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1. Diffraction of phase-modulated broad-band femtosecond pulses

In recent years the linear regime of propagation of optical pulses with broad-band spectrum in air and transparent media attracts a considerable attention. The attosecond pulses in UV region as well as femtosecond pulses with time duration 5-15 fs in optical and IR region admit from one to several optical cycles under their envelopes. For such pulses, as it was pointed in [1], the diffraction is not paraxial and on few diffraction lengths their shape takes parabolic form. In the theory developed in [2, 3] it is shown that the non-paraxiality depends on the spectral width Δκz=Δω/νgr of the pulse. Thus, one additional possibility appears: Is it possible to use phase-modulated broad-band femtosecond pulses with many cycles under their envelope to obtain Fraunhofer type diffraction? In this paper we will numerically investigate Fraunhofer type diffraction of phase-modulated 25 fs laser pulses which admit at least 10 cycles at level e-1 of the maximum. This study gives us knowledge about the limits and applicability of paraxiality in the frames of femtosecond optics. The linear scalar non-paraxial amplitude equation, governing the evolution of laser pulses in isotropic dispersive media, is written in Local type coordinates [4]:



where A(x,y,τ,z) is the slowly varying amplitude function of the pulse envelope, ν is the group velocity, β=κ0ν2gr|κ"| is a parameter characterizing the second order of linear dispersion of the medium, z = z and τ=t-z/ν.The last two terms in Eq.(1) are called non-paraxial terms, because the rest part gives paraxial spatio-temporal optics. This equation can be solved by using the Fourier method.

We studied the evolution of phase-modulated 25 fs laser pulses with a Gaussian spatio-temporal profile in air. The initial condition for numerical investigation of equation (1) is:



where r0 is the transverse size (the spot of the pulse), τ0 is the temporal size and b=1+ia is a complex number, where a is the linear frequency modulation (chirp of the pulse). Our numerical results have shown that:

  • The non-paraxial equation in Local coordinate system (1) gives the correct profile deformation of optical pulses but inverted with respect to the axis Oz;
  • The real spatial Fraunhofer diffraction of femtosecond and attosecond laser pulses is obtained when the non-paraxial equation is solved in Galilean or Laboratory coordinate system.

In our investigations we have shown that the Fraunhofer type diffraction is not a characteristic only to attosecond or femtosecond pulses with one or few optical cycles under the envelope. There are different methods to obtain broad-band laser pulses with many cycles under the envelope. For examples: nonlinear self-phase modulation, optical grating etc.

We demonstrated that even in the case of phase-modulated femtosecond pulses with many cycles under the envelope Fraunhofer type diffraction can be observed in the experiments.


2. Optimization of the regime of self-compression for fs pulses in silica and air

One of the main task of modern laser physics is to obtain ultra short laser pulses. There are different optical schemes for self-compression based on pair diffraction grating or other optical elements during propagation of pulses in linear regime. The shortening of pulse can be obtained also by using nonlinear mechanisms such as phase self-modulation (PSM), when a pulse reaches critical power for nonlinear regime in dispersion mediums. The most efficient schemes for compression require carful analyse for collection of possibly optical elements, linear and nonlinear effects to obtain maximal compression. We examine by numerical simulations two different mechanisms for self-compression of femtosecond pulses- in fused silica and air. In the first case we consider self-compression of a laser pulse during its nonlinear propagation in the anomalous dispersion region of a glass. The numerical experiments are performed for pulses with time duration 100 fs at wavelength 800 nm. In the second case we considered compression of initial chirped 25 fs pulse. Our numerical results show, that after only one pass trough optical diffraction grating the chirped pulse self-compress in nonlinear regime and obtain X wave deformation. Investigated of propagation of phase-modulated femtosecond pulses with the use of the lens as a phase corrector provides substantially different dynamic depending on the dispersion and coefficient of nonlinearity. It appears that during the evolution of the pulse in a medium with normal linear dispersion, initially shows a strong focusing, and is then diffracted at several diffraction lengths. In case of anomalous dispersion the pulse is compressed significantly.

During the evolution of a pulse in a medium with a positive linear dispersion, we initially observe a self - focusing. After the focus of the lens, the paraxiality of the diffraction is preserved and the pulse diffracted in several diffraction lengths without wave front distortion. The propagation in a medium with negative dispersion the initial phase modulation increases the compression of the pulse. Numerical modelling of the optical pulses in a nonlinear regime of propagation with initial phase modulation caused by different optical elements has important application in the modern optical engineering.