152
M. F. Ciappina et al.
description of the systems of rate equations, as well as all the parameters needed for
their numerical implementation. Atomic units e = m e = = 1, with the speed of
light c = 1/α, where α is the fine structure constant, are used unless stated otherwise.
8.2 Theory
8.2.1 Qualitative Analysis
We rely on the theoretical approach introduced in [34]. It roots on the following
physical considerations:
1. A gas in the laser focus is kept at sufficiently low pressure, so that neither
nonlinear propagation and plasma effects in the medium, nor electron-ion collisions,
play any relevant role. For instance, at a concentration of atoms n 0 = 10
14 cm
−3 and
an average ionic charge ¯
z = 20, the plasma frequency is of the order of ω p 10
12 s
−1 ,
so that no plasma oscillations take place during the interaction with a laser pulse of
100-fs or shorter time duration. The energies of the laser-ionized photoelectrons ε e
are of the order of the ponderomotive energy, so that in the ultrarelativistic case we
are interested in results:
ε e
1 +
a
2
0 (1 + ρ 2 )
2
c
2
a 0 c
2
,
(8.1)
where a 0 is the dimensionless field amplitude defined as:
a 0 =
E 0
ωc
,
(8.2)
with E 0 and ω being the laser field amplitude and frequency, respectively. Equation
(8.1) gives energies ε e ≈ 4 × 10
6 eV and ε e ≈ 4 × 10
8 eV for I = 10
20 W/cm
2 and
I = 10
24 W/cm
2 , respectively. In (8.1) ρ defines the laser field ellipticity, with ρ = 0
(ρ = ±1) for linear (elliptical) polarization. At such energies, cross-sections of elastic and inelastic electron ion collisions are comparable to that of bremsstrahlung,
with none of them exceeding 10
−20 cm
2 , even at the lower boundary of the intensity
interval (for numerical values and formulas see e.g. [35, 36]). As a result, the mean
free path of electrons exceeds the laser focal size (which is limited by several microns
for the tight focusing necessary to reach the highest values of intensity, see Sect. 2.4)
by several orders in magnitude. At the same time, the number of atoms in the interaction volume, estimated assuming the diffraction-limit focusing N a n 0 λ
3
≈ 10
2 ,
remains sufficiently high to reliably record the distribution in charge states with a
typical TOF ion detector.
M. F. Ciappina et al.
description of the systems of rate equations, as well as all the parameters needed for
their numerical implementation. Atomic units e = m e = = 1, with the speed of
light c = 1/α, where α is the fine structure constant, are used unless stated otherwise.
8.2 Theory
8.2.1 Qualitative Analysis
We rely on the theoretical approach introduced in [34]. It roots on the following
physical considerations:
1. A gas in the laser focus is kept at sufficiently low pressure, so that neither
nonlinear propagation and plasma effects in the medium, nor electron-ion collisions,
play any relevant role. For instance, at a concentration of atoms n 0 = 10
14 cm
−3 and
an average ionic charge ¯
z = 20, the plasma frequency is of the order of ω p 10
12 s
−1 ,
so that no plasma oscillations take place during the interaction with a laser pulse of
100-fs or shorter time duration. The energies of the laser-ionized photoelectrons ε e
are of the order of the ponderomotive energy, so that in the ultrarelativistic case we
are interested in results:
ε e
1 +
a
2
0 (1 + ρ 2 )
2
c
2
a 0 c
2
,
(8.1)
where a 0 is the dimensionless field amplitude defined as:
a 0 =
E 0
ωc
,
(8.2)
with E 0 and ω being the laser field amplitude and frequency, respectively. Equation
(8.1) gives energies ε e ≈ 4 × 10
6 eV and ε e ≈ 4 × 10
8 eV for I = 10
20 W/cm
2 and
I = 10
24 W/cm
2 , respectively. In (8.1) ρ defines the laser field ellipticity, with ρ = 0
(ρ = ±1) for linear (elliptical) polarization. At such energies, cross-sections of elastic and inelastic electron ion collisions are comparable to that of bremsstrahlung,
with none of them exceeding 10
−20 cm
2 , even at the lower boundary of the intensity
interval (for numerical values and formulas see e.g. [35, 36]). As a result, the mean
free path of electrons exceeds the laser focal size (which is limited by several microns
for the tight focusing necessary to reach the highest values of intensity, see Sect. 2.4)
by several orders in magnitude. At the same time, the number of atoms in the interaction volume, estimated assuming the diffraction-limit focusing N a n 0 λ
3
≈ 10
2 ,
remains sufficiently high to reliably record the distribution in charge states with a
typical TOF ion detector.