Correcting flat mirrors with surface stress: Analytical stress fields

Thin mirrors, important for next-generation space telescopes, are difficult to accurately fabricate. One approach is to fabricate a mirror using traditional methods, then to bend the mirror using surface stress to correct residual height errors. We present two surface stress fields that correct any height error field in thin flat plates. For round plates, we represent these as linear combinations of Zernike polynomials. We show that equibiaxial stress, a common and easy-to-generate state of stress, cannot generally be used to make exact corrections. All three components of the surface stress are needed for exact corrections. We describe a process to design an equibiaxial stress field to make approximate corrections in round plates. Finally, we apply the three stress fields to simulate flattening of a measured glass wafer with 3.64 μm root-mean-squared (RMS) height error. Using our chosen equibiaxial stress field, the residual error is 0.34 μm RMS. In comparison, using all three stress components, the correction is exact and the required RMS stress is about 2.5× smaller than when using equibiaxial stress only. We compare the deformation with a finite element model and find agreement within 10 nm RMS in all three cases.