Nicolas Cherroret

Coherent multiple scattering of light in (2+1) dimensions

Keywords: numbers: 42, 25, Dd, 72, 15, Rn

We formulate a multiple scattering theory of light in media spatially disordered along two directions and homogeneous along the third one, without making any paraxial approximation on the wave equation and fully treating the vector character of light. With this formalism, we calculate the distribution of transverse momenta of a beam as it evolves along the optical axis, and unveil a phenomenon not captured by the paraxial equation: a cross-over from a scalar to a vector regime, visible in the coherent backscattering peak as polarization gets randomized.

I. INTRODUCTION

When light propagates mostly along a given direction z in a weakly heterogeneous medium, its evolution is in first approximation governed by the paraxial wave equation. The latter has the same structure as the time-dependent Schrödinger equation, the longitudinal coordinate z playing the role of time [START_REF] Cherroret | Nonlinear Fiber Optics[END_REF][START_REF] Rosanov | Spatial Hysteresis and Optical Patterns[END_REF]. In Kerr media this idea was recently applied to study optical analogues of quantum matter-wave phenomena, like Bose condensation [START_REF] Sun | [END_REF]4] or superfluidity [5,6]. In a similar spirit, in spatially disordered materials paraxial light propagation was exploited to observe the coherent backscattering (CBS) effect without interface [7], in a configuration that reproduces cold-atomic setups [8,9]. In the same context, transverse Anderson localization of optical wave packets [10] was reported in the paraxial limit [11,12]. A major drawback of the paraxial equation, however, is that it neglects the vector nature of light. In presence of disorder, how polarization effects manifest themselves beyond the paraxial approximation and when the latter breaks down is not well known. Clarifying these questions is essential to clearly identify the regime where a scalar description of light can be used in experiments that operate close to the paraxial limit. More generally, understanding the role of the vector nature of light in heterogeneous media is required for a proper characterization of disorder-induced interference effects like CBS, known to be sensitive to polarization in general [13,14]. The question is even crucial for Anderson localization of light, which was shown to be severely altered in random ensembles of scatterers when vector effects are accounted for [15][16][17].

In this paper, we develop a general theory of multiple scattering of light in media disordered along two transverse directions x and y and homogeneous along the third one z (the optical axis). This theory does not rely on the paraxial approximation, but is constructed from the exact Helmholtz wave equation. As a concrete example, we calculate the disorder-average distribution |E(k ⊥ , z)| 2 * Electronic address: cherroret@lkb.upmc.fr FIG. 1: We consider the propagation of a quasi plane-wave beam of transverse wave vector k0 and polarization through a medium spatially disordered along x and y and homogeneous along the optical axis z. Light is detected on a polarization channel belonging to the plane (x, y).

of transverse momenta of an incident plane-wave beam of transverse momentum k 0 as it propagates along z, see Fig. 1. In this configuration, it was theoretically shown in the framework of the paraxial (or Schrödinger) equation that this distribution evolves toward a pedestal ring of radius |k 0 | due to the diffusive randomization of momenta, on top of which a CBS peak arises around k ⊥ = -k 0 [8,18]. This phenomenon was recently observed experimentally in photorefractive materials [7]. By revisiting it within a full vector treatment, we discover an interesting effect beyond the paraxial approximation: the momentum distribution crosses-over from a scalar to a vector regime as z exceeds a characteristic scale z p , identified as the time over which the polarization direction is randomized. As long as z z p , the polarization remains fixed, the paraxial equation applies and the physics of CBS, in particular, is the one of a scalar wave. In strong contrast, when z z p the paraxial theory breaks down and the CBS peak starts to non-trivially depend on polarization. In particular, for a circularly-polarized incident beam the CBS peak is visible only from light of opposite circular polarization, as imposed by the reciprocity principle. We also find that the scale z p strongly depends on the injection angle of the incident beam (θ in Fig. 1), thus offering a convenient way of switching from a scalar to a vector regime for light in a 2D disordered environment.

II. OPTICAL MOMENTUM DISTRIBUTION

Our starting point is the Helmholtz equation for the complex electric field E(r) = E(x, y, z):

∆δ ij -∇ i ∇ j + k 2 δ ij -V (r ⊥ )δ ij E j (r) = 0, (1)
where k 2 = ω 2 ¯ /c 2 , with ω the light carrier frequency. The disorder "potential" V (r ⊥ ) = -k 2 δ (r ⊥ )/¯ stems from spatial fluctuations δ (r ⊥ ) of the refractive index around an average value ¯ . Disorder is only along r ⊥ = (x, y), not along z. In Eq. ( 1) and in the following, summation over repeated indices j = x, y, z is implied.

For simplicity, we choose the disorder to be Gaussian distributed and uncorrelated of strength γ:

V (r ⊥ )V (r ⊥ ) ≡ k 4 δ (r ⊥ )δ (r ⊥ ) ¯ 2 = k 4 γδ(r ⊥ -r ⊥ ), (2)
with δ (r ⊥ ) = 0. The main goal of the paper is to evaluate the disorder-average distribution | • E(k ⊥ , z)| 2 of transverse momenta as a function of the longitudinal coordinate z. In this definition, the output light beam, at z, is detected in the direction k ⊥ and along a polarization axis that lies in the plane (x, y), see Fig. 1. We model the incident beam at z = 0, E i (k ⊥ , z = 0) = A(k ⊥ ) i , by a field distribution A(k ⊥ ) peaked around k ⊥ = k 0 and polarized along the complex unit vector perpendicular to k (Fig. 1). The field at z is given by:

E j (k ⊥ , z) = 2ik d 2 k ⊥ (2π) 2 k ⊥ |G ij (z)|k ⊥ A(k ⊥ ) i , (3)
where G ij is the Green tensor of the Helmholtz equation [START_REF] Cherroret | Nonlinear Fiber Optics[END_REF]. The tensor 2ikG ij (z) can be interpreted as the medium's transmission coefficient from z = 0 to z [19]. In order to obtain the momentum distribution, we average the square modulus of Eq. (3) contracted with , introducing the Fourier transform

G ij (z) ≡ ∞ -∞ dk z /(2π)G ij (k z )e ikzz .
This gives [8]:

| • E(k ⊥ , z)| 2 = 4k 2 ∞ 0 dk z 2π ∞ -∞ dq z 2π d 2 k ⊥ (2π) 2 |A(k ⊥ )| 2 × * i j k ⊥ |G ij (k + z )|k ⊥ k ⊥ |G † kl (k - z )|k ⊥ * k l e iqzz , (4)
where k ± z ≡ k z ± q z /2 and we resorted to translation invariance in plane (x, y) to remove one integral over k ⊥ . We now assume a normalized, incident plane wave,

|A(k ⊥ )| 2 (2π) 2 δ(k ⊥ -k 0 )
, and decompose the average product of two Green tensors in a standard way [START_REF] Sheng | Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena[END_REF]:

| • E(k ⊥ , z)| 2 = 4k 2 * i j * k l × ∞ 0 dk z 2π G iα (k 0 , k z )G * kβ (k 0 , k z )G jγ (k ⊥ , k z )G * lδ (k ⊥ , k z ) × ∞ -∞ dq z 2π k ⊥ , k 0 |Γ αβ,γδ (k z , q z )|k 0 , k ⊥ e iqzz . (5)
In writing Eq. ( 5), we introduced four average Green tensors defined through the relation

k ⊥ |G ij (k z )|k ⊥ = δ(k ⊥ -k ⊥ )G ij (k ⊥ , k z
) implied by translation invariance on average in the plane (x, y). We also dropped the q z dependence of these tensors, which is justified in the largez limit [START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF] (the so-called diffusive regime, see below for the precise condition). Physically, the first two Green tensors describe the average propagation of the incoming beam at k 0 to the first scattering event in the material, and the last two ones the average propagation of the outgoing beam at k ⊥ from the last scattering event to the detection point. The structure tensor Γ, on the other hand, contains all information about the multiple scattering process from the first to the last scattering event. Evaluating the momentum distribution requires the computation of the tensors G and Γ. We execute this program in the next two sections.

III. AVERAGE GREEN TENSOR IN (2+1) DIMENSIONS

We first examine the average Green tensor G(k ⊥ , k z ), given by the vector Dyson equation [13,[START_REF] Lagendijk | [END_REF]23]

G(k ⊥ , k z ) = G (0) (k ⊥ , k z ) -1 -Σ(k ⊥ , k z ) -1 , (6)
where G (0) is the free-space, transverse Green tensor (here and in the following, we neglect near-field effects):

G (0) ij (k ⊥ , k z ) = δ ij -ki kj k 2 -k 2 ⊥ -k 2 z + i0 + , (7)
where k ≡ (k ⊥ , k z ). In this paper we assume a weakly disordered material, so that the Born approximation can be used for calculating the self-energy tensor Σ ij . For a delta-correlated potential, Eq. ( 2), this gives [START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF] Σ

ij (k z ) = d 2 k ⊥ (2π) 2 δ ij -k i k j k 2 -k 2 ⊥ -k 2 z + i0 + . (8)
Let us comment on the physical meaning of the longitudinal wave vector k z . At weak disorder, the Green tensors G(k 0 , k z ) in the general formulation (5) of the momentum distribution are peaked around k 2 z = k 2 -k 2 0 . Therefore, the parameter kz ≡ k z /k 1 -k 0 /k, between 0 and 1, quantifies the deviations of the incident beam from paraxiality. Σ ij (k z ) and G ij (k ⊥ , k z ) can be found for any value of k z by exact calculation of the integral in Eq. ( 8) and diagonalization of Eq. ( 6). We give their expressions in the appendix as they are rather cumbersome. They will be needed for the proper evaluation of the structure factor presented in Sec. IV. For the present discussion, however, let us focus our attention on the value of G for kz 1. In this limit, the imaginary part of the self-energy ImΣ ij (k z ) -δ ij γk 4 /4 and:

G ij (k ⊥ , k z ) δ ij -ki kj k 2 -k 2 ⊥ -k 2 z + ik/z s , (9)
where z s ≡ 4/(γk 3 ). By Fourier transforming Eq. ( 9) with respect to k z and inserting the result into Eq. ( 3), we infer:

E(k ⊥ , z) E(k ⊥ , z = 0)e ikz-z/2zs , (10)
which describes the usual depletion of the incident mode due to scattering as it evolves along z [START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF], at a rate governed by the effective scattering time z s . Eq. ( 9) also provides the regime of validity of the Born approximation:

|k 2 -k 2 z | ∼ k 2 0 k/z s .
This weak-disorder condition can be rewritten as the familiar k 0 1 [START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF], where ≡ k0 z s is the mean free path, proportional to the effective transverse velocity k0 of the incident beam.

