André É Berg

AND David Cohen

Guillaume Dujardin

Lie-Trotter Splitting for the Nonlinear Stochastic Manakov System

Keywords: AMS Classification, 65C30, 65C50, 65J08, 60H15, 60M15, 60-08, 35Q55 Keywords, Stochastic partial differential equations, Stochastic Manakov equation, Coupled system of stochastic nonlinear Schrödinger equations, Numerical schemes, Splitting scheme, Lie-Trotter scheme, Strong convergence, Convergence in probability, Almost sure convergence, Convergence rates, Blowup

published or not. The documents may come

Introduction

The Internet and its many areas of applications and dependencies create a huge demand for faster optical communication systems. One of the current limiting factors of high bit rate transmissions is dispersive effects which accumulate over long distances [START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation[END_REF]. One of these limiting factors is due to polarization mode dispersion (PMD) which follows from birefringence in the optical fibers. This effect in turn may vary due to e. g. core geometry, non-uniform anisotropy, or mechanical distortions from point-like pressure or twisting. These restrictive factors can together be modeled as random influences leading to the Manakov PMD equation and its limiting equation, the stochastic Manakov equation, see for instance [START_REF] Wai | Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence[END_REF][START_REF] De Bouard | A diffusion approximation theorem for a nonlinear PDE with application to random birefringent optical fibers[END_REF] for details. A precise definition of the stochastic Manakov equation is given below. This stochastic partial differential equation (SPDE) thus serves as a model to study long distance light propagation in random optical fibers.

Let us now discuss recent literature on the numerical analysis of the stochastic Manakov equation. The work [START_REF] Gazeau | Numerical simulation of nonlinear pulse propagation in optical fibers with randomly varying birefringence[END_REF] (see also [START_REF] Gazeau | Analyse de modèles mathématiques pour la propagation de la lumière dans les fibres optiques en présence de biréfringence aléatoire[END_REF]) numerically studies the impact of noise on Manakov solitons and soliton wave-train propagation by the following time integrators: the nonlinearly implicit Crank-Nicolson scheme, the linearly implicit relaxation scheme, and an explicit split-step scheme (the Lie-Trotter scheme). The paper [START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation[END_REF] (see also [START_REF] Gazeau | Analyse de modèles mathématiques pour la propagation de la lumière dans les fibres optiques en présence de biréfringence aléatoire[END_REF]) proves that the order of convergence in probability of the Crank-Nicolson scheme is 1{2. In addition, it is shown that this numerical integrator preserves the L 2 -norm as does the exact solution to the stochastic Manakov equation (see below for details). Furthermore, it is numerically observed in the reference [START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation[END_REF] that the almost-sure order of convergence of the relaxation scheme and the split-step scheme is 1{2 ´. To the best of our knowledge, no proofs for these orders of convergence exist. Finally, the recent reference [START_REF] Berg | Exponential integrators for the stochastic Manakov equation[END_REF] proves, among other things, that the order of convergence in probability of an exponential integrator is 1{2.

The main goal of this article is to analyse a linearly implicit version of the Lie-Trotter integrator for an efficient time integration of the stochastic Manakov system. This numerical integrator is an application of the classical deterministic Lie-Trotter splitting from [START_REF] Trotter | On the product of semi-groups of operators[END_REF] to the present stochastic setting. The outline of the paper is as follows. The numerical integrator is described in Section 2. In Section 3, we theoretically confirm that this time integrator applied to the Manakov system with a truncated Lipschitz nonlinear term has the same order of convergence as that of the nonlinearly implicit Crank-Nicolson scheme from [START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation[END_REF] and that of the exponential integrator from [START_REF] Berg | Exponential integrators for the stochastic Manakov equation[END_REF]. This is achieved in Theorem 3, which is the main theoretical result of this paper. As a consequence, we prove in Section 4 that the order of convergence of the scheme applied to the Manakov system with untruncated nonlinearity is 1{2 ´in probability (see Proposition 4) and almost surely (see Proposition 5). Finally, Section 5 is devoted to numerical experiments. In particular, we illustrate numerically the strong order of convergence of the scheme applied to the Manakov system, its order of convergence in probability and its order of almost-sure convergence. In addition to comparing its qualitative properties (L 2 -norm preservation) and its computational cost with other numerical methods from the literature applied to the stochastic Manakov equation, we use this new scheme to investigate the existence of a critical powerlaw exponent for the stochastic Manakov system, which is a theoretical open problem [START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation[END_REF] at the time of writing.

A Lie-Trotter scheme for the nonlinear stochastic Manakov system

In this section, we set notation, we introduce the stochastic Manakov equation and the Lie-Trotter splitting scheme that we analyse and use in the next sections.

Let pΩ, F, Pq be a probability space on which a three-dimensional standard Brownian motion W ptq :" pW 1 ptq, W 2 ptq, W 3 ptqq is defined. We endow this probability space with the complete filtration F t generated by W ptq.

Following [START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation[END_REF], we write the nonlinear stochastic Manakov system as [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF] idX `B2

x X dt `i?

γ 3 ÿ k"1 σ k B x X ˝dW k `|X| 2 X dt " 0,
where X " Xpt, xq " pX 1 , X 2 q is the unknown vector-valued function with values in C 2 , ˝denotes the Stratonovich product, γ ě 0 measures the intensity of the noise, |X| 2 " |X 1 | 2 `|X 2 | 2 is the nonlinear coupling, and σ 1 , σ 2 and σ 3 are the classical Pauli matrices defined by

σ 1 " ˆ0 1 1 0 ˙, σ 2 "
ˆ0 ´i i 0 ˙, and σ 3 " ˆ1 0 0 ´1˙. The mild form of the stochastic Manakov equation ( 1) reads [START_REF] Berg | Exponential integrators for the stochastic Manakov equation[END_REF] Xptq " U pt, 0qX 0 `i ż t 0 U pt, sqF pXpsqq ds, where X 0 " pX 0,1 , X 0,2 q denotes the initial value of the problem, U pt, sq for t ě s with s, t P R `is the random unitary propagator defined as the unique solution to the linear part of (1), and F pXq " |X| 2 X. Let p ě 1. We define L p :" L p pRq :" pL p pR; Cqq 2 the Lebesgue spaces of functions with values in C 2 . We equip L 2 with the real scalar product pu, vq 2 "

2 ÿ j"1 Re ˆżR u j v j dx ˙.
Further, for m P N, we denote by H m :" H m pRq the space of functions in L 2 with their m first derivatives in L 2 , for which we denote the corresponding norm by ¨ H m " ¨ m . Just as for the classical cubic Schrödinger equation, the L 2 -norm of the exact solution to the stochastic Manakov system (1) is almost surely preserved:

Xptq L 2 " X 0 L 2
for all t P r0, τ ˚r, where τ ˚ą 0 is a stopping time, see [START_REF] De Bouard | A diffusion approximation theorem for a nonlinear PDE with application to random birefringent optical fibers[END_REF] for details. This is not the case for the evolution of the Hamiltonian, or total energy

HpXq :" 1 2 ż R ˇˇˇB X Bx ˇˇˇ2 dx ´1 4 ż R |X| 4 dx, where |X| 4 " p|X| 2 q 2 " p|X 1 | 2 `|X 2 | 2 q 2 , as shown in [6, Lemma 3.1].
For the time-integration of the system (1), one has to face two issues. First, the linear part of this SPDE generates a stochastic group which is not easy to compute. In particular, since the Pauli matrices do not commute, it is not the product of the stochastic semi-groups associated to each Brownian motion with the group generated by iB 2

x . Second, the nonlinear coupling term |X| 2 X often leads to implicit numerical methods that are costly to solve, see for instance the Crank-Nicolson scheme from [START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation[END_REF].

Therefore, we numerically approximate solutions to the stochastic Manakov equation ( 1) with the Lie-Trotter splitting scheme

(3) X n`1 " U h,n`1 ˆXn `i ż t n`1 tn F pY n psqq ds ˙,
where h ą 0 denotes the stepsize,

t n " nh for nonnegative integers n, U h,n`1 " `Id `1 2 H h,n ˘´1 `Id ´1 2 H h,n ˘
with Id the identity operator and H h,n " ´ihI 2 B 2 x `aγh

3 ÿ k"1 σ k χ n k B x .
Here, I 2 is the 2 ˆ2 identity matrix and ? hχ n k " W k ppn `1qhq ´Wk pnhq, for k " 1, 2, 3, are i.i.d. Wiener increments. Furthermore, Y n is the exact solution to the nonlinear differential equation i dY `F pY q dt " 0 with initial value X n at time t " t n . Iterating the recurrence given by (3), one obtains the discrete mild form of the Lie-Trotter splitting scheme (4)

X n " U n,0 h X 0 `i n´1 ÿ l"0 U n,l h ż t l`1 t l F pY l psqq ds,
where U n,l h :" U h,n ¨. . . ¨Uh,l`1 with U 0,0 h " Id if needed. As the exact solution to the SPDE (1), we have that the Lie-Trotter scheme also preserves the L 2 -norm almost surely:

Lemma 1.
d dt |Y n ptq| 2 " d dt `|Y n 1 ptq| 2 `|Y n 2 ptq| 2 ˘2 " 4 `|Y n 1 ptq| 2 `|Y n 2 ptq| 2 ˘Re ˆY n 1 ptq d dt Y n 1 ptq `Y n 2 ptq d dt Y n 2 ptq " 0,
pointwise in x P R. In particular, we have |Y n pt n q| 2 " |Y n pt n`1 q| 2 . Integrating this last identity over R yields Y n pt n q 2 L 2 " Y n pt n`1 q 2 L 2 . Using the above, the fact that U h,n`1 is an isometry over L 2 , see for instance [2, Appendix 5], and the definition of Y n , one then obtains

X n`1 L 2 " X n `i ż t n`1 tn F pY n psqq ds L 2 " Y n pt n`1 q L 2 " Y n pt n q L 2 " X n L 2 .

Convergence analysis of the Lie-Trotter splitting scheme

In this section, we consider the convergence analysis of the Lie-Trotter splitting (3) where we have a globally Lipschitz continuous and bounded nonlinearity in [START_REF] Berg | Exponential integrators for the stochastic Manakov equation[END_REF]. This is the case for instance, when one introduces a cut-off function for the cubic nonlinearity present in (1): Let R ą 0 and θ P C 8 pR `q, with θ ě 0, supppθq Ă r0, 2s and θ " 1 on r0, 1s. For x ě 0, we set θ R pxq " θp x R q and define F R pXq " θ R p X 2 1 q|X| 2 X. We next present some properties of the function F R as well as of the numerical solution, given by the Lie-Trotter splitting (3), of the SPDE (2) with the cut-off nonlinearity F R . Lemma 2. There exists a positive constant C such that for all R ą 0 and all X P H 1 ,

F R pXq 1 ď CR 3{2 .
Furthermore, for all R ą 0, the function F R is globally Lipschitz continuous in H 1 , with corresponding Lipschitz constant L R . The map F R also sends bounded subsets of H 2 to bounded subsets of H 2 , resp. H 6 to H 6 . Finally, the numerical solution of (2), with the cut-off nonlinearity F R , given by the Lie-Trotter splitting scheme (3) is almost surely bounded in H m for all m P t1, 2, 6u: For all X 0 P H m , for all T ą 0, there exists a positive constant Cp X 0 m , T, L R q such that for all integer N large enough, for all n " 1, . . . , N , one has

X n m `sup tnďsďt n`1 Y n psq m ď Cp X 0 m , T, L R q a.s.
Proof. We only highlight parts of the proofs.

In order to show that F R is globally Lipschitz continuous from H 1 to H 1 , one first observes that F R is of class C 8 and vanishes outside the ball

B H 1 2R ˆBH 1 2R Ă B H 1 2R ?
2 , where H 1 " H 1 pR; Cq. Moreover, the derivative of F R is bounded on H 1 . The mean value theorem then implies that F R is globally Lipschitz continuous, and we denote by L R the corresponding Lipschitz constant.

