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

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

6.1.2 The dielectric

6.1.2 The dielectric function of a solid including a contribution of free carriers The charge displacement x together with the charge density ρ can now be used to express the polarization P of a given material containing 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 contributing effects. In our case, as we are interested 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) Including 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 defined 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 together with equations 6.10 and 6.11 to ωlaser = ωplasma = ρcrite 2 ɛ0ɛrestm ∗ from the expression above we directly get the critical plasma charge density ρcrit = ɛ0ɛrestm ∗ (6.12) e 2 · ωlaser (6.13) ɛrest can be written as the normal refractive index 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 intensity distribution in space and time I(r, t) = The power P as function of time is then from this integral 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 resulting 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 together 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 final expression for the evaluation of the data 41 √ π √2 √ π √2 (6.20) (6.21)

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