### 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 of the strain tensor
with
where are the Fourier–Bessel coefficients of the pressure distribution, and , , , and have been defined in the preceding Section 8. Moreover,
(see Equation (8.105) for ), so that the gradient of is

#### 9.3.1 Case of the bulk material

Owing to the orthogonality relations of , we get

where
and, obviously,
(N.B. and are orthogonal in the integration). We have, finally,
And for the spectral density,

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

For the flat beam, b = 9.1 cm,
For the mesa beam,  = 10.7 cm,
For the beam, w = 3.5 cm,
In Figures 63, 64, 65, and 66, one can see the distribution of in Ex1, Ex2, the mesa beam, and Ex3, respectively.

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 , and depending on specific parameters and . The components of the gradient are as follows, with the notation already introduced.

so that a volume integration gives
with
The Saint-Venant correction to only has a derivative, so that
And the global result is

#### 9.3.3 Numerical results

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

For the mode with w = 3.5 cm:
And for the mesa mode:
We see how the infinite mirror model underestimates the noise by a factor of two for the last case.