Tailored diffraction asymmetries from spatially odd-symmetric phase gratings

Phase gratings, each period of which consists of even numbers of equal-width and equal-thickness elements, can be devised to attain asymmetric diffraction patterns. We show that engineering of offset refractive indices in different elements exhibiting a spatially odd symmetry leads to the elimination of a single diffraction order (directional elimination), while further manipulations on offset refractive indices may lead to the elimination of all odd or even orders (grouped elimination) or all orders but one or two selected (directional selection). These intriguing effects arise from destructive interference between diffracted amplitudes contributed by paired or successive elements, and violation of Friedel's law in such transparent gratings is an effect of higher order multiple scattering.