154
M. F. Ciappina et al.
relativistic intensities. This considerable extension of the nonrelativistic regime of
tunneling toward ultra-high intensities, roughly six orders in magnitude above that
corresponding to a 0 = 1, where the electron motion becomes fully relativistic, can
be qualitatively explained in the following way. The ionization step can be viewed
as a detachment of the electron from its atomic orbital. This event takes place at a
distance of the order of the tunnel barrier width:
b =
I p
E 0
a B .
(8.5)
The electron travels this distance b during the time given by (8.3a), which is a small
fraction of the laser period: the respective time can be estimated as:
τ sb ω =
I p ω
√
2E 0
≡
γ
2
1.
(8.6)
Here we introduce the well-known Keldysh parameter γ [37]. During this time
interval the electron cannot be accelerated to relativistic kinetic energies, so that
the magnetic component of the Lorentz force remains a small correction, unless the
electron has had a relativistic velocity already in the beginning of its sub-barrier
motion. The latter is only possible for I p c
2 . As a result, at ξ
2
1 (see (8.4a) and
(8.4b)) the process of the electron’s escape from an atom (ion) proceeds within the
nonrelativistically domain.
4. Finally, note that ionization takes place in a deep tunneling regime, when the
Keldysh parameter (8.6) is small. For a laser with wavelength λ 1 μm the value
of (8.6) varies from γ 10
−2 for I = 10
20 W/cm
2 and I p = 10
3 eV to γ < 10
−3
for I = 10
24 W/cm
2 and I p = 3 × 10
4 eV. This means that instantaneous static-field
ionization rates are entirely applicable for the calculation of the ionization probability.
For the following, it will be useful to relate the laser electric field amplitude to the
value of intensity expressed in units of 10
20 W/cm
2 as:
E 0 =
53.4
√ I
1 + ρ 2
,
(8.7)
where ρ is the laser field ellipticity. Throughout this contribution we set ρ = 0, i.e. we
particularly consider the linear polarized case.
8.2.2 Tunneling Ionization Rates Calculation
Under the assumptions formulated in the previous subsection, the ionization rate of
an ionic level with a residual charge z, i.e. z = 1 for neutral atoms and z = N if all
the electrons are removed (ionization of an A
(N −1)+ ion), with the effective principal
quantum number [30, 31]
M. F. Ciappina et al.
relativistic intensities. This considerable extension of the nonrelativistic regime of
tunneling toward ultra-high intensities, roughly six orders in magnitude above that
corresponding to a 0 = 1, where the electron motion becomes fully relativistic, can
be qualitatively explained in the following way. The ionization step can be viewed
as a detachment of the electron from its atomic orbital. This event takes place at a
distance of the order of the tunnel barrier width:
b =
I p
E 0
a B .
(8.5)
The electron travels this distance b during the time given by (8.3a), which is a small
fraction of the laser period: the respective time can be estimated as:
τ sb ω =
I p ω
√
2E 0
≡
γ
2
1.
(8.6)
Here we introduce the well-known Keldysh parameter γ [37]. During this time
interval the electron cannot be accelerated to relativistic kinetic energies, so that
the magnetic component of the Lorentz force remains a small correction, unless the
electron has had a relativistic velocity already in the beginning of its sub-barrier
motion. The latter is only possible for I p c
2 . As a result, at ξ
2
1 (see (8.4a) and
(8.4b)) the process of the electron’s escape from an atom (ion) proceeds within the
nonrelativistically domain.
4. Finally, note that ionization takes place in a deep tunneling regime, when the
Keldysh parameter (8.6) is small. For a laser with wavelength λ 1 μm the value
of (8.6) varies from γ 10
−2 for I = 10
20 W/cm
2 and I p = 10
3 eV to γ < 10
−3
for I = 10
24 W/cm
2 and I p = 3 × 10
4 eV. This means that instantaneous static-field
ionization rates are entirely applicable for the calculation of the ionization probability.
For the following, it will be useful to relate the laser electric field amplitude to the
value of intensity expressed in units of 10
20 W/cm
2 as:
E 0 =
53.4
√ I
1 + ρ 2
,
(8.7)
where ρ is the laser field ellipticity. Throughout this contribution we set ρ = 0, i.e. we
particularly consider the linear polarized case.
8.2.2 Tunneling Ionization Rates Calculation
Under the assumptions formulated in the previous subsection, the ionization rate of
an ionic level with a residual charge z, i.e. z = 1 for neutral atoms and z = N if all
the electrons are removed (ionization of an A
(N −1)+ ion), with the effective principal
quantum number [30, 31]