In order to show that F R sends bounded subsets of H 2 to bounded subsets of H 2 , one uses the definition of θ R and the fact that H 2 is an algebra to get

F R pXq 2 2 " F R ppX 1 , X 2 qq 2 2 ď C ˇˇθ R p X 2 1 q ˇˇ2 ´ |X 1 | 2 X 1 `|X 2 | 2 X 1 2 H 2 ` |X 1 | 2 X 2 `|X 2 | 2 X 2 2 H 2 ď C ` X 1 6 H 2 ` X 2 4 H 2 X 1 2 H 2 ` X 1 4 H 2 X 2 2 H 2 ` X 2 6 H 2 ˘,
which is bounded if pX 1 , X 2 q 2 is bounded. Finally, when considering the SPDE (2) with the cut-off Lipschitz nonlinearity F R , it is classical to show, using for instance Picard's iterations and Grönwall's lemma, that the Lie-Trotter splitting scheme is almost surely bounded in H m , with m P t1, 2, 6u.

With the above preparation, we can now show strong convergence of the Lie-Trotter splitting scheme when applied to the stochastic Manakov equation ( 2) with a cut-off nonlinearity F R . Theorem 3. Let R ą 0, T ě 0, N P N, h " T {N , p ě 1, and X 0 P H 6 . Consider the stochastic Manakov equation (2) with the cut-off nonlinearity F R . Then, the Lie-Trotter splitting scheme (3) has strong order of convergence 1{2: There exists h 0 ą 0 such that @h P p0, h 0 q, E " max n"0,1,...,N

X n ´Xpt n q 2p H 1  ď Ch p ,
where C " Cp X 0 6 , T, L R , p, γq.

Proof. For ease of presentation, in the proof below, we remove the subscript R in the stochastic processes X R ptq and X n R .

Let us denote the difference X n ´Xpt n q by e n . Using the mild equations ( 2) and ( 4), one gets

e n 1 " U n,0 h X 0 `i n´1 ÿ l"0 U n,l h ż t l`1 t l F R pY l psqq ds ´U pt n , 0qX 0 ´i ż tn 0 U pt n , sqF R pXpsqq ds 1 ď `Un,0 h ´U pt n , 0q ˘X0 1 ` n´1 ÿ l"0 ż t l`1 t l ´Un,l h F R pY l psqq ´U pt n , sqF R pXpsqq ¯ds 1 ": I n 1 `In 2 .
We begin by estimating the term I n 2 using the following decomposition

I n 2 " n´1 ÿ l"0 ż t l`1 t l U n,l h F R pY l psqq ´U pt n , t l qF R pY l psqq `U pt n , t l qF R pY l psqq ´U pt n , sqF R pY l psqq `U pt n , sqF R pY l psqq ´U pt n , sqF R pX l q `U pt n , sqF R pX l q ´U pt n , sqF R pXpt l qq `U pt n , sqF R pXpt l qq ´U pt n , sqF R pXpsqq ds 1 ď n´1 ÿ l"0 ż t l`1 t l ´Un,l h ´U pt n , t l q ¯FR pY l psqq ds 1 ` n´1 ÿ l"0 ż t l`1 t l pU pt n , t l q ´U pt n , sqq F R pY l psqq ds 1 ` n´1 ÿ l"0 ż t l`1 t l U pt n , sq `FR pY l psqq ´FR pX l q ˘ds 1 ` n´1 ÿ l"0 ż t l`1 t l U pt n , sq `FR pX l q ´FR pXpt l qq ˘ds 1 ` n´1 ÿ l"0 ż t l`1 t l U pt n , sq pF R pXpt l qq ´FR pXpsqqq ds 1 ": J n 1 `Jn 2 `Jn 3 `Jn 4 `Jn 5 .
In estimating these five terms, we repeatedly make use of the facts that F R is globally Lipschitz continuous, and that U n,l h and U pt, sq are isometries on H 1 , see [START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation[END_REF][START_REF] Berg | Exponential integrators for the stochastic Manakov equation[END_REF]. In order to bound the first term, J n 1 , we first define

Y ˚ptq " N ´1 ÿ l"0
1 rt l ,t l`1 q ptqY l ptq.

We then use [12, Proposition 2.2] (strong convergence for linear problems, i. e when F R " 0) and Lemma 2 (the almost sure boundedness of the Lie-Trotter splitting scheme in H 6 ) to conclude that @t P r0, T s, E " max n"0,1,...,N ´1 max l"0,1,...,n ´Un,l h ´U pt n , t l q ¯FR pY ˚ptqq

2p 1  ď Ch p .
With this, we get through Hölder's inequality that

E " max n"0,1,...,N pJ n 1 q 2p  ď E » -max n"0,1,...,N ˜n´1 ÿ l"0 ż t l`1 t l ´Un,l h ´U pt n , t l q ¯FR pY l psqq 1 ds ¸2p fi fl ď E « max n"0,1,...,N ´1 ˆż T 0 max l"0,1,...,n ´Un,l h ´U pt n , t l q ¯FR pY ˚psqq 1 ds ˙2p ff ď T 2p´1 ż T 0 E " max n"0,1,...,N ´1 max l"0,1,...,n ´Un,l h ´U pt n , t l q ¯FR pY ˚ptqq 2p 1  dt ď C 1 p X 0 6 , T, L R , p, γq h p .
Using the isometry property of U pt n , sq and Hölder's inequality, we obtain

E " max n"0,1,...,N pJ n 2 q 2p  ď E » -max n"0,1,...,N ˜n´1 ÿ l"0 ż t l`1 t l U pt n , sq pU ps, t l q ´Idq F R pY l psqq 1 ds ¸2p fi fl ď Ch 2p E » - - ¨˜N´1 ÿ l"0 1 2p 2p´1 ¸2p´1 2p ˜N´1 ÿ l"0 sup t l ďsďt l`1 pId ´U ps, t l qq F R pY l psqq 2p 1 ¸1 2p '2p fi ffi fl ď Ch 2p N 2p´1 N ´1 ÿ l"0 E " sup t l ďsďt l`1 pId ´U ps, t l qq F R pY l psqq 2p 1  . (5)
In order to estimate the above expectation, we write this term as

E " sup t l ďsďt l`1
pId ´U ps, t l qq pF R pY l pt l qq ´FR pY l pt l qq `FR pY l psqqq

2p 1  ď CE " sup t l ďsďt l`1 pId ´U ps, t l qq F R pY l pt l qq 2p 1  `CE " sup t l ďsďt l`1
pId ´U ps, t l qq `FR pY l psqq ´FR pY l pt l qq ˘

using the triangle inequality. The first term in the equation above is the exact solution to the linear SPDE idZptq B2 Zptq Bx 2 dt `i?

γ 3 ÿ k"1 σ k BZptq Bx ˝dW k ptq " 0 with initial value F R pY l pt l qq at initial time t l
which has the mild Ito form

Zptq ´FR pY l pt l qq " pSpt ´tl q ´Idq F R pY l pt l qq `i?

γ 3 ÿ k"1 ż t t l Spt ´uqσ k B x Zpuq dW k puq,
where Sptq is the group solution to the free Schrödinger equation. Owning at the regularity property of the group S (see for instance the first inequality in the proof of [10, Lemma 4.2.1]), the fact that the numerical solution Y l is bounded, that F R sends bounded sets from H 2 to H 2 , and Burkholder-Davis-Gundy's inequality (for the second term), one obtains the following bound

E " sup t l ďsďt l`1 pId ´U ps, t l qq F R pY l pt l qq 2p 1  ď Ch p .
Using the fact that the random propagator U is an isometry, that F R is Lipschitz continuous, and Y l is solution to a Lipschitz differential equation, one gets the estimate

E " sup t l ďsďt l`1 pId ´U ps, t l qq `FR pY l psqq ´FR pY l pt l qq ˘ 2p 1  ď E " sup t l ďsďt l`1 F R pY l psqq ´FR pY l pt l qq 2p 1  ď Ch 2
for the second term in [START_REF] De Bouard | A diffusion approximation theorem for a nonlinear PDE with application to random birefringent optical fibers[END_REF]. Combining the estimates above, one finally arrives at

E " max n"0,1,...,N pJ n 2 q 2p  ď Ch 2p N 2p´1 N h p ď Ch p .
For the third term, by definition of Y l and Lemma 2, we get

E " max n"0,1,...,N pJ n 3 q 2p  ď E » -max n"0,1,...,N ˜n´1 ÿ l"0 ż t l`1 t l U pt n , sq `FR pY l psqq ´FR pX l q ˘ 1 ds ¸2p fi fl ď E » -max n"0,1,...,N ˜LR n´1 ÿ l"0 ż t l`1 t l ż s t l F R pY l prqq dr 1 ds ¸2p fi fl ď C 3 pT, L R , pqh 2p .
For the estimation of the fourth term, one may use the original error term involving e n :

E " max n"0,1,...,N pJ n 4 q 2p  ď E » -max n"0,1,...,N ˜n´1 ÿ l"0 ż t l`1 t l U pt n , sq `FR pX l q ´FR pXpt l qq ˘ 1 ds ¸2p fi fl ď E » -max n"0,1,...,N ˜LR n´1 ÿ l"0 ż t l`1 t l X l ´Xpt l q 1 ds ¸2p fi fl ď pL R T q 2p E " max n"0,1,...,N p e n 1 q 2p  .
For the fifth term we use Hölder's inequality and [START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation[END_REF]Lemma 5.4] (temporal regularity of the mild solution), which yields

E " max n"0,1,...,N pJ n 5 q 2p  ď E » -max n"0,1,...,N ˜n´1 ÿ l"0 ż t l`1 t l U pt n , sq pF R pXpt l qq ´FR pXpsqqq 1 ds ¸2p fi fl ď E » -max n"0,1,...,N ˜LR h n´1 ÿ l"0 sup t l ďsďt l`1 Xpt l q ´Xpsq 1 ¸2p fi fl ď pL R hq 2p E » - - ¨˜N´1 ÿ l"0 1 2p 2p´1 ¸2p´1 2p ˜N´1 ÿ l"0 sup t l ďsďt l`1 Xpt l q ´Xpsq 2p 1 ¸1 2p '2p fi ffi fl " pL R hq 2p N 2p´1 N ´1 ÿ l"0 E " sup t l ďsďt l`1 Xpt l q ´Xpsq 2p 1  ď pL R hq 2p N 2p Cp X 0 2 , T, p, R, γqh p " C 5 p X 0 2 , T, L R , p, γqh p .
All together, with another use of [12, Proposition 2.2] for bounding the term I n 1 , we thus obtain

E " max n"0,1,...,N e n 2p 1  ď CE " max n"0,1,...,N I n 1 2p 1  `pL R T q 2p E " max n"0,1,...,N e n 2p 1  `C1 h p `C2 h p `C3 h 2p `C5 h p ď Ch p `pL R T q 2p E " max n"0,1,...,N e n 2p 1  .
Now, as in the proof of [2, Theorem 2], for T " T 1 small enough, i. e. such that pL R T q 2p ă 1, the inequality above gives

E " max n"0,1,...,N e n 2p 1  ď C 1 ´pL R T q 2p h p ,
on r0, T 1 s. In order to iterate this procedure, we impose, if necessary, that h is small enough (or, equivalently, that N is big enough), to ensure that T 1 can be chosen as before and as some integer multiple of h (say T 1 " rh for some positive integer r), while T is some multiple integer of T 1 (say KT 1 " T for some positive integer K). To obtain a bound for the error on the longer time interval r0, T s, we iterate the procedure above by choosing T 2 " 2T 1 and estimate the error on the interval rT 1 , T 2 s. We repeat this procedure, K times, up to the final time T . This can be done since the above error estimates are uniform on the intervals rT k , T k`1 s for k " 0, . . . , K ´1 (with a slight abuse of notation for the time interval):

E " max rT k ,T k`1 s X n ´Xk pt n q 2p 1  ď C E h p
where C E is the error constant obtained above, t n " nh are discrete times in rT k , T k`1 s, X 0 ptq :" Xptq is the exact solution with initial value X 0 , X k ptq denotes the exact solution with initial value p X k at time T k " kT 1 " pkrqh " t kr , and p X k " X kr corresponds to numerical solutions at time T k for k " 0, . . . , K ´1. For the total error, we thus obtain (details are only written for the first two intervals)

E " max n"0,1,...,N e n 2p 1  "E " max r0,T s X n ´Xpt n q 2p 1  ď CE " max r0,T 1 s X n ´Xpt n q 2p 1  `CE " max rT 1 ,T 2 s X n ´Xpt n q 2p 1  `. . . `CE " max rT K´1 ,T K s X n ´Xpt n q 2p 1  ďC E h p `CE h p `CL E " p X 1 ´XpT 1 q 2p 1  `. . . ď C E h p `CL C E h p `. . . ďC E h p `CL C E h p `C2 L C E h p `¨¨¨`C K´1 L C E h p ď Ch p ,
where C L is the Lipschitz constant of the exact flow of ( 1) from H 1 to itself and the last constant C is independent of N and h with N h " T for N big enough. This concludes the proof of the theorem.

