Under specific circumstances the fractional Talbot effect can be described by simplified equations. We have obtained simplified analytic phase-factor equations to describe the relation between the pure-phase factors and their fractional Talbot distances behind a binary amplitude grating with an opening ratio of (1/M). We explain how these simple equations are obtained from the regularly rearranged neighboring phase differences. We point out that any intensity distribution with an irreducible opening ratio (M N/M) (M N < M, where M N and M are positive integers) generated by such an amplitude grating can be described by similar phase-factor equations. It is interesting to note that an amplitude grating with additional arbitrary phase modulation can also generate pure-phase distributions at the fractional Talbot distance. We have applied these analytic phase-factor equations to neighboring (0, π) phase-modulated amplitude gratings and have analytically derived a new set of simple phase-factor equations for Talbot array illumination in this case. Experimental verification of our theoretical results is given.
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