Rémi Buffe

Takéo Takahashi

Controllability of a simplified fluid-structure interaction system

Keywords: Null controllability, Navier-Stokes systems, Carleman estimates 2010 Mathematics Subject Classification. 76D05, 35Q30, 93B05, 93B07, 93C10

We are interested by the controllability of a fluid-structure interaction system where the fluid is viscous and incompressible and where the structure is elastic and located on a part of the boundary of the fluid's domain. In this article, we simplify this system by considering a linearization and by replacing the wave/plate equation for the structure by a heat equation. We show that the corresponding system coupling the Stokes equations with a heat equation at its boundary is null-controllable. The proof is based on Carleman estimates and interpolation inequalities. One of the Carleman estimates corresponds to the case of Ventcel boundary conditions. This work can be seen as a first step to handle the real system where the structure is modeled by the wave or the plate equation.

Introduction

Fluid-structure interaction systems are important systems for many applications such as aerodynamics, medicine (for instance the study of the motion of the blood in veins or in arteries), biology (animal locomotion in a fluid), civil engineering (design of bridges), naval architecture (design of boats and submarines), etc. Moreover, their mathematical studies can be very challenging due to several difficulties: in particular, the complexity of fluid equations such as the Navier-Stokes system, the strong coupling between the fluid system and the structure system and the free-boundary corresponding to the structure displacement.

In this article we consider a simplified fluid-structure interaction system. The corresponding system without simplification has been proposed in [START_REF] Quarteroni | Computational vascular fluid dynamics: problems, models and methods[END_REF] as a model for the blood flow in a vessel. It writes as follows: we denote by I the torus (in order to consider periodic boundary conditions):

I := R/(2πZ),
and for any deformation : I → (-1, ∞), we consider the corresponding fluid domain Ω = {(x 1 , x 2 ) ∈ I × R ; x 2 ∈ (0, 1 + (x 1 ))} .

(1.1)

Then the equations of motion are

           ∂ t w + (w • ∇)w -div T(w, π) = 0 t > 0, x ∈ Ω (t)
, div w = 0 t > 0, x ∈ Ω (t) , w(t, x 1 , 1 + (t, x 1 )) = (∂ t )(t, x 1 )e 2 t > 0, x 1 ∈ I,

w = 0 t > 0, x ∈ Γ 0 , ∂ tt + α 1 ∂ 4 x1 -α 2 ∂ 2 x1 -δ∂ t ∂ 2 x1 = -H (w, π) t > 0, x 1 ∈ I, (1.2)
where Γ 0 = I × {0}, Γ 1 = I × {1}.

In the above system, we have used the following notations: (e 1 , e 2 ) is the canonical basis of R 2 and T(w, π) = 2D(w) -πI 2 , D(w) = 1 2 (∇w + (∇w) * ) ,

H (w, π)(t, x 1 ) = (1 + |∂ x1 | 2 ) 1/2 [T(w, π)n] (t, x 1 , 1 + (t, x 1 )) • e 2 .

(1.4)

The two first lines of (1.2) correspond to the Navier-Stokes system for the fluid velocity w and the pressure π. The last line of (1.2) is a beam equation satisfied by the deformation . We have used the standard noslip boundary conditions (third and forth equations). To simplify, we assume that the viscosity of the fluid is constant and equal to 1. The vector fields n corresponds to the unit exterior normal to Ω (t) . This system has been studied by many authors: [START_REF] Chambolle | Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate[END_REF] (existence of weak solutions), [START_REF] Veiga | On the existence of strong solutions to a coupled fluid-structure evolution problem[END_REF], [START_REF] Lequeurre | Existence of strong solutions to a fluid-structure system[END_REF], [START_REF] Grandmont | Existence of global strong solutions to a beam-fluid interaction system[END_REF] and [START_REF] Maity | Lp theory for the interaction between the incompressible Navier-Stokes system and a damped plate[END_REF] (existence of strong solutions), [START_REF] Raymond | Feedback stabilization of a fluid-structure model[END_REF] (stabilization of strong solutions), [START_REF] Badra | Feedback boundary stabilization of 2D fluid-structure interaction systems[END_REF] (stabilization of weak solutions around a stationary state). There are also some works in the case δ = 0, that is without damping on the beam equation: the existence of weak solutions is proved in [START_REF] Grandmont | Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate[END_REF] and in [START_REF] Muha | Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls[END_REF] (see also [START_REF] Čanić | Fluid-structure interaction in hemodynamics: modeling, analysis, and numerical simulation[END_REF]). In [START_REF] Grandmont | Existence of local strong solutions to fluidbeam and fluid-rod interaction systems[END_REF], the existence of local strong solutions is obtained for a structure described by either a wave equation (α 1 = δ = 0 and α 2 > 0) or a beam equation with inertia of rotation (α 1 > 0, α 2 = δ = 0 and with an additional term -∂ ttss ). In [START_REF] Badra | Gevrey regularity for a system coupling the Navier-Stokes system with a beam equation[END_REF] and [START_REF] Badra | Gevrey regularity for a system coupling the navier-stokes system with a beam: the non-flat case[END_REF], the authors show the existence and uniqueness of strong solutions in the case α 1 > 0, α 2 0 and δ = 0. Using similar techniques they also analyze the case of the wave equation (α 1 = δ = 0 and α 2 > 0) in [START_REF] Badra | Analyticity of the semigroup associated with a Stokes-wave interaction system and application to the system of interaction between a viscous incompressible fluid and an elastic structure[END_REF] showing in particular that the semigroup of the linearized system is analytic. Let us mention also some results for more complex models: [START_REF] Lengeler | Weak solutions for an incompressible Newtonian fluid interacting with a Koiter type shell[END_REF][START_REF] Lengeler | Weak solutions for an incompressible, generalized Newtonian fluid interacting with a linearly elastic Koiter type shell[END_REF] (linear elastic Koiter shell), [START_REF] Muha | Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls[END_REF] (dynamic pressure boundary conditions), [START_REF] Muha | A nonlinear, 3D fluid-structure interaction problem driven by the timedependent dynamic pressure data: a constructive existence proof[END_REF][START_REF] Muha | Fluid-structure interaction between an incompressible, viscous 3D fluid and an elastic shell with nonlinear Koiter membrane energy[END_REF] (3D cylindrical domain with nonlinear elastic cylindrical Koiter shell), [START_REF] Trifunović | Existence of a weak solution to the fluid-structure interaction problem in 3D[END_REF] and [START_REF] Trifunović | Weak solution to the incompressible viscous fluid and a thermoelastic plate interaction problem in 3D[END_REF] (nonlinear elastic and thermoelastic plate equations), [START_REF] Maity | Existence of strong solutions for a system of interaction between a compressible viscous fluid and a wave equation[END_REF], [START_REF] Maity | Existence and uniqueness of strong solutions for the system of interaction between a compressible Navier-Stokes-Fourier fluid and a damped plate equation[END_REF] (compressible fluids), etc.

The advantage of the damping in the beam equation is that the term -δ∂ t ∂ 2 x1 is a structural damping so that the corresponding beam equation becomes a parabolic equation (see, for instance, [START_REF] Shu | Proof of extensions of two conjectures on structural damping for elastic systems[END_REF]). In this work,

Ω Γ 0 -- -- Γ 1 ("beam") 0 2π 1 Figure 1: Our geometry
we consider a simplified model associated with (1.2). We neglect the deformation of the fluid domain due to the elastic déformation and we also linearized the Navier-Stokes system by only considering the Stokes system. Moreover, we replace the damped beam equation by a heat equation. By setting

Ω = I × (0, 1)
(see Figure 1), we are thus considering the following system

               ∂ t w -∆w + ∇π = 1 ω f in (0, T ) × Ω, div w = 0 in (0, T ) × Ω, w = 0 on (0, T ) × Γ 0 , w = ζe 2 on (0, T ) × Γ 1 , ∂ t ζ -∂ x1x1 ζ = -T(w, π)n • e 2 in (0, T ) × I, w(0, •) = w 0 in Ω, ζ(t, 0) = ζ 0 in I.
(1.5)

In the above system, ζ corresponds to the displacement velocity ∂ t in (1.2) and we do not consider anymore the displacement position . We have added a control f localized in the fluid domain, in an arbitrary small nonempty open set ω of Ω. Our goal is to show the null-controllability of this simplified system and to do this, as it is standard (see, for instance, [51, Theorem 11.2.1, p.357]), we prove an observability inequality on the adjoint system:

               ∂ t u -∆u + ∇p = 0 in (0, T ) × Ω, div u = 0 in (0, T ) × Ω, u = 0 on (0, T ) × Γ 0 , u = ηe 2 on (0, T ) × Γ 1 , ∂ t η -∂ x1x1 η = -T(u, p)n |Γ1 • e 2 in (0, T ) × I, u(0, •) = u 0 in Ω, η(0, •) = η 0 in I. (1.6)
We set

L 2 0 (I) := f ∈ L 2 (I) ; 2π 0 f (x 1 ) dx 1 = 0
and we define the space

H := [u, η] ∈ L 2 (Ω) × L 2 0 (I) ; u 2 = 0 on Γ 0 , u 2 = η on Γ 1 , div u = 0 in Ω . (1.7)
Then, our main result states as follows:

Theorem 1.1. Let γ > 1 and let ω be a nonempty open set of Ω. Then, there exists C 0 > 0 such that for any T ∈ (0, 1) and for any u 0 , η 0 ∈ H, the solution [u, η] of (1.6) satisfies

[u(T, •), η(T, •)] 2 H C 0 exp C 0 T γ (0,T )×ω |u| 2 dtdx.
In particular, for any w 0 , ζ 0 ∈ H and for any T > 0, there exists a control f ∈ L 2 ((0,

T ) × ω) such that the solution [w, ζ] of (1.5) satisfies w(T, •) = 0, ζ(T, •) = 0.
Remark 1.2. As explained above, Theorem 1.1 can be seen as a first control result on a simplified model. We expect to extend some of the tools developed here to handle the control properties of the system (1.2) in future works. The controllability properties of fluid-structure interaction systems have been tackled mainly in the case where the structure is a rigid body (see, [START_REF] Ramaswamy | Remark on the global null controllability for a viscous Burgers-particle system with particle supported control[END_REF], [START_REF] Doubova | Some control results for simplified one-dimensional models of fluid-solid interaction[END_REF], [START_REF] Liu | Single input controllability of a simplified fluid-structure interaction model[END_REF], [START_REF] Imanuvilov | Exact controllability of a fluid-rigid body system[END_REF], [START_REF] Boulakia | Local null controllability of a two-dimensional fluid-structure interaction problem[END_REF], [START_REF] Boulakia | Local null controllability of a fluid-solid interaction problem in dimension 3[END_REF], [START_REF] Roy | Local null controllability of a rigid body moving into a Boussinesq flow[END_REF], [START_REF] Imene | Local null controllability of a fluid-rigid body interaction problem with Navier slip boundary conditions[END_REF], [START_REF] Cîndea | Particle supported control of a fluidparticle system[END_REF], etc.). In [START_REF] Sourav | Observability and unique continuation of the adjoint of a linearized simplified compressible fluid-structure model in a 2D channel[END_REF], the author shows an observability inequality for the adjoint of a linearized and simplified fluid-structure interaction system in the case of a compressible viscous fluid and of a damped beam. Note that the corresponding control problem involves two controls, one for the fluid and one for the structure.

For the stabilization of fluid-structure interaction systems, one can quote some results: [START_REF] Raymond | Feedback stabilization of a fluid-structure model[END_REF], [START_REF] Badra | Feedback boundary stabilization of 2D fluid-structure interaction systems[END_REF] (for the case of a damped beam), [START_REF] Takahashi | Stabilization of a fluid-rigid body system[END_REF], [START_REF] Roy | Stabilization of a rigid body moving in a compressible viscous fluid[END_REF], [START_REF] Badra | Feedback stabilization of a simplified 1d fluid-particle system[END_REF], [START_REF] Badra | Feedback stabilization of a fluid-rigid body interaction system[END_REF] (for the case of a rigid body).

Remark 1.3. The method proposed here to control a system involving the Stokes equations is quite different from the method used in a large part of the literature for the controllability of fluid systems. In general, the method is based on "global Carleman inequalities" (see, for instance [START_REF] Fernández-Cara | Local exact controllability of the Navier-Stokes system[END_REF][START_REF] Yu | On exact controllability for the Navier-Stokes equations[END_REF]). Here, we follow another strategy as in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF]. Such a method is based on local Carleman inequalities for an "augmented" elliptic operator, from which one deduces a spectral inequality, in the spirit of [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Jerison | Nodal sets of sums of eigenfunctions[END_REF]. However, as it is pointed out in [START_REF] Chaves-Silva | Spectral inequality and optimal cost of controllability for the Stokes system[END_REF], unique continuation property does not hold for the augmented operator in the direction of the additional variable, due to the pressure. We then use an adaptation of the original strategy of [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF][START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the nullcontrollability of parabolic systems[END_REF] in our context. This type of spectral inequality has already been used in the context of fluids in [START_REF] Buffe | Spectral inequality for an Oseen operator in a two dimensional channel[END_REF]. We also recall that one can use Theorem 1.1 to handle nonlinear controllability issues by applying the general method proposed in [START_REF] Liu | Single input controllability of a simplified fluid-structure interaction model[END_REF].

Remark 1.4. Using the particular geometry considered here, we can simplify the adjoint system. First on Γ 1 , n = e 2 and using (1.3), we deduce

-T(u, p)n • e 2 = -2∂ x2 u 2 + p = 2∂ x1 u 1 + p = p, (1.8)
since u 1 (x 1 , 1) = 0 for x 1 ∈ I. Moreover, using the incompressibility of the fluid and the boundary conditions, we deduce that

0 = Ω div u dx = I η dx 1 .
Using this condition on the heat equation on the boundary and (1.8) yields

I p(x 1 , 1) dx 1 = 0. (1.9)
In particular, in contrast with the standard Stokes system, the pressure is not determined up to a constant.

