### 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)
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.

Table 6: Thermal aberrations from coating and bulk absorption
 results (coating abs.) w = 2 cm flat b = 9.1 cm w = 3.5 cm Curvature radius 5.8 km W 167 km W 478 km W Coupling losses 4.3 ×10–2/W2 4.4 ×10–3/W2 6.7 ×10–3/W2 results (bulk abs.) w = 2 cm flat b = 9.1 cm w = 3.5 cm Curvature radius 22 km W 327 km W 937 km W Coupling losses 3.0 ×10–3/W2 2.2 ×10–3/W2 1.8 ×10–3/W2