Mathias Legrand Gem

Dongying Jiang

Christophe Pierre

Steven W Shaw

Nonlinear Normal Modes of a Rotating Shaft Based on the Invariant Manifold Method

Keywords: Invariant manifold method, Nonlinear normal modes, Oilwhip, Oil-whirl, Shaft-bearing system

The nonlinear normal mode methodology is generalized to the study of a rotating shaft supported by two short journal bearings. For rotating shafts, nonlinearities are generated by forces arising from the supporting hydraulic bearings. In this study, the rotating shaft is represented by a linear beam, while a simplified bearing model is employed so that the nonlinear supporting forces can be expressed analytically. The equations of motion of the coupled shaft-bearings system are constructed using the Craig-Bampton method of component mode synthesis, producing a model with as few as six degrees of freedom (d.o.f.). Using an invariant manifold approach, the individual nonlinear normal modes of the shaft-bearings system are then constructed, yielding a single-d.o.f. reduced-order model for each nonlinear mode. This requires a generalized formulation for the manifolds, since the system features damping as well as gyroscopic and nonconservative circulatory terms. The nonlinear modes are calculated numerically using a nonlinear Galerkin method that is able to capture large amplitude motions. The shaft response from the nonlinear mode model is shown to match extremely well the simulations from the reference Craig-Bampton model.

Many rotating structures are supported by devices that are inherently nonlinear, e.g., journal bearings. The dynamic analysis of nonlinear rotating systems has been the subject of a number of studies, for example, those of [START_REF] Yamauchi | The nonlinear vibration of flexible rotors, first report, development of a new analysis technique[END_REF] or Kim and Noah 1 (1991a), who employed the method of harmonic balance. [START_REF] Choi | Nonlinear steady state response of a rotor-support system[END_REF] added discrete Fourier transform procedures to the harmonic balance method and also included subharmonic response components. Kim and Noah 2 (1991b) used dynamic condensation techniques in conjunction with the harmonic balance method, in order to reduce the size of the system models. The objective of the present work is to develop reduced-order models of nonlinear shaft-bearings systems using the invariant manifold-based nonlinear normal mode methodology.

Nonlinear normal modes (NNM) provide a general framework for the construction of reduced-order models for nonlinear systems. The concept of NNM was first introduced by [START_REF] Rosenberg | On nonlinear vibration of systems with many degrees of freedom[END_REF] with the study of conservative, symmetric, nonlinear systems. A NNM was defined in the configuration space so that its application was strictly limited to systems without gyroscopic effects and damping. [START_REF] Shaw | Nonlinear normal modes and invariant manifolds[END_REF] extended the definition of nonlinear normal modes using invariant manifold techniques, wherein a NNM is defined as a two-dimensional invariant surface in the phase space, which is tangent to the hyperplane that represents the corresponding mode of the linearized model. The NNM response is thus captured by a single-d.o.f nonlinear oscillator. Using this more general definition, a systematic construction method for NNM has been proposed by [START_REF] Boivin | Nonlinear modal analysis of structural systems using invariant manifolds[END_REF] and [START_REF] Pesheck | Reduced order modeling of nonlinear structural systems using nonlinear normal modes and invariant manifolds[END_REF] for nonlinear systems with quadratic and cubic nonlinearities, including systems with a large number of d.o.f. and for large amplitude motions. [START_REF] Nayfeh | On nonlinear modes continuous systems[END_REF] also used the invariant manifold approach to construct NNM using perturbation methods for weakly nonlinear systems.

In this article, the NNM are constructed for an established model of a nonlinear shaft-bearings system, which consists of a linear rotating shaft supported by short nonlinear journal bearings at its two ends. Since the linearized system is gyroscopic, damped, and (nonconservative) circulatory-thus featuring a non-symmetric stiffness matrix-an extension of the invariant manifold approach is required in order to accommodate these effects. The NNM are shown to provide very accurate reducedorder models of the shaft-bearings system.

The article is organized as follows. In the first section, the mathematical model of the shaft-bearings system is derived using component mode synthesis (CMS). Then, in the second section, the vibration modes of the linearized system are investigated. Finally, in the third section, the theory of NNM is extended to systems whose linearized counterpart is gyroscopic, nonconservative, and circulatory, and the individual NNM invariant manifolds of the nonlinear shaft-bearings system are calculated.

THE ROTATING SHAFT-BEARINGS SYSTEM

A diagram of the system of interest, a rotating shaft supported by short journal bearings at its two ends, is shown in Figure 1. A Rayleigh beam with uniform cross-section properties is used to model the shaft, which is assumed to be perfectly balanced. The forces created by the oil film in the bearings are nonlinear and can be represented as nonlinear boundary conditions for the beam.

In Figure 1, the inertial frame, R XYZ , is fixed in space and the Y-axis passes through the centers of the bearings at the two ends. The shaft is defined by its length L = 1 m, outer diameter D 2 = 0.0592 m, and inner diameter D 1 = 0.02 m. The nominal clearance between the bearing and the shaft is set as c = (D 3 -D 2 )/2 = 5.1•10 -5 m, where D 3 is the inner diameter for both bearings. The length of the bearing is L b = 0.0285 m, and the dynamic viscosity of the oil film is chosen as µ = 0.0068 N•s/m 2 .

FIGURE 1

Schematic of the shaft-bearings system.

The Shaft Model

From the Rayleigh beam theory, the kinetic energy, T, and strain energy, U, of the rotating shaft are given by, respectively:

T = ρA 2 L 0 (u 2 + ẇ2 )dy + ρI 2 L 0 ∂ u ∂y 2 + ∂ ẇ ∂y 2 dy + ρIL 2 -2ρI L 0 ∂ u ∂y ∂w ∂y dy [1] U = EI 2 L 0 ∂ 2 u ∂y 2 2 + ∂ 2 w ∂y 2 2 dy [2]
where u(y, t) and w(y, t) are the displacements of the points on the neutral axis of the shaft in the X and Z directions, respectively, is the constant angular velocity of the shaft, A is the shaft's cross-sectional area, and I is the second area moment of inertia of the cross section. The material parameters for the shaft are its Young's modulus E = 2.1•10 11 Pa and mass density ρ = 7800 kg/m 3 .

In order to obtain an efficient discretized model for the rotating shaft, the method of component mode synthesis developed by [START_REF] Craig | Coupling of substructures for dynamic analyses[END_REF] is applied, where the shaft is the linear substructure constrained at its ends by the nonlinear bearings. This allows for the nonlinear effects of the supporting bearings to be captured solely by the d.o.f. corresponding to the static constraint modes. The shaft displacements u and w are thus each expanded as a linear combination of the modes of free vibration of the shaft pinned at both ends and two constraint modes, each corresponding to a rigid body motion of the shaft induced by a unit displacement at one of its ends:

u(y, t) = m i=1 i (y)a i (t) w(y, t) = m i=1 i (y) b i (t) [3] where i (y) =        sin iπ y L i = 1, . . . , m -2 y/L i= m -1 1 -y/L i = m
The first (m -2) expansion functions are the mode shapes of the simply supported shaft and the final two are the static constraint modes. We substitute the expansion functions for u and w into Hamilton's principle,

t 2 t 1 (δT -δU + δW)dt = 0 ∀(t 1 , t 2 ) [4]
where δW is the virtual work done by the gravity force and the nonlinear supporting forces of the two bearings. The resulting discretized model is given as: where

[M 1 ]{ẍ} + [G 1 ]{ẋ} + [K 1 ]{x} = {F} [5]
{x} = {a} {b} , [M 1 ] = ρS [A] [0] [0] [A] + ρI [B] [0] [0] [B] , [G 1 ] = 2ρI [0] [B] -[B] [0] , [K 1 ] = EI [C] [0] [0] [C]
Here {a} and {b} are m-vectors of the generalized coordinates generated by the CMS formulation, a i and b i , respectively. The The matrices [M 1 ], [G 1 ], and [K 1 ], are the inertia, gyroscopic, and stiffness matrices, respectively. Note that {x} is defined with respect to the inertial frame R XYZ . The force vector {F} has the form:

{F} = -ρgA L 0 { (y)}dy + {F 2 (x, ẋ)} = {F 1 } + {F 2 (x, ẋ)} [6]
where the first term is the gravitational load, and the second term {F 2 } is the supporting force at the journal bearing, which is considered next.

The Bearing Forces

In Figure 2, R XYZ is the inertial frame and is the angular velocity, defined in Figure 1. The nominal thickness of the oil film is h, e is the eccentricity between the bearing axis and the shaft axis, and φ is the attitude angle of the line connecting the bearing and shaft centers with respect to the Z-axis. The horizontal and vertical displacements of the center of the shaft (the journal) in the bearing are denoted as x j and z j , respectively (j for journal). The thickness of the oil film, h, can be expressed as:

h = c -z j cos (θ + φ) + x j sin (θ + φ) [7]
where c is the nominal clearance between the shaft and the bearing. Based on Reynolds' equation, the pressure of the oil film can be modeled as :

∂ ∂y h 3 6µ ∂p ∂y + 1 R 2 ∂ ∂θ h 3 6µ ∂p ∂θ = ∂h ∂θ + 2 ∂h ∂t [8]
In Equation ( 8), µ is the fluid viscosity, R is the outer radius of the beam, and p is the fluid film pressure. The short bearing assumption implies that R 2 is preponderant in Equation ( 8), so that the second term on the left-hand side in Reynolds' equation can be neglected. This yields:

h 3 6µ ∂ 2 p ∂y 2 = (z j sin(θ + φ) + x j cos(θ + φ)) -2(ż j cos(θ + φ) -ẋj sin(θ + φ)) [9]
The boundary conditions over y are p(θ, 0) = p(θ , L b ) = 0, where L b is the length of the bearing and θ is integrated over [0, π] instead of [0, 2π ] due to cavitation effects described by [START_REF] Vance | Rotordynamics of Turbomachinery[END_REF]. Then the two forces created by the pressure field are the integrals of the pressure over the fluid-film surface contact.

A dimensionless analysis of the problem leads to the following definitions:

Z j = z j c , X j = x j c , Żj = żj c , Ẋj = ẋj c [10] H = 1 -Z j cos(θ + φ) + X j sin(θ + φ)
where H is the dimensionless fluid-film thickness. Then, the resultant forces F X and F Z in the X and Z directions can be obtained as (see [START_REF] Lee | Vibration Analysis of Rotors[END_REF]:

F X = - µRL 3 2c 2 π 0 Z j sin(θ + φ) + X j cos(θ + φ) -2( Żj cos(θ + φ) -Ẋj sin(θ + φ)) H 3 sin(θ + φ)dθ [11a] F Z = µRL 3 2c 2 π 0 Z j sin(θ + φ) + X j cos(θ + φ) -2( Żj cos(θ + φ) -Ẋj sin(θ + φ)) H 3 cos(θ + φ)dθ [11b]
Both forces are nonlinear in the displacements and velocities of the shaft's ends, involving complicated integrals that can be obtained analytically using commercial mathematical software such as Maple R .

THE LINEARIZED MODEL

Complex modal analysis is applied to the system linearized about its equilibrium position to determine the natural modes of vibration and the shaft's first critical speed.