The outline of the article is as follows: in Section 2, we show how to obtain the observability inequality stated in Theorem 1.1 from a spectral inequality. Such a result is quite standard, but here we show that for a self-adjoint operator, we do not need the usual assumption that is made on the eigenvalues of the main operator. Then using this general result, we are reduced to show a spectral inequality that we state in Section 3 along with the functional framework. The spectral inequality is itself the consequence of an interpolation inequality that we obtain in Section 5. One of the main difficulties to obtain such an inequality comes from the fact that we need to estimate the pressure. Section 4 is devoted to such an estimate which is one of the main parts of this article. The proof of the spectral inequality and thus of Theorem 1.1 is obtained at the end of Section 5. In the appendix, we show an interpolation estimate for the Ventcel boundary condition that is mainly a consequence of a Carleman estimate obtained in [START_REF] Buffe | Stabilization of the wave equation with Ventcel boundary condition[END_REF]. Notation 1.5. In the whole paper, we use C as a generic positive constant that does not depend on the other terms of the inequality. The value of the constant C may change from one appearance to another. We also use the notation X Y if there exists a constant C > 0 such that we have the inequality X CY .

From a spectral inequality to the null-controllability

This section is devoted to a "classical" result showing that a spectral inequality implies the final-state observability and thus the null-controllability. The proof follows closely the proof in [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the nullcontrollability of parabolic systems[END_REF][START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF] and we only prove it here for sake of completeness and also to show that we do not need any assumption on the asymptotic behavior of the spectrum of the operator (which is used in the above references).

More precisely, we assume here that A : D(A) → H is a positive self-adjoint operator with compact resolvents in an Hilbert space H. We denote by (λ j ) the nondecreasing sequence of eigenvalues and by (w j ) an orthonormal basis of H composed by eigenvectors of A: Aw j = λ j w j for j 1. We also consider a control operator B ∈ L(U, H).

Theorem 2.1. Assume the above hypotheses. Assume moreover the existence of S 0 > 0, C > 0 and κ ∈ C ∞ 0 (0, S 0 ) such that that for any Λ > 0, and for any

(a j ) j ∈ C N , λj Λ |a j | 2 Ce C √ Λ S0 0 κ 2 (s) λj Λ a j cosh(s √ λ j )B * w (j) 2 U ds. (2.1)
Then for all γ > 1, there exists C 0 > 0 such that for any T ∈ (0, 1) and for any z 0 ∈ H,

e -T A z 0 2 H C 0 exp C 0 T γ T 0 B * e -tA z 0 2 U dt. (2.2)
We recall that relation (2.2) implies the null-controllability of the system

   dθ dt + Aθ = Bg in (0, T ), θ(0) = θ 0 ∈ H. (2.3)

Controllability of the first modes

We define H Λ = span {w j , λ j Λ} , Π Λ : H → H Λ the orthogonal projection.

We are interested here by the control problem

   dθ dt + Aθ = Π Λ Bg in (0, τ ), θ(0) = θ 0 ∈ H Λ , (2.4)
for some τ > 0. We consider the linear operator

G Λ := S0 0 κ 2 (s) cosh(s √ A)Π Λ BB * cosh(s √ A)Π Λ ds. From (2.1), G Λ ∈ L(H Λ ) is symmetric, positive and invertible with G -1 Λ L(HΛ) Ce C √ Λ .
We set

σ := 2 - 1 γ ∈ (1, 2).
From [26, Lemma A.1], there exists e ∈ C ∞ (R) such that for some constants c j

supp e = [0, 1], (2.5
)

| e(z)| c 1 e -c2|z| 1/σ if Im(z) 0, (2.6
)

| e(z)| c 3 e -c4|z| 1/σ if z ∈ iR -. (2.7)
From (2.7), we have that e(-iτ A) ∈ L(H Λ ) is invertible and

e(-iτ A) -1 L(HΛ) 1 c 3 e c4(Λτ ) 1/σ . (2.8) Then we define h Λ (s) := - 1 2 κ 2 (s)B * cosh(s √ A)G -1 Λ e(-iτ A) -1 e -τ A θ 0 (s ∈ R). We have that h Λ ∈ C ∞ 0 (R, U) with supp h Λ ⊂ (0, S 0 ) and h Λ L ∞ (R,U ) Ce C √ Λ+c4(Λτ ) 1/σ θ 0 H . (2.9)
Thus, h Λ ∈ Hol(C; U) and

h Λ (z) U Ce C √ Λ+c4(Λτ ) 1/σ e S0|Im(z)| θ 0 H . (2.10)
As in [START_REF] Russell | A unified boundary controllability theory for hyperbolic and parabolic partial differential equations[END_REF], we introduce

Q Λ ∈ Hol(C; U) such that Q Λ (-iz 2 ) = h Λ (iz) + h Λ (-iz) (z ∈ C).
We deduce from the above relation and (2.10) that

Q Λ (z) U Ce C √ Λ+c4(Λτ ) 1/σ e S0 √ |z| θ 0 H . (2.11) We define g Λ ∈ Hol(C; U) by g Λ (z) := e(τ z)Q Λ (z).
From (2.5), (2.6) and (2.11), we have

g Λ (z) U Ce C √ Λ+c4(Λτ ) 1/σ e S0 √ |z| e τ | Im z| θ 0 H (z ∈ C) (2.12) and g Λ (z) U Ce C √ Λ+c4(Λτ ) 1/σ e S0 √ |z| e -c2τ 1/σ |z| 1/σ θ 0 H if Im z 0. (2.13)
Since σ < 2, we can use a Paley-Wiener type theorem (see [START_REF] Léautaud | Spectral inequalities for non-selfadjoint elliptic operators and application to the nullcontrollability of parabolic systems[END_REF]Proposition A.3]) and deduce the existence of

g Λ ∈ C ∞ 0 ((0, τ ); U) such that g Λ (z) = g Λ (z).
In particular, from (2.13) and the Laplace method, for all t ∈ (0, τ ),

|g Λ (t)| g Λ L 1 (R;U ) Ce C √ Λ+c4(Λτ ) 1/σ + C τ 1/(2-σ) θ 0 H . (2.14)
Now, for any j such that λ j Λ,

τ 0 e -(τ -t)A Bg Λ (τ -t) dt, w j H = (B g Λ (-iλ j ), w j ) H = e(-iτ A)B h Λ (i λ j ) + h Λ (-i λ j ) , w j H = e(-iτ A)B S0 0 h Λ (s)2 cosh(s λ j ) ds, w j H = -e -τ A θ 0 , w j H
so that the solution θ of (2.4) with the control g(t) = g Λ (τ -t) satisfies θ(τ ) = 0. By a duality argument and (2.14), this implies that

Π Λ e -τ A z 0 2 C 1 exp C 1 1 τ γ + √ Λ + (Λτ ) γ/(2γ-1) τ 0 B * Π Λ e -tA z 0 2 U dt. (2.15)

Proof of Theorem 2.1

We are now in a position to prove Theorem 2.1, adapting the method of [START_REF] Miller | A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups[END_REF].

Proof of Theorem 2.1. We set z(t) = e -tA z 0 .

Assume T (1) > 0, τ > 0, and T (2) = T (1) + τ.

From (2.15)

Π Λ z T (2) 2 C 1 exp C 1 1 (ετ ) γ + √ Λ + (Λετ ) γ/(2γ-1) T (2)
T (2) -ετ

B * Π Λ z(t) 2 U dt. (2.16) We set Λ = 1 (ετ ) 1+γ
(2.17) so that for τ, ε ∈ (0, 1),

(2.16) becomes 2ρ(τ ) Π Λ z T (2) 2 T (2)
T (2) -ετ

B * Π Λ z(t) 2 U dt (2.18) with ρ(τ ) := 1 2C 1 exp - 3C 1 (ετ ) γ . (2.19)
Then from (2.18), we deduce

ρ(τ ) z T (2) 2 T (2)
T (2) -ετ

B * z(t) 2 U dt + C z T (1) 2 H ετ e -2Λτ (1-ε) + ρ(τ ) z T (1) 2 H e -2Λτ (2.20)
For ε > 0 small enough, the above relation yields

ρ(τ ) z T (2) 2 H T (2)
T

B * z(t)

2 U dt + ρ(τ /2) z T (1) 2 H (2.21) Assume T ∈ (0, 1).
Then for all k 0, (2.21) implies

ρ T 2 k+1 z T 2 k 2 H T 2 k T 2 k+1 B * z(t) 2 U dt + ρ T 2 k+2 z T 2 k+1 2 H (2.22)
and thus

ρ T 2 z (T ) 2 H T 0 B * z(t) 2 U dt. (2.23)
Thus for some constant C 2 > 0,

z (T ) 2 H C 2 exp C 2 T γ T 0 B * z(t) 2 U dt. (2.24)
3 Functional framework and spectral inequality

In order to prove Theorem 1.1, we are going to apply Theorem 2.1. In this section, we first give the functional framework associated with (1.6). Then we write the spectral inequality that will be proven in the remaining part of the article.

Functional framework

We recall that H is defined by (1.7). We also define

V := [u, η] ∈ H 1 (Ω) × H 1 (I) ∩ H ; u 1 = 0 on ∂Ω .
We denote by P 0 the orthogonal projection from L 2 (Ω) × L 2 0 (I) onto H. We now define the linear operator

A 0 : D(A 0 ) ⊂ H → H by D(A 0 ) := V ∩ H 2 (Ω) × H 2 (I) , (3.1)
and for u, η ∈ D(A 0 ), we set

A 0 u η := P 0   ∆u ∂ 2 x1 η   . (3.2)
Then one can check that (1.6) writes

dz dt = A 0 z in (0, T ), z(0) = z 0 . with z = [u, η], z 0 = [u 0 , η 0 ].
In the next proposition, we show in particular that A 0 is the infinitesimal generator of a semigroup so that z(t) = e tA0 z 0 for t 0. Proof. By definition of A 0 , we have for any

[u, η], [v, ζ] ∈ D(A 0 ), A 0 u η , v ζ H = Ω ∆u • v dx + 2π 0 ∂ 2 x1 η ζ dx 1 = - Ω ∇u : ∇v dx - 2π 0 (∂ x1 η) (∂ x1 ζ) dx 1 .
Thus A 0 is symmetric and negative (by using the Poincaré inequalities).

In order to show that A 0 is self-adjoint it is sufficient to show that it is onto. Assume [f, g] ∈ H and let us solve the equation

-A 0 u η = [f, g]. (3.3)
Multiplying the above equation by [v, ζ] ∈ V leads to the weak formulation

Ω ∇u : ∇v dx + 2π 0 (∂ x1 η) (∂ x1 ζ) dx 1 = Ω f • v dx + 2π 0 gζ dx 1 ([v, ζ] ∈ V). (3.4)
Using the Poincaré inequalities, we see that we can apply the Riesz theorem and deduce the existence and uniqueness of

[u, η] ∈ V solution of (3.4). Then if v ∈ C ∞ c (Ω) with div v = 0 and ζ = 0 in (3.4), we obtain that ∇u : ∇v dx = Ω f • v dx (v ∈ C ∞ c (Ω), div v = 0). (3.5)
Using the De Rham theorem, we deduce the existence of p such that

         -∆u + ∇p = f in Ω, div u = 0 in Ω, u = 0 on Γ 0 u = ηe 2 on Γ 1 . (3.6)
Using the elliptic regularity of the Stokes system, we deduce that (u, p)

∈ H 3/2 (Ω) × H 1/2 (Ω). Multiplying the first above equation by v, with [v, ζ] ∈ V, we deduce that for any ζ ∈ H 1 (I) ∩ L 2 0 (I), 2π 0 (∂ x1 η) (∂ x1 ζ) dx 1 = -p |Γ1 , ζ + 2π 0 gζ dx 1 . (3.7) Since p |Γ1 = ([pI 3 -∇u] n) |Γ1 ∈ H -1/2 (I), we deduce that η ∈ H 3/2 (I) and from (3.6) that (u, p) ∈ H 2 (Ω) × H 1 (Ω). Thus p |Γ1 ∈ H 1/2 (I)
, and from (3.7), we deduce that η ∈ H 2 (I). We conclude that [u, η] ∈ D(A 0 ) and satisfies (3.3). The fact that A 0 has compact resolvents is coming from the compact embedding of H 2 into L 2 for bounded domains.

In particular, the eigenvalues λ j > 0 of -A 0 satisfy λ j → ∞ and there exists

u (j) η (j) j orthonormal basis of H (3.8)
composed by eigenvectors of A 0 :

-A 0 u (j) η (j) = λ j u (j) η (j) (3.9)
The above system can be written as

               -∆u (j) + ∇p (j) = λ j u (j) div u (j) = 0 u (j) = 0 on Γ 0 u (j) = η (j) e 2 on Γ 1 -∂ 2 x1 η (j) -p (j) = λ j η (j) in I (3.10)
and more precisely as

                   -λ j u (j) 1 -(∂ 2 x1 + ∂ 2 x2 )u (j) 1 + ∂ x1 p (j) = 0 -λ j u (j) 2 -(∂ 2 x1 + ∂ 2 x2 )u (j) 2 + ∂ x2 p (j) = 0 ∂ x1 u (j) 1 + ∂ x2 u (j) 2 = 0 u (j) 1 = 0 on ∂Ω u (j) 2 = 0 on Γ 0 -λ j u (j) 2 -∂ 2 x1 u (j) 2 = p (j) on Γ 1 .
(3.11)

Spectral inequality

We are now in a position to state the spectral inequality for the operator A 0 defined in the previous section.

Theorem 3.2. Let ω 0 be a nonempty open subset of Ω and S 0 > 0. There exist C > 0 and κ ∈ C ∞ 0 (0, S 0 ) such that for any Λ > 0, and for any

(a j ) j ∈ C N , λj Λ |a j | 2 Ce C √ Λ S0 0 κ 2 (s) λj Λ a j cosh(s √ λ j )u (j) 2 L 2 (ω)
ds.

