A detailed comparison of the original Gerchberg-Saxton and the Yang-Gu algorithms for the reconstruction of model images from two intensity measurements in a nonunitary transform system is presented. The Yang-Gu algorithm is a generalization of the Gerchberg-Saxton algorithm and is effective in solving the general amplitude-phase-retrieval problem in any linear unitary or nonunitary transform system. For a unitary transform system the Yang-Gu algorithm is identical to the Gerchberg-Saxton algorithm. The reconstruction of images from data corrupted with random noise is also investigated. The simulation results show that the Yang-Gu algorithm is relatively insensitive to the presence of noise in data. In all cases studied the Yang-Gu algorithm always resulted in a highly accurate recovered phase.

Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison

Recently the combination of the Gerchberg–Saxton (GS) algorithm and a fractional Fourier transform was proposed to implement beam shaping in the fractional Fourier domain [ Zalevsky , Opt. Lett.21, 842 (1996)]. We generalize this idea to deal with the problem of phase retrieval from two intensity measurements in a fractional Fourier transform system. The relevant equations for determining the unknown phases are derived, based on the general theory of amplitude–phase retrieval in an optical system. The unitarity condition of the fractional Fourier transform in a practical optical system with finite aperture is discussed. For different fractional orders P, the phase retrieval of several typical model images is studied in detail. A comparison of the GS and our algorithms is given, based on numerical simulations. It follows that our algorithm can offer the desired phase in all cases considered. However, the GS algorithm may fail when the transform system is nonunitary.

Numerical investigation of phase retrieval in a fractional Fourier transform

Publisher Summary This chapter emphasizes several aspects of the theory developed that may be used for solving the modern problems of physical electronics that are not associated directly with axially symmetric electron mirrors. One of these aspects is a special choice of the basic set of particular solutions of the near-axial trajectory equation. This choice is based on the character of the equation singularity, and not on the initial conditions of particle motion. It allows one to consider in the same way the qualitatively different systems-electron mirrors, mirror microscopes, and emission systems. The concept of arbitrary and guiding particles is the other aspect. This idea may be extended, without serious difficulty, into arbitrary static electron-optical systems with curved axes and relativistic particles. Thus, the perspective for development of a general theory of charged-particle nonstationary flux focusing and flux control by virtue of static electromagnetic fields is open. This problem refers to a poorly developed field of physical electronics, which is located at the junction of static and dynamical electron optics. At the same time, in the techniques and practice of physical experiments the use of nonstationary fluxes is widened, in particular, owing to the implementation of lasers into electronics.

Theory of Electron Mirrors and Cathode Lenses

Publisher Summary This chapter discusses the theory of classic mechanics and describes the electron trajectories and aberrations in the generalized position and momentum representations and is derived by the Hamiltonian function and the canonical equations. This Hamiltonian approach may be regarded as canonical aberration theory and has several advantages. The chapter also discusses canonical aberration theory and canonical electron beam optics. A brief account of conventional aberration theory, using Lagrangian representation, in a rotationally symmetric electromagnetic system is also provided. Canonical aberration theory is discussed, using Hamiltonian representation, also in a rotationally symmetric electromagnetic system. Higher order aberration in a round lens is derived followed by higher order aberration in electromagnetic multipoles. The chapter also discusses canonical electron beam optics which is divided into three parts: (1) electron beam optics in round lenses; (2) electron beam optics in a combined system consisting of round, quadrupole and octopole lenses; and (3) electron beam optical computations in round electrostatic lenses.

Canonical theory in electron optics

Publisher Summary This chapter discusses the mathematical and software support for the computer-aided design (CAD) system of emission-imaging electron-optical systems (EOS). Computer-aided design is based on mathematical simulation that combines algorithmic, programming, and hardware tools to provide a deeper insight into the objective properties and laws of the objects or processes under study. The chapter gives an example of solving problems in order to investigate and to design emission EOS that is one of the specific tasks of mathematical and technical physics. The chapter outlines the ways of building the aberration model of cathode lenses and, on its basis, determines some practically important functionals of emission systems. In addition, the chapter discusses variational analysis of parametric optimization and synthesis problems. A general formulation of cathode-lens optimization problems is provided and the numerical aspects of the main stages for computing and optimization of emission systems are elaborated. General principles of mathematical and software support for electron-optical system computer-aided design and the approaches to applied program packages (APP) development are discussed. The technique of computer-aided generation of EFIR APP algorithms reported the algorithms featuring a natural input language and advanced service provide effective calculations and the optimization of emission systems. The analyzed results of numerical experiments carried out to solve model and practical problems are presented.

Emission-Imaging Electron-Optical System Design

In the present paper, the relativistic first order chromatic and third order geometrical aberration formulas, which are different from those in ref. [1], have been derived for a combined focusing-deflection system with superimposed field, and all aberration coefficients have been given in explicit form. With the help of these aberration formulas, the deflection aberrations of a combined system with multi-stage deflectors have been discussed in detail.

On the relativistic aberration theory of a combined focusing-deflection system with multi-stage deflectors

Variational theory of aberrations in cathode lenses

Publisher Summary This chapter discusses the aberrations of cathode lenses with combined electrostatic and magnetic fields and provides the general formulae for the aberration coefficients of cathode lenses at an arbitrary image plane. It shows that these aberration coefficients can be transformed into a linear combination of Glaser's well-known geometrical and chromatic aberration coefficient. It is considered that the aberrations in both wide electron beam system (cathode lens) and narrow electron beam system (conventional lens) can be treated by a universal variational theory. The chapter explains the relationship between these two systems. The chapter also explains the argument that Glaser's aberration theory can be generalized for both narrow and wide electron beam systems. As an extension, the more general condition, where the cathode is immersed in both magnetic and transversal electric fields, has been considered in the chapter.

Ximen Ji-Ye

Papers

On the relativistic aberration theory of a combined focusing-deflection system with multi-stage deflectors

Variational theory of aberrations in cathode lenses

Variational Theory of Aberrations in Cathode Lenses

Aberration theory for combined electromagnetic focusing-deflection systems with curvilinear axes

On the aberration theory for wide and narrow electron beams in a combined electromagnetic focusing system possessing a spherical cathode