Philippe Langlois

Rafife Nheili

Christophe Denis

Numerical Reproducibility: Feasibility Issues

Keywords: Numerical reproducibility, floating-point arithmetic, finite element simulation, open Telemac-Mascaret, Tomawac

Floating-point arithmetic may introduce failures of the numerical reproducibility between a priori similar sequential and parallel executions of HPC simulation. We present how to apply some existing techniques to part of hydrodynamic finite element simulations. We analyze how easy these techniques allow us to recover its numerical reproducibility.

I. MOTIVATIONS

Exascale high performance computing will soon allow us to process 10 18 floating-point operations per second and so to enable the numerical simulation of larger, more complex and more sensitive physical phenomena. This huge amount of computing power proceeds from the massive and heterogeneous parallelism of our machines: current HPC computers consist in more than one million of computing units. Failures of the numerical reproducibility of some HPC simulations have been reported in several sensitive application domains as is energy [START_REF] Villa | Effects of floating-point non-associativity on numerical computations on massively multithreaded systems[END_REF], dynamical weather science [START_REF] He | Using accurate arithmetics to improve numerical reproducibility and stability in parallel applications[END_REF], dynamical molecular [START_REF] Taufer | Improving numerical reproducibility and stability in large-scale numerical simulations on gpus[END_REF] or dynamical fluid [START_REF] Robey | In search of numerical consistency in parallel programming[END_REF]. Indeed HPC is performed in a finite precision world where for instance, the floating-point addition is no more an associative operation. So computed values depend on the order of the operations and may be non-reproducible between a priori similar executions.

We illustrate a reproducibility failure case with a HPC code for the simulation at the industrial scale of free-surface flows in 1D-2D-3D hydrodynamic. The open Telemac-Mascaret suite is an integrated set of open source Fortran 90 modules [5]. It includes more than 300 000 lines of code issue from a 20 years international collaboration and declares 4000 registered users. The 2D-simulation of the Malpasset dam break (433 dead people and huge damage in 1959) is performed with a finite element resolution of the Saint-Venant equations. Unknowns are the water depth (H) and its velocity (U,V). The parallel resolution uses a subdomain decomposition of the triangular Numerical reproducibility is necessary to debug or to validate the computed simulation. Reproducibility may also be mandatory to satisfy legal agreements. The comparison between trustful sequential results and parallel ones is a common practice to check the correctness of the parallel implementation. How to conclude for our previous example? Of course, non reproducible executions may be the symptom of remaining bugs. But how to fix them if errors cannot be reproduced? In most cases, such failures damage the confidence in the whole numerical simulation process.

The operation order uncertainty for consecutive executions of a given binary file explains this non reproducible behavior. It appears both in parallel or in sequential+vectorized (SIMD) environments. One source is the parallel loop reduction provided by SIMD, openMP, MPI or GPU programming. The number of computing units p modifies the partial computed values before the reduction. Even for a given p, the computed reduced value depends on the dynamic scheduling of the reduction (openMP, MPI, GPU). One other source is the memory data alignment. This appears both in the parallel and the sequential case. Different p modifies the data alignment to the cache line boundary. Hence the vectorized iteration loop and the prologue or the epilogue parts apply to different numerical values. Sequential cases where this alignment depends on external events coming from the operating system have even been reported [START_REF] Corden | Consistency of Floating-Point Results using the Intel Compiler or Why doesn't my application always give the same answer?[END_REF] .

The reproducibility requirement is not a portability issue. Portability problems comes when one source code yields different binaries, i.e. binaries with different numerical properties. In our scope, the main portability parameters are the compilers, their options, the libraries, the few freedom spaces let by the IEEE-754 floating-point standard (rounding modes, double rounding, fused multiply and add operator) and the targeted computing unit. As mentioned before, reproducibility may fail for a given set of these portability parameters.

As already pointed out by [START_REF] Demmel | Fast reproducible floating-point summation[END_REF], the reproducibility requirement is not a claim for accuracy. Numerical reproducibility is getting bitwise identical results for every p-parallel run, p ≥ 1. Full accuracy, or accuracy up to the computing precision, is getting bitwise exact result. [START_REF] Demmel | Fast reproducible floating-point summation[END_REF] introduces reproducible but not necessarily accurate summation algorithms. Of course, improving the computed result accuracy up to the IEEE-754 correct rounding guarantees its numerical reproducibility.

In this paper, we study the feasibility issues of numerical reproducibility. Do existing techniques easily provide reproducibility to large software used for industrial scale simulations? In Section II we present one test case from the previously introduced hydrodynamic simulation code Telemac. In Section III we apply three existing scenarios to recover the numerical reproducibility of a simulation. A first conclusion is that all presented solutions succeed. This motivates us to describe how easy, or not, these reproducible simulations have been derived from the original code. Some recent algorithms specifically designed to ensure reproducible summation surprisingly appears to be, at least in our context, both the less easily applicable solution and the most expensive one. The chosen test case is a real simulation at the industrial scale. So the presented results are significant and encouraging. Nevertheless it only covers a small part of a general open Telemac-Mascaret treatment and a large amount of work remains to be produced. This also justifies this preliminary study that will help to identify the best choices for the future steps towards trustful and reproducible large scale simulations.

II. ONE TEST CASE AND ITS ANALYSIS

A. The test case and the original floating-point computation

We consider Tomawac, the Telemac's module for the wave propagation simulation in coastal areas. It solves a transport equation, a first order PDE changed as ODEs along its characteristic curves [START_REF] Hervouet | Hydrodynamics of free surface flows: Modelling with the finite element method[END_REF]. This turns to solve a diagonal linear system where the second member comes from a finite element assembly. Unknowns here are the significant wave height, the mean wave frequency and direction. We solve the Tomawac's Nice test case which simulates the effect of high-speed ferry waves when approaching the Mediterranean Nice harbour. The relative variations between the sequential execution (FPAss) and the p-parallel (FPAss p ) ones are measured at every mesh node for every time step for p = 2, 4, 8, 16. The non reproducibility of these computations is illustrated with Figure 1 which displays max |FPAss p -FPAss|/|FPAss|. The dashed line plots the computing precision (IEEE-754 binary64).

B. Analysis and solutions

The main computing step is the finite element assembly of the elementary contributions W e to every mesh node. For every mesh element e that contains the i node, it consists in the accumulation:

X(i) = elements e W e (i). (1)
The parallel resolution divides the mesh into subdomains that are distributed over the computing units. Subdomain decomposition introduces inner and interface nodes, the latter belonging to a common boundary between several subdomains, i.e. between several computing units. Inner node values X(i) are computed as previously while the interface nodes needs a two step accumulation: Relation (1) applied over the elements of one subdomain d k gives X d k sub (i). Then communications between the subdomains that contain the interface node i allows us to compute the final contribution:

X(i) = subdomains d k X d k sub (i). (2)
The parallel assembly step adds the communication and the reduction (2) of the subdomain's X d k sub (i) for every interface node i. In practice, every subdomain uses a local table that defines its communication scheme. For instance subdomain d k knows how much, which i and to which d k , it has to send X d k sub (i) and to receive X d k sub (i). The Tomawac's implementation of this communication scheme introduces a different accumulation order of a given i according to the subdomains. For p = 3 subdomains, the d 0 's computation of X(i) is:

X(i) = X d0 sub (i) + X d1 sub (i) + X d2
sub (i), while the d 1 and d 2 ones are:

X(i) = X d1 sub (i) + X d0 sub (i) + X d2 sub (i), !" !"# $$# %&# "!# "# &# ' '# ()*+,'-# +# *+# for idp = 1, ndp for ielem = 1, nelem i = IKLE(ielem,idp) X(i) = X(i) + W(ielem, idp)
Figure 2. Assembly step at the inner node i and its associated numberings.

and

X(i) = X d2 sub (i) + X d0 sub (i) + X d1 sub (i).
Hence the floating-point computed X(i) may differ over the subdomains. Let us remark that the dynamic scheduling of classical reductions may also generate the same differences. To ensure the solution continuity between the subdomains, Tomawac introduces one more communication step to share the maximum value of every X(i) (this choice being justified by physical reasons). As a side effect, this ensures the reproducibility between repeated simulations at a given p (which is not the case with the dynamic reduction). Nevertheless this is not enough to recover the simulation reproducibility for different p: in the previous relations, X(i) depends on the value p.

