D Carpentier

T Jourdan
email: thomas.jourdan@cea.fr

Y Le Bouar

M.-C Marinica

Effect of saddle point anisotropy of point defects on their absorption by dislocations and cavities

Keywords: kinetic Monte Carlo, Diffusion, Irradiation, Dislocation, Cavity

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

Introduction

Supersaturation of point defects in metals occurs, for example, after plastic deformation [START_REF] Goods | The nucleation of cavities by plastic deformation[END_REF], hydrogen charging [START_REF] Gavriljuk | Hydrogeninduced equilibrium vacancies in fcc iron-base alloys[END_REF][START_REF] Neeraj | Hydrogen embrittlement of ferritic steels: Observations on deformation microstructure, nanoscale dimples and failure by nanovoiding[END_REF], solid or liquid state quenching [START_REF] Takamura | Quenched-in vacancies and quenching strains in gold[END_REF][START_REF] Hirsch | Dislocation loops in quenched aluminum[END_REF][START_REF] Thomas | Defects in aluminum quenched from the liquid state[END_REF] and irradiation [START_REF] Greenwood | The role of vacancies and dislocations in the nucleation and growth of gas bubbles in irradiatied fissile materials[END_REF]. This supersaturation leads to the formation of vacancy clusters such as voids, stacking fault tetrahedra and dislocation loops and, in the case of irradiation, self-interstitial clusters. These clusters are sinks for point defects and grow, thereby affecting the macroscopic properties of materials.

Fluxes of point defects to the sinks of the microstructure strongly depend on the elastic field created by the sinks, which modify the energy landscape. Striking examples of the role of elastic interactions between sinks and point defects on the final microstructure come from materials under irradiation. Swelling [START_REF] Greenwood | The role of vacancies and dislocations in the nucleation and growth of gas bubbles in irradiatied fissile materials[END_REF] and irradiation creep [START_REF] Heald | Steady-state irradiation creep[END_REF][START_REF] Mansur | Irradiation creep by climb-enabled glide of dislocations resulting from preferred absorption of point defects[END_REF] are due to a slight imbalance of absorption of point defects (self-interstitial atoms (SIAs) and vacancies) by the different sinks of the microstructure. In the so-called "dislocation bias model", swelling is due to the preferential absorption of interstitials by dislocations, which is responsible for a net flux of vacancies to cavities [START_REF] Brailsford | The rate theory of swelling due to void growth in irradiated metals[END_REF]. In this model, cavities are assumed to be neutral sinks, which means that they have no absorption bias for interstitials. Some models for irradiation creep, such as SIPA (Stress Induced Preferred Absorption), rely on the difference in climb velocity of different dislocation populations, depending on their orientation with respect to the applied stress. In such models, the climb velocity depends not only on the applied stress, but also on the stress field created by the dislocations [START_REF] Heald | Steady-state irradiation creep[END_REF]. In order to simulate irradiation induced phenomena such as swelling and creep, a proper description of the effect of the elastic field created by sinks on point defect migration is therefore crucial.

The simulation of long term microstructures under irradiation is conveniently performed by rate equation cluster dynamics [START_REF] Golubov | Grouping method for the approximate solution of a kinetic equation describing the evolution of point-defect clusters[END_REF][START_REF] Hardouin-Duparc | Microstructure modelling of ferritic alloys under high flux 1 MeV electron irradiations[END_REF][START_REF] Xu | Combining in situ transmission electron microscopy irradiation experiments with cluster dynamics modeling to study nanoscale defect agglomeration in structural materials[END_REF][START_REF] Jourdan | Efficient simulation of kinetics of radiation induced defects: A cluster dynamics approach[END_REF]. In this kind of mean-field model, the variation of the migration energy along the point defect trajectory cannot be taken into account explicitly. The effect of elastic interactions on point defect diffusion is found in the sink strengths for SIAs (k 2 i ) and vacancies (k 2 v ). The bias, which is defined as the relative difference between the sink strengths k 2 i and k 2 v , quantifies to what extent a sink preferentially absorbs SIAs or vacancies, depending on its sign.

A large amount of data exists about sink strengths and bias values, but the scattering is rather high. Assuming that dislocations are the only biased sinks, the dislocation bias can be inferred from experimental swelling data, using standard rate theory models [START_REF] Makin | The void swelling behaviour of electron irradiated type 316 austenitic steel[END_REF][START_REF] Walters | The electron irradiation of pure Fe-Cr-Ni alloys in temperature range 400 to 700 • C[END_REF][START_REF] Golubov | On recoil-energy-dependent defect accumulation in pure copper Part II. Theoretical treatment[END_REF][START_REF] Okita | A critical test of the classical rate theory for void swelling[END_REF][START_REF] Chang | Dislocation bias factors in fcc copper derived from atomistic calculations[END_REF]. However, swelling is a combination of the bias and the fraction of freely migrating defects, which is not precisely known for ion and neutron irradiations. Therefore, there is some uncertainty about the bias values obtained by such methods. Depending on the experimental data and the assumptions of the rate theory model, the dislocation bias values typically range between 0.01 and 0.35. Sink strengths and biases can also be computed. A common way is to solve the drift-diffusion equation for the concentration of point defects around a sink. This method was used for the first calculations, mostly for simple geometries and simple description of the sinkpoint defect interaction [START_REF] Woo | A Finite-difference Calculation of Point Defect Migration into a Dislocation Loop[END_REF][START_REF] Bullough | The interaction energy between interstitial atoms and dislocations and its relevance to irradiation damage processes[END_REF][START_REF] Dubinko | Numerical evaluation of the dislocation loop bias[END_REF][START_REF] Jourdan | Influence of dislocation and dislocation loop biases on microstructures simulated by rate equation cluster dynamics[END_REF]. For more complex cases, phase field [START_REF] Rouchette | Quantitative phase field model for dislocation sink strength calculations[END_REF] and object kinetic Monte Carlo (OKMC) simulations have been used [START_REF] Heinisch | The effects of one-dimensional glide on the reaction kinetics of interstitial clusters[END_REF][START_REF] Malerba | Object kinetic Monte Carlo study of sink strengths[END_REF][START_REF] Jansson | Sink strength calculations of dislocations and loops using OKMC[END_REF].

OKMC methods are particularly handy to take into account the effect of elastic interactions at stable and saddle points [START_REF] Sivak | Kinetic Monte-Carlo simulation of self-point defect diffusion in dislocation elastic fields in bcc iron and vanadium[END_REF][START_REF] Vattré | Nonrandom walk diffusion enhances the sink strength of semicoherent interfaces[END_REF].

The influence of elastic interactions at saddle position on the value of sink strength was emphasized by Dederichs and Schroeder [START_REF] Dederichs | Anisotropic diffusion in stress fields[END_REF]. Although this effect had been discussed previously [START_REF] Koehler | Diffusion of lattice defects in a stress field[END_REF][START_REF] Birnbaum | The effect of diffusivity gradients on diffusion to dislocations[END_REF][START_REF] Schroeder | Diffusion reactions in long range potentials[END_REF], these authors also suggested that the anisotropy of point defects at saddle position could have an effect on the sink strength. Such an effect was later confirmed for straight dislocations [START_REF] Tomé | Point-defect diffusion in a strained crystal[END_REF][START_REF] Chen | Anisotropic diffusion of point defects to edge dislocations[END_REF][START_REF] Skinner | Shape effect in the drift diffusion of point defects into straight dislocations[END_REF][START_REF] Borodin | The effect of diffusion anisotropy on dislocation bias and irradiation creep in cubic lattice materials[END_REF], infinitesimal dislocation loops [START_REF] Woo | Intrinsic bias differential between vacancy loops and interstitial loops[END_REF] and voids [START_REF] Borodin | Void bias factors due to the anisotropy of the point defect diffusion[END_REF]. However, all these works contain approximations to make calculations tractable, so the values of sink strengths significantly vary from one study to the other [START_REF] Borodin | The effect of diffusion anisotropy on dislocation bias and irradiation creep in cubic lattice materials[END_REF]. In addition, the elastic dipoles of point defects are based on empirical potential calculations, which are not always in agreement with first-principles calculations.

Only recently, the effect of saddle point anisotropy has been shown for the sink strength of semi-coherent interfaces, using OKMC simulations parameterized with density functional theory (DFT) calculations [START_REF] Vattré | Nonrandom walk diffusion enhances the sink strength of semicoherent interfaces[END_REF].

In the present work, we perform OKMC simulations to study the sink strengths of straight dislocations and spherical cavities in pure aluminum, in order to assess the role of saddle point anisotropy. To that purpose, the elastic interactions between sinks and point defects are modeled explicitly in the OKMC code. Point defects are represented by their elastic dipole tensors computed by DFT calculations.

This paper is structured as follows. Section 2 describes the method used to calculate sink strengths. Section 3 presents the study of the straight dislocation.

The case of a spherical cavity is treated in section 4.

Methods

OKMC simulations and sink strength calculation

Sink strengths are calculated with an OKMC code [START_REF] Vattré | Nonrandom walk diffusion enhances the sink strength of semicoherent interfaces[END_REF], allowing the simulation of many point defect trajectories in an efficient way, and making it possible to account for the point defects properties and elastic interactions between point defects and sinks [START_REF] Sivak | Kinetic Monte-Carlo simulation of self-point defect diffusion in dislocation elastic fields in bcc iron and vanadium[END_REF][START_REF] Subramanian | Method to account for arbitrary strains in kinetic Monte Carlo simulations[END_REF][START_REF] Vattré | Nonrandom walk diffusion enhances the sink strength of semicoherent interfaces[END_REF]. A single type of sink is introduced in the simula- To increment the simulation time, a residence time algorithm is used [START_REF] Gillespie | A general method for numerically simulating the stochastic time evolution of coupled and chemical reactions[END_REF][START_REF] Bortz | A new algorithm for Monte Carlo simulation of Ising spin systems[END_REF].