(3.12)

In order to prove Theorem 3.2, we define for s ∈ (0, S 0 ) and x ∈ Ω,

U (s, x) := λj Λ a j cosh( √ λ j s)u (j) (x), P (s, x) := λj Λ a j cosh( √ λ j s)p (j) (x) + c P (s) (3.13)
and the domains Z := (0, S 0 ) × Ω = (0, S 0 ) × I × (0, 1), J i := (0, S 0 ) × Γ i = (0, S 0 ) × I × {i} (i = 0, 1).

(3.14)

From (3.11), we deduce that

               -∂ 2 s U 1 -(∂ 2 x1 + ∂ 2 x2 )U 1 + ∂ x1 P = 0 in Z, -∂ 2 s U 2 -(∂ 2 x1 + ∂ 2 x2 )U 2 + ∂ x2 P = 0 in Z, ∂ x1 U 1 + ∂ x2 U 2 = 0 in Z, U 1 = 0 on J 0 ∪ J 1 U 2 = 0 on J 0 , -∂ 2 s U 2 -∂ 2 x1 U 2 = P -m I (P ) on J 1 .
(3.15)

In the above system, we write

m I (P ) := 1 2π 2π 0 P (x 1 , 1) dx 1
and by using this notation in the last equation of (3.15), we can replace the pressure that should satisfies a relation of the form (1.9) by the pressure P defined up to a function c P of s. In that way, we can, in what follows, impose another condition on P (typically that its mean on an open set is zero).

To show Theorem 3.2, we first truncate U and P in a neighborhood of {s = s 0 }, with

s 0 := S 0 2 .
We thus consider χ ∈ C ∞ 0 ((0, S 0 )), satisfying 0 χ 1 and

χ(s) = 1 if |s -s 0 | S 0 /8, 0 if |s -s 0 | S 0 /6. (3.16)
We work with the following localized solutions u(s, x 1 , x 2 ) := χ(s)U (s, x 1 , x 2 ), p(s, x 1 , x 2 ) := χ(s)P (s,

x 1 , x 2 ) (3.17) that satisfy                -∂ 2 s u 1 -(∂ 2 x1 + ∂ 2 x2 )u 1 + ∂ x1 p = f 1 in Z -∂ 2 s u 2 -(∂ 2 x1 + ∂ 2 x2 )u 2 + ∂ x2 p = f 2 in Z ∂ x1 u 1 + ∂ x2 u 2 = 0 in Z u 1 = 0 on J 0 ∪ J 1 u 2 = 0 on J 0 -∂ 2 s u 2 -∂ 2 x1 u 2 = f 3 + p -m I (p) on J 1 , (3.18)
where

f 1 := -χ U 1 -2χ ∂ s U 1 , f 2 := -χ U 2 -2χ ∂ s U 2 , f 3 := -χ (U 2 ) | J 1 -2χ (∂ s U 2 ) | J 1 . (3.19)
We also have that

u = 0 and p = 0 if s / ∈ 1 3 S 0 , 2 3 S 0 . (3.20)
As usual, we can use the three first equations to obtain the following equation for the pressure:

-∆p = -(∂ 2 x1 + ∂ 2 x2 )p = ∂ x1 f 1 + ∂ x2 f 2 = 0. (3.21)

A global observability estimate on the pressure

In this section, we prove a global estimate on the pressure. We first introduce our weight and the corresponding conjugated operators. We then state our main result, that is Theorem 4.1. Then we show a first estimate on the pressure involving high frequency pressure terms at the boundary. Such terms are then estimated by showing some a priori estimates and this allows us to prove Theorem 4.1.

Choice of the weight and conjugated operators

Let us consider a nonempty open set ω 0 such that ω 0 ⊂ ω. Let λ > 0, τ > 0.

Then we consider ψ ∈ C ∞ (Ω; R + ), such that

ψ(x 1 , x 2 ) = 1 -x 2 in a neighborhood of {x 2 = 1}, and ψ(x 1 , x 2 ) = x 2 in a neighborhood of {x 2 = 0} (4.1)
and such that all its critical points belong to ω 0 :

∇ ψ(x) = 0 =⇒ x ∈ ω 0 . (4.2)
We set ϕ(s, x) := e λ ψ(x)-(s-s0) 2 , ϕ 0 (s) := e -(s-s0) 2 . (4.3)

Note that with our above choices,

ϕ 0 (s) = ϕ(s, •, 0) = ϕ(s, •, 1) = min x∈Ω ϕ(s, x).
We recall that we define (u, p) from (U, P ) by (3.17) (truncation in s) and that the source f i are defined by (3.19). We then define v := e τ ϕ u, q := e τ ϕ p, g i := e τ ϕ f i (i ∈ {1, . . . , 3}).

In order to take into account the dependence in s, we write

z = (s, x) = (s, x 1 , x 2 ) ∈ Z, ∇ z =   ∂ s ∂ x1 ∂ x2   , ∆ z = ∂ 2 s + ∂ 2 x1 + ∂ 2 x2 ,
and their tangential counterparts

∇ s,x1 = ∂ s ∂ x1 , ∆ s,x1 = ∂ 2 s + ∂ 2 x1 .
We keep our previous notation

∇ = ∂ x1 ∂ x2 , ∆ = ∂ 2 x1 + ∂ 2 x2 .
The equations satisfied by v and q can be written with the introduction of the following conjugated operators:

Q ϕ := -e τ ϕ ∆ z e -τ ϕ = -∆ z + 2τ ∇ z ϕ • ∇ z -τ 2 |∇ z ϕ| 2 + τ (∆ z ϕ), (4.5)
D ϕ := -e τ ϕ ∆e -τ ϕ = -∆ + 2τ (∇ϕ) • ∇ -τ 2 |∇ϕ| 2 + τ (∆ϕ), (4.6)
S ϕ := -e τ ϕ0 ∆ s,x1 e -τ ϕ0 = -∆ s,x1 + 2τ (∂ s ϕ 0 )∂ s -τ 2 (∂ s ϕ 0 ) 2 + τ (∂ 2 s ϕ 0 ). (4.7)
Then we deduce from (3.18) and (3.21) the following conjugated system:

               Q ϕ v 1 + e τ ϕ ∂ x1 p = g 1 in Z, Q ϕ v 2 + e τ ϕ ∂ x2 p = g 2 in Z, D ϕ q = 0 in Z, v 1 = 0 on J 0 ∪ J 1 , v 2 = 0 on J 0 , S ϕ v 2 = g 3 + q -m I (q) on J 1 . (4.8)
We define h i by

h i := e τ ϕ0 f i (i = 1, 2, 3). (4.9)
The main result of this section is the following result.

Theorem 4.1. There exist λ 0 = λ 0 ( ψ, S 0 ) > 0 and τ 0 = τ 0 ( ψ, S 0 ) > 0 such that for any λ λ 0 and τ τ 0 , there exists C = C(λ, ψ, S 0 ) > 0 such that for any Λ > 0 and for any (a j ) j ∈ C N , the function q defined by (3.13), (3.17) and (4.4) satisfies

τ 3 q 2 L 2 (Z) + τ ∇q 2 L 2 (Z) + τ 3 |q -m I (q)| 2 L 2 (J1) + τ |∇q| 2 L 2 ((0,S0)×∂Ω) C τ 3 q 2 L 2 ((0,S0)×ω0) + τ ∇q 2 L 2 ((0,S0)×ω0) + τ ∂ x1 h 1 2 L 2 (Z) + ∂ x1 h 2 2 L 2 (Z) + |∂ x1 h 3 | 2 L 2 (J1)
. (4.10)

We prove this theorem in the remainder of this section.

A first estimate on the pressure

In order to prove Theorem 4.1, we exploit that q satisfies the third equation of (4.8), where D ϕ is defined by (4.6). Since we do not have any boundary condition, we need to split the boundary value of q into high and low frequencies. More precisely, for Q ∈ H 2 (Ω), we introduce the Fourier coefficients of the trace of Q :

a k (Q) :=     1 2π 2π 0 Q(x 1 , 0)e -ikx1 dx 1 1 2π 2π 0 Q(x 1 , 1)e -ikx1 dx 1     (k ∈ Z). (4.11)
We then define the sets of low tangential frequencies and high tangential frequencies:

LF τ := {k ∈ Z, k 2 τ 2 2 inf |∂ x2 ϕ| 2 }, HF τ := {k ∈ Z, k 2 > τ 2 2 inf |∂ x2 ϕ| 2 }.
In the above definition, the infimum of ∂ x2 ϕ is taken for x ∈ ∂Ω and s ∈ [0, S 0 ]. Due to (4.1) and (4.3), we have inf |∂ x2 ϕ| = λe -s 2 0 .

Proposition 4.2. There exist λ 0 = λ 0 ( ψ, S 0 ) > 0 and τ 0 = τ 0 ( ψ, S 0 ) > 0 such that for any λ λ 0 and τ τ 0 , there exists C = C(λ, ψ, S 0 ) > 0 such that for any s ∈ [0, S 0 ], and any Q ∈ H 2 (Ω),

τ 3 Q 2 L 2 (Ω) + τ ∇Q 2 L 2 (Ω) + τ |∂ x2 Q| 2 L 2 (∂Ω) + k∈LFτ τ (τ 2 + k 2 )|a k (Q)| 2 C D ϕ Q 2 L 2 (Ω) + τ 3 Q 2 L 2 (ω0) + τ ∇Q 2 L 2 (ω0) + k∈HFτ τ (τ 2 + k 2 )|a k (Q)| 2 . (4.12)
Proof. We can decompose the operator D ϕ (see (4.6)) as follows D ϕ = S + A + R, where

S = -∆ -τ 2 |∇ϕ| 2 , A = 2τ ∇ϕ • ∇ + 2τ (∆ϕ), R = -τ (∆ϕ).
Then, after some standard computation, we can obtain that

Ω (SQ)(AQ)dx = τ Ω 2∇ 2 ϕ(∇Q, ∇Q) + ∆ϕ|∇Q| 2 dx + τ 3 Ω 2∇ 2 ϕ(∇ϕ, ∇ϕ) -|∇ϕ| 2 ∆ϕ |Q| 2 dx -τ Ω (∆ 2 ϕ)|Q| 2 dx + B, (4.13)
where

∇ 2 ϕ = ∂ 2 ϕ ∂x i ∂x j i,j
and where B corresponds to the boundary terms:

B = -2τ ∂Ω ∂ n Q(∇ϕ • ∇Q)dΓ + τ ∂Ω (∂ n ϕ)|∇Q| 2 dΓ -2τ ∂Ω (∆ϕ)(∂ n Q)QdΓ + τ ∂Ω (∂ n ∆ϕ)Q 2 dΓ -τ 3 ∂Ω |∇ϕ| 2 (∂ n ϕ)Q 2 dΓ.
Using (4.1), we have that ∂ n ϕ < 0 and we can simplify the above quantity:

B = τ ∂Ω |∂ x2 ϕ| (∂ x2 Q) 2 dΓ -τ ∂Ω |∂ x2 ϕ| (∂ x1 Q) 2 dΓ -2τ ∂Ω (∂ 2 x2 ϕ)(n • e 2 )(∂ x2 Q)QdΓ + τ ∂Ω (∂ 3 x2 ϕ)(n • e 2 )Q 2 dΓ + τ 3 ∂Ω |∂ x2 ϕ| 3 Q 2 dΓ.
Combining the above relation with (4.3), there exists τ 1 = τ 1 (S 0 ) > 0 such that for any τ τ 1 , we have

B 1 2 τ λϕ 0 ∂Ω (∂ x2 Q) 2 dΓ -τ λϕ 0 ∂Ω (∂ x1 Q) 2 dΓ + 3 4 τ 3 λ 3 ϕ 3 0 ∂Ω Q 2 dΓ 1 2 τ λϕ 0 ∂Ω (∂ x2 Q) 2 dΓ + 2πτ λϕ 0 k∈Z 3 4 τ 2 λ 2 ϕ 2 0 -k 2 |a k (Q)| 2 . (4.14)
Using (4.3) and (4.2), there exist

C 1 = C 1 ( ψ), C 2 = C 2 ( ψ), τ 2 = τ 2 (S 0 , ψ) and λ 1 = λ 1 ( ψ) such that for λ λ 1 and τ τ 2 , τ Ω 2∇ 2 ϕ(∇Q, ∇Q) + ∆ϕ|∇Q| 2 dx + τ 3 Ω 2∇ 2 ϕ(∇ϕ, ∇ϕ) -|∇ϕ| 2 ∆ϕ |Q| 2 dx -τ Ω (∆ 2 ϕ)|Q| 2 dx C 1 Ω τ λ 2 ϕ |∇Q| 2 + τ 3 λ 4 ϕ 3 |Q| 2 dx -C 2 ω0 τ λ 2 ϕ |∇Q| 2 + τ 3 λ 4 ϕ 3 |Q| 2 dx. (4.15)
Finally, combining (4.13), (4.14) and (4.15), we deduce the existence of λ 0 > 0 and τ 0 > 0 such that for any λ λ 0 and τ τ 0 , there exist C 3 = C 3 (λ, ψ, S 0 ) > 0 and C 4 = C 4 (λ, ψ, S 0 ) > 0 such that

D ϕ Q 2 L 2 (Ω) 1 2 (S + A)Q 2 L 2 (Ω) -RQ 2 L 2 (Ω) Re (SQ, AQ) L 2 (Ω) -RQ 2 L 2 (0,1) C 3 Ω τ |∇Q| 2 + τ 3 |Q| 2 dx + τ ∂Ω (∂ x2 Q) 2 dΓ + τ k∈LFτ (τ 2 + k 2 )|a k (Q)| 2 -C 4 ω0 τ |∇Q| 2 + τ 3 |Q| 2 dx + τ k∈HFτ k 2 |a k (Q)| 2 .
Applying the above result to Q = q solution of (4.8) and integrating into (0, S 0 ), we deduce that

τ 3 q 2 L 2 (Z) + τ ∇q 2 L 2 (Z) + τ |∂ x2 q| 2 L 2 ((0,S0)×∂Ω) + k∈LFτ τ (τ 2 + k 2 )|a k (q)| 2 L 2 (0,S0) C τ 3 q 2 L 2 ((0,S0)×ω0) + τ ∇q 2 L 2 ((0,S0)×ω0) + k∈HFτ τ (τ 2 + k 2 )|a k (q)| 2 L 2 (0,S0) . (4.16)
We recall that q(s, •) ≡ 0 if |s -s 0 | S 0 /6 due to the support of χ (see (3.16)). Next, we will estimate the high tangential frequencies of the pressure.