Let us remind that Eq. ( 9) was obtained assuming kz 1, i.e. quasi-paraxiality. This can be confirmed a posteriori by noticing that Eq. ( 9) indeed coincides with the solution of the paraxial wave equation. The latter is traditionally obtained by writing E(r) = E(r)e ikz and assuming slow variations of the envelope along z, |∂ 2 E/∂z 2 | k|∂E/∂z|, and of the permittivity in the plane (x, y),

|∇ ⊥ | k [1, 2]
. Under these conditions, the envelope E is mostly transverse and the Helmholtz equation ( 1) simplifies

∆δ ij + 2ikδ ij ∂ ∂z -V (r ⊥ )δ ij E j (r) = 0, (11)
which mimicks a time-dependent Schrödinger equation where z plays the role of time. By applying to Eq. ( 11) the scalar version of the Green function technique introduced above [8], we recover, in the Born approximation, that

E(k ⊥ , z) = E(k ⊥ , z = 0) exp(-z/2z s ).

IV. STRUCTURE FACTOR IN (2+1) DIMENSIONS A. Classical contribution

Under the weak-disorder condition k 0 1, the structure factor has two well-known contributions: the series of ladder and of crossed diagrams, which respectively describe classical diffusion and coherent backscattering. We first focus on the ladder series. It gives a contribution Γ (L) to Γ, which obeys the Bethe-Salpether equation [13,[START_REF] Sheng | Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena[END_REF][START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF][START_REF] Lagendijk | [END_REF][23]:

Γ (L) αβ,γδ (k z , q z , q) = γk 4 δ αγ δ βδ + γk 4 Γ (L) mn,γδ (k z , q z , q) × d 2 k (2π) 2 G αm (k + , k + z )G * βn (k -, k - z ), (12)
where k ± = k ± q/2 and we introduced the short-hand notation

Γ (L) αβ,γδ (k z , q z , q) ≡ k + ⊥ , k - 0 |Γ (L) αβ,γδ (k z , q z )|k + 0 , k - ⊥ . ( 13
)
The Bethe-Salpether equation is shown digrammatically in Fig. 2, which also indicates the conventions used for wave vectors and polarization indices. Notice that the object ( 13) is slightly more general than the structure factor in Eq. ( 4), because it involves incoming and outgoing wave vectors k ± 0 = k 0 ± q/2 and k ± ⊥ = k ⊥ ± q/2 instead of simply k 0 and k ⊥ . The additional variable q will be useful later for evaluating the crossed contribution. The definition (13) moreover expresses that Γ (L) αβ,γδ (k z , q z , q) does not explicitly depend on k 0 and k ⊥ [this is why we could take Γ (L) out of the integral over k in Eq. ( 12)]. Such property is a consequence of our model of uncorrelated disorder, Eq. ( 2), which scatters light isotropically.

The momentum distribution (5) involves the ladder structure factor at q = 0. To find it, we set q = 0 in Eq. ( 12) and introduce the tensor

Q αβ,γδ (k z , q z ) ≡ γk 4 d 2 k (2π) 2 G αγ (k , k + z )G * βδ (k , k - z ) (14)
as well as I αβ,γδ ≡ γk 4 δ αγ δ βδ . Eq. ( 12) thus reads

Γ (L) = I + Q • Γ (L) = γk 4 ∞ m=0 Q m . ( 15
)
The infinite sum can be carried out by diagonalizing Q in terms of orthogonal projectors Π (n) and associated eigenvalues λ n [24]:

Q = n λ n Π (n) , (16)
where

Π (n) •Π (n ) = δ nn and n Π (n) = 1.
Substituting the formal decomposition (16) for Q in Eq. ( 15), we obtain:

Γ (L) = n γk 4 1 -λ n Π (n) . ( 17
)
The problem thus reduces to finding the eigenmodes (λ n , Π (n) ). This task requires to use the complete expression of the average Green tensor, Eq. (41) of the FIG. 3: Factors 1 -λn(kz, qz = 0) of the mode decomposition (17). The mode n = 6 fulfills 1 -λ6(kz, 0) = 0 for all kz. It thus always controls the large z (small qz) limit of the structure factor (18). In the experimentally-relevant limit where kz is close to 1, the two modes n = 1 and 2 obey 1 -λ1,2(kz, 0) 1, so they also contribute in general. The modes n = 3, 4 and 5 are, on the other hand, strongly attenuated in this regime.

appendix. Due to the complex anisotropic structure of the latter, the diagonalization of the tensor Q is considerably more difficult than in usual 3D isotropic disordered media, which are rotationally invariant on average [13].

We have been able to derive the modes (λ n , Π (n) ) in the diffusive regime z z s , equivalently q z z s 1, by unfolding the coordinates of Q αβ,γδ onto a 9 × 9 matrix to diagonalize. The polarization space of this matrix can be decomposed into 6 eigensubspaces, three of them associated with a twice-degenerate eigenvalue. The exact expressions of the (λ n , Π (n) ) are rather cumbersome so they are reported in the appendix for clarity. The eigenvalues are of the form

λ n (k z , q z ) = λ n (k z , 0) -iq z µ n (k z ), such that Γ (L) αβ,γδ (k z , q z , 0) = 6 n=1 γk 4 Π (n) αβ,γδ (k z ) 1 -λ n (k z , 0) + iq z µ n (k z ) . (18)
In the diffusive regime q z z s 1, Eq. ( 18) is dominated by the smallest factors 1-λ n (k z , 0). These quantities are displayed in Fig. 3 as a function of kz = k z /k, from the paraxial regime ( kz 1) to the limit of grazing incidence ( kz 1). The plots reveal two remarkable properties. First, the mode n = 6 satisfies 1 -λ 6 (k z , 0) = 0 whatever kz . This implies a diffusive behavior at large distances, Γ (L) (k z , q z , 0) ∝ 1/(iq z ), which is needed to guarantee the conservation of normalization [25]. It is worth noticing that keeping the complete expression (41) of G is absolutely crucial to recover this property. Second, in the vicinity of the paraxial regime, kz 1, two additional modes, n = 1 and 2, have very small denominators 1 -λ 1,2 (k z , 0) and therefore persist up to very long times. This phenomenon, which constitutes the main finding of the paper, signals a cross-over in light propa-gation, occuring around a characteristic scale z p that will be discussed in detail in Sec. V. In short, for z z p the modes n = 1 and 2 are negligible, whereas for z z p they must be accounted for, which qualitatively modifies the properties of the structure factor. In the strict limit kz = 1 (beam aligned along the optical axis), we even have 1 -λ 6 (k z , 0) = 1 -λ 1 (k z , 0) = 1 -λ 2 (k z , 0) = 0, so that the three modes n = 1, 2 and 6 equally contribute to Eq. ( 18) up to arbitrary large z.

B. CBS contribution

The CBS interference peak is described by the series of crossed diagrams denoted by Γ (C) . By virtue of reciprocity, Γ (C) is related to Γ (L) through [START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF] k

⊥ , k 0 |Γ (C) αβ,γδ (k z , q z )|k 0 , k ⊥ = k ⊥ , -k ⊥ |Γ (L) αδ,γβ (k z , q z )|k 0 , -k 0 ( 19
)
as illustrated diagrammatically in Fig. 2. According to Eq. ( 13), we thus obtain the CBS contribution from Γ

(L)
αβ,γδ (k z , q z , q), by interchanging β and δ and setting q = k ⊥ + k 0 . This requires to solve Eq. ( 12) for q = 0. We have accomplished this task in the diffusive regime |q|z s 1. This leads to an extra dependence of the λ n on q 2 (the q 2 corrections to the projectors Π (n) are, on the other hand, negligible):

Γ (C) αβ,γδ (k z , q z ) = Γ (L) αδ,γβ (k z , q z , k 0 + k ⊥ ) = (20) 6 n=1 γk 4 Π (n) αδ,γβ (k z ) 1 -λ n (k z , 0) + iq z µ n (k z ) + ν n (k z )(k ⊥ + k 0 ) 2 .
The explicit expression of the new factors ν n (k z ) is given in the appendix for the three modes n = 1, 2 and 6, relevant at large z near the paraxial limit.

V. RESULTS

A. The paraxial regime revisited

We now derive the explicit expression of the momentum distribution (5) for a typical setup where the incident beam is almost aligned with the optical axis ( k0 1). In this configuration, it is sufficient to use Eq. ( 9) to evaluate the four average Green tensors in Eq. ( 5). The first two ones are strongly peaked around k 2 z k 2 -k 2 0 , which allows us to approximate Γ αβ,γδ (k z , q z ) Γ αβ,γδ ( k 2 -k 2 0 , q z ) Γ αβ,γδ (k, q z ). The integral over k z then applies to the product of four Green tensors only

∞ 0 dk z 2π G iα (k 0 , k z )G * kβ (k 0 , k z )G jγ (k ⊥ , k z )G * lδ (k ⊥ , k z ) = z s /2k 2 (2k/z s ) 2 + (k 2 ⊥ -k 2 0 ) 2 δ iα δ kβ δ jγ δ lδ . (21)
We dropped here the tensorial parts of the type ki kα of the Green tensors, which are negligible once contracted with the incoming and outgoing polarization vectors and . Let us now deal with the structure factor in Eq. ( 5). As seen in Sec. IV A, when k0 1 (corresponding to kz 1) only the three modes n = 1, 2 and 6 contribute to Γ. Eqs. ( 18) and ( 20) then lead to:

∞ -∞ dq z 2π k ⊥ , k 0 |Γ αβ,γδ (k z , q z )|k 0 , k ⊥ e iqzz (22)
(γk 4 ) 2 4k n=1,2,6 e -zn/zs Π (n) αβ,γδ + Π (n) αδ,γβ e -Dn(k ⊥ +k0) 2 z .
The first term inside the brackets is the ladder contribution, while the second describes the CBS peak.

The projectors Π (n) and their corresponding lifetimes

z n ≡ µ n (k z )/[1 -λ n (k z , 0)
] take a particularly simple form in the regime kz 1 considered here:

Π (6) αβ,γδ 1 2 δ αβ δ γδ , z 6 = ∞ Π (1) αβ,γδ 1 2 (δ αγ δ βδ -δ αδ δ βγ ), z 1 4z s 3 k4 0 (23) Π (2) αβ,γδ 1 2 (δ αγ δ βδ + δ αδ δ βγ -δ αβ δ γδ ), z 2 8z s 5 k4 0 .
The quantities D n = ν n (k z )/µ n (k z ) in Eq. ( 22) are the diffusion coefficients of each mode. When kz 1, they all equal (see appendix)

D n k2 0 z s 2 ≡ D. ( 24
)
In this expression, we recall that z s is the effective mean free time and k0 is the transverse velocity of the incident beam. The factor 2, finally, refers to the dimensionality of the disorder. Making use of Eqs. ( 21), ( 22), ( 23) and ( 24), we rewrite Eq. ( 5) as:

| • E(k ⊥ , z)| 2 = 8k z s F( , , z, k ⊥ ) (2k/z s ) 2 + (k 2 ⊥ -k 2 0 ) 2 , ( 25
)
where the function

F = F L + F C splits into a diffusive contribution, F L ( , , z) = 1 2 1 + (| • * | 2 -| • | 2 )e -z/z1 +(| • * | 2 + | • | 2 -1)e -z/z2 , ( 26
)
and a contribution of the coherent backscattering peak:

F C ( , , z, k ⊥ ) = 1 2 | • | 2 + (| • * | 2 -1)e -z/z1 +(| • * | 2 -| • | 2 + 1)e -z/z2 e -D(k ⊥ +k0) 2 z . ( 27
)
Eqs. (25,26,27) for the momentum distribution constitute the main result of the paper. The distribution (25) is sketched in the upper-right inset of Fig. 4 in the plane (k x , k y ). It consists of a diffusive ring of radial width 1/z s , on top of which the CBS peak stands around k ⊥ = -k 0 [7,8]. This peak has a narrow width 1/ √ Dz that decreases with z at a rate controlled by the diffusion coefficient D [8,9,26,27].

