Alexander D. Dymnikov

OPTIMAL MAGNETIC AND ELECTROSTATIC RUSSIAN QUADRUPLET MICROPROBE LENS SYSTEM WITH HIGH DEMAGNIFICATION

Abstract The goal for the microbeam is a lateral resolution in the nanometer range. Usually one needs a very small object diaphragm to obtain this which gives problems with scattering and is difficult to manufacture. Focusing systems with very high demagnification solve these problems. In this paper a focusing system consisting of a magnetic or electrostatic Russian Quadruplet (RQ) working in the second mode of excitation and with high positive demagnification (up to 10000) is investigated in detail. In this system it is possible to vary the focal distance and the demagnification within a wide range, changing the distance between the second and the third lenses. We call this RQ the Separated Russian Quadruplet (SRQ). A general method for optimising the quadrupole configuration is described and a numerical optimization has been performed. We consider the differential equation of motion of the particles accurate to terms of third order inclusive. The minimum spot size, the appropriate radii of the diaphragms and the distance between them are found. It is shown that for 1 μm object and aperture diaphragms it is possible to obtain 1 nm resolution.

The matrizant method for an optimal synthesis of electrostatic multiple cylinder lenses

Abstract The matrizant method for the synthesis of electrostatic systems with axial symmetry, consisting of multiple cylinder lenses and producing an optimal beam quality, is presented. This method includes: the matrix method for the description of motion of a charged particle beam in the nonlinear approximation; the mathematical analytical model of the axial potential distribution; the method of the moments of the particle distribution function over the whole totality of the phase coordinates for finding the averaged radius of the beam; and the integral equation method to solve Laplaces equation for obtaining the parameters of the physical model which has the same axial potential distribution as the mathematical model. The developed matrizant method is applied to minimize the beam spot size for a given beam current in a focusing system consisting of multi-cylinder lenses with equal diameter. The desired object and image plane positions are satisfied before evaluating the merit function, but are not a part of the set of constraints for the optimization.

The optimal construction of an electrostatic quadruplet as focusing microprobe system

Abstract A general analytical and numerical technique for finding high-quality probe-forming systems is proposed and developed. This optimisation procedure includes mathematical and physical modeling based on the matrizant method and on the boundary element method, respectively. The Separated Russian Quadruplet with electrostatic quadrupole lenses consisting of four copper rods as the microprobe focusing system is numerically simulated. The influence of the rod diameter and the energy spread on the beam spot size is investigated for different emittances.

MATRIX METHODS IN PERIODIC FOCUSING SYSTEMS

A linear differential equation with periodic driving matrix P in the n-dimensional phase space, the matrizant R(P) of this equation and an envelope matrix σ, representing the simultaneous transmission of an ensemble of trajectories, are considered. Three new n×n matrices are introduced: the oscillating antisymmetric matrix, the amplitude matrix and the phase orthogonal matrix, elements of which are derived as functions of the envelope and the driving matrices. The Courant–Snyder parametrization for n=2 in periodic systems is generalized to an arbitrary n. The generalized multiplicative representation of the matrizant R(P) is derived via the amplitude and phase matrices. For the particular case n=2 the Courant–Snyder representation is obtained.

Influence of the rod diameter of electrostatic quadrupole lenses on the axial field and on the minimum spot size in nonlinear microprobes

Different constructions of practical electrostatic lenses are numerically investigated. The electric field in electrostatic quadrupole lenses consisting of four copper rods depends, for a given lens aperture, on the value of the rod diameter. Furthermore, the aberration coefficients of electrostatic systems depend on the axial partial derivative of the electric field which, for example, influences the beam spot size, i.e., the microprobe resolution. We apply an integral formulation to the given set of polarized conductors in vacuum and in absence of space charge. These equations are numerically solved using an accurate version of the boundary element method in order to obtain the potential and the field distributions. The axial distributions of the electric field derivatives are found for different diameters of the rods. The developed Matrizant method is then applied to minimize the beam spot size -- for a fixed beam current -- in focusing systems consisting of electrostatic quadrupole lenses, using the computed axial distributions.

Numerical synthesis of an optimal microprobe focusing system

Abstract An electrostatic microprobe focusing system including a separated Russian quadruplet and two slits (or diaphragms) is numerically synthesized, taking into consideration all geometrical aberrations of the third order. The matrizant method is applied to minimize the beam spot size for a fixed beam current. For quadrupoles consisting of four copper rods, the electric field inside of the quadruplet is computed using an accurate version of the boundary charge method. The minimum spot size, the appropriate radii of the diaphragms, the distance between them, the effective length and the polarisation for different diameters of the rods are found.

High-frequency focusing system for nuclear microprobes

Abstract The possibility of designing a high-frequency microprobe (RF-microprobe) for ions of a few MeV energy is studied. The first-order focusing of the pulsed beam in an RF-microprobe lens system is considered. Three systems giving the same demagnification are compared: the Russian quadruplet, the superconducting magnetic solenoid and the RF-lens. The two last lenses are investigated as systems consisting of one or two lenses (doublets). The frequency and the amplitude of the electric field of RF-lenses, which have the same geometry and demagnification as the Russian quadruplet and solenoid system, are determined.

Charged particle microprobes with mininum beam spot size for a given beam current

In a nuclear microprobe the beam (usually 1–3 MeV protons) is focused on the target to a lateral spot size of micrometer dimensions, using a focusing lens and at least two beam emittance-defining apertures. This type of beam has gained an increased interest during the last decade as an analytical instrument. In this paper the limits to high spatial resolution in ion microprobes and the requirements for its achievement are presented. Different focusing systems with quadrupole and rotational symmetry are studied and compared. The optimal geometry, the minimum spot size and the appropriate optimal radii of two apertures and the distance between them for a given emittance are found.

Matrizant method for an optimal synthesis of nonlinear magnetic and electrostatic focusing systems for a given beam current

The Matrizant method for the synthesis of magnetic and electrostatic focusing systems, producing an optimal beam quality (a minimum spot size for a given beam current), is proposed. This method includes: the matrix method for the description of motion of a charged particle beam in the nonlinear approximation on the base of a general relativistic theory of charged particle beam motion along a curved optical axis, developed by one of the authors; the analytical model of the axial field or their derivatives; the method of the moments of the particle distribution function over the whole totality of the phase coordinates for finding the averaged radius of the beam; and the integral equation method to solve Laplaces equation for obtaining the parameters of the physical model which has the same axial field as the mathematical model. As an example, the developed Matrizant method is applied to minimize the beam spot size for a given beam emittance in a focusing system consisting of multiple cylinder lenses with equal diameter.

Optimal electrostatic axisymmetric microprobe focusing system

Abstract This paper investigates the possibility of utilising an electrostatic multiple cylinder lens to focus the beam in microprobes. Multiple cylinders reduce the lens potentials which are required. We use a new analytical model of the axial potential distribution varying the parameters of this distribution and the size of the object and aperture diaphragms to obtain the minimum spot size at the specimen for a given beam emittance and finding by this way the optimal parameters of the axial potential distribution. For synthesising the optimal physical model we applied an accurate version of the Integral Equation Method to solve Laplaces equation in order to obtain the parameters of the physical model which has the same axial potential distribution as the optimal analytical model.