### 8.4 Finite mirrors

The treatment of a mirror as an infinite medium can be accepted when the readout beam is a sharp one, of width much smaller than the mirror’s dimensions. But it becomes highly questionable when the spatial extension of the mode is comparable to those dimensions, which is precisely the case when high-order LG modes or mesa beams are considered. This is why an analytic method for computing in the case of finite size mirrors has been developed in BHV, then amended in [28] for one boundary condition. We change slightly the coordinate system with respect to the preceding thermal studies. The mirror is assumed to fill the cylindrical volume defined by , . The reflecting face (intracavity) is assumed to be at . As in the thermal studies, we take a displacement vector under the form of an FB series
where and are unknown functions of , and are constants to be determined.

#### 8.4.1 Equilibrium equations

which yields
The general solution of which is
This allows one to compute in terms of and to substitute it in Equation (8.67), so that one gets
The solution of which is
where and are two more arbitrary constants. Now is determined by

#### 8.4.2 Boundary conditions

The boundary conditions we assume are

• No shear on the cylindrical edge, i.e.,
This can be satisfied by requiring to be a strictly positive zero of . The family of all such zeroes defines a family of functions complete and orthogonal on , on which any function of may be expanded as a FB series. Note that this family is different from the families encountered in thermal studies. In particular, the orthogonality relation is simpler:
• No shear on the two circular faces, i.e.,
• The given pressure on the reflecting face:
where is a pressure distribution normalized to an integrated force of 1 N, identical to the normalized optical intensity function .
• No pressure on the rear face:
• No radial stress on the edge:

The pressure distribution can, as usual, be expanded on the complete orthogonal family :

Owing to the norm of the functions , the FB coefficients are now defined by
We have already encountered this kind of integral. The general result for an mode is
and for a flat mode
For a mesa mode, numerical integration is necessary. The and components of the stress tensor are obtained as FB series:
making clear that the first boundary condition is satisfied. In more detail, we have
The boundary conditions on the faces provide four equations allowing one to determine the constants and . We have
with the notation and . This was found by [5]. At this point, [28] pointed out that the component of spatial frequency zero of the pressure was not taken into account (recall that the are the nonzero solutions of ). The preceding displacement vector has a zero average on the strained face. One must now consider the resulting force acting on the body under the uniform pressure
producing a force of 1 N after integration on the disk. But this force produces an acceleration, which should be added to the Navier–Cauchy equations (8.67) (recall that the mirrors of GW interferometers are practically free in the longitudinal degree of freedom in the detection band). This effect can be taken into account by an extra displacement of the form
This extra displacement contributes only axial stress
all other extra stress components being null. The equilibrium equations remain satisfied for
as remarked by [28]. Now, the sum of the displacement (8.66) and of the extra displacement (8.92) satisfies all boundary conditions except the vanishing of the radial stress on the edge. We have for the FB component of the radial stress
But, due to the fact that , and after substituting the explicit expressions of and , we get
It is numerically easy to check that this function of is not very different from linear. It has a vanishing average. Therefore, it is possible to find an approximate solution using the Saint-Venant principle once more. We add to our displacement one more extra displacement giving a linear edge stress compensating the preceding. The second extra displacement is of the form
such that the equilibrium equations (8.67) are identically satisfied. It contributes only radial stress, thus leaving unchanged the boundary conditions, except for a radial contribution
As usual, we require a minimum residual stress on the edge, which amounts to having null resulting mean force and torque on the edge. If we define
then the values of and are
The explicit expression (8.96) allows one to compute and . One finds
with
and
In summary, the total displacement is now
with
and

#### 8.4.3 Strain energy

The strain tensor is now of the form

where the component is computed from , whereas the component is computed from . Now the strain energy density is defined by
being the trace of the strain tensor. Thus, integrated strain energy , i.e., our target, is
The squares of the strain tensor components obviously contain, in general, squares of the main strain, squares of the extra strains and cross products. However, in the integral, cross products vanish, so that the total internal energy is the sum of two contributions
These can be computed separately. We have
The dimension of is J N. And for the second contribution, we have
with
For our three reference examples, using Virgo mirrors, we get, with a loss angle of 10–6,
• , w = 2 cm
• flat mode, b = 9.1 cm
• mesa mode,  = 10.7 cm
• , w = 3.5 cm
It is clear that for modes widely spread on the mirror surface, the Saint-Venant correction becomes important. Moreover, if we compare to the values found in the infinite mirror approximation, we see that the first example was underestimated by about 7%, the flat mode by 17%, and the third by a factor of 3. We also see the discrepancy (11%) between the flat estimation and the mesa beam. This leads us to be cautious with the foregoing estimations. Figure 58 summarizes the gain in thermal noise obtained with respect to the current situation on Virgo input mirrors for several beams having 1 ppm clipping losses.

#### 8.4.4 Explicit displacement and strain tensor

In Section 9, we shall need the explicit expressions of the displacement vector and particularly of the trace of the strain tensor. We have, after the preceding calculations for the FB components of the main displacement,

with the notation
In the same way
The derivative of is needed as well,
and the combination is still
For the extra displacement we have
with the notation
This allows one to plot the (virtually) deformed solid (see Figure 59) in our three examples. We have amplified the displacement by a large factor, to give a better idea of the shape.

We find the FB components of the main strain tensor to be

This gives, in particular, the FB component of the trace of the strain tensor
with
And for the extra contributions, we get
This allows us to map the energy density in the material (see Figures 60, 61, and 62).