B. Cross-over from a scalar to a vector regime

We now come to the central result of the paper: Eqs. ( 26) and ( 27) highlight a cross-over between two well distinct regimes. When z z 1,2 ∼ z s / k4 0 first, we have

F( , , z z 1,2 , k ⊥ ) = | • * | 2 1 + e -D(k ⊥ +k0) 2 z , (28)
corresponding to a structure factor Γ (L) αβ,γδ ∝ Π (1) + Π (2) + Π (6) = δ αγ δ βδ .

(

) 29
This result characterizes the behavior of a scalar wave, and coincides with the prediction of the paraxial wave equation ( 11): the complex polarization vector stays fixed during the multiple-scattering process (α = γ and β = δ), as signaled by the prefactor | • * | 2 which varies from 0 to 1 depending on the polarization detection setup. We show in the first two rows of Table I the values of

F L = F C (k ⊥ = -k 0 ) = | • * | 2 in
four main experimental setups where linearly polarized light is analyzed along the parallel (l l) or perpendicular (l ⊥ l) channels, or where circularly polarized light is analyzed in channels of same (σ σ) or opposite (σ ⊥ σ) polarization. As intuition suggests, the diffusive ring and the CBS peak are visible in the l l and σ σ channels only ( = ). The opposite situation z z 1,2 , on the contrary, cannot be described by the paraxial wave equation. In this limit, we find

F( , , z z 1,2 , k ⊥ ) = 1 2 1 + | • | 2 e -D(k ⊥ +k0) 2 z , (30)
corresponding to a structure factor

Γ (L) αβ,γδ ∝ Π (6) = 1 2 δ αβ δ γδ . (31)
The values of

F L = 1/2 and F C (k ⊥ = -k 0 ) = | • | 2 /2
in the four polarization channels are displayed in the last two rows of Table I: the diffusive signal is visible with the same probability in all channels. This indicates that polarization has been randomized by the multiple scattering process. The CBS peak, on the other hand, as a high visibility in the channel l l and, more unexpectedly, in the channel σ ⊥ σ ( = * ): for circularly-polarized incident light the CBS peak is visible only from light of opposite circular polarization. This unusual property is a consequence of the vector nature of light, on which we will come back to in Sec. V C below.

channels σ σ σ ⊥ σ l l l ⊥ l FL(z z1,2) 1 0 1 0 FC(z z1,2) 1 0 1 0 FL(z z1,2) 1/2 1/2 1/2 1/2 FC(z z1,2) 0 1/2 1/2 0 TABLE I:
Values of the diffusive, FL [Eq. ( 26)] and CBS, FC [Eq. ( 27)], contributions of the momentum distribution in the four polarization channels, for z z1,2 (scalar regime) and z z1,2 (vector regime).

For completeness, we plot in Fig. 4 the magnitude of the diffusive and CBS contributions F L , Eq. ( 26), and 27), as a function of z in the four polarization channels. The ratio F C (k ⊥ = -k 0 )/F L is the contrast of the CBS peak. It is also shown in the figure, together with the full 2D shape of the momentum distribution (insets). Note that in the channels σ ⊥ σ and l ⊥ l, the CBS peak and the diffusive ring are both very small when z z 1,2 [they scale as z/z 1,2 , see Eqs. ( 26) and ( 27)], but their ratio is finite. In particular, in the channel l ⊥ l, the CBS contrast equals z 2 /z 1 -1 = 1/5.

F C (k ⊥ = -k 0 ), Eq. (

C. Reciprocity theorem in (2+1) dimensions

In the vector regime z z 1,2 , we have seen that the CBS peak is only visible in channels l l and, more surprisingly, σ ⊥ σ. This somewhat counter-intuitive behavior can be understood by general symmetry arguments based on time-reversal symmetry and homogeneity of the medium along z, as we now show.

Consider the wave amplitude A 1→N (k 0 , ; k ⊥ , ; k z ) associated with a multiple scattering sequence 1 → N from the initial state (k 0 , ) to the final state (k ⊥ , ), with k z conserved. The CBS interference is constructed by pairing this amplitude with its counter-propagating partner A N→1 (k 0 , ; k ⊥ , ; k z ). To find the condition of constructive interference, we first apply the reciprocity theorem pertained to time-reversal symmetry [START_REF] Akkermans | Mesoscopic Physics of Electrons and Photons[END_REF]28]:

A N→1 (k 0 , ; k ⊥ , ; k z ) = A 1→N (-k ⊥ , * ; -k 0 , * ; -k z ).
(32) Owing to the independence of disorder on the longitudinal coordinate z, the right-hand-side of Eq. ( 32) is also parity-symmetric with respect to the variable k z . The CBS interference is therefore constructive provided

A 1→N (k 0 , ; k ⊥ , ; k z ) = A 1→N (-k ⊥ , * ; -k 0 , * ; k z ).
This equality is obviously satisfied when:

k ⊥ = -k 0 , = * , (33)
i.e. the CBS peak is fully contrasted in the channels l l and σ ⊥ σ, as found in the previous section. 27), in the four channels. Bottom panel: contrast of the CBS peak, FC/FL (curves σ ⊥ σ and l l overlap). The insets display the shape of the momentum distribution in the plane (kx, ky), in three configurations where the CBS peak is fully contrasted, partially contrasted and not present. Note that in the channels σ ⊥ σ and l ⊥ l, the CBS peak and the diffusive ring are both very small when z z1,2, but their ratio is finite.

D. Total distribution and norm conservation

It is also interesting to evaluate the total momentum distribution, |E(k ⊥ , z)| 2 , which does not require any particular polarization detection setup. This quantity is readily obtained from Eq. ( 25) by summing over outgoing polarization vectors . Using that

| • | 2 = | • * | 2 = 1, we infer: |E(k ⊥ , z)| 2 = 8k z s 1 (2k/z s ) 2 + (k 2 ⊥ -k 2 0 ) 2 × 1 + 1 2 e -D(k ⊥ +k0) 2 z (1 -e -z/z1 + 2e -z/z2 ) . ( 34
)
The CBS contrast measured from the total distribution thus varies from 1 in the scalar regime z z 1,2 , to 1/2 in the vector regime z z 1,2 . These two values are easily understood from the results in Table I: the diffusive ring and the CBS peak are present in the same two polarization channels at small z, whereas at large z the ring shows up in all channels and the CBS peak in only half of them. Notice, finally, that if we neglect the (small) contribution of CBS, Eq. (34) fulfills, as required, the conservation of normalization for all z:

d 2 k ⊥ (2π) 2 |E(k ⊥ , z)| 2 = 1. ( 35
) E. Physical interpretation of z1,2
Let us finally comment on the two characteristic times z 1,2 . They are both on the order of

z p ≡ z s k4 0 ∼ z s θ 4 , ( 36
)
where θ is the angle made by the incident beam with the optical axis (see Fig. 1). We have seen above that this time scale separates a scalar regime where polarization is fixed, from a vector regime where polarization is randomized. z p can therefore be interpreted as the time needed to randomize the direction of polarization in (2+1) dimensions. This interpretation can be confirmed by the following qualitative argument. The incident beam, of wave vector k = (k 0 , k z ), has the polarization . After the first scattering event on a refractive-index fluctuation, the light is scattered into a direction k and acquires a (unit) polarization which fulfills [29] =

-( k • ) k 1 -|k • | 2 . ( 37
)
In the right-hand side, the dot product k

• ≡ k • ⊥ + z kz ∼ k0 , since | kz | ∼ | ⊥ | ∼ 1 and | z | ∼ | k | ∼ k0 ≡ k 0 /
k for an incident beam almost along z. By expanding Eq. (37) to leading order in k0 1, we find that the change in polarization in the plane (x, y), ∆ ⊥ ≡ ⊥ -⊥ , is on the order of

|∆ ⊥ | ∼ k2 0 . (38)
After a random walk of N = z/z s scattering events, the polarization subsequently changes by an amount

|∆ ⊥ (N )| ∼ √ N |∆ ⊥ | ∼ √ N k2
0 . This change becomes on the order of 1 when

N ∼ 1 k4 0 ⇔ z ∼ z s k4 0 ≡ z p , (39)
which confirms the physical interpretation of z p given above. The existence of the two time scales z 1 and z 2 can be understood from Eq. ( 26): in the channels of linear polarization, F L ∝ 1±e -z/z2 , while in the channels of circular polarization, F L ∝ 1 ± e -z/z1 . We conclude that z 2 (resp. z 1 ) is the time needed to randomize the polarization of a linearly (resp. circularly) polarized beam. We can estimate z p for parameters used in state-ofthe-art experiments. In the recent work [7] for instance, λ 532 nm and light is detected at z 20 mm. We estimate the largest k0 used in this paper to k0 0.17, and the mean free path 15 µm. This yields z s = / k0 90 µm, and thus z p = z s / k4 0 10 cm. This value is not unreasonably larger than z. In view of detecting the cross-over discussed in the present paper, z p could be decreased by using a slightly larger incident angle or a stronger disorder.

VI. CONCLUSION

We have developed a general theory of multiple scattering of light in (2+1) dimensions. Starting from the exact Helmholtz equation, we have computed the full vector intensity tensors associated with classical diffusion and with the CBS effect.

Our results demonstrate that in this geometry multiply scattered light experiences a cross-over as it propagates along the effective time axis z. This cross-over takes place around a characteristic z p that corresponds to the time needed for light to randomize its polarization direction. Due to the peculiar anisotropic structure of the medium, z p is much larger than the scattering time (unlike in usual 3D disordered media where they are comparable): it varies with the inverse of the fourth power of the incident beam's angle and is inversely proportional to the disorder strength. z p thus becomes smaller and smaller as the beam is less and less paraxial, and when disorder fluctuations increase. As long as z z p , the multiple scattering process is the one of a scalar wave, and is well captured by the paraxial wave equation. When z exceeds z p , light starts to behave as a vector wave and the paraxial equation breaks down. In particular, the CBS interference peak becomes only visible in polarization configurations where light initially linearly (resp. circularly) polarized is detected along the same (resp. the opposite) channel.

Although in this paper we have focused on the optical momentum distribution, our approach is very general and can be applied to the calculation of various physical observables for light in disordered media of dimension (2+1). Furthermore, as it fully accounts for the coupling between polarization and the spatial structure of the field, our theory could be used to unveil the so-called spin-orbit interactions of light in the presence of disorder, known to show up in heterogeneous media and for non-paraxial beams [30,31]. From an experimental point of view finally, probing light transport over the scalar-tovector cross-over discussed in this paper would be highly interesting at the onset of Anderson localization, to clarify whether the latter still exists beyond z p .

