Undersampling errors in measuring the moments of images aberrated by turbulence

Theoretical analysis and computer simulation show that it is possible to measure accurately the second moment of image irradiance distributions with considerable undersampling. The measurement error is found by applying the moment theorem to the sampled distributions. Sampling with small elements is compared to sampling with extended elements, and it is shown that the latter is always more accurate. Error calculations for a Gaussian beam and for a square aperture are presented. For a Gaussian the sampling interval can be as large as 2.3sigma (sigma being the variance) with a 5% error. Simulation of atmospherically aberrated images shows that although the functional form of the irradiance distribution changes considerably, the measurement errors decrease due to the increase in the effective sampling rate, which is given by the ratio between the spot size and the sampling interval.