Before introducing three possible scenarios to ensure the numerical reproducibility, we present a practical but important assembly step feature. The well known implementation difficulty of these steps comes from the management of the node numbering. Finite element assembly introduces several for a node and tables from another. Figure 2 describes it: every triangular element ielem has ndp=3 nodes. Assembly loops over the domain or the subdomain elements, and then over the element nodes. The IKLE table returns the global number i of the node related to the contribution W e (i) in [START_REF] Villa | Effects of floating-point non-associativity on numerical computations on massively multithreaded systems[END_REF]. Let us note it defines indirect accumulations, i.e. two consecutive iterations of the inner loop do not apply to the same i value.

III. EVALUATION OF THREE REPRODUCIBLE SOLUTIONS

We propose to evaluate the following three existing techniques that provide reproducibility of this assembly step. Since here this latter is not ill-conditioned, the compensated summation of [START_REF] Ogita | Accurate sum and dot product[END_REF] yields a correctly rounded accumulation, and so an accurate and reproducible computation. Another choice are recent Demmel and Nguyen's reproducible sums [START_REF] Demmel | Fast reproducible floating-point summation[END_REF]. The latest Telemac release introduces integer conversions and so exact accumulations [5].

We use the following notations to measure the accuracy and the reproducibility of these solutions. A s denotes a sequential assembly algorithm and A p its p-parallel version. With

max rel (A 1 , A 2 ) = |A 1 -A 2 |/|A 2 |
, the measures of the A p accuracy compared to the original sequential FPAss s and of the reproducibility between parallel executions are respectively: acc = max rel (A p , FPAss s ), rep = max rel (A p , A s ).

A. Accurate compensated assembly

Compensated summation consists in computing every rounding error generated by the successive floating-point additions and to accumulate them in an error term. The addition rounding error is actually a floating-point number that can be exactly computed with the Knuth-Moller's 2Sum algorithm or (in base 2) with the Dekker-Kahan's Fast2Sum one, e.g. see [START_REF] Muller | Handbook of Floating-Point Arithmetic[END_REF]. The rounding-error accumulation is performed in parallel along the main accumulation. Hence the computation of the error term does not need the knowledge of the whole sum entries to start. This error term accumulation performed in floating-point arithmetic again generates rounding-errors but the same principle iteratively applies if necessary. Finally this error term is added to the main sum value and so compensates it. Ogita-Rump-Oishi prove that one compensation iteration improves the sum's accuracy as if this sum was computed in twice the computing precision u [START_REF] Ogita | Accurate sum and dot product[END_REF]. For condition numbers less than 1/u and reasonable sum lengths, the compensated algorithm returns a correctly rounded sum. It is our case here.

Compensated assembly steps are easy to derive for both inner and interface node assembly steps. Accumulation (1) is now written as:

[X(i), E(i)] = Compensated Sum elements e W e (i), where E(i) is the accumulated error term associated to elements e W e (i). The compensated assembly of an inner node is:

X(i) + E(i),
while the interface node communications encompass the pairs

[X d k sub (i), E d k (i)].
The compensated assembly of an interface node now is:

X(i) + G(i),
where the error term G(i) carefully takes into account the two step accumulation as follows. i) F k,k (i) is the rounding error generated by the partial accumulation in (2) over two consecutive subdomains d k and d k that share the interface node i:

X d k sub (i) + X d k sub (i).
ii) The whole error term in (2) is then computed, according to the parenthesis order, as:

G(i) = subdomains d k ,d k (E d k (i) + E d k (i)) + F k,k (i).
This compensated assembly fits well the existing one. It follows the iteration flow defined in Figure 2 for the inner nodes (Relation (1)) and for the interface ones (Relation (2)). We already noted the very slight modification of the communications between the computing units that now exchange the

(i), E d k (i)],
and not only its first one. Nevertheless no supplementary communication step is introduced here.

Tomawac recovers numerical reproducibility with this compensated assembly. Figure 3 exhibits that all parallel executions return the same simulation results up to the computing precision (rep for p = 2, 4, 8, 16). Since compensation also improves the accuracy, the acc plot actually measures the original accuracy of the floating point assembly FPAss s . It measures about one digit enhancement which is consistent with the error bound of the classical floating point accumulation. Compensation introduces a small amount of supplementary computation but we did not measure any significant increase of the whole simulation time.

B. Reproducible algorithm based assembly

Demmel and Nguyen have recently introduced algorithms to compute reproducible sums, i.e. bitwise identical results independently on the summation order [START_REF] Demmel | Fast reproducible floating-point summation[END_REF]. These solutions derive from Rump-Ogita-Oishi's AccSum and FastAccSum algorithms [START_REF] Rump | Ultimately fast accurate summation[END_REF]. A pre-rounding step returns K shrunks over the entries such that the floating-point sum of each shrunk is exact. Defining the right shrunk cutting width is the keypoint of this error-free vector transformation of the entries: it depends on the maximum absolute value of the entries and of their numbers. These ReproSum and FastReprodSum algorithms are parallel K-fold processes where K is a priori chosen; the larger K, the more accurate sum. In practice, they introduce two reductions (max and sum).

The main difficulty to integrate these algorithms within the assembly step in Tomawac is the need for maxima of the inner point and the interface point contributions, respectively max e W e (i) in (1), and one other maximum value for (2) we detailled a few lines further. As we already mentioned it, the assembly loop (Figure 2) consists in indirect accumulations. In this scope, ReprodSum and FastReprodSum introduces a significant volume of supplementary computations and communications. One more previous run of the assembly double loop is necessary to identify the maximum values for every inner node i (and also generates some extra-storage Maximum Relative Error vs. Sequential Computation

ReprodAssK2 P=1 ReprodAssK2 P=2 ReprodAssK2 P=4 ReprodAssK2 P=8 ReprodAssK2 P=16 ReprodAssK1 P=1 ReprodAssK1 P=2 ReprodAssK1 P=4 ReprodAssK1 P=8 ReprodAssK1 P=16 ; Figure 5.
Accuracy of the reproducible based assembly for K = 1, 2 compared to the compensated one. Mean frequency wave, Tomawac (Nice) on every computing unit). The interface node assembly is even more costly since every computing unit needs to use the same maximum value before computing both Relation (1) and Relation [START_REF] He | Using accurate arithmetics to improve numerical reproducibility and stability in parallel applications[END_REF]. For every subdomain d k , max e k ∈d k W e k (i) is first identified and reduced over the domain before computing every X d k sub (i) with Relation (1). Relation (2) is finally computed after a last classical communication step of the X d k sub (i). Hence the interface node assembly adds one other previous run similar to the Relation (2) implementation and the reduction of the identified partial maximum values.

