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

3. DYCOEC Feb. 5-6, 2008 3 Choice of nonlinear medium Theory: Kerr medium n = n 0 + n 2 I Experiment: liquid crystals Liquid Crystal Light Valves (LCLV) Photorefractive crystals alkali vapors, esp. Na

4. DYCOEC Feb. 5-6, 2008 4 Coupling between photon spin and atomic spin: production of “orientation” w in atomic ground state (Zeeman pumping) Nonlinearity in Na vapor: spin-1/2 model m j =-1/2m j =+1/2  1     P N 2   1 Nonlinear (complex) susceptibility:     (1 – w(E))     (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

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

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

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

8. 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

9. DYCOEC Feb. 5-6, 2008 9 Polarization instability (perfect) pitchfork bifurcation very low threshold  angle between input polarization and main axis of /8-plate two equivalent states  obs. polarizer mediummirror analyzer  plate

10. DYCOEC Feb. 5-6, 2008 10 Rotated polarizer (  30` 5o5o perturbed pitchfork bifurcation Increased threshold of bistability “Negative branch” preferred

11. DYCOEC Feb. 5-6, 2008 11 The complementary case (  “Positive branch“ preferred

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

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

14. 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

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

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

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

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

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

20. DYCOEC Feb. 5-6, 2008 20 (Unexpected?) result: family of solitons background suppressed with LP family of solitons *) “higher order solitons” “excited states of soliton” S1S2S3S4 Note: Observed quantity (intensity) is not the state variable! *) Many predictions for 1D-systems M. Pesch et al., PRL 95, 143906 (2005)

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

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

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

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

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

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

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

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

29. 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

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

31. 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)

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

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

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

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

36. DYCOEC Feb. 5-6, 2008 36 The team and its supporters 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

37. DYCOEC Feb. 5-6, 2008 37

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

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

40. DYCOEC Feb. 5-6, 2008 40 Contraction of domains Parameter: Input power 

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

42. DYCOEC Feb. 5-6, 2008 42 Time-dependence of domains i()i() c()c()  c (P)

43. DYCOEC Feb. 5-6, 2008 43 Time-dependence of domains i()i() c()c()  c (P)

44. DYCOEC Feb. 5-6, 2008 44 Time-dependence of domains i()i() c()c()  c (P)

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

46. DYCOEC Feb. 5-6, 2008 46 SuS 2,1 +SiSAS 2,1 +SiSExperiment unstablestable Phase selection on the square grid

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

48. DYCOEC Feb. 5-6, 2008 48 P PcPc qcqc q P P qq q eine Wellenzahl Wellenzahlband Origin: phase sensitive cubic coupling

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

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

51. DYCOEC Feb. 5-6, 2008 51 Experimental access to Fourier space ffffd “Fourier filter ” mirror Fourier space real space image of nonlinear medium nonlinear medium “far field” or “near field” Camera

52. DYCOEC Feb. 5-6, 2008 52 Marginal stability curve linear stability analysisexperiment M. Pesch et al., Phys. Rev. E68, 016209 (2003).