8.1 The Fluctuation-Dissipation theorem and Levin’s generalized coordinate method

We are interested in the spectral density of thermal noise. There is a general derivation of this spectral density, based on the Fluctuation-Dissipation (FD) theorem of Callen and Welton [9]. For an elementary dynamic system described by a degree of freedom x and any driving force F, one can consider the resulting velocity &tidle;v = iω &tidle;x, and compute a mechanical impedance as &tidle; 𝒵 = &tidle;v∕F. Then, the power spectral density of displacement is (this is the FD theorem)
4kBT Sx(f) = ---2--ℜe [𝒵 ]. (8.1 ) ω
We can now address the problem of internal degrees of freedom in the mirrors. Internal elastic waves eventually distort the reflecting surface, causing a phase noise. We have already discussed how to obtain the information on the surface relevant to the beam. Let uz(t,x,y) be the z component of the displacement vector of matter at the surface of the mirror. The equivalent displacement (generalized coordinate x) is, as usual, the axial displacement, averaged by the intensity profile,
∫ ∫ Z (t) = uz(t,x,y)I(x,y )dxdy, (8.2 )
where I(x,y) is the normalized light-intensity distribution in the readout beam. We now follow the method proposed by Levin [26]. Let F (t) be the corresponding driving force. The interaction energy is
ℰ = − F (t)Z (t) (8.3 )
∫ ∫ ℰ = uz(t,x,y )F (t)I(x,y)dxdy, (8.4 )
where the displacement u may be thought of as being caused by the pressure distribution F × I. We address now the case of low frequencies. This case is very relevant, because resonances of mirrors are at relatively high frequencies (several kHz) and the region where internal thermal noise is disturbing lies long before the first resonance, in the low frequency regime. Thus, although a general knowledge of internal thermal noise is useful, it is nevertheless extremely interesting to have the low frequency tail. This can be obtained as follows. If we consider a force F (t) = Feiωt oscillating at very low frequency, the frequency will be lower than the cutoff for any standing waves. The pressure F × I will produce an oscillating stationary displacement u, of the form
i(ωt−ϕ) uz(t,x,y) = e u(x,y ). (8.5 )
This is equivalent to neglecting inertial forces in the motion of matter. The phase ϕ represents a retardation effect that dissipation may cause; in the case of thermoelastic dissipation, we know that ϕ can be considered very small and independent of the frequency (at least in the GW detection band). In the Fourier domain, this is
uz(ω,x, y) = (1 − iϕ)uz (x, y). (8.6 )
The impedance is
∫ ∫ 𝒵 (f) = iω (1 −-iϕ-)---uz(x,y-)I-(x,-y)dxdy- (8.7 ) F
so that
∫ ∫ uz (x, y)F × I(x, y)dxdy ℜe [𝒵 ] = ω ϕ-----------F-2------------- (8.8 )
, where the numerator of the fraction appears as the elastic energy stored in the solid stressed by the pressure distribution F × I. The strain energy is defined in classical elasticity theory by
∫ ∫ W = 1- uz(x,y)p(x,y )dxdy, (8.9 ) 2
where p(x,y) is the pressure distribution causing the displacement u (x,y) z at the surface on which it is applied. Thus, we can write for the spectral density of displacement,
4k T W Sx (f) = --B---ϕ--2. (8.10 ) πf F
In fact, W is proportional to F 2, so that U ≡ W ∕F 2 is the strain energy for a static pressure normalized to 1 N. The spectral density of displacement takes the general (low frequency) form
4kBT-- Sx (f ) = πf ϕU. (8.11 )
The problem is reduced to the computation of U and this is the main point of Levin’s approach. This can be difficult in the general case of an arbitrary solid, but numerical finite-element codes are able to give more-or-less accurate estimates. However, it is possible to obtain analytic solutions in the case of axisymmetry.
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