### 5.1 Transient temperature field

We consider the time-dependent heat equation with an internal source of heat (bulk absorption of light)
and a surface source on the coating. The Fourier equation is
where is the density of the material and its specific heat. accounts for the dissipation of light
passing through the substrate. As in the static case, it is independent on because the attenuation of the
beam is extremely weak; moreover, we assume it is independent of time. In a recycling interferometer, we
would expect the transient lens to change the reflectance of the cavity, the recycling rate, in
turn, and eventually the source , so that is a function of . But we consider here
an isolated mirror. The case of a complex system with feedback can be treated sequentially
(see [19]) using any propagation code, but this is beyond our present goal. For instance, we assume
some servo loop acting to maintain a constant power flow. The boundary conditions are still
where accounts for the dissipation of light on the reflective coating. The next two boundary conditions
are
and
We already know the steady state solution for coating or bulk dissipation in the general form
where and have the same definition as in Equation (3.11).
satisfies all boundary conditions plus the inhomogeneous heat equation. Thus, it is a special solution of
Equation (5.1). We now look for a general time-dependent solution of the homogeneous heat equation
and satisfying homogeneous boundary conditions. We search this transient temperature field under the form
so that the general solution of Equation (5.1) will be
Now, the time functions and must satisfy

and
whose solutions are
where the time constants are, respectively,
with the main time constant
(about 0.8 h for a regular Virgo mirror substrate). The boundary conditions on the faces lead to the
following equations:
In terms of the reduced radiation constant , the first equation is of the form
This equation admits an infinite discrete family of solutions , giving us . In
the same way, we have , where the constants are all solutions of
Owing again to the Sturm–Liouville theorem, the functions form a
complete orthogonal set, and a basis for symmetric functions of defined in .
Functions form also a complete and orthogonal set for antisymmetric
functions on . The two sets are obviously mutually orthogonal. Moreover, we have
with the normalization constant
and in the same way
with
At this point, all constants have been determined, except and . This is done depending on the
initial conditions on the temperature. The total temperature field being
Thus, the initial temperature is
and if we require heating from room temperature, i.e., , considering the orthogonality of the
functions , we are led to the equation
giving
which completes the determination. Finally, the temperature field is
It is now possible to specialize the result to the two cases of coating and bulk absorption.

#### 5.1.1 Transient temperature from coating absorption

In the case of coating absorption, we know from Equation (3.18) specialized to axisymmetry that

(). After some elementary algebra, we have
so that the final result for the transient temperature field is
The transient thermal lens is
Figures 36, 37, and 38 show the time evolution of the thermal lens. The time constant for temperature
evolution is very long, several hours, but the time constant of the focal length of the thermal lens is much
shorter because the final profile of the lens is reached within about 1/2 h, after which the temperature
keeps growing uniformly without changing the gradients.
#### 5.1.2 Transient temperature from bulk absorption

In the case of internal absorption, we have

We have, obviously, and, after some algebra, we find
so that the temperature field is
and the transient thermal lens is
The time evolution curves are practically identical to those shown in Figures 36, 37, and 38. See, for
example, Figure 43; in the same way, the final lens profile is reached long before the temperature reaches
the steady state.