Estimates in the high frequency regime

We define Y := (0, S 0 ) × (0, 1), I i := (0, S 0 ) × {i}, i = 0, 1.

We recall that (u, p) is defined by (3.17) and (f 1 , f 2 , f 3 ) is defined by (3.19). We define

(u k , p k , f k 1 , f k 2 , f k 3 ) the Fourier coefficients of (u, p, f 1 , f 2 , f 3 ) in the x 1 direction. For instance u k (s, x 2 ) := 1 2π 2π 0 u(s, x 1 , x 2 )e -ikx1 dx 1 ((s, x 2 ) ∈ Y ).
Finally, with τ > 0 and ϕ 0 defined by (4.3), we set

w k = e τ ϕ0 u k , π k = e τ ϕ0 p k , h k i = e τ ϕ0 f k i (i = 1, 2, 3). ( 4

.17)

Note that h k i are the Fourier coefficients of the functions h i defined by (4.9). Since ϕ 0 only depends on s, and using (4.3), (4.4), we have

a k (q) = (π k ) |I0 (π k ) |I1 . (4.18)
Let us define the following conjugated operators:

Q k,ϕ0 := e τ ϕ0 (-∂ 2 s -∂ 2 x2 + k 2 )e -τ ϕ0 = -∂ 2 s -∂ 2 x2 + k 2 + 2τ ϕ 0 ∂ s -τ 2 (ϕ 0 ) 2 + τ ϕ 0 , S k,ϕ0 := e τ ϕ0 (-∂ 2 s + k 2 )e -τ ϕ0 = -∂ 2 s + k 2 + 2τ ϕ 0 ∂ s -τ 2 (ϕ 0 ) 2 + τ ϕ 0 .
Then, for k ∈ Z, (3.18) transforms into

               Q k,ϕ0 w k 1 + ikπ k = h k 1 in Y, Q k,ϕ0 w k 2 + ∂ x2 π k = h k 2 in Y, div k w k = 0 in Y, w k 1 = 0 on I 0 ∪ I 1 , w k 2 = 0 on I 0 , S k,ϕ w k 2 = h k 3 + π k on I 1 , (4.19)
where

div k f 1 f 2 = ikf 1 + ∂ x2 f 2 .
The relation (3.20) yields

w k = 0 and π k = 0 if s / ∈ 1 3 S 0 , 2 3 S 0 . (4.20)
Proposition 4.3. If the solution of (4.19) satisfies (4.20), then there exist λ 3 = λ 3 (S 0 ) > 0 and C(S 0 ) > 0 such that for any λ λ 3 and k ∈ HF τ ,

π k L 2 (I0∪I1) C (k 2 + τ 2 ) 1/4 h k 1 L 2 (Y ) + h k 2 L 2 (Y ) + h k 3 L 2 (I1) .
Proof. We multiply the first line (4.

Y |∂ s w k | 2 dy + Y |∂ x2 w k | 2 dy + k 2 Y |w k | 2 dy -τ 2 Y (ϕ 0 ) 2 |w k | 2 dy + I1 |∂ s w k 2 | 2 ds + k 2 I1 |w k 2 | 2 ds -τ 2 I1 (ϕ 0 ) 2 |w k 2 | 2 ds = Re Y h k 1 w k 1 dy + Re Y h k 2 w k 2 dy + Re I1 h k 3 w k 2 ds. (4.21)
Now, since k ∈ HF τ , we have

k 2 > τ 2 2 inf |∂ x2 ϕ| 2 = τ 2 2 λ 2 e -2s 2 0 .
On the other hand, sup

[0,S0] |ϕ 0 | S 0 .
From the two previous relations, we deduce the existence of λ 3 = λ 3 (S 0 ) > 0 such that for λ λ 3 and for

k ∈ HF τ k 2 -τ 2 |ϕ 0 | 2 1 2 (τ 2 + k 2 ).
Combining the above relation and (4.21) yields

(τ 2 + k 2 ) ∂ s w k , ∂ x2 w k 2 L 2 (Y ) + (τ 2 + k 2 ) 2 w k 2 L 2 (Y ) + (τ 2 + k 2 ) ∂ s w k 2 2 L 2 (I1) + (τ 2 + k 2 ) 2 w k 2 2 L 2 (I1) h k 1 2 L 2 (Y ) + h k 2 2 L 2 (Y ) + h k 3 2 L 2 (I1) . (4.22)
Now, we write (4.19) under the form

               -∂ 2 s -∂ 2 x2 + k 2 w k 1 + ikπ k = H k 1 in Y, -∂ 2 s -∂ 2 x2 + k 2 w k 2 + ∂ x2 π k = H k 2 in Y, div k w k = 0 in Y, w k 1 = 0 on I 0 ∪ I 1 , w k 2 = 0 on I 0 , -∂ 2 s + k 2 w k 2 = H k 3 + π k on I 1 , (4.23)
where

H k 1 := -2τ ϕ 0 ∂ s w k 1 + τ 2 (ϕ 0 ) 2 w k 1 -τ ϕ 0 w k 1 + h k 1 , H k 2 := -2τ ϕ 0 ∂ s w k 2 + τ 2 (ϕ 0 ) 2 w k 2 -τ ϕ 0 w k 2 + h k 2 , H k 3 := -2τ ϕ 0 ∂ s w k 2 + τ 2 (ϕ 0 ) 2 w k 2 -τ ϕ 0 w k 2 + +h k 3 .
From (4.22), we deduce

H k 1 L 2 (Y ) + H k 2 L 2 (Y ) + H k 3 L 2 (I1) h k 1 L 2 (Y ) + h k 2 L 2 (Y ) + h k 3 L 2 (I1) . (4.24)
We multiply the first line (4.23) by -∂ 2 s w k 1 , the second line (4.23) by -∂ 2 s w k 2 , and the last line (4.23) by -∂ 2 s w k 2 . Integrating by parts, summing up and using (4.24) yield

∂ 2 s w k 2 L 2 (Y ) + ∂ s ∂ x2 w k 2 L 2 (Y ) + k 2 ∂ s w k 2 L 2 (Y ) + ∂ 2 s w k 2 2 L 2 (I1) + k 2 ∂ s w k 2 2 L 2 (I1) h k 1 2 L 2 (Y ) + h k 2 2 L 2 (Y ) + h k 3 2 L 2 (I1) . (4.25)
Next, we write (4.23) under the form

                 -∂ 2 x2 + k 2 w k 1 + ikπ k = H k 1 in Y, -∂ 2 x2 + k 2 w k 2 + ∂ x2 π k = H k 2 in Y, div k w k = 0 in Y, w k 1 = 0 on I 0 ∪ I 1 , w k 2 = 0 on I 0 , k 2 w k 2 = H k 3 + π k on I 1 , (4.26)
where

H k 1 := H k 1 + ∂ 2 s w k 1 , H k 2 := H k 2 + ∂ 2 s w k 2 , H k 3 := H k 3 + ∂ 2 s w k 1 .
From (4.24) and (4.25), we deduce

H k 1 L 2 (Y ) + H k 2 L 2 (Y ) + H k 3 L 2 (I1) h k 1 L 2 (Y ) + h k 2 L 2 (Y ) + h k 3 L 2 (I1) . (4.27)
The first two lines of (4.26) can be written as

∇ e ikx1 π k = e ikx1 H k 1 H k 2 + ∆ e ikx1 w k (4.28)
and thus

∇ e ikx1 π k H -1 (Ω) e ikx1 H k 1 H k 2 H -1 (Ω) + ∇ e ikx1 w k L 2 (Ω) (4.29)
and using that k = 0, we deduce from the above estimate that

e ikx1 π k H -1 (Ω) e ikx1 H k 1 H k 2 H -1 (Ω) + ∇ e ikx1 w k L 2 (Ω) . ( 4
H k 1 L 2 (0,1) + H k 2 L 2 (0,1) + k ∇ e ikx1 w k L 2 (Ω) (4.31)
and thus, with (4.27) and (4.22),

k π k L 2 (Y ) h k 1 L 2 (Y ) + h k 2 L 2 (Y ) + h k 3 L 2 (I1) . (4.32)
On the other hand, differentiating the divergence equation of system (4.26) with respect to x 2 and using (4.22), (4.27) yield

∂ 2 x2 w k 2 L 2 (Y ) + ∂ x2 π k L 2 (Y ) h k 1 L 2 (Y ) + h k 2 L 2 (Y ) + h k 3 L 2 (I1) .
Then, combining the above relation with (4.32) and with a trace inequality, we deduce

π k L 2 (I0∪I1) 1 k 1/2 h k 1 L 2 (Y ) + h k 2 L 2 (Y ) + h k 3 L 2 (I1) .
Using that k ∈ HF τ , we deduce the result.

Proof of Theorem 4.1

From Proposition 4.3 and from (4.18), for k ∈ HF τ

(τ 2 + k 2 ) a k (q) 2 L 2 (0,S0) k 2 h k 1 2 L 2 (Y ) + h k 2 2 L 2 (Y ) + h k 3 2 L 2 (I1)
and thus, with the Parceval formula,

k∈HFτ (τ 2 + k 2 )|a k (q)| 2 L 2 (0,S0) ∂ x1 h 1 2 L 2 (Z) + ∂ x1 h 2 2 L 2 (Z) + |∂ x1 h 3 | 2 L 2 (J1) ,
where we have used (4.17) and (4.9). Combining this estimate with (4.16) we finally obtain the sought result.

Proof of the spectral inequality

The proof of the spectral inequality, that is (3.12) is based on interpolation estimates. More precisely, it will be a consequence of Theorem 5.3 stated below. In order to show such a result, we first recall some interpolation inequalities available in the literature and then we combine them with the global pressure estimates, that is Theorem 4.1 to show Theorem 5.3. The last part of this section is devoted to the proof of the spectral inequality from the interpolation inequality. First, we need the following notation for this section:

O 0 := s 0 - S 0 6 , s 0 + S 0 6 × ω 0 , O := s 0 - S 0 5 , s 0 + S 0 5 × ω, (5.1)
Z := s 0 - S 0 10 , s 0 + S 0 10 × Ω, J 1 := s 0 - S 0 10 , s 0 + S 0 10 × Γ 1 , (5.2)
Z := s 0 - S 0 9 , s 0 + S 0 9 × Ω, J 1 := s 0 - S 0 9 , s 0 + S 0 9 × Γ 1 . (5.3)
Note that Z ⊂ Z and J 1 ⊂ J 1 .

Estimates on the velocity

The two components of the velocity satisfy different boundary conditions (see (3.15)). We start by an estimate on the first component U 1 that satisfies homogeneous Dirichlet boundary conditions. For the proof of this result, we refer to relation (1) in Section 3 of [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF].

Theorem 5.1. There exist C 1 > 0 and µ 1 ∈ (0, 1) such that for all w ∈ H 2 (Z) such that w

| J 0 ∪J 1 = 0, w H 1 ( Z) C 1 w 1-µ1 H 1 ( Z) ∆ z w L 2 ( Z) + w L 2 (O0) µ1 . (5.4)
Note that Theorem 5.1 is stated with an H 1 observation in [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF], but we can transform it into an L 2 observation as in (5.4) by using a cut-off function and integrations by parts.

For the estimate of U 2 , we note that it satisfies a Ventcel boundary condition on J 1 and the Dirichlet boundary condition on J 0 . Hence, we use the following result, which is basically a consequence of a Carleman estimate obtained in [START_REF] Buffe | Stabilization of the wave equation with Ventcel boundary condition[END_REF]. However for the sake of completeness, we prove the next result in Appendix A.

Theorem 5.2. There exist C 2 > 0 and µ 2 ∈ (0, 1) such that

w H 1 ( Z) + |w| H 1 ( J1) C 2 w H 1 ( Z) + |w| H 1 ( J1) 1-µ2 × ∆ z w L 2 ( Z) + (∂ x2 w) | J 1 -∆ x1,s w | J 1 L 2 ( J1) + w L 2 (O0) µ2 . (5.5)
for all w ∈ H 2 (Z) such that w | J 0 = 0 and w | J 1 ∈ H 2 (J 1 ).

Note that both Theorem 5.1 and Theorem 5.2 hold for µ 3 ∈ (0, 1) such that µ 3 µ 1 and µ 3 µ 2 (with a modification of the constants C 1 and C 2 ). Thus, with µ 3 = min(µ 1 , µ 2 ) and an adequate constant C 3 , we can apply Theorem 5.1 with w = U 1 and Theorem 5.2 with w = U 2 , where U satisfies (3.15) and we deduce

U H 1 ( Z) + |U 2 | H 1 ( J1) C 3 U H 1 ( Z) + |U 2 | H 1 ( J1) 1-µ3 × ∇P L 2 ( Z) + P |J1 -m I (P ) L 2 ( J1) + U L 2 (O0) µ3 . (5.6)
We have used here the fact that on J 1 ,

∂ x2 U 2 = -∂ x1 U 1 = 0.
We are going now to combine the above estimate with the estimates of the pressure terms obtained in Section 4.

