Based on the scattering theory and the Green function method, a dynamical theory is given for calculating the diffraction of deeply-etched gratings with a stratified structure substrate. The key of our method is that the patterned grating structure is considered as a perturbation to the unpatterned stratified structure rather than to vacuum. Using the first-order Born approximation and in the Fresnel diffraction region, we obtain a simple analytical expression, which can be used to calculating the scattering intensity of deeply-etched circular binary Fresnel zone plates with a stratified substrate (MDECBFZPs). The numerical results show that the focusing intensity at the foci of the MDCBFZP changes periodically with the etching depth and the thickness of the substrate film. Our results are in good agreement with FDTD simulations.
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