8 Towards Laser Intensity Calibration Using High-Field Ionization
153
2. We a priori assume multiple ionization to be sequential, so that electrons leave an
atom independently, i.e. one after another. This assumption relies on two inequalities.
First, the travel time of the electron under the potential barrier
τ sb
b
v 0
≈
I p
√
2E 0
,
(8.3a)
where b = I p /E 0 is the barrier width and v 0 =
2I p is the characteristic electron
velocity in a bound state with ionization potential I p , is much longer than the electronelectron or electron-ion correlation time for the same bound state
τ corr
a B
v 0
≈
1
2I p
,
(8.3b)
i.e. τ sb τ corr . In (8.3b) a B 1/
2I p is the characteristic size of the bound state
(Bohr radius). The ratio of these two times results then:
τ corr
τ sb
E 0
E ch
≡ F, E ch = (2I p )
3/2
,
(8.3c)
where E ch is the characteristic electric field strength at the Bohr orbit of the bound
state, and F is known as the reduced field [30]. For tunneling or multiphoton ionization the reduced field is always numerically small, i.e. F 1. Furthermore, its
typical value decreases with increase of both the atomic ionization potential and laser
intensity, as we will demonstrate below.
3. Although the electron motion after the ionization step quickly becomes ultrarelativistic, the tunneling itself proceeds nonrelativistically, as long as the ionization
potential is small compared to the rest energy of the electron, i.e. I p c
2 . Quantitatively, relativistic effects in laser-induced tunneling are determined by the value of
the parameter [30, 32]
ξ
2
= 1 −
1
2
2 + 8 −
, , =
c
2
− I p
c 2 .
(8.4a)
In all the cases we study here, ξ remains small, i.e.
ξ
2
≈
2I p
3c 2 1.
(8.4b)
As an example, let us consider the ground 1s state of the Ar
17+ ion, with I p =
4426 eV ≈ 163 a.u. Below we show that intensities I ≈ 2 × 10
21 W/cm
2 are required
to ionize this state producing bare Ar
18+ ions. In this case results ξ
2
= 0.0058,
showing that the relativistic effect on the tunneling remains on the level of 1% or
less. For intensities I ≈ 10
24 W/cm
2 and I p ≈ 30 keV (see Fig. 8.1) ξ
2
≈ 0.04, so that
the nonrelativistic approximation remains quite accurate even at such otherwise ultra
153
2. We a priori assume multiple ionization to be sequential, so that electrons leave an
atom independently, i.e. one after another. This assumption relies on two inequalities.
First, the travel time of the electron under the potential barrier
τ sb
b
v 0
≈
I p
√
2E 0
,
(8.3a)
where b = I p /E 0 is the barrier width and v 0 =
2I p is the characteristic electron
velocity in a bound state with ionization potential I p , is much longer than the electronelectron or electron-ion correlation time for the same bound state
τ corr
a B
v 0
≈
1
2I p
,
(8.3b)
i.e. τ sb τ corr . In (8.3b) a B 1/
2I p is the characteristic size of the bound state
(Bohr radius). The ratio of these two times results then:
τ corr
τ sb
E 0
E ch
≡ F, E ch = (2I p )
3/2
,
(8.3c)
where E ch is the characteristic electric field strength at the Bohr orbit of the bound
state, and F is known as the reduced field [30]. For tunneling or multiphoton ionization the reduced field is always numerically small, i.e. F 1. Furthermore, its
typical value decreases with increase of both the atomic ionization potential and laser
intensity, as we will demonstrate below.
3. Although the electron motion after the ionization step quickly becomes ultrarelativistic, the tunneling itself proceeds nonrelativistically, as long as the ionization
potential is small compared to the rest energy of the electron, i.e. I p c
2 . Quantitatively, relativistic effects in laser-induced tunneling are determined by the value of
the parameter [30, 32]
ξ
2
= 1 −
1
2
2 + 8 −
, , =
c
2
− I p
c 2 .
(8.4a)
In all the cases we study here, ξ remains small, i.e.
ξ
2
≈
2I p
3c 2 1.
(8.4b)
As an example, let us consider the ground 1s state of the Ar
17+ ion, with I p =
4426 eV ≈ 163 a.u. Below we show that intensities I ≈ 2 × 10
21 W/cm
2 are required
to ionize this state producing bare Ar
18+ ions. In this case results ξ
2
= 0.0058,
showing that the relativistic effect on the tunneling remains on the level of 1% or
less. For intensities I ≈ 10
24 W/cm
2 and I p ≈ 30 keV (see Fig. 8.1) ξ
2
≈ 0.04, so that
the nonrelativistic approximation remains quite accurate even at such otherwise ultra