FIG. 2 :

 2 FIG.2: Top: diagrammatic representation of the Bethe-Salpether equation(12) for the ladder series Γ(L) . Upper solid lines symbolize the Green tensor G, and lower dashed lines its complex conjugate. Vertical dotted lines refer to the correlation function of the disorder potential, Eq. (2). Bottom: reciprocity relation (32) between ladder and crossed series.

FIG. 4 :

 4 FIG. 4: Top panel: Diffusive contribution FL to the momentum distribution as a function of z, Eq. (26), in the four polarization channels. Middle panel : CBS contribution FC at k ⊥ = -k0, Eq. (27), in the four channels. Bottom panel: contrast of the CBS peak, FC/FL (curves σ ⊥ σ and l l overlap). The insets display the shape of the momentum distribution in the plane (kx, ky), in three configurations where the CBS peak is fully contrasted, partially contrasted and not present. Note that in the channels σ ⊥ σ and l ⊥ l, the CBS peak and the diffusive ring are both very small when z z1,2, but their ratio is finite.

Acknowledgments

The author thanks the Agence Nationale de la Recherche (grant ANR-14-CE26-0032 LOVE) for finan-cial support, and Dominique Delande for illuminating discussions and many advice.

Appendix

In this appendix we provide the full expressions of various tensors, valid for any value of k z from the paraxial regime ( kz = k z /k 1) to the regime of grazing incidence ( kz = k z /k 0).

The imaginary part of Σ, first, follows from Eq. ( 8):

The self-energy tensor is anisotropic, which stems from the symmetry axis z of the problem. Note that while the self energy does not depend on k ⊥ , as is expected for a delta-correlated potential in the plane (x, y), it does depend on k z because no average along the longitudinal direction is involved, unlike in the conventional scattering theory of three-dimensional (3D) disordered media. The usual expression of the self energy expected for that case, ImΣ ij (k z ) = -δ ij γk 4 /6 [START_REF] Lagendijk | [END_REF], is recovered by averaging Eq. ( 40) over k2 z . Once the self energy is known, the average Green tensor follows from a diagonalization of the Dyson equation ( 6) with respect to polarization indices. This procedure leads to:

where ImΣ

Apart from the Born approximation, Eq. (41) constitutes the exact expression of the average Green tensor in disordered media of dimension (2+1). It contains three terms which, in general, must be kept to describe the full multiple scattering problem in this geometry. In particular, working with the exact expression of G ij is essential to guarantee the conservation of normalization at all z and to obtain the expressions of the structure factor given in Sec. V.

We then provide the expressions of the projectors Π

αβ,γδ on the eigensubspaces of the structure factor Γ (L) (k z , q z , q = 0), and of their corresponding eigenvalues λ n . In the diffusive regime q z z s 1, we find:

It is easy to check that

Close to the paraxial regime, all terms of the type δ iz , i = α, β, γ or δ are negligible. In particular, Π (1) , Π (2) and Π (6) reduce to Eq. ( 23).

The eigenvalues λ n ≡ λ n (k z , q z , q = 0) read, as a function of kz and q z :

The coefficients of the terms in iq z define the µ n (k z ) factors in Eqs. (18) and [START_REF] Sheng | Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena[END_REF]. The eigenvalues λ 2 , λ 3 and λ 4 are twice degenerate, all the other are non-degenerate. We finally provide the factors ν n (k z ) in [START_REF] Sheng | Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena[END_REF] for the three relevant modes n = 1, 2 and 6 :

Close to the paraxial regime kz 1, the three coefficients D n = ν n (k z )/µ n (k z ) (n = 1, 2 and 6) reduce to Eq. ( 24).


B Eatrice Labat

Sandrine Morin-Grognet

Fabien Gaudie

Lucia Bertolini-Forno

Olivier Thoumire

Jean-Pierre Vannier

Guy Ladam

Hassan Atmani

Synergistic influence of topomimetic and chondroitin sulfate-based treatments on osteogenic potential of Ti-6Al-4V

Keywords: titanium alloy, topography, chondroitin sulfate, genipin, pre-osteoblasts

We combined topographical and chemical surface modifications of Ti-6Al-4V (TA6V) to improve its osteogenic potential. By acid-etching, we first generated topomimetic surface features resembling, in size and roughness, bone cavities left by osteoclasts. Next, we coated these surfaces with biomimetic Layer-by-Layer films (LbL), composed of chondroitin sulfate A and poly-L-lysine that were mechanically tuned after a post-treatment with genipin. The structural impact of each surface processing step was thoroughly inspected. The desired nano/microrough topographies of TA6V were maintained upon LbL deposition. Whereas no significant promotion of adhesion and proliferation of MC3T3-E1 preosteoblasts were detected after independent or combined modifications of the topography and the chemical composition of the substrates, osteogenic maturation was promoted when both surface treatments were combined, as was evidenced by significant long-term matrix mineralization. The results open promising route toward improved osseointegration of titanium-based implants.

INTRODUCTION

Titanium alloys, in particular Ti-6Al-4V (TA6V), are biomaterials of choice in dentistry and orthopaedic surgery for their appropriate mechanical properties but, they are classified as bioinert materials because of their low capability to promote intimate contact with host bone tissue, required for optimal implant osseointegration. Therefore, strategies have been explored to render them bioactive through topographical or chemical surface modifications mimicking the bone tissue microenvironment.

To texture the surfaces with a well-defined nano-, submicro-, or micrometric roughness, i.e. in the relevant range capable to influence the response of bone cells, blasting, oxidation, and acid etching (AE) are commonly used techniques. [START_REF] Sykaras | Implant materials, designs, and surface topographies: Their effect on osseointegration. A literature review[END_REF][START_REF] Anselme | Statistical demonstration of the relative effect of surface chemistry and roughness on human osteoblast short-term adhesion[END_REF][START_REF] Giordano | A new chemical etching process to improve endosseous implant osseointegration: In vitro evaluation on human osteoblast-like cells[END_REF][START_REF] Wennerberg | Effects of titanium surface topography on bone integration: A systematic review[END_REF] Nanoscale topographies, for instance, were reported to improve bone-cell/biomaterial interactions. [START_REF] Ergun | Increased osteoblast adhesion on nanograined hydroxyapatite and tricalcium phosphate containing calcium titanate[END_REF][START_REF] De Oliveira | Enhancement of in vitro osteogenesis on titanium by chemically produced nanotopography[END_REF] Elsewhere, smoothest surfaces were reported to promote osteoblast anchoring and proliferation, while roughest ones favored differentiated cell morphologies and matrix mineralization. [START_REF] Gough | Osteoblast attachment and mineralized nodule formation on rough and smooth 45S5 bioactive glass monoliths[END_REF][START_REF] Zhao | Requirement for both micron-and submicron scale structure for synergistic responses of osteoblasts to substrate surface energy and topography[END_REF] However, the existence of a unique range of roughness promoting osteogenesis is controversial as far as bone cells are rather sensitive to combinations of nano-and Correspondence to: Dr B. Labat; e-mail: beatrice.labat@univ-rouen.fr C Layer-by-Layer (LbL) films (or polyelectrolyte multilayers) are widely studied, versatile polymer coatings allowing to modify surfaces of any chemical nature and geometry. [START_REF] Decher | Fuzzy nanoassemblies: Toward layered polymeric multicomposites[END_REF] The buildup of LbL films consists in successive depositions of oppositely charged polyelectrolytes mainly through electrostatic interactions. This produces nano-to micrometric coatings with different potential functionalities. Many biofunctional, biological, and biomimetic compounds, such as proteins and polysaccharides, can be used as components of LbL films, for instance atop Ti-based biomaterials. [START_REF] Hu | Regulation of the differentiation of mesenchymal stem cells in vitro and osteogenesis in vivo by microenvironmental modification of titanium alloy surfaces[END_REF][START_REF] Huang | Fabrication and in vitro evaluation of the collagen/hyaluronic acid PEM coating crosslinked with functionalized RGD peptide on titanium[END_REF] In particular, the use of such coatings as reservoirs for local controlled release of bioactive compounds (growth factors, hormones) and/or therapeutics drugs (antibiotics, antifungals, anticancerous, and anti-inflammatory molecules) in diseased tissues, is a growing field of interest. [START_REF] Pavlukhina | Polymer assemblies for controlled delivery of bioactive molecules from surfaces[END_REF] Some works have reported on LbL films composed of the anionic GAG chondroitin sulfate A (CSA) associated with cationic poly-L-lysine (PLL). [START_REF] Gaudie `re | Mechano-chemical control of cell behaviour by elastomer templates coated with biomimetic Layerby-Layer nanofilms[END_REF][START_REF] Heng | Adhesion, proliferation, and gene expression profile of human umbilical vein endothelial cells cultured on bilayered polyelectrolyte coatings composed of glycosaminoglycans[END_REF][START_REF] Grohmann | Colloidal force spectroscopy and cell biological investigations on biomimetic polyelectrolyte multilayer coatings composed of chondroitin sulphate and heparin[END_REF][START_REF] Tezcaner | Polyelectrolyte multilayer films as substrates for photoreceptor cells[END_REF] However, GAG-based LbL films are generally too soft because of their hydration such that they mismatch the physiological stiffness of cell microenvironment. Cross-linking methods were developed in order to optimize their mechanical properties, but current crosslinkers such as glutaraldehyde or EDC/NHS are potentially cytotoxic if they remain in their free form, [START_REF] Hao | Impact of carbondiimide crosslinker used for magnetic carbon nanotube medi-ated GFP plasmid delivery[END_REF][START_REF] Sung | In vitro evaluation of cytotoxicity of a naturally occurring cross-linking reagent for biological tissue fixation[END_REF] therefore there is a need for biocompatible alternatives. Recently, genipin (GnP), a biological compound extracted from Gardenia fruit, was reported to cross-link LbL films [START_REF] Hillberg | Effect of genipin crosslinking on the cellular adhesion properties of layer-by-layer assembled polyelectrolyte films[END_REF][START_REF] Chaubaroux | Collagen-based fibrillar multilayer films crosslinked by a natural agent[END_REF][START_REF] Gaudie `re | Genipin-crosslinked Layer-by-Layer assemblies: Biocompatible microenvironments to direct bone cell fate[END_REF] and its biocompatibility was demonstrated in numerous works. Of note, rare papers related apoptotic or antiproliferative effects of genipin treatments, but with cancer cell lines. [START_REF] Kim | Genipin, a constituent of Gardenia jasminoides Ellis, induces apoptosis and inhibits invasion in MDA-MB-231 breast cancer cells[END_REF][START_REF] Hyeonseok | Induction of apoptosis by genipin inhibits cell proliferation in AGS human gastric cancer cells via Egr1/p21 signaling pathway[END_REF] Henceforward, based on the upmost promising literature, we implemented this natural cross-linking agent in our strategy. We reported in a previous work, that it significantly improved the mechanical properties of our CSA/PLL LbL films and favored adhesion, proliferation, and differentiation of pre-osteoblasts cells. [START_REF] Gaudie `re | Genipin-crosslinked Layer-by-Layer assemblies: Biocompatible microenvironments to direct bone cell fate[END_REF] Here, inspired by the challenge of bone surface mimicry, we combined (i) topographical modification of TA6V substrates exhibiting nano-to microscale roughness, with (ii) further deposition of biomimetic CSA/PLL LbL films mimicking the native bone extracellular matrix. We characterized the topography, the composition and the wettability of the modified surfaces, and the bioactivity was assessed in terms of cell morphology, proliferation and early/late differentiation of MC3T3-E1 pre-osteoblasts.

