COUPLED PARAXIAL WAVE EQUATIONS IN RANDOM MEDIA IN THE WHITE-NOISE REGIME

Transcription

1 Submitted to the Annals of Applied Probability COUPLED PARAXIAL WAVE EQUATIONS IN RANDOM MEDIA IN THE WHITE-NOISE REGIME By Josselin Garnier and Knut Sølna Université Paris VII and University of California at Irvine In this paper the reflection and transmission of waves by a threedimensional random medium is studied in a high-frequency, whitenoise, and paraxial regime. The limit system derives from the acoustic wave equations and is described by a coupled system of random Schrödinger equations driven by a Brownian field whose covariance is determined by the two-point statistics of the fluctuations of the random medium. For the reflected and transmitted fields the associated Wigner distributions and the autocorrelation functions are determined by a closed system of transport equations. The Wigner distribution is then used to describe the enhanced backscattering phenomenon for the reflected field. 1. Introduction. The paraxial wave equation, in homogeneous or in random media, is a model for waves that is used in many applications, for instance in the context of communication and imaging [18]. It describes the evolution of waves that propagate along a particular direction by neglecting backscattering. Its simplicity, compared to the full three-dimensional wave equation, enables analysis of many important phenomena, such as laser beam spreading [9, 11, 23], time reversal in random media [5, 10], underwater acoustics [24] or migration problems in geophysics [6]. The derivation of the paraxial model in homogeneous media is well understood, and it can also be justified in heterogeneous media when the medium parameters have small fluctuations [2, 4]. However, the derivation is not so clear in the limit case in which the medium fluctuations are described by a white noise. The two main motivations for studying the white-noise paraxial wave equation are i) it is natural in many applications where the correlation radius of the medium is much smaller than the beam width, ii) it allows for the use of Itô s stochastic calculus, which in turn enables the closure of the hierarchy of moment equations. A classic conjecture is that the limit equation should take the form of the random Schrödinger equation studied in particular in [8]. The proof of the convergence of the solution of the wave AMS 2000 subject classifications: 60H15, 35R60, 74J20 Keywords and phrases: waves in random media, parabolic approximation, diffusionapproximation 1

2 2 J. GARNIER AND K. SØLNA equation in random media to the solution of the white-noise paraxial equation was obtained in the case of stratified weakly fluctuating media in [1]. In our paper we consider the transmission and reflection of acoustic waves by a slab of medium whose parameters have three-dimensional random fluctuations and whose end is either transparent or is a strong interface. This model is particularly interesting in the context of optical coherence tomography [25]. We analyze a wave propagation regime in which the parabolic paraxial) and white-noise approximations are valid. In this regime we obtain a system of coupled Schrödinger equations driven by a Brownian field that fully determines the statistics of the transmitted and reflected waves. As a corollary we compute explicitly the two-point statistics of the transmitted and reflected waves. These results show that the often used independent approach for the reflected wave in which the statistics of the forward- and backward-propagating waves are assumed to be independent [25]) is valid if and only if the transverse correlation radius of the fluctuations of the random medium is smaller than the initial beam width. Finally, we use the coupled Schrödinger equations to give a rigorous account for the enhanced backscattering or weak localization phenomenon [21] and we compute explicitly the enhancement factor and the shape of the enhanced backscattering cone. 2. The Transmission and Reflection Operators. We consider linear acoustic waves propagating in 1 + d spatial dimensions with heterogeneous and random medium fluctuations. The governing equations are 2.1) ρz, x) u t + p = F, 1 p Kz, x) t + u = 0, where p is the pressure field, u is the velocity field, ρ is the density of the medium, K is the bulk modulus of the medium, and z, x) R R d are the space coordinates. The source is modeled by the forcing term F. We consider in this paper the situation in which a random slab occupying the interval z 0, L) is sandwiched in between two homogeneous half-spaces. The source, F, is located outside of the slab at z = z 0, z 0 > L. We shall refer to waves propagating in the direction with a positive z component as rightpropagating waves. The medium fluctuations in the random slab 0, L) vary rapidly in space while the background medium is constant. We normalize the background bulk modulus K b and density ρ b in the slab to one, so that the background speed c b = K b /ρ b and impedance Z b = K b ρ b are also equal to one. The medium is assumed to be matched at the right boundary z = L. We consider a possible mismatch at the boundary z = 0 and denote

3 COUPLED PARAXIAL WAVE EQUATIONS IN RANDOM MEDIA 3 the medium parameters in the half-space z < 0 by ρ 0 and K 0 : 1 K 1 Kz, x) = 0 if z 0, ρ 0 if z 0, 1 + ε 3 ν z, x ) if z 0, L), ρz, x) = 1 if z 0, L), ε 2 ε 2 1 if z L, 1 if z L, with ε a small parameter. The random field νz, x) models the medium fluctuations and we assume that it is stationary and that it satisfies strong mixing conditions in z. The amplitude ε 3 of the noise has been chosen so as to obtain an effective regime of order one when ε goes to zero. That is, if the magnitude of the fluctuations is smaller than this, then the wave would propagate as if the medium were homogeneous, while if the magnitude is much larger, then the wave would not penetrate the slab. The scaling that we consider here corresponds to the physically most interesting situation. We moreover consider a scaling regime where the spatial support of the source in the transverse direction), or equivalently the initial beam width, is of order ε 2. This means that the transverse scale of the source and the one of the spatial fluctuations of the medium are of the same order. Remember that the Rayleigh length for a beam with initial beam width r 0 and central wavenumber k 0 is of the order of k 0 r0 2 in absence of random fluctuations the Rayleigh length is the distance from beam waist where the mode area is doubled by diffraction). In order to get a Rayleigh length of order one, we assume that the central wavenumber of the source is of order ε 4. In this regime the source has the form t 2.2) Ft, z, x) = f ε 4, x ) ε 2 δz z 0 )e z, where e z is the unit vector pointing in the z-direction, z 0 > L is the source position, and ft, x) is the normalized source shape function. We re-scale x/ε 2 x and set 2.3) p ε t, z, x) = pt, z, ε 2 x). The reader should keep in mind that thus, in the discussion below, when we refer to the transversal spatial parameter x it corresponds to ε 2 x in the original coordinates. We first consider the wave field in the homogeneous half-space z 0, which allows us to discuss the standard or homogeneous medium paraxial wave equation. The wave field satisfies the wave equation with the wave speed c 0 = K 0 /ρ 0. Following [17] in a different scaling) we make the