Patching the estimates together

Combining the previous estimates, we can now prove the following result.

Theorem 5.3. There exist C > 0 and µ ∈ (0, 1) such that for any Λ > 0 and for any (a j ) j ∈ C N , the function U defined by (3.13) satisfies

U H 1 ( Z) + |U 2 | H 1 ( J1) C U H 2 (Z) + |U 2 | H 2 (J1) 1-µ U µ H 2 (O0) .
Proof. We start with the estimate (4.10), where we recall that q is given by (3.17), (4.4) and h i , i = 1, 2, 3 by (4.9):

τ 3 e τ ϕ χP 2 L 2 (Z) + τ e τ ϕ χ∇P 2 L 2 (Z) + τ 3 |e τ ϕ0 χ(P -m I (P ))| 2 L 2 (J1) + τ |e τ ϕ0 χ∂ x1 P | 2 L 2 (J1) + τ |e τ ϕ0 χ∂ x2 P | 2 L 2 (J1) C τ 3 e τ ϕ P 2 L 2 (O0) + τ e τ ϕ ∇P 2 L 2 (O0) + τ e τ ϕ0 ∂ x1 f 1 2 L 2 (Z) + τ e τ ϕ0 ∂ x1 f 2 2 L 2 (Z) + τ |e τ ϕ0 ∂ x1 f 3 | 2 L 2 (J1) . (5.7)
Note that f 1 , f 2 (respectively f 3 ) are supported in

e τ ϕ0 ∂ x1 f 1 L 2 (Z) + e τ ϕ0 ∂ x1 f 2 L 2 (Z) + |e τ ϕ0 ∂ x1 f 3 | L 2 (J1) Ce τ e -S 2 0 36 U H 2 (Z) + |U 2 | H 2 (J1) .
(5.9)

In Z (respectively in J 1 ), χ(s) ≡ 1, and

inf Z ϕ = inf Z ϕ 0 = inf J1 ϕ 0 = e -S 2 0
81 .

(5.10)

Combining (5.7), (5.9) and (5.10), there exist τ 4 , c 1 , c 2 > 0 such that for all τ τ 4 , we have

∇P 2 L 2 ( Z) + |(P -m I (P ))| 2 L 2 ( J1) e c1τ P 2 L 2 (O0) + ∇P 2 L 2 (O0) + e -c2τ U H 2 (Z) + |U 2 | H 2 (J1) . (5.11)
On the other hand, we deduce from (5.6) and a Young inequality that

U H 1 ( Z) + |U 2 | H 1 ( J1) C 4 e - µ 3 c 2 2(1-µ 3 ) τ U H 1 ( Z) + |U 2 | H 1 ( J1) + e c 2 2 τ ∇P L 2 ( Z) + P |J1 -m I (P ) L 2 ( J1) + U L 2 (O0)
and combining this relation with (5.11), we deduce the existence of c 3 , c 4 > 0 such that for all τ τ 4

U H 1 ( Z) + |U 2 | H 1 ( J1) e -c3τ U H 2 (Z) + |U 2 | H 2 (J1) + e c4τ P 2 L 2 (O0) + ∇P 2 L 2 (O0) + U L 2 (O0)
Now, we can use c P in (3.13) so that for all s ∈ [0, S 0 ], ω0 P (s, x) dx = 0 and using the Poincaré-Wirtinger inequality, we deduce that

P 2 L 2 (O0) ∇P 2 L 2 (O0) ∆ z U L 2 (O0) .
We deduce that for some constants c 5 , c 6 > 0, for all τ τ 4 ,

U H 1 ( Z) + |U 2 | H 1 ( J1) e -c5τ U H 2 (Z) + |U 2 | H 2 (J1) + e c6τ U H 2 (O0) .
Optimizing this inequality with respect to τ τ 4 (see, for instance, [START_REF] Buffe | Stabilization of the wave equation with Ventcel boundary condition[END_REF]Lemma 8.4]) allows us to conclude the proof of Theorem 5.3.

From the interpolation inequality to the spectral inequality

Using Theorem 5.3, we are now in a position to prove Theorem 3.2. This inequality combined with Theorem 2.1 yields the main result of the article (Theorem 1.1).

Proof of Theorem 3.2. From (3.8) and (3.1), we deduce that

U 2 H 1 ( Z) + |U 2 | 2 H 1 ( J1) U 2 L 2 ( Z) + |U 2 | 2 L 2 ( J1) s0+ S 0 10 s0- S 0 10 λj Λ |a j | 2 cosh( √ λ j s) 2 ds λj Λ |a j | 2 and U 2 H 2 (Z) + |U 2 | 2 H 2 (J1) e C √ Λ λj Λ |a j | 2
Combining Theorem 5.3 with the previous relations, we deduce that A Proof of Theorem 5.2

A.1 A Carleman estimate

The proof of Theorem 5.2 is mainly based on a Carleman estimate obtained in [START_REF] Buffe | Stabilization of the wave equation with Ventcel boundary condition[END_REF] that we recall here. We recall that Z, J 1 , J 0 are defined by (3.14) whereas Z and J 1 are defined by (5.2). In what follows, we consider z 0 ∈ J 1 , an open neighborhood V of z 0 in Z and a weight function ϕ ∈ C ∞ (V ). For any σ ∈ R, we define

p ϕ,σ (z, ξ, τ ) = |ξ| 2 -τ 2 |∇ z ϕ(z)| 2 -σ 2 + 2iτ ξ • ∇ z ϕ(z), (z ∈ V, ξ ∈ R 3 , τ ∈ R).
It is the principal symbol of the conjugated operator associated with -∆ z -σ 2 , that is, of the operator

P ϕ,σ = -e τ ϕ ∆ z + σ 2 e -τ ϕ = -∆ z + 2τ ∇ z ϕ • ∇ z -τ 2 |∇ z ϕ| 2 + τ (∆ z ϕ) -σ 2 .
We assume the following hypotheses on ϕ: sub-ellipticity on V , that is the existence of τ 0 > 0 such that for any z ∈ V , ξ ∈ R 3 , |σ| 1 and τ τ 0 |σ|, p ϕ,σ (z, ξ, τ ) = 0 =⇒ 1 2i {p ϕ,σ , p ϕ,σ } (z, ξ, τ ) > 0.

(A.1)

and the two following conditions to handle Ventcel boundary conditions (see [11, conditions (23) and ( 24)]):

∇ z ϕ = 0 in V , and sup

V ∩ J1 |∇ s,x1 ϕ| ν 0 inf V |∂ x2 ϕ| , (A.2)
for ν 0 > 0 small enough. We recall that the Poisson bracket is defined by p (1) , p (2) = 3 j=1 p (1) ∂ξ j p (2) ∂z j -p (2) ∂ξ j p (1) ∂z j

where we set here z 1 = s, z 2 = x 1 and z 3 = x 2 to simplify. Then we have the following result proved in [START_REF] Buffe | Stabilization of the wave equation with Ventcel boundary condition[END_REF]:

Theorem A.1. Assume z 0 ∈ J 1 and V is an open neighborhood of z 0 in Z. Assume also that ϕ ∈ C ∞ (V ) satisfies the conditions (A.1)) and (A.2) for ν 0 small enough. Then, there exist τ 0 > 0 and C > 0 such that for all |σ| 1, for all τ τ 0 |σ| and for all w ∈ C ∞ 0 (V ), First to precise the above statement, by w ∈ C ∞ 0 (V ) we mean that w is the restriction of a C ∞ function with compact support in V 0 where V 0 is an open set of R × I × R such that V 0 ∩ Z = V . Second, we use the above result in the case σ = 1, so that by taking τ 0 large enough, we obtain that for all τ τ 0 and for all w ∈ C ∞ 0 (V ), .

We consider z 0 = (s * , x * 1 , 1) ∈ J 1 ,

V := (s * -δ, s * + δ) × (x * 1 -δ, x * 1 + δ) × (1 -δ, 1] ,
with δ ∈ (0, 1) small enough such that (s * -δ, s * + δ) ⊂ s 0 -S 0 9 , s 0 + S 0 9

(see (5.3)). We also define z * = (s * , x * 1 , 0). Then we define ψ(z) := |z -z * | λ , ϕ = e -λψ .

Lemma A.2. There exists λ 0 > 0 such that for any λ λ 0 , the weight function ϕ satisfies (A.1) and (A.2) on V for some τ 0 = τ 0 (λ).

Proposition 3 . 1 .

 31 The operator A 0 defined by (3.1)-(3.2) has compact resolvents, and is self-adjoint negative on H.

Z 8 :

 8 = (supp χ ) × I × (0, 1) (respectively in J 8 := (supp χ ) × I × {1}), and since supp χ ⊂ [s 0 -S 0 /6, s 0 + S 0/6]

2 H 2 (U 2 L 2 (

 2222 O0) . Using a cut-off function and integrations by parts, we deduce λj Λ |a j | 2 e C √ Λ O) . and thus Theorem 3.2.

τ 3 e τ ϕ w 2 L 2 2 L 2 2 L 2 2 L 2 2 L 2 2 L 2 2 L 2

 22222222222222 (V ) + τ e τ ϕ ∇ z w (V ) + τ 3 e τ ϕ w |J1 (V ∩ J1) + τ e τ ϕ ∇ s,x1 w |J1 (V ∩ J1) + τ e τ ϕ ∂ x2 w |J1 (V ∩ J1) C e τ ϕ (-∆ z -σ 2 )w (V ) + τ e τ ϕ (∂ x2 w |J1 -∆ s,x1 w |J1 (V ∩ J1) .

τ 3 e τ ϕ w 2 L 2 (V ) + τ e τ ϕ ∇ z w 2 L 2 (V ) + τ 3 e τ ϕ w |J1 2 L 2 ( 2 L 2 ( 2 L 2 ( 2 L 2 ( 2 L 2 (

 22222222222222 V ∩ J1) + τ e τ ϕ ∇ s,x1 w |J1 V ∩ J1) + τ e τ ϕ ∂ x2 w |J1 V ∩ J1) C e τ ϕ ∆ z w V ) + τ e τ ϕ ∂ x2 w |J1 -∆ s,x1 w |J1 V ∩ J1) . (A.3)A.2 Interpolation estimates for the Ventcel boundary conditionUsing the Carleman inequality of the previous section, one can deduce, in a classical way, an interpolation inequality. First let us define the weight function that we are going to use.We consider the following norms on R × I × R:s, x 1 , x 2 λ := s 2

  19) by w k 1 , the second line (4.19) by w k 2 , and the last line (4.19) by w k 2 . Integrating by parts and summing up yield

Proof. We assume that λ 1 in all what follows. First, since ∇ z ψ = 0 in V , we deduce the first point of (A.2). For the second point of (A.2), we first notice that inf

We thus deduce

|∇ s,x1 ϕ| sup

ϕ = e -λ and inf

.

Consequently there exists λ 1 such that the second point of (A.2) holds for λ λ 1 . For (A.1), we compute the Poisson bracket:

From (A.4), there exist positive constants independent of λ such that

In particular there exist C > 0 and λ 0 λ 1 , such that for λ λ 0 ,

and there exists τ 0 = τ 0 (λ) such that for τ τ 0 |σ|,

From now on, the value of λ shall be kept fixed. We define, for β > 0,

Proof. Standard computation shows the existence of r 3 > 1 such that

We consider r 2 ∈ (1, r 3 ). We can also check the existence of r 1 ∈ (0, δ) such that

We consider two cut-off functions χ

Let us consider v ∈ C ∞ (Z) and let us apply the Carleman estimate (A.3) to w = χ 0 χ 1 v ∈ C ∞ 0 (V ). In the right-hand side of this estimate, we have

Note that in supp χ 0 χ 1 , ϕ C 3 := e -λ (1-r1) . The two last terms in the above relation are included in

and on this set, ϕ C 3 or in

and on this set, ϕ C 1 := e -λr2 . Therefore,

and similarly,

There exists r 4 > 0 such that

Then on the set

we have χ 0 χ 1 = 1 and ϕ C 2 := e -λ sup V ψ , with C 2 ∈ (C 1 , C 3 ). Combining this with (A.3), (A.6) and (A.7), we deduce that for all τ τ 0 ,

Optimizing this inequality with respect to τ yields the interpolation inequality for v smooth. A density argument permits to conclude the proof of Lemma A.3.

Proof of Theorem 5.2. By a compactness argument, one can deduce from Lemma A.3 an interpolation result on a neighborhood of J 1 . Then we combine this with classical interpolation estimates (see [START_REF] Lebeau | Contrôle exact de l'équation de la chaleur[END_REF]) in the interior and at the boundary J 0 where Dirichlet boundary condition hold to conclude. A similar proof is done in [START_REF] Buffe | Stabilization of the wave equation with Ventcel boundary condition[END_REF] (see Lemma 8.3).

.


Olivier Lévêque
email: olivier.leveque@institutoptique.fr

Romain Duverger

Hervé Sauer

Caroline Kulcsár

AND François Goudail

Influence of high numerical aperture on depth-of-field enhancing phase mask optimization in localization microscopy

INTRODUCTION

The depth-of-field (DoF) of localization microscopes is limited by the natural DoF of the objective, which is very short since the numerical aperture (NA) must be large to increase lateral resolution. Several approaches have been investigated in order to improve the DoF, involving multi-focusing techniques [START_REF] Ram | High Accuracy 3D Quantum Dot Tracking with Multifocal Plane Microscopy for the Study of Fast Intracellular Dynamics in Live Cells[END_REF] or phase masks [START_REF] Pavani | Three-dimensional, singlemolecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function[END_REF][START_REF] Shechtman | Optimal Point Spread Function Design for 3D Imaging[END_REF]. Among them, we have proposed a method based on annular binary phase masks, which are easy to manufacture. After proper optimization and appropriate signal processing, they have been shown to be able to improve the DoF by a factor 3 [START_REF] Lévêque | Can phase masks extend depth-of-field in localization microscopy?[END_REF][START_REF] Lévêque | Co-designed annular binary phase masks for depth-of-field extension in single-molecule localization microscopy[END_REF].