MATERIALS AND METHODS

Substrates preparation and AE TA6Vfoils (Goodfellow, original Ra 40 nm) were cut into 15 3 15 mm 2 pieces. Before use, all samples were cleaned by sonication, and degreased successively in acetone and ethanol, each for 10 min, with intermediate rinsing steps with ultrapure water, and finally dried under a flow of argon. With the aim of nano-and micro-structuring the surfaces, AE of the TA6V substrates was performed into concentrated H 2 SO 4 (50% in water) for 1 h at 608C. The corresponding loss of material was controlled by weighing the substrates before and after the AE treatment. In the following, the nontreated and acid-etched substrates will be denoted as TA6V and TA6V-AE, respectively.

Polyelectrolyte solutions, LbL assembly, and genipin cross-linking Solutions of CSA (20-30 kDa, Sigma) and PLL (40-60 kDa, Sigma) at 1 mg mL [START_REF] Huang | Fabrication and in vitro evaluation of the collagen/hyaluronic acid PEM coating crosslinked with functionalized RGD peptide on titanium[END_REF] , and solutions of poly(ethyleneimine) (PEI, branched, 750 kDa, 50 wt % solution in water, Sigma) at 5 mg mL 21 , were prepared in saline Tris/NaCl buffer (10 mM Tris, 150 mM NaCl, pH 7.4). Polycationic PEI was always adsorbed as a precursor layer prior to LbL film buildup. To verify that the PEI layer was uniformly adsorbed on the substrates, we labeled PEI molecules with fluorescein isothiocyanate (FITC) fluorescent probe (Sigma) through condensation of the isothiocyanate groups and the primary amines of PEI for 12 h. PEI FITC solution was dialyzed against DI water for 72 h in order to remove unbound FITC. We then immersed TA6V and TA6V-AE substrates into a PEI FITC solution (5 mg mL [START_REF] Huang | Fabrication and in vitro evaluation of the collagen/hyaluronic acid PEM coating crosslinked with functionalized RGD peptide on titanium[END_REF] ) for 30 min, and rinsed them three times prior to observation with a fluorescence microscope (Zeiss Axio Scope A1).

The buildup of PEI/(CSA/PLL) 6 or PEI/(CSA/PLL) 6 /CSA LbL architectures onto TA6V and TA6V-AE substrates was performed as follows: after the deposition of PEI with the same protocol as for PEI FITC , CSA and PLL were alternatively adsorbed onto the substrates by means of 10-min immersions into the corresponding solutions, followed by three rinses with Tris/NaCl buffer. For the post cross-linking with GnP, we immersed the PEM-coated substrates into solutions of 0.25% In situ LbL assembly assessment by QCM-D We followed the buildup of our LbL films on top of Ti-based surfaces by quartz crystal microbalance with dissipation monitoring (QCM-D) using a D300 system (Q-Sense, Sweden) equipped with a QAFC302 flow chamber and QSX310 Ti-coated quartz crystal sensors. QCM-D consists in measuring the resonance frequency shifts Df and the dissipation factor changes DD of the quartz crystal sensor upon material deposition. [START_REF] Voinova | Viscoelastic acoustic response of layered polymer films at fluid-solid interfaces: Continuum mechanics approach[END_REF] Df depends on the total oscillating mass, including coupled water, therefore providing a "wet mass" measurement. The LbL buildup was performed by successive injections of polyelectrolyte solutions (5 mL) and rinsing solution (5 mL) through the flow chamber, and monitored in situ. Df was measured at the fundamental frequency around 5 MHz and at the 3rd (15 MHz), 5th (25 MHz), and 7th (35 MHz) overtones. The fundamental frequency was discarded for it is often affected by contact with the mounting frame. [START_REF] H€ | Variations in coupled water, viscoelastic properties, and film thickness of a Mefp-1 protein film during adsorption and cross-linking: A quartz crystal microbalance with dissipation monitoring, ellipsometry, and surface plasmon resonance study[END_REF] Analysis of the raw data was carried out using the Q-Tools software provided with the instrument. A Voigt viscoelastic model was used to derive adsorbed masses per unit area and thicknesses assuming a film density of 1.1 g cm [START_REF] Gaudie `re | Mechano-chemical control of cell behaviour by elastomer templates coated with biomimetic Layerby-Layer nanofilms[END_REF] . [START_REF] Trist An | Tunable Protein-Resistance of Polycation-Terminated Polyelectrolyte Multilayers[END_REF] Surface characterization: Topography, roughness, wettability Scanning electron microscopy. Before LbL deposition, we inspected the TA6V surfaces before and after AE using a JEOL scanning electron microscope Neoscope JCM 5000 without sample preparation. For LbL-coated TA6V and TA6V-AE, we immersed the substrates in a 2.5% glutaraldehyde solution for 30 min, dehydrated them into two successive acetone baths, and extensively washed them with ultrapure water. Samples were stored at 2808C until used.

Prior to Scanning Electron Microscopy (SEM) inspection, the samples were sputter-coated with gold by using a JFC 1300 Autofine coater.

Epifluorescence microscopy. To ascertain the homogeneous coating of TA6V and TA6V-AE substrates with GnP-crosslinked LbL films, we took advantage of the strong autofluorescence of amino-coupled GnP. [START_REF] Hwang | Effect of genipin crosslinking on the optical spectral properties and structures of collagen hydrogels[END_REF] Observations were carried out using a Zeiss Axio Scope A1 fluorescence microscope equipped with a TRITC filter. Static contact angle measurements. We assessed the wettability of all samples by measuring static contact angles with a Digidrop apparatus (GBX instruments -France). Before measurements, the samples were briefly rinsed with ultrapure water and dried by blowing argon. At least three 5-mL drops of Tris/NaCl buffer were generated and deposited on each substrate using an automatic micrometric syringe. For each drop, the static contact angle was determined thrice semi-manually from an image captured by a digital camera 2 s after deposition.

Cell culture conditions MC3T3-E1 pre-osteoblast cells (subclone 4) were used for the biological assessment of our substrates. Cells were routinely cultured with a-MEM supplemented with 10% foetal bovine serum (FBS), 100 mg mL 21 streptomycin, 100 mM penicillin, 2 mM L-glutamine (standard culture medium) in 75 cm 2 tissue culture flasks, and incubated in a humidified atmosphere with 5% CO2 at 378C. When subconfluent, cells were enzymatically detached using 0.25% trypsin/1 mM EDTA.

Cell morphology MC3T3-E1 pre-osteoblasts were seeded at a density of 2 3 10 4 cell cm 22 onto the substrates. After 3 h, cells were rinsed with cold PBS, fixed with a 3% formaldehyde solution for 15 min, permeabilized for 5 min in 1% PBS-Triton X-100 and treated with 1% PBS-BSA for 30 min. Then, cells were stained during 1 h for F-actin cytoskeleton with FITCconjugated phalloidin (10 lg mL [START_REF] Huang | Fabrication and in vitro evaluation of the collagen/hyaluronic acid PEM coating crosslinked with functionalized RGD peptide on titanium[END_REF] ) and for nuclei with DAPI (2.5 lg mL 21 ). Finally, preparations were mounted with Mowiol 4-88 overnight at 148C and observed with a Zeiss Axio Scope A1 epifluorescence microscope.

Cell proliferation MC3T3-E1 proliferation was indirectly assessed by using

Alamar BlueV R (AB) assay. AB assay is an oxidation-reduction indicator of the metabolic activity of viable cells during proliferation. Cells were seeded at a 1 3 10 4 cells cm [START_REF] Pavlukhina | Polymer assemblies for controlled delivery of bioactive molecules from surfaces[END_REF] density onto each substrate and cultured for 2 and 9 days (D2 and D9). At each time point, culture medium was replaced by a 10% (v/v) AB containing culture medium for 4 h incubation. Then, absorbance was measured at 570 and 600 nm and the percentage of AB reduction was determined in accordance to the manufacturer's instructions.

Pre-osteoblasts early/late differentiation For both early and late MC3T3-E1 differentiation, cells were seeded at a 5 3 104 cell cm 22 density in the standard culture medium. After 2 days, the medium was replaced by an osteogenic culture medium, i.e. standard medium supplemented with 10 mM b-glycerophosphate and 50 mg mL 21 L-ascorbic acid-2-phosphate.

Alkaline phosphatase assay. Alkaline phosphatase (ALP) activity, an early marker for osteoblast differentiation, was determined for MC3T3-E1 cultured atop each substrate for 7 and 14 days (D7 and D14). At the desired time, cells were collected, subjected to three freeze-thaw cycles, and finally lysed with 0.1% Triton X-100. Then, ALP activity in cell lysates was determined by measuring the UV-Vis absorbance of p-nitrophenol yielded by ALP-induced hydrolysis of p-nitrophenol phosphate. Cell lysates were also used to quantify the total intracellular proteins with a micro-BCA protein kit (Pierce) in order to normalize ALP activity to total protein concentration.

Confocal Raman microspectroscopy. Mineralization of the osteoid, the organic extracellular matrix produced by osteoblasts, is a late marker for osteoblast maturation. Confocal Raman microspectroscopy was used (i) to identify the chemical nature of the calcium phosphate nodules and (ii) for chemical mapping of the mineralized areas. All the measurements were performed on dried samples. The setup was composed of a LabRam HR Raman spectrometer (Horiba Jobin-Yvon) with a 600 lines mm 21 grating coupled to a confocal microscope (Model BX41, Olympus) via optical fibers. The excitation of Raman scattering was operated with a helium-neon laser at a wavelength of 632.8 nm. A confocal pinhole (diam. 250 lm) before the entrance slit rejected the Raman signal from out-of-focus planes. The laser beam was focused on the sample by means of a 350LWD or a 3100 microscope objective. Integration times were set to produce Raman spectra with good signal-to-noise ratio.

Statistical analysis

All surface characterizations and cell experiments were performed in triplicate and quantitative results were presented as mean 6 SEM. We applied the nonparametric Mann-Whitney-Wilcoxon test for comparisons between unrelated groups and significance level was assigned at p <0.05.

