Phase optimization algorithm for 3D particle localization with large axial depth

We propose an optimization algorithm based on Fresnel approximation (FA) imaging to optimize an extended-axial-depth point spread function (PSF) for 3D particle localization. The transfer function efficiency of the PSF is improved by repeatedly imposing constraints in the object plane, the spatial domain, and the Fourier domain. During the iterative calculation, the effective photon number or Cramer-Rao lower bound is used as the termination condition of the iteration. The algorithm allows flexible adjustment of the peak intensity ratio of the two main lobes. Moreover, the transfer function efficiency can be balanced by increasing the weight of the modulation function of the expected PSF at each axial position. The twin-Airy (TA) PSF optimized by the FA optimization algorithm does not require complex post-processing, whereas post-processing is an essential step for the unoptimized TA-PSF. The optimization algorithm is significant for extended-axial-depth PSFs used for 3D particle localization, as it improves localization precision and temporal resolution.