Intrinsic Localized Modes in Nonlinear Photonic Crystals

1. Intrinsic Localized Modes in Nonlinear Photonic Crystals Sergei Mingaleev Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine In collaboration with Kurt Busch # and Yuri Kivshar $ # Institut für Theorie der Kondensierten Materie University ofKarlsruhe, Germany $ Nonlinear Physics Group, Australian National University Canberra, Australia NATO ARW -- Erice, Sicily

2. Credits: • Development of the Wannier functions approach for PCs: • with Kurt Busch, Matthias Schillinger, Daniel Hermann (all from Uni-Karlsruhe, Germany), and Antonio Garcia-Martin (Inst. Microelectr. Madrid, Spain). • Investigations on Nonlinear Photonic Crystals: • withYuri S. Kivshar (ANU, Australia). • Investigations on Nonlinear Schrödinger Models with Long-Range Interactions: • with Yuri B. Gaididei (BITP, Ukraine), Peter L. Christiansen (DTU, Denmark), Franz Mertens (Uni-Bayreuth, Germany), Magnus Johansson (Linköping Uni, Sweeen), and Kim Rasmussen (Los Alamos Lab, USA). NATO ARW -- Erice, Sicily

3. Outline of the talk • Systems: • -- Brief overview of the photonic crystals… • (semiconductors + liquid crystals=?) • Modeling tools: • -- Wannier functions approach… • Results: • -- Single nonlinear cavity – frequency shift… • -- Two coupled nonlinear cavities – symmetry breaking… • -- Intrinsic Localized Modes in PC waveguides – bistability… • -- 2D Intrinsic Localized Modes -- stabilization?.. NATO ARW -- Erice, Sicily

4. Photonic Crystals: 15 years of research About 3500 papers altogether… Phys. Rev. Lett. 58 (1987): - S. John - E. Yablonovitch Kurt Busch et al.(Ed.),Photonic Crystals: Advances in Design, Fabrication, and Characterization, (Wiley-VCH, 2004), 360 pages. NATO ARW -- Erice, Sicily

5. Existence of complete 3D photonic band-gaps… • Applications of Complete Band-Gaps: • Quantum optics !!! • A perfect photonic isolator (highly directional microwave antennas, etc) NATO ARW -- Erice, Sicily

6. Fabrication of 3D photonic crystals: is still difficult… Opal structure – relatively cheap, but the bandgap is very small; there are big problems with introducing defects… Promising alternative – inverse opal structure… Woodpile structure – very good properties, but fabrication from semiconductors is very expensive… MIT - quasi-2D structure… S. John - spiral structure… NATO ARW -- Erice, Sicily

7. Fabrication of 3D photonic crystals by a photo-polymerization… Markus Deubel et al. Daniel Meisel et al. University of Karlsruhe, Germany -- No complete band-gaps due to low refractive index! NATO ARW -- Erice, Sicily

8. Fabrication of 2D PC’s: on the way to mass production… Real 2D crystals – electrochemical etching of the Macroporous Silicon: - Limited defect tunability… - Possibility to infiltrate something… - Possibility to modulate hole radius… 2D photonic crystal slabs: - On substrate… - Self-supporting in air… Problems: - Limitation on the hole radius due to out-of-plane losses… NATO ARW -- Erice, Sicily

9. 2D photonic crystal circuits… NATO ARW -- Erice, Sicily

10. Photonic MicropolisDreamby the Joannopoulos Research Group at MIT… NATO ARW -- Erice, Sicily

11. Nonlinear Photonic Crystals: • Type of nonlinearity:χ(2)orχ(3) • Periodic modulation: c or both,c and e • Geometry: Dimensionality, Symmetry, Heterogeneity, Defects… V. Berger, Phys. Rev. Lett. 81, 4136 (1998) N.G.R. Broderick et al., Phys. Rev. Lett. 84, 4345 (2000) NATO ARW -- Erice, Sicily

12. Maxwell’s equations… Maxwell’s equations for E-polarized light in 2D photonic crystals can be reduced, for the monochromatic light to the eigenvalue problem whose solutions, for perfectly periodic PCs, are Bloch modes En,k(r) with frequencies wn,k. NATO ARW -- Erice, Sicily

13. Example of a 2D photonic crystal: square-lattice of air holes… We assume that the radius of air holes r=0.475a, where a is a lattice constant. In this case PC exhibits large 20% PBG for the E-polarized light with the frequencies (wa/2pc)=0.238 to 0.291. NATO ARW -- Erice, Sicily

14. PCs with defects and nonlinearity… In the case when one or several pores of the 2D photonic crystal are infiltrated by the same nonlinear liquid crystal or polymer where (r)=1 inside pores and(r)=0 outside, and positions of the infiltrated pores are given by Rm. NATO ARW -- Erice, Sicily