The Equilibrium Position

The dimensionless equilibrium position of the shaft center is given by [START_REF] Ocvirck | Short bearing approximation for full journal bearings[END_REF] as:

Z e = -ε e cos φ e X e = ε e sin φ e
where φ e = arctan π 1 -ε 2 e 4ε e [12] and the dimensionless eccentricity ε e = e e /c is the solution of:

µN f R 2c 2 L 3 b = (1 -ε 2 ) 2 πε π 2 (1 -ε 2 ) + 16ε 2 [13]
where N (with = 2πN) is the angular frequency and f is the load, which in this study is given by the weight of the beam. position, the stiffness matrix (K 2 ) (which is not symmetric) and the damping matrix (D 2 ) can be obtained as:

K 2IJ = ∂F I ∂J and D 2IJ = ∂F I ∂ J with I = (X, Z) and J = (X, Z)
where Z = Z j -Z e and X = X j -X e . The eight dimensionless coefficients for these matrices are given by [START_REF] Lee | Vibration Analysis of Rotors[END_REF] as follows:

K ZZ = - 4(π 2 + (32 + π 2 )ε 2 + 2(16 -π 2 )ε 4 ) 1 -ε 2 .Q(ε) K ZX = π (π 2 + (32 + π 2 )ε 2 + 2(16 -π 2 )ε 4 ) ε √ 1 -ε 2 .Q(ε) K XZ = - π (π 2 -2π 2 ε 2 -(16 -π 2 )ε 4 ) ε √ 1 -ε 2 .Q(ε) K XX = -4(2π 2 + (16 -π 2 )ε 2 ).Q(ε) C ZZ = - 2π (π 2 + 2(24 -π 2 )ε 2 + π 2 ε 4 ) ε √ 1 -ε 2 .Q(ε) C ZX = 8(π 2 -2(8 -π 2 )ε 2 ).Q(ε) C XZ = C ZX C XX = - 2π √ 1 -ε 2 (π 2 -2(8 -π 2 )ε 2 ) ε .Q(ε) Q(ε) = 1 (π 2 (1 -ε 2 ) + 16ε 2 ) 3/2
The relationships between the dimensionless and the dimensional coefficients are:

K IJ = k ij •c f and C IJ = c ij •c• f
The Linear Modes In order to study the modes of free vibration of this system, Equation ( 5) is linearized with respect to the equilibrium position. The linear system can be written in the following general form:

[M]{ẍ} + [C]{ẋ} + [K]{x} = {0} [14]
where

[M] = [M 1 ], [C] = [G 1 ] -[D 2 ], [K] = [K 1 ] -[K 2 ],
and {x} is defined relative to the equilibrium position.

Complex modal analysis can be applied to the state space formulation of the system. Defining the velocity vector as {y} = {ẋ}, Equation ( 14) can be written in first-order form as:

{ż} = [D]{z} [15]
where

{z} = {x} {y} and [D] = [O] [I] -[M] -1 [K] -[M] -1 [C]
Since the matrix [D] is real, those eigensolutions that are complex must occur in conjugate pairs. Since the NNM are to be defined in terms of real modal coordinates, a traditional complex mode transformation matrix would lead to difficulties in constructing the corresponding invariant manifolds. Thus, herein a real modal transformation is chosen that renders the matrix [D] block diagonal. More details on this procedure are given by [START_REF] Hirsch | Differential Equations, Dynamical Systems, and Linear Algebra[END_REF]. This is achieved by constructing the matrix [Z ], whose columns are the real eigenvectors and the real and the imaginary parts of the complex eigenvectors, as follows:

[Z ] = [ z 1 , z 1 , . . . , z p , z p , z 2p+1 , . . . , z 2N ] [16]
where N is the total number of d.o.f. for the rotating shaft system described by Equation ( 5), and is equal to 2 m. Letting {z(t)} = [Z ]{η(t)}, Equation ( 15) is transformed into:

{ η(t)} = [ ]{η(t)}, [17] with [ ] =                  λ 1 λ 1 -λ 1 λ 1 0 . . . λ p λ p -λ p λ p λ 2p+1 0 . . . λ 2N                 
The resulting first-order equations are at worst pairwise coupled.

Those equations that are uncoupled are associated with the real eigenvalues and correspond to overdamped modes, whose dynamics occur in a one-dimensional linear subspace. Motions in these modes are decaying and non-oscillatory. The modes associated with the complex conjugate pairs of eigenvalues correspond to the 2 × 2 diagonal blocks of [ ] and are for underdamped, oscillatory modes. The dynamics of these modes occur in two-dimensional subspaces. Motions in the underdamped modes consist of decaying oscillations, which are described by pairs of first-order differential equations:

   ṡi = -ζ i ω i s i + ω i 1 -ζ 2 i t i ṫi = -ω i 1 -ζ 2 i s i -ζ i ω i t i i = 1, . . . , p [18]
where ω i is the undamped natural frequency of the mode and ζ i its modal damping ratio, both of which are defined from λ i = -ζ i ω i and λ i = ω i 1 -ζ 2 i . Also, s i = η 2i-1 and t i = η 2i are the components of {η} associated with the ith underdamped mode.

Stability of The Linear Model

In the Lyapunov sense, the system is unstable if the real part of any eigenvalue is positive. The linear motion associated with such an instability becomes infinitely large with time and thus linear theory breaks down. The eight coefficients of the linear model given by [START_REF] Lee | Vibration Analysis of Rotors[END_REF] are strongly linked to the stability of the system, especially the cross-coupled stiffness terms. In this study we have examined the stability of the shaft-bearings system using two, four, and six normal modes of vibration to describe the shaft motion, corresponding, respectively, to shaftbearings models with six, eight, and ten d.o.f.s. The results for these three models have been found to be in close agreement, indicating that a single normal mode in each direction is sufficient to represent the shaft's motion. This makes sense since the shaft is balanced and thus the first beam mode is dominant. Therefore, the six-d.o.f. model is employed in the remainder of the paper. Figure 4 shows that the linearized system is unstable when the angular velocity of the shaft is larger than 375 rad/s.

Linear Mode Motions

As the convergence study indicated, a six-d.o.f. model is sufficient to capture accurately the lower oscillating modes of the system. Recall that this model includes one free vibration normal mode and two static constraint modes for the shaft in each of the X and Z directions. The eigenanalysis was carried out in the state space in terms of the shaft angular velocity.

It was found that up to approximately = 550 rad/s, the shaft-bearings system possesses four, one-dimensional overdamped modes, corresponding to real eigenvalues, and four, two-dimensional oscillatory modes, corresponding to complex eigenvalues. Two of the four overdamped modes become a single two-dimensional oscillatory mode as grows beyond 550 rad/s, and the last two overdamped modes turn into a sixth oscillatory mode for > 700 rad/s as shown in Figures 5a and5b. In the following, we focus on the oscillatory motions at the rotation speed = 100 rad/s. Since both real and imaginary parts are very close together for the two first modes, their undamped natural frequency and modal damping ratio are very close too, as depicted in Figures 6 and7.

The physical nature of the modes can be depicted as follows.

In Figures 6789, curves are drawn for each oscillating mode in the X and Z coordinates that display the displacements of both ends of the beam as they move in the bearings during a modal motion. The accompanying sketches of the shaft show it as straight, whereas in reality it deflects in its first vibration mode (although its amplitude is relatively small compared to the bearing deflections).

Comparison Between Nonlinear and Linear Models

Using a very small disturbance from the equilibrium position as initial conditions for both the linear and nonlinear models, the time simulation results are seen to be very close in Figure 10. For = 100 rad/s, which lies in the stable range, after a short time the beam settles back to its equilibrium position.

As seen in Figure 11, for a much larger shaft rotation speed, = 2550 rad/s, the linear model is unstable, as the angular velocity is then greater than the critical speed, c ≈ 375 rad/s, whereas the nonlinear model reaches a limit cycle corresponding to oil whirl detailed by [START_REF] Muszynska | Whirl and whip rotor/bearing stability problems[END_REF]. For a lightly loaded shaft, the whirl occurs at a frequency that is exactly half the shaft's angular velocity, and here the whirl motion frequency is 1225 rad/s. The thick black line in Figure 11 represents the largest displacement of the center point of the shaft at the bearing location that is kinematically allowed. As shown in Figure 11, the oil whirl phenomenon cannot be predicted by linearized stability theory.

FIGURE 5a

The largest four real parts of the eigenvalues versus .

FIGURE 5b

Imaginary part of the eigenvalues versus .

FIGURE 6

First linear mode shape at = 100 rad/s: ω 1 = 50.647, ζ 1 = 0.1166.

NONLINEAR NORMAL MODES

For a linear system, an important property of a modal motion is that if one knows the motion of a single generalized coordinate, then the motions of all others coordinates are specified by that mode's eigenvector. The same property is used for the nonlinear case, wherein a motion in a nonlinear normal mode is defined such that all displacements and velocities are functionally related to a single displacement-velocity pair. In a NNM the nonlinear system thus behaves essentially as a single-d.o.f. oscillator. Even though NNM are defined using a linear normal mode concept, the similarities stop there. Many results obtained by NNM have no counterparts in linear systems, such as amplitude-dependant frequencies and internal resonances (see [START_REF] Nayfeh | Nonlinear Oscillations[END_REF]), where two NNM exchange energy during the motion.

Using the concept of invariant manifolds, [START_REF] Shaw | Nonlinear normal modes and invariant manifolds[END_REF] defined a NNM as a motion that lies on a two-dimensional invariant manifold in the system's phase space. Here "invariant" indicates that any motion initiated on the manifold will remain on it for all times. A single displacement-velocity pair is chosen as master coordinates, which characterize the individual nonlinear mode motion. All the remaining "slave" coordinates are parameterized by these two master coordinates and give the constrained conditions. Previous work on the invariant manifold method has been restricted to systems with diagonal stiffness and damping matrices in the modal coordinates space. However, in order to construct NNM for the rotating shaft system, a new formulation had to be introduced for obtaining the PDE governing the invariant manifold. This new formulation is more general and can be applied to systems with nonproportional damping

NNM Formulation

In this section we use Equation ( 5) written with respect to the equilibrium position. The equations of motion become:

[M]{ẍ} + [C]{ẋ} + [K]{x} = {F NL (x, ẋ)} [19]
and

{F NL (x, ẋ)} = {F 2 (x + x e , ẋ)} -[K 2 ]{x} -[D 2 ]{ẋ} + {F 1 }-[K 1 ]{x e }
where {x} is defined with respect to the equilibrium position {x e }.

All the matrices and forces in this equation have been defined in previous sections. Equation ( 19) are transformed following the procedure of Equation ( 15) and the linear transformation of Equation ( 17) is carried out. It is assumed that the real linear transformation can be written as follows: Then, the new equations of motion are:

[ ] =          α 1 β 1 β 2 α 2 . . . α 2N-1 β 2N-1 β 2N α 2N          [20] α k = α k+1 β k = -β k+1 k = 1, 3, . . . , p -1, α k = α k+1 β k = β k+1 = 0 k = p + 1, p + 3, . . . , 2N -1
                 ṡ1 ṫ1 . . . ṡN ṫN                  =          α 1 β 1 β 2 α 2 . . . α 2N-1 β 2N-1 β 2N α 2N          .                  s 1 t 1 . . . s N t N                  +                  f 1 f 2 . . . f 2N-1 f 2N                  [21]
with nonlinear forces:

{f} = [Z ] -1 {O} {F NL ({x(η, η)}, {ẋ(η, η)})}
The new set of master coordinates is defined as:

s k = a cos φ t k = a sin φ [22]
where s k and t k are one pair of real linear coordinates defined in Equation ( 18), chosen as one of the oscillating modes.

