1. What Are Solitons, Why Are They Interesting And How Do They Occur in Optics? George Stegeman KFUPM Chair Professor Professor Emeritus College of Optics and Photonics, Un. Central Florida, USA Material Requirement: The phase velocity of a beam (finite width in space or time) must depend on the field amplitude of the wave! High Power Low Power courtesy of Moti Segev

2. Space: Broadening by Diffraction Time: Broadening by Group Velocity Dispersion All Wave Phenomena: A Beam Spreads in Time and Space on Propagation Broadening + Narrowing Via a Nonlinear Effect = Soliton (Self-Trapped beam) Spatial/Temporal Soliton 1.An optical soliton is a shape invariant self-trapped beam of light or a self-induced waveguide 2.Solitons occur frequently in nature in all nonlinear wave phenomena 3.Contribution of Optics: Controlled Experiments

3. Solitons Summary solitons are common in nature and science any nonlinear mechanism leading to beam narrowing will give bright solitons, beams whose shape repeats after1 soliton period! solitons are the modes of nonlinear (high intensity) optics robustness (stay localized through small perturbations) unique collision and interaction properties Kerr media no energy loss to radiation fields number of solitons conserved exhibit both wave-like and particle-like properties I(x) x Δn(x) x I(x)  Δn(x) = n 2 I(x) Δn(x) traps beam Saturating nonlinearities small energy loss to radiation fields depending on geometry, number of solitons can be either conserved or not conserved. Self-consistency Condition

4. 1D Bright Spatial Soliton Optical Kerr Effect → Self-Focusing: n(I)=n 0 +n 2 I, n 2 >0 Soliton Properties 1.No change in shape on propagation 2.V p (soliton) < V p (I  0) 3.Flat (plane wave) phase front 4.Nonlinear phase shift  z (not obvious) Soliton! Diffraction in space Phase front Diffraction in 1D only! Self-focusing x z n 2 >0 I(x)I(x) V p (I>0 ) V p (I  0) V p (I  0)>V p (I>0) phase velocity:

5. Soliton John Scott Russell in 1834 was riding a horse along a narrow and shallow canal in Scotland when he observed a “rounded smooth well-defined heap of water” propagating “without change of form or diminuation of speed” First “Published” Scientific Record of Solitons Russell, J. S., 1838, Report of committee on waves. Report of the 7-th Meeting of British Association for the Advancement of Science, London, John Murray, 417-496.

6. Union Canal, Edinburgh, 12 July 1995. Soliton on an Aqueduct

7. Solitons in Oceans: The “Rogue” Wave N. Akhmediev, A. Ankiewicz, and M. Taki, “Waves that appear from nowhere and disappear without a trace”, Physics Letters, A 373 (2009) 675–678.

8. Soliton Sightings by Weather Satellites and/or Weather Planes

9. Optical Solitons Spatial Temporal Spatio-Temporal Homogeneous MediaDiscrete Media 1D, 2D Propagating Solitons Cavity Solitons Media Local Non-local Photorefractive Kerr  n=n 2 I Kerr-like Quadratic Gain Media Liquid Crystals

10. Optical Solitons Temporal Solitons in Fibers Spatial Solitons 1D Discrete Spatial Solitons 1D Spatial Solitons 2D nonlinearity NOT Kerr Two color solitons Quadratic nonlinearity Supported by Kerr nonlinearity  n NL = n 2 I h eff  Field distribution along x-axis fixed by waveguide mode n2n2 n1n1 n2>n1n2>n1 Field distribution along x-axis fixed by waveguide mode n2n2 n1n1 n2>n1n2>n1

11. Nonlinear Wave Equation Slowly varying phase and amplitude approximation (SVEA,1 st order perturbation theory) depends on nonlinear mechanism diffraction nonlinearity Group velocity dispersion Nonlinear Mode Spatial soliton Shape invariance + Zero diffraction and/or dispersion Plane Wave Solution?  Unstable mode  Filamentation

12. 1D Kerr Solitons:  n NL = n 2 I= n 2,E |E| 2 Bright Soliton, n 2 >0 2(w 0,T 0 ) All other nonlinearities do NOT lead to analytical solutions and must be found numerically! x, T Invariant shape on propagation Nonlinear phase shift “Nonlinear Schrödinger Equation” “NLSE” diffraction nonlinearity dispersionnonlinearity

13. Stability of Kerr Self-Trapped Beams in 2D? Diffraction length Nonlinear length (  /2) h w0w0 1 D Waveguide Case constant 2 D Bulk Medium Case w0w0 No Kerr solitons in 2D! BUT,2D solitons stable in other forms of nonlinearity Fluctuation in power leads to either diffraction or narrowing dominating

14. Higher Order Solitons - Previously discussed solitons were N=1 solitons where - Higher Order solitons obtained from Inverse Scattering or Darboux transforms 0.2 0.4 1.0 0.8 0.6 0 -10 10 4 2 0 Intensity N=3 Need to refine “consistency condition”. Soliton shape must reproduce itself every soliton period!

