Cesare Tronci

Emanuele Tassi

Philip J Morrison

Energy-Casimir stability of hybrid Vlasov-MHD models

published or not. The documents may come

Introduction

Several plasma systems comprise an energetic species interacting with a fluid bulk usually described by magnetohydrodynamics (MHD). For example, in fusion plasmas, the fusion reactions produce energetic alpha particles that escape the usual fluid closures and require a kinetic theory description. Another example is provided by the interaction of Earth's magnetosphere with the energetic solar wind.

The interaction of an energetic component with a MHD plasma can be described by hybrid kinetic-fluid models that couple the energetic particle kinetics with the MHD equations. Although nonlinear hybrid kinetic-fluid models already appeared in the 70's [START_REF] Freidberg | Vlasov-fluid model for studying gross stability of high-β plasmas[END_REF], the very first nonlinear hybrid MHD models were proposed in the early 90's [START_REF] Cheng | A kinetic-magnetohydrodynamic model for low-frequency phenomena[END_REF][START_REF] Fu | Recent progress in linear and nonlinear studies of toroidal Alfvén eigenmodes[END_REF][START_REF] Fu | Nonlinear hybrid simulation of the toroidicity-induced Alfvén eigenmode[END_REF][START_REF] Park | Three-dimensional hybrid gyrokinetic-magnetohydrodynamics simulation[END_REF], when it was recognized that two different kinetic-MHD coupling schemes can be consistently formulated. When the coupling occurs by Lorentz force terms in the MHD fluid equation, the hybrid model is known as the current-coupling scheme (CCS); on the other hand, when the coupling occurs by the kinetic pressure tensor, the corresponding model is referred to as the pressurecoupling scheme (PCS). Moreover, each scheme may involve different possible variants to describe the energetic particle kinetics: drift-kinetic, gyro-kinetic and full Vlasov-type equations. While the first two are most used in current computer simulations [START_REF] Kim | Hybrid kinetic-MHD simulations in general geometry[END_REF][START_REF] Park | Plasma simulation studies using multilevel physics models[END_REF][START_REF] Takahashi | Kinetic effects of energetic particles on resistive MHD stability[END_REF], the simple mathematical form of the Vlasov equations can be advantageous in analytical studies such as that presented in this work. For an example of how the Vlasov equation is employed in hybrid MHD simulations, see [START_REF] Kim | Preliminary simulations of FLR effects on RFP tearing modes[END_REF].

While energy conservation for the CCS was first verified in [START_REF] Todo | Magnetohydrodynamic Vlasov simulation of the toroidal Alfvén eigenmode[END_REF], the PCS suffers (in its original formulation) from not conserving the total energy; consequently, unphysical energy sources may drive spurious instabilities [START_REF] Tronci | Hybrid Vlasov-MHD models: Hamiltonian vs. non-Hamiltonian[END_REF]. The formulation of an energy-conserving PCS variant was presented in [START_REF] Tronci | Hamiltonian approach to hybrid plasma models[END_REF], upon adopting the Vlasov description for the energetic particles. In this case, while the fluid momentum equation is unaltered by energy conservation, the accompanying kinetic equation is modified by the presence of terms that arise from expressing the energetic particle kinetics in the frame of the MHD bulk. The energy-conserving PCS of [START_REF] Tronci | Hamiltonian approach to hybrid plasma models[END_REF] was obtained by applying standard methods in the theory of noncanonical Hamiltonian systems. More particularly, the Hamiltonian structures of MHD [START_REF] Morrison | Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics[END_REF][START_REF] Holm | Noncanonical Hamiltonian formulation of ideal magnetohydrodynamics[END_REF][START_REF] Morrison | Poisson brackets for fluids and plasmas[END_REF][START_REF] Marsden | Hamiltonian systems with symmetry, coadjoint orbits and plasma physics[END_REF][START_REF] Morrison | Hamiltonian description of the ideal fluid[END_REF][START_REF] Morrison | Hamiltonian and action principle formulations of plasma physics[END_REF][START_REF] Morrison | On Hamiltonian and action principle formulations of plasma dynamics[END_REF] and of the Vlasov equation [START_REF] Morrison | The Maxwell-Vlasov equations as a continuous Hamiltonian system[END_REF][START_REF] Morrison | Poisson brackets for fluids and plasmas[END_REF][START_REF] Marsden | The Hamiltonian structure of the Maxwell-Vlasov equations[END_REF][START_REF] Marsden | Hamiltonian systems with symmetry, coadjoint orbits and plasma physics[END_REF][START_REF] Morrison | Hamiltonian description of the ideal fluid[END_REF][START_REF] Morrison | Hamiltonian and action principle formulations of plasma physics[END_REF][START_REF] Morrison | On Hamiltonian and action principle formulations of plasma dynamics[END_REF] were combined to obtain the Hamiltonian structure of the CCS and of the new Hamiltonian PCS. Then, since the energy-conserving PCS is also Hamiltonian, the general theory offers the opportunity to investigate stability properties by applying the energy-Casimir method [START_REF] Morrison | Energy stability analysis for a hybrid fluid-kinetic plasma model in Nonlinear Physical Systems -Spectral Analysis, Stability and Bifurcations[END_REF][START_REF] Holm | Nonlinear stability of fluid and plasma equilibria[END_REF][START_REF] Morrison | Hamiltonian description of the ideal fluid[END_REF], a method that is a descendent of the Lyapunov method known as thermodynamic stability in the early plasma literature [START_REF] Kruskal | On the Stability of Plasma in Static Equilibrium[END_REF][START_REF] Fowler | Thermodynamics of unstable plasmas[END_REF][START_REF] Brizard | Thermodynamic Constraints Applied to Tokamaks[END_REF]. This method has two main advantages: it is relatively easy to implement and it gives a stronger notion of stability than that based on linear analysis, for it can lead to mathematically rigorous proofs of nonlinear stability and nonlinear bounds on growth. However, the expense paid for this is that the conditions are generally less definitive. Also, the energy-Casimir method is effective when applied to Hamiltonian systems possessing a large number of Casimirs, which is the case for the CCS and the PCS when the dynamics is restricted to a plane, an approximation that can be relevant for both laboratory and astrophysical plasmas. This paper applies the energy-Casimir method to both the CCS and the Hamiltonian PCS in order to analyze the stability properties of these two systems in planar geometry.

In the remainder of this Introduction we describe the hybrid models in Sec. 1.1 and give a brief description of the energy-Casimir method in Sec. 1.2. Then, in Sec. 2 we consider the CCS. Here we present the Casimir invariants, obtain equilibria from the energy-Casimir variational principle, and then proceed to stability results. The analogous material is presented in Sec. 3 for the PCS. Finally, in Sec. 4 we conclude.

Hybrid Vlasov-MHD in three and two dimensions

The nonlinear equations of the CCS and PCS appeared first in [START_REF] Park | Three-dimensional hybrid gyrokinetic-magnetohydrodynamics simulation[END_REF][START_REF] Cheng | A kinetic-magnetohydrodynamic model for low-frequency phenomena[END_REF], where they were derived by direct manipulations on the separate equations for MHD and energetic particle kinetics. Later, the CCS and the energy-conserving PCS were derived by operating on the Poisson brackets for MHD and the Vlasov equation [START_REF] Tronci | Hamiltonian approach to hybrid plasma models[END_REF]. In planar geometry, one can study two possible configurations depending on whether the magnetic field lies on the plane or points vertically. Due to the analogy with reduced MHD [START_REF] Hazeltine | Generalized Poisson brackets and nonlinear Liapunov stability-application to reduced MHD[END_REF][START_REF] Holm | Nonlinear stability of fluid and plasma equilibria[END_REF][START_REF] Morrison | Hamiltonian formulation of reduced magnetohydrodynamics[END_REF], here we choose to study the first case as it is more physically relevant. This section presents explicitly the nonlinear equations of Vlasov-MHD models in both the CCS and the Hamiltonian PCS variants.