RESULTS

Surface characterization of bare and acid-etched substrates AE had a strong impact on the TA6V substrates: (i) visually, they lost their metallic luster, (ii) their mass was reduced by ca. 4% upon dissolution, and (iii) they were much roughened as seen on AFM and SEM images (Fig. 1). Roughness values Ra derived from AFM data were varied over one order of magnitude in the biologically relevant range, from 40 6 5 nm for TA6V (in accordance with the supplier's data given in section "Substrates preparation and AE") to 334 6 28 nm for TA6V-AE. AFM [Fig. 1(A)] and highest magnification SEM [Fig. 1(B), right] analyses were consistent, showing relatively smooth surfaces for TA6V samples, and nanostructured morphologies with numerous submicrometric pits for TA6V-AE samples. Interestingly, larger scale SEM images revealed periodic lateral micro-ranged cavities

Surface characterization of LbL-coated TA6V and TA6V-AE substrates

In previous works, we have extensively characterized the buildup mechanism and the structure of PEI/(CSA/PLL) i (with i the number of layer pairs) LbL films onto various substrates, [START_REF] Gaudie `re | Mechano-chemical control of cell behaviour by elastomer templates coated with biomimetic Layerby-Layer nanofilms[END_REF][START_REF] Gaudie `re | Genipin-crosslinked Layer-by-Layer assemblies: Biocompatible microenvironments to direct bone cell fate[END_REF][START_REF] Abdelkebir | Evidence of different growth regimes coexisting within biomimetic Layer-by-Layer films[END_REF] but not onto Ti-based surfaces. Therefore, we validated the assembly of the films onto Ticoated QCM-D sensors viewed as more realistic models of our TA6V substrates than the Au-coated sensors used previously. Figure 2(A andB) respectively show the corresponding variations of the raw QCM-D parameters, and their conver-sion into hydrodynamic thickness and adsorbed mass during the buildup of a typical PEI/(CSA/PLL) 6 /CSA film. We obtained an exponential growth regime during the building of LbL films. The hydrodynamic thickness and the hydrated mass of the films were about 180 nm and 20 mg cm 2 , respectively. Also, from the AFM data reported in Figure 2(C), the films displayed an islet-like topography.

We verified the presence and the uniformity of the precursor layer atop the TA6V and TA6V-AE substrates by using PEI FITC . Fluorescence microscopy images reported in Figure 3 confirmed that the substrates were fully covered with the polymer, with apparently preserved topography for the AE samples (bottom-right image).

We then characterized the TA6V and TA6V-AE substrates after the deposition of PEI/(CSA/PLL) 6 6 CSA films and after the post-treatment with GnP. From the typical islet-like topography observed on 2D AFM images [Fig. 4(A), top], TA6V substrates were homogeneously covered by LbL films. The presence of the films on TA6V-AE could not be evidenced by 2D AFM images because of the intrinsic roughness of the substrates. 3D AFM images [Fig. 4(A), bottom] and SEM images [Fig. 4(B)] showed that the dual topography of the substrates was maintained upon the deposition of LbL films, meaning that the polymers conformed to the surface, and did not fill up the pits and cavities. We also verified that the final topographies were similar whatever the CSA or PLL terminal layer of the films. The strong autofluorescence of amino-coupled GnP [Fig. 4(C)] (i) showed that the cross-linking treatment of the films with GnP was successful, (ii) confirmed the continuous coverage of the substrates and (iii) corroborated that the desired topography combining nano-and microscale roughness, was preserved.

Figure 5(A) reports the average roughness Ra of the substrates, as derived from 2D AFM images. Similar values of 40-50 nm were measured for both bare and all LbLcoated TA6V, whereas values in the range 350-450 nm were obtained for the corresponding TA6V-AE substrates. The AE treatment clearly increased the hydrophilic character of the bare substrates, with contact angles decreased from 518 to 238 [Fig. 5(B)]. All the substrates coated with native LbL films had similar hydrophilic character, with contact angles between 208 and 308. A slight decrease of the wettability was observed after treatment of the LbL-coated substrates with GnP.

Cell morphology

For anchorage-dependent mammalian cells such as preosteoblasts, optimal adhesion is a required and crucial step prior to any subsequent cell process and even, for cell survival. Therefore we inspected the morphology, which is a direct indicator of the quality of adhesion, of MC3T3-E1 pre-osteoblast cells on top of our substrates. DAPI-labelled nuclei and FITC-phalloidin-labelled actin filaments organization were observed 3 h after seeding by means of epifluorescence microscopy (Fig. 6). Both bare TA6V and bare TA6V-AE substrates were favorable to the adhesion of cells, as indicated by their mostly polygonal and well-spread morphology, with slight differences. Whereas cells exhibited a dense cortical actin network onto nano-rough TA6V, the actin network was rather diffuse atop nano/microrough TA6V-AE (Fig. 6, left). After 3 h of culture atop TA6V and TA6V-AE substrates coated with native (non-cross-linked) CSA/PLL LbL films, cells poorly adhered and remained rather rounded. In contrast, initial adhesion was much enhanced on top of the GnP-cross-linked films, leading to well-spread, polygonal cells with organized actin filament cytoskeleton, whatever the topography of the substrate and the nature of the outer, CSA or PLL layer. Contrary to bare substrates, there was no visible dependence of cell morphology on the outer layer of the film, indicating that the favorable mechanical properties provided by the treatment with GnP constituted the main factor of the cell adhesion improvement. At this stage, it comes out that cell adhesion was governed by a complex interplay of topography, chemistry, and mechanics of the substrates.

Cell proliferation

Proliferation of MC3T3-E1 cells was assessed indirectly by using the Alamar Blue (AB) assay at D2 and D9 (Fig. 7). Consistently with the cell adhesion data, AE treatment had little impact on cell proliferation in the absence of LbL films, whereas native CSA/PLL films strongly reduced the cell proliferation at short-term (D2) on top of TA6V. Surprisingly, the proliferation at D2 on top of AE substrates coated with native films was unaffected despite poor initial adhesion. At longer term (D9), the native films were less favorable to cell proliferation than the bare substrates. In line with cell adhesion data, cell growth at both short and long terms were fully preserved on top of GnP-stiffened films whatever the underlying topography.

Cell differentiation

We inspected the potential osteoconductive properties of our substrates at early (D7, D14) and late (5 weeks) stages. MC3T3-E1 cells were plated at a high cell density (5 3 10 4 cell cm [START_REF] Pavlukhina | Polymer assemblies for controlled delivery of bioactive molecules from surfaces[END_REF] ) in order to reduce their proliferation, and subsequently, to favor their differentiation. They were cultured in a minimal osteogenic medium (ascorbic acid and bglycerophosphate).

We followed the cultures over 14 days and ALP was assayed at D7 and D14 [Fig. 8(A)]. The ALP activity was clearly decreased by ca. 50% atop bare TA6V-AE compared to bare TA6V. The ALP activity was still more dramatically reduced on top of native LbL films, consistently with the previous data showing that cell processes are impaired by native LbL films. In the presence of GnP-cross-linked films, the ALP activity at D7 was far less decreased on TA6V, and was clearly enhanced on TA6V-AE compared to the corresponding bare substrates. For all the substrates, we could thus almost preserve the potential of differentiation induction of bare TA6V as far as the LbL films were GnP-crosslinked. For most of the substrates, we noted an overall reduction of ALP activity at D14. As a result, the optimal substrates for early differentiation were the TA6V-AE substrates covered with GnP-treated LbL films whatever the ending layer, and the TA6V substrates covered with CSAending, but not PLL-ending, GnP-treated LbL films.

Next, we inspected the capability of MC3T3-E1 to mineralize their organic extracellular matrix (osteoid) atop TA6V and TA6V-AE substrates coated with GnP-treated films, by means of Raman microspectroscopy analyses, after 5 weeks of culture. Importantly, to ascertain that the detected mineralization was exclusively cell-mediated, we immersed a series of acellular substrates in the same osteogenic culture medium as that for cell-seeded ones. Indeed, it has been shown that the presence of b-glycerophosphate (b-GP) in the culture medium could induce a non-cell-mediated mineral precipitation because of the cleavage of phosphate from b-GP. [START_REF] Khouja | Calcium and orthophosphate deposits in vitro do not imply osteoblast mediated mineralization: Mineralization by beta-glycerophosphate in the absence of osteoblasts[END_REF] Whereas no mineralization was observed in the absence of cells, we detected the presence of (i) sparse mineralized nodules with Raman signatures containing the m 1 PO 4 3-(major band), m 2 PO [START_REF] Giordano | A new chemical etching process to improve endosseous implant osseointegration: In vitro evaluation on human osteoblast-like cells[END_REF] 4 , m 3 PO 3- 4 , m 4 PO 3- 4 , and m 1 CO 3 2-(1070 cm 21 ) bands related to B-type carbonated hydroxyapatite (HA), that is the major mineral phase of the mineralized ECM produced by mature osteoblasts [Fig. 8(B-a)], and (ii) a uniform underlying ECM, characterized by the intense symmetric ring breathing Raman band of phenylalanine at 1003 cm [START_REF] Huang | Fabrication and in vitro evaluation of the collagen/hyaluronic acid PEM coating crosslinked with functionalized RGD peptide on titanium[END_REF] [Fig. 8(B-b)].

As a first approach to compare the potentials of the various substrates in terms of late mineralization induction, we thoroughly explored their surface to localize the largest mineralized nodules. We then established chemical mappings of the corresponding areas, by following the intensity of the major specific band of HA at 960 cm [START_REF] Huang | Fabrication and in vitro evaluation of the collagen/hyaluronic acid PEM coating crosslinked with functionalized RGD peptide on titanium[END_REF] [Fig. 8(C)], and we compared the morphologies and sizes of the nodules. The largest and most uniform cell-induced HA deposits were detected on top of the AE substrates coated with GnPtreated LbL films, which were thus confirmed as the best promoters of late osteogenic differentiation.

DISCUSSION

In this study, we proposed biomimetic surface modifications of TA6V by combining (i) topomimetic features that imitate bone surface cavities left after osteoclast resorption and (ii) ECM-mimetic LbL films that involve chondroitin sulfate A. Such a combination revealed a promising strategy that preserved the intrinsic osteoconductive properties of TA6V and that would improve its osteogenic potential since the adsorbed LbL films will ultimately serve as reservoirs for local delivery of bioactive compound. [START_REF] Volodkin | LbL films as reservoirs for bioactive molecules[END_REF][START_REF] Huang | Enhancement of local bone remodeling in osteoporotic rabbits by biomimic multilayered structures on Ti6Al4V implants[END_REF] Surface topography of an implant is one of the key parameters affecting cellular events that has been investigated for long, for improving bone bonding. Some recent papers reported patterned surfaces exhibiting well-defined geometries (grooves, ridges, pits, posts, pillars or stars. . .) to assess cell behavior from various lineages, [START_REF] Lee | Geometric guidance of integrin mediated traction stress during stem cell differentiation[END_REF][START_REF] Giulio | Surface topography enhances differentiation of mesenchymal stem cells towards osteogenic and adipogenic lineages[END_REF] while some others proposed more physiologically relevant surfaces with biologically inspired features (nano/microfibrils, combined nano/microroughness. . .) mimicking the topography and the roughness of the natural tissues for tissue engineering purpose. [START_REF] Lumetti | The response of osteoblastic MC3T3-E1 cells to micro-and nanotextured, hydrophilic and bioactive titanium surfaces[END_REF][START_REF] Syverud | Controlling the elastic modulus of cellulose nanofibril hydrogels-scaffolds with potential in tissue engineering[END_REF] Here, by AE TA6V, we generated uniformly distributed periodic lateral microscaled cavities with combined nano and microscaled features. As explained by Variola et al. [START_REF] Variola | Tailoring the surface properties of Ti6Al4V by controlled chemical oxidation[END_REF] such coexistence of nano and microtextures must result from different rates of etching for the aand b-grains composing the TA6V alloy: etching of a-grains produces "nanopits", while more readily etching of b-grains leads to both nanopits and large cavities. This dual topography closely resembles the complex morphology of osteoclast resorption depressions [START_REF] Davies | Bone bonding at natural and biomaterial surfaces[END_REF] (known as Howship's lacunae), which constitute the natural microenvironment of osteoblasts during bone remodelling. [START_REF] Zinger | Differential regulation of osteoblasts by substrate microstructural features[END_REF] Therefore TA6V-AE substrates appeared of particular relevance in view of the activation of osteoblast cells at implant surfaces.

