1. Non- paraxiality and femtosecond optics Lubomir M. Kovachev Institute of Electronics, Bulgarian Academy of Sciences Laboratory of Nonlinear and Fiber Optics Nonlinear physics. Theory and Experiment. V 2008

2. Paraxial optics of a laser beam Solution in (x, y, z) space Analytical solution for initial Gaussian beam Initial conditions - Gaussian beam

3. z=0 z=z diff

4. Numerical solution using FFT technique. Paraxial optics. Laser beam on 800 nm (z diff =k 0 r 0 2 = 7.85 cm; r 0 = 100µm) Initial condition z=0 z=1/3 z=2/3 z=1;z diff =7.85 cm

5. z=0 z=1/3 z=2/3 z=1=z diff Phase modulated (by lens) Gaussian beam a-radius of the lens, f- focus distance d 0 - thickness in the centrum S eff - effective area of the laser spot f=200 cm a=1,27 cm Seff=0.2

6. Paraxial optics of a laser pulse. From ns to 200-300 ps time duration Dimensionless analyze: In air, gases and metal vapors t 0 >100-200 fs ; β<<1 - Negligible dispersion.

7. Nonlinear paraxial optics Nonlinear paraxial equation: Initial conditions: 1) nonlinear regime near to critical γ~ 1.2 2) nonlinear regime γ=1.7

8. 1) nonlinear regime near to critical γ~ 1.2

9. 2) Nonlinear regime γ=1.7

10. References 1. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, "Self-channeling of high-peak-power femtosecond laser pulses in Air, Opt. Lett. 20, 73-75, 1995. 2. E. T. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysyrowich, "Conical emission from self-guided femtosecond pulses", Opt. Lett, 21, 62, 1996. 3. A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov, "Moving focus in the propagation of ultrashort laser pulses in air", Opt. Lett., 22, 304-306, 1997. 4. L. Wöste, C. Wedekind, H. Wille, P. Rairroux, B. Stein, S. Nikolov, C. Werner, S. Niedermeier, F. Ronnenberger, H. Schillinger, and R. Sauerbry, "Femtosecond Atmospheric Lamp", Laser und Optoelektronik 29, 51, 1997. 5. H. R. Lange, G. Grillon, J.F. Ripoche, M. A. Franco, B. Lamouroux, B. S. Prade, A. Mysyrowicz, E. T. Nibbering, and A. Chiron, "Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding?", Opt. lett. 23, 120-122, 1998. Non-collapsed regime of propagation of fsec pulses

11. Nonlinear pulse propagation of fsec optical pulses Three basic new experimental effects 1. Spectral, time and spatial modulation 2. Arrest of the collapse 3. Self-channeling

12. Extension of the paraxial model for ultra short pulses and single-cycle pulses ? Expectations: Self-focusing to be compensated by plasma induced defocusing or high order nonlinear terms - Periodical fluctuation of the profile. Experiment: 1)No fluctuations - Stable profile 2) Self- guiding without ionization

13. Arrest of the collapse and self-channeling in absence of ionization G. Méchian, C. D'Amico, Y. -B. André, S. Tzortzakis, M. Franco, B. Prade, A. Mysyrowicz, A. Couarion, E. Salmon, R. Sauerbrey, "Range of plasma filaments created in air by a multi-terawatt femtosecond laser", Opt. Comm. 247, 171, 2005. G. Méchian, A. Couarion, Y. -B. André, C. D'Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, A. Couarion, R. Sauerbrey, "Long range self-channeling of infrared laser pulse in air: a new propagation regime without ionization", Appl. Phys. B 79, 379, 2004.

14. C. Ruiz, J. San Roman, C. Mendez, V.Diaz, L.Plaja, I.Arias, and L.Roso, ”Observation of Spontaneous Self-Channeling of Light in Air below the Collapse Threshold”, Phys. Rev. Lett. 95, 053905, 2005. Self-Channeling of Light in Linear Regime ?? (Femtosecond pulses) Saving the Spatio -Temporal Paraxial Model – linear and nonlinear X waves?? 1)X-waves - J 0 Bessel functions – infinite energy 2) X-waves - Delta functions in (k x, k y ) space. Experiment: 1.Self-Channeling is observed for spectrally - limited (regular) pulses 2. “Wave type” diffraction for single- cycle pulses.

