Wave Turbulence and Condensation in an Optical Experiment S. Residori, U. Bortolozzo Institut Non Linéaire de Nice, CNRS, France S. Nazarenko, J. Laurie Mathematics Institute, University of Warwick, UK
Outline Optical Wave Turbulence: a brief introduction Experiment on optical wave turbulence: - light propagation in nematic liquid crystals - WT experiment and inverse cascade MI and solitons - Theory and numerics Conclusions
Wave Turbulence WT: the statistical study of a large number of waves which interact through a weak nonlinearity A wave turbulence state is accompanied by dissipation and requires external source of energy to sustain it ex: waves in the sea excited by wind or ships WT in Optics?
NLS equation i "# "z + 1 2k 0 $ % 2 #+ & # 2 # = 0 α > 0 α < 0 focusing defocusing describes the propagation of almost monochromatic light beams in nonlinear media. D=1 completely integrable α > 0 stable solitons D 2 not integrable; it has three integrals of motion N = ψ 2 dr number of particles or wave action H = ψ 2 1 2 α ψ 4 dr energy M = 1 2 i ( ψ ψ* ψ * ψ) dr momentum can be taken zero in homogeneous turbulence
Optical Wave Turbulence D =2 i t E + γe ( ) + xx ( 2 2 + yy )E + α E 2 E = 0 inertial range ω k = k 2 dispersion relation n k = A k 2 N = n k dk E = ω k n k dk H = ω k A k A k * d k α 4π 2 A * k A * A A δ k1 ( k+k k k )dkdk k k 1 2 3 1 dk 2 dk 3 2 3 4 wave interaction, N and E conserved
Kinetic equation and power law solutions n k t + 2γ k n k = α2 4π ( n k1 n k2 n k3 + n k n k1 n k2 n k n k1 n k3 n k n k1 n k2 ) δ( ω k + ω k1 ω k2 ω k3 )δ( k + k k k 1 2 3 )dk 1 dk 2 d k 3 Q = flux of particles N P = flux of energy E P = Q = 0 Thermodynamic equilibrium T n k = µ + ω k Kolmogorov-Zakharov solutions Q = const P = 0 n k = a 1 Q 1 3 ω k 2 3 Q = 0 P = const n k = a 2 P 1 3 ω k 1 Mixed solutions (Zakharov-Newell 1992) Q 0 P 0 n k = n k ( P,Q,T,µ,k) = T ( ) µ + ω k + Φ ω k
Direct and inverse cascade α<0 (defocusing) Inverse cascade of particles : photon condensation α>0 (focusing) Filament formation : solitons
WT versus nonlinear optics WT many waves weak nonlinearity NLO a few waves high nonlinearity KZ spectra photon condensation modulational instability solitons vortex wave soliton interactions. phase conjugation second harmonic generation optical parametric amplification... Our aim: to create a state of optical wave turbulence (photon fluid) Optics : the inverse cascade of particles is associated with the condensation of light
Optical WT experiment Setup based on light propagation in liquid crystals low dissipation - weak nonlinearity Experimental evidence of inverse cascade Solitons Theory and numerics
Liquid crystals Nematic phase Molecular reorientation ne=1.7 no=1.5 E #"!" Focusing nonlinearity
Optical nonlinearity and solitons in LC I FT kw cm 2
Solitons in nematics: nematicons nematicons : balance diffraction/focusing nonlinearity of LC NLS E v X Y Z
Solitons in nematics: nematicons nematicons : balance diffraction/focusing nonlinearity of LC NLS E v X Y Z
WT experiment in nematics Nonlinearity is provided by the reorientational Kerr effect of the LC A k 0 + + + + + + + + ( ) "! r,z z V d=50 µm - - - - - - - - - V=2.5 V z has the role of time The spatial distribution of the reorientation angle satisfies the steady equation ( ) A K 2 θ Δε RF πd V 2 θ + ε 0 2 4 n 2 2 e n o 2 = 0 external electric field optical field
Model equations i z A = 1 2k 0 2 A k 0 ( n 2 2 e n ) o θa 2 ( ) A K 2 θ Δε RF πd V 2 θ + ε 0 2 4 n 2 2 e n o 2 = 0 rescaled variables l " = #K 2$% d V
Experimental setup LASER OB P 1 L 1 SLM L 2 A PH W P L 3 L 4 x LC z Input beam wavelength : 473 nm size : 30 µm x 1.8 mm intensity : 200 µw PC observations are done by looking at the scattered light from the top of the cell
Prepare the beam at z=0 Initial condition A(z=0) Qin Input beam @ 473 nm laminar shaped : 30 µm x 1.8 mm weak NL low input intensity : 200 µw k in k d k x SLM p = 37 ± 10µm - spatial modulation at high wavenumbers - random phase dissipative scale 10 µm πk d l ξ = 2Δε V given by radiation losses and elastic coupling
Evolution of the field uniform phase random phase Linear evolution: transverse periodicity induces periodic recurrences along z (Talbot effect) Nonlinear evolution: the wavelength increases as the field propagates along z (smoothing)
Intensity profiles
Inverse cascade during the propagation the field evolves towards lower wavenumbers : we observe an inverse cascade in the spectrum of the wave intensity I k 2 k -1/5 z=4.2 mm 10 0 z=0 mm 10 1 10 2 k [mm -1 ]
Modulational instability and solitons x [mm] 1.0 0.5 0 2 4 6 8 z [mm] inverse cascade: the intensity goes toward the homogeneous solution (k=0) MI: the solution at k=0 is unstable 4000 a single large soliton at long times I (gray values) 3000 2000 7.5 mm 1000 0 4.5 mm 0.3 mm -0.4-0.2 0 0.2 0.4 x [mm]
Energy dissipation 1 <I>/I in 0.95 0.9 inverse cascade 0.85 solitons 0.8 0 1 2 3 4 5 6 7 8 z (mm)
PDF of the intensity "#!$ 1 %&'' (&'' )&'' 0.1 solitons 0.01 0 2000 4000 6000! - intermittency due to the presence of coherent structures - relation with rogue waves?
