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Adaptive optics (AO) systems deliver high-resolution images that may be idealfor precisely measuring positions of stars (i.e. astrometry) if the system hasstable and well-calibrated geometric optical distortions. A calibration unit,equipped with back-illuminated pinhole mask, can be utilized to measureinstrumental optical distortions. AO systems on the largest ground-basedtelescopes, such as the W. M. Keck Observatory and the Thirty Meter Telescoperequire pinhole positions known to 20 nm to achieve an astrometric precision of0.001 of a resolution element. We characterize a photo-lithographic pinholemask and explore the systematic errors that result from different experimentalsetups. We characterized the nonlinear geometric distortion of a simple imagingsystem using the mask; and we measured 857 nm RMS of optical distortion with afinal residual of 39 nm (equivalent to 20 {\mu}as for TMT). We use a sixthorder bivariate Legendre polynomial to model the optical distortion and allowthe reference positions of the individual pinholes to vary. The nonlineardeviations in the pinhole pattern with respect to the manufacturing design of asquare pattern are 47.2 nm +/- 4.5 nm (random) +/- 10.8 nm (systematic) over anarea of 1788 mm$^2$. These deviations reflect the additional error induced whenassuming the pinhole mask is manufactured perfectly square. We also find thatordered mask distortions are significantly more difficult to characterize thanrandom mask distortions as the ordered distortions can alias into opticalcamera distortion. Future design simulations for astrometric calibration unitsshould include ordered mask distortions. We conclude that photo-lithographicpinhole masks are >10 times better than the pinhole masks deployed in firstgeneration AO systems and are sufficient to meet the distortion calibrationrequirements for the upcoming thirty meter class telescopes.

Geometric Distortion Calibration with Photo-lithographic Pinhole Masks for High-Precision Astrometry