At a given time t, the time step is given by ∆t = -ln(r 1 )/Γ tot where r 1 is a random number chosen in ]0, 1] and Γ tot is the sum of the frequencies of all N e possible events, i. e. Γ tot = Ne-1 i=0 Γ i . The possible events are the creation of a point defect due to irradiation (frequency Γ 0 = G 0 ) or an atomic jump from a stable position to a neighboring one (frequency Γ i , i = 1, . . . , N e -1). The chosen event j is such that

j-1 i=0 Γ i < r 2 Γ tot ≤ j i=0 Γ i , with r 2
a random number chosen in ]0, 1]. Frequencies of atomic jumps are given by

Γ i = ν 0 exp(-∆E i /(k B T ))
, with ν 0 the attempt frequency assumed to be the same for all jumps, k B is the Boltzmann constant, T the temperature and ∆E i =

E sad i -E sta i
the difference of energy between the saddle point of the jump and the initial stable position.

The energy of point defects at stable point E sta i and at saddle point E sad i are given by

E sta i = - j,k P sta i,jk ǫ jk (r sta i ) (1)
E sad i = E m - j,k P sad i,jk ǫ jk (r sad i ), (2)
where E m is the migration energy without elastic interactions and P sta and P sad are elastic dipole tensors (P-tensors) describing the point defects at stable and saddle positions, respectively [START_REF] Kröner | Die Versetzung als elementare Eigenspannungsquelle[END_REF][START_REF] Siems | Mechanical interactions of point defects[END_REF]. The saddle position r sad is simply considered as the midpoint along the reaction coordinate between the two stable positions involved in the jump. The strain, written ǫ, is generated by the sink. Since we focus on the effect of saddle point anisotropy, we consider that P-tensors do not depend on the local strain, i. e. we neglect polarisability effects [START_REF] Kröner | Die Versetzung als elementare Eigenspannungsquelle[END_REF]. Other energy terms, such that higher order terms in the multipole expansion [START_REF] Siems | Mechanical interactions of point defects[END_REF] and image interactions for voids [START_REF] Wolfer | Segregation of point defects by internal stress fields[END_REF][START_REF] Ryazanov | Void bias factor in materials with weak cubic anisotropy[END_REF], are also neglected.

In our simulations, unless otherwise specified, the calculations are performed in isotropic elasticity, using available analytical expressions of the strain produced by the sink. This assumption is expected to be reasonable in the case of face centered cubic (FCC) aluminum, studied here, because the elastic moduli tensor of this material is only weakly anisotropic (see Tab. 1). This makes it possible to focus on the effect of point defects anisotropy only. However, to check the influence of the anisotropy of the elastic moduli tensor on the sink strength, simulations are also performed in anisotropic elasticity, using the values of elas-tic constants given in [START_REF] Tallon | Temperature dependence of the elastic constants of aluminum[END_REF]. In that case, a Fast Fourier Transform (FFT)-based method is used to compute the strain field at mechanical equilibrium [START_REF] Finel | Phase field methods and dislocations[END_REF][START_REF] Hu | Solute segregation and coherent nucleation and growth near a dislocation -a phase-field model integrating defect and phase microstructures[END_REF][START_REF] Rodney | Phase field methods and dislocations[END_REF].

Elastic moduli tensor terms and Zener anisotropy ratio [START_REF] Tallon | Temperature dependence of the elastic constants of aluminum[END_REF] C In the mean field rate theory, the sink strength k 2 defines the ability of a sink to absorb point defects. In this formalism, the evolution equation of the average number of defects N is given by

dN dt = G 0 -k 2 DN , (3)
with D the point defect diffusion coefficient, which reads

D = αa 2 0 ν 0 exp - E m k B T . (4)
In this equation, α = 2 3 for SIAs ( 100 dumbbells) and α = 1 for vacancies, and a 0 is the lattice parameter of the FCC lattice.

The absorption of point defects by the sink is expressed in Eq. (3) through the second term in the right-hand side. When the steady state is reached, the sink strength can be computed by

k 2 = G 0 DN . (5)
Two sink strength values are obtained, k 2 SIA for SIAs and k 2 vac for vacancies. From these two values, the sink bias B can be calculated by [START_REF] Heald | Point defect behaviour in irradiated materials[END_REF]

B = k 2 SIA -k 2 vac k 2 SIA . (6)
This parameter describes the ability of the sink to absorb more SIAs than vacancies.

In the OKMC simulations, the number of point defects in the box N is recorded every 10 4 steps. At the end of the simulation, the average number of defects at steady state N is computed by averaging the recorded values. The sink strengths and bias values are then computed using Eqs. ( 5)- [START_REF] Thomas | Defects in aluminum quenched from the liquid state[END_REF]. The simulated times are chosen to obtain sufficient accuracy on the value of N , and thus on k 2 .

A block-averaging procedure [START_REF] Flyvbjerg | Error estimates on averages of correlated data[END_REF] is used to compute the error estimates σ on the sink strength and bias. In the following, the error bars in figures correspond to ±3σ.

Representation of point defects

To compute interactions between the sink and the point defects, the Ptensors at stable points P sta and at saddle points P sad are needed. The values of P-tensors for SIAs and vacancies in aluminum are computed by DFT calculations. The DFT simulation cell for SIA/vacancy contains 256 ± 1 atoms.

Calculations are performed with VASP code using the projector augmented wave framework [START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF][START_REF] Kresse | Ab initio molecular-dynamics simulation of the liquidmetal amorphous-semiconductor transition in germanium[END_REF][START_REF] Kresse | Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set[END_REF][START_REF] Kresse | Efficient iterative schemes for ab initio totalenergy calculations using a plane-wave basis set[END_REF]. The plane wave energy cutoff is set to 450 eV and the Hermite-Gaussian broadening-width for Brillouin zone integration is 0.2 eV. The calculations are performed including the s semi-core states [Ne]3s 2 3p 1 . The exchange correlation energy is evaluated using the Perdew-Burke-Ernzerhof (PBE)

Generalized Gradient Approximation (GGA) [START_REF] Perdew | Generalized gradient approximation made simple[END_REF]. The k-point grid mesh is set to shifted 3 × 3 × 3 grid. Each configuration is relaxed using the gradient conjugate technique. The climbing image method [START_REF] Henkelman | A climbing image nudged elastic band method for finding saddle points and minimum energy paths[END_REF] using 15 images is used in order to localize the saddle point. In relaxed configurations, minimum or saddle point, the forces on each atom are lower than 0.001 eV/ Å. With such settings the relative error in the energy and P-tensor components is estimated to be less than 0.5% and 4%, respectively, from converged values [START_REF] Vattré | Nonrandom walk diffusion enhances the sink strength of semicoherent interfaces[END_REF].

For the configurations illustrated in Tab. 2, the following results are obtained:

1. For SIAs, migration energy of E m = 0.105 eV, and

• at stable point, dumbbell along [100]:

P sta SIA,ref =      19.652 0 0 0 18.518 0 0 0 18.518      eV (7)
• at saddle point, for a [100]-to-[010] migration path:

P sad SIA,ref =      19.498 1.133 0 1.133 19.498 0 0 0 19.034      eV (8)
2. For vacancies, migration energy of E m = 0.605 eV and

• at stable point:

P sta vac,ref =      -3.238 0 0 0 -3.238 0 0 0 -3.238      eV (9)
• at saddle point, for a migration along the [110] direction:

P sad vac,ref =      -2.866 -0.080 0 -0.080 -2.866 0 0 0 1.000      eV (10)
These values show that SIAs are weakly anisotropic at stable and saddle points, while vacancies are perfectly isotropic at stable point and strongly anisotropic at saddle point. The anisotropy of SIAs at stable point is in excellent agreement with the experimental value |P 11 -

P 22 | = 1.1 ± 0.3 eV [60].
The relaxation volumes at stable point for interstitials and vacancies, deduced from the trace of the dipole tensor through ∆V sta = Tr (P ) /3K with K the bulk modulus, seem slightly overestimated: they are equal to -0.40 Ω for vacancies and 2.35 Ω for interstitials (Ω is the atomic volume), while experimental values range from -0.05 to -0.38 Ω for vacancies and are equal to 1.9 ± 0.4 Ω for interstitials [START_REF] Schilling | Self-interstitial atoms in metals[END_REF][START_REF] Wolfer | 1.01 -Fundamental properties of defects in metals[END_REF].

Stable position Saddle position

SIA vac

Table 2: Schematic representations of stable and saddle positions of point defects in FCC lattice, with orientations corresponding to the P-tensors in Eqs. ( 7) to [START_REF] Brailsford | The rate theory of swelling due to void growth in irradiated metals[END_REF]. Here, black spheres represent the regular lattice atomic positions, red spheres represent atomic positions outside of lattice nodes and gray cubes represent vacancies.

To understand which properties of point defects have a major influence on the sink strengths, different simulation cases are built, using the previous Ptensors or simplified versions of them. The cases are given in Tab. 3.

The first case (case 0, P sta 0 = 0 and P sad 0 = 0) is considered as a reference to validate the method used in this work, by comparing the calculated sink strengths to the corresponding analytical solutions. Indeed, for the boundary conditions considered here, the analytical expressions only exist when the sink strain field in not considered, i. e. when the point defects do not interact with the sink.

An accurate assessment of the effects of elastic interactions on the sink strength is done using the real point defect description (case 2, P sta The following two approximations 2 ′ and 2 ′′ are used to identify the point defect properties influencing the sink strengths. Case 2 ′ corresponds to point defects with isotropic dipole tensors at saddle point but with the same relaxation volume as for case 2:

P sta 2 ′ = P sta ref ( 11
)
P sad 2 ′ = 1 3 Tr P sad ref I, ( 12
)
where I is the identity tensor. Such an approximation permits to enlighten the effect of saddle point anisotropy by comparing results to case 2. For case 2 ′′ , a further approximation is used: dipole tensors are taken isotropic at stable point, with the same relaxation volume as cases 2 and 2 ′ :

P sta 2 ′′ = 1 3 Tr P sta ref I (13)
P sad 2 ′′ = 1 3 Tr P sad ref I. (14)
Therefore, by comparing cases 2 ′ and 2 ′′ , the effect of anisotropy at stable point can be evaluated.

