3.3 Thermal distortions in the steady state
We assume from now on that the temperature field is axisymmetric. Due to the temperature field, at the
same time, thermal expansion of the material causes a change of the shape of the mirror and a distortion of
the reflecting face. If we call
the displacement vector, i.e., the difference between the coordinates of
a given atom before and after heating, the classical relevant quantities are the strain tensor
and the stress tensor
. In the presence of a temperature field
, the two tensors are
related by the generalized Hooke law for isotropic media via the Lamé coefficients
:
where
is the stress temperature modulus and
the trace of the strain tensor.
is the Kronecker
tensor. In cylindrical coordinates
, assuming cylindrical symmetry, the displacement vector has
only two components,
and
. Then the strain tensor has four components:
The relation (3.87) is, in detail,
where, as above,
is the excess temperature field given by the generic FB expansion
The stress tensor must obey the homogeneous divergence equation in the absence of external applied forces
(static equilibrium), i.e., a special case of the Navier–Cauchy equations,
Moreover, the following boundary conditions must be satisfied:
It is more convenient, at the end of the calculations, to express the results in terms of the Poisson ratio
and Young’s modulus
, using the correspondence
where
is the linear thermal expansion coefficient.
3.3.1 Thermal expansion from thermalization on the coating
In the case of a heat source localized on the reflecting face, so that the temperature field
is known,
it is possible to satisfy Equation (3.91) and all but one of the boundary conditions in Equation (3.92) by a
displacement vector of the form
where
is a function to be determined. It can be checked that
. The equilibrium equations
then reduce to
We have
But let us recall that
is harmonic [see Equation (3.1)], so that
and consequently the first equilibrium equation (3.96) is identically satisfied
Now we have
Therefore, in order to satisfy the second equilibrium equation (3.96), we take
where
is an arbitrary constant. The stress component
is now explicitly known
making clear that
, and since it has been shown that
, we have, simply,
Two more boundary conditions are satisfied. At this point, the only remaining unsatisfied condition is the
vanishing of
on the edge. Indeed, we have
or explicitly
so that
It is easy to check numerically that the function
is almost linear for any type of beam.
Therefore, it can be almost cancelled by an opposite linear stress on the edge. Such a stress can be induced
by the following extra displacement vector:
where
and
are arbitrary constants. One can check that this vector firstly satisfies the equilibrium
equations (3.91), secondly has identically null stress components
and
, and finally produces a
radial edge stress:
The constants
and
can now be chosen in order to minimize the quadratic error:
After using the classical mean-squares formulas, this cancels the mean force and the mean torque on the
edge:
The complete displacement
now satisfies the Navier–Cauchy equations, all constraints on the
faces, and induces null mean force and torque on the edge. Owing to the principle of de Saint-Venant [38],
we can conclude that the displacement is correct almost everywhere in the bulk material, except possibly
near the edge. But any effective optical beam has a vanishing intensity near the edge in order to prevent
diffraction losses, so that the solution
is relevant for our purpose. If we use Young’s modulus
, the
Poisson ratio
and the linear thermal expansion coefficient
instead of
, we find
with
. The FB coefficients
are identical to the
computed in Section 3.1.3
and
With the same notation, we have, explicitly, the components of the displacement:
The displacement vector is defined up to a constant vector. We have chosen the constant in such a way that
the displacement is zero at
. We can see in Figure 23 the global deformation of the mirror in
the cases Ex1, Ex2 and Ex3.
For the deformation of the coating, we have
or, in detail,
with
(The displacement has now been taken to be zero at the center of the reflecting face), and
The geometrical effects of heating (see Figure 23) are mainly a thermally-induced aberration due to the
change of the reflecting surface by
, then a change of the optical path through the
substrate by a quantity
(
being the nominal refractive index), which can be directly included in the thermal lens expression
[Equation (3.40)], which has the same dependence on temperature. Note that the Saint-Venant
correction contributes a constant (independent on
), so that we can ignore it in a lensing study.
Thus, the global thermal lens is identical to the result found in Section 3, up to the correction
Estimations of the weighted curvature of the distorted surface are obtained with the same technique as in
Section 3. For Laguerre–Gauss modes, we obtain
For a flat mode, this is
where the coefficients
have been defined by Equations (3.70) and (3.75).
We can see in Figure 24 the distorted reflecting face of the mirror in our three examples. The results in
terms of curvature radius are
- Ex1 (
, w = 2 cm):
= 5,842 mW
- Ex2 (Flat, b = 9.1 cm):
= 165,485 mW
- Ex3 (
, w = 3.5 cm):
= 477,565 mW .
We see in the case of axisymmetry that the use of unconventional modes (either flat or high-order LG)
allows one to dramatically reduce spurious thermal effects in mirrors to be installed in advanced GW
detectors, where high light-power flows are planned. Up to two orders of magnitude can be gained with
respect to the present Virgo configuration for thermal lensing, or for thermal deformation of the coating. As
in the thermal lens Section 3, we give some results (see Table 4) for LG modes having the same (1 ppm)
clipping losses.
