data Acquisition, Methodology and Physically Based Rendering[END_REF]). They concern both monuments [START_REF] Callet | Natural Lighting, Gilts and Polychromy of Notre-Dame de Paris Cathedral[END_REF]) and historical artefacts [START_REF] Robin | Virtual Replication of the Idalion Tablet[END_REF]). The method is very rigorous from a physical realism point of view, but is too complex to be implemented in an augmented reality application, where the computing power of the devices involved is often limited.

Concerning mobile applications for heritage enhancement, initiatives have multiplied since the advent of modern smartphones [START_REF] Lesaffre | Les applications mobiles de musées et de sites patrimoniaux en France : quelles propositions de médiation ?[END_REF]) but few of them offer realistic visualization functionalities with dynamic lighting. Among the most high-profile projects is the Gunzo project, which aims to reproduce the Abbey of Cluny [START_REF] Père | Le projet Gunzo reconstruit (virtuellement) l'église abbatiale de Cluny[END_REF]), or the Abbey of Jumièges (Jumièges 3D ( 2018)). However, most of these applications are not in real time and present only pre-calculated images, inducing a strong constraint: the user can only observe around a fixed point without being able to move. Immersion, and therefore the impact on the visitor, is then limited.

EXPERIMENTATIONS

Our goal was to create an application that removes the limitations of previous achievements. The heritage to be valued at Germolles is a courtly and bucolic wall decoration, that of the dressing-room of the dukes' daughter-in-law, Margaret of Bavaria, alternating white letters (initials of the Dukes' first names) and thistles (symbols of protection and fidelity) now of the same white colour but which were once metallic and golden in appearance, all on a green background. The application must be able to reproduce the supposed visual sensation of this decor, under different lighting conditions and without restricting the user's freedom of movement. The main constraint is that the chosen platform, a tablet, does not allow physical calculations as advanced as those presented in the previous section, it is for example impossible to perform spectral calculations. It was therefore necessary to find a compromise between physically realistic results and real-time computations. At the same time, this project also allowed to test whether physical realism is a necessary condition for such a mediation.

In order to be convincing, the rendering of the virtual decor integrates three determining models: materials, ambient lighting, and night lighting detailed in the following.

Materials

The materials model we used is the one present in the Unity engine. It is a micro-facet reflectance function (based on Walter et al. ( 2007)) whose objective is not an exact physical reproduction of materials at the molecular scale, but only the reproduction of their visual appearance. It has the advantage of being very widespread, easily parameterizable, while being physically plausible. The different parameters of metallic character, colour, and roughness can be modulated by textures. This model also allows to alter the local geometry of the surface to add mesoscopic details. For ease of editing, we used the Substance Designer software to design the materials in collaboration with the managers of the château. Three materials were designed, for the letters, thistles, and background. There is no certainty about the composition of the original materials, only hypotheses. The model had to be parameterized to test these hypotheses visually and iteratively (Degrigny et al. (2020)).

Ambient lighting

The ambient lighting model is based on a well-known technique in image synthesis: Image-Based Lighting. This technique consists in using a panoramic HDR image from the real environment to illuminate the virtual decor to be enhanced [START_REF] Debevec | Rendering synthetic objects into real scenes: Bridging traditional and image-based graphics with global illumination and high dynamic range photography[END_REF]). In the application, we want to visualize the decor under four different weather conditions for each of the four seasons, at two times of the day, i.e. a total of 32 HDR images. These images were captured with a Canon Powershot 230S camera whose internal software has been modified to allow bracketing for HDR reconstruction. Multiple shots were taken using a photographic stand and then assembled into a panorama with the PTGui Pro software. Figure 1 shows examples of reconstructed panoramas, in spring and for two weather conditions. However, in this project, the HDR panorama alone does not faithfully render the lighting in the room. The main light source, the original open window of the north wall in figure 1, is under-represented and does not contribute sufficiently to the appearance of the decor. We have therefore added an additional rectangular light source to the scene, representing a virtual version of the window in the room. Unfortunately, Unity engine does not allow the use of non-uniform area light sources. We therefore chose to opt for an approximation based on point sources, positioned on the plane of the source and whose characteristics (colour and position) are defined by importance sampling an HDR image of the window. Figure 2 shows a representation of this sampling with the corresponding image.