Convergence in probability and almost surely in the non-Lipschitz case

Using the same strategy as in [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF][START_REF] Berg | Exponential integrators for the stochastic Manakov equation[END_REF], one can show convergence in probability of order 1{2 and almost sure convergence of order 1{2 ´for the Lie-Trotter splitting scheme (3) when applied to the stochastic Manakov equation (1) with the original cubic nonlinearity. Proposition 4. Let X 0 P H 6 and T ą 0. Denote by τ ˚" τ ˚pX 0 , ωq the maximum stopping time for the existence of a strong adapted solution, denoted by Xptq, of the stochastic Manakov equation [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF]. For all stopping time τ ă τ ˚^T a.s. there exists h 0 ą 0 such that we have

@h P p0, h 0 q, lim CÑ8 P ˆmax 0ďnďNτ X n ´Xpt n q H 1 ě Ch 1{2 ˙" 0,
where X n denotes the numerical solution given by the Lie-Trotter splitting scheme (3) with time step h and N τ " r τ h u. Proof. For R ą 0, let us denote by X R , resp. X n R , the exact, resp. numerical, solutions to the stochastic Manakov equation ( 2) with a truncated nonlinearity F R .

Fix X 0 P H 6 , T ą 0, ε P p0, 1q. Let τ be a stopping time such that a.s. τ ă τ ˚^T .

By [6, Theorem 1.2] there exists an R 0 ą 1 such that P ˜sup tPr0,τ s

Xptq 1 ě R 0 ´1¸ď ε{2.
Observe that one has the inclusion " max 0ďnďNτ

X n ´Xpt n q 1 ě ε * Ă " max 0ďnďNτ Xpt n q 1 ě R 0 ´1* Y ˆ" max 0ďnďNτ X n ´Xpt n q 1 ě ε * X " max 0ďnďNτ Xpt n q 1 ă R 0 ´1*˙.
Taking probabilities, we obtain

P ˆ" max 0ďnďNτ X n ´Xpt n q 1 ě ε *ď ε{2 `P ˆ" max 0ďnďNτ X n ´Xpt n q 1 ě ε * X " max 0ďnďNτ Xpt n q 1 ă R 0 ´1*˙.
In order to estimate the terms on the right-hand side, we define the random variable n ε :" mintn P t0, . . . , N τ u : X n ´Xpt n q 1 ě εu, with the convention that n ε " N τ `1 if the set is empty. If max 0ďnďNτ

Xpt n q 1 ă R 0 ´1 then we have by the triangle inequality

max 0ďnďnε´1 X n 1 " max 0ďnďnε´1 X n ´Xpt n q `Xpt n q 1 ď ε `R0 ´1 ď R 0 .
From the definition of the Lie-Trotter splitting scheme (3), the isometry property of U h,n`1 , the fact that F below is the cubic nonlinearity, and Lemma 2 follows

X nε 1 " Y nε´1 pt nε q 1 ď Y nε´1 pt nε´1 q 1 `ż tn ε t nε´1 F pY nε´1 psqq 1 ds ď X nε´1 1 `C ż tn ε t nε´1 Y nε´1 psq 3 1 ds a.s. (7)
Next, we give an explicit bound on Y nε´1 psq for t nε´1 ď s ď t nε , in order to show that X nε is also explicitly bounded. To do so, we use the explicit form of Y nε´1 , as solution to the ODE system idY `|Y | 2 Y dt " 0. First, we recall that |Y 1 pt, xq| 2 `|Y 2 pt, xq| 2 is pointwise preserved by the flow, see the proof of Lemma 1. This implies in particular that, for all s P rt nε´1 , t nε s,

Y nε´1 psq 2 L 2 " Y nε´1 pt nε´1 q 2 L 2 a.s.
Moreover, this preservation property allows for the exact solution of the ODE on pt nε´1 , t nε q to be written for all s P rt nε´1 , t nε s as

Y nε´1 psq " e ips´t nε´1 q|Y nε´1 pt nε´1 q| 2 ˆY nε´1 1 pt nε´1 q Y nε´1 2 pt nε´1 q ˙.
Then, the first spatial derivative of Y nε´1 can be computed as follows

B x Y nε´1 psq " e ips´t nε´1 q|Y nε´1 pt nε´1 q| 2 "ˆB x Y nε´1 1 pt nε´1 q B x Y nε´1 2 pt nε´1 q 2ips ´tnε´1 qRe ´Y nε´1 1 pt nε´1 qB x Y nε´1 1 pt nε´1 q `Y nε´1 2 pt nε´1 qB x Y nε´1 2 pt nε´1 q ¯ˆY nε´1 1 pt nε´1 q Y nε´1 2 pt nε´1 q ˙
Using Cauchy-Schwarz inequality and the fact that H 1 is an algebra, one obtains that, for some constant C ą 0,

B x Y nε´1 psq 2 L 2 ď 2 B x Y nε´1 pt nε´1 q 2 L 2 ´1 `C2 ps ´tn ε´1 q 2 Y nε´1 pt nε´1 q 4 1

¯.

Using the above estimates and the definition of the H 1 norm, one arrives at the following bound

Y nε´1 psq 2 1 ď 2 Y nε´1 pt nε´1 q 2 1 ´1 `C2 ps ´tn ε´1 q 2 Y nε´1 pt nε´1 q 4 1 ¯.
Taking the square root of the above and using the fact that ? a 2 `b2 ď a `b for positive real numbers a, b, one gets

Y nε´1 psq 1 ď ? 2 Y nε´1 pt nε´1 q 1 ´1 `C|s ´tnε´1 | Y nε´1 pt nε´1 q 2 1 ¯.
Now, as Y nε´1 pt nε´1 q " X nε´1 is bounded in H 1 by R 0 , if we assume that h is small enough to ensure that Ch `?2R 0 p1 `ChR 2 0 q ˘3 ď R 0 , then we have X n " X n 2R 0 for 0 ď n ď n ε by [START_REF] Debussche | Numerical simulation of focusing stochastic nonlinear Schrödinger equations[END_REF].

If n ε ď N τ , then X nε 2R 0 ´X2R 0 pt nε q 1 ě ε thanks to the definition of n ε . Therefore we get max 0ďnďNτ X n 2R 0 ´X2R 0 pt n q 1 ě ε. Furthermore, by definition of n ε , we have

" max 0ďnďNτ X n ´Xpt n q 1 ě ε * X tn ε ą N τ u " ∅.
We then deduce that

" max 0ďnďNτ X n ´Xpt n q 1 ě ε * " " max 0ďnďNτ X n ´Xpt n q 1 ě ε * X tn ε ď N τ u .
Combining the above, using Markov's inequality as well as the strong error estimates from Theorem 3, since τ ă T a.s., there exists C ą 0 such that

P ˆmax 0ďnďNτ X n ´Xpt n q 1 ě ε, n ε ď N τ , max 0ďnďNτ Xpt n q 1 ă R 0 ´1ď P ˆmax 0ďnďNτ X n 2R 0 ´X2R 0 pt n q 1 ě ε ˙ď 1 ε 2p E " max 0ďnďNτ X n 2R 0 ´X2R 0 pt n q 2p 1  ď 1 ε 2p Ch p .
This last term is smaller than ε{2 for h small enough. All together we obtain

P ˆmax 0ďnďNτ X n ´Xpt n q 1 ě ε ˙ď ε 2 `ε 2 " ε,
and thus convergence in probability.

To get the order of convergence in probability, we choose R 1 ě R 0 ´1 such that for all h ą 0 small enough, P ˆmax 0ďnďNτ

X n 1 ě R 1 ˙ď ε 2 .
As above, for all positive real number C, we have

" max 0ďnďNτ X n ´Xpt n q 1 ě Ch 1{2 * Ă " max 0ďnďNτ Xpt n q 1 ě R 1 * Y ˆ" max 0ďnďNτ X n ´Xpt n q 1 ě Ch 1{2 * X " max 0ďnďNτ Xpt n q 1 ă R 1 * ˙.
Taking probabilities and using Markov's inequality as well as the strong error estimate from Theorem 3, we obtain

P ˆ" max 0ďnďNτ X n ´Xpt n q 1 ě Ch 1{2 *˙ď ε 2 `P ˆ" max 0ďnďNτ X n 4R 1 ´X4R 1 pt n q 1 ě Ch 1{2 *ď ε 2 `Kp X 0 6 , T, 4R 1 , p, γq C 2p ,
since τ ď T almost surely. For C large enough, we infer

P ˆ" max 0ďnďNτ X n ´Xpt n q 1 ě Ch 1{2 *˙ď ε 2 `ε 2 " ε,
uniformly for h ă h 0 . This shows that the order of convergence in probability of the Lie-Trotter splitting scheme is 1{2.

Using the results above, one arrives at the following proposition, which establishes that the Lie-Trotter splitting scheme has almost sure convergence order 1{2

´.

Proposition 5. Under the assumptions of Proposition 4, for all δ P `0, 1 2 ˘and T ą 0, there exists a random variable K δ pT q such that for all stopping times τ with τ ă τ ˚^T , we have max n"0,...,Nτ

X n pωq ´Xpt n , ωq H 1 ď K δ pT, ωqh δ P ´a.s., for h ą 0 small enough.

Proof. The proof uses similar arguments as the corresponding proof in [START_REF] Berg | Exponential integrators for the stochastic Manakov equation[END_REF]. Let τ be a stopping time such that τ ă τ ˚^T almost surely. Fix R ą 0 and p ą 1. Using the strong error estimate from Theorem 3 and Markov's inequality, one gets positive h 0 and C, which depend on T but not on τ itself, such that

@h P p0, h 0 q, P ˆmax 0ďnďNτ X n R ´XR pt n q 1 ą h δ ˙ď Ch pp1´2δq .
Using [14, Lemma 2.8], one then obtains that, choosing p ě 1 sufficiently large to ensure that pp1 ´2δq ą 1, there exists a positive random variable K δ pR, γ, T, p, ¨q such that (8) P ´a.s., @h P p0, h 0 q, max 0ďnďNτ X n R ´XR pt n q 1 ď K δ pR, γ, T, p, ωqh δ .

After this preliminary observation, we shall proceed as in the proof of Proposition 4. We know that, since τ ă τ ˚a.s., there exists a random variable R 0 such that sup 0ďtďτ Xptq 1 ď R 0 pωq P ´a.s..

Let now ε P p0, 1q and h small enough (h ď 3R ´2 0 pωq). Assume by contradiction that max 0ďnďNτ X n ´Xpt n q 1 ě ε.

Define n ε :" mintn : X n ´Xpt n q 1 ě εu. By definition of R 0 and h, we have that X n 1 ď R 0 a.s. for 0 ď n ă n ε ´1. Hence, X nε 1 ď 2R 0 and so the numerical solution equals to the numerical solution of the truncated equation X n " X n 2R 0 for n " 0, 1, . . . , n ε . We thus obtain that max 0ďnďNτ X n 2R 0 ´X2R 0 pt n q 1 ě ε for h small enough. This contradicts (8) with R " 2R 0 . Therefore, we have almost sure convergence.

To get the order of almost sure convergence, we proceed similarly as in the proof of Proposition 4. From the above, we have for ω in a set of probability one and all ε ą 0, there exists h 0 ą 0 such that for all h ď h 0 , max 0ďnďNτ X n ´Xpt n q 1 ď ε. Thus, there exists

R 1 pωq ą R 0 pωq such that max 0ďnďNτ X n 1 ď R 1 pωq. If now h ď 3R ´2 1 :" h 0 , we obtain from (8) that max 0ďnďNτ X n ´Xpt n q 1 " max 0ďnďNτ X n R 1 ´XR 1 pt n q 1 ď K δ pR 1 , γ, T, p, ωqh δ .
This shows that the order of a.s. convergence of the exponential integrator is 1 2 ´.

Numerical experiments

In this section, we present various numerical experiments in order to illustrate the main properties of the Lie-Trotter scheme (3), denoted by LT below, and to compare it with other time integrators from the literature for the stochastic Manakov system. We start by numerically illustrating various types of convergence (strong, in probability, and almostsuerly) of various time integrators. Then, we focus on the preservation of the L 2 -norm and computational costs. Finally, we consider stochastic evolution of deterministric solitons and we discuss the possible occurrence of blowup of solutions to the stochastic Manakov equation.

