2. Theoretical framework
2.3. Field-driven ionization
When an atom is exposed to a laser field with photon energy ~ω larger than the atoms ionization energy Ip, the atom may be ionized by absorption of a single photon. Upon such single-photon ionization, an electron is liberated with excess energy ~ω − Ip. If the photon energy is below the ionization energy, absorption of a single photon is in- sufficient to lift a bound electron into the continuum. However, when applying strong laser fields, ionization is still possible. In this case the typical picture to describe the ionization mechanism depends strongly on the laser intensity I0. For low intensities, where the strength of the laser electric field is small compared to the field that binds the electron to its ion, the absorption of n photons is necessary to overcome the binding potential (n~ω > Ip), see Fig. 2.6(a). This multiphoton ionization can be described conve- niently by means of perturbation theory that predicts an ionization rate proportional to In
0.
The perturbative picture breaks down when the electric field strengths of laser and ion become comparable. In this case, the effective potential Veff = Vion+ Vlas formed by the combined ionic potential Vion and the potential Vlas mediated by the laser field, can be substantially deformed such that a finite barrier is formed, seeFig. 2.6(b). The probability for a bound electron to tunnel through this barrier depends mainly on the shape of the po- tential and particularly on the barrier width. For arbitrary shapes the respective ionization rate can be derived within WKB approximation. For atomic tunneling a widely utilized tunneling rate has been derived by Ammosov, Delone and Krainov [95], where the re- spective ADK rate6 ΓADK ∼ exp−25/2Ip3/2
3E
scales highly nonlinear with the field-strength.
2 Intensity [W/cm ] Keldysh parameter tunneling multiphoton 1011 1012 1013 1014 1015 101 100 10-1 102 103 Energy Position multiphoton tunneling (a) (b) (c) Position roi (streaking) roi (strong-field)
Figure 2.6. Photoionization regimes. (a) Multiphoton regime, where electrons are liberated from the ionic potential (dashed black) by absorption of multiple photons (red arrows). The laser field only acts as a weak perturbation to the ionic field, resulting in an almost unperturbed effective potential (solid black). (b) Tunneling regime, where the strong laser field deforms the ionic potential creating a finite tunneling barrier in the effective potential. (c) Keldysh parameter κ in dependence of laser intensity for Ip= 9 eV for two wavelength (as indicated). The dashed
gray line separates the multiphoton regime (κ 1) and the tunneling regime (κ. 1). Gray areas indicate the regions of interest (roi) for the strong-field and streaking scenarios considered in this work.
2.3. Field-driven ionization
Atomic photoemission is often described following the famous work of Keldysh [1] that connects the regimes of multiphoton ionization and ionization via field-induced tunneling. Introducing the dimensionless Keldysh parameter
κ = s
Ip 2Up
, (2.79)
that sets the relevant energy scales of atom (Ip) and laser field (Up) into proportion, allows to roughly estimate the relevant regime for a specific scenario. Typically, ionization is treated in the multiphoton regime for κ 1, while tunneling is considered for κ. 1.
Near-field driven ionization in dielectric nanospheres
In M3C, atomic ionization is considered for a convenient description of the ionization of the initially neutral dielectric spheres. Potential ionization sites are sampled randomly within the sphere volume and the ionization probability is determined from the local near-field. For the scenarios investigated in this work, the XUV near-field is considered to drive single-photon ionization with the respective rate scaling proportional to the local instantaneous intensity. Ionization by the combined NIR near-field and the nonlinear mean-field is treated in the tunneling picture and the ionization probability is determined from the ADK rate calculated from the local instantaneous near-field. These assumptions are substantiated by the respective Keldysh parameters at the relevant laser wavelength and for the considered intensities, see Fig. 2.6(c). Here, an effective ionization energy of Ip = 9 eV is assumed to approximate the wide band gap of the SiO2 nanoparticles (cf. Fig. 2.7). For the strong-field simulations, NIR pulses with peak intensities around 1013W/cm2 are considered, corresponding to a Keldysh parameter close to unity when taking into account a field enhancement of (≈ 2) at the nanosphere surface (cf. red curve and right shaded area in Fig. 2.6(c)). For the attosecond streaking simulations, both fields have intensities of around 1012W/cm2. For these intensities NIR-driven tunneling can be safely neglected due to vanishing ionization probabilities. Ionization from the XUV near-field can be treated in the multiphoton regime as the respective Keldysh parameter is > 100 (cf. blue curve and left shaded area inFig. 2.6(c)).
-140 -120 -100 -80 -60 -40 -20 0
Binding energy [eV]
log
yield [arb. units]
10 2 3 4 5 6 Si Auger (LVV) Si L3,2 secondary electrons SiO valences2
Figure 2.7. Photoelectron spectrum mea- sured for 50 nm SiO2 nanoparticles under soft X-rays at photon energy ~ω = 137.9 eV. The vertical dashed line indicates the effec- tive ionization energy of around 8.5 − 9 eV as considered for the semi-classical simulations. Adapted from [96].
2.3.1. Tunnel ionization
For the intensities considered in this work, the description of tunnel ionization in the dielectric nanospheres is restricted to tunneling from the surface into the surrounding vacuum. Tunneling within the volume can be neglected, as the local field strengths within the spheres are substantially lower and the tunneling probabilities vanish due to the highly nonlinear ADK rate. In the atomic case, the starting point of a classical electron trajectory is typically chosen as the classical tunneling exit xte= Ip/|Elas| (see dashed black curve and blue arrow in Fig. 2.8) and their statistical weight is determined by the tunneling probability calculated from the lased field.
For an atom that is located close to the surface within a dielectric material (seeFig. 2.8) the case is fundamentally different. Here, the effective potential that provides the tunneling barrier is determined by the local near-field instead of the laser field alone. When considering only an enhanced linear near-field, the effective potential exhibits a steeper slope in the vacuum region (compare solid red to dashed black curve in Fig. 2.8). This results in (i) a shorter classical tunneling exit and (ii) an increased tunneling probability. For this reason, tunneling can be expected to be most pronounced at localized field hot spots at the nanosphere surface. Taking into account the full near-field (linear near-field & mean-field, see blue curve in Fig. 2.8) can result in a reduced tunneling probability and may even lead to complete quenching of tunnel ionization if the mean-field becomes comparable to the linear near-field. Most importantly, the simple straight-forward approach to determine the classical tunneling exit and the tunneling rate from the laser field breaks down. Therefore, in the simulations the tunneling path is sampled along the effective near-field (cf. blue curve in Fig. 2.8(b)) and the tunneling exit and tunneling rate are determined from the average field along the path.
classical tunneling exit vacuum dielectric Energy Position
(a) (b) dielectric vacuum
classical tunneling exit Radial position T angential position tunneling path
Figure 2.8. Schematic representation of tunneling from the surface of a dielectric. (a) The dashed black curve reflects the effective potential for the atomic case (Coulomb + Laser), cf.
Fig. 1.1. Solid red and blue curves represent the effective potentials when considering the enhanced
linear (Mie) near-field and the full near-field for an atom located near the surface of a dielectric, respectively. (b) Visualization of the potential landscape at the surface of a dielectric in a 2D-cut (false color plot). The blue curve illustrates the tunneling path ending at the classical tunneling