Night lighting

The application must also be able to simulate candlelight at night. In contrast to the ambient lighting in the previous section, in this case the light source is dynamic and cannot be simulated with HDR images. We have chosen to take up the work of Bridault-Louchez et al. (2008) and port it to the tablet. This technique is based on the use of a candle flame photometric solid applied to a point light source in order to modulate its intensity in a non-uniform manner. On the tablet used for the application, however, it is not possible to simulate the dynamics of air flow to obtain the flickering of the flame, which is an obvious clue of this type of lighting. In order to reproduce this effect, we chose to alter the photometric solid with small random rotations (Degrigny and Farrugia (2020)).

RESULTS

The application itself is a classical mobile augmented reality application. Depending on the season, outdoor weather conditions and time of day (32 options in total), the user chooses the most suitable ambient lighting conditions. The tablet is pointed towards the decor (only the west wall of the dressing-room is currently being considered). By pressing on the touch screen, the reconstructed virtual decor is gradually superimposed on the real decor (directly seen by the tablet camera). No placement is imposed in the room, the user is completely free to move around, the virtual view of the decor being calculated and adjusted in real time. Figure 3 shows an example of lighting with the environment map only (a), and with the environment map plus the light source of the window area (b), showing the addition of the latter. The chosen ambience is a sunny summer morning. The application is only shown, during the tour, after the description of the paintings, the history of their rediscovery and conservation. This is a user experience designed to help visitors understand the visual rendering of the paintings beyond words. The speech used remains accessible despite the technological contribution. However, visitors are made aware of the possibilities and limits of augmented reality, while insisting on the hypothetical nature of the approach, even if it is based on historical and scientific knowledge accompanied by cutting-edge research on the light-matter interaction.

CONCLUSION

This article presents a mobile augmented reality application that provides a historical visualization of a medieval decor. This application, currently used on site, is one of the few to allow dynamic and realistic lighting of virtual elements on a general public device, without restricting the user's freedom of movement or requiring intrusive elements on the site. It can still be improved on many points: it is, for example, possible to carry out the process of capturing ambient lighting with the tablet via a dedicated function within the application itself, without external intervention.

The effects of the application on the dynamics of the visits are noticeable. The visualisation of the decor in its original state via augmented reality creates an effect of surprise and renewed attention from the audience. The appreciation by the public, although difficult to quantify, is enthusiastic. The next step could be a more formal evaluation of this appreciation, through user experience and dedicated instrumentation of the application in order to collect relevant indicators.

  An example of two HDR panoramic image in Margaret of Bavaria's dressing-room taken on a rainy day (a) and on a sunny day (b), both in spring.

  An image of the window of the north wall (a) and point source sampling (b). The colour of the point light sources reflects the dominant blue colour of the window.

  The virtual decor lit with environment map only (a) and with environment map and area light (b).

ACKNOWLEDGEMENTS

We would like to thank Matthieu Pinette, owner of the Château de Germolles, for allowing us access to the site.


Rafael Colares
email: rafael.colares@uca.fr

Hervé Kerivin
email: herve.kerivin@uca.fr

Annegret Wagler
email: annegret.wagler@uca.fr

An extended formulation for the Constraint Routing and Spectrum Assignment Problem in Elastic Optical Networks *

The emergence of Elastic Optical Networks allowed a more flexible spectrum allocation for routing traffic demands within telecommunication networks. From this context arises the Routing and Spectrum Assignment problem, which consists of routing a given set of origin-destination traffic demands and assigning them to contiguous spectrum frequencies such that no frequency slot is assigned to more than one demand within a network link. This work deals with the variant where each demand route must additionally satisfy a maximal-length constraint. In this paper we propose a compact extended formulation for the Constrained Routing and Spectrum Assignment Problem. We show that our extended formulation is theoretically stronger than formulations known in the literature. Experimental results demonstrate the efficiency of our approach.

Introduction

Modern optical networks represent a crucial infrastructure for our information society. To transmit signals, light is used as a communication medium between sending and receiving nodes. For over two decades, Wavelength-Division Multiplexing (WDM) has been the most popular technology used in optical networks, where different wavelengths are used to simultaneously transmit signals over a single optical fiber. Hereby, the wavelengths have to be selected from a rather coarse fixed grid of frequencies specified by the United Nations agency ITU (International Telecommunication Union), which leads to an inefficient use of spectral resources.

In response to the continuous growth of data traffic volumes in communication networks, a new generation of optical networks, called Elastic Optical Networks (EONs), has been introduced to enhance the spectrum efficiency and to enlarge the network capacity [START_REF] Jinno | Spectrum-efficient and scalable elastic optical path network: architecture, benefits, and enabling technologies[END_REF]. In EONs, the frequency spectrum of an optical fiber is divided into many narrow frequency slots, and any sequence of contiguous slots can form a channel to create an optical connection between sending and receiving nodes, called lightpath (an optical connection represented by a route and a channel). That way, EONs enable capacity gain by allocating minimum required bandwidth to every traffic demand thanks to a finer spectrum granularity than in the traditional WDM networks.

