3.1 Steady temperature field

3.1.1 Coating absorption

Let us briefly recall that in the steady state with no internal source of heat, the Heat (Fourier) equation reads:

( ) 2 1 1 2 2 ∂r + r∂r + r2 ∂φ + ∂z T (r,φ, z) = 0 (3.1 )
so that the temperature field is a harmonic function. A harmonic function is, for example,
±kz inφ Tn(r,φ,z) = e Jn (kr)e (3.2 )
where {Jn(x),n = 0,1,..} are Bessel functions of the first kind, n is an arbitrary integer and k an arbitrary constant. We assume the readout beam reflected by the coating has an intensity distribution of the form
I(r,ϕ) = In(r)cos(n φ + φn), (3.3 )
(φ n is an arbitrary constant) and thus having a simple angular parity (possibly one term of a Fourier series). Therefore, we define the temperature field as
T (r,φ,z) = e±kzJ (kr )cos(nφ + φ ). (3.4 ) n n n
The boundary conditions describe the heat flows on the faces and on the edge. The powerful light beam is assumed to be reflected at z = − h∕2. If K denotes the thermal conductivity and T0 the (homogeneous) temperature of the surrounding walls of the vacuum vessel, and if we consider T (r,φ,z) n as the excess of temperature with respect to T0, we have, at thermal equilibrium, the following balance on the irradiated face;
[ ] ∂Tn-(r,φ,z-) ( 4 4) − K ∂z = − σf [T0 + Tn(r,φ,− h∕2 )] − T0 + 𝜖In(r)cos(n φ + φn), (3.5 ) z=−h∕2
where the left-hand side represents the power lost by the substrate and the first term of the right-hand side represents the power flow of thermal radiation according to Stefan’s law. σf is the Stefan–Boltzmann (SB) constant (σ ∼ 5.6710− 8 Wm − 1K− 4 f) corrected for the emissivity of silica; accounting for a possible special processing of the edge (e.g., some thin metallic layer), we shall allow different values of the SB constant on the faces, σf, and on the edge, σedge. The second term of the right-hand side is the density of power received from the light beam. 𝜖 represents the relative loss at reflection. After linearization (i.e., assuming Tn ≪ T0), we get
[ ] − K ∂Tn(r,φ,-z) = − 4σfT 3Tn(r,φ,− h∕2 ) + 𝜖In(r,φ). (3.6 ) ∂z z=− h∕2 0
On the opposite face, we have a similar condition (the radiation flow is in the opposite direction)
[ ] ∂Tn-(r,φ,z-) 3 − K ∂z = 4σfT0Tn (r,φ, h∕2), (3.7 ) z=h∕2
whereas on the edge, the boundary condition is
[∂Tn (r,φ,z )] − K ----------- = 4σedgeT 30Tn(r,φ, a). (3.8 ) ∂r r=a
This last condition is relative to the radial function and gives
− KkJ ′n(ka) = 4σedgeT30Jn(ka ) (3.9 )
or, by introducing the reduced radiation constant 3 χe ≡ 4σedgeT 0a∕K,  
kaJ ′(ka) + χeJn (ka) = 0. (3.10 ) n
An equation of the form
ζJ′n(ζ) + χeJn(ζ) = 0 (3.11 )
has an infinite and discrete family of solutions {ζn,s,s = 1,2,...}. Therefore, a sufficiently general solution of the Heat equation having a given angular parity will be taken as a series,
∑ Tn (r,φ,z) = Tn,s(z)Jn (ζn,sr∕a)cos(nφ + φn). (3.12 ) s>0
Moreover, after the Sturm–Liouville theorem, the family of functions {Jn (ζn,sr∕a),s = 1,2, ...} is orthogonal and complete on the interval [0,a]. The normalization factor is (see, for instance, Equation (11.4.5) in [1])
∫ 1 1 ( 2 2 2) 2 xJn (ζp,sx)Jn(ζp,s′x)dx = δs,s′--2-- χe + ζn,s − n Jn(ζn,s) . (3.13 ) 0 2ζn,s
The first consequence is that it is possible to express the radial intensity function In(r ) of integrated power P in the form of a Fourier–Bessel (FB) series,
∑ I (r) = -P-- p J (ζ r ∕a). (3.14 ) n πa2 n,s n n,s s>0
The dimensionless FB coefficients {pn,s,s = 1,2,...} being obtained by
2πζ2 ∫ a pn,s = ---(----------n,s)--------- In(r)Jn(ζn,sr∕a )rdr. (3.15 ) P χ2e + ζ2n,s − n2 Jn (ζn,s)2 0
Now the longitudinal function is of the form
Tn,s(z) = An,s exp(ζn,sz∕a ) + Bn,s exp(− ζn,sz∕a) (3.16 )
and the constants A n,s and B n,s are determined by the boundary conditions (3.6View Equation) and (3.7View Equation). Finally, one finds
−ζn,s(h−z)∕a −ζn,sz∕a T (z ) = -𝜖P--p e− ζn,sh∕2a (ζn,s-−-χ)e-----------+-(ζn,s-+-χ)e-------, (3.17 ) n,s πKa n,s (ζn,s + χ)2 − (ζn,s − χ)2e−2ζn,sh∕a
where χ is the reduced radiation constant for faces (i.e., 3 χ ≡ 4σfT 0a∕K). This completely determines the temperature field through Equation (3.12View Equation), once the zeroes ζn,s are known by solving Equation (3.11View Equation), and once the coefficients pn,s are computed by the integration (3.15View Equation). This last point will be treated in Section 3.1.3 below. We can write Equation (3.17View Equation) in a more compact form exhibiting the symmetric and antisymmetric parts,
[ ] T (z) = -𝜖P--- p cosh(ζn,sz∕a-)− sinh-(ζn,sz∕a)- (3.18 ) n,s 2πKa n,s d1,n,s d2,n,s
with the following definitions (used in all parts of this review),
{ d = ζ sinh γ + χ cosh γ 1,n,s n,s n,s n,s , (3.19 ) d2,n,s = ζn,s coshγn,s + χ sinh γn,s
where γn,s ≡ ζn,sh ∕2a. Note that if the heat source is located on the opposite face of the mirror, as in the case (to be treated later) of a thermal compensation beam, the preceding formula becomes simply
[ ] T (z) = -𝜖P---p cosh-(ζn,sz∕a) + sinh(ζn,sz-∕a) . (3.20 ) n,s 2πKa n,s d1,n,s d2,n,s