4 4 J. GARNIER AND K. SØLNA ansatz for the pressure field p ε t, z, x) = 1 2π 0 = ǎε 0 z ǎ ε 0k, z, x)e ik z ε 4 k, z, x)eik z ε 4 + ˇb ε 0 z + ˇb ε 0k, z, x)e ik z ε 4 ) e ic 0k t ε 4 dk, k, z, x)e ik where ǎ ε 0 and ˇb ε 0 are the complex amplitudes of the right- and left-propagating modes respectively, see Figure 1. They satisfy the coupled mode equations ǎ ε 0 z = i 2k xǎ ε 0 + e 2ik z ε 4 i 2k xˇb ε 0, ˇb ε 0 z = e2ik z ε 4, z ε 4 i 2k xǎ ε 0 i 2k xˇb ε 0, where x is the transverse Laplacian. In the limit ε 0, the cross terms proportional to e ±2ik z ε 4 ) average out to zero and we get the two uncoupled paraxial wave equations ǎ ε 0 z = i 2k xǎ ε 0, ˇb ε 0 z = i 2k xˇb ε 0. Taking into account the fact that there is no source in the half-space z 0, and therefore no right-going wave, we obtain the general expression of the wave in the left homogeneous half-space 2.4) p ε t, z, x) = 1 2π 1 2π ˇbε 0 k, z, x)e ik z ε 4 e ic 0k t ε 4 dk, z 0. Similarly, the wave fields in the homogeneous regions L, z 0 ) and z 0, ) have the forms p ε 1 2π ǎε t, z, x) = 2 k, z, x)e ik z ε 4 e ik t ε 4 dk, z > z 0, ǎ ε z 1 k, z, x)eik ε 4 + ˇb ε z ) 1 k, z, x)e ik ε 4 e ik t ε 4 dk, z L, z 0 ). Here we used the fact that there is no source and therefore no left-going wave in the region z > z 0, see Figure 1. We can also use the jump conditions across the source position z = z 0 to obtain the relations 2.5) 2.6) ˇbε 1 k, z 0, x) = 1 2 eik z 0 ε 4 ǎ ε 2k, z 0, x) ǎ ε 1k, z 0, x) = 1 2 e ik z 0 ε 4 ˇfk, x), ˇfk, x). By solving the paraxial wave equation for ˇb ε 1, we obtain the expression for the complex amplitude of the wave impinging on the random slab at z = L: 2.7) 2.8) ˇbε 1 k, L, x) = e ik z 0 ε 4 ˇb inc k, x), ˇbinc k, x) = 1 ˆfk, 22π) d κ)e i 2k κ 2 L z 0 )+iκ x dκ,

5 COUPLED PARAXIAL WAVE EQUATIONS IN RANDOM MEDIA 5 where the Fourier transforms are defined by 2.9) ˇfk, x) = ft, x)e ikt dt, ˆfk, κ) = ˇfk, x)e iκ x dx. The pressure field in the region z 0, L) can be written as: p ε t, z, x) = 1 ǎ ε k, z, x)e ik z ε 2π 4 + ˇb ε k, z, x)e ik ) z ε 4 e ik t ε 4 dk, 0 = ǎε z k, z, x)eik z ε 4 + ˇb ε z k, z, x)e ik The complex amplitudes ǎ ε and ˇb ε of the right- and left-propagating modes are given explicitly by 1 ǎ ε k, z, x) = 2ε 4 p ε t, z, x)e ik t ε 4 dt + 1 p ε ) t t, z, x)eik ε 2ik z 4 dt e ik z ε 4, 1 ˇbε k, z, x) = 2ε 4 p ε t, z, x)e ik t ε 4 dt 1 p ε ) t t, z, x)eik ε 2ik z 4 dt e ik z ε 4. We obtain the following mode coupling equations 2.10) ǎ ε ik z = 2.11) 2ε ν z ε 2, x ) ˇb ε z = z ik z ) e2ik ε 4 2ε ν ε 2, x z ε 4. + i ) 2k x ǎ ε + e 2ik z ik z ) ε 4 2ε ν ε 2, x + i ) 2k x ˇbε, + i ) ik z ) 2k x ǎ ε 2ε ν ε 2, x + i ) 2k x ˇbε. This system is valid in z 0, L) and it is complemented with the following boundary conditions at z = 0 and z = L: 2.12) 2.13) ˇbε k, z = L, x) = e ik z 0 ε 4 ˇb inc k, x), ǎ ε k, z = 0, x) = R 0ˇbε k, z = 0, x), where R 0 = Z 0 1)/Z 0 + 1) is the reflection coefficient of the interface at z = 0 and Z 0 = K 0 ρ 0 is the impedance of the left homogeneous half-space. These boundary conditions are obtained from the continuity relations of the fields p ε and e z u ε at z = 0 and z = L. The continuity relations also give the expressions for the complex amplitudes of the transmitted field ˆb ε 0 in the region z < 0 and of the reflected field in the region z > L: 2.14) 2.15) ˆbε 0 k, z = 0, x) = T 0ˇbε k, z = 0, x), â ε 1k, z = L, x) = ǎ ε k, z = L, x),

6 6 J. GARNIER AND K. SØLNA ǎ 00) = 0 ǎ ε 0) ǎ ε L) ǎ ε 1L) ǎ ε 1z 0) ǎ ε 2z 0) ˇb ε 00) ˇb ε 0) ˇbε L) ˇb ε 1L) ˇb ε 1z 0) ˇb ε 2z 0) = 0 z 0 L z 0 Fig 1. Boundary conditions for the modes in the presence of an interface at z = 0, a random slab 0, L), and a source at z = z 0. where T 0 = 2Z 1/2 0 /1 + Z 0 ) is the transmission coefficient of the interface at z = 0. If there is no impedance contrast Z 0 = 1, then T 0 = 1 and R 0 = 0 and the boundary condition 2.13) reads ǎ ε k, z = 0, x) = 0. This is the radiation condition expressing the fact that there is no wave incoming from. We now make use of an invariant imbedding step and introduce transmission and reflection operators. First, we define the lateral Fourier modes 2.16) â ε k, z, κ) = ǎ ε k, z, x)e iκ x dx, ˆbε k, z, κ) = ˇbε k, z, x)e iκ x dx, and make the ansatz 2.17) ˆbε k, 0, κ) = â ε k, z, κ) = T ε k, z, κ, κ )ˆb ε k, z, κ )dκ, R ε k, z, κ, κ )ˆb ε k, z, κ )dκ. Using the mode coupling equations ) we find that the operators T ε and R ε satisfy d dz R ε k, z, κ, κ ) = e 2ikz ε 4 L ε k, z, κ, κ ) +e 2ikz ε 4 R ε k, z, κ, κ 1 ) L ε k, z, κ 1, κ 2 ) R ε k, z, κ 2, κ ) dκ 1 dκ 2 + L ε k, z, κ, κ 1 ) R ε k, z, κ 1, κ )dκ ) + R ε k, z, κ, κ 1 ) L ε k, z, κ1, κ ) dκ 1, d dz T ε k, z, κ, κ ) = T ε k, z, κ, κ 1 ) L ε k, z, κ1, κ ) dκ ) +e 2ikz ε 4 T ε k, z, κ, κ 1 ) L ε k, z, κ 1, κ 2 ) R ε k, z, κ 2, κ ) dκ 1 dκ 2,

