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Ionization of matter, i.e. the liberation of bound electrons by a laser field, is strongly dependent on the peak amplitude of the light field oscillation. Not only the ionization yield increases with intensity but also the dominant process changes for higher intensities. Figure 2.7 visualizes these different ionization regimes. For low intensities, ionization of a medium can be understood in terms of multiphoton ionization (MPI), where the absorption of n

photons is required to overcome the binding potential n_~ω > Ip (as illustrated in Fig. 2.7

a)). Perturbation theory predicts that the rate of multi-photon ionization scales with the corresponding power of the laser intensity wMPI ∝I₀n, where n is the number of absorbed photons. As the electric field strength approaches or even overcomes the Coulomb field which binds the electron to the parent ion, the ionization will take place in a distinctively different manner. As schematically shown in Fig. 2.7 b) and c), the atomic potential is strongly bent by the laser field enabling either a mixture of MPI and tunneling ionization or pure tunneling.

20 2. Theoretical background

a)

b)

Figure 2.8: a) Illustration of the simple man’s model for ATI and HHG. The electron is ionized by the laser field (1). The electron is accelerated in the laser field (2). Upon recollision the electron can either scatter (3a) or recombine (3b). b) Schematic ATI energy spectrum for a few-cycle laser pulse.

Ionization by intense laser fields is referred to as above-threshold-ionization (ATI). The name already suggests that the laser field can transfer more energy to the liberated elec- tron than necessary for ionization. Under such conditions the electron has an appreciable probability to tunnel through the effective potential barrier (tunnel regime) or can even leave the atom freely (above the barrier regime). To characterize the dynamics the Keldysh parameter is usually employed which relates the strength of the atomic potential to the driving field [35]: γ = ω p meIp |eE| = s Ip 2Up . (2.38)

Here me and e denote the electron’s mass and charge, while Ip is the ionization potential

and Up the ponderomotive potential. At high intensities γ <1 and tunneling ionization is

usually the dominant process, while for γ > 1 multi-photon ionization needs to be taken into account. There are a number of theoretical approaches to model the ionization rate. Often Keldysh-like theories like the Ammosov-Delone-Krainov (ADK) model are employed in the tunneling regime [35, 36]. This ADK model is derived for linearly polarized laser light in which the pulse length is assumed to be much longer than an optical cycle. The ionization rate is given by:

w=Nexp −4 √ 2meI 3/2 p 3_~eE0 . (2.39)

Hereeandme is the electron charge and mass, respectively. N is a slowly varying function,

which depends on the ionization potential Ip and average laser intensity.

The dynamics of the ionized electron wave packet will clearly depend on the time of birth as it is subject to the (time-dependent) electric fields. This is the realm of the so

2.2 Ionization of atoms and solids by strong light fields 21

called three-step or simple man’s model [37]. The main assumption of this model is that the acceleration of the electron is dominated by the external laser field. This can be justified by the strong laser field itself and furthermore by a large enough distance between tunneled electron and parent ion. Fig. 2.8 a) shows an illustration of the processes. Integrating the classical equation of motion with only the external laser field as a driving force, one finds that there are two types of trajectories. Direct electrons leave the atom without return. Varying the emission time, classical mechanics predict a maximum kinetic energy of 2 Up

of such direct electrons. All other trajectories will return to the parent ion. Here they can either be scattered or recombine. The recombination will produce high energetic photons which will be described in the next section. If the electron scatters, it will be further accelerated in the laser field potentially gaining more energy. Taking into account only the acceleration in the laser field, the highest kinetic energy of a back-scattered photoelectron was found to be Ec = 10.0007 Up by integration of the classical equations of motion [38].

Busuladˇzi´c et al. proposed a modified version of this cutoff law which takes the influence of the atomic potential into account and therefore yields more accurate results especially for lower intensities [39]:

Ec= 10.0007 Up+ 0.538 Ip. (2.40)

The shape of an ATI-photoelectron spectrum heavily depends on intensity, wavelength and laser pulse length. The occurrence of trajectories yielding the same final kinetic energy results in interference. Sub- and intercycle interferences create the typical ATI peaks in the spectrum being more pronounced for long pulses. For few-cycle laser fields, the highest energetic trajectories will only occur once during a single pulse. Thus, the high-energy part of the spectrum usually features a plateau which extends up to the highest observable energies (cutoff). Fig. 2.8 b) shows such a spectrum schematically.