Discrimination and estimation of incoherent sources under misalignment

Spatially resolving two incoherent point sources whose separation is well below the diffraction limit dictated by classical optics has recently been shown possible using techniques that decompose the incoming radiation into orthogonal transverse modes. Such a demultiplexing procedure, however, must be perfectly calibrated to the transverse profile of the incoming light as any misalignment of the modes effectively restores the diffraction limit for small source separations. We study by how much can one mitigate such an effect at the level of measurement which, after being imperfectly demultiplexed due to inevitable misalignment, may still be partially corrected by linearly transforming the relevant dominating transverse modes. We consider two complementary tasks: the estimation of the separation between the two sources and the discrimination between one and two incoherent point sources. We show that, although one cannot fully restore super-resolving powers even when the value of the misalignment is perfectly known, its negative impact on the ultimate sensitivity can be significantly reduced. In the case of estimation we analytically determine the exact relation between the minimal resolvable separation as a function of misalignment whereas for discrimination we analytically determine the relation between misalignment and the probability of error, as well as numerically determine how the latter scales in the limit of long interrogation times.

Automated summary

This study examines the impact of misalignment errors on super-resolution imaging through demultiplexing processes. Although full restoration of super-resolution capabilities is not possible, the negative impact of misalignment on sensitivity can be significantly reduced by partially correcting the dominant transverse modes. In estimation and discrimination tasks involving two incoherent point sources, the study reveals the exact relations between misalignment and the minimal resolvable separation, as well as the probability of error, especially in the case of long interrogation times.