II. Investigation of ionization process 65
6. Hydrogen atom in a strong laser field 115
6.3. Ionization rates
Consider now the dependence of the ionization rates for the H atom on frequency and intensity of the laser field. Figure 6.8 shows the results of Floquet calculations performed for laser intensities up to 1014W/cm2 using the laser wavelengthsλ= 400,532,650,800 and 1064 nm. As can be seen from Fig. 6.8, the dependence of the ionization rates on intensity exhibits resonant structures occurring due to the multiphoton coupling of the ground state to Stark-shifted excited states. For a given wavelength the ionization rate primarily increases with increasing radiation intensity, but the slope (on the log scale) becomes smaller. Furthermore, for a given intensity the ionization rate mainly decreases with increasing radiation wavelength and tends to the corresponding quasistatic ion-ization rate. The latter is very well approximated by the Popov-Peremolov-Terent’ev (PPT) formula (5.83) for smaller intensities, whereas for higher intensities PPT rates evidently overestimate the correct quasistatic rates.
0.05 0.1 0.15 0.2 Photon energy, a.u.
10-15 10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3
Ionization rate, a.u.
Floquet, 5×1013 W/cm2 Floquet, 2×1013 W/cm2 Floquet, 5×1012 W/cm2 Popruzhenko, 5×1013 W/cm2 Popruzhenko, 2×1013 W/cm2 Popruzhenko, 5×1012 W/cm2
Figure 6.9.: Ionization rates of an H atom vs. photon energy. Ionization rates obtained by solving the Floquet equations for different radiation intensities are compared to ionization rates obtained by using an analytical formula proposed by Popruzhenkoet al.[189]). Dashed curves indicate the results obtained with Eq. (6.6).
A strong dependence of the ionization rates on the laser wavelength and a substantial deviation from the quasistatic limit (apparent in Fig. 6.8 for small intensities) indicate the multiphoton character of the ionization process. As the intensity increases, the ionization rates depend less on the laser wavelength and the deviation from the quasistatic rate gets smaller. This indicates that the ionization gradually changes its character from multiphoton to quasistatic ionization.
Figure 6.9 presents another view on this transition. It shows Floquet rates for three different radiation intensities plotted as a function of photon energy. As in the previous study, REMPI resonances are clearly visible and the ionization rate may vary by orders of magnitude in a small range of photon energies. Note the evident shift (in the range 0.1 to 0.2 a.u.) of the REMPI peaks towards higher photon energies with increasing radiation intensity, especially well visible for the strong resonances with positions at about 0.19 a.u. Although it makes it difficult to find universal behavior of ionization rates, a general trend is well apparent. Whereas for an intensity of 5×1012W/cm2 the ionization rate substantially decreases (by 7 orders of magnitude) with decreasing
0 1 2 3 4 5 6 7 8 9 10 Intensity, 1013 W/cm2
10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3
Ionization rate, a.u.
Floquet TDSE K-SFA B-SFA L-SFA V-SFA
Figure 6.10.: Ionization rates of an H atom for an 800 nm laser field vs. laser intensity.
Floquet results are compared with ionization rates yielded by the TDSE method, L-gauge SFA (L-SFA), L-gauge SFA with Krainov’s Coulomb correction (K-SFA), V-gauge SFA (V-SFA), and V-gauge SFA with the Coulomb correction of A. Beckeret al. (B-SFA).
photon energy, the slope of the decrease diminishes with increasing radiation intensity.
Surprisingly, for sufficiently low photon energies a very accurate estimation of the slope dependence on photon energy and intensity is provided by a simple correction Ccor to the quasistatic rates ΓQS,
Γcor=CcorΓQS, Ccor = exp 2
3F n
1−g(γ)o
, (6.6)
where the function g, defined as g(γ) = 3
2γ
1 + 1 2γ2
arcsinhγ−
p1 +γ2 2γ
, (6.7)
has been introduced by Perelomov et al.[95]. A comparison with the recently proposed Coulomb-corrected short-range ionization rates [189] shows a reasonable agreement in the whole range of the photon energies, although the resonances are not described and
0 1 2 3 4 5 6 7 8 9 10 Intensity, 1013 W/cm2
10-7 10-6 10-5 10-4 10-3
Ionization rate, a.u.
Floquet TDSE K-SFA B-SFA L-SFA V-SFA
Figure 6.11.: The same as Fig. 6.10 but for an 400 nm laser field.
a slight underestimation can be observed for smaller intensities in the low-energy part of Fig. 6.9, most likely due to the matching technique applied in [189, 190]3.
It is instructive to compare the Floquet ionization rates with the ones obtained using the TDSE method (as discussed in Sec. 6.1.1) and various SFA implementations. Such a comparison is presented in Fig. 6.10 for an 800 nm laser field. Besides the traditional L-gauge and V-gauge SFA, L-gauge SFA with the Coulomb correction of Krainov [192]
as well as V-gauge SFA with the Coulomb correction of A. Beckeret al. proposed in [168]
are presented as well.
The largest deviation from the Floquet results (about 3 orders of magnitude) is found for the result of the V-gauge SFA. The L-gauge SFA results are roughly two orders of magnitude smaller than the Floquet ionization rates, whereas the Krainov and A.
Beckeret al. corrections yield ionization rates which are in a very good agreement with the exact result. Finally, the agreement with ionization rates provided by the TDSE
3The present calculation uses the Coulomb correction defined by Eq. (25) in [189] instead of its ap-proximate version given by Eq. (5) in [190]. The latter yields by about a factor of 2 smaller results for γ ≈ 1. Note, the numerical results presented in [190] were also obtained using the Coulomb correction defined by Eq. (25) [191].
method is almost excellent disregarding small deviations caused by resonances.
Since SFA models do not take into account excited bound states of the H atom, the agreement of the Coulomb corrected SFA models with the exact result becomes worser for those laser parameters at which ionization occurs via an intermediate resonant state. An example of such a situation is presented in Fig. 6.11 where the same type of comparison as in Fig. 6.10 is carried out, but for an 400 nm radiation field. The Floquet ionization rate exhibits a pronounced hump at intensities about 2.5×1013W/cm2 which is completely absent in all SFA versions. A more comprehensive analysis shows that this hump is a consequence of (4+1) REMPI through the 4d state. Since the TDSE method takes into account both bound and continuum states, the TDSE rates are again in an excellent agreement with the Floquet rates.