Schrödinger equation with random potential

Transcription

1 Lille January, 213 Schrödinger equation with random potential Josselin Garnier Université Paris VII

2 Lille January, 213 Physical derivation of the paraxial wave equation Wave equation in homogeneous medium: z 2 + u 1 tu c 2 = 2 Fourier transform ûω, x, z = ut,x,ze iωt dt Helmholtz equation in homogeneous medium: 2 z + û+ ω2 c 2 û = with initial/incoming and boundary/radiation conditions. A particular solution: plane wave going into the z direction û = e i ω c z. Ansatz: slowly-varying envelope around a plane wave going into the z direction ûω,x,z = e i ω c zˆφω,x,z 2 z ˆφ+2i ω c zˆφ+ ˆφ = Forward-scattering approximation in direction z ˆφ slowly varying in z: 2 ω z ˆφ+2i zˆφ+ ˆφ = c with initial condition ˆφω,x,z = = ˆφ ω,x.

3 Mathematical derivation: scaling assumptions 2 z + û+ ω2 c 2 1+µx,z û = 1 the correlation length of the medium l c is smaller than the propagation distance L: ε 2 = l c L 2 the transverse width of the source R and the correlation length of the medium l c are of the same order: R L ε2 3 the wavelength λ = 2πc /ω is smaller than the propagation distance high-frequency regime: λ L ε4 Motivation: In the absence of random fluctuations, if one wants diffraction effects of order one for the propagation distance L: 2i ω c zˆφ }{{} + ˆφ }{{} 1 λ L 1 R 2 = then one must have Lλ R 2 = Lλ L 2 ε 4 = λ /L ε 4. Lille January, 213

4 [1] J. Garnier and K. Sølna, Ann. Appl. Probab. 19, Mathematical derivation: sketch of proof Consider the time-harmonic form of the scalar wave equation x = x,z 2 z + û+ ω2 c 2 1+µx,z û =. Consider the paraxial regime λ l c L. More precisely, in the scaled regime the function ˆφ ε defined by ω ω ε 4, µx,z ε3 µ x ε 2, z ε 2, û ε ω,x,z = e i ωz ε 4 c ˆφε ω ε 4, x ε 2,z satisfies 2 z ˆφ ε + 1 ε 4 2i ω c zˆφε + ˆφε + ω2 c 2 1 ε µ x, z ˆφε ε 2 =. In the regime ε 1, the forward-scattering approximation in direction z is valid and ˆφ = lim ε ˆφε satisfies the Itô-Schrödinger equation in Stratonovich form [1] 2i ω ω 2 zˆφ+ ˆφ+ Ḃx,zˆφ = c Bx,z is a Brownian field, i.e. a Gaussian process with mean zero and covariance function E[Bx,zBx,z ] = z z γx x, with γx = E[µ,µx,z]dz. c 2

5 Paraxial wave propagation x Propagation L z Emission Propagation direction z. Initial condition located in the plane z =. Inhomogeneous medium with local celerity c 2 x,z = The field ut,x,z, t R, x,z R 3, reads as: ut,x,z = 1 e iωt ûω,x,zdω 2π with ûω,x,z = e i ω c zˆφω,x,z, where ˆφ is solution of: c 2 1+µx,z. 2i ω ω 2 ˆφz + ˆφ+ Ḃx,zˆφ =, c c 2 ˆφω,x,z = = ˆφ ω,x =transverse Laplacian w.r.t. x. Bx, z = Gaussian process with mean zero and covariance function E[Bx,zBx,z ] = z z γx x, with γx = E[µ,µx,z]dz.

6 Transmitted wave - homogeneous case Solve by Fourier transform in space: 2i ω c zˆφ+ ˆφ =, ˆφω,x,z = = ˆφ x, ˆφx,z = DFT ˇφκ,z = LS zˇφ = ic κ 2 ˆφx, z IFT ˇφκ,z 2ω ˇφ ˆφω,x,z = ˇφ ω,κ = 1 ˇφ 2π 2 ω,κexpi κ x c κ 2 2ω z dκ, ˆφ ω,xexp iκ xdx Consider 2i ω c zˆφ+ ˆφ =, ˆφω,x,z = = exp We get 1 ˆφω,x,z = exp 1+ 2ic z ωr 2 Gaussian beam with radius Rz = r 1+ 4c2 z2 ω 2 r 4 r 2. x ic z ωr 2 x 2 r 2

7 Transmitted wave - random case In Stratonovich form: 2i ω ω 2 ˆφz + ˆφ+ Ḃx,zˆφ = c c 2 dˆφ = ic 2ω ˆφdz + iω 2c ˆφ dbx,z In Itô s form: dˆφ = ic 2ω ˆφdz + iω 2c ˆφdBx,z ω 2 γ 8c 2 ˆφdz Analysis of the moments of the transmitted wave. The nth order moments satisfy closed form equations.

