Modeling thermoelastic distortion of optics using elastodynamic reciprocity

Thermoelastic distortion resulting from optical absorption by transmissive and reflective optics can cause unacceptable changes in optical systems that employ high-power beams. In advanced-generation laser-interferometric gravitational wave detectors, for example, optical absorption is expected to result in wavefront distortions that would compromise the sensitivity of the detector, thus necessitating the use of adaptive thermal compensation. Unfortunately, these systems have long thermal time constants, and so predictive feed-forward control systems could be required, but the finite-element analysis is computationally expensive. We describe here the use of the Betti-Maxwell elastodynamic reciprocity theorem to calculate the response of linear elastic bodies (optics) to heating that has arbitrary spatial distribution. We demonstrate, using a simple example, that it can yield accurate results in computational times that are significantly less than those required for finite-element analyses.

Figure 1

A radial cross section of the cylindrical optic, showing the mesh used for the FEM. The mesh consisted of 32000 nodes, and is finest on the heated top surface of the test mass.

Figure 2

Comparison of the surface distortion calculated using the elastodynamic FEM, $u_{\mathrm{FEM}}$, the sum of the first six Zernike components $u_{\mathrm{Z}}$, and the sum of the first six orthonormalized LG components $u_{\mathrm{LG}}$.

Figure 3

Comparison of $u_{\mathrm{FEM}}$ and the sum of the 17 dominant orthonormalized-HG components $u^T$ for a heat flux offset of 10 mm.

Figure 4

(a) A plot of $u_{\mathrm{FEM}}$ and $u^T$ calculated for the on-axis heating using the six lowest-order orthonormalized-LG functions. (b) A plot of $u_{\mathrm{FEM}}-u^T$.

Figure 5

(a) A plot of $u_{\mathrm{FEM}}$ and $u^T$ calculated for the off-axis heating using the 17 dominant orthonormalized-HG functions. (b) A plot of $u_{\mathrm{FEM}}-u^T$.

Figure 6

Plot of the mean-weighted-squared difference between $u_{\mathrm{FEM}}$ and the sum of the first six orthonormalized-LG components as a function of $r_0$.

Figure 7

Plot of the mean-weighted-squared difference between $u_{\mathrm{FEM}}$ and the sum of the first six orthonormalized-HG components as a function of $r_0$.

Figure 8

The 17 dominant orthonormalized-HG functions.