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- Threshold,
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Characterization of the laser induced damage threshold of mirrors in ...

(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 illum**in**ation. The left picture shows a **damage** site created with our setup on a HR Mirror, **the** dimensions **of** **the** image correspond to 180x135µm. The right picture shows **damage** from an old output coupler **of** an high power ns-system, **the** dimensions **of** **the** image correspond to 1600x1400 µm. A common approximation reduces this sum to three dom**in**ant 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 **in**termediate decay states have been neglected, as **the** primary relaxation time constant is already **of** **the** order **of** **the** pulse duration. Thus **the**ir contribution to **the** electron density **in** **the** CB by re excitation should be m**in**or. dN(t) dt = WP i + WAv − WR (2.3) Describ**in**g **the** photo ionization rate is usually done by **the** already mentioned Keldysh **the**ory [48]. It is common to simplify **the** problem by assum**in**g that MPI dom**in**ates, 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 **of** **the** order m and I(t) is **the** **laser** **in**tensity. Recent publications suggest that this approximation is not sufficient for treat**in**g 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] toge**the**r with **the** electron ion collision probability and a given band gap energy. The (2.4) f**in**al 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 **of** **the** number **of** electrons with a given k**in**etic energy 10

WAv = σ EG · N(t) · I(t) = α · N(t) · I(t) (2.5) Counteract**in**g **the**se ionization processes are different recomb**in**ation channels for conduction band electrons. As **the**se channels may have different time constants, a decay term WR with an effective time constant τeff is **in**troduced.[52, 5] WR = N(t) The complete rate equation describ**in**g **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 **in**tensity where **the**re is a N(t, I) greater or equal Ncr. Def**in****in**g a peak CB electron density ˆ N **the** **threshold** condition can be written as ˆN(I) = Ncr Mero et al. have done a fit **of** this model to experimental data us**in**g **the** parameters from equation 2.7 as fit parameters. Figure 2.7 shows **the**se data and **the** result**in**g fit (2.8) parameters for different materials are presented **in** table 2.1. The effective time constant result**in**g 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 **of** **the** presented model to experimental data. [5] 11

- Page 1 and 2: BACHELOR THESIS Characterization of
- Page 3 and 4: Characterization of the laser induc
- Page 5 and 6: 4 Presentation of Results 31 4.1 Co
- Page 7 and 8: 2.9 Damage threshold fluence as a f
- Page 9 and 10: Chapter 1 Introduction 1.1 About th
- Page 11 and 12: 1.2 Goal of this Thesis The goal of
- Page 13 and 14: observable under a visible light mi
- Page 15 and 16: Figure 2.3: Schematic diagram of th
- Page 17: Figure 2.5: Laser induced damage th
- Page 21 and 22: number of consecutive pulses on one
- Page 23 and 24: lattice 8 or its deformations shoul
- Page 25 and 26: Figure 2.11: Dependence of the lase
- Page 27 and 28: Chapter 3 Materials and Methods 3.1
- Page 29 and 30: Figure 3.3: FROG-trace taken at the
- Page 31 and 32: Meilhaus RedLab 1208LS USB analog-d
- Page 33 and 34: (a) ←→ 100µm (b) ←→ 100µm
- Page 35 and 36: Aσ denotes here the area in the be
- Page 37 and 38: Aσ(x) = π 2ln(2) · dAx1 · 1 +
- Page 39 and 40: Chapter 4 Presentation of Results 4
- Page 41 and 42: 4.2 Measured damage threshold of fu
- Page 43 and 44: we examined the irradiated spots wi
- Page 45 and 46: 5.2 Outlook 5.2.1 Improvement of th
- Page 47 and 48: Chapter 6 Appendix 6.1 Appendix A:
- Page 49 and 50: 6.2 Appendix B: Derivation of the e
- Page 51 and 52: List of References [1] P. J. W. C.
- Page 53 and 54: [20] R. Szipöcs, K. Ferencz, C. Sp
- Page 55 and 56: [44] D. von der Linde, K. Sokolowsk
- Page 57: using frequency-resolved optical ga