Nevertheless the reproducibility failure of the floating point assembly is thus repaired for all considered parallel executions. Figure 4 exhibits it for K = 1 and K = 2. Figure 5 presents the accuracy behavior of these solutions for K = 1 and 2 compared to the accurate compensated assembly. As expected, the K = 1 case provides less accurate results, without being here significantly interesting even compared to the original floating point ones. It also illustrates that no difference appears for K = 2 between the ReprodSum based assembly and the accurate compensated one.

C. Integer converted assembly

The last solution towards reproducible assembly step is the integer conversion the latest Telemac release provides [5]. It consists in one 8 byte integer conversion of every floating point value concerned by assembly computations. Integer accumulations are exact as long as no overflow occurs and their conversions back to floating point values yield reproducible computed results. This 8 byte integer conversion is simple. We denote max(IN T K8) the maximum of these 8 byte integer values. A N -length floating point vector X is converted into the integer vector:

IX = IN T (X × Q 8 (X, N )), (3)
where the scaling factor Q 8 is chosen to avoid the overflow while summing X. We have [5]:

Q 8 (X, N ) = max(IN T K8) N × max(|X|) .
This conversion introduces two approximation levels. The first one comes from the difference between the floatingpoint discretisation and the integer one. We illustrate it with one example in decimal. Let N = 10 and X ∈ [10 2 , 10 6 [ (componentwisely); so max(X) = 10 6 . Let IX ∈ [0, 10 8 [ (componentwisely) thus max(IX) = 10 8 . The conversion IX = X × Q applies here with Q = 10. So every floating point value with more than one fractional digit suffers from a 10 -1 absolute error of conversion. This corresponds to a relative error from 10 -3 to 10 -7 for the whole X range that acts as data errors before the summation. Even if the sum is exactly computed, cancellation is well known to magnify the data errors. Hence no accuracy bound better than 10 -4 can be expected after such (fancy) conversion. The floating point evaluation of the scaling factor Q 8 introduces a second approximation level. Indeed evaluation (3) suffers from rounding errors in the Q 8 evaluation, in the multiplication and then in the division on the way back to floating point values. Of course, these errors are smaller than the previous discretisation ones but occur in every conversion [START_REF] Taufer | Improving numerical reproducibility and stability in large-scale numerical simulations on gpus[END_REF]. Nevertheless changing Q 8 to its next larger power of the base, i.e. to 2 α with 2 α-1 < Q 8 ≤ 2 α , in binary arithmetic, is a slight modification that avoids this approximation error. In this case, it is clear that no difference appears between the exact floating-point sum and its integer counterpart as long as X × 2 α ∈ IN T K8.

There is no difficulty to apply this integer conversion to the assembly process in Tomawac. The pre and post conversions are applied on-the-fly while accumulating inner nodes in [START_REF] Villa | Effects of floating-point non-associativity on numerical computations on massively multithreaded systems[END_REF]. Interface assembly step only needs to communicate the integer values to evaluate (2).

Integer conversion yields the expected reproducibility of the simulation as exhibited with Figure 6. It does not appear a significant loss of accuracy compared to the previous solutions. The Tomawac computation is a favourable case for this integer conversion strategy. Indeed few terms are accumulated for every node i and their ranges remain small enough to avoid suffering a lot from the discretisation error.

IV. CONCLUSION AND FUTURE WORK

We have been able to recover the numerical reproducibility of the Tomawac simulation with the three presented solutions. In this context we can summarize their current feasibility as follows. The compensated solution appears to be the easiest one to apply and provides accurate results for a low computing over-cost. The integer conversion provided in Tomawac is also easy to derive and introduces a low over-cost. Even if it is definitely not redhibitory here, this solution may introduce a risky loss of accuracy in other cases. A careful error analysis should be performed before other applications. The solution that uses the reproducible sums is efficient but applies less easily to our case and introduces a significant communication over-cost. Let us mentioned that the latest evolution of these algorithms only introduces one reduction step [START_REF] Demmel | Fast reproducible floating-point summation[END_REF] and so should introduce less communication cost in our context.

A preliminary step has been to identify what parts of such large computation generate failures of the numerical reproducibility while changing the number of computing resources. Assembly is an inevitable step in the ubiquitous finite element resolution. The chosen simulation was simple enough to successfully only focus on this assembly step. Future work will tackle the linear system resolution which is the second main step of such simulations. The efficiency, in term of computing time overcost, will be of most interest for this next stage.

Figure 1 .

 1 Figure 1. Floating-point assembly. Mean frequency wave, Tomawac (Nice)

Figure 3 .

 3 Figure 3. Compensated assembly. Mean frequency wave, Tomawac (Nice)

Figure 4 .

 4 Figure 4. Reproducible based assembly. Mean frequency wave, Tomawac (Nice)

Figure 6 .

 6 Figure 6. Integer assembly. Mean frequency wave, Tomawac (Nice)

Table I .

 I REPRODUCIBILITY FAILURE OF THE DAM BREAK SIMULATION

		The sequential run	a 64 procs run	a 128 procs run
	depth H	0.3500122E-01	0.2748817E-01	0.1327634E-01
	velocity U	0.4029747E-02	0.4935279E-02	0.4512116E-02
	velocity V	0.7570773E-02	0.3422730E-02	0.7545233E-02

element mesh (26000 elements, 53000 nodes, 2200 seconds with a 2 sec. time step). Table I exhibits the non-reproducibility of some randomly chosen point values with respect to the number of computing units: 1 vs. 64 vs. 128. Most of the parallel results share nothing more than the order of magnitude compared to the sequential result.

V. ACKNOWLEDGMENT Authors thank J.-M. Hervouet, EDR R&D (Chatou, France) for his valuable comments and his strong support to this work.


Kevin Bourhis
email: k.bourhis@argolight.com

Jonathan Massera

Laeticia Petit

Heikki Ihalainen

Alexandre Fargues

Thierry Cardinal

Leena Hupa

Mikko Hupa

Marc Dussauze

Vincent Rodriguez

Catherine Boussard- Plédel

Bruno Bureau

Claire Roiland

Monica Ferraris

Influence of P 2 O 5 and Al 2 O 3 content on the structure of erbium-doped borosilicate glasses and on their physical, thermal, optical and luminescence properties

Keywords: A. Glasses, C. Infrared spectroscopy, C. Nuclear magnetic resonance (NMR), D. Optical properties, D. Luminescence