Finally, a traditional approximation in the calculation of sink strengths consists in ignoring the differences in point defect properties at stable and saddle points. Defect properties are given solely by the relaxation volume at stable point ∆V sta [START_REF] Heald | Point defect behaviour in irradiated materials[END_REF][START_REF] Heald | The preferential trapping of interstitials at dislocations[END_REF][START_REF] Dubinko | Numerical evaluation of the dislocation loop bias[END_REF][START_REF] Jiang | Effect of grain boundary stresses on sink strength[END_REF][START_REF] Jourdan | Influence of dislocation and dislocation loop biases on microstructures simulated by rate equation cluster dynamics[END_REF]. This is tantamount to using the following dipole tensors (case 1):

P sta 1 = P sad 1 = 1 3 Tr P sta ref I. ( 15
)

Straight dislocation

Case definition

The simulation box contains two dislocations, each dislocation having the following characteristics: line vector l = 1 As illustrated in Fig. 2, due to the periodic boundary conditions, the con-figuration is equivalent to a 2D array of dislocations in the xy-plane. To calculate the strain field, we have to sum along x the strain fields of infinite ycolumns of dislocations with alternate Burgers vectors, noted ǫ + and ǫ -for e x • b > 0 and e x • b < 0, respectively. The strain field generated by an infinite row of dislocations ǫ ± is given in [START_REF] Kuykendall | Conditional convergence in two-dimensional dislocation dynamics[END_REF] and [START_REF] Hirth | Theory of Dislocations[END_REF]. The complete strain field is then given by summing the contributions of neighboring boxes ǫ(x, y)

= N k=-N (ǫ + (x + k • d x , y) + ǫ -(x + k • d x , y
)), using a sufficiently large value for N to reach convergence. We have used N = 1, which gives a relative error on the strain lower than 10 -3 .

Sink strength and bias

Values of the sink efficiency Z = k 2 /ρ and of the bias for dislocation densities from 5 • 10 13 m -2 to 10 16 m -2 , encompassing the densities in irradiated materials [START_REF] Brager | Defect development in neutron irradiated type 312 stainless steel[END_REF], are shown in Fig. 3.

The reliability of our approach is proved by the good agreement between the results obtained in the case of non interacting defects (case 0, yellow squares) and the analytical expression [START_REF] Nichols | On the estimation of sink-absorption terms in reaction-ratetheory analysis of radiation damage[END_REF]. We have verified that the small differences are due to the different boundary conditions at the capture radius r c . Indeed, the analytical treatment assumes that the concentration of point defects is zero at exactly r = r c , while for the OKMC simulations, the defects are only removed after the last atomic jump that enters the capture region r ≤ r c , so the concentration is zero for some value lower than r c . As proposed in [START_REF] Hou | Modification on theory of sink strength: An Object Kinetic Monte Carlo study[END_REF][START_REF] Heinisch | The effects of one-dimensional glide on the reaction kinetics of interstitial clusters[END_REF][START_REF] Rouchette | Sink efficiency calculation of dislocations in irradiated materials by phase-field modelling[END_REF], this difference can be accounted for by using an effective capture radius r c slightly smaller than r c in the analytical formula to mimic the OKMC configuration.

When the complete P-tensors are used (case 2 -real defects, gray downpointing triangles), sink strengths for vacancies are twice higher than the values obtained when the elastic interactions are neglected, for the highest dislocation densities sudied. This increase is even more important for SIAs than for vacancies, and the difference increases with the dislocation density. This leads to a strongly positive bias value, increasing with the dislocation density.

To identify the properties of point defects responsible for the sink strength variation, the dipole tensors are modified. By removing the anisotropy of the elastic dipole tensors at the saddle point (case 2 ′ ), the sink strength is reduced for both types of point defects. The decrease is particularly important for vacancies, which are very anisotropic at saddle point. This shows that the saddle point anisotropy is responsible for a significant variation of the sink strength, through the decrease of the migration barriers near the dislocation.

The influence of stable point anisotropy can be assessed by comparing the latter results for case 2 ′ to the results for case 2 ′′ (defects simplified at stable and saddle points). Removing the anisotropy at stable points does not modify the sink strength values, and consequently leaves the bias unchanged. This shows that the anisotropy at stable point has no influence on the sink strength and sink bias in aluminum. This is due to the fact that, in this material, the anisotropy at stable point for SIAs is quite small, while vacancies are isotropic.

In Fig. 3 we also show the sink strength obtained when point defects have the same properties at stable and saddle points based on the relaxation volume at stable point (case 1, blue disks). The sink strengths are indeed quite different from the ones obtained for case 2. For SIAs, the sink strength is close to the one obtained for cases 2 ′ and 2 ′′ , due to the fact that the relaxation volumes at saddle points are similar: for case 1, [START_REF] Ham | Stress-Assisted Precipitation on Dislocations[END_REF][START_REF] Margvelashvili | Influence of an elastic field of a dislocation on steady-state diffusion fluxes of point defects[END_REF][START_REF] Wolfer | Diffusion of vacancies and interstitials to edge dislocations[END_REF][START_REF] Rauh | On the Diffusion Process of Point Defects in the Stress Field of Edge Dislocations[END_REF]. In Fig. 3 we plot the solution provided by Rauh and Simon [START_REF] Rauh | On the Diffusion Process of Point Defects in the Stress Field of Edge Dislocations[END_REF], which corresponds to the case of prescribed concentration on the system boundary (other expressions of the sink strengths correspond to slightly different boundary conditions, see Ref. [START_REF] Jourdan | Influence of dislocation and dislocation loop biases on microstructures simulated by rate equation cluster dynamics[END_REF] for a discussion of this point and a comparison of the formulas). It is shown that the results agree qualitatively but the analytical bias values are smaller than the OKMC values. Reasons for this discrepancy are given in Appendix A.

These results reveal the strong effect of elastic interactions, and more particularly show that the saddle point anisotropy of point defects is a key parameter for the variation of the sink strength and bias.

Analysis and discussion

In the previous section we stated that the decreasing migration barriers near the dislocation, due to the saddle point anisotropy, leads to a significant increase in the sink strengths, especially for the vacancy. These numerical results are compared with previous analytical calculations, using various approximations.

These previous studies were done in copper. However, due to similar point defect properties in aluminum and copper, trends are expected to be alike in both materials. The first estimation of the effect of saddle point anisotropy was obtained by Skinner and Woo [START_REF] Skinner | Shape effect in the drift diffusion of point defects into straight dislocations[END_REF]. To make the calculations tractable, they transformed the diffusion problem into a cylindrically symmetric one and assumed that the diffusion coefficient only depends on one component of the dipole tensor. They concluded that the effect was small and led to a slight increase in the sink strengths for vacancies and interstitials. With an arbitrary choice of the point defect anisotropy and diffusion tensor, Chen showed that the effect of anisotropy could be described by an effective relaxation volume [START_REF] Chen | Anisotropic diffusion of point defects to edge dislocations[END_REF].

The increase of the relaxation volume due to the anisotropy at saddle point is such that the effect on the sink strength is rather large. Borodin and Ryazanov did not resort to some of the assumptions made by Skinner and Woo [START_REF] Borodin | The effect of diffusion anisotropy on dislocation bias and irradiation creep in cubic lattice materials[END_REF] and obtained a few percents decrease of the sink strength for interstitials and an increase of about 20 % for vacancies. More recently, Sivak and Sivak [START_REF] Sivak | Efficiency of dislocations as sinks of radiation defects in fcc copper crystal[END_REF] showed with OKMC calculations that the anisotropy of vacancies at saddle point is responsible for complex variations of the sink strength with the dislocation type.

Here, by exactly taking into account the interaction between the dislocation and the point defect at saddle position, we show that an increase of about 20 % for interstitials and 100 % for vacancies can be obtained with respect to an isotropic defect.

We have also shown that the anisotropy of interstitials at stable point does not change the bias value. Several authors have pointed out that if the concentration at the boundary of the simulation cell is imposed, fluxes, and therefore sink strengths, do not depend on the stable point energy [START_REF] Dederichs | Anisotropic diffusion in stress fields[END_REF][START_REF] Skinner | Shape effect in the drift diffusion of point defects into straight dislocations[END_REF][START_REF] Borodin | The effect of diffusion anisotropy on dislocation bias and irradiation creep in cubic lattice materials[END_REF]. Therefore, in this case, no effect of defect anisotropy at stable point is expected. Here, the situation is different, since we impose the creation rate of defects and measure the sink strength by the average concentration in the simulation cell (Eq. ( 5)).

The concentration depends on the stable point energy, so the sink strength can potentially depend on the point defect anisotropy at stable position [START_REF] Rouchette | Influence of shape anisotropy of self-interstitials on dislocation sink efficiencies in Zr: Multiscale modeling[END_REF]. However, the anisotropy of interstitials is so low that the difference in energy between the isotropic and the anisotropic cases is negligible, leading to no appreciable effect on the sink strength.

To have a better understanding of the influence of the saddle point anisotropy,

v = v 2 1 + v 2 2 + v 3 3 .
the migration paths of point defects to the dislocations can be extracted from OKMC simulations. The fluxes of point defects are recorded through dividing surfaces between stable atomic positions during OKMC simulations. From these data, average migration paths are obtained. The result is represented in Fig. 4.

Results for point defects with isotropic saddle points (case 2 ′ ) are shown in Fig. 4 a-b. As expected, SIAs migrate and reach the dislocations through their dilated sides and are repelled from the compressed sides. On the contrary, vacancies mostly flow to the compressed sides. The difference is less clear than for interstitials, due to the lower magnitude of the elastic interactions.

When the saddle point anisotropy is taken into account (case 2), we obtain the fluxes in Fig. 4 c-d. These graphs show that the saddle point anisotropy not only has an influence on the sink strength but also on the point defects trajectories. Vacancies now migrate to the dislocations to enter through the lateral sides along the Burgers vector direction (±e x ). The difference is not so visible for SIAs, as the relative importance of the deviatoric component of their dipole tensors is smaller than for vacancies.

In the case of vacancies, the saddle point anisotropy also gives rise to nonzero component of the flux along the dislocation line direction. Note, however, that by symmetry the flux averaged in the xy-plane is zero in this direction. The fluxes along the dislocation line direction can be explained using the diffusion tensor, whose terms can be written [START_REF] Dederichs | Anisotropic diffusion in stress fields[END_REF]

D ij (r) = 1 2 ν 0 h h i h j exp - E sad h (r) -E sta (r) k B T , (16)
where (see Eqs. ( 1)-( 2))

E sta (r) = - k,l P sta kl ǫ kl (r) (17)
E sad h (r) = E m - k,l P sad h,kl ǫ kl (r + h/2). (18)
In these equations, h refers to the possible jumps from a stable position r to the neighboring one r + h. These possible jumps are represented in Fig. 5.

