3D+1 Lorentz type soliton in air
  Up to now the long range filaments have been considered as a balance between Kerr focusing and defocusing by plasma generation in the nonlinear focus. However, it is difficult to apply the above explanation of filamentation in far-field zone. There are basically two main characteristics which remain the same at these distances - the super broad spectrum and the width of the core, while the power in a stable filament drops to the critical value for self-focusing. At such power the plasma and higher-order Kerr terms are too small to prevent self-focusing. We suggest here a new mechanism for stable soliton pulse propagation in far-away zone, where the power of the laser pulse is slightly above the critical one, and the pulse comprises super-broad spectra. For such pulses the diffraction is not paraxial and an initially symmetric Gaussian pulse takes parabolic form at several diffraction lengths . The stable soliton propagation appears as a balance between the divergent parabolic type diffraction of broadband optical pulses and the convergent nonlinear refractive index due to the intensity profile. We investigate more precisely the nonlinear third order polarization, using into account the carrier-to envelope phase. This additional phase transforms the third harmonic term to THz or GHz one, depending on the spectral width of the pulse. 
Lubomir Kovachev
physics.optics nlin.PS nlin.SI