Upon introducing the Vlasov phase-space density f (x, p, t) and its first three moments

n = f d 3 p , K = pf d 3 p , P = ppf d 3 p ,
(here, p is the kinetic momentum coordinate) the nonlinear equations for the CCS and the Hamiltonian PCS are respectively:

ρ ∂u ∂t + u • ∇u = -∇p + q h nu -a h K + µ -1 0 J × B , (1)
∂f ∂t + p m h • ∇f + q h p m h -u × B • ∂f ∂p = 0 , (2)
∂ρ ∂t + ∇ • (ρu) = 0 , ∂B ∂t = ∇ × (u × B) , (3)
and

ρ ∂u ∂t + u • ∇u = -∇p -m h -1 ∇ • P + µ -1 0 J × B (4) ∂f ∂t + p m h + u • ∇f + p × a h B -∇ × u -p • ∇u • ∂f ∂p = 0 , (5)
∂ρ ∂t + ∇ • (ρu) = 0 , ∂B ∂t = ∇ × (u × B) , (6)
where a h = q h /m h is the kinetic particle mass-to-charge ratio and we have used the ordinary relation J = µ -1 0 ∇ × B. Both systems preserve the total energy

H = 1 2 ρ|u| 2 d 3 x + 1 2m h f |p| 2 d 3 x d 3 p + ρ U(ρ) d 3 x + 1 2µ 0 |B| 2 d 3 x , (7)
where U(ρ) denotes the MHD internal energy so that the scalar pressure is expressed as p = ρ 2 U (ρ). The non conservative version of the PCS occurring in the plasma literature is given by ( 4) and ( 6), while equation ( 5) is replaced by [START_REF] Andreussi | Hamiltonian magnetohydrodynamics: Symmetric Formulation, Casimir Invariants, and Equilibrium Variational Principles[END_REF].

In order to study the simplest case, while retaining essential physical features of these models, we consider the incompressible limit in two-dimensional geometry. In particular, we assume that the magnetic and velocity fields lie in the x -y plane of a Cartesian coordinate system, and we assume translational invariance along the z-direction. Consequently, we replace the first of ( 3) and ( 6) by the divergenceless condition ∇ • u = 0 and write

B = ∇ ⊥ A × ẑ , u = ∇ ⊥ φ × ẑ , and ω = ẑ • ∇ × u = -∆φ , (8)
where Aẑ is the vector potential, φ is the stream function, and ω is the usual vorticity of the MHD fluid, the subscript ⊥ denotes planar components, and ∆ denotes the perpendicular Laplacian without the subscript ⊥. Then, after introducing the notation A(x ⊥ ) = A(x) dz, we have the following equations for the planar incompressible CCS:

∂ω ∂t = [φ, ω] + µ -1 0 J -a h K z , A , ∂A ∂t = [φ, A] (9)
∂ f ∂t + p ⊥ m h • ∇ ⊥ f + q h p ⊥ m h • ∇ ⊥ A + [φ, A] ∂ f ∂p z -a h p z ∇ ⊥ A • ∂ f ∂p ⊥ = 0 (10) where [a, b] = ∂ x a ∂ y b-∂ y a ∂ x b
is the canonical Poisson bracket, φ = -∆ -1 ω and J = -∆A is the current density, which is directed perpendicular to the x -y plane. By proceeding analogously, we also find the corresponding equations of motion for the planar incompressible (Hamiltonian) PCS:

∂ω ∂t = [φ, ω] + µ -1 0 [J, A] + ẑ • ∇ ⊥ × (∇ ⊥ • P ⊥ ) , ∂A ∂t = [φ, A] , (11)
∂ f ∂t + f , φ + p ⊥ m h • ∇ ⊥ f - p ⊥ m h • ẑ × ∇ ⊥ ∇ ⊥ φ - p z m h ∇ ⊥ A • ∂ f ∂p ⊥ - p ⊥ m h • ∇ ⊥ A ∂ f ∂p z = 0 , (12)
where we have defined P ⊥ = p ⊥ p ⊥ f d 3 p. The explicit derivation of the above planar models is presented in the Appendix, which makes use of the corresponding Hamiltonian structures. Both models conserve the following expression of the total energy:

H = 1 2 ω(-∆ -1 )ω - 1 µ 0 A ∆A + 1 m h f |p| 2 d 3 p d 2 x . ( 13
)
As outlined in the previous Section, the Hamiltonian nature of the CCS and the PCS presented here allows a Lyapunov stability study that is made possible by direct application of the energy-Casimir method [START_REF] Morrison | Energy stability analysis for a hybrid fluid-kinetic plasma model in Nonlinear Physical Systems -Spectral Analysis, Stability and Bifurcations[END_REF][START_REF] Holm | Nonlinear stability of fluid and plasma equilibria[END_REF]. This procedure is discussed in the next Section. Unless otherwise specified, the remainder of this paper will refer to the Hamiltonian PCS simply as the PCS and we shall drop the hat symbol for simplicity of notation. Moreover, we shall set all physical constants equal to 1, as they can always be restored without essential difficulties.

Stability and the energy-Casimir method

Consider a dynamical system ż = V (not necessarily Hamiltonian) defined on some space Z, where z ∈ Z can be a point or a trajectory, ' ˙' denotes time derivative, and V is an autonomous vector field defined on Z. Equilibria of such system are solutions z e that satisfy V (z e ) = 0. According to the standard definition of stability, an equilibrium point z e of a dynamical system is said to be stable if, for any neighborhood N of z e there exists a subneighborhood S ⊂ N of z e such that if z, an initial condition, is in S then the trajectory z(t) ∈ N for all time t > 0.

The linear problem associated with the above dynamical system reads δ ż = DV (z e ) • δz and is obtained by expanding V (z e + δz) to first order. If δz remains in N , then the system is said to be linearly stable, and to distinguish this kind of stability from that with dynamics under the full vector field V one adds the adjective nonlinear to describe the latter.

Assuming a solution of the form δz = ẑ exp(λt) the linear problem becomes (DV -λ id) • ẑ = 0, where id is the identity operator. The spectrum of DV , σ(DV ), is the set composed of λ ∈ C for which the linear operator DV -λ id has no inverse, and an equilibrium point is said to be spectrally stable if, for every λ ∈ σ(DV ), λ lies in the left half complex plane, including the ordinate. This latter case, in which λ is purely imaginary, corresponds to pure oscillations, a case that is sometimes called neutral stability. This is the only kind of spectral stability possessed by Hamiltonian systems.

We remark that there are logical relations between linear, nonlinear, spectral stability but these can be subtle, particularly in infinite dimensions. Some descriptions of implications between the different kinds of stability can be found, for instance, in Refs. [START_REF] Morrison | Hamiltonian description of the ideal fluid[END_REF][START_REF] Holm | Nonlinear stability of fluid and plasma equilibria[END_REF]. The present work concentrates on the particular kind of stability in Hamiltonian systems that we refer to as energy-Casimir stability.

For Hamiltonian systems, the vector field of our dynamical system is generated by a Poisson bracket so that the equations of motion have the form ż = {z, H}. In finite dimension, one can write żi = J ij ∂ j H, where J is the Poisson bivector (cosymplectic form). Recalling the above definition of equilibrium point, it follows then that, for Hamiltonian systems, equilibrium points satisfy dH ∈ Ker(J). A consequence of the Poisson bracket identities is the Lie-Darboux theorem, which implies Ker(J) is spanned by Casimir invariants, which satisfy {C, f } = 0, for all functions f (although there are serious unresolved issues with this theorem for infinite-dimensional systems (see, e.g., Ref. [START_REF] Yoshida | Singular Casimir elements of the Euler equation and equilibrium points[END_REF])). Equilibria of Hamiltonian systems are then critical points of an invariant energy function

F = H + C
and this fact is very useful for establishing stability criteria.

For finite-dimensional systems, definiteness (positive or negative) of the quadratic form δ 2 F(z e ; δz) assures both linear and nonlinear stability. For linear systems δ 2 F is invariant, in fact it is the Hamiltonian for the linear dynamics, and its definiteness means the phase space region surrounding the equilibrium point is foliated by nested invariants sets that are topologically spheres. The interior of these sets can thus serve as the subneighborhoods S in the above definition of stability. For nonlinear systems F is invariant and, under mild smoothness conditions, δ 2 F determines the topological character of the level sets of F in a sufficiently small neighborhood of the equilibrium point z e . This guarantees that for any neighborhood N there is an S ⊂ N , determined by some level set of F, within which the flow must remain.

