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:
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.
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.