1. Observation of Raman Self-Focusing in an Alkali Vapor Cell Nicholas Proite, Brett Unks, Tyler Green, and Professor Deniz Yavuz

2. Self-Focusing Effect  Non-linear effect due to the intensity dependent refractive index generated by  (3)  Mechanism by which optical spatial solitons are formed

3. Single Photon vs Raman Systems  E  EpEp  EsEs

4. Propagation Equations in Raman System  EpEp  EsEs

5. Our System F=0, 1, 2, 3 EPEP ESES optical pumping laser F=1 F=2  ~ 1 MHz 85 GHz 87 Rb D2 Line

6. General Procedure of Experiment Pinhole Photodiode

7. General Procedure of Experiment Pinhole Photodiode

8. General Procedure of Experiment Pinhole Photodiode

9. Intensity x (mm) (a) (b) The peak intensity for a freely propagating beam is normalized to 1. FocusedDe-Focused  = 2   0.25MHz  = 2   -0.25MHz Experimental Results Simulation

10. Experimental Results normalized transmission  (MHz)

11. Thank you References: 1)DD Yavuz, Phys Rev A,75, 041802, (2007). 2)N. A. Proite, B. E. Unks, J. T. Green, and D. D. Yavuz, Phys. Rev. A, 77, 023819 (2008).

13. General Procedure of Experiment

16. What is a Soliton? Normal Gaussian Beam: z x,y II xx

17. What is a Soliton? Soliton: z x,y I x I x

18. Maxwell’s Equation inside a medium with no charge or current density: Gaussian Beam Propagation in a Medium

19. Paraxial Wave Equation in a Linear Medium Using the relation:

20. Paraxial Wave Equation in a Linear Medium  ''   ''  '  n  ''  Loss (or gain) of medium Index of refraction E

21. Paraxial Wave Equation in a Non-Linear Medium  As the strength of the beam is increased polarization of the medium is no longer linear; we must introduce higher order susceptibilities: In an isotropic medium: E

22. How will non-linear terms affect beam propagation?  Non-Linear Schrödinger’s Equation E

23. How will non-linear terms affect beam propagation?  Non-Linear Schrödinger’s Equation The solution (with one transverse dimension ‘x’): x Sech(x) E

24. Raman System (a 3 rd order non-linear process)  EpEp  EsEs Transitions may be one photon forbidden, but by using the intermediate state associated with we can couple them.

25. Atomic Raman System using Rubidium 87 Rb D2-line (F' = 0,1,2,3) (F = 1) 5 2 P 3/2 5 2 S 1/2   EsEs EpEp | a > | b > (F = 2) Only dipole transitions are considered here HoHo H int | i >

26. Atomic Raman System using Rubidium 87 Rb D2-line (F' = 0,1,2,3) (F = 1) 5 2 P 3/2 5 2 S 1/2   EsEs EpEp | a > | b > (F = 2) | i > Key Assumption: Large one photon detuning where

27. Propagation Equation in a Raman Medium How can we get this equation in terms of quantities we know?

28. Propagation Equation in a Raman Medium Nonlinear part of the propagation equations: Use expectation value of polarization operator to find polarization term.

29. Interpreting the coupled propagation equations

30. For ‘p’ beam:

31. Non-linear Refractive Index nn zz Phase Front

32. Self-Trapping and Solitons SolitonFreely Propagating beam x (  m) Intensity (Wcm -2 ) 500 -500 0.4 z (m) 9001600 Parameters

33. Soliton Stability Beam Intensity Refractive Index Peak Refractive Index ~ 6.7x10 -6 x (  m) Intensity (Wcm -2 ) 500 -500 0.4 z (m) 1600

34. Soliton Stability (Vakhitov, Kolokolov criterion)  Power (W) Assume the electric fields are identical to reduce to one non- linear equation. Assume electric field takes the form of a field which only accumulates phase with z. The corresponding propagation constant is . Stability Condition:

35. Soliton Dynamics Soliton attraction: Intensity y x

36. Soliton Dynamics Soliton repulsion: Intensity y x

37. Soliton Dynamics Soliton fusion: Intensity y x

38. Index Waveguides |E| 2 x n = 3.2 n = 3.4

39. Index Waveguides |E| 2 x n = 3.2 n = 3.4 n = 3.2 n = 3.6 |E| 2 x

40. Soliton Interactions Relative Phase: 0 x (  m) 500-500 z (m) 1 Intensity (W cm -2 ) 4000 Beam Intensity Refractive Index