In the case of an isotropic saddle point (case 2 ′ ), P sad h,kl = P sad kl . In addition, if we assume that ǫ kl (r + h/2) ≈ ǫ kl (r), the saddle point energy E sad h (r) no longer depends on the jump and the diffusion tensor becomes

D ij (r) = 1 2 ν 0 exp - E sad (r) -E sta (r) k B T h h i h j . (19)
Since

h h i h j = 0 if i = j, (20)
an isotropic saddle point leads to a diagonal diffusion tensor. This conclusion also holds for a non uniform strain field, if it is expanded to first order in h.

On the contrary, this simplification cannot be made when the saddle point dipole tensor is anisotropic. The diffusion tensor terms D ij then depend on

h h i h j exp - P sad h,kl ǫ kl (r + h/2) k B T ≈ h h i h j exp - P sad h,kl ǫ kl (r) k B T . (21)
Because the strain has a different coupling with each jump h, the lattice symmetry is lowered and the different jumps in Fig. 5 no longer cancel each other out.

Non-zero off-diagonal terms, in particular D xz and D yz , can therefore appear and lead to flux components along the dislocation line direction.

The presence of such a flux component may have some implications on the effect of external stress on dislocation sink strengths. It has been found that the point defect anisotropy at saddle point induces a stress dependence of the dislocation sink strength, known as the "stress-induced preferential absorption due to anisotropic diffusion" (SIPA-AD) [START_REF] Tomé | Point-defect diffusion in a strained crystal[END_REF][START_REF] Skinner | Shape effect in the drift diffusion of point defects into straight dislocations[END_REF][START_REF] Borodin | The effect of diffusion anisotropy on dislocation bias and irradiation creep in cubic lattice materials[END_REF]. This phenomenon has been shown to be more than one order of magnitude larger than the usual SIPA effect, noted SIPA-I, due to the polarisability of point defects [START_REF] Skinner | Shape effect in the drift diffusion of point defects into straight dislocations[END_REF][START_REF] Woo | Irradiation creep due to elastodiffusion[END_REF]. Assessing the relative importance of both mechanisms is crucial in order to describe properly irradiation creep for example. For a uniaxial stress, SIPA-AD manifests itself through a dependency of the sink strength on the direction of the stress with respect to the dislocation line. For the sake of simplicity, let us consider the migration of the vacancy in an xy-plane perpendicular to the dislocation line.

The saddle point energy reads

E sad (ǫ a ) = E sad (0) -P sad 1 ǫ a x ′ x ′ -P sad 2 ǫ a y ′ y ′ -P sad 3 ǫ a zz , (22)
where ǫ a is the applied strain, expressed here in the basis (e x ′ , e y ′ , e z ) where P sad is diagonal, and P sad 1 = -2.946 eV, P sad 2 = -2.786 eV and P sad 3 = 1.000 eV are the three eigenvalues of P sad (see Eq. ( 10)). If a tensile stress is applied along the dislocation line (e z ), the energy becomes

E sad (ǫ a ) = E sad (0) -2.946νǫ a -2.786νǫ a -1.000ǫ a . (23)
If the stress is in the plane, for example along e x ′ , it reads

E sad (ǫ a ) = E sad (0) + 2.946νǫ a -2.786νǫ a + 1.000ǫ a . ( 24
)
Therefore it is easily seen that providing vacancies migrate in a plane normal to the dislocation line, a tensile stress along the line will decrease the migration barrier and thus increase the sink strength, whereas a tensile stress in the jump direction will have the opposite effect. If the vacancy does not migrate in this plane, the effect can be lower. In the works dealing with SIPA-AD, the large magnitude of the SIPA-AD effect has been obtained by assuming that point defect fluxes are perpendicular to the dislocation line [START_REF] Skinner | Shape effect in the drift diffusion of point defects into straight dislocations[END_REF][START_REF] Woo | Irradiation creep due to elastodiffusion[END_REF]. However, we have seen that when a three-dimensional model is used, saddle point anisotropy leads to fluxes with components along the dislocation line. Therefore, it is not clear whether the magnitude of the SIPA-AD will remain as large compared to SIPA-I. Additional work is needed to evaluate precisely the importance of both mechanisms, by studying specifically different stress levels and directions [START_REF] Sivak | Effect of external stresses on efficiency of dislocation sinks in BCC (Fe, V) and FCC (Cu) crystals[END_REF].

Importance of elastic moduli tensor anisotropy

Point defect anisotropy at saddle point has been identified as one of the main parameters influencing the sink strength value in isotropic elasticity. Although aluminum is a weakly anisotropic material (A = 1.21, see Tab. 1), it can be worth investigating the effect of the anisotropy of the elastic moduli tensor on the sink strength.

The results obtained for the dislocation bias are presented in Fig. 6. In this figure, both results are obtained with calculations using the analytical solution in Fourier space for the strain field [START_REF] Finel | Phase field methods and dislocations[END_REF][START_REF] Hu | Solute segregation and coherent nucleation and growth near a dislocation -a phase-field model integrating defect and phase microstructures[END_REF][START_REF] Rodney | Phase field methods and dislocations[END_REF]. With anisotropic elasticity, only a small variation of the bias values can be observed. It is much less important than the variation induced by the saddle point anisotropy. The influence of the elastic moduli tensor anisotropy is therefore far less important than the one of the point defect anisotropy. Consequently, as far as aluminum is concerned, the calculations can be done with isotropic elasticity. A similar conclusion was drawn in the case of iron, for which it was seen that the change in the sink strength is lower than 10% when elastic constants are changed from their values at 0 K (A = 2.3) to the ones at 1000 K (A = 4.5) [START_REF] Sivak | Dislocation sinks efficiency for self-point defects in iron and vanadium crystals[END_REF].

Spherical cavity

Case definition

To compute the cavity sink strengths and bias, the cubic simulation box of size d contains a single centered spherical cavity, as illustrated in Fig. 7.

This cavity is a perfect sink with a constant radius r c , with a capture radius of d reac = r c + r PD where r PD is the point defect radius (r PD = 0.16 nm). The cavity density only depends on the box dimensions and is given by ρ = 1/d 3 , so different densities can be studied by varying the box dimension d. The strain field generated by a cavity is calculated after the expressions given in [80,[START_REF] Borodin | Void bias factors due to the anisotropy of the point defect diffusion[END_REF]:

ǫ ij (r) = -2γ rc 4µ r c r 3 δ ij - 3x i x j r 2 . ( 25
)
In this equation, µ is the shear modulus of the material, the coordinates in the box x, y, z are noted x 1 , x 2 , x 3 for convenience, with

r = x 2 1 + x 2 2 + x 2 3 ,
and γ is the surface tension, set to the typical value of 1 J.m -2 [START_REF] Borodin | Void bias factors due to the anisotropy of the point defect diffusion[END_REF].

From Eq. ( 25), it can be seen that Tr (ǫ) = 0. As a consequence, for defects having an isotropic dipole tensor P = P 0 I, the energy induced by the elastic field is E = -P 0 Tr (ǫ) = 0. Therefore when both vacancies and SIAs have isotropic dipole tensors, e.g. in the usual approximation of case 1, the bias is zero, and the cavity is a neutral sink.

The simulations involving defects with isotropic dipole tensors at both stable and saddle points (case 1 and case 2 ′′ ) are then equivalent to case 0, with no interactions. Consequently, only the cases of non interacting defects (case 0), real defects (case 2) and defects simplified at saddle points (case 2 ′ ) are considered here. The cavity sink strength and bias are calculated for a cavity of radius r c = 1 nm, for a density between 10 21 and 10 23 cavities per m 3 , corresponding to densities commonly measured in irradiated materials [START_REF] Brager | Defect development in neutron irradiated type 312 stainless steel[END_REF]. The sink efficiencies Z are then calculated by

Sink strength and bias

Z = k 2 4πd reac ρ . ( 26
)
Results are presented in Fig. 8.

As in the case of dislocations, a good agreement is obtained between the sink strengths for non interacting defects (case 0, yellow squares) and the analytical expression for the cavity sink strength [START_REF] Nichols | On the estimation of sink-absorption terms in reaction-ratetheory analysis of radiation damage[END_REF], validating our approach. The small difference can be explained by the boundary conditions at the cavity surface, in the same way as for the dislocation.

It is particularly interesting to notice how the elastic interactions influence the sink strength values. The results obtained for real defects (case 2, gray downpointing triangles) show that the interactions increase the cavity sink strength for SIAs, but slightly reduce the sink strength for vacancies as compared to the case without elastic interactions. This yields a positive bias of 0.3 in average for the densities studied. The bias weakly depends on the cavity density, because of the short range of interactions. This bias value shows that the cavity cannot be considered as a neutral sink when the elastic interactions are taken into account.

The sink strengths obtained for point defects with isotropic saddle points (case 2 ′ , green triangles) are equal to the ones obtained for non interacting defects, so the bias is zero. Therefore, results show that the saddle point anisotropy of point defects alone is responsible for the increase of the bias value. The stable point anisotropy does not influence the sink strength nor the bias, which is consistent with the results obtained for the dislocation.

Fig. 9 presents the cavity bias for a density of 3.74 10 21 m -3 , for cavity radii from 1 nm to 5 nm, with consideration of point defects anisotropy (case 2). The cavity bias decreases when the cavity radius increases due to the decrease in 1/r c in Eq. ( 25), but even for a radius of 5 nm the bias is still above 0.1, which indicates that the cavity is a biased sink.

Analysis and discussion

In the same way as for dislocations, cavity sink strengths have been determined by analytical calculations, taking into account the diffusion anisotropy [START_REF] Borodin | Void bias factors due to the anisotropy of the point defect diffusion[END_REF].

