8.3 Infinite mirrors, noise in coating

The dielectric coatings required to transform a polished blank of silica into a mirror are deposited by successive layers and form a region at the reflecting surface whose mechanical parameters are different from the bulk material. There is also a large difference for the loss angle Φ coating compared to the substrate’s. We shall treat the coating as a layer of thickness δC, having a specific Young’s modulus YC and Poisson ratio σC, and corresponding Lamé coefficients λC ,μC. The coating is assumed to be located in the region [− δC ≤ z ≤ 0], so that the interface substrate/coating is located at z = 0. For z > 0, we have the expression (8.12View Equation) of the displacement vector (with (α, β) possibly different from the preceding solution (8.18View Equation) at the end). The following solution of the Navier–Cauchy equations for the displacement vector in the coating is easily found
∫ ∞ u (r,z) = A (k,z)J (kr )kdk (8.41 ) r 0 C 1
∫ ∞ ur(r,z ) = BC (k,z)J0(kr)kdk. (8.42 ) 0
But in the present case, the functions AC and BC are more complicated, taking into account two finite boundaries
λ + μ AC (k,z) = ---C----C---kz [γ1 cosh(kz) + γ2sinh(kz )] + γ3 sinh (kz) + γ4 cosh(kz) (8.43 ) 2(λC + 2μC )
( λ + 3μ ) ( λ + 3 μ ) BC (k,z) = ---C-----C--γ1 − γ3 cosh(kz) + --C------C--γ2 − γ4 sinh(kz ) 2(λC + 2μC ) 2(λC + 2μC ) λC + μC − 2(λ--+-2-μ-)kz [γ1sinh(kz) + γ2cosh (kz)]. (8.44 ) C C
The four arbitrary constant γ1,2,3,4 can be related to α and β of the preceding solution (8.12View Equation) by requiring continuity of the displacements and of the pressure components at the interface z = 0. Then the boundary conditions are
{ Θzz (r,− δC) = − I(r) Θ (r,− δ ) = 0 , (8.45 ) rz C
which allow one to compute (α,β ) and find the complete solution. The exact solution is complicated and of little interest because we are in a case where δC (tens of microns) is small compared to the w parameter of the beam (a few cm). A solution at first order in kδ c is therefore sufficient. We first find the energy stored in the bulk. It would be difficult to use the same method as in the preceding case (a surface integral). Instead we use the definition of the energy density
1[ 2 ( 2 2 2 )] w (r,z) = 2 λE (r,z) + 2μ Err(r,z) + Eϕϕ(r,z) + 2E rz(r,z) . (8.46 )
We rewrite using the form
w(r,z) = --------Y--------{ σE (r,z)2 + (1 − 2σ) [(E (r,z) + E (r,z))2 2 (1 + σ )(1 − 2σ ) rr ϕϕ − 2E (r,z)E (r,z) + E (r,z)2]} (8.47 ) rr ϕϕ zz
and we integrate over the volume [0 ≤ ϕ ≤ 2π] × [0 ≤ r ≤ ∞ ] × [0 ≤ z ≤ ∞ ]. It is easy to see that the crossed term ErrE ϕϕ does not contribute in an r integral, being the derivative of a function null at r = 0 and r = ∞. By using the closure relation
∫ ∞ δ(k − k′) Jν(kr)Jν(k ′r)rdr = --------- (8.48 ) 0 k
one finally obtains
[ ] 1-−-σ2- (1 +-σ)(1 −-2σ)- Y--1-+-σC- 2 U = 2π Y ϖ0 + 2πδC Y (1 − σC ) 1 − 2σ + σC YC 1 + σ ϖ1 + ′(δC ) (8.49 )
with the notation
∫ ∞ ϖ1 = I&tidle;2 (k )kdk. (8.50 ) 0
Note that, due to the Plancherel theorem (or to the closure relation in the direct space), this is also
∫ ∞ ϖ1 = I2(r)rdr. (8.51 ) 0
Now the energy stored in the coating can be computed in the same way, by integrating the energy density
wC(r,z) = --------YC---------{σC E (r,z )2 + (1 − 2σC )[(Err(r,z) + Eϕϕ(r,z))2 2(1 + σC )(1 − 2σC) − 2E (r,z)E (r,z) + E (r,z)2]} (8.52 ) rr ϕϕ zz

within the volume [0 ≤ ϕ ≤ 2π ] × [0 ≤ r ≤ ∞ ] × [− δC ≤ z ≤ 0]. But at first order in δC, it is sufficient to take

