1. Mathematical Ideas in Nonlinear Optics: Guided Waves in Inhomogenous Nonlinear Media July 19-23, 2004, Edinburgh Nonlocal Solitons Ole Bang, Nikola I. Nikolov, and Peter L. Christiansen COM Centre, Informatics and Mathematical Modelling, and Physics Department Technical University of Denmark, Lyngby, Denmark Wieslaw Krolikowski, Darran Edmundson, and Dragomir Neshev Laser Physics Centre, ANU Supercomputer Facility, and Nonlinear Physics Group Australian National University, Canberra, Australia John Wyller Department of Mathematical Sciences, Agricultural University of Norway Department of Mathematical Sciences, Agricultural University of Norway Jens Juul Rasmussen Risoe National Laboratory, Roskilde, Denmark

2. Mathematical Ideas in Nonlinear Optics: Guided Waves in Inhomogenous Nonlinear Media July 19-23, 2004, Edinburgh Physical systems exhibiting nonlocal nonlinear response Systems involving transport effects –heat conduction in materials with thermal nonlinearity -light-induced diffusion of molecules or atoms in atomic vapours -Drift/diffusion of photoexcited charges in photorefractives Propagation of electromagnetic waves in plasma Many body interaction with finite scattering parameter in Bose-Einstein condensates Molecular re-orientation in liquid crystals Parametric wave-mixing

3. Mathematical Ideas in Nonlinear Optics: Guided Waves in Inhomogenous Nonlinear Media July 19-23, 2004, Edinburgh Impact of nonlocality on beam propagation Impact of nonlocality on beam propagation Theoretical predictions: Theoretical predictions: –nonlocality arrests collapse (catastrophic self-focusing) of optical beams in self- focusing media and enables formation of stable 2D solitons [Turitsyn, 1985] –Nonlocality-induced long-range interaction enables attraction of out-of-phase bright solitons [Kolchugina, Mironov, Sergeev, 1980] –Nonlocality can suppress MI in focusing media [Litvak, Mironov, Fraiman, Yunakovskii, 1975] Experimental observations: Experimental observations: –Stabilization of 2D beams in atomic vapors due to atomic diffusion removing excitation from the interaction region [Suter, Blasberg,1993] –Assanto and his group observed this attraction in nematic liquid crystals [Peccianti, Brzdkiewicz, Assanto, 2002] –Assanto and his group observed suppression of MI in nematic liquid crystals [Peccianti, Conti, Assanto, 2003]

4. Mathematical Ideas in Nonlinear Optics: Guided Waves in Inhomogenous Nonlinear Media July 19-23, 2004, Edinburgh Nonlocal model Nonlocal model u(x,y,z) – slowly varying field amplitude diffractionnonlocal nonlinearityext. confinement 1D V(r) - confining potential (waveguide) s =±1 determines the type of nonlinearity R(r) - nonlocal response function Determined by the physical process Determined by the physical process

5. Mathematical Ideas in Nonlinear Optics: Guided Waves in Inhomogenous Nonlinear Media July 19-23, 2004, Edinburgh Nonlocal model Nonlocal model The relative width of the response function and the intensity profile determines 4 regimes Local NLS Strongly nonlocal Weakly nonlocal

6. Mathematical Ideas in Nonlinear Optics: Guided Waves in Inhomogenous Nonlinear Media July 19-23, 2004, Edinburgh Modulational instability in nonlocal media Modulational instability in nonlocal media Modulational instability (MI) signifies the exponential growth of a weak perturbation of the amplitude of a plane wave as it propagates The gain leads to amplification of sidebands, which breaks up the otherwise uniform wave front - filamentation MI may act as a precursor for the formation of bright solitons Stable dark solitons requires absence of MI of the constant intensity background MI has been identified in fluids, plasma, nonlinear optics, discrete nonlinear systems, such as molecular chains and waveguide arrays Gaussian response Self-focusing (s=+1) σ=0.1 σ=1.0

