Modèles stochastiques pour la propagation dans les fibres optiques

Transcription

1 Modèles stochastiques pour la propagation dans les fibres optiques A. de Bouard CMAP, Ecole Polytechnique joint works with R. Belaouar, A. Debussche, M. Gazeau

2 Nonlinear fiber optics in communications A very efficient way to transmit information : high data rates (up to six time faster than satellites) Explosion of the field in the 1990 s, due to development of Erbium-doped amplifiers and lasers Wavelength division multiplexing capacities of more than 1 Tb/s

3 Optical fibers Glass or plastic fiber that carries light along its length ; Light is kept in the core by total internal reflection : the fiber acts as a waveguide

4 Single mode/multimode fibers Core diameter may be less than 10 times the wavelength of the propagation light use of Maxwell equations

5 Nonlinear fiber optics plays an important role in the design of such high capacity systems Agrawall : Applications of nonlinear fiber optics Limiting factors to be taken into account : I Chromatic dispersion (monomode fibers) use of nonlinearity, or more efficiently, of dispersion managed fibers I Polarization mode dispersion (PMD) : linked to birefringence

6 The NLS equation in fiber optics Maxwell equations for the electromagnetic field E : electric field 1 2 E c 2 t 2 + E (dive) = 1 2 P ε 0 c 2 t 2 P : polarization (depends nonlinearly on E) Properties of the material : isotropic centrosymmetric t 1 ε 0 P( x, t) = χ (1) (x, t t 1 ) E(t 1 )dt 1 t + χ (3) (x, t t 1, t t 2, t t 3 )( E(t 1 ). E(t 2 )) E(t 3 )dt 1 dt 2 dt 3

7 Weakly nonlinear WKB expansion : E = ε E 0 + ε 2 E1 + ε 3 E E j (x, z, t) = U j (x, t) [ E j (εz, ε 2 z, εt)e i(k 0z ω 0 t) ] + cc (convolution in time) Newell, Moloney, Nonlinear Optics

8 Assumptions : monomode fiber : only one mode Û(x, ω) is confined polarization preserving Then compatibility conditions for E 1 and E 2 give the NLS equation for E 0 : i ζ E k 0 2 E 0 τ 2 + ω 0 c n 2 E 0 2 E 0 = 0 with τ = ε(t k 0 z), ζ = ε2 z. Many effects have been neglected... The sign of k 0 (GVD) determines whether the medium is focusing or defocusing.

9 NLS : mathematical properties One dimensional cubic nonlinear Schrödinger equation (dimensionless form) : i t u + λ 2 x u + u 2 u = 0, x R, λ = ±1 Integrable by inverse scattering (Zakharov-Shabat, 1972) explicit soliton solutions Focusing case (λ=1) : bright solitons with u(t, x) = A ω (x 2vt s)e i(vx v 2 t+ωt+θ) A ω (x) = 2ω sech( ωx) Defocusing case (λ=-1) : No localized solution ; dark solitons with v ρ, lim x u(t, x) = ρ. v = 0 : black soliton

10 Soliton profiles

11 Conserved quantities : Energy : N(t) = u(t) 2 dx = N(0) Hamiltonian : H(t) = u(t) 2 dx λ 2 u(t) 4 dx = H(0) Allow to show propagation of H 1 solutions (N(t) and H(t) finite) for all time, or along all the fiber Remark : Global propagation also holds if only N(0) finite (Strichartz estimates : dispersion inequalities in space + convolution inequalities in time)

12 Remark : not always true ; 1-D focusing quintic NLS equation (not physical) i t u + 2 x u + u 4 u = 0 Explicit solution which blows up at t = 1 : x 2 4(1 t) e i 1 t u(t, x) = e i 3 1/4 1 t Critical L 2 case : the scale change cosh 2x 1 t u µ (t, x) = µ 1/2 u( t µ 2, x µ ) preserves both the equation and the L 2 norm. Same phenomenon for the 2-D cubic NLS equation (but no explicit solution)

13 4 NLS critique time=0.8 σ =

14 Dispersion managed fibers In DM fibers, the group velocity dispersion varies with the distance along the fiber NLS equation is modified by (dimensionless form) with i t u D(t) 2 x u + u 2 u = 0 D(t) = d(t) + d m + m(t) d m : residual dispersion (small) ; m : random process with zero mean d(z) z

