Spatial gap solitons in dynamically induced and engineered waveguide arrays

1. Spatial gap solitons in dynamically induced and engineered waveguide arrays Andrey A. Sukhorukov Dragomir Neshev Yuri S. Kivshar Nonlinear Physics Group, RSPhysSE,Australian National University, Canberra www.rsphysse.anu.edu.au/nonlinear Wieslaw Krolikowski Laser Physics Centre, RSPhysSE www.rsphysse.anu.edu.au/nonlinear

2. Research Topic SOLITONS • Diffraction of optical beam • Nonlinear self-action www.rsphysse.anu.edu.au/nonlinear

3. Odd mode - stable Even mode - unstable 4um 4m 4  m 1  m Al0.24Ga0.76As 1.5  m Al0.18Ga0.82As 1.5  m Al0.24Ga0.76As 4.0  m Discrete solitons in waveguide arrays Experiments by the groups of Silberberg, Aitchison, and Stegeman. • Broken translational symmetry: • Peierls-Nabarro potential • Soliton trapping at high powers • Stable bound states www.rsphysse.anu.edu.au/nonlinear

4. Spatial gap solitons D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, Phys. Rev. Lett. 90, 053902 (2003). www.rsphysse.anu.edu.au/nonlinear

5. c Optically-induced lattices SBN • Optically-induced lattices created in a photorefractive crystal by interference of two or more beams • Strong electro-optic anisotropy:r33=1340pm/V r13=67pm/V v o z Recent results by the groups of Segev & Christodoulides. e www.rsphysse.anu.edu.au/nonlinear

6. Band-gaps in lattices Total internal reflection gap Bragg-reflection gaps www.rsphysse.anu.edu.au/nonlinear

7. Floquet-Bloch theory Bloch waves – linear eigenmodes of the periodic potential: Kb is the Bloch wave number, Kz is the wavevector component along the waveguides, h is the spatial period, n is the band index. www.rsphysse.anu.edu.au/nonlinear

8. Spatial Floquet-Bloch Solitons Continuous nonlinear Schrodinger equation for the Floquet-Bloch wave envelope: Solitons appear near the gap edges where D>0 www.rsphysse.anu.edu.au/nonlinear

9. Diffraction in a periodic grating Exact solution in the linear regime: Bloch spectrum B(Kb,n) characterizes decomposition into Bloch-waves. In the far field, the spatial profile maps the beam spectrum: www.rsphysse.anu.edu.au/nonlinear

10. Model equation Nonlinear Schrodinger equation for the electric field envelope: G is the nonlinear response, and V(x) is the refractive index profile www.rsphysse.anu.edu.au/nonlinear

11. Bloch-wave spectroscopy Diffraction of a Gaussian beam, incident at a Bragg angle. Theory: dispersion of Bloch waves Experiment: diffracted beam profile www.rsphysse.anu.edu.au/nonlinear

12. Bloch-wave spectroscopy Angle of incidence Bragg angle www.rsphysse.anu.edu.au/nonlinear

13. Nonlinear band interactions www.rsphysse.anu.edu.au/nonlinear

14. Beam profile at the crystal output vs. nonlinearity www.rsphysse.anu.edu.au/nonlinear

15. Bloch-Wave self-focusing Near the Bragg angle Intensity www.rsphysse.anu.edu.au/nonlinear

16. Bloch-Wave self-focusing Below the Bragg angle Intensity www.rsphysse.anu.edu.au/nonlinear

17. Gap engineering in binary arrays • an array of two types of coupled waveguides, • an optical superlattice induced by two overlapping mutually incoherent interference patterns. www.rsphysse.anu.edu.au/nonlinear

18. Discrete coupled-mode equations Tight-binding approximation - seek solution as a superposition of guided modes. Equations for the mode amplitudes:  defines the detuning between the propagation constants of the A and B-type guided modes,  characterizes the relative coupling strength between the neighboring wells. www.rsphysse.anu.edu.au/nonlinear

19. Linear Modes in Binary Arrays www.rsphysse.anu.edu.au/nonlinear

20. Discrete Gap Solitons Symmetries: A – unstable (dashed) B – stable (solid)unstable (dotted) www.rsphysse.anu.edu.au/nonlinear