41. Soliton Interactions Relative Phase:  x (  m) 500-500 z (m) 1 Intensity (W cm -2 ) 2000 Beam Intensity Refractive Index

42. Soliton Interactions Relative Phase: 1.8  x (  m) 400-400 z (m).8 Intensity (W cm -2 ) 3000 Beam Intensity Refractive Index

43. Possible Application 1 1 1 1 0 0 1 0 0 AND gate 0 0 0

44. Experimental Observations of Self-Focusing and Self- Defocusing

45. F=0, 1, 2, 3 EPEP ESES optical pumping laser F=1 F=2  85 GHz

46. Experimental Observations of Self-Focusing and Self- Defocusing Pinhole Photodiode

47. Experimental Observations of Self-Focusing and Self- Defocusing Photodiode

48. Experimental Observations of Self-Focusing and Self- Defocusing Photodiode

49. Intensity x (mm) (a) (b) The peak intensity for a freely propagating beam is normalized to 1. FocusedDe-Focused  =2   0.25MHz  =2   -0.25MHz Experimental Results Simulation

50. Experimental Results normalized transmission Detuning (MHz)

51. Acknowledgments and References Thank you to Brett Unks, Nick Proite, Dan Sikes, and Deniz Yavuz for their helpful suggestions. (And David Hover for letting me use his computer) References: 1)DD Yavuz, Phys Rev A,75, 041802, (2007). 2)Stegeman, Sevev, Science, 256 1518, (1999). 3)NG Vakhitov, AA Kolokolov, Sov. Radiophys. 16,1020, (1986). 4)NA Proite, BE Unks, JT Green, DD Yavuz, (Recently Submitted). 5)MY Shverdin, DD Yavuz, DR Walker, Phys. Rev. A, 69, 031801, (2004).

53. Paraxial Wave Equation in a Medium Using the relation: The real and imaginary parts of  (  ' and  '' respectively) reveal much about the behavior of the beam as it propagates through the medium. Loss Propagation constant 

54. Raman System Rb D2-line F' = 0,1,2,3 (F = 1) 5 2 P 3/2 5 2 S 1/2   EsEs EpEp | a > | b > (F = 2) Only dipole transitions are considered here HoHo H int | i >

55. Raman System Rb D2-line F' = 0,1,2,3 (F = 1) 5 2 P 3/2 5 2 S 1/2   EsEs EpEp | a > | b > (F = 2) | i > Assumptions: 1)Only dipole transitions allowed 2)Large one photon detuning 3)  <<  b -  a 4)Terms varying faster than  are integrated out

56. Raman System Rb D2-line F' = 0,1,2,3 (F = 1) 5 2 P 3/2 5 2 S 1/2   EsEs EpEp | a > | b > (F = 2) | i > Assumptions: 1)Only dipole transitions allowed 2)Large one photon detuning 3)  <<  b -  a 4)Terms varying faster than  are integrated out whereand H effective

57. Non-linear Refractive Index

59. nn

60. Acknowledgements and References

61. Effective Hamiltonian Finding the eigenvalues of this effective Hamiltonian and expressing in terms of Bloch vectors we can find the density matrix elements. New eigenvector smoothly coupled to the ground state. The eigenvector is shifted from |a> because of the interaction with the incident wave: where  

62. Effective Hamiltonian Finding the eigenvalues of this effective Hamiltonian and expressing in terms of Bloch vectors we can find the density matrix elements. New eigenvector smoothly coupled to the ground state. The eigenvector is shifted from |a> because of the interaction with the incident wave: where This gives us the following expressions for the density matrix elements:

63. Propagation Equation Propagation equation for the qth frequency component of E:

64. Propagation Equation Propagation equation for the qth frequency component of E: where, Start with

65. Propagation Equation Making the same assumptions as in the derivation of the effective Hamiltonian and assuming only significant coupling is between E s and E p, the polarization expectation values in frequency space for E s and E p are: Plug these expressions into the propagation equation:

66. Propagation Equation Make assumptions:

67. Propagation Equation Make assumptions:

68. Soliton Stability (Vakhitov, Kolokolov criterion)  Power (W) Assume the electric fields are identical to reduce to one non- linear equation. Assume electric field takes the form of a field which only accumulates phase with z. The corresponding propagation constant is . Stability Condition: