9.2 Case of infinite mirrors

Let us recall the results obtained in the preceding chapter on standard thermal noise. Under beam pressure, the displacement vector is
∫ ∞ ur(r,z) = α(k)[kz − 1 + 2σ ]exp(− kz)J1(kz)kdk (9.17 ) 0
∫ ∞ uz(r,z) = α(k)[kz + 2 − 2σ ]exp(− kz)J0(kz)kdk (9.18 ) 0
so that
∫ ∞ 2 E (r,z) = div⃗u(r,z) = − 2(1 − 2σ) α(k )exp(− kz)J0(kz )k dk. (9.19 ) 0
The function u(k) is determined by the virtual pressure distribution I(r) (normalized beam intensity). Namely,
1-+-σ-&tidle;I(k)- α(k) = − Y k , (9.20 )
where &tidle; I(k) is the Hankel transform of I(r). As a result,
∫ 2(1 − 2σ)(1 + σ) ∞ E (r,z) = − -------Y--------- &tidle;I(k) exp(− kz)J0(kr)kdk, (9.21 ) 0
which shows, in passing, that
2(1 − 2σ )(1 + σ ) E(r,0) = − --------Y--------I(r). (9.22 )
Thus, wee can already foresee that in the case of an ideally flat beam the gradient will involve Dirac distributions; therefore, the volume integration of its square will be problematic. Let us compute the gradient of E:
∂E 2(1 − 2 σ)(1 + σ)∫ ∞ ----= ----------------- &tidle;I(k)exp (− kz )J1(kr)k2dk (9.23 ) ∂r Y 0
∂E 2(1 − 2σ )(1 + σ )∫ ∞ ----= ----------------- I&tidle;(k )exp(− kz)J0(kr )k2dk. (9.24 ) ∂z Y 0
Now, using the closure relation,
∫ ∞ ′ δ(k − k′) Jν(kr)Jν(k r)rdr = ---k----- (9.25 ) 0
for ν = 0,1. It is now possible to carry out the volume integration:
∫ ∫ ∫ ∞ ∞ ⃗ 2 (1 −-2-σ)2(1 +-σ-)2 ∞ &tidle; 2 2 2π rdr dz (gradE ) = 8π Y 2 I(k) k dk (9.26 ) 0 0 0
so that
2 2 W = 8π KT-α--(1 +-σ-)-ϖ (9.27 ) ρ2C2 2
with
∫ ∞ &tidle; 2 2 ϖ2 = I(k) k dk. (9.28 ) 0
This expression shows that the function &tidle;I(k) must have an asymptotic evanescence strictly faster than k− 3∕2 for the integral to converge. This is a strong requirement on the Hankel transform of the pressure distribution.

9.2.1 Gaussian beams

For a Laguerre–Gauss mode LGn,m of width parameter w, we have seen that

1 I&tidle;n,m(k) = ---e−yLm (y)Lm+n (y) (y ≡ k2w2 ∕8) (9.29 ) 2 π
giving
ϖ2,n,m = ---√1----g2,n,m, (9.30 ) 2π πw3
where g 2,n,m are numerical factors (see Table 18).


Table 18: Some numerical values of g2,n,m
  m 0 1 2 3 4 5
n              
0   1 .75 .64 .57 .53 .49
1   .44 .39 .36 .33 .31 .30
2   .33 .31 .29 .27 .26 .25
3   .28 .26 .25 .24 .23 .22
4   .24 .23 .22 .21 .21 .20
5   .22 .21 .20 .20 .19 .19

Thus,

∫ 2 2 (gr⃗adE )2dV = 4-(1-−√2σ-)-(1-+-σ-)-g2,n,m, (9.31 ) πY 2w3
so that the spectral density of thermoelastic noise is, using Equations (9.1View Equation) and (9.16View Equation),
4kBKT 2α2 (1 + σ )2 Sx (f) = ---2√---2--2-2--3--g2,n,m. (9.32 ) π πρ C f w
This result was found (in the case of n = m = 0) first by Braginsky et al. [7] using their own formalism, then by Liu et al. [28], using our approach. For silica parameters and for Ex1 (the LG 0,0 mode with w = 2 cm), one finds
−3 ϖ2,0,0 = 11, 224 m (9.33 )
and
[ ] Sx (f)1∕2 = 8.53 × 10− 20 1-Hz- mHz −1∕2, (9.34 ) f
which is lower than the standard thermal noise, but still significant. For Ex3 (the LG 5,5 mode with w = 3.5 cm), we have
ϖ2,5,5 = 398 m −3 (9.35 )
[ ] Sx(f)1∕2 = 1.61 × 10 −20 1-Hz- mHz −1∕2. (9.36 ) f