3.1.2 Bulk absorption

Let us now assume that the heat source results from the loss of optical power by the beam inside the mirror substrate due to weak absorption. The beam intensity propagating inside the substrate is, strictly speaking, of the form

I (r,z) = I (r) exp[− β(z + h∕2)], (3.21 )
where I(r) is the incoming intensity. However, the linear absorption β is assumed to be so weak that there is no significant change in amplitude of the beam along the optical path. Thus the heat source in the bulk material is βI (r) (Wm−3). For any given angular parity, the Heat equation becomes
( ) 2 1 1 2 2 β ∂r + -∂r + -2 ∂φ + ∂z T (r,φ, z) = − --In(r)cos(n φ + φn). (3.22 ) r r K
We will look for a solution of the same form as Equation (3.12View Equation). The boundary condition on the edge is identical to Equation (3.8View Equation), so that the family of orthogonal functions is unchanged. The coefficients pn,s allowing expansion of the intensity function are also identical. Now we shall express the relevant solution as the sum of a special solution of Equation (3.22View Equation) and a more general solution of the homogeneous equation (identical to Equation (3.1View Equation)). Using the Bessel differential equation, the special solution of Equation (3.22View Equation) is found with
βP--∑ pn,s- Tn,1(r,φ ) = πK ζ2 Jn(ζn,sr∕a) cos(n φ + φn ). (3.23 ) s>0 n,s
The solution of the homogeneous equation will be symmetric in z, owing to the independence of the heat source in z,
∑ Tn,2(r,φ) = An,scosh (ζn,sz∕a)Jn(ζn,sr ∕a)cos(nφ + φn ). (3.24 ) s>0

