High order aberration calculations of a quadrupole–octupole corrector using a differential algebra method
Introduction
The resolution limiting spherical and chromatic aberrations of static rotationally symmetric electron lenses are unavoidable in the absence of space charge and flight reversal [1]. To improve the resolution of electron microscopes, quadrupole–octupole correctors are necessary to compensate for both the chromatic and the spherical aberrations, while a hexapole corrector suffices to eliminate the spherical aberration which is the dominant resolution-limiting aberration at accelerating voltages larger than about 100 kV [2]. By employing the quadrupole–octupole correctors, the aberration correction of round electron lenses is proven successful [3], [4]. With the successful correction of the primary aberration, the high order aberration calculation of round lenses with quadrupole–octupole correctors become dominant. Due to the complexity of such systems, it is very difficult to calculate the high aberration coefficients with the Aberration Integrals methods.
Liu et al. derived a set of formulae for computing geometrical aberration coefficients up to the third order, and the first order chromatic aberration coefficients for systems containing electrostatic and magnetic round, quadrupole, hexapole and octupole lenses and deflectors [5]. As well the fifth and even higher order geometrical aberrations and the chromatic aberrations of third order were analyzed and a full list of formulae for the fifth order aberration coefficients of round lenses in reasonably usable form at last became available [6], [7], [8]. Lencová extended exact ray tracing to cover the new requirements of aberration-corrected instruments [9].
Differential algebraic (DA) method is a powerful and promising technique in computer numerical analysis. When applied to nonlinear dynamics systems, the arbitrary high-order transfer properties of the systems can be computed directly with high precision [10]. DA method presents a straightforward way to compute nonlinearity to arbitrary orders, only by tracing a reference ray. Furthermore, the DA method is always accurate, limited only by machine precision and algorithm error independent of the order of the aberrations. In the previous work of our group, DA methods have been introduced into the electron optics [11], [12]. Further a local analytical expression has been constructed with high accuracy interpolation of the field, which is calculated numerically for a given position of the reference electron ray. An advantage of such a way of the built-in interpolation method is that it is relatively easy to use, in comparison with the expansion of the axial function into a series of the Hermite functions [13], [14]; this method that is independent of the axial field functions, can directly use the potential values of nodal points which are surrounding the given point. Thus, up to the fifth order aberrations for a round electron lens, the combined focusing-deflection systems and hexapole correctors have been solved effectively [15], [16], [17].
In this work, the quadrupole and octupole fields are calculated by the finite element method (FEM) and the reduced quadrupole and octupole potentials at mesh points are obtained [18], [19]. Moreover, the DA theory, operation method and algorithm are investigated in detail and a local analytical expression of the numerically computed quadrupole–octupole field which can be adopted as DA extension numbers will be introduced into the up to the fifth order aberrations calculations. Only by tracing a reference ray, up to the fifth order aberrations of a quadrupole–octupole corrector are achieved. Finally, a practical quadrupole–octupole corrector is analyzed and discussed as an example using the developed DA software in this paper.
Section snippets
DA methods for high order aberrations calculations of a quadrupole–octupole corrector
The properties of a charged particle system can be described by a transfer map as follow:where rf denotes the final positions and slopes of a charged particle, and r0 denotes the initial positions and slopes of the charged particle. δ contains other systemic interesting parameters such as energy spread. Thus ∂ℜ/∂r0 denotes the geometric aberrations with respect to initial conditions, while ∂ℜ/∂δ is corresponding to the chromatic aberrations. DA method presents a straightforward way to
An example for calculating a quadrupole–octupole corrector
A software package has been developed by implementing the above theory with C++ . And up to the fifth order aberrations of a quadrupole–octupole corrector can be calculated using the DA methods presented in this paper. If need be, it can be conveniently extended to the higher order by improving the interpolation algorithm. The approach is demonstrated on an example of a quadrupole–octupole corrector employed for correction of the axial spherical and chromatic aberrations of round lenses
Conclusions
In this paper, the principle of differential algebraic (DA) methods is applied to round lenses with quadrupole–octupole correctors. The high order aberrations of such systems can be achieved by DA method. The DA method applicable to engineering design is presented where local analytical expressions of quadrupole and octupole field variables are constructed using the numerical computed results of FEM or FDM. The corresponding program has been developed to calculate the fifth order aberrations of
Acknowledgments
We are indebted to Tiantong Tang, Professor of Xi'an Jiaotong University, for long-term advice. This work is sponsored by the National Natural Science Foundation of China (Grant no. 61671372), and the authors would like to appreciate sincerely for these supports.
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