With different boundary conditions as those used in our simulations, and under the assumptions that the cavity is small with respect to the simulation cell and that the strain is low, the expression obtained for the sink strength is remarkably simple: it depends on the components of the dipole tensor and varies as 1/r c . The bias values obtained from these expressions are represented in Fig. 9. This analytical expression is in agreement with our simulation results for the larger cavities, but the values differ for the small radii, for which the strain field is stronger. For our simulation parameters, these calculations show that for a cavity of radius 1 nm, the sink strength for vacancies is reduced by a factor 0.69 when the anisotropy at saddle point is taken into account, while the one for SIAs increases by a factor 1.29. In our simulations, the reduction for vacancies is less marked, whereas the value for SIAs is very close. Therefore, the bias value is higher in the analytical approach (B = 0.46). More in-depth investigation of the effect of saddle point anisotropy can be obtained by extracting point defect trajectories. They are represented in Fig. 10 in the xy-plane, for both isotropic (case 2 ′ , a-b) and anisotropic (case 2, cd) saddle points. In the case of isotropic saddle point, the average fluxes of both SIAs and vacancies are normal to the cavity. This corresponds to the fluxes that would be obtained without interactions between point defects and the sink. When the saddle point is anisotropic, trajectories of SIAs are not markedly changed. On the contrary, trajectories of vacancies are curved close to the cavity. Fig. 11 presents the probability of absorption on the surface of the cavity for a radius of 1 nm. These results confirm that the probability of absorption of SIAs is almost homogeneous on the cavity surface, with a small increase along the 111 directions. On the contrary, Fig. 11-b shows that the probability of absorption of vacancies is strongly anisotropic. Vacancies are absorbed almost only through the 100 directions. In a similar way, anisotropic concentration fields around the cavity were found by Borodin et al. [START_REF] Borodin | Void bias factors due to the anisotropy of the point defect diffusion[END_REF]. Qualitatively, it can be explained by the curvature of the point defect trajectories (Fig. 10-d), which converge to the 100 directions. In general, as emphasized in the case of the dislocation, vacancies tend to migrate more easily if the strain along the jump direction is negative (such defects are called "F-type defects" by Woo [START_REF] Woo | Intrinsic bias differential between vacancy loops and interstitial loops[END_REF]).

Tangential strain due to the cavity is negative, while normal strain is positive. Therefore, defects tend to migrate tangentially close to the cavity, which reduces the point defect absorption and the sink strength. This effect is more or less strong, depending on the orientation of jumps with respect to the strain field of the cavity. This lattice effect is a signature of the saddle point anisotropy.

In order to more precisely assess this trajectory curvature effect very near to the cavity, more involved simulations, including other energy terms that may be important at low distance [START_REF] Wolfer | Segregation of point defects by internal stress fields[END_REF], should be performed.

Conclusion

In this work, OKMC simulations have been performed to evaluate sink Similar results are expected for other FCC metals, as dipole tensors have the same features [START_REF] Vattré | Nonrandom walk diffusion enhances the sink strength of semicoherent interfaces[END_REF]. These results have implications on swelling and irradiation creep rates, and also suggest that the saddle point anisotropy of vacancies could play a significant role in many FCC based materials, in particular when solidification, heat treatments and/or deformation processes have led to a supersaturation of vacancies. In addition, this anisotropy could be of importance in coherent multi-phase materials, such as nickel based superalloys, in which vacancies diffuse in an inhomogeneous strain field resulting from both the internal multiphase microstructure and the thermo-mechanical treatment. In this section we compare the OKMC results for case 1 (spherical inclusions), for straight dislocations, with the analytical calculation provided by Rauh and Simon [START_REF] Rauh | On the Diffusion Process of Point Defects in the Stress Field of Edge Dislocations[END_REF]. This analytical study corresponds to a single dislocation with a prescribed concentration at a distance R from the dislocation position. The effect of the strain field of other dislocations is therefore not taken into account.

In order to explain the discrepancy observed in Fig. 3, the drift-diffusion equation has also been solved on two different cases (Fig. A.12), using the finite element (FE) method. More details about the methodology can be found in Ref. [START_REF] Jourdan | Influence of dislocation and dislocation loop biases on microstructures simulated by rate equation cluster dynamics[END_REF]. In the first case (Fig. A.12-(a)), the concentration is imposed at the outer boundary of the computation domain, corresponding to the analytical treatment from [START_REF] Rauh | On the Diffusion Process of Point Defects in the Stress Field of Edge Dislocations[END_REF]. The sink strength is deduced from the flux to the dislocation. In the second case (Fig. A.12-(b)), a source term is imposed, corresponding to our OKMC simulations. The sink strength is deduced from the average concentration of defects, using Eq. ( 5). In both cases, the effect of the strain field of surrounding dislocations is taken into account. On the contrary, FE simulations with imposed concentration tends to the an-

Results are compared in

  tion box. It can be a dislocation or a spherical cavity. The sinks are considered as immobile and remain unchanged after absorption of defects. The temperature is set to 300 K. At this temperature, thermal equilibrium concentration of point defects is far smaller than the concentration imposed by irradiation. Therefore, thermal generation of point defects by the sinks is neglected. SIAs and vacancies are considered separately, in dedicated simulations, thus no recombination is possible. They are generated uniformly at a constant creation rate G 0 (in s -1 ), and migrate inside the box by performing atomic jumps until they are absorbed by the sink. The migrating point defects do not react with each other to form clusters and no long-range interactions between point defects are considered. Periodic boundary conditions are used in all 3 dimensions. Point defects are considered as absorbed by the sink when the distance d between the sink center and the point defect verifies d ≤ d reac where d reac is the reaction distance depending on the nature of the sink.

2 = P sta ref and P sad 2 =

 22 P sad ref ).

√ 6 [ 1 √ 3 [

 613 1 1 2], vector normal to the glide plane n = 1 1 1] and Burgers vector b = ± a0 2 [ 1 1 0] ( b = a0 √ 2

  corresponds to a perfect straight edge dislocation in the aluminum FCC lattice. The capture radius of the straight dislocations is set to r c = 2 b , which means that for each dislocation d reac = 2 b [65, 20]. The two edge dislocations of opposite Burgers vectors are introduced in an orthorombic simulation box, according to the configuration in Fig. 1. Therefore, the dislocation density only depends on the box dimensions according to ρ = 2 dx•dy , so different densities can be studied by varying the box dimensions(keeping the aspect ratio constant). The lattice is rotated to ensure that the dislocation line vector is aligned with the z-direction of the box, while the Burgers vectors are aligned with the x-direction. Such a configuration ensures that the strain is continuous at the simulation box borders. Other configurations could have been studied, a few have been tested and the configuration choice does not alter the method nor the generality of our results.

Figure 1 :

 1 Figure 1: Configuration of the simulation box containing two edge dislocations, with opposite Burgers vectors.

Figure 2 :

 2 Figure 2: Schematic representation of the calculation of the strain with periodic boundary conditions. The strains ǫ + and ǫ -are generated by the columns associated to the edge dislocations with the Burgers vectors b such that ex • b > 0 and ex • b < 0, respectively.

Figure 3 :

 3 Figure 3: Straight dislocation sink efficiency for (a) SIAs and (b) vacancies, (c) bias. The error bars are displayed for all values, some of them are too small to be visible.

Figure 4 :

 4 Figure 4: Average fluxes of (a) SIAs and (b) vacancies with isotropic saddle points (case 2 ′ ), and (c) SIAs and (d) vacancies with real P-tensors (case 2). The flux v = (v 1 , v 2 , v 3 ) is recorded through surfaces between stable positions. The color corresponds to the component of the flux in the z-direction (v 3 / v ), and the lines represent the flux in the xy plane vxy = (v 1 , v 2 ). The width of the lines is here proportional to the norm of the flux

Figure 5 :

 5 Figure 5: Lattice stable position (gray sphere) and its first nearest neighbor positions (white spheres). Representation (a) in the xy plane and (b) in the yz plane of the simulation box.

  defects (2) with anisotropic elasticity Real defects (2) with isotropic elasticity

Figure 6 :

 6 Figure 6: Dislocation bias obtained for real defects (case 2) with interpolation of the analytical solution in Fourier space for the strain, with isotropic and anisotropic elasticity.

Figure 7 :

 7 Figure 7: Configuration of the simulation box containing a spherical cavity.

Figure 8 :

 8 Figure 8: Cavity sink efficiency for (a) SIAs and (b) vacancies, (c) bias, for a cavity of radius 1 nm.

Figure 9 :

 9 Figure9: Cavity bias B as a function of the cavity radius rc, for real defects (case 2), and the analytical expression computed from[START_REF] Borodin | Void bias factors due to the anisotropy of the point defect diffusion[END_REF]. The cavity density is 3.74 10 21 m -3 .

Figure 10 :

 10 Figure 10: Average fluxes of (a) SIAs and (b) vacancies with isotropic saddle points (case 2 ′ ), and (c) SIAs and (d) vacancies with real P-tensors (case 2). The same convention is applied as in Fig. 4.

  strengths and bias values of a straight dislocation and of a spherical cavity in aluminum. The elastic interactions between the sink and the migrating point defects are explicitly modeled and the point defects are represented by their elastic dipole tensors (P-tensors), which are computed by DFT calculations. The influence of the saddle point anisotropy on the sink strengths and bias has been quantitatively assessed. Results show that the values of sink strengths and bias are strongly increased

Figure 11 :

 11 Figure 11: Probability density of absorption p of (a) SIAs and (b) vacancies (case 2) on the surface of the cavity, for a cavity radius of 1 nm.

  Appendix A. Comparison of OKMC results and analytical calculations for straight dislocations (case 1)

  Fig. A.13. A very good agreement is obtained between FE simulations with imposed creation rate and OKMC simulations.

Table 1 :

 1 Elastic constants of aluminum at 300 K. The isotropic elastic constants are calculated from the tensor terms by the Voigt average[START_REF] Hirth | Theory of Dislocations[END_REF].

	11 (GPa) C 12 (GPa) C 44 (GPa)	A = 2C 44 / (C 11 -C 12 )
	106.51	60.38	27.8	1.21
		Constants for isotropic elasticity
	Poisson's ratio ν	Shear modulus µ (GPa)
	0.35			25.91

Table 3 :

 3 ). This

						SIA				vacancy
	Case 0	Non interacting defects				P sta SIA = 0 P sad SIA = 0				P sta vac = 0 P sad vac = 0
	Case 1	Spherical inclu-sions		P sta SIA = 18.896 I P sad SIA = 18.896 I 				P sta vac = -3.238 I P sad vac = -3.238 I
			P sta SIA =	   	19.652 0	0 18.518	0 0	   	P sta vac = -3.238 I 	
	Case 2	Real defects	P sad SIA =	    	0 19.498 1.133 0 1.133 19.498	18.518 0 0     	P sad vac =	   	0 -2.866 -0.080 0 -0.080 -2.866	1.000 0 0    
					0	0	19.034 
	Case 2 ′ Defects simplified at saddle points	P sta SIA =	   	19.652 0 0	0 18.518 0	0 0 18.518    		P sta vac = -3.238 I P sad vac = -1.577 I
			P sad SIA = 19.343 I
	Case 2 ′′	Defects simplified at stable and sad-dle points		P sad SIA = 18.896 I P sad SIA = 19.343 I				P sad vac = -3.238 I P sad vac = -1.577 I

P-tensors for the different simulation cases. The values are given in eV, for the configuration illustrated in Tab. 2. For vacancies, as they are isotropic at stable point, cases 2 ′ and 2 ′′ are identical.

