SMR 1673/27 AUTUMN COLLEGE ON PLASMA PHYSICS 5-30 September 2005 Nonlinear Wave Dynamics in Nonlocal Media J.J. Rasmussen Risoe National Laboratory Denmark
Nonlinear Wave Dynamics in Nonlocal Media Jens Juul Rasmussen 1 O. Bang 2 W.Z. Królikowski 3,J. Wyller 4 N.I. Nikolov 1,2, D. Neshev 3, and D. Edmundson 3, P.L. Christiansen 2 jens.juul.rasmussen@risoe.dk 1 Risø National Laboratory, OPL-128 DK-4000 Roskilde, Denmark 2 Technical University of Denmark, DK-2800 Kgs. Lyngby 3 Australian National University, Canberra ACT 0200 4 Agricultural University of Norway, N-1432 Ås New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 1/2
Contents Nonlocal nonlinear media; examples Modulational instability of plane waves; NL suppress MI Interaction of solitons; dark solitons Wave collapse/self-focusing; NL prevents blow-up Conclusions New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 2/2
Nonlocal nonlinear interaction Spatial nonlocality: response in a point depends on the intensity in its vicinity Nonlocality results from a transport process: atom diffusion, heat transfer, drift of charges. Generic feature in many nonlinear systems: plasma, atom gasses, liquid crystals, Bose-Einstein condensates Stationary solutions in quadratic nonlinear (χ (2) ) media are equivalent to stationary solutions in nonlocal Kerr-media New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 3/2
References W. Krolikowski and O. Bang, Phys. Rev. E 63, 016610 (2001) W. Krolikowski, O. Bang, J.J. Rasmussen, and J. Wyller, Phys. Rev. E 64, 016612 (2001) O. Bang, W. Krolikowski, J. Wyller, J.J. Rasmussen, Phys. Rev. E 66, 046619 (2002) J. Wyller, W. Krolikowski, O. Bang, J.J. Rasmussen, Phys. Rev. E 66, 066615 (2002) N.I. Nikolov, D. Neshev, O. Bang, and W. Krolikowski, Phys. Rev. E 68, 036614 (2003) N.I. Nikolov, W. Królikowski, O. Bang, J. J. Rasmussen, and P.L.Christiansen, Optics Lett. 29, 286-288, (2004) D. Briedis, D.E. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, Optics Express 13, 435 (2005) W. Królikowski, O. Bang, N.I. Nikolov, D. Neshev, J. Wyller, J.J. Rasmussen, and D. Edmundson, J.Opt. B 6 S288 (2004) New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 4/2
Model equations Optical beam propagating along the z-axis of a nonlinear medium; ψ( r, z) is the slowly varying amplitude E( r, z) =ψ( r, z)exp(ikz iωt)+c.c. New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 5/2
Model equations E( r, z) =ψ( r, z)exp(ikz iωt)+c.c. Refractive index change N(I); I( r, z) = ψ( r, z) 2 N(I) =s R( ξ r)i( ξ,z)d ξ, s = 1(s = 1): focusing (defocusing) nonlinearity R( r) =R(r), r = r real, localized: R( r)d r =1. New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 5/2
Model equations E( r, z) =ψ( r, z)exp(ikz iωt)+c.c. N(I) =s R( ξ r)i( ξ,z)d ξ, R( r) =R(r), r = r real, localized: R( r)d r =1. Nonlocal nonlinear Schrödinger (NLS) equation i z ψ + 1 2 2 ψ + N(I)ψ =0, New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 5/2
Model equations E( r, z) =ψ( r, z)exp(ikz iωt)+c.c. N(I) =s R( ξ r)i( ξ,z)d ξ, R( r) =R(r), r = r real, localized: R( r)d r =1. i z ψ + 1 2 2 ψ + N(I)ψ =0, Singular response, R(r) =δ( r ), : N(I) =si( r, z):, i z ψ + 1 2 2 ψ + sψ ψ 2 =0 s = 1(s = 1): focusing (defocusing) nonlinearity New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 5/2
Degrees of nonlocality (a) R(x'-x) (b) R(x'-x) Weakly nonlocal: N(I) =I + γ 2 I, γ = 1 2 r 2 R(r)d r I(x') 0 x x' I(x') 0 x x' Strongly nonlocal: N(I) =sr(r)p, P = Id r (c) R(x'-x) (d) R(x'-x) I(x') I(x') 0 x x' 0 x x' New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 6/2