To optimize these masks and characterize their performance, the defocus wavefront aberration is in general modeled as a quadratic function of the pupil coordinates [START_REF] Goodman | Introduction to Fourier optics[END_REF]. However, this model, which we will call "quadratic approximation" (QA), does not include two essential features of microscope objectives used for localization microscopy: extremely high NA and possible mismatch between nominal immersion liquid and true sample refractive indices. There exists more accurate scalar or vectorial models that better account for these imaging characteristics. Among them, the "Gibson & Lanni" (GL) scalar model [START_REF] Gibson | Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy[END_REF] is significantly more accurate than the QA model for computing the actual point spread function (PSF) variation with defocus, while being much more easy to tackle than more complex vectorialbased models [START_REF] Aguet | A maximum-likelihood formalism for sub-resolution axial localization of fluorescent nanoparticles[END_REF][START_REF] Braat | Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system[END_REF]. An important question in practice is to determine if masks optimized with the approximate QA model are also efficient in the more realistic world described by the GL model. The purpose of the present article is to investigate this issue. A major difficulty is that in the QA model, the effect of defocus can be entirely described with a single reduced parameter ψ (the peak-to-valley quadratic defocus wavefront error), whereas in the GL model, it depends on several parameters (NA, mismatches between real and nominal focusing depths and refractive indices). To address this difficulty, we first show that although the DoF predicted by the QA and the GL models are significantly different, they can be connected by a simple scaling factor that depends on the NA and on the sample refractive index. Second, we compare the optimization of DoF enhancing annular binary phase masks for these equivalent configurations in the QA and GL models. We show that the resulting optimal masks are similar and yield similar localization performance. This result is important since it makes it possible to use generic phase masks that are nearly optimal for many different configurations of NA, focusing depth and index mismatch. The article is organized as follows. In Section 2, we define the defocused microscope setup investigated in this paper and its parameters. In Section 3, we introduce DoF enhancing with annular binary phase masks and their optimization in the framework of the QA model. In Section 4, we determine the equivalent expressions of the DoF for the QA and GL models as a function of imaging parameters. In Section 5, we show that for these equivalent DoF settings, the optimal masks in the QA and GL models are similar and yield similar localization performance. Section 6 is devoted to conclusions and perspectives to this work.

DESCRIPTION OF MICROSCOPE CONFIGURATION IN THE PRESENCE OF DEFOCUS

We consider in this paper an immersion microscope objective (associated with the microscope tube lens if it is infinity corrected) that gives an image with lateral magnification M and which is assumed to be, as usual for such lens, object-space telecentric, aplanatic (i.e., it fulfills the Abbe sine condition), and with negligible on-axis and field aberrations when used in its nominal configuration. As shown on the left side of Fig. 1, the objective is nominally designed to work immersed in a liquid of refractive index n i with a given coverslip and a sample observed just below the coverslip. DoF extension will be used to image objects that stand below the coverslip in a medium of refractive index n s (see the right side of Fig. 1). In this section and the next one, we will assume that n i = n s ; possible index mismatch between the two mediums will be considered in subsequent sections. Let us denote the 3D position of a fluorescent emitter in the observed sample by the coordinates θ = (x p , y p , z p ), and the 2D position of a point of observation in the image plane (i.e., on the image sensor) by the coordinates (x, y). Due to object-space telecentricity, the magnification M does not depend on the z p value. Assuming the field aberrations are negligible, this leads to a space-invariant (x, y) response where the irradiance in the image plane of a point like emitter can be written:

f (x, y, θ) = h z p x -Mx p , y -My p (1)
where h z p (x, y) is the point spread function (PSF) of the microscope objective, i.e., the image-plane irradiance of an emitter centered at lateral position (x p , y p ) = (0, 0) and located at a depth z p . Due to the large value of the magnification |M|, the image-space numerical aperture, denoted by NA i , is linked to the object-space numerical aperture, denoted by NA, by the Abbe sine condition: NA i = NA /|M|. Since NA i is small, classical paraxial Fourier Optics can be applied to compute the PSF in the image plane [START_REF] Goodman | Introduction to Fourier optics[END_REF]:

h z p (x, y) ∝ FT 2D Π Disk exp i Φ z p x NA i λ , y NA i λ 2 ( 2
)
where the operator FT 2D denotes the 2D Fourier transform, λ is the wavelength of the collected light, Π Disk is equal to 1 inside the exit pupil disk (of normalized radius 1) and 0 elsewhere, and Φ z p is the exit pupil phase function. Π Disk and Φ z p are assumed to be 2D circular symmetric functions of the normalized exit pupil coordinates. Note that h z p (x, y) can be expressed using the object-space NA by replacing NA i /λ by NA /(|M|λ). We assume that the function f (x, y, θ) is normalized so that f (x, y, θ) dx dy = 1. The pupil phase function is given by

Φ z p (r) = 2π λ W z p (r) (3)
where W z p (r) denotes the optical path difference (OPD) between the focused and defocused spherical wavefronts at a normalized radius r in the exit pupil. The above mentioned QA and GL models correspond to different expressions of W z p (r).

DOF ENHANCING PHASE MASK OPTIMIZATION

In this section, we describe the optimization of DoF enhancing phase masks when the usual QA model is used for representing the PSF variation with defocus.

A. Defocus within quadratic approximation model

For an instrument limited by diffraction in the presence of pure defocus, one often uses the following quadratic approximation of the OPD [START_REF] Goodman | Introduction to Fourier optics[END_REF]:

W QA (r) = ψr 2 . ( 4
)
Note that to simplify notation, we have dropped the subscript z p in the expression of the OPD. In Eq. ( 4), the parameter ψ represents the value of the first non-zero expansion coefficient of the OPD between the focused and defocused spherical wavefronts at the edge of the pupil and is suitable for low or moderate NA. We assume in this section no mismatch between sample and immersion medium indices. The parameter ψ has the following expression [START_REF] Goodman | Introduction to Fourier optics[END_REF][START_REF] Aguet | Super-Resolution Fluorescence Microscopy Based on Physical Models[END_REF]:

ψ = NA 2 2n s z p -z focus (5)
where n s = n i is the matched refractive index of the sample and immersion media, z p is the actual position of the fluorescent emitter along the optical axis and z focus the position at which the instrument is focused. If z p = z focus , then W QA (r) = 0, the wavefront error vanishes, and the PSF is described by the well known Airy pattern centered at the ideal image position:

f Airy (x, y, θ) ∝ J 1 2π NA |M|λ x -|M|x p 2 + y -|M|y p 2 2π NA |M|λ x -|M|x p 2 + y -|M|y p 2 2 (6)
where J 1 denotes the Bessel function of the first kind of order 1.

Let us denote by [z min , z max ] the targeted DoF range, that is, the range of values of z p within which we want the localization to be accurate. In the QA model, since the deviation of the defocused PSF from the Airy pattern depends only on the defocus parameter ψ, the targeted DoF range can be entirely specified with this reduced parameter. Indeed, in order to have the sharpest PSF, we have to minimize the maximum value of |ψ|, i.e., of the phase aberration function Φ QA (r), reached when z p varies over the targeted DoF range. Obviously, this is obtained by setting the focus point at the middle distance:

z opt focus = z max + z min 2 . ( 7
)
With this setting, the maximum absolute value of ψ is reached at the ends of the DoF range. Indeed, from Eq. ( 5), the defocus parameter is equal to -ψ max at z p = z min and to ψ max at z p = z max , with

ψ max = NA 2 4n s (z max -z min ) . (8)
Within the targeted DoF range, the defocus parameter ψ thus varies in the interval [-ψ max , ψ max ].

As an illustration, we have represented in Fig. 2.a the variation of the PSF profile as a function of ψ, for the microscope configuration described in Table 1. Note that since the PSF has circular symmetry, a cross-section along any line passing through its center is sufficient to represent its properties. The left-side vertical axis in Fig. 2.a represents the defocus parameter ψ expressed in units of the wavelength λ of the collected light, the graph being drawn for ψ ∈ [-1.5λ, 1.5λ]. The right-side vertical axis represents the corresponding axial emitter position z p . The relationship between these two quantities is given by Eq. ( 5) with z focus = z opt focus . We observe on the graph that the farther the emitter is from the in-focus plane (i.e., the larger |ψ| is), the more the PSF spreads out with its central lobe getting fainter. This greatly limits the localization accuracy of out-offocus fluorescent emitters.

B. Localization accuracy evaluation using the Cramér-Rao bound

The primary interest in localization microscopy is the localization accuracy of fluorescent emitters over a given range of defocus values. This accuracy can be expressed by the Cramér-Rao bound (CRB), which is a lower bound on the estimation variance that can be reached with an unbiased localization algorithm [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation theory[END_REF].

In the following, we will rather consider its square root, denoted by RCRB, since this value has the dimension of a standard deviation and is a length that will be expressed in nanometers in this article.

The value of the RCRB depends on the nature and the level of the noise sources that corrupt the observed images [START_REF] Ober | Localization Accuracy in Single-Molecule Microscopy[END_REF]. The three main sources of noise occurring in modern sensors are photon noise (i.e., shot noise), dark noise, and read-out noise. With high-performance cameras, the latter two noise sources

Simulation parameters Symbols Values

Fluorophore position in the plane (x p , y p ) (0, 0) µm can often be considered negligible [START_REF] Aguet | A maximum-likelihood formalism for sub-resolution axial localization of fluorescent nanoparticles[END_REF]. Photon noise can arise from the useful fluorescence signal (i.e., the light emitted by the excited molecule) or from the scene background. The level of background noise depends on the application, and in the present paper, for the sake of simplicity, we will neglect it, considering only the shot noise due to the light emitted by the excited molecule. Note, however, that our methodology and our main conclusions can be easily generalized to the presence of background shot noise. As a reminder, θ = (x p , y p , z p ) is the 3D position of a fluorescent emitter and (x, y) is the 2D position of its image on the sensor. Due to the quasi-circular symmetry of the 2D pixelated spatial distribution of irradiance (around its centroid) on the sensor for any value of z p , the non-diagonal terms of the Fisher information matrix [START_REF] Kay | Fundamentals of Statistical Signal Processing: Estimation theory[END_REF] are negligible. The RCRB values along the x and y-axis directions are therefore simply equal to the inverse square root of its diagonal terms. According to Ref. [START_REF] Lévêque | Can phase masks extend depth-of-field in localization microscopy?[END_REF], the RCRB also depends on the sub-pixel position of the PSF relatively to the pixel grid (i.e., the "phasing"). The largest value of the RCRB (which corresponds to the worst localization precision) is obtained when the PSF is centered on a pixel of the sensor. In this case, and in the presence of emitter shot noise only, the RCRB along the x and y-axis directions has the same value and can be calculated as follows [START_REF] Lévêque | Can phase masks extend depth-of-field in localization microscopy?[END_REF]:

RCRB =        N 0 P ∑ i=-P P ∑ j=-P (j+ 1 2 )∆xy (j-1 2 )∆xy ∆ f i (y, θ 0 ) dy 2 (i+ 1 2 )∆xy (i-1 2 )∆xy (j+ 1 2 )∆xy (j-1 2 )∆xy f (x, y, θ 0 ) dx dy        -1 2 (9)
where N 0 denotes the total number of photo-electrons expected in the squared observed image of (2P + 1) 2 pixels, ∆ xy the pitch of the pixel grid (assuming a fill factor of 100%) and ∆ f i (y, θ 0 ) the difference between the PSF values at the opposite edges of the pixels in row i:

∆ f i (y, θ 0 ) = f i - 1 2 ∆ xy , y, θ 0 -f i + 1 2 ∆ xy , y, θ 0 (10)
with θ 0 = (0, 0, z p ). It is clear from Eq. ( 9) that the accuracy with which an emitter can be located depends on the shape of the PSF, and thus on the value of the defocus parameter ψ. Indeed, we have represented in Fig. 2.c (blue solid line) the value of the RCRB as a function of ψ. We observe that the RCRB

(a) (b) (c) Fig. 2.
Variation of the PSF profile, in the QA framework, as a function of the defocus parameter ψ ∈ [-1.5λ, 1.5λ] or the axial emitter coordinates z p (a) for an aberration-free diffraction-limited system with a circular aperture, (b) for the same optical system using an optimal binary phase mask. (c) Curves of the localization accuracy (i.e., the RCRB value) without (blue solid line) or with (red dashed line) phase mask. The microscope configuration and the simulation parameters are given in Table 1 (with here

n i = n s = 1.33).
increases very quickly as the PSF degrades because the DoF of the system is small. We will quantify this degradation precisely in Section 4.B.

