(a) fs damage ←→ 18µm (b) ns damage ←→ 160µm Figure 2.6: Qualitative comparison of typical fs damage and ns damage. Both pictures were taken with the same microscope with the same illumination. The left picture shows a damage site created with our setup on a HR Mirror, the dimensions ofthe image correspond to 180x135µm. The right picture shows damage from an old output coupler of an high power ns-system, the dimensions ofthe image correspond to 1600x1400 µm. A common approximation reduces this sum to three dominant terms: WP i, the photo ionization rate, WAv, the avalanche ionization rate and WR, the effective relaxation rate[5, 52] (see equation 2.3). Possible intermediate decay states have been neglected, as the primary relaxation time constant is already ofthe order ofthe pulse duration. Thus their contribution to the electron density inthe CB by re excitation should be minor. dN(t) dt = WP i + WAv − WR (2.3) Describing the photo ionization rate is usually done by the already mentioned Keldysh theory [48]. It is common to simplify the problem by assuming that MPI dominates, this yielded reasonable results.[2, 5, 17] For the case of solely MPI the expression for the photo ionization rate is shown in equation 2.4, where βm ist the multi photon absorption coefficient ofthe order m and I(t) is thelaserintensity. Recent publications suggest that this approximation is not sufficient for treating this problem and the complete Keldysh expression has to be used.[52, 4] WP i = βm · I(t) m The avalanche ionization rate was deduced from a Fokker-Planck equation 7 [53, 2] together with the electron ion collision probability and a given band gap energy. The (2.4) final result is shown in equation 2.5, where σ represents the absorption cross section of a lattice ion and EG denotes the band gap energy. 7 an equation that describes the evolution ofthe number of electrons with a given kinetic energy 10

WAv = σ EG · N(t) · I(t) = α · N(t) · I(t) (2.5) Counteracting these ionization processes are different recombination channels for conduction band electrons. As these channels may have different time constants, a decay term WR with an effective time constant τeff is introduced.[52, 5] WR = N(t) The complete rate equation describing the system is given in equation 2.7. τeff dN(t) dt = βm · I(t) m + α · N(t) · I(t) − N(t) As discussed, breakdown is assumed to occur as soon as the critical electron density in τeff (2.6) (2.7) the CB is reached. Typically, a value of NCr ≈ 10 21 cm −3 is used.[46, 23, 28, 2, 47, 31, 4] Therefore a solution for equation 2.7 has to be found and evaluated for the lowest intensity where there is a N(t, I) greater or equal Ncr. Defining a peak CB electron density ˆ N thethreshold condition can be written as ˆN(I) = Ncr Mero et al. have done a fit of this model to experimental data using the parameters from equation 2.7 as fit parameters. Figure 2.7 shows these data and the resulting fit (2.8) parameters for different materials are presented in table 2.1. The effective time constant resulting from the fit, at least in case of SiO2, agrees well with a time constant measured in a pump probe experiment.[43] Figure 2.7: An example fit ofthe presented model to experimental data. [5] 11