For infinite-dimensional systems, such as those of interest here, the situation is considerably more complicated. In the first place, even if δ 2 F has definite sign, this does not imply that an extremal point that satisfies δF = 0 is in fact an extremum (maximum or minimum). Moreover, for both linear and nonlinear stability one requires δ 2 F to lead to a norm for defining open sets. This leads to a second and for nonlinear stability an often formidable complication: even if one is able to construct a norm from δ 2 F, a rigorous proof of nonlinear stability requires that the solution to the dynamical system actually exists in this norm. For most of the infinite-dimensional systems of interest in plasma physics, in particular, global existence results are not available. Even in cases when existence results are available, it can turn out that more than one choice of norm may be constructed and a given equilibrium can be stable in one norm and not in another; in this case a physical determination must be made about what is important. The reader interested in seeing what makes up a rigorous application of the energy-Casimir method is referred to Refs. [START_REF] Batt | Linear stability of stationary solutions of the Vlasov-Poisson system in three dimensions[END_REF][START_REF] Rein | Nonlinear stability for the Vlasov-Poisson system -the energy-Casimir method[END_REF].

In this paper we will derive criteria for which δ 2 F is positive definite for our hybrid models, following the practice in the physics literature (e.g. [START_REF] Hazeltine | Generalized Poisson brackets and nonlinear Liapunov stability-application to reduced MHD[END_REF][START_REF] Holm | Nonlinear stability of fluid and plasma equilibria[END_REF][START_REF] Morrison | Spontaneous symmetry breaking and neutral stability in the noncanonical Hamiltonian formalism[END_REF][START_REF] Morrison | Hamiltonian description of the ideal fluid[END_REF]). We will refer to this kind of stability as energy-Casimir stability. We remark that this stability implies linear stability [START_REF] Morrison | Hamiltonian description of the ideal fluid[END_REF][START_REF] Holm | Nonlinear stability of fluid and plasma equilibria[END_REF] but, for infinite-dimensional systems nonlinear stability is subject to the caveats mentioned above.

Finally, we find it opportune to recall that the classical δW criterion conceived and adopted for ideal MHD stability analysis [START_REF] Bernstein | An energy principle for hydromagnetic stability problems[END_REF] is intimately related to a energy-Casimir stability. Indeed, the variation of the potential energy in terms of the displacement vector, which is denoted as δW , is related (see, e.g., Ref. [START_REF] Morrison | Hamiltonian description of the ideal fluid[END_REF][START_REF] Holm | Nonlinear stability of fluid and plasma equilibria[END_REF]) to the second variation δ 2 F, so that stability in terms of δW implies energy-Casimir stability with respect to δ 2 F. The connection between the two criteria can be extended to reduced models obtained from MHD, as in the case e.g. of compressible reduced MHD [START_REF] Morrison | Stability of compressible reduced magnetohydrodynamic equilibria -Analogy with magnetorotational instability[END_REF].

Stability of the planar CCS

This section presents the stability analysis of the planar incompressible CCS model ( 10)- [START_REF] Freidberg | Vlasov-fluid model for studying gross stability of high-β plasmas[END_REF]. After reviewing the corresponding Casimirs in both three and two dimensions, these quantities will be used to provide energy-Casimir stability conditions.

Helicity and Casimirs

In three dimensions, the CCS equations ( 1)-(3) were shown in Ref. [START_REF] Holm | Euler-Poincaré formulation of hybrid plasma models[END_REF] to possess the following conservation law for the cross-helicity invariant:

d dt u • B d 3 x = 0 ,
which is accompanied by the usual conservation of magnetic helicity, that is B • A d 3 x. Moreover, collisionless kinetic equations are well known [START_REF] Holm | Nonlinear stability of fluid and plasma equilibria[END_REF] to possess an infinite family of invariants that are expressed as

C V [f ] = Λ(f ) d 3 x d 3 p , ( 14
)
where Λ is an arbitrary function of one variable. The above functional is also preserved by the hybrid CCS.

In planar geometry, the above conservation laws produce a more general family of invariants for the incompressible CCS. This family of invariants can be combined into a single functional

C = C V [f ] + C MHD [ω, A] , with C MHD [ω, A] = ωΦ(A) + Ψ(A) d 2 x ,
where C MHD has arbitrary functions Φ and Ψ, and integrations are now over two-dimensional domains. The Casimir C MHD for ideal MHD in planar geometry was obtained in Ref. [START_REF] Morrison | Hamiltonian formulation of reduced magnetohydrodynamics[END_REF],

where it was shown that the two terms are descendent from the conservation of cross-helicity and magnetic helicity of three-dimensional MHD, respectively.

Equilibria via the first variation

As we shall see, the stability analysis for the planar CCS is actually straightforward due to the direct sum structure of the Poisson bracket underlying CCS [START_REF] Tronci | Hamiltonian approach to hybrid plasma models[END_REF]. Upon following the prescriptions in Section 1.2, the equilibria are identified by the variational principle

δF = δH + δC MHD + δC V = 0 .
Since the total energy (13) can also be split as

H = H V + H MHD , with H MHD [ω, A] = 1 2 ω(-∆ -1 )ω -A ∆A d 2 x , H V [f ] = 1 2 f |p| 2 d 2 x d 3 p ,
the equilibria are found from

δF MHD [ω, A] + δF V [f ] = 0 ,
where

F MHD = H MHD + C MHD and F V = H V + C V .
Because the Hamiltonian and Casimirs are additive, this implies

δF MHD [ω, A] = 0 , δF V [f ] = 0 .
Therefore, the equilibrium states for the hybrid CCS obtained from the variational principle are simply given by MHD equilibria coexisting with Vlasov equilibria. Upon computing

δF V = Λ + |p| 2 /2 δf d 2 xd 2 p and δF MHD = Φ(A) -(∆ -1 )ω δω d 2 x + -∆A + ωΦ (A) + Ψ (A) δA d 2 x ,
the following equilibrium conditions are obtained:

φ e + Φ(A e ) = 0 , (15)
∆A e -ω e Φ (A e ) -Ψ (A e ) = 0 , (16)

1 2 |p| 2 + Λ (f e ) = 0 . ( 17
)
where the index e is used to denote quantities in the equilibrium configuration. All the above equilibrium relations are well known and have been extensively discussed in [START_REF] Hazeltine | Generalized Poisson brackets and nonlinear Liapunov stability-application to reduced MHD[END_REF][START_REF] Holm | Nonlinear stability of fluid and plasma equilibria[END_REF][START_REF] Morrison | δ 2 F : A generalized energy principle for determining linear and nonlinear stability[END_REF]. In particular, the first equation expresses the fact that, at equilibrium, streamlines are magnetic flux functions. The second equation is a generalization of the Grad-Shafranov equation1 that includes flow, while the third implies (upon assuming an invertible Λ ) isotropic equilibria of the type f e = f e (|p| 2 /2).

In [START_REF] Morrison | δ 2 F : A generalized energy principle for determining linear and nonlinear stability[END_REF] (and later in [START_REF] Throumoulopoulos | Ideal magnetohydrodynamic equilibria with helical symmetry and incompressible flows[END_REF]) it was shown that the transformation

χ(A e ) := Ae 1 -(Φ (A)) 2 dA (18)
transforms ( 15) and ( 16) into ∆χ = F (χ), where F = χ Φ • A(χ). Using ( 8) and [START_REF] Holm | Euler-Poincaré formulation of hybrid plasma models[END_REF] gives

Φ = - u e B e (19)
where u e = |u e | and B e = |B e |, we see that associated with every sub-Alfvénic MHD equilibrium with flow there corresponds a static equilibrium state.

Energy-Casimir stability via the second variation

The calculation of the second variation δ 2 F proceeds analogously to those in the previous section. Indeed, the same arguments lead to the stability criterion

δ 2 F MHD [ω, A] > 0 and δ 2 F V [f ] > 0 .
Then, stability of the hybrid CCS requires that both the stability conditions for ideal MHD (first inequality) and for the Vlasov kinetic equation (second inequality) are satisfied. More explicitly, upon computing

δ 2 F MHD = |∇ ⊥ δφ -∇ ⊥ δΦ| 2 + 1 -(Φ ) 2 |∇ ⊥ δA| 2 d 2 x + ωΦ + Ψ + Φ ∆Φ (δA) 2 d 2 x . (20)
From ( 20) the following sufficient conditions for positive definiteness are immediate:

|Φ | 2 < 1 , (21)
ω Φ + Ψ + Φ ∆Φ > 0 , (22)
Λ > 0 . (23)
In order to add some physical insight to expression [START_REF] Marsden | Hamiltonian systems with symmetry, coadjoint orbits and plasma physics[END_REF] and conditions ( 21)-( 23), we rewrite (20) as in Ref. [START_REF] Morrison | δ 2 F : A generalized energy principle for determining linear and nonlinear stability[END_REF], viz.