15 Case m=0 (no random perturbation) : Lot of physical/numerical studies Zharnitsky, Grenier, Jones, Turitsyn, Ph. D 2001 : On large propagation distances, rescaled solution close to solution of an averaged equation : with i t v + d m 2 x v + Q (v, v, v) = 0 Q(v 1, v 2, v 3, t) = T 1 (t)(t (t)v 1.T (t)v 2.T (t)v 3 ) and T (t)u 0 solution of with u(0) = u 0. i t u + d(t) 2 x u = 0 DM soliton = ground state of the averaged system (d m 0)

16 Asymptotic evolution of the pulse from Turitsyn et. al. Opt. Comm Fig. 1. Evolution over one compensation period of DM soliton shown in normal Žbottom. and logarithmic Župper picture. scales. In the leading order the dynamics is self-similar Žbottom. and is given by Eqs. Ž37.. It is seen that the dips appear at the beginning Žend. and in the middle of the periodic cell. Dispersion map dž z. s d q ² d : s "5 q 0.15, cž z. s 1. An important feature of the soliton carrier signal Žin general, chirped return-to-zero ŽRZ. formatted data w7x. is that it can be described by a few main parameters, like pulse width, peak power, chirp parameter, and spectral width Žthe latter can be

17 Case of random perturbations : Garnier, Opt. Comm., 2002 : collective coordinate approach on the complete model We consider : d m = d(t) = 0 m(t) = β(t) is a white noise i.e β is Gaussian and β(t) β(s) = σ 0 δ(t s) natural if the correlation length of m is much less than the deterministic characteristic length Mathematical notations : idu + 2 x u dβ + u 2 udt = 0, x R, t > 0 β = β(t) : real valued Brownian motion 2 x u dβ : Stratonovich product

18 Global propagation of solutions : Marty, PhD Thesis, 2005 : case u 2 u replaced by f ( u 2 )u, f smooth and bounded ; regular solutions. AdB, A. Debussche, JFA 2010 : Global propagation of solutions if N(0) is finite No Hamiltonian conservation use of Strichartz estimates Representation of the solution : u(t, x) = S(t, 0)u 0 + i t 0 S(t, s) u 2 u(s)ds with S(t, s) = e i(β(t) β(s)) given by ( ) 1 1/2 S(t, s)ϕ(x) = 4πi(β(t) β(s)) e i x y 2 4(β(t) β(s)) ϕ(y)dy not a convolution in time.

19 Diffusion-approximation : Continuity of the solution w.r. to the Brownian paths m centered stationary process + classical ergodic assumptions v ε solution of i t v + εm(t) 2 x v + ε 2 v 2 v = 0 then u ε (t, x) = v( t ε 2, x) converges in law as ε goes to zero to the solution of idu + σ 0 2 x u dβ + u 2 u dt = 0 with the same initial state, and with σ 2 0 = E[m(0)m(t)]dt Also true with the addition of (small) residual dispersion ε 2 d m.

20 Remarks : A. Debussche, Y. Tsutsumi, JMPA 2011 : Global propagation (for finite N(0)) still true for idu + 2 x u dβ + u 4 u dt = 0 contrary to the deterministic case. Make use of local smoothing properties of the linear equation. scaling argument : β(t) Brownian Motion λ 1/2 β(t/λ) is also a B.M. u 8 u should be the critical nonlinearity in L 2 (invariant by the L 2 scaling) Global propagation in the cubic case still true with d m 0, hence d(t) 0. Long distance behavior?

21 Numerical simulations R. Belaouar, AdB, A. Debussche : u n j approximation of u(nδt, jδx) Splitting + spectral Finite differences in time and space Crank-Nicolson : i δt (un+1 j uj n) + χn δt (u n+1/2 j+1 2u n+1/2 j + u n+1/2 j 1 ) = 1 2 ( un j 2 + u n+1 j 2 )u n+1/2 j (χ n ) : family of independent N (0, 1) Relaxation scheme C. Besse, 1998 i δt (un+1 j uj n) + χn δt (u n+1/2 j+1 2u n+1/2 j 1 2 (φn 1/2 j + φ n+1/2 j ) = uj n 2 + u n+1/2 j 1 ) = φ n+1/2 j u n+1/2 j