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Materials doped with rare-earth (RE) (e.g. Nd 3+ , Er 3+ , Yb 3+ , Tm 3+ ) are of crucial importance in optoelectronics and are widely deployed in fiber amplifiers and solid-state lasers [START_REF] Saleh | Fundamentals of Photonics[END_REF][START_REF] Becker | Erbium-Doped Fiber Amplifiers: Fundamentals and Technology[END_REF]. The coincidence between the Er 3+ emission band around 1530 nm and the principal low-loss window in the absorption spectrum of silicate-based optical fibers has been the main driving force behind much recent work on erbium-doped silica optical fibers and waveguides [START_REF] Becker | Erbium-Doped Fiber Amplifiers: Fundamentals and Technology[END_REF][START_REF] Kenyon | [END_REF]. Intense efforts are on-going worldwide to increase the performances of the Er-doped silica-based fibers used as solidstate lasers and amplifiers. These performances are governed by the relevant electronic and optical characteristics of the active ion, such as absorption and emission cross sections, spectral shapes of the emission and absorption bands, excited state lifetimes, as well as the ion-ion interactions. The host material also has an important impact on all these properties [4,5]. Most of the work reported on Er 3+ -doped fiber lasers and amplifiers has used high silica-content glasses co-doped with various combinations of Ge, P, and Al atoms [START_REF] Becker | Erbium-Doped Fiber Amplifiers: Fundamentals and Technology[END_REF][START_REF] Kenyon | [END_REF]5]. The main limitations of Er-doped silicate fibers originate from the low solubility of the erbium ions resulting in the formation of ErEr clusters for the highest erbium concentrations. ErEr energy transfers most often result in the non-radiative relaxation of the ion pair leading to a quenching of the luminescence at high erbium contents. To prevent such clustering, the fiber core can be additionally doped with alumina or phosphorus pentoxide [6]. It was shown that silica-based glasses should contain at least 8-10 Al or 15 P atoms per rare-earth ion (mainly Yb 3+ and Nd 3+ ) to prevent RE clustering [710]. Moreover, these oxides have also a significant impact on the silica network [START_REF] Arai | [END_REF]. Du et al. showed that the introduction of aluminum cations in sodium silicate glasses leads to the formation of negatively-charged [AlØ 4 ] - groups, where Ø denotes a bridging-oxygen (bO) atom12. A charge compensation mechanism occurs within the glass structure resulting in the transformation of the silicate units with nonbridging oxygen atoms (noted Si-nbO) into units with bridging oxygen atoms (noted Si-bO). The Na + ions, previously neutralizing the Si-nbO atoms, are transferred to the [AlØ 4 ] -tetrahedra [12,13].

In Na 2 O-B 2 O 3 -SiO 2 (NBS) glasses the charge excess is preferentially compensated by the borate groups and in a lesser extent by the silicate groups, as evidenced by IR [12][13][14][15][START_REF] Moncke | Proc. Fifth Int. Conf. on Borate Glasses, Crystals and Melts Glass Tech[END_REF][START_REF] Massera | [END_REF] and NMR spectroscopies [12,18].

Co-doping silica glass simultaneously with Al 2 O 3 and P 2 O 5 presents an interesting cooperative behavior. Likhachev et al. showed that the rare-earth dissolution in the ternary Al 2 O 3 P 2 O 5 SiO 2 host systems was higher than in the binary P 2 O 5 SiO 2 or Al 2 O 3 SiO 2 glasses [19]. Although doping with Al 2 O 3 or P 2 O 5 increases the refractive index (RI) of silica glasses, codoping the silica glass with Al 2 O 3 and P 2 O 5 leads to the formation of a glass with a lower RI than that of silica, that the authors related to the formation of AlPO 4 -like units in the glass network [6,20,21]. The authors stated that such units are created at the expense of [PØ 4 ] -and [AlØ 4 ] -units, resulting in the formation of AlOP linkages with similar bond strength than those in SiOSi linkages [20]. This offers the possibility of doping the glass with higher concentrations of P 2 O 5 and Al 2 O 3 to improve the RE dissolution, thus limiting the RE clustering in the fiber while maintaining a low refractive index. To the best of our knowledge, there have been very few studies on the influence of P 2 O 5 and Al 2 O 3 on the properties of glasses with a low silica-content.

In this paper, erbium-doped sodium-borosilicate glasses with different Al 2 O 3 /P 2 O 5 ratios have been prepared to better understand the role of P 2 O 5 and Al 2 O 3 on the Er 3+ emission properties.

The main parameters affecting the luminescence at 1530 nm (absorption properties at 980 nm and 1530 nm, concentration of hydroxyl groups) have been investigated and related to structural modifications of the glass network induced by the introduction of P 2 O 5 and/or Al 2 O 3 .

Experimental

Sample preparation

Glasses having the composition 50SiO 1.

Before melting, the batches were treated at 400 °C for 15 h to completely decompose the raw materials. The batches were then melted in ambient atmosphere in alumina crucibles for 30 min between 1425°C and 1475°C, depending on the glass composition. After quenching on a brass mold, the glasses were annealed for 3 h at 40 o C below their respective glass transition temperature (T g ) to remove the internal stress. Finally the samples were cut, ground and polished. The glass compositions were checked by electron-probe micro-analysis and were found to be in accordance with the theoretical ones, within the accuracy of measurement (~ 0.1 wt%). The Er 3+ concentration is almost the same in all glasses except for the glass matrix for which it is slightly higher.

Optical measurements

A fully automated Metricon, model 2010 prism-coupled refractometer was used to measure the refractive indices (RI) at 1312 nm, n 1312 . The accuracy of the measurement is estimated to be ± 0.003. Each measurement was repeated 5 times and at least 3 measurements were performed on different areas of the glass. When the standard deviation (SD) of the measurements was superior to 0.003, the SD was used as uncertainty.

The UV-Vis-NIR absorption spectra were measured on 2 mm-thick samples with a doublebeam spectrophotometer (CARY 5000 UV-VIS-NIR) over the 250-2500 nm spectral region. The absorption spectra over the 2500-4000 cm -1 range were performed on a FTIR Bruker Alpha-T spectrometer with a spectral resolution of 2 cm -1 . The measurements were performed at room temperature and were corrected for the Fresnel losses, which are estimated to be about 7.73-7.67% at 1312 nm (extreme RI values = 1.502 and 1.493). The absorption cross-sections,  abs (), were calculated from the absorption spectra using Equation [START_REF] Saleh | Fundamentals of Photonics[END_REF].

 abs () = 2.303log(I 0 /I)/(NL) = ()/L, (1)
where log(I 0 /I) is the absorbance, N the rare earth ion concentration, L the thickness of the sample and  abs () the absorption coefficient.

Thermal and physical measurements

The glass transition (T g ) and glass crystallization (T x ) temperatures were measured by differential thermal analysis (Mettler Toledo TGA/SDTA851) at a heating rate of 15 o C.min -1 . The T g was taken at the inflection point of the endotherm, as obtained by taking the first derivative of the DSC curve. The T g was determined with an accuracy of 2 o C. The T x was taken at the maximum point of the exotherm point. The T x was determined with an accuracy of 5 o C, and was hardly or not distinguishable in some cases.

The density of the bulk glasses was measured after Archimedes' method in distilled water as an immersion fluid. The accuracy was better than 0.3 %.

Luminescence measurements

The emission spectra in the 14001700 nm range were measured at room temperature using an AOC (Applied Optronics Corp.) laser diode excitation source emitting at 980 nm, an Edinburgh Instruments monochromator (M300) and a liquid nitrogen cooled germanium detector (ADC 403L).

The emission spectra of the samples were collected on powder between quartz plates to avoid reabsorption.

Infrared spectra measurements

The reflexion infrared spectra were obtained using a Fourier-transform spectrometer (Nicolet 6700 FTIR), by utilizing a 12° off-normal reflection attachment and a high-reflectivity mirror as reference. A proper combination of DTGS detector and beam-splitters (germanium-coated KBr beam splitter or hybrid FIR beam splitter) allowed the measurement of continuous reflectance spectra in the 1007000 cm -1 range. Each spectrum represents the average of 200 scans at 1 cm -1 resolution. The spectrometer was purged with dry air to minimize atmospheric CO 2 and water vapor. The reflectance data were analyzed by the Kramers-Krönig transformation to obtain frequency-dependent phase angle between reflected and incident wave. The reflectivity and phase angle spectra were subsequently employed to calculate the optical and dielectric properties of glasses [22]. Infrared data reported in this work are in the form of the absorption coefficient spectra, a, calculated from the relation,

 = 4k = 2 '' /n, ( 2
)
where n and k are the frequency-dependent real and imaginary parts of the refractive index, respectively,  '' is the imaginary part of the dielectric constant and  is the frequency.

