9.1 Introduction

The Brownian motion of matter inside the substrates is not the only cause of noise in the optical readout. There is another cause due to temperature fluctuations in a finite volume of material. These fluctuations are called thermodynamic and can couple with strain via the thermal dilatation constant α, eventually producing random motions of the surface. A good way to model this kind of noise is to start from the general thermodynamic formulas detailed by Landau and Lifshitz [24Jump To The Next Citation Point] and use the Levin approach already presented. As in the preceding Section 8, we will consider the low frequency tail of the spectral density of the effective motion of the surface (i.e., the readout noise) as depending on the energy dissipated when the body is under a virtual pressure having the same profile as the optical beam and excited at low frequency. In this case, the spectral density is still of the form (Levin’s formula)
4k T Sx (f) = --B---W, (9.1 ) ω2
where W is the average dissipated energy. For the standard thermal noise, we had W = 2U ωΦ as average dissipated energy, Φ being a global loss angle and U the static strain energy. But now W must be interpreted as the energy dissipated via coupling of the strain with the temperature field in the bulk. Obviously, the temperature field itself depends on the strain field. Using the same approach as used in [28Jump To The Next Citation Point], we first solve the static linear elastic problem (done in the preceding Section 8), then we compute the resulting temperature field and use it to compute the dissipated energy. For computing the dissipated energy, we use the time dependence of the entropy. The variations of the entropy density S are related to the heat flux ⃗q by requiring conservation of the energy in the body
∂S T ---= − div(⃗q), (9.2 ) ∂t
where ⃗q = − KgradT, K being the thermal conductivity of the material (cf. Landau and Lifshitz [24]). Or, as well,
∂S- 1- ∂t = − T div(⃗q). (9.3 )
Therefore, total entropy variation in the body is
dStot ∫ 1 ----- = − --div(⃗q)dV, (9.4 ) dt T
where the integral is extended to the whole body. This is
∫ ∫ ( ) dStot -⃗q 1- dt = − divT dV + ⃗q.grad T dV. (9.5 )
Neglecting the heat flow at the surface of the body, the first integral vanishes, and we have
∫ dStot= − -1-⃗q.gradT dV. (9.6 ) dt T2
But using the definition of ⃗q, this is
∫ dStot = K--(gradT )2dV, (9.7 ) dt T 2
so that the energy variation is
∫ W = T dStot = K-(gradT )2dV. (9.8 ) dt T
We shall say now that the temperature gradient field is caused by the small deformations of the body that we computed earlier, while T is the mean temperature. This becomes
∫ dStot K-- 2 W = T dt = T (gradδT )dV. (9.9 )
where we have replaced T by a δT in the gradient for more clarity. On the other hand, it is well known (cf. Landau-Lifshitz) that the total entropy is the sum of two terms, one being the entropy in the reference state, and a second one proportional to the trace E of the strain tensor
S = S + νE, (9.10 ) 0
ν being the thermoelastic coefficient, so that there is, in the bulk material, a power source given by
P = T dS-= νT dE-, (9.11 ) dt dt
where E is the trace of Eik. The resulting temperature field obeys the Heat (Fourier) equation
(ρC ∂ − K Δ)δT = νT dE-. (9.12 ) t dt
The trace of the strain tensor Eik found in the preceding Section 8 is, in any case, a harmonic function, so that there is a trivial solution
νT- δT = ρC E. (9.13 )
The boundary conditions (null heat flux on the surfaces) are considered satisfied in time average (δT is assumed oscillating at a few tens of Hz). In fact, they are exactly satisfied on the circular edge of the mirror. Now we reach the relevant equation for the dissipated energy
2 ∫ W = K--ν-T (gradE )2dV. (9.14 ) ρ2C2
ν is related to the linear dilatation coefficient α by
ν = --αY---, (9.15 ) 1 − 2σ
where Y is Young’s modulus and σ the Poisson ratio. Finally,
[ αY ]2 ∫ W = KT ------------ (gradE )2dV (9.16 ) (1 − 2σ )ρC
(see [28Jump To The Next Citation Point]). We have, after the preceding Section 8 on standard thermal noise, everything we need to compute W.
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