  1 3 Tr P sad SIA = 18.896 eV and for cases 2 ′ and 2 ′′ , 1 3 Tr P sad SIA = 19.343 eV. On the contrary, the relaxation volumes at saddle points for vacancies vary a lot between case 1 and cases 2 ′ and 2 ′′ :

	for case 1, 1 3 Tr P sad vac	= -3.238 eV and for cases 2 ′ and 2 ′′ , 1 3 Tr P sad vac	=
	-1.577 eV. Consequently, the interactions are stronger for the vacancies in case
	1 than in cases 2

′ and 2 ′′ , giving a stronger sink strength. The dislocation bias obtained in case 1 is close to the one obtained for real dipole tensors (case 2), but this is completely incidental. Finally, various analytical expressions exist in the literature for case 1, but for isolated dislocations and different boundary conditions

alytical solution for low dislocation densities, whereas they depart from each other as the dislocation density increases. This means that the discrepancy between the analytical solution and the OKMC results mostly comes from the different boundary conditions, whereas at high dislocation densities the effect of surrounding dislocation strain fields also plays a role in the difference. (a) Γ D,1 Γ D,2 Γ N


Olivier A Martin
email: olivier.martin@lam.fr

Carlos M Correia

Eric Gendron

Gérard Rousset

Damien Gratadour

Fabrice Vidal

Tim J Morris

Alastair G Basden

Richard M Myers

Benoit Neichel

T Fusco

Point spread function reconstruction validated using on-sky CANARY data in multiobject adaptive optics mode

Keywords: Multi-object adaptive optics, Canary, PSF reconstruction

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

INTRODUCTION

For studies of formation and evolution of early galaxies, multiplexed observations of several objects need be carried out at once using integral field spectrographs over a large Field-of-View (FoV). The need to correct only a limited number of discrete directions over the field led to the concept of Multi-object adaptive optics (MOAO) which uses separate narrow field wavefront correctors for each of several objects. To solve optical issues encountered on both wide FoV and large dimension systems, the MOAO concept consists in putting Wave-Front Sensors (WFS) ahead the Deformable Mirror (DM) in the optical path. We adopt thus an open-loop configuration since there is no longer feedback between DM and WFS. Such a design has been proposed for Eagle ( 2 ), the first multi-object Integral Field Units (IFU) proposed for the European-Extremely Large Telescope (E-ELT) ( 3 ), with a large FoV of 5 to 10 arc-minutes required for science observations ( 4 ). Today, Mosaic is considered the new MOAO-assisted multi-object IFU for the E-ELT ( 5). The development of Mosaic will include a High-Multiplex Mode (HMM), seeing-limited or coarsely-sampled using the Ground Layer Adaptive Optics (GLAO) available on the E-ELT using the M4 ( 6 ) pre-focal adaptive mirror, for observing from 100 to 250 objects. In addition, it will also include a High-Definition Mode (HDM) for observing from 10 to 20 faint galaxies with MOAO-corrected IFUs.

The feasibility of designing such a multi-channel instrument has to be demonstrated on-sky mainly for two reasons. The first relates to the reliability and high-fidelity control of DMs in open-loop configuration. Whereas the recursive nature of a closed-loop hides potential inaccuracies in the linear relationship between WFS and DM -the interaction matrix -in open-loop the absence of feedback puts to light ever-present deviations, DM creeping, variable gains, hysteresis to cite only a few. We build on the past experiments which have demonstrated operational open-loop DM control, e.g. Volts, [START_REF] Andersen | VOLT: the Victoria Open Loop Testbed[END_REF] Villages, [START_REF] Gavel | Villages: an on-sky visible wavelength astronomy AO experiment using a MEMS deformable mirror[END_REF] Canary 9 and Raven. [START_REF] Lavigne | Design and test results of the calibration unit for the MOAO demonstrator RAVEN[END_REF] The second point relates to atmospheric tomography using both NGSs and LGSs, blending signals of different nature into a single estimation step.

In order to progress on these topics Canary has been designed by an international consortium, involving the LESIA and the Durham University starting as soon as 2007. The demonstrator was installed in 2010 at the William Herschel Telescope (WHT) at La Palma, in the Canary islands. It has provided proof of our capability to handle the open-loop configuration and specify the on-sky performance, either using NGSs only ( 11), or mixed NGS/LGS ( 12 ).

In Sect. 2 we provide a simplified analytic expression of the residual phase in the science directions. With it we can quantify the main contributors to the residuals, as tomography or system bandwidth. A successful attempt at PSF reconstruction in MOAO will represent a crucial milestone for a comprehensive understanding of the error budget.

In Sect. 3 laying out two methods to reconstruct the PSF: i) defining the analytic expression of the Optical Transfer Function (OTF), computed from off-science guide-star directions telemetry. It relies on the tomographic reconstructor and on calibration matrices -the same that are used during real-time operation -leading us to dub the method RTC-based PSFR. ii) a reference method that uses Truth Sensor (TS) telemetry looking at bright science objects and thus directly measuring the residual phase. The implementation is very similar to the one proposed by Véran ( 1 ), with slight modifications required for MOAO systems. In normal operational conditions such sensor won't be usable since MOAO systems will generally observe very faint targets.

The comparison is made on over 450 on-sky Canary data sets, acquired in September 2013. We report in Sect. 4, Strehl ratio (SR) and Full Width at Half Maximum (FWHM) evaluated on reconstructed PSF as function of sky quantities. We provide a statistical analysis to evaluate biases and variances using the RTC-based and TS-based method. We conclude in Sect. 5

MOAO SYSTEM MODELING

Analytic expression of the residual phase

We illustrate in Fig. 1 a schematic diagram of a single science channel MOAO system with two GS. Each of these sensing channels has a WFS, assumed to be a Shack-Hartman; it measures the uncompensated phase a αj in a direction α j . This WFS delivers noisy slopes assumed to be a linear function of the input phase through G -the Zernike interaction matrix, calibrated on the bench, which gives the WFS slopes from the Zernike modal coefficients according to

s α = Ga α + η α + r α , (1)
with s α the concatenation of WFS measurements, a α the Zernike modes corresponding to the parallel phase, η α and r α the additive measurement noise and aliasing respectively.

The WFS slopes s α are used to estimate the slopes s β in direction β through the tomographic reconstructor R defined in slopes space ( 13). The latter is calibrated on-sky. The estimation of the on-axis phase a β is then

WFS1 WFS2 a α1 a α2 a β 1 s β 1 + + η β 1 r β 1 + + s α2 s α1 η α2 η α1 r α2 r α1 + + - TS RTC ε β 1 + a ⊥β 1 m β 1 DM 1
Figure 1: Description by block functions of a elementary MOAO system using two sensing WFS to compensate the turbulence into one science direction. See text for a more complete description.

given by:

a β = CRs α ≃ CRGa α + CRη α + CRr α (2)
where we use C -the pseudo-inverse of G -to reconstruct the pupil-plane phase out of slopes. We further assume CG ≃ I, the identity matrix.

The DM produces a corrective phase m β , updated iteratively from a β . To reproduce the total system fractional delay 1 + ∆ t , with ∆ t calibrated on on the bench to be a fraction of 0.45 of a frame, we combine values of a β acquired at time t -1 and t -2. Mathematically, we have

m β (t) = (1 -g) × m β (t -1) + g ∆ t a β (t -2) + (1 -∆ t ) a β (t -1) . (3)
Applying the Z-transform to Eq. 3 gives

mβ (z) = hol (z) ˜ a β (z), (4)
where z = e -2iπν/νe , ν e the temporal sampling frequency and hol (z) the controller transfer function defined as

hol (z) = g × ∆ t + (1 -∆ t )z z(z -1 + g) . (5)
We can now ascribe the phase residual as

ε β = ε β + a ⊥β , (6)
where ε β is the in-band residual leftover after DM compensation

ε β = a β -m β . ( 7
)
whereas a ⊥β is the DM fitting error.

Combining Eqs. 2 4, 6 and 7, the Z-transform of ε β , noted εβ (z), becomes

εβ = ã β -hol CRGã α -hol CRη α -hol CRr α + ã⊥β . ( 8
)
where the term ã β -hol CRGã α is the aniso-servo error that combines both tomographic reconstruction and system temporal delay errors. For on-sky Canary performance diagnostic purposes, we have split this aniso-servo error into a sum of two components: a pure tomographic error plus a pure temporal error. We define the tomographic error as follows:

ẽ β = ã β -CRGã α , (9)
which allows us to rewrite Eq. 8 as

εβ = ẽ β Tomography + hrej CRGã α Servo-Lag -hol CRη α Noise -hol CRr α Aliasing + ã⊥β Fitting , (10)
where hrej , represented in Fig. 2, is the controller's rejection transfer function

hrej (z) = 1 -hol (z). ( 11
)
If we assume all the terms involved in Eq. 10 are independent, we get the residual phase WF error

σ 2 ε σ 2 ε ≃ σ 2 Tomography + σ 2 Servo + σ 2 Noise + σ 2 Alias + σ 2 Fitting , (12)
where the WF error is given by summing over the variance of individual modes. To obtain Eq. 12 from Eq. 10 the following approximations have been done:

• We have split the aniso-servo error as a sum of a tomographic error and a pure temporal one. We have thus neglected the correlation between ẽ β and hrej CRGã α . The rejection function hrej acts as a high pas filter according to Fig. 2, the signal hrej CRGã α is thus a high temporal frequency signal. According to, [START_REF] Martin | Démonstration de l'optique adapative multi-objet avec étoiles lasers par Canary[END_REF] the tomographic error ẽ β contains a large portion of low temporal frequencies. To a first degree, considering the statistical independence of Fourier modes, it is reasonable to neglect this correlation

• For the same reasons, we neglect the correlation between the aliasing hol CRr α , mostly composed by low temporal frequencies, and the servo-lag error hrej CRGã α .

