156
M. F. Ciappina et al.
owing to the fact that all states with relatively small ionization potentials appear to
be quickly depleted at high intensities. Abbreviated systems of rate equations for
argon, krypton, and xenon are presented in the Appendices.
8.2.3 Intensity-Dependent Ionization Offset
Before discussing the numerical solution of the system (8.11a)–(8.11c), which is done
in the next section, we obtain an estimate for the maximal charge state which can be
achieved via tunneling ionization for a given value of laser intensity. The purpose
of this approximate calculation is twofold. First, it allows for approximately finding,
without stringent numerical calculations, the interval of charge states expected to be
observed if the value of intensity is known, and vice versa. Secondly, and not less
relevant, it will help us to considerably reduce the number of terms in the system
of equations defined in (8.11a)–(8.11c) by omitting those ones which, at a given
intensity, correspond either to levels which are ionized instantaneously, so that for
them c z = 0 or, instead, to those which stay almost unaffected for the laser field,
i.e. c z ≈ 1. For high-Z ions, where the number of rate equations becomes excessively
large, this markedly simplifies the actual numerical calculations.
In order to estimate the maximal charge number which can be produced at a given
intensity with a high probability, we neglect the (l, m) dependence in (8.9a) and
consider the total ionization probability per laser period for an s state at the maximal
intensity (to simplify notations, we set the time such that the maximum is achieved
at t = 0):
W =
2π
ω
w(ν, 0, 0; 0) = πC
2
κ K 0 2
2ν+2 F
1−2ν exp
−
2
3F
,
(8.12)
where K 0 = I p /ω is the multiquantum parameter [30, 31] and the reduced field
F, see (8.10), is calculated for the peak amplitude value E 0 of the laser electric
field. The exponential factor dominates the intensity dependence of the rate (8.12),
while the prefactor plays a very minor role in the estimation of the ionization offset
value. For a given value of intensity (and therefore of E 0 ) we estimate the threshold
ionization potential I
∗
p (I), such that the probability of ionization per laser cycle is of
the order of unity, W 1. This condition is essentially approximate as it ignores the
pulse duration, the (l, m) dependence and the ionization saturation effect. However,
thanks to the exponential factor in (8.12), which changes very rapidly with variations
of the laser field amplitude, it provides a logarithmically accurate estimate:
F
∗
=
2
3
ln
−1
2
2ν+2 C
2
κ π K 0 (F
∗
)
1−2ν
.
(8.13)
M. F. Ciappina et al.
owing to the fact that all states with relatively small ionization potentials appear to
be quickly depleted at high intensities. Abbreviated systems of rate equations for
argon, krypton, and xenon are presented in the Appendices.
8.2.3 Intensity-Dependent Ionization Offset
Before discussing the numerical solution of the system (8.11a)–(8.11c), which is done
in the next section, we obtain an estimate for the maximal charge state which can be
achieved via tunneling ionization for a given value of laser intensity. The purpose
of this approximate calculation is twofold. First, it allows for approximately finding,
without stringent numerical calculations, the interval of charge states expected to be
observed if the value of intensity is known, and vice versa. Secondly, and not less
relevant, it will help us to considerably reduce the number of terms in the system
of equations defined in (8.11a)–(8.11c) by omitting those ones which, at a given
intensity, correspond either to levels which are ionized instantaneously, so that for
them c z = 0 or, instead, to those which stay almost unaffected for the laser field,
i.e. c z ≈ 1. For high-Z ions, where the number of rate equations becomes excessively
large, this markedly simplifies the actual numerical calculations.
In order to estimate the maximal charge number which can be produced at a given
intensity with a high probability, we neglect the (l, m) dependence in (8.9a) and
consider the total ionization probability per laser period for an s state at the maximal
intensity (to simplify notations, we set the time such that the maximum is achieved
at t = 0):
W =
2π
ω
w(ν, 0, 0; 0) = πC
2
κ K 0 2
2ν+2 F
1−2ν exp
−
2
3F
,
(8.12)
where K 0 = I p /ω is the multiquantum parameter [30, 31] and the reduced field
F, see (8.10), is calculated for the peak amplitude value E 0 of the laser electric
field. The exponential factor dominates the intensity dependence of the rate (8.12),
while the prefactor plays a very minor role in the estimation of the ionization offset
value. For a given value of intensity (and therefore of E 0 ) we estimate the threshold
ionization potential I
∗
p (I), such that the probability of ionization per laser cycle is of
the order of unity, W 1. This condition is essentially approximate as it ignores the
pulse duration, the (l, m) dependence and the ionization saturation effect. However,
thanks to the exponential factor in (8.12), which changes very rapidly with variations
of the laser field amplitude, it provides a logarithmically accurate estimate:
F
∗
=
2
3
ln
−1
2
2ν+2 C
2
κ π K 0 (F
∗
)
1−2ν
.
(8.13)