Advantages of Multi-photon Detection Revealed by Fisher Information In Resolving Incoherent Sources

With the progress of optical detection technology, the classical diffraction limit raised a hundred years ago has been continuously broken through. In previous experiments within fluorescence sources, one of the techniques used is detecting auto-correlation functions. The image is reconstructed by single-photon and multi-photon detecting intensities. By taking the joint intensity of more than one single-photon detectors into consideration, the point spread function of each photon emitter can be resolved even when the distance of their central position is below the classical diffraction limit. However, the measurement precision is not considered. In actual detecting process the detectors are imperfect, they have quantum detecting efficiency $\eta$ and even can not count the photon numbers during one absorbing process. Therefore, we analyze the detecting intensity of each detector separately and use the total Fisher information to depict the resolution. Higher Fisher information is obtained when second order correlation function is considered and higher detecting efficiency is beneficial to the Fisher information enhancement, meanwhile the Cram\`er-Rao bound for an effective distance resolving is lower than the classical diffraction limits. Furthermore, when the emitted photons are coherent states, the resolution ability can be enhanced with infinite single-photon detectors.