Let us note that, in order to be able to consider soliton solutions (see below for precise details), we slightly modify the SPDE (1) with a factor 1{2 in front of the second order spatial derivative:

(9) idX `1 2 B 2 x X dt `i? γ 3 ÿ k"1 σ k B x X ˝dW k `|X| 2 X dt " 0.
We also introduce the following numerical schemes in order to compare their performance:

' The nonlinearly implicit Crank-Nicolson scheme from [START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation[END_REF] (CN) X n`1 " X n ´Hh,n X n`1{2 `ihGpX n , X n`1 q, where GpX n , X n`1 q " 1 2 p|X n | 2 `|X n`1 | 2 q X n`1{2 and X n`1{2 " X n`1 `Xn 2 . ' The exponential integrator studied in [START_REF] Berg | Exponential integrators for the stochastic Manakov equation[END_REF] (EXP)

X n`1 " U h,n pX n `ihF pX n qq .

' The relaxation scheme presented in [START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation[END_REF] (Relax)

i `Xn`1 ´Xn ˘`H h,n ˆXn`1 `Xn 2 ˙`Φ n`1{2 ˆXn`1 `Xn 2 ˙" 0, where Φ n`1{2 " 2|X n | 2 ´Φn´1{2 with Φ ´1{2 " |X 0 | 2 .
Observe that the definitions of the quantities H h,n and U h,n have to be modified in order to take into account the factor 1{2 in equation [START_REF] Garnier | Statistical analysis of pulse propagation driven by polarizationmode dispersion[END_REF]. Equation ( 9) allows the formation of solitons in the deterministic case (i. e. for γ " 0). Such solutions are given by the initial value [START_REF] Gazeau | Numerical simulation of nonlinear pulse propagation in optical fibers with randomly varying birefringence[END_REF][START_REF] Hasegawa | Effect of polarization mode dispersion in optical soliton transmission in fibers[END_REF] (10) X 0 pη, κ, α, τ, θ, φ 1 , φ 2 q " Xpx, 0q " ˆcospθ{2q exppiφ 1 q sinpθ{2q exppiφ 2 q ˙η sechpηxq expp´iκpx ´τ q `iαq, where η, κ, α, τ, θ, φ 1 , φ 2 P R, and take the form (11) Xpx, tq " ˆcospθ{2q exppiφ 1 q sinpθ{2q exppiφ 2 q ˙η sechpηpx ´τ ptqqq exp p´iκpx ´τ ptqq `iαptqq , where τ ptq " τ 0 ´κt, αptq " α 0 `1 2 pη 2 `κ2 q t, and τ 0 , α 0 P R.

For the numerical experiments we follow a few standards that will hold unless stated otherwise. We consider the SPDE (9) with γ " 1 on a bounded interval r´a, as with a sufficiently large a ą 0, with homogeneous Dirichlet boundary conditions, and over the time interval r0, T s, T ą 0. The spatial discretisation is done by uniform finite differences with mesh size denoted by ∆x. The initial condition for the SPDE is given by equation [START_REF] Gazeau | Analyse de modèles mathématiques pour la propagation de la lumière dans les fibres optiques en présence de biréfringence aléatoire[END_REF] with the parameters α " τ " φ 1 " φ 2 " κ " 0 and θ " π{4, η " 1. Lastly, all experiments will per sample, whenever possible, use a common Brownian motion for each numerical scheme and time discretization.

Strong convergence.

In this subsection we will numerically demonstrate the meansquare orders of convergence of the four numerical schemes seen above. The strong order of convergence has been shown to be 1{2, in the case of a cut-off nonlinearity, for the Lie-Trotter splitting scheme, the exponential integrator, and the Crank-Nicolson scheme in Theorem 3 above, [2, Theorem 2], and [12, Proposition 3.4], respectively.

To illustrate these strong orders of convergence, we consider discretizations of equation ( 9) with the following parameters: a " 50, ∆x " 0.05, T " 1, N " 2 k , k " 10, 11, . . . , 16, and h " T {N . We approximate the exact solution using a reference solution, denoted by r X, simulated using the Lie-Trotter splitting scheme with N ref " 2 18 and time step size

h ref " T {N ref .
We then compute the mean-square errors e N :" E " max n"1,...,N

X n ´r Xpt n q 2 1  at the coarse time grid tt n u N n"1 " tnhu N n"1 . The expectations are approximated using 300 samples (we have checked that this number of samples was enough). The results are presented in Figure 1, where it can be observed that all four numerical schemes demonstrate the mean-square rate of convergence 1{2. Restating a definition from [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF], we say that a numerical scheme converges in probability with order δ, in H 1 , if for all small enough h ą 0, lim CÑ8 P ˜sup nPt1,2,...,N u X n ´Xpt n q 1 ě Ch δ ¸" 0, or, equivalently, if for all ε P p0, 1s, there exists Cpεq ą 0 such that

P ˜sup nPt1,2,...,N u X n ´Xpt n q 1 ě Cpεqh δ ¸ă ε.
Numerically, we investigate the order in probability by using the equation ( 12) max nPt1,2,...,N u

X n ´r Xpt n q 1 ě Ch δ ,
where r X denotes a reference solution. We then either study the proportion of samples, P , fulfilling equation [START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation[END_REF] for given C and δ, or estimate the constant C for given δ and proportion of samples P . This is to say, when estimating P for given δ, h, and C, we then observe whether P Ñ 0 for the given δ as h Ñ 0 and C increases. Or, when estimating C for given δ, h, and the proportion P allowed by the sample size, we then observe whether the range of obtained C is small or not. In order to allow for comparison, we normalize this range via Cpδ, h, P q " Cpδ, h, P q max ĥ Cpδ, ĥ, 0q , which forces max h Cpδ, h, 0q " 1.

In the first numerical experiment we simulate 36 samples (on a 12 core computer) using the Lie-Trotter splitting scheme using a pseudospectral spatial discretization using 2 10 Fourier modes and we take the parameters γ " 9, a " 50, T " 1{2, N " 2 k , k " 12, 14, . . . , 20, and h " T {N . We approximate the exact solution using a reference solution, r X, simulated using the Lie-Trotter splitting scheme with N ref " 2 24 and time step size

h ref " T {N ref .
We then estimate the proportion P of samples fulfilling [START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation[END_REF] for each given h, δ " 0.4, 0.5, 0.6, and C " 10 c for c " 0, .5, 1. The results are presented in Figure 2. In this figure, one clearly sees how the proportion of samples P quickly goes to zero for δ ď 1{2 and an increasing C. Furthermore, this property does not hold for δ ą 1{2. This numerical experiment thus confirms that the order of convergence in probability of the Lie-Trotter scheme is 1{2. Observe that the reason for choosing a larger value of γ in the model ( 9) is for being able to perform such computations in reasonable times with reasonable large values of C and N . We have performed similar experiments for the other time integrators and have obtained alike results.

In the second numerical experiment we simulate 300 samples using all four schemes and considering the parameters a " 50, ∆x " 0.05, T " 1, N " 2 k , k " 10, 11, . . . , 16, and h " T {N . We approximate the exact solution using a reference solution, r X, simulated using the Lie-Trotter splitting scheme with N ref " 2 18 and time step size h ref " T {N ref . Using the obtained samples, we estimate the normalized ranges C for the given h, δ " 0.3, 0.4, 0.5 and P P p0, 1s. The results are presented in Figure 3. In this figure, one clearly sees how the ranges C becomes smaller when δ Ñ 1{2. This figure illustrates that all four numerical schemes converge in probability with at least order 1{2.

5.3.

Almost-sure convergence. In this subsection, with the help of two numerical experiments, we numerically demonstrate the orders of almost-sure convergence of the four numerical schemes. This order has been shown to be 1{2 ´for the exponential integrator and the Crank-Nicolson scheme in [2, Proposition 4], and [START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation[END_REF]Theorem 1.3], respectively as well as for the Lie-Trotter scheme in Proposition 5.

Restating the definition, we say that a numerical scheme converges almost-surely with order δ if for all δ P p0, δq, there exists a random variable K δ pT q such that one has max n"0,...,Nτ X n pωq ´Xpt n , ωq H 1 ď K δ pT, ωqh δ P ´a.s., for h ą 0 small enough, where τ is a stopping time. For ease of presentation we take T " 1. Numerically, we can investigate the order of almost-sure convergence by first computing the sample errors, with respect to a reference solution r X, e N pωq :" max n"1,...,N

X n pωq ´r Xpω, t n q 1 at the coarse time grid tt n u N n"1 " tnhu N n"1 , where h " 1{N . For a fixed sample, we can then estimate the constant K δ pT, ωq, according to the formula min tK δ pT, ωq P R `: K δ pT, ωqh δ ě e N pωqu for all considered time step sizes h. For a sufficiently large span of h, we would then expect that the distance [START_REF] Hasegawa | Effect of polarization mode dispersion in optical soliton transmission in fibers[END_REF] e δ " max h |K δ pT, ωqh δ ´eN pωq| would be minimized for the correct order δ.

Let us now use the above and investigate e δ from (13) for all four numerical schemes. We simulate 300 samples and consider the following parameters a " 50, ∆x " 0.2, T " 1, N " 2 k , k " 8, 11, . . . , 16, and h " T {N . We approximate the exact solution using a reference solution, r X, simulated using the Lie-Trotter splitting scheme with N ref " 2 18 and time step size h ref " T {N ref . We then estimate the constant K δ pT, ωq for each sample, and compute the mean, median and standard deviation of e δ in equation [START_REF] Hasegawa | Effect of polarization mode dispersion in optical soliton transmission in fibers[END_REF]. The results are presented in Table 1. We see that δ " 1{2 minimizes the mean, median and the deviation of e δ for all four numerical schemes. This confirms the theoretical results that the order of almost-sure convergence is 1{2´for the Lie-Trotter scheme, Crank-Nicolson scheme, and exponential integrator.

By treating the e δ as random variables dependent on the choice of δ and numerical scheme, we can statistically compare their expected values via one-sided paired t-tests. We do this by pairing e 0.5 and e δ , δ " 0.4, 0.45, 0.55, 0.6 for each numerical scheme, and test the hypothesis pairs # H 0 : Ere 0.5 s " Ere δ s H 1 : Ere 0.5 s ă Ere δ s.

The p-values obtained by the t-tests is the probability, under the assumption of the null hypothesis H 0 , to observe a set of observations at least as extreme as those tested. This means that if one obtains a p-value smaller than a chosen significance level, typically 5% or 1%, one may reject the null hypothesis H 0 in favor of the alternative hypothesis, H 1 . For 12 of the 16 combinations of e δ and numerical schemes, we obtain p-values which are at most approximately 10 ´11 . When comparing e 0.5 with e 0.55 for the four numerical schemes, we obtain p-values of approximately 0.35 to 0.37. We may therefore safely reject the null hypotheses in favor of the alternative hypotheses, with the exception of Ere 0.5 s ă Ere 0.55 s. This leads to the conclusion that δ close to 1{2 minimizes the mean of the chosen e δ , for all four numerical schemes. Combining this with the results in Table 1 thus illustrates that all time integrators converge almost-surely with at least order δ " 1{2.

In the second numerical experiment, we illustrate the behavior of how each individual sample of the Lie-Trotter splitting scheme converges in H 1 as h Ñ 0. To do this, we take the parameters a " 50, ∆x " 1{16, T " 1, N " 2 k , k " 4, 6, . . . , 18, and h " T {N . The real part of the obtained numerical solutions are presented in Figure 4. Solutions computed with larger time steps h are displayed with lighter gray, the red color is used for the reference solution r X. In this figure, one can clearly observe that not only the heights Mean δ " 0.4 δ " 0.45 δ " 0.5 δ " 0.55 δ " 0. 1. Mean, median and standard deviation of (13) obtained for each scheme, and δ " 0.4, 0.45, 0.5, 0.55, 0.

- of the oscillations will converge properly, but also that the offsets of the peaks, caused by coarse time stepping, will be lessen as the number of time steps increases.

5.4.

Preservation of the L 2 -norm. In this subsection we numerically illustrate the preservation of the L 2 -norm for the above time integrators. The numerical schemes which have been shown to preserve the L 2 -norm are the Lie-Trotter splitting scheme (in Lemma 1 above) and the Crank-Nicolson scheme (in [12, Proposition 3.1]). In contrast, the exponential integrator does not preserve the L 2 -norm, see [START_REF] Berg | Exponential integrators for the stochastic Manakov equation[END_REF]Ch. 4].