XPS spectra measurements

The X-ray photoelectron spectra were obtained using a PHI 5000 VersaProbe (PHI Electronics) spectrometer. Monochromatic Al Kα radiation (1486.6 eV) was operated under a residual pressure of 5×10 -9 mbar. The spectrometer was calibrated using the photoemission lines of Au (Au 4 f 7/2 = 83.9 eV, with reference to the Fermi level). Survey spectra were recorded from 0 to 1400 eV with a pass energy of 187.5 eV using 1 eV steps and four scans. High resolution spectra of the O1s, C1s and Si2p bands were recorded with a pass energy of 23.50 eV. Scan accumulation enabled to reach a satisfactory signal/noise ratio. Binding energies have been corrected using adventitious hydrocarbon carbon (C1s peak at 284.8 eV) as a reference.

11 B MAS NMR measurements

Solid State 11 B MAS NMR experiments were carried out with a Bruker Avance 600 spectrometer working at 192 MHz for 11 B. The spectra were acquired under magic angle spinning (MAS) at a spinning rate of 30 kHz using a 2.5 mm diameter rotor. A 50 kHz radio-frequence field was used for excitation, ensuring an homogeneous and selective excitation of the central transition.

A spin echo sequence was used, synchronized on the spinning speed, in order to avoid probe signal and distortion of the baseline. Thus, the delay between two pulses was 33 µs. Besides, it is known that both boron families, the [3] B and [4] B contributions, have different transverse relaxation times (T 2 ). In order to get some quantitative results, the T 2 were properly measured, T 2 ( [3] B) = 16 ms, and T 2 ( [4] B) = 10 ms, and the integrated intensities of each component were corrected following the equation I 0 = I(t).exp(t/T 2 ). The chemical shift scale was calibrated with respect to a 1 M B 2 O 3 solution. The spectra were reconstructed using the DMFIT software. The model chosen takes into account the spinning sidebands.

The Dell-Bray-Xiao model

The structure of alkaline borosilicate glasses can be described by the Dell, Bray and Xiao (DBX) model as described in [23]. Using the molar ratios R = SiO 2 /B 2 O 3 and K = Na 2 O/B 2 O 3 , this model is used to find the fractions of three (N 3 ) and four (N 4 ) coordinated boron in each glass after Equations 36.

 For 0  R  0.5 Na 2 O converts symmetrical trigonal [3] B into diborate [4] B units. In this range, N 4 is given by

N 4 = R (3)
 For 0.5  R  R max (where R max = 1/2 + K/16), a borosilicate network starts to form.

Additional Na 2 O in excess of R = 0.5 is used to destroy diborate groups and creates four loose [4] B units per diborate. Every [4] B unit is then bonded to four Q 4 tetrahedra to form a reedmergnerite (½(Na 2 O.B 2 O 3 .8SiO 2 )) group. In this range, N 4 is given by

N 4 = R (4)  For R max  R  R d1 (where R d1 = 1/2 + K/4),
N 4 = R max (5)
Note that for all values of R, N 3 is given by

N 3 = 1-N 4 (6)
In

N 4 = R' = (Na 2 O-Al 2 O 3 )/B 2 O 3 (= R' max ) (7)
The Q 3 and Q 4 silicate units proportion can thus be calculated from Equation 7:

Q 3 = (SiO 2 -2Na 2 O exc )/SiO 2 ; Q 4 = 1-Q 3 (8)
where Na 2 O exc is defined as Na 2 O-(Al 2 O 3 +2R' max ×B 2 O 3 )

Results

Physical properties

The evolutions of the refractive index at 1312 nm and of the glass transition temperature as a function of x are shown in Fig. 1a and 1b, respectively. The addition of P 2 O 5 (x = 0) or Al 2 O 3 (x =

3) in the glass matrix leads to a decrease of the refractive index. The error bar for x = 0 is quite large since a wide dispersion of the refractive index values was observed for this composition. It can be related to the low solubility of P 2 O 5 in silicate-based glasses in agreement with [26]. The refractive index can be considered as almost constant on the composition range, with slightly higher values for x close to 1.5.

The T g for x = 0 is lowered while it is almost equal to the matrix one for x = 3. A continuous increase of the T g can be seen when x increases from 0 to 3. The difference (T x -T g ) is presented in the inset of Fig. 1b. For the matrix glass, no crystallization peak was detected. The presence of P 2 O 5 (x = 0) or Al 2 O 3 (x = 3) gives rise to a maximum of (T x -T g ) at around 260 °C and 220 °C, respectively. This difference progressively decreases between x = 0 and 3 and reaches a local minimum at around 110 °C for x = 1.67.

UV-Vis-NIR absorption spectra

A typical absorption spectrum in the UV-Vis-Near IR range of an Er 3+ -doped glass (matrix) is depicted in Fig. 2a. We found that the UV absorption edge is not clearly affected by x and that the cut-off wavelength remains almost constant at  315 nm. In the 2501600 nm range, several bands characteristic of the Er 3+ ion transitions from the ground state to various excited levels are observed.

The absorption bands at 980 nm and 1530 nm are represented in Fig. 2b and2c, respectively. The shape of the band at 980 nm (inset of Fig. 2b) remains almost unchanged while some modifications are observed at 1490 nm and 1530 nm (inset of Fig. 2c) with the addition of P The absorption spectra recorded between 3000 and 4000 cm -1 are reported in Fig. 3 for some glasses (matrix, x = 0, 1.5 and 3). The spectra show a broad absorption band peaking at around 3500 cm -1 usually attributed to OH groups in several oxide glasses [27]. The free OH content can be estimated from the absorption coefficient at 3500 cm -1 , , using Eq. 9

[OH] = N A / (9) where N A is the Avogadro constant and ε is the molar absorptivity of the free OH groups in the glass, taken equal to ε = 49.1×10 3 cm 2 /mol [28].

The values of  and [OH] for the different glass compositions are listed in Table 1. They are comparable to those reported in several silicate glasses [29,30].

[OH] is close to 56×10 19 ions.cm -3 for most of the glasses, with slightly higher values for the glasses with x = 0.5 and 1. It has to be noted that [OH] in the P/Al-doped glasses is higher to the one of the matrix glass.

Luminescence properties

The emission spectra after excitation at 980 nm for some glasses (matrix, x = 0, 1.5 and 3) are depicted in Fig. 4a. They arise from the 4 I 13/2  4 I 15/2 transition typical of the Er 3+ emission in oxide glasses. As seen for the absorption bands at 1530 nm, the shape of the emission bands depends on the glass composition. The spectral bandwidth remains almost constant upon the composition modification (32-35 nm). The changes in the shape of the emission band (inset of Fig. 4a) were found to be more pronounced in the case of the glasses with 0  x  1.5. Fig. 4b presents the variation of the integrated area of the emission band at 1530 nm, A Em (1530), as a function of x.

