Effect of a periodic magnetic field on phase matching condition in second harmonic generation at interactions of laser-plasma

In this paper, the process of second harmonic generation (SHG) is studied in underdense plasma in the presence of a periodic magnetic field. It is shown that the difference of momentums of photon of second harmonic (ℏk→(2ω)) and two photons of main wave (2ℏk→(ω)) can be provided by momentum of everyone of Fourier components of periodic magnetic field so that momentum of n  th Fourier component can be chosen by nℏk→0=ℏk→2−2ℏk→1. It is also proven that the highest efficiency of second harmonic generation will be provided by the first Fourier component of periodic magnetic field (k→0=k→2−2k→1). It is revealed that periodic magnetic field can produce longitudinal waves at (ω,nk→0+k→1) and (2ω,2k→1) as well.