Maxence Cassier
email: maxence.cassier@ensta-paristech.fr

Christophe Hazard
email: christophe.hazard@ensta-paristech.fr

Space-time focusing of acoustic waves on unknown scatterers

Consider a propagative medium, possibly inhomogeneous, containing some scatterers whose positions are unknown. Using an array of transmit-receive transducers, how can one generate a wave that would focus in space and time near one of the scatterers, that is, a wave whose energy would confine near the scatterer during a short time? The answer proposed in the present paper is based on the so-called DORT method (French acronym for: decomposition of the time reversal operator) which has led to numerous applications owing to the related space-focusing properties in the frequency domain, i.e., for time-harmonic waves. This method essentially consists in a singular value decomposition (SVD) of the scattering operator, that is, the operator which maps the input signals sent to the transducers to the measure of the scattered wave. By introducing a particular SVD related to the symmetry of the scattering operator, we show how to synchronize the time-harmonic signals derived from the DORT method to achieve space-time focusing. We consider the case of the scalar wave equation and we make use of an asymptotic model for small sound-soft scatterers, usually called the Foldy-Lax model. In this context, several mathematical and numerical arguments that support our idea are explored.

Introduction 1.General context and objective

Time reversal (TR) techniques have been extensively studied during the last decades, in particular owing to the related focusing properties. The simplest illustration of these properties consists in the following classical TR experiment (see, e.g., [START_REF] Fink | Acoustic time-reversal mirrors[END_REF]). In a first step, an acoustic source emits a sound which propagates in the surrounding medium. The acoustic field is then measured and recorded by an array of transmitreceive transducers. In a second step, all the recordings are time-reversed and then used as input signals for the transducers of the array, generally called a time reversal mirror (TRM) because of this time reversal step. The acoustic wave emitted by the TRM propagates back in the same medium towards the original source. If this source fills a small region and if the initial sound is a short pulse, one can observe that a great part of the acoustic energy of the back-propagating wave will be concentrated near the original source at a given time. In this sense, space-time focusing is achieved. The focus quality will be high if the TRM surrounds completely the source (full aperture case) and if the number of transducers is large enough. On the other hand, the focusing will generally deteriorate if the number of transducers becomes smaller or if the aperture of the TRM is reduced, that is, if the angular area occupied by the TRM decreases (partial aperture case).

In such a TR experiment, the source is active in the sense that it emits the original sound which is then recorded by the TRM. In the present paper, we are no longer concerned by active sources, but rather by unknown scatterers which behave passively only by their ability to reflect an incident wave. Hence the way to retrieve some information on the scatterers is to test the medium by means of the array of

Mathematical formulation of the problem

In the present paper, we consider a reference medium, possibly inhomogeneous in a bounded region, filling the whole space R d (with d > 1). This medium is characterized by a sound velocity function c = c(x) for x ∈ R d . We denote by G = G(x, y, t) the time-dependent Green's function of the acoustic wave equation, which is solution to

1 c 2 (x) ∂ 2 G(x, y, t) ∂t 2 -∆ x G(x, y, t) = δ(x -y) ⊗ δ(t),
where δ stands for the Dirac measure at the origin, either in space or time. Moreover G is assumed to be causal in the sense that G(x, y, t) = 0 for all (x, y) ∈ R 2d and t < 0 (which ensures the uniqueness of G). Recall that for a homogeneous medium, i.e., c ≡ 1, this function is given by

G hom (x, y, t) =        H(t -|x -y|) 2π t 2 -|x -y| 2 if d = 2, δ(t -|x -y|) 4π |x -y| if d = 3, ( 1
)
where H is the Heaviside function.

We consider an array of N point-like transducers located at x n for n = 1, . . . , N which can emit incident waves of the form

v inc (x, t) := N n=1 (q n t ⋆ G(x, x n , •))(t) = N n=1 R q n (t ′ ) G(x, x n , t -t ′ ) dt ′ (2)
where q in (t) := (q 1 (t), . . . , q N (t)) ⊤ represents the vector composed of the input signals which are applied to the transducers (the symbol ⊤ indicates transposition) and t ⋆ denotes the convolution with respect to time.

We suppose that our reference medium is perturbed by the presence of a family of P sound-soft scatterers S p ⊂ R d for p = 1, . . . , P. In this case, when the same signals q in (t) are sent to the transducers, we observe instead of the incident wave (2) a perturbed wave v inc + v sc where v sc represents the scattered wave, that is, the part of the wave due to the presence of the scatterers, which is the causal solution to 1 c 2 (x)

∂ 2 v sc ∂t 2 -∆v sc = 0 in R d \ ∪ P p=1 S p , (3)
v sc = -v inc on ∪ P p=1 ∂S p . ( 4
)
This wave is finally measured by the transducers, which yields a vector composed of N functions which is denoted by q out (t) := (v sc (x 1 , t), . . . , v sc (x N , t)) ⊤ . The linear operator S : q in -→ q out is called here the scattering operator. Note that this definition differs slightly from the scattering operator involved in mathematical physics.

In this paper, we use a spectral representation of S, which amounts to representing all time-dependent signals as superpositions of time-harmonic signals. Thus we introduce the Fourier transform F with respect to time defined by

f (ω) := Ff (ω) := +∞ -∞ f (t) e +iωt dt
whose inverse is given by

f (t) = F -1 f (t) = 1 2π +∞ -∞ f (ω) e -iωt dω.
Note that in the case where f is real-valued, its Fourier transform satisfies the symmetry property

f (-ω) = f (ω) so that f (t) = 1 π Re +∞ 0 f (ω) e -iωt dω. (5)
Define qin (ω) := Fq in (ω) = (q 1 (ω), . . . , qN (ω)) ⊤ . For all ω, qin (ω) represents the time-harmonic input signals which emit a time-harmonic incident wave

vinc (x, ω) := (Fv inc (x, •))(ω) = N n=1 qn (ω) Ĝ(x, x n , ω), (6)
where Ĝ = Ĝ(x, y, ω) denotes the Fourier transform of G, that is, the time-harmonic Green's function of the reference medium. Its expression is well-known for a homogeneous medium (c ≡ 1) :

Ĝhom

(x, y, ω) =      i 4 H (1) 0 (ω|x -y|) if d = 2, e iω|x-y| 4π|x -y| if d = 3,
where H

(1) 0

is the Hankel function of the first kind and order 0. The interaction of vinc (x, ω) with the scatterers produces a scattered wave vsc (x, ω) = (Fv sc (x, •))(ω) which is finally measured by the transducers. This yields the time-harmonic scattering operator

S ω : qin (ω) -→ qout (ω) := (v sc (x 1 , ω), . . . , vsc (x N , ω)) ⊤ . ( 7
)
We can sum up the link between the time-dependent and time-harmonic scattering operators by the equality

S = F -1 S F, (8)
where S denotes the operator defined from the family { S ω ; ω ∈ R} by the simple relation ( S q)(ω) = S ω ( q(ω)).

The question we address in the present paper is the following: from the only knowledge of the operator S, how can one generate a wave that focuses selectively in space and time on one of the scatterers? The answer we propose is the subject of section 3. It is based on the DORT method presented in the next section.

2 Space focusing in the frequency domain

The scattering operator for small scatterers

As mentioned in the introduction, the spatial focusing properties related to the DORT method hold for small enough scatterers. That is why we use in the sequel an asymptotic model in the frequency domain, which enables us to replace the family {S p ⊂ R d ; p = 1, . . . , P } by a family of P point-like scatterers located at y 1 , . . . , y P . The model we use, often called the Foldy-Lax model [START_REF] Foldy | The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers[END_REF], has the advantage to take into account an approximation of the interactions between the scatterers. It is widely used by physicists. For instance, [START_REF] Marengo | Subspace-based localization and inverse scattering of multiply scattering point targets[END_REF] presents an application of the MUSIC method for estimating the locations and reflectivities of point-like inhomogeneities. A mathematical justification of the Foldy-Lax model was recently proposed in [START_REF] Cassier | Multiple scattering of acoustic waves by small sound-soft obstacles in two dimensions: mathematical justification of the Foldy-Lax model[END_REF] for the two-dimensional problem. This model is based on the fact that in the case of one single sound-soft scatterer (P = 1) whose size is small compared to the wavelength, the time-harmonic scattered field vsc behaves like the field emitted by a point source (see [START_REF] Martin | Multiple Scattering. Interaction of Time-Harmonic Waves with N Obstacles[END_REF]). More precisely, one has vsc (x, ω) ≈ σ(ω) vinc (y, ω) Ĝ(x, y, ω), where y ∈ R d is the location of the scatterer and σ(ω) ∈ C is its reflectivity coefficient. For instance, in a homogeneous medium (c ≡ 1), if the scatterer is circular (d = 2) or spherical (d = 3) with radius ε, we have

σ(ω) = -4i/H (1) 0 (ωε) if d = 2, 4πε if d = 3. (9)
For several obstacles (P > 1), the scattered field appears as a superposition of similar behaviours, vsc (x, ω)

≈ P p=1 σ p (ω) vp (ω) Ĝ(x, y p , ω) (10)
where vp (ω) represents the exciting field on the p-th scatterer. Choosing vp (ω) = vinc (y p , ω) as for a single scatterer amounts to neglecting the interactions between the scatterers. The Foldy-Lax model consists in correcting the latter choice by adding the waves scattered by all the other obstacles, i.e., vp (ω) = vinc (y p , ω) + q =p σ q (ω) vq (ω) Ĝ(y p , y q , ω) for p = 1, . . . , P.

If we denote by V ω and V inc ω the vectors of C P with components vp (ω) and vinc (y p , ω) respectively, this coupling between the exciting fields can be written equivalently as

I -H ω Σ ω V ω = V inc ω , (11)
where Σ ω is the P × P diagonal matrix composed of the reflectivity coefficients, i.e., (Σ ω ) pp := σ p (ω), and H ω is the P × P matrix defined by ( H ω ) pq := Ĝ(y p , y q , ω) if q = p and ( H ω ) pp := 0.

The use of the Foldy-Lax model yields a simple expression for the time-harmonic scattering operator [START_REF] Cassier | Multiple scattering of acoustic waves by small sound-soft obstacles in two dimensions: mathematical justification of the Foldy-Lax model[END_REF]. Indeed, applying [START_REF] Foldy | The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers[END_REF]

to x = x n for n = 1 to N yields qout (ω) = G ⊤ ω Σ ω V ω , (12)
where G ω denotes the P × N matrix defined by ( G ω ) pn := Ĝ(x n , y p , ω), and we recall that the superscript ⊤ indicates transposition. Moreover, noticing from (6) that the righthand side of (11) writes as

V inc ω = G ω qin (ω), we have V ω = (I -H ω Σ ω ) -1 G ω qin (ω)
. Substituting this expression in [START_REF] Hazard | Selective acoustic focusing using time-harmonic reversal mirrors[END_REF] yields finally

S ω = G ⊤ ω (Σ -1 ω -H ω ) -1 G ω . ( 13
)
This product of three matrices can be interpreted as the succession of three steps in the scattering process. First G ω represents the propagation of the incident wave from the transducers to the scatterers. Then (Σ -1 ω -H ω ) -1 describes the reflection on the scatterers (note that this matrix reduces to Σ ω if H ω is neglected, which amounts to neglecting the interaction between the scatterers). And finally G ⊤ ω corresponds to back-propagation from the scatterers to the transducers.

