Observation of two-dimensional Anderson localization of light in disordered optical fibers with nonlocal nonlinearity

Transcription

1 Observation of two-dimensional Anderson localization of light in disordered optical fibers with nonlocal nonlinearity Claudio Conti Institute for Complex Systems National Research Council ISC-CNR Rome (IT)

2 ISC CNR Dep Physics Sapienza (IT) Marco Leonetti Viola Folli University of Wisconsin Milwuakee Salman Karbasi & Arash Mafi

3 Outline Introduction Experiments Disordered fiber Multi-color disorder localized states Power dependent localization length Evidence of a nonlocal effect Theory

4 Above a certain amount of disorder no transport is possible Anderson localization The reason: localized states due to disorder

5 Literature Observation of Anderson localization in Nonlinear Optics Y. Lahini et al. PRL 100, (2008) T. Schwartz, G. Bartal, S. Fishman, M. Segev, Nature 446, 52 (2007) Bose-Einstein condensation J. Billy et al. Nature 453, 891 (2008) G. Roati et al. Nature 453, 895 (2008) S. S. Kondov, Science 66, 334 (2011) Linear disordered media (optics) M. Storzer, P. Gross, C. M. Aegerter, G. Maret, PRL 96, (2006) A. A. Chabanov, M. Stoytchev, A. Z. Genack, Nature 404, 850 (2000) T. Sperling at al, Nature Photonics 7, 48 (2013)

6 1D Bosons (BEC) Billy et al. & Roati et al. Nature 2008 Localization length versus strenght of disorder

7 TRANSVERSE Anderson Loc T. Schwartz, G. Bartal, S. Fishman, M. Segev, Nature 446, 52 (2007) INDEX CONTRAST PROPAGATION 1cm

8 The simplest model

9 Linearly localized states Gaussian potential Negative eigenvalues Decays as Link between localization length and eigenvalue See book Lifshitz, Gredskul, Pastur Introduction to theory of disordered systems Also CC, PRA 86, (2012) E=-5 V0=4

10 The statistical distribution of eigenvalues There is a tail of negative energies corresponding to exponentially highly localized states Distribution of negative eigenvalues V0=2 The localization length decreases as the Inverse square root of the energy, hence the localization length decreases with the amount of disorder (as observed experimentally) Energy E

11 Including nonlinearity

12 Nonlinearity : theory Effect of nonlinearity and disorder Scattering theory (Gredeskul, Kivshar & others) Chaos (Flach & others) Perturbation theory on Lyapunov exponents (Fishman & others) Spin glass theory (Leuzzi, Conti & others) Self consistent approaches (Tureci & others) Scaling laws (Skipetrov & others) FDTD (Sebbah, Conti & others) Many others... COMPARISON WITH EXPERIMENTAL DATA IS LIMITED The simplest thing to do: Measure the localization length Versus nonlinearity

13 Question: Disorder induced states are un coupled (absence of transport, Anderson regime) What happens in the presence of a long range interaction? Hypothesis: localized states interact We want an experimental evidence!

14 Transverse localization in 2D fibers Our experiments on transverse localization in two dimensional fibers

15 Mixture of PS and PPMA Index contrast 0.1 Propagation >7 cm pieces of PMMA and pieces of PS randomly mixed and fused together n(ps)=1.59 n(pmma)=1.49

16 Calculated mode Observed mode 250 microns

17 Multicolor transverse Anderson-localization - we excite several localizations at different wavelengths simultaneously

18 FIBER TYPICAL OUTPUT SETUP We Wemeasure measurethe the localization localization profile profileatatany any spectral peak spectral peak

19 Output Spectrum Input spectrum At Atany anyspatial spatiallocation location there are several there are several localized localizedmodes modesatat different differentfrequencies frequencies 5mm

20 Nonlinear regime - at any wavelength we study the localization profile Vs power

21 5 mm Mode profile at 820nm Mode at 820nm Mode at 835nm We Weobserve observe focalization focalizationofofany anyofof the thelocalized localizedmode mode when incresing when incresingpower power

22 2D SELF-FOCUSING of Anderson localizations Experiments Localization length Versus Intensity (50 modes) Numerically calculated bound states of the 2D-NLS with Gaussian disorder Theory from the variational approach Folli, Conti, OL 2011 Conti, PRA, 2012

23 Action at a distance between Anderson localizations in nonlinear nonlocal media - thermal nonlinearity is nonlocal

24 MODIFIED SETUP

25 Probe Anderson mode (532nm) 20 microns Pump Anderson Mode (800nm)

26 The size of the probe changes with the pump power! Probe Anderson mode (532nm) PUMP PROBE 25 microns DISPLACEMENT LOCALIZATION

27 The migration of the multicolor Anderson localization A form of transport in the Anderson regime

28 Density map of localizations We count the states in any spatial location 300 microns y x Here localizations FIBER OUTPUT 25 microns

29 Density map of locs Vs power y x 25 microns

30

31 Leonetti et al, Phys. Rev. Lett., 112: (2014)

32 THEORY

33 (transverse) Anderson localization in nonlocal media Link between localization length and power

34 Comparison with experiments At low power : linear trend

35 Modelling the action at a distance Using collective coordinates in the highly nonlocal approximation

36 Equation for the positions Ehrenfest theorem (CC, PRE 72, ) The position of the localization p varies because of nonlinearity The localizations are incoherent Pairwise interaction potential Leonetti et al, Phys. Rev. Lett., 112: (2014)

37 Comparison with experiments Pump and probe Anderson states We consider two states Leonetti et al, Phys. Rev. Lett., 112: (2014)

38 Conclusions Nonlinearity and nonlocality in 2D disorder fibers Action at a distance Transport in the Anderson regime Incoherent Anderson states Variational theoretical approaches THANKS! Leonetti et al, Phys. Rev. Lett., 112: (2014) ; Nature Communications 5, 5534 (2014)

39 Calculated vectorial modes Index distribution Many modes at any spatial location

40 Which the origin of the observed nonlinear focusing? - it's thermal - but thermal is defocusing!

41 Time dynamics of the focusing effect Timescale is compatible with thermal effects (PMMA and PS absorb the infrared light)

42 Finite Element Modelling of temperature profile in the fiber

43 FEM SIMULATION OF THE THERMAL DIFFUSION EQUATION PMMA PS

44 TEMPERATURE IS DIFFERENT IN THE TWO MATERIALS!!!!!! (Temperature is localized)

45 Our interpretation Due to the different thermal properties, temperature increases in the PMMA more than in PS This increases the index contrast and hence the strength of disorder and reduces the localization length SO WE HAVE: A FOCUSING ACTION DUE A DEFOCUSING THERMAL NONLINEARITY!!!! (because of disorder...)

46 Comparison with ordered fibers PPMA PS PS+PMMA input Here Wavelength 1064nm

47 The effect of nonlinearity on the 2D Anderson localization profile T. Schwartz, G. Bartal, S. Fishman, M. Segev, Nature 446, 52 (2007) LOCAL NONLINEARITY