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 Z (ω) is given, as usual, by the average of the lens, weighted by the normalized intensity profile I(r),
∫ a Z (ω) = 2π Z (ω, r)I(r)rdr. (6.10 ) 0
We have
∑ Z (ω, r) = Z (ω)J (k r) (6.11 ) s 0 s s
1 ∑ I(r) = ---2 psJ0(ksr) (6.12 ) πa s
so that the equivalent displacement is
∑ Z (ω ) = [1 + (χe∕ζs)2]J0 (ζs)2psZs(ω ). (6.13 ) s

Asymptotic regime

The time constant 2 τd ≡ ρCa ∕K is about 10 h. Therefore, it is clear that for frequencies in the target GW band (more than a few Hz), we have

2 2 ξs = ζs + 2iπτdf ∼ 2iπτdf. (6.14 )
Because the FB series is converging, the values of s at which the real part becomes comparable to the imaginary are never reached. If we adopt the preceding approximation, we have
dn 𝜖P(ω ) Zs (ω ) ∼ − i---------ps, (6.15 ) dT πρC ω
which allows one to compute the asymptotic equivalent displacement. In Figure 56View Image we have plotted the transfer function |Z (ω )∕𝜖P (ω )| 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
Z (ω, r) = − idn-𝜖P-(ω) I(r), (6.16 ) dT πρC ω
where I(r) is the normalized intensity of the beam. In words, the thermal lens is proportional to the beam intensity, with a phase lag of π∕2. The conclusions are identical for the case of bulk absorption. The asymptotic formulas are identical up to the change 𝜖 → βh.
View Image

Figure 56: Coating absorption: transfer function from power to displacement. Dashed line: asymptotic regime

For the case of the heat source on the reflective coating, we get the following values. For an LG0,0 mode with w = 2 cm, we have

[ ] Z-(f)-∼ 2.7 × 10− 10 1-Hz- m ∕W; (6.17 ) 𝜖P (f) f
for a flat mode of width 9.1 cm, we have
[ ] Z (f) − 11 1 Hz ------∼ 1.3 × 10 ----- m ∕W; (6.18 ) 𝜖P (f) f
and for an LG5,5 mode with w = 3.5 cm, we have
[ ] Z-(f)- −12 1-Hz- 𝜖P(f ) ∼ 8.5 × 10 f m∕W. (6.19 )
The results are quasi-identical for bulk absorption (see Figure 57View Image).
View Image

Figure 57: Bulk absorption: transfer function from power to displacement. Dashed line: asymptotic regime.

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