7 COUPLED PARAXIAL WAVE EQUATIONS IN RANDOM MEDIA 7 where we have defined 2.20) L ε k, z, κ 1, κ 2 ) = i 2k κ 1 2 δκ 1 κ 2 ) + ik z ) 22π) d ε ˆν ε 2, κ 1 κ 2. with ˆνz, κ) the partial Fourier transform in x) of νz, x). This system is complemented with the initial conditions at z = 0: R ε k, z = 0, κ, κ ) = R 0 δκ κ ), T ε k, z = 0, κ, κ ) = T 0 δκ κ ). The transmission and reflection operators evaluated at z = L carry all the relevant information about the random medium from the point of view of the transmitted and reflected waves, which are our main quantities of interest. Our objective in the next sections is to characterize the transmitted wave field p ε trs, x) = p ε z 0 + ε 4 s, z = 0, x) ) = T ε 2π) d+1 k, L, κ, κ )ˆb inc k, κ )dκ e iκ x ks) dκdk, and the reflected wave field p ε refs, x) = p ε z 0 + L + ε 4 s, z = L, x) ) = R ε 2π) d+1 k, L, κ, κ )ˆb inc k, κ )dκ e iκ x ks) dκdk. Note that the wave field fronts are observed on the time scale of the source and around their respective expected arrival times z 0 for the transmitted wave, and L+z 0 for the reflected wave, which corresponds to the travel time to go from the source at z = z 0 to the interface at z = 0 and back at the surface at z = L). 3. The Random Schrödinger Model Statement of the Main Result. We consider the transmitted and reflected fields p ε tr and p ε tr defined by ) and use diffusion approximation theorems to identify a random Schrödinger model that derives from a family of convergence determining moments. The main result is the following. Proposition 3.1. The processes p ε trs, x), p ε ref s, x)) s R,x R d converge in distribution as ε 0 in the space C 0 R, L 2 R d, R 2 )) L 2 R, L 2 R d, R 2 ))

8 8 J. GARNIER AND K. SØLNA to the limit process p tr s, x), p ref s, x)) s R,x R d 3.1) 3.2) p tr s, x) = 1 2π p ref s, x) = 1 2π Ť k, L, x, x )ˇb inc k, x )dx e iks dk, Řk, L, x, x )ˇb inc k, x )dx e iks dk. Here C 0 R, L 2 R d, R 2 )) is the space of continuous functions in s) with values in L 2 R d, R 2 ) and L 2 R, L 2 R d, R 2 )) = L 2 R R d, R 2 ). The operators Ť k, z, x, x ) and Řk, z, x, x ) are the solutions of the following Itô- Schrödinger diffusion models 3.3) 3.4) dť k, z, x, x ) = i 2k x Ť k, z, x, x )dz + ik 2 Ť k, z, x, x ) dbz, x ), dřk, z, x, x ) = i 2k x + x ) Řk, z, x, x ) dz with the initial conditions + ik 2 Řk, z, x, x ) dbz, x) + dbz, x ) ), Ť k, 0, x, x ) = T 0 δx x ), Řk, 0, x, x ) = R 0 δx x ). The symbol stands for the Stratonovich stochastic integral, Bz, x) is a real-valued Brownian field with covariance 3.5) E[Bz 1, x 1 )Bz 2, x 2 )] = min{z 1, z 2 }C 0 x 1 x 2 ), and we have used the notations 3.6) 3.7) Cz, x) = E[νz + z, x + x)νz, x )], C 0 x) = Cz, x)dz. The moments of the finite-dimensional distributions also converge, in the sense that 3.8) [ q q ] E p ε trs j, x j ) m j p ε ref s j, x j ) m j [ ε 0 q q ] E p tr s j, x j ) m j p ref s j, x j ) m j, for any q, q N, s j ),...,q R q, s j ),..., q R q, x j ),...,q R dq, x j ),..., q R d q, m j ),...,q N q, and m j ),..., q N q.

9 COUPLED PARAXIAL WAVE EQUATIONS IN RANDOM MEDIA 9 In [8] the existence and uniqueness has been established for the random process V k z, x) = Ť k, z, x, x )φx )dx, for a test function φ with unit L 2 R d )-norm, in the case T 0 = 1. It is shown that the process V k z, x) is a continuous Markov diffusion process on the unit ball of L 2 R d, C). The moment equations moreover satisfy a closed system at each order [14]. The analysis can be readily extended to the pair p tr, p ref ) defined in terms of Ť, Ř) and can be carried out jointly for all frequencies k in an interval bounded away from 0 and infinity. We then get that the processes p tr and p ref have constant L 2 R R d )-norms, so that the conservation of energy relation holds: 3.9) p tr s, x) 2 + p ref s, x) 2 dsdx = b inc s, x) 2 dsdx, where we have also used the identity R T 2 0 = 1. The main steps of the proof of Proposition 3.1 given in the next section are 1) Tightness and a priori estimates for p ε tr, p ε ref ), 2) Convergence of the finite-dimensional distributions of the process p ε tr, p ε ref ), using the convergence of specific moments of the reflection and transmission operators T ε and R ε. It is important to note that the reflection and transmission operators T ε and R ε themselves do not converge to T and R, but only certain moments expectations of products of components with distinct frequencies), which are those needed to ensure the convergence of the transmitted and reflected fields. This approach is similar to the one used to prove the O Doherty-Anstey theory in one-dimensional random media [7, 13] and to study the second-order statistics of the wave backscattered by a threedimensional random medium [16, 17]. The limit moments are characterized by the system A.2) of coupled equations, which gives the practical way to compute all the moments of the reflected and transmitted wave fields. Some specific applications will be given in Sections ) Use of Itô s lemma for Hilbert-space-valued processes [19, Theorem 2.4] in order to check that the specific moments of the reflection and transmission operators T and R given by ) are solutions of the system A.2) of coupled equations Proof of Proposition 3.1. This section is devoted to the proof of Proposition 3.1. We shall use a technique similar to the one presented in [13] in the case of randomly layered media.