Remark 1. It is readily seen that the scattering operator is symmetric (but not hermitian) in the sense that S ω = S ⊤ ω (but S ω = S * ω where S * ω is the adjoint matrix, i.e., the conjugate transpose matrix) since both matrices Σ ω and H ω are symmetric. This can be seen as a consequence of the reciprocity principle.

The DORT method

The DORT method [START_REF] Prada | Detection and imaging in complex media with the D.O.R.T. method[END_REF][START_REF] Prada | Eigenmodes of the time reversal operator: a solution to selective focusing in multiple-target media[END_REF][START_REF] Prada | Decomposition of the time reversal operator: detection and selective focusing on two scatterers[END_REF] consists in a singular value decomposition (SVD) of the scattering operator S ω :

S ω = V ω D ω W * ω (14)
where D ω is a N × N diagonal matrix with nonnegative real numbers on the diagonal, whereas V ω and W ω are unitary N × N matrices. The diagonal entries of D ω are known as the singular values of S ω and the columns of V ω and W ω are called the left and right singular vectors, respectively.

A real number λ ≥ 0 is a singular value of S ω if and only if there exist unit vectors v and ŵ in C N such that

S ω ŵ = λ v and S * ω v = λ ŵ. ( 15
)
Vectors v and ŵ are respectively left and right singular vectors of S ω .

Remark 2. The SVD of S ω is not unique (actually D ω is unique but not V ω and W ω ). This is obvious if at least one eigenvalue is degenerate, that is, if the dimension of the associated space of right (respectively, left) singular vectors is greater than one. On the other hand, for a simple (or non-degenerate) singular value, the right and left singular vectors are unique up to a multiplication by a unit complex number. Indeed it is readily seen that if a pair (v, ŵ) satisfies [START_REF] Martin | Multiple Scattering. Interaction of Time-Harmonic Waves with N Obstacles[END_REF], then so does (e iφ v, e iφ ŵ) for all φ ∈ R.

Remark 3. The singular values of S ω are the square roots of the eigenvalues of the positive hermitian matrix S * ω S ω and the right singular vectors of S ω coincide with the eigenvectors of S * ω S ω . The initial presentation of the DORT method (see, e.g., [START_REF] Prada | Eigenmodes of the time reversal operator: a solution to selective focusing in multiple-target media[END_REF][START_REF] Prada | Decomposition of the time reversal operator: detection and selective focusing on two scatterers[END_REF]) consisted in the diagonalization of the latter matrix, called the time reversal operator because it was derived from an iterative TR process. The basic loop of this process consists of the following steps: for given input signals q, the array of transducers first emit an incident wave; the associated scattered wave in then measured, which yields S ω q; and finally this measure is time-reversed, which amounts to a conjugation in the frequency domain (see [START_REF] Prada | Eigenmodes of the time reversal operator: a solution to selective focusing in multiple-target media[END_REF]). The time-reversed measure can then be used to re-emit a new incident wave. The time reversal operator T ω describes two successive iterations of this loop, that is,

T ω q := S ω S ω q = S ω S ω q so T ω = S ω S ω = S * ω S ω ,
where the last equality follows from the symmetry of S ω .

The main interest of the DORT method lies in the selective focusing properties related to the singular vectors of S ω . In our context of point-like scatterers, it is well understood [START_REF] Hazard | Selective acoustic focusing using time-harmonic reversal mirrors[END_REF][START_REF] Prada | Eigenmodes of the time reversal operator: a solution to selective focusing in multiple-target media[END_REF][START_REF] Prada | Decomposition of the time reversal operator: detection and selective focusing on two scatterers[END_REF] that if the reference medium is homogeneous and if the scatterers are distant enough from each other (they are often referred to as ideally resolved in this case), then 1. the number of scatterers P is equal to the number of nonzero singular values of S ω (provided that P ≤ N and that the rank of G ω is equal to P, which excludes some exceptional situations such as scatterers located symmetrically with respect to the array);

Figures 1 to 3 illustrate these results by some numerical experiments in the two-dimensional case. We first consider the case of a homogeneous medium (i.e., c ≡ 1) filling the whole plane R2 . The part of the plane represented on each figure is the square [0, 2] × [0, 2]. Our array is composed of 128 transducers (N = 128) regularly distributed along the segment {0} × [0, 2] which corresponds to the left edge of each figure. Two scatterers (P = 2), with respective diameters 2 × 10 -3 and 10 -3 , are located at (1.2, 0.3) and (1.7, 1.9) respectively. They are represented by white disks (not to scale). The time-harmonic scattering operator has two non-zeros singular values in this case. For the circular frequency ω = 40, that is, a wavelength λ = 2π/ω ≈ 0.16, Figure 1 shows the modulus of the incident field generated by the associated singular vectors. We clearly see the spatial focusing effect: the field becomes maximal in an elliptical area centered at each scatterer, generally called the focal spot. If ℓ denotes the length of the array (here ℓ = 2) and d is the distance between the array and the considered scatterer, the transverse radius of this area is given by λd/ℓ (Rayleigh resolution formula, see, e.g., [START_REF] Born | Principles of Optics[END_REF]) provided that λ ≪ ℓ ≪ d. For instance λd/ℓ ≈ 0, 1 for the biggest scatterer, which is consistent with the observed focal spot though the assumption ℓ ≪ d is not fulfilled.

In Figures 2 and3, we consider the case where a scattering medium is located between the array and the scatterers. Here our scattering medium consists of 500 point-like scatterers with same diameter 2 × 10 -4 (represented by black points) randomly displayed in the rectangle [0.3, 0.8] × [0, 2]. In this situation, the Green's function of the reference medium appears as a perturbation of the Green's function of the homogeneous medium: Ĝ(x, y, ω) = Ĝhom (x, y, ω) + Ĝsc (x, y, ω)

where the perturbation Ĝsc (x, y, ω) can be computed by the Foldy-Lax model described in section 2.1, which requires us to invert the 500 × 500 matrix involved in equation [START_REF] Garnier | Travel time estimation by cross correlation of noisy signals[END_REF]. Figure 2 represents again the modulus of the incident field generated by the singular vectors associated with the non-zero eigenvalues of S ω for the same circular frequency ω = 40 as in Figure 1. We see that the presence of the scattering medium has damaged considerably the spatial focusing effect. Focal spots are still observed, but their intensity is reduced significantly, for the acoustic field seems to be trapped in the scattering medium. Figure 3 shows that for the higher circular frequency ω = 90, spatial focusing is recovered.

The physical notions of ballistic and diffusive regimes (see, e.g., [START_REF] Elaloufi | Diffusive-to-ballistic transition in dynamic light transmission through thin scattering slabs: a radiative transfer approach[END_REF]) can help us to interpret these numerical observations. We have to compare the thickness L of the scattering medium with the mean free path which is given here by

L fp := 4ω N |σ(ω)| 2
where N is the number of scatterers per unit surface and σ(ω) is the reflectivity coefficient defined in [START_REF] Fink | Acoustic time-reversal mirrors[END_REF]. The so-called ballistic regime corresponds to the case where L ≪ L fp . In such a situation, when a plane wave crosses the scattering medium, its direction of propagation is not much affected and the perturbation of the plane wave essentially consists in the specular reflections on the scatterers. Higher order successive reflections are negligible. On the other hand, when L ≫ L fp , the latter may become significant, which leads to resonant phenomena inside the scattering medium. In such a diffusive regime, the structure of a wave crossing the medium is strongly perturbed. In the case of Figure 2 (ω = 40), we have L fp ≈ 0.28 whereas L = 0.5, which rather corresponds to the diffusive regime. In the case of Figure 3 (ω = 90), we have L fp ≈ 0.47, which corresponds to an intermediate regime. In particular, resonances are still observed in the scattering medium. By increasing the frequency further, we would enter the ballistic regime and the focus quality would improve.

3 Space-time focusing

From frequency domain to time domain

A phase synchronization issue.

We arrive now at the main purpose of the paper, which is to take advantage of the selective focusing properties of the DORT method in the frequency domain in order to construct a time-dependent wave which focuses in space and time on one given scatterer chosen as a target. Suppose that for all frequency ω in a given frequency band [ω 1 , ω 2 ] (which depends on the specifications of the transducers), we know a triplet (λ(ω), v(ω), ŵ(ω)) of singular elements of the scattering operator corresponding to our target: λ(ω) > 0 is a singular value, which is assumed simple for simplicity, and the associated left and right singular vectors v(ω) and ŵ(ω) are unit vectors of C N such that

S ω ŵ(ω) = λ(ω) v(ω) and S * ω v(ω) = λ(ω) ŵ(ω). ( 16
)
The fact that these singular elements correspond to our target means that the time-harmonic wave generated by the right singular vector ŵ(ω) focuses spatially on this target. As this spatial focusing occurs for all ω ∈ [ω 1 , ω 2 ], it holds true for any frequency superposition of these waves. So if we consider a pair (χ, φ) of real-valued functions defined on [ω 1 , ω 2 ], the time-dependent input signal defined by

q φ (t) := 1 π Re ω2 ω1 χ(ω) e iφ(ω) ŵ(ω) e -iωt dω (17)
generates a time-dependent wave which focuses in space on our target. But can we choose these functions so that the above superposition of time-harmonic signals yields also time focusing? Instead of searching for a pair (χ, φ), we assume here that χ is known and we restrict ourselves to the determination of the phase function φ : [ω 1 , ω 2 ] → R (which justifies the fact that χ is not involved in the notation q φ above).

In other words, the issue we are faced with is to synchronize the singular vectors of S ω in order to achieve space-time focusing (recall that e iφ(ω) ŵ(ω) is a right singular vector of S ω whatever φ(ω), see remark 2).

Continuity assumptions in the frequency domain.

From a practical point of view, the signals q φ (t) we are interested in must have a finite duration of emission. However this is incompatible with the expression (17) of q φ (t). Indeed, from [START_REF] Borcea | Theory and applications of time reversal and interferometric imaging[END_REF], it is clear that the Fourier transform qφ of q φ is given by

qφ (ω) =    χ(ω) e iφ(ω) ŵ(ω) if ω ∈ [ω 1 , ω 2 ], 0 if ω ∈ R + \ [ω 1 , ω 2 ], qφ (-ω) if ω ∈ R -. (18)
In particular qφ has a compact support in the frequency domain, so q φ cannot be compactly supported in the time domain since it is an entire function of t. However q φ (t) may be close to 0 outside a bounded interval provided that qφ (ω) is regular enough (recall that the Riemann-Lebesgue theorem tells us that if

t n q φ (t) ∈ L 1 (R), then qφ (ω) ∈ C n (R))
. In such a case, ( 17) can be considered as an approximation of a compactly supported signal.

It is then natural to assume at least that qφ (ω) ∈ C 0 (R). This means on the one hand that χ and φ both belong to C 0 ([ω 1 , ω 2 ]) and χ(ω 1 ) = χ(ω 2 ) = 0, on the other hand that ŵ(ω) depends also continuously on ω ∈ [ω 1 , ω 2 ]. This assumption is not restrictive. Indeed we can assume without loss of generality that our family of singular elements is continuous in the sense that λ(ω), v(ω) and ŵ(ω) are continuous functions of ω. One can always find such a family. This follows from classical arguments of perturbation theory [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF] using the fact that S ω is a continuous (even C ∞ ) function of ω (which derives from the regularity of the time-harmonic Green's function Ĝ(•, •, ω)).

