Number Phase Relation for Elastic Scattering in Optics

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Abstract
An elastically scattering optical object impresses a phase shift on the incident photon or electron beam. From this phase shift the scattering probability is derived and is used to determine the minimum number of incident particles necessary to detect the object. Then the accuracy is derived with which a phase shift can be measured at best from a given number of recorded particles. The results can be interpreted as a direct consequence of the number-phase uncertainty relation. In practice, the detection of elastically scattering objects always requires very much longer illumination times than expected from geometric optics, if the phase shifts are smaller than π.