Optimal conditions for the generation of moderate-order harmonics of a short-wave field by helium atoms

It is shown that under optimal conditions the generation of the third, fifth, seventh, and ninth harmonics of the short-wave, ultraviolet or vacuum ultraviolet, laser field by helium atoms is mainly due to transitions between bound states, and the maximum energy of the harmonics is achieved under conditions of their resonant multiphoton excitation. In this case, the optima for the generation of the third, fifth, seventh, and ninth harmonics of the field correspond to three-, four-, five-, and six-photon resonances, and the optimal value of the peak field intensity, depending on the harmonic order, varies from $2.5\ifmmode\times\else\texttimes\fi{}{10}^{14}\phantom{\rule{0.16em}{0ex}}\mathrm{W}/\mathrm{c}{\mathrm{m}}^{2}$ to $1.2\ifmmode\times\else\texttimes\fi{}{10}^{15}\phantom{\rule{0.16em}{0ex}}\mathrm{W}/\mathrm{c}{\mathrm{m}}^{2}$. The total probability of excitation and ionization of an atom at the end of a laser pulse under corresponding conditions exceeds 1/2. With such intensity and not-too-high frequency of the laser field, the Stark effect turns out to be very significant, which allows it to be tuned in resonance with an arbitrary excited state of the atom by changing the intensity of the field without changing its frequency. It is shown that for the laser field parameters maximizing the energy of the $N\mathrm{th}$ harmonic, 3 \ensuremath{\le} $N$ \ensuremath{\le} 9, this harmonic dominates in the dipole acceleration spectrum. At the same time, the amplitudes of the remaining harmonics increase as the harmonic order approaches $N$. In particular, under the conditions maximizing the yield of the ninth harmonic, the harmonic amplitudes increase when moving from the third harmonic to the fifth and then to the seventh and ninth harmonics, and the harmonics form an attosecond pulse train. At the same time, under the conditions maximizing the yield of the third harmonic, its amplitude in the dipole acceleration spectrum exceeds not only the amplitudes of the other harmonics, but also the amplitude of the atomic response at the frequency of the driving field.