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Non- paraxiality and femtosecond optics

Institute of Electronics, Bulgarian Academy of Sciences Laboratory of Nonlinear and Fiber Optics. Non- paraxiality and femtosecond optics. Lubomir M. Kovachev. Nonlinear physics. Theory and Experiment. V 2008. Paraxial optics of a laser beam. Solution in (x, y, z) space.

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Non- paraxiality and femtosecond optics

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

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

  3. z=zdiff z=0

  4. Numerical solution using FFT technique. Paraxial optics. Laser beam on 800 nm (zdiff=k0r02= 7.85 cm; r0= 100µm) Initial condition z=0 z=1/3 z=2/3 z=1;zdiff=7.85 cm

  5. Phase modulated (by lens) Gaussian beam a-radius of the lens, f- focus distance d0- thickness in the centrum Seff- effective area of the laser spot a=1,27 cm Seff=0.2 f=200 cm z=0 z=1/3 z=2/3 z=1=zdiff

  6. Paraxial optics of a laser pulse. From ns to 200-300 ps time duration Dimensionless analyze: In air, gases and metal vaporst0>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 regimeγ=1.7

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

  9. 2) Nonlinear regime γ=1.7

  10. References Non-collapsed regime of propagation of fsec pulses 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 Optoelektronik29, 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.

  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: • 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. Self-Channeling of Light in Linear Regime ?? (Femtosecond pulses) 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. • Saving the Spatio -Temporal Paraxial Model – • linear and nonlinear X waves?? • X-waves - J0 Bessel functions – infinite energy • 2) X-waves - Delta functions in (kx, ky) space. • Experiment: • Self-Channeling is observed for spectrally - limited (regular) pulses • 2. “Wave type” diffraction for single- cycle pulses.

  15. Something happens in FS region?? Wanted for new model to explain: 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 Three basic new nonlinear experimentally confirmed effects: 3. Spectral, time and spatial modulation 4. Arrest of the collapse 5. Self-channeling

  16. Non-paraxial model 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.

  17. 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:

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

  19. SVEA in Galilean coordinate frames

  20. Constants

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

  22. SVEA in dimensionless coordinates Laboratory Galilean

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

  24. Laboratory frame Galilean frame Solutions in kx ky kz space : where

  25. Fundamental solutions of the linear SWEA

  26. Fundamental linear solutions of SVEAfor media with dispersion: Fundamental solutions of VLAE for media without dispersion:

  27. 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: αδ2=1; β1=2.1X10-5

  28. 43 ps (long pulse) αδ2=6; β1=2.1X10-5

  29. Light Bullet (330 fs) α=785; δ2=1; β1=2.1X10-5

  30. Light Disk (33 fs) α=78,5; δ2=100; β1=2.1X10-5

  31. Analytical solution of SVEA (when β1<<1)and VLAE for initial Gaussian LB (δ=1) (Lab coordinate)

  32. Analytical solution of SVEA (when β1<<1)and VLAE for initial Gaussian LB (δ=1)

  33. Shaping of LB on one zdifpulse=k02r4/z0 length 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).

  34. Analytical solution of SVEA (when β1<<1)and VLAE for initial Gaussian LB (δ=1) (Galilean coordinate)

  35. Analytical solution of SVEA (when β1<<1)and VLAE for initial Gaussian LB (δ=1) (Galilean coordinate)

  36. Analytical solution of SVEA (when β1<<1)and VLAE for initial Gaussian LB (δ=1) (Galilean coordinate)

  37. 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.

  38. 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

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

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

  41. T=0 t=3Pi

  42. Analytical solution of SVEA (when β1<<1) and VLAE for initial Gaussian LB in single-cycle regime (δ=1 and α=2).

  43. Conclusion (linear regime) • 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 zdifpulse=k02W4/z0

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

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

  46. 2)Nonlinear regime γ=1.7

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