A Em (1530) has a similar evolution as (980) and A Abs (1530). A Em (1530) strongly decreases when x increases, with a local minimum at x ~ 1.5. The glasses with x ≤ 1 exhibit a larger emission intensity at 1530 nm than the matrix glass, whereas the glasses with x > 2 exhibit a lower emission intensity.

Structural characterization

The IR spectra in the 250-1600 cm -1 range are depicted in Fig. 5a-c. The spectra are composed of broad bands situated in the 400700, 8001200 cm -1 and 12001500 cm -1 ranges. The spectra are dominated by an intense band at 1030 cm -1 accompanied by shoulders near 900 cm -1 and 1150 cm -1 . The high-frequency domain exhibits two bands at 1290 cm -1 and 1385 cm -1 with a shoulder at around 1470 cm -1 . In the low-frequency part, a rather intense band at 450 cm -1 and a less intense one at around 710 cm -1 are visible. As seen in Fig. 5a, the introduction of P 2 O 5 /Al 2 O 3 (x = 0 or 3) in the matrix glass strongly modifies the spectra. Though the position of the most intense band at 1030 cm -1 is not significantly affected by the compositional modifications, its width slightly increases on both low-and high-frequency sides. The intensity of the bands in the 12001600 cm -1 range decreases depending on the glass composition, a minimum amplitude of these bands being observed for the glass with x = 0. In the low frequency part, the intensity of the bands at 450 and 710 cm -1 increases, the band at 450 cm -1 also becomes sharper for the P Comparable evolutions are noticed when x increases from 0 to 1.5 and when x decreases from 3 to 1.5. When x increases from 0 to 1.5 or decreases from 3 to 1.5, a widening of the main band at 1030 cm -1 is observed when approaching the composition x = 1.5 for which the molar ratio Al/P = 1. The intensity of the bands in the range 12001600 cm -1 increases while the amplitude of the bands at 450 cm -1 and 710 cm -1 slightly decreases when x increases from 0 to 1.5 and when x decreases from 3 to 1.5 (which is less evident for the band at 710 cm -1 ). It has to be noted that the maximum of the band at 450 cm -1 is shifted to higher wavenumbers when x comes close to 1.5.

Fig. 6a presents the Si2p XPS spectra of some glasses (matrix, x = 0, 1.5 and 3). The Si2p spectrum of the glass matrix is composed of a wide band peaking at  102.8 eV. The addition of P 2 O 5 (x = 0) and/or Al 2 O 3 (x = 3) leads to a shift of the band to higher binding energy (BE). Fig. 6b represents the evolution of the Si2p band maximum as a function of x. When x increases from 0 to 1, the position of the Si2p remains almost constant, then shifts to lower BE when x increases up to 1.67. When x further increases from 1.67 to 3, the Si2p band shifts to higher BE.

The 11 B MAS-NMR spectra of some glasses (matrix, x = 0, 1.5 and 3) are presented in Fig. 7a. The two peaks centered at 12 and -1 ppm are attributed to [3] B and [4] B, respectively, in agreement with [18,31]. The relative values for both sites as a function of x are presented in Fig. 7b.

The [4] B fraction reaches its highest value (60%) for x = 0, decreases to 53% when x increases from 0 to 1.5 (close to the level of the matrix glass) and finally increases when x increases from 1.5 to 3.

The three (N 3 ) and four (N 4 ) coordinated boron were estimated using the Dell-Bray-Xiao model as explained in the experimental section. Table 2 lists the N 4 , N 3 , Q 4 and Q 3 proportion calculated according to the Dell-Bray-Xiao model. Since R max  R'  R d1 , the [4] B fraction is expected to be a constant which is in disagreement with the measurement. The DBX model also predicts that the Q 4 fraction increases linearly when x increases (see Fig. 8).

Discussion

Evolution of the refractive index and of the glass transition temperature

The refractive index depends on the overall polarizability of the glass matrix. ] -units, giving rise to a glass with a higher T g for x = 3

Evolution of the glass structure as a function of x

The structure of sodium-borosilicate glasses is generally described as mixed silicate and borate networks [18,3237]. Borate groups appear to exist largely as polymerized trigonal [3] B and tetragonal ( [4] B) units and silicate units as tetrahedral SiO 4 units (Q n ), where n is the number of bO atoms [12,17,3842].

In the IR spectrum of the glass matrix (Fig. 5a), the main band at 1030 cm -1 is associated to the SiO asymmetric stretching vibrations of the Q n groups, superposed with the asymmetric stretching vibrational modes of SiOB and BOB bridges at around 1080 cm -1 and 1250 cm -1 [33,35,4044]. The shoulder in the 900-1000 cm -1 range is related to the symmetric stretching of the SiO [4] B and [3] B O [4] B vibrations modes superposed to the asymmetric stretching of the [BØ 4 ] - tetrahedra [16,4042,44,45]. The two bands at 1270 cm -1 , 1385 cm -1 and the shoulder at 1470 cm -1 are related to the asymmetric stretching vibrations of various [3] B units in different ring and nonring configurations [13,34,35,[START_REF] Chryssikos | Proc. Int. Symp. Glass Sci. Tech[END_REF]45]. The band of low intensity at 710 cm -1 is related to the bending modes of the SiOSi, SiOB and BOB linkages and can be superposed to the bending vibrations of [3] B units [13,22,34]. The low-frequency envelope in the 400550 cm ] network isolated by Al 3+ cations [48,49]. The presence of such units in the glass network was suspected due to an improvement of the optical properties such as fluorescence enhancement or the decrease of the refractive index of the glass [6,8,[START_REF] Arai | [END_REF]. Here, the P 2 O 5 -and Al 2 O 3 -containing glass networks are suspected to contain simple isolated AlOP linkages rather than crystalline AlPO 4 units. Such units exhibit infrared absorption bands located at 730 cm -1 and at 1130 cm -1 according to [48,49], which eventually contribute to the appearance and increase in intensity of shoulders on both sides of the main absorption band (Fig. 5).Since AlOP linkages in AlPO 4 units have similar bond strengths than the SiO 4 units [20] their presence in the glass network is expect to lead to a small increase in T g (Fig. 1b). It is also supported by the strong decrease in the (T x -T g ) difference at around x ~ 1.5 (Inset of Fig. 1b), indicating the high tendency of the glass x=1.5 to crystallization that may be due to a higher number of AlOP linkages.

The DBX model was modified to take into account the formation of such linkages by simulating the formation of AlPO 4 units. In this hypothesis the simultaneous presence of Al 2 O 3 and P 2 O 5 leads to the formation of two AlPO 4 units, so that there is one more Na 2 O free to interact with the borosilicate network. Therefore, R', defined in Equation 8, can be modified as follows:

R* = (Na 2 O-Al 2 O 3 -P 2 O 5 +2×AlPO 4 )/B 2 O 3 (10)
The Q 3 and Q 4 silicate units proportion can thus be calculated from Equation 11:

Q 3 = (SiO 2 -2Na 2 O exc )/SiO 2 = 2×(R*-R max )/K; Q 4 = 1-Q 3 ( 11
)
It can be seen in Fig. 8 that the modified DBX model simulating the formation of AlPO 4 groups shows modification of the silicate network evidenced by XPS spectroscopy, but is able neither to give a quantitative analysis of the Q n group distribution not a good description of the borate network changes induced by the increase of x. Because phase separation can exist between the borate and the silicate network at the microscopic scale, it is possible to think that P 2 O 5 and/or Al 2 O 3 enter preferentially in the silicate or in the borate network, giving rise to discrepancies between the measured and predicted structure.