The arbitrary constants An,s are determined by the boundary condition (3.7View Equation. Boundary condition (3.6View Equation) disappears, being identical to the preceding, due to symmetry. One finally finds a series analogous to Equation (3.12View Equation), except that the longitudinal function Tn,s(z) is now

βP pn,s [ χ cosh(ζn,sz∕a )] Tn,s(z) = -----2-- 1 − --------------- . (3.25 ) πK ζn,s d1,n,s

3.1.3 Fourier–Bessel expansion of the readout beam intensity

We address now the central point of the calculation of the FB coefficients pn,s of the intensity. The LGm,n mode of integrated power P has the following intensity function

Im(n)(r,φ) = 2μnR (mn)(r)cos(nφ + φn )2, (3.26 )
with μn ≡ 1∕(1 + δn,0), and
( ) ( ) ( ) (n) 2P m! 2r2 n (n) 2r2 2 2r2 Rm (r) = ---2--------- --2- Lm --2- exp --2- , (3.27 ) πw (n + m )! w w w
where the functions L (pq)(X ) are the generalized Laguerre polynomials. The function can be split into two terms of simple angular parity,
I(n)(r,φ) = μ R (n)(r) + μ R (n)cos(2n φ + 2φ ), (3.28 ) m n m n m n
so that the FB series of the intensity will be two-fold,
∑ ∑ I(n)(r, φ) = p J (ζ r∕a) + p J (ζ r∕a )cos(2nφ + 2φ ) (3.29 ) m 0,s 0 0,s n,s 2n n,s n s>0 s>0
where {ζ0,s} are all solutions of
ζJ ′(ζ) + χeJ0 (ζ) = 0, (3.30 ) 0
whereas {ζn,s} are all solutions of
ζJ′2n(ζ) + χeJ2n (ζ) = 0. (3.31 )
The p0,s are given by [see Equation (3.15View Equation)]
2πζ20,s ∫ a p0,s = ----2----2--------2- R(mn)(r)J0 (ζ0,sr∕a)rdr. (3.32 ) P (χe + ζ0,s)J0(ζ0,s) 0

In fact, the intensity is necessarily negligible near the edge, so that the upper bound of the integral may be replaced by + ∞ without appreciably changing the result. The integral then becomes explicitly computable, and one finds

ζ20,s p0,s = --2----2---------2e−y0,sLm (y0,s)Ln+m (y0,s), (3.33 ) (χe + ζ0,s)J0(ζ0,s)


ζ2 w2 y0,s ≡ -0,s-- (3.34 ) 8a2

and the functions LN (X ) are the (ordinary) Laguerre polynomials. In the same way, the coefficients pn,s are obtained by

2πζ2n,s ∫ a pn,s = ----2----2------2---------2 R (mn)(r)J2n(ζ2n,sr∕a )rdr. (3.35 ) P (χe + ζn,s − 4n )J2n(ζn,s) 0

We can again replace the upper bound by + ∞, which allows explicit calculation,

ζ2n,s m! pn,s = --2----2------2---------2---------e−yn,s(yn,s)nL (nm)(yn,s)2 (3.36 ) (χe + ζn,s − 4n )J2p(ζp,s) (n + m )!

with the notation

ζ2n,sw2 yn,s ≡ ----2-. (3.37 ) 8a

In the latter case we see that the Hankel transform maps the square of a Laguerre–Gauss function onto the same function with a different argument, up to a scaling factor. The temperature field is now completely known. Let us add that the Fourier–Bessel series are rapidly convergent so that the reconstruction of the intensity is obtained with excellent accuracy with only 50 terms. In Figure 8View Image we show the difference between the original intensity and the reconstructed one.

