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:

so that the temperature field is a harmonic function. A harmonic function is, for example,
where are Bessel functions of the first kind, is an arbitrary integer and an arbitrary constant. We assume the readout beam reflected by the coating has an intensity distribution of the form
( 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
The boundary conditions describe the heat flows on the faces and on the edge. The powerful light beam is assumed to be reflected at . If denotes the thermal conductivity and the (homogeneous) temperature of the surrounding walls of the vacuum vessel, and if we consider as the excess of temperature with respect to , we have, at thermal equilibrium, the following balance on the irradiated face;
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. is the Stefan–Boltzmann (SB) constant () 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, , and on the edge, . The second term of the right-hand side is the density of power received from the light beam. represents the relative loss at reflection. After linearization (i.e., assuming ), we get
On the opposite face, we have a similar condition (the radiation flow is in the opposite direction)
whereas on the edge, the boundary condition is
This last condition is relative to the radial function and gives
or, by introducing the reduced radiation constant ,
An equation of the form
has an infinite and discrete family of solutions . Therefore, a sufficiently general solution of the Heat equation having a given angular parity will be taken as a series,
Moreover, after the Sturm–Liouville theorem, the family of functions is orthogonal and complete on the interval . The normalization factor is (see, for instance, Equation (11.4.5) in [1])
The first consequence is that it is possible to express the radial intensity function of integrated power in the form of a Fourier–Bessel (FB) series,
The dimensionless FB coefficients being obtained by
Now the longitudinal function is of the form
and the constants and are determined by the boundary conditions (3.6) and (3.7). Finally, one finds
where is the reduced radiation constant for faces (i.e., ). This completely determines the temperature field through Equation (3.12), once the zeroes are known by solving Equation (3.11), and once the coefficients are computed by the integration (3.15). This last point will be treated in Section 3.1.3 below. We can write Equation (3.17) in a more compact form exhibiting the symmetric and antisymmetric parts,
with the following definitions (used in all parts of this review),
where . 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

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

where is the incoming intensity. However, the linear absorption 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 (Wm). For any given angular parity, the Heat equation becomes
We will look for a solution of the same form as Equation (3.12). The boundary condition on the edge is identical to Equation (3.8), so that the family of orthogonal functions is unchanged. The coefficients 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 (3.22) and a more general solution of the homogeneous equation (identical to Equation (3.1)). Using the Bessel differential equation, the special solution of Equation (3.22) is found with
The solution of the homogeneous equation will be symmetric in , owing to the independence of the heat source in ,

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

3.1.3 Fourier–Bessel expansion of the readout beam intensity

We address now the central point of the calculation of the FB coefficients of the intensity. The mode of integrated power has the following intensity function

with , and
where the functions are the generalized Laguerre polynomials. The function can be split into two terms of simple angular parity,
so that the FB series of the intensity will be two-fold,
where are all solutions of
whereas are all solutions of
The are given by [see Equation (3.15)]

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

where

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

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

with the notation

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 8 we show the difference between the original intensity and the reconstructed one.

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

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 (2.16) via a numerical integration.

See Figure 9 for the reconstructed intensity profile of a Laguerre–Gauss mode .

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 half diameter 0.175 m thickness 0.1 m density 2,202 kg m–3 thermal conductivity 1.38 Wm–1 K–1 specific heat cap. 745 J kg–1 K–1 thermal expansion coef. 5.4 ×10–7 K–1 linear absorption 10–5 m–1 Young’s modulus 7.3 ×1010 Nm–2 Poisson ratio 0.17 dimensionless thermal refractive ind. 1.1 ×10–5 K–1

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

The temperature map of the coating is shown in Figure 16 (heating by coating absorption).

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

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