9.2 Case of infinite mirrors
Let us recall the results obtained in the preceding chapter on standard thermal noise. Under beam
pressure, the displacement vector is
so that
The function
is determined by the virtual pressure distribution
(normalized beam intensity).
Namely,
where
is the Hankel transform of
. As a result,
which shows, in passing, that
Thus, wee can already foresee that in the case of an ideally flat beam the gradient will involve Dirac
distributions; therefore, the volume integration of its square will be problematic. Let us compute the
gradient of
:
Now, using the closure relation,
for
. It is now possible to carry out the volume integration:
so that
with
This expression shows that the function
must have an asymptotic evanescence strictly faster than
for the integral to converge. This is a strong requirement on the Hankel transform of the pressure
distribution.
9.2.1 Gaussian beams
For a Laguerre–Gauss mode
of width parameter
, we have seen that
giving
where
are numerical factors (see Table 18).
Table 18: Some numerical values of 
|
m |
0 |
1 |
2 |
3 |
4 |
5 |
n |
|
|
|
|
|
|
|
0 |
|
1 |
.75 |
.64 |
.57 |
.53 |
.49 |
1 |
|
.44 |
.39 |
.36 |
.33 |
.31 |
.30 |
2 |
|
.33 |
.31 |
.29 |
.27 |
.26 |
.25 |
3 |
|
.28 |
.26 |
.25 |
.24 |
.23 |
.22 |
4 |
|
.24 |
.23 |
.22 |
.21 |
.21 |
.20 |
5 |
|
.22 |
.21 |
.20 |
.20 |
.19 |
.19 |
|
Thus,
so that the spectral density of thermoelastic noise is, using Equations (9.1) and (9.16),
This result was found (in the case of
) first by Braginsky et al. [7] using their own formalism,
then by Liu et al. [28], using our approach. For silica parameters and for Ex1 (the
mode with
w = 2 cm), one finds
and
which is lower than the standard thermal noise, but still significant. For Ex3 (the
mode with
w = 3.5 cm), we have
9.2.2 Flat beams
If we now consider a flat beam modeled by its ideal representation
we have the Hankel transform
which shows that the requirement on the decreasing rate for large
is not fulfilled,
having
asymptotic behavior in
, just below the limit. Therefore, it is impossible to use the crude flat
model, the integral
being divergent. If we want to have an evaluation, we must carry out
a numerical integration with the mesa intensity profile. We find (for our particular model)
seeming to indicate a strong reduction factor of the SD of noise, namely 0.44 with respect to Ex3. This
result is due to the fact that the
mode has a number of rings causing many local gradients.
This was pointed out by Agresti ([2]) in the case of the
mode. Anyway this mode is
unwanted, as well as other
modes, because we wish to avoid sharp central power peaks
.
9.2.3 Thermoelastic noise in the coating
We apply the same strategy for all coating calculations. The gradient of the trace of the strain tensor can be
integrated on the surface
giving
with
(
takes the value one when
). Contrary to
, which is not so sensitive to
parameters,
can take values quite different from one. For instance, if we assume the parameters of
fused silica for the substrate, and (
,
) for the coating, we have
.
has the following definition
so that the energy
is (taking into account special values for the coating material)
In the case of
modes, we obtain
where
are numerical factors, the first ones being given by Table 19.
Table 19: Some numerical values of 
|
m |
0 |
1 |
2 |
3 |
4 |
5 |
n |
|
|
|
|
|
|
|
0 |
|
1 |
1.5 |
1.72 |
1.86 |
1.96 |
2.05 |
1 |
|
.50 |
.81 |
.98 |
1.10 |
1.20 |
1.27 |
2 |
|
.37 |
.62 |
.77 |
.88 |
.96 |
1.03 |
3 |
|
.31 |
.53 |
.66 |
.76 |
.83 |
.90 |
4 |
|
.27 |
.46 |
.58 |
.67 |
.75 |
.81 |
5 |
|
.25 |
.42 |
.53 |
.61 |
.68 |
.74 |
|
It seems clear that, as already mentioned, the modes
, having a sharp peak on the axis, become
worse and worse as the order
increases. On the other hand, the reduction factor for the noise in the
best cases is much less than for the Brownian thermal noise.
9.2.4 Scaling laws
This section offers an opportunity to summarize the various coefficients encountered in the
parts of this noise study. Several authors (see [29
] for his discussion) have remarked on the
dependence of the various noises encountered on the integrals we have denoted
.
- Brownian noise, substrate:
- Brownian noise, coating:
- Thermoelastic noise, substrate:
- Thermoelastic noise, coating:
. However, in this case there is a more refined analysis [29],
taking into account the heat flow. Attention must be paid to this theory (see also [17, 8]).
However, the approximate character of the semi-infinite–mirror approach reduces its practical
interest.
We have given these integrals in the case of different
modes. In particular, Table 20 gives the
values for our four examples. These can be used to derive figures of merit.
9.2.5 Numerical results
We give briefly some figures regarding our three reference situations. We take the parameter of fused silica
for the substrate, and the parameters of
for the coating ([36]), namely,
,
,
. The thickness of the coating is assumed to be 25 µm. For a
mode of waist 2 cm, we obtain
- Spectral density of thermoelastic noise in the substrate:
Spectral density of noise in the coating:
We see that the large difference in parameters overcompensates for the difference in volume. For an
mode of waist 3.5 cm:
- Spectral density of thermoelastic noise in the substrate:
Spectral density of noise in the coating:
The reduction factor is about five for the substrate and only 3.5 for the coating. We see that the
large difference in parameters overcompensates for the difference in volume. For a mesa mode:
- Spectral density of thermoelastic noise in the substrate:
- Spectral density of noise in the coating:
The reduction factor with respect to Ex1 is about 12 for the substrate and 26 for the coating. This kind of
mode is obviously the best regarding this kind of noise.