Aliasing in a Hartmann wavefront sensor at x-ray wavelengths

The Hartmann Sensor is a simple and well-established method to interrogate wavefront quality. Recently the Hartmann sensor has been used at very short wavelengths, including the extreme UV. Here we consider the Hartmann sensor and its ability to measure the wavefront of an x-ray beam. We use both analytic methods and a wave-optics, Fresnel-diffraction simulation. The Hartmann sensor samples the wavefront, which means that it is susceptible to aliasing (the non-linear phenomenon where high-spatial frequency components are incorrectly measured as low-spatial frequency components). Our analysis shows that aliasing is more severe in the Hartmann sensor than in the corresponding (optical) Shack-Hartmann. Aliasing worsens as Hartmann hole size shrinks. The wave-optics simulations show that for reasonable optics-polishing errors and Hartmann mask design, aliasing errors can be of the same magnitude as the phase that is to be measured.