162
M. F. Ciappina et al.
nonrelativistic static rate, (8.9a), for argon, krypton and xenon in different ranges of
laser intensities. We employ an adaptive stepsize Runge-Kutta scheme, based on the
Numerical Recipes (for more details see [43]).
In a fully accurate calculation, one should start the simulation from a neutral
atom and take into account all possible trees of ionic states which develop starting
from this initial condition. A simple analysis shows, however, that the number of
possible pathways grows enormously as the maximal charge number increases. Even
assuming that (a) there is no ionization from inner shells until the outer shell is not
fully striped out of electrons, (b) ionization proceeds without excitation of the residual
remaining ion and (c) one neglects the fine structure of electronic terms, we obtain
that the number of pathways for ionization of argon is equal to P = 1 for initial
states Ar
N + with N ≥ 14, P = 6 for N = 12, P = 28 for N = 8, etc. Calculation
of P for the initial configuration 1s
2 2s
2 2 p
6 (corresponding to neutral neon, Ar
8+ ,
etc.), based on these simplifications, is presented in Appendix A.
The analytical estimates of previous Sections allow to dramatically reduce the
complexity of the problem. This is so by assuming that ionization starts from ionic
charge states whose ionization potential can be estimated from (8.14), while all the
outermost levels have been quickly ionized before the peak value of intensity has been
attained. Thus, our numerical calculation intends to show the ionization dynamics of
levels with ionization potentials not very much dissimilar of those given by (8.14).
Such computation will verify the accuracy of the analytic estimate and show how
quickly ionization saturation can be achieved at a fixed laser intensity.
For clarity, we suppose that an experiment aims probing laser intensities in a given
fix interval, e.g. I = 10
21
− 10
23 W/cm
2 at the center of the focus (see Fig. 8.2).
According to Fig. 8.1, in this interval one would expect significant production of
Ar
18+ (fully stripped argon) and Kr
28+ –Kr
34+ ions. Ionic states with lower ionization potentials will be quickly ionized during the pulse intensity growth, while the
probability of production of Kr
35+ –Kr
36+ will remain negligibly small in this range
of intensities. In order to check these qualitative predictions, we solve the associated
systems of rate equations for Ar and for a spatially homogeneous laser pulse linearly
polarized (ρ = 0). In this case F(t) takes the form:
F(t) = E L f (t) cos(ωt),
(8.24)
where the pulse envelope is defined as f (t) = sin
2
ωt
2n p
, with n p the total number
of optical cycles. In our simulations we use ω = 0.0455 a.u. and n p = 10, that
correspond to a laser wavelength λ = 1 μm and a total pulse length T ∼ 33 fs,
respectively.
We consider the following assumptions:
(a) The initial state is chosen to have considerably lower ionization potential
than I
∗
p (I max ), but high enough to minimize the number of rate equations in the
system. For actual calculations, we set up for the initial condition a gas of A
k+
with k determined from (8.14) with intensity 2 orders in magnitude below the peak
intensity under consideration. As an example, considering ionization by pulses with
M. F. Ciappina et al.
nonrelativistic static rate, (8.9a), for argon, krypton and xenon in different ranges of
laser intensities. We employ an adaptive stepsize Runge-Kutta scheme, based on the
Numerical Recipes (for more details see [43]).
In a fully accurate calculation, one should start the simulation from a neutral
atom and take into account all possible trees of ionic states which develop starting
from this initial condition. A simple analysis shows, however, that the number of
possible pathways grows enormously as the maximal charge number increases. Even
assuming that (a) there is no ionization from inner shells until the outer shell is not
fully striped out of electrons, (b) ionization proceeds without excitation of the residual
remaining ion and (c) one neglects the fine structure of electronic terms, we obtain
that the number of pathways for ionization of argon is equal to P = 1 for initial
states Ar
N + with N ≥ 14, P = 6 for N = 12, P = 28 for N = 8, etc. Calculation
of P for the initial configuration 1s
2 2s
2 2 p
6 (corresponding to neutral neon, Ar
8+ ,
etc.), based on these simplifications, is presented in Appendix A.
The analytical estimates of previous Sections allow to dramatically reduce the
complexity of the problem. This is so by assuming that ionization starts from ionic
charge states whose ionization potential can be estimated from (8.14), while all the
outermost levels have been quickly ionized before the peak value of intensity has been
attained. Thus, our numerical calculation intends to show the ionization dynamics of
levels with ionization potentials not very much dissimilar of those given by (8.14).
Such computation will verify the accuracy of the analytic estimate and show how
quickly ionization saturation can be achieved at a fixed laser intensity.
For clarity, we suppose that an experiment aims probing laser intensities in a given
fix interval, e.g. I = 10
21
− 10
23 W/cm
2 at the center of the focus (see Fig. 8.2).
According to Fig. 8.1, in this interval one would expect significant production of
Ar
18+ (fully stripped argon) and Kr
28+ –Kr
34+ ions. Ionic states with lower ionization potentials will be quickly ionized during the pulse intensity growth, while the
probability of production of Kr
35+ –Kr
36+ will remain negligibly small in this range
of intensities. In order to check these qualitative predictions, we solve the associated
systems of rate equations for Ar and for a spatially homogeneous laser pulse linearly
polarized (ρ = 0). In this case F(t) takes the form:
F(t) = E L f (t) cos(ωt),
(8.24)
where the pulse envelope is defined as f (t) = sin
2
ωt
2n p
, with n p the total number
of optical cycles. In our simulations we use ω = 0.0455 a.u. and n p = 10, that
correspond to a laser wavelength λ = 1 μm and a total pulse length T ∼ 33 fs,
respectively.
We consider the following assumptions:
(a) The initial state is chosen to have considerably lower ionization potential
than I
∗
p (I max ), but high enough to minimize the number of rate equations in the
system. For actual calculations, we set up for the initial condition a gas of A
k+
with k determined from (8.14) with intensity 2 orders in magnitude below the peak
intensity under consideration. As an example, considering ionization by pulses with