δ 2 F MHD = ∇ ⊥ δφ + ∇ ⊥ u e B e δA 2 + 1 - u 2 e B 2 e |∇ ⊥ δA| 2 (24)
+ (δA) 2 B 2 e 1 - u 2 e B 2 e B e × ẑ • ∇ ⊥ J || -B e × ẑ • ∇ ⊥ u 2 e B 2 e B e × ẑ • ∇ ⊥ B 2 e 2B 2 e d 2 x .
The first term of (24) vanishes for Alfvénic motions, i.e., motions to neighboring equilibrium states [START_REF] Morrison | Spontaneous symmetry breaking and neutral stability in the noncanonical Hamiltonian formalism[END_REF], the second term represents line bending energy, reduced by the presence of an equilibrium flow. The first term of the second line can be recognized as the usual kink term, also modified by the presence of flow, while the last term is interchange-like, that involves curvature and ram pressure u 2 e , taking the role of pressure, which does not occur in low-β reduced MHD. As follows from Eq. ( 19), the inequality [START_REF] Morrison | The Maxwell-Vlasov equations as a continuous Hamiltonian system[END_REF] implies that one of the conditions for stability is that the equilibrium flow be sub-Alfvénic. Condition ( 22) is a sufficient condition for stability, but a more in-depth analysis would show that the line bending term can add a positive contribution. (See e.g. Ref. [START_REF] Andreussi | Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability -theory[END_REF]). Lastly, condition ( 23) is the monotonicity-isotropy condition on the distribution function (see e.g. [START_REF] Morrison | Free energy expressions for Vlasov equilibria[END_REF][START_REF] Morrison | Variational principle and stability of nonmonotonic Vlasov-Poisson equilibria[END_REF]).

While the stability analysis of the hybrid CCS does not add any new physical feature to the well-studied stability properties of ideal MHD and the usual convexity relation [START_REF] Morrison | δ 2 F : A generalized energy principle for determining linear and nonlinear stability[END_REF] for Vlasov stability, the hybrid PCS leads to quite different results. These differences are the subject of the remainder of this paper.

Stability of the planar PCS

This section contains the stability analysis of the planar PCS model [START_REF] Fu | Nonlinear hybrid simulation of the toroidicity-induced Alfvén eigenmode[END_REF] and [START_REF] Hazeltine | Generalized Poisson brackets and nonlinear Liapunov stability-application to reduced MHD[END_REF]. After reviewing the corresponding Casimirs in both three and two dimensions, these quantities will be used to provide stability conditions. Although the discussion of Casimirs and equilibria for this case can also be found in Ref. [START_REF] Morrison | Energy stability analysis for a hybrid fluid-kinetic plasma model in Nonlinear Physical Systems -Spectral Analysis, Stability and Bifurcations[END_REF], here we present new stability conditions that recover the usual condition [START_REF] Morrison | δ 2 F : A generalized energy principle for determining linear and nonlinear stability[END_REF] for noncollisional kinetics.

Helicity and Casimirs

In three dimensions, the hybrid PCS of ( 4) and ( 6) preserves the usual magnetic helicity as well as the Vlasov invariant [START_REF] Holm | Nonlinear stability of fluid and plasma equilibria[END_REF]. On the other hand, the usual expression for the crosshelicity is not conserved and a velocity shift is required for the conservation of a modified cross-helicity, which reads [START_REF] Holm | Euler-Poincaré formulation of hybrid plasma models[END_REF]

d dt B • (u + ρ -1 K) d 3 x = 0.
Then, in two dimensions, one obtains the following infinite set of Casimir invariants for the incompressible PCS:

C = C V [f ] + C MHD [ω, A] -Φ(A) ẑ • ∇ × K ⊥ d 2 x
where K ⊥ := p ⊥ f d 3 p. The proof of the conservation relation Ċ = 0 was presented in Ref. [START_REF] Morrison | Energy stability analysis for a hybrid fluid-kinetic plasma model in Nonlinear Physical Systems -Spectral Analysis, Stability and Bifurcations[END_REF], by using the Poisson bracket structure. The last term above couples the parallel vector potential A to the Vlasov distribution f and, as we shall see, this has a substantial effect on the equilibrium configurations and their stability conditions.

Equilibria and hybrid Grad-Shafranov relation

Consider now the equilibrium conditions for the incompressible PCS in planar geometry. Following again the general theory of Sec. 1.2, we set δF = 0. Unlike the CCS, the equilibrium condition for the PCS do not split. They read

δF MHD [ω, A] + δF V [f ] + δ d 2 x Φ(A) ẑ • p ⊥ × ∇ ⊥ f d 3 p = 0 .
Upon computing the variation of the new term,

δ d 2 x Φ(A) ẑ • p ⊥ × ∇ ⊥ f d 3 p = d 2 x δAΦ ẑ • p ⊥ × ∇ ⊥ f d 3 p (25) - δf ẑ • p ⊥ × ∇ ⊥ Φ d 2 x d 3 p ,
the equilibrium conditions are seen to be

φ e + Φ(A e ) = 0 (26)
-

∆A e + (ω e -ẑ • ∇ ⊥ × K ⊥ ) Φ (A e ) + Ψ (A e ) = 0 (27) ẑ • ∇ ⊥ Φ(A e ) × p ⊥ + |p| 2 2 + Λ (f e ) = 0 . ( 28
)
Using ∇ ⊥ φ e = ẑ × u e , ( 26) and ( 28) imply

1 2 |p + u e | 2 - 1 2 |u e | 2 + Λ = 0 , (29)
so that assuming an invertible Λ , we obtain the following class of anisotropic equilibria:

f e =f e 1 2 |p + u e | 2 - 1 2 |u e | 2 .
Next, an explicit computation [START_REF] Morrison | Energy stability analysis for a hybrid fluid-kinetic plasma model in Nonlinear Physical Systems -Spectral Analysis, Stability and Bifurcations[END_REF] yields

ẑ • ∇ × p ⊥ f e d 3 p = -∇ ⊥ • (n∇ ⊥ Φ) = -ẑ • ∇ ⊥ × (nu e )
so that Eq. ( 27) reads

-∆A e + Φ (A e ) ∇ ⊥ • (1 + n e )∇ ⊥ Φ(A e ) + Ψ (A e ) = 0 , (30)
which we shall call the hybrid Grad-Shafranov equation.

Recognize that the special case Φ(A e ) = -A e (Alfvén equilibrium) takes the relation ( 30) to

∇ ⊥ • (n e ∇ ⊥ A e ) + Ψ (A e ) = ẑ • ∇ ⊥ × (n e u e ) + Ψ (A e ) = 0 =⇒ A e = A e ẑ • ∇ ⊥ × (n e u e ) ,
while the case Φ(A e ) = 0 produces the usual static Grad-Shafranov equilibria of ideal MHD [START_REF] Hazeltine | Generalized Poisson brackets and nonlinear Liapunov stability-application to reduced MHD[END_REF][START_REF] Holm | Nonlinear stability of fluid and plasma equilibria[END_REF]. For Alfvén one-dimensional equilibria with Φ(A e ) = -A e (x), one can study cases that allow reducing the problem (30) to quadratures. In particular, if n e = n e (A 2 e ) equation ( 30) reads ∂ x (n e (A 2 e )A e ) + Ψ = 0, where prime denotes differentiation with respect to the argument. This equation arises as an Euler-Lagrange equation with Lagrangian L(A e , A e ) = N e (A 2 e )/2-Ψ(A e ), where N e denotes the antiderivative of n e . Then, the variational structure guarantees conservation of the total energy E = A e ∂L/∂A e -L = A 2

e N e -N e /2 + Ψ, which in turn can be written as A 2

e N e -N e /2 = κ(A e ) = E -Ψ. If the function κ is invertible, then the problem can be reduced to quadratures in the form A e = κ -1 (E -Ψ). We notice that for the case of Maxwellian distributions f e (x, p) ∝ exp(|u e | 2 /2 -|p + u e | 2 /2), one has n e (A 2 e ) = n 0 exp(-A 2 e /2) with constant n 0 . This yields κ(A ) = n 0 (A 2 + 1) exp(-A 2 /2). The function κ in this case is not invertible due to the presence of a local maximum at A e = 1. This is a manifestation of the resonance between the MHD and the kinetic component.

Having characterized the equilibria of the planar PCS, we can consider the energy-Casimir conditions for stability, which we do in the next section.

