Integer, fractional and fractal Talbot effects

Self-images of a grating with period a, illuminated by light of wavelength ̧, are produced at distances z that are rational multiples p /q of the Talbot distance zT = a / ̧; each unit cell of a Talbot image consists of q superposed images of the grating. The phases of these individual images depend on the Gauss sums studied in number theory and are given explicitly in closed form; this simpliŽ es calculations of the Talbot images. In t̀ransverse’ planes, perpendicular to the incident light, and with z = z /zT irrational, the intensity in the Talbot images is a fractal whose graph has dimension 2. In l̀ongitudinal’ planes, parallel to the incident light, and almost all oblique planes, the intensity is a fractal whose graph has dimension 4. In certain special diagonal planes, the fractal dimension is 4. Talbot images are sharp only in the paraxial approximation ̧ /a ® 0 and when the numberN of illuminated slits tends to inŽ nity. The universal form of the post-paraxial smoothing of the edge of the slit images is determined. An exact calculation gives the spatially averaged non-paraxial blurring within Talbot planes and defocusing between Talbot planes. Similar calculations are given for the blurring and defocusing produced by Ž nite N. Experiments with a Ronchi grating conŽ rm the existence of the longitudinal fractal, and the transverse Talbot fractal at the golden distance z = (35/)/2, within the expected resolutions.