Aberration Figures Research Articles

Electrode length effects on the performance of the electrostatic quadrupole triplet lens

The present work aims to better understand the relationship between the electrode length of the electrostatic quadrupole triplet lens (EQTL) and its optical properties, as well as the possibility of increasing efficiency by selecting the electrode length that produces the best results, as this topic has not been discussed in previous studies. The effects of electrode length on the performance of the quadrupole triplet lens (QTL) were studied. The quadrupole triplet lens (QTL) electrode voltage combinations were found to produce the stigmatic image, and the effects of electrode length on the image’s characteristics were examined. Two lenses with electrode lengths L of 80 mm and 90 mm were designed, and the comparisons were made to show the effects on the electrode voltages, electron beam trajectories, and aberration figures. The findings indicated that a 12.5% increase in electrode length resulted in a 60% increase in aberration figures for Δy in point object-to-point image focusing mode and an 80% increase for Δz in parallel object-to-point image focusing mode.

Effects of Diameter between Electrodes on Properties of Electrostatic Quadrupole Deflector

Abstract: This study investigates the impact of varying the electrostatic quadrupole deflector diameter (EQDD) on the image when charged particles undergo deflection. A comparison is made between aberration figures, the kinetic energy of the electron beam, and the flight time concerning different EQDD values. The results showed that the most effective image is created at the image plane when the EQDD is double the radius of the electrostatic quadrupole deflector electrode (r). Keywords: Charge particle, Electrostatic quadrupole deflector, SIMION 8.1.

The Effect of the Distance between the Cylindrical Poles of the Electrostatic Quadrupole Deflector on the Trajectory of Charged Particles

The impact of the electrostatic quadrupole deflector diameter (E.Q.D.D) on the image characteristics and efficiency of the 90°quadrupole deflector was explored. For electrostatic quadrupole deflectors (E.Q.D) of various diameters, a comparison of aberration figures, electron beam kinetic energy, and fight time was done. The results revealed that when the E.Q.D.D of the E.Q.D was two and a half times greater than the electrode radius, the best image could be created at the image plane (r).

Effects of Electrode Beam Radius on Properties of four-poles Electrostatic Double Lens for Point-to-Point Focus

This study investigates the effect of selection the electric pole radius of the four poles electrostatic doublets lens on the optical property. Also, a comparison between the optical property and performance of two lenses with different electrode radius was made. The results showed that the electric pole diameter of the lens plays main function in the imaging characteristics of the four poles electrostatic doublets lens. The results show that the aberration figures can be enhanced when the electrode diameter of the quadrupole lens is equal to the lens aperture, while the chromatic aberration can be improved by choosing the electrode diameter according to the equation.

Simulation and Optimization of the Optical Properties of an Einzel lens

The simulation and optimization calculations were carried out for an einzel lens consists of three overlap cylindrical electrodes with the same length but with different electrode diameters under point-to-point focusing condition. The effects of the overlap distance (d) between the central and the outer cylindrical electrodes on the optical properties of an einzel lens were investigated. The results showed that the increasing of the overlap distance (d) will improve the aberration figures of the images, where by increasing the overlap distance (d) from 2 mm to 6 mm the dimensions of the image were reduced by about 50% for both y and z-directions. The results appeared that both the dimensions of the lens, and the aberrations can be reduced at the same time without affecting the homogeneous distribution of an electron-beam kinetic energy of at the image plane.

Comparison between the imaging properties of two different types of electrostatic quadrupole doublet lenses

The comparisons between the optical properties of two types of electrostatic quadrupole lenses with the same dimensions and operation conditions are made. The optimum voltages of the electrodes which give the stigmatic image for both focusing planes with are calculated. The calculations are carried out for point-to-point focusing mode. The effects of changing the electrode shape on the optical properties of the lenses are investigated, where the magnification, electron-beam trajectories, and aberration figures are studied, and the comparisons are made. The effects of variation the object and image positions with respect to lens are studied and the comparison between the results for the two lens systems is done. The results show that the electrode shape of the lens play important role in the imaging properties of the electrostatic quadrupole doublet lens systems, especially in the aberrations figures at the image plane. The b-type electrostatic quadrupole doublet lens system gives the best imaging properties in comparison with a-type.

