Image reconstruction from electron and X-ray diffraction patterns using iterative algorithms: experiment and simulation

The hybrid input–output iterative algorithm, which solves the phase problem for scattering from non-periodic objects, is reviewed for application to X-ray and electron diffraction data. Desirable convex constraints, including the sign of the scattering potential for electrons, and compact support, are discussed. The cases of complex and real exit-face wavefunctions, strong and weak phase objects, various supports, and the use of coherent focussed radiation are reviewed. Reconstruction of general complex objects requires accurate knowledge of the support, which should consist of two holes or a triangle in an opaque mask. The support boundaries should be as sharp as possible. Strong phase objects without absorption can be recovered if the support consists of one hole, is accurately known and has sufficiently sharp boundaries. Real and weak phase objects with absorption can be recovered without accurate knowledge of the support area if the support boundaries are sufficiently sharp and the support consists of one or more holes. A sign constraint on the scattering potential is used to recover weak phase objects. The experimental realization of theoretically desirable support conditions is discussed. A two-stage method of finding the support for complex objects is proposed. Experimental results from applying the Gerchberg–Saxton–Fienup HiO-algorithm to coherent electron diffraction patterns are presented, using specially made e-beam lithographed support structures. Images with a resolution of about 5 nm are thus recovered from the intensities alone in coherent electron diffraction patterns from non-periodic objects. Limitations of the present experiments are identified and suggestions made for development of both X-ray and electron work.