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Sodium vapor in a single-mirror feedback scheme: a paradigm of self-organizing systems in optics

Sodium vapor in a single-mirror feedback scheme: a paradigm of self-organizing systems in optics. W. Lange Institut fuer Angewandte Physik Univ. of Muenster (Germany) w.lange@uni-muenster.de. “ Single-mirror“ system: basic setup. Firth 1990, d’Alessandro Firth 1991,1992. nonlinear medium.

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Sodium vapor in a single-mirror feedback scheme: a paradigm of self-organizing systems in optics

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  1. Sodium vapor in a single-mirror feedback scheme: a paradigm of self-organizing systems in optics W. Lange Institut fuer Angewandte Physik Univ. of Muenster (Germany) w.lange@uni-muenster.de

  2. “Single-mirror“ system: basic setup Firth 1990, d’Alessandro Firth 1991,1992 nonlinear medium mirror laser beam Talbot effect • spatial coupling via diffraction and reflection • nonlinearity and spatial coupling spatially separated DYCOEC Feb. 5-6, 2008

  3. Choice of nonlinear medium Theory:Kerr mediumn = n0 + n2I Experiment: liquid crystals Liquid Crystal Light Valves (LCLV) Photorefractive crystals alkali vapors, esp. Na DYCOEC Feb. 5-6, 2008

  4. P N 2 s s G G 1 1 - + g m =-1/2 m =+1/2 j j Nonlinearity in Na vapor: spin-1/2 model Coupling between photon spin and atomic spin: production of “orientation” win atomic ground state (Zeeman pumping) Nonlinear (complex) susceptibility: c+ = co (1 – w(E)) c- = co (1 + w(E)) No Zeeman pumping in linearly polarized light – but polarization instability Polarization very critical – add polarizing element in feedback loop Orientation very sensitive to magnetic field – introduce longitudinal and transverse components DYCOEC Feb. 5-6, 2008

  5. Stripes (“rolls”) Squares Hexagons (pos. and neg.) Self-induced patterns Transitions between pos. and neg. hexagons via rolls and squares DYCOEC Feb. 5-6, 2008

  6. 12-fold Quasipatterns 8-fold Aumann et al., Phys. Rev. E 66, 046220 (2002) R. Herrero et al., PRL 82, 4657 (1999) DYCOEC Feb. 5-6, 2008

  7. Superstructures square subgrid hexagonal subgrid Two slightly different wave numbers involved E. Große Westhoff et al., Phys. Rev. E67, 025203 (2003) DYCOEC Feb. 5-6, 2008

  8. Self-induced patterns • Observed phenomena reproduced in simulations semiquantitatively • Linear stability analysis available • Weakly nonlinear analysis in most cases • Gaussian beam reduces “aspect ratio”, but usually has little influence on patterns DYCOEC Feb. 5-6, 2008

  9. medium mirror two equivalent states obs. analyzer polarizer l/8-plate very low threshold Polarization instability r = 0 r angle between input polarization and main axis of l/8-plate (perfect) pitchfork bifurcation DYCOEC Feb. 5-6, 2008

  10. 30` 5o “Negative branch” preferred Increased threshold of bistability Rotated polarizer (r>0) perturbed pitchfork bifurcation DYCOEC Feb. 5-6, 2008

  11. The complementary case (r<0) “Positive branch“ preferred DYCOEC Feb. 5-6, 2008

  12. Polarization fronts In switch-on experiments spontaneous formation of polarization fronts Analyzer adjusted for minimum intensity in region with (a) negative or (b) positive rotation Analyzer adjusted to suppress input beam Dark line indicates Ising front DYCOEC Feb. 5-6, 2008

  13. Circular domains In “holding beam” system sits on “disadvantaged branch” System is locally brought to complementary state by “address beam” of suitable polarization, i. e. domainsareignited. Evolution after switching off the address beam? pump rate of “holding beam” DYCOEC Feb. 5-6, 2008

  14. Front dynamics • straight fronts are stable • circular domains contract: “curvature driven contraction” (not in 1D) Case of equivalent states DYCOEC Feb. 5-6, 2008

  15. Domain contraction DYCOEC Feb. 5-6, 2008

  16. Fronts between nonequivalent states r nonvanishing The ‘preferred‘ state expands: “pressure driven expansion” Simulation gi(r) (also determined experimentally) DYCOEC Feb. 5-6, 2008

  17. r=10° r=9° r=5° r=-0° r=-5° Evolution of circular domains (simul.) Expansion and contraction can balance But:Equilibrium is not stable Stabilization of a domain requiresadditional mechanism DYCOEC Feb. 5-6, 2008

  18. Circular domains: switching experiment polarization of addressing beam • “domain” can be switched on and off by an addressing beam • direction of switching determined by the polarization of addressingbeam • bistable behavior intensity ofaddressing beam time In detection: projection on linear pol. state such that holding beam is suppressed “domain” ignited stable stationary “domain” “domain” extinguished Transverse (feedback) soliton DYCOEC Feb. 5-6, 2008

  19. Repetition of the experiment (second soliton observed much easier!) DYCOEC Feb. 5-6, 2008

  20. (Unexpected?) result: family of solitons S1 S2 S3 S4 background suppressed with LP family of solitons*) “higher order solitons” “excited states of soliton” *) Many predictions for 1D-systems M. Pesch et al., PRL 95, 143906 (2005) Note: Observed quantity (intensity) is not the state variable! DYCOEC Feb. 5-6, 2008

