Integer, fractional and fractal Talbot effects

Abstract

Self-images of a grating with period a, illuminated by light of wavelength λ, are produced at distances z that are rational multiples p/q of the Talbot distance z _T = a ²/λ; 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 simplifies calculations of the Talbot images. In ‘transverse’ planes, perpendicular to the incident light, and with ζ = z/z _T irrational, the intensity in the Talbot images is a fractal whose graph has dimension . In ‘longitudinal’ planes, parallel to the incident light, and almost all oblique planes, the intensity is a fractal whose graph has dimension . In certain special diagonal planes, the fractal dimension is . Talbot images are sharp only in the paraxial approximation λ/a → O and when the number N of illuminated slits tends to infinity. 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 finite N. Experiments with a Ronchi grating confirm the existence of the longitudinal fractal, and the transverse Talbot fractal at the golden distance ζ = (3 − 5^1/2)/2, within the expected resolutions.