10 10 J. GARNIER AND K. SØLNA Step 1. A priori estimates. From ) we can check that, for any k, the integral ǎ ε k, z, x) 2 ˇb ε k, z, x) 2 dx, is conserved in z. Applying this conservation relation at z = 0 and z = L, and taking into account the boundary conditions ), we obtain ǎ ε k, L, x) 2 dx + 1 R 2 0) ˇb ε k, 0, x) 2 dx = ˇb inc k, x) 2 dx. Using now ) and the identity R T 0 2 = 1 we obtain 3.10) ǎ ε k, L, x) 2 dx + ˇb ε 0k, 0, x) 2 dx = ˇb inc k, x) 2 dx, which expresses the fact that the power of the incoming wave is fully recovered by the transmitted and reflected waves. Integrating in k and using Parseval s equality establishes the total energy conservation relation 3.11) p ε refs, x) 2 dxds + p ε trs, x) 2 dxds = b inc s, x) 2 dxds. We first state a priori estimates for our quantities of interest. Lemma 3.1. There exists C > 0 such that, uniformly in ε and in s 0, s 1, 3.12) p ε trs 0, x) 2 dx C and p ε trs 1, x) p ε trs 0, x) 2 dx C s 1 s 0. The same estimate holds true for p ε ref. Proof. Using the Sobolev s embedding L R) H 1 R), there exists a constant C sob such that, for any x, sup p ε trs, x) 2 C sob p ε tr, x) H 1 R,R) = C sob 1 + k 2 ) ˇb ε k, 0, x) 2 dk, s 2π where we have also used Parseval s equality. Integrating in x and using the conservation equation 3.10) yields the first result of the lemma: sup p ε trs, x) 2 dx sup p ε trs, x) 2 dx C sob 1+k 2 ) ˇb inc k, x) 2 dxdk. s s 2π By Cauchy-Schwarz inequality, we have p ε trs 1, x) p ε trs 0, x) 2 s1 = p ε tr s, x)ds s s 0 s 1 s 0 2 s1 s 0 p ε tr s s, x) 2 ds. s1 ds pε tr s 0 s s, x) 2 ds

11 COUPLED PARAXIAL WAVE EQUATIONS IN RANDOM MEDIA 11 The integral in x of the last term in the inequality can be bounded uniformly as above. The reflected field can be analyzed in the same way, which completes the proof. Step 2. The moments of the finite-dimensional distribution of p ε trs, x), p ε ref s, x)) converge to those of p tr s, x), p ref s, x)). The general moment 3.8) of p ε trs, x) can be expressed as the multiple integral q q ] E[ p ε trs j, x j ) m j p ε ref s j, x j ) m 1 j = 2π) N+M)d+1) q m h q dκ m h h,jdκ h,j dk h,j d κ h,jd κ h,j d k h,j h=1 h=1 ˆbinc ) k h,j, κ h,j)e iκ h,j x h k h,j s h ) ˆbinc ) k h,j, κ h,j)e i κ h,j x h k h,j s h ) h,j h,j [ E T ε k h,j, L, κ h,j, κ h,j) R ε ] k h,j, L, κ h,j, κ h,j), h,j h,j for N = q h=1 m h and M = q h=1 m h. Therefore, the convergence of the general moment of the transmitted and reflected wave fields in the whitenoise limit will follow from the convergence of the following specific moments E[I ε L)] of the transmission and reflection operators, where 3.13) I ε L) = N T ε M k j, L, κ j, κ j) R ε k j, L, κ j, κ j). We call these moments specific because we restrict our attention to the case in which the frequencies k j, k j are all distinct. We use diffusion approximation theorems to obtain equations for the moments I ε in the limit ε 0. In Appendix A we show that lim E[ I ε L) ] [ N = E ε 0 T k j, L, κ j, κ j) M ] R k j, L, κ j, κ j), when the right-hand side expectation is taken with respect to the following

12 12 J. GARNIER AND K. SØLNA Itô-Schrödinger model for the transmission and reflection operators: d T k, z, κ, κ ) = i κ ) + ik 22π) d 2k T k, z, κ, κ )dz k2 C 0 0) 8 T k, z, κ, κ 1 )d ˆBz, κ 1 κ )dκ 1, T k, z, κ, κ )dz d Rk, z, κ, κ ) = i κ 2 + κ 2 ) Rk, z, κ, κ )dz k2 C 0 0) Rk, z, κ, κ )dz 2k 4 k2 42π) d Ĉ 0 κ 1 ) Rk, z, κ κ 1, κ κ 1 )dκ 1 dz + ik Rk, 22π) d z, κ, κ 1 ) d ˆBz, κ 1 κ ) + Rk, z, κ 1, κ ) d ˆBz, ) 3.15) κ κ 1 ) dκ 1, with the initial conditions T k, 0, κ, κ ) = T 0 δκ κ ) and Rk, 0, κ, κ ) = R 0 δκ κ ). Here we have used the the notations ) and the Brownian field ˆB is the partial Fourier transform of the field B so that it has the following operator valued spatial covariance 3.16) E[ ˆBz 1, κ 1 ) ˆBz 2, κ 2 )] = min{z 1, z 2 }2π) dĉ0κ 1 )δκ 1 + κ 2 ), where 3.17) Ĉ0κ) = Cz, x)e iκ x dxdz. R d Consider next the reflection operator in the original spatial variables : Ť k, z, x, x ) = 1 2π) d e iκ x κ x ) 3.18) T k, z, κ, κ )dκdκ, 3.19) Řk, z, x, x ) = 1 2π) d e iκ x κ x ) Rk, z, κ, κ )dκdκ. Then we find that this operator is weakly characterized by the following Itô-Schrödinger diffusion dť k, z, x, x ) = i 2k x Ť k, z, x, x )dz k2 C 0 0) Ť k, z, x, x )dz 8 + ik 2 Ť k, z, x, x )dbz, x ), dřk, z, x, x ) = i 2k x + x ) Řk, z, x, x )dz k2 C 0 0) + C 0 x x)) Řk, z, x, x )dz 4 + ik 2 Řk, z, x, x ) dbz, x) + dbz, x ) ).

