8 Towards Laser Intensity Calibration Using High-Field Ionization
151
and dominant mechanism. Simultaneously, deeper electronic shells are stripped out
by the laser: helium is fully ionized at I ≈ 5 × 10
16 W/cm
2 and neon at I ≈ 10
20
W/cm
2 . Thus probing higher intensities with ionization requires a target consisting
of high-Z atoms [22, 23].
A substantial number of experimental works considered multiple ionization of
atoms by intense optical and infrared laser radiation. As far as ultrahigh intensities are
concerned, Walker et al. [24, 25] and Yamakawa et al. [26, 27] considered ionization
of noble gases at intensities up to 10
19 W/cm
2 , aiming to validate the tunneling
ionization theory, assuming the value of intensity known with certain degree of
accuracy. Although in their calculations they did not consider effects of the intensity
distribution over the laser focus and simply estimated the maximal ionic charge
number using the tunneling rate formulas [28–32], a fairly good agreement with
the experimental data was demonstrated. Particularly for the case of [26, 27] the
utilization of a mobile slit to limit the intensity-dependent volume expansion, placed
on the central axis of the time of flight (TOF) tube (see e.g. [33] for details), made
possible to detect ions only coming from the tight-focused region (the most intense
one) and in this way improving the comparison with the theoretical models. In other
high field experiments within the tunneling regime, the ionization yield of He was
employed to confirm focusing intensities of over 10
20 W/cm
2 with an excellent
precision, less than 20%, when compared with optical measurements of the laser
pulse [33].
In this contribution, we suggest using a setup similar to the one implemented
in [24–27] in the opposite way: the unknown value of the peak intensity and, to
some extent, of the spatial intensity distribution in the focus can be reconstructed,
in principle, from the distributions of ionic charge states. This idea has been rather
widely spread in the strong field community (see, e.g. [22, 23]) with, however, little
emphasis on both the experimental constraints needed for its realization and to the
accuracy which can be achieved within this method. Additionally, to the best of our
knowledge, a systematic and exhaustive theoretical study of this approach, joint with
its limitations and advantages, is still missing.
In order to test the feasibility of this idea, we employ the theory of sequential
tunnel ionization and numerically obtain the charge spectra by varying the laser peak
value. Our article is organized in the following way. Section 8.2 presents qualitative
estimates justifying the theory of single-particle nonrelativistic sequential tunneling,
which applies for the problem under study up to intensities of I ≈ 10
24 W/cm
2 . Here,
a brief summary of the tunneling ionization theory is given as well, in the part which
relates to the total ionization rates. Using an analytic formulae for the rate of tunneling
ionization, a simple estimate for the off-set ionization potential, corresponding to the
highest charge state present in the spectrum, is presented. In Sect. 8.3 we formulate the
system of rate equations, based on the ionization rates, that govern the laser ionization
of argon, krypton and xenon. Essential simplifications of the full system, making it
accessible for fast and easy to handle numerical computations, are introduced and
discussed here. Results of numerical calculations are presented as well and compared
to the theoretical predictions of Sect. 8.2. Finally, the last section contains a brief
conclusion. As a matter of completeness, we include in the Appendices a complete
151
and dominant mechanism. Simultaneously, deeper electronic shells are stripped out
by the laser: helium is fully ionized at I ≈ 5 × 10
16 W/cm
2 and neon at I ≈ 10
20
W/cm
2 . Thus probing higher intensities with ionization requires a target consisting
of high-Z atoms [22, 23].
A substantial number of experimental works considered multiple ionization of
atoms by intense optical and infrared laser radiation. As far as ultrahigh intensities are
concerned, Walker et al. [24, 25] and Yamakawa et al. [26, 27] considered ionization
of noble gases at intensities up to 10
19 W/cm
2 , aiming to validate the tunneling
ionization theory, assuming the value of intensity known with certain degree of
accuracy. Although in their calculations they did not consider effects of the intensity
distribution over the laser focus and simply estimated the maximal ionic charge
number using the tunneling rate formulas [28–32], a fairly good agreement with
the experimental data was demonstrated. Particularly for the case of [26, 27] the
utilization of a mobile slit to limit the intensity-dependent volume expansion, placed
on the central axis of the time of flight (TOF) tube (see e.g. [33] for details), made
possible to detect ions only coming from the tight-focused region (the most intense
one) and in this way improving the comparison with the theoretical models. In other
high field experiments within the tunneling regime, the ionization yield of He was
employed to confirm focusing intensities of over 10
20 W/cm
2 with an excellent
precision, less than 20%, when compared with optical measurements of the laser
pulse [33].
In this contribution, we suggest using a setup similar to the one implemented
in [24–27] in the opposite way: the unknown value of the peak intensity and, to
some extent, of the spatial intensity distribution in the focus can be reconstructed,
in principle, from the distributions of ionic charge states. This idea has been rather
widely spread in the strong field community (see, e.g. [22, 23]) with, however, little
emphasis on both the experimental constraints needed for its realization and to the
accuracy which can be achieved within this method. Additionally, to the best of our
knowledge, a systematic and exhaustive theoretical study of this approach, joint with
its limitations and advantages, is still missing.
In order to test the feasibility of this idea, we employ the theory of sequential
tunnel ionization and numerically obtain the charge spectra by varying the laser peak
value. Our article is organized in the following way. Section 8.2 presents qualitative
estimates justifying the theory of single-particle nonrelativistic sequential tunneling,
which applies for the problem under study up to intensities of I ≈ 10
24 W/cm
2 . Here,
a brief summary of the tunneling ionization theory is given as well, in the part which
relates to the total ionization rates. Using an analytic formulae for the rate of tunneling
ionization, a simple estimate for the off-set ionization potential, corresponding to the
highest charge state present in the spectrum, is presented. In Sect. 8.3 we formulate the
system of rate equations, based on the ionization rates, that govern the laser ionization
of argon, krypton and xenon. Essential simplifications of the full system, making it
accessible for fast and easy to handle numerical computations, are introduced and
discussed here. Results of numerical calculations are presented as well and compared
to the theoretical predictions of Sect. 8.2. Finally, the last section contains a brief
conclusion. As a matter of completeness, we include in the Appendices a complete