Phase matching and the spectrum of high-order harmonics generated in an extended medium

We present analytical and numerical calculations of the power and angular distributions of high-order harmonics generated by limited laser beams in extended media with a dispersion weak when compared with the geometric dispersion. When the thickness of the harmonic-generating layer is large, the spectrum of high-order harmonics differs considerably from the spectrum of a single-atom response. In particular, two plateaus may arise in the spectrum of high-order harmonics owing to the off-axially phase-matched generation of certain harmonics (the generation of a part of the angular spectrum of these harmonics is phase-matched). The efficient off-axially phase-matched harmonic generation is provided mainly by amplitude modulation of the high-frequency response of a medium to the pump field in the cross section of the beam, which broadens the radiation pattern of harmonic emission. Off-axially phase-matched harmonic generation in Gaussian beams is possible with any geometric dispersion (with an arbitrarily tight focusing). The results of our calculations agree well with the experimental data on high-order harmonic generation in thick gas layers available from the literature.