13 COUPLED PARAXIAL WAVE EQUATIONS IN RANDOM MEDIA 13 In Stratonovich form this diffusion model becomes 3.3). This proves the last statement of the proposition the convergence of the moments). Step 3. Convergence of p ε tr, p ε ref ) to p tr, p ref ) in C 0 R, L 2 wr d, R 2 )) L 2 wr, L 2 wr d, R 2 )). Here L 2 w is the L 2 space equipped with the weak topology. Lemma 3.1 shows that the process p ε tr, p ε ref ) is tight in C0 R, L 2 wr d, R 2 )). Moreover, the first estimate in the lemma shows that, for any L 2 R d, R)-functions φ,ψ, the random processes Xφs) ε = p ε trs, x)φx)dx, Yψs) ε = p ε refs, x)ψx)dx, are uniformly bounded. Therefore, the finite-dimensional distributions are characterized by the moments of the form [ q E Xφ ε j s j ) m j q ] Yψ ε j s j ) m j, where q, q N, m j, m j N, s j, s j R, φ j, ψ j L 2 R d, R). These moments can be written as mutiple integrals [ q E Xφ ε j s j ) m j q q m h ] Yψ ε j s j ) m j = dκ h,jdκ h,j dk h,j 1 2π) N+M)d+1) h=1 h=1 ˆbinc k h,j, κ h,j) ˆφ ) h κ h,j )e ik h,js h q m h d κ h,jd κ h,j d k h,j h,j h,j [ E T ε k h,j, L, κ h,j, κ h,j) R ε ] k h,j, L, κ h,j, κ h,j), h,j h,j ˆbinc k h,j, κ h,j) ˆψ h κ h,j )e i k h,j s h ) for N = q h=1 m h and M = q h=1 m h. Note that only specific moments of quantities of the form 3.13) appear i.e., moments of products of the transmission and reflection operators at distinct k). The convergence of these specific moments therefore implies the convergence of the finite-dimensional distributions, hence the weak convergence in C 0 R, L 2 wr d, R 2 )). Furthermore, the estimate 3.11) shows that the processes are tight in L 2 wr, L 2 wr d, R 2 )) the unit ball is compact in the weak topology). This proves the weak convergence in L 2 wr, L 2 wr d, R 2 )).

14 14 J. GARNIER AND K. SØLNA Step 4. Convergence of p ε tr, p ε ref ) to p tr, p ref ) in C 0 R, L 2 R d, R 2 )) L 2 R, L 2 R d, R 2 )). Here L 2 is the L 2 space equipped with the strong topology. Since the convergence has been proved in the weak topology, it is sufficient to show that the L 2 -norm is preserved. On the one hand, the L 2 -norms of the processes p ε tr, p ε ref ) are deterministic, independent of ε, and given by 3.11). On the other hand, the limit process p tr, p ref ) has constant L 2 -norm given by 3.9). This proves the first statement of the proposition and completes its proof. 4. The Wigner Distributions. We first introduce the dimensionless autocorrelation function C of the fluctuations of the random medium z Cz, x) = σ 2 C, x ), l z l x where σ is the standard deviation of the fluctuations of the random medium, l z respectively l x ) is the longitudinal respectively transverse) correlation radius of the medium. With this representation we have C 0 x) = σ 2 l z C 0 x l x ), Ĉ0u) = σ 2 l z l d xĉ0ul x ). We assume next that the power spectral density Ĉ0u) decays fast enough so that u 2 Ĉ 0 u)du is finite. This means that the autocorrelation function C 0 x) is at least twice differentiable at x = 0, which corresponds to a smooth random medium. For simplicity, we assume also that the random fluctuations are isotropic in the transverse directions, in the sense that the autocorrelation function C 0 x) depends only on x The Wigner Distribution of the Transmitted Wave. We now consider two frequencies k 1 and k 2 in a frequency band centered at k and we define the two-frequency Wigner distribution of the transmission operator by 4.1) Wk T 1,k 2 z, x, x, q, q ) = [ E Ť k 1, z, Ť k 2, z, k x + y ), k1 2 k x y ), k2 2 e iq y+q y ) k k1 k k2 x + y ) ) 2 x y ) )] dydy. 2

15 COUPLED PARAXIAL WAVE EQUATIONS IN RANDOM MEDIA 15 Using the stochastic equation 3.3) and Itô s formula, we find that the Wigner distribution satisfies the closed system Wk T 1,k 2 + q z k x W k T 1,k 2 = C 00)k1 2 + k2 2 ) Wk T 8 1,k 2 + k 1k 2 42π) d Ĉ 0 u) Wk T 1,k 2 z, x, x, q, q 1 k k + )u )e iu x ) k k2 k k1 du, 2 k1 k2 starting from W T k 1,k 2 z = 0, x, x, q, q ) = T 2 0 4π 2 k 1 k 2 /k 2 ) d/2 δx x )δq + q ). It is possible to solve this system and to find an integral representation of the two-frequency Wigner distribution by using the approach of [9]. However, we aim at focusing on spatial aspects in the next sections, and we shall simplify the algebra by assuming that the bandwidth B of the incoming wave with carrier wavenumber k 0 is small. To describe this regime it is convenient to introduce 4.2) β = σ2 k 2 0 Ll z 4, α = L k 0 lx 2, α 0 = L k 0 r0 2, where r 0 is the initial beam width, β describes the intensity of random scattering, while α and α 0 represent the intensities of lateral scattering on respectively the scales of the medium variations and the input beam. Note that these parameters correspond to inverse Fresnel numbers up to a factor 2π) relative to respectively the lateral medium correlation scale and the aperture, below we shall consider explicitly the case with small medium Fresnel number. We assume that the bandwidth B of the incoming wave is small in the sense that 4.3) B B c, B c := k 0 min 1, α 1, α 1 0, β 1). In this regime we can approximate the two frequency Wigner distribution by its behavior at the carrier. That is, if k 1, k 2 lie in the spectrum of the incoming wave, the two-frequency Wigner distribution W T k 1,k 2 can be approximated by the simplified Wigner distribution W T that depends only on the carrier wavenumber k 0 and not on the lag k 1 k 2. This Wigner distribution can be written in the form W T z, x, x, q, q ) = T 2 0 2π) d W T z L, x r 0, x r 0, ql x, q l x ),