Stability conditions and kinetic particle effects

As before, we compute the second variation, giving

δ 2 F = δ 2 F MHD + Λ (δf ) 2 d 2 xd 3 p + (ẑ • p ⊥ × ∇ ⊥ f ) Φ (A)(δA) 2 d 2 xd 3 p -2 Λ 1 Λ δf (∇ ⊥ Φ δA + Φ ∇ ⊥ δA) • ẑ × p ⊥ d 2 xd 3 p = δ 2 F MHD + (ẑ • p ⊥ × ∇ ⊥ f ) Φ (A)(δA) 2 d 2 xd 3 p + 1 2 Λ δf - 2 Λ ∇ ⊥ Φ δA • ẑ × p ⊥ 2 d 2 xd 3 p + 1 2 Λ δf - 2 Λ Φ ∇ ⊥ δA • ẑ × p ⊥ 2 d 2 xd 3 p -2 1 Λ p ⊥ × ∇ ⊥ Φ δA 2 d 2 xd 3 p -2 1 Λ Φ p ⊥ × ∇ ⊥ δA 2 d 2 xd 3 p = δ 2 F MHD + (ẑ • p ⊥ × ∇ ⊥ f ) Φ (A)(δA) 2 d 2 xd 3 p + 1 2 Λ δf - 2 Λ ∇ ⊥ Φ δA • ẑ × p ⊥ 2 d 2 xd 3 p + 1 2 Λ δf - 2 Λ Φ ∇ ⊥ δA • ẑ × p ⊥ 2 d 2 xd 3 p -2 (Tr Π ⊥ )|∇ ⊥ Φ | 2 (δA) 2 d 2 x + 2 1 Λ (p ⊥ • ∇ ⊥ Φ ) 2 (δA) 2 d 2 xd 3 p -2 Φ 2 (Tr Π ⊥ )|∇ ⊥ δA| 2 d 2 x + 2 1 Λ Φ 2 (p ⊥ • ∇ ⊥ δA) 2 d 2 xd 3 p
where Π ⊥ := (Λ ) -1 p ⊥ p ⊥ d 3 p. Thus, the PCS stability conditions are

|Φ | 2 < 1 1 + 2 Tr Π ⊥ (31) ω Φ + Ψ + Φ ∆Φ -2|∇ ⊥ Φ | 2 Tr Π ⊥ > 0 (32) Λ > 0, (33)
where ω = ω -ẑ • ∇ ⊥ × K is a shifted vorticity accounting for the contributions of both the MHD and the kinetic components.

Notice that φ e = -Φ(A e ) implies u e = -Φ (A e )B e , so that the first stability condition reads B 2 e ≥ (1 + 2 Tr Π ⊥ ) u 2 e . It follows from this expression that, for instance in the presence of a Maxwellian equilibrium distribution function, a condition stronger than ( 21) is required on the equilibrium flow for stability. Compared to the case of reduced MHD or to the CCS, it turns out indeed, that a sub-Alfvénic equilibrium flow is no longer sufficient for stability in the presence of a population of kinetic particles, even if the latter follows a Maxwellian distribution.

The condition [START_REF] Morrison | Energy stability analysis for a hybrid fluid-kinetic plasma model in Nonlinear Physical Systems -Spectral Analysis, Stability and Bifurcations[END_REF], on the other hand, can be reformulated as

1 B 2 e 1 - u 2 e B 2 e B e × ẑ • ∇ ⊥ J || -B e × ẑ • ∇ ⊥ u 2 e B 2 e B e × ẑ • ∇ ⊥ B 2 e 2B 2 e + u e B e B e × ẑ B 2 e • ∇ ⊥ (ẑ • ∇ × K) -2 ∇ ⊥ u e B e 2
Tr Π ⊥ > 0.

In the first line of the inequality (34) one can recognize the terms originated from δ 2 F M HD and already discussed in the case of the CCS. The remaining terms, on the other hand, are peculiar to the PCS stability conditions and show an influence of the presence of the kinetic population. In particular, in the first term of the second line, one is able to identify a further possible source of instability analogous to the kink instability but with the role of the current density played by the vorticity associated with the mean flow of the kinetic species. The last term, on the other hand is destabilizing and depends on the second order moments of the equilibrium distribution function.

Finally, upon deriving the equilibrium relation [START_REF] Morrison | Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magnetohydrodynamics[END_REF] with respect to p, we obtain

Λ = - 1 f e =⇒ f e < 0 and Π ⊥ = -f e p ⊥ p ⊥ d 3 p ,
which states that distributions such that f e = -ν 2 f e for some constant ν (e.g. Gaussian distributions) are stable equilibria.

Conclusions

The Hamiltonian structures of two hybrid kinetic-MHD models have been presented and their energy-Casimir stability properties have been analyzed. The two adopted models are examples of the two paradigmatic coupling schemes adopted for hybrid models, i.e., the CCS and the PCS. For the latter, in particular, the model we investigated corresponds to a modification of the one derived in Ref. [START_REF] Cheng | A kinetic-magnetohydrodynamic model for low-frequency phenomena[END_REF], in such a way that the resulting hybrid model is energy-conserving and possesses a Hamiltonian structure. The Casimir invariants for such models have been described and compared with those of ideal MHD in two and three dimensions. Equilibria obtained by extremizing δF and the stability conditions corresponding to the definiteness of δ 2 F have then been derived and discussed.

In the case of the CCS, equilibrium relations and stability conditions derived with the energy-Casimir method correspond to those that would be obtained by considering independently 2D planar MHD and the Vlasov equation. In particular, we identified (by imposing definiteness of δ 2 F) the conditions of having a sub-Alfvénic MHD flow, the convexity of the distribution function, as well as the stability with respect to the ideal kink instability and to an interchange-like instability.

A more interesting situation occurs for the PCS. In this case, setting to zero the first variation of the free energy functional yields, among the equilibrium equations, a generalized hybrid Grad-Shafranov equation, which accounts for the presence of both the MHD flow and the kinetic particle population. With regard to stability, it emerges that the presence of a kinetic particle population leads to a different condition on the MHD equilibrium speed, whereas the convexity condition on the distribution function remains valid. In the case of a Maxwellian distribution for the kinetic species, the MHD flow has indeed an upper bound corresponding to the Alfvén speed diminished by a quantity proportional to the equilibrium pressure of the kinetic population.

As complement to the above results, we have provided an Appendix containing derivations of the planar CCS and PCS models and of their Hamiltonian structures.

Acknowledgements.

A Explicit derivation of planar Vlasov-MHD

This Appendix presents the explicit derivation of both CCS and Hamiltonian PCS in planar geometry.

A.1 Planar current-coupling scheme

Consider the incompressible CCS equations (1)-(3). These comprise a Hamiltonian system with Hamiltonian [START_REF] Cheng | A kinetic-magnetohydrodynamic model for low-frequency phenomena[END_REF] and Poisson bracket [START_REF] Tronci | Hamiltonian approach to hybrid plasma models[END_REF] {F,

G} CCS = m • δF δm • ∇ δG δm - δG δm • ∇ δF δm d 3 x + f δF δf , δG δf + q h B • ∂ ∂p δF δf × ∂ ∂p δG δf d 3 x d 3 p + q h f B • δF δm × δG δm - δF δm × ∂ ∂p δG δf + δG δm × ∂ ∂p δF δf d 3 x d 3 p + B • δF δm × ∇ × δG δB - δG δm × ∇ × δF δB d 3 x -ρ δF δm • ∇ δG δρ - δG δm • ∇ δF δρ d 3 x , (35)
where m = ρu is the fluid momentum, as it is expressed in terms of the velocity u and the density ρ. Then, we use the formula

B = ∇ × (Aẑ) = ∇ ⊥ A × ẑ together with the variational relation δF δm = ∇ × δF δω ẑ = ∇ ⊥ δF δω × ẑ
where ω = ẑ • ∇ × m. (See e.g. [START_REF] Andreussi | MHD Equilibrium Variational Principles with Symmetry[END_REF][START_REF] Andreussi | Hamiltonian magnetohydrodynamics: Symmetric Formulation, Casimir Invariants, and Equilibrium Variational Principles[END_REF] for discussion of such relations.) With these formulas in mind we compute the following term appearing in the fourth line of (35): Then, upon using the total energy (13), one obtains

B • δF δm × ∇ × δG δB d 3 x = curl(Aẑ) • curl δF δω ẑ × δG δA d 3 x = - δG δA • curl δF δω ẑ × curl(Aẑ) d 3 x = -A δF δω ,
∂ t ω =[φ, ω] + J -p z f d 3 p dz , A , ∂ t A = [φ, A] ∂ t f +p • ∂f ∂x + p ⊥ • ∇ ⊥ A + [φ, A] ∂f ∂v z -v z ∇ ⊥ A ∂f ∂p ⊥ = 0 .
Notice that the longitudinal coordinate z plays no role in the dynamics and thus it can be integrated out to yield the equations ( 9)- [START_REF] Fu | Recent progress in linear and nonlinear studies of toroidal Alfvén eigenmodes[END_REF].