The equations of coherence are:

         ṡk = ṡk ⇔ ȧ cos φ -a φ sin φ = α 2k-1 a cos φ + β 2k-1 a sin φ + f 2k-1 ṫk = ṫk ⇔ ȧ sin φ + a φ cos φ = β 2k a cos φ + α 2k a sin φ + f 2k [23]
           ȧ = α 2k-1 a cos 2 φ + α 2k a sin 2 φ + f 2k-1 cos φ + f 2k sin φ φ = -β 2k-1 + (α 2k -α 2k-1 ) sin φ cos φ - f 2k-1 sin φ a + f 2k cos φ a [24]
Following the definition of the invariant manifold, all the other d.o.f. are constrained in the (a,φ) coordinates:

s i = P i (a, φ) t i = Q i (a, φ) i = 1, . . . , N i = k [25]
The time-independent partial differential equations in P i and Q i governing the invariant manifold geometry can be written by combining Equations ( 21) and ( 25):

       ṡi = ∂P i ∂a ȧ + ∂P i ∂φ φ = α 2i-1 s i + β 2i-1 t i + f 2i-1 i = 1, . . . , N i = k ṫi = ∂Q i ∂a ȧ + ∂Q i ∂φ φ = β 2i s i + α 2i t i + f 2i [26]
In those equations, ȧ and φ have to be replaced by expressions (24). A computation is necessary to approximate the manifold. As P i and Q i must be periodic in φ due to the transformed coordinates, they are expanded as a double series in the amplitude

FIGURE 10

Trajectory of one end of the shaft for a very small initial disturbance and = 100 rad/s.

FIGURE 11

Trajectory of one end of the shaft for a very small initial disturbance and = 2550 rad/s.

FIGURE 12

Internal resonance between mode 1 and mode 2: imaginary and real parts of the eigenvalues with as a parameter.

FIGURE 13

First nonlinear normal mode invariant manifold: P 4 and Q 4 versus amplitude a and phase φ.

FIGURE 14

Motion on the first nonlinear normal mode: t 4 versus s 1 and t 1 obtained using both the NNM reduced-order model and the full system.

FIGURE 15

Sixth nonlinear normal mode invariant manifold: P 4 and Q 4 versus amplitude a and phase φ. and the phase:

           P i (a, φ) = N a l=1 N φ m=1 C l,m i T l,m (a, φ) Q i (a, φ) = N a l=1 N φ m=1 D l,m i U l,m (a, φ) i = 1, . . . , N i = k [27]
where C l,m i and D l,m i are the unknown expansion coefficients and T l,m and U l,m are the known shape functions, typically products of functions of a over [0, a max ] and φ over [0, 2π ]. These functions are harmonic cosinus and sinus functions in φ and piecewise first-degree polynomials in a. N a and N φ are the number of expansion functions used in the a and φ directions, respectively. The more of these functions that are used, the more accurate the approximation is. Expansions ( 27) are substituted into Equation ( 26) and a Galerkin projection (multiplication by test function and integration over the chosen domain, i.e., [0, a max ] × [0, 2π]) is carried out. This yields a set of nonlinear algebraic equations in the coefficients C and D which, when solved, produces a solution that minimizes the error for the given expansion in a least squares sense.

FIGURE 16

Motion on the sixth nonlinear normal mode: t 4 versus time obtained using both the NNM reduced-order model and the full system.

Results and Discussion

The above single-NNM formulation cannot handle systems with internal resonances, for which a multi-nonlinear manifold formulation would be needed. Thus, it is necessary to determine under what conditions an internal resonance can occur for the system.

An internal resonance can occur if there exists positive or negative integers m 1 , m 2 , . . . , m n such that m 1 ω 1 +m 2 ω 2 +• • •+ m n ω n = 0. In this case, strong nonlinear interactions between the corresponding modes may take place. To check for the presence of internal resonances in the shaft-bearings system, the loci of the eigenvalues of the linearized system are plotted in the complex plane, with the angular speed as a parameter, as shown in Figure 12.

Here varies from 0 to 4000 rad/s, and four distinct branches of complex conjugate pairs of oscillatory eigenvalues are observed (the other modes fall outside the scale of the plot). In order to avoid a one-to-one internal resonance, the eigenvalues must not be close to one another. Note that such a resonance precisely takes place between mode 1 and 2 for approximately between 0 and 200 rad/s. It was found that in this region, the single-mode manifolds could not be calculated, and a two-mode manifold would be needed to capture the internal resonance. Hence this region was avoided.

In order to avoid the above internal resonance, nonlinear manifolds were calculated for = 300 rad/s. Figure 13 shows components of the invariant manifold for the first nonlinear oscillating normal mode. Namely, the contributions P 4 and Q 4 of the fourth linear mode are depicted in terms of the amplitude a and phase φ. For a motion initiated on this invariant manifold, Figure 14 shows the nearly perfect agreement between the time histories calculated from the NNM single-d.o.f. reduced-order model and the full shaft bearings modal, Equation (5). Note that the latter is another possible view of a manifold, directly in the (s,t)-space instead of the (a,φ)-space. It is important to notice that using the new NNM formulation and the real linear transformation, nonlinear manifolds can be calculated for both overdamped and oscillatory normal modes. In every computation, the nonlinear manifolds have been constructed with N a = N φ = 20, and the dynamics on these manifolds are compared to the exact solution, using a fourth order Runge-Kutta method time-integration of these first-order differential equations with the same initial conditions.

Figures 15a and15b depict the contributions P 4 and Q 4 of the fourth linear mode to the sixth nonlinear overdamped mode. Differences between the NNM reduced-order model and the exact solution are negligeable as illustrated in Figure 16, for a motion initiated on the sixth NNM manifold.

CONCLUSIONS

The nonlinear normal mode invariant manifold approach has been successfully extended to describe the vibration behavior of nonlinear mechanical systems with general damping, gyroscopic, and stiffness matrices. This new formulation is based on a real linear transformation and a new set of coordinates in amplitude and phase. The nonlinear Galerkin method used to solve for the invariant manifold geometry gives very good results and is quite fast. The methodology has been applied successfully to the construction of the NNM of a rotating shaft with nonlinear bearings.

Although similar results could be obtained using classical methods for nonlinear systems (e.g., harmonic balance, multiple scales), the present approach produces systematically very accurate and minimal NNM reduced-order models. It holds significant promise for the study of realistic shaft models with mass imbalance, physical phenomena such as oil whirl, and internal resonances using multi-mode manifolds.

NOMENCLATURE

FIGURE 2

 2 FIGURE 2Schematic of the journal bearing model.

[

  ,yy (y)] T [ ,yy (y)]dy

  FIGURE 3Locus of the equilibrium position versus .

FIGURE 4

 4 FIGURE 4Stability of the linear model: largest real part of the eigenvalues versus .

FIGURE 7

 7 FIGURE 7Second linear mode shape at = 100 rad/s: ω 2 = 50.676, ζ 2 = 0.1112.

FIGURE 8

 8 FIGURE 8Third linear mode shape at = 100 rad/s: ω 3 = 775.495, ζ 3 = 0.0457.

FIGURE 9

 9 FIGURE 9Fourth linear mode shape at = 100 rad/s: ω 4 = 775.684, ζ 4 = 0.0561.

  ], [G 1 ] and [K 1 ] mass, gyroscopic, and stiffness matrices of the shaft, respectively[K 2 ], [D 2 ]stiffness and damping matrices obtained by linearizing the nonlinear forces created by the oil film, respectively

		ε = e/c	dimensionless eccentricity
		{η}	vector defined by {s 1 ,t 1 , . . . ,s N ,t N }
		f	load (weight of the shaft), 186.42 N
		{f}	vector of the bearing nonlinear forces in
			the (s,t)-space
		{F NL }	vector of the bearing nonlinear forces
		h	oil film thickness
		I	second moment of inertia of the cross sec-
			tion, 1.19.10 -6 m 4
		L	length of the shaft, 1 m
		L b	length of the bearing, 2.86.10 -2 m
		N = /2π	rotational frequency of the shaft
			angular velocity of the shaft, rad/s
		p	pressure in the oil film
		φ	attitude angle
		R = D2/2	outer radius of the shaft, 2.96.10 -2 m
		ρ	mass density of the shaft, 7800 kg•m -3
		µ	viscosity, 6.8.10 -3 N•s/m 3
		u, w	displacement of the shaft in X and Z
			direction, respectively
		x j , z j	coordinates of the center of the shaft in
			the bearing
		{x}	generalized coordinates vector
		[M 1
	A	area of the cross section of the shaft,
		2.438•10 -3 m 2
	D1	inner diameter of the shaft, 2.10 -2 m
	D2	outer diameter of the shaft, 5.92.10 -2 m
	D3	inner diameter of the bearing, 5.92.10 -2 m
	c	nominal clearance, 51.10 -6 m
	e	eccenticity
	E	young modulus, 2.1.10 11 Pa


Carlos Montes
email: carlos.montes@unice.fr

Marc De Micheli

Pierre Aschieri

Downconversion efficiency in backward optical parametric oscillators

Keywords: Coherence gain, Nonlinear optical waveguides, Parametric oscillators, Parametric downconversion, Semiconductor nonlinear optics

In optical parametric oscillators based on periodically poled (PP) nonlinear quadratic materials, choosing a submicronic poling period allows generating the phase-matched signal and idler waves in opposite directions. In this case a distributed feedback mechanism is established, which enables optical parametric oscillation without the need of external mirrors or surface coatings. A backward mirrorless optical parametric oscillator (BMOPO), has been experimentally realized using a periodically-poled KTiOPO4 (PPKTP) crystal with sub-µm periodicity as a quasi-phase-matched (QPM) nonlinear medium. A remarkable property of the BMOPO is the strong asymmetry in spectral bandwidth between the signal and the idler pulses: the bandwidth of the co-propagating forward wave is comparable to the pump bandwidth, whereas that of the backward wave is typically one to two orders of magnitude narrower. It has been recently demonstrated both experimentally and numerically for a pump wave exhibiting deterministic phase modulation in the PPKTP BMOPO. This property is studied here in a guided wave configuration for broad incoherent pump pulses with bandwidth up to 4.7 THz. We demonstrate that in this case the higher the incoherence of the pump, the larger the amount of coherence transfer. The group velocities of the forward propagating waves can be perfectly matched by an appropriate waveguide profile in order to optimize the coherence transfer to the backward propagating wave. However, a nanometric QPM periodicity is required for perfect group velocity matching of the pump with the forward idler wave. Up to now such short periods have been only achieved by epitaxy in GaN. In order to have long enough waveguides we have studied this process in a PPGaN waveguide presenting stitching errors, which are very likely to occur during the fabrication process of the waveguide. We model a BMOPO in a fragmented GaN waveguide consisting of a sequence of submicronic PP elements separated by uniformly polarized connection sections representing the stitching errors. We have shown that in these fragmented waveguides the coherence transfer is nonintuitivelly almost as high as for a single piece PP waveguide because the generated coherent phase of the backscattered wave locks the phases of the forward propagating waves in such a way that the dynamics is almost insensitive to the short connection sections. We analyse on a large pump bandwidth range the downconversion efficiency versus the stochastic pump bandwidth for two different fragmented PPGaN BMOPOs and we compare it to the one piece periodically polarized PPGaN BMOPO.