16 16 J. GARNIER AND K. SØLNA where W T satisfies 4.4) WT ζ + αl x r 0 q x W T = β 2π) d [ Ĉ 0 u) W T q u ) W T q )] du, starting from W T ζ = 0, x, x, q, q ) = δx x )δq + q ). We remark that that these are in fact the classical equations of radiative transport for angularly resolved wave energy density [22]. In this context we have that βc 0 0) is the total scattering cross-section, βĉ0 ) the differential scattering crosssection describing coupling of modes depending on their relative propagation directions and αl x /r 0 )q a transport or velocity vector. It is clear from this equation that diffractive effects characterized by the term q x ) are of order one if αl x /r 0 1 or equivalently k 0 r 0 l x L. In terms of Fresnel numbers, diffractive effects are of order one if α e L k 0 a 2 e 1, where we have defined the effective aperture a e by: a e = l x r 0. In Section 5 we shall see that a physically important and mathematically interesting regime corresponds to: α 0 α e 1 α, so that from the point of view of the medium Fresnel number we are in a Fraunhofer diffraction scaling, while from the point of view of the source Fresnel number we are in a Fresnel diffraction scaling and finally from the point of view of the effective Fresnel number, 1/2πα e ), we are in a general or scalar diffraction theory setup. By taking a Fourier transform in q and x, we obtain a transport equation that can be integrated and we find the following integral representation for W T : 4.5) W T ζ, x, x, q, q ) = T 0 2 4π 2 α) d e iq +q) η 1 i r 0 αlx x x)+qζ ) η 2 e β ζ 0 C 0η 1 +η 2 ζ ) C 0 0)dζ dη 1 dη 2. This expression will be used in Section 5.1 to compute and discuss the twopoint statistics of the transmitted field.

17 COUPLED PARAXIAL WAVE EQUATIONS IN RANDOM MEDIA The Wigner Distribution for the Reflected Wave. We define the twofrequency Wigner distribution of the reflection operator by Wk R 1,k 2 z, x, x, q, q ) = e iq y+q y ) [ k E Ř k 1, z, x + y ) k, x + y ) ) k1 2 k1 2 k Ř k 2, z, x y ) k, x y ) )] 4.6) dydy. k2 2 k2 2 If the bandwidth of the incoming wave satisfies 4.3) and if k 1, k 2 lie in the spectrum of the wave, then we find by using 3.4) that the two-frequency Wigner distribution Wk R 1,k 2 can be approximated by the simplified Wigner distribution W R that depends only on the carrier wavenumber k 0 and not on the lag k 1 k 2. This Wigner distribution satisfies the closed system W R + q x W R + q x W R = k2 0 z k 0 k 0 42π) d Ĉ 0 u) [ W R z, x, x, q u, q ) + W R z, x, x, q, q u ) +2W R z, x, x, q 1 2 u, q 1 2 u ) cos u x x ) ) 2W R z, x, x, q 1 2 u, q u ) cos u x x ) ) 2W R z, x, x, q, q )] du, starting from W R z = 0, x, x, q, q ) = R 2 0 2π)d δx x )δq + q ). We now cast the Wigner distribution in a suitable dimensionless form. We consider the following Fourier transform V R of the Wigner distribution W R : W R z, x, x, q, q ) = 1 2π) d V R z, q + q ), q q, s e is x x) ds, 2 which we introduce because the stationary maps that we will identify in Lemma 4.1, in the asymptotic regime α, have simple representations in this new frame. Note also that this ansatz incorporates the fact that W R does not depend on x + x, only on x x, q, and q, which follows since we model the random medium as being stationary. The Fourier-transformed operator V R z, q, r, s) has the form V R z, q, r, s) = R 2 0πl x ) d e iz r s k 0 V R z ) L, ql x, rl x, sl x,

18 18 J. GARNIER AND K. SØLNA where V R is the solution of the dimensionless system V R ζ = β 2π) d 4.7) Ĉ 0 u) [V R ζ, q 1 ) 2 u, r u, s e iαs uζ +V R ζ, q 1 2 u, r + u, s ) e iαs uζ + V R ζ, q 1 2 u, r, s u ) e iαr uζ +V R ζ, q 1 2 u, r, s + u ) e iαr uζ 2V R ζ, q, r, s ) V R ζ, q 1 ) 2 u, r u, s + u e iα[r s) u u 2 ]ζ V R ζ, q 1 ) ] 2 u, r u, s u e iα[r+s) u+ u 2 ]ζ du, starting from V R ζ, q, r, s) = δq). The parameters α and β are given by 4.2). We want now to analyze the regime in which the transverse correlation length l x of the medium is smaller than the beam width r 0. More exactly, we assume from now on in this section that 1) r 0 l x, which means that the transverse correlation length of the medium is small, 2) k 0 r 0 l x L, which means that diffractive effects are of order one. These two conditions are equivalent to α 0 α e 1 α. Note that in the previous section we established the fact that diffraction plays a role for the transmitted wave for a propagation distance L of the order of k 0 r 0 l x, which is smaller than the usual Rayleigh length k 0 r0 2. This is a well-known result [18], and we shall deduce it for the reflected field in the analytic framework that we have set forth. The rapid transverse variations regime is particularly interesting to study because W R has a multi-scale behavior. In 4.7) this regime gives rise to rapid phases. The following proposition describes the asymptotic behavior of V R as α. The presence of singular layers at r = 0 and at s = 0 requires particular attention and is responsible for instance for the enhanced backscattering phenomenon studied in Section 5.3. The situation with α large corresponds to a strong diffraction situation, at the scale of the lateral medium fluctuations. In general part 1) in Lemma 4.1) the intensity of the reflection operator decays exponentially according to the parameter βc 0 0) corresponding to the total scattering cross section. This decay follows from a partial loss of coherence by random forward scattering. However, as articulated in parts 2) and 3) of the lemma below the coupling of wave modes depends on the full medium autocorrelation function if we look at nearby specular reflection or small spatial offset frequencies. This coupling will be

19 COUPLED PARAXIAL WAVE EQUATIONS IN RANDOM MEDIA 19 important when we analyze enhanced backscattering in Section 5.3. Lemma ) For any r 0, s 0: 4.8) V R ζ, q, r, s) α δq)e 2βC 00)ζ. 2) For any s 0 we have V R ζ, q, r α solution of 4.9) V R r ζ = 2β 2π) d and is given explicitly by 4.10) Vr R ζ, q) = 1 2π) d, s) α V R r ζ, q) where V R r ζ, q) is [ Ĉ 0 u) Vr ζ, R q 1 ) 2 u cos r uζ ) Vr R ) ] ζ, q du, e iq u e β ζ 0 C 0 u 2 +rζ )+C 0 u 2 rζ ) 2C 0 0)dζ du. Similarly, for any r 0 we have V R ζ, q, r, s α ) α Vs R ζ, q). 3) For any r and s we have 4.11) V R ζ, q, r α, s ) α Vr R ζ, q) + Vs R ζ, q) δq)e 2βC00)ζ. α Proof. In case 1), the rapid phases cancel the contributions of all but the term V R ζ, q, r, s) in 4.7), and we get V R ζ = 2 β 2π) d Ĉ 0 u)v R du = 2βC 0 0)V R, which gives 4.8). In case 2), we obtain in the limit α the simplified system ṼR r ζ = β 2π) d +ṼR r [ Ĉ 0 u) Ṽr ζ, R q 1 ) 2 u, s u e ir uζ ζ, q 1 2 u, s + u ) e ir uζ 2ṼR r ζ, q, s ) ] du. We then Fourier transform this equation in q and s, and we obtain that the solution does not depend on s, that it satisfies 4.9), and that it is given by 4.10). In case 3) we obtain the simplified system for V R r,s ζ, q ) = limα V R ζ, q, r α, s α) : V R r,s ζ = 2β 2π) d +V R r [ Ĉ 0 u) Vs ζ, R q 1 ) 2 u cos s uζ ) ζ, q 1 2 u ) cos r uζ ) V R r,s ζ, q ) ] du.

