F Vernay

Z Sabsabi

H Kachkachi
email: hamid.kachkachi@univ-perp.fr

AC susceptibility of an assembly of nanomagnets: combined effects of surface anisotropy and dipolar interactions

We compute the AC susceptibility of a weakly dipolar-interacting monodisperse assembly of magnetic nanoclusters with oriented anisotropy. For this purpose we first compute the relaxation rate in a longitudinal magnetic field of a single nanomagnet taking account of both dipolar interactions in the case of dilute assemblies and surface anisotropy. We then study the behavior of the real and imaginary components of the AC susceptibility as functions of temperature, frequency, surface anisotropy and inter-particle interactions. We find that the surface anisotropy induces an upward shift of the temperature at the maximum of the AC susceptibility components and that its effects may be tuned so as to screen out the effects of interactions. The phenomenological Vogel-Fulcher law for the effect of dipolar interaction on the relaxation rate is revisited within our formalism and a semi-analytical expression is given for the effective temperature is given in terms of inter alia the applied field, surface anisotropy and dipolar interaction.

I. INTRODUCTION

The dynamics of magnetic systems in the form of nanoclusters (nanoparticles or nanomagnets) assemblies is a rather challenging issue from the standpoint of fundamental physics as it requires a simultaneous investigation of both long-range inter-cluster interactions and the intricacies of inhomogeneous magnetism taking place inside the clusters. Even for the equilibrium properties, the problem is of a tremendous difficulty especially if one tries to take account of the internal structure of the cluster by regarding it as a many spin system. In fact, only advanced numerical approaches may offer a way out, though with a limited success inasmuch as one considers the effect of surface anisotropy and its interplay with the inter-cluster dipolar interactions. Recently, this issue has been tackled 1,2 to some extent by representing each nanocluster by an effective macroscopic model [3][4][5][6] with an energy potential whose coefficients are functions of the cluster's characteristics (size, shape, lattice crystal, spin-spin interactions). It was shown that the magnetic properties of an assembly may be improved by a tailored variation of the assembly parameters, such as its concentration and geometry, and the clusters intrinsic characteristics such as the size and shape. In this work, we investigate the joint effect of inter-cluster interactions and surface anisotropy on the dynamic behavior of the assembly, in the case of low concentration and not too strong surface effects. For this we study the AC susceptibility with a variable measuring frequency.

AC susceptibility of an assembly of magnetic nanoclusters has been studied by many authors during the last decades, experimentalists and theorists, by varying the applied magnetic field, temperature and frequency. [7][8][9][10][11][12][13][14][15][16][17][START_REF] Spinu | Etudes des propriétés dynamiques d'assemblées de nanoparticule magnétiques[END_REF][START_REF] Dormann | [END_REF][20][21][22][23] These studies have greatly contributed to improve our understanding of the superparamagnetic behavior of such systems and to provide estimates of their physical parameters. In particular, the size study 22 of AC susceptibility, together with Mössbauer spectroscopy, of diluted and concentrated assemblies of maghemite nanoclusters dispersed in polymer, has revealed the important role of surface effects. On the theoretical side, it is the first time that the joint effects of inter-cluster interactions and surface anisotropy on the AC susceptibility are considered in a single study.

According to Debye's model applied to assemblies of magnetic nanoclusters, 7 the AC susceptibility is given by χ (ω) = χ eq / 1 + iωΓ -1 , where χ eq is the static or equilibrium susceptibility, ω the frequency and Γ the clusters relaxation rate (inverse of relaxation time). This model describes the absorption by a single mode of the electromagnetic energy provided by the applied field. The dynamics of this mode is rather slow and characterized by the longitudinal relaxation time τ = Γ -1 corresponding to the population inversion from the blocked state to the superparamagnetic state. This transition corresponds on average to the crossing by each cluster's magnetic moment of its energy barrier. Therefore, in order to compute the AC susceptibility, one has to compute the longitudinal relaxation rate of a nanocluster in the assembly (described by an effective model) in a magnetic field.

The paper is organized as follows: Section II is devoted to the presentation of the model and the statement of the problem. This Section closes with a brief summary of the results for the equilibrium susceptibility obtained in Ref.

2 as a function of the applied field, temperature, surface anisotropy and including the contribution of long-range dipolar interaction. The formulas for the AC susceptibility are then derived in Section III: we first describe in details the evaluation of single nanocluster's relaxation rate Γ with both a uniaxial and a cubic anisotropy representing the surface effects; by using Debye's model the semianalytical form of the AC susceptibility is then given at the end of the Section. In Section IV we deal with the main focus of the present work, namely the study of the effect of surface anisotropy on the AC susceptibility and its competition with dipolar inter-particle interactions.

The paper ends with a discussion of the Vogel-Fulcher law and concluding remarks.

II. ENERGY AND EQUILIBRIUM SUSCEPTIBILITY A. Nanoparticle assembly

We consider a monodisperse and textured assembly of N ferromagnetic nanoclusters each carrying a magnetic moment m i = m i s i , i = 1, • • • , N of magnitude m and direction s i , with |s i | = 1. Each magnetic moment has a uniaxial easy axis e aligned along the z direction. The energy of a magnetic moment m i interacting with all the other magnetic moments within the assembly, in a magnetic field H = He h , reads (after multiplying by

-β = -1/k B T ) E i = E (0) i + E DDI i , (1)
where the first contribution

E (0) i = x i s i • e h + A (s i
) is the energy of the free nanocluster at site i, comprising the Zeeman energy and the anisotropy contributions from the core and surface. A (s i ) is a function that depends on the anisotropy model and is given by

A(s i ) =      σ i (s i • e i ) 2 , OSP, σ i (s i • e i ) 2 -ζ 2 s 4 i,x + s 4 i,y + s 4 i,z
, EOPS.

(2) OSP and EOSP stand respectively for One-spin problem and Effective One-spin problem which are macroscopic models used for representing the magnetic state of the nanocluster. 2 In the present case, we restrict ourselves to the situation where the uniaxial anisotropy axis is aligned along the z direction, i.e. with a common axis with the cubic anisotropy. This assumption makes the analytical calculations somewhat simpler and the physical interpretation more transparent, but it does not represent a significant discrepancy with regard to the real situation. Indeed, the uniaxial anisotropy considered in Eq. ( 2) is in fact an effective anisotropy that takes account of both the magneto-crystalline and shape anisotropy. In typical nanoparticle assemblies this effective anisotropy is rather strong, especially for elongated nanoparticles. As such a small tilting of the cubic anisotropy with respect to the axis of the effective uniaxial anisotropy should not change the results in a significant way. For a more general situation with an arbitrary orientation of the cubic anisotropy axes with respect to the uniaxial anisotropy axis, one can write the cubic contribution in a different reference frame (x ′ , y ′ , z ′ ) and then introduce in Eq. (2) a rotation matrix such that s i,α ′ = β=x,y,z R αβ s i,β , as was done in a different context in Ref. 24.

The second term in Eq. ( 1) is the dipole-dipole interaction (DDI) between nanoclusters which can be writ-ten as

E DDI i = ξ j<i s i • D ij • s j , where D ij is the DDI tensor D ij ≡ 1 r 3 ij (3e ij e ij -1
), with r ij = r ir j and e ij = r ij /r ij is the unit vector along the i-j bond.

For convenience, we have introduced the following dimensionless parameters

x ≡ mH k B T , σ ≡ K 2 V k B T , ζ ≡ K 4 K 2 , ξ ≡ µ 0 4π m 2 /a 3 k B T
together with the DDI coefficient ξ ≡ ξC (0,0) . C (0,0) = -4π D z -1 3 and D z is the demagnetizing factor along the z axis. K 2 , K 4 are the constants of the uniaxial and cubic anisotropy, respectively. a is the "super-lattice" parameter or the inter-particle distance in the assembly whose particles are supposed to occupy a simple cubic (SC) lattice. Yet, we stress that a generalization to other super-lattices (FCC, BCC, ...) is rather straightforward. One should simply re-evaluate the lattice sums C (0,0) for the given super-lattice. Similarly, one could easily mimic a disordered spatial arrangement by an evaluation of C (0,0) in the case of a randomly depleted lattice. However, for the sake of clarity and to keep our discussion simple we will consider the SC case in the rest of this paper.

The (dimensionless) DDI field Ξ i acting on the magnetic moment m i reads

Ξ i = ξ j D ij • s j . (3)
Later we make use of the spin average Ξ 2 i, 0

, where

Ξ i, = Ξ i • e i is the longitudinal component of Ξ i , which is defined by Ξ 2 i, 0 ≡ 1 4π ˆ  j ds j   Ξ 2 i, e j E (0) j . (4)
The average 0 is defined with respect to the Gibbs probability distribution containing only the energy contributions pertaining to a free cluster. Finally, the spin average of the transverse component of Ξ i can be obtained from the identity

Ξ 2 i,⊥ 0 = Ξ 2 i 0 -Ξ 2 i, 0
.

B. Statement of the problem

In the present work we shall be concerned with the study of the combined effects of surface anisotropy and dipolar interactions on the dynamic susceptibility of an assembly of monodisperse nanoclusters with oriented uniaxial anisotropy. The cubic anisotropy which stems from spin non-collinearities on the cluster's surface is assumed to have its axes parallel to the crystal axes. We then derive analytical formulas in several cases of low field (x ≪ 1), high-energy barrier (σ ≫ 1), small surface anisotropy (|ζ| < 1) and weak DDI (ξ ≪ 1). In particular, for the calculation of the spin averages (4) and kindred ones we will drop all terms of orders higher than 2. For this reason, it turns out that the calculation of such averages can be done with good approximation with only the uniaxial anisotropy contribution in the Gibbs probability distribution. 2 The final results are expressed in the end in terms of the following well known averages (obtained in the absence of a magnetic field) s α i 0 = 0, and

s α j s β k 0 = 1 3 (1 -S j2 )δ αβ + S j2 e α j e β j δ jk (5)
with 15,25 S il

(σ i ) ≃      (l-1)!! (2l+1)!! ( σi 2 ) l/2 + . . . , σ i ≪ 1, 1 -l(l+1) 4σi + . . . , σ i ≫ 1. (6)

C. Equilibrium susceptibility

For a weakly interacting assembly of nanoclusters described with the help of the EOSP model, the equilibrium susceptibility reads (to first order in ξ) χ eq x, σ, ζ, ξ ≃ χ eq free + ξχ eq int (7) where χ eq free is the equilibrium (linear) susceptibility of the non-interacting assembly in the limit of high anisotropy energy barrier 2,25

χ eq free (x, σ, ζ) = 2χ ⊥ 0 σ χ (1)
free + 3χ

(3) free x 2 , (8) χ (1)
free = 1 -

1 σ + ζ σ -1 + 2 σ , χ (3) free
= 1 3 -1 + 2 σ + ζ σ 2 - 5 σ .
Here χ ⊥ 0 is the transverse equilibrium susceptibility per spin at zero temperature in the absence of a bias field

χ ⊥ 0 ≡ µ 0 m 2 2K 2 V .
This can be obtained from Eq. (3.86) of Ref. 26 upon setting the field to zero. The contribution of DDI to the equilibrium susceptibility is given by 2

χ eq int (x, σ, ζ) = 2χ ⊥ 0 σ χ (1) int + 3χ (3) int x 2 , (9)
χ (1) int = 1 - 2 σ -2 1 - 3 σ ζ σ , χ (3)
int = -

4 3 1 - 3 σ - 3ζ σ .
In the sequel, all susceptibilities will be measured in units of χ ⊥ 0 .

