8.2 Infinite mirrors noise in the substrate

We use the linear elasticity theory already summarized by Equations (3.87View Equation) and (3.91View Equation). A first approach, outlined in [5Jump To The Next Citation Point], consists in neglecting edge effects and treating the mirror as a plane interface between vacuum and material with no limiting transverse dimension. In this case, the elastic equilibrium equations (3.91View Equation) are satisfied by a displacement vector of the form [5Jump To The Next Citation Point]
( ( λ+2μ ) − kz { ur(r,z) = (α − -λ+μ β + βkz) e J1 (kr ) ( --μ- −kz (8.12 ) uz(r,z) = α + λ+ μβ + βkz e J0(kr),
where α,β and k are arbitrary constants. Therefore, a more general solution is based on integrals having the form of Hankel transforms
( ( ) { ∫ ∞ λ+2μ − kz ur(r,z) = 0 (α (k ) − λ+μ β(k) + β(k)kz) e J1 (kr )kdk , (8.13 ) ( u (r,z) = ∫ ∞ α (k ) + -μ-β(k) + β (k )kz e−kzJ (kr)kdk z 0 λ+μ 0
where α(k) and β (k) are now unknown functions. We can compute the relevant stress components
∫ ∞ −kz 2 Θrz (r,z ) = 2 μ (β (k ) − α (k) − β (k)kz)e J1(kr)k dk (8.14 ) 0
and
∫ ∞ Θ (r,z) = − 2μ (α (k) + β (k)kz)e−kzJ (kr)k2dk. (8.15 ) zz 0 0
The boundary condition Θrz(r,z = 0) = 0 is satisfied by taking α = β. A second boundary condition is
Θzz(r,z = 0) = I(r), (8.16 )
where I(r) is, as seen above, the normalized intensity of the readout beam. Therefore, we have
∫ ∞ − 2μ α (k )J0(kr)k2dk = I(r). (8.17 ) 0
Inverting the Hankel transform, we find
kα(k ) = −-1-&tidle;I(k), (8.18 ) 2 μ
where &tidle;I(k) is the Hankel transform of the intensity
∫ ∞ I&tidle;(k ) = I(r)J0(kr)rdr. (8.19 ) 0
The surface displacement is now, with α = β,
λ + 2μ ∫ ∞ uz(r,z = 0) = ------- α(k)J0(kr)kdk λ + μ 0 ∫ -λ-+-2-μ-- ∞ &tidle; = − 2μ(λ + μ ) I (k )J0(kr)dk 2 ∫ 0∞ = − 2(1 −-σ-)- &tidle;I(k)J (kr)dk, (8.20 ) Y 0 0
where we have switched from Lamé coefficients to the Poisson ratio σ and Young’s modulus Y. The strain energy U is given by
∫ ∫ 1- 2π ∞ U = − 2 dϕ rdrI(r)uz(r,0), (8.21 ) 0 0
so that we get the general formula
1 − σ2 U = 2π --Y---ϖ0, (8.22 )
where
∫ ∞ 2 ϖ0 ≡ &tidle;I(k) dk. (8.23 ) 0
It is possible to get explicit results for Laguerre–Gauss and ideally flat beams. The Bessel transforms of the intensity profiles have already been calculated in Section 3.1.3. We have, for an LGn,m beam of width w,
1 k2w2 &tidle;In,m (k) = ---e−yLm (y)Ln+m (y), (y ≡ ----) (8.24 ) 2π 8
and for an ideally flat beam of radius b
&tidle; J1-(kb-) Iflat(k) = πkb . (8.25 )
And now
∫ ∞ 2 1 ϖ0,n,m = I&tidle;n,m(k) dk = --3∕2--g0,n,m, (8.26 ) 0 4π w
where g0,n,m are numerical factors. The final result is
2 U = -1√ −-σ--g . (8.27 ) n,m 2 πY w 0,n,m
The result for n = 0,m = 0 is first given in [5Jump To The Next Citation Point]. Table (15) gives an idea of the numerical values of g0,n,m.


Table 15: Some numerical values of g0,n,m
  m 0 1 2 3 4 5
n              
0   1 .60 .46 .39 .34 .31
1   .69 .50 .41 .36 .32 .29
2   .57 .44 .37 .33 .30 .28
3   .50 .40 .35 .31 .29 .27
4   .46 .37 .33 .30 .27 .26
5   .43 .35 .31 .28 .26 .25

For larger values of n and m, an increasingly good approximation is the asymptotic value

g0,n,m ∼ (2m + n + 1 )− 1∕2. (8.28 )
For an ideally flat mode, we get
∫ ∞ 4 ϖ0,F = &tidle;Iflat(k )2dk = ---3- (8.29 ) 0 3π b
so that
8(1 −-σ2)- U flat = 3π2bY . (8.30 )
These results are useful in order to compare with a more accurate calculation taking into account the finite dimensions of mirrors. For instance, with (Ex1) with a mirror (Virgo size) having a 35 cm diameter and an LG 0,0 mode of width 2 cm (Virgo input mirror), we get
−1 ϖ0 = 2.245 m (8.31 )
U0,0 = 1.88 × 10 −10 JN −2 (8.32 )
giving a noise linear spectral density of equivalent displacement:
[ ]1∕2 S1∕2(f) = 9.95 × 10−19 1-Hz- mHz −1∕2. (8.33 ) x f
Then (Ex2) for a flat beam of radius 9.1 cm gives
−1 ϖ0 = 0.473 (mesa : 0.427 ) m (8.34 )
U = 3.95 × 10 −11 JN −2 (8.35 ) flat
[ ] 1∕2 −19 1-Hz- 1∕2 −1∕2 Sx (f) = 4.56 × 10 f mHz . (8.36 )
For the mesa beam, we have
[ ]1∕2 S1∕2(f) = 4.34 × 10−19 1-Hz- mHz −1∕2 (8.37 ) x f
and for an LG 5,5 beam of width 3.5 cm (Ex3), this is
ϖ0 = 0.321 m −1 (8.38 )
U55 = 2.68 × 10− 11 JN −2 (8.39 )
[ 1 Hz]1∕2 S1x∕2(f) = 3.76 × 10−19 ----- mHz −1∕2. (8.40 ) f
We will see in Section 8.4 that the first estimate is good (sharp spot on the center, far from the edge), whereas the last two (widely spread light power) are far from reality. This is why the infinite mirror approximation must be used with care.
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