To operate EONs, the so-called Routing and Spectrum Assignment (RSA) problem has to be solved which consists of establishing the lightpaths (represented by a route and a channel) for a set of traffic demands (given as sending and receiving nodes and required slot numbers), thereby optimizing some objective function. To comply with ITU regulations, the following constraints need to be respected when dealing with the RSA problem:

1. spectrum continuity: the frequency slots allocated to a demand remain the same on all the links of a route; 2. spectrum contiguity: the frequency slots allocated to a demand must be contiguous; 3. non-overlapping spectrum: on each link of the network, a frequency slot can be allocated to at most one demand.

In addition, technical properties further force that the length of a route must not exceed the transmission reach of the optical signal which leads to the Constraint Routing and Spectrum Assignment (CRSA) problem, see Section 2 for details and examples.

The RSA problem has started to receive a lot of attention over the last few years. It has been shown to be NP-hard [START_REF] Christodoulopoulos | Elastic bandwidth allocation in flexible OFDM based optical networks[END_REF][START_REF] Wang | A study of the routing and spectrum allocation in spectrum-sliced elastic optical path networks[END_REF] and remains hard even when the optical network is a path [START_REF] Shirazipourazad | On routing and spectrum allocation in spectrum sliced optical networks[END_REF] which makes the RSA problem much harder than the WDM problem which is polynomially solvable on paths, see e.g. [START_REF] Fayez | Spectrum Assignment in Mesh Elastic Optical Networks[END_REF].

To solve the RSA problem, various approaches have been studied in the literature, based on different Integer Linear Programming (ILP) models. Hereby, detailed models aiming at precisely describing all technological aspects of EONs and being able to handle various criteria for optimization typically suffer from tractability issues resulting from their greater complexity such that the tendency is to use simplified or restricted models.

The majority of the existing models uses an edge-path formulation where for each demand, variables are associated either with all possible routing paths or with all possible lightpaths for this demand. One characteristic of this formulation is, therefore, an exponential number of variables issued from the total number of all feasible paths between origin-destination pairs in the network, which grows exponentially with the size of the network and the number of demands.

To bypass the exponential number of variables, edge-path formulations with a precomputed subset of all possible paths per demand have been studied e.g. in [START_REF] Klinkowski | An overview of routing methods in optical burst switching networks[END_REF][START_REF] Klinkowski | Routing and Spectrum Assignment in Spectrum Sliced Elastic Optical Path Network[END_REF][START_REF] Velasco | Modeling the routing and spectrum allocation problem for flexgrid optical networks[END_REF][START_REF] Zotkiewicz | Optimization models for flexgrid elastic optical networks[END_REF], see [START_REF] Zotkiewicz | Optimization models for flexgrid elastic optical networks[END_REF] for an overview. However, such formulations cannot guarantee optimality of the solutions in general (as only a precomputed subset of paths is considered and, thus, a restricted problem solved). In order to be able to find optimal solutions of the RSA problem w.r.t. any objective function with the help of an edge-path formulation, all possible paths have to be taken into account. As the explicit models are far too big for computation, it is in order to apply column-generation methods. However, computational results from e.g. [START_REF] Klinkowski | Spectrum allocation problem in elastic optical networks -a branchand-price approach[END_REF][START_REF] Klinkowski | A Simulated Annealing Heuristic for a Branch and Price-Based Routing and Spectrum Allocation Algorithm in Elastic Optical Networks[END_REF][START_REF] Ruiz | Column generation algorithm for RSA problems in flexgrid optical networks[END_REF] show that the size of the instances that can be solved that way is rather limited.

An alternative to edge-path formulations is to use edge-node formulations that have the advantage to be compact in terms of the number of variables and constraints, but have the disadvantage that the routing is rather involved and less intuitive. Only few authors made use of this type of model, as Cai et al. [START_REF] Cai | Novel Node-Arc Model and Multiiteration Heuristics for Static Routing and Spectrum Assignment in Elastic Optical Networks[END_REF], Velasco et al. [START_REF] Velasco | Modeling the routing and spectrum allocation problem for flexgrid optical networks[END_REF], Zotkiewiez et al. [START_REF] Zotkiewicz | Optimization models for flexgrid elastic optical networks[END_REF], and Jia et al. who used in [START_REF] Jia | An integrated ILP model for Routing, Modulation Level and Spectrum Allocation in the next generation DCN[END_REF] an edge-node formulation to treat a more general problem.