∫ ∞ Ucoating = 2πδC wC (r,0)rdr (8.53 ) 0
and we find
(1 + σ )(1 − 2 σ) Ucoating = 2πδC ---------------Ω1ϖ1, (8.54 ) Y
where
[ ] -1 −-2σC-- ----(1-+-σC)Y----- (1-+-σ)YC-- Ω1 = 2(1 − σC) (1 + σ)(1 − 2σ)YC + (1 + σC )Y (8.55 )
(Ω takes the value 1 when Y = YC and σ = σC).

8.3.1 Coating Brownian thermal noise: LG modes

In the case of an LGn,m mode, we get

1 ϖ1,n,m = 2-π2w2 g1,n,m, (8.56 )
where g1,n,m is a numerical factor. Table 16 gives some values of g1,n,m.


Table 16: Some numerical values of g1,n,m
  m 0 1 2 3 4 5
n              
0   1 .50 .34 .27 .22 .19
1   .50 .31 .23 .19 .16 .14
2   .38 .25 .19 .16 .14 .12
3   .31 .21 .17 .14 .12 .11
4   .27 .19 .15 .13 .11 .10
5   .25 .17 .14 .12 .11 .10

The strain energy stored in the coating is

U = δ (1 +-σ)(1 −-2σ)Ω g . (8.57 ) n,m,coating C πY w2 1 1,n,m
Thus, the ratio between the main energy in the substrate and that in the preceding coating is
ϱ ≡ Un,m,coating-= √2-δC-1-−-2σ-Ω g1,n,m-. (8.58 ) Un,m,substrate π w 1 − σ 1 g0,n,m

A similar result has been also reported by [32] and [18] in the case of (n = 0, m = 0) at the limit Ω1 = 1. For reasonable parameters, Ω1 is not so different from one. In the case of the Virgo cavity input mirrors (Ex1), assuming a stack 25 µm thick, and elastic parameters (YC ∼ 1.4 × 1011 Nm −2, σ ∼ 0.23 C), Ω ∼ 0.93 1 and we get

ϱ ∼ 10 −3, (8.59 )
and, for an LG5,5 mode of width w = 3.5 cm (Ex3),
ϱ ∼ 2.4 × 10 −4. (8.60 )

8.3.2 Coating Brownian thermal noise: Flat modes

The Fourier transform of the flat mode pressure was found in Section 3.1.3:

&tidle; J1-(kb-) Iflat(k) = πkb . (8.61 )
Thus, we have
∫ 1 ∞ J1(x)2 1 ϖ1,F = --2-2 ------dx = --2-2- (8.62 ) π b 0 x 2π b
and the coating strain energy is
(1-+-σ)(1 −-2σ)- -δC- UF,coating = Y Ω1 πb2 (8.63 )
with the ratio ϱ now
ϱ = UF,coating-= 3π-δC1-−-2σ-. (8.64 ) U flat 8 b 1 − σ
For the flat mode of (Ex2), we find
ϱ ∼ 2.6 × 10 −4. (8.65 )

Here are some numerical values for comparison. For the LG5,5 mode, we have ϖ1,5,5 ∼ 4.14 m−2, For the flat mode (b = 9.1 cm), we have ϖ ∼ 1,flat 6.12 m−2 (to be discarded, as the sharp edge effect becomes spurious, see the next value), ϖ1,mesa ∼ 4.52 m−2 (numerical integration), and for the Gauss–Bessel mode of Figure 5View Image, this is ϖ1,GB ∼ 3.56 m− 2. A value of 2.34 m−2 is reported by [4] after an optimization process involving an expansion on LG modes. It is probably possible to have a not too different result by a fine tuning of the conical mode’s parameters.


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