Introduction

In optical nonlinear quadratic materials, the periodically poled (PP) technique allows to build optical parametric oscillators (cf. Fig. 1). It is possible to choose a submicronic poling period such that the phase-matched signal and idler waves are generated in opposite directions [START_REF] Ding | Backward optical parametric oscillators and amplifiers[END_REF] correspondig to configurations of Fig. 1

(b)(c).

In these cases, a distributed feedback mechanism is established, which enables optical parametric oscillation without the need of external mirrors or surface coatings. The backward mirrorless optical parametric oscillator (BMOPO) has been experimentally realized in the configuration of Fig. 1(b) using a periodically-poled KTiOPO 4 (PPKTP) crystal with 800 nm periodicity as a quasi-phase-matched (QPM) nonlinear medium [START_REF] Canalias | Mirrorless optical parametric oscillator[END_REF]. One of the remarkable properties of the BMOPO is the strong asymmetry in spectral bandwidth between the signal and the idler pulses: the bandwidth of the co-propagating forward wave is comparable to the pump bandwidth, whereas that of the backward wave is typically one to two orders of magnitude narrower. This has been recently demonstrated both experimentally and numerically for a pump wave exhibiting deterministic phase modulation in the PPKTP BMOPO for relatively narrowband (1.2 THz) linearly-chirped pump pulses [START_REF] Strömqvist | Coherent phasemodulation transfer in counterpropagating parametric down-conversion[END_REF] and relatively broadband (4 THz) pump pulses too [START_REF] Strömqvist | Temporal coherence in mirrorless optical parametric oscillators[END_REF]. In accordance with the convectioninduced phase-locking mechanism [START_REF] Picozzi | Parametric Three-Wave Soliton Generated from Incoherent Light[END_REF][START_REF] Picozzi | Coherence properties of the parametric three-wave interaction driven from an incoherent[END_REF][START_REF] Montes | Ultra-coherent output from an incoherent cw-pumped singly resonant optical parametric oscillator[END_REF][START_REF] Picozzi | Influence of dispersion on the resonant interaction between three incoherent waves[END_REF][START_REF] Montes | Coherent signal from incoherently cw-pumped singly resonant Ti:LiNbO3 integrated optical parametric oscillators[END_REF], the phase modulation in the pump is coherently transferred to the forward parametric wave, while the backward wave retains a narrow bandwidth and high coherence. This property is studied here for broad incoherent pump pulses up to 4.7 THz. We demonstrate that in this case the higher the incoherence of the pump, the larger the amount of coherence transfer. The group velocities of the forward propagating waves can be perfectly matched by an approriate waveguide profile in order to optimize the coherence transfer to the backward propagating wave, corresponding to the configuration of Fig. 1(c). However, a nanometric QPM periodicity is required for perfect group velocity matching of the pump and the forward propagating idler wave. Up to now, this can be only achieved by epitaxy in a GaN material. Asymmetric GaN waveguides grown on (0001) sapphire by molecular beam epitaxy or by metal-organic chemical vapor deposition are promising structures for applications in nonlinear optics. Periodically poled GaN (PPGaN) was used for second harmonic generation [START_REF] Chowdhury | Secondharmonic generation in periodically poled GaN[END_REF] and the possibility to grow submicronic PP GaN films [START_REF] Pezzagna | Submicron periodic poling and chemical patterning of GaN[END_REF] opens the door for parametric counterpropagating oscillation thanks to reasonably large second order susceptibility (χ (2) 33 = 10.6 pm/V measured in [START_REF] Pezzagna | Refractive indices and elasto-optic coefficients of GaN studied by optical waveguiding[END_REF]), comparable to KTP. We have studied this process in a PPGaN waveguide presenting stitching errors, which are very likely to occur during the fabrication process of the waveguide. Indeed, the realization of centimeter long devices presenting a submicrometer periodicity is not possible without moving the sample under the writing beam. This displacement introduces the stitching faults. The device is then formed by a sequence of PP poled elements of about 200 µm long (PPGaN bricks) separated by uniformly polarized connection sections of several micrometers long representing the stitching errors. We have shown that the coherence transfer is nonintuitivelly almost as high as for a single piece periodically polarized waveguide [START_REF] Montes | Backward optical parametric efficiency in quasi-phase-matched GaN waveguide presenting stitching faults[END_REF], because the generated coherent phase of the backscattered wave locks the phases of the forward propagating waves in such a way that the dynamics is almost insensitive to the short connection sections. Here we analyse the downconversion efficiency versus the stochastic pump bandwidth for two different fragmented PPGaN BMOPOs in a large frequency pump bandwidth range and compare it to the one piece periodically polarized PPGaN BMOPO. The BMOPO dynamics is numerically simulated with a new numerical scheme that solves the coupled wave equations in the counterpropagating configuration in the presence of group-velocity dispersion (GVD) by combining the trajectory method for the nonlinear three-wave interaction, extensively used for the treatment of stimulated Brillouin backscattering problems [START_REF] Montes | Dissipative three-wave structures in stimulated backscattering. I. A subluminous solitary attractor[END_REF][START_REF] Montes | Dissipative three-wave structures in stimulated backscattering. II. Superluminous and subluminous solitons[END_REF], with fast Fourier transformation (FFT) to account for the group-velocity dispersion effects in the spectral domain [START_REF] Montes | Model for Coherence Transfer in a Backward Optical Parametric Oscillator[END_REF][START_REF] Strömqvist | Temporal coherence in mirrorless optical parametric oscillators[END_REF].

BMOPO model in a fragmented waveguide

In a BMOPO, the pump wave is down-converted into forward-propagating and backward-propagating parametric waves, in accordance with energy conservation, ω p = ω f + ω b and counterpropagating quasi-phase matching, k p -k f + k b = K G , where j = p, f, b denote the pump, the forward and the backward parametric waves, respectively. K G = 2π/Λ G is the grating vector given by Λ G , the modulation period of the second-order nonlinear coefficient. The coherence transfer effect to the backward wave is especially pronounced when the pump wavelength and the modulation period of the nonlinear medium are chosen so that the group velocities of the pump and the forward wave are exactly matched [START_REF] Picozzi | Coherence properties of the parametric three-wave interaction driven from an incoherent[END_REF]. This configuration may be achieved in a GaN waveguide with a group velocity dispersion modified by a particular profile of the refractive index, a technique widely used in fiber to compensate the material dispersion with the waveguide dispersion [START_REF] Monerie | Propagation in doubly clad single-mode fibers[END_REF]. Then it is able to exactly match the pump and the idler group velocities and the counterpropagating wave will be the signal, corresponding to Fig. 1(c). However, this requires a short PP period Λ G of hundreds of nm in length, which may be realized in the GaN structure thanks to epitaxy [START_REF] Pezzagna | Submicron periodic poling and chemical patterning of GaN[END_REF][START_REF] Pezzagna | Refractive indices and elasto-optic coefficients of GaN studied by optical waveguiding[END_REF]. For z-polarized waves in PPGaN, matching group velocities can be achieved by designing the experiment so that the pump and the forward wave are on different sides of the maximum on the group-velocity curve shown in Fig. 2. This shape of group-velocity curve may be obtained in the guided PPGaN structure composed of a strip-loaded GaN waveguide of 2 µm width and 1.8 µm depth on a AlInN buffer which has been realized on sapphire substrate by epitaxy. The Sellmeier relation given in [START_REF] Pezzagna | Refractive indices and elasto-optic coefficients of GaN studied by optical waveguiding[END_REF] is used for material dispersion, and finite waveguide dispersion corrections [START_REF] Monerie | Propagation in doubly clad single-mode fibers[END_REF] yield the corresponding group-velocity curve of Fig. 2.

Figure 2: Group velocity for the z-polarized waves in GaN as function of the wavelength, calculated from the Sellmeier expansions in [START_REF] Pezzagna | Refractive indices and elasto-optic coefficients of GaN studied by optical waveguiding[END_REF] with a particular profile of the refractive index of the guide [START_REF] Monerie | Propagation in doubly clad single-mode fibers[END_REF]. The triangles mark the configuration of perfectely matched pump and idler group velocities (|vp -vi| = 0), and are labeled P for the pump, S for the signal, and I for the idler.

The pump intensity required to reach threshold for BMOPO operation, I p,th , can be estimated from the plane-wave monochromatic model for the undepleted pump parametric approximation [START_REF] Ding | Backward optical parametric oscillators and amplifiers[END_REF] as

I p,th = cε 0 n p n f n b λ f λ b /2L 2 d 2 ef f
, where c is the speed of light in vacuum, ε 0 is the permittivity of free space, n j denotes the refractive indices, λ j denotes the backward and forward wavelengths, L is the nonlinear interaction length and d ef f = 7 pm/V is the effective quadratic nonlinearity.

Numerical simulations of the nonlinear three-wave interactions in a BMOPO solve the coupled wave equations in the slowly-varying-envelope approximation. The parameters used in the model are those of a periodically poled GaN waveguide (PPGaN) grown by epitaxy on sapphire substrates [START_REF] Chowdhury | Secondharmonic generation in periodically poled GaN[END_REF] and presenting submicronic periodicity of the χ (2) coefficient [START_REF] Pezzagna | Submicron periodic poling and chemical patterning of GaN[END_REF], which measure in [START_REF] Pezzagna | Refractive indices and elasto-optic coefficients of GaN studied by optical waveguiding[END_REF], yields

d ef f = (2/π)χ
(2) 33 = 7 pm/V. The field amplitudes, A j (j = p, s, i), of the pump and the parametric backward signal wave and forward idler wave, evolve with the coupled equations:

(∂ t + v p ∂ x + γ p + iβ p ∂ tt ) A p = -σ p (x)A s A i (∂ t -v s ∂ x + γ s + iβ s ∂ tt ) A s = σ s (x)A p A * i (1) (∂ t + v i ∂ x + γ i + iβ i ∂ tt ) A i = σ i (x)A p A *
s , where σ j (x) = 2πd ef f v j /λ j n j are the nonlinear coupling coefficients in the PP elements, while σ j (x) = 0 within the uniformly polarized gaps, γ j and β j ≡ v j β 2,j /2 are the damping and dispersion coefficients. The input parameters in the model are the properties of the nonlinear medium and the pump amplitude at the input face, A p (x = 0, t), generating outputs of A p (x = L, t), A i (x = L, t) and A s (x = 0, t). The nonlinear counterpropagation dynamics in the quadratic BMOPO in the presence of group-velocity dispersion, which introduces secondorder time derivatives, makes use of the numerical scheme which combines the trajectories method for the treatment of the nonlinear counterpropagatig threewave interaction with fast Fourier transformation (FFT) to account for the groupvelocity dispersion effects in the spectral domain [START_REF] Montes | Model for Coherence Transfer in a Backward Optical Parametric Oscillator[END_REF]. The nonlinear medium is a GaN structured ensemble composed by a train of N periodically-poled GaN elements of length ℓ disposed in a straight row as shown in Fig. 3.