Table 4: Curvature radii from thermal expansion due to coating absorption for modes having 1 ppm
clipping losses
order  |
w [cm] |
of th. aberr. [km W] |
(0,0) |
6.65 |
77 |
(0,1) |
5.56 |
149 |
(1,0) |
6.06 |
141 |
(0,2) |
4.93 |
210 |
(1,1) |
5.23 |
201 |
(2,0) |
5.65 |
197 |
(0,3) |
4.49 |
258 |
(1,2) |
4.70 |
250 |
(2,1) |
4.97 |
250 |
(3,0) |
5.35 |
250 |
(0,4) |
4.17 |
299 |
(1,3) |
4.32 |
290 |
(2,2) |
4.52 |
292 |
(3,1) |
4.76 |
298 |
(4,0) |
5.11 |
300 |
(0,5) |
3.91 |
332 |
(1,4) |
4.03 |
324 |
(2,3) |
4.18 |
326 |
(3,2) |
4.36 |
332 |
(4,1) |
4.58 |
342 |
(5,0) |
4.91 |
348 |
|
3.3.2 Thermal expansion from internal absorption
When the linear absorption of light through the bulk material results in an internal heat source, the
temperature field is no longer harmonic, and we are bound to solve explicitly the thermo-elastic
equations (3.91) and (3.92). As seen earlier, the case of internal absorption leads to a symmetric
temperature field. However, we shall derive the general thermoelastic solution, which will also prove useful
in Section 4 below.
The temperature field is assumed to be of the generic form (
)
We consider a displacement vector of the form
The equilibrium equations (3.91) reduce to a system of ordinary differential equations,
so that, by a basic combination of these two, we get
a solution of which is
where
and
are arbitrary constants. By substituting in Equation (3.129) we get
A solution of which is
where
and
are two arbitrary constants and
is a special solution of
We can now find
from Equation (3.132)
The boundary conditions,
lead to the system (we return to the Poisson ratio and to the linear thermal expansion coefficient)
with
. It is easy to combine these equations to find
with the notation
We have also used the symmetrized coefficients (even and odd parts)
The radial stress is
with
It can be checked that
gives a null contribution to the mean force and to the mean torque on the
edge, i.e.,
Therefore these two mean moments can be computed with
. We have
In the special case of bulk absorption, we have, due to symmetry,
and
The explicit expression for functions
and
is, finally,
Two boundary conditions have been forgotten. We need vanishing
and
on the edge
.
However, a numerical investigation shows that
is practically constant, having an average value of
In the same spirit as in the preceding case (Saint-Venant correction) we can add an extra displacement
which induces null
and
extra stresses, trivially satisfies the equilibrium equations, and
produces a constant
. Now the
stress component is antisymmetric with respect to
so that it is zero at
with a vanishing average value on the edge
. Moreover, it
can be checked that the values taken on the edge are weak compared to other places and other components.
Therefore, the sum
satisfies exactly the equilibrium equations, exactly the boundary conditions on the faces, and on average on
the edge. The displacement vector at
represents the deformation of the reflecting face. We have
One can see in Figure 25 the distorted shape of the mirror in three situations. The thermally-induced
curvature radius can be computed as usual. (See Figure 26 for the profiles of the reflecting surface in three
situations and the best fitted paraboloid.) For our three examples, we obtain the following
figures
- Ex1 (
, w = 2 cm):
= 22 km W
- Ex2 (flat, b = 9.1 cm):
= 325 km W (mesa: 361 km W)
- Ex3 (
, w = 3.5 cm):
= 937 km W
See Table 5 for several other LG modes.
Table 5: Curvature radii from thermal expansion due to bulk absorption for modes having 1 ppm
clipping losses
order  |
[cm] |
of th. aberr. [km W] |
(0,0) |
6.65 |
165 |
(0,1) |
5.56 |
318 |
(1,0) |
6.06 |
290 |
(0,2) |
4.93 |
440 |
(1,1) |
5.23 |
410 |
(2,0) |
5.65 |
404 |
(0,3) |
4.49 |
533 |
(1,2) |
4.70 |
507 |
(2,1) |
4.97 |
504 |
(3,0) |
5.35 |
512 |
(0,4) |
4.17 |
613 |
(1,3) |
4.32 |
586 |
(2,2) |
4.52 |
585 |
(3,1) |
4.76 |
597 |
(4,0) |
5.11 |
615 |
(0,5) |
3.91 |
677 |
(1,4) |
4.03 |
654 |
(2,3) |
4.18 |
650 |
(3,2) |
4.36 |
661 |
(4,1) |
4.58 |
684 |
(5,0) |
4.91 |
713 |
|
In Table 6 we give numerical results for our three examples.