A.2 Planar pressure-coupling scheme

Consider the incompressible PCS equations ( 4)-( 6). These comprise a Hamiltonian system with Hamiltonian [START_REF] Cheng | A kinetic-magnetohydrodynamic model for low-frequency phenomena[END_REF] Then, upon using the total energy (13), one obtains

∂ t ω = [ψ, ω] + [J, A] + ẑ • ∇ × ∇ ⊥ • p ⊥ p ⊥ ϕ d 3 p , ∂ t A = [ψ, A] (43)
∂f ∂t + [f, ψ] + p • ∂f ∂x ⊥ -((ẑ • p ⊥ × ∇ ⊥ )∇ ⊥ ψ -v z ∇ ⊥ A) • ∂f ∂p ⊥ -p ⊥ • ∇ ⊥ A ∂f ∂v z = 0 (44)
Again, notice that the longitudinal coordinate z plays no role in the dynamics and thus it can be integrated out to yield the equations ( 11)- [START_REF] Hazeltine | Generalized Poisson brackets and nonlinear Liapunov stability-application to reduced MHD[END_REF].

  The authors acknowledge financial support by the Leverhulme Trust (Research Project Grant No. 2014-112). Also, CT acknowledges partial support by the Institute of Mathematics and its Applications (Grant No. SGS27/13) and by the London Mathematical Society (Grants No. 31320 & 41371). ET acknowledges financial support received from the Agence Nationale de la Recherche (ANR GYPSI n. 2010 BLAN 941 03) and from the CNRS through the PEPS project GEOPLASMA2. PJM was supported by U.S. Dept. of Energy Contract # DE-FG05-80ET-53088.

  δG δA d 3 x where we have used δG/δA = ẑ • δG/δA and we have introduced the planar bracket [k, h] = ∂ x k∂ y h -∂ y k∂ x h. More importantly, we have

														B •	δF δm	×	δG δm	= 0
	and
		f B •	δF δm	×	∂ ∂p	δG δf	d 3 xd 3 p = -f	∂ ∂p z	δG δf	∇ ⊥	δF δω	• ∇ ⊥ A × ẑd 3 x
																= -f	∂ ∂p z	δG δf	ẑ • ∇ ⊥	δF δω	× ∇ ⊥ Ad 3 x
																= f	∂ ∂p	δG δf	A,	δF δω	d 3 x.
	In addition, one computes
	f B •	∂ ∂p	δF δf	×	∂ ∂p	δG δf	d 3 x = f ∇ ⊥ A • ẑ ×	∂ ∂p	δF δf	×	∂ ∂p	δG δf	d 3 x
													= f ∇ ⊥ A •	∂ ∂p z	δG δf	∂ ∂p ⊥	δF δf	-	∂ ∂p z	δF δf	∂ ∂p ⊥	δG δf	d 3 x
	Thus, replacing the above results in (35) yields the following Poisson bracket of the planar
	incompressible CCS:
	{F, G} = ω	δF δω	,	δG δω	d 3 x + A		δF δω	,	δG δA	-	δG δω	,	δF δA	d 3 x
		+ f		∂ ∂v z	δG δf	δF δω	, A -	∂ ∂v z	δF δf	δG δω	, A d 3 xd 3 p
		+ f				δF δf	,	δG δf	+ ∇ ⊥ A •	∂ ∂p z	δG δf	∂ ∂p ⊥	δF δf	-	∂ ∂p z	δF δf	∂ ∂p ⊥	δG δf	d 3 xd 3 p

  and Poisson bracket[START_REF] Tronci | Hamiltonian approach to hybrid plasma models[END_REF] {F, G} P CS = m • The remaining terms in the bracket are computed as in the previous section of this Appendix. Then, the Poisson bracket for the incompressible hybrid PCS reads as

															δF δm	• ∇	δG δm	-	δG δm	• ∇	δF δm	d 3 x
			+ f		δF δf	,	δG δf	+ q h B •	∂ ∂p	δF δf	×	∂ ∂p	δG δf	d 3 x d 3 p
			+ f		δF δf	, p •	δG δm	-	δG δf	, p •	δF δm	d 3 x d 3 p
			+ B •		δF		× ∇ ×	δG δB	-	δG δm	× ∇ ×	δF δB	d 3 x
			-ρ	δF δm	• ∇	δG δρ	-	δG δm	• ∇	δF δρ	d 3 x .	(36)
	In 2D, the terms in the third row become
	f	δF δf	, p •	δG δm	d 3 x d 3 p = f	δF δf	, p • ∇ ⊥	δG δω	× ẑ d 3 x d 3 p	(37)
																=	δG δω	ẑ • ∇ × p ⊥	δF δf	, f d 3 x d 3 p	(38)
																= f	δF δf	, ẑ • p ⊥ × ∇ ⊥	δG δω	d 3 x d 3 p.	(39)
	{F, G} = ω	δF δω	,	δG δω		d 3 x + A		δF δω	,	δG δA	-	δG δω	,	δF δA	d 3 x	(40)
	+ f		δF δf	, ẑ • p ⊥ × ∇ ⊥	δG δω	-	δG δf	, ẑ • p ⊥ × ∇ ⊥	δF δω	d 3 xd 3 p	(41)
	+ f			δF δf	,	δG δf		+ ∇ ⊥ A •	∂ ∂v z	δG δf	∂ ∂p ⊥	δF δf	-	∂ ∂v z	δF δf	∂ ∂p ⊥	δG δf	d 3 xd 3 p (42)

While the Grad-Shafranov equation is traditionally written in toroidal geometry, equation (16) simply represents a slab version of the original. Following reference[START_REF] Holm | Nonlinear stability of fluid and plasma equilibria[END_REF], we shall also call[START_REF] Kim | Preliminary simulations of FLR effects on RFP tearing modes[END_REF] and its hybrid generalisation (see following sections) Grad-Shafranov equation.


Trung Hien Nguyen

Kevin Lenglé

Alexandre Bazin

Laurent Bramerie

Christophe Peucheret

Mathilde Gay

Olivier Sentieys

Jean-Claude Simon

Rama Raj

Fabrice Raineri

Phase-Preserving Power Limiting Function Using InP on SOI Photonic Crystal Nanocavity

Keywords: Photonic integrated circuits, photonic crystal nanocavity, power-limiter, optical signal processing

published or not. The documents may come

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I. INTRODUCTION

ll-optical signal processing relies on the availability of nonlinear platforms able to operate at high speed with compact size and low power consumption. For future photonic circuits, InP over silicon-on-insulator (SOI) hybrid technology is an extremely promising solution as it combines CMOS compatibility with the optoelectronic properties of III-V materials. The III-V/SOI hybrid technology has recently allowed the demonstration of lasers [START_REF] Halioua | Hybrid III-V semiconductor/silicon nanolaser[END_REF], amplifiers [START_REF] Park | A hybrid AlGaInAs-silicon evanescent amplifier[END_REF], modulators, flip-flops [START_REF] Van Thourhout | Nanophotonic devices for optical interconnect[END_REF], wavelength converters [START_REF] Kumar | Ultrafast and bias-free all-optical wavelength conversion using III-V-on-silicon technology[END_REF], modulation format converters [START_REF] Kumar | 10 Gbit/s all-optical NRZ-OOK to RZ-OOK format conversion in an ultra-small III-V-on silicon microdisk fabricated in a CMOS pilot line[END_REF] and 2R-regenerators, including power-limiters [START_REF] Mechet | All-optical low-power 2R regeneration of 10 Gb/s NRZ signals using a III-V on SOI microdisk laser[END_REF].

Thanks to their small achievable size with strong optical confinement and enhanced nonlinearity originating from their resonant behavior, photonic crystal (PhC) nanocavities are considered to be promising building blocks for high-density photonic integrated circuits. In particular, all-optical switching has been recently demonstrated using an InP on SOI PhC Manuscript received Y X th , 2014. This work was supported by the European Union (Copernicus project), the Contrat de Plan Etat-Région Ponant and the French Ministry of Research.

We acknowledge I. Sagnes and G. Beaudoin for material growth, R. Braive for participating in the processing of the samples.

