Wave packet in the phase problem in optics and ptychography

At present, ptychography seems to be the most natural and efficient method for approaching the diffraction-limited optical resolution. The general setup of a ptychoscope does not contain refracting or focusing elements and includes a coherent illumination source, a translation stage for displacement of a macroscopic object, and a detector for recording transmitted or reflected radiation from the object, which is connected to a computer for processing diffractograms. In classical optics, the main problem with achieving high spatial resolution is the correction and elimination of aberrations in optical systems, whereas the spatial resolution in ptychography mainly depends on the reliability of recording and computer processing diffractograms with large numerical apertures. After a brief introduction to the history and current state of ptychography, the wave-packet method for calculating the wave field on a detector in the far field and for a large numerical aperture is considered in detail. This gives a relation between fields on the object and on the detector, which underlies the ePIE (extended Ptychography Iterative Engine) algorithms for recovering images used in practice. The realization of algorithms involves operations with functions defined in certain domains (coordinate networks) of the direct space and Fourier space related to the object and detector. The size of and steps involved in such networks are strictly related to the object size, its distance from the detector, and the numerical aperture. The programs developed in this paper are used to refine the limits of applicability of the paraxial approximation (Fresnel integrals) in calculations of the field on the detector. Simulations of images obtained by the ptychography method are presented.