22 Semi-discrete Crank-Nicolson : i u n+1 u n δt + χ n δt u n+1/ ( u n 2 + u n+1 2 )u n+1/2 = 0 has order one in time : ( lim P max C n=0,, T δt ) u n u(t n ) H 1 Cδt = 0 uniformly in δt, provided u 0 H 7 ; in addition, for any δ < 1 there is a r.v. K δ such that max u n u(t n ) H n=0,, T 1 K δ (δt) δ δt R. Marty, 2011 : Same result for Strang splitting and Lipschitz nonlinearity

23 Ideas of proof : convergence of the scheme : compactness method use of a cut-off Lipschitz nonlinearity linear case : i v n+1 v n δt + χ n δt v n+1/2 = 0 ˆv(nδt, ξ) ˆv n (ξ) 2 = 1 e imn(ξ) 2 ˆv 0 (ξ) 2 with M n (ξ) = n [ δt δtχ k ξ 2 2 arctan( 2 χ k ξ 2 )] k=0 which implies (martingale inequalities) ( E max n=0,, T δt ) v n v(t n ) 2 L C 2 α (δt) 2α E( v 0 2 H ) 6α

24 Crank Nicolson Erreurs logarithmiques Deterministe Bruit blanc log erreur Entier n tel que h=t/n

25 Polarization mode dispersion A phenomenon due to birefringence : the light is a vector field and components of E travel with different group velocities In realistic situations, birefringence parameters non uniform ; random variations in orientations on a length scale of 100m with zero average Residual effects pulse spreading, called polarization mode dispersion (PMD)

26 Model for random variations of birefringence : Way-Menyuk, 1994, 1996 : i A z + bσa + ib Σ A t + d 0 2 A 2 t A 2 A + (A σ 3 A) σ 3 A ( A 1 A 2 2 A 2 A2 1 ) = 0, b = n : birefringence strength, b = db dω and Σ is a 2 2 matrix with random coefficients depending on z Assume (random) orientation angle θ such that dθ dz = 2εα(z), ε2 = l c l b2 α : real valued centered Markov process, with unique ergodic invariant measure l c correlation length of α + averaging over fast oscillations Manakov PMD equation

27 Manakov PMD equation Garnier-Marty, 2006 : i X t (t, x)+ d 0 2 X 2 x 2 (t, x)+ 8 9 X 2 X = ib σ(t) X x 1 (N N ), 6 with σ = ( ν1 2 ν 2 2 ) 2 ν 1 ν 2 2ν 1 ν 2 ν ν 2 2 = σ 1 m 1 (t)+σ 2 m 2 (t)+σ 3 m 3 (t) and ν is obtained as dν = i γ c (σ 1 ν(t) dw 1 (t) + σ 2 ν(t) dw 2 (t)) + iγ s σ 3 ν(t)dt σ i : Pauli matrices N : cubic term with random coefficients depending on ν N : same with averaged coefficients w.r.t. invariant measure of ν

28 PMD correlation length much smaller than deterministic length z 0 small parameter ε such that correlation length of order one fiber length of order ε 2 Then, setting X ε (t, x) = 1 ε X( t, x ε 2 ε ) and σ ε(t) = σ( t ), we get the ε 2 evolution i Xε t (t, x) + d 0 2 with 2 X ε ib (t, x) + x 2 ɛ σ ε(t) Xε x (t, x) + F ν ε(t)(x ε ) = 0 F νε(t)(x ε ) = 8 9 X 2 X (Nε (X) N(X) ) and N ε is the same as N with ν(t) replaced by ν ε (t) = ν( t ε 2 )

29 Diffusion approximation : Garnier-Marty, Wave Motion, 2006, linear case AdB, M. Gazeau, to appear in Ann. Appl. Prob., full model Under standard assumptions on the original (C 2 valued) process ν defining (m 1, m 2, m 3 ) : Existence of a unique global adapted square solution in L 2 for all positive ε Moreover, if X 0 H 3, then for any stopping time τ with τ < τ a.s., the process X ε τ (stopped at τ) converges to X τ in distribution in C([0, T ], H 1 ), where X τ is the unique local solution of ( d0 2 X idx + 2 x ) 9 X 2 X dt + i γ 3 σ k x X dw k = 0 k=1

30 numerical simulations Mean amplitude of each component 8 x Initial state : Manakov soliton

31 Evolution of the energy /pulse width (in mean) 5.86 x

32 Same simulation with different initial state

33 Evolution of the energy /pulse width (in mean)