Solitons in Nonlinear Photonic Lattices

Transcription

1 Solitons in Nonlinear Photonic Lattices Moti Segev Physics Department, Technion Israel Institute of Technology Jason W. Fleischer Princeton) Oren Cohen Univ. of Colorado) Hrvoje Buljan Univ. of Zagreb) Tal Carmon Caltech) Guy Bartal Barak Freedman Ofer Manela Tal Schwartz Physics Department, Technion Nikolaos K. Efremidis Jared Hudock Demetrios N. Christodoulides School of Optics/CREOL, University of Central Florida

2 Outline Nonlinear lattices in science: general problem of wave propagation, role of optics Lattice solitons Optical induction of nonlinear photonic lattices 1D spatial gap lattice solitons D lattice solitons Vorte-ring lattice solitons ; higher-band vortices Multi-band vector lattice solitons Random-phase lattice solitons Brillouin zone spectroscopy of linear and nonlinear photonic lattices Photonic Quasi-Crystals Conclusions

3 Nonlinear Lattices in Science Coupled anharmonic oscillators Optics E. Fermi, J. Pasta, and S. Ulam Los Alamos Report LA-1940 (1955). Biology: phonon energy in α-helices Advantages: Easily control input Directly image output Myoglobin 1D: D.N. Christodoulides et al., Opt. Lett. 13, 794 (1988). H.S. Eisenberg et al., PRL 81, 3383 (1998). A.S. Davydov, J. Theor. Biol. 38, 559 (1973). A. Xie et al., PRL 84, 5436 (000). D: N.K. Efremidis et al., PRE 66, (00). J.W. Fleischer et al., Nature 4, 147 (003). Atomic chain Potential applications (e.g. photonics) Charge density waves in transition metals B.I. Swanson et al., PRL 8, 388 (000). Spin waves in antiferromagnets U.T. Schwartz et al., PRL 83, 3 (1999). D.N. Christodoulides et al., PRL 87, (001).

4 Nonlinear Lattices in Science Coupled anharmonic oscillators Optics E. Fermi, J. Pasta, and S. Ulam Los Alamos Report LA-1940 (1955). Biology: phonon energy in α-helices Myoglobin Beam Envelope Diffraction Kerr NL Periodic inde ψ i + ψ + n z 1 ψ ψ + n Array ( r ) ψ = 0 Nonlinear Schrödinger equation A.S. Davydov, J. Theor. Biol. 38, 559 (1973). A. Xie et al., PRL 84, 5436 (000). Atomic chain CDW waves in transition metals B.I. Swanson et al., PRL 8, 388 (000). Spin waves in antiferromagnets U.T. Schwartz et al., PRL 83, 3 (1999).

5 Nonlinear Lattices in Science Coupled anharmonic oscillators Optics E. Fermi, J. Pasta, and S. Ulam Los Alamos Report LA-1940 (1955). Biology: phonon energy in α-helices Myoglobin Beam Envelope Diffraction Kerr NL Periodic inde ψ i + ψ + n z 1 ψ ψ + n Array ( r ) ψ = 0 Nonlinear Schrödinger equation A.S. Davydov, J. Theor. Biol. 38, 559 (1973). A. Xie et al., PRL 84, 5436 (000). Bose-Einstein Condensates Gross-Pitaevskii (NLS) equation ~~~~~ ~~~ ~~~~ ~ ~~ Atomic chain Mean-field interactions Periodic potential CDW waves in transition metals B.I. Swanson et al., PRL 8, 388 (000). Spin waves in antiferromagnets U.T. Schwartz et al., PRL 83, 3 (1999). iħ ψ = ħ ψ + U t m ψ + V ()ψ A. Trombettoni and A. Smerzi, PRL (001). B. Eiermann et al., PRL 9, (004). 0 ψ r

6 Linear Waves in Periodic Media i ψ + z ψ + V () = 0 ψ iβ z (, z) = Φ( ) e Homogeneous: V = 0 Lattice: V(+d) = V(d) Translational symmetry Fourier basis β { e ik } [ β k /k ] z k Quadratic wavefront (Bloch) momentum k θ = k π/d z θ k k z Periodicity Floquet-Bloch basis u ( + d) u ( ) k = β k () { u ik } k e k π/d