8 First-order moment First, we describe the mean or coherent field M 1, ω,x,z = E[ˆφω,x,z]. M 1, z ω,x,z = ic 2ω xm 1, ω,x,z ω2 γ M 1, ω,x,z, 8c 2 so that M 1, ω,x,l = ˆφ homo ω,x,lexp ω2 γl 8c 2, where ˆφ homo is the wave solution in the homogeneous medium. The exponential frequency-dependent damping reflects the fact that the wave field becomes incoherent as it propagates into the medium.

9 Second-order moment The second-order moment satisfies M 1,1 z M 1,1 ω,x,y,z = E [ˆφω,x,zˆφω,y,z ]. = ic x y M1,1 ω2 γ γx y M1,1. 2ω 4c The mid-point-offset second-order moment M 1,1 ω,x,y,z = E [ˆφ ω,x+ y ] 2,zˆφ y ω,x 2,z satisfies M 1,1 z = ic ω x y M1,1 ω2 γ γy M1,1 4c 2 The second-order moment can also be expressed in terms of the Wigner transform Wω,x,κ,z = e iκ y M1,1 ω,x,y,z dy The Wigner transform satisfies the transport equation W z + c ω κ xw = ω2 4c 2 ˇγκ Wκ Wκ κ dκ 2π2

10 Resolution of the transport equation W z + c ω κ xw = ω2 4c 2 2π2 Take Fourier transform ˇWξ,y,z = 1 2π 2 It solves Take shift It solves We find Wx,κ,ze iξ x+iκ y dxdκ = ˇγκ Wκ Wκ κ dκ ˇW z + c ω ξ ˇW y = ω2 γy γ ˇW 4c 2 Wξ,y,z = W ξ,yexp ˇWξ,y,z = ˇW ξ,y ξ c z ω Wξ,y,z = ˇW ξ,y +ξ c z ω,z W z = ω2 c z γ y +ξ γ W 4c 2 ω [ ω2 4c 2 z exp [ ω2 4c 2 M 1,1 x,y,z e iξ x dx γ γ y +ξ c z ] dz ω z γ γ y ξ c z ] dz ω

11 The solution to is Wx,κ,z = 1 2π 2 W z + c ω κ xw = ω2 4c 2 2π2 e iξ x κ y ˇW ξ,y ξ c z ω ˇγκ Wκ Wκ κ dκ exp [ ω2 4c 2 z γ γ y ξ c z ] dz dξdy ω The second-order moment is M 1,1 ω,x,y,z = 1 e iξ x c z 2π 2 ˇW ξ,y ξ ω exp [ ω2 4c 2 z γ γ y ξ c z ] dz dξ ω

12 With the initial condition ˆφx,z = = exp x 2 /r 2 : M 1,1 x,y,z = = exp[ 2 x 2 /r 2 x 2 /2r 2 ], and then 2 W x,κ = 2πre 2 2 x r 2 κ 2 r 2 2 M 1,1 ω,x+ y 2,x y 2,L exp = r2 8π 1 y ξlc 2 r2 ξ 2 2r 2 ω 8, ˇW ξ,y = πr2 2 ξ 2 r 2 y 2 2 e 8 2r 2 e iξ x + ω2 4c 2 With the initial condition ˆφx,z = = δx x : M 1,1 x,y,z = = δx x δy, L γ y +ξ zc ω W x,κ = δx x, ˇW ξ,y = δye iξ x γdz dξ and then M 1,1 ω,x+ y 2,x y 2,L = ω 2 [ 2ω 2πc L 2ei c L y x x exp }{{} homogeneous case ω2 L 4c 2 1 ] γ γysds dξ

13 With the initial condition ˆφx,z = = exp x 2 /r 2. For relatively strong scattering ω2 L 4c 2 E[ˆφ ω,x+ y ˆφ ω,x 2,L y ] 2,L where γx = γ γ 2 2 x 2 +o x 2. γ 1, then E[ˆφω,x,L] and = r2 8π e iξ x exp exp γ 2ω 2 8c 2 1 ξlc y 2 r2 ξ 2 2r 2 ω 8 y 2 L+y ξ L2 c ω + ξ 2 L 3 c 2 3ω 2 dξ By integrating in ξ: E [ˆφ ω,x+ y ˆφ ω,x 2,L y ] 2,L = r dexp 2 x 2 rl rl y 2 x y +i 2 2ρL 2 χl 2 The beam radius rl and the correlation radius ρl are k = ω/c rl = r 1+ 4L2 + γ 2L 3 L 3/2, k 2r4 3r 2 ρl = r 1+ 4L2 k 2 r4 1+ k2 r2 γ 2L γ 2L 2 k 2 r4 + γ 2L 3 3r γ 2L 3 3r 2 + k2 γ2 2 L4 48 L 1/2

