8.5 Coating Brownian thermal noise: finite mirrors

We again consider the coating as a very thin, but finite, layer having a different loss angle. Thus, the contributions of the coating and of the bulk material to overall noise must be computed separately. For the coating, we consider (as in the case of semi-infinite mirrors) a new solution of the Navier–Cauchy equations in the region [-δC,0]; then the continuity of the displacements and the continuity of the pressures (Θijnj) at the interfaces z = 0 and z = − δC, allow us to completely determine the new solution. The energy density is
[ ( w (r,z ) = 1-λC E (r,0)2 + 2μc (Err(r,0) + Eϕ,ϕ(r,0))2 2 )] − 2Err(r,0)E ϕ,ϕ(r,0) + Ezz(r,z)2 (8.152 )
Due to the orthogonality of functions J0(ksr), we get, after some algebra,
[ ] δC 1 + σC 1 ∑ p2 ( 1 + σ )2 U0,coating = ---2----------- -s2J0(ζs)2 (1 − 2σC )Δ2s + ------- X2s (8.153 ) 2πa 1 − σC YC s Δ s 1 + σC
with the notation
X ≡ (1 − 2σ )(1 − q )2 + 4q x2. (8.154 ) s s s s
In the case of an homogeneous medium (λC = λ, μC = μ), this reduces to
δ (1 + σ )∑ p2 [ ] U0,coating,hom = -C---2---- --s2J0(ζs)2 16q2sx4s + (1 − 2σ)(1 − qs)4 . (8.155 ) πa Y s Δ s
Note that the radial stress on the edge of the coating is
[ ] -1--∑ YC(1-+-σ-)Xs- Θrr(a,0) = − πa2 psJ0(ζs) σC + Y(1 + σC )Δs . (8.156 ) s
But we must add the contribution of the Saint-Venant correction. In the same way, we consider a new solution of the Navier–Cauchy equations connected to the preceding Saint-Venant correction. The components of the strain tensor in the coating are then
ΔErr = --1--[σ + 6S (1 − σ)] (8.157 ) πa2Y
ΔE ϕϕ = --1---[σ + 6S (1 − σ)] (8.158 ) πa2Y
ΔEzz = − --1--------1-----[2YC (σ + 6S(1 − σ)) + Y (1 + σC )(1 − 2σC)] (8.159 ) πa2Y YC(1 − σC )
ΔErz = 0. (8.160 )
After integrating the energy density over the volume πa2δC, one finds
2 1-+-24σS--+-72(1-−-σ)S-- ΔUcoating = δC 2πa2Y Ω2 (8.161 )
with the notation
Y 2(1 + σC )(1 − 2σC) + 2Y 2[σ + 6(1 − σ)S ]2 Ω2 = -------------------------C------------2----. (8.162 ) Y YC (1 − σ )[1 + 24σS + 72 (1 − σ )S ]
Ω2 takes the value one in the homogeneous case (when λC = λ and μC = μ). Finally, U = U + ΔU coating 0,coating coating. At this point, it is very interesting to compare the figures obtained within the infinite medium approximation and the finite mirror formula. In Table 17, we give a few examples (in the homogeneous case) showing that the infinite mirror approximation is almost acceptable for peaked power distributions, but quite bad for wide beams, and especially for coating energy. Thus, optimization work based on infinite mirrors are, at minimum, questionable.


Table 17: Comparison infinite/finite mirror Strain Energy (SE)
3 examples coating SE [J N–2] bulk SE [J N–2]
Ex1: LG00, w = 2 cm
U∞ 2.10 ×10–13 1.88 ×10–10
U 2.38 ×10–13 2.02 ×10–10
Ex2: flat
U∞ 1.02 ×10–14 3.95 ×10–11
U 2.07×10–14 3.28 ×10–11
Ex3: LG55, w = 3.5 cm
U∞ 6.87 ×10–15 2.68 ×10–11
U 7.47×10–15 8.64 ×10–12


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