For this numerical experiments, we consider the parameters a " 50, ∆x " 0.2, T " 2, N " 2 14 , and h " T {N . We simulate 100 samples and for each sample and each scheme compute the maximum drift in the L 2 -norm,

max n"1,...,N log 10 ˇˇ X n L 2 ´ X 0 L 2 ˇˇ.
The maximum drifts of these samples are presented in Figure 5. In this figure, it can be EXP LT CN Relax -

Figure 5. Log of the maximum drift in the L 2 -norm for each time integrators.

observed that one has a preservation up to the 14'th decimal for all numerical schemes but the exponential integrator.

5.5. Computational costs. The goal of the present numerical experiment is to compare the computational costs of the above time integrators. To do this, we consider the following parameters a " 50, γ " 2, ∆x " 0.25, T " 1, N " 2 k , k " 8, 9, . . . , 14, and h " T {N . We approximate the exact solution using a reference solution, denoted by r X and simulated using the Lie-Trotter splitting scheme with N ref " 2 16 and time step size h ref " T {N ref , and compute the mean-square errors

e N :" E " X N ´r XpT q 2 1  .
The expectations present in e N are approximated using 100 samples. The mean computational times for all time integrators are presented in Figure 6.

In this figure, one can see that the Crank-Nicolson scheme is significantly slower than the other three schemes, which is to be expected due to the implicit calculations. In addition, it seems that the exponential integrator has a slight advantage when it comes to computational time. 5.6. Comparison of the evolution of solitons. In this subsection, we investigate the impact of noise on soliton solutions to the deterministic Manakov system. We do this by simulating equation ( 9) with the Lie-Trotter splitting scheme (3) for different levels of noise. We observe the evolutions of the H 1 -norm, Hamiltonian, the mass center, and the pulse width. We also inspect the evolution of the profiles of the solitons. The mass center (or barycenter, or time displacement) and the pulse width (or viriel), see e.g. [START_REF] Garnier | Statistical analysis of pulse propagation driven by polarizationmode dispersion[END_REF][START_REF] Gazeau | Analyse de modèles mathématiques pour la propagation de la lumière dans les fibres optiques en présence de biréfringence aléatoire[END_REF], are respectively defined as

T c ptq " xxyptq " ż R x|Xpt, xq| 2 d x ż R |Xpt, xq| 2 d x and V ptq " xx 2 yptq " ż R px ´xxyptqq 2 |Xpt, xq| 2 d x ż R |Xpt, xq| 2 d x .
In the deterministic case, the soliton [START_REF] Gazeau | Numerical simulation of nonlinear pulse propagation in optical fibers with randomly varying birefringence[END_REF] preserves a number of properties, including H 1 -norm, the Hamiltonian, and the pulse width. Some of the preserved properties can be seen in the lemma below. Lemma 6. The soliton [START_REF] Gazeau | Numerical simulation of nonlinear pulse propagation in optical fibers with randomly varying birefringence[END_REF], obtained by equation (9) with γ " 0 and initial value [START_REF] Gazeau | Analyse de modèles mathématiques pour la propagation de la lumière dans les fibres optiques en présence de biréfringence aléatoire[END_REF], preserves the following quantities:

(1) HpXp¨, tqq, where H is the Hamiltonian

HpXq " 1 2 ż R ˇˇˇB Bx X ˇˇˇ2 d x ´1 4 ż R |X| 4 d x.
(2) The norm Xp¨, tq (5) The pulse width V ptq.

Proof. To show that the first four quantities are preserved along the soliton [START_REF] Gazeau | Numerical simulation of nonlinear pulse propagation in optical fibers with randomly varying birefringence[END_REF], one inserts the definition of the soliton into these quantities and uses the translation invariance of the Lebesgue measure.

Set 1 2 3 η 1 1.2 1.5 θ π{3 π{2 ´π{2 φ 1 π{4 ´π{4 4π{5 φ 2 π{2 π{4 ´π{2 κ 2 3 4
Table 2. Initial value coefficients.

Along the soliton [START_REF] Gazeau | Numerical simulation of nonlinear pulse propagation in optical fibers with randomly varying birefringence[END_REF], the evolution of the mass center T c is given by

T c ptq " ż R x|Xpt, xq| 2 d x ż R |Xpt, xq| 2 d x " ż R pηz `τ0 ´κtq| sechpzq| 2 d z 2 " τ 0 ´κt.
It then follows that the pulse width satisfies

V ptq " ż R px ´Tc ptqq 2 |Xpt, xq| 2 d x ż R |Xpt, xq| 2 d x " π 2 12η 2 .
With the following experiments we highlight how some of these quantities evolve in the stochastic case considered in this paper (γ ą 0), and present a visual comparison, using the Lie-Trotter splitting scheme and a pseudospectral spatial discretization. We consider three instances of equation ( 9) with periodic boundary conditions and the noise coefficients γ " 0, γ " 1, and γ " 1{20. We use three initial values for each choice of γ in the form of equation [START_REF] Gazeau | Analyse de modèles mathématiques pour la propagation de la lumière dans les fibres optiques en présence de biréfringence aléatoire[END_REF], with the coefficients seen in Table 2. Further, we use a " 20π, 2 14 Fourier modes, T " 10, N " 2 10 and h " T {N . In the stochastic cases (γ " 1 and γ " 1{20) we simulate one sample each using one common Brownian motion. The H 1 -norm, Hamiltonian, the mass center, and the pulse width can be seen in Figure 7. In addition to this, the evolution of the first component of some of the numerical solutions can be seen in Figure 8.

When γ " 0, we see how the soliton produces the expected drift of the mass center. In addition, hardly visible in these figures, is the fact that the Lie-Trotter splitting scheme does not exactly preserve the H 1 -norm, Hamiltonian, or the pulse width of the soliton. Their respective evolutions instead oscillate around their starting values, with the amplitude of the oscillation decreasing as h decreases. When γ ą 0, it can be clearly observed that the presence of the noise prevents the preservation of the H 1 -norm, Hamiltonian and the pulse width of solitons. Furthermore, it can be noted how the pulse width only varies slightly for γ " 1{20. Finally, the conjecture posed in [START_REF] Gazeau | Numerical simulation of nonlinear pulse propagation in optical fibers with randomly varying birefringence[END_REF], stating that the soliton is stable and not strongly destroyed for small noise and short distances, seems to hold. . Time evolution various quantities of the deterministic ( * ) and stochastic (˝and for γ " 1 and γ " 1{20 respectively) numerical approximations of equation [START_REF] Garnier | Statistical analysis of pulse propagation driven by polarizationmode dispersion[END_REF]. Top to bottom: the initial values are equation [START_REF] Gazeau | Analyse de modèles mathématiques pour la propagation de la lumière dans les fibres optiques en présence de biréfringence aléatoire[END_REF] with coefficient set 1, 2, and 3, from Table 2. where u " upx, tq, is a complex valued random process, with t ě 0 and x P R d , ∆u "

d ÿ j"1 B 2 u Bx 2 j
denotes the Laplacian in R d , σ a positive real numbers, β " βptq is a real valued standard Brownian motion, and u 0 is a given initial value, see for instance [START_REF] Cohen | Exponential integrators for nonlinear Schrödinger equations with white noise dispersion[END_REF]. Solutions to this SPDE may blowup in finite time, see details below, depending on the choice of the power-law σ. We therefore introduce a general power-law nonlinearity in the stochastic Figure 8. Evolution of the square of the modulus of the first components, X 1 , up to t " 2.5 (left) and t " 10 (right) of the deterministic (bottom) and stochastic (γ " 1 top and γ " 1{20 middle) numerical approximations of ( 9) with initial value equation ( 10) and coefficient set 3 from Table 2.

Manakov equation ( 9) and consider

(14) idX `1 2 B 2 x X dt `i? γ 3 ÿ k"1 σ k B x X ˝dW k `|X| 2σ X dt " 0,
where σ P R `. The aim of the following numerical experiments is to numerically investigate possible blowup of the stochastic Manakov equation with power-law nonlinearity.

Following the convention of e. g. [START_REF] Debussche | Numerical simulation of focusing stochastic nonlinear Schrödinger equations[END_REF], given u 0 P H 1 and ω P Ω, we define the blowup time of the process X by τ pu 0 , ωq " inftτ P r0, `8s : lim tÑτ Xp¨, t, ωq 1 " `8, Xp0q " u 0 u.

We say that an exponent σ crit is critical for equation ( 14) if on the one hand τ pu 0 , ωq " `8 for all σ ă σ crit and all u 0 and on the other hand τ pu 0 , ωq ă `8 for all σ ą σ crit for some u 0 . The exponents σ ă σ crit and σ ą σ crit would be called subcritical exponents and supercritical exponents respectively. It has been shown that the NLSw has solutions in H 1 for dimension d " 1 and σ " 2, [8, Theorem 2.2], and for σ ă 2{d in any dimension, [START_REF] De Bouard | The nonlinear Schrödinger equation with white noise dispersion[END_REF]Theorem 2.3]. It is also conjectured, see [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF], that the critical exponent in the stochastic case is σ crit " 4{d, twice that of the deterministic case. Extensive numerical experiments on the NLSw are presented in [START_REF] Belaouar | Numerical analysis of the nonlinear Schrödinger equation with white noise dispersion[END_REF][START_REF] Cohen | Exponential integrators for nonlinear Schrödinger equations with white noise dispersion[END_REF][START_REF] Berg | Numerical study of nonlinear Schrödinger equations with white noise dispersion[END_REF]. Identification of critical exponents for the stochastic Manakov equation ( 14) is still an open problem, as discussed in [START_REF] Gazeau | Probability and pathwise order of convergence of a semidiscrete scheme for the stochastic Manakov equation[END_REF] for the cubic case (σ " 1).

Let us first investigate possible blowup of solutions to equation ( 14) in the cubic case, i. e. when σ " 1, for γ " 0 (deterministic case) and γ " 1 by observing how the H 1 -norms evolve over a sufficiently long time interval. These simulations use the following four initial values: The initial value given by the soliton [START_REF] Gazeau | Analyse de modèles mathématiques pour la propagation de la lumière dans les fibres optiques en présence de biréfringence aléatoire[END_REF] with the usual parameters given at the end of the introduction of Section 5, a sum of solitons [START_REF] Gazeau | Analyse de modèles mathématiques pour la propagation de la lumière dans les fibres optiques en présence de biréfringence aléatoire[END_REF] (see [START_REF] Gazeau | Numerical simulation of nonlinear pulse propagation in optical fibers with randomly varying birefringence[END_REF]) with arbitrarily chosen coefficients [START_REF] Trotter | On the product of semi-groups of operators[END_REF] X 0,2 " X 0 p5, 0, 0, 0, π{4, 0, 0q `X0 p1, π{3, 0, 0, π{4, 3, 0q, a Gaussian initial value (16) X 0,3 " ˆ3 exp p´10x 2 q 2 exp p´5x 2 q ˙, and a modification of equation [START_REF] Gazeau | Analyse de modèles mathématiques pour la propagation de la lumière dans les fibres optiques en présence de biréfringence aléatoire[END_REF] (17) X 0,4 " X 0 p1, 0, 0, 0, π{4, 0, 0q `ˆcospxq expp´x 2 q 0 ˙.

In order to avoid too long computational times, we perform this numerical experiment using a periodic boundary condition and a pseudospectral spatial discretization. We consider numerical discretizations with the following parameters: a " 20π, M " 2 13 , 2 15 Fourier modes, T " 40, N " 2 k , k " 15, 17, 19, and h " T {N . We then, at the time grids tt n u N n"0 " tnhu N n"0 , compute the mean of the H 1 -norms of the numerical solutions, H n " E r X n H 1 s . The expectations are approximated using 48 samples. The results of these numerical experiments are presented in Figure 9 (deterministic case) and Figure 10.

From these figures, it is clear that no indication of blowup is present, for any of the chosen discretizations or simulated samples. Had τ pu 0 , ωq ă `8 for any of the samples, we would have expected H n to increase as M and N increased.