As noticed in [START_REF] Hadhbi | A novel integer linear programming model for routing and spectrum assignment in optical networks[END_REF], the models from [START_REF] Cai | Novel Node-Arc Model and Multiiteration Heuristics for Static Routing and Spectrum Assignment in Elastic Optical Networks[END_REF][START_REF] Velasco | Modeling the routing and spectrum allocation problem for flexgrid optical networks[END_REF][START_REF] Zotkiewicz | Optimization models for flexgrid elastic optical networks[END_REF] are incomplete as their feasible region is a superset of all feasible solutions of the RSA problem and can, thus, handle only some objective functions. The first complete edge-node formulation presented in [START_REF] Hadhbi | A novel integer linear programming model for routing and spectrum assignment in optical networks[END_REF] exactly encodes the feasible solutions, but requires an exponential number of constraints to ensure proper routings. Moreover, in [START_REF] Hadhbi | A novel integer linear programming model for routing and spectrum assignment in optical networks[END_REF] a procedure is given to separate the exponentially-sized families of constraints in polynomial time, which makes the model computationally competitive with the compact but incomplete models from [START_REF] Cai | Novel Node-Arc Model and Multiiteration Heuristics for Static Routing and Spectrum Assignment in Elastic Optical Networks[END_REF][START_REF] Velasco | Modeling the routing and spectrum allocation problem for flexgrid optical networks[END_REF][START_REF] Zotkiewicz | Optimization models for flexgrid elastic optical networks[END_REF]. The computational tests in [START_REF] Hadhbi | A novel integer linear programming model for routing and spectrum assignment in optical networks[END_REF] were performed on a set of instances that resembles those used in [START_REF] Cai | Novel Node-Arc Model and Multiiteration Heuristics for Static Routing and Spectrum Assignment in Elastic Optical Networks[END_REF][START_REF] Velasco | Modeling the routing and spectrum allocation problem for flexgrid optical networks[END_REF][START_REF] Zotkiewicz | Optimization models for flexgrid elastic optical networks[END_REF]. However, the used instances are rather small and it is expected that the running time to optimally solve real-sized instances of the RSA problem (e.g., for the French national network operator Orange) will be drastically increase with all existing edge-node formulations.

Our contribution is to develop the edge-node formulations from [START_REF] Hadhbi | A novel integer linear programming model for routing and spectrum assignment in optical networks[END_REF] further to a compact model that enables solving real-sized instances of the CRSA problem in reasonable time.

For that, we propose an extended formulation in an auxiliary network constructed from the original optical network and an appropriate modification of involved variables to simplify the way to encode routings, see Section 3. We show that our extended formulation is theoretically stronger than the original formulation from [START_REF] Hadhbi | A novel integer linear programming model for routing and spectrum assignment in optical networks[END_REF]. Experimental results presented in Section 5 demonstrate the efficiency of our approach.

We close with some concluding remarks and lines of future research.

Problem Definition

In this section, we formally define the CRSA problem by describing in detail the input and the desired output of the CRSA problem together with the studied objective functions.

As input of the CRSA problem, we are given

• an optical spectrum S = {1, . . . , s} of available frequency slots;

• an optical network, represented as an undirected, loopless, connected graph G = (V, E) that may have parallel edges (if parallel optical fibers are installed between two nodes), and for each edge e ∈ E its length e ∈ R + , • a multiset K of demands where each demand k ∈ K is specified by

-an origin node o k ∈ V and a destination node d k ∈ V \ {o k }, -a requested number w k ∈ N + of slots, and -a transmission reach ¯ k ∈ R + .
The task is to determine for each demand k ∈ K a lightpath composed of an (o k , d k )-path P k in G respecting the transmission reach ¯ k and a channel S k ⊂ S of w k consecutive frequency slots (spectrum contiguity) that is available on all edges of P k (spectrum continuity) and disjoint from the channels S k of all other demands k ∈ K routed along an edge of P k (non-overlapping spectrum), thereby minimizing some objective function. Note that we focus on the case where all demands must be satisfied.

Hence, the desired output of the CRSA problem is, for each demand k ∈ K, a lightpath composed of

• an (o k ,d k )-path P k in G with e∈E(P k ) l e ≤ ¯ k , • a subset S k ⊂ {1, . . . , s} of w k consecutive slots with S k ∩ S k = ∅ for each demand k ∈ K routed along an edge e ∈ E(P k ).
This output can be given in terms of a matrix M ∈ N |E|×s with M e,s = k if slot s ∈ S is allocated to demand k ∈ K on edge e ∈ E, 0 otherwise (i.e., slot s is not used by any demand on edge e).