7 Linear Waves in Periodic Media The modes (Floquet-Bloch waves) are etended waves Transmission spectrum is divided into bands Mode characterization: Band inde and Bloch wavenumber k {-π/d < k < π/d} -π/d β a d 1 b c π/d k Brillouin zone 1 V FB modes of single-mode WG array k = 0 k = π/d 0 Consider array where each WG supports a single guided mode (For individual WG, higher modes would radiate away) 1 st band nd band a d b c

8 Linear Transport in Lattices Normal of wavefront defines direction of transport (ray): θ = z k = k Divergence of rays gives diffraction: z n=1 n= -π/d β k z k π/d Homogeneous ψ D k z k n=3 Periodic Normal Diffraction + Anomalous Eisenberg et al., PRL 85, 1863 (000).

9 Linear Transport in Lattices Normal of wavefront defines direction of transport (ray): θ = z k = k Divergence of rays gives diffraction: z n=1 n= -π/d β k z k π/d Homogeneous ψ D k z k n=3 Periodic General definition of transport: Effective mass in lattice Dispersion of temporal pulse v gr 1 1 m eff ε k D 1 ω v gr = k = ω ω k ω grating

10 Transport in Lattices Normal of wavefront defines direction of transport (ray): θ = z k = k Divergence of rays gives diffraction: z n=1 n= -π/d β k z k π/d Homogeneous ψ D k z k n=3 Periodic Normal: self-focusing nonlinearity D.N. Christodoulides et al., Opt. Lett. 13, 794 (1988). H.S. Eisenberg et al., PRL 81, 3383 (1998). Soliton formation with focusing NL Anomalous: self-defocusing nonlinearity Y.S. Kivshar, Opt. Lett. 18, 1147 (1993). J.W. Fleischer et al., PRL 90, 0390 (003).

11 Linear transport of a wave-packet in a lattice Lattice transport Homogeneous system Develops two lobes Discrete Model: den i + C( En 1+ En + 1) = 0 dz Transverse displacement Gaussian profile stays Gaussian Coupling between nearest-neighbors n-1 n n+1 d Analogous to mass-spring system A.L. Jones, J. Opt. Soc. Am. 55, 61 (1965). Treats only bound states decay between sites No radiation modes Tight-binding approimation Continuous models are more general

12 Non-linear Transport in Lattices Propagation distance Transverse displacement Discrete soliton Propagation distance den i + C( En 1+ En + 1) + γ En En = 0 dz Lattice ( discrete ) solitons: when nonlinearity balances diffraction In-phase k = 0 Christodoulides et al. (1988). Eisenberg et al., (1998). on-ais Staggered (GAP) k = π/d Kivshar, (1993). Fleischer et al., (003) θ = k k k k z Edge of Brillouin zone

13 Lattice ( discrete ) solitons Propagation distance Transverse displacement Discrete soliton Propagation distance den i + C( En 1+ En + 1) + γ En En = 0 dz In-phase k = 0 D.N. Christodoulides et al. (1988). H.S. Eisenberg et al., (1998). Gap (Staggered) k = π/d Y. Kivshar, (1993). J.W. Fleischer et al., (003) β Lattice soliton is bound state of its own self-induced defect -π/d π/d k β β β β + β β β β β β β β β β β β

14 Zoology of Lattice Solitons β -π/d More recent: π/d k In-phase k = 0 D.N. Christodoulides and R.I. Joseph, Opt. Lett. 13, 794 (1988). H.S. Eisenberg et al. PRL 81, 3383 (1998). Gap (Staggered) k = π/d D: J.W. Fleischer et al., Nature 4, 147 (003) Y.S. Kivshar, Opt. Lett. 18, 1147 (1993). J.W. Fleischer et al. PRL 90, 390 (003) - Lattice solitons in quadratic media Stegeman s group Lattice solitons in liquid crystals Assanto s group Modulation instability Stegeman s group 004 Twisted (dipole) 0 < k << π/d Darmanyan et al., Sov. Phys. JETP 86, 68 (1998) Neshev et al., Opt. Lett. 9, 710 (003) D: Yang et al., Opt. Lett. 9, 166 (004) Vector at k =0 Darmanyan et al., Phys. Rev. E, 57, 350 (1998) Meier et al., PRL 91, (003). D: Chen et al., Opt. Lett. 9, 1656 (004) Theory Eperiment

