Chromatic Aberration Coefficients Research Articles

Statistical error analysis of electrode construction for electrostatic spline lenses

It is easy to calculate approximately the electrode shapes for a given axial potential distribution by neglecting its high-order derivatives. The lens with the electrodes determined this way has different optical properties than that with the original axial potential distribution. These differences were calculated for a large number of randomly generated samples with axial potential distributions in the form of cubic splines. The differences were found to be about ±5%–10% for the focal lengths and the chromatic aberration coefficients. The spherical aberration coefficients for the constructed electrode systems usually have smaller values than those of the original lens models, but the differences are larger. The conclusion is that this simple electrode construction method is usually accurate enough for practical lens design.

Chromatic aberration of symmetric two-cylinder electrostatic lenses

Accurate calculations of the axial chromatic aberration coefficients of symmetric two-cylinder bipotential electrostatic lenses for three different gap values (0.10D, 0.50D, and 1.0D) are presented. This is an extension of the first-order properties and the third-order spherical aberration coefficients published by Harting and Read.

Optical properties of a symmetrical slit electron lens

Optical properties, such as focal length, and defects of a symmetrical and parallel three-slit plane electrode lens are presented in diagrammatic form with aperture gap widths and applied potential differences as parameters. In calculating these properties as potential distributions on the optical axis, the analytical expression determined by G.D. Archard (1954) was applied. Calculated results show that the aperture gaps have considerable influence on the focal length, F, and spherical aberration coefficient, CS, values, but little effect on the chromatic aberration coefficient, CC. For example, typical values of CSF-1 and CCF-1 are of the order 4.2*102 and 2.5 respectively, for F=10 and infinite magnification, when the aperture gaps of the three electrodes are each equal to half the spacing between the outer electrodes.

High resolution patterning system with a single bore objective lens

An electron beam microfabrication tool for use in nanometer scale device research has been built incorporating a single bore objective lens. The sample resides in a micrometer-controlled in-lens stage which is an integral part of the boreless lower polepiece. This piece is removed to change samples, a straightforward operation due to the relaxed alignment tolerances between the two polepieces. The focal length of the lens, which has a 4 mm upper bore and a 4 mm gap, is 1.7 mm, with spherical and chromatic aberration coefficients taking on values of 1.0 and 1.2 mm, respectively. At the 40 kV operating voltage of the system with a tungsten hairpin source, 5 pA can be focused into a 50 Å diameter probe. Using ultrasonic liftoff with polymethyl methacrylate (PMMA) resist, we have obtained two-dimensional arrays of 200 Å gold:palladium particles on 500 Å centers on bulk GaAs. As a test of the overall system, which includes a computer-controlled pattern generator, a page of alphabetic characters was written in PMMA on a silicon nitride membrane at a data density of 1010 characters/cm2.

Design of an electrostatic optical system for ion beam lithography

An electrostatic optical system has been designed to produce a submicron ion beam probe. The main task is to design a low spherical and low chromatic aberration accelerating–focusing lens for the ion gun and to design a combined focusing–deflection system with low deflection aberrations. The gun lens aberrations are minimized by using a parameter optimization method. Very low chromatic aberration coefficients of 3.1 or 6.9 mm are obtained for a working distance of 3 or 6 mm and for an acceleration ratio of 10. An analysis is made of the combined electrostatic focusing–deflection system using MOL concept. A practical electrostatic MOL system was developed which gives good overall deflection aberration performances. It has a theoretical resolution of 73 nm for a beam half‐angle of 5 mrad over a 0.8×0.8 mm scanning area and the resolution can be upward to 20 nm after dynamic correction of the deflection aberrations.

