### 8.2 Infinite mirrors noise in the substrate

We use the linear elasticity theory already summarized by Equations (3.87) and (3.91). A first approach, outlined in [5], consists in neglecting edge effects and treating the mirror as a plane interface between vacuum and material with no limiting transverse dimension. In this case, the elastic equilibrium equations (3.91) are satisfied by a displacement vector of the form [5]
where and are arbitrary constants. Therefore, a more general solution is based on integrals having the form of Hankel transforms
where and are now unknown functions. We can compute the relevant stress components
and
The boundary condition is satisfied by taking . A second boundary condition is
where is, as seen above, the normalized intensity of the readout beam. Therefore, we have
Inverting the Hankel transform, we find
where is the Hankel transform of the intensity
The surface displacement is now, with ,
where we have switched from Lamé coefficients to the Poisson ratio and Young’s modulus . The strain energy is given by
so that we get the general formula
where
It is possible to get explicit results for Laguerre–Gauss and ideally flat beams. The Bessel transforms of the intensity profiles have already been calculated in Section 3.1.3. We have, for an beam of width ,
and for an ideally flat beam of radius
And now
where are numerical factors. The final result is
The result for is first given in [5]. Table (15) gives an idea of the numerical values of .

Table 15: Some numerical values of
 m 0 1 2 3 4 5 n 0 1 .60 .46 .39 .34 .31 1 .69 .50 .41 .36 .32 .29 2 .57 .44 .37 .33 .30 .28 3 .50 .40 .35 .31 .29 .27 4 .46 .37 .33 .30 .27 .26 5 .43 .35 .31 .28 .26 .25

For larger values of and , an increasingly good approximation is the asymptotic value

For an ideally flat mode, we get
so that
These results are useful in order to compare with a more accurate calculation taking into account the finite dimensions of mirrors. For instance, with (Ex1) with a mirror (Virgo size) having a 35 cm diameter and an mode of width 2 cm (Virgo input mirror), we get
giving a noise linear spectral density of equivalent displacement:
Then (Ex2) for a flat beam of radius 9.1 cm gives
For the mesa beam, we have
and for an beam of width 3.5 cm (Ex3), this is
We will see in Section 8.4 that the first estimate is good (sharp spot on the center, far from the edge), whereas the last two (widely spread light power) are far from reality. This is why the infinite mirror approximation must be used with care.