15 Finding Lattice Solitons: the self-consistency method Homogeneous The soliton is a guided mode of its own induced waveguide Intensity Lattice The soliton is a localized mode of the full potential (lattice + induced defect) Intensity I n Guided mode n Idea: Askar yan, 196 Formulation: Snyder et al First use: Mitchell, Segev, Christodoulides 1997 Localized mode + V () + I Φ = β Φ Full potential (lattice + induced defect) I = Φ n = n(i) O. Cohen et al. PRL 91, (003).

16 Self-Consistency: nd -band lattice soliton The soliton is a localized mode of the full potential (lattice + induced defect) Single radiation mode from second-band O. Cohen et al. PRL 91, (003).

17 Optical Solitons in Nonlinear Waveguide Arrays D.N. Christodoulides and R.I. Joseph, Opt. Lett. 13, 794 (1988). T. Pertsch et al., OSA Trends in Optics and Photonics 80 (00). H.S. Eisenberg, Y. Silberberg, R. Morandotti, A.R. Boyd, and J.S. Aitchison, PRL 81, 3383 (1998). Motivation Previous eperimental configurations: Fied waveguide arrays 1D topologies Self-focusing nonlinearity only certain classes of solitons Challenges: Reconfigurable lattices D lattices collisions, angular momentum All classes of solitons allowed

18 Transition to D How? Direct manufacturing, etching - difficult - possible on microwave scale, but linear - unknown process to date Naturally-occurring D structure - atomic scale -ray (again, linear) - unlikely - none found so far All-optical - induction technique (à la holography) material { Strong nonlinear response solitons at low intensity (mw) Strength and sign of nonlinearity adjustable Dynamically adjustable lattice spacing and potential well depth

19 Optically-induced Waveguide Arrays Requirements Array propagates linearly Probe feels periodic potential, NL self-focusing Want strong optical anisotropy e.g. photorefractive (SBN-75) crystal c-ais Create WGs by interfering pairs of plane waves polarize to c-ais V Polarize probe to c-ais Apply Voltage to c-ais I grating :I probe ~ 5-10:1 Sharpens array Nonlinearity: n ~ 10-3 NL requires very low power, but has slow response time V > 0 focus, V < 0 defocus

20 Optical Induction of D Array of D Waveguides Interfere pairs of plane waves to create dynamic D array Crystal output 15mW in each plane wave I grating :I probe = 5:1 Spatial scale of individual waveguides is small, but must consider diffraction of broad (in-phase) plane waves. Representative square array WG diam = 7µm D=11µm Strong anisotropy of crystal allows distinction between induced lattice (waveguide array) and signal beam

21 On-ais Propagation in D NL Photonic Lattice Lattice Diffraction (00V) D Lattice Soliton (800V) Soliton simulation 8 reduction of I probe Interference with plane waves (800V) J.W. Fleischer et al., Nature 4, 147 (003) 8 red. of I probe at same voltage Interferogram

22 D Propagation at Edge of Brillouin Zone Lattice Diffraction (-00V) D Lattice Soliton (-800V) Soliton simulation 8 reduction of I probe Interference with plane waves (-800V) J.W. Fleischer et al., Nature 4, 147 (003) 8 red. of I probe at same voltage Interferogram

23 Spatial Gap Solitons From 1 st band with defocusing nonlinearity β 1D: J.W. Fleischer et al., PRL 90, 0390 (003). -π/d π/d k D: J.W. Fleischer et al., Nature 4, 147 (003) β β β β β β β β From nd band with focusing nonlinearity β -π/d π/d k β + β D. Mandelik et al., PRL 90, (003). β β β β β β

