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Characterization of the laser induced damage threshold of mirrors in ...

Characterization of the laser induced damage threshold of mirrors in ...

Material m α fit [ cm2

Material m α fit [ cm2 J ] βfit m [ cm2m−3 ·fs m−1 J m ] τ fit eff [fs] τmeasured[fs] T iO2 3 34 1.5 · 10 23 120 T a2O5 3 11 6.7 · 10 24 490 HfO2 4 10 2.9 · 10 25 1050 Al2O3 5 12 2.3 · 10 24 220 SiO2 6 8 9.9 · 10 25 220 250[43] Table 2.1: Resolved fit parameters for eq. 2.7 by Mero et al. for different thin films [5] Multiple pulse femtosecond breakdown It has been shown experimentally that the damage threshold for multiple pulse exposures is lower compared to single pulse illumination.[11, 27, 7, 6] This is explained by accumulation of electrons in long lived intermediate states. For instance self-trapped excitons(STE)[54] and subsequently color centers can be generated at standard laser parameters[55]. These excitons can have life times up to months at room temperature.[56] Similar to the single pulse breakdown the multiple pulse fs breakdown can be described by rate equations, only that intermediate states have to be taken into account. Each intermediate state contributes an excitation from that state into the CB and the decay from the CB into that state. Equation 2.9 describes this case, K denotes the number of contributing states. dNCB(t) dt = βm · I(t) m + α · NCB · I(t) − NCB τCB→V B + K i=1 σi · Ni · I(t) mi − NCB τCB→i A set of coupled rate equations is required, one equation for every contributing state. Equation 2.10 shows an example for a rate equation of an intermediate state. In this example there are terms for the decay into the VB and decay from the CB and excitation into the CB. It should be kept in mind that, depending on the state, this equation system can have an almost arbitrary amount of terms. dNStateX(t) dt = −σ · NStateX(t) · I(t) mStateX + NCB(t) τCB→StateX − NStateX(t) τStateX→V B The total system of ordinary differential equations can be solved numerically, for (2.9) (2.10) instance by a Runge Kutta algorithm which is included in any standard math program like Matlab or Mathematica. Having a solution for the CB electron density, the threshold fluence is derived analogous to the single pulse case. With this model, Mero et al. explained the dependence of the threshold fluence on the 12

number of consecutive pulses on one spot. In order to model the measured T a2O5 layer they introduced two additional states, one requiring single photon absorption and another one requiring three photons for re-excitation into the conduction band.[6] Their collected data and a fit of this model on their data is shown in figure 2.8. In the next section there will be further discussion about multiple pulse effects and the possible utilization of this model Figure 2.8: Damage threshold fluence as a function of the number of illumination pulses. The solid line is the result of simulations using a similar model to equations 2.9 and 2.10. The measured material was a single T a2O5 film deposited on fused silica. The repetition rate was 1kHz. [6] 2.3 Dependence of the damage threshold on different parameters This section will give a phenomenological overview about the different dependencies of the damage threshold of wide band gap materials. 2.3.1 Dependence on the pulse duration In the discussion of the damage theory we already stated that for pulse durations below some picoseconds the physical mechanism for damage changes. This was originally concluded after numerous experimental studies investigated the dependence of the critical breakdown fluence FT h on the pulse duration τP .[2, 28, 11] Figure 2.2 displays the data from Stuart et al..[2] Above some picoseconds the threshold fluence is approximately proportional to √ τP , while below that pulse duration it deviates significantly from that proportionality. 13

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