III. AC SUSCEPTIBILITY

The dynamic response of the EOSP assembly can be studied with the help of the AC susceptibility. For an arbitrary angle ψ between the (common) easy axis and the field direction, the effective susceptibility may be written as χ = χ cos 2 ψ + χ ⊥ sin 2 ψ.

Shliomis and Stepanov 27 proposed a simple Debye form for χ(ω) which can be generalized to describe the effect of a longitudinal bias field by writing

χ = χ (T, H) 1 + iωτ cos 2 ψ + χ ⊥ (T, H) 1 + iωτ ⊥ sin 2 ψ, (10)
where τ and τ ⊥ are appropriate longitudinal (inter-well) and transverse (intra-well) relaxation times; χ (T, H) and χ ⊥ (T, H) are respectively the longitudinal and transverse components of the equilibrium susceptibility.

For an assembly with oriented anisotropy in a longitudinal field (ψ = 0), we may assume that the transverse response is instantaneous, i.e. τ ⊥ = 0. In this case the AC susceptibility is given by Eq. ( 10) or using τ = Γ -1 and χ = χ eq defined in Eq. ( 7),

χ x, σ, ζ, ξ, η = χ eq 1 + iωΓ -1 . (11)
Next, we introduce the reduced frequency

η x, σ, ζ, ξ, λ ≡ ωτ = (ωτ D ) (τ D Γ) -1 , (12)
with λ being the damping parameter. Γ x, σ, ζ, ξ, λ is the relaxation rate of an EOSP nanocluster weakly interacting within the assembly; τ D = (λγ gyr. H K ) -1 is the free diffusion time, H K = 2K 2 V /M the (uniaxial) anisotropy field, and γ gyr. ≃ 1.76 × 10 11 (T.s) -1 the gyromagnetic ratio. For example, for cobalt particles the anisotropy field is H K ∼ 0.3 T, and for λ = 0.1 -10, τ D ∼ 2 × 10 -10 -2 × 10 -12 s.

At this point, the only missing ingredient to evaluate the susceptibility in Eq. ( 11) is the relaxation rate. Therefore, the next Section is devoted to the calculation of the relaxation rate Γ x, σ, ζ, ξ, λ .

A. Relaxation rate

Here we derive an expression for the relaxation rate of a weakly interacting EOSP nanocluster.

In Ref. 11 Jönsson and Garcia-Palacios derived the following approximate expression for Γ

Γ ≃ Γ 0 1 + 1 2 Ξ 2 0 + 1 4 F (α) Ξ 2 ⊥ 0 . (13)
This takes account of the various approximations stated earlier inasmuch as the general spin averages ... are replaced by their analogs ... 0 defined in Eq. ( 4). Γ 0 is the relaxation rate in the absence of DDI. The function F (α) is given by 28

F (α) = 1 + 2(2α 2 e) 1/(2α 2 ) γ(1 + 1 2α 2 , 1 2α 2 ), (14)
with γ(a, z) = ´z 0 dt t a-1 e -t the incomplete gamma function, and where α = λ √ σ. In Ref. 11 the free-particle relaxation rate Γ 0 was given in the absence of the applied magnetic field, i.e. τ D Γ 0 = 2 √ π σ 1/2 e -σ . A more general expression for the free-particle relaxation rate in a longitudinal magnetic field is the Néel-Brown formula 29

τ D Γ NB = σ 1/2 1 -h 2 √ π × (1 + h) e -σ(1+h) 2 + (1 -h) e -σ(1-h) 2 , ( 15
)
with h ≡ x/2σ. Setting h = 0 recovers the previous expression.

The relaxation rate (15) has to be generalized for the present purposes in order to take into account surface anisotropy, in addition to the magnetic field as well as the core anisotropy.

For intermediate-to-high damping Langer's approach allows us to compute the relaxation rate Γ of a system with many degrees of freedom related with its transition from a metastable state through a saddle point [30][31][32][33][34][35]

Γ = |κ| 2π Zs Z m , (16)
where Z m and Zs are respectively the partition functions in the vicinity of the energy metastable minimum and the saddle point. The two partition functions are computed using a quadratic expansion of the energy at the corresponding stationary states. The attempt frequency κ is computed upon linearizing the dynamical equation around the saddle point, diagonalizing the resulting matrix and selecting its negative eigenvalue. 30,31 The dynamics of a single magnetic moment is governed by the (damped) Landau-Lifshitz equation and Langer's (or Néel-Brown) expression renders the relaxation rate for its escape from the minimum (θ (m) , ϕ (m) ) through the saddle point θ (s) , ϕ (s) , in the limit of intermediate-to-high damping. Owing to the approximations adopted in this work, especially the smallness of the surface anisotropy with respect to the uniaxial anisotropy (|ζ| < 1), the energy potential of the non-interacting cluster presents two global minima that are mainly defined by the uniaxial anisotropy, as is shown in Fig. 1 (in zero field), while the surface anisotropy induces saddle points at the equator. In the present case, changing the sign of ζ does not affect the loci of the minima but those of the saddle points are rotated by π/4 around the z axis. The overall shape of the energy landscape remains, though, quite similar. The global minima are θ (m) = 0, π with uniaxial symmetry around the z axis. Then, we have

Z m ≃ 2π 2σ (1 -ζ -h) e E (0) m , (17)
where

E (0) m = 2σ × 1 4 (2 -ζ -4h)
is the energy at the metastable minimum θ (m) = π.

There are four equivalent escape routes (saddle points) related to each other by a rotational symmetry with respect to the azimuthal angle ϕ and their loci depend on the sign of ζ. Indeed, for ζ > 0 we have ϕ

(s) = π 4 , 3π 4 , 5π 4 , 7π 4 and cos θ (s) = 2 + ζ 3ζ sin φ 3 - 2 + ζ 9ζ cos φ 3 (18)
with cos φ = 9hζ 1/2 (2+ζ) 3/2 . For ζ < 0 the saddle points are given by ϕ (s) = 0, π 2 , π, 3π 2 and

cos θ (s) = h 4ζ + √ ∆ 1/3 + h 4ζ - √ ∆ 1/3 (19)
with

∆ = h 4ζ 2 -1+ζ 6ζ 3 .
For a small magnetic field h, the azimuthal angle at the saddle point remains close to the equator while an expansion of Eq. ( 18) yields θ (s) ≃ π 2 + 2h 2+ζ . It is worth mentioning that the symmetry breaking of the continuous rotation around ϕ, induced by the introduction of a cubic anisotropy, appears as soon as ζ assumes a finite value. However, for very small values of ζ the energy surface around the saddle points remains flat rendering the quadratic expansion of the energy at the saddle point questionable [see Fig. 2 below]. As a consequence Langer's approach does not apply in such situations, as was emphasized earlier. 36,37 Next, expanding the energy at the saddle points for ζ > 0 and ζ < 0 (with not too small |ζ|) we obtain the following generic expression for the relaxation rate (upon mul-tiplying by the symmetry factor 4) Γ 0 = Γ (π,0)→(θ (s) ,ϕ (s) )

τ D Γ 0 = 4 × |κ| 2π sin θ (s) 2σ (1 -ζ -h) µ (s) 1 µ (s) 2 e ∆E (0) . (20)
The attempt frequency κ, as a function of the damping parameter λ, is given by the general expression

κ = λ 2 × µ (s) 2 + µ (s) 1 - µ (s) 2 + µ (s) 1 2 -4 1 + 1 λ 2 µ (s) 1 µ (s) 2 (21) where µ (s)
i , i = 1, 2 are the eigenvalues of the energy quadratic form near the saddle point, with respect to the variables θ, ϕ, respectively. These, together with the energy at the saddle point, are given by

E (0) s = 2σ h cos θ (s) + 1 2 cos 2 θ (s) -ζ 8 sin 4 θ (s) + 2 cos 4 θ (s) , µ (s) 1 = 2σ × -1 4 4h cos θ (s) + (4 -ζ) cos 2θ (s) -3ζ cos 4θ (s) , µ (s) 2 = 2σ -ζ sin 4 θ (s) .
for ζ > 0.

As the energy landscape remains globally the same by changing ζ → -ζ, only the energy at the saddle points and the eigenvalues change, yet the overall form of the relaxation rate is still given by Eqs. ( 20) and ( 21) with the following substitutions

E (0) s = 2σ h cos θ (s) + 1 2 cos 2 θ (s) -ζ 4 cos 4 θ (s) + sin 4 θ (s) , µ (s) 1 = 2σ -h cos θ (s) -cos 2θ (s) + ζ cos 4θ (s) , µ (s) 2 = 2σ ζ sin 4 θ (s) .
for ζ < 0.

Finally, the energy barrier ∆E (0) in Eq. ( 20) is defined as

∆E (0) = E (0) s -E (0) m .
In the limit of zero field (h = 0) and for ζ > 0, for instance, µ

(s) 1 /2σ → (ζ + 2) /2, µ (s) 2 /2σ → ζ, E (s) 0 /2σ → -ζ/8
so that the relaxation rate in (20) reduces to the result obtained in Ref. 37, normalized with respect to the Néel's free-diffusion relaxation time 17

τ N = m 2αγkBT = στ D .
Two remarks are in order:

-There are two limits to the range of ζ (> 0). First, ζ must not exceed some value that marks the limit of validity of the EOSP model. From numerical calculations, 4,6 this has been evaluated to ∼ 0.25 for an SC lattice and ∼ 0.35 for an FCC lattice.

The second limit stems from the fact that the analytical expressions obtained above for Γ within Langer's approach cannot be continued to ζ = 0 because the saddle points created by the cubic contribution to the anisotropy disappear at the uniaxial anisotropy limit. The lower limit on ζ can be obtained by setting to zero the first derivative of Γ with respect to ζ and numerically solving the ensuing equation. Doing so, we find that for σ = 15...25, for instance, ζ crit is of the order of 0.1.