In the case of an ideally flat mode of radius b, the integral (3.15View Equation) is trivial, and we have the following FB coefficients:

2a ζ0,s pflat,s = -----2----2---2-----J1(ζ0,sb∕a). (3.38 ) b (χ + ζ0,s)J 0(ζ0,s)
The reconstruction of the flat mode from a limited number of Fourier–Bessel coefficients is not perfect, owing to the generation of high frequencies by the sharp edges. But the heat field itself is rapidly convergent because of the regularizing effect of integration, so that even with a small number of terms, the FB series are accurate. In cases where the ideally flat model is forbidden, the FB coefficients must be computed using Equation (2.16View Equation) via a numerical integration.
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Figure 8: Error in intensity reconstruction (50 Fourier–Bessel terms) for LG0,0, w = 2 cm (black curve) and LG5,5, w = 3.5 cm (red curve); Cut: φ = 0

See Figure 9View Image for the reconstructed intensity profile of a Laguerre–Gauss mode LG5,5.

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Figure 9: Reconstructed intensity (FB series) for LG5,5 mode (φ = 0)

3.1.4 Numerical results on temperature fields

If we assume a mirror of the Virgo input mirrors size, made of synthetic silica, we can take the parameters of Table 1.

Table 1: Physical constants used in this paper
Symbol Parameter Value units
a half diameter 0.175 m
h thickness 0.1 m
ρ density 2,202 kg m–3
K thermal conductivity 1.38 Wm–1 K–1
C specific heat cap. 745 J kg–1 K–1
α thermal expansion coef. 5.4 ×10–7 K–1
β linear absorption 10–5 m–1
Y Young’s modulus 7.3 ×1010 Nm–2
σ Poisson ratio 0.17 dimensionless
dn∕dT thermal refractive ind. 1.1 ×10–5 K–1

A cut of the temperature field in the φ = 0 plane can be seen in Figures 10View Image, 12View Image, and 14View Image for heating by coating absorption, and in Figures 11View Image, 13View Image, and 15View Image for heating by dissipation in the substrate.

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Figure 10: Temperature field in the substrate, 1 W dissipated in the coating, mode LG 0,0, w = 2 cm (φ = 0) [logarithmic scale]
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Figure 11: Temperature field in the substrate, 1 W dissipated in the bulk substrate, mode LG 0,0, w = 2 cm (φ = 0) [logarithmic scale]
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Figure 12: Temperature field in the substrate, 1 W dissipated in the coating, flat mode, b = 9.1 cm (φ = 0) [logarithmic scale]
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Figure 13: Temperature field in the substrate, 1 W dissipated in the bulk substrate, flat mode, b = 9.1 cm (φ = 0) [logarithmic scale]
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Figure 14: Temperature field in the substrate, 1 W dissipated in the coating, mode LG5,5, w = 3.5 cm (φ = 0) [logarithmic scale]
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Figure 15: Temperature field in the substrate, 1 W dissipated in the bulk substrate, mode LG5,5, w = 3.5 cm (φ = 0) [logarithmic scale]

The temperature map of the coating z = − h ∕2 is shown in Figure 16View Image (heating by coating absorption).

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Figure 16: Nonaxisymmetric mode LG5,5, w = 3.5 cm, temperature on the coating (coating absorption) (z = − h∕2)

The temperature map of the meridian plane (z = 0) of the substrate (heating by internal dissipation) is shown in Figure 17View Image, where one can see the effect of thermal conduction, which generates a practically axisymmetric temperature field, despite the cos2 signature of the incoming light beam.

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Figure 17: Nonaxisymmetric mode LG5,5, w = 3.5 cm. Temperature in the meridian plane (bulk absorption) (z = 0)

The dependence of the temperature field on the longitudinal variable z is shown in Figures 18View Image and 19View Image.

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Figure 18: Temperature at various depths in the substrate (coating absorption) case of LG5,5 (φ = 0)
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Figure 19: Temperature at various depths in the substrate (flat mode, bulk absorption) (φ = 0)

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