C. Enhancing DoF with phase masks

As proposed in Ref. [START_REF] Lévêque | Co-designed annular binary phase masks for depth-of-field extension in single-molecule localization microscopy[END_REF], a solution to improve the RCRB over the targeted DoF range is to introduce in its aperture stop a phase function Φ mask to modify the PSF of the microscope objective, such that Eq. ( 3) becomes:

Φ z p (r, ρ) = 2π λ W z p (r) + Φ mask (r, ρ) ( 11
)
where the vector ρ denotes the parameters that define the mask. Note that although the results presented in this article are limited to phase masks with circular symmetry, they can easily be extended to the many types of phase masks proposed in the literature to improve the DoF [START_REF] Dowski | Extended depth of field through wavefront coding[END_REF][START_REF] Sherif | Phase plate to extend the depth of field of incoherent hybrid imaging systems[END_REF][START_REF] Sauceda | High focal depth with fractionalpower wave fronts[END_REF][START_REF] Yang | Optimized phase pupil masks for extended depth of field[END_REF][START_REF] Caron | Polynomial phase masks for extending the depth of field of a microscope[END_REF][START_REF] Zhou | Rational phase mask to extend the depth of field in optical-digital hybrid imaging systems[END_REF]. As already mentioned, annular binary phase masks are easy to manufacture and they have been shown to extend DoF in localization microscopy [START_REF] Lévêque | Can phase masks extend depth-of-field in localization microscopy?[END_REF][START_REF] Lévêque | Co-designed annular binary phase masks for depth-of-field extension in single-molecule localization microscopy[END_REF]. This type of mask is based on concentric rings implementing static spatial phase modulations of alternatively 0 and π radians at a nominal wavelength λ as illustrated in Fig. 3. An N-ring phase mask is entirely characterized by the vector ρ = (ρ 1 , ρ 2 , • • • , ρ N-1 ) of the normalized radii of each ring. The last radius is at the edge of the aperture, i.e., ρ N = 1. Our goal is to find the optimal mask, that is, the optimal parameter vector ρ opt , that minimizes the RCRB over the defocus range. In other words,

ρ opt = arg min ρ J(ρ) (12)
with

J(ρ) = max ψ∈[-ψ max ,ψ max ] RCRB(ρ, ψ) ( 13
)
where we have explicitly indicated the dependence of the RCRB in ρ and ψ. For illustration, let us consider the optimal annular binary phase mask for a targeted DoF range of ψ max = 1.5λ obtained by solving Eq. ( 12) as explained in Ref. [START_REF] Lévêque | Can phase masks extend depth-of-field in localization microscopy?[END_REF]. The phase modulation of this mask is illustrated in Fig. 3. We have represented the variation of the PSF profile using this optimal mask in Fig. 2.b and the corresponding RCRB as a red dashed line in Fig. 2.c. We note that when a mask is used, the RCRB is much more invariant to defocus and significantly smaller for large defocus values. This mask has been optimized under the assumption that the defocus-dependent OPD, and thus the PSF, is given by the QA model. As mentioned previously, this model is physically suitable for diffraction-limited microscope objectives with moderate values of NA and a perfect match between refractive indices. For higher NA values and a possible mismatch between the refractive indices, this model therefore becomes inadequate. There exists more accurate physical imaging models for representing outof-focus emitters in real microscopy applications [START_REF] Braat | Extended Nijboer-Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system[END_REF][START_REF] Aguet | Super-Resolution Fluorescence Microscopy Based on Physical Models[END_REF][START_REF] Chomik | Déconvolution 3D orientée vers la reconstruction d'objets biologiques observés en microscopie optique de fluorescence[END_REF][START_REF] Janssen | Extended Nijboer-Zernike approach for the computation of optical point-spread functions[END_REF]. We will focus on the Gibson & Lanni model, a simple and efficient scalar model to describe the PSF of high NA immersion microscope objectives [START_REF] Gibson | Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy[END_REF]. The question we will investigate is whether this more realistic model leads to different optimal masks and different DoF enhancing performance.

THE GIBSON & LANNI MODEL FOR OUT-OF-FOCUS EMITTERS

Modern microscope objectives are designed with high NA and provide optimal imaging conditions for a point-source located immediately below the coverslip (i.e., z p = 0). However, these conditions are no longer met as soon as the emitters are located a few micrometers below the coverslip since extremely high NA and refractive index mismatch can give rise to non-negligible spherical aberrations of third or higher orders.

A. The Gibson & Lanni model

We use the GL model to better account for the imaging characteristics described above [START_REF] Gibson | Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy[END_REF]. This model considers that the imaging conditions are ideal if the emitter is located just below the coverslip and the optical parameters (including the coverslip thickness and index) have their nominal values. All the observed aberrations thus arise from factors external to the microscope objective, such as displacement of the emitter or variation of the optical parameters. In the present article, we assume that the values of the refractive indices and the microscope settings are perfectly known, and that the coverslip has its nominal properties. Under these assumptions, the OPD or wavefront error has, according to Ref. [START_REF] Gibson | Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy[END_REF], the following expression:

W GL (r) = d n i   1 - NA r n i 2 -1   -z p n s   1 - NA r n s 2 -1   , (14)
where n s , n i , z p and d have been defined in Section 2 and in Fig. 1, and r is the normalized radial coordinate on the entrance or exit pupil that ranges between 0 and 1. Note that if z p = 0 and d = 0 (i.e., the emitter is located immediately below the coverslip and in-focus), then W GL (r) = 0 and the PSF is described by the Airy pattern, see Eq. [START_REF] Goodman | Introduction to Fourier optics[END_REF]. We can also note that the QA model in Eq. ( 4) corresponds to an approximation of the GL model defined in Eq. ( 14) when n i = n s and NA 1. Let us assume that the targeted DoF is [z min , z max ] as illustrated in Fig. 1. The parameter d can be adjusted to focus the microscope at any point within the sample, and the question is how to set this focus in order to have the sharpest PSF over the whole DoF range. This question is more difficult to answer than in the QA case since there is no reduced parameter which entirely characterizes the OPD expression of Eq. ( 14). To solve this problem, we choose to represent the global amplitude of the OPD by its quadratic deviation σ GL over the whole pupil. We call it the root mean square (RMS) wavefront error, and it is precisely defined in Appendix A. To have the sharpest PSF, the parameter d must be chosen so as to minimize the maximum value of σ GL over the targeted DoF range. In Appendix B, we determine this optimal value d opt , and show that it leads to focus the microscope at the distance z p = z opt focus , defined in Eq. [START_REF] Gibson | Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy[END_REF]. Interestingly, the optimal focusing distance is thus the same for the QA and GL models, i.e., halfway between z min and z max .

B. The conditions of approximate equivalence between the GL and QA models

In this section, we compare the imaging properties predicted by the QA and the GL models. We have represented in Fig. 4.b the variation of the PSF profile as a function of the axial emitter position z p when d = d opt . The microscope configuration and the simulation parameters are given in Table 1, and n i = n s = 1.33. It is observed that this graph is quite similar to the graph obtained with the QA model represented in Fig. 4.a (this is the same figure as Fig. 2.a, it has been reproduced here to facilitate comparison). The main difference between Fig. 4.b and Fig. 4.a is a global "shrinking" of the PSF variation with the GL model. This means that the natural DoF predicted by the GL model is smaller than the one predicted by the QA model.

To confirm this interpretation, we have represented in Fig. 4.d, the Strehl ratio for the QA model (red solid line) and the GL model (blue solid line). The Strehl ratio is defined as the ratio between the maximal value of the PSF at depth z p and the maximal value of the Airy pattern. We notice in Fig. 4.d that the curves corresponding to the QA and GL models have similar shapes and can be related by a simple scaling factor. In other words, it appears that the PSF provided by the GL model for a given value of z p is quasi-equal to the PSF given by the QA model, but for a value of ψ different from that defined in Eq. ( 5). We show in Appendices C and D that this "equivalent" value of ψ is equal to:

ψ * = κ ψ z p -z opt focus (
It is easily seen that ψ * is always larger than that the value of ψ given by the QA model in Eq. ( 5), which means that the actual defocus in the GL model is larger than the one predicted by the QA model. The difference between these two values increases with NA.

To illustrate this result, we have represented in Fig. Let us now consider the case where n i = n s . We have plotted in Fig. 5 the same graphs as in Fig. 4.b, 4.c and 4.d, but for n i = 1.52 and n s = 1.33. It is seen by comparing the blue and red solid curves in Fig. 5.c that most of the deviation between the Strehl ratios of the QA and GL models still amounts to a scaling. However, the fact that n i = n s introduces two differences compared with the previous case. The first one is that the Strehl ratio of the GL model at the best focus point is no longer equal to one (see the blue solid curve in Fig. 5.c). This is due to the fact that when n i = n s , the RMS wavefront error is non zero even at z p = z focus (see Eq. ( 27)). The expression of the correcting factor β on the Strehl ratio is given in Eq. (34). The second difference is that the PSF variation is now slightly asymmetric with respect to the focus point. This is due to the fact that in case of index mismatch, the wavefront error W GL (r) over the pupil, defined in Eq. ( 14), has an asymmetrical behavior around z focus even if the RMS wavefront error is perfectly symmetric, as shown by Eq. ( 27).

We have represented in Fig. 5.b the variation of the PSF of the QA model as a function of the equivalent parameter ψ * and multiplied by the correcting factor β, which is equal to β = 0.89 in our case. We have also plotted in Fig. 5.c the corresponding value of the Strehl ratio multiplied by β (red dashed curve). It is The microscope configuration and the simulation parameters are described in Table 1.

seen that the variations of the PSF and the Strehl ratio multiplied by β are quasi-identical to those of the GL model, albeit for the slight dissymmetry on either side of the focus point.

If we substitute z p for z max (or z min ) in Eq. ( 15), a targeted DoF range [z min , z max ] in the GL model is thus equivalent to a range [-ψ * max , ψ * max ] in the QA model, with

ψ * max = κ ψ z max -z min 2 . ( 17
)
Note that this expression does not depend on the index n i of the immersion medium. This means that the apparent scaling between the QA and GL models, observed by comparing Fig. 5.a and Fig. 5.b for example, comes from the high value of the NA in the medium of refractive index n s and not from the refractive index mismatch between the immersion and sample media. This mismatch, on the other hand, explains the drop of the Strehl ratio and the dissymmetry of the PSF shape with respect to the focus plane.

As a conclusion, the main benefit of switching from the QA model to the GL model is to predict the actual axial shrinking of the PSF, and therefore of the DoF, in real configurations with high NA and possible index mismatch between the sample and the immersion media. In other words, it avoid overestimating the DoF range by a too optimistic model.

C. Equivalence of GL and QA models in the presence of phase mask

Let us now assume that a DoF-enhancing phase mask is present in the aperture stop of the microscope. As seen in Eq. ( 11), this amounts to adding a phase function which depends on the pupil coordinates r but which is independent from the emitter position z p . Hence, if a configuration corresponding to z p in the GL model is equivalent to a configuration corresponding to ψ * in the QA model -that is, they yield "equivalent" aberration wavefronts in the pupil -these aberration wavefronts should remain equivalent when using a phase mask. In consequence, we may expect that the conclusions obtained so far without mask can be extrapolated in the presence of a phase mask.

In order to verify this conjecture, we have plotted in Fig. 6 the same graphs as presented in Fig. 4 when the localization microscope is equipped with the mask shown in Fig. 3, and for n i = 1.52 and n s = 1.33. In order to better appreciate the variation of the PSF profile with defocus, we have changed the color bar for better contrast. We draw the same conclusions as established previously: the main difference between Fig. 6.a (QA model) and Fig. 6.b (GL model) is a global "shrinking" of the PSF variation. We have represented in Fig. 6.c the variation of the PSF of the QA model as a function of the equivalent parameter ψ * defined in Eq. ( 15), and displayed in Fig. 6.d the variations of the Strehl ratio for the configurations illustrated Fig. 6.a, Fig. 6.b and Fig. 6.c. We thus verify that even when using a phase mask, the variations of the PSF and of the Strehl ratio as a function of the equivalent parameter ψ * are still very similar to those of the GL model.

In the following section, we study the consequences of this approximate equivalence relation on the optimization of the DoF enhancing phase masks. Namely, we compare the masks optimized for equivalent QA and GL configurations to determine if their performance are also equivalent.

EQUIVALENCE OF DOF ENHANCING MASKS OPTI-MIZED IN THE FRAMEWORK OF QA AND GL MOD-ELS

The CRB-based phase mask optimization problem is defined in Eq. ( 12) and Eq. ( 13). It consists in minimizing the criterion J(ρ), which is the maximum value of the CRB over the targeted defocus range. Since the expressions of the PSF are different in the QA and GL models, the expressions of the RCRB are also different. We thus need to define the two following criteria:

J QA (ρ) = max ψ∈[-ψ max ,ψ max ]
RCRB QA (ρ, ψ) , The microscope configuration and the simulation parameters are described in Table 1. The microscope is equipped with the phase mask illustrated in Fig. 3 and its configuration and the simulation parameters are described in Table 1.

J GL (ρ) = max z p ∈[z min ,z max ] RCRB GL (ρ, z p ) , (18)

Fig. 7.

Optimization landscape of 2-ring binary phase masks computed for ψ max = 1λ. We use the microscope configuration and the simulation parameters described in Table 1.

where RCRB QA and RCRB GL are the values of the RCRB computed from the PSF expressions given by the QA model and the GL model respectively. In this section, we compare the masks obtained by optimizing these two criteria.

For this comparison, we select equivalent configurations, that is, we choose a value of ψ max in the QA model, and consider the equivalent targeted DoF range in the GL model, assuming for simplicity that z min = 0, and choosing the value of z max such that ψ * max = ψ max , where ψ * max is defined in Eq. ( 38). We will consider two possible combinations of refractive indices. In the first one, the indices of the immersion and sample media are equal: n i = n s = 1.33. In the second one, they are different: n i = 1.52 and n s = 1.33. The other parameters of the imaging system are given in Table 1.

Let us first consider a targeted DoF range ψ max = 1λ and annular binary masks with two rings (i.e., N = 2), that are defined by a single parameter denoted by ρ 1 . We have represented in Fig. 7 the values of J QA (ρ 1 ) (blue solid curve), and of J GL (ρ 1 ) when n i = n s (red solid curve) and when n i = n s (yellow solid curve), as a function of ρ 1 . The optimal values of ρ 1 correspond to the positions of the global minimum of each curve. It is observed that the curves are close to each other and that their global minima occur at very similar positions. Hence, it can be concluded that the 2-ring annular binary phase masks optimized in the framework of the QA and GL models are almost identical.