In addition, the selected set of lightpaths is supposed to minimize a chosen objective function, for instance:

O 1 : minimize the sum of hops in paths (where the term hops refers to the number of edges in a path P k ), O 2 : minimize the sum of the total length of paths (taking the edge weights l e into account), O 3 : minimize the maximal used slot position (and, thus, the width of the subspectrum of S used for the spectrum assignment), O 4 : minimize the sum of the maximal used slot positions over all demands.

Note that the two objective functions O 1 and O 2 are only related to the routing (provided that a feasible spectrum assignment within S exists for this routing), whereas the other two objective functions O 3 and O 4 seek for the most efficient spectrum assignments over all possible routings. Example 1. Consider the following small instance of the CRSA problem, given by a spectrum of width s = 7, the network G shown in Figure 1 as indicated, and the following set K of demands:

k o k → d k w k ¯ k 1 a → c 2 4 2 a → d 1 4 3 b → f 2 4 4 b → e 1 4 5 d → f 3 4
An optimal solution w.r.

M 1 =               1 2 3 4 5 6 7 ab 3 3 4 af 1 1 3 3 2 4 bc cd cf 1 1 de 2 df 5 5 5 ef 2 4               M 2 =               1 2 3 4 5 6 7 ab 1 1 2 af bc 1 1 3 3 2 4 cd 2 4 cf 3 3 de 5 5 5 4 df ef 5 5 5               M 3 =               1 2 3 4 5 6 7 ab 4 af 1 1 2 4 bc 3 3 cd cf 1 1 3 3 de 2 df 5 5 5 ef 2 4               M 4 =               1 2 3 4 5 6 7 ab 4 af 2 4 1 1 bc 3 3 cd cf 3 3 1 1 de 2 df 5 5 5 ef 2 4              

Extended Formulation

Our extended formulation is based on the Integer Linear Programming (ILP) formulation proposed by [START_REF] Hadhbi | A novel integer linear programming model for routing and spectrum assignment in optical networks[END_REF], hereafter denoted by RSA-BASE. This formulation mainly uses the following three sets of binary variables (other variables may be added according to the choice of the objective function) in order to model the RSA problem. For each demand k ∈ K and each edge e ∈ E, variables x k e ∈ {0, 1} are used to indicate whether or not demand k is routed through edge e. For each demand k ∈ K and each slot s ∈ {1, . . . , s}, variables z k s ∈ {0, 1} expresses the fact of whether or not the slot s is the last slot of the channel assigned to demand k. Finally, for each demand k ∈ K, each slot s ∈ {1, . . . , s} and each edge e ∈ E, variables t sk e ∈ {0, 1} indicates whether or not demand k uses the slot s on edge E. Due to page limit the reader is referred to [START_REF] Hadhbi | A novel integer linear programming model for routing and spectrum assignment in optical networks[END_REF] for further details on formulation RSA-BASE.

The extended formulation proposed in this paper consists of constructing a directed graph G from the network graph G as well as combining these three sets of variables into a single set of flow variables. The directed graph G is constructed by simply creating an in-going arc (u, v) and an out-going arc (v, u) for each edge uv in E(G). Then for each demand k ∈ K, each slot s ∈ {1, . . . , s} and each arc a ∈ A(G ), we define the binary variable f sk a as follows.

f sk a = 1, if demand k is routed through arc a and s is its last channel slot 0, otherwise.

It follows that every variable x, z and t from the formulation RSA-BASE can be described as a linear function of the above defined variables f . Indeed, we have that

x k uv = s s=1 f sk (u,v) + f sk (v,u) ∀uv ∈ E, k ∈ K, (1)
z k s = a∈δ + (o k ) f sk a ∀k ∈ K, s ∈ {1, . . . , s}, (2)
t sk uv = min{s,s+w k -1} i=s f ik (u,v) + f ik (v,u) ∀uv ∈ E, k ∈ K, s ∈ {1, . . . , s}. (3)
By applying this variable transformation we obtain the extended formulation hereafter denoted RSA-EXT. In this way, we can now perceive the RSA as a multi-commodity flow problem where each commodity is associated with a demand.

Notice that while formulation RSA-BASE requires s|K||E| + s|K| + |E||K| variables, formulation RSA-EXT requires 2(s|K||E|) variables. Therefore RSA-EXT uses less than twice the number of variables employed in RSA-BASE. Moreover, RSA-BASE and RSA-EXT uses the same number of constraints. Finally, RSA-BASE and RSA-EXT are equally strong in the sense that the optimal solutions of their linear relaxations have, by definition, the same value.