T.H. Nguyen, K. Lengle, L. Bramerie, C. Peucheret, M. Gay, J.C. Simon are with Université Européenne de Bretagne (UEB), CNRS-Foton Laboratory (UMR 6082) / Université de Rennes 1, Enssat, BP 80518, 22305 Lannion Cedex, France (e-mail: trung-hien.nguyen@enssat.fr).

O. Sentieys is with INRIA/ IRISA, Campus Beaulieu, 35000 Rennes, France.

A. Bazin, R. Raj, F. Raineri are with Laboratoire de Photonique et de Nanostructures (CNRS UPR 20), route de Nozay, 91460 Marcoussis, France.

F. Raineri is with Universite Paris Diderot, Sorbonne Paris Cité, 75207 Paris Cedex 13, France. nanocavity, and applied to wavelength conversion up to 20 Gbit/s [START_REF] Lengle | Modulation contrast optimization for wavelength conversion of a 20 Gbit/s data signal in hybrid InP/SOI photonic crystal nanocavity[END_REF]. However, no detailed analysis of the gate's operational potential as an optical power-limiter for telecommunication signals has been attempted so far.

Over the last decade, there has been a renewed interest for coherent optical communication systems thanks to their improved receiver sensitivity and spectral efficiency, and ability to mitigate transmission impairments in the digital domain [START_REF] Ip | Coherent detection in optical fiber systems[END_REF]. In particular, the quadrature phase-shift-keying (QPSK) format has been the object of intensive investigations and is now employed in commercial 100-Gbit/s transmission solutions. However, the conversion of amplified spontaneous emission (ASE) noise by the transmission fiber Kerr nonlinearity is known to cause nonlinear phase noise (NPN) [START_REF] Gordon | Phase noise in photonic communication systems using linear amplifiers[END_REF], which may severely degrade QPSK signals, and whose compensation in the digital domain results in increased complexity for the receiver. Instead, amplitude limiters can be used to reduce the accumulation of NPN along the link. It is therefore essential to show that the powerlimiting function we presented in [START_REF] Lengle | Modulation contrast optimization for wavelength conversion of a 20 Gbit/s data signal in hybrid InP/SOI photonic crystal nanocavity[END_REF] is compatible with QPSK signals with negligible modifications of the phase information.

Phase-preserving amplitude regeneration has been demonstrated using, among others, parametric processes in highly nonlinear fibers [START_REF] Matsumoto | Performance improvement of DPSK signal transmission by a phase-preserving amplitude limiter[END_REF] or nonlinear amplifying loop mirrors [START_REF] Boscolo | Design of Raman-based nonlinear loop mirror for alloptical 2R regeneration of differential phase-shift-keying transmission[END_REF][START_REF] Stephan | Phase-preserving amplitude regeneration in DPSK transmission systems using a nonlinear amplifying loop mirror[END_REF], which are practically limited by a weak integration capacity. Other techniques using saturable absorbers [START_REF] Bramerie | All-optical 2R regeneration with a vertical microcavity-based saturable absorber[END_REF] suffer from high power requirements. As far as PhCs are concerned, some theoretical studies have been conducted on the power-limiting function [START_REF] Bahl | Nonlinear optical effects in a two-dimensional photonic crystal containing one-dimensional Kerr defects[END_REF], without experimental demonstration so far, nor consideration on the phase.

In this letter, a phase-preserving all-optical power-limiter function is experimentally demonstrated by using a PhC nanocavity based on III-V/SOI hybrid technology. This cavity incorporates a material optimized for ultrafast operation. The limiter performance is investigated with different artificial amplitude noise levels for non return-to-zero (NRZ) QPSK signals at 20 Gbit/s and estimated in terms of bit error ratio (BER) and phase distribution of the multilevel modulation signal. Improvements of the receiver sensitivity demonstrate the effectiveness of the nanocavity used as a power limiter. Furthermore, amplitude regeneration by nanocavity switching is reported for the first time for a QPSK signal with coherent detection, showing no substantial phase distortion, thus The passive device, presented in Fig. 1, is composed of two optical levels. The first level consists of a single mode SOI wire waveguide, while the second one is an InP-based PhC nanocavity [START_REF] Karle | Heterogeneous integration and precise alignment of InP-based photonic crystal lasers to complementary metal-oxide semiconductor fabricated silicon-on-insulator wire waveguides[END_REF]. These two levels are separated by a low-index layer (made from silica and benzocyclobutene), which preserves the vertical optical confinement within the SOI waveguide and the PhC cavity. Optical coupling between the two levels, ensured by the penetration of the evanescent tails of the optical modes into the other level, has been measured to be up to 90% in a similar structure [START_REF] Halioua | Hybrid III-V semiconductor/silicon nanolaser[END_REF]. The studied nanocavity is a "nanobeam" cavity [START_REF] Foresi | Photonic-bandgap microcavities in optical waveguides[END_REF][START_REF] Bazin | Design of silica encapsulated high-Q photonic crystal nanobeam cavity[END_REF], which is formed by drilling holes (targeted radius r = 120 nm, lattice constant around 350 nm) in a single mode InP wire waveguide (650 nm × 260 nm × 15 µm) where a single InGaAs quantum well (8 nm) is grown at both horizontal surfaces. The distance between the holes is subtly varied (around 300 nm) [START_REF] Bazin | Design of silica encapsulated high-Q photonic crystal nanobeam cavity[END_REF] in order to obtain a Gaussian apodisation of the electromagnetic field profile. To avoid heating, which usually hampers operation at high bit rates, the cavity is fully encapsulated in SiO 2 in order to increase heat evacuation. The SOI waveguide, with an initial width of 400 nm, is tapered up to a width of 10 µm so as to etch grating couplers at each extremity to allow coupling with cleaved-facet single-mode optical fibers [START_REF] Taillaert | Grating couplers for coupling between optical fibers and nanophotonic waveguides[END_REF]. Optimal coupling at 1550 nm is obtained with a fiber angle of 10°. This tapered structure leads to a total device length of 8 mm.

The cavity was first characterized under static operation in order to locate the transmission resonance (Fig. 2(a)). A lowpower continuous wave (CW) signal was coupled into the silicon waveguide via the grating coupler and the transmission was measured as a function of wavelength. The Q-factor of the cavity is around 500. The minimum fiber-to-fiber insertion loss is 12 dB and mainly originates from fiber-to-waveguide coupling loss (25% estimated coupling efficiency) since the coupling from the SOI waveguide to the InP-based PhC cavity itself is very efficient [START_REF] Halioua | Hybrid III-V semiconductor/silicon nanolaser[END_REF].

