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Application of differential algebraic method to the aberration analysis of curvilinear-axis electron optical systems

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The differential algebraic (DA) method is introduced into the aberration analysis of curvilinear-axis electron optic systems. In the rotational local orthogonal (RLO) coordinate system established around the optical axis, the general trajectory equations for electrons are derived, and a further form of equations in which x″ and y″ are decoupled are given. The expansion of fields in the RLO coordinate system is studied. By Mathematica®, the scalar potential in the RLO coordinate system can be expanded to arbitrary order, in which the independent coefficients (i.e. field distribution functions of the curvilinear-axis system) are determined by the field distribution near the curvilinear axis, and the dependent coefficients are derived according to the Laplace equation. The aberration analysis by the DA method in a curvilinear-axis system can be done by ray tracing in n D v, and the results are DA quantities in which aberrations up to n-th order, along with the Gaussian properties, are included simultaneously with very high accuracy. As an example, the performance of a 15″ CDT (Colour Display Tube) is calculated using the DA method. A curved trajectory under the deflection fields is chosen as the optical axis and the DA method is used to calculate aberrations up to 4th order. As examples, the spot defocusing and beam mis-convergence of such a system are calculated and their patterns on a spherical screen are shown. It is concluded that the DA method is an effective and smart numerical method for the analysis and design of curvilinear-axis electron optical systems.

Keywords: Differential algebra; automatic differentiation; electron optics

Document Type: Original Article


Affiliations: 1: Microelectronics Research Centre, Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK 2: School of Electronic & Information Engineering, Xi'an Jiaotong University, Xi'an, 710049, P. R. China

Publication date: July 1, 2002


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