Reciprocal vector theory for diffractive self-imaging

A reciprocal vector theory for analysis of the Talbot effect of periodic objects is proposed. Using this method we deduce a general condition for determining the Talbot distance. Talbot distances of some typical arrays (a rectangular array, a centered-square array, and a hexagonal array) are derived from this condition. Further, the fractional Talbot effect of a one-dimensional grating, a square array, a centered-square array, and a hexagonal array is analyzed and some simple analytical expressions for calculation of the complex amplitude distribution at any fractional Talbot plane are deduced. Based on these formulas, we design some Talbot array illuminators with a high compression ratio. Finally, some computer-simulated results consistent with the theoretical analysis are given.