Simmulations of Sommerfeld and Brillouin precursors in the medium with frequency dispersion using numerical method of solving wave equations

The wave equation (2+1) taking into account the frequency dispersion of the material for TE polarization is written in a form in which the second time derivative included under the sign of the convolution integral describing the electric induction. An algorithm for solving such equations using explicit finite difference scheme is developed. The solution of the wave equation, describing the propagation of ultrashort pulse duration of 3.36 fs (in the spectral range 274-806 nm) in a planar waveguide made of quartz glass at the center wavelength of 532 nm, is different from solutions of this equation without dispersion for 3% and from solutions of Maxwell's equations derived by FDTD-method in the program FullWAVE, for 6%. A detailed analysis of the obtained solutions for a femtosecond pulse with a sharp initial front resulted in the discovery of precursors, which come to the observation point before the main pulse and the intensity is 100 times smaller. Moreover, the calculated time delay Sommerfeld precursor differs from the theoretical to 20%, And the Brillouin precursor of 3%.