Computer programs for designing multipole electron and ion optical systems

New computer programs have been developed for designing multipole electron and ion beam systems. The charge density method is used to calculate the axial field distribution for electrostatic quadrupole and octopole lenses and deflectors, dipole deflectors, and rotated magnetic quadrupole lenses. The programs are described and some comparisons against analytic models are presented. An existing finite element program is used so that electrostatic round lenses may also be included in the analysis. An optical properties program uses a new compact aberration theory developed by the authors to calculate the first‐order focusing and deflection properties, third‐order geometrical and first‐order chromatic aberrations, for any combination of the elements already mentioned. The desired focusing is obtained by using an iterative least squares routine to focus the round and quadrupole lenses. An analysis of crossed electrostatic and magnetic quadrupole lenses is presented showing how the first‐order axial chromatic aberration coefficient can be reduced to less than 0.1 mm, making the size of the chromatic disk negligible.

Design parameters of an unconventional probe-forming lens

The design parameters of an electron probe-forming lens with a zero-bore flat-face single pole piece have been considered under nonsaturation conditions. From the measured axial flux density distribution of an experimental single-pole-piece lens, the maximum flux density, the spherical and chromatic aberration coefficients, the focal length and the object position have been investigated in terms of the diameter of the pole's flat face, which is the effective geometrical parameter. Empirical equations relating the geometrical electron-optical parameters have been put forward and compared with those of a different experimental lens reported in the literature. The focal properties of the probe-forming lens have also been investigated in the telescopic mode of operation. The results should be useful for preliminary design considerations of an unconventional electron probe-forming lens with zero-bore flat-face single pole piece for various types of electron-optical instruments according to the operational requirements.

High-performance two-electrode electron gun lenses: Beyond the Butler gun

High-performance two-electrode electron gun lens designs with lower spherical and chromatic aberration coefficients than the Butler gun are presented. It is shown that the Butler gun is a specific solution to the general class of one-interval polynomial lenses of degree three. It is found that the higher-order (> 3) polynomial lenses have equipotential angles at the saddle points which differ from 54°44′08″. Reductions in aberration disks of 60% are possible using the new lenses instead of the Butler gun. Retarding-mode performance of the lenses is considered for potential application to low voltage microscopy. A simple design of a scanning gun microscope is considered.

Elemental imaging and resolution in energy-filtered conventional electron microscopy

Energy-filtered transmission electron microscopy (EFEM) was used to image the distributions of uranium and carbon in uranyl acetate stained catalase crystals. The spatial resolution obtained from inelastic C K-edge and U O 4.5-edge images, determined from the highest-order reflection in the computed diffraction pattern, was 3.4 nm for both carbon and uranium. The resolution limit imposed by the delocalization of inelastic scattering was estimated from cross-section measurements to be 0.6 nm for U and 0.2 nm for C. Considering both delocalization and the effects of microscope aberrations, for an objective lens chromatic aberration coefficient of 2.8 mm and 10 eV energy window, the calculated resolutions are 2.0 nm for C and 1.2 nm for U. The effects of plural inelastic and elastic-inelastic scattering were sufficiently large to show crystalline structure in unprocessed pre-edge inelastic images. Previously suggested methods for eliminating these artifacts were applied to obtaine the compositional information in the catalase EFEM images.

Unipotential electrostatic lenses: Paraxial properties and aberrations of focal length and focal point

An experimental study of electrostatic electron lenses as a function of geometrical and electrical parameters is described. The lenses are of the symmetrical three-electrode unipotential type. The parameters are the thickness of the center electrode and the interelectrode spacing, both relative to the center electrode aperture diameter, and the ratio of lens voltage to cathode voltage. The lens properties are characterized in terms of the focal length and focal distance, and the spherical and chromatic aberrations of these quantities. In general, the principal surfaces of a lens are not plane, and the aberrations of focal length and focal distance are not the same. Expressions are derived relating the focal length and focal distance aberrations to the spherical and chromatic imaging aberration coefficients Cs and Cc, and the magnification aberrations. The advantages of formulating the lens properties in terms of focal length and focal distance and their aberrations, and the usefulness of the data presented here, are illustrated with several examples.