Modulational Instability Plane wave ψ( r, z) = ρ 0 exp(i k 0 r iβz), ρ 0 > 0, β = 1 2 k2 0 sρ 0, New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 7/2
Modulational Instability ψ( r, z) = ρ 0 exp(i k 0 r iβz), ρ 0 > 0, β = 1 2 k2 0 sρ 0, Perturbation to the plane wave solution ψ(r,z)=[ ρ 0 + a 1 ( r, z)+cc]exp(i( k 0 r βz)), a 1 ( r, z) = ã 1 ( k)exp(i k r + λz)d k New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 7/2
Modulational Instability ψ( r, z) = ρ 0 exp(i k 0 r iβz), ρ 0 > 0, β = 1 2 k2 0 sρ 0, ψ(r,z)=[ ρ 0 + a 1 ( r, z)+cc]exp(i( k 0 r βz)), a 1 ( r, z) = ã 1 ( k)exp(i k r + λz)d k Dispersion relation [ k λ 2 = k 2 2 ] ρ 0 s 4ρ R( k), 0 New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 7/2
Modulational Instability ψ( r, z) = ρ 0 exp(i k 0 r iβz), ρ 0 > 0, β = 1 2 k2 0 sρ 0, ψ(r,z)=[ ρ 0 + a 1 ( r, z)+cc]exp(i( k 0 r βz)), a 1 ( r, z) = ã 1 ( k)exp(i k r + λz)d k [ k λ 2 = k 2 2 ] ρ 0 s 4ρ R( k), 0 k = k spatial frequency, R( k )= R( r )exp(i k r)d r Fourier spectrum of R(r). New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 7/2
Modulational Instability ψ( r, z) = ρ 0 exp(ik 0 r iβz), ρ 0 > 0, β = 1 2 k2 0 sρ 0, ψ(r,z)=[ ρ 0 + a 1 ( r, z)+cc]exp(i( k 0 r βz)), a 1 ( r, z) = ã 1 ( k)exp(i k r + λz)d k [ k λ 2 = k 2 2 ] ρ 0 s 4ρ R( k), 0 R( k ) positive definite: s =1MI; s = 1 stability R( k ) sign indefinite: s =1MI; s = 1 possibility for MI! New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 7/2
λ λ MI in focusing media (a) (b) ρ =1 0 σ =0.5 G ρ =1 0 σ =0.5 σ =10 G σ =1 G σ =10 σ =20 σ =1 MI gain for ρ 0 =1, s =1 [ ] (a) Gaussian: R (x) = 1 σ π exp x2 σ with R (k) =exp [ 1 2 4 σ2 k 2] (b) Rectangular: R (x) = 1 2σ for x σ and R (x) =0for x >σ R (k) = sin(kσ) kσ New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 8/2
MI in focusing media σ= 10-3 MI evolution ρ 0 =1, s =1 Gaussian response profiles: [ ] R (x) = 1 σ π exp σ x2 2 Experimental verification Peccianti et al. PRE 68 025602 (2003) σ= 1 New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 9/2
λ MI in de-focusing media ρ =1 0 σ=20 σ=10 s=-1 σ=5 R (x) = R (k) = 1 2σ ; x σ 0; x >σ sin (kσ) kσ MI in de-focusing nonlocal media with rectangular response function New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 10/2
MI in de-focusing media 1.0 0.8 wave number k 0.6 σ=5 σ=20 σ=35 0.4 0.2 0.0 0 5 10 15 20 25 30 35 nonlocality parameter σ Wave number for maximum gain versus σ. sρ 0 = 1 and rectangular response function. New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 11/2
Dark solitons: interaction i z u + 2 xu + nu =0, n(i) = R(x τ)i(τ)dτ, R(x) =(2σ) 1 exp( x /σ): n σ 2 x n 2 = u 2, u 2 and 1 n profiles 1.2 0.8 0.4 0 ( a ) ( b ) 10 0 10 10 0 10 Transverse coordinate Dark solitons: u(x, z) =u(x)exp(iλz) u(0)=0, π phase jump at x =0 Equivalent χ (2) -solitons Single dark soliton (a) and bound state (b). Nonlocal effects smear out n, creating an effective waveguide. New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 12/2
Dark solitons: interaction Evolution of bound state of two nonlocal solitons Separation x 2σ Evolution of dark soliton in χ (2) - media and in nonlocal media New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 13/2
Dark solitons: interaction Interaction of dark nonlocal solitons. a) Phase jump = π, σ =2and x 0 = 5.5, 4.0, 2.5 b) Phase jump =0.95π, σ = 0.1, 1.0, 2.0, and x 0 = 2.5. c) Intensity gap width 7.5 and σ = 0.1, 3.0, 6.0. Critical separation x 0c 2σ New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 14/2
