VUV Spectroscopy of Atoms, Molecules and Surfaces

10 VUV light generation: possibilities and limitations
The situation covered by the two-step model is called the tunneling regime
and is characterized by a value γ1, is called the ”ordinary ATI” regime, for which the proces of ATI
dominates that of HHG [35]. In this regime the fast oscillations of the electric
field prevent an atomic electron from escaping by tunneling. It sees on the
average a zero field, resulting in an ionization probability that is uniform in
time [50]. In the tunneling regime the periodicity of maximum ionization,
i.e. harmonic generation, equal to half an electric field cycle, explains the
periodicity in the frequency spectrum of the emitted harmonic radiation,
being equal to twice the frequency of the fundamental laser light [50]. The
two ways of ionization may be compared with emptying a bowl of water by
shaking it at a high frequency or by tilting it at a low frequency, respectively
[51]. Tunneling may be recognized as a smearing of the ATI peak structure,
but the boundary between the two regimes, on which most studies have been
performed, is not well defined [35].
In practice, the high-harmonic photon-energy cut-off is often limited by
the saturation intensity at which the medium is fully ionized rather than by
the intensity that is physically provided by the laser. This means that in
order to obtain the highest possible harmonic orders, the rise time (in most
cases equivalent to the duration) of the laser pulses must be shortest possible.
As a consequence of the Fourier transform limit, imposing a minimum
value to the product of the laser bandwidth and pulse duration [52], and the
fact that the harmonic generation process introduces additional bandwidth
broadening [53], there is an upper (lower) limit to the pulse duration (bandwidth)
of the harmonics that can be generated at a desired minimum photon
energy. Or stated the other way around: there is an upper limit to the photon
energy that can be reached with a desired maximum bandwidth. The saturation
intensity may be determined experimentally from a measurement of the
yield of high-order harmonics of a given order as a function of laser intensity,
resulting in an increase which levels off when ionization becomes dominant
[44]. Such measurements have only been performed for a few different combinations
of pulse durations and wavelengths and only a very limited effort
has been directed towards establishing eventual scaling relations [54]. Theoretically,
saturation intensities can be calculated in the perturbative regime
from a knowledge of the N-photon generalized photoionization cross sections,
which, however, are not easily calculated [55]. An attempt to estimate the
generalized cross sections of heavier atoms by scaling from those known for
the neutral hydrogen atom resulted in an order of magnitude disagreement
with experimental data [56]. As a consequence, a relation for the photon
energy that can be obtained as a function pulse duration and wavelength is
at present not readily available.