We did not explore possible AE-induced chemical modification of the TA6V surface. It has been shown that AE treatment starts with the dissolution of the external passive oxide layer and then converts the underlying metallic Ti into titanium hydride TiH 2 . [START_REF] Ban | Surface modification of titanium by etching in concentrated sulfuric acid[END_REF] We assume that the subsequent uniform coverage of surfaces by LbL films should limit the impact of such chemical variations on cells behavior. To validate the possible buildup of PEI/(CSA/PLL) 6 /CSA LbL films atop Ti-based surfaces and to characterize their physico-chemical structure, we used Ti-coated QCM-D sensors. We obtained very similar dimensions and viscoelastic character to films described onto Au-coated sensors. [START_REF] Abdelkebir | Evidence of different growth regimes coexisting within biomimetic Layer-by-Layer films[END_REF] These films displayed islet-like topography as those built up previously onto smooth PDMS and glass substrates (Ra 30 nm). [START_REF] Gaudie `re | Mechano-chemical control of cell behaviour by elastomer templates coated with biomimetic Layerby-Layer nanofilms[END_REF][START_REF] Abdelkebir | Evidence of different growth regimes coexisting within biomimetic Layer-by-Layer films[END_REF] These data confirmed that the structure of PEI/(CSA/PLL) 6 6 CSA films was maintained whatever the chemical nature of the underlying substrate, certainly because of the use of a PEI precursor layer. Trybała et al. [START_REF] Trybała | The effect of anchoring PEI layer on the build-up of polyelectrolyte multilayer films at homogeneous and heterogeneous surfaces[END_REF] reported that a continuous precursor layer of branched PEI formed an anchoring network that allows the standardization of almost any surface material. We then checked that all our substrates were fully covered with the PEI polymer and preserved the topography for the AE samples. Based on these control experiments, we assumed that subsequent LbL coatings should have the same structural features atop our Ti-based substrates as onto the other PEI-activated substrates previously inspected.

Native CSA-containing LbL films are known to be cellrepellent because of very low elasticity. [START_REF] Gaudie `re | Genipin-crosslinked Layer-by-Layer assemblies: Biocompatible microenvironments to direct bone cell fate[END_REF][START_REF] Boudou | Multiple functionalities of polyelectrolyte multilayer films: New biomedical applications[END_REF] Therefore, we stiffened the films with genipin, a biocompatible natural cross-linking agent. [START_REF] Poursamar | The effects of crosslinkers on physical, mechanical, and cytotoxic properties of gelatin sponge prepared via in-situ gas foaming method as a tissue engineering scaffold[END_REF] In our previous work, we were able to measure the surface elastic moduli of similar cross-linked films built up onto flat glass slides by AFM nanoindentation and we ascertained that the GnP treatment stiffened the films. [START_REF] Gaudie `re | Genipin-crosslinked Layer-by-Layer assemblies: Biocompatible microenvironments to direct bone cell fate[END_REF] Here, we could not perform such measurements because of the substrate roughness, but we can reasonably assume that our surfaces were similarly stiffened upon the GnP treatment. Genipin treatment of (CSA/PLL)based LbL films would likely generate semi-IPN architectures since it reacts with amino-containing components, which is the case for PLL. Therefore, GnP would likely produce a cross-linked PLL-GnP network in which CSA molecules are free to diffuse. We assumed that these semi-IPNs would provide biomimetic rheological properties since the natural cell micro-environment is composed of interlaced self-assembled biopolymers. [START_REF] Tong | Engineering interpenetrating network hydrogels as biomimetic cell niche with independently tunable biochemical and mechanical properties[END_REF] The roughness obtained for our substrates provided quantitative confirmation that LbL coatings and further GnP treatment did not alter the desired topographies and conformed the surfaces. Also, AE treatments rendered the bare surfaces hydrophilic. This is consistent with the Wenzel model supporting that there is a threshold of 908 below which an increase of roughness increases the surface wettability and above which a converse effect is observed. [START_REF] Wenzel | Resistance of solid surfaces to wetting by water[END_REF] All the substrates coated with native LbL films had similar contact angles which is in accordance with a previous work reporting a decrease of the contact angle of Ti-based substrates upon coating PLL or other polyelectrolytes. [START_REF] Park | Use of polyelectrolyte thin films to modulate osteoblast response to microstructured titanium surfaces[END_REF] We noted a slight decrease of the wettability after GnP treatment possibly because of the conversion of primary amines into more hydrophobic tertiary amines upon their coupling to GnP. [START_REF] Gopalakrishnan | A new green thermoplastic polymer with improved hydrophobic character[END_REF] Overall, all the LbL-coated substrates were hydrophilic and thus, likely favorable to cell adhesion, as surfaces with water contact angles below 608 are known to promote cell adhesion. [START_REF] Vogler | Water and the acute biological response to surfaces[END_REF] Although the influence of topography onto short-term cell events like adhesion remains to be fully elucidated, it was already much documented in the literature. Generally, nanotextured surfaces were reported to promote the cytoskeleton and membrane receptors organization, [START_REF] Lipski | The effect of silica nanoparticle-modified surfaces on cell morphology, cytoskeletal organization and function[END_REF] whereas microtextured surfaces can withdraw the favorable effects of nanofeatures on cell adhesion. [START_REF] Richert | Surface nanopatterning to control cell growth[END_REF] Our results were rather consistent with this general scheme since the cortical actin network of MC3T3-E1 cells cultured onto nanorough TA6V was dense while atop nano/microrough TA6V-AE, cells exhibited a rather diffuse actin network. When the Ti-based substrates were covered with native LbL films, we observed a poor cell adhesion with rather rounded cells. These results were somewhat expected since GAG-containing films are known to be highly hydrated particularly because of the presence of CSA that can bind up to 30 water molecules per disaccharide unit. [START_REF] Servaty | Hydration of polymeric components of cartilage-an infrared spectroscopic study on hyaluronic acid and chondroitin sulfate[END_REF] Of note, GAGs represent a few part of ECM by mass (5%), but they turn into hydrogels when they are hydrated, representing 70% volume of the ECM. [START_REF] Hodde | Glycosaminoglycan content of small intestinal submucosa[END_REF] Therefore, CSA-containing LbL films display poor mechanical properties. In addition, as a polyanion, CSA exhibits negatively charged moieties that are exposed to negatively charged cell membrane, likely generating repulsive interactions. Although polycationic PLL is known to favor cell adhesion through electrostatic interactions, one can note here, that cell adhesion was not improved by PLLending, compared to CSA-ending films, which suggests that film hydration was the main factor of the poor initial adhesion observed. Yet, we previously reported that, despite similar limited anchoring at initial stage, MC3T3-E1 cells managed to adhere onto CSA/PLL LbL films at longer term. [START_REF] Gaudie `re | Mechano-chemical control of cell behaviour by elastomer templates coated with biomimetic Layerby-Layer nanofilms[END_REF] As anticipated, the post-treatment of the films with genipin clearly improved cell adhesion with well-spread cells exhibiting a typical polygonal osteogenic morphology. Indeed, the intrinsic mechanical properties of such semi-IPN films likely matched those that pre-osteoblasts experience in their natural micro-environment. It is well admitted that the matrix stiffness can direct lineage specification of mesenchymal stem cells. [START_REF] Engler | Matrix elasticity directs stem cell lineage specification[END_REF][START_REF] Janson | Extracellular matrix elasticity and topography: Material-based cues that affect cell function via conserved mechanisms[END_REF][START_REF] Watt | Role of the extracellular matrix in regulating stem cell fate[END_REF][START_REF] Trappmann | How cells sense extracellular matrix stiffness: A material's perspective[END_REF] In this line, anchoring cells and osteoblasts in particular, must adhere to mechanobiological relevant ECM-like substrates to survive and to preserve their phenotype and not de/trans-differentiate. Cell-ECM interactions are mainly mediated by integrins that translate external mechanical signals (i.e. the matrix stiffness), through the membrane, to intracellular actin-binding molecules. In particular, cells probe their mechanical environment by pulling on it and adapt accordingly by modulating their cytoskeleton. Stiff microenvironments exert resistance to cellular traction that leads to intracellular actin selfassembly and consequently, generates stress fibers that transmit the external mechanical signals to nucleus to direct lineage specification of cells. As a result, cells differentiate according to a mechanotransduction process that is matrix stiffness-dependent, which was probably the case for MC3T3-E1 in the present work. Proliferation of MC3T3-E1 cells was investigated over 9 days of culture in a standard culture medium. In line with cell adhesion data and with our previous work, [START_REF] Gaudie `re | Genipin-crosslinked Layer-by-Layer assemblies: Biocompatible microenvironments to direct bone cell fate[END_REF] cell growth at both short and midterms were fully preserved on top of GnP-stiffened films whatever the underlying topography. Altogether, these results highlighted that (i) the substrate stiffness was the major factor governing the initial (adhesion at 3 h) and long term (proliferation at D9) cell behavior, (ii) the topography remained a minor factor, as it controlled only intermediately (at D2) the proliferation of cells on top of the native LbL films, while (iii) no clear effect of the surface chemistry could be detected.

Alkaline phosphatase (ALP) is a dephosphorylating enzyme, commonly used as an early marker of osteoblast maturation, necessary for the ultimate mineralization of the osteoid. [START_REF] Kuboki | Time-dependent changes of collagen cross-links and their precursors in the culture of osteogenic cells[END_REF] ALP activity first increases during the differentiation induction step, and then decreases when the mature osteoblasts initiate mineralization of the osteoid. In this work, we observed that ALP activity was decreased atop bare TA6V-AE compared to bare TA6V. This is in accordance with others works reporting a reduction of ALP activity for osteoblasts cultured atop AE Ti-based substrates. [START_REF] Park | Use of polyelectrolyte thin films to modulate osteoblast response to microstructured titanium surfaces[END_REF][START_REF] Zhao | The influence of hierarchical hybrid micro/nano-textured titanium surface with titania nanotubes on osteoblast functions[END_REF] In the presence of LbL films, we could preserve the potential of differentiation induction of bare TA6V as far as the LbL films were GnP-cross-linked. Furthermore, the overall reduction of ALP activity at D14 indicated the progress of preosteoblasts toward the mineralization induction phase.