A particular singular value decomposition.

The phase choice we propose is related to the symmetry of the scattering operator (see remark 1). Indeed, this property implies that there exists a particular SVD, called the symmetric singular value decomposition (SSVD) or Takagi's factorization [START_REF] Horn | Matrix Analysis[END_REF], which writes as

S ω = U ω D ω U * ω (19)
where D ω is the diagonal matrix composed of the singular values and U ω is unitary. This decomposition is a particular version of ( 14) with W ω = V ω = U ω , i.e., the right and left singular vectors are conjugate of each other. From a practical point of view, the way to obtain the SSVD from a given SVD is straightforward, especially for simple singular values. Indeed, as S ω is symmetric, conjugating (16) yields

S * ω ŵ(ω) = λ(ω) v(ω) and S ω v(ω) = λ(ω) ŵ(ω).
This shows that if (v(ω), ŵ(ω)) is a pair of left and right singular vectors, then so is ( ŵ(ω), v(ω)). For a simple singular value λ(ω), we deduce from remark 2 that there exists θ(ω) ∈ R such that ( ŵ(ω), v(ω)) = e iθ(ω) (v(ω), ŵ(ω)).

As a consequence,

S ω ŵ(ω) = e -iθ(ω) λ(ω) ŵ(ω).
This relation tells us how to change the phase of our right singular vector in order to obtain a singular vector of the SSVD [START_REF] Prada | Detection and imaging in complex media with the D.O.R.T. method[END_REF]. Indeed by setting ŵs (ω) := e iθ(ω)/2 ŵ(ω), we see that

S ω ŵs (ω) = λ(ω) ŵs (ω), (20)
which means that ŵs (ω) is a singular vector of the SSVD. Note that contrary to the singular vectors of a SVD (see remark 2), ŵs (ω) is defined uniquely up to a multiplication by -1.

In the sequel our aim is to show that if we choose this particular singular vector ŵs (ω) instead of an arbitrary one ŵ(ω) in the definition [START_REF] Mordant | Highly resolved detection and selective focusing in a waveguide using the D.O.R.T. method[END_REF] of q φ (t), then the optimal choice for φ is either φ ≡ 0 or φ ≡ π/2. We begin in the next section by proving that both phase functions lead to the minimum of a function which describes a time reversal process. But do they yield an optimal focusing? A ideal way to give a mathematical answer to this question would be to prove that these constant phase functions lead to the minimum of a function which would quantify the focus quality. Unfortunately, we were not able to find such a function. The only mathematical answer we can give is based on the connection between these constant phase functions and the time reversal experiment mentioned in the introduction. It is the subject of §3.3. We finally present in §3.4 some numerical results which confirm the space-time focusing effect.

Optimality of a time reversal gap functional

Hereafter we denote by J the time reversal transformation with origin t = 0 : for every real-valued function f = f (t), function Jf is simply given by Jf (t) := f (-t). Using the Fourier transform, it is readily seen that FJf (ω) = Ff (ω), or equivalently

J = F -1 J F where J f := f . ( 21
)
This is nothing but the well-known fact that time reversal becomes complex conjugation in the frequency domain.

Our idea of using the SSVD originates from a heuristic argument which can be explained as follows. Suppose that the array emits several short pulse waves, respectively at times t j for j = 1, . . . , J, that propagate along different paths leading to the aimed scatterer. Let T j denote the travel time related to the j-th path. After the interaction of these incident waves with the scatterer, suppose that the associated scattered wave propagate back to the array along the same respective paths. Thus they will reach the array respectively at times t ′ j := t j + 2T j for j = 1, . . . , J. Obviously, the incident pulse waves will arrive at the same time t * on the scatterer only if t j + T j = t * for all j. In this case, the times of emission and reception become symmetric with respect to t * in the sense that t ′ jt * = t *t j for all j. More precisely, up to a scaling factor related to the reflectivity of the scatterer, the input and the output are time-reversed with respect to the focus time t * .

This very simple model prompted us to search for phase functions such that this time reversal property could be nearly satisfied. This is the subject of Theorem 4 below which leads us to the SSVD in the case where the aimed focus time is t * = 0. Actually the connection between this time reversal property and the SSVD can be seen directly in the characterization [START_REF] Prada | Mémoire d'Habilitation à Diriger les Recherches[END_REF] of the singular vectors of the SSVD, which can be written S ω ŵs (ω) = λ(ω) J ŵs (ω). This relation means that when the input signal sent to the transducers is ŵs (ω), then the measure of the scattered field is, up to a positive real factor λ(ω), the time reversed input signal. As relation [START_REF] Prada | Mémoire d'Habilitation à Diriger les Recherches[END_REF] holds for all frequencies ω ∈ [ω 1 , ω 2 ], all time-harmonic signals are time-reversed with respect to the same origin t = 0. To a certain extent, this means that working with singular vectors of the SSVD amounts to synchronizing the time-harmonic emitted waves at time t = 0. In the above comments, one can replace ŵs (ω) by i ŵs (ω). The only change is that the positive real factors become negative. These are the two possible choices which yield a synchronization at t = 0. Theorem 4. Suppose that (λ(ω), ŵs (ω Then the minimum of G(µ, φ) is reached at (µ + , φ + ) and (µ -, φ -) where

)) ∈ R + × C N ; ω ∈ [ω 1 , ω 2 ] is a continuous family of singular elements of the SSVD of S ω . Let χ ∈ C 0 ([ω 1 , ω 2 ]) chosen such that χ(ω 1 ) = χ(ω 2 ) = 0. For all (µ, φ) ∈ R × C 0 ([ω 1 , ω 2 ]), define G(µ, φ) := (S -µ J)q φ 2 Rt; C N := R (S -µ J)q φ (t)
µ ± := ± ω2 ω1 λ(ω) χ(ω) 2 dω ω2 ω1 χ(ω) 2 dω -1
, φ + ≡ 0 and φ -≡ π 2 .

Proof. Thanks to ( 8) and ( 21), Parseval's identity yields

G(µ, φ) = 1 2π ( S -µ J) qφ 2 Rω; C N = 1 2π a φ µ 2 -2b φ µ + c φ , (22)
where qφ := Fq φ and a φ := qφ Note that the expression of a φ follows from the fact that J is an isometry. From ( 18) and ( 20), we deduce

a φ = 2 ω2 ω1 χ(ω) 2 dω, b φ = 2 ω2 ω1 λ(ω) χ(ω) 2 cos(2φ(ω)) dω and c φ = 2 ω2 ω1 λ(ω) 2 χ(ω) 2 dω.
For a fixed φ, the minimum of the second-order polynomial in the right-hand side of ( 22) is reached at

µ φ := b φ /a φ . Moreover G(µ φ , φ) = 1 2π c φ - b 2 φ a φ .
As a φ and c φ do not actually depend on φ, the minimum of this function is reached for the maximum of b 2 φ , that is, when cos(2φ(ω)) = ±1. Hence φ(ω) = kπ/2 for some integer k. And of course it is sufficient to consider both cases k = 0 and k = 1 since the other cases correspond to the same signals q φ (t) or their opposites.

Function G(µ, φ) is a time reversal gap functional since it quantifies the gap between the measure of the scattered field and the time-reversed input signal rescaled by a real factor. If λ(ω) was constant in the whole frequency band [ω 1 , ω 2 ], say λ(ω) = λ c , we would have µ ± = ±λ c and G(µ ± , φ ± ) = 0. Hence q φ± (t) would be an eigenvector of the operator J S associated with the eigenvalue ±λ c . In this case, the TR property satisfied in the frequency domain holds true in the time domain: the measure of the time-dependent scattered field is, up to a real factor ±λ c , the time-reversed input signal. Theorem 4 tells us that in the general case, the phase functions φ + ≡ 0 and φ -≡ π/2 bring us as close as possible to this situation: q φ± (t) is close to some kind of eigenvector of J S. But of course, this is not really an eigenvector since the time reversal gap ε := G(µ ± , φ ± ) does not vanish in general. It can be interpreted by means of the notion of pseudospectra [START_REF] Trefethen | Spectra and Pseudospectra[END_REF]: µ ± belongs to the ε-pseudospectrum of J S and q φ± (t) is a corresponding ε-pseudoeigenvector.

Space-time focusing: from active sources to passive scatterers

In the previous section, the link between Theorem 4 and space-time focusing is only based on a heuristic argument. The purpose of this section is to reinforce this link by a mathematical argument: we show that under certain conditions, the optimal signals of Theorem 4 yield the same space-time refocusing as the time reversal experiment for active sources which is mentioned in the introduction.

Mathematical interpretation of the time reversal experiment.

Suppose that at some point y ∈ R d , an active point-like source emits a short pulse defined by

v (ℓ) act (x, t) := 1 π Re ω2 ω1 χ act (ω) (-iω) ℓ Ĝ(x, y, ω) e -iωt dω, (23)
where ℓ ∈ N and χ act is a cutoff function chosen as in Theorem 4. This wave can be seen as a spectral truncation of a time-derivative of the time-dependent Green's function since v

(ℓ) act (x, t) = (∂ ℓ /∂t ℓ )G(x, y, t) if (ω 1 , ω 2 ) =
(0, +∞) and χ act ≡ 1 (see ( 5)). Except in this latter situation, v (ℓ) act is not causal but it can be considered as an approximation of a causal function for a large enough frequency band (ω 1 , ω 2 ). The measure of v (ℓ) act by the array is then given by

q (ℓ) (t) = 1 π Re ω2 ω1 χ act (ω) (-iω) ℓ ĝ(ω) e -iωt dω where ĝ(ω) := ( Ĝ(x 1 , y, ω), . . . , Ĝ(x N , y, ω)) ⊤ . ( 24
)
The idea of the TR experiment consists in using the time-reversed measure The fact that this wave propagates back to the original source can be understood easily in the case of a full aperture spherical array located far enough from the inhomogeneities of the medium. Assume that the family of points x n is distributed regularly on the surface S R := {z ∈ R d ; z = R}. Hence, in the above formula, the sum can be interpreted as an approximation of an integral on S R , that is,

J q (ℓ) (t) = q (ℓ) (-t) = 1 π Re ω2 ω1 χ act (ω) (iω) ℓ ĝ(ω) e -iωt dω (
N n=1 Ĝ(x n , y, ω) Ĝ(x, x n , ω) ≈ C SR Ĝ(z, y, ω) Ĝ(x, z, ω) dS(z),
where C > 0 depends on N and the space dimension d. The asymptotic behaviour of the latter integral for large R follows from the Helmholtz-Kirchhoff formula [START_REF] Garnier | Travel time estimation by cross correlation of noisy signals[END_REF]:

SR Ĝ(z, y, ω) Ĝ(x, z, ω) dS(z) R→+∞ ≈ Ĝ(x, y, ω) -Ĝ(x, y, ω) 2iω for x ≪ R.
As a consequence,

v (ℓ) back (x, t) ≈ C 2π Re ω2 ω1 χ act (ω) (iω) ℓ-1 Ĝ(x, y, ω) -Ĝ(x, y, ω) e -iωt dω.
Going back to the expression (23

) of v (ℓ) act , we conclude that v (ℓ) back (x, t) ≈ C 2 (-1) ℓ-1 v (ℓ-1) act (x, t) -v (ℓ-1)
act (x, -t) .

