Considerations on an optical test of Popper's experiment

We present a detailed analysis of a previously published realization of Popper's experiment using entangled ghost imaging. Our analysis, which is based on optical diffraction integrals, shows that, for the configuration previously described, the transverse spread of an unmeasured particle (the signal photon here) does not increase in inverse proportion to the width of its virtual confinement when its entangled twin (the idler photon) is confined in a transverse dimension by a physical slit. Rather, we show that the spread of the unmeasured particle carries no dependence on the width of its virtual confinement in the published configuration. Instead, it spreads geometrically at a rate determined by the numerical aperture of the ghost imaging system. We further propose an alternative configuration for which the spread of the unmeasured particle does increase in inverse proportion to the width of its virtual confinement. We discuss the relation of these results to the predictions of Popper.