- Text
- Threshold,
- Laser,
- Beam,
- Measurement,
- Pulses,
- Fluence,
- Measurements,
- Dielectric,
- Dependence,
- Electron,
- Characterization,
- Induced,
- Mirrors

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

6.1.2 The dielectric function **of** a solid **in**clud**in**g a contribution **of** free carriers The charge displacement x toge**the**r with **the** charge density ρ can now be used to express **the** polarization P **of** a given material conta**in****in**g free carriers as P = ɛ0 · χfree · E = x · e · ρ (6.6) This results **in** an expression for **the** dielectric susceptibility **of** **the** free electron χfree χfree = − xeρ ɛ0E ρe2 1 = · ɛ0m∗ ω(ω + i 1 τp ) The dielectric function **of** an arbitrary material is given as first order approximation as ɛ = ɛ0(1 + (6.7) N χi) (6.8) where N denotes **the** number **of** contribut**in**g effects. In our case, as we are **in**terested **in** **the** additional contribution **of** free carriers we can reduce this expression to i=0 ɛ = ɛ0(1 + χrest + χfree) = ɛ0(ɛrest + χfree) = ɛ0ɛrest(1 + χfree ) (6.9) Includ**in**g equation 6.7 this leads to ɛ = ɛ0ɛrest(1 − ρe 2 τ 2 p ɛ0ɛrestm ∗ (ω 2 τ 2 p + 1) + i ɛrest ρe2τp ɛ0ɛrestm∗ω(ω 2τ 2 p + 1) 6.1.3 The plasma frequency and plasma charge density The plasma frequency can be def**in**ed as [60] ) (6.10) Re(ɛ(ω)) = 0 (6.11) **in** our case **of** NIR **laser** pulses we can assume w 2 · τ 2 p >> 1. This leads toge**the**r with equations 6.10 and 6.11 to ω**laser** = ωplasma = ρcrite 2 ɛ0ɛrestm ∗ from **the** expression above we directly get **the** critical plasma charge density ρcrit = ɛ0ɛrestm ∗ (6.12) e 2 · ω**laser** (6.13) ɛrest can be written as **the** normal refractive **in**dex **of** **the** material squared. 40

6.2 Appendix B: Derivation **of** **the** evaluation expression 6.2.1 The power measured via **the** power meter We assume a pulse Gaussian **in**tensity distribution **in** space and time I(r, t) = The power P as function **of** time is **the**n from this **in**tegral we get P (t) = Î · e−2 r2 σ 2 · e −2fract2 τ 2 I(r, t)dA = P (t) = Î · 2π · σ2 4 The power meter **in** use only measures **the** time averaged power ¯ P result**in**g **in** ¯P = 1 ∆t t+∆t t ¯P = frep · Î · σ2 · τ · π√ π 2 √ 2 (6.14) I(r, t) · r · dr · dφ (6.15) · e−2 t2 τ 2 (6.16) P (t)dt (6.17) 6.2.2 The peak fluence as a function **of** **the** time averaged power The fluence J is given as toge**the**r with equation 6.14 we get J(r) = J(r) = Î · τ · Its obvious, that **the** peak fluence ˆ J = J(r = 0) ˆJ = Î · τ · (6.18) I(t, r)dt (6.19) From equation 6.18 and equation 6.21 we get **the** f**in**al expression for **the** evaluation **of** **the** data 41 √ π √2 √ π √2 (6.20) (6.21)

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