Simulations for this segmented structure of length L seg = Nℓ + (N -1) εℓ , where εℓ is the short random length of the gaps linking the PPGaN bricks, will be compared to the case of a continuos medium of length L= Nℓ. Two segmented structures have been simulated: I) For L = 64 ℓ = 1.25 cm, each element has a length ℓ = 195 µm. The segmented BMOPO will have a length L seg = 1.32 cm with 63 random gaps εℓ = εℓ + δε ℓ where δε ℓ is random uniformly distributed in the interval [-∆ε ℓ , +∆ε ℓ ] with εℓ = 10.8 µm for the mean value and ∆ε ℓ / εℓ = 0.15. II) For L = 48 ℓ = 1.25 cm, each element has a length ℓ = 260 µm. The segmented BMOPO will have a length L seg = 1.30 cm with 47 random gaps εℓ = εℓ + δε ℓ where δε ℓ is random uniformly distributed in the interval [-∆ε ℓ , +∆ε ℓ ] with εℓ = 10.8 µm for the mean value and ∆ε ℓ / εℓ = 0.15. For both cases, the waveguide has a section S eff = 3.6 µm 2 .

We choose the three QPM waves marked on the Fig. 2, and we model BMOPO operation with a broadband stochastic pump pulse with a FWHM temporal length of 150 ps, and where the spectral FWHM width ∆ν p (0) centered at λ p = 1.06µm varies in a large spectral bandwidth range [1.79 THz -4.7 THz], with n p = 2.30856, v p /c = 0.41337, β 2,p = 0.286 ps 2 /m; and with λ s = 1.56242µm, n s = 2.26580, v s /c = 0.41872, β 2,s = 0.0647 ps 2 /m; λ i = 3.29791µm, n i = 2.12353, v i /c = 0.41337, β 2,i = -0.418 ps 2 /m. The QPM period as low as Λ G = 335 nm may be realized in GaN by epitaxy [START_REF] Pezzagna | Refractive indices and elasto-optic coefficients of GaN studied by optical waveguiding[END_REF].

The pump pulse is issued from a Gaussian stochastic process translationally invariant with zero mean A p (x = 0, t) and exponential autocorrelation function

A p (x = 0, t ′ + t)A * p (x = 0, t) = A 0 p 2 exp (-|t|/τ c )
, where τ c = 1/π∆ν p is the correlation time. Note that, while the pump phase fluctuations are absorbed by the idler wave thanks to the phase-locking mechanism, the generation of the signal still remains affected by the intensity fluctuations of the pump. In the presence of small group velocity difference between the pump and the signal, which is the case for the forward OPOs corresponding to Fig. 1(a), pump intensity fluctuations are partly transferred to the signal component [START_REF] Montes | Ultra-coherent output from an incoherent cw-pumped singly resonant optical parametric oscillator[END_REF][START_REF] Montes | Coherent signal from incoherently cw-pumped singly resonant Ti:LiNbO3 integrated optical parametric oscillators[END_REF]. Conversely, in the presence of the strong group velocity difference of the backward OPO, corresponding to configurations of Fig. 1(b)(c), intensity fluctuations of the pump are averaged by the strong convection effect, which thus leads to an enhancement of the coherence of the backward wave [START_REF] Strömqvist | Temporal coherence in mirrorless optical parametric oscillators[END_REF]. The higher the incoherence of the pump, the larger the amount of coherence transfer as studied in section V.B of Ref. [START_REF] Picozzi | Coherence properties of the parametric three-wave interaction driven from an incoherent[END_REF]. Stochastic BMOPO operation in both continuous and segmented structures are simulated with a pump pulse of I p = 2 GW/cm 2 intensity, the BMOPO threshold intensity being I p,th = 0.937 GW/cm 2 . The peak power yields 72 W, a pulsed energy of 10.8 nJ, and a fluence of 0.3 J/cm 2 which remains below the damage threshold for GaN [START_REF] Wang | Femtosecond pulsed laser-induced periodic surface structures on GaN/sapphire[END_REF][START_REF] Ozono | High-speed ablation etching of GaN semiconductor using femtosecond laser[END_REF]. The pump and the parametric spectra for the 64 times segmented BMOPO are illustrated in Fig 4, As we can see, the higher the incoherence of the pump (larger spectral bandwidth), the larger the amount of coherence transfer to the signal wave. The pump depletion or parametric downconversion efficiency versus initial pump bandwidth for the periodically polarized continuous waveguide and for both fragmented BMO-POs are shown in We can see that these efficiencies are of the same order for the three BMOPOs and tender to decrease when the initial pump bandwidth increases. The more interesting domain is detailed (with high discretization) in This rough discretization gives rise to the serrated curves, which are more evident in the narrower input bandwidth region. The fluctuations in the efficiency comes from the complex shape of the stochastic input spectrum, composed of several high amplitude peaks. It becomes difficult to associate a single input bandwidth parameter by fitting a smooth Gaussian profile. This difficulty was avoided for the deterministic pump broadening through pump chirping of Ref. [START_REF] Strömqvist | Temporal coherence in mirrorless optical parametric oscillators[END_REF], as we can remark in the more smoothed curve of Fig. 5. Nonintuitivelly, the stitched BMOPO is as efficient as the uniform quasi phase matched guided BMOPO. Its efficiency turns to be about that of the one piece periodically polarized PPGaN BMOPO, because the generated coherent phase of the backscattered wave locks the phases of the forward propagating waves in such a way that the dynamics is almost insensitive to the short uniformly polarized connection sections. The phase-locking behaviour inside the PPGaN bricks is illustrated in Fig 11. To avoid other nonlinearities, such as stimulated Raman scattering, which is the most harmful nonlinearity competing with the parametric downconversion [START_REF] Canalias | Mirrorless optical parametric oscillator[END_REF], we can further decrease the pump intensity. With I p = 1.5 GW/cm 2 and a fluence of 0.225 J/cm 2 , we obtain 15% pump depletion and a coherence gain ∆ν p (L)/∆ν s (0) = 173.

As expected from the convection-induced phase-locking mechanism, even for the segmented stochastic BMOPO, the stochastic phase modulation in the pump is essentially transferred to the forward idler, while the phase of the backward signal is almost constant. This is further illustrated in Fig. 11 where the phase distributions of the waves are plotted inside the interaction medium for the maximum nonlinear interaction at time t = 254 ps. The stochastic idler phase approximately follows the phase of the pump, whereas the signal phase is almost invariant throughout the BMOPO sample.

Summary

We have numerically simulated the dynamics in fragmented BMOPOs and realistic PPGaN waveguides by using stochastic pump pulses from 0.1 THz to 4.7 THz input bandwidth. It has been shown that mirrorless optical parametric oscillation occurs with reasonable pump power and that the coherence transfer ∆ν p (L)/∆ν s (0) may attain more than 330 in a 1.25 cm length waveguide. Parametric three-wave interaction is a reversible process and downconversion in one PP element could be destroyed in the adjacent one if the phase shift produced in the gap reverses the downconversion, as is the case for a standard forward OPO. Quite surprisingly in the backward BMOPO, the structure presenting stitching errors is almost as efficient as a perfect one, like that presented in [START_REF] Strömqvist | Temporal coherence in mirrorless optical parametric oscillators[END_REF], because the coherent phase of the backward wave locks the phases in such a way that the dynamics is almost insensitive to the random length of the uniformly polarized junctions.

Figure 1 :

 1 Figure 1: Wave vector diagrams (momentum conservation) for the non-degenerate three-wave interaction in an OPO : from top to bottom, (a) three wave forward configuration; (b) singly backward idler configuration; (c) singly backward signal configuration; and (d) doubly backward (signal and idler) configuration. As we can see the QPM grating show a decreasing phase-reversal period for the nonlinear susceptibilty represented by the bold broken lines under each configuration.

Figure 3 :

 3 Figure 3: Quasi-phase-matching (QPM) is obtained by a phase-reversal sequence of the ferroelectric domains with a grating periodicity ΛQP M . The segmented backward BMOPO is composed by a train of periodically poled GaN elements ℓ separated by uniformly polarized connection sections εℓ of random lenght representing stitching errors.

  Fig 5 and Fig 6, for three different starting pump bandwidths.

Figure 4 :

 4 Figure 4: BMOPO (segmented in 64 elements) for a stochastic pump input of FWHM ∆νp(0) = 3.64 THz : (a) Output incoherent pump spectrum fitted by a Gaussian envelope of FWHM ∆νp(L) = 5.06 THz, (b) backward signal spectrum with ∆νs(0) = 26.75 GHz, and (c) forward idler spectrum with a Gaussian fitting of FWHM ∆νi(L) = 3.06 THz. The coherence gain attains ∆νp/∆νs = 189 for a pump depletion or efficiency of 0.1945.

Fig 7 ,

 7 Fig 8, Fig 9, and Fig 10.

  Fig 7 in the broad

Figure 5 :

 5 Figure 5: BMOPO (segmented in 64 elements) for a stochastic pump input of FWHM ∆νp(0) = 4.20 THz : (a) Output incoherent pump spectrum fitted by a Gaussian envelope of FWHM ∆νp(L) = 5.35 THz, (b) backward signal spectrum with ∆νs(0) = 20.25 GHz, and (c) forward idler spectrum with a Gaussian fitting of FWHM ∆νi(L) = 4.08 THz. The coherence gain attains ∆νp/∆νs = 264, for a pump depletion or efficiency of 0.1133.

Figure 6 :

 6 Figure 6: BMOPO (segmented in 64 elements) for a stochastic pump input of FWHM ∆νp(0) = 4.28 THz : (a) Output incoherent pump spectrum fitted by a Gaussian envelope of FWHM ∆νp(L) = 5.35 THz, (b) backward signal spectrum with ∆νs(0) = 15.87 GHz, and (c) forward idler spectrum with a Gaussian fitting of FWHM ∆νi(L) = 4.06 THz. The coherence gain attains ∆νp/∆νs = 337 for a pump depletion or efficiency of 0.0843.

Figure 7 :

 7 Figure 7: Pump depletion 1 -Ip(L)/Ip(0) versus input pump spectral width, for both segmented BMOPOs (48 segments and 64 segments) and for the one piece periodically polarized PPGaN BMOPO.

Figure 8 :

 8 Figure 8: Pump depletion 1 -Ip(L)/Ip(0) versus input pump spectral width, for the segmented BMOPO of 64 segments L(64seg) compared to the one piece periodically polarized PPGaN BMOPO L(cont).

Figure 9 :

 9 Figure 9: Pump depletion 1 -Ip(L)/Ip(0) versus input pump spectral width, for the segmented BMOPO of 48 segments L(48seg) compared to the one piece periodically polarized PPGaN BMOPO L(cont).

Figure 10 :

 10 Figure 10: Pump depletion 1 -Ip(L)/Ip(0) versus input pump spectral width for the BMOPO of 64 segments in a large spectral bandwidth range [0.1 THz -4.7 THz].

Figure 11 :

 11 Figure 11: Phase distributions inside the stochastic segmented GaN BMOPO for the pump, the forward idler (locked to the pump), and the backward signal (almost constant), at the maximum of the interaction (at time t = 254 ps).