14 General moment We define the n,m order moment of ˆφ by M n,m ω,x 1,...,x n,y 1,...,y m,z [ n = E j=1 ˆφω,x j,z m k=1 ] ˆφω,y k,z, and it then follows from Itô s formula that M n,m solves the closed-form equation M n,m z = ic n 2ω x j j=1 m k=1 y k M n,m ω2 8c 2 V n,m M n,m, where we have introduced the imaginary potential V n,m = γx j x k + γy j y k 2 γx j y k 1 j,k n 1 j,k m 1 j n,1 k m This is a Schrödinger equation with a deterministic imaginary potential in n + md dimensions and it is not exactly solvable in general. However, the equations for the first- and second-order moments can be readily solved. Analysis of the fourth-order moment scintillation.

15 Analysis of time reversal experiment

16 Time reversal experiment cf. A. Tourin, M. Fink, and A. Derode, Multiple scattering of sound, Waves Random Media 1 2, R31-R6. Experimental set-up for a time-reversal experiment through a multiple-scattering medium: a first step, the source sends a pulse through the sample, the transmitted wave is recorded by the TRM. b second step, the multiply scattered signals have been time-reverted, they are retransmitted by the TRM, and S records the reconstructed pressure field.

17 Time reversal experiment cf. A. Tourin, M. Fink, and A. Derode, Multiple scattering of sound, Waves Random Media 1 2, R31-R6. Experimental set-up for a time-reversal experiment through a multiple-scattering medium: a first step, the source sends a pulse through the sample, the transmitted wave is recorded by the TRM. b second step, the multiply scattered signals have been time-reverted, they are retransmitted by the TRM, and S records the reconstructed pressure field.

18 Time-reversal experiment

19 We introduce the fundamental solution Ĝ ω,x,z,x,z : 2i ω c z Ĝ+ Ĝ+ ω2 c 2 starting from Ĝ ω,x,z = z,x,z = δx x. Ḃx,zĜ =, In a homogeneous medium µ, B the fundamental solution is Ĝ ω,x,z,x,z exp iω x x 2 2c z z =. z z 2iπc ω In a random medium: E [ Ĝ ω,x,z,x,z ] = Ĝ ω,x,z,x,z exp γω2 z z 8c 2 where γx = E[µ,µx,z]dz = Strong damping of the coherent wave., E [ Ĝ ω,x,z,x,z Ĝ ω,x,z,x,z ] = Ĝ ω,x,z,x,z Ĝ ω,x,z,x,z exp γ 2x x ω 2 z z 4c 2, where γ 2 x = 1 γ γxsds = Lateral decoherence properties.

20 Time reversal setup x Mirror L z Emission Reception 1 Forward emission. We emit at z = : u t,x = 1 2π e iωt û ω,xdω. We receive and record at z = L, y mirror: u F t,y,l = 1 e iω c L tˆφf ω,y,ldω 2π where ˆφ F ω,y,l = Ĝ ω,y,l,x, û ω,xdx We shift the time origin: ũ F t,y,l := u F t L,y,L c = 1 2π e iωtˆφf ω,y,ldω

21 2 Time reversal. - The signal is recorded at the mirror with limited spatial support. - The recorded signal is time-reversed: u B t,y,l = ũ F t,y,l χ M y = 1 e iωtˆφf ω,y,ldω χ M y 2π where χ M is the spatial truncation function of the mirror. If the mirror is square with side length D, χ M y = 1 [ D/2,D/2] [ D/2,D/2] y. Taking the complex conjugate u B is real-valued: u B t,y,l = 1 e iωtˆφb ω,y,ldω 2π with ˆφ B ω,y,l = ˆφ F ω,y,lχ M y = Ĝ ω,y,l,x, û ω,xdxχ M y The signal u B t,y,l is emitted from the mirror.

22 x Mirror L z Reception Emission 3 Reception of the backward signal. We emit at z = L: u B t,x,l = 1 e iωtˆφb ω,x,ldω. 2π We receive at z = : u B t,x, = 1 e iω c L tˆφb ω,x,dω. 2π We shift the time origin: ũ B t,x, := u B t L,x, = 1 e iωtˆφb ω,x,dω c 2π where ˆφ B ω,x, = Ĝ ω,x,,y,l ˆφB ω,y,ldy = Ĝ ω,x,,y,l Ĝ ω,y,l,x, û ω,x χ M ydx dy Reciprocity relation: Ĝ ω,x,,y,l = Ĝ ω,y,l,x,.

