### 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 compared to the substrate’s. We shall treat the coating as a layer of thickness , having a specific Young’s modulus and Poisson ratio , and corresponding Lamé coefficients . The coating is assumed to be located in the region , so that the interface substrate/coating is located at . For , we have the expression (8.12) of the displacement vector (with () possibly different from the preceding solution (8.18) at the end). The following solution of the Navier–Cauchy equations for the displacement vector in the coating is easily found
But in the present case, the functions and are more complicated, taking into account two finite boundaries
The four arbitrary constant can be related to and of the preceding solution (8.12) by requiring continuity of the displacements and of the pressure components at the interface . Then the boundary conditions are
which allow one to compute and find the complete solution. The exact solution is complicated and of little interest because we are in a case where (tens of microns) is small compared to the parameter of the beam (a few cm). A solution at first order in 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
We rewrite using the form
and we integrate over the volume . It is easy to see that the crossed term does not contribute in an integral, being the derivative of a function null at and . By using the closure relation
one finally obtains
with the notation
Note that, due to the Plancherel theorem (or to the closure relation in the direct space), this is also
Now the energy stored in the coating can be computed in the same way, by integrating the energy density

within the volume . But at first order in , it is sufficient to take

and we find
where
( takes the value 1 when and ).

#### 8.3.1 Coating Brownian thermal noise: LG modes

In the case of an mode, we get

where is a numerical factor. Table 16 gives some values of .

Table 16: Some numerical values of
 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

Thus, the ratio between the main energy in the substrate and that in the preceding coating is

A similar result has been also reported by [32] and [18] in the case of () at the limit . For reasonable parameters, 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 (, ), and we get

and, for an mode of width w = 3.5 cm (Ex3),

#### 8.3.2 Coating Brownian thermal noise: Flat modes

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

Thus, we have
and the coating strain energy is
with the ratio now
For the flat mode of (Ex2), we find

Here are some numerical values for comparison. For the mode, we have  4.14 m, For the flat mode (b = 9.1 cm), we have  6.12 m (to be discarded, as the sharp edge effect becomes spurious, see the next value), 4.52 m (numerical integration), and for the Gauss–Bessel mode of Figure 5, this is  3.56 m. A value of 2.34 m 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.