Na + , [PØ 4 ] 0 and [AlØ 4 ] -not only act on the silicate network, as predicted by Dell and co-workers, but also on the borate network.

The two [3] B and [4] B coordinations can be combined to form different groups like boroxol rings, orthoborate, pentaborate, tetraborate, metaborate and diborate groups, etc. The [4] B units give rise to tetrahedral network features in the glass structure, where BO 4 units are always negatively charged ([BØ 4 ] -), whereas [3] B units may be present as neutral

[BØ 3 ] 0 , charged units [BØ 2 O] -, [BØO 2 ] 2-or even [BO 3 ] 2-) [50].
P 2 O 5 has been assumed to form only neutral [PØ 4 ] 0 units, so that the number of nbO in the glass should be unchanged for the matrix glass and for x = 0. Since an increase of the Si BE and of the [4] B content is observed, it is possible to conclude that some part of the nbO atoms born by the silicate network is transferred to the borate network. A fraction of [3] B units may thus be present as charged [BØ 2 O] -(metaborate) units. Given the large variation in the [4] B fraction, it seems reasonable to suggest that phosphate groups enters preferentially the borate network. One cannot exclude the formation of [PØ 3 O] -units requiring Na + ions for charge compensation, which would slightly increase the amount of nbO atoms.

On the aluminate-rich side for x = 3, the [AlØ 4 ] -groups formed when Al 2 O 3 is introduced are charge-compensated by some Na + ions, leading to a higher amount of bO atoms in the glass network. The slight increase of [4] B fraction compared to the large XPS shift to high BE may indicate that the created bO atoms are born in a large extent by the silicate network, and in a lesser extent by the [3] B units under the form of neutral [BØ 3 ] 0 units.

When P 2 O 5 and Al 2 O 3 are mixed the silicate network is depolymerized as indicated by the downshift of the Si BE, while the equilibrium [4] B  [3] B is shifted to the right. Since the formation of AlPO 4 units is not a satisfying explanation, one may suggest that when only one dopant (P 2 O 5 or Al 2 O 3 ) is introduced, it is preferentially present in the phase with which it has an affinity. In the case of the mixed P 2 O 5 /Al 2 O 3 glasses the assumptions made above do not explain the observed tendencies of the IR, XPS and NMR measurements. When approaching the composition x = 1.5 the number of nbOs should be reduced compared to the aluminate-rich phase, and should be increased compared to the phosphate-rich phase. That is why it is possible that phosphate and aluminate group interact at the frontier of the silicate and borate phase domains, thus locally affecting both networks.

Other hypothesis may also explain some features, such as the formation of alumina-rich domains, as suggested by Monteil et al. by molecular dynamics [START_REF] Monteil | [END_REF]. Correlative multi-nuclei NMR studies may help solving these issues.

Influence of the structure on the emission properties

As seen in Fig. 2b-e and 4, the addition of Al 2 O 3 and/or P 2 O 5 has a strong impact on the absorption and emission properties of Er 3+ ions. The changes in the shapes and intensities of these bands tend to indicate that the Al and P atoms most probably participate to the second coordination shell around the Er 3+ ions, while being progressively introduced in the borosilicate glass matrix.

Similar changes in the surrounding environment of Er 3+ after the introduction of P 2 O 5 in sodium aluminosilicate glasses have been reported elsewhere [52]. The decrease of the absorption crosssections at 980 nm and of the absorption intensity at 1530 nm (Fig. 2d and2e) traduces a lowering of the 4f-4f transitions probabilities when x increases from 0 to 3. Therefore the Er 3+ site is expected to be less non-centrosymmetric with the progressive addition of Al than with the Al 2 O 3 addition confirming the strong role of phosphorus on the Er 3+ site distortion.

The decrease of the relative intensity of the emission band at 1490 nm over the one at 1530 nm when the Al 2 O 3 concentration increases may be explained by a modification of the electronic population of the 4 I 13/2 sub-levels favoring the sub-level at 1530 nm at the expense of the one at 1490 nm. This is also confirmed by the fact that the glasses with x ~ 1 have a higher emission intensity at 1530 nm than the glass with x = 3 while having a similar absorption cross-section at 980 nm but a higher OH content. The local maximum of the absorption cross-section for x ~ 1.5 (Fig. 2d and 2e) is most probably related to the formation of AlOP linkages in the network, which are also believed to have a significant Er 3+ site distorting role. However, while the glass with x = 1.5 exhibits a similar absorption cross-section at 980 nm and a similar OH content that the glass x = 0, the glass x = 1.5 exhibits however a lower emission at 1530nm. This is not fully understood and still under investigation. The low emission at 1530 nm of the glass with x = 3 can be related to the high content in hydroxyl groups as seen in Fig. 3, since it is well know that OH hydroxyl groups are serious quenchers of Er 3+ ions luminescence 54,55. Theoretical calculations by the Judd-Ofelt analysis are in progress in order to get a better description of the Er 3+ coordination environment.

Conclusion

The goal of this study was to evaluate the influence of P 2 O 5 and Al 2 O 3 content on the structure of erbium-doped sodium-borosilicate glasses and also on their physical, thermal, optical and luminescence properties. The addition of P 2 O 5 or Al 2 O 3 at the expense of B 2 O 3 and Na 2 O results in an increase of the polymerization degree of the silicate network while the fraction of [4] B increases, with various ratios of neutral and charged [3] B species. These changes in the structure have been related to changes in T g , n, absorption cross-sections at 1530 nm and 980 nm and also in the emission at 1530 nm. Al and P are strongly suspected to modify the coordination shell of Er 3+ , thus affecting the optical and luminescence properties. To conclude, the present study has demonstrated the better efficiency of phosphorous rather than aluminium in the declustering process of erbium ions in sodium borosilicate glasses with a low silica content. (Inset: T x -T g difference. For the matrix glass, no crystallization peak was detected). Solid line is a guide for the eye. 11 B MAS NMR spectra of the glasses matrix, x = 0, 1.5 and 3, (b) Evolution of the [3] B and [4] B as a function of x. Solid line is a guide for the eye.

Figure captions

  2 O 5 and/or Al 2 O 3 . The spectral bandwidth is almost constant in both cases (21-23 nm at 980 nm and 24-26 nm at 1530 nm). The evolution of the absorption cross-sections at 980 nm,  abs (980), and of the integrated area at 1530 nm, A Abs (1530), are plotted respectively in Fig 2d and 2e, respectively. They have very similar evolutions: the introduction of P 2 O 5 (x = 0) increases (980) and A Abs (1530) while these values are similar to the matrix ones when Al 2 O 3 (x = 3) is introduced. Both  abs (980) and A Abs (1530) decrease when x increases from 0 to 3. A slight increase of these values is noticed for x ~ 1.5.

2 O 5 /

 25 Al 2 O 3 -containing glasses compared to that of the matrix. The intensity of the bands of the glass for x = 1.5 are intermediate between the ones of the -matrix glass and the ones of the P 2 O 5 /Al 2 O 3 -containing glasses.

  P and Al are less polarizable than Na. Therefore the observed decrease of refractive index might be due to the partial replacement of Na 2 O by P 2 O 5 or Al 2 O 3 . The lower T g for x = 0 than for x = 3 can be explained by the number of bonding oxygen atoms born by the P or Al atoms. When P 2 O 5 is introduced in the glass (x = 0) [PØ 3 O] 0 units are formed, whereas when Al 2 O 3 is introduced in the glass (x = 3) [AlØ 4 ] -groups are formed. The 3bonded [PØ 3 O] 0 units, with one terminal atom, weaken the network compared to the 4-bonded [AlØ 4

Fig. 1 .

