Integer, fractional and fractal Talbot effects

TLDR

Self-images of a grating with period a illuminated by light of wavelength λ appear at distances that are rational multiples of the Talbot distance z_T = a²/λ, with each unit cell comprising q superposed images, and these images are sharp only under the paraxial approximation λ/a → 0 and for an infinite number of illuminated slits. An exact calculation gives the spatially averaged non‑paraxial blurring within Talbot planes and defocusing between Talbot planes. The study shows that the phases of the Talbot images are given by closed‑form Gauss sums, enabling simplified calculations, and that the intensity in transverse, longitudinal, and certain diagonal planes exhibits fractal behavior with specific dimensions, while post‑paraxial smoothing and finite‑slit effects are universally described, and experimental results with a Ronchi grating confirm the predicted longitudinal and transverse fractal structures at the golden distance.

Abstract

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 − 51/2)/2, within the expected resolutions.