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 ζ 0 2×10 -5 4×10 -5 6×10 -5 8×10 -5 1×10 -4 τ D Γ 0 0.
-Because of the non-axial symmetry (owing to the presence of surface cubic anisotropy) considered here, the relaxation rate depends in a non trivial way on the damping parameter. Consequently, the longitudinal response (in-phase and out-of-phase) are damping-dependent.

In Fig. 2 we plot the relaxation rate for both ζ > 0 and ζ < 0 as a function of ζ and different values of the applied field h, for σ = 15. In this case, as mentioned above, the relaxation rate computed within our approach is only valid for 0.1 < |ζ| < 1. For smaller values of |ζ| Langer's approach is no longer valid and the relaxation rate is given by the Néel-Brown formula (15) which does not depend on ζ. This is shown by the dashed lines in Fig. 2. As it can be expected, the relaxation rate that includes the cubic anisotropy is larger than the Néel-Brown relaxation rate since the creation of saddle points increases the probability of escaping from the metastable state.

Next, if we write the relaxation rates given by Eq. ( 20) in the form

Γ 0 (h, σ, ζ, λ) = Γ ǫ p (h, σ, ζ, λ) e ∆E (0) ǫ (ζ) (22)
with ǫ = + for ζ > 0 and ǫ =for ζ < 0 we can study the behavior of the ratio of the prefactors and the difference of the energy barriers as the field is varied. The corresponding plots are given in the inset in Fig. 2. We see that the ratio of the prefactors is a decreasing function of h while the difference of the energy barriers is an increasing function thereof. This implies that there is a competition between the prefactor-dominated dynamics and the relaxation through the energy-barrier crossing, or in other words, between the dynamics dominated respectively by the fluctuations of the transverse and the longitudinal components of the magnetic moment.

Caution is necessary when trying to compare the expression of the relaxation rate Γ 0 (h, σ, ζ, λ) derived here in the presence of both surface effects (ζ = 0) and DDI (ξ = 0) with the relaxation rate obtained, in the absence of the cubic anisotropy, by other authors. 11,38 Indeed, in the presence of an arbitrary magnetic field, one cannot simply set ζ = 0 in our expressions because these have been derived using Langer's approach that relies on the validity of the quadratic expansion of the energy at the minima and saddle points ; a validity that breaks down for rather small (but non vanishing) values of ζ. From a mathematical point of view, setting ζ to zero in Eqs. (18, 19), for example, leads to a singularity. Now, for the assembly we use the spin averages

Ξ 2 i, 0 and Ξ 2 i,⊥ 0 obtained in Ref.
11 for a monodisperse assembly on a SC lattice and in the absence of an external magnetic field

Ξ 2 i, 0 = ξ 2 3 [(1 -S 2 ) R + 3S 2 T ] , Ξ 2 i,⊥ 0 = ξ 2 3 [(2 + S 2 ) R -3S 2 T ] , (23)
where S 2 is defined in Eq. ( 6). R and T are lattice sums given by R = 2 j =i r -6 ij , T = j =i (e • D ij e) 2 . For a simple cubic lattice we have, in the thermodynamic limit, R ≃ 16.8, T ≃ 13.4.

Therefore, using Eqs. ( 14), (20), and (23) in Eq. ( 13) we obtain the relaxation rate for an assembly of interacting clusters within the EOSP approach

Γ (h, σ, ζ, λ, ξ) ≃ Γ 0 (h, σ, ζ, λ) 1 + ξ 2 6 S (λ) . ( 24
)
where S (λ) is defined by

S (λ) = (1 + F (λ)) R+(3T -R) 1 - F (λ) 2 S 2 . ( 25
)
Alternatively, using η 0 = ωΓ -1 0 , we can also rewrite Eq. ( 12) as

η (h, σ, ζ, ξ, λ) = η 0 (h, σ, ζ, λ) 1 + ξ 2 6 S (λ) . (26)

B. AC susceptibility

We rewrite the AC susceptibility (11) separating its real and imaginary parts χ h, σ, ζ, ξ, η = χ ′iχ ′′ with

χ ′ = χ eq 1 1 + η 2 , χ ′′ = χ eq η 1 + η 2 . (27)
Now, we substitute for χ eq and η their respective expressions ( 7) and ( 26), taking account of DDI and surface anisotropy contributions given above. We obtain

χ ′ ≃ χ ′ free + ξ 1 + η 2 0 Λ (1) + ξ η 2 0 1 + η 2 0 Λ (2) , χ ′′ = χ ′′ free + ξη 0 1 + η 2 0 Λ (1) + ξ 1 -η 2 0 1 + η 2 0 Λ (2) ,
where we have defined the in-phase and out-of-phase susceptibilities in the absence of DDI

χ ′ free (h, σ, ζ, λ) ≡ χ eq free 1 + η 2 0 , χ ′′ free (h, σ, ζ, λ) ≡ η 0 χ eq free 1 + η 2 0
together with the 1 st -and 2 nd -order DDI contributions

Λ (1) ≡ χ eq int C (0,0) , Λ (2)
≡ χ eq free 3 S (λ) .

χ eq free and χ eq int are given by Eqs. (8, 9) and η 0 = ωΓ -1 0 by Eq. (20).

IV. RESULTS

A. Noninteracting assembly of OSP nanomagnets

Using our formalism we first reproduce the well known results for the in-phase and out-of-phase susceptibilities for an assembly of noninteracting nanomagnets with uniaxial anisotropy, in zero DC field. 17,26 In Fig. 3 we plot the in-phase (left) and out-of-phase (right) susceptibilities as functions of 1/σ ∝ T for zero field (x = 0) and different frequencies. On the left we have also included the equilibrium susceptibility χ eq free (ζ = 0), represented by the solid line.

The appearance in χ ′ and χ ′′ of a maximum at some particular temperature T max and the displacement of the latter to the right (higher temperatures) upon increasing the measuring frequency is already well understood and explained in details, e.g. in Ref. 17. In particular, the maximum of χ ′ is formed as a result of the competition between the blocking effect (namely the decrease of the relaxation rate) and the increase of χ eq as the temperature decreases. At low temperature, the relaxation time is longer than the measuring time t m = 2π/ω and thereby over a large number of cycles of the AC field, the over-barrier switching probability is nearly zero and the response consists mainly of intra-well rotations. As T increases the clusters magnetic moments start to depart from their respective energy minima due to thermal fluctuations. Then, over the same number of cycles of the AC field the switching probability acquires a non negligible value. The response starts to increase with increasing temperature within a range where the thermallyactivated mechanism of over-barrier crossing is not yet efficient enough, leading to a considerable delay of the response with respect to the excitation. This leads to a considerable out-of-phase response χ ′′ as witnessed by the increase of the latter, see Fig. 3 (right). At higher temperatures, the over-barrier crossing mechanism becomes so efficient that the magnetic moments instantaneously distribute themselves among the various energy minima, in phase with the probing field. At much higher temperatures, the distribution of the magnetic moments reaches its equilibrium state and the χ ′ curves become independent of the measuring frequency and superimpose on the equilibrium linear susceptibility χ eq , and correspondingly χ ′′ tends to zero.

The displacement of T max is easily understood from the expression of the latter as a function of the measuring frequency ν m . Indeed, this temperature is related with the over-barrier rotation process whose relaxation time is approximately given by the simple Arrhenius law τ = τ 0 exp (∆E/k B T ), where ∆E is the effective energy barrier and τ 0 ∼ 10 -12 -10 -9 s the characteristic time of the intra-well dynamics. At T = T max we can write τ ≃ t m , i.e. the measuring time (∼ 100 s for a commercial SQUID), and this then leads to

T max = ∆E k B × ln -1 τ m τ 0 . ( 28
)
From this relation, one can easily infer the increase of T max as the measuring frequency ν m = τ -1 m increases. From the physical viewpoint, with higher ν m one probes on average more probable (with higher relaxation rate) switching processes and this is in effect induced by an increase in temperature.

B. Noninteracting assembly: effects of surface anisotropy

Now, to investigate the effect of surface anisotropy on the AC susceptibility we can compute the real and imaginary components of the latter as functions of temperature, for different values of the parameter ζ > 0.

We have observed that the maxima of both χ ′ and χ ′′ shifts toward higher temperatures as ζ increases. Indeed, setting to zero the first derivative of χ ′ with respect to temperature and setting T = T max in the ensuing equation, we can solve the latter for T max as a function of the other parameters, especially ζ. We indeed find a monotonously increasing function of ζ. Intuitively this result appears to be at variance with the fact that since the cubic (surface) anisotropy creates saddle points it leads to an increase of the relaxation rate and thereby to a decrease of T max . However, as mentioned earlier the location of the maximum of the dynamic response, while it does depend on the energy barriers, it is strongly dependent on the equilibrium response (i.e. χ eq ) which is rather different for the pure uniaxial case (ζ = 0). More precisely, χ eq is a decreasing function of ζ and thereby when ζ increases the dynamic response requires higher temperatures to reach its maximum, thus leading to an increasing T max for increasing ζ.

C. Effects of inter-particle interactions in the absence of surface anisotropy

The effect of DDI on the AC susceptibility has been widely investigated by many groups. 11,12,16,[START_REF] Spinu | Etudes des propriétés dynamiques d'assemblées de nanoparticule magnétiques[END_REF][39][40][41][42][43][44][45][46][47] In Ref. 47 the authors provide a short review of the situation regarding the effect of DDI on the maximum of χ ′ and χ ′′ and their shift in temperature as the DDI inten- sity is varied and the assembly shape changed from oblate to prolate. It was argued that the discrepancy of conclusions found in the literature as to whether the DDI shift the maximum of χ ′ and χ ′′ towards higher or lower temperatures resides in many reasons, mostly related with the effects of damping, the shape of the (assembly) sample, and anisotropy. Here we use the same formalism and approximations and obviously confirm the same results. Therefore, we shall not repeat the conclusions of the previous work.