A focused ion beam system for submicron lithography

A two-lens optical system for a focused ion beam system has been designed, using a four-electrode accelerating lens as the condenser lens and an Einzel lens as the objective lens. A 1 nA beam current in a 0.055–0.1 μm beam spot is obtained under the following conditions: 20 μA/sr angular current intensity, 10 eV beam energy spread, 30–60 kV acceleration voltage, and 4 mrad object side beam half angle. The chromatic aberration coefficient of the condenser lens is reduced by picking out the dominant design parameters and optimizing them under the design constraints. The four-electrode accelerating lens has an object-side chromatic aberration coefficient Cco=11 mm at the object side focal length fo=25 mm at infinite magnification for 30 kV acceleration voltage. The Einzel lens operating in the acceleration mode has an image-side chromatic aberration coefficient Cci=70 mm at the image-side focal length fi=50 mm. A focused ion beam system is developed using the designed lenses and a Ga ion source. A pattern as fine as 0.15 μm with excellent definition is obtained.

Weighted order doubling in the computation of chromatic aberration coefficients

The essentially Hamiltonian process of order doubling is incorporated into a Lagrangian method for the determination of the chromatic aberration coefficients of symmetric systems composed of homogeneous lenses and mirrors. In dealing with chromatic aberration coefficients, the prescription for weighted truncation of power series has been found to be of great value. Accordingly, in this paper we are initially concerned with the general principles of order doubling when weighted truncation is employed. These considerations give rise to the idea of weighted order doubling, which follows simply from a condition corresponding to Fermat’s principle. In the context of aberration theory, it is shown that, by using only the knowledge of a general ray path that includes terms up to order m in a chromatic coordinate, say ω, the configuration of this general ray at the initial or final base surface can be determined (by means of a characteristic function) including terms of order 2m + 1 in ω. A process is devised that incorporates these ideas into an efficient scheme for the determination of the Taylor coefficients of characteristic functions. As expected, the explicit inclusion of the chromatic dependence gives methods that are far superior to the corresponding monochromatic procedures.

Weighted truncation of power series and the computation of chromatic aberration coefficients

In keeping with the recent growth of interest in the use of analytic computational methods in the analysis of the behavior of optical systems, this paper reports on the incorporation of chromatic dependence into these schemes. A novel prescription for the truncation of power series in several variables (which is also of value outside this context) is introduced in order to avoid much unnecessary computation. Subsequently, a process is devised for computing chromatic aberration coefficients to arbitrary orders for symmetric systems composed of homogeneous lenses and mirrors. It is found that, in less than the time required for the analytical analysis of a system at two distinct wavelengths, this process can yield the required results for the entire visible spectrum and beyond, if necessary.

Chromatic coordinates in aberration theory

In the computation of the chromatic-aberration coefficients of a specified optical system, it is crucial that the chromatic behavior of the materials in the system be closely approximated by truncated Taylor series in one coordinate. The form of this chromatic coordinate determines the quality of the resulting approximations and is therefore of central importance. Four different forms are proposed, each being appropriate to different circumstances. The approximation of dispersive properties by polynomials in these coordinates is considered in detail. Although this method was devised solely for use in the context of aberration theory, it is found to be among the most effective means for the approximation of dispersion curves in general.

Some properties of single pole piece objective electron lenses

From the measured magnetic field distributions at constant excitation, the objective focal properties for two experimental single pole piece electron lenses of equal dimensions, one with a spherical pole and the other with a cylindrical pole are computed at different diameters of the axial bore in the pole piece. The electron optical properties of the two objective lenses including their spherical and chromatic aberration coefficients are compared. The effect of the pole piece shape, its axial bore size and its orientation with respect to the incoming electron beam on the objective focal properties are discussed in detail. The results should be useful in the design and operation of single pole piece objective lenses for the transmission and scanning transmission electron microscope.