Model equations : long-wave limit k in l ξ <<1 modified 1D-NLS
1D modified NLS in non dimensional form WT theory i ψ z = ψ 2 y 1 2 2α ψ ψ 2 1 2α 2 ψ 2 2 ψ y 2 = H ψ * This equation conserves energy and the number of photons: H = ψ 2 1 y 4α ψ 2 + 1 N = ψ 2 dx 4α 2 ψ 2 y 2 dx admits wave solutions ψ( x,z) = a k e i ( ω kz Kx) ω k = k 2 Weak NL a k =a k (z)
Canonical transformation ω 1 = ω 3 ω 1 + ω 2 ω 3 + ω 4 4 - wave interactions 4 - wave int. not possible in 1D ω 1 + ω 2 = ω 3 + ω 4 ω 3 k 3 k 4 ω 1 k 1 k 2 0 k 6 - wave interactions
Kinetic equation n k z = 9π 128α 6 f k12345 δ( k + k 1 + k 2 k 3 k 4 k 5 )δ( ω k + ω k1 + ω k2 ω k3 ω k 4 ω k5 )dk 1 dk 2 dk 3 dk 4 dk 5 n k = c k c k * 1 f k12345 = n k n 1 n 2 n 3 n 4 n 5 + 1 + 1 1 1 1 n k n 1 n 2 n 3 n 4 n 5 6-wave interaction KZ spectra n k k 1 n k k 3 5 direct energy cascade inverse wave action cascade 2 k 1 5 I k inverse intensity cascade
Numerical simulations : inverse cascade WAVE ACTION LIGHT INTENSITY 10 0 z = 0 mm z = 0 mm 10-1 n k k -3/5 I k 2 10 0 k -1/5 10-2 z = 63 mm 10-3 10 1 10 2 k [mm -1 ] 10-1 z = 63 mm 10 1 10 2 k [mm -1 ] n k = A k 2 2 " 2 % I k = $ A ' # & k I k 2 z=4.2 mm k -1/5 n k k 3 5 2 k 1 5 I k 10 0 z=0 mm 10 1 10 2 k [mm -1 ]
Numerical simulations : solitons
Numerical simulations : PDF PDF(I/<I>) 10 0 10-1 10-2 0 mm 31 mm 63 mm exp(-i/<i>) 10-3 0 2 4 6 8 10 12 I/<I>
Bogoliubov dispersion relation The homogeneous solution ( ) = A 0 e i a z A0 2 z 2 α is unstable the dispersion law is given by ω 2 = k 4 1+ A 0 2 α 2 k A 2 0 α 2
Coherent solitons over random waves 2000 Wave Soliton Intensity[a.u.] 1500 1000 500 0 0 2 4 6 8 10 x[mm]
Waves and solitons n k [a.u.] 10 3 10 2 10 1 10 0 10-1 Total Wave Soliton k -3/5 k -1 ln(kl) -1/5 k -2 10 0 10 1 10 2 k [mm -1 ] Initial Condition inverse cascade is soliton dominated direct cascade is dominated by random waves Long-time simulation with a low intensity initial condition
Soliton behaviors a) passing through each other b) merging c) dissipating
WT Life Cycle
Conclusions evidence of inverse cascade in a optical WT experiment explanation through a WT theory approach - 1D modified NLSE breakdown of WT theory : solitons through MI Further investigations experiment in larger LC systems, solitons/wave interactions 2D optical WT short-wave limit