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Multiple filamentation of intense laser beams Gadi Fibich Tel Aviv University

Multiple filamentation of intense laser beams Gadi Fibich Tel Aviv University. Boaz Ilan, University of Colorado at Boulder Audrius Dubietis and Gintaras Tamosauskas, Vilnius University, Lithuania Arie Zigler and Shmuel Eisenmann, Hebrew University, Israel. Laser propagation in Kerr medium.

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Multiple filamentation of intense laser beams Gadi Fibich Tel Aviv University

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  1. Multiple filamentation of intense laser beamsGadi FibichTel Aviv University • Boaz Ilan, University of Colorado at Boulder • Audrius Dubietis and Gintaras Tamosauskas, Vilnius University, Lithuania • Arie Zigler and Shmuel Eisenmann, Hebrew University, Israel

  2. Laser propagation in Kerr medium

  3. Vector NL Helmholtz model (cw)

  4. Simplifying assumptions • Beam remains linearly polarized E = (E1(x,y,z), 0, 0) • Slowly varying envelope E1 = (x,y,z) exp(i k0 z) • Paraxial approximation

  5. 2D cubic NLS • Initial value problem in z • Competition: self-focusing nonlinearity versus diffraction • Solutions can become singular (collapse) in finite propagation distance z = Zcr if their input power P is above a critical power Pcr (Kelley, 1965), i.e.,

  6. Multiple filamentation (MF) • If P>10Pcr, a single input beam can break into several long and narrow filaments • Complete breakup of cylindrical symmetry

  7. Breakup of cylindrical symmetry • Assume input beam is cylindrically-symmetric • Since NLS preserves cylindrical symmetry • But, cylindrical symmetry does break down in MF • Which mechanism leads to breakup of cylindrical symmetry in MF?

  8. Standard explanation (Bespalov and Talanov, 1966) • Physical input beam has noise e.g. • Noise breaks up cylindrical symmetry • Plane waves solutions of the NLS are linearly unstable (MI) • Conclusion: MF is caused by noise in input beam

  9. Weakness of MI analysis • Unlike plane waves, laser beams have finite power as well as transverse dependence • Numerical support? • Not possible in the sixties

  10. G. Fibich, and B. Ilan Optics Letters, 2001 and Physica D, 2001

  11. Testing the Bespalov-Talanov model • Solve the NLS • Input beam is Gaussian with 10% noise and P = 15Pcr

  12. Blowup while approaching a symmetric profile Even at P=15Pcr, noise does not lead to MF in the NLS Z=0 Z=0.026

  13. Model for noise-induced MF NLS with saturating nonlinearity (accounts for plasma defocusing) Initial condition: cylindrically-symmetric Gaussian profile + noise

  14. Typical simulation (P=15Pcr) Ring/crater is unstable

  15. Noise-induced MF • MF pattern is random • No control over number and location of filaments • Disadvantage in applications (e.g., eye surgery, remote sensing)

  16. Can we have a deterministic MF?

  17. Vectorial effects and MF • NLS is a scalar model for linearly-polarized beams • More comprehensive model – vector nonlinear Helmholtz equations for E = (E1, E2, E3) • Linear polarization state E = (E1,0,0) at z=0 leads to breakup of cylindrical symmetry • Preferred direction • Can this lead to a deterministic MF?

  18. Linear polarization - analysis • vector Helmholtz equations for E = (E1,E2,E3)

  19. Derivation of scalar model • Small nonparaxiality parameter • Linearly-polarized input beam E = (E1,0,0) at z=0

  20. NLS with vectorial effects • Can reduce vector Helmholtz equations to a scalar perturbed NLS for  = |E1|:

  21. Vectorial effects and MF • Vectorial effects lead to a deterministic breakup of cylindrical symmetry with a preferred direction • Can it lead to a deterministic MF?

  22. Simulations • Cylindrically-symmetric linearly-polarized Gaussian beams 0 = c exp(-r2) • f = 0.05, = 0.5 • No noise!

  23. P = 4Pcr Ring/crater is unstable Splitting along x-axis

  24. P = 10Pcr Splitting along y-axis

  25. P = 20Pcr more than two filaments

  26. What about circular polarization?

  27. G. Fibich, and B. Ilan Physical Review Letters, 2002 and Physical Review E, 2003

  28. Circular polarization and MF • Circular polarization has no preferred direction • If input beam is cylindrically-symmetric, it will remain so during propagation (i.e., no MF) • Can small deviations from circular polarization lead to MF?

  29. Standard model (Close et al., 66) • Neglects E3 while keeping the coupling to the weaker E- component

  30. Circular polarization - analysis • vector Helmholtz equations for E = (E+,E-,E3)

  31. Derivation of scalar model • Small nonparaxiality parameter • Nearly circularly-polarized input beam at z=0

  32. NLS for nearly circularly-polarized beams • Can reduce vector Helmholtz equations to the new model: • Isotropic to O(f2)

  33. Circular polarization and MF • New model is isotropic to O(f2) • Neglected symmetry-breaking terms are O(f2) • Conclusion – small deviations from circular polarization unlikely to lead to MF

  34. Back to Linear Polarization

  35. Testing the vectorial explanation • Vectorial effects breaks up cylindrical symmetry while inducing a preferred direction of input beam polarization • If MF pattern is caused by vectorial effects, it should be deterministic and rotate with direction of input beam polarization

  36. Dubietis, G. Tamosauskas, G. Fibich, and B. Ilan, • Optics Letters, 2004

  37. First experimental test of vectorial explanation for MF • Observe a deterministic MF pattern • MF pattern does not rotate with direction of input beam polarization • MF not caused by vectorial effects • Possible explanation: collapse is arrested by plasma defocusing when vectorial effects are still too small to cause MF

  38. So, how can we have a deterministic MF?

  39. Ellipticity and MF • Use elliptic input beams 0 = c exp(-x2/a2-y2/b2) • Deterministic breakup of cylindrical symmetry with a preferred direction • Can it lead to deterministic MF?

  40. Possible MF patterns • Solution preserves the symmetries x -x and y -y y x

  41. Simulations with elliptic input beams • NLS with NL saturation • elliptic initial conditions 0 = c exp(-x2/a2-y2/b2) • P = 66Pcr • No noise!

  42. e = 1.09 • central filament • filament pair along minor axis • filament pair along major axis

  43. e = 2.2 • central filament • quadruple of filaments • very weak filament pair along major axis

  44. All 4 filament types observed numerically

  45. Experiments • Ultrashort (170 fs) laser pulses • Input beam ellipticity is b/a = 2.2 • ``Clean’’ input beam • Measure MF pattern after propagation of 3.1cmin water

  46. P=4.8Pcr • Single filament

  47. P=7Pcr • Additional filament pair along major axis

  48. P=18Pcr • Additional filament pair along minor axis

  49. P=23 Pcr • Additional quadruple of filaments

  50. All 4 filament types observed experimentally

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