INTEGER, FRACTIONAL AND FRACTAL TALBOT EFFECTS

Self-images of a grating with period a, illuminated by light of wavele ngth lambda, are produced at distances z that are rational multiples p /q of the Talbot distance z(T)=a(2)/lambda; each unit cell of a Talbot image consists of q superposed images of the grating. The phases of t hese individual images depend on the Gauss sums studied in number theo ry and are given explicitly in closed form; this simplifies calculatio ns of the Talbot images. In 'transverse' planes, perpendicular to the incident light, and with zeta=z/z(T) irrational, the intensity in the Talbot images is a fractal whose graph has dimension 3/2. In 'longitud inal' planes, parallel to the incident light, and almost all oblique p lanes, the intensity is a fractal whose graph has dimension 7/4 In Cer tain special diagonal planes, the fractal dimension is 5/4. Talbot ima ges are sharp only in the paraxial approximation lambda/a-->0 and when the number N of illuminated slits tends to infinity. The universal fo rm of the post-paraxial smoothing of the edge of the slit images is de termined. An exact calculation gives the spatially averaged non-paraxi al blurring within Talbot planes and defocusing between Talbot planes. Similar calculations are given for the blurring and defocusing produc ed by finite N. Experiments with a Ronchi grating confirm the existenc e of the longitudinal fractal, and the transverse Talbot fractal at th e golden distance zeta=(3-5(1/2))/2, within the expected resolutions.