Characteristic functions of Hartmann-Shack wavefront sensors and laser-ray-tracing aberrometers

It is shown that the aberration estimated at any point of the pupil using wavefront slope aberrometers such as Hartmann-Shack wavefront sensors or laser ray tracers is a spatial average of the actual aberration weighted by a characteristic function that depends on the aberrometer design and on the estimation procedure. This characteristic function, whose explicit form is given here for wavefront slope aberrometers using either modal or zonal estimators, may be useful in analyzing some basic aspects of the aberrometer performance. It is also instrumental in establishing the links between the statistical properties of the actual and the estimated aberrations. Explicit formulas are given to show in terms of this function how the bias arises in the first- and second-order statistics of the retrieved aberrations. This approach is mathematically equivalent to the analysis of the effects of modal coupling (cross-coupling and aliasing). It may provide, however, some complementary insight.