7. Mathematical Ideas in Nonlinear Optics: Guided Waves in Inhomogenous Nonlinear Media July 19-23, 2004, Edinburgh Modulational instability in nonlocal media Modulational instability in nonlocal media Linearization of propagation equation around plane wave solution Plane wave solution and dispersion relation Equation for the perturbation growth Sign indefinite spectrum –S–S–S–S = +1 => MI –S–S–S–S = -1 => Possibility for MI Positive definite spectrum –S = +1 => MI –S = -1 => Stability

8. Mathematical Ideas in Nonlinear Optics: Guided Waves in Inhomogenous Nonlinear Media July 19-23, 2004, Edinburgh Modulational instability in nonlocal media Modulational instability in nonlocal media Self-focusing Gaussian response Self-defocusing rectangular response

9. Mathematical Ideas in Nonlinear Optics: Guided Waves in Inhomogenous Nonlinear Media July 19-23, 2004, Edinburgh Beam collapse in nonlocal medium Beam collapse in nonlocal medium Conserved Power P and Hamiltonian H Fourier approach Gradient norm bounded from above => No collapse

10. Mathematical Ideas in Nonlinear Optics: Guided Waves in Inhomogenous Nonlinear Media July 19-23, 2004, Edinburgh Beam stabilization via nonlocality Beam stabilization via nonlocality Stabilizing role of the nonlocality can be illustrated by considering properties of stationary solutions (2D). Using Gaussian response function and gaussian soliton ansatz: the variational approach gives the relation between soliton power and propagation constant the variational approach gives the relation between soliton power and propagation constant

11. Mathematical Ideas in Nonlinear Optics: Guided Waves in Inhomogenous Nonlinear Media July 19-23, 2004, Edinburgh Interaction of dark nonlocal solitons Interaction of dark nonlocal solitons Dark solitons in local nonlinear media always repel Nonlocality induces long-range attraction of dark solitons and leads to the formation of their bound states CW beam passing through phase mask with 2 opposite phase jumps (a-b) and through a thin wire (c)

12. Mathematical Ideas in Nonlinear Optics: Guided Waves in Inhomogenous Nonlinear Media July 19-23, 2004, Edinburgh Nonlocality-assisted stabilization of vortices Nonlocality-assisted stabilization of vortices Vortex beams in local nonlinear media always disintegrate Nonlocality induces long-range attraction keeps the vortex together Gaussian response and a charge 1 vortex: σ=0 σ=1 σ=10

13. Mathematical Ideas in Nonlinear Optics: Guided Waves in Inhomogenous Nonlinear Media July 19-23, 2004, Edinburgh Nonlocality-assisted vortex stabilization - II Nonlocality-assisted vortex stabilization - II More accurate variational solution: –Stable over exceptionally long distances. Nonlocality =1 Vortex width 0 =10

14. Mathematical Ideas in Nonlinear Optics: Guided Waves in Inhomogenous Nonlinear Media July 19-23, 2004, Edinburgh Solitons in weakly nonlocal media Solitons in weakly nonlocal media Both types of solitons are stable The weakly nonlocal 1D model can be solved analytically weak nonlocality

15. Mathematical Ideas in Nonlinear Optics: Guided Waves in Inhomogenous Nonlinear Media July 19-23, 2004, EdinburghConclusions Nonlocality in nonlinear media arrests collapse of multidimensional beams and stabilizes solitons arrests collapse of multidimensional beams and stabilizes solitons stabilizes propagation of vortex beams in focusing media stabilizes propagation of vortex beams in focusing media suppresses MI of plane waves in focusing media and may induce MI in defocusing media suppresses MI of plane waves in focusing media and may induce MI in defocusing media induces long-range attraction of dark and out-of-phase bright solitons and enables formation of bound states induces long-range attraction of dark and out-of-phase bright solitons and enables formation of bound states Nonlocal nonlinear media support formation of many novel stable bright and dark solitons and their bound states Nonlocality provides nice physical picture of parametric wave interaction, for example predicting novel quadratic soliton solutions