15. Zoology of Spatial Soliton Systems Soliton Type# Soliton ParametersCritical Trade-Off 1D Kerr1*1* Diffraction vs self-focusing 1D & 2D Saturating Kerr 1*1* Diffraction vs self-focusing 1D & 2D Quadratic2†2† Diffraction vs self-focusing 1D & 2D Photorefractive 1*1* Diffraction vs self-focusing 1D & 2D Liquid Crystals 1*1* Diffraction vs self-focusing 1D & 2D Dissipative0Diffraction vs self-focusing + Gain (e.g. SOA) vs loss † Two of peak intensity, width and wavevector mismatch * Peak intensity or width 1D & 2D Discrete Arrays of coupled waveguides 0, 1, 2Discrete diffraction vs self-focusing (or defocusing)

16. White Light (Incoherent) Photorefractive Solitons 12  m Self-Trapped Output Beam with Voltage Applied 82  m Diffracted Output Beam 14  m Input Beam M. Mitchell and M. Segev, Nature, 387, 880 (1997) But aren’t solitons supposed to be coherent beams? Most are, BUT that is NOT a necessary condition! Why? Because the nonlinear index change required depends on intensity I i.e.  n  |E| 2 not E 2 ! No coherence required!

17. Optical Bullets: Spatio-Temporal Solitons Electromagnetic pulses that do not spread in time and space Require: dispersion length (time)  diffraction length (space)  nonlinear length Characteristic Lengths tt x

18. 600 400 200 0 05 10 15 20 25 Propagation Distance Pulse Duration (fs) Dispersion 05 10 15 20 25 Propagation Distance 200 0 300 100 Beam Waist (  m) Diffraction along soliton dimension Diffractive Broadening Dispersive Broadening Spatiotemporal Soliton ”Light Bullet Quasi-1D Optical Bullets: Frank Wise’s Group x y z x y

19. Particle or Wave? Kerr Nonlinearity: Remains Highly Spatially Localized Number of Particles Conserved on Collision Diffraction Interference Refraction BOTH!

20. Coherent Kerr Soliton Collisions: Particles or Waves? Phase  1 Phase  2  =0  =   =  /2  =3  /2 1.Number of solitons in = Number of solitons out particle-like behavior 2.For  0,  also wave-like behavior - energy exchange occurs via nonlinear mixing Incoherent Soliton Interaction

21. Soliton Collisions  Soliton “Birth”: Non-Kerr Media horizontal colliding angle 0.9 0 in vertical plane not collided center to center (vertical center to center separation 10  m) Soliton birth – a third soliton appears! Observed at Output

22. Control gain versus loss by adjusting width of electrode strips Diffraction vs self-focusing + Gain (e.g. SOA) vs loss Dissipative Solitons: AlGaAs Semiconductor Optical Amplifier

23. Waveguide Arrays: Discrete Solitons Discrete diffraction

24. Theoretical prediction: Nonlinear surface waves exist above a power threshold! Discrete Spatial Surface Solitons Input power is increased slowly and output from array is recorded Single channel soliton >50% of power at output In input channel Without normalization Observation plane Input beam Single channel excitation

25. 1.Two discrete interface solitons with power thresholds propagate along 1D interfaces 2.In 1D, two different surface soliton families exist with peaks on or near the boundary channels. One family experiences an attractive potential near the boundary, and the second a repulsive potential. 3.Single channel excitation can lead to the excitation of single channel solitons peaked on channels different from the excitation channel. Interface Solitons Between Two Dissimilar Arrays

26. 2D Edge and Corner Discrete Solitons Corner soliton Edge soliton K.G. Makris, J. Hudock, D.N. Christodoulides, G.I. Stegeman M. Segev et. al, Opt. Lett. 31, 2774-6 (2006).

27. Experiment: A. Szameit, et. al., Phys. Rev. Lett., 98, 173903 (2007); Z. Chen, et. al., Phys. Rev. Lett., 98, 123903 (2007) 2D Edge and Corner Discrete Solitons: Experiment Theory Experiment Power Soliton Intensity Profile Excitation channel Discrete Diffraction Edge Soliton

28. Solitons Summary solitons are common in nature and science any nonlinear mechanism leading to beam narrowing will give bright solitons, beams whose shape on propagation is either constant or repeats after 1 soliton period! they arise due to a balance between diffraction (or dispersion) and nonlinearity in both homogeneous and discrete media. Dissipative solitons also require a balance between gain and loss. solitons are the modes (not eigenmodes) of nonlinear (high intensity) optics an important property is robustness (stay localized through small perturbations) unique collision and interaction properties Kerr media no energy loss to radiation fields number of solitons conserved exhibit both wave-like and particle-like properties Saturating nonlinearities small energy loss to radiation fields depending on geometry, number of solitons can be either conserved or not conserved. Solitons force you to give up certain ideas which govern linear optics!!