Optimization of an electrostatic quadrupole doublet focusing systems

The imaging properties of an electrostatic quadrupole doublet lens were analyzed with the aid of computer simulation. The optimal electrode voltages which lead to stigmatic image in both planes of the quadrupole doublet lens with minimum spot size at position sensitive detector (PSD) were found for two operation modes: point-to-point focusing and parallel-to-point focusing. The optical properties as: Magnifications, spot sizes in the image plane and aberration figures were discussed. The results showed that the focusing of the lens was strong in the xy-plane in comparison with the focusing in the xz-plane. The distortion of the image was greater when the image position will be close to the lens in comparison with object position. Also, the imaging properties were very sensitive to the lunching angle of the electron-beam.

Wavefront and ray-density plots using seventh-order matrices

The optimization of an optical system benefits greatly from a study of its aberrations and an identification of each of its elements’ contribution to the overall aberration figures. The matrix formalism developed by the author was the object of a previous paper and allows the expression of image-space coordinates as high-order polynomials of object-space coordinates. In this paper we approach the question of aberrations, both through the evaluation of the wavefront evolution along the system and its departure from the ideal spherical shape and the use of ray density plots. Using seventh-order matrix modelling, we can calculate the optical path between any two points of a ray as it travels along the optical system and we define the wavefront as the locus of the points with common optical path length; the results are presented on the form of traces of the wavefront on the tangential plane, although the formalism would also permit sagittal plane plots. Ray density plots are obtained by actual derivation of the seventh-order polynomials.

Third‐order aberration theory of Wien filters for monochromators and aberration correctors

Third-order aberrations at the first and the second focus planes of double focus Wien filters are derived in terms of the following electric and magnetic field components--dipole: E1, B1; quadrupole: E2, B2; hexapole: E3, B3 and octupole: E4, B4. The aberration coefficients are expressed under the second-order geometrical aberration free conditions of E2 = -(m + 2)E1/8R, B2 = -mB1/8R and E3R2/E1 - B3R2/B1 = m/16, where m is an arbitrary value common to all equations. Aberration figures under the conditions of zero x- and y-axes values show very small probe size and similar patterns to those obtained using a previous numerical simulation [G. Martinez & K. Tsuno (2004) Ultramicroscopy, 100, 105-114]. Round beam conditions are obtained when B3 = 5m2B1/144R2 and (E4/E1 - B4/B1)R3 = -29m2/1152. In this special case, aberration figures contain only chromatic and aperture aberrations at the second focus. The chromatic aberrations become zero when m = 2 and aperture aberrations become zero when m = 1.101 and 10.899 at the second focus. Negative chromatic aberrations are obtained when m < 2 and negative aperture aberrations for m < 1.101. The Wien filter functions not only as a monochromator but also as a corrector of both chromatic and aperture aberrations. There are two advantages in using a Wien filter aberration corrector. First, there is the simplicity that derives from it being a single component aberration correction system. Secondly, the aberration in the off-axis region varies very little from the on-axis figures. These characteristics make the corrector very easy to operate.

Numerical calculation of electron optical aberrations

A numerical method for the accurate calculation of aberration figures in systems with a straight optical axis is concisely outlined. The method involves computing a fundamental set of paraxial rays, and then directly ray-tracing a judiciously chosen set of deviations from these paraxial rays, that enable the direct plotting of aberration figures. This method avoids using complicated aberration integrals, and can directly handle the effects of high-order aberrations and asymmetry errors. A fast general predictor–corrector method for the numerical ray-tracing is also presented.

High-order combined geometric-chromatic aberrations of magnetic electron lenses using differential algebra

In this paper, high-order geometric and chromatic aberrations of magnetic electron lenses are dealt with together by using the differential algebraic method. High-order geometric-chromatic aberration analytic expressions are obtained and arbitrary order combined aberrations can be calculated. As an example, a Glaser's bell-shaped magnetic electron lens has been studied and the second- and third-order combined geometric-chromatic aberration coefficients have been calculated. The aberration figures of the combined second-order aberrations are also presented. This developed method can be of great utility in high-order combined aberration analysis and correction for high-resolution charged particle optical systems.

Aberrations caused by mechanical misalignments in electrostatic quadrupole lens systems

Abstract Image aberrations resulting from small misalignments in quadrupole lenses multiplets have been analysed. Analytical formulas for the coefficients of the beam displacement, astigmatism and coma associated with misalignments in a general quadrupole lens system have been derived. Numerical computations of systems of three and four quadrupole lenses have also been carried out. The aberration figures obtained for systems with and without a mechanical defect are compared. The aberration coefficients that have been obtained can be used for estimating tolerance limits for lens misalignments.

