Hartmann vs reverse Hartmann test: a Fourier optics point of view

The Shack-Hartmann Wavefront Sensor (WFS) is well-known in the fields of optical metrology, wavefront sensing in astronomy, and ophthalmologic control applications. The purpose of this communication is to bring new insights on the historical Hartmann test and to compare it with the less known reverse Hartmann test, where the locations of the pupil mask and observed image planes are exchanged. Both tests can actually be interpreted by using the formalism of Fourier optics, i.e. Fraunhofer diffraction for the Shack-Hartmann and Fresnel diffraction in the reverse configuration. The principles of these models are firstly described in the communication. The results of numerical simulations are then presented, allowing comparing both optical arrangements from the Fourier optics point of view, in terms of achievable wavefront measurement accuracy. They show that a WFS based on the reverse Hartmann test may globally achieve the same performance as the classical Shack-Hartmann.