Analysis of general double periodic structure diffraction phenomena based on the ambiguity function

We present a general configuration made up of two arbitrary periodic structures and a thin lens under partially coherent illumination. The separation of these elements along the optic axis can be adjusted to meet some special requirements (system conditions) to form a meaningful pattern on the observation screen. An elegant analytic expression is deduced to describe the intensity distribution based on the ambiguity function approach. Two extremes of illumination are considered. The Talbot self-imaging effect, the moiré effect of double-grating superposition, the classical Lau effect, the generalized Lau effect, incoherent grating–lens imaging, and misfocus diffraction convolution are discussed.