9.2.2 Flat beams

If we now consider a flat beam modeled by its ideal representation

{ 1∕πb2 (r < b) I(r) = (9.37 ) 0 (r ≥ b),
we have the Hankel transform
&tidle;I(k) = J1(kb), (9.38 ) πkb
which shows that the requirement on the decreasing rate for large k is not fulfilled, J ν(k) having asymptotic behavior in −1∕2 k, just below the limit. Therefore, it is impossible to use the crude flat model, the integral ϖ2 being divergent. If we want to have an evaluation, we must carry out a numerical integration with the mesa intensity profile. We find (for our particular model)
ϖ ∼ 77 m −3, (9.39 ) 2
seeming to indicate a strong reduction factor of the SD of noise, namely 0.44 with respect to Ex3. This result is due to the fact that the LG5,5 mode has a number of rings causing many local gradients. This was pointed out by Agresti ([2]) in the case of the LG 0,5 mode. Anyway this mode is unwanted, as well as other LG0,m modes, because we wish to avoid sharp central power peaks .

9.2.3 Thermoelastic noise in the coating

We apply the same strategy for all coating calculations. The gradient of the trace of the strain tensor can be integrated on the surface z = 0 giving

∫ ∞ 8(1 + σ)2(1 − 2σ)2 ∇E2rdr = ------------------Ω3 ϖ3 (9.40 ) 0 Y2
with
[ ] ( )2 [ ]2 Ω3 ≡ 1- (1-−-σ-)(1-−-2σC-)- + ----1------ 1 − 2σC + -Y- (1-+-σC-)(1 −-2σC)- (9.41 ) 2 (1 − σC )(1 − 2σ) 4(1 − σC)2 YC (1 + σ)(1 − 2σC )
(Ω3 takes the value one when Y = YC,σ = σC). Contrary to Ω1, which is not so sensitive to parameters, Ω 3 can take values quite different from one. For instance, if we assume the parameters of fused silica for the substrate, and (11 −2 YC ∼ 1.4 × 10 Nm, σC ∼ 0.23) for the coating, we have Ω3 ∼ 0.59. ϖ3 has the following definition
∫ ∞ ϖ3 ≡ &tidle;I(k)2k3dk, (9.42 ) 0
so that the energy W is (taking into account special values for the coating material)
( α (1 + σ ))2 W = 16πKC T -C-------C-- Ω3 ϖ3δC . (9.43 ) ρCCC
In the case of LG n,m modes, we obtain
2 ϖ3,n,m = π2w4-g3,n,m, (9.44 )
where g3,n,m are numerical factors, the first ones being given by Table 19.


Table 19: Some numerical values of g3,n,m
  m 0 1 2 3 4 5
n              
0   1 1.5 1.72 1.86 1.96 2.05
1   .50 .81 .98 1.10 1.20 1.27
2   .37 .62 .77 .88 .96 1.03
3   .31 .53 .66 .76 .83 .90
4   .27 .46 .58 .67 .75 .81
5   .25 .42 .53 .61 .68 .74

It seems clear that, as already mentioned, the modes LG0,m, having a sharp peak on the axis, become worse and worse as the order m increases. On the other hand, the reduction factor for the noise in the best cases is much less than for the Brownian thermal noise.

9.2.4 Scaling laws

This section offers an opportunity to summarize the various coefficients encountered in the parts of this noise study. Several authors (see [29Jump To The Next Citation Point] for his discussion) have remarked on the dependence of the various noises encountered on the integrals we have denoted ϖm, {m ∈ ℕ}.

∫ ∞ ϖm = &tidle;I(k)kmdk (9.45 ) 0

We have given these integrals in the case of different LGmn modes. In particular, Table 20 gives the values for our four examples. These can be used to derive figures of merit.


Table 20: Some values of ϖn
n LG 00 w = 2 cm LG 55 w = 3.5 cm flat b = 9.1 cm mesa b f = 10.7 cm units
0 2.245 0.321 0.473 0.426 m–1
1 126.65 4.13 6.12 4.52 m–2
2 1.122 ×104 398 * 76.7 m–3
3 1.27 ×106 105 * 1800 m–4

9.2.5 Numerical results

We give briefly some figures regarding our three reference situations. We take the parameter of fused silica for the substrate, and the parameters of TA2O5 for the coating ([36]), namely, YC = 1.41011 Pa, σC = 0.23, ρC × CC = 2.1 × 106 J∕m3 ∕K. The thickness of the coating is assumed to be 25 µm. For a TEM 0,0 mode of waist 2 cm, we obtain

We see that the large difference in parameters overcompensates for the difference in volume. For an LG55 mode of waist 3.5 cm:

The reduction factor is about five for the substrate and only 3.5 for the coating. We see that the large difference in parameters overcompensates for the difference in volume. For a mesa mode:

The reduction factor with respect to Ex1 is about 12 for the substrate and 26 for the coating. This kind of mode is obviously the best regarding this kind of noise.


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