### 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
so that
The function is determined by the virtual pressure distribution (normalized beam intensity). Namely,
where is the Hankel transform of . As a result,
which shows, in passing, that
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 :
Now, using the closure relation,
for . It is now possible to carry out the volume integration:
so that
with
This expression shows that the function must have an asymptotic evanescence strictly faster than 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 of width parameter , we have seen that

giving
where are numerical factors (see Table 18).

Table 18: Some numerical values of
 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,

so that the spectral density of thermoelastic noise is, using Equations (9.1) and (9.16),
This result was found (in the case of ) 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 mode with w = 2 cm), one finds
and
which is lower than the standard thermal noise, but still significant. For Ex3 (the mode with w = 3.5 cm), we have

#### 9.2.2 Flat beams

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

we have the Hankel transform
which shows that the requirement on the decreasing rate for large is not fulfilled, having asymptotic behavior in , just below the limit. Therefore, it is impossible to use the crude flat model, the integral 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)
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 mode has a number of rings causing many local gradients. This was pointed out by Agresti ([2]) in the case of the mode. Anyway this mode is unwanted, as well as other 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 giving

with
( takes the value one when ). Contrary to , which is not so sensitive to parameters, can take values quite different from one. For instance, if we assume the parameters of fused silica for the substrate, and (, ) for the coating, we have . has the following definition
so that the energy is (taking into account special values for the coating material)
In the case of modes, we obtain
where are numerical factors, the first ones being given by Table 19.

Table 19: Some numerical values of
 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 , having a sharp peak on the axis, become worse and worse as the order 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 [29] for his discussion) have remarked on the dependence of the various noises encountered on the integrals we have denoted .

• Brownian noise, substrate:
• Brownian noise, coating:
• Thermoelastic noise, substrate:
• Thermoelastic noise, coating: . However, in this case there is a more refined analysis [29], taking into account the heat flow. Attention must be paid to this theory (see also [178]). However, the approximate character of the semi-infinite–mirror approach reduces its practical interest.

We have given these integrals in the case of different 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
 n w = 2 cm w = 3.5 cm flat b = 9.1 cm mesa  = 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 for the coating ([36]), namely, , , . The thickness of the coating is assumed to be 25 µm. For a mode of waist 2 cm, we obtain

• Spectral density of thermoelastic noise in the substrate:
Spectral density of noise in the coating:

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

• Spectral density of thermoelastic noise in the substrate:
Spectral density of noise in the coating:

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:

• Spectral density of thermoelastic noise in the substrate:
• Spectral density of noise in the coating:

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.