24 Vorte-Ring Lattice Solitons Advantage of D: solitons with angular momentum But lattice breaks continuous rotational symmetry of homogeneous medium in general, angular momentum in lattice not conserved On-site Off-site Prediction of vorte lattice solitons: Malomed & Keverkidis, PRE 001 Yang & Musslimani, Opt. Lett. 003 Diffraction Eperiments: Intensity J.W. Fleischer et al., PRL 9, (004) * see also Neshev et al. PRL 9, (004). Phase π/ π 0, π 3π/ π/ π 0, π 3π/

25 nd Band vorte lattice soliton D Lattice Transmission spectrum X M X In the nd band the edge of BZ is at the X points Soliton intensity Phase structure array of counter-rotating vortices Diffraction intensity Manela, Cohen, Bartal, Fleischer, and Segev. Opt. Lett. 17, 049 (004)

26 Evolution from a ring to a nd Band vorte lattice soliton Ecitation (input) Input intensity Input phase (interferogram) Power spectrum Diffraction Soliton intensity Output Soliton phase (interferogram) Power spectrum

27 1 st band vorte vs. nd band vorte Ecitation (input) diffraction Soliton 1 st band k-space Phase information (interferogram) nd band Bartal, Manela, Cohen, Fleischer, and Segev, submitted to PRL, Feb. 005

28 Multiband vector lattice solitons β Intensity Self-consistency method -π/d k π/d n Fundamental + nd -band mode self-trap as vector soliton Localized modes Full potential (lattice + induced defect) + V Φ () + I i = βiφi I = c i Φi i = { n, k } i Have vector soliton with components from different bands, each with same curvature O. Cohen et al. PRL 91, (003). [similar work: Sukhorukov et al., PRL 91, (003)]

29 Multi-band vector lattice solitons Self-focusing nonlinearity β 1 Richer dynamics than traditional solid-state (with transitions between passive bands) -π/d π/d k Both bands contribute actively to NL Bands interact with each other (through I) O. Cohen et al. PRL 91, (003). 1 st 40 0 Linear diffraction Vector soliton Individual with NL Noisy intensities at input nd BPM Results z 150 Intensities at output (on unperturbed sols.)

30 Multi-band vector lattice solitons Self-focusing nonlinearity Self-defocusing nonlinearity -π/d β 1 π/d k β k 1 -π/d π/d 1 st 40 0 Linear diffraction Vector soliton Individual with NL BPM Results 50 Linear diffraction Vector soliton Individual with NL nd z z 150

31 Spatially-incoherent lattice solitons All previous research on lattice solitons performed with coherent waves Phase is perfectly correlated (space & time) staggered, vorte Propagating waves undergo multiple reflections Interference phenomena drive dynamics greatly affected by coherence of waves l c Eplore with statistical (spatially-incoherent) light Correlation length l c vs. lattice spacing d Random nature of light vs. periodic constraint d ψ (, z, t) ψ *(, z, t) τ 1 f ( ) = f ( + d) (Mutual correlation) (Bloch s theorem) Have lattice soliton with randomly-varying phase front

32 System: partially coherent light in nonlinear waveguide arrays Spatially incoherent light beam 1 Periodically modulated noninstantaneous nonlinear medium z State of the system (intensity and statistics at a given z) B( * 1,, z) = E(1,z,t)E (,z,t) Equation of motion for the mutual coherence function: B i + z k n 0 1 k Diffraction B B + { δn[b(,,z)] δn[b(, Nonlinear term k n 0 {p( 1 ) τ m Periodicity,z)]}B( p( 1, )}B +,z) = 0

33 Typical results: Random-phase lattice solitons Transmission spectrum Power spectrum β I Fourier l c -π/d π/d k I Floquet -Bloch k π/d Note l s () = l s (+md) Interpretation using modal theory n Many modes low l c Slowest-decaying mode higher l c Mutual coherence/correlation function H. Buljan et al., Physical Review Letters 9, 3901 (004). µ = ψ n () ψ ( + δ ) n ()( + δ ) I I X

34 Typical results: Random-phase lattice solitons Transmission spectrum Power spectrum β I Fourier l c -π/d π/d k I Floquet -Bloch k π/d Note l s () = l s (+md) Intensity profiles, power spectra, and statistics (coherence properties) all conform to the lattice periodicity Mutual coherence/correlation function H. Buljan et al., Physical Review Letters 9, 3901 (004). µ = ψ n () ψ ( + δ ) n ()( + δ ) I I X

