New theoretical and experimental results in fresnel optics with applications to matter-wave and X-ray interferometry

Abstract

We present new methods and formulae for calculating the image amplitude and image spatial power spectral density produced by monochromatic point-source illumination of a finite (and/or infinite) periodic complex transmission grating. At specific finite-width resonances the image amplitude is seen to display periodic complex amplitude self-imaging of the grating, with interlaced alias images. The finite width grating resonances (as a function of spatial frequency) are broadened (from zero width) and displaced in frequency relative to those produced by an infinite grating, although the finite resonance width relative to illumination wavelength variation persists with infinite gratings. In the Fresnel domain the self images are generalizations of the Talbot and von Lau effects, while in the Fraunhofer to Fresnel transition domain, our formulae demonstrate the formation of these structures from Fraunhofer diffraction order side-lobes. Using these results, design criteria are provided for constructing lens-free three-grating interferometers with spatially diffuse illumination and detection. Such interferometers have a wide variety of applications for both X-rays and matter-waves, including a phase sensitive imaging device and/or narrow-band interference filter. For wavelengths in the Angstrom to sub-Angstrom range they feature high throughput and ease of fabrication. Experimental results using light with such an interferometer are presented. Our results conclusively demonstrate interference and image aliasing in such a device with spatially diffuse illumination and detection. The experiment is readily reproducible in any undergraduate physics laboratory.