Nevertheless, Fig. 4 shows that as the shape of the assembly changes from prolate to oblate, we obtain an opposite shift in temperature in both the maximum of χ ′ and χ ′′ and also in the corresponding T max . In the case of isotropic samples, such as cubes, the lattice sum C (0,0) vanishes leading to a DDI coefficient ξ = 0. Therefore, no shift is observed and the DDI do not contribute, within the present approach. For prolate and oblate samples, both shifts are explained by the fact that the equilibrium susceptibility increases with DDI in a prolate sample whereas it decreases in an oblate sample. More importantly, it is seen that the effect of DDI is more pronounced in the oblate case because there the DDI are in competition with the uniaxial anisotropy and thus strongly contribute to suppress the equilibrium susceptibility. The effect of damping, while remaining secondary as compared to that of the assembly shape, seems to be somewhat more pronounced in the case of prolate samples. This may be due again to the fact that in the prolate case the increase of χ eq with DDI is slower than its decrease for the oblate shape. As such, χ ′ and χ ′′ , and more so for χ ′ , are more sensitive to the change of the relaxation rate which then starts to prevail, and which does depend on damping.

D. DDI versus surface effects

Now we are ready to investigate the interplay between inter-particle DDI and intrinsic surface anisotropy. We only present the case of ζ > 0 in which surface (cubic) anisotropy favors the magnetic alignment along the cube diagonals. In order to deal with the case ζ < 0 one has to use the corresponding relaxation rate, as discussed in Section III A. Yet, as shown in Fig. 2 the behavior of the relaxation rate for ζ < 0 is qualitatively the same as that for ζ > 0 and that even quantitatively the difference is not really significant. Therefore, in the remaining part of the paper we will focus our discussion on ζ > 0.

We have systematically analyzed χ ′ and χ ′′ for various values of the surface anisotropy coefficient ζ, for both prolate and oblate assemblies. We have observed the upward shift of T max as ζ increases and the downward shift of the maximum of χ ′ and χ ′′ , as already discussed earlier. However, owing to the fact that the effect of increasing ζ is to draw the particle's magnetic moment towards the cube diagonals, it basically plays the same role in a prolate sample where the magnetization is enhanced along the z axis, or in an oblate sample where the magnetization is enhanced in the xy plane.

The effect of increasing the strength of DDI alone is shown in Fig. 4. In the case of a prolate sample, we have observed a shift of the maximum toward lower temperatures, in the absence of surface anisotropy. Note again that this is not the ζ = 0 limit of the expressions of Section III A. It is simply the OSP model with the relaxation rate (15). The effect of frequency observed by Lee et al. 48 is similar to the behavior that we observe here: T max increases and χ ′ decreases. Furthermore, the fact that T max increases as the concentration increases is in line with what we observe for oblate samples and corresponds to the type of samples investigated by Lee et al. Despite the relative success of our model in interpreting the experimental data, one has to be careful as not to push the comparison too far because our approach has been derived for textured monodisperse assemblies and, more importantly, is perturbative and thus inherently restricted to weak DDI. This is in general not the case in experiments where the assemblies are often random and rather dense. In such cases (especially high densities), a more quantitative comparison with experiments can only be accessible with the help of numerical investigations 23,49 .

In Fig. 5 we present a specific case in order to highlight the competing effects of surface and dipolar interaction on the susceptibility. The curves are obtained for ξ = 0.008 and small (and increasing) surface anisotropy parameter ζ. These results show that the surface anisotropy, in the present case of positive ζ, has the opposite effect to that of DDI. More precisely, this implies that surface effects can screen out the effect of DDI and the other way round. This confirms the results of Ref. 2 for equilibrium properties for both negative and positive ζ.

E. Discussion

Very often the experimental results related with the dynamics of an assembly of DDI-coupled nanoparticles are analyzed with the help of the Vogel-Fulcher law 38,46,48,[50][51][52]

Γ = τ -1 0 e ∆E k B (T -θ VF ) (29)
where ν 0 = τ -1 0 ≃ 10 9 -10 12 Hz, θ VF represents an effective temperature supposed to include the DDI correction and ∆E is the energy barrier, which reads ∆E = K 2 V in the case of uniaxial anisotropy and zero field. The main concern with this phenomenological formula is to provide an interpretation of the parameter θ VF on physical grounds. Accordingly, in Ref. 50. there is a discussion of a few approaches in this regard. For instance, it is shown how the work of Shtrikman and Wohlfarth 53 leads to an expression of θ VF in terms of the applied magnetic field and how the work by Déjardin 38 yields an expression in terms of the DDI coupling. In the work of Landi itself θ VF is expressed in terms of the inter-particle distance and other parameters such as the particles magnetic moment and the uniaxial-anisotropy energy.

Here we show that our formalism is in full agreement with the previous results and further extends them along the following lines: i) surface anisotropy, ii) particles spatial distribution and shape of the assembly, iii) damping parameter.

In Eq. ( 24) the factor Γ 0 (h, σ, ζ, λ) depends on the applied field, surface anisotropy and damping, together with other parameters, as is seen in Eq. (22)

Γ (h, σ, ζ, λ, ξ) ≃ Γ (h = 0, σ, λ) e σ 1 + σζ 4 + ξ 2 6 S (30
) where S (λ) is defined in Eq. (25). Note that Γ (h = 0, σ, λ) is given in the second line of Eq. ( 6) in Ref. 37. Now, an expansion of Eq. ( 29) with respect to θ VF /T yields 50

Γ = τ -1 0 e ∆E k B (T -θ VF ) ≃ τ -1 0 e σ 1 + σ θ VF T
which is of the same form as our expression (30). Next, using Néel's approximation with a constant prefactor τ -1 0 , thus ignoring any dependence on temperature, damping and applied field, Γ (h = 0, σ, λ) can be identified with τ -1 0 . Then, we can further identify the terms between parentheses leading to the following expression for θ VF (in Néel's approximation)

θ VF T = ζ 4 + 1 6σ ξ 2 S . (31)
This expression provides a somewhat "microscopic" description of the phenomenological parameter θ V F in terms of the inter-particle interactions, the surface anisotropy and damping. Indeed, the last term in (31), which is similar to the one derived in Ref. 50, includes both the damping parameter and the shape of the assembly, owing to the expression of S (λ) [see Eq. ( 25)]. In addition, we note that ξ is proportional to the assembly concentration 2 C V and thereby to a -3 , a being the inter-particle distance. Therefore, we expect that in the absence of surface anisotropy, θ VF scales as θ VF ∼ C 2 V ∼ a -6 . In Ref. 46 experimental estimates of θ VF are given for an assembly of Ni nanoparticles with varying concentration. A comparison of Eq. ( 31) with the corresponding data is given in Fig. 6.

On the other hand, the first term in Eq. ( 31) accounts for the contribution from surface anisotropy. As discussed earlier, in practice it should be possible to adjust the assembly characteristics (assembly shape, particles size and underlying material) so as to achieve to some extent a compensation between surface effects and the DDI contribution. This could in principle suppress the dependence of θ VF on the assembly concentration. In addition, the term in ζ can also be used to extract from the experimental data an estimate of the surface anisotropy coefficient ζ by reading off the intercept from the plot in Fig. 6.

In the most often encountered situation where the particles anisotropy is modeled with an effective uniaxial anisotropy of constant K eff , as would apply for elongated particles, dropping the ζ term, the effective temperature θ VF explicitly reads (as a function of the assembly con- centration C V )

k B θ VF = µ 0 4π 2 M 2 s V 2 K eff V S 6 × C 2 V . (32)
For example, consider a monodisperse assembly of spherical cobalt nanoparticles of 3 nm in diameter with M s ≃ 1.4 × 10 6 J.T -1 .m -3 , K eff ≃ 5 × 10 5 J.m -3 , and C V ≃ 1%. Then, if the assembly is assumed to be in the form of a box-shaped sample with its particles arranged into a simple cubic lattice, the lattice sums R and T were given earlier in the thermodynamic limit. Then, using F (λ) ≃ 1 and S 2 ≃ 1, the factor S evaluates to S ≃ 45. This yields θ VF ≃ 0.05 K, which is small compared to the particle's blocking temperature T B ≃ 14 K. However, one should keep in mind that θ VF scales with the particle's volume.

It is worth emphasizing the fact that θ VF given by Eq. ( 32) is independent of temperature, as can be often encountered in the literature. However, if we take account of surface anisotropy, Eq. (31) shows that the phenomenological parameter θ VF is in fact a linear function of temperature via the term in ζ. This can be understood by noting that surface anisotropy, which is of cubic nature in the EOPS model, drastically modifies the energy potential and thereby affects the dynamics of the particle's magnetization. As a consequence, the effect of DDI becomes strongly dependent on the thermal fluctuations and the elementary switching processes they induce.

V. CONCLUSION

We have studied the combined effects of surface anisotropy and dipolar inter-cluster interactions on the dynamic response of a mono-disperse assembly of magnetic nanoclusters with textured anisotropy. We have derived semi-analytical expressions for the in-phase and out-of-phase components of the AC susceptibility as functions of temperature, applied field, surface anisotropy, damping, frequency, and (weak) dipolar interactions. If we ignore the surface anisotropy, we recover the well known results of frequency-and interaction-induced shift in both the maximum of χ ′ and χ ′′ and of the temperature T max thereat, taking into account the effect of the assembly shape (oblate or prolate). In the presence of surface anisotropy we have derived and used a semianalytical expression for the relaxation time and investigated the effect of surface (cubic) anisotropy. We have done so in the limit of small field, high uniaxial anisotropy barrier and weak surface anisotropy. The expressions obtained for the small ζ show that the relaxation rate or the switching probability increases with surface anisotropy, but the equilibrium susceptibility decreases, thus leading to an overall upward shift of T max . When the interparticle interactions are switched on, a competition sets in between the latter and surface anisotropy that may lead, in adequately prepared samples, to a mutual compensation of the two effects.

Finally, our results for the relaxation rate have been analyzed in connection with the so-called Vogel-Fulcher law and an expression for the ad hoc effective temperature has been given in terms of the inter-particle dipolar interactions, the intra-particle surface anisotropy and the damping parameter, in addition to the other physical parameters such as the applied magnetic field and uniaxial anisotropy.

Figure 1 :

 1 Figure 1: Energy landscape at zero field in the limit of a large uniaxial anisotropy, for ζ > 0 (left), and ζ < 0 (right).

Figure 3 :

 3 Figure 3: (a) χ ′ (left) and (b) χ ′′ for a free assembly (ξ = 0) within the OSP model, i.e. without surface anisotropy (ζ = 0), for different frequencies f ≡ ωτD/(2π).

Figure 4 :

 4 Figure 4: (a) χ ′ and (b) χ ′′ for an interacting prolate (10 × 10 × 20) assembly with varying DDI strength ξ, for the frequency f ≡ ωτD/(2π) = 0.01 in the absence of an external field h = 0, in the high damping regime λ = 10. Same plots in (c) and (d) for an oblate (20 × 20 × 5) assembly.

Figure 5 :

 5 Figure 5: χ ′ for an interacting prolate (10 × 10 × 20) assembly with a fixed DDI strength ξ = 0.008 and varying surface anisotropy coefficient ζ, for the frequency f ≡ ωτD/(2π) = 0.01. h = 0.