23 Finally, we receive the signal ũ B t,x, = 1 2π e iωtˆφb ω,x,dω with ˆφ B ω,x, = Γω,y,x,x,Lû ω,x χ M ydx dy where Γω,y,x,x,L := Ĝ ω,y,l,x, Ĝ ω,y,l,x, In a homogeneous medium, Γ is explicitly known the double integral can be computed. In random media: the statistical distribution of Γ is important.

24 Homogeneous medium µ We have Γω,y,x,x ω 2 ω,l = c L y x+x x x 2. 2πc L 2e i If u t,x = ftδ x, then ˆφ ω 2 B ω,x, = 2πc L ˆfωe i ω x 2 2 2Lc ωx ˆχ M. c L In case of a square mirror: where r c ω = 2πLc ωd In the time domain: ˆφ B ω,x, = ωd2 2πc L ˆfωe i ω x 2 πx1 2Lc sinc 2 r c ω and sincs = sins/s. sinc πx2 r c ω if u t,x = cosω tvtδ x with bandwidthv ω, then: u B t,x, = ω D 2 2πc L cosω tv t x 2 πx1 πx2 sinc sinc 2 2Lc r c r c where r c = 2πLc ω D = λ L D.

25 Time-reversal experiment

26 Random medium With: u t,x = δxft, û ω,x = δxˆfω. E[φ B ω,x,] = Γω,y,x,x,Lδx χ M ydx dy = 2 γ2 Γω,y,x,,L homo e ω xl 4c 2 χ M ydy ω2γ 2xL 4c = φ B ω,x, homo e 2 ω 2 L 4c 2 γ 1 φ B ω,x, homo e ω 2 γ2 L 24c 2 x 2 Therefore E[φ B ω,x,] = ω 2 2πc L ˆfωe i ω x 2 x1 x2 2 2Lc sinc sinc exp x 2 r c r c 2ra 2 }{{} homogeneous case where r c = 2πLc ωd and r a = 12c ω γ 2 L.

27 In the time domain: u t,x = cosω tvtδx with bandwidthv ω : E[u B t,x,] =cosω tv t x 2 sinc 2Lc x1 r c sinc x2 r c exp x 2 2r 2 a where r c = 2πLc ω D = λ L D and r a = 12c ω γ2 L = λ 3 π γ 2 L. If r c < r a : Same result as in homogeneous medium If r c > r a : E[u B ] spot thiner than in homogeneous medium! The radius of the focal spot is r eff = λ L D eff with the effective aperture: D 2 eff = D 2 +π 2 γ 2 L 3.

28 But: the result holds true in average averaging over all possible realizations of the medium!

29 Frequency correlation of Γ We have shown that: E[φ B ω,x,] = Γω,y,x,x,Lû ω,x χ M ydx dy The frequency autocorrelation function of W: E[Γω +h/2,l,...γω h/2,l,...] = γ c ω,h,l,... with γ c ω,h,l,... h. The width ω c of γ c decays to zero with L. Thus E[Γω 1,L,...Γω 2,L,...] E[Γω 1,L,...]E[Γω 2,L,...] as soon as ω 1 ω 2 ω c. Similarly E[φ B ω 1,x,φ B ω 2,x,] E[φ B ω 1,x,]E[φ B ω 2,x,] as soon as ω 1 ω 2 ω c.

30 In the time domain: E[u B t,x] = 1 2π Self-averaging in time E [ u B t,x 2] = 1 2π 2E e iωt E[φ B ω,x]dω, [ e iωt φ B ω,xdω 2 ] Provided the bandwidth B of u is larger than ω c : E [ u B t,x 2] 1 = e iω 1+ω 2 t E[φ 2π 2 B ω 1,xφ B ω 2,x]dω 1 dω 2 B ω c 1 e iω 1+ω 2 t E[φ 2π 2 B ω 1,x]E[φ B ω 2,x]dω 1 dω 2 = E[u B t,x] 2 [ ub Thus Varu B t,x := E t,x E[u B t,x] ] 2 B ω c. This implies that, for any δ > : P u B t,x E[u B t,x] > δ Varu Bt,x δ 2 The frequency decorrelation implies the self-averaging in time. B ω c

31 Conclusion Spatial time-reversal refocusing enhanced by randomness: - in the homogeneous case, the diameter of the focal spot λl/d is determined by the aperture of the focusing cone the waves propagate in a straightforward manner. - in the random case, the focal spot is smaller because it is determined by an effective focusing cone induced by multi-pathing physical interpretation. Statistical stability of the refocused focal spot ensured by the frequency decorrelation of the product of two Green s functions Γ. Applications in imaging: interferometric or cross correlation imaging techniques in seismology and medical imaging.