For a Laguerre–Gauss mode of width parameter , we have seen that

giving where are numerical factors (see Table 18).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 haveIf 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 .

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.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.

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.

n | w = 2 cm | w = 3.5 cm | flat b = 9.1 cm | mesa = 10.7 cm | units |

0 | 2.245 | 0.321 | 0.473 | 0.426 | m^{–1} |

1 | 126.65 | 4.13 | 6.12 | 4.52 | m^{–2} |

2 | 1.122 ×10^{4} |
398 | * | 76.7 | m^{–3} |

3 | 1.27 ×10^{6} |
10^{5} |
* | 1800 | m^{–4} |

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.

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