Improvement of electrostatic lenses for ion beam lithography

A systematic investigation of a large number of practically feasible axial potential distributions was carried out with a subsequent reconstruction of the electrode systems producing such distributions. By using parametrized analytical functions and different curve-fitting techniques we were able to find many potential distributions with both small spherical and chromatic aberrations. A cubic spline curve-fitting algorithm proved to be especially effective. The reconstruction of the electrode systems resulted in several new electrostatic ion lenses with excellent optical properties. As an example, a four-electrode electrostatic lens is presented with a voltage ratio of Vmax/Vmin =9, spherical aberration coefficient Cs0/f0 =0.92 and chromatic aberration coefficient Cc0/f0 =0.55 (both coefficients are referred to the object, related to the object side focal length and calculated for the case of infinite magnification). With an acceptance angle α0=5 mrad and magnification M=2.58 a 10.5 nm aberration disk diameter can be achieved. If α0=2 mrad, the disk diameter is 3 nm.

Design of objective lens for 200-kV high-resolution electron microscopes

Developement of computer technology provides much improvements on electron microscopy, such as simulation of images, reconstruction of images and automatic controll of microscopes (auto-focussing and auto-correction of astigmatism) and design of electron microscope lenses by using a finite element method (FEM). In this investigation, procedures for simulating the optical properties of objective lenses of HREM and the characteristics of the new lens for HREM at 200 kV are described.The process for designing the objective lens is divided into three stages. Stage 1 is the process for estimating the optical properties of the lens. Firstly, calculation by FEM is made for simulating the axial magnetic field distributions Bzc of the lens. Secondly, electron ray trajectory is numerically calculated by using Bzc. And lastly, using Bzc and ray trajectory, spherical and chromatic aberration coefficients Cs and Cc are numerically calculated. Above calculations are repeated by changing the shape of lens until! to find an optimum aberration coefficients.

Optimisation of magnetic lens with pole-piece of finite inside diameter

The relation between the coefficients of spherical and chromatic aberration of round magnetic lenses and the flux distribution over a cylindrical surface that touches the pole-piece at some point is studied. The magnetic flux density distribution over the cylindrical surface is approximated by a polygonal line, and the successive approximation is used to determine the optimum field distributions which make either of the aberration coefficients minimum. The minimum spherical and chromatic aberration coefficients and the magnetic flux density corresponding to these are calculated. It is found that the aberration coefficients become larger as the minimum inside diameter of the pole-piece becomes larger.

Achromatic proton microbeams

A doublet of achromatic quadrupole lenses has been tested as a probe-forming lens for high-energy ions. Each lens has length 6 cm and diameter 14 cm, and contains a magnetic quadrupole of maximum pole tip field strength 1.0 T and bore 3.40 mm, and an electric quadrupole of maximum pole potential 3.0 kV and bore 1.05 mm. Maximum achromatic operating energy is calculated to be 29 MeV for protons and 410 keV for singly charged antimony ions. To test the doublet, backscattering of 600 keV protons is measured from a tungsten wire as the beam is deflected by dipole excitation of the second lens. In preliminary measurements, a chromatic aberration coefficient of less than 1.2 cm and beam width of 4 μm are found, compared to 20 cm and 4 μm for operation with no electric quadrupole voltage. If their parasitic aberrations can be reduced, lenses of this type offer prospects for over coming the problems of chromatic aberration which limit the resolution of microprobes used for proton-induced X-ray emission or Rutherford backscattering measurements, as well as of kilovolt energy systems utilizing field ionization sources. Experimental methods of characterizing and compensating parasitic aberrations of first and second order will be presented.

An achromatic quadrupole lens doublet for positive ions

The resolution and chromatic aberration of a doublet of achromatic quadrupole lenses have been experimentally measured. Each achromatic lens is of length 5.99 cm and comprises an electric quadrupole of bore 1.05 mm and a magnetic quadrupole of bore 3.40 mm. In preliminary tests with 600-keV protons, a chromatic aberration coefficient of less than 1.2 cm and beam width of 4 μ are measured, compared to 20 cm and 4 μ for the same doublet operated in a solely magnetic mode.