For a Laguerre–Gauss mode of width parameter , we have seen that
Thus, using their own formalism, then by Liu et al. , using our approach. For silica parameters and for Ex1 (the mode with w = 2 cm), one finds w = 3.5 cm), we have
If we now consider a flat beam modeled by its ideal representation) 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 .
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
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.
This section offers an opportunity to summarize the various coefficients encountered in the parts of this noise study. Several authors (see  for his discussion) have remarked on the dependence of the various noises encountered on the integrals we have denoted .
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.
|n||w = 2 cm||w = 3.5 cm||flat b = 9.1 cm||mesa = 10.7 cm||units|
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 (), namely, , , . The thickness of the coating is assumed to be 25 µm. For a mode of waist 2 cm, we obtain
We see that the large difference in parameters overcompensates for the difference in volume. For an mode of waist 3.5 cm:
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:
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.
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