Formulation improvement

In this section we describe how formulation RSA-EXT can be reinforced and hence stronger than RSA-BASE.

Reinforcement of length constraints

In RSA-BASE, the maximum reach of a demand routed path is ensured through the following length constraints

e∈E(G) l e x k e ≤ ¯ k ∀k ∈ K, which become a∈A(G ) s s=1 l a f sk a ≤ ¯ k ∀k ∈ K, (4)
with the variable transformation described in Section 3. Consider now the following disaggregated length inequalities.

a∈A(G ) l a f sk a ≤ ¯ k a∈δ + (o k ) f sk a ∀k ∈ K, s ∈ {1, . . . , s}. (5)
Proposition 2. The disaggregated length inequalities (5) are valid inequalities.

Proof. For a given demand k and slot s, 0 ≤ a∈δ + (o k ) f sk a ≤ 1 since only one path is to be found for routing k. If a∈δ + (o k ) f sk a = 0, then demand k cannot be routed through channel {s-w k +1, . . . , s} and hence variable f sk a must equal 0 for any a ∈ A(G ). In return, if a∈δ + (o k ) f sk a = 1, then the path used for routing demand k should respect its maximum reach ¯ k .

Notice that by summing up inequalities (5) for each s ∈ {1, . . . , s} one obtains the original length constraints (4). Moreover, it is easy to see that there exists some fractional solutions that satisfy the original inequalities (4) but violate the proposed inequalities [START_REF] Jia | An integrated ILP model for Routing, Modulation Level and Spectrum Allocation in the next generation DCN[END_REF]. The following example illustrates such situation.

Consider the following RSA instance where the topology G is depicted in Figure 2 and all arcs have length 1. Let s = 2, and let K be composed of a single demand k going from node 1 to node 2, w k = 1 and L k = 2. Clearly the fractional solution proposed in Figure 3 verifies the original length constraints (4) but is cutoff by inequalities [START_REF] Jia | An integrated ILP model for Routing, Modulation Level and Spectrum Allocation in the next generation DCN[END_REF]. For this reason, we replace the original length constraints (4) by the proposed disaggregated length constraints (5) in the RSA-EXT formulation.

Making the extended formulation compact

In order to ensure path-continuity (i.e., the fact that the path leaving o k reaches d k for each demand k ∈ K) and to eliminate any cycle in the demand route, formulation RSA-BASE requires an exponential number of constraints. Next, we show how in RSA-EXT these constraints can either be dropped or replaced with a polynomial number of inequalities in order to obtain a compact extended formulation.

Ensuring path continuity

The demand's path continuity is ensured in RSA-BASE through the following constraints e∈δ(S)

x k e ≥ 1, ∀k ∈ K, S ⊆ V, o k ∈ S, d k / ∈ S, which become a∈δ + (S) s s=1 f sk a + a∈δ -(S) s s=1 f sk a ≥ 1, ∀k ∈ K, S ⊆ V, o k ∈ S, d k / ∈ S, (6)
with the variable transformation proposed in Section 3. Consider now the following classic flow conservation constraints. Proof. For a given subset S ⊆ V \ {d k }, o k ∈ S, and a demand k ∈ K, summing up equations [START_REF] Klinkowski | An overview of routing methods in optical burst switching networks[END_REF] for each v ∈ S yields the equation

a∈δ + (v) s s=1 f sk a - a∈δ -(v) s s=1 f sk a =      1, if v = o k 0, if v ∈ V \ {o k , d k } -1, if v = d k ∀k ∈ K, v ∈ V. (7
a∈δ + (S) s s=1 f sk a - a∈δ -(S) s s=1 f sk a = 1.
Since every variable is required to be nonnegative, a∈δ + (S) s s=1 f sk a ≥ 1. It follows directly that

a∈δ + (S) s s=1 f sk a + a∈δ -(S) s s=1 f sk a ≥ a∈δ + (S) s s=1 f sk a ≥ 1, ∀k ∈ K, S ⊆ V, o k ∈ S, d k / ∈ S.
Consider now the following disaggregated flow conservation constraints [START_REF] Klinkowski | Spectrum allocation problem in elastic optical networks -a branchand-price approach[END_REF].

a∈δ + (v) f sk a - a∈δ -(v) f sk a = 0 ∀k ∈ K, v ∈ V \ {o k , d k }, s ∈ {1, . . . , s}. (8)
Proposition 5. The disaggregated flow conservation constraints (8) are valid.

