On the Theory of the Reconstruction of a Weak Phase-amplitude Object from Its Images, Especially in Electron Microscopy: I. Isoplanatic imaging

Abstract

In this paper we discuss the reconstruction of a weak phase-amplitude object from its intensity image. This problem occurs in electron microscopy where the weak object approximation holds for not too low accelerating voltages (E ≳ 100 keV). Isoplanatic imaging is assumed. The illumination is supposed to be quasi-monochromatic and coherent. This reconstruction problem has often been treated in the literature [11, 12]. The assumption which has tacitly been made in these articles is that the background wavefunction is real. This is not true in general. E.g., when tilted illumination is used and/or non-isoplanatic aberrations are present the background wavefunction is complex. This makes a reconsideration of the traditional approaches necessary. The quantities to be reconstructed are the phase shift and amplitude attenuation due to the object. The reconstruction proceeds in two steps. The first step is essentially the determination of the complex amplitude in the exit pupil from the intensity distribution in the image plane. For a mixed phase-amplitude object we need two exposures in order to calculate this complex amplitude. For a pure phase or amplitude object one exposure is sufficient. The complex amplitude arises as the (unique) solution of two coupled Volterra integral equations. The traditional difficulties (Blaschke factors) do not occur. In the second step the phase shift and amplitude attenuation are determined from the complex amplitude in the plane of the exit pupil. Phase shift and amplitude attenuation can be determined with arbitrary accuracy if noise may be neglected.