In this expression, v (ℓ-1) act (x, t) approximates a time-derivative of the Green's function, which is causal, whereas the other term v (ℓ-1) act (x, -t) represents the time-reversed of this function and then appears as an approximation of an anti-causal function. As a consequence,

v (ℓ) back (x, t) ≈      - C 2 v (ℓ-1) act (x, -t) if t < 0, (-1) ℓ-1 C 2 v (ℓ-1) act (x, t) if t > 0.
This means that for t < 0, v

back behaves like a converging wave which propagates towards the initial source y and refocuses at this point at time t = 0, whereas for t > 0, v (ℓ) back diverges from y. This gives us a mathematical interpretation of the TR experiment.

Back to passive scatterers.

Let us now come back to our problem and suppose for simplicity that there is only one scatterer located at y. In this case, the expression ( 13) of the time-harmonic scattering operator simplifies as

S ω = σ(ω) ĝ(ω) ĝ(ω) ⊤ ,
where σ(ω) is the reflectivity coefficient of the scatterer (see ( 9)) and ĝ(ω) is defined in (24). This shows in particular that the rank of S ω is 1. Moreover,

S ω e i arg(σ(ω))/2 ĝ(ω) = |σ(ω)| ĝ(ω) 2 C N e i arg(σ(ω))/2 ĝ(ω) , which means that λ(ω) := |σ(ω)| ĝ(ω) 2 C N and ŵs (ω) := e -i arg(σ(ω))/2 ĝ(ω) ĝ(ω) C N
are singular elements of the SSVD of S ω . From ( 9), we infer that arg(σ(ω)) either vanishes (if d = 3) or is small (if d = 2 : the asymptotic of H

(1) 0

for small arguments shows that arg(σ(ω)) = O((ln ε) -1 )). Hence ŵs (ω) ≈ ĝ(ω)/ ĝ(ω) C N . Thus the respective behaviours of the optimal signals of Theorem 4 are given by

q φ± (t) ≈ 1 π Re ω2 ω1 χ(ω) e iφ± ĝ(ω) ĝ(ω) C N e -iωt dω,
where we recall that φ + ≡ 0 and φ -≡ π/2. Comparing the latter formula with (25), we see that if both cutoff functions χ and χ act satisfy

χ(ω) = χ act (ω) ω ℓ ĝ(ω) C N ,
then, up to a change of sign, q φ+ (t) (respectively, q φ-(t)) behaves like q (ℓ) (-t) for even ℓ (respectively, odd ℓ). We conclude that the wave emitted by an optimal signal of Theorem 4 is similar to the backpropagating wave produced by the TR experiment when the initial wave is some spectral truncation of a time-derivative of the Green's function. In short, our method for passive scatterers yields some kind of optimal space-time focusing. This is justified above for one single scatterer. Using similar arguments as in [START_REF] Hazard | Selective acoustic focusing using time-harmonic reversal mirrors[END_REF], it is easy to see that this holds true for distant enough scatterers. The next section illustrates these results and also shows that the focus quality decreases if the scatterers become close.

Numerical results

We finally present some numerical experiments which confirm the space-time focusing effect related to the optimal signals of Theorem 4. We consider the same geometrical configuration as in section 2.2. We use the family of singular vectors of the SSVD of S ω associated with the dominant eigenvalue (which Figure 4: In the case of a homogeneous medium, modulus of the field generated by the optimal signal q φ+ (t) of Theorem 4 at times t = -2.0, -1.4, -1.0, -0.6, 0.0 and 0.4 (the field is rescaled so that its modulus belongs to [0, 1]). focus in space on the biggest scatterer, located at (1.2,0.3)). The cutoff function we have chosen is a truncated Gaussian function given by

χ(ω) = 1 [ω1,ω2] (ω) exp (ω -ω 0 ) 2 2γ 2
where [ω 1 , ω 2 ] = [START_REF] Prada | Mémoire d'Habilitation à Diriger les Recherches[END_REF]100], ω 0 = 60 and γ = 10. Note that strictly speaking, this function is not continuous at ω 1 and ω 2 as assumed in Theorem 4. However the Gaussian function is very small at these points (about 3 × 10 -4 ).

Figure 4 shows the evolution of the wave generated by the optimal signal q φ+ (t) of Theorem 4 in the case of a homogeneous medium. The space-time focusing appears very clearly: the array seems to emit an angular portion of a radial wave which propagates towards the target and concentrates near it at time t = 0. In this situation, the structure of the signal is very simple: Figure 6 (left) shows that all the transducers emit approximatively the same signal starting at different times which only depend on the distances between the transducers and the target.

Figure 5 illustrates the influence of the same scattering medium as in section 2.2, i.e., 500 point-like scatterers randomly displayed in the rectangle [0.3, 0.8] × [0, 2]. Again space-time focusing is confirmed. The focal spot observed at time t = 0 is even slightly smaller than in the homogeneous case, which is the well-known super-resolution effect of a scattering medium (see, e.g., [START_REF] Bal | Time reversal and refocusing in random media[END_REF][START_REF] Borcea | Theory and applications of time reversal and interferometric imaging[END_REF]). On the other hand, the structure of the wave is strongly perturbed. Resonance phenomena are observed inside the scattering medium, which slow down the propagation of the wave. In order to compensate these effects, a part of the wave is emitted earlier so that the various components meet at the same time t = 0 near the target. This is shown in Figure 6 (right) where we recognize a main front whose shape is similar to the homogeneous case, but which is now preceded by an unstructured contribution.

The above comments remain valid if we use the optimal signal q φ-(t) of Theorem 4 instead of q φ+ (t). Actually the differences between both cases are not so easy to detect. Figure 7 shows a zoom on the focal spots observed at t = 0 in the case of a homogeneous medium. We see that both focal spots are about the same size. They seem to be shifted by a quarter wavelength, which is consistent with the fact that in the frequency domain, both signals are in phase quadrature.

In order to quantify the focus quality, we propose an energy criterion which simply consists in comparing the acoustic energy concentrated in a given vicinity of the target with the total energy emitted by the array of transducers. Let u = u(x, t) denote the wave emitted in the reference medium by the array for some input signal q(t) (see [START_REF] Ben Amar | Numerical simulation of time-harmonic reversal mirrors for acoustic waves[END_REF]). For a given box B ⊂ R d around our target, we define a focus quality function by

Q B (t) := E B (t) E tot where E B (t) := 1 2 B ∂u ∂t (x, t) 2 + |∇u(x, t)| 2 dx
represents the local acoustic energy contained in B at time t (see ( 27)) and E tot stands for the total energy emitted by the array. As our array consists of point-like transducers, the fact that E tot is finite is not obvious. As a matter of fact, we show in the appendix that during the emission, this energy is infinite, but becomes finite as soon as the transducers become silent (provided that the input signals are regular enough). A suitable expression of E tot is given in Proposition 5. Figure 8 compares the focus quality functions computed for two signals: on the one hand, the optimal signal q φ+ (t) of Theorem 4 and on the other hand, the signal (25) of the TR experiment with ℓ = 0 and χ act = χ. On the left, we consider the same configuration as in Figure 4, that is, the case of the homogeneous medium. We see that both curves coincide, which is consistent with the results of section 3.3. The maximum of Q B (t) is reached at time t = 0 with 44 % of the total energy concentrated in the box represented in Figure 7. Note that in our model, the array emits the same wave on both sides, so Q B (t) cannot exceed 50 %. The case of the scattering medium of Figure 5 is represented in the middle part. Both curves nearly coincide, which confirms again the results of section 3.3, but here the maximum of Q B (t) is only 15 %, which shows that a great part of the energy is 'lost' in the scattering medium. Finally, in the right part, we consider again the scattering medium but the second obstacle is moved closer to the target. Of course, Q B (t) is unchanged for the signal of the TR experiment since it does not depend on the second obstacle. On the other hand, the focus quality decreases for the optimal signal For each case, the dashed line corresponds to the optimal signal q φ+ (t) of Theorem 4 and the solid line, the signal (25) of the TR experiment with ℓ = 0 and χ act = χ. q φ+ (t) (maximum at 12.5 %). This is a well-known effect of the DORT method, which maximizes the energy of the scattered wave and then leads to illuminate all obstacles when they are close to each other.

Conclusion

In the present paper, we have shown that the use of the symmetric singular value decomposition (SSVD) of the time-harmonic scattering operator allows us to synchronize its singular vectors in the frequency domain in order to achieve space-time focusing. We have developed several theoretical and numerical arguments which support this assertion. In particular, we have seen that for distant enough scatterers, the choice of the SSVD for unknown passive scatterers yields the same refocusing as time reversal for an active source, which can be considered as an optimal refocusing. Nevertheless two criticisms can be made.

On the one hand, from a practical point of view, the principle of DORT method is based on the measure of the scattered wave. As mentioned in the introduction, this means that one has to measure separately the incident and the perturbed waves (that is, in the absence and in the presence of the target) and compute their difference. In many situations, it is impossible to make this double measure. However when the medium is weakly inhomogeneous and when the array is located on one side of the medium (open aperture case), only a tiny part of the incident wave propagates back to the array so that one can consider that one measures directly the scattered field.

On the other hand, from a mathematical point of view, it is quite frustrating to be unable to relate the use of the SSVD to a mathematical criterion which should evaluate the quality of space-time focusing. The energy criterion proposed in section 3.4 seems very natural. But can we affirm that according to this criterion or another focusing criterion, the best signals are those of Theorem 4? The question remains open.

Appendix: Energy emitted by point sources

Let Ω be a domain of R d , bounded or not, with boundary Γ and u = u(x, t) a causal solution to the acoustic wave equation

∂ 2 u ∂t 2 -∆u = f in Ω × R,
for a given causal function f = f (x, t). By multiplying this equation by ∂u/∂t and integrating over Ω, we deduce classically from Green's formula the energy conservation law:

d dt E Ω (t) = P Ω (t) + F Γ (t), (26)
where E Ω (t) denotes the acoustic energy at time t, P Ω (t) is the power produced by the source f and F Γ (t) is the incoming energy flux across Γ, which are defined respectively by

E Ω (t) := 1 2 Ω ∂u ∂t (x, t) 2 + |∇u(x, t)| 2 dx, (27)
P Ω (t) := Ω f (x, t) ∂u ∂t (x, t) dx and (28)
F Γ (t) := Γ ∂u ∂ν (x, t) ∂u ∂t (x, t) dγ x , (29)
where ν is the unit normal on Γ pointing outside Ω. Formula (26) holds provided that u is regular enough so that the definitions above make sense, for instance if f ∈ L 2 (Ω × R). The purpose of this appendix is to deal with the case where f represents the excitation produced by point sources located at x n for n = 1, . . . , N, i.e.,

f (x, t) = N n=1 q n (t) δ(x -x n ),
where functions q n are assumed causal. In this case, definitions ( 27)-( 29) are not adapted. Indeed, because of the singular behaviour of u near the sources, the integrals on Ω diverge, which means that the usual acoustic energy is infinite for point sources. However we show below that this energy becomes finite as soon as the sources become silent. Our aim is to exhibit an explicit expression of this energy. Surprisingly, we did not succeed in finding one in the literature! We consider only the cases of dimensions 2 and 3. Formula (2) provides us the field produced by point sources. Thanks to the expression (1) of the Green's function in a homogeneous medium, it can be written more explicitly as

u = N n=1
u n where (30)

u n (x, t) :=        max{t,|x-xn|} |x-xn| q n (t -τ ) 2π τ 2 -|x -x n | 2 dτ if d = 2, q n (t -|x -x n |) 4π|x -x n | if d = 3. (31)
We see here that if q n is regular enough, then u n has a similar regularity outside a vicinity of x n . Hence, outside a collection of small balls B n (ε) := {x ∈ R d ; |xx n | ≤ ε}, energy conservation (26) holds. We shall use an integrated version of this equation:

E Ω(ε) (t) = t 0 F Γ(ε) (s) ds, (32)
where Ω(ε

) := R d \ ∪ n=1,N B n (ε) and Γ(ε) := ∂Ω(ε) = ∪ n=1,N ∂B n (ε) (note that P Ω(ε) (t) = 0 since f vanishes in Ω(ε)).
The following proposition tells us that we can pass to the limit ε → 0 in this equation when the sources become silent.