Proof. For any demand k ∈ K and any node v ∈ V \ {o k , d k }, the demand route leaves node v with channel {s -w k + 1, . . . , s} if and only if the route enters it with the same channel (recall spectrum continuity).

Notice that the disaggregated flow constraints [START_REF] Klinkowski | Spectrum allocation problem in elastic optical networks -a branchand-price approach[END_REF] are stronger than the initially proposed flow constraints [START_REF] Klinkowski | An overview of routing methods in optical burst switching networks[END_REF]. Indeed, summing up constraints (8) for s ∈ {1, . . . , s} yields the former constraints [START_REF] Klinkowski | An overview of routing methods in optical burst switching networks[END_REF]. For this reason, we replace the path continuity constraints (6) by the disaggregated flow constraints (8) in RSA-EXT formulation.

Dropping cycle-elimination constraints

In order to prevent cycles in the route of the demands, RSA-BASE includes an exponential number of cycle-elimination constraints. Next we show that for the objective functions studied in this paper, such constraints can be dropped, and whenever a solution including cycles is found, a simple post-processing is capable of finding a solution of at most the same cost without any cycles in polynomial time.

Cycles may appear in a demand route either attached to it or detached from it. For cycles attached to the route (e.g., Figure 4), including a polynomial sized family of degree constraints suffices for excluding such solutions. Proposition 6. The following degree constraints are valid:

a∈δ + (v) s s=1 f sk a ≤ 1 ∀k ∈ K, v ∈ V. (9)
Proof. For any demand k ∈ K, there can exist at most one arc leaving node v ∈ V with at most one channel.

For excluding cycles that are detached from the demand's route (e.g., Figure 5), an exponential number of inequalities is required in formulation RSA-BASE. For the objective functions studied here however such inequalities can be dropped, making formulation RSA-EXT compact. The next proposition describes how to obtain a feasible solution from a solution with cycles. The obtained solution has at most the same cost as the cyclic one. Proposition 7. Given linear objective function where every variable f sk a implies a cost c sk a ≥ 0, any solution to the RSA containing cycles can be transformed into an acyclic solution of at most the same cost as the former solution.

Proof. Let f be a solution of the RSA-EXT formulation where the route of a given demand k ∈ K includes a cycle detached from its main path (e.g., Figure 5). Let C denote the set of arcs composing a cycle in the route of demand k . Then, since every variable cost c sk a and every arc length is nonnegative, the solution f defined as

f sk a = f sk a ∀k ∈ K \ k , a ∈ A(G ), s ∈ {1, . . . , s}, f sk a = f sk a ∀a ∈ A(G ) \ C, s ∈ {1, . . . , s}, f sk a = 0 ∀a ∈ C, s ∈ {1, . . . , s},
is a feasible solution without cycle C that costs at most the cost of solution f .

It follows from Proposition 7 that whenever RSA-EXT yields a solution containing cycles, a postprocessing method is capable of providing an acyclic solution of at most the same cost in polynomial time. Such postprocessing involves a simple labeling algorithm. Our formulation is therefore compact and solves the RSA problem for all the objective functions studied.

Variable elimination

In order to boost the performances of the proposed formulation, a preprocessing method for eliminating variables may be applied.

Proof. If the shortest path from o k to d k passing through arc a ∈ A(G ) has length strictly greater than ¯ k , then arc a can never be used for routing demand k.

Computing the shortest path between two nodes s and t passing through an arc (u, v) can be done in polynomial time by computing the shortest path from s to u and from v to t. In order to identify the variables in Proposition 8 that can be eliminated, we apply this preprocessing method for every demand k ∈ K and every arc (u, v) ∈ A(G ). Such preprocessing is hereafter denoted as length preprocessing.

Since G is supposed to be connected, there always exists a path between any two given nodes in G . However, when applying the variable elimination described in Proposition 8, one might trouble this property. Let D k ⊆ A(G ) denote the set of arcs that are unable to route demand k due to length restrictions (i.e., the arcs identified through length preprocessing). Now, consider the practical graph for demand k ∈ K defined as

G k = (V (G ), A(G ) \ D k ). Graph G k is not
Proof. If there is no path from o k to d k passing through arc a, then arc a can never be used for routing demand k.

For identifying the variables that can be eliminated as a consequence of Proposition 9, it suffices to construct graphs G k and run a preprocessing similar to length preprocessing.

Computational results

In order to confirm the relevance of our approach, in this section we evaluate the computational performances achieved with the proposed extended compact formulation. For this, three network topologies are investigated: Spain, NSF and German. Spain topology consists of 5 nodes, 7 edges and 30 slots. NSF topology has 9 nodes, 13 edges and 120 slots. German topology consists of 17 nodes, 25 edges and 140 slots. For each network topology three sets of randomly generated demands are evaluated.