15. Computational methods… • Numerically exact: • Finite Difference Method • Green Function Approach • Plane Waves Expansion • Bloch Modes Expansion • Wannier Functions Expansion • Approximate: • Coupled-Mode Theory • Tight-Binding Model • Multiple-Scale Approach NATO ARW -- Erice, Sicily

16. Expansion of electromagnetic fileld into a complete set of Basis Functions… Writing electromagnetic field as a sum of some Basis functions, (r), we derive a set of discrete equations: where NATO ARW -- Erice, Sicily

17. Choice of the Basis Functions… • Plane waves: Requires hundreds of functions for modeling a perfectly periodic PC, and several thousands of functions for modeling a simple PC cavity. All coefficients are dense matrices! • Bloch modes: Very good for modeling periodic nonlinear waves. However, requires hundreds of functions for modeling intrinsic localized modes. All coefficients are dense matrices, except of C= due to orthogonality! • Localized defect modes: Very good in some cases, but incomplete! • Wannier functions: NATO ARW -- Erice, Sicily

18. Photonic Wannier functions… When we know the Bloch modes, one can create the complete basis of the orthogonal localized Wannier functions for a given PC: where choice of the phases qn,k is a major problem… Topical Review on Wannier functions in PCs: K. Busch et al., J. Phys.: Condens. Matt., in press (2003) NATO ARW -- Erice, Sicily

19. Single cavity: Linear localized mode… NATO ARW -- Erice, Sicily

20. Single cavity: Non-Linear localized mode… NATO ARW -- Erice, Sicily

21. Two coupled cavities:Linear localized modes… A A B B NATO ARW -- Erice, Sicily

22. Two coupled cavities:Non-Linear localized modes… y x NATO ARW -- Erice, Sicily

23. Photonic Crystal Waveguides… NATO ARW -- Erice, Sicily

24. Intrinsic Localized modes in Photonic Crystal Waveguides… Modes for NATO ARW -- Erice, Sicily

25. Another example of solitons in Photonic Crystal Waveguides… P S.F. Mingaleev, Yu.S. Kivshar, and R.A. Sammut, Phys. Rev. E 62, 5777 (2000) NATO ARW -- Erice, Sicily

26. Interplay of nonlinearity and long-range interactions in NLS models… • Yu.B. Gaididei et al., • Phys. Rev. E 55, 6141 (1997). • Effects of ultra-long-range interactions: • P.L. Christiansen et al., • Eur. Phys. J. B 19, 545 (2001). Switching between intrinsic localized mode and mobile soliton: M. Johansson et al, Phys. Rev. E 57, 4739 (1998). NATO ARW -- Erice, Sicily

27. Intrinsic localized modes in waveguide bends… Interplay of nonlinearity, long-range interactions and geometry in NLS models: • Yu.B. Gaididei et al., • Phys. Rev. E 62, R53 (2000). • P.L. Cristiansen et al., • J. Phys.: Condens. Matter 13, • 1181 (2001). NATO ARW -- Erice, Sicily

28. Transmission through waveguide bends is very sensitive to geometry… NATO ARW -- Erice, Sicily

29. Solitons in add-drop filters Switching Fusion Splitting Conversion Generation P.L. Chu et al., J. Opt. Soc. Am. B 12, 898 (1995) NATO ARW -- Erice, Sicily

30. What about localization length of intrinsic localized modes?.. Soliton in W1-1 waveguide with y x We define localization length as: NATO ARW -- Erice, Sicily

31. Multi-dimensional solitons?Yes! But what about stability at low intensities? Tunable 2D PCs using liquid-crystal infiltration: S.W. Leonard et al., Phys. Rev. B 61, R2389 (2000). NATO ARW -- Erice, Sicily

32. Stable low-intensity 2D solitonsin 2D photonic crystals: S.F. Mingaleev and Yu.S. Kivshar, Phys. Rev. Lett. 86, 5474 (2001) NATO ARW -- Erice, Sicily

33. Conclusions… • Intrinsic Localized Modes and other properties of Nonlinear Photonic Crystals can be accurately and very efficiently studied using Wannier functions approach. • Infiltration of semiconductor photonic crystals (with high  to open complete band gaps) by liquid crystals or polymers (with high ) is perhaps one of the most promising ways for observations of Intrinsic Localized Modes in Photonic Crystals. • Intrinsic Localized Modes in Photonic Crystals are described by lattice models with essentially long-range coupling. As a result they manifest a number of qualitative effects such as bistability, symmetry-breaking instability, stabilization of low-amplitude 2D solitons… NATO ARW -- Erice, Sicily