Let us now generalize this result to larger DoF ranges. For that purpose, we need to consider annular binary phase masks with larger number of rings. Indeed, as shown in Ref. [START_REF] Lévêque | Co-designed annular binary phase masks for depth-of-field extension in single-molecule localization microscopy[END_REF], large DoF ranges require a larger number of rings to reach optimal localization accuracy. Therefore, in our optimization process, we consider that N can be as large as 5. Note, however, that in some cases, the performance levels-off for a smaller value of N, in which case the optimal mask has less than five rings. We consider three different values of the targeted DoF range: ψ max = 1λ, 1.5λ, and 2λ. To optimize the criteria, which are highly non-convex, we use the particle swarm global optimization algorithm described in Ref. [START_REF] Kennedy | Particle swarm optimization[END_REF].

Table 2 synthesizes the results obtained for each considered value of ψ max (first column) and combination of optical indices (n i = n s or n i = n s , second column). The third column represents the optimal masks for the GL model, defined by the parameter vector ρ GL opt , and the corresponding value of the criterion J GL (ρ GL opt ), which is the maximum value of the RCRB GL within the DoF range. It is observed that as the targeted DoF range ψ max increases, the value of J GL (ρ GL opt ) increases, since the DoF extension problem becomes more difficult. It is also noticed that it is slightly larger in the case of n i = n s , due to the PSF alteration induced by the refractive index mismatch. The fourth column of Table 2 represents the optimal mask for the QA model, defined by the parameter vector ρ QA opt , and the corresponding value of the criterion J QA (ρ QA opt ). By comparison with the masks in the third column, it is seen that the masks that are optimal for QA and GL models are different but bear some similarities. In particular, the radii of their main rings are similar. The question is: are these masks equivalent in terms of DoF extension performance?

To answer this question, let us consider the following scenario: a mask has been optimized with the simple QA model, for a targeted DoF range ψ max , and is used in a system with high NA and possibly index mismatch, which parameters (NA, refractive indices, distances) are such that ψ * max = ψ max . The value of the localization criterion obtained in this case, J GL (ρ QA opt ), is given in the fifth column of Table 2. By comparing the values in the third and fifth columns of the table, it is seen that J GL (ρ QA opt ) > J GL (ρ GL opt ), which is expected since the mask ρ QA opt is not optimal for the GL model. However, the values of J GL (ρ QA opt ) and J GL (ρ GL opt ) are quite close, which means that even if the mask has been optimized with the QA model, it is still close to optimality for the GL model. It is also observed in Table 2 that the departure from optimality increases with the value of ψ max and with the index mismatch. However, even in the worst case, which corresponds to ψ max = 2λ and n i = n s , one has J GL (ρ GL opt ) = 10.25 nm and J GL (ρ QA opt ) = 11.45 nm, which corresponds to a relative difference of only 12%.

As a summary, the DoF enhancing masks optimized in the framework of the QA model are also close to optimality in the framework of the GL model, provided optimization in the QA case has been done with the "equivalent" defocus parameter ψ * max defined in Eq. ( 17). This result is important since it allows the use of generic masks targeting a given DoF range ψ * max without needing precise knowledge of NA, n i and n s . Indeed, let us consider various microscope configurations having different values of refractive indices, NA, and imaging depth, but corresponding to the same value of the reduced defocus parameter ψ * max . The optimal masks in all these configurations yield nearly equal performance, and they are similar to the mask optimal in the framework of the QA model for a DoF range equal to [-ψ * max , ψ * max ].

CONCLUSION

In this article we have compared the DoF extending performance of phase masks optimized in the framework of the simple QA model and of the GL model that more accurately represents imaging with high NA and refractive index mismatch. We have first demonstrated that the validity of the QA model can be extended by using a scaled DoF parameter derived from the GL model. This parameter is based on the equivalence of the RMS wavefront errors. Then, we have shown that for these equivalent configurations, the optimal masks in the GL model and in the QA model are similar and yield similar localization performance. This result is important since it allows the use of generic masks without needing precise knowledge of NA, n i and n s . This work has many interesting perspectives. Since the annular binary phase masks are easy to manufacture, the main one would be to validate the conclusions of this paper experimentally. Moreover, the equivalent DoF parameter defined in this paper is totally independent of the type of phase mask used in the system. It would thus be interesting to apply the proposed 2. Optimal multi-ring binary phase masks and their performance based on the criteria defined in Eq. ( 18). The microscope configurations and the simulation parameters are described in Table 1.

approach to other types of phase masks, such as those aimed at 3D localization [START_REF] Shechtman | Optimal Point Spread Function Design for 3D Imaging[END_REF][START_REF] Pavani | High-efficiency rotating point spread functions[END_REF].

APPENDICES A. RMS WAVEFRONT ERROR IN THE GL MODEL

By definition, the RMS wavefront error, denoted by W(r) with r the normalized radius in the pupil, is defined as

σ 2 W = 1 1 0 r dr 1 0 W(r) -W 2 r dr = 2 1 0 W(r) -W 2 r dr (19)
where the average value W is defined as

W = 1 1 0 r dr 1 0 W(r)r dr = 2 1 0 W(r)r dr . (20)
For the GL model, by substituting the OPD expression given by Eq. ( 14) in Eq. ( 19), the RMS wavefront error has the following expression:

σ 2 GL (z p , d) = z 2 p κ 2 s + d 2 κ 2 i -2 z p d c 2 si ( 21
)
where the coefficients κ 2 s , κ 2 i and c 2 si are defined as

           κ 2 s = 2 1 0 h s (r) -h s 2 r dr κ 2 i = 2 1 0 h i (r) -h i 2 r dr c 2 si = 2 1 0 h s (r) -h s h i (r) -h i r dr (22)
with

h k = 2 1 0 h k (r)r dr and h k (r) = n 2 k -NA 2 r 2 , (23)
and k represents the subscript letter "s" or "i" used in Eq. [START_REF] Pavani | High-efficiency rotating point spread functions[END_REF].

B. OPTIMAL PARAMETER d IN THE GL MODEL

Let us assume that the targeted DoF range is [z min , z max ]. We want to adjust the parameter d so as to minimize the maximum of the RMS wavefront error σ GL , defined in Eq. ( 21), over the targeted DoF range. This leads solving the following problem:

d opt = arg min d max z p ∈[z min ,z max ] σ 2 GL (z p , d) . (24)
Since σ 2 GL (z p , d) is a second degree polynomial in the variable z p with positive coefficient for the second order term (see Eq. ( 21)), its maximum value, for a given d, is reached at the extremities of the range, that is, for z p = z min or z max . Thus, the problem described in Eq. ( 24) can be rewritten as:

σ 2 GL (z min , d opt ) = σ 2 GL (z max , d opt ) . (25)
Solving this problem, the optimal value of d is obtained as:

d opt = z min + z max 2 κ 2 s c 2 si . (26)
By substituting Eq. (26) into Eq. ( 21) and simplifying, we obtain:

σ 2 GL (z p , d opt ) = κ 2 s z p -z focus 2 + αz 2 focus ( 27
)
with

z focus = z min + z max 2 and α = κ 2 s κ 2 i c 4 si -1 . (28)

C. SCALING BETWEEN THE QA AND GL MODELS

It is interesting to compare Eq. ( 27) to the expression of the RMS wavefront error in the QA model, obtained by substituting the OPD expression given by Eq. ( 4) and Eq. ( 5) into Eq. ( 19):

σ 2 QA = ψ 2 12 = κ 2 QA z p -z focus 2 (29) with κ 2 QA = NA 4 48n 2 s . (30)
One observes that Eq. ( 27) and Eq. ( 29) are both quadratic functions of the variable ∆z = (z pz focus ), but they bear two important differences. The first one is that the coefficients of the quadratic term are different in the two equations. For the QA model, it is κ QA , and for the GL model, it is κ s whose expression is given in Eq. [START_REF] Pavani | High-efficiency rotating point spread functions[END_REF]. With the help of a symbolic math software, we were able to determine the first terms of the Taylor series expansion of κ

Note that the first term is exactly κ 2 QA , which shows that for a low NA, both models predict, as expected, the same behavior. However, in the general case, the QA and GL models lead to the same value of the RMS wavefront error for different depth variations ∆z, given by κ

It is easily seen that this factor starts at 1 for low (zero) NA and significantly grows as NA increases. This means that when NA is large, the QA model yields a too optimistic value of the DoF extension. According to the GL model, this value has to be divided by the the factor η as NA increases. This is clearly seen by comparing Fig. 4.a and Fig. 4.b for example. Interestingly, this scaling coefficient depends only on n s (through κ s ), and not on n i . It is thus independent of the index mismatch between the immersion and sample media: it is due only to the high value of NA in the medium with refractive index n s .

The second difference difference between Eq. ( 27) and Eq. ( 29) is the presence of an offset term equal to α κ 2 s z 2 focus in σ 2 GL . According to the Cauchy-Schwartz inequality, κ s κ i ≥ c 2 si , the factor α is thus positive or zero. Moreover, this term vanishes when n i = n s , since in this case α = 0, and increases with index mismatch and the value of z focus .

To illustrate the effects of these two differences, let us consider the Strehl ratio, denoted by S, which is the ratio between the maximal values of the actual PSF and of the Airy pattern for the same aperture. A rule of thumb widely used in optical system design states that if the RMS wavefront error σ is not too large, the Strehl ratio can be expressed as [START_REF] Mahajan | Strehl ratio for primary aberrations in terms of their aberration variance[END_REF]:

D. EQUIVALENT VALUE OF ψ

It can be noted that even if n s = n i , the offset term in Eq. ( 27) is very small for the parameters used in our simulations of localization microscopy: the standard values of n s and of n i in

Replacing z p with z max or z min in Eq. ( 36), one can also define the equivalent targeted DoF range as

ψ * max = κ ψ z max -z min 2 . ( 38
)
Eq. (38) makes it possible to establish a simple correspondence between the targeted DoF ranges in the GL and QA models.

Fig. 1 .

 1 Fig. 1. Schematic drawing of the nominal and depth-extended object-space configurations of the immersion microscope objective. ABC is a ray originating from a point source located just below the coverslip when d has its nominal 0 value. PQRS is a ray from a point source located within the targeted DoF range [z min , z max ] when d = 0.

Fig. 3 .

 3 Fig. 3. Example of a 5-ring annular binary phase mask. This mask is optimal for a targeted DoF range of ψ max = 1.5λ in the QA framework.

  4.c the variation of the PSF profile computed with the QA model as a function of ψ * (this is a scaled version of Fig. 4.a). It is clear that Fig. 4.c is now very similar to Fig. 4.b: with this definition of the equivalent defocus parameter ψ * , the PSF profiles given by the QA and the GL models are quasi-equal. This result is confirmed by plotting the corresponding values of the Strehl ratio in Fig. 4.d (red dashed curve): it is nearly superimposed on the GL model (blue solid curve).

Fig. 4 .

 4 Fig. 4. (a) PSF profile as a function of the defocus parameter ψ ∈ [-1.5λ, 1.5λ] using the QA model (same graph as Fig. 2.a). (b) PSF profile as a function of the emitter position z p with the GL model (n i = n s = 1.33). (c) PSF profile as a function of the equivalent defocus parameter ψ * using the QA model. (d) Variation of the Strehl ratio for the optical systems illustrated in (a), (b) and (c).The microscope configuration and the simulation parameters are described in Table1.

Fig. 5 .

 5 Fig. 5. (a) PSF profile as a function of the emitter position z p with the GL model when n i = 1.52 and n s = 1.33. (b) PSF profile, multiplied by the correcting factor β defined in Eq. (34), as a function of the equivalent defocus parameter ψ * using the QA model. (c) Variation of the Strehl ratio for the optical systems illustrated in (a) and (b).The microscope configuration and the simulation parameters are described in Table1.

Fig. 6 .

 6 Fig. 6. (a) PSF profile as a function of the defocus parameter ψ ∈ [-1.5λ, 1.5λ] using the QA model. (b) PSF profile as a function of the emitter position z p with the GL model (n i = 1.52 and n s = 1.33). (c) PSF profile as a function of the equivalent defocus parameter ψ * using the QA model. (d) Variation of the Strehl ratio for the optical systems illustrated in (a), (b) and (c).The microscope is equipped with the phase mask illustrated in Fig.3and its configuration and the simulation parameters are described in Table1.

S exp - 2π λ σ 2 . ( 33 )Fig. 5 .

 2335 Fig. 5.c. shows an excellent agreement with this predicted behavior.

Table 1 .

 1 Microscope configuration and simulation parameters used in the present paper.

	Fluorophore wavelength emission	λ	700 nm
	Total number of photo-electrons	N 0	500 ph.e -
	PSF image length	2P + 1	21 pixels
	Pixel length	∆ xy	10 µm
	Object numerical aperture	NA	1.3
	Lateral magnification	M	60
	Biological sample refractive index	n s	1.33
	Immersion refractive index *	n i	1.33 or 1.52

*

We consider two types of nominal immersion medium: water or oil.

15)

	where	κ ψ	NA 2 2n s	+	NA 4 8n 3 s	+	11 NA 6 192n 5 s	+	25 NA 8 s 768n 7	.

  2 s with respect to (NA /n s ) 2 :

	κ 2 s	NA 4 48n 2 s	1 +	NA 2 2n 2 s	+	7 NA 4 24n 4 s	+	3 NA 6 s 16n 6	.

  Table1lead to α = 0.0062, and since emitters lie only a few micrometers below the coverslip, z focus is small. If this offset term is neglected, Eq. (27) becomes:σ 2 GL (z p , d opt ) κ 2 s z pz focusBy equalizing this expression with the RMS wavefront error of the QA model defined in Eq. (29), we can define the value ψ * of the defocus parameter in the QA model that is "equivalent" to an emitter position z p in the GL model:ψ * = κ ψ z pz focus ,

										2	(35)
											(36)
	with
	κ ψ = 2	√	3 κ s	NA 2 2n s	+	NA 4 8n 3 s	+	11 NA 6 192n 5 s	+	25 NA 8 s 768n 7	.

Acknowledgments. This work has received the support of the

French research group GdR ISIS of the CNRS through the Projet de Recherche Exploratoire MASK.

Disclosures.

The authors declare no conflicts of interest.