Expanding on the above results, we perform two numerical experiments where we vary the exponent σ " 2, 3, 4 in the stochastic Manakov equation with a power-law nonlinearity Evolution of H 1 -norm of ( 14) for γ " 0 using the four initial values ( 10), ( 15), ( 16), and (17) (left to right, top to bottom). Pink ˆ: M " 2 13 , grey ˛: M " 2 15 . Darker lines implies larger N . Figure 10. Evolution of the mean of H 1 -norms γ " 1 using the four initial values ( 10), ( 15), [START_REF] Wai | Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence[END_REF], and (17) (left to right, top to bottom). Pink ˆ: M " 2 13 , grey ˛: M " 2 15 . Darker lines implies larger N . [START_REF] Printems | On the discretization in time of parabolic stochastic partial differential equations[END_REF] and the coefficient γ " 0 and γ " 1 (again comparing deterministic and stochastic results). As in the previous experiment we perform these numerical experiments using a periodic boundary condition and a pseudospectral spatial discretization. In order to further limit the computational time, we only consider the initial value [START_REF] Trotter | On the product of semi-groups of operators[END_REF], abort the simulations if X n H 1 ą 500, and simulate only one sample per combination of σ and γ. The common parameters for the following two experiments are T " 0.01, N " 2 k , k " 16, 17, and h " T {N . In order to verify possible blowup we also vary the width of the interval between the two experiments, by using a " 20π, and M " 2 16 , 2 17 Fourier modes in the first experiment and a " 40π, and M " 2 17 , 2 18 Fourier modes in the second experiment. For the stochastic samples we use one common Brownian motion.

The results can be seen in Figure 11 (for the first experiment with a " 20π) and in Figure 12 (for the second experiment with a " 40π). We see that the H 1 -norm increases sharply before exceeding 500 in all cases but for the stochastic (γ " 1) processes with σ " 2. Further, we see that taking a finer discretization or wider interval neither prevents nor delays these sharp increases. This is a clear evidence of blowup, for 5 of the 6 combinations of γ and σ, but we have two notable observations to make. The first is that the presence of noise (γ " 1) either delays or completely prevents blowup when σ " 2. The second is that an insufficient number of Fourier modes will fail to properly reflect the rapid increase in the H 1 -norm, as seen in both deterministic (γ " 0) cases with σ " 4. Other numerical experiments, not shown here, using finer time and spatial discretizations or the initial value ( 16) produce similar results. Evolution of H 1 -norm of ( 14) using the initial value (15), a " 40π, γ " 0 (left) or γ " 1 (right), and σ " 2, 3, 4 (top, middle, bottom). Pink M " 2 17 , grey ˛: M " 2 18 . Darker lines implies larger N .

Having confirmed that blowup may occur, and that the discretization parameters of the previous experiment is sufficient to observe it, we now simulate 48 samples with σ " 1, 1.25, 1.5, 1.75, 2., 2.25, 2.5, 2.9, 3, 3.5, 4 and the parameters T " 0.01, N " 2 17 , h " T {N , a " 20π, and M " 2 17 Fourier modes. As in the two previous experiments we terminate calculations if X n H 1 ą 500. The results are presented in Figure 13 and Figure 14. We observe that a rapid increase in the H 1 -norm is present for most samples. For σ " 2.5, we suspect that the spatial discretisation may perhaps not be fine enough to reach blowup. We also observe that, as expected, blowup for each sample is reached earlier for larger σ.

The results for the deterministic problem (γ " 0) are presented in Figure 15 (with a focus on values of σ between 1 and 2.25).

With these preliminary numerical experiments, we formulate the following two conjectures regarding the Manakov equation ( 9) in dimension d " 1: First, in the deterministic case (γ " 0), blowup occurs when σ ě 2, see left columns of Figure 11, Figure 12, and Evolution of H 1 -norms of ( 14) with γ " 0 and σ " 1, 1.25, 1.5, 1.75, 2., 2.25 (in order left to right, top to bottom) using the initial value [START_REF] Trotter | On the product of semi-groups of operators[END_REF].

Figure 1 . 5 . 2 .

 152 Figure 1. Strong rates of convergence.

Figure 2 .

 2 Figure 2. Proportion of samples fulfilling (12) for the Lie-Trotter splitting scheme.

  Relaxation scheme.

Figure 3 .

 3 Figure 3. The ranges C of each numerical schemes obtained with δ " 0.3, 0.4, 0.5.

Figure 4 .

 4 Figure 4. Almost sure convergence with the Lie-Trotter splitting scheme.

Figure 6 .

 6 Figure 6. Mean-square errors at time T " 1 compared to mean computational times.

Figure 7

 7 Figure 7. Time evolution various quantities of the deterministic ( * ) and stochastic (˝and for γ " 1 and γ " 1{20 respectively) numerical approximations of equation[START_REF] Garnier | Statistical analysis of pulse propagation driven by polarizationmode dispersion[END_REF]. Top to bottom: the initial values are equation[START_REF] Gazeau | Analyse de modèles mathématiques pour la propagation de la lumière dans les fibres optiques en présence de biréfringence aléatoire[END_REF] with coefficient set 1, 2, and 3, from Table2.

5. 7 .

 7 Conjecture on the critical exponent. A stochastic partial differential equation related to the stochastic Manakov equation (1) or (9) is the nonlinear Schrödinger equation with white noise dispersion (NLSw) in dimension d # idu `∆u ˝dβ `|u| 2σ u dt " 0 up0q " u 0 ,

Figure 9 .

 9 Figure 9. Evolution of H 1 -norm of (14) for γ " 0 using the four initial values (10), (15),[START_REF] Wai | Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence[END_REF], and (17) (left to right, top to bottom). Pink ˆ: M " 2 13 , grey ˛: M " 215 . Darker lines implies larger N .

Figure 11 .

 11 Figure 11. Evolution of H 1 -norm of (14) using the initial value (15), a " 20π, γ " 0 (left) or γ " 1 (right), and σ " 2, 3, 4 (top, middle, bottom). Pink ˆ: M " 216 , grey ˛: M " 2 17 . Darker lines implies larger N .

Figure 12 .

 12 Figure 12. Evolution of H 1 -norm of (14) using the initial value (15), a " 40π, γ " 0 (left) or γ " 1 (right), and σ " 2, 3, 4 (top, middle, bottom). Pink M " 2 17 , grey ˛: M " 2 18 . Darker lines implies larger N .

Figure 15 .

 15 Figure 15. Second, in the stochastic case (γ ą 0), blowup occurs when σ ą 2, see right columns of Figure 11, Figure 12, and Figure 13 and Figure 14.

6 .Figure 13 .

 613 Figure 13. Evolution of H 1 -norms of (14) with σ " 1, 1.25, 1.5, 1.75, 2., 2.25 (in order left to right, top to bottom) using the initial value[START_REF] Trotter | On the product of semi-groups of operators[END_REF].

Figure 14 .

 14 Figure 14. Evolution of H 1 -norms of (14) with σ " 2.5, 2.9, 3, 3.5, 4 (in order left to right, top to bottom) using the initial value[START_REF] Trotter | On the product of semi-groups of operators[END_REF].

Figure 15 .

 15 Figure 15. Evolution of H 1 -norms of (14) with γ " 0 and σ " 1, 1.25, 1.5, 1.75, 2., 2.25 (in order left to right, top to bottom) using the initial value[START_REF] Trotter | On the product of semi-groups of operators[END_REF].

  The Lie-Trotter splitting scheme (3) preserves the L 2 -norm almost surely. Proof. Choose n P N such that the scheme is well-defined at step n `1. By definition of Y n , since Y n 1 and Y n 2 solve pointwise in x P R the following differential equations, d dt Y 1 ptq " ´ip|Y 1 ptq| 2 `|Y 2 ptq| 2 qY 1 ptq and d dt Y 2 ptq " ´ip|Y 1 ptq| 2 `|Y 2 ptq| 2 qY 2 ptq,

over rt n , t n`1 s with Y n pt n q " X n , we have for all t P rt n , t n`1 s,

  ).

	9.34					9.42
	9.32					9.4
	9.3					9.38
	9.28					9.36
	9.26					9.34
	9.24					9.32
	9.22					9.3
	0	0.002	0.004	0.006	0.008	0.01	0	0.002	0.004	0.006	0.008	0.01
	0	0.002	0.004	0.006	0.008	0.01

Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87 Umeå, Sweden

Email address: andre.berglund@umu.se University of Gothenburg, Chalmers, Sweden Email address: david.cohen@chalmers.se Inria, Univ. Lille, CNRS, UMR 8524 -Laboratoire Paul Painlevé F-59000 Email address: guillaume.dujardin@inria.fr


Arnaud Recoquillay

Tom Druet

Simon Nehr

Margaux Horpin

Olivier Mesnil

Bastien Chapuis

Guillaume Laffont

Oscar D'almeida

Guided wave imaging of composite plates using passive acquisitions by fiber Bragg gratings

Keywords: Elastic guided waves, composite plates, Fiber Bragg Gratings, Delay and Sum, Excitelet, passive imaging, Structural Health Monitoring

In this paper are presented imaging results of defects in composite plates using guided wavebased algorithms such as Delay and Sum and Excitelet. Those algorithms are applied to passive data, for which the signal corresponding to each emitter-receiver couple is recovered thanks to the cross-correlation of the ambient noise measured simultaneously by the two sensors. The transition to passive imaging allow the use of lighter sensors unable to emit ultrasonic waves such as Fiber Bragg Gratings sensors on optical fibers, which are used in this study. The imaging results presented here show the feasibility of active and passive imaging in composite plates using Fiber Bragg Gratings as receivers, reducing the impact of the acquisition system on the structure in the context of Structural Health Monitoring.

I. INTRODUCTION

Over the last decade, an increasing interest appeared for guided wave-based imaging methods in the context of Non Destructive Testing (NDT) as well as of Structural Health Monitoring (SHM). Indeed, guided waves enable the inspection of a wide area of a structure with just a few sensors, reducing the inspection time and cost. In particular, the use of less sensors allows to embed them in the structure, making SHM possible. However, SHM systems using classical sensors and algorithms may still be too big of a burden for an industrial structure such as an aircraft. Hence the need of reducing this burden, for example by reducing the energy consumption of the sensors or their weight. To do so, passive acquisition methods seem a promising solution: they consist in recovering the impulse response of the system composed of the medium and the sensors by treating the ambient noise measured simultaneously at all sensors. The most common technique is the noise cross-correlation [START_REF] Lobkis | On the emergence of the Green's function in the correlations of a diffuse field[END_REF], which was used succesfully in various fields (Chehami et al., 2014; Davy et al., 2016; Roux et al., 2004; Sabra et al., 2007 , 2008; Shapiro and Campillo, 2004). For structures subject to vibrations, it is then possible to recover data to which classical imaging algorithms can be applied without emitting energy. Furthermore, sensors such as Fiber Bragg Gratings (FBG) on optical fibers, which are unable to emit waves, can be used [START_REF] Betz | Acousto-ultrasonic sensing using fiber Bragg gratings[END_REF]. Those sensors have previously been successfully used in combination with piezoelectric transducers (PZT) for active a) arnaud.recoquillay@cea.fr data acquisition [START_REF] Betz | Acousto-ultrasonic sensing using fiber Bragg gratings[END_REF]Takeda et al., 2005),

30
and more recently for passive data acquisition using noise cross correlation to recover the active signal first in seismology (Zeng et al., 2017) and then at ultrasonic frequencies for metallic plates [START_REF] Druet | Passive guided waves measurements using fiber Bragg 525 gratings sensors[END_REF].

The aim of this paper is to present passive guided-

35

wave imaging in composite panels using only Fiber Bragg Gratings. Indeed, as passive methods do not need the emission of ultrasonic waves into the domain, this kind of imaging can be done with classic PZT sensors but also FBGs, opening the way of SHM systems of very low intru-40 siveness. In a first section is briefly recalled the principle of passive acquisition, before explaining the basis of Fiber Bragg Gratings sensors. Then, two guided wave imaging techniques are presented, namely Delay and Sum and Excitelet before presenting ultrasonic wave measurement using FBGs. In the last section, experimental results obtained with these two algorithms on passive acquisitions for CFRP plates are presented. The first experiment was conducted using PZT sensors to check the feasibility of passive imaging in composite plates, and the full system, 50 using only FBG sensors, is tested in the second experiment.

II. ACTIVE AND PASSIVE ACQUISITION

The first step of any imaging algorithm is the data acquisition. For classical guided wave imaging algorithms 55 such as Delay and Sum, this step consists in successively emitting a wave with an actuator and measuring the signal with all other sensors. This method is called, in the following, active acquisition. Another way to obtain sensors simultaneously and then process those signals to reconstruct the data used in the imaging algorithm. This method is called, in the following, passive acquisition.

