### 7.1 Dynamic surface distortion caused by coating absorption

The temperature field is known from Section 6 (see Equation (6.6)), so that we get
which, with Equation (7.23), gives the complete solution.

#### 7.1.1 Mean displacement

We find the contribution to phase noise by computing, as usual, the mean equivalent displacement by

I(r) being the normalized intensity profile of the beam. This yields

#### 7.1.2 Under cutoff regime

The dimensionless parameter is very small. With the current parameters, is slightly larger than
one, so that

so that in the GW band (far from mirror resonances), we can take the first-order approximation of the
precedent functions with respect to . The elastic wave regime begins when the frequency exceeds a
value (cutoff) such that some and become imaginary. A study of the resonance modes can
then be addressed, but this requires a careful treatment of the boundary conditions on the edge. We
consider here only the case where the frequency is below the cutoff, so that a simple theory neglecting the
edge is relevant and may be considered small. In particular, we find from 0.1 Hz to 1 kHz
This gives for our three examples the following transfer functions for the displacement noise. mode
w = 2 cm:
Flat mode (b = 9.1 cm) or mesa mode ( = 10.7 cm):
mode w = 3.5 cm:
Below 0.1 Hz we are in the regime where the displacements are corrected by the locking system.
Above 1 kHz the displacement noise is negligible compared to shot noise and that in the present
situation.