35 Incoherent lattice solitons Eperimental setup 1. Optical induction of lattice 3. Imaging into CCD camera a. Real space. Spatially incoherent (probe) beam 4f system b. Fourier space laser Rotating diffuser Spatial filter aperture controls power spectrum f f Optical Fourier transform

36 Fourier space and diffusion reflection Input beam Array beams define Bragg angles input No diffuser (coherent) With diffuser (incoherent) nd nd 1 st zone nd nd Fourier space Points from array Brillouin zones Homogeneous Diffraction in homogeneous medium determined by λ L λ vs w. λ l c Real Space output Diffraction coherent incoherent

37 Incoherent lattice solitons Eperimental results Input Diffraction Random-phase lattice soliton Real space 1-peak -peak d = 11.5 microns laser Rotating diffuser nd nd Brillouin zone ecitement!! correlation length l c ~ d 1 st zone nd nd Brillouin zones Incoherent probe Soliton output

38 Varying the spatial correlation distance Correlation length l c ~ d Input Real space Input Fourier space Low intensity Diffraction Soliton Real space Soliton Fourier space Correlation length l c ~1. d d = 11.5 microns Input Real space Input Fourier space Zero voltage Diffraction Output Real space Output Fourier space Lattice off

39 Gap Random-Phase Lattice Solitons (under self-defocusing nonlinearity) Input Real space Input Fourier space Low intensity Diffraction Soliton Real space Soliton Fourier space What happens if we apply a self-focusing nonlinearity on an input with a square hole in the spectrum?

40 General technique: Bloch-wave spectroscopy Eperimental lattice Theoretical Brillouin zones Spatially-incoherent probe beam 4-fold symmetry 3 rd 3 rd 3 rd 3 rd nd nd 1 st zone nd n 3 rd 3d rd 3 rd 3 rd 3-fold symmetry Trigonal lattice wide in both real space and Fourier space Image of output Made in Israel

41 Photonic lattices with defects Our eperiments Photonic Crystal Fibers Real space K-space P.S. Russell, Science 99, 58 (003). No defect Far-field dispersion Real space Positive defect Guided Modes Negative defect

42 Bloch-wave spectroscopy: Nonlinear effect nd nd 1 st zone nd nd k z De-focusing Focusing π/d π/d k Nonlinearity: Echanges energy between linear modes Sensitive to band curvature Maps regions of different transport Nonlinearity off Self focusing nonlinearity De-focusing nonlinearity

43 Photonic Quasi-Crystals Penrose Tiling Interference of 5 plane waves a Linear Diffraction b 30µm 35µm Under self-focusing b a 85µm Towards solitons in Quasi-Crystals Theory Eperiment 35µm

44 The k-space picture of photonic Quasi-Crystals Quasi Brillouin Zones Theory Eperiment Same scattering at different spots Statistical similarity Crystal

45 Summary: recent group progress on waves in nonlinear photonic lattices Theory: self-consistency method + multi-band lattice solitons [PRL 91, (003)] analysis of D lattice solitons [PRL 91, (003)] grating-mediated waveguiding + first eperiment [PRL 93, (004)] incoherent (random-phase) lattice solitons [PRL 9, 3901 (004)] second-band vorte lattice solitons [Opt. Lett. 17, 049 (004)] First eperimental demonstration, in any medium 1D spatial gap lattice solitons [PRL 90, 0390 (00).] D in-phase and gap lattice solitons [Nature 4, 147 (003).] vorte-ring lattice solitons [PRL 9, (004).] incoherent (random-phase) lattice solitons [Nature, Feb. 005] D second-band (vorte) lattice solitons [submitted to PRL, Feb. 005] random-phase gap lattice solitons [in preparation, Feb. 005] Technique of Bloch-wave spectroscopy of photonic lattice [submitted to PRL, Dec. 004] Brand new results: photonic quasi-crystals, Penrose Tiling, etc.

46 Summary Induction technique allows D lattices Optics allows direct and k-space imaging Optical physics + general physics using optics Theory and eperiment ongoing