A study of small probe formation in a field emission gun TEM/STEM

A Philips EM 400T equipped with a field emission gun (FEG) has been studied to determine the spatial and angular current distribution in the electron microprobe at the specimen level. The field emission gun provides a high brightness source capable of producing electron probes with diameters of several nanometers and total currents of 0.05 - 50 nA. The advantage of the TEM/STEM over the dedicated STEM in this study is its ability to produce real-space images of the probe at the specimen plane. Detailed information about the probe current distribution can be experimentally obtained and applied to various small-probe techniques (i.e. HRAEM, STEM, microdiffraction) in order to determine their spatial resolution.The probe diameters listed in Table 1 were measured from high magnification images of the focussed probes which showed sharply defined outlines of the aberration figures. Additional measurements were obtained by scanning the probe across a small fixed entrance aperture to an EELS spectrometer, and also by an edge resolution method similar to that used by Venables and Janssen.

The twin lens system - A high resolution objective lens at 300 kV

Point resolution of better than 0.2 nm is today achievable in medium voltage electron microscopes. Crucial for this achievement is the objective lens design which has to provide sufficiently low spherical and chromatic aberration figures. In addition to the lens itself, its integration into the overall microscope optics has to ensure a reproducible and accurate alignment.Cleaver has shown that symmetrical lenses have lower aberrations than their asymmetrical counterparts by a factor of 1.5 or 2. Theoretical optimization of this lens at 300 kV can be carried out using a first-order finite element program, as developed by Munro. The symmetrical single-field condenser objective with saturated pole-piece tips produces particularly good values with Cs = 1.0 mm and Cc = 1.4 mm at 300 kV. In this lens the specimen is placed in the centre of the magnetic field.

Determination of the Characteristics of a Field Emission Source by Geometrical and Wave Optical Methods

The radius of the image figure of a field emission source is determined by the aberrations of the total electron optical system and is usually given by adding the individual aberration and image figures in quardrature where rO is the intrinsic source radius, rs and rc are the radius of aberration figures given by spherical and chromatic aberration respectively, and rd is the radius given by diffraction, all referred to object space. Different authors have used different image planes and ambiguity has been introduced. The plafie with the minimum image figure (optimum plane) should be used and can be determined by geometrical methods only when diffraction effects are negligible; otherwise, a wave optical treatment should be used.

The diffraction theory of optical aberrations*)Part I: General discussion of the geometrical aberrations

In this paper the geometrical aberrations of a rotationally symmetrical optical system are dealt with in a new way, suggested by the wave-optical treatment of the image errors to be published subsequently. A spherical surface is introduced with its centre at the Gaussian image (situated at a radial distance σ from the axis) and passing through the centre of the exit-pupil. The actual wave-front deviates by an amount V (σ, γ, ) from this sphere, where γ andare plane polar co-ordinates upon the latter. The aberration function” V is expanded in terms γn cosm, whereas usually the characteristic function analogous to V is expanded in terms containing γn cosm . Thus a single aberration is defined now by an aberration function blnmσ2l+mγn cos m. As a consequence of this new definition the discussion of the aberration figures is simplified considerably. Also a simple classification of image errors of all orders is found, according to which the value of m determines the general type to which a given single aberration belongs (spherical aberration for m = 0, coma for m = 1, astigmatism for m = 2).

The diffraction theory of optical aberrations: Part I: General discussion of the geometrical aberrations

In this paper the geometrical aberrations of a rotationally symmetrical optical system are dealt with in a new way, suggested by the wave-optical treatment of the image errors to be published subsequently. A spherical surface is introduced with its centre at the Gaussian image (situated at a radial distance σ from the axis) and passing through the centre of the exit-pupil. The actual wave-front deviates by an amount V (σ, γ, ) from this sphere, where γ andare plane polar co-ordinates upon the latter. The aberration function” V is expanded in terms γ n cos m , whereas usually the characteristic function analogous to V is expanded in terms containing γ n cos m . Thus a single aberration is defined now by an aberration function b lnm σ 2l+m γ n cos m . As a consequence of this new definition the discussion of the aberration figures is simplified considerably. Also a simple classification of image errors of all orders is found, according to which the value of m determines the general type to which a given single aberration belongs (spherical aberration for m = 0, coma for m = 1, astigmatism for m = 2).