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Convolution expression for partially coherent beams that are partially blocked by an opaque obstacle is derived in detail in a paraxial $ABCD$ optical system. The examples are showed to how the formula can be applied to study the self-healing for different beams and different obstructions. In particular, this approach can be used to conveniently deal with any partially coherent Schell-model beams partially blocked by any shape obstacle. Furthermore, the self-healing ability characterization based on the similarity degree of intensity to intensity is proposed. Our method not only can be used to estimate the self-healing ability of beams in theory, but also can be applied to characterize the self-healing ability of beams in experiment. It offers a good and convenient method for investigating or comparing the self-healing ability of different beams, including fully coherent beams and partially coherent beams.
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Wu et al. (2017) studied this question.
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