9.3 Case of finite mirrors

In the case of finite mirrors, the model developed for standard thermal noise provides the explicit expressions for the trace E of the strain tensor
E (r,z ) = E0(r,z) + ΔE (r,z) (9.52 )
with
2(1 − 2σ )(1 + σ )∑ p [ ] E0 (r,z) = − ----------------- --sJ0(ζsr∕a) Qse −ζsz∕a − Rse ζsz∕a , (9.53 ) πa2Y s>0 Δs
where ps are the Fourier–Bessel coefficients of the pressure distribution, and Δs, qs, xs, Qs and Rs have been defined in the preceding Section 8. Moreover,
--1--- ΔE (r,z) = − (1 − 2σ)πa2Y [1 − 12S − (1 − 24S )z∕h] (9.54 )
(see Equation (8.105View Equation) for S), so that the gradient of E is
∂E 2 (1 − 2 σ)(1 + σ )∑ p ζ [ ] ---0 = ----------------- -s-sJ1(ksr) Qse−ksz − Rseksz (9.55 ) ∂r πa3Y s>0 Δs
∑ [ ] ∂E0- = 2(1-−-2-σ)(1 +-σ)- psζsJ0(ksr) Qse−ksz + Rseksz (9.56 ) ∂z πa3Y s>0 Δs
∂ΔE-- 1-−-2σ- ∂z = πa2hY (1 − 24S ). (9.57 )

9.3.1 Case of the bulk material

Owing to the orthogonality relations of J ν(ζsr∕a ), we get

∫ 2 2 (g⃗radE )2dV = 4-(1-−-2σ-)-(1-+-σ-)-∑ W (9.58 ) 0 πa3Y 2 s s>0
where
p2sζs 2 Ws = -Δ2-J0(ζs) (1 − qs) s[ 2 ] × (1 − qs)(1 − qs) + 8qsxs(1 − qs + xs) (9.59 )
and, obviously,
∫ 2 (g⃗radΔE )2dV = (1 −-2σ)-(1 − 24S )2 (9.60 ) πa2hY 2
(N.B. ⃗ grad ΔE and ⃗ gradE0 are orthogonal in the r integration). We have, finally,
2 [ ∑ ] W = -4KT--α-- (1 + σ)2 Ws + (1 − 24S )2 a-- . (9.61 ) πa3 ρ2C2 s>0 4h
And for the spectral density,
2 2 [ ] S (f) = 4kBKT----α-- (1 + σ)2 ∑ W + (1 − 24S )2 a- . (9.62 ) x π3a3ρ2C2f 2 s 4h m>0

For Gaussian beams, we substitute the ps’s in the preceding formulas. For the parameters corresponding to Virgo input mirrors (LG0,0, w = 2 cm) we find

[ ] S1x∕2(f) = 8.83 × 10−20 mHz − 1∕2 1-Hz- . (9.63 ) f
For the flat beam, b = 9.1 cm,
[ ] S1∕2(f) = 1.87 × 10−20 mHz − 1∕2 1-Hz- . (9.64 ) x f
For the mesa beam, b f = 10.7 cm,
[ ] S1∕2(f) = 1.49 × 10−20 mHz − 1∕2 1-Hz- . (9.65 ) x f
For the LG5,5 beam, w = 3.5 cm,
[1 Hz ] S1x∕2(f) = 1.72 × 10−20 mHz − 1∕2 ----- . (9.66 ) f
In Figures 63View Image, 64View Image, 65View Image, and 66View Image, one can see the distribution of (g⃗radE )2 in Ex1, Ex2, the mesa beam, and Ex3, respectively.
View Image

Figure 63: Distribution of ⃗ 2 (∇E ) in the case of a LG0,0 mode (w = 2 cm). (Logarithmic scale, arbitrary units)
View Image

Figure 64: Distribution of ⃗ 2 (∇E ) in the case of a flat mode (b = 9.1 cm). (Logarithmic scale, arbitrary units)
View Image

Figure 65: Distribution of ⃗ 2 (∇E ) in the case of a mesa mode (bf = 10.7 cm). (Logarithmic scale, arbitrary units)
View Image

Figure 66: Distribution of the square gradient of the trace of the strain tensor in the case of an LG5,5 mode (w = 3.5 cm). (Logarithmic scale, arbitrary units)

In the case of the flat beam, one should note the peaks at the location of the sharp edges of the intensity distribution. This was the cause of the divergence of the infinite mirror approach. However, the estimation for the flat and mesa beams are not very different.

9.3.2 Case of the coatings

The components of the gradient of the trace of the strain tensor on the reflecting surface can be obtained from the model developed for the Brownian thermal noise. We consider a new solution of the Navier–Cauchy equations, matched to the bulk solution at z = 0, and depending on specific parameters YC and σC. The components of the gradient are as follows, with the notation already introduced.

∂E--= (1-+-σ)(1-−-2σC-)∑ psζs [2(1 − σ)(1 − q2+ 4q x )] J (k r) (9.67 ) ∂r πa3Y (1 − σC) Δs s s s 1 s s
∂E (1 + σ)(1 − 2σC )∑ psζs[ Y (1 + σC ) ] ----= ----3------------ ---- ----------Δs + Xs J0(ksr), (9.68 ) ∂z πa Y (1 − σC) s Δs YC (1 + σ )
so that a volume integration gives
∫ a (1 + σ )2(1 − 2σ )2∑ p2ζ2J (ζ )2 2πδC (⃗∇E )2rdr = δc----4-2--------C2-- --ss--02-s--Vs (9.69 ) 0 πa Y (1 − σC ) s Δ s
with
[ ]2 V ≡ 4(1 − σ2)2(1 − q2 + 4q x )2 + Y-(1 +-σC)Δ + X . (9.70 ) s s s s YC (1 + σ) s s
The Saint-Venant correction to E only has a z derivative, so that
∫ a 2 2 2 πδC (⃗∇ ΔE )2rdr = δC 4(1-−-2σC-)-[σ-+-12S-(1-−-σ)]-. (9.71 ) 0 πa2h2Y 2(1 − σC)2
And the global result is
( )2 [ ∑ 2 2 2 W = K T --α--- -----δC------ (1 + σ )2 psζsJ0(ζs)-V C ρC CC πa4 (1 − σC )2 Δ2s s 2 ] s + 4a--(σ + 12S (1 − σ))2 . (9.72 ) h2

9.3.3 Numerical results

With the coating parameters already given, we get the following results for the coating noise. For the LG00 mode with w = 2 cm waist:

[ ] S1∕2(f) = 1.54 × 10−19 mHz − 1∕2 1-Hz- . (9.73 ) x f
For the LG55 mode with w = 3.5 cm:
[ ] 1∕2 −20 − 1∕2 1 Hz Sx (f) = 4.33 × 10 mHz --f-- . (9.74 )
And for the mesa mode:
[ ] S1∕2(f) = 8.66 × 10−21 mHz − 1∕2 1-Hz- . (9.75 ) x f
We see how the infinite mirror model underestimates the noise by a factor of two for the last case.


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