A few algorithms exist to retrieve the data from the ambient noise. The most known, which we will explain in more details and use in the following, is the ambient noise cross-correlation [START_REF] Lobkis | On the emergence of the Green's function in the correlations of a diffuse field[END_REF]). Another possibility is the passive inverse filter [START_REF] Gallot | A passive inverse filter for Green's function retrieval[END_REF], which is well suited when many receivers are used. For example, this technique has been used for guided wave tomography [START_REF] Druet | Tomographie passive par ondes guidées pour des applications de contrôle santé intégré[END_REF][START_REF] Druet | Passive guided wave tomography for structural health monitoring[END_REF], for which more sensors need to be used to obtain a relevant image of the domain. As the passive inverse filter is based on the inversion of the impulse response matrix, it is a global process, which reconstructs all active signals from all ambient noise simultaneously. This seems to allow a better reconstruction, in particular if the ambient noise does not have a good distribution in space or frequency domain [START_REF] Gallot | A passive inverse filter for Green's function retrieval[END_REF], when many sensors are involved. Let us also mention the correlation of the coda of the noise cross-correlation (Stehly et al., 2008), which may enhance the data's reconstruction from the noise cross-correlation when many sensors are used, by enhancing the spatial distribution of the noise.

In this paper, we focus on the use of the crosscorrelation of the ambient noise as this method has been more widely used until now and works well for any number of sensors. Its principle and hypotheses are recalled in the following. For more details, the reader may refer to [START_REF] Garnier | Passive sensor imaging using cross correlations of noisy signals in a scattering medium[END_REF]Lobkis and Weaver, 90 2001; Roux et al., 2005). As the aim of this paper is the ultrasonic imaging of plates, we restrict ourselves to a domain in which linear elastic waves propagate. To the author's best knowledge, the method for elastic waves has not yet been fully justified theoretically. Hence the justification is given for scalar acoustic waves. Suppose that the ambient noise ϕ is a space-time stationary random field that is also delta correlated in space, which means that, for all t, τ > 0 and points x and y, the following relation holds

100 ϕ(x, t)ϕ(y, τ ) = K(x)δ(x -y)F (t -τ ), (1)
where • denotes the ensemble average, K characterizes the spatial support of the sources and F is the normalized time correlation function, its Fourier transform being linked to the spectral energy density of the ambient noise. We suppose in the following that the sources are "well distributed" in space which means, depending on the nature of the domain Ω, that K ≡ 1 over a sufficient sub-domain (for more details, see [START_REF] Garnier | Passive sensor imaging using cross correlations of noisy signals in a scattering medium[END_REF]).

Let us now consider two sensors, at points x and y acquiring the ambient noise within the time interval [0, T ], T > 0. Those signals are denoted V x and V y . The cross-correlation C of these two signals is then

C(x, y, t; T ) = 1 T T 0 V x (τ )V y (τ + t) dτ . (2)
Let H xy be the impulse response of the system formed of the finite domain, the sensor at x acting as a source 115 and the sensor at y acting as a receiver. Suppose that the noise sources have a good distribution in space, that is K ≡ 1, then lim

T →+∞ dC dt (x, y, t; T ) ∝ F * [H xy (•) -H xy (-•)] (t), ( 3
)
where * denotes the convolution operation in time and • indicates the variable over which the operator, here the 120 convolution one, is taken. The convolution in time with F in (3) denotes the influence of the spectral density of the ambient noise: the impulse response's reconstruction strongly depends on the time correlation of the noise, and so on the spectral energy density of the ambient noise.

125

In the following, as elastic waves propagate in the domain of interest, we make the assumption that (3) is still relevant in this case. On a practical point of view, the quality of the data reconstruction will depend on a few parameters: first, the acquisition time T should be taken 130 large enough so that the cross-correlation converges. As will be seen in our applications, acquisition times of the order of at least one second should be used. The spatial repartition of the noise is also important: if the noise is not propagating in the axis going from x to y, then the 135 two signals V x and V y are not correlated and the reconstruction will be poor. Likewise, if the energy is only propagating from x to y and not from y to x, for example if the domain has a low reverberation, then only the causal part of the signal will be correctly reconstructed.

140

As already stressed out, the frequency distribution of the noise is also to take into account, so that the signals' energy is sufficient for the frequencies of interest. Let us remark that to obtain (3), the assumption has been made that the transfer function of the sensor acting as a source 145 or as a receiver is the same, which in general is true.

The combination of ambient noise cross-correlation and Fiber Bragg Gratings sensors, described in [START_REF] Druet | Passive guided waves measurements using fiber Bragg 525 gratings sensors[END_REF], allows for the determination of the pitchcatch response, that is the response of the structure for a 150 given sollicitation at one sensor and measured by another sensor, between two FBG sensors as if one was used as an emitter. This offers, on one hand, all the advantages of purely optical fiber sensing, such as dense wavelength multiplexing of the sensing points, immunity to and no 155 generation of Electro-Magnetic Interferences, compatibility with harsh environments such as those involving high or low (cryogenic) temperatures, shock waves (detonic) and also ionizing radiations, as well as the possibility to drastically reduce the intrusiveness of the instrumenta-160 tion of any kind of structure thanks to reduced wirings. On the other hand, guided elastic waves offer a fine diagnosis of the structure by enabling the detection, localization and sizing of defects as well as a full structure coverage.

III. GUIDED WAVE IMAGING TECHNIQUES

Many existing guided-wave based techniques could be used in combination with passive acquisitions. The only restrictions are the poor control over the reconstructed field, which prevent modal selection and, depending on the application, the frequency range for which the noise has enough energy. In this paper, two classical methods are tested, namely Delay and Sum (DAS) [START_REF] Michaels | Detection, localization and characterization of damage in plates with an in situ array of spatially distributed ultrasonic sensors[END_REF] and Excitelet [START_REF] Quaegebeur | Dispersion-based imaging for structural health monitoring using sparse and compact arrays[END_REF], as they were already tested in various cases for active acquisitions [START_REF] Kulakovskyi | Defect imaging in layered composite plates and honeycomb sandwich structures using sparse piezoelectric transducers network[END_REF]. Both algorithms were considered as they are representative of guided wave imaging techniques, using simple (DAS) or complex (Excitelet) models. Their principle is briefly recalled hereafter.

In both cases, the algorithm is used for detecting a single defect in the plate. Suppose that reference data were acquired in a sound state, denoted s ref for each emitter-receiver pair. Then, in the case of DAS, the idea is, for each point x of the inspected area, to affect the value of the envelope of the residual signal corresponding to the time of flight of the guided mode of interest going from the emitter to the point x and then to the receiver. Indeed, if a defect is at this point x and the mode is not completely converted into another one, then a wave should be measured by the receiver at this time.

By summing over all pairs, the coherence of the data is verified and the points at which there is a higher probability of presence of the defect are highlighted. More precisely, the Delay and Sum algorithm is the following:

1. Computation of the envelope of the residual signal for each pair:

sres k (t) = |H(s insp k (t) -s ref k (t))|, (4)
where H denotes the Hilbert transform.

2. For each point x of the inspected area, computation of the corresponding time of flight for each pair:

T k (x) = |x -x e k | c g (θ(x e k , x)) + |x r k -x| c g (θ(x, x r k )) , (5)
where x e k (x), respectively x r k (x), denotes the position of the emitter, respectively the receiver, for the k-th signal, θ(x, y) is the angle formed by y -x and the positive horizontal axis and c g denotes the energy velocity of the mode of interest, which will be discussed in the following.

3. Extraction of the value of the envelope corresponding to the time of flight and summation over all signals:

P DAS (x) = N -1 k=0 sres k (T k (x)). (6)
The strong hypothesis, and the main weakness of this method, is that it considers one single mode, propagating at a fixed energy velocity. It means that no mode 220 conversion by the defect is taken into account, and that other emitted modes may perturb the imaging process. Furthermore, it is well known that guided waves have a dispersive behavior, which means that their energy velocity depends on the frequency. To use this method, it 225 is then recommended to consider a frequency range for which the dispersion of the mode of interest is low, or to consider a narrow-band excitation, which will reduce the resolution as the support in time of the signal will be large.

230

The Excitelet algorithm is based on the same ideas but compares, thanks to a correlation, the residual signal to the theoretical one obtained when propagating the wave obtained for a given source from the emitter to the considered point and then to the receiver. More precisely, 235 the steps of the algorithm are the following:

1. Computation of the residual signal for each pair:

s res k (t) = s insp k (t) -s ref k (t). (7)
2. Computation of the theoretical signal for each pair: as the defect is considered as a point scatterer, the theoretical field scattered by this point is equal, in 240 the frequency regime, to the Green's function of the plate time the incident field emitted by the source:

s th k (t) = F -1 [V r k (x, •)V e k (x, •)] (t), (8)
where F -1 is the inverse Fourier Transform, and V e k and V r k are respectively the wave emitted by 245 the emitter and measured by the receiver of the k-th pair in the frequency regime. More precisely,

V e k (x, ω) = R 2 G(x, y, ω)f (y) dy U (ω), (9)
V r k (x, ω) = R 2 g(y) T G(y, x, ω) dy, (10)
G being the Green's function of the problem, f the source function in space, g the measurement function and U the Fourier transform of the excitation 250 signal. Note that f and g are implicitly supposed compactly supported on the support of the sensor, giving meaning to the integrals. Note also that, as the considered problem is the elastodynamic one, G is a matrix whereas f and g are vectors but the 255 signal is scalar in (8).

3. Computation of the correlation between the residual signal and the theoretical one and summation over all signals:

P Excitelet (x) = N k=0 R s res k (t)s th k (t) dt. ( 11
)
Both algorithms use the same inputs, that is signals at the inspection time and reference ones. Unlike DAS, Excitelet takes into account the guided waves dispersion through the computation of the theoretical signal. However, as the model used is more precise, a good knowledge of the parameters of the sample is mandatory. The use of a correlation in time also implies that the emitted signal should be short to increase the resolution. The computation of the theoretical signal may be time and memory-consuming in complex cases such as composite materials. To reduce it, a far-field approximation of the Green's function may be used in expressions (10) or only some modal components of the Green's function can be selected. With those two approximations, one obtains

G(x, y, ω) ≈ H (i) 1 (k m (ω, θ)|x-y|), where
H (i)
1 is the Hankel function of first order and i-th kind, i = 1, 2 depending on the convention used in the Fourier transform, k m is the wavenumber of the considered mode and θ is the angle of the vector x -y.

Finally, let us point out that, for passive acquisitions, the Fourier transform of the excitation signal U can be imposed in post processing, enabling the optimization of the imaging results.

In all the following, the imaging results correspond to normalized indicator functions, that is P a (x)/ max P a , where a is either DAS or Excitelet.

IV. ULTRASONIC WAVE MEASUREMENT USING FIBER BRAGG GRATINGS

Usually employed to measure temperature and strain, Fiber Bragg gratings are increasingly attracting researchers studying ultrasonic guided waves-based imaging methods for SHM applications. Indeed, their low intrusiveness and multiplexing capabilities (both spectrally and temporally) make FBGs an attractive alternative to piezoelectric elements as acoustic receivers. While dense wavelength multiplexing of tens of FBGs on a single optical fiber is commonly used in Telecommunications and even in Sensing applications, its immediate transposition to the case of FBGs used as acoustic receivers is still challenging. Active or passive acquisitions of ultrasonic signals using FBGs sensors require both a sampling frequency ranging from kHz to MHz levels while preserving their multiplexing capabilities and a high sensitivity to the strain variations induced by the ultrasonic signals propagating across composite plates. In this study, we have developed a monitoring system dedicated to wavelength-multiplexed FBG used as acoustic receivers. The principle of this measurement system relies on an edge filtering technique. The basic idea of the technique is the use of a tunable laser source per FBG acoustic receiver. For a given Bragg wavelength, the lasing wavelength emitted by an external cavity wavelength-tunable laser source is adjusted to the midpoint of one edge of the Bragg peak. Any shift of the Bragg wavelength will modulate the transmitted/reflected optical power. This demodulation scheme is commonly referred to as "edge filtering" technique [START_REF] Melle | A passive wavelength demodulation system for guided-wave bragg grating sen-545 sors[END_REF]. Hence the sampling rate at which each FBG is demodulated does not depend on the acquisition speed of a given spectral analyzer but rather simply on the band pass of both the pigtailed photodetectors and the associated 320 transimpedance electronic. This straightforward demodulation method allows adjusting easily the measurement frequency by simply changing the cutoff frequency of the transimpedance electronic but at the expense of the rather low degree of integration due to the need of one 325 tunable laser source per FBG acoustic receiver. Multiplexing the FBG acoustic receivers simply requires to cascade several tunable lasers and to inject their optical signals to the fiber containing the sensors thanks to an allfiber wavelength multiplexer. A similar component po-330 sitioned before the photodetectors separates the signals reflected by each FBG. In this study, a transimpedance circuit with cutoff frequencies of 1 MHz has been realized in order to handle four photodetectors simultaneously.

