A note on the Petzval field curvature in electron-optical systems

1. The Petzval field curvature produced in a compound glass optical system of axial symmetry is given by a well-known formula. If the system consists of a number of media, of refractive indices n0, n1, n2, …, having spherical faces whose radii of curvature are r0, r1, r2, …, the formula is [see for example, (1) or (2)]and this is in common use amongst those concerned with the design of optical equipment. The analogous integral expression in the electron optical case has not, however, received the attention it deserves, in spite of the developments in electron optics during the past decade. In 1935 Glaser(3) presented the third-order error theory of an axially symmetric electron optical system.

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Method of calculating the aberrations of soft X-ray and vacuum ultraviolet optical systems

Journal of Synchrotron Radiation ◽

10.1107/s1600577519007823 ◽

2019 ◽

Vol 26 (5) ◽

pp. 1558-1564

Author(s):

Yiqing Cao ◽

Zhijuan Shen ◽

Zhixia Zheng

Keyword(s):

Optical System ◽

Vacuum Ultraviolet ◽

Optical Systems ◽

Third Order ◽

Order Accuracy ◽

X Ray ◽

Plane Symmetric ◽

The Third ◽

Aberration Theory ◽

Ray Coordinates

Based on the the third-order aberration theory of plane-symmetric optical systems, this paper studies the effect on aberrations of the second-order accuracy of aperture-ray coordinates and the extrinsic aberrations of this kind of optical system; their calculation expressions are derived. The resultant aberration expressions are then applied to calculate the aberrations of two design examples of soft X-ray and vacuum ultraviolet (XUV) optical systems; images are compared with ray-tracing results using SHADOW software to validate the aberration expressions. The study shows that the accuracy of the aberration expressions is satisfactory.

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The geometrical aberrations of general electron optical systems II. The primary (third order) aberrations of orthogonal systems, and the secondary (fifth order) aberrations of round systems

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences ◽

10.1098/rsta.1965.0014 ◽

1965 ◽

Vol 257 (1086) ◽

pp. 523-552 ◽

Cited By ~ 15

Keyword(s):

Second Order ◽

Order Perturbation ◽

Optical Systems ◽

Characteristic Functions ◽

Electron Optics ◽

Third Order ◽

Variation Of Parameters ◽

General Kind ◽

Fifth Order ◽

Orthogonal Systems

The theory of characteristic functions, developed by Sturrock for electron optics, is used to calculate the primary aberrations of rectilinear orthogonal systems of the most general kind. In the second part, the secondary aberrations of round systems are calculated with the aid of Sturrock’s second-order perturbