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
The Navier–Cauchy equations read
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).