20 20 J. GARNIER AND K. SØLNA Using the equation 4.9) satisfied by V R s and V R r, we get V R r,s ζ = VR r ζ + VR [ ] s ζ + 2βC 00) Vr R + Vs R Vr,s R, which yields 4.11). 5. Two-point Statistics of the Transmitted and Reflected Fields The Transmitted Field. The results of Section 4.1 allow us to compute the two-point statistics of the transmitted field. We assume that a) the pulse has carrier frequency k 0 and it is narrowband in the sense that it satisfies 4.3), b) the input beam spatial profile is Gaussian with radius r 0, 5.1) b inc t, x) = f 0 t)e ik0t exp x 2 ) r0 2, suppressing the complex conjugate part here and below). c) the transverse correlation radius l x of the random fluctuations of the medium is much smaller than r 0 and k 0 r 0 l x L. As we have discussed after 4.4), this last condition ensures that diffractive effects are of order one. Under a-b), we find that the autocorrelation function of the transmitted field defined by [ A tr s, t, x, y) = lim E p ε tr s + t ε 0 2, x + y ) p ε tr s t 2 2, x y ) ] 2 has the form 5.2) A tr s, t, x, y) = T 2 0 f 0 Under a-c), we obtain s + t 2 e 1 ηl 2r 0 2 +y 2 r2 0 η 2 k 0 8 )f 0 s t ) e ik 0t r π e iη x k 0 2 e 4 L C 0 0 η z +y k 0 ) d/2 ) C 0 0)dz dη. 5.3) A tr s, t, x, y) = T0 2 f 0 s + t )f 0 s t ) e ik 0t r π e r 2 0 η 2 y 2 8 2r 0 2 e iη x k 0 2 e 4 ) d/2 L ) C 0 0 η z +y C k 0 0)dz 0 dη.

21 COUPLED PARAXIAL WAVE EQUATIONS IN RANDOM MEDIA 21 If moreover, random scattering is strong, in the sense that β 1, or equivalently k 2 0 C 00)L 1, and if the autocorrelation function of the random fluctuations of the medium is twice differentiable at zero: 5.4) C 0 x) = C 0 0) D 2 x 2 +o x 2 ), D = 1 d C 00) = σ2 l z dlx 2 C 0 0), then we obtain that the autocorrelation function has the Gaussian shape 5.5) A tr s, t, x, y) = T 2 0 f 0 s + t 2 )f 0 s t 2 exp 2 x 2 r T L) 2 ) e ik 0t r 0 r T L) ) d y 2 2ρ T L) 2 + i x y ) χ T L) 2. The beam radius r T L), the correlation radius ρ T L) and the parameter χ T L) are characterized by 5.6) r T L) = r DL3, 5.7) 5.8) ρ T L) = r 0 χ T L) = 3r DL3 3r k2 0 r2 0 DL 4 + k2 0 D2 L 4 48 r T L) k0 DL 2 /2, where we have taken into account that k 0 r0 2 L in the considered regime. Note in particular that the beam width increases at the anomalous rate L 3/2 which was first obtained in the physical literature in Ref. [12] and confirmed mathematically in Ref. [10]). Furthermore, the lateral correlation radius decays to zero, which means that the beam becomes partially coherent. We finally remark that these results hold true in the case with a smooth random medium with C 0 twice differentiable at 0 as in 5.4)), the situation with a rough random medium will be addressed elsewhere The Reflected Field. Under a-b), the limit autocorrelation function of the reflected field defined by [ p ε ref A ref s, t, x, y) = lim ε 0 E s + t 2, x + y ) p ε ref s t 2 2, x y ) ] 2 has the form A ref s, t, x, y) = R 2 0f 0 s + t )f 0 s t ) e ik 0t r 2 ) d/ π 5.9), dη 1 dη 2 dη 3 e il k 0 η 3 η 2 r2 0 8 η η 2 2η 1 2 ) e iη 3 x+iy η 1 + η 2 2 ) V R 1, η 1, η 2, η 3 ),

22 22 J. GARNIER AND K. SØLNA where the function V R is the solution of the system 4.7). Under a-c) the asymptotic behavior of the function V R is determined by Lemma 4.1 and we find that the autocorrelation is given by 5.10) A ref s, t, x, y) = R 2 0f 0 s + t )f 0 s t ) e ik 0t r π e r 2 0 η 2 y 2 8 2r 0 2 e iη x k 0 2 e 4 2L 0 ) d/2 C 0 η z k 0 +y) C 0 0)dz dη. This is exactly the form 5.3) of the autocorrelation function of the transmitted wave, upon the substitution 2L for L. This shows that we would have obtained the same result if we had assumed that the backward propagation was independent of the forward propagation. The independent approach is valid in the regime in which the beam width is much larger than the transverse correlation radius of the fluctuations of the random medium, but it is not valid in other regimes, as can be seen by comparing the full expressions 5.2) and 5.9) Enhanced Backscattering. The comparison of the autocorrelation function of the reflected wave and that of the transmitted wave for the propagation distance 2L shows that, in the regime α 1, there is no coherent effect building up between the forward and backward propagations. However, corrective terms show that there are some residual effects. In particular, we would like to show that the reflected intensity exhibits a singular picture in a very narrow cone, of angular width of order α 1, around the backscattered direction. This phenomenon called enhanced backscattering or weak localization is widely discussed in the physical literature [3, 21]. The physical observation is that, for an incoming quasi-monochromatic quasi-plane wave, the mean reflected power has a local maximum in the backscattered direction, which is twice as large as the mean reflected power in the other directions. In this section, we assume that the incoming wave has the form b inc t, x) = ft)e ik 0t g inc x), that it is narrowband in the sense that it satisfies 4.3), and that it is nearly a plane wave, in the sense that ĝ inc κ) is concentrated at some κ inc assumed to be different from the normal incident vector 0). By concentrated we mean that the angular width of the incoming beam is smaller than α 1. The

