7 Dynamic Surface Distortion

We address here the motion of the reflecting surface of a mirror receiving a time-varying power flow creating a time-varying temperature and finally time-varying stresses. For the temperature field, the problem was solved in the preceding Section 6. We take the generic form

∑ T(ω, r,z) = Ts(ω, z)J0(ksr). (7.1 ) s
And we search for a displacement vector in the (already used) form
{ ∑ ur(ω, r,z) = ∑ sAs (ω,z)J1(ksr) us(ω, r,z) = Bs (ω,z)J0(ksr) (7.2 ) s
with, as usual, ks ≡ ζs∕a. The equilibrium equations must be modified with respect to Equation (3.91View Equation) to take into account inertial effects
{ ∂ Θ + (Θ − Θ )∕r + ∂ Θ = − ρω2u r rr rr ϕϕ z r2z r (7.3 ) (∂r + 1∕r)Θrz + ∂zΘzz = − ρω uz
with the expression of the displacement vector, and the expression of thermoelastic stresses, this gives
[ 2] μ ∂2− k2+ ρω-- As − (λ + μ )ks(∂zBs + ksAs) + ksνTs = 0 (7.4 ) z s μ
[ 2] μ ∂2 − k2 + ρω-- Bs + (λ + μ)∂z(∂zBs + ksAs ) − ν∂zTs = 0. (7.5 ) z s μ
By taking ∂z(7.4View Equation)+ks(7.5View Equation) , we get
[ ] 2 2 ρω2 ∂z − ks + ---- (∂zAs + ksBs ). (7.6 ) μ
The general solution of which is
∂zAs + ksBs = ksCs cosh(κT,sz) + ksDs sinh(κT,sz), (7.7 )
where Cs,Ds are arbitrary constants, and where the transverse wave number κT,s is defined as
∘ (---)2-------- ζs ρ-ω2 κT,s = a − μ (7.8 )
. Equation (7.4View Equation) can then be written as
[ ] 2 2 ρω2 λ + μ ν ∂ z − k s +------ As = ------∂z(∂zAs + ksBs) − ks------Ts, (7.9 ) λ + 2μ λ + 2μ λ + μ
which allows one to find As:
As(ω, z) = Ms sinh(κL,sz) + Ns cosh(κL,sz) λ + μ ksκT,s ν + --------2------2--[Cs sinh (κT,sz) +Ds cosh(κT,sz)] − ks-----𝒯s, (7.10 ) λ + 2μ κT,s − κ L,s λ + μ
where M s and N s are two more arbitrary constants, 𝒯 s is a special solution of
[∂2 − κ2 ]𝒯s = Ts (7.11 ) z L,s
and the longitudinal wave number κL,s is defined as
∘ ----------------- ( )2 2 κL,s = ζs − -ρω---. (7.12 ) a λ + 2μ
Once As is found, one obtains Bs:
( 2 ) B (ω, z) = 1 − λ-+-μ-----κT,s--- [C cosh (κ z) + D sinh(κ z)] s λ + 2μκ2T,s − κ2L,s s T,s s T,s κ ν − -L,s-(Ms cosh(κL,sz) + Ns sinh (κL,sz)) + ------∂z𝒯s. (7.13 ) ks λ + μ
The boundary conditions on the faces allow one to determine the arbitrary constants. It is convenient to introduce the dimensionless parameter
ρω2a2 (1 + σ ) xs ≡ -------2-----. (7.14 ) Y ζs
Then, we have
1 + σ 2xs [ κL,s ] Cs = α---------- (1 − xs )sinh ϕL,se′s +----cosh ϕL,sksos (7.15 ) 1 − σ D1,s ks
[ ] D = α1-+-σ-2xs- (1 − x )cosh ϕ o′ + κL,s sinh ϕ k e (7.16 ) s 1 − σ D2,s s L,s s ks L,s s s
[ ] 1 + σ 1 κT,s ′ Ms = α --------- (1 − xs)cosh ϕT,sksos + ---- sinh ϕT,ses (7.17 ) 1 − σ D1,s ks
[ ] 1 + σ 1 κT,s ′ Ns = α1-−-σ-D--- (1 − xs)sinh ϕT,skses + -k--cosh ϕT,sos (7.18 ) 2,s s
with
κ κ D1,s = -L,s-T,s-sinh ϕT,scosh ϕL,s − (1 − xs)2 sinh ϕL,scosh ϕT,s (7.19 ) k2s
D = κL,sκT,s-sinh ϕ cosh ϕ − (1 − x )2 sinh ϕ cosh ϕ (7.20 ) 2,s k2s L,s T,s s T,s L,s
and
ϕL,s = κL,sh∕2, ϕT,s = κT,sh ∕2. (7.21 )
e ,e′s,o ,o′ s s s are defined as in Equation (3.148View Equation). The surface distortion is given by
∑ Z (ω, r) = Bs(ω)J0(ksr) (7.22 ) s
and we have
[ 1 + σ (1 − xs)sinh ϕL,se′s + κL,s-coshϕL,sksos Bs (ω,− h∕2 ) = α------xs cosh ϕT,s----------------------ks-------------- 1 − σ D1,s ′ κL,s- ] − sinh ϕ (1-−-xs)cosh-ϕL,sos +-ks-sinh-ϕL,skses- . (7.23 ) T,s D2,s
We have seen that the dynamic temperature field at frequencies within the GW band is practically proportional to the beam intensity profile, and consequently negligible on the edge. This is why we neglect the boundary conditions for the edge stresses here. At this point, we can specialize for the two cases of coating/bulk heating.

 7.1 Dynamic surface distortion caused by coating absorption
  7.1.1 Mean displacement
  7.1.2 Under cutoff regime

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