Tables 1-4 provide a sample of the performances obtained with each formulation using each of the objective functions described in Section 2. All the experiments were performed using the state-of-the-art MIP solver CPLEX 12.10 on a computer equipped with a 1.60 GHz Intel Core i5-8265U processor and 16 Gb RAM. A time limit of two hours was imposed in each run. For each formulation, the total time in seconds required for the optimization is displayed under column CPU. If the time limit of two hours is exceeded, the remaining gap percentage is displayed under column gap. The number of nodes explored in the enumeration tree is given under column node. The absence of results for some of the instances indicates that the model breaks down and no feasible solution was found within the two hours time limit. On all tested instances, we could either prove optimality faster or the remaining gap by the end of the time limit was smaller using formulation RSA-EXT

Figure 1 :

 1 Figure 1: The network G used in Example 1.

Figure 2 :Figure 3 :

 23 Figure 2: Topology G .

) Proposition 3 .Proposition 4 .

 34 The classic flow conservation constraints[START_REF] Klinkowski | An overview of routing methods in optical burst switching networks[END_REF] are valid.Proof. If v = o k (resp. d k ), then the demand path must leave (resp. enter) node v through some channel. If v ∈ V \ {o k , d k }, then the demand route can only leave (and must leave) node v if the route enters it. The path continuity constraints (6) are dominated by the classic flow conservation constraints[START_REF] Klinkowski | An overview of routing methods in optical burst switching networks[END_REF].

Figure 4 :Figure 5 :

 45 Figure 4: The flow path of an (s, t)-demand with a cycle attached to it.

Proposition 8 .

 8 If, for a given demand k ∈ K, the shortest path from o k to d k passing through arc a ∈ A(G ) has length strictly greater than ¯ k , then f sk a = 0 ∀s ∈ {1, . . . , s}.

Proposition 9 .

 9 necessarily a connected graph and hence the following proposition holds. If there is no path in G k from o k to d k passing through arc a ∈ A(G k ), then f sk a = 0 ∀s ∈ {1, . . . , s}.

Table 1 :

 1 Comparison between formulations RSA-BASE (left) and RSA-EXT (right) for objective function O 1 .

	Network |K|	CPU UB	LB gap node	CPU	UB	LB gap node
	Spain	15	2.7 18	18	0	0	0.2	18	18	0	0
	Spain	20	6.3 24	24	0	0	0.3	24	24	0	0
	Spain	25	38.6 37	37	0	21	0.9	37	37	0	0
	NSF	30 4167.4 69	69	0	165 16.5	69	69	0	0
	NSF	40 2468.0 90	90	0	0	38.0	90	90	0	0
	NSF	50	7200	-	98	-	-37.9	98	98	0	0
	German	40	7200	-93.7	-	-	51.6	95	95	0	0
	German	50	5194.6 82	82	0	0	43.3	82	82	0	0
	German	60	7200	-	176	-	-77.4 180 180	0	0

* This work was supported by the French National Research Agency grant ANR-17-CE25-0006, project FLEXOPTIM.

than when using RSA-BASE. For objectives O 1 and O 2 , the extended formulation could solve all tested instances to optimality within less than 1.5 minute, while the original formulation struggled substantially more. Indeed, 5 instances could not be solved to optimality within the time limit of 2 hours. Considering the instances that could be solved by both formulations, RSA-EXT was in average 65.7 times faster than RSA-BASE. For objectives O 3 and O 4 , RSA-BASE could only solve the instances tested over Spain network, and for all other topologies it could not even find a feasible solution. RSA-EXT, in return, only failed to find a feasible solution for the largest instance tested (i.e., German, 60 demands) with objective O 3 . Considering the instances that could be solved by both formulations, RSA-EXT was in average 4742 times faster than RSA-BASE. Such computational results strongly confirm the efficiency of the proposed extended formulation.

Concluding remarks

Objective functions related to efficient spectrum assignments were found to be much harder to optimize than the objective functions related to routing aspects. This has probably with the fact that when trying to fit all demands within a smaller subspectrum width, the non-overlapping requirements become increasingly restrictive and hence the interactions between demands grow to be progressively conflicting. Moreover, such spectrum related objectives induce a great degree of symmetry in the MILP. Symmetry-breaking techniques might be useful to be considered in these cases. As the number of demands to be routed increases, the root node becomes progressively harder to be solved. A promising idea is to consider other methods (e.g., Lagrangian relaxation) for obtaining good approximations of the linear relaxation values.