23 COUPLED PARAXIAL WAVE EQUATIONS IN RANDOM MEDIA 23 reflected signal in the direction κ 0 is ˇp ε refs, κ 0 ) = p ε refs, x)e iκ0 x dx = 1 R ε k, L, κ 0, κ )ˆb inc k, κ )e iks dk. 2π The moment of the square modulus of ˇp ε ref s, κ 0) only involves specific moments of quantities of the form 3.13) with distinct k). Therefore this moment converges to the one of the limit process ˇp ref s, κ 0 ) defined as the Fourier transform in x of p ref s, x) given by 3.2). This means that the mean reflected intensity in the direction κ 0 converges to [ E ˇp ε refs, κ 0 ) 2] ε 0 R 2 0 fs) 2 I R κ 0 ), I R κ 0 ) = 2 d l d x V R 1, κ 0 κ ) 1 l x, κ 0 + κ 2 1)l x, 0 ĝ inc κ 1) 2 dκ 1. Using the fact that ĝ inc κ) is concentrated at κ inc, we get 5.11) I R κ 0 ) = P V R 1, κ 0 κ ) inc l x, κ 0 + κ inc )l x, 0, 2 where P = 2 d lx d ĝinc κ 1 ) 2 dκ 1. This formula gives the mean reflected intensity in the direction κ 0 and is valid for arbitrary values of α and β. Let us consider the regime α 1. The mean reflected intensity far enough from the backscattered direction κ inc is of the form I R κ 0 ) = P V R 0 1, κ 0 κ inc 2 l x ), for κ 0 + κ inc l x α 1, where we have used the second point of Lemma 4.1. In a narrow angular cone around the backscattered direction κ inc, the reflected intensity is locally larger: [ ] I R κ inc + α 1 κ) = P V0 R 1, κ inc l x ) + Vκl R x 1, κ inc l x ), where we have used the third point of Lemma 4.1. If we assume, additionally, that β 1, then we have I R κ 0 ) = P πdβ) d/2 e κ 0 κ inc 2 l 2 x 4Dβ, for κ 0 + κ inc l x α 1 where D is the dimensionless version of D given by 5.4): D = σ 2 l z lx 2 D. This formula gives the width of the diffusion cone around the specular direction κ inc : 5.12) κ spec = 2 Dβ l x = Dσk0 Llz l x = DLk 0.

24 24 J. GARNIER AND K. SØLNA On the top of this broad cone, we have a narrow cone of relative maximum equal to 2 centered along the backscattered direction κ inc : I R κ inc + α 1 κ) = P πdβ) d/2 e κ inc 2 l 2 x ] Dβ [1 + e Dβ 3 κ 2 lx 2. This shows that the width of the enhanced backscattering cone is 5.13) κ EBC = 3 = 2 3lx l x Dβα Dσ lz L = DL 3 Note that the angular width θ EBC = κ EBC /k 0 of the cone is proportional to the wavelength, as predicted by physical arguments based on diagrammatic expansions [21]. Acknowledgements. This work was supported by ONR grant N and DARPA grant N K Sølna was supported by NSF grant DMS and the Sloan Foundation. APPENDIX A: DERIVATION OF THE LIMIT MOMENT EQUATIONS The purpose of this appendix is to compute the limit of the expectation of 3.13) as ε 0, for distinct frequencies k j, k j. Using 2.18) and 2.19) we find di ε dz z) = N { + A.1) N l=1 j T ε M k l, z, κ l, κ l) R ε k l, z, κ l, κ l) l=1 T ε k j, z, κ j, κ 1 ) L ε k j, z, κ 1, κ j)dκ 1 2ik j z + e ε 4 M N l=1 T ε k j, z, κ j, κ 1 ) L ε k j, z, κ 1, κ 2 ) R ε k j, z, κ 2, κ j) dκ 1 dκ 2 } T ε k l, z, κ l, κ l) {e 2i kj z ε 4 L ε k j, z, κ j, κ j) M l=1 j R ε k l, z, κ l, κ l) 2i k j z + e ε 4 R ε k j, z, κ j, κ 1 ) L ε k j, z, κ 1, κ 2 ) R ε k j, z, κ 2, κ j)dκ 1 dκ 2 + L ε k j, z, κ j, κ 1 ) R ε k j, z, κ 1, κ j)dκ 1 + R ε k j, z, κ j, κ 1 ) L ε k j, z, κ 1, κ j)dκ 1 }.

25 COUPLED PARAXIAL WAVE EQUATIONS IN RANDOM MEDIA 25 We next apply the diffusion approximation to get limit equations for the moments, see [13] for background material on and related applications of the diffusion approximation. Observe that the random coefficients are rapidly fluctuating in view of 2.20). Those coefficients that are of order ε 1 are centered and fluctuate on the scale ε 2, moreover they are assumed to be rapidly mixing, giving a white-noise scaling situation. Moreover, the rapid phase terms exp±2ikz/ε 4 ) lead to some cancellations between interacting terms. Here, the fact that the frequencies are distinct plays a key role. As a consequence, by applying diffusion approximation results, we obtain that the equations for the moments E[I ε ] in the limit ε 0: Īz) = lim ε 0 E[I ε z)]. We obtain from A.1) that Ī solves a system of integro-differential equations A.2) dī dz z) = i N κ j k j M κ j 2 + κ j 2 )Īz) k j )Īz) 1 82π) d C 00) N M kj k j 2 Ĉ 0 κ) 8 { N M k j k l Īκ j κ, κ l + κ) + 2 k j 2 Ī κ j κ, κ j κ) l j M + k j kl Ī κj κ, κ l κ) + Ī κ l κ, κ j κ) l j ) +Ī κ j κ, κ l + κ) + Ī κ j κ, κ l + κ) N M + 2 k j kl Īκ j κ, κ l κ) + Īκ j κ, κ l + κ)) } dκ, l=1 where we only write the shifted arguments for Ī. The initial conditions are Īk, k, κ, κ, κ, κ, z = 0) = T0 N N δκ j κ M j )RM 0 δ κ j κ j). Using in particular the relation [ z1 z2 ] E λ 1 s 1, κ 1 )λ 2 s 2, κ 2 )d ˆBs 1, κ 1 )d ˆBs 2, κ 2 )dκ 1 dκ 2 0 = 0 minz1,z 2 ) we can then verify that 0 [ N Īz) = E E [λ 1 s, κ)λ 2 s, κ)] 2π) dĉ0κ) dsdκ, T k j, L, κ j, κ j) M ] R k j, L, κ j, κ j),

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