Raffaella Moretti

Pierre-Louis Leger

Valérie C Besson

Zsolt Csaba

Julien Pansiot

Lorena Di Criscio

Andrea Gentili

Luigi Titomanlio

Philippe Bonnin

Olivier Baud

Christiane Charriaut-Marlangue
email: christiane.marlangue@gmail.com

microglial modulation after focal ischemia in the neonatal mouse brain

Keywords: Neonatal mice, Focal ischemia, Microglia, M2 microglia, Microcirculation

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

Background

Perinatal/neonatal arterial stroke is a cerebrovascular event occurring around the time of birth, with pathological or radiological evidence of focal arterial infarction mainly affecting the middle cerebral arterial territory, with an incidence of 1/2800 to 1/5000 live births. There is currently no evidence-based treatment for neonates with stroke [START_REF] Chabrier | New insights (and new interrogations) in perinatal arterial ischemic stroke[END_REF].

Decreased regional cerebral blood flow (CBF) is the principal factor determining the topography of tissue injury after cerebral ischemia and/or hypoxia-ischemia (HI) in the mature and immature rodents [START_REF] Charriaut-Marlangue | Brief update on hemodynamic responses in animal models of neonatal stroke and hypoxiaischemia[END_REF]. Partially maintained blood flow in the penumbral tissue after stroke is primarily due to the recruitment of collateral circulation in the brain. Collateral recruitment involves nitric oxide (NO)-dependent vasodilatation, and the effects of endogenous NO may be enhanced by preventing cGMP degradation by phosphodiesterases (PDEs). In particular, sildenafil and tadalafil, two potent selective PDE-5 inhibitors, prolong the action of cGMP in multiple vascular territories [START_REF] Wallis | Tissue distribution of phosphodiesterase families and the effects of sildenafil on tissue cyclic nucleotides, platelet function, and the contractile responses of trabeculae carneae and aortic rings in vitro[END_REF]. Interestingly, sildenafil treatment, by increasing CBF, reduced HI damage in the P7 rat brain [START_REF] Charriaut-Marlangue | Sildenafil mediates blood-flow redistribution and neuroprotection after neonatal hypoxia-ischemia[END_REF]. Sildenafil has been reported to promote neurorestoration in rat models of stroke as measured by neurogenesis, synaptogenesis, and angiogenesis [START_REF] Bednar | The role of sildenafil in the treatment of stroke[END_REF] and to reduce proliferation of astrocytes and microglia in chronic inflammatory diseases [START_REF] Nunes | Sildenafil (Viagra(R)) down regulates cytokines and prevents demyelination in a cuprizone-induced MS mouse model[END_REF].

While microglia are considered as conductors of the inflammatory response in the central nervous system (CNS), their contribution to the progression of ischemic stroke remains debated [START_REF] Iadecola | The immunology of stroke: from mechanisms to translation[END_REF]. Microglia/macrophages may transition between inflammatory and healing phenotypes in response to specific activation cues (review in [START_REF] Murray | Macrophage activation and polarization: nomenclature and experimental guidelines[END_REF]). While the M1 or classic phenotype is neuroinflammatory and neurotoxic, the M2 phenotype supports the recovery or neural tissue and is characterized by the production of anti-inflammatory mediators and neurotrophic factors [START_REF] Taylor | Microglial responses after ischemic stroke and intracerebral hemorrhage[END_REF]. Although the action of sildenafil on neuroinflammation is not fully understood, the cGMP-PDE inhibitor zaprinast was shown to significantly reduce the inflammatory response of astrocytes and microglia/macrophages in adult rats submitted to focal traumatic brain injury [START_REF] Pifarre | Cyclic GMP phosphodiesterase inhibition alters the glial inflammatory response, reduces oxidative stress and cell death and increases angiogenesis following focal brain injury[END_REF].

Revealing molecular mechanisms and pharmacological treatments that promote the transition of microglia/ macrophages towards a phenotype that facilitates the repair of the neonatal brain after ischemic damage is of therapeutic interest. The goal of this study is to examine if sildenafil citrate (1) mediates CBF and reduces lesion damage and (2) modifies the modulation of microglia/ macrophage between M1 and M2 phenotypes in P9 mice subjected to ischemia after permanent middle cerebral artery occlusion (pMCAo).

Methods

Ethics statement

All experiments were carried out (license A75-19-01, French Department of Agriculture) in accordance with the European Committee's Council Directive and performed to comply with the ethical guidelines of the Robert Debre Hospital Research Council Review Board and have been approved by the local ethic committee (Paris7, France).

Ischemia

pMCAo was carried out under isoflurane anesthesia in 30 % O 2 and 70 % N 2 O in P9 C57Bl/6 mouse (Janvier, Le Genest-St. Isle, France; 4.6 ± 0.6 g, n = 148). Pups were sacrificed at 72 h and/or 8 days after pMCAo. Two investigators, both blinded to the treatment group, determined the size of the lesion in each animal [START_REF] Charriaut-Marlangue | Inhaled nitric oxide reduces brain damage by collateral recruitment in a neonatal stroke model[END_REF].

Drug treatment

In the first set of experiments, animals were randomly divided into five groups and treated with either PBS or a single dose of sildenafil citrate (0.5, 2.5, 10, and 15 mg/kg; Pfizer, France), given intraperitoneally (i.p.) 5 min after pMCAo. In the second set of experiments, animals were randomly divided into three groups and treated with either PBS or a single dose of sildenafil citrate (0.5 and 10 mg/kg, i.p.) 5 min after pMCAo (see Additional file 1: Figure S1 for an outline of the experimental procedure).

cGMP measurement

Competitive enzyme immunoassay (Cayman Chemical Company, Ann Arbor, MI, USA) was used to quantify cGMP in the forebrain, according to the manufacturer's instructions. Whole brains at P9 were harvested 1 and 3 h after the administration of sildenafil (0.5 and/or 10 mg/kg) and immediately frozen at -80 °C until measurements were performed.

Ultrasonographic brain imaging

Thermoregulated mice (n = 6 per group) were subjected to ultrasound measurements under inhaled isoflurane anesthesia (0.8 % in air via a facemask) using an echograph (ACUSON S3000, Siemens, Erlangen, Germany) equipped with a 14.5-MHz linear transducer (14L5 SP) [START_REF] Leger | Early recruitment of cerebral microcirculation by neuronal nitric oxide synthase inhibition in a juvenile ischemic rat model[END_REF]. Heart rate and time-average mean blood flow velocities (mBFVs) were measured in both intracranial carotid arteries (ICA) and the basilar trunk (BT) at baseline and 1 h after pMCAo and PBS and sildenafil (10 mg/kg) treatment.

Laser speckle contrast (LSC) imaging

Imaging was carried out using a full-field laser perfusion imager (FLPI2, Moor Instruments Ltd., Axminster, UK) [START_REF] Leger | Early recruitment of cerebral microcirculation by neuronal nitric oxide synthase inhibition in a juvenile ischemic rat model[END_REF]. Briefly, thermoregulated and anesthetized (with 1 % isoflurane inhalation in air via a facemask) mouse pups (n = 6 per group) were placed in lateral decubitus and the skin incised to provide access to the skull. Vegetable oil was applied to avoid surface dryness. Speckle images (760 × 568 pixels) were collected at 0.25 Hz (4 ms exposure time) and recorded at baseline, after MCA electrocoagulation and 5, 15, 60, and 90 min after treatment. Blood fluxes were measured in three cortical regions of interest (ROI, 50 × 50 pixels), ROI 1 present in the core of the infarct and ROI 2 and ROI 3 in the proximal and distal penumbra, respectively. Fluxes in the ROIs were expressed in arbitrary units using a 16-color palette.

Real-time quantitative reverse transcriptase polymerase chain reaction (qRT-PCR)

Animals were sacrificed at 72 h and 8 days after pMCAo (n = 8 per group). The harvested tissues correspond to the parietal cortex, including the ischemic core (pale zone) and penumbra at 72 h, and the penumbra at 8 days as the pale zone being a cavity. Total RNA was extracted using RNeasy lipid tissue mini kit (Qiagen, Courtaboeuf, France). RNA purity and quality were assessed by spectrophotometry with the NanoDrop™ apparatus (Thermo Scientific, Wilmington, DE, USA). Total RNA (1-2 μg) was subjected to reverse transcription using the iScript™ cDNA synthesis kit (Bio-Rad, Marnes-la-Coquette, France). RT-qPCR was performed in duplicates for each sample using SYBR Green Supermix (Bio-Rad) for 40 cycles with a 2-step program (5 s of denaturation at 96 °C and 10 s of annealing at 60 °C). Amplification specificity was assessed by melting curve analysis. Primers were designed using Primer3 software. Sequences and NCBI references are provided in Additional file 2: Table S1. Abundance of transcripts of interest was expressed relative to the expression of the reference gene, glyceraldehyde 3-phosphate dehydrogenase (GAPDH). Analyses were performed with the Bio-Rad CFX Manager 2.1 software.

Immunohistochemistry

Coronal 16-μm thick paraffin sections at the MCA and hippocampal level (corresponding to 3.27 and 4.83 mm-atlas of the developing mouse brain at P6) were stained for mouse anti-GFAP (Sigma-Aldrich G3893; 1:500 dilution), goat anti-Iba1 (Abcam ab5076; 1:500), rabbit anti-Iba1 (Wako, 019-19741; 1:1000), goat anti-MRC1 (MMR/CD206 polyclonal Goat IgG, R&D system, AF2535; 1:400), rabbit anti-COX-2 (Abcam; 1:200), goat anti-arginase-1 (Santa Cruz Biotechnology, Inc., sc-18354; 1:200), rabbit anti-Glut1 (54-KDa form, Millipore, Molsheim, France; 1:700), and tomato lectin (Vector Laboratories, 1:500). For immunofluorescence, secondary antibodies coupled with the green marker Fluoroprobe S488 (Interchim, Montluçon, France) or the red fluorescent marker cyanine 3 (Jackson Immuno Research laboratories, West grove, PA) were used. For confocal imaging, sections were analyzed using a Leica TCS SP8 confocal scanning system (Leica Microsystems, Wetzlar, Germany) equipped with 405-nm diode, 488-nm argon, and 561-nm DPSS lasers. A series of optical sections separated by 0.3 μm was collected using the ×63 HC PL APO CS2 objective (numerical aperture 1.40). For each optical section, double-or triple-immunofluorescence images were acquired in sequential mode to avoid potential contamination by fluorescence emission overlap. The orthogonal sectioning and 3D viewing were produced by the Leica Application Suite X-software (Leica Microsystems). Images were equally adjusted for brightness and contrast, and composite illustrations were built in Adobe Photoshop CS3 (Adobe Systems, San Jose, CA).

Cell quantification

GFAP and Iba-1-positive cells were counted in a blind manner in the penumbra region on three coronal sections for animal at 72 h and 8 days after ischemia and in age-matched controls using a ×20 objective. Iba-1-Arg-1 and Iba1-COX-2 double-positive cells were counted in the same regions with a ×20 lens and then merged images were created with Fiji software (http://fiji.sc/Fiji).

Statistical analysis

Values are expressed as mean ± SD (BF measures and lesion volume) and mean ± SEM (RT-qPCR, immunohistochemistry, and immunofluorescence). BFV and CBF values measured in the different ROIs were compared using repeated measures ANOVA and a post hoc Newman-Keuls test to analyze differences between the two groups. One-way ANOVA followed by a Newman-Keuls post hoc test was used for the difference between two groups (RT-qPCR, immuno-labeled cells, and lesion volumes). Statistical analyses were performed using GraphPad PRISM version 5.0 (GraphPad Software, San Diego, CA).