+

  (ζ) ≃ -σ + σζ/4 and upon expanding in ζ we obtain

2 Figure 6 :

 26 Figure 6: θVF against the assembly concentration. (stars) Experimental data from Masunaga et al. 46 and (full line) fit of Eq. (31).

  Figure 2: Relaxation rate as a function of the (negative and positive) parameter ζ, for three values of the (reduced) applied field h: Full lines correspond to the relaxation rate defined in Eq. (22), while the dashed lines are plots of the ζ-independent Néel-Brown relaxation rate (15) for different values of the magnetic field. Insets: Ratio of the two prefactors Γ + p (h, σ, ζ, λ) /Γ - p (h, σ, ζ, λ), as defined in Eq. (22), and energy barrier difference against h, for σ = 15 and ζ = 0.2.

	Prefactor ratio	0.56 0.57 0.58 0.59	1 h ζ=0.2 0.2	Energy barrier diff.	0 0.8 0.9 1	h 0.1 ζ=0.2	0.2
							h=0.07
							h=0.05
							h=0.02

Acknowledgments

F.V. and H.K. would like to thank Denis Ledue for discussions and useful comments in the early stage of this work.


F Mady

J-B Duchez

Y Mebrouk

Keywords: Erbium-doped fiber amplifiers, Modeling, Optical fibers, Radiation effects, Radiation-induced attenuation

This work investigates the local interplay between radiation-induced darkening and pump-induced bleaching in erbium-doped silica optical fibers. Thanks to proper characterization conditions using short fiber samples, we demonstrate that degradation levels can be tuned reversibly between equilibrium levels by varying the pump power and/or the dose rate while the dose is being accumulated. This novel finding inspires a local physical model of the radiation-induced degradation of erbium-doped fibers in active conditions. This model is shown to reproduce all empirical features, including dose rate and pump power dependencies. This validated local model could be implemented in long fiber amplifier simulations.

I. INTRODUCTION

rbium-doped fibers amplifiers (EDFA) operating around 1.55 µm have proved to be a reliable technology, notably in ground fiber telecommunication networks. They also represent an attractive option to design compact and efficient optical amplifiers to be used in the harsh space environment, as in satellite fiber optic gyroscopes (FOG) or in power fiber laser sources required for future optical inter-satellite links (OISL) or remote sensing (LIDAR). The development of such systems is though limited by the fact that erbium-doped silicabased fibers (EDF) are damaged by ionizing radiations. An excess optical loss indeed builds-up due to the formation of color centers upon ionization effects, this effect being referred to as 'darkening'. A rather significant amount of publications has addressed the radiation-resistance of EDF and EDFA for more than 20 years [START_REF] Williams | Radiation effects in erbiumdoped optical fibers[END_REF][START_REF] Williams | Space radiation effects on erbium-doped fiber devices: sources, amplifiers and passive measurements[END_REF][START_REF] Berné | A model for the prediction of EDFA gain in a space radiation environment[END_REF][START_REF] Caussanel | Extrapolation of radiation-induced EDFA gain degradation at space dose rate[END_REF][START_REF] Girard | Proton-and gamma-induced effects on erbium-doped optical fibers[END_REF][START_REF] Tortech | Core versus cladding effects of proton irradiation on erbium-doped optical fiber: micro-luminescence study[END_REF][START_REF] Zotov | Radiation-resistant erbiumdoped fiber for spacecraft applications[END_REF][START_REF] Zotov | Radiationresistant Er-doped fibers: optimization of pump wavelength[END_REF][START_REF] Lezius | Radiation-induced absorption in rare earth doped optical fibers[END_REF][START_REF] Gilard | Theoretical explanation of enhanced low dose rate sensitivity in erbium-doped optical fibers[END_REF][START_REF] Thomas | Radiation-resistant erbiumdoped-nanoparticles optical fiber for space applications[END_REF][START_REF] Girard | Radiation hardening techniques for Er/Yb doped optical fibers and amplifiers for space applications[END_REF][START_REF] Girard | Radiation-hard erbium optical fiber and fiber amplifier for both low-and high-dose missions[END_REF], and several technological routes have been shown to greatly improve their radiation tolerance. Hardening recipes are based on cerium co-doping [START_REF] Girard | Radiation hardening techniques for Er/Yb doped optical fibers and amplifiers for space applications[END_REF][START_REF] Girard | Radiation-hard erbium optical fiber and fiber amplifier for both low-and high-dose missions[END_REF] (the benefit of cerium has been known for at least 40 years [START_REF] Friebele | Radiation protection of fiber optic materials: effects of cerium doping on the radiation-induced absorption[END_REF]), on the drastic reduction of aluminum content through nanoparticle doping process [START_REF] Thomas | Radiation-resistant erbiumdoped-nanoparticles optical fiber for space applications[END_REF] (aluminum greatly enhances the formation of color centers), or on hydrogen loading coupled with hermetic fiber coating [START_REF] Zotov | Radiation-resistant erbiumdoped fiber for spacecraft applications[END_REF][START_REF] Zotov | Radiationresistant Er-doped fibers: optimization of pump wavelength[END_REF][START_REF] Girard | Radiation hardening techniques for Er/Yb doped optical fibers and amplifiers for space applications[END_REF][START_REF] Girard | Radiation-hard erbium optical fiber and fiber amplifier for both low-and high-dose missions[END_REF]. Thanks to this knowledge, EDF-based FOG have been flying around the Earth already. However, mechanisms of the radiation-induced darkening of EDF remain unclear. Whereas it is widely admitted that RIA is related to the host matrix absorption and color centers, some basic questions are still not clearly elucidated: what is the exact nature of color centers? What are the intrinsic silica-or dopant-related defects acting as precursors? How do precursors compete as regards trapping of charge carriers? How do color centers interact with the lights that propagate in the fiber? What are their thermal activation and photo-ionization energies? Etc… Such precise questions are crucial to gain the detailed understanding required to introduce relevant prediction/explanation frameworks (modeling). Models have been tentatively proposed to account for the dose-rate and total dose degradation dependence [START_REF] Berné | A model for the prediction of EDFA gain in a space radiation environment[END_REF][START_REF] Gilard | Theoretical explanation of enhanced low dose rate sensitivity in erbium-doped optical fibers[END_REF], but they rely on semi-empirical descriptions (power law dependence of the color center generation rate) and therefore lack in physical basis. In addition, a most basic mitigation agent, namely the pump, has not received the attention it deserves. It is well known that light at pump wavelength (most often around 980 nm) is responsible for photo-bleaching (PB) of darkened EDF [START_REF] Zotov | Radiationresistant Er-doped fibers: optimization of pump wavelength[END_REF][START_REF] Peng | Photoannealing effects for erbium doped fiber sources after gamma irradiation tests by using 532 nm and 976 nm lasers[END_REF], but pump power is too often not specified or not taken into account in data interpretation. A proper analysis of the pump-induced PB is however of crucial importance, first because the degradation of EDF is generally assessed by the measurement of the socalled radiation-induced attenuation (RIA) at pump or signal wavelengths, which requires of course the guidance of a light that does not act as a probe only, and second because the pump is naturally present in active operation conditions. Actually, the 1 to 10 m EDF lengths used in most of cited studies are a pitfall for PB characterization. They are chosen to comply with operation conditions, but result in an inhomogeneous distribution of the pump power along the fiber and so in a distributed PB efficiency that makes the pump effect -if sought through the length-integrated RIA -highly extrinsic (length-dependent). Therefore realistic amplifier lengths do not offer convenient conditions to investigate properly the sole role of PB. In what follows, we use short fiber pieces to demonstrate new remarkable features in the interplay between pump-induced PB and radiation-induced darkening. Highlighted mechanisms are included in a physical model of the local EDF darkening, as a function of the dose rate and pump power. This model is validated against data.

II. EXPERIMENTAL

The measurement setup, sketched in Fig. 1, is similar to the Equilibrium Degradation Levels in Irradiated and Pumped Erbium-Doped Optical Fibers Franck Mady, Jean-Bernard Duchez, Yasmine Mebrouk and Mourad Benabdesselam E one presented in [START_REF] Duchez | Interplay between photo-and radiation-induced darkening in ytterbiumdoped fibers[END_REF], except that the 633 nm probe laser is not used here. The fiber under test (FUT) is in-core pumped at 976 nm (temperature and fiber-Bragg grating stabilized monomode laser diode) while being homogeneously irradiated in the narrow beam of an X-ray generator (Inel XRG 3500, copper anode, 45 kV accelerating voltage). The FUT is spliced to passive single mode fibers (SMF) at both ends. A large 3 mmthick lead lid, in which a rectangular opening can be adjusted to the FUT length, protects both SMF and splices. This lid efficiently screens soft x-rays and makes unwanted losses that could appear in splices and SMF negligible compared to the RIA of the FUT. Two InGaAs photodiodes served as powermeters (PM1 and PM4) to monitor launched and transmitted pump powers. Records made with PM1, which approximately corresponds to 1% of the power supplied by the laser diode, are properly corrected from losses in the SMF, multiplexor and in the connecting splice to provide the actual power launched in the FUT (pump power indicated below well correspond to actual values). Dose rates given to the FUT are monitored by an ionization chamber (PTW 23342 with an UnidosE electrometer) placed in the beam, right below the FUT. The 976 nm RIA is calculated according to:

€ RIA 976 = - 10 L × ln(10) ln P out (t) P in (t) × P in (0) P out (0) ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ , in dB m -1 .
In this relationship, L is the FUT length in m, P in (t) and P out (t) are input and output powers at time t, respectively. P in is actually constant (P in (t) = P in (0), fluctuation limited to 2%). We investigated a commercial off-the-shelf (COTS), Er80-4/125, and a tailor-made alumino-silicate EDF, M04, fabricated and drawn in our laboratory by the standard MCVD and solution doping techniques. The Er80-4/125 fiber is commercialized by Liekki (n-light) as a very highly doped erbium fiber designed for fiber lasers and amplifiers (80 dB m -1 at 1530 nm, erbium ion concentration 3×10 19 cm -3 ). The 1530 nm absorption of the M04 fiber, designed to have a rather similar erbium content, has been measured to 72.3 dB m -1 (erbium ion concentration 2.8×10 19 cm -3 ). We used very short fiber pieces to avoid amplification of spontaneous emission (ASE) and ensure a uniform distribution of both pump power and inversion ratio of Er 3+ ions before irradiation. For instance, the pump power was always virtually constant throughout 2 cm-long samples (varying by less than 0.8%). In long fibers, ASE powers are not negligible and their profiles, together with those of pump power, excitation ratio, and therefore PB efficiency, strongly depend on L and P in . By taking short fiber pieces, we free the problem from extrinsic spatial variations to retain a well-defined power as the unique pump parameter. Profile effects, that additionally accompany changes in P in or L and are necessarily integrated when the RIA is averaged over several meters, are thus ruled out. Builtin losses from the 976 nm laser diode to PM4, deduced from initial attenuation (in the presence of a FUT but before RIA develops), is about 3.05 dB in all experiments. Fig. 1. Experimental setup, similar to that of reference [START_REF] Duchez | Interplay between photo-and radiation-induced darkening in ytterbiumdoped fibers[END_REF] except that the visible RIA probe at 633 nm is not used in this study. PM are power-meters, SMF are passive single mode fibers (commercial SM980) and WDM are wavelength division multiplexors. A lead lid protects splices and SMF.