• The correlation between the aliasing hol CRr α and the tomographic error ẽ β is considered as negligible as well. The aliasing is propagated through the tomographic reconstructor and the MOAO loop hol , we can consider it as a residual, as the tomographic error is. We consider thus this cross-correlation between residual as negligible relative to the main errors in Eq. 12.

• We further neglect the correlation between the high order modes ã⊥β above the DM cut-off frequency and the parallel ones included into the tomographic and servo-lag errors.

• We assume the DM is well represented by Zernike modes, using the Zernike interaction matrix calibrated on the bench.

In Sect. 4.1, we show how our modelling of ε β fits quite well to the measurements over a large set of data. A posteriori, we conclude these approximations are not critical for reconstructing the PSF at the Canary scale.

Analytic expression of the TS measurements

On Canary we have a Truth Sensor, i.e. an extra WFS that measures the residual phase in a single science direction on-axis. In practice, observing faint targets precludes the use of such sensor to probe the residual turbulence in the science directions of interest. As a technical demonstrator, Canary observes star asterisms composed of a bright central star, surrounded by up to two or three off-axis stars, usually separated by less than one arc-minute from the center [START_REF] Vidal | Analysis of on-sky MOAO performance of CANARY using natural guide stars[END_REF][START_REF] Morris | CANARY phase B: on-sky open-loop tomographic LGS AO results[END_REF] ). The Canary TS provides an estimation of the residual phase, and is involved for calibration and diagnostic purposes. The vector of slopes s β -see Fig. 1 -provides a measurement of the residual phase with additional noise and aliasing terms

s β = Gε β + η β + r β . (13)
Using the Zernike reconstruction matrix C, we get

ε β = Cs β ≃ ε β + Cη β + Cr β . (14)
Pursuing as before we get

˜ ε β ≃ ẽ β + hrej CRGã α -hol CRη α ε β +Cη β + C (r β -hol Rr α ) rε β . ( 15
)
Equation 15 shows that the TS measures a residual aliasing term, i.e. the difference between the pure TS aliasing rβ and the post-tomographic term hol Rr α . We define r ε β as this aliasing residual measured by the TS during observation as:

r ε β ≃ r β -Rr α . (16)
In Eq. 16, we do not consider the temporal filtering of the off-axis aliasing as it appears in Eq. 15. Since hol operates as a low pass filter, it means we neglect the temporal high frequencies of the aliasing.

MOAO PSF-R METHODS 3.1 OTF decomposition

The Optical Transfer Function (OTF) of an AO system is commonly modelled as a multiplication of the telescope OTF including static aberrations, and an AO OTF of the residuals ( 1). This latter is then split into a in-band parallel part, OTF ε β over spatial frequencies the AO systems has acted upon and a fitting orthogonal part, OTF ⊥β that gathers all the spatial frequencies above the DM cut-off frequency. The final OTF εβ , can be written as follows:

OTF εβ = OTF NCPA β × OTF Static β × OTF ⊥β × OTF ε β , (17)
where OTF NCPA β is the telescope OTF together with the Non Common Path Aberrations (NCPA) in directions β. It is mandatory to consider the real telescope OTF only once, which is already included into OTF NCPA β . We have to derive others terms in Eq. 17 in freeing them from this telescope OTF.

In tomographic systems the challenge is to be able to estimate OTF ε β without actual measurements of the residual phase on those directions. In the following, we propose to compare two different techniques:

• RTC-based method: it consists in estimating the residual phase measured by the TS when the loop is engaged. To do that, we generalize calculations presented in Sect. 2 to estimate the covariance matrix of the residual modes and then OTF ε β . See Sect. 3.2.1 for details.

The RTC-based method requires the off-axis measurements, the tomographic reconstructor R and the turbulence parameters as well. The latter are obtained from the Learn 3 steps approach, see paper 9909-163 William Herschel site characterization using Canary on-sky data in that conference. This is a profiling method based on the Learn & Apply technique ( 13) that consists in splitting the ground and altitude layers identification. We want to emphasise that the RTC-based PSFr handles information coming from the RTC that has been either acquired during the sky observation or calibrated on the bench.

• TS-based method: this is the adaptation of the Véran's method ( 1 ) to MOAO systems. It will actually not be available for MOAO systems observing faint science objects. It is used in Canary for demonstrating operational MOAO plus diagnostics as the one presented here. Using however the TS, we will be able to get the best reconstructed PSF as possible on Canary .

This method requires the time-series of TS measurements. To manage noise and aliasing from TS, it also needs the tomographic reconstructor and a model of the aliasing covariance matrix in loop disengaged. See Sect. 3.2.2 for details.

Determination of OTF ε β

We use the approach from Véran ( 1 ). The contribution of the mirror compensation into the residual phase from the long-exposure OTF is given by :

OTF ε β (ρ/λ) = e -0.5× Dφε β (ρ) , (18)
where D φε β is the residual phase structure function which when averaged over the telescope pupil becomes :

Dφε β (ρ) = P φ ε β (r) -φ ε β (ρ + r)
2 P (r)P (ρ + r)dr P P (r)P (ρ + r)dr

, ( 19
)
where P is the telescope pupil domain and P (r) the pupil function. We evaluate these integrals for 36 Zernike polynomials (piston excluded) which match the spatial frequencies the Canary DM is able to correct for ( 11).

φ ε β (r) = 36 i=2 ε β Z i (r). ( 20
)
The structure function then becomes :

Dφε β (ρ) = n i,j ε β (i)ε β (j) U ij (ρ), (21)
where :

U ij (ρ) = P P (r)P (ρ + r)dr (Z i (r) -Z i (r + ρ)) × (Z j (r) -Z j (ρ + r)) d 2 r P P (r)P (ρ + r)d 2 r (22)
The estimation of the PSF boils down to estimating the covariance matrix of the residual phase expressed on the Zernike modal basis Σ ε β ε β (i, j) = ε β (i)ε β t (j) .

RTC-based PSF/R

The RTC-based method consists in determining ε β from the off-axis WFS telemetry. Eq. 10 gives an analytic expression of the Z-transform of the residual modes ε β . In subtracting off the orthogonal part a ⊥β and in transforming back to the real space, we get :

ε β ≃ e β + a ε β + CR(h ol * η α ) + CRr α , (23)
where h ol the impulsive response of the MOAO system, (. * .) the convolution product and a ε α is the output of a first order low-pass filter defined by :

ãε β (z) = hrej (z) × CRGã α (z), ( 24
)
where hrej is given by Eq. 5 and Eq. 11. From Eq. 23 and Eq. 9, assuming all the terms involved in Eq. 23 are independent processes, we get the following expression of Σ ε β ε β :

Σ ε β ε β = Σ e β e β + Σ a ε β a ε β + hol 2 × CRΣ ηαηα R t C t + CRΣ rαrα R t C t , (25)
where

• Σ e β e β = e β e β t is the tomographic error covariance matrix. Without a TS, we cannot determine the value of e β . Nevertheless, from the turbulence parameters retrieved using the Learn 3 steps approach, we have a model of this covariance matrix ( 15 ).

• Σ a ε α a ε α = a ε α a ε α
t is the covariance matrix of the filtered atmospheric parallel modes by the MOAO system. It is readily derived from the time-series of the off-axis WFS measurements and parameters from the loop during the observation.

• Σ ηαηα = η α η α t is the noise covariance matrix of off-axis WFS telemetry, identified by a parabolic interpolation, near to a zero-delay, of temporal auto-covariance of s α • Σ rαrα is the covariance matrix of the off-axis aliasing. It is computed from the model of the spatial covariance of slopes based on outputs from the Learn 3 steps approach.

TS-based method

The SF in Eq. 21 requires the covariance matrix of parallel modes Σ ε β ε β estimated from ε β (ρ, t). Using Eq. 14, we get :

Σ ε β ε β ≃ Σ ε β ε β + CΣ ηβηβ C t + CΣ r ε β r ε β C + Σ ε β r ε β C t + CΣ r ε β ε β . ( 26
)
The matrix Σ ε β ε β can be readily estimated from the times series of s β (ρ, t). The noise covariance matrix Σ ηβηβ is estimated from the temporal auto-correlation function of s β . To determine the covariance matrix of the aliasing Σ r ε β r ε β and the cross-term Σ ε β r ε β , we consider Eq. 16. The covariance matrix Σ r ε

β r ε β = r ε β r ε β t is
derived from the following expression :

Σ r ε β r ε β = Σ rβrβ + RΣ rαrα R t -Σ rβrα R t -RΣ rαrβ , (27)
where the matrix Σ rβrα is the concatenation of the covariance matrices of off/on axis aliasing in loop disengaged.

To determine the cross-term Σ ε β r ε β , we consider Eq. 10 that gives the analytic expression of ε β . Assuming the independence of terms given in Eq. 10, the TS aliasing r ε β is only correlated with ε β . We thus have :

Σ ε β r ε β ≃ C Rr α r ε β t C t . (28)
Combining Eq. 28 with Eq. 16 leads thus to:

Σ ε β r ε β ≃ C Rr α (r β -Rr α ) t C t ≃ C RΣ t rαrβ -RΣ rαrα R t C t . (29)
Finally, in combining Eq. 26, Eq. 27 and Eq. 29, one gets the final expression of Σ

ε β ε β Σ ε β ε β ≃ Σ ε β ε β -CΣ ηβηβ C t -CΣ rβrβ C t + CRΣ rαrα R t C t . ( 30
)
Covariance matrices of aliasing required in Eq. 30 are derived analytically using the turbulence and system parameters, see paper 9909-163 in that conference.

Determination of fitting and static OTFs

Static OTF, OTF Static β

In addition to NCPA, we have to deal with common static aberrations a β in the science directions. These are not strongly mitigated by the loop as it is done for closed-loop systems. In MOAO, they must be calibrated on-sky and compensated by a DM offset. Even if the science object are too faint for WF sensing, the Canary experiment has demonstrated it is feasible to offset the telescope towards a bright enough star to get the required calibration ( 16).