Results

Setup of the animal model and effect of sildenafil on CBF and lesion size

In the first set of experiments (Additional file 1: Figure S1), animals were assessed by color-coded pulsed Doppler ultrasound imaging at baseline and after pMCAo. As compared to basal mBFV, pMCAo induced a significant transient decrease in mBFV in the left ICA and a consecutive significant increase in mBFV in the basilar trunk (BT) 15 min after pMCAo (Additional file 1: Figure S2A). One hour after pMCAo, no significant change in mBFV was observed in the three large arteries (data not shown). One hour after pMCAo and sildenafil administration (10 mg/kg, n = 8 each) mBFVs were not modified in the three large arteries as compared to a single PBS administration (Fig. 1a). Heart rates in the sildenafil-treated animals (395 ± 67 bpm) were not significantly different from PBS-treated animals (428 ± 66 bpm) at this time point. No difference was observed between the different doses of sildenafil on mBFVs measured in the large arteries (data not shown), except for the 15 mg/ kg dose (Additional file 1: Figure S2B).

Representative patterns and values of cortical BF using LSC imaging are shown in Fig. 1b,c. Immediately after pMCAo, BF dropped in the ROI 1 (core) to a mean value around 40-42 % of basal BF in both groups of animals (n = 6 each) and was reduced further to 33-36 % of basal at 90 min after pMCAo. A similar drop in BF was measured in the ROI 2 and in a less severe manner in the ROI 3. BF was further reduced at 90 min after pMCAo, in particular in ROI 3. As expected, sildenafil treatment (10 mg/kg) significantly increased the concentration of intracellular cGMP to 1.91 ± 0.38 and 1.87 ± 0.31 pmol/mL at 1 and 3 h after administration, respectively, as compared to PBS-treated animals (0.08 ± 0.02 pmol/mL, p < 0.001) (Additional file 1: Figure S3). No statistical difference in cGMP concentration was found between the different doses used. Nevertheless, sildenafil treatment did not significantly modify the BF profile in the three distinct ROIs at any time point.

Mean cortical infarct volume at 72 h after pMCAo was 12.5 ± 3.0 % (n = 14, one animal died) in PBS-treated animals. Sildenafil at the three lower doses tested (0.5, 2.5, and 10 mg/kg) did not change the mean infarct volume (11.0 ± 3.0 %, n = 7; 12.2 ± 3.1 %, n = 8; and 13.4 ± 2.3 %, n = 7, respectively, N.S.), whereas it increased both infarct volume (21.6 ± 4.6 %, n = 7, p < 0.001, Fig. 2a) and mortality (3/10 animals died) at 15 mg/kg, suggesting toxicity at high doses (Fig. 2a). Therefore, the 15-mg/kg dose was not further used in the following experiments. Eight days after pMCAo, lesion volume significantly increased up to 23.8 ± 7.5 % (n = 16, p < 0.01 vs 72 h) in PBS-treated animals and became cystic. Sildenafil significantly counteracted this increase in a dose-dependent manner to 16.4 ± 4.8 % (sildenafil 0.5 mg/kg; n = 13; p < 0.01) and 11.0 ± 4.8 % (sildenafil 10 mg/kg; n = 13; p < 0.001) (Fig. 2b).

Effect of sildenafil treatment on glial activation

Reactive astrogliosis manifested by a significant increase in GFAP immunoreactivity was clearly observed around the lesion in PBS-treated animals at 72 h after pMCAo (Additional file 1: Figure S4 and Additional file 1: Figure S5A,B). Surprisingly, sildenafil significantly increased the prevalence of GFAP + cells at 10 mg/kg. Conversely, sildenafil significantly reduced the number of GFAP + cells 8 days after pMCAo as compared to PBS-treated animals. No significant difference was observed between PBS and 0.5 mg/kg sildenafil treatment at either time point (Additional file 1: Figure S4).

A high abundance of microglia/macrophages (stained by tomato lectin (TL)) was detected in the penumbra (including the white matter) at 72 h. Sildenafil treatment significantly decreased the number of TL + -cells at 10 but not 0.5 mg/kg (Fig. 3a and Additional file 1: Figure S5C,D). At this time point, cells positive for the M1-like marker COX-2 + [START_REF] Chhor | Characterization of phenotype markers and neuronotoxic potential of polarised primary microglia in vitro[END_REF] were found in the ischemic core in PBStreated animals, whereas they were mostly observed in the penumbra in 10 mg/kg (but not 0.5 mg/kg) sildenafiltreated animals (Fig. 3b-d). In contrast, 8 days after pMCAo the number of microglia/macrophages stained by Iba-1 were significantly reduced by sildenafil treatment (0.5 and/or 10 mg/kg dose) (Fig. 6b).

We then focused on the phenotype of microglial cells at both transcriptional (Figs. 4 and5) and translational (Figs. 6 and7) levels in the cortical tissue obtained from PBS-and sildenafil-treated animals. Seventy-two hours after pMCAo and PBS treatments, most of the markers (M1-and M2-like) were increased with the exception of ptgs2 (COX-2). As compared to PBS, sildenafil (10 mg/kg) did not change the expression of M1-like (ptgs2, CD32 and CD86) markers 72 h after ischemia. In contrast, a significant increase was measured for CD32 and CD86 8 days after pMCAo (Fig. 4). At 72 h after pMCAo, sildenafil treatment (10 mg/kg) significantly decreased mRNA expression of M2-like markers (CD206, IL1-Rn, and IL4-Ra) but not Arg-1 and Lgals3. In contrast, most of these markers (CD206, Arg-1, Lgals3 and IL1-Rn) were increased 8 days after pMCAo (Fig. 5).

To further address the effect of sildenafil treatment on M1 and M2 markers, we examined the co-localization of Iba-1 + (a marker for both M1 and M2 microglia/macrophages) with either the M2-like marker Arg-1 (Fig. 6a, upper panel) or the M1-like marker COX-2 (Fig. 6a, lower panel) in the cortical penumbra. While a global decrease of Iba-1 + microglia/macrophage was observed by sildenafil irrespective of dosage (Fig. 6b), an increase of Iba-1 + Arg-1 + double-positive M2 microglia/macrophage was observed at 10 but not 0.5 mg/kg (Fig. 6c), suggesting that a high dosage of sildenafil was required for microglia/ macrophages to obtain the M2 status. Interestingly, sildenafil decreased the number of Iba-1 + COX-2 + double-positive M1 microglia/macrophages irrespective of dosage (Fig. 6d), suggesting it does not only affect phenotype polarization and that a lower dose threshold is required for inhibition of neurotoxic M1-type than for activation of neuroprotective M2-type microglia/ macrophages.

Co-expression of M1-and M2-like markers in perivascular microglia/macrophages

The macrophage mannose receptor MRC-1 (CD206), a marker for M2-type microglia/macrophages, was mainly detected in the leptomeninges from where they entered the neocortex along the penetrating arterioles in the penumbral tissue at 72 h (Fig. 7) and 8 days after pMCAo (data not shown). On confocal images, we observed the co-localization of MRC-1 with the M1 marker COX-2 (Fig. 7a) indicating either a mixed phenotype or lack of specificity of this marker. MRC-1 expression was present in macrophages/microglia localized around microvessels labeled with the endothelial marker Glut-1 (stained with 54 kDa Glut-1 protein, which regulates import of glucose from the blood to the brain across the endothelial cells of the blood-brain barrier) and present in the penumbra (Fig. 7b and Additional file 1: Figure S6). A few COX-2 + /Arg-1 + -cells (2-4 cells per brain section) were still detected in the leptomeninges in both PBS-(Fig. 7c) and sildenafil-(10 mg/kg, data not shown) treated animals 8 days after pMCAo.

Discussion

In this study, we show that brain lesions induced by pMCAo in P9 mouse pups evolve between 72 h and 8 days post-ischemia and that the selective inhibition of PDE-5 by sildenafil dose-dependently reduces lesion extent at 8 days post-pMCAo. The protective action of sildenafil appeared to involve a reduction in the overall number of microglia/macrophages in the late phase of lesion development as well as their polarization towards a neuroprotective M2 phenotype.

Whereas lesion volumes decrease after pMCAo in adult mice between 24 h and 7 days [START_REF] Poittevin | Diabetic microangiopathy: impact of impaired cerebral vasoreactivity and delayed angiogenesis after permanent middle cerebral artery occlusion on stroke damage and cerebral repair in mice[END_REF], they increased between 3 and 8 days in neonatal mice, as previously shown in neonatal rat [START_REF] Renolleau | A model of transient unilateral focal ischemia with reperfusion in the P7 neonatal rat: morphological changes indicative of apoptosis[END_REF]. Whether this difference reflects upon continued tissue destruction even in late phases after occlusion or reduced growth potential of the neonatal brain is yet unclear. Sildenafil treatment reduced lesion size at 8 days post-pMCAO, similar to that found at 72 h, suggesting that sildenafil prevents lesion enlargement. We previously reported that sildenafil treatment, associated with an early significant increase in cerebral blood flow (CBF), reduces hypoxic-ischemic damage in the P7 rat brain [START_REF] Charriaut-Marlangue | Sildenafil mediates blood-flow redistribution and neuroprotection after neonatal hypoxia-ischemia[END_REF]. By using both US and LSC imaging, we were here unable to observe early hemodynamic changes in either sildenafil-or PBStreated mice. A high variability in the number of leptomeningeal and/or pial vessels may explain the hemodynamic differences observed between the rat and mouse. Indeed, all ischemic models in the rat combine transient occlusion of two arteries at the same time, either pMCAo with transient occlusion of one or both common carotid arteries [START_REF] Charriaut-Marlangue | Brief update on hemodynamic responses in animal models of neonatal stroke and hypoxiaischemia[END_REF][START_REF] Bonnin | Impact of intracranial blood-flow redistribution on stroke size during ischemiareperfusion in 7-day-old rats[END_REF] or blockade of the past external carotid artery-internal carotid artery bifurcation (See figure on previous page.) Fig. 5 Gene expression of microglia/macrophage M2-like markers in PBS-and sildenafil (Sil)-treated animals, 72 h and 8 days after pMCAo. M2 markers were measured in the ipsilateral (IL) and contralateral (CL) side in PBS-(plain bars) and sil-10 (hatched bars) animals. Data are mean ± SEM. Data were assessed via an ANOVA, and when significant, the results of the Newman-Keuls post-test are shown; *p < 0.05, **p < 0.01, and ***p < 0.001, compared to PBS [START_REF] Ashwal | L-NAME reduces infarct volume in a filament model of transient middle cerebral artery occlusion in the rat pup[END_REF][START_REF] Derugin | Neonatal reversible focal cerebral ischemia: a new model[END_REF]. In contrast, a single pMCAo is sufficient to create ischemic lesions in mice [START_REF] Tsuji | A novel reproducible model of neonatal stroke in mice: comparison with a hypoxiaischemia model[END_REF]. Together, this suggests that combined occlusion of carotid and MC arteries is necessary to drop the BF in the ipsilateral hemisphere sufficiently for anastomoses to no longer be efficient despite the circle of Willis and to create a lesion in the rat. As BF was only measured up to 90 min after pMCAo, we cannot exclude the contribution of late collateral opening and patency. Together, our data suggest that sildenafil, although given shortly after pMCAo, may have other neuroprotective effects that may develop at later time points of recovery.