III. RESULTS

A typical experiment is reported in Fig. 2 for the Er80 fiber. A unique L = 2 cm-long fiber sample has been irradiated and pumped continuously for almost 3 days. The dose rate D' and the pump power P in have been switched between various constant values, as indicated in graphs. The upper part of the figure displays the evolution of the RIA at 976 nm whereas the lower part gives the cumulated ionizing dose. The experiment can be divided into 3 parts: A, B and C. In A, the pump power was constant (P in = 132 mW). The dose rate has been first set at 58 krad/min and then decreased by steps down to 39, 20, 10.5, 5, 2.6, 1.3 and 0 krad/min before being raised again to 5 and 10.5 krad/min. In the B region, the dose rate was constant (D' = 10.5 krad/min) but P in was switched from 132 to 167.8 mW, 48.2 mW, and raised back to 132 mW. The pump power was again dropped to 48 mW to enter the C region where we started with D' = 0 before switching to 20, 39, and 58 krad min -1 . Fig. 3 reports the record of another 6-day irradiation and pumping sequence for the same fiber (Er80) but a new sample of length L = 0.5 cm. Cumulated doses at the end of the various steps are now indicated in the RIA graph. This experiment is divided is 5 regions (A-E), each being characterized by a fixed input pump power (74.7-178.5 mW) and a similar sequence of decreasing dose rates in the 55-10 krad min -1 range. Regions A and C include long photobleaching periods, of about 2 and 3 days respectively. The fiber sample is then submitted to the pump only (no x rays), at the same power (132 mW), but starting from different degradation levels (see arrows). Fig 4 displays the result of a similar 10-hour experiment made on a 2 cm-long sample of our laboratory-made M04 fiber. RIA of Fig. 2 to 4 show positive or negative short overshoots that correspond to natural transients betraying rapid pump power changes (switches). Positive overshoots mark sudden power rises whereas negatives ones reveal power drops. In these figures, the RIA is always calculated relative to the initial fiber absorption at 976 nm before irradiation, which is mainly due non-inverted erbium ions. Since the first applied pump power is always 132 mW, displayed RIA are constantly estimated with respect to the pristine absorption at 132 mW, which is much lower than the nominal small signal absorption for our 2 and 0.5 cm-long samples (about 0.39 dB m -1 pristine loss at 976 nm and 132 mW, the small signal value being close to 30 dB m -1 at this wavelength for the Er80-4/125 fiber). If the pristine absorption is still virtually equal to 0.39 dB m -1 for pump powers > 132 mW, it can be perceptibly greater at pump powers < 132 mW (1.05 dB m -1 at 48.2 mW). As a result, RIA of are slightly overestimated at pump powers below 132 mW, but the error is lower than 1 dB m -1 for the lowest power we used (48.2 mW).

A. Equilibrium degradation levels (EDL)

In experiments of Fig. 234, each couple (P in ,D') was maintained a sufficiently long time to demonstrate that the RIA can reach a steady-state level (hereafter termed equilibrium degradation level, EDL). This original finding, made possible by the use of short EDF samples, well illustrates that a pump-induced PB effect exists. PB competes with radiation-induced darkening and balances it after a transient evolution, as confirmed by the fact that EDL are raised when the dose rate is increased and/or pump power is decreased. After its initial steep increase, the RIA decreases in zone A of Fig. 234(transmission recovery) while the total dose increases continuously! This uncommon behavior is obviously enabled by PB, but it would not be observed if the dose rate was switched before EDL are reached. EDL formed under irradiation are entirely determined by the couple (P in ,D'). They neither depend on the cumulated dose, i.e. the exposure history of the sample, nor on the starting RIA level. For instance, the couple (132 mW, 10.5 krad/min) is formed twice in region A and once in region B of Fig. 2, from various total doses and different initial RIA levels, but it always results in the same EDL around 30 dB m -1 despite a > 20 h temporal gap. In Fig. 2 and4, dashed horizontal lines connect EDL obtained at identical pump/dose rate couples. The fact that existence and properties of EDL are established for two commercial and tailor-made EDF, fabricated in different units, probably with different co-dopants indicates that they result from most basic mechanisms (the unknown Er80 fiber composition is probably based on a alumino-silicate matrix also, even if additional co-dopants are present).

B. Zero-dose rate EDL

EDL obtained in the absence of irradiation are particular in some respects. First, they appear after very long transient periods, lower pump powers leading to longer transients (see e.g. photo-bleaching periods in regions A and C of Fig. 2). Much swifter kinetics, dominated by the dose rate, is achieved under irradiation. Second, they remain equal to zero when the pump is applied on a pristine sample. Indeed, EDF are not subjected to photo-darkening (PD) like ytterbium-doped fibers [START_REF] Duchez | Interplay between photo-and radiation-induced darkening in ytterbiumdoped fibers[END_REF]. We submitted for instance a fresh 2 cm-long sample of Er80 to PD by pumping at 132 mW for more than 64 hours without observing any noticeable photo-induced loss. In the absence of irradiation, non-zero EDL are only reached when starting from some previous RIA level, i.e. after a certain dose (case of photo-bleaching, Fig. 2 and3), and in that sense zerodose rate EDL depend on the initial darkening state. However, photo-bleaching periods followed at P in = 132 mW in regions A and C of Fig. 3, which start from different RIA levels but tend towards the same EDL around 6 dB m -1 , show that zerodose rate EDL are independent of the initial degradation state provided it is not zero. The increase of the pump power from 132 to 178.5 mW at the onset of region D (between t ~ 117 and 118.5 h in Fig. 3) does not lead to further decrease of this EDL. Therefore, zero-dose rate EDL neither depend on the pump power. This conclusion could not be drawn from photobleaching periods reported in Fig. 2, because of the very long transient at P in = 48.2 mW (region C) which is not completed when x rays are turned on again (at t ~ 55 h). In this case, the zero-dose rate EDL is expected to be slightly below 13 dB m -1 . The fact that photo-bleaching does not decrease the RIA down to zero but to a zero-dose rate EDL independent of the pump power and of the initial RIA level strongly suggest that part of color centers is not bleachable.

C. EDL dependence on dose rate and pump power

EDLs extracted from Fig. 2 to 4 are plotted in Fig. 5 (resp. 6) as a function of the dose rate (resp. pump power) at fixed pump powers (resp. dose rate). EDL variations are all well fitted by power laws in D' a or P in b . For a given fiber and length, the dose rate exponent a is quite independent of pump power. For Er80 samples, a = 0.22-0.23 for L = 0.5 cm, and 0.17-18 for L = 2 cm (Fig. 5). Conversely, the pump power exponent b lies between -0.38 and -0.39 whatever the dose rate for Er80 and L = 0.5 cm and its value decreases slightly down to -0.44 when L = 2 cm. Variations in a and b values are more pronounced when passing from the Er80 to the M04 fiber. For L = 2 cm and P in = 132 mW, the dose rate exponent of M04 is for instance twice that of Er80 (a = 0.34 instead of 0.17, see Fig. 5). The higher sensitivity of a and b to the fiber type, by comparison to extrinsic parameters such as dose rate, pump power or length, may indicate that these exponents primarily reflect intrinsic mechanisms whose efficiency vary with composition or doping level. A power law dependence of the RIA on the dose rate D' is not -a priori -an original finding, since the RIA was empirically found to vary as D' (1 -f) D f , where D is the dose and f an exponent [START_REF] Berné | A model for the prediction of EDFA gain in a space radiation environment[END_REF][START_REF] Caussanel | Extrapolation of radiation-induced EDFA gain degradation at space dose rate[END_REF][START_REF] Griscom | Radiation-induced defects in glasses: origin of power-lax dependence of concentration on dose[END_REF] (f is close to 0.8 in [START_REF] Caussanel | Extrapolation of radiation-induced EDFA gain degradation at space dose rate[END_REF]). However, we deal here with new reference levels, i.e. EDL, which do not depend on the dose (RIA is frozen at the EDL, even under continuous irradiation at constant P in and D'). We have therefore EDL∝(D' a ×P in b ), the dose exponent being zero.

D. Intermittent Pumping

Stabilization of the RIA at EDL is a direct consequence of continuous pumping. The modification of this property in case of intermittent pumping is now addressed, for two reasons. First, intermittent pumping can be considered more representative of actual operation conditions in space. To save energy, the pump should be turned on at some required times only (when the EDFA-based system is needed) rather than continuously. Since 976 nm photons do not only probe RIA but also affect it, it is also necessary to limit pump ignition to the bare minimum (discrete measurements) and assess whether the 'intrinsic' 976 nm RIA can be estimated reliably.