For MOAO PSF-R purpose, we determine a β from the TS measurements using the Zernike reconstruction matrix C, and considering noise and aliasing as zero-mean processes, as follows:

a β = C s β . ( 31
)
Equation 17 involves a static OTF Static β term. To avoid including multiple times the telescope OTF which is factored in every term as well as in Eq. 31, precaution must be taken to divide the OTFs by that of the telescope 17 by doing : OTF Static β (ρ/λ) = P P (r)P (r + ρ) × e -i aβ 2 (r) × e -i aβ 2 (ρ+r) dr OTF tel (ρ/λ) ,

where : OTF tel (ρ/λ) = P P (r)P (r + ρ)dr

and applying a threshold for values of ρ nearing the telescope pupil.

Determination of OTF ⊥β

The DM fitting OTF ⊥β is computed analytically from the perpendicular phase structure function that comes from the Power Spectrum Density (PSD) of spatial frequencies above the DM cut-off frequency, which scales with (d/r 0 ) 5/3 . The proportionality has been calibrated using end-to-end simulations ( 11). We get σ 2 Fitting = 0.3125 × (d/r 0 ) 5/3 for the Canary Adonis DM.

PSF-R RESULTS USING ON-SKY CANARY DATA 4.1 Statistics on reconstruction performance

Figure 3 depicts results on reconstructed Strehl-ratio (SR) and Full Width at Half Maximum (FWHM) as function of the corresponding sky quantities. Each point is a result of the PSF-R performed on a single data set acquired in September 13, 15, 17 and 18 2013, where Canary was operating is phase B configuration, observing a four stars asterism, with a central bright star and with four Rayleigh LGS focused at 21 km (see. [START_REF] Morris | CANARY phase B: on-sky open-loop tomographic LGS AO results[END_REF] for observation details). We report in Tab. 2 statistics of the reconstruction in terms of SR and FWHM.

The FWHM is determined from fitting the reconstructed PSF to five-parameters Moffat function :

f (x, y, I 0 , α x , α y , β, θ) = I 0 1 + x cos(θ) -y sin(θ) α x 2 + x sin(θ) -y cos(θ) α y 2 -β , (34)
where I 0 is the PSF amplitude, α x and α y coefficients of spreading along x and y directions, β the skewness coefficient and θ the rotation coefficient. The FWHM is then given by :

FWHM = 2 × (2 1/β -1) × (α 2 x + α 2 y ) 1/2 . ( 35
)
We obtained 95% of correlation on the SR over 450 data sets with TS-based method -table 2. The reconstruction is affected by a bias of 0.41% (SR are ranged between 0 and 100% here), which is under the error bars of few percent on the H-band SR estimation directly on the science images. In addition, we get a 1-σ standard deviation of 3.52% of SR, over a large swath of observation conditions. In terms of FWHM, we get 92.5% of correlation, a bias of 2.71 mas and a 1-σ standard deviation of 14.9 mas. On the WHT, we get about 80 mas for λ/D in H-band. The high correlation obtained on SR and FWHM with the reconstructed PSFs confirm the accurateness of the profiling estimation routines and system calibration.

From table. 2, the RTC-based method achieves satisfactory results considering the approximations done to determine the PSF when comparing with the TS-based method. On the SR, we get 92.3% of correlation,a bias of 1.29% and a 1-σ standard-deviation of 4.82%. On the FWHM, we get 88.3% of correlation, a bias of 2.94 mas and a 1-σ standard deviation of 19.9 mas. As noticed on Fig. 3, we get a larger scattering than using the TS-based method. The higher bias with the RTC-based method than with the TS-based cannot be properly interpreted since those bias are smaller than the expected errors bars on the sky SR estimation.

An important point to notice is the FWHM is similarly well reconstructed with both methods. It means the static aberrations, that spread out the PSF core and contributes significantly to the FWHM, are well taken into account and well calibrated. It means methods developed for static aberrations calibration are robust and accurate enough to reach the level of PSF reconstruction ensured with the TS-based method.

RTC-based FWHM [mas]

Data set 00h15m36s on September 13 2013

We focus now on a particular data set acquired at 00h15m36s on September 13th 2013. Canary was operating in MOAO mode based on four LGS and three NGS. During 2048 temporal frames, sampled at 150 Hz, the RTC has provided synchronized telemetry coming from WFS and imaging camera.

We report in Fig. 4 both sky and reconstructed PSFs. On the sky PSF, we measure a H-band SR of 24.0% and a FWHM of 107.5 mas. With the Ts-based method, with a SR of 24.05% and a FWHM of 111.4 mas. Then, with the RTC-based method, we reach 24.5% of SR and 118.8 mas of FWHM. The TS-based method achieves 0.2% and 3.4% of relative error on respectively the SR and FWHM reconstruction, while the RTCbased method reaches 2% and 10.5% of relative error. In addition, Fig. 4 illustrates the PSF core elongation is well reconstructed whatever the method. In Fig. 5 we report the relative error on the Ensquared Energy (EE), the radial average of the PSFs and the OTFs. One can notice the quality of the reconstruction is not only a matter of SR and FWHM. We stay below 6% of relative error on the EE. On the OTF, we reconstruct more energy from the low spatial frequencies that translates a faster decreasing of the reconstructed PSF core than the sky, as we noticed on the radial PSF curves between 0.2 and 0.5 arcsec. We believe it is related to assumptions on the DM: the phase is represented by Zernike modes and not by DM influence functions. This effect is minimized when using a reconstruction matrix calibrated on the bench.

Relative error on EE [%]

CONCLUSIONS

We have laid out a PSF-R algorithm tailored to open-loop, tomographic MOAO systems and compared it to a reference method using the Truth-Sensor and to on-sky metrics (SR, FWHM, EE) obtained directly from the science detector.

We have processed over 450 data sets of telemetry from the Canary MOAO demonstrator. We report 94% and 95% of correlation between the reconstructed and sky SR in using respectively the RTC-based and TSbased method. On the FHWM, we have respectively 90% and 94% of correlation. Bias on reconstructed values are lower than errors bars on sky values for both methods on both SR and FWHM. We get also very similar standard-deviation on reconstructed values as well for both methods.

This work shows that fine-detailed system calibration and modeling can be operated to achieve such high levels of correlation on SR and FWHM. We emphasis particularly on the reliability and efficiency of tools implemented for turbulence characterization based on the Learn & Apply approach (see paper 9909-163), and static aberrations calibration as well, since we are able to reproduce the asymmetric shape of the PSF caused by these static aberrations.

We now plan to improve the reconstruction method by reviewing the approximations/assumptions by evaluating cross-terms using end-to-end simulations and by integrating measurable matrices from the bench on the reconstruction process. For instance, we can tackle bias and standard-deviation on FWHM estimation in using directly the DM influence functions calibrated on bench.

We will then extend results to ELT-sized systems particularly in view of the LTAO instrument Harmoni. We aim to overview existing PSF reconstruction methods and compare each other to the RTC-based approach in terms of accuracy of reconstruction, and describe which one is more tailored for a given science case.

Figure 2 :

 2 Figure 2: Bode diagrams of hrej (left) and hol (right) for three different values of gain, ∆t = 0.45 frames and νe = 150 Hz.

Figure 3 :

 3 Figure 3: Left: TS-based/RTC-based reconstructed SR as function of H-band images SR. and histograms of the relative error on the SR. Right: TS-based/RTC-based reconstructed FWHM as function of H-band images FWHM. and histograms of the relative error on the FWHM.

Figure 4 :

 4 Figure 4: From left to right: reconstructed PSF using the RTC-based method, the TS-based and acquired on-sky in H-band (logarithmic scale). The reconstructed PSFs include an additional Gaussian noise whose the standard-deviation is measured from the corner of the sky PSF, outside the inner tangent circle to the edges of the image.

Figure 5 :

 5 Figure 5: Top-Left:From left to right Relative error on the Ensquared Energy computed on the reconstructed PSF. Radial average of both reconstructed/sky PSFs as function of the angular separation from center. Radial average of both reconstructed/sky OTFs as function of the normalized frequency.

Table 1 :

 1 Notation

		Science directions β
	a β	Atmospheric parallel phase on which the system is acting
	a⊥β	Atmospheric high spatial frequencies not corrected
	εβ	Residual modes
	ε β	residual of the DM correction
	mβ	DM modes
	sβ	Concatenation of all WFS measurements in science directions
	ηβ	Noise contribution into WFS measurements
	r ε β	Aliasing contribution into WFS measurements in loop engaged
	rβ	Aliasing contribution into WFS measurements in loop disengaged
		Sensing directions α
	sα	Concatenation of all WFS measurements in sensing directions
	a α	Measured modes by WFS in sensing directions
	ηα	Noise contribution into WFS measurements
	rα	Aliasing contribution into WFS measurements
		Matrices
	C	Zernike reconstruction matrix (slopes to Zernike): calibrated on bench
	G	Zernike interaction matrix (Zernike to slopes): calibrated on bench

R

Tomographic reconstructor (slopes α to slopes β): calibrated on-sky using the Learn& Apply approach Zi(r) i th Zernike polynomials. System g Loop gain ∆t Latency of the system in frames calibrated on bench at 0.45 frame. νe Sampling frequency of the RTC, 150 Hz Miscellaneous x Average of x Σxy Cross-covariance matrix of x and y x Estimation of x x(z) Z-transform of x(t)

Table 2 :

 2 Statistics on PSF-R performed using either the RTC-based or the TS-based method. They are given for an amount of 450 Canary MOAO data sets acquired in September 2013. SR are ranged between 0 and 100%.

	Method	Criteria	Mean Std Correlation [%]
	RTC-based	∆SR = SR rtc -SR sky [%]	1.29	4.82	92.3
		∆FWHM = FWHM rtc -FWHM sky [mas]	2.94	19.9	88.3
	TS-based	∆SR = SR ts -SR sky [%]	0.41	3.52	95.0
		∆FWHM = FWHM ts -FWHM sky [mas]	2.71	14.9	92.5

ACKNOWLEDGMENTS

The research leading to these results received the support of the A*MIDEX project (no. ANR-11-IDEX-0001-02) funded by the Investissements d'Avenir French Government program, managed by the French National Research Agency (ANR). This work is also supported CNRS, INSU, Observatoire de Paris, Université Paris Diderot-Paris 7 and European Commission (Fp7 Infrastructures 2012-1, OPTICON Grant 312430, WP1).