The impact of inflammation on post-injury outcome in the developing brain is increasingly recognized. In our pMCAo model in the P9 mouse, strong astrocytic and microglial responses were detected 72 h after injury and persisted for more than 1 week (personal data). We show that sildenafil potentiates reactive astrogliosis 72 h after pMCAo, in agreement with a previous report following focal cryolesions onto the cortex in adult rats treated with zaprinast and killed 3 days thereafter [START_REF] Taylor | Microglial responses after ischemic stroke and intracerebral hemorrhage[END_REF]. Zaprinast was expected to accelerate the formation of the glial scar and the regeneration of the injured tissue [START_REF] Pifarre | Cyclic GMP phosphodiesterase inhibition alters the glial inflammatory response, reduces oxidative stress and cell death and increases angiogenesis following focal brain injury[END_REF]. Sildenafil similarly increased GFAP immunoreactivity in adult mice submitted to a cortical cryolesions [START_REF] Prado | Metallothioneins I/II are involved in the neuroprotective effect of sildenafil in focal brain injury[END_REF]. After brain injury, NO-dependent cGMP formation can occur in the astrocytes following NO synthase-2 induction in activated astrocytes and infiltrating macrophages [START_REF] Brahmachari | Induction of glial fibrillary acidic protein expression in astrocytes by nitric oxide[END_REF][START_REF] Murphy | Production of nitric oxide by glial cells: regulation and potential roles in the CNS[END_REF]. Natriuretic peptides (NPs) comprising atrial (ANP), brain NP (BNP), and C-type (CNP) share the same intracellular signal transduction pathways with cGMP/cGMP-dependent protein kinase (cGK) as well as the NO pathway. The NPs/NO/cGMP/cGK pathway has been reported to increase BF and to be critical for neovascularization in vivo [START_REF] Yamahara | Significance and therapeutic potential of the natriuretic peptides/cGMP/ cGMP-dependent protein kinase pathway in vascular regeneration[END_REF]. Interestingly, the content of ANP has been reported to increase in reactive astrocytes 3 days after injury in the white matter (WM) from autopsied human brain specimens after brain infarction and upregulation of the ANP/cGMP pathway (ANP is a vasorelaxant) may contribute to an increase in BF in response to the BF reduction in the infarcted area [START_REF] Nogami | Immunohistochemistry of atrial natriuretic peptide in brain infarction[END_REF]. The increase of GFAP + cells observed here 3 days after pMCAo might be related to an increase in ANP and subsequent BF that we did not evaluate and could further explain the delayed reduction in astrogliosis observed at 8 days after pMCAo.

Another important finding of this study is that sildenafil reduces the recruitment and activation of microglia/ macrophages in the penumbral tissue at both 72 h and 8 days after pMCAo, as found for zaprinast in the cortical cryolesion model [START_REF] Taylor | Microglial responses after ischemic stroke and intracerebral hemorrhage[END_REF]. Microglial activation may be beneficial or harmful depending on polarization towards a pro-inflammatory M1 or an anti-inflammatory and pro-healing M2 phenotype [START_REF] Fumagalli | The ischemic environment drives microglia and macrophage function[END_REF]. Information regarding the expression of M1 and M2 markers and their temporal and spatial evolution in the ischemic brain are largely lacking. Changes in mRNA expression of M1 and M2 markers were maximally increased between 3-5 and 14 days after transient MCAo in the adult C57BL/6 mice [START_REF] Hu | Microglia/macrophage polarization dynamics reveal novel mechanism of injury expansion after focal cerebral ischemia[END_REF].

In our neonatal mouse model, we observed an impact of sildenafil treatment on macrophage/microglia activation and the expression of M1-and M2-like markers. Whereas sildenafil only increased M1-like genes 8 days after ischemia, it decreased M2-like genes expression at the same time point. Cyclooxygenases (COX-1 and COX-2), which produce vasodilatory prostaglandins [START_REF] Funk | Prostaglandins and leukotrienes: advances in eicosanoid biology[END_REF], contribute to neurovascular coupling [START_REF] Stefanovic | Modulatory role of cyclooxygenase-2 in cerebrovascular coupling[END_REF] and could contribute to vasodilation in the penumbra and a subsequent absence of lesion extension in animals treated with sildenafil. Furthermore, cerebrovascular dilations to hypercapnia were demonstrated to be prostanoid dependent and nitric oxide independent in the newborn pig brain [START_REF] Leffler | Prostanoids and pial arteriolar diameter in hypotensive newborn pigs[END_REF][START_REF] Willis | NO and prostanoids: age dependence of hypercapniaand histamine-induced dilations of pig pial arterioles[END_REF]. The combined increase of potential vasodilatory mediators in astrocytes and COX-2 + cells in the penumbra at 72 h could, therefore, induce patency and/or recruitment of collaterals leading to a reduced damage 5 days after (at P17).

A significant upregulation of M2-like mRNA was detected after sildenafil treatment. In the adult rodent, M1 markers (iNOS, CD11b, CD16, and CD32) were still elevated at 14 days after stroke, while M2 markers (CD206, Arg-1, Ym1/2 and IL-10) were decreased at 7 days after ischemia [START_REF] Xia | Selective modulation of microglia polarization to M2 phenotype for stroke treatment[END_REF]. The M2 phenotype has a stronger capacity to elicit phagocytosis of dead neurons to avoid secondary inflammatory response and promote tissue regeneration, probably by the ANP/cGMP/cKG pathway [START_REF] Miro-Mur | Immature monocytes recruited to the ischemic mouse brain differentiate into macrophages with features of alternative activation[END_REF]. We suggest a shift in polarization towards the M2 phenotype may contribute to the ability of sildenafil to prevent the late-stage extension of lesions in our model. We also found MRC-1 (CD-206) protein in infiltrating macrophages (in the leptomeninges), as previously reported after hypoxia-ischemia in the P7 rat [START_REF] Bonestroo | Cerebral and hepatic inflammatory response after neonatal hypoxiaischemia in newborn rats[END_REF]; however, we further demonstrate that MRC-1 + /CD206 perivascular microglia/macrophages entered the brain via penetrating microvessels. In contrast CD-206 + cells were exclusively located in the ischemic core in adult mice 24 h after pMCAo [START_REF] Perego | Temporal pattern of expression and colocalization of microglia/macrophage phenotype markers following brain ischemic injury in mice[END_REF]. Interestingly, mixed transitional (Mtran) phenotype has been introduced to characterize double-labeled M1-M2 macrophage/microglia following traumatic brain injury [START_REF] Kumar | Microglia/ macrophage polarization dynamics following traumatic brain injury[END_REF]. Such MRC-1 + /COX-2 + and Arg-1 + /COX-2 + macrophage/microglia in the leptomeninges may also represent a transitional Mtran phenotype in our ischemic model.

Conclusions

In conclusion, our study suggests that sildenafil may prevent the progression of the ischemic lesions in neonatal mice through the modulation of inflammatory pathways including microglial/macrophage polarization. Further investigation is needed to elucidate mechanisms that govern the change in microglia/macrophage phenotype and the regulation of lesion size in the neonatal rodent brain. As the use of sildenafil is indicated for idiopathic pulmonary hypertension [START_REF] Fraisse | Acute pulmonary hypertension in infants and children: cGMP-related drugs[END_REF], it could be explored for immunomodulatory stroke treatment in children.

Fig. 1 Fig. 2

 12 Fig. 1 Redistribution of blood flow after permanent MCAo (pMCAo) in C57BL/6 P9 mice using ultrasound Doppler and laser speckle contrast (LSC) imaging. a Variation in mean blood flow velocity (mBFV) in the large cerebral arteries (lICA left ICA, rICA right ICA, BT basilar trunk) 1 h after pMCAo and administration of PBS and/or sildenafil (10 mg/kg; n = 6 per group). b, c Representative color-coded LSC images (b) and time-course of cortical blood flow (c) taken in basal, upon MCAo, shortly after sildenafil treatment (Tr, PBS, and/or sildenafil, Sil, 10 mg/kg), and after 5, 15, 60, and 90 min in three ROIs (ROI 1, core; ROI 2 and ROI 3 close and far away from the penumbra, respectively). BF is expressed in arbitrary units (A.U.) according to the color-coded scale (0 to 1500)

Fig. 4

 4 Fig. 4 Gene expression of microglia/macrophage M1-like markers in PBS-and sildenafil (Sil)-treated animals, 72 h and 8 days after pMCAo. M1 markers were measured in the ipsilateral (IL) and contralateral (CL) sides in PBS (plain bars) and sil-10 (hatched bars) animals. Data are mean ± SEM. Data were assessed via an ANOVA, and when significant, the results of the Newman-Keuls post-test are shown; *p < 0.05, and ***p < 0.001, compared to PBS

Fig. 6

 6 Fig. 6 Effect of sildenafil on the number of Iba-1-positive microglia and M2 and M1 phenotypes 8 days after pMCAo. a Typical example of double fluorescence labeling in the cortical penumbra for Iba-1 and Arg-1 and Iba-1 and COX-2 cells in a PBS-treated animal. Note that almost Iba-1 + cells are co-stained with COX-2 marker (white arrows), which is not the case for double-stained Iba-1 + -Arg-1 + cells. b-d Quantification of the number of Iba-1 + , Iba-1 + -COX-2 + , and Iba-1 + -Arg-1 + microglia cells in PBS-treated and sildenafil-treated (0.5 and 10 mg/kg) animals. Data are mean ± SEM. Data were assessed via an ANOVA, and when significant, the results of the Newman-Keuls post-test are shown. *p < 0.05, **p < 0.01, sildenafil vs PBS treatment

Fig. 7

 7 Fig. 7 3D confocal analysis of double-labeled microglia and vessels after pMCAo and PBS and/or sildenafil (0.5 mg/kg) treatment. a Co-localization of M1 (COX-2 in red) and M2 (MRC-1 in green) markers in the leptomeningeal membranes, 72 h after ischemia. b MRC-1 (in red) is present in the perivascular microglia (in green). c Co-localization of M1 (COX-2 in green) and M2 (Arg-1 in red) markers in the leptomeningeal membranes 8 days after pMCAo, in a PBS-treated animal. Scale bar represents 10 μm (a-c)

Raffaella Moretti, Pierre-Louis Leger, Valérie C. Besson, Zsolt Csaba, Julien Pansiot, et al.. Sildenafil, a cyclic GMP phosphodiesterase inhibitor, induces microglial modulation after focal ischemia in the neonatal mouse brain. Journal of Neuroinflammation, 2016, 10.1186/s12974-016-0560-4. inserm-

Acknowledgements

All the authors are indebted to Dr. E. Camerer (INSERM U970) for critical revision and editing this manuscript.

Additional files

Additional file 1: Figures S1-S6. Figure S1. Outline of the experimental procedure in P9 C57Bl/6 mice subjected to pMCAo. S1. Primer sequences, protein targets, and NCBI references. (PDF 306 kb)

Competing interests

The authors declare that they have no competing interests.

Authors' contributions RM, CCM, and OB conceived and designed the study. RM, PLL, and VB conducted the experimental protocols. RM, LD, AG, and JP performed the histology, histochemistry, and qPCR. ZC performed the confocal analysis. RM, LD, and PB performed the ultrasound imaging. RM and PLL performed the laser speckle contrast imaging. RM, PLL, VB, PB, OB, and CCM analyzed and interpreted data. CCM, RM, and OB wrote the manuscript with input from all authors. All authors read and approved the final manuscript.

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