7.1 Dynamic surface distortion caused by coating absorption

The temperature field is known from Section 6 (see Equation (6.6View Equation)), so that we get
𝜖P (ω)ps 1 1 es = ----------2----2- -------------- (7.24 ) 2πKa κ s − κ L,s ξstanh ηs + χ
𝜖P-(ω)ps ---1-----------1------ os = − 2πKa κ2s − κ2 ξscothηs + χ (7.25 ) L,s
′ 𝜖P (ω)ps κ 1 es = − -2πKa---κ2-−-κ2--ξ--+-χ-tanh-η- (7.26 ) s L,s s s
𝜖P(ω )ps κ 1 o′s = ---------2----2---------------, (7.27 ) 2πKa κs − κL,sξs + χ cothηs
which, with Equation (7.23View Equation), gives the complete solution.

7.1.1 Mean displacement

We find the contribution to phase noise by computing, as usual, the mean equivalent displacement by

∫ a Z (ω) = 2π uz(ω,r,− h∕2)I(r)rdr. (7.28 ) 0
I(r) being the normalized intensity profile of the beam. This yields
∑ Z (ω ) = Bs (ω, − h∕2)ps[1 + (χe ∕ζs)2]J20(ζs). (7.29 ) s

7.1.2 Under cutoff regime

The dimensionless parameter x s is very small. With the current parameters, ζ 1 is slightly larger than one, so that

[ f ]2 ∀sxs < 4.3 × 10 −8 ----- , (7.30 ) 1 Hz
so that in the GW band (far from mirror resonances), we can take the first-order approximation of the precedent functions with respect to xs. The elastic wave regime begins when the frequency exceeds a value (cutoff) such that some κL,s and κT,s become imaginary. A study of the resonance modes can then be addressed, but this requires a careful treatment of the boundary conditions on the edge. We consider here only the case where the frequency is below the cutoff, so that a simple theory neglecting the edge is relevant and xs may be considered small. In particular, we find from 0.1 Hz to 1 kHz
α(1 + σ )𝜖P (ω) ( 2sinh2γs cosh2γs ) Bs (ω,− h∕2) = − i---------2----- ----2-------2------2- . (7.31 ) π ρCa ω sinh γscosh γs − γs
This gives for our three examples the following transfer functions for the displacement noise. LG0,0 mode w = 2 cm:
[ ] − 11 1 Hz |Z(ω )∕ 𝜖P(ω )| = 2 × 10 m ∕W --f-- . (7.32 )
Flat mode (b = 9.1 cm) or mesa mode (bf = 10.7 cm):
[ ] |Z(ω )∕𝜖P(ω )| = 1.2 × 10−11 m ∕W 1-Hz- . (7.33 ) f
LG5,5 mode w = 3.5 cm:
[1 Hz ] |Z(ω )∕𝜖P(ω )| = 1.1 × 10−11 m ∕W ----- . (7.34 ) f
Below 0.1 Hz we are in the regime where the displacements are corrected by the locking system. Above 1 kHz the displacement noise is negligible compared to shot noise and that in the present situation.


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