1. Pattern Formation Induced by Modulation Instability in Nonlinear Optical System Ming-Feng Shih( 石明豐 ) Department of Physics National Taiwan University

2. Alan Turing recognized that the formation of organized structures can arise from the interplay between reaction and diffusion. Such structures are universal and form in a variety of different systems. 1. Introduction

3. Nonlinear Optical Medium

4. Phenomenon well known: Modulation instability Light

5. The spatial period of the pattern is decided by the balance between diffraction and self-focusing. Diffraction Self-focusing

9. with No threshold

10. Phenomena to be explored: How about incoherent light? (First proposed by M. Soljacic et al., PRL v.84, 467(2000)) Why? 1. Incoherent optical soliton. (1997) 2. Closely related to the BEC around the critical temperature. Incoherent light different : “ fast ” phase variation, or phase cannot be defined “ exactly ” which sets the material requirement. 2. Modulation instability with incoherent light Correlation length

11. “ Slow ” materials

12. Coherent Density Approach: Assuming partially incoherent light consists of many coherent but mutually incoherent light fields, each field propagates with angle  with respective to z -direction. Nonlinearity  n is a function of I +  +.......  G(  )

14. Threshold coherence exists for a fixed nonlinear strength.

15. D. Kip, et al, Science 290, 495 (2000) Degree of coherence: Low  High The result is reasonable: Incoherent light diffracts more than coherent light. Therefore it requires higher nonlinearity to form MI pattern. Threshold coherence

16. Vortex light beam carrying orbital angular momentum l=0 l=1 Plane of constant phase Intensity

17. In self-focusing media, vortex ring is unstable due to azimuthal instability V. Tikhonenko, et al, J. Opt. Soc. Am. B 12, 2046 (1995); Phys. Rev. Lett. 76, 2698 (1996). D.V. Skryabin and W. J. Firth, Phys. Rev. Lett. 79, 2450 (1997); Phys. Rev. E 58, 3916 (1998). Single-charge : l = 1 Double-charge : l = 2

18. Experimental Observations a. Single-charge vortices : l = 1 Coherent Partially incoherent Speckle pattern Input Output(2.5kV)

19. coherence: High  Low Simulation Experiment PRL v. 92, 043904 (2004)

20. Interaction between Optical Spatial Solitons In-phase  out-of-phase

21. Meng et al. OPTICS LETTERS / Vol. 22, No. 7 / April 1, 1997 Input Diffraction indivisual in-phase   out-of-phase

22. Coherent Density Approach: Assuming partially incoherent light consists of many coherent but mutually incoherent light fields, each field propagates with angle  with respective to z -direction. Nonlinearity  n is a function of I +  +.......  G(  )

23. Soliton interaction is controlled by the coherence Note: two beams as a whole are made partially incoherent, but the relative phase  between the two parts is fixed.

24. In Phase Threshold at    0.0028   

25.  out-of-phase Threshold at  0 =0.0022

26. Why the coherence affects the interaction? We use the in-phase interaction as an example to illustrate. +  + +.......  G(  ) Larger separation, 2d, means smaller threshold value of  0 ! smaller   larger    d

27. d=10 d=12   th =0.0022  0, th =0.0018 0.0022 x 10 / 12 = 0.183  out-of-phase Similar for the in-phase interaction.

28. In-phase Coherent Partially incoherent

29. Partially incoherent less coherent more coherent  out-of-phase Highlighted by Optics in 2005 by Optics and Photonics News (OSA magazine) PRL v.94, 063904(2005)

30. 0 sec10 sec20 sec30 sec40 sec 293 m μ 572 mW/cm 2 0.75 kV 3. coherent MI with time-varying noise

31. Time=t1 t2 t3........ In instantaneous nonlinear self-focusing media

32. Time=t1 t2 t3........ In noninstantaneous nonlinear self-focusing media

33. with assuming nonlinearity of relaxation type: For a wave and We have

34. 1.When  (noise is static), no difference for instantaneous and noninstantaneous media. 2If  is large enough, h 2 is leveled off and contains no peak.

35. Increasing the material response time can arrest the MI: PRL v88, 133902, Apr. 2002 0.7 kV 5mm (a) (b) 572 mW/cm 2 143 mW/cm 2 57.2 mW/cm 2 371.8 mW/cm 2 0.9 kV 5mm 57.2 mW/cm 2 286 mW/cm 2 143 mW/cm 2 572 mW/cm 2 smalllarge τ 293 m μ

37. optical intensity smalllarge

39. In self-defocusing medium, MI cannot happen. Propagation However, MI still can happen for moving pattern.

40. Output face of the crystal: self-defocusing nonlinearity is on Input face of the crystal

42. L Feedback

43. 0% 9% 16% 28% With feedback percentage equal to

44. And of course, when the MI forms, it moves with time.