The noise limiter function relies on the self-switching of the cavity under the action of modulated signal intensity. A powerdependent refractive index change shifts the resonance of the cavity towards shorter wavelengths via a change in the carrier density. This frequency shift is schematically represented in Fig. 2(b) to illustrate the principle. If the signal has intensity fluctuations, the dynamic shift will be different depending on the instantaneous intensity of the pulse (shown by the blue or red transmission profiles). The wavelength shift is estimated to be less than 1 nm under our experimental conditions and enables power limitation, provided that the signal is positioned on the shorter wavelength side of the resonance. The shifted resonance is actually also deeper compared to the initial static resonance emphasizing the transmission loss [START_REF] Lengle | Modulation contrast optimization for wavelength conversion of a 20 Gbit/s data signal in hybrid InP/SOI photonic crystal nanocavity[END_REF]. By tuning the signal to a suitable wavelength, a transmission function such as the one represented in Fig. 2(c) can be obtained, allowing intensity noise limitation to be performed. Note that, the resonance wavelength can be appropriately tuned with respect to the channel by a suitable thermal control of the device. The response time of the device was evaluated to be as short as 30 ps [START_REF] Bazin | Ultrafast all-optical switching and error-free 10 Gbit/s wavelength conversion in hybrid InP-silicon on insulator nanocavities using surface quantum wells[END_REF] thanks to the engineered enhancement of nonradiative recombination of carriers, both through surface growth and by patterning the material with dimensions smaller than the carrier diffusion length. The power-limiter function was demonstrated using the implementation of Fig. 3. To evaluate the potential of the device as a power limiter, intensity fluctuations were artificially added to a 10-Gbaud QPSK signal. Intensity noise was generated by externally modulating the output of a CW laser with a pseudo-random binary sequence (PRBS) of length 2 7 -1 at 10 Gbit/s. This optical noisy source was then modulated in an IQ modulator driven by two decorrelated PRBSs with lengths of 2 15 -1 and 2 23 -1 obtained from arbitrary waveform generators (AWGs), resulting in a 10-Gbaud NRZ-QPSK signal. The noisy modulated signal had its state-ofpolarisation aligned to the TE mode of the input waveguide before being injected into the component at a slightly blue shifted wavelength compared to the cavity resonance. A variable optical attenuator (VOA) followed by an erbiumdoped fiber amplifier (EDFA) and a 3-nm optical bandpass filter (BPF) allowed the adjustment of the optical signal-to-noise ratio (OSNR) at the receiver input. An optical 90° hybrid (100-Gbit/s DP-QPSK integrated receiver) mixed the QPSK signal and the local oscillator (LO). To simplify the setup, a homodyne configuration was employed, with the same laser (linewidth of about 100 kHz) used at the transmitter and as LO in the receiver. The in-phase and quadrature components were then detected by balanced photodiodes with a bandwidth of 32 GHz. Finally, the electrical output signals were measured by a real-time oscilloscope with electrical bandwidth of 16 GHz. After data acquisition at 20 GSample/s, postprocessing was performed offline using the Matlab environment. Re-timing of the samples was realized with a well-known adaptive finite impulse response (FIR) filter [START_REF] Proakis | Digital Communications[END_REF] acting as equalizer. After convergence of the feedback loop of the equalizer, the following processes were performed with one sample per symbol after decimating the symbol rate to 10 GSymbol/s. The offline processing was applied to blocks of 2×10 5 samples. The widely-used Viterbi and Viterbi algorithm for carrier phase estimation was performed on blocks of 50 samples, enabling to reconstruct the constellations. Moreover, the phase ambiguity was alleviated by processing the first 5000 samples to minimize the BER. Finally, the decoded bit sequence was compared to the transmitted one to determine the BER over 4 millions samples. Fig. 4 shows examples of constellations of the received QPSK signal for an OSNR of 13 dB (defined over a 0.1-nm noise bandwidth). Compared to the reference constellation in Fig. 4(a), the effect of the added amplitude fluctuations manifests itself as a splitting of each phase modulation state into two lobes in the constellation diagram, as shown in Fig. 4(b). After passing through the power-limiter, the regenerated constellation (Fig. 4(c)) clearly indicates the effect of intensity noise suppression by merging the split lobes again, even though some residual intensity noise remains compared to the reference constellation.

Input signal

Input power Transmission Transmission function

Output signal

Input signal

Input power Transmission Transmission function

Output signal

In order to investigate quantitatively the impact of the noise level, the phase modulation was removed by elevating the processed complex signal samples to the fourth power, so that the phase distributions of all symbols are superimposed at the same angle. The noise magnification induced by the 4 th power exponentiation was then compensated by inverse operations (division by four for the phase noise and one-fourth power operation for the amplitude noise). The amplitude and phase distributions of the samples can then be calculated. The amplitude distributions are plotted in Fig. 4(d), in the case of the degraded signal and in Fig. 4(e) in the regenerated case. To be able to evaluate the functionality of the powerlimiter, an estimated SNR, SNR est , is used to quantify the noise level. This parameter is defined as SNR est = 10×log(|µ|²/σ 2 ), where µ and σ 2 are the average and variance values of the squared modulus of the processed signal samples. SNR est is calculated to be 9.8 dB before regeneration and 12.2 dB in the regenerated case, showing an amplitude noise reduction of 2.4 dB.

The impact of the regenerator on the phase of the signal was investigated next for an OSNR at the receiver of 13 dB and for the same input noise level of SNR est = 9.8 dB. The phase variation was evaluated by the standard deviation of the phase distribution relatively to the average value. The resulting histograms of the phase distribution obtained before and after regeneration using 2×10 5 symbols are depicted in Fig. 5(a) and (b), respectively. Fig. 5(c) shows the evolution of the phase distribution standard deviation as a function of amplitude SNR based noise level, without regeneration in grey and with regeneration in black. It can be seen that the added amplitude noise level is accompanied with an increase in phase noise variance. However, the standard deviation of the phase noise after the cavity is only marginally lower (about 0.02 at high amplitude noise levels) than that of the unregenerated signal, which proves the phase-preservation of the regenerated signal. The amplitude-to-phase noise transfer induced by the amplitude noise reduction process is therefore very small.

To assess this quality improvement, BER measurements were performed on signals with the same amplitude noise level as in Fig. 5. The theoretical BER curve of a QPSK signal is plotted in Fig. 6 (green diamonds) based on the analysis in [START_REF] Proakis | Digital Communications[END_REF]. The experimental back-to-back measurement (blue stars) matches the theoretical curve with minor differences. First, BER measurements were performed when the transmitter was tuned away from the resonance without additional amplitude noise (red empty circles). This result demonstrates that the transmission through the component does not have any impact on the signal compared to the back-to-back case. Next, the impact of the device away from its resonance on a signal with added amplitude noise was assessed by comparing the resulting BER curve (red filled circles) with that obtained by replacing the device with an attenuator having the same insertion loss (grey filled triangles). Both curves are quite similar, which implies that the effect of the device on the BER performance is negligible away from the resonance. Finally, the measurement was performed on the degraded signal after being switched through the cavity resonance (black cross symbols). This result points out a penalty reduction of up to 6 dB at a BER of 10 -3 compared to the degraded signal, demonstrating the efficiency of the amplitude noise limiter for phase encoded modulation formats. These results were obtained with a sub-milliwatt coupled peak power. Finally, the efficiency of the device was investigated as a function of amplitude noise level. BER measurements were performed for different artificial amplitude noise levels and a received OSNR of 10 dB, as shown in Fig. 7, with (black diamond), and without (grey circle) the device. The BER is severely degraded at high-amplitude noise levels compared to low noise levels. However it can be noticed that, over the studied noise level range, the BER is improved by one order of magnitude after the cavity and that the improvement remains constant regardless of the input amplitude noise level. At a noise level SNR est =13 dB, performance similar to the back-toback case is obtained after regeneration, confirming the effectiveness of the power-limiter.

IV. CONCLUSION

In this letter, an experimental analysis of the power-limiting properties of an optical gate based on a hybrid InP PhC nanocavity on an SOI substrate is reported. Effective intensity noise reduction was obtained for 20-Gbit/s NRZ -QPSK signals. The phase preserving capability of the power-limiter was demonstrated, with up to 6 dB power penalty reduction at a BER of 10 -3 .

Fig. 1 .

 1 Fig. 1. (a) Device structure and (b) scanning electron microscope picture of the fabricated component.

Fig. 2 .Fig. 3 .

 23 Fig. 2. (a) Measured transmission of the cavity (resolution 0.01 nm). (b) Principle of the power limiter function. (c) Transmission function.III. SYSTEM ASSESSMENTAWGEDFABPF PC

Fig. 4 .

 4 Fig. 4. Experimental constellation diagrams for OSNR at the receiver of 13 dB: (a) Reference without additional noise. (b) Degraded signal with additional noise. (c) Regenerated signal through the nanocavity. Amplitude distributions of (d) the degraded signal and (e) the regenerated signal.

Fig. 5 .

 5 Fig. 5. Impact of the limiter on the phase (for OSNR at the receiver of 13 dB). (a) Phase histogram of the degraded signal. (b) Phase histogram of the regenerated signal. (c) Evolution of the standard deviation of the phase with the imposed amplitude noise level.

Fig. 6 .

 6 Fig. 6. BER curves of the power-limiter for 20-Gbit/s QPSK (for artificial amplitude noise level SNRest=9.8 dB).

Fig. 7 .

 7 Fig. 7. Evolution of the BER as a function of the amplitude noise level for a 20-Gbit/s QPSK signal with and without power limitation (for an OSNR at the receiver of 10 dB).

Injection fiber (10°) Grating coupler Si waveguide PhC cavity with surface QWs SiO 2 overcladding for thermal dissipation Si substrate SiO 2 buffer Bonding layer (BCB) Injection fiber (10°) Grating coupler Si waveguide PhC cavity with surface QWs SiO 2 overcladding for thermal dissipation Si substrate SiO 2 buffer Bonding layer (BCB)

	demonstrating the compatibility of the scheme with this
	Phase-Preserving Power Limiting Function advanced optical modulation format.
	Using InP on SOI Photonic Crystal Nanocavity II. DEVICE DESCRIPTION AND POWER LIMITING PRINCIPLE
	T. H. Nguyen, K. Lengle, A. Bazin, L. Bramerie, C. Peucheret, M. Gay, O. Sentieys, J. C. Simon,
	R. Raj and F. Raineri.
	A