### 6.2 Equivalent displacement noise

The induced dynamic thermal lens produces a dynamic excess phase after crossing the mirror’s
substrate. This dynamic phase is analogous to a displacement. We assume this equivalent displacement is
not desired and call it “displacement noise”. This equivalent displacement is given, as usual, by the
average of the lens, weighted by the normalized intensity profile ,
We have
and
so that the equivalent displacement is

#### Asymptotic regime

The time constant is about 10 h. Therefore, it is clear that for frequencies in the target
GW band (more than a few Hz), we have

Because the FB series is converging, the values of at which the real part becomes comparable to the
imaginary are never reached. If we adopt the preceding approximation, we have
which allows one to compute the asymptotic equivalent displacement. In Figure 56 we have plotted the
transfer function relating displacement fluctuations to power fluctuations, making clear
that the asymptotic regime (dashed line) is fully valid for frequencies larger than 10 mHz.
On the other hand, we see that in the asymptotic regime, the dynamic thermal lens is simply
where is the normalized intensity of the beam. In words, the thermal lens is proportional to the beam
intensity, with a phase lag of . The conclusions are identical for the case of bulk absorption. The
asymptotic formulas are identical up to the change .
For the case of the heat source on the reflective coating, we get the following values. For an
mode with w = 2 cm, we have

for a flat mode of width 9.1 cm, we have
and for an mode with w = 3.5 cm, we have
The results are quasi-identical for bulk absorption (see Figure 57).