Proposition 5. Suppose that all functions q n belong to C 2 (R) and have a compact support contained in [0, T ] for some T > 0. Then for all t > T, the limit lim ε→0 E Ω(ε) (t) exists and is independent of t. It is given by

E tot = N n=1 N m=1
E nm where

E nn :=        T 0 T τ q ′ n (s -τ ) q n (s) ds dτ 2π τ if d = 2, T 0 q ′ n (s) 2 4π ds if d = 3, E nm := T 0 q n (s) ∂u m ∂t (x n , s) ds for n = m.
Remark 6. Using Parseval's identity, these expressions are easily converted to the frequency domain.

We obtain

E nn :=        -1 2π T 0 1 τ d dτ F -1 |q n | 2 (τ ) dτ if d = 2, 1 4π 2 +∞ 0 |ω qn (ω)| 2 dω if d = 3, E nm := - 1 π Re +∞ 0 iω Ĝ(x m , x n , ω) qm (ω) qn (ω) dω for n = m.
In the above formulas, E nn represents the energy produced by the source located at x n if it was alone, whereas the other terms E nm for n = m stand for the interactions between the sources. In order to prove Proposition 5, we need to know the behaviour of u n near x n , which is summarized in the following lemma.

Lemma 7. Suppose that q n ∈ C 2 (R) is causal. Then u n ∈ C 2 ((R d \ B n (ε)) × R) for all ε > 0.
Moreover, when ε tends to 0, the following asymptotic expansions hold uniformly for x ∈ ∂B n (ε) and t ∈ [0, T ] for any given T > 0 :

∂u n ∂t (x, t) =        1 2π max{t,ε} ε q ′ n (t-τ ) τ 2 τ 2 -ε 2 dτ + q ′ n (t) +O(ε) if d = 2, q ′ n (t) 4πε - q ′′ n (t) 4π + o(1) if d = 3, ∂u n ∂ν (x, t) =      q n (t) 2πε + O(1) if d = 2, q n (t) 4πε 2 + O(1) if d = 3.
Moreover, if d = 2, the integral involved in the expression of ∂u n /∂t is O(| log ε|).

Proof. The case d = 3 is easy. From (31), we see that u n ∈ C 2 ((R 3 \ B n (ε)) × R) and for x ∈ ∂B n (ε), we have

∂u n ∂t (x, t) = q ′ n (t -ε) 4πε and ∂u n ∂ν (x, t) = q ′ n (t -ε) 4πε + q n (t -ε) 4πε 2 .
Using a Taylor expansion of q n (tε) and q ′ n (tε), the conclusion follows. The case d = 2 is more intricate. The C 2 -regularity of u n is easily deduced from its expression (31), which shows in addition that for x ∈ ∂B n (ε),

∂u n ∂t (x, t) = 1 2π tε ε q ′ n (t -τ ) ρ ε (τ ) dτ
where ρ ε (τ ) := √ τ 2ε 2 and t ε := max{t, ε}. Noticing that ρ ′ ε (τ ) = τ /ρ ε (τ ) and integrating by part, we deduce that

∂u n ∂t (x, t) = 1 2π tε ε q ′ n (t -τ ) τ 2 + q ′′ n (t -τ ) τ ρ ε (τ ) dτ. ( 33
)
The first term in the integral yields the dominant contribution, which is O(| log ε|) because The first integral of the right-hand side is equal to q ′ n (tε) = q ′ n (t) + O(ε) whereas the second one is O(ε) since |ρ ε (τ )/τ -1| ≤ ε 2 /τ 2 . This gives the asymptotic behaviour of ∂u n /∂t.

We proceed similarly for ∂u n /∂ν starting from an expression of u n (x, t) similar to (33) (with q ′ n and q ′′ n replaced by q n and q ′ n , respectively). Taking the normal derivative and using again an integration by parts, it is easily seen that

∂u n ∂ν (x, t) = ε 2π tε ε 3 q n (t -τ ) τ 4 + 3 q ′ n (t -τ ) τ 3 + q ′′ n (t -τ ) τ 2 ρ ε (τ ) dτ.
By the same argument as above, the last term in the integral yields a contribution O(ε| log ε|) whereas the second one is O(1). The dominant contribution is given by the first term. Indeed we have This completes the proof.

Proof of Proposition 5. From (30) and the definition (29) of the energy flux, the energy conservation law (32) writes as Finally, if n = m, in the three-dimensional case, we have

E Ω(ε) (t) = N n,
E n nn (t, ε) = t 0 q n (s) q ′ n (s) 4πε
q n (s) q ′′ n (s) 4π

+ o(1) ds = q n (t) 2 8πε + t 0 q ′ n (s) 2 4π ds -q n (t) q ′ n (t) 4π

+ o(1).

Thus we see that for all time t such that q n (t) = 0, this quantity diverges as O(ε -1 ) when ε tends to 0. On the other hand, when q n (t) = 0, it has a finite limit given by the second term. Similarly, in the two-dimensional case, Lemma 7 shows that E n nn (t, ε) = t ε s ε q ′ n (s-τ ) q n (s) 2π τ 2 ρ ε (τ ) dτ ds + t 0 q ′ n (s) q n (s) 2π ds + O(ε| log ε|).

The second integral of the right-hand side is equal to q n (t) 2 /(4π). The first one can be rewritten as t ε t τ (q ′ n (sτ )q ′ n (s)) q n (s) ds + q n (t) 2q n (τ ) 2 2 ρ ε (τ ) 2π τ 2 dτ.

Hence this quantity diverges as O(| log ε|) if q n (t) = 0, since On the other hand, when q n (t) = 0, Lebesgue's dominated convergence theorem shows that it tends to t 0 t τ (q ′ n (sτ )q ′ n (s)) q n (s) ds -q n (τ ) 2 2 dτ 2π τ , which can be rewritten in the simpler form shown in the proposition.

Figure 1 :

 1 Figure 1: Modulus of the incident field generated by the right singular vectors of S ω associated with the dominant (left) and second (right) singular values. Case of a homogeneous medium with ω = 40.

Figure 2 :

 2 Figure 2: Same as Figure 1 in the case of a scattering medium with ω = 40.

Figure 3 :

 3 Figure 3: Same as Figure 2 for ω = 90.

2 C

 2 N dt where • C N denotes the Euclidean norm in C N and q φ (t) e iφ(ω) ŵs (ω) e -iωt dω.

2

  Rω; C N , b φ := Re( S qφ , J qφ ) Rω; C N and c φ := S qφ 2 Rω; C N .

  n , y, ω) Ĝ(x, x n , ω) e -iωt dω.

Figure 5 :

 5 Figure 5: Same as Figure 4 in the case of a scattering medium.

Figure 6 :

 6 Figure6: Representation of the optimal signal q φ+ (t) of Theorem 4 (rescaled in [-1, +1]) in the homogeneous medium (left) and in the presence of the scattering medium (right). The horizontal coordinate is time and the vertical one is the transducer's number.

Figure 7 :

 7 Figure 7: In the case of a homogeneous medium, zoom in the box [1.0, 1.4] × [0.18, 0.42] of the focal spots obtained at time t = 0 with the optimal signals q φ+ (t) (left) and q φ-(t) (right) of Theorem 4.

Figure 8 :

 8 Figure 8: Representation of the focus quality function Q B (t) for t ∈ [-2, +2] and the box B of Figure 7 in three configurations: the homogeneous medium of Figure 4 (left), the scattering medium of Figure 5 (middle) and the same scattering medium with the second scatterer moved at point (0.5, 1.6) (right).For each case, the dashed line corresponds to the optimal signal q φ+ (t) of Theorem 4 and the solid line, the signal (25) of the TR experiment with ℓ = 0 and χ act = χ.

  since ρ ε (τ ) ≤ τ for all τ ≥ ε. To evaluate the second one, rewrite it astε ε q ′′ n (tτ ) τ ρ ε (τ ) dτ = tε ε q ′′ n (tτ ) dτ + tε ε q ′′ n (tτ ) ρ ε (τ ) τ -1 dτ.

  tε ε 3q n (t-τ ) τ 4 ρ ε (τ ) dτ = q n (t) tε ε 3ρ ε (τ ) τ 4 dτ + tε ε 3(q n (t-τ )-q n (t)) τ 4 ρ ε (τ ) dτ,where the last integral is O(ε -1 ) (since |q n (tτ )q n (t)| ≤ C τ for some C > 0) and

  ρ ε (τ ) + τ ) dτ =log ε + O(1).

  Lemma 7 shows that if p = n and p = m, thenE p nm (t, ε) = O(ε d-1 ). Moreover if n = m, then E m nm (t, ε) = O(ε) if d = 3 (respectively, E m nm (t, ε) = O(ε| log ε|) if d = 2) whereas

			m,p=1	E p nm (t, ε) where
	E p nm (t, ε) :=	0	t	∂Bp(ε)	∂u n ∂ν	(x, s)	∂u m ∂t	(x, s) dγ x ds.
	E n nm (t, ε) =	0	t	q n (s)	∂u m ∂t	(x

n , s) ds + O(ε).

when these singular values are simple, each right singular vector (column of W ω ) associated with each nonzero singular value generates a wave which focuses selectively on each scatterer.


O Marcelot

P Magnan

Comparison between TCAD simulated and measured carrier lifetime in CMOS photodiodes, using the Open Circuit Voltage Decay method

The control and the prediction of minority carrier lifetime are crucial for the design of photodiodes, especially for CMOS image sensors, as signal electrons must be captured before recombination. Analytic models have been developed but do not allow accurate and reliable lifetime estimations according to complex photodiode architecture. In this work, we show for the first time that mixed mode TCAD simulations give accurate and reliable results on realistic photodiodes design. To achieve this conclusion, we have performed measurements and simulations on two different photodiodes using the Open Circuit Voltage Decay method.

1.

Introduction

The recombination lifetime of photo-generated carriers is an important parameter in photodiode design. Indeed, considering photo-generated carriers, we might wonder: in what time interval will the excess carriers recombine before being eventually captured by the cathode? This question is crucial in image sensors, especially if we want to realize very sensitive pixels. The recombination process at the origin of the signal carrier density decay is characterized by the carrier lifetime.

Maintaining a long recombination lifetime is crucial in the photo-detection, as we want to capture as much photo-generated carriers as possible in some areas, and avoid collecting them in others (e.g. to avoid crosstalk). It is thus necessary to measure and predict accurately the recombination lifetimes. In this way, recombination lifetime models have been developed. These models are based on high dopant concentration [START_REF] Swirhun | Electron Devices Meeting[END_REF] (around 1x10 17 at.cm -3 ) or on moderate dopant concentration [START_REF] Fossum | [END_REF][START_REF] Glunz | Conference Record of the Twenty-Eighth IEEE[END_REF] (from 1x10 14 at.cm -3 ). However, these analytical models are unsuitable for 2D/3D photodiode structures with various doping layers such as the one used in CMOS image sensors (Fig. 1).

TCAD simulations may become a solution to estimate the average lifetime in complex structure. In order to demonstrate that TCAD is reliable and a powerful tool in lifetime estimations, we propose here to compare recombination lifetime results with TCAD simulations.

Several lifetime measurement techniques have been studied [START_REF] Schroder | Semiconductor Material and Device Characterization[END_REF][START_REF] Schroder | [END_REF][6], implying optical, or current excitations. As we want to characterize photodiodes, we have used the Open-Circuit-Voltage-Decay (OCVD) method. The OCVD technique was originally introduced by Gossick [7] and Lederhandler [START_REF] Lederhandler | Proc. IRE[END_REF] in 1955. This technique involves the creation of excess carriers in a PN junction using a brief forward current excitation, and then a monitoring of the voltage across the junction after the abrupt termination of the excitation [7][START_REF] Lederhandler | Proc. IRE[END_REF][START_REF] Davies | Proc. IEEE[END_REF][START_REF] Choo | [END_REF][11][12][START_REF] Totterdell | Proc. IEE[END_REF][START_REF] Joardar | [END_REF]. Then others authors changed the forward current excitation by an optical pulse [12,[15][16][17].

In the following, basics of the OCVD theory are presented in section 2, and then OCVD method is used to measure or to simulate photodiodes in order to extract carrier lifetimes in section 3. Discussions of the results are included in the section 3.

Summary of the OCVD theory

The junction voltage V(t) across the photodiode decreases when the excitation is switched off. It has been shown that the voltage expression as a function of time may be expressed as following [START_REF] Schroder | Semiconductor Material and Device Characterization[END_REF][START_REF] Lederhandler | Proc. IRE[END_REF]18]:

( ) ( ) 1 • © § τ + ≈ r t erfc ln q kT V t V 0 ( 1
)
where is the recombination lifetime, and erfc(x) = 1 -erf(x) the complementary error function. This function shows a rapid decay followed by a linear slope. With the approximation t greater than IJ r the lifetime is deduced from the gradient of the V(t) vs t waveform, thanks to the following equation [START_REF] Schroder | Semiconductor Material and Device Characterization[END_REF]12]:

( ) dt t dV q kT r - = τ (2)
which is derived under the simplifying assumption that recombination is dominated by quasineutral region recombination with the simple exponential voltage dependence exp(qV/kT). For the recombination in space charge region of forward biased junction, the dependence becomes exp(qV/nkT) where n is the diode ideality factor, between 1 and 2. Under high level injection, the lifetime is given by:

( ) dt t dV q kT 2 r - = τ (3)
The high and low level injection curves usually show two different slopes [START_REF] Schroder | Semiconductor Material and Device Characterization[END_REF]11,12,18,19], as illustrated in the Fig. 2.

3.

Experimental results, TCAD and discussion

3.1

Forward current induced OCVD N + P photodiodes have been fabricated using a 0.35µm process, with two different epitaxies on Czochralski (Cz) p-type wafers. The first epitaxy (sample A) is a 5 µm p-type silicon doped at 1.0x10 15 B + /cm 3 (see Fig. 1), and the second one (sample B) is a 10 µm p-type silicon doped at 1.0x10 14 B + /cm 3 . The device under test is a 5.5 µm side square Nwell photodiode surrounded by Pwell.

The experimental set-up is shown in Fig. 3. The photodiodes are contacted with a Cascade prober which is connected to the MOSFET switch using coaxial cables.

In order to avoid any perturbation from the oscilloscope [12,19], this experiment was performed with a unity gain buffer (TCL272), providing a very high input impedance of 10 12 Ohm.

The OCVD results of the tested photodiodes are presented in the Fig. 4.

Both measurements show at t=0 a voltage step due to the ohmic voltage drop in the diode that is observed when the current flow ceases, as described by Schuster [20]. The voltage drop is more important on the sample B due to a higher resistive epitaxy. Then, a region of fast decay can be observed until t=100µs, which could be attributed to a high level injection regime [START_REF] Schroder | Semiconductor Material and Device Characterization[END_REF]6,[11][12] or to higher decay modes. Indeed, our samples are made of epitaxy and various layers, and the initial fast decay may be interpreted as a redistribution of carrier flows characterized by higher decay modes. After t=100 µs, the voltage decay vs. the time is stabilized and linear, characterized by the effective lifetime. The lifetime computation is done in this part of the curve and gives the results listed in the Table 1.

The epitaxial layer the most highly resistive (sample B) gives the longest lifetime. The difference between the two samples is hardly visible because the Nwell area is surrounded by Pwell and only has its backside in contact with the substrate.

This experiment is simulated on the Synopsys Sentaurus environment, using the mixed mode of Sentaurus Device. The mixed-mode capability of Sentaurus Device allows for the simulation of a circuit which can combine Sentaurus devices with other devices based on compact models (SPICE). The lifetime model used in Sentaurus is the one proposed by Fossum in 1976 [START_REF] Fossum | [END_REF], and described by the following relation with IJ n0 =10 -5 s and N 0A =10 16 cm -3 :

( )

A A n A n N N N 0 0 1+ τ = τ (4)
This model was built from experimental measurement of lifetime as a function of resistivity of grown silicon ingots. In all simulations are activated the SRH recombination model with doping dependence, the "Auger recombination" model and the "surface SRH" parameter between silicon and oxide. The photodiodes are simulated from SIMS doping profiles in 2D to reduce computational time and memory constraints (Fig. 5). The photodiode anode is connected to a resistor Rs, connected to the source of an N channel MOS Transistor (NMOST) which is the switch of our experiment. The spice parameters of the N-MOST are adjusted in order to keep the same ratio between N-MOST and photodiode capacitance in the TCAD study and in the experiment.

As in the experiment, we use Vd=0.6 V to polarize the photodiodes in the forward direction, and we apply a square pulse on Vg. Anode voltage versus time is plotted in the Fig. 6.

TCAD simulations do not reproduce the short voltage drop when the switch is turned off. This is due to the fact that the anode current i s is low when the switch is turned off. Indeed, i s in the range of nano-Ampere does not allow a visible voltage drop

s s R i V × = ∆
(R s represents the ohmic component of the circuit made of a resistor and of the MOSFET resistor). The anode current i s is low because only one (instead of several ones in parallel) photodiode is simulated in 2D.

From the start to approximately 100 µs, a fast decay process is visible on both samples. In order to try to clarify the root cause by the fast decay process, two distribution profiles of the carrier recombination rates are plotted: 1 µs and 200 µs after the current extinction (Fig. 7), i.e. during and after the fast decay process. In the epitaxy layer, the carrier recombination rate is 100 times higher during the fast decay process than in the stabilized phase. It means that the fast decay is mainly due to a high level injection.

As in the experiment, the slope of the curve representing the sample A is higher during this regime. Regarding the sample B, the fast decay regime looks longer than in the sample A. Indeed, the lower slope of the curve is showing us a slower carrier recombination process which delays the stabilized phase.

In the stabilized and linear regime, the voltage decay is used to estimate the lifetime in a same way than in the experimental configuration (Table 1).

Simulated lifetimes are slightly underestimated by less than 20%. TCAD simulations give lifetime estimations in a good agreement with the experimental results, and it shows the good capability of TCAD to reproduce the forward current induced OCVD method.

3.2

Photo induced OCVD

Measurements on same photodiodes are performed using the Cascade prober, however the excitation is replaced by a LED light. A white LED NSPW500BS is put in the cascade prober, above the tested chip. This LED has rise and fall times about 140 ns, and is fast enough to unable a lifetime measurement. As for the forward current induced OCVD the voltage across the photodiodes is measured using the unity gain buffer.

The LED is pulsed at 100 Hz and biased at 5 V. The measurement results are plotted in Fig. 8.

On both samples, we may observe a very small voltage drop at the time of the light extinction. Then a fast decay regime is visible, which is longer on the sample B. After 100 µs, the curve slope is weaker and stabilized, and a lifetime estimation is performed, using a linear interpolation (Table 2).The lifetimes measured are slightly higher compared to the ones measured with the forward current induced OCVD. Indeed, this photo induced method gives estimates of the carrier lifetime that are generally longer than those obtained from the forward induced OCVD [12,19].

The experiment is as well simulated with the TCAD tools. The switch is connected to the anode of the photodiode, and Vgate=Vdrain=0 V. The photodiodes are illuminated with a 580 nm wavelength light at a power of 200 mW/cm². OCVD resulting curves are plotted in the Fig. 9.

At the extinction of the light, no voltage drop is visible as in the previous forward current OCVD simulation. Then, we observe a fast decay regime until 100 µs for the sample A and until 300 µs for the sample B. As in the previous simulation, the slope of the curve representing the sample B is lower and involves a longer fast decay regime. Finally, the stabilized regime takes place and is used to estimate the lifetime. Some linear interpolations in the linear regime give the lifetimes in the Table 2.

Only a 6% mismatch on the lifetime estimation is visible: lifetimes estimations from TCAD simulations are quite in a good agreement with the experimental values. TCAD simulations are able to reproduce the photo induced OCVD experiment and confirm that the Fossum model used in TCAD gives good approximation of carriers lifetime [START_REF] Fossum | [END_REF]21] according to a complex 2D doping distributions.

Conclusion

We have successfully measured the minority carrier lifetime in CMOS photodiodes, using the OCVD method. The results show a limited increase of the minority carrier lifetime with the epitaxy resistivity because the cathode of the photodiode is surrounded by a Pwell. These experimental results were reproduced with TCAD tools using the mixed mode, and confirm the capability of TCAD to model the carrier lifetime in complex photodiodes. Both simulation, of forward current induced OCVD method and photo induced OCVD method are in good agreement with experimental data and lifetime mismatch is less than 20% in CMOS photodiodes.

Fig. 1 :

 1 Fig.1: example of simplified CMOS image sensor photodiode unsuitable for analytic lifetime calculation (cross-section view).

Fig. 2 :

 2 Fig. 2: example of open-circuit voltage decay characteristic showing a high level and a low level injection region [11].

Fig. 3 :

 3 Fig. 3: experimental read-out setup. A unity gain buffer is used between the oscilloscope and the photodiodes.

Fig. 4 :

 4 Fig. 4: forward current induced OCVD on photodiodes on 1.0x10 15 B + /cm 3 epitaxy (sample A) and on 1.0x10 14 B + /cm 3 epitaxy (sample B). Linear interpolations are shown.

Fig. 5 :

 5 Fig. 5: TCAD circuit used to simulate the OCVD experiment. The photodiode is simulated from SIMS profiles, and the NMOS is simulated from SPICE parameters. Vd=0.6 V, Vb=V_cathode=0 V.

Fig. 6 :

 6 Fig. 6: TCAD simulations of the forward current induced OCVD, and linear interpolations in the stabilized phase. The switch is turned off at t=0 s.

Fig. 7 :

 7 Fig. 7: TCAD distributions of the carrier recombination rates of sample A (epitaxy 1.0x10 15 B + /cm 3 ). Left plot: 1 µs after the end of the forward current. Right plot: 200 µs after the end of the forward current.

Fig. 8 :

 8 Fig. 8: photo induced OCVD on photodiodes on 1.0x10 15 B + /cm 3 epitaxy (sample A) and on 1.0x10 14 B + /cm 3 epitaxy (sample B). Linear interpolations are shown.

Fig. 9 :

 9 Fig. 9: TCAD simulations of the photo induced OCVD, and linear interpolations. The light is switched off at t=0 s

Table 1 :

 1 calculated lifetimes from TCAD and experimental forward current-OCVD results

	Measurement	TCAD

Table 2 :

 2 calculated lifetimes from TCAD and experimental photo-OCVD results.

	Measurement	TCAD


Eric Amar

Estimates L r -L s for solutions of the ∂ equation in strictly pseudo convex domains in C n

We prove estimates for solutions of the ∂u = ω equation in a strictly pseudo convex domain Ω in C n . For instance if the (p, q) current ω has its coefficients in L r (Ω) with 1 ≤ r < 2(n + 1) then there is a solution u in L s (Ω) with

. We also have BM O and Lipschitz estimates for r ≥ 2(n + 1). These results were already done by S. Krantz [12] in the case of (0, 1) forms and just for the L r -L s part by L. Ma and S. Vassiliadou [14]for general (p, q) forms. To get the complete result we propose another approach, based on Carleson measures of order α introduced and studied in [4] and on the subordination lemma [5].

1 Introduction.

Let Ω be a bounded strictly pseudo convex domain with smooth C ∞ boundary. We shall denote these domains as s.p.c. domains in the sequel.

Ovrelid [START_REF] Ovrelid | Integral representation formulas and L p estimates for the ∂ equation[END_REF] proved that if we have a (p, q) current ω, ∂ closed in Ω and such that its coefficients are in L r (Ω) then there is a (p, q -1) current u solution of the equation ∂u = ω and with coefficients still in L r (Ω). Let us define a norm on these currents : ω ∈ L r (p,q) (Ω), ω = Then Ovrelid proved that u r < C ω r , where the constant C does not depend on ω.

In the case of r = ∞, this was done before by Lieb [START_REF] Lieb | Die cauchy-riemannschen differentialgleichungen auf streng pseudokonvexen gebieten. beschrnkte lsungen[END_REF] and Romanov and Henkin [START_REF] Romanov | Exact hölder estimates of the solutions of the δ-equation[END_REF] proved that still for r = ∞, there is a solution u in the space Lipschitz 1/2. In the book of Henkin and Leiterer [START_REF] Henkin | Theory of functions on complex manifolds[END_REF] we can find precise references for these topics.

The L p results were strongly improved by Krantz [START_REF] Krantz | Optimal Lipschitz and L p regularity for the equation ∂u = f on stongly pseudoconvex domains[END_REF] in the case of (0, 1) forms and the aim of this work is to generalise Krantz results to the case of (p, q) forms as a consequence of results on Carleson measures of order α.

A more general case was done by L. Ma and S. Vassiliadou [START_REF] Ma | L p estimates for Cauchy-Riemann operator on q-convex intersections in C n[END_REF] on q-convex intersections in C n , but only for the L r -L s part, the Lipschitz one is not treated in their work. (Thanks to the referee who signals me this nice paper.) 1 Moreover, in the case of bounded convex domains of finite type, these results are already known, done by K. Diederich, B. Fischer and J-E. Fornaess [START_REF] Diederich | Hölder estimates on convex domains of finite type[END_REF], A. Cumenge [START_REF] Cumenge | Sharp estimates for ∂ on convex domains of finite type[END_REF] and B. Fischer [START_REF] Fischer | L p estimates on convex domains of finite type[END_REF].

So in the case of strictly convex domains, theorem 1.2 can also be seen as a corollary of their results, but for general strictly pseudo convex domains this is not the case and of course their proofs are much more involved than this one.

I shall reproof the L r -L s part of this theorem and prove the BMO and Lipschitz one by another approach.

We already got this kind of results in [START_REF] Amar | Extension de formes ∂b fermées et solutions de l'équation ∂b u = f[END_REF] by use of Skoda's kernels [START_REF] Skoda | Valeurs au bord pour les solutions de l'opérateur d¨et caractérisation des zéros de la classe de Nevanlinna[END_REF] but we where dealing with boundary values instead of inside ones. Nevertheless using Skoda results we shall prove the following theorem, where A B means that there is a constant C > 0 independent of A and B such that A ≤ CB.

Theorem 1.1 Let Ω be a s.p.c. domain in C n then for 1 < r < 2n + 2 we have ∀ω ∈ L r (p,q) (Ω), ∂ω = 0, ∃u ∈ L s (p,q-1) (Ω) :

: ∂u = ω, u L s (Ω) ω L r (Ω) ,
for any s such that

1 s > 1 r - 1 2(n + 1)
.

We shall also generalise Krantz theorem [START_REF] Krantz | Optimal Lipschitz and L p regularity for the equation ∂u = f on stongly pseudoconvex domains[END_REF] to (p, q) forms :

Theorem 1.2
Let Ω be a s.p.c. domain in C n then for 1 < r < 2n + 2 we have

• ∀ω ∈ L r (p,q) (Ω), ∂ω = 0, ∃u ∈ L s (p,q-1) (Ω) :: ∂u = ω, u L s (Ω) ω L r (Ω) , with 1 s = 1 r - 1 2(n + 1)
.

• For r = 2n + 2 we have ∃u ∈ BMO (p,q) (Ω) :

: ∂u = ω, u BM O(Ω) ω L 2n+2 (Ω) .
If ω is a (p, 1) form we have also :

• for r = 1, ∃u ∈ L s,∞ (p,0) (Ω) :: ∂u = ω, u L s,∞ (Ω) ω L 1 (Ω) with 1 s = 1 - 1 2(n + 1)
.

• for r > 2n + 2, ∃u ∈ Γ β (p,0) (Ω) :: ∂u = ω, u Γ β (Ω) ω L r (Ω) ,
where β = 1 -2n + 2 r and Γ β is an anisotropic Lipschitz class of functions. Moreover the solution u is linear on the data ω.

The classes BMO(Ω) and Γ β (Ω) will be defined later. The space L s,∞ (p,0) (Ω) is the Lorentz space [START_REF] Bergh | Interpolation Spaces[END_REF]. This theorem is stronger than theorem 1.1 because here, in the case 1 ≤ r < 2(n + 1) we get the result for the end point s such that

1 s = 1 r - 1 2(n + 1)
.

Of course if u ∈ L s (p,q-1) (Ω) for s > r then u ∈ L r (p,q-1) (Ω) hence we also have an strong improvement to Ovrelid's theorem.

Because the class Lipschitz 1/2 is contained in Γ 1 (Ω) we see that we recover the Romanov-Henkin result when r = ∞ in the case of (p, 1) forms.

Even if they do not appear in the statement, the Carleson measures of order α, A. Bonami and I introduced in [START_REF] Amar | Mesure de Carleson d'ordre α et solution au bord de l'équation ∂[END_REF], are at the heart of this proof.

2 Proof of the first theorem.

Let Ω be a s.p.c. in C n , defined by the function ρ ∈ C ∞ (C n ), i.e. Ω := {z ∈ C n :: ρ(z) < 0} and ∀z ∈ ∂Ω, ∂ρ(z) = 0.

Let Ω ′ := {(z, w) ∈ C n ×C :: ρ ′ (z, w) := ρ(z) + |w| 2 < 0} and lift a current ω to Ω ′ this way :

ω ′ (z, w) := ω(z). Lemma 2.1 Let Ω be a s.p.c. domain in C n , with the above notations we have ω ∈ L r (p,q) (Ω) ⇒ ω ′ (z, w) ∈ L r (p,q) (∂Ω ′ ).
Proof. This is an instance of the subordination principle [START_REF] Amar | Suites d'interpolation pour les classes de Bergman de la boule et du polydisque de C n[END_REF], [START_REF] Amar | A subordination principle[END_REF]. Let f (z) ∈ L r (Ω) and set f ′ (z, w) := f (z) in Ω ′ , then, by the main lemma in [START_REF] Amar | A subordination principle[END_REF], p. 6,

f ′ r L r (∂Ω ′ ) := ∂Ω ′ |f ′ (z, w)| r dσ(z, w) = Ω |f (z)| r -ρ(z) + |gradρ(z)| 2 4 { |w| 2 =-ρ(z) d |w|}dm(z),
where d |w| is the normalized Lebesgue measure [START_REF] Amar | A subordination principle[END_REF] on the circle

|w| 2 = -ρ(z). Because Ω is compact, we have ∀z ∈ Ω, -ρ(z) + |gradρ(z)| 2 4 ≤ C(ρ) < ∞ hence we have f ′ r L r (∂Ω ′ ) ≤ C(ρ) Ω |f (z)| r dm(z) = C(ρ) f L r (Ω) .
It remains to apply this taking for f any coefficient of ω.

Proof of theorem 1.1. Since Ω is a s.p.c. domain so is Ω ′ by the subordination lemma [START_REF] Amar | A subordination principle[END_REF]. By use of lemma 2.1 we have that ω ′ ∈ L r (p,q) (∂Ω ′ ) and still ∂ω ′ = 0, hence we can apply Skoda's theorem 2 in [START_REF] Skoda | Valeurs au bord pour les solutions de l'opérateur d¨et caractérisation des zéros de la classe de Nevanlinna[END_REF] to get that there is a solution u ′ of ∂b u ′ = ω ′ such that

u ′ ∈ L s (p,q-1) (∂Ω ′ ) with 1 s > 1 r - 1 2(n + 1)
.

We have

u ′ (z, w) = I,J a ′ I,J (z, w)dz I ∧ dz J .
Because ω ′ does not depend on w we have that the coefficients of u ′ are holomorphic in w, hence we can set (recall that u ′ is defined on ∂Ω ′ ) ∀z ∈ Ω, a I,J (z) :=

|w| 2 =-ρ(z) a ′ I,J (z, w)d |w| and u(z) := I,J a I,J (z)dz I ∧ dz J ,
then exactly as in [START_REF] Amar | Extension de fonctions holomorphes et courants[END_REF] we still have ∂u = ω in Ω.

Moreover the subordination lemma [START_REF] Amar | A subordination principle[END_REF] gives again u ∈ L s (p,q-1) (Ω), because u ′ ∈ L s (p,q-1) (∂Ω ′ ).

3 Carleson measures of order α.

For Ω a s.p.c. domain in C n , let V 0 (Ω) be the space of bounded measures in Ω, and V 1 (Ω) the space of Carleson measures in Ω as defined for instance in [START_REF] Amar | Mesure de Carleson d'ordre α et solution au bord de l'équation ∂[END_REF]. We know that these spaces form a interpolating scale for the real method [START_REF] Amar | Mesure de Carleson d'ordre α et solution au bord de l'équation ∂[END_REF], and we set

V α (Ω) := (V 0 , V 1 ) (α,∞) ; W α (Ω) := (V 0 , V 1 ) (α,p) with p = 1 1 -α .
Recall that a (p, q) form ω is in W α (p,q) (Ω) (resp. V α (p,q) (Ω) ) if its coefficients and the coefficients of ω ∧ ∂ρ √ -ρ are measures in W α (Ω) (resp. V α (Ω) ) see [START_REF] Amar | Mesure de Carleson d'ordre α et solution au bord de l'équation ∂[END_REF] and [START_REF] Andersson | Estimates of solutions of the H p and BMOA corona problem[END_REF].