16. Something happens in FS region?? Wanted for new model to explain: 3. Spectral, time and spatial modulation 4. Arrest of the collapse 5. Self-channeling Three basic new nonlinear experimentally confirmed effects: 1. Relative Self -Guiding in Linear Regime. 2. “Wave type” diffraction for single - cycle pulses. Optical cycle ~2 fs ; pulses with 4-8 fs duration

17. 1. L. M. Kovachev, "Optical Vortices in dispersive nonlinear Kerr-type media", Int. J. of Math. and Math. Sc. (IJMMS) 18, 949 (2004). 2. L. M. Kovachev and L. M. Ivanov, "Vortex solitons in dispersive nonlinear Kerr type media", Nonlinear Optics Applications, Editors: M. A. Karpiez, A. D. Boardman, G. I. Stegeman, Proc. of SPIE. 5949, 594907, 2005. 3. L. M. Kovachev, L. I. Pavlov, L. M. Ivanov and D. Y. Dakova, “Optical filaments and optical bullets in dispersive nonlinear media”, Journal of Russian Laser Research 27, 185- 203, 2006 4. L.M.Kovachev, “Collapse arrest and wave-guiding of femtosecond pulses”, Optics Express, Vol. 15, Issue 16, pp. 10318-10323 (August 2007). 5. L. M. Kovachev, “Beyond spatio - temporal model in the femtosecond optics”, Journal of Mod. Optics (2008), in press. Non-paraxial model

18. Introducing the amplitude function of the electrical field and the amplitude function of the Fourier presentation of the electrical field The next nonlinear equation of the amplitudes is obtained: Convergence of the series: I. Number of cycles; II. Media density:

19. SVEA in laboratory coordinate frame or V. Karpman, M.Jain and N. Tzoar, D. Christodoulides and R.Joseph, N. Akhmediev and A. Ankewich, Boyd……

20. SVEA in Galilean coordinate frames

21. Constants

22. Dimensionless parameters 1. Determine number of cycles under envelope with precise 2π 2. Determine relation between transverse and longitudinal initial profile of the pulse 3. Determine the relation between diffraction and dispersion length 4. 5. Nonlinear constant Constant connected with nonlinear addition to group velocity

23. SVEA in dimensionless coordinates Laboratory Galilean

24. Linear Amplitude equation in media with dispersion (SVEA) Laboratory : Galilean: Linear Amplitude Equation in Vacuum (VLAE) In air

25. Laboratory frame Galilean frame Solutions in k x k y k z space : where

26. Fundamental solutions of the linear SWEA

27. Fundamental linear solutions of SVEA for media with dispersion: Fundamental solutions of VLAE for media without dispersion:

28. Evolution of long pulses in air (linear regime, 260 ps and 43 ps) Light source form Ti:sapphire laser, waist on level e -1 : 1) 260 ps: αδ 2 =1; β 1 =2.1X10 -5

29. 43 ps (long pulse) αδ 2 =6; β 1 =2.1X10 -5

30. Light Bullet (330 fs) α=785; δ 2 =1; β 1 =2.1X10 -5

31. Light Disk (33 fs) α=78,5; δ 2 =100; β 1 =2.1X10 -5

32. Analytical solution of SVEA (when β 1 <<1) and VLAE for initial Gaussian LB (δ=1) (Lab coordinate)

35. Analytical solution of SVEA (when β 1 <<1) and VLAE for initial Gaussian LB (δ=1)

36. Gaussian shape of the solution when t=0. The surface |A(x,y=0,z; t=0) | is plotted. Deformation of the Gaussian bullet with 330 fs time duration obtained from exact solution of VLAE. The surface |A(x,y=0,z; t=785) | is plotted. The waist grows by factor sqrt(2) over normalized time-distance t=z=785, while the amplitude decreases with A=1/sqrt(2). Shaping of LB on one z dif pulse =k 0 2 r 4 /z 0 length

