Wave Turbulence and Condensation in an Optical Experiment

Transcription

1 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

2 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

3 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?

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5 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 α ψ 4 dr energy M = 1 2 i ( ψ ψ* ψ * ψ) dr momentum can be taken zero in homogeneous turbulence

6 Optical Wave Turbulence D =2 i t E + γe ( ) + xx ( 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 dk 2 dk wave interaction, N and E conserved

7 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 )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

8 Direct and inverse cascade α<0 (defocusing) Inverse cascade of particles : photon condensation α>0 (focusing) Filament formation : solitons

9 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

10 Optical WT experiment Setup based on light propagation in liquid crystals low dissipation - weak nonlinearity Experimental evidence of inverse cascade Solitons Theory and numerics

11 Liquid crystals Nematic phase Molecular reorientation ne=1.7 no=1.5 E #"!" Focusing nonlinearity

12 Optical nonlinearity and solitons in LC I FT kw cm 2

13 Solitons in nematics: nematicons nematicons : balance diffraction/focusing nonlinearity of LC NLS E v X Y Z

14 Solitons in nematics: nematicons nematicons : balance diffraction/focusing nonlinearity of LC NLS E v X Y Z

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16 WT experiment in nematics Nonlinearity is provided by the reorientational Kerr effect of the LC A k ( ) "! 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 n 2 2 e n o 2 = 0 external electric field optical field

17 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 n 2 2 e n o 2 = 0 rescaled variables l " = #K 2$% d V

18 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

19 Prepare the beam at z=0 Initial condition A(z=0) Qin Input 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

20 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)

21 Intensity profiles

22 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 k [mm -1 ]

23 Modulational instability and solitons x [mm] 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) mm mm 0.3 mm x [mm]

24 Energy dissipation 1 <I>/I in inverse cascade 0.85 solitons z (mm)

25 PDF of the intensity "#!$ 1 %&'' (&'' )&'' 0.1 solitons ! - intermittency due to the presence of coherent structures - relation with rogue waves?

26 Model equations : long-wave limit k in l ξ <<1 modified 1D-NLS

27 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 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)

28 Canonical transformation ω 1 = ω 3 ω 1 + ω 2 ω 3 + ω 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

29 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 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

30 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 k -1/ z = 63 mm k [mm -1 ] 10-1 z = 63 mm 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 k 1 5 I k 10 0 z=0 mm k [mm -1 ]

31 Numerical simulations : solitons

32 Numerical simulations : PDF PDF(I/<I>) mm 31 mm 63 mm exp(-i/<i>) I/<I>

33 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

34 Coherent solitons over random waves 2000 Wave Soliton Intensity[a.u.] x[mm]

35 Waves and solitons n k [a.u.] Total Wave Soliton k -3/5 k -1 ln(kl) -1/5 k 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

36 Soliton behaviors a) passing through each other b) merging c) dissipating

37 WT Life Cycle

38 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

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