Intermittent pumping tests were made on fresh 2 cm-long pieces of the Er80 fiber at various dose rates. Results are illustrated in Fig. 7. The lower plot has been obtained under 40, 60 and 10 krad min -1 continuous irradiations, by turning on the pump at 132 mW for 5 s every 15 minutes (pump ignition time ratio 0.56%), or every hour (0.14%). We set the record frequency to get one data point per second. RIA levels are higher than EDL found in Fig. 2 at this dose rates and pump power (40, 36 and 30 dB m -1 respectively), but the pump effect is still strong. It prevents the RIA from continuous increase and leads to its stabilization at a mean level around 100-120 dB m -1 at the three dose rates and both pump ignition periods. We again reduced the pump impact by dropping the pump power down to 7.7 mW and the ignition time ratio to 0.06 and 0.03 % (2 s ignition every 1 or 2 hours). We then obtained the upper plot of Fig. 7 at 40 and 60 krad min -1 dose rates. Despite this further reduction, PB is still sufficient to make the RIA stabilize, now at a mean level around 260-270 dB m -1 . In intermittent pumping conditions, the EDL still exists under the form of mean steady-state levels. Such mean EDL tend to increase with the dose rate, but very slowly compared with the case of continuous pumping (at least within the 10-60 krad min -1 range explored in Fig. 7). Mean EDL are markedly enhanced at low pump power. They are almost independent of the ignition period (time between two successive pump ignitions), but highly sensitive to the duration of the pump operation. The RIA drops indeed very rapidly upon pump lighting, as evidenced by steep falls in Fig. 7. The fall rate tremendously increase with the starting RIA level: about 28 dB m -1 s -1 around the mean EDL at 270 dB m -1 (upper plot at 7.7 mW), but 'only' 14 dB m -1 s -1 around 110 dB m -1 (lower plot at 132 mW) for a pump power which is though >14 times greater. To maintain a low RIA, it is therefore much more efficient to increase the pump operation time than its ignition frequency. Even short, low-frequency and low-power pump actions induce efficient RIA mitigation. As a result, measuring the 'intrinsic' RIA at 976 nm turns out to be a really hard task! At the very low dose rates of spacebased applications, RIA will develop very slowly and will not be restored significantly between successive pump operations. Successive ignitions should keep degradation at a low level provided their duration and period are properly optimized. We finally estimated the room temperature annealing (RTA) by stopping x-rays and waiting for 9 h before the last RIA record. RTA is more significant when it starts from the 270 dB m -1 level (upper plot, almost -40 % in 9 h). The relative annealing is 4 times smaller when starting from 90 dB m -1 (lower plot). In Fig. 2 to 4, RIA levels were such that RTA was weak: with significant RTA, we could not reach the same EDL, as we did three times in 21 h at 132 mW and 10.5 krad min -1 (Fig. 2).

IV. MODELING AND DOSE RATE EXTRAPOLATION

A. Local physical model

The physical model of EDF degradation is adapted from that proposed for ytterbium-doped fibers (YDF) in [START_REF] Mady | A physical model of the photo-and radiation-induced degradation of ytterbiumdoped silica optical fibers[END_REF] by introducing two specificities, namely the absence of photodarkening and the existence of a non-bleachable RIA. We also already proposed that radiation-induced ionization effects, which start by electron-hole pair excitation, are somewhat similar in YDF [START_REF] Mady | Thermoluminescence characterization of traps involved in the photodarkening of ytterbiumdoped silice fibers[END_REF] and EDF [START_REF] Mebrouk | Experimental evidence of Er 3+ ion reduction in the radiation-induced degradation of erbium-doped silica fibers[END_REF]. Both notably include reduction of Yb 3+ or Er 3+ ions into divalent ions (due to free electron capture), and trapping of free holes on silica-or dopant-related precursors to form color centers. Spectrallyresolved thermoluminescence analyses, showing the typical emissions of trivalent ions, revealed that radiation-induced Yb 2+ or Er 2+ ions are the dominant recombination centers when trapped holes are released by thermal annealing (recombination re-forming trivalent ions) [START_REF] Mady | Thermoluminescence characterization of traps involved in the photodarkening of ytterbiumdoped silice fibers[END_REF][START_REF] Mebrouk | Experimental evidence of Er 3+ ion reduction in the radiation-induced degradation of erbium-doped silica fibers[END_REF]. These hypotheses inspired the darkening/bleaching scheme of Fig. 8. 1.62 eV). Retaining cooperative ion-pair de-excitation as the energy supplier of pump-induced PB enabled us to explain various experimental features in YDF [START_REF] Mady | Thermoluminescence characterization of traps involved in the photodarkening of ytterbiumdoped silice fibers[END_REF]. In EDF however, first modeling attempts showed this hypothesis does not correctly render the observed EDL dependence on pump power. In EDFA operating at 1530 nm, upconversion is a detrimental effect causing parasitic losses. It is normally weak (neglected in [START_REF] Berné | A model for the prediction of EDFA gain in a space radiation environment[END_REF] for instance) at low or moderate pump powers. It probably contributes to PB at high power levels, but its consideration adds much complexity in the equations and their treatment. For simplicity, we suppose that two-photon absorption is the dominant source of PB. This rough limitation offers future refinement opportunities. Fig. 8 presents a minimal model including only two discrete hole-trapping levels. Other possible electron traps besides Er 3+ ions are neglected, as well as thermal release in agreement with preceding conclusions. It would be more realistic to include a distribution of gap levels because of the variety of centers identified in silicate or alumino-silicate glasses and of the inhomogeneity of such materials. This would introduce a significant complexity that would again interfere with the validation of the crucial mechanisms that are basically sufficient to explain the observed behaviors.

Transitions of Fig. 8 can be rationalized by the set of coupled non-linear differential equations ( 1)-( 5) defining the local darkening/bleaching evolution model.

€ dn c dt = g 0 -δ n c (N -n), (1)
€ dn dt = δ n c (N -n) -γ h v n, (2)
€

dh 1 dt = β 1 h v (H 1 -h 1 ) -σ PB P in 2 N v h 1 , (3)
€ dh 2 dt = β 2 h v (H 2 -h 2 ), (4)
€

dh v dt = g 0 -β 1 h v (H 1 -h 1 ) -β 2 h v (H 2 -h 2 ) + σ PB P in 2 N v h 1 -γ h v n. (5)
In ( 1)-( 4), n c and h v are the densities of electrons in the CB and of holes in the VB, respectively. N V is the equivalent density of states in the VB. The photo-bleaching rate of shallow traps is taken proportional to the square of the pump power P in , as usual for two-photon processes, and the proportionality coefficient is σ PB . The latter includes among others the photo-ionization cross-section of trapped-hole color centers. Capture and recombination coefficients of free carriers (β 1,2 , δ and γ) are commonly described as the product of the thermal velocity of free carriers in their band by the trapping or recombination cross section.

Color centers associated with shallow and deep trapped-hole states produce overlapping optical absorption bands in the near infra-red range so they both contribute to the 976 nm RIA. Calculations show that transient band populations n c and h v are always negligible (by far) so n and h 1 + h 2 are virtually identical. The computed RIA is proportional to the density of color centers, h 1 + h 2 (or n), which is the output quantity to be compared with data. Time-resolved RIA and EDL have been simulated by solving (1)-( 5) numerically with the Scilab® software (all densities being initially equal to zero).

B. Validation against data

We first simulated parts of reported experiments to check the capability of the model of reproducing basic experimental features. Comparison is relevant since we confront at this stage a local model, rigorously valid for a fiber length tending to zero, to data measured in very short fiber samples.

Calculations have been made with the following material parameters: N = 10 × (H 1 +H 2 ) = 3×10 19 cm -3 (case of the Er80 fiber), H 1 = 0.9×(H 1 +H 2 ), γ = δ = 1.26×10 -11 cm 3 s -1 , β 1 = 1.24×10 -11 cm 3 s -1 , β 2 = 1.26×10 -13 cm 3 s -1 , and σ PB N v = 4×10 -7 mW -2 s -1 . Orders of magnitude for trapping and recombination coefficients are consistent with those expected for trapping and recombination cross-sections (10 -21 -10 -12 cm 2 [START_REF] Dimaria | Defects and impurities in thermal SiO2[END_REF]) and for the thermal velocity of holes in silica (significantly lower than that of electrons which is usually estimated to 10 7 cm s -1 ). They are also similar to those used in [START_REF] Mady | Golbal view on dose rate effects in silica-based fibers and devices damaged by radiation-induced carrier trapping[END_REF] where we applied a generalized model to germaniumdoped optical fibers. Given these values, the order of magnitude of σ PB N v was determined to reproduce correctly the experimental time scale. We used the unique set of parameters above for all results in Fig. 9-12, to allow comparison between the various simulations. A qualitative agreement with data is actually always obtained, even if these parameters are varied across several decades. Dose rates in krad min -1 have been converted into pair generation rate g 0 in cm -3 s -1 thanks to the silica pair generation efficiency G ~ 5.18×10 15 pairs cm -3 krad -1 . Computed RIA is presented in terms of the proportion of Er 3+ to Er 2+ ion conversion given by the ratio r = n/N. With chosen parameters, this ratio can be converted into dB m -1 according to RIA(dB m -1 ) = σ CC ×3×10 22 ×r/ln [START_REF] Gilard | Theoretical explanation of enhanced low dose rate sensitivity in erbium-doped optical fibers[END_REF], where σ CC is the total optical absorption cross-section by color centers at 976 nm. This cross-section is not known, but taking σ CC = 10 -19 cm 2 as a typical order of magnitude gives a correspondence of 13 dB m -1 for each per cent of Er 3+ to Er 2+ ion conversion. Inset: EDL vs pump power at 10.5 krad min -1 and its power law fit. Predicted conversion rates range between 5 to 10 percents. This order of magnitude is consistent with the 10 to 30 % estimates, according to the glass composition, given in [START_REF] Dimaria | Defects and impurities in thermal SiO2[END_REF] (aluminum enhances conversion). In this reference, conversion ratios were obtained after a 300 krad dose only, but in the absence of pump and hence of photo-induced mitigation. Simulations of Fig. 11 have been done to check the model predictions with regards to zero-dose rate EDL. After irradiation at 10.5 krad min -1 under 132 mW pumping to reach a significant RIA level, the system was left to the sole action of the pump at 48.2, 132 and 178.5 mW to make it tend to the zero-dose rate EDL. As observed in experiments, this EDL is not zero: deep hole traps well introduce a non-bleachable RIA background. The second pump power switch (from 132 to 178.5 mW) happens when the EDL is already established, and it does not induce any further bleaching. This point is in full agreement with the behavior highlighted in Fig. 3, region C (around 117 h, same powers involved). Modeled mechanisms are therefore sufficient to account for the independence of zero-dose rate EDL on the pump power. In Fig. 11, the EDL could seem to be higher under 48.2 mW than under 132 or 178.5 mW pumping. In fact the RIA always falls to the same limit, the decay is just much slower at 48.2 mW and PB is markedly accelerated when the pump power is raised to 132 mW. This may explain the apparent difference between the ill extrapolated zero-dose rate EDL marked in zones A and C of Fig. 2 at 13 and 18 dB m -1 .

Fig. 11. RIA resulting from the simulation of zero-dose rate EDL after irradiation. Numeric labels: dose rates and pump powers.