21. Discretness and soliton mobility Strongly localized conventional discrete solitons become trapped [R. Morandotti et. al., Phys. Rev. Lett. 83, 2726 (1999)]. Maximum localization of discrete gap solitons is inversely proportional to the gap width: Discretness-induced trapping of gap solitons is only possible in wide gaps. www.rsphysse.anu.edu.au/nonlinear

22. Soliton motion and trapping www.rsphysse.anu.edu.au/nonlinear

23. Resonance with the gap edge, similar to fiber Bragg grating solitons. Barashenkov et al., PRL 80, 5117 (1998) Resonance with the internal-reflection gap. Sukhorukov & Kivshar, PRL 87, 083901 (2001) Soliton Stability Small amplitude perturbations on top of exact soliton profile u0 Im()0 indicates Instability www.rsphysse.anu.edu.au/nonlinear

24. Discrete vs. gap solitons www.rsphysse.anu.edu.au/nonlinear

25. Excitation of Gap Solitons Input Gaussian beam: Excited Bloch modes have opposite signs of group velocities and diffraction coefficients. Optimal excitation: 0.6. www.rsphysse.anu.edu.au/nonlinear

26. Excitation Dynamics and Switching Excitation of discrete gap solitons by inclined input beams with the peak intensities (a) 0.3, (b) 0.75, (c) 0.9, (d) 5. www.rsphysse.anu.edu.au/nonlinear

27. Results with continuous model Structure parameters: d1=4m, d2=2.5m, ds=5m. n=1.5 10-3 www.rsphysse.anu.edu.au/nonlinear

28. Holographic generation scheme Excitation of discrete gap solitons by two interfering Gaussian beams: (a) self-focusing and soliton formation, (b) self-defocusing when the interference maxima are located at narrow and wide waveguides, respectively; (c) soliton instability at high powers; and (d) soliton steering due to power imbalance. www.rsphysse.anu.edu.au/nonlinear

29. Limitation of mutual focusingthrough inter-band resonances www.rsphysse.anu.edu.au/nonlinear

30. Incoherent interactions Co-propagating beams with different polarizations or detuned frequencies. Bloch-wave envelopes near the band edges:  are the nonlinear coupling coefficients www.rsphysse.anu.edu.au/nonlinear

31. Engineered inter-band interactions Normalized self- and cross-phase modulation nonlinear coefficents between the gap edges 1=+ and 2=--. www.rsphysse.anu.edu.au/nonlinear

32. Multi-gap soliton waveguides Power vs. propagation constant for discrete solitons. Eigenvalues of the guided modes supported by the discrete solitons localized in the complimentary gap. www.rsphysse.anu.edu.au/nonlinear

33. Collision of discrete and gap solitons www.rsphysse.anu.edu.au/nonlinear

34. Multi-gap discrete solitons Powers of the soliton components in the (a) internal reflection and (b) and Bragg-reflection gaps. Dashed lines mark solutions exhibiting symmetry-breaking instability. www.rsphysse.anu.edu.au/nonlinear

35. Symmetry breaking instability www.rsphysse.anu.edu.au/nonlinear

36. Photonic Crystal Waveguides Discrete gap solitons can be formed in coupled-resonator optical waveguides in photonic crystals. S. F. Mingaleev, Yu. S. Kivshar, R. A. Sammut, Phys. Rev.E62, 5777 (2000). D. N. Christodoulides & N. K. Efremidis, Opt. Lett. 27, 568 (2002). Group of F. Lederer (unpublished) www.rsphysse.anu.edu.au/nonlinear

37. Discrete soliton networks D.N. Christodoulides and E.D. Eugenieva, Phys. Rev. Lett. 87, 233901 (2001) www.rsphysse.anu.edu.au/nonlinear

38. BEC in Optical Lattices Atomic Bose-Einstein condensate forms at extremely low temperatures. Effective nonlinearity appears due to scattering. Discrete solitons can be formed in optical lattices. www.rsphysse.anu.edu.au/nonlinear

39. Conclusions We have discussed beam diffraction, self-focusing, and soliton formation in optically-induced lattices and waveguide arrays. • Localization in periodic structures is due to total internal reflection or Bragg-reflection • Discrete gap solitons resemble conventional discrete and fiber Bragg solitons • Gap solitons can be excited by a single beam or through two-beam mutual focusing • Multi-gap solitons can form due to coupling between different bands www.rsphysse.anu.edu.au/nonlinear