37. Analytical solution of SVEA (when β 1 <<1) and VLAE for initial Gaussian LB (δ=1) (Galilean coordinate)

40. Fig. 5. Shaping of Gaussian pulse obtained from exact solution of VLAE in Galilean coordinates. The surface A(x; y = 0; z=0; t= 785) is plotted. The spot grows by factor sqrt(2) over the same normalized time t = 785 while the pulse remains initial position z = 0, as it can be expected from Galilean invariance.

41. Linear Amplitude equation in media with dispersion (SVEA). Laboratory : Galilean: Linear Amplitude Equation in Vacuum (VLAE). Analytical (Galilean invariant ) solution of 3D+1 Wave equation. In air

42. 2. Comparison between the solutions of Wave Equation and SVEA in single-cycle regime

43. Evolution of Gaussian amplitudude envelope of the electrical field in dynamics of wave equation. Single – cycle regime

44. t=3Pi T=0

46. Analytical solution of SVEA (when β 1 <<1) and VLAE for initial Gaussian LB in single-cycle regime (δ=1 and α=2).

47. Conclusion (linear regime) 1. Fundamental solutions k space of SVEA and VLAE are obtained 2. Analytical non-paraxial solution for initial Gaussian LB. 3. Relative Self Guiding for LB and LD (α>>1) in linear regime. 4. “Wave type” diffraction for single - cycle pulses (α~1-3). 5. New formula for diffraction length of optical pulses is confirmed from analytical solution z dif pulse =k 0 2 W 4 /z 0

48. Nonlinear paraxial optics Nonlinear paraxial equation: Initial conditions: 1) nonlinear regime near to critical γ~ 1.2 2) nonlinear regime γ=1.7

49. 1) nonlinear regime near to critical γ~ 1.2

50. 2) Nonlinear regime γ=1.7

51. Nonlinear non-parxial regime. Laboratory frames Galilean

52. Dynamics of long optical pulses governed by the non - paraxial equation Nonlinear regime γ=2 (x,y plane) of long Gaussian pulse. Regime similar to laser beam.

53. Dynamics of long optical pulses governed by the non - paraxial equation Nonlinear regime γ=2 Longitudinal x, z plane of the same long Gaussian pulse. Large longitudinal spatial and spectral modulation of the pulse is observed.

54. 1/ Optical bullet in nonlinear regime γ=1.4. Arrest of the collapse.

55. 2/ OPTICAL DISK in nonlinear regime γ=2.25 NONLINEAR WAVEGUIDING.

56. 1/ Long optical pulse: The self-focusing regime is similar to the regime of laser beam and the collapse distance is equal to that of a cw wave. The new result here is that in this regime it is possible to obtain longitudinal spatial modulation and spectral enlargement of long pulse. 2/ Light bullet: We observe significant enlargement of the collapse distance (collapse arrest) and weak self-focusing near the critical power without pedestal. 3/ Optical pulse with small longitudinal and large transverse size (light disk): nonlinear wave-guiding. Conclusion - Nonlinear regime

57. Something happens in FS region?? Wanted for new model to explain: √ 3. Spectral, time and spatial modulation of long pulse √ 4. Arrest of the collapse of light bullets √ 5. Self-channeling of light disk Three basic new nonlinear effects: √ 1. Relative Self Guiding in Linear Regime of light disk. √ 2. “Wave type” diffraction for single - cycle pulses.

58. Експеримент - 800 nm: Ti-Sapphire laser 30 fs; 100 μm – леща: Мощност- 1.10 9 W пикова мощност на импулса 1X10 13 W/cm 2 ~2-3 P kr H. Hasegawa, L.I. Pavlov,.... z=0z=12 z diff