V. DISCUSSION AND CONCLUSION

This work deals with RIA mitigation by 976 nm pumping in EDF. The underlying PB effect had been reported already [START_REF] Zotov | Radiationresistant Er-doped fibers: optimization of pump wavelength[END_REF][START_REF] Lezius | Radiation-induced absorption in rare earth doped optical fibers[END_REF][START_REF] Gilard | Theoretical explanation of enhanced low dose rate sensitivity in erbium-doped optical fibers[END_REF][START_REF] Thomas | Radiation-resistant erbiumdoped-nanoparticles optical fiber for space applications[END_REF][START_REF] Girard | Radiation hardening techniques for Er/Yb doped optical fibers and amplifiers for space applications[END_REF][START_REF] Girard | Radiation-hard erbium optical fiber and fiber amplifier for both low-and high-dose missions[END_REF][START_REF] Friebele | Radiation protection of fiber optic materials: effects of cerium doping on the radiation-induced absorption[END_REF][START_REF] Peng | Photoannealing effects for erbium doped fiber sources after gamma irradiation tests by using 532 nm and 976 nm lasers[END_REF] but not examined in detail. Proper characterization conditions using very short fibers samples have been proposed to liberate observables from spatial distribution effects and elucidate basic mechanisms. Our results demonstrate a novel and important property: due to pump-induced PB, the local RIA can be tuned reversibly between equilibrium degradation levels (EDL) by varying the local pump power P in and/or the dose rate D' while the dose is being accumulated. EDL are found to vary as D' a and P in b , with a = 0.17 to 0.34 and b = -0.44 to -0.26 for investigated EDF. Intermittent pumping also leads to the formation of mean EDL determined by pump power and operation duration. Basically, EDL could provide a most relevant reference to assess the radiation-resistance of EDF. They set the maximum (worst) local degradation level achievable at given (P in ,D'). Optimizing EDL, rather than lower dose-dependent levels, focuses the discussion on 'rates' (pump energy rate and dose rate), while offering the best safety margin. These features had never been noticed to date because studies were based on standard fiber amplifier lengths. At fixed launched power and dose rate, regions of long EDF experiencing higher pump power (Fig. 1) reach their EDL rapidly, whereas RIA in lower pump power regions cannot stabilize readily due to their high EDL. This kinetics difference is reinforced with the passage of time, since lower pump power means faster RIA build-up, higher pump losses, even less efficient PB and enhanced EDL… Thus it takes much more time for length-integrated RIA to reach a stationary level, and this considerable delay probably explains why EDL have not been observed in long fibers. Local EDL increase as the fiber is exposed to irradiation, due to the development of pump power loss (P in decreases). EDF regions with initially high (resp. low) pump power will experience more (resp. less) efficient PB so their RIA, together with their local EDL, will increase slowly (resp. rapidly). In other words, lower EDL are less rapidly damaged (i.e. raised). In spacebased applications, the small dose rate will not only make initial EDL low, but it will also ensure a very slow increase of this EDL as well as of the RIA towards it. Low EDL also require high pump power, but the notion of 'high' power is only relative to the dose rate magnitude. It is defined as a power inducing efficient PB, i.e. capable of photo-ionizing color centers at a frequency comparable to that of their formation (set by D'). Under low dose rate conditions, 'high' pump powers can be low actually. Therefore efficient PB and low EDL may be the case throughout the fiber, even if a simple one-directional pumping scheme is used. As a result, the low 976 nm RIA level predicted by the local model (Fig. 12) at a space dose rate can be probably projected for long fibers: EDFA should not be significantly damaged in space, whatever the total duration and dose (since EDL do not depend on the dose). This conclusion is opposite to that of enhanced low dose rate sensitivity (ELDRS) demonstrated in [START_REF] Gilard | Theoretical explanation of enhanced low dose rate sensitivity in erbium-doped optical fibers[END_REF], which anticipates higher degradation levels in space than in accelerated tests made on Earth. Our experimental results did not exhibit ELDRS. However, we established that ELDRS is the most usual form of dose rate effects inherent in trappingrecombination models when detrapping is neglected [START_REF] Mady | Golbal view on dose rate effects in silica-based fibers and devices damaged by radiation-induced carrier trapping[END_REF]. The model of Fig. 8 is thus capable of yielding ELDRS if pump power and PB are weak and observation of ELDRS is not inconsistent with our model, depending on experimental conditions (small launched pump power and long fibers). Since we have now a validated local degradation model at our disposal, the next step is to implement it into EDFA simulations to take pump, ASE and signal distribution effects into account and develop predictive simulators for realistic EDFA. Then, we will be able to confirm the qualitative arguments developed in this conclusion and explore the conditions enabling ELDRS.

Fig. 4 .

 4 Fig. 4. Temporal evolution of the RIA under various pump powers/dose rates (indicated on graph) applied continuously on the same 2 cm-long piece of the M04 laboratory-made EDF. Cumulated doses reached at the end of each step are indicated.

Fig. 3 .

 3 Fig. 3. Temporal evolution of the RIA under various pump power/dose rate (indicated on graph) applied continuously on the same 0.5 cm-long piece of the COTS Er80-4/125 fiber. Cumulated doses reached at the end of each step are indicated.

Fig. 2 .

 2 Fig. 2. Temporal evolution of the RIA under various pump power/dose rate couples (upper part), and the corresponding evolution of the cumulated dose in silica (lower part), applied continuously on the same 2 cm-long piece of the COTS Er80-4/125 fiber.

Fig. 5 .

 5 Fig. 5. EDL as a function of dose rate at fixed pump powers between 48.2 and 178.5 mW, and power law fits in dashed lines. Fiber types, sample lengths, pump powers and fit equations are indicated on graphs.

Fig. 6 .

 6 Fig.6. EDL as a function of input pump power at fixed dose rates between 10 and 55 krad min -1 , and power law fits in dashed lines. Fiber types, sample lengths, dose rates and fit equations or exponents are indicated on graphs.

Fig. 7 .

 7 Fig. 7. RIA at 976 nm measured by short pump ignition periods. Lower plot: Pin = 132 mW during 5 s every 15 min or 1 h. Upper plot: Pin = 7.7 mW during 2 s, pump ignition period = 1 or 2 h at 60 krad min -1 and 2 h at 40 krad min -1 .

Fig. 8 .

 8 Fig. 8. Energy level scheme of the proposed model. Dashed arrows = trapping/recombination steps. Solid arrows = effective energy transitions between initial and final states.The ionizing radiation creates electron-hole pairs at a rate g 0 , representative of the dose rate. The total ionizing dose thus corresponds to the total density of generated pairs given by g 0 ×t irr , where t irr is the irradiation time. Electrons injected in the conduction band (CB) can be trapped by Er 3+ ions (initial density N) to form Er 2+ ions in density n (δ transitions). Holes in the VB are trapped by gap states to form color centers (H 1,2 precursor states occupied by h 1,2 holes, β 1,2 transitions). These gap states should be associated with defects of the host matrix, i.e. silica point defects or dopant-related centers (due to, e.g., aluminum, phosphorus…). We assume photo-bleaching (PB) results from the photo-ionization of trapped holes to the VB (σ PB transition) and from their subsequent recombination on Er 2+ ions (γ transition). PB only concerns shallowest trappedhole states (density h 1 ). 'Optically disconnected' deep hole traps (h 2 ) are introduced to account for the non-bleachable part of the RIA. Actually, even shallow traps are likely too deep to enable direct ionization by single pump photons (1.27 eV), because efficient photo-ionization requires about twice the thermal activation energy which is typically around or greater than 1 eV. Three mechanisms could then contribute to PB: a two-pump photon absorption (2.54 eV), the absorption of green photons emitted after pump-induced upconversion of Er 3+ ions ( 4 S 3/2 to 4 I 15/2 radiative de-excitation, 2.28 eV), or cooperative de-excitation of 2 close inverted Er 3+ ions upon spontaneous 4 I 13/2 to 4 I 15/2 1530 nm relaxation (2 × 0.81 eV =

Fig. 9 .

 9 Fig. 9. Simulation of the first irradiation sequence of Fig. 2 (zone A) and a variant showing that EDLs given by the model are well reproducible for identical (Pin, D') couples. Numeric plot labels: dose rates in krad min -1 . Inset: EDL as a function of the dose rate and its power law fit.

Fig. 10 .

 10 Fig. 10. Simulation of irradiation sequences of Fig. 2: end of zone A, zone B and beginning of zone C (dose rates and pump powers indicated in plots).Inset: EDL vs pump power at 10.5 krad min -1 and its power law fit.

Fig. 9

 9 Fig.9was obtained by simulating the first 6 steps of Fig.2, zone A, with the same sequence of decreasing dose rates (from 60 down to 2.6 krad min-1 ). It also shows the result of a different series where the last two dose rates have been raised back to the third and second values to show that simulated RIA well returns to the same EDL at identical dose rates. Fig.10presents the simulation of the end of zone A, zone B, and of the beginning of zone C of Fig.2. It mainly aims at showing that the modeled system behaves as real EDF when pump power switches are applied. As experimental RIA, the computed degradation well exhibits EDL that are only determined by the couple (P in , D'). The power law dependence of the EDL on D' and P in are qualitatively reproduced (see insets of Fig.9 and 10). Computed dose rate and pump exponents, a = 0.27 and b = -0.44, are rather close to experimental ones for the Er80 fiber. Predicted conversion rates range between 5 to 10 percents. This order of magnitude is consistent with the 10 to 30 % estimates, according to the glass composition, given in[START_REF] Dimaria | Defects and impurities in thermal SiO2[END_REF] (aluminum enhances conversion). In this reference, conversion ratios were obtained after a 300 krad dose only, but in the absence of pump and hence of photo-induced mitigation. Simulations of Fig.11have been done to check the model predictions with regards to zero-dose rate EDL. After irradiation at 10.5 krad min -1 under 132 mW pumping to reach a significant RIA level, the system was left to the sole action of the pump at 48.2, 132 and 178.5 mW to make it tend to the zero-dose rate EDL. As observed in experiments, this EDL is not zero: deep hole traps well introduce a non-bleachable RIA background. The second pump power switch (from 132 to 178.5 mW) happens when the EDL is already established, and it does not induce any further bleaching. This point is in full agreement with the behavior highlighted in Fig.3, region C (around 117 h, same powers involved). Modeled mechanisms are therefore sufficient to account for the independence of zero-dose rate EDL on the pump power. In Fig.11, the EDL could seem to be higher under 48.2 mW than under 132 or 178.5 mW pumping. In fact the RIA always falls to the same limit, the decay is just much slower at 48.2 mW and PB is markedly accelerated when the pump power is raised to 132 mW. This may explain the apparent difference between the ill extrapolated zero-dose rate EDL marked in zones A and C of Fig.2at 13 and 18 dB m -1 .

Manuscript received This work falls within the framework of the HAPoLO (098-AO11-2408) and PARADYSIO (ANR-11-JS04-007) projects supported by the AirbusGroup Foundation and the French Research National Agency (ANR).