  21. Spatially resolved Stokes parameters Rotation represents orientation! (for low absorption) M. Pesch, PhD thesis, Muenster 2006 (unpublished) DYCOEC Feb. 5-6, 2008

  22. Positive Solitons “initial (background) state” “target state” DYCOEC Feb. 5-6, 2008

  23. Negative Solitons “initial state” “target state” DYCOEC Feb. 5-6, 2008

  24. Comparison with simulations experiment numerical simulations for Gaussian beam DYCOEC Feb. 5-6, 2008

  25. Comparison: medium power – high power Soliton “sits” on modulated background – homogeneous background not required DYCOEC Feb. 5-6, 2008

  26. Dynamics of domain wall low power high power M. Pesch et al., PRL 99, 153902 (2007) DYCOEC Feb. 5-6, 2008

  27. Shape of initial domain Strong diffraction patterns for high power! Fronts interact with intensity and phase gradients Solitons occur when pronounced diffraction patterns are present: self-interaction of circular front by diffraction prevents contraction DYCOEC Feb. 5-6, 2008

  28. Bifurcation diagram DYCOEC Feb. 5-6, 2008

  29. The mechanism Curvature-driven contraction + (pressure- driven expansion) + diffraction = transverse soliton Enhancement of diffraction by modulation insta-bility or its precursors required DYCOEC Feb. 5-6, 2008

  30. pattern formation High power behavior Zero crossing of gc? ? DYCOEC Feb. 5-6, 2008

  31. Labyrinths • “Negative contraction” • Distances determined by Talbot effect • Limitations by Gaussian beam J. Schüttler, PhD thesis, Muenster 2007 (unpublished), J. Schüttler et al. (submitted) DYCOEC Feb. 5-6, 2008

  32. Target patterns and spirals (observed by sampling method) Occurs in oblique magnetic field, but only in phase gradient produced by self-induced lens (Gaussian beam) F. Huneus et al., Phys. Rev. E 73, 016215 (2006) Spirals = azimuthally disturbed target patterns F. Huneus, PhD thesis, Muenster 2006 (unpublished) DYCOEC Feb. 5-6, 2008

  33. Coexistence between spirals and solitons Solitons do not need astationarybackground DYCOEC Feb. 5-6, 2008

  34. Simulation E. Schöbel, diploma thesis, Münster 2006 (unpublished) DYCOEC Feb. 5-6, 2008

  35. Conclusions • System displaysvast variety of phenomena • (Relatively)simple (microscopic) model • Simulationsagree with (nearly) all observations semiquantitatively • Some analysis, but more in-depth theoretical work welcome • Small aspect ratio • New phenomena due to phase and intensitygradientsin Gaussian beam; beam divergence and convergence need attention DYCOEC Feb. 5-6, 2008

  36. Thorsten Ackemann(- 2005; now: Strathclyde Univ.) Andreas Aumann(-1999; now: consultant) Edgar Große Westhoff(-2001; now: product manager) Florian Huneus (-2006; now: optical engineering) Matthias Pesch(-2006; now: optical engineering) Burkhard Schäpers (-2001; now: banking, risk analysis) Jens Schüttler(-2007; now: optical engineering) Several diploma students Support by Deutsche Forschungsgemeinschaft Guests: Ramon Herrero (Barcelona) Yurij Logvin (Minsk) Igor Babushkin (Minsk/Berlin) Cooperations: Damian Gomila (Palma) Willie Firth (Glasgow) Gian Luca Oppo (Glasgow) Stimulus by Pierre Coullet The team and its supporters DYCOEC Feb. 5-6, 2008

  37. DYCOEC Feb. 5-6, 2008

  38. Plane wave simulations of w (large int.) Hex. down S 2 Hex. up S 3 S 4 S 1 DYCOEC Feb. 5-6, 2008

  39. Three-dimensional plot (low input power) Direct comparison with experiment not possible! DYCOEC Feb. 5-6, 2008

  40. Contraction of domains Parameter: Input power r = 0 DYCOEC Feb. 5-6, 2008

  41. unobserved New type of soliton exp. sim. New family DYCOEC Feb. 5-6, 2008

  42. Time-dependence of domains gi(r) gc(r) gc(P) DYCOEC Feb. 5-6, 2008

  43. Time-dependence of domains gi(r) gc(r) gc(P) DYCOEC Feb. 5-6, 2008

  44. Time-dependence of domains gi(r) gc(r) gc(P) DYCOEC Feb. 5-6, 2008

  45. Contracting domains (simulation) pattern formation DYCOEC Feb. 5-6, 2008

  46. Phase selection on the square grid Experiment SuS2,1+SiS AS2,1+SiS unstable stable DYCOEC Feb. 5-6, 2008

  47. The great mystery • Angle in the compressed grid: 41.9o (exp.) • Wave vectors have equal length for 41.4o • Occurs far above threshold • Requires slightly divergent laser beam (phase gradient) General problem: structures in nonplanar situations DYCOEC Feb. 5-6, 2008

  48. Origin: phase sensitive cubic coupling P P P Pc Dq qc q q Wellenzahlband eine Wellenzahl DYCOEC Feb. 5-6, 2008

  49. Patterns on polarized branches Intensity of Fourier mode Patterns + Patterns - Input power Waveplate rotation DYCOEC Feb. 5-6, 2008

  50. Bistable behavior threshold rotation of polarization Positive branch Negative branch Variable: rotation of waveplate DYCOEC Feb. 5-6, 2008

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