Acid-etched TA6V substrates coated with GnP-treated LbL films were the best promoters of late osteogenic differentiation as they displayed the largest and most uniform HA deposits. Other works also reported enhanced differentiation of osteoblasts on top of substrates combining nanoand microscaled roughness. [START_REF] Gittens | The effects of combined micron-/submicron-scale surface roughness and nanoscale features on cell proliferation and differentiation[END_REF][START_REF] Zhao | The influence of hierarchical hybrid micro/nano-textured titanium surface with titania nanotubes on osteoblast functions[END_REF] Based on the work of Zinger et al. reporting that osteoblasts aggregated within 100 mm cavities of sand blasted/acid-etched substrates, [START_REF] Zinger | Differential regulation of osteoblasts by substrate microstructural features[END_REF] we assumed that cells might have clustered within the topomimetic depressions generated by the AE treatment, mimicking the Howship's lacunae-like cavities that are left by osteoclasts, favoring cell-cell interactions required for cell differentiation. Also, the presence of CSA in our LbL coatings might have promoted the maturation of MC3T3-E1, as was the case on Ti-based implants coated with chondroitin sulfate and collagen. [START_REF] Bierbaum | Collageneous matrix coatings on titanium implants modified with decorin and chondroitin sulfate: Characterization and influence on osteoblastic cells[END_REF] Noteworthy, we also analyzed our substrates with the current and widely used Von Kossa method to stain the extent of mineralization. But, this method reveals all calcium-containing deposits, and not specifically calcium phosphates. Indeed, by Raman microspectroscopy, we detected some calcium oxalate (CaOx) deposits sequestered in the extracellular matrix, originating from the foetal bovine serum added in the culture medium. We evidenced the specific peaks of weddelite (dehydrated CaOx) and whewellite (monohydrated CaOx) at 1468 cm 21 for 1493 cm [START_REF] Huang | Fabrication and in vitro evaluation of the collagen/hyaluronic acid PEM coating crosslinked with functionalized RGD peptide on titanium[END_REF] , respectively. Therefore the results of Von Kossa staining were not enough reliable to be reported here. Generally, we consider that Raman microspectroscopy is a more accurate technique than the common nonspecific Von Kossa or Alizarin Red calcium staining methods, which may overestimate the extent of mineralization and even, retrieve false positives. [START_REF] Pedraza | Calcium oxalate crystals in fetal bovine serum: Implications for cell culture, phagocytosis and biomineralization studies in vitro[END_REF][START_REF] Hashimoto | In situ Raman imaging of osteoblastic mineralization[END_REF]

CONCLUSION

In this work, by using AE, we generated topomimetic surfaces of TA6V substrates exhibiting cavities that are bioinspired from Howship's lacunas with global microsized and local nanosized features. Then, by coating these substrates with CSA-based LbL films, we produced physiologically relevant surfaces, mimicking organic extracellular matrix of bone. Whereas short-and mid-term MC3T3-E1 events (adhesion and proliferation) were governed mainly by the stiffness of the substrate with a clear promotive effect of GnP-cross-linked films, the main factor of long-term cell differentiation on these films was the topography of the underlying substrate, with improved osteoconductive properties of the nano-and microtextured acid-etched TA6V compared to pristine TA6V.

As a whole, the nano/microtopography and chondroitin sulfate-based LbL films acted synergistically to preserve osteogenic properties of TA6V substrates. Even though no crucial influence of the nature of the outer CSA or PLL layer was detected, the potential of the LbL architectures as reservoirs of endogenous molecules or exogenous bioactive compounds and therapeutics (growth factors, antiinflammatory or bactericidal drugs) opens a promising route toward further sophisticated implants allowing the optimization the osseointegration processes.

FIGURE 1 .

 1 FIGURE 1. Surface analysis of Ti-6Al-4V substrates before (TA6V) and after (TA6V-AE) acid-etching (AE). (A) 3D AFM images. (B) SEM images at two magnifications. Surfaces were roughened upon AE treatment with bone-lacunae-like cavities of ca. 50-100 mm in size (white delimited areas-bottom left in B) and nanopits (bottom in A, and bottom right in B).

(

  ca. 50-100 mm in size), uniformly distributed over AE-treated substrates [Fig. 1(B), bottom left), white-delimited areas.

FIGURE 2 .

 2 FIGURE 2. (A) Typical evolutions of the normalized resonance frequencies and dissipation factors for the 3rd, 5th, and 7th overtones (n 5 3, 5, 7) during the buildup of a PEI/(CSA/PLL) 6 /CSA film at pH 7.4 and Tris/NaCl buffer (10 mM Tris, 150 mM NaCl) onto a QSX310 Ti-coated quartz crystal sensor. (B) Conversion of the raw data shown in (A) into hydrodynamic thickness d QCM-D and hydrated mass Q QCM-D according to the Voigt viscoelastic model. (C) 2D and 3D AFM images of the Ti-coated QCM-D sensors before and after LbL film deposition: (left) bare sensor, (middle) PEI/(CSA/PLL) 6 , (right) PEI/(CSA/PLL) 6 /CSA. Average roughness values Ra were derived from 50 3 50 lm 2 areas.

FIGURE 3 .

 3 FIGURE 3. Fluorescence microscopy images of TA6V and TA6V-AE substrates coated with (left) PEI and (right) PEI FITC . PEI FITC continuously covered the substrates.

FIGURE 4 .

 4 FIGURE 4. Qualitative surface characterization of the LbL-coated TA6V and TA6V-AE substrates. (A) 2D and 3D contact mode AFM micrographs in hydrated conditions after deposition of a PEI/(CSA/PLL) 6 or a PEI/(CSA/PLL) 6 /CSA film. (B) SEM views of the former substrates. (C) Fluorescence microscopy images of PEI/(CSA/PLL) 6 films after treatment with GnP. Continuous LbL coatings were obtained, without alteration of the underlying topography.

FIGURE 5 .

 5 FIGURE 5. Quantitative surface characterization of bare and LbLcoated TA6V and TA6V-AE substrates. LbL films consisted in native and GnP-cross-linked PEI/(CSA/PLL) 6 6 CSA architectures. (A) Average roughness Ra derived from 50 3 50 mm 2 2D AFM images. (B) Static contact angles measured with the Tris/NaCl buffer (5 mL droplets) at room temperature. Results were expressed in mean 6 SD. Statistical differences set for p < 0.05 (*).

FIGURE 6 .

 6 FIGURE 6. Morphology of MC3T3-E1 pre-osteoblasts cultured for 3 h onto bare and LbL-coated TA6V and TA6V-AE substrates as observed by epifluorescence of DAPI (nuclei) and FITC-phalloidin (actin). LbL films consisted in native and GnP-cross-linked PEI/(CSA/PLL) 6 architectures. Similar results were obtained with PEI/(CSA/PLL) 6 /CSA architectures (see Supporting Information).

FIGURE 7 .

 7 FIGURE 7. Proliferation of MC3T3-E1 pre-osteoblasts cultured for 9 days onto bare and LbL-coated TA6V and TA6V-AE substrates. LbL films consisted of native or GnP-cross-linked PEI/(CSA/PLL) 6 6 CSA architectures. Alamar Blue (AB) reduction was assayed after 2 days (D2) and 9 days (D9). The results are presented in % of AB reduction and expressed as mean 6 SEM. * p < 0.05 with respect to the corresponding native LbL films.

FIGURE 8 .

 8 FIGURE 8. Early and late differentiation of MC3T3-E1 cultured in an osteogenic culture medium. (A) ALP activity for cells cultured atop bare (TA6V) and acid-etched (TA6V-AE) substrates, coated with PEI-(CSA/PLL) 6 or PEI-(CSA/PLL) 6 -CSA (noted -PLL and -CSA respectively) in their native state or cross-linked with GnP (noted -GnP). ALP was assayed at D7 and D14. Results are presented as mean 6 SD. * p < 0.05 with respect to the corresponding native LbL films. # p < 0.05 with respect to the corresponding non-AE substrates. (B) Typical Raman spectra of (a) the mineralized nodules found atop AE substrates coated with GnP-treated films and (b) the extracellular matrix produced cells. The major Raman bands associated with hydroxyapatite are assigned by arrow heads and phenylalanine (Phe, ring breathing) by an asterisk. (C) Corresponding optical micrographs and 2D Raman spectral mapping (in green) of the band m1 PO 4

TABLE I .

 I Nomenclature and Description of the Substrates Prepared GnPGnP (w/v -Wako Chemicals) in PBS, corresponding to 11 mM, for 16 h at room temperature and rinsed them with PBS. Then, all the substrates were UV-sterilized for 30 min. Table I reports the 10 types of substrates prepared.

	Nomenclature	Description
	TA6V	Nontreated Ti-6Al-4V
	TA6V-PLL	Nontreated Ti-6Al-4V 1 PEI/(CSA/PLL) 6
	TA6V-CSA	Nontreated Ti-6Al-4V 1 PEI/(CSA/PLL) 6 /CSA
	TA6V-PLL-GnP	Nontreated Ti-6Al-4V 1 PEI/(CSA/PLL) 6 1 GnP
	TA6V-CSA-GnP	Nontreated Ti-6Al-4V 1 PEI/(CSA/PLL) 6 /CSA 1 GnP
	TA6V-AE	Acid-etched Ti-6Al-4V
	TA6V-AE-PLL	Acid-etched Ti-6Al-4V 1 PEI/(CSA/PLL) 6
	TA6V-AE-CSA	Acid-etched Ti-6Al-4V 1 PEI/(CSA/PLL) 6 /CSA
	TA6V-AE-PLL-Gnp	Acid-etched Ti-6Al-4V 1 PEI/(CSA/PLL) 6 1 GnP
	TA6V-AE-CSA-GnP	Acid-etched Ti-6Al-4V 1 PEI/(CSA/PLL) 6 /CSA 1

LABAT ET AL.SYNERGISTIC INFLUENCE ON OSTEOGENIC POTENTIAL OF TA6V

LABAT ET AL.SYNERGISTIC INFLUENCE ON OSTEOGENIC POTENTIAL OF TA6V

-at 960 cm 21 . ORIGINAL ARTICLE JOURNAL OF BIOMEDICAL MATERIALS RESEARCH A | MONTH 2016 VOL 00A, ISSUE 00

ACKNOWLEDGMENTS

This work was partially supported by the Grand Evreux Agglom eration (GEA) and the Conseil G en eral de l'Eure, France. The authors want to thank Beatrice Foulon (GPM-CNRS UMR 6634, University of Rouen, France) for her help in the cutting of TA6V foils and also the License and Master students who have been involved in this work, in a way or another (Jeremy Beloncle, Helena Abed and Soror Mahieddine).