 1 Fig. 1. (a) Refractive index (n 1312 ) at 1312 nm and (b) Glass transition temperature (T g ) versus x

Fig. 2 .

 2 Fig. 2. (a) UV-visible-IR absorption spectrum of the Er-doped borosilicate matrix glass. Absorption cross-section spectra at (b) 980 nm and (c) 1530 nm corresponding respectively to the 4 I 15/2  4 I 11/2

Fig. 3 .

 3 Fig. 3. IR absorption spectra of some Er-doped borosilicate glasses with various contents of P 2 O 5 and Al 2 O 3 in the 30004000 cm -1 range.

Fig. 4 .

 4 Fig. 4. (a) Emission spectra of some Er-doped borosilicate glasses with various contents of P 2 O 5 and Al 2 O 3 ( exc = 980 nm) (Inset: normalized spectra) and (b) Emission integral A Em (1530) at 1530

Fig. 5 .

 5 Fig. 5. Normalized IR reflexion spectra of the glasses (matrix, x = 0, 1.5 and 3).

Fig. 6 .

 6 Fig.6. (a) Normalized XPS Si2p spectra of Er-doped borosilicate glasses with various contents of

Fig. 7 .

 7 Fig. 7. (a)11 B MAS NMR spectra of the glasses matrix, x = 0, 1.5 and 3, (b) Evolution of the[3] B

Fig. 8 . 33 FIG. 6

 8336 Fig. 8. Evolution of the silicate fractions (Q 3 and Q 4 ) calculated from the Dell-Bray-Xiao model

  2 -21.82Na 2 O-28.06B 2 O 3 -0.12Er 2 O 3 (matrix) and 50SiO 2 -20.51Na 2 O-26.36B 2 O 3 -0.12Er 2 O 3 -xAl 2 O 3 -(3-x)P 2 O 5 (x = 03 mol%) were prepared using NaH 2 PO 4 .H 2 O (Merck, 99%), H 3 BO 3 (Aldrich, 99.99%), Na 2 CO 3 (Sigma-Aldrich, >99.5%), Al 2 O 3 (Sigma-Aldrich, >99%), SiO 2 (Sigma-Aldrich, 99%), Er 2 O 3 (MV Laboratory, 99.999%) as raw materials in powder form. For the P/Al co-doped glasses, a fraction of the B 2 O 3 and Na 2 O content is substituted by P 2 O 5 and/or Al 2 O 3 . The compositions of the studied glasses are listed in Table

  N 4 is considered as constant and independent of the Na 2 O content. The model assumes that all Na 2 O in excess of R max are absorbed by reedmergnerite (½(Na 2 O.B 2 O 3 .8SiO 2 )) groups and form nbO atoms on the silicate tetrahedra. N 4 is given by

  -1 arises from the charged units brought by P 2 O 5 and Al 2 O 3 is similar independently for x = 0 or 3. However the presence probability of such units in the glass is low. Several authors reported the formation of AlPO 4 structural units when P 2 O 5 and Al 2 O 3 are introduced in pure silica glass network. The pure

	AlPO 4 crystal consists of a [PO 4 3-

  2 O 3 at the expense of P 2 O 5 . As the Pauling electronegativity of P (2.19) is higher than that of Al (1.61) [53, it is reasonable to think that the strong PO bonds have a higher influence on the Er 3+ site than the AlO bonds increasing the probability of the 4f-4f transition. It is interesting to notice that the changes in the Er 3+ absorption and emission band at 1530 nm are more pronounced with the addition of P 2 O 5

Table captions Table 1 .

 captions1 Glass composition, label, glass density, absorption coefficient at 3500 cm -1 , OH group concentration

Table 2

 2

	. Values of R (= Na 2 O/ B 2 O 3 ), R' (= (Na 2 O-Al 2 O 3 )/B 2 O 3 ), R* (= (Na 2 O-P 2 O 5 -
	Al 2 O 3 +2*AlPO 4 )/B 2 O 3 ), and N 3 , N 4 , Q 3 and Q 4 fractions calculated by the Dell-Bray-Xiao (DBX)
	model before and after modification.

Table 1 1 2

 12

	3
	4
	5
	6
	7
	8
	9
	10
	11
	12

Table 2

 2

	DBX model	Modified DBX model

Acknowledgments

A part of the research leading to these results has received funding from the European Union Seventh Framework Programme FP7/2007-2013 under grant agreement n°264526 through the GlaCERCo Marie-Curie ITN project. The Finnish Funding Agency for Technology and Innovation (TEKES) is acknowledged by nLIGHT for financial support. The Academy of Finland is gratefully acknowledged for the financial support of J.M.. The help of S. Guastella for the XPS measurements and of A. Venturella for the FTIR measurements is greatly acknowledged.

FIG. 8

rocking motion of SiOSi linkages as well as deformation modes of various borate species [13,22,35,45,46]. The introduction of P 2 O 5 (x = 0) or Al 2 O 3 (x = 3) in the glass matrix leads to the formation of [PØ 3 O] 0 and [AlØ 4 ] -units, which present typical absorption bands at  970 cm -1 [47] and at ~ 900 cm -1 [13], respectively, corresponding to the vibrations of the SiO(Al,P) linkages.

Given the low P 2 O 5 fraction (max. 3 mol%), phosphate atoms are expected to be mainly under the form of [PØ 3 O] 0 tetrahedra (no pyrophosphate or POP linkage), in analogy with phosphosilicate glasses [24]. The introduction of such units have an impact on the structure of the glass: the addition of P 2 O 5 or Al 2 O 3 at the expense of Na 2 O and B 2 O 3 leads to a glass network with a larger amount of bO atoms on the silicate groups than in the glass matrix as suspected from the shift of the Si2p BE band to higher energy which is in agreement with the Dell-Bray-Xiao model (See Fig. 8). One can notice that the addition of P 2 O 5 or Al 2 O 3 also leads to an increase in [4] B (Fig. 7b). As explained in [1216,18], these changes in the glass structure are suspected to be related to a charge compensation mechanism taking place to neutralize the negative charges of the [AlØ 4 ] -units. These changes in the glass network result in a larger amount of OH groups as shown in Fig. 3 When x increases from 0 to 1.5 or decreases from 3 to 1.5, the Si2p band shifts to lower BE indicating the formation of a less polymerized silicate network. This hypothesis is also comforted by the shift of the IR band at 470 cm -1 when x increases from 0 to 1.5 or decreases from 3 to 1.5. It also leads to a decrease in [4] B (Fig. 7b).

These changes in the glass structure are not in agreement with the evolutions predicted by the Dell-Bray-Xiao model. The Q 4 fraction is expected to increase linearly as illustrated in Fig. 8 and N 4 are supposed to be x independent in disagreement with our measurements. The discrepancy between the prediction of the DBS model and our results, can be explained by the fact that P 2 O 5 not only leads to tri-bonded [PØ 3 O] 0 units but also to charged [PØ 2 O 2 ] -units. Such units thus have the same behavior as [AlØ 4 ] -units and also need to be charge-compensated when present in the glass.

Assuming that the P 2 O 5 is only present under the form of [PØ 2 O 2 ] -units, the amount of negatively-FIG. 7