Wave Collapse I Local NLS: Collapse for D 2 C. Sulem and P. Sulem, The Nonlinear Schrödinger Equation, Springer-Verlag, Berlin (1999) D =2: necessary and sufficient condition: Collapse (blow-up) for P>P sol D =3: Sufficient condition H < 0 Kuznetsov et al Physica D 87, 273 (1995) New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 15/2
Wave Collapse I Nonlocal NLS: collapse is suppressed: S.K. Turitsyn, Teor. Mat. Fiz. 64, 226 (1985) V.M. Pérez-García et al. Phys Rev E 62, 4300 (2000) New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 15/2
Wave Collapse I General case: symmetric non-singular response function: Conservations of power, P, and Hamiltonian, H (N(I) =s R( ξ r)i( ξ, z)d ξ) P = Id r, H = ψ 2 2 1 2 NId r. New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 15/2
Wave Collapse I General case: symmetric non-singular response function: P = Id r, H = ψ 2 2 1 NId r. 2 Using: a b a b a b and f( x)d x f( x) d x H = ψ 2 2 1 NId r ψ 2 2 1 2 2 NId r New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 15/2
Wave Collapse I General case: symmetric non-singular response function: P = Id r, H = ψ 2 2 1 NId r. 2 H = ψ 2 2 1 NId r ψ 2 2 1 2 2 NId r Fourier space NId r = 1 (2π) D R( k) Ĩ( k) 2 d k New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 15/2
Wave Collapse II NId r 1 (2π) D R 0 = R(0) 1 (2π) D R( k) Ĩ( k) 2 d k P 2 R 0 R( k) d k, P Id r Demands R( k) is integrable ψ 2 2 H + 1 2 R 0P 2 New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 16/2
Wave Collapse II NId r 1 (2π) D R 0 = R(0) 1 (2π) D R( k) Ĩ( k) 2 d k P 2 R 0 R( k) d k, P Id r ψ 2 2 H + 1 2 R 0P 2 ψ 2 2 is bounded from above no blow-up! Quasi collapse: smallest width width of R(r). Width w Pσ P/2π+Hσ 2 New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 16/2
2D Beam Stability Soliton solution: ψ( r, z) =φ(r)exp(iλz). Variational results for Gaussian response R( r )= ( 1/πσ 2) D 2 exp ( r 2 /σ 2) and trial function φ(r) =α exp[ (r/β) 2 ] Different degrees of nonlocality, σ=0, 0.2, 0.4, and 0.6. dp dλ > 0 implies stability New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 17/2
3D Beam Stability 3D variational results with Gaussian response and trial function. σ=0, 0.2, 0.4, and 0.6. Dashed line threshold for instability New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 18/2
Stabilized vortex beam Gaussian-Laguerre vortex beam ψ( r) =r exp( (r/r 0 ) 2 exp(iθ) (charge 1, r 0 =1) in a self-focusing nonlocal medium (Gaussian response function). (a) σ =0,(b)σ =1, (c) σ =10. New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 19/2
Stabilized vortex beam Propagation of a charge m = 1 vortex beam in a nonlocal medium. Gaussian response function (a) σ =0.1, (b) σ =10. Briedis et al. Optics Express 13 435 (2005) New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 20/2
Conclusions Nonlocal effects have a strong influence on the nonlinear evolution The modulational instability of plane waves is suppressed for positive definite R(k) Higher order unstable bands appear for sign indefinite R(k) and unstable bands may appear in de-focusing media. Nonlocality influences soliton interactions and may introduce 2(n)-soliton bound states. Nonlocality prevents blow-up, stabilize 2 and 3-D solitons; quasi collapse is possible. New Trends in Nonlinear Physics, ICTP, September 17, 2005 Back Next p. 21/2