A (p, q) form is in L r (p,q) (Ω) if just its coefficients are in L r (Ω). Let Ω ′ := {(z, w) ∈ C n ×C :: ρ ′ (z, w) := ρ(z) + |w| 2 < 0} and lift a current ω to Ω ′ as before : ω ′ (z, w) := ω(z). Our first result links L r estimates to Carleson α ones.

Theorem 3.1 Let Ω be a s.p.c. domain in C n then we have ω ∈ L r (p,q) (Ω) ⇒ ω ′ (z, w) := ω(z) ∈ W α (p,q) (Ω ′ ) with α = 1 r ′ + 1 2(n + 1)
.

Proof.

Let U ′ := N j=1 Q ′ (ζ ′ j , h j ) ∩ ∂Ω ′ be an open set in ∂Ω ′ and T (U ′ ) = N j=1
Q ′ (ζ ′ j , h j ) be its associated "tent" set inside [START_REF] Amar | Mesure de Carleson d'ordre α et solution au bord de l'équation ∂[END_REF] ; in order to see that a measure dµ = f dm, with m the Lebesgue measure in C n , belongs to V α (Ω ′ ) we have to show, see [START_REF] Amar | Mesure de Carleson d'ordre α et solution au bord de l'équation ∂[END_REF],

T (U ′ ) |f (z ′ )| dm(z ′ ) ≤ C |U ′ | α where |U ′ | := σ(U ′ )
is the Lebesgue measure of U ′ on ∂Ω, and with a constant C independent of U ′ . Because we are dealing with (p, q) currents here, this means that we have to estimate

A := T (U ′ ) |ω(z)| -ρ ′ (z, w) dm(z, w) with ρ ′ (z, w) := ρ(z) + |w| 2 is equivalent to the distance of (z, w) ∈ Ω ′ to the boundary ∂Ω ′ . Back to A, A := T (U ′ ) |ω(z)| -ρ ′ (z, w) dm(z, w) ≤ N j=1 Q ′ j |ω(z)| -ρ ′ (z, w) dm(z, w). The Carleson window Q ′ j is equivalent to the product (Q ′ j ∩ ∂Ω ′ )×[h j ] ν j with [h j ] ν j the real segment of length h j supported by the real normal ν j to ∂Ω ′ at ζ ′ j . Set h := max j=1,...,N h j , we shall replace Q ′ j by Q ′′ j := (Q ′ j ∩ ∂Ω ′ )×[h] ν j .

So we have

A ≤ N j=1 Q ′ j |ω(z)| -ρ ′ (z, w) dm(z, w) ≤ N j=1 Q ′′ j |ω(z)| -ρ ′ (z, w) dm(z, w),
where now all the depths have the same value h. Hence by Fubini we have

A ≤ h 0 1 √ t { U ′ t |ω(z)| dσ(z, w)}dt with U ′ t := N j=1 Q ′′ j ∩ ∂Ω ′ t and ∂Ω ′ t := {(z, w) ∈ Ω ′ :: ρ(z) + |w| 2 = -t}.
We can estimate the inner integral by Hölder

U ′ t |ω(z)| dσ(z, w) ≤ U ′ t |ω(z)| r dσ(z, w) 1/r U ′ t dσ(z, w) 1/r ′ (3.1) but U ′ t |ω(z)| r dσ(z, w) ≤ ∂Ωt |ω(z)| r dσ(z, w) ≤ C(ρ) Ωt |ω(z)| r { |w| 2 =-ρ(z)-t d |w|}dm(z)
where d |w| is the normalized Lebesgue measure on the circle |w| 2 = -ρ(z) -t. Hence, with Ω t := {z ∈ Ω :: ρ(z) < -t},

U ′ t |ω(z)| r dσ(z, w) ≤ C(ρ) Ωt |ω(z)| r dm(z) = C(ρ) ω r L r (Ω) .
For the last factor of (3.1) we have

U ′ t dσ(z, w) = σ(U ′ t ) σ(U ′ ), so A ≤ h 0 1 √ t { U ′ t |ω(z)| dσ(z, w)}dt ω L r (Ω) (σ(U ′ )) 1/r ′ h 0 dt √ t = 1 2 ω L r (Ω) √ hσ(U ′ ) 1/r ′ . Recall that σ(Q ′ j ) ≃ h (n+1) j then we have √ h = max j h j max σ(Q ′ j ) 1/2(n+1) ≤ σ( N j=1 Q ′ j ∩ ∂Ω ′ ) 1/2(n+1)
, so finally we get

A := T (U ′ ) |ω(z)| √ -ρ ′ (z,w) dm(z, w) ω L r (Ω) σ(U ′ ) 1 r ′ + 1 2(n+1) . This means that |ω(z)| -ρ ′ (z, w) is a Carleson measure in Ω ′ of order α with α = 1 r ′ + 1 2(n + 1)
.

To get a usual Carleson measure, we need α = 1 hence 1

r ′ + 1 2(n + 1) = 1 ⇐⇒ r = 2(n + 1).
We have by theorem 1 in [START_REF] Amar | Mesure de Carleson d'ordre α et solution au bord de l'équation ∂[END_REF], written in our situation, that if µ ∈ V α (Ω ′ ) then P 0 * (µ) ∈ L r,∞ (∂Ω ′ ), where P 0 * (µ) is the "balayage" of µ by the Hardy Littlewood kernel P 0 t . Hence we have that the linear operator P 0 * sends V α 0 (Ω ′ ) to L r 0 ,∞ (∂Ω ′ ), and V α 1 (Ω ′ ) to L r 1 ,∞ (∂Ω ′ ) with, as usual,

α j = 1 - 1 r j . This means that f ∈ L r (Ω) ⇒ µ := f / -ρ ′ dm ∈ V α (Ω ′ ) ⇒ P 0 * (µ) ∈ L s,∞ (∂Ω ′ ) with control of the norms.
So we have a linear operator T such that, with r 0 < r 1 ,

T : L r 0 (Ω) → L s 0 ,∞ (∂Ω ′ ), with 1 s 0 = 1 r 0 - 1 2(n + 1)
; But this implies by theorem 2 in [START_REF] Amar | Mesure de Carleson d'ordre α et solution au bord de l'équation ∂[END_REF], that µ := f / -ρ ′ dm ∈ W α (Ω ′ ).

T : L r 1 (Ω) → L s 1 ,∞ (∂Ω ′ ), with 1 s 1 = 1 r 1 - 1 2(n +
4 The main result.

Let Ω be a domain in C n defined by the function ρ as above ; define Ω ′ ⊂ C n+1 the lifted domain : we shall define the anisotropic class Γ β (∂Ω ′ ) as in [START_REF] Amar | Mesure de Carleson d'ordre α et solution au bord de l'équation ∂[END_REF] ; we say that a vector field X on ∂Ω ′ is admissible if X is of class C k and at any point of ζ ∈ ∂Ω ′ , X(ζ) belongs to the complex tangent space of ∂Ω ′ at ζ.

We say that u ∈ Γ β (∂Ω ′ ) if u is bounded on ∂Ω ′ and u belongs to the usual Lipschitz Λ β/2 (∂Ω ′ ), where ∂Ω ′ is viewed as a real manifold, and on any integral curve of an admissible vector field, t ∈ [0, 1] → γ(t) ∈ ∂Ω ′ , the function u • γ belongs to Λ β (0, 1).

We can now define the class Γ β (Ω) : take a function u defined in Ω and lift it as u ′ (z, w)

:= u(z) in Ω ′ ; then u ∈ Γ β (Ω) if u ′ ∈ Γ β (∂Ω ′ ). We have that u ∈ Γ β (Ω) implies that u ∈ L ∞ (Ω) and u ∈ Λ β/2 (Ω) with a Lipschitz constant uniform in Ω.
The same way we define function u ∈ BMO(Ω) if u ′ ∈ BMO(∂Ω ′ ). We have that u ∈ BMO(Ω) implies that u ∈ r≥1 L r (Ω). Now we are in position to prove our main result. Theorem 4.1 Let Ω be a s.p.c. domain in C n then for 1 < r < 2n + 2 we have ∀ω ∈ L r (p,q) (Ω), ∂ω = 0, ∃u ∈ L s (p,q-1) (Ω) :

: ∂u = ω, u L s (Ω) ω L r (Ω) , with 1 s = 1 r - 1 2(n + 1)
.

For r = 2n + 2 we have ∃u ∈ BMO (p,q) (Ω) :

: ∂u = ω, u BM O(Ω) ω L 2n+2 (Ω) . If ω is a (p, 1) form we have also : for r = 1, we have ∃u ∈ L s,∞ (p,0) (Ω) :: ∂u = ω, u L s,∞ (Ω) ω L 1 (Ω) with 1 s = 1 - 1 2(n + 1) . for r > 2n + 2 we have ∃u ∈ Γ β (p,0) (Ω) :: ∂u = ω, u Γ β (Ω) ω L r (Ω) ,
where β = 1 -2(n + 1) r and Γ β is an anisotropic Lipschitz class of functions. Moreover the solution u is linear on the data ω.

Proof. By use of theorem 3.1 we have that ω

′ ∈ W α (p,q) (Ω ′ ) with α = 1 r ′ + 1 2(n + 1)
where Ω ′ is still s.p.c. [START_REF] Amar | A subordination principle[END_REF],

hence we can apply the theorem 7 in [START_REF] Amar | Mesure de Carleson d'ordre α et solution au bord de l'équation ∂[END_REF] if ω is a (p, 1) current or the generalisation to (p, q) current done in theorem 4.1 in [START_REF] Andersson | Estimates of solutions of the H p and BMOA corona problem[END_REF] to get that there is a solution u ′ of ∂b u ′ = ω ′ such that u ′ ∈ L s (p,q-1) (∂Ω ′ ) with

1 s = 1 -α = 1 r - 1 2(n + 1)
.

Because ω ′ does not depend on w we have that the coefficients of u ′ are holomorphic in w hence with u ′ (z, w) = I,J a ′ I,J (z, w)dz I ∧ dz J we can set (recall that u ′ is defined on ∂Ω ′ ) ∀z ∈ Ω, a I,J (z) :=

|w| 2 =-ρ(z)
a ′ I,J (z, w)d |w| and we set also u(z) := I,J a I,J (z, w)dz I ∧ dz J , then exactly as in [START_REF] Amar | Extension de fonctions holomorphes et courants[END_REF] we still have ∂u = ω in Ω. Moreover the subordination lemma [START_REF] Amar | A subordination principle[END_REF], gives us u ∈ L s (p,q-1) (Ω). The last two results came directly from [START_REF] Amar | Mesure de Carleson d'ordre α et solution au bord de l'équation ∂[END_REF], theorem 7 and theorem 8 with the fact that we apply them in Ω ′ ⊂ C n+1 so we have from theorem 8 that β = 2(n + 1)(α -1). and not only for

1 s > 1 r - 1 2(n + 1)
.

  |I|=p,|J|=q ω I,J dz I ∧ dz J ⇒ ω r r

Remark 4 . 2

 42 In the range 1 < r < 2n + 2 theorem 1.2 is stronger than theorem 1.1 because we get