V. EXPERIMENTAL RESULTS

335

A. Passive acquisition using piezoelectric transducers

A first experiment was carried out using classic piezoelectric transducers of diameter 18 mm in order to check the feasibility of using passive acquisitions to image composite plates. The used sample was a 1000 × 600 × 5.75 340 mm 3 CFRP plate having 21 plies, all oriented in the same direction. Its material parameters are given in table I.

Table I. Material parameters of the CFRP plates The ambient noise was created thanks to a compressed air flow moved randomly over the whole surface of the plate during the acquisition. Eight piezoelectric 345 transducers were placed on a circle of radius 200 mm. This radius was chosen so that reflections on the border of the plate are measured after an incident wave coming from the opposite sensor for frequencies around 20 kHz, the measured ambient noise having a significant spec-350 tral energy density between 10 kHz and 80 kHz. The ambient noise was measured simultaneously on the eight sensors for 10 seconds at a sampling rate of 1 MHz. As already mentioned in section III, the quality of the image obtained with DAS or Excitelet is strongly linked to energy velocity is taken at the central frequency of the considered filter. Synthetic defects were obtained using 365 a pair of magnets of diameter 32 mm, placed for the first defect at (0, 5), which is a slightly off-centered position, and at (-40, 40) for the second one. In all the following images, the black dashed lines are contour lines corresponding to the value 0.9, added as an illustration. The green circle is the true shape of the defect whereas the black cross is the location of the maximum amplitude of the cartography. First, the imaging results for DAS are shown in figures 1 and 2.

E1 = E2 (MPa) E3 (MPa) ν12 ν13 = ν23 G12 (MPa) G13 = G23 (
In both cases, the localizations of the defect are well 375 retrieved. It should be noted that the image has a better accuracy in figure 2 than in figure 1, that is the high amplitude spot is smaller in figure 2, enabling a better local- ization of the defect. This comes from the low dispersion of the A 0 mode in this frequency range, inducing a bet-380 ter resolution for a bigger bandwidth as the wavepacket is shorter in time.

The imaging results corresponding to Excitelet are presented figures 3 and 4. The localization is again of good quality, even though the 0.9 contour is wider in 385 this case. This may come from a greater sensitiveness to the noise in the passive reconstruction: as will be seen in V B, Excitelet is more precise as long as the data are of good quality. Finally, in table II is given the distance between the center of the defect and the maximum of 390 the cartography. Except for the case of low frequency DAS, the distance is lower than the radius of the defect, which is the best result possible given the point scatterer assumption of the algorithms.

395

B. Passive acquisition using Fiber Bragg Gratings on optical fibers

The acquisition using FBG sensors was done on a 400 1010 × 610 × 2.2 mm 3 CFRP plate having 8 plies, all oriented in the same direction. Its material parameters are the same as the previous plate, given in table I. Eight FBG sensors were placed on the surface of the plate on a circle of radius 150 mm centered at the middle of the plate. Due to the used acquisition device, the signals of only four FBG could be recorded simultaneously. As the ambient noise needs to be recorded simultaneously on the sensors to recover the impulse response from the cross-correlation, this means that only a part of the data 410 was obtained (see figure 5). Only 24 different signals can then be reconstructed, where 56 different pairs exist and were used in the previous section. Note that all FBGs are oriented toward the center of the circle in order to maximize their sensitivity in the region of interest (Betz 415 et al., 2003). Defects were again simulated thanks to a pair of magnets of diameter 32 mm, placed at (-44, 82) for the first position and (-32, -24) for the second one. The noise was measured during 50 seconds as to ensure a good reconstruction of the signals. The Fiber Bragg 420 Gratings are 5 mm long with a reflectivity greater than 97%. They were photowritten in a polyimide-coated ger- Figure 5. Scheme of the acquisition setup using FBGs manosilicate singlemode optical fiber using a KrF laser emitting at 248 nm and a Talbot interferometer. The images obtained using DAS are presented figures 6 and 7. In this case, the obtained images do not enable to recover the position of the defects. An approximate position of the first defect may be recovered for both bandwidth, but the second one is not retrieved. In both cases, a strong noise exists in the images. Let us recall that less than half the data used in the previous section were available in this case, which may explain for a great part the poor results. Applying Excitelet, figures 8 and 9, the results are more satisfying: even though there still is noise in the images, the locations of the defect are properly reconstructed, with a better accuracy for the central frequency of 45 kHz as expected. Indeed, in figure 8, the algorithm detects the presence of a defect but its position and size are not retrieved. This result is the first demonstration of the possibility of using FBG sensors with a passive acquisition for guided wave imaging of composite panels. The imaging results obtained with both algorithms are of the same quality using FBG or PZT, even though less than half the signals used in the PZT case could be acquired with the FBGs. It is interesting to note that both algorithms may be used with passive data as DAS is easy to use and does not require a high knowledge of the wave propagation in the plate whereas Excitelet requires a higher knowledge leading to a higher accuracy.

450

Once again, the distance between the center of the defect and the maximum of the cartography is summarized in table III. In this case, as expected, the maximum is in general far from the defect. It should nevertheless be noted that the results are generally better for the high 455 frequency filter, which is expected, and that the results ar particularly good in this case for Excitelet.

Before concluding, the imaging results of Excitelet are plotted in figure 10 for various noise acquisition time. It can be seen that the image's resolution increases with 460 the acquisition duration due to the convergence of the cross-correlation of the ambient noise. It seems that, with the current system, an ambient noise acquisition of 30 seconds is necessary to have a sufficient signal reconstruction, enabling the localization of a defect. But 465 this value depends on various factors, among which are: the frequency content of the noise source, the frequency dependent transfer function of the sensors and of the acquisition system. For example, stronger analogical filtering of the noise around the bandwidth of interest, or equivalently a noise with a bandwidth closer to the one of interest, can help to have a better resolution in the reconstruction. As the noise source depends strongly on the application and is a parameter on which the user has no control, tailoring the acquisition system towards the expected frequency content of the noise may be a satisfying solution.

VI. CONCLUSION

In this article has been shown the feasibility to image 480 composite CFRP plate with elastic guided wave imaging algorithms on passive acquisitions thanks to FBG sensors. Many parameters need to be thoroughly studied to determine in which configurations this approach can be used and several ways of improvement are being investi-485 gated: first of all, not all signals could be reconstructed with the current acquisition system, limited to only 4 parallel FBG channels. The development of a system with more channels is undergoing at CEA and will lead to improved imaging results in the future. Another pa-490 rameter is the angular sensitivity of the FBG sensors: the measurement depends on the angle between the incident wave on the gratings and its direction. Even though iment, the imaging algorithms used does not take that into account. They should be adapted to take into account this variable sensitivity. Finally, other methods to reconstruct the active signal from the ambient noise could be tested, such as the passive inverse filter [START_REF] Gallot | A passive inverse filter for Green's function retrieval[END_REF], which may improve the reconstruction, especially when many sensors are used.

This first demonstration of a guided-wave-based imaging of composite panels using only FBG sensors opens the way to lightweight SHM systems with a very 505 low intrusiveness. Other declinations can be anticipated for configurations that highly benefit of the other advan- tages of the FBGs, such as their high temperature or radiation resistance. The Journal of the Acoustical Society of America 116(4), 1995-2003. Roux, P., Sabra, K. G., Kuperman, W. A., and Roux, A. (2005).

"Ambient noise cross correlation in free space: Theoretical approach," The Journal of the Acoustical Society of America 117(1), 79-84. Sabra, K. G., Conti, S., Roux, P., and Kuperman, W. (2007).

"Passive in vivo elastography from skeletal muscle noise," Applied physics letters 90(19), 194101.

Sabra, K. G., Srivastava, A., Lanza di Scalea, F., Bartoli, I., Rizzo, P., and Conti, S. (2008). "Structural health monitoring by extraction of coherent guided waves from diffuse fields," The Journal of the Acoustical Society of America 123(1), EL8-EL13. Shapiro, N. M., and Campillo, M. (2004). "Emergence of broadband Rayleigh waves from correlations of the ambient seismic noise," Geophysical Research Letters 31(7). Stehly, L., Campillo, M., Froment, B., and Weaver, R. L. (2008).

"Reconstructing Green's function by correlation of the coda of the correlation (C3) of ambient seismic noise," Journal of Geophysical Research: Solid Earth 113(B11). Zeng, X., Lancelle, C., Thurber, C., Fratta, D., Wang, H., Lord, N., Chalari, A., and Clarke, A. (2017). "Properties of noise crosscorrelation functions obtained from a distributed acoustic sensing 585 array at Garner Valley, California," Bulletin of the Seismological Society of America 107(2), 603-610.

  the k-th emitter-receiver pair, 0 ≤ k < N . Note that s ref k (t) = C * F * H ref x k y k with the notations of (3), where H ref x k y k is the impulse response of the reference system for a source at x k and a receiver at y k and C is a filter that can be added in the post-processing of the reconstructed signals. The signals corresponding to the state at the time of inspection are denoted s insp k . Residual signals are computed by subtracting s ref k to s insp k

355

  the excitation signal. Here the signal reconstructed from the ambient noise cross-correlation has been filtered to obtain signal characteristics appropriate for those imaging algorithms. More precisely, two different filters were tested: a Hann filter of central frequency 20 kHz and of 360 bandwidth 20 kHz and one of central frequency 45 kHz and of bandwidth 70 kHz. For both algorithms, only the A 0 mode was considered and, in the case of DAS, the

  Figure 1. (color online) DAS applied to passive acquisitions with PZT, frequency range: 10-30 kHz.

Figure 2 .

 2 Figure 2. (color online) DAS applied to passive acquisitions with PZT, frequency range: 10-80 kHz.

Figure 3 .

 3 Figure 3. (color online) Excitelet applied to passive acquisitions with PZT, frequency range: 10-30 kHz.

Figure 4 .

 4 Figure 4. (color online) Excitelet applied to passive acquisitions with PZT, frequency range: 10-80 kHz.

Figure 6 .

 6 Figure 6. (color online) DAS applied to passive acquisitions with FBG, frequency range: 10-30 kHz.

Figure 7 .

 7 Figure 7. (color online) DAS applied to passive acquisitions with FBG, frequency range: 10-80 kHz.

Figure 8 .

 8 Figure 8. (color online) Excitelet applied to passive acquisitions with FBG, frequency range: 10-30 kHz.

  Figure 10. (color online) Excitelet applied to reconstructed signals for various noise acquisition durations.

  Takeda, N., Okabe, Y., Kuwahara, J., Kojima, S., and Ogisu, T. (2005). "Development of smart composite structures with small-diameter fiber Bragg grating sensors for damage detection: Quantitative evaluation of delamination length in CFRP lami-580 nates using Lamb wave sensing," Composites science and technology 65(15-16), 2575-2587.

Table II .

 II Distance of the maximum to the center of the defect, PZT case

	Algorithm	case	wavelength (mm)	distance (mm)
	DAS	low frequency, defect 1	31	25
		low frequency, defect 2	31	18
		high frequency, defect 1	20	10
		high frequency, defect 2	20	13
	Excitelet	low frequency, defect 1	31	5
		low frequency, defect 2	31	7
		high frequency, defect 1	20	10
		high frequency, defect 2	20	16

Table III .

 III Distance of the maximum to the center of the defect, FBG case

	Algorithm	case	wavelength (mm)	distance (mm)
	DAS	low frequency, defect 1	31	79
		low frequency, defect 2	31	167
		high frequency, defect 1	20	6
		high frequency, defect 2	20	235
	Excitelet	low frequency, defect 1	31	57
		low frequency, defect 2	31	80
		high frequency, defect 1	20	3
		high frequency, defect 2	20	23

J. Acoust. Soc. Am. / October 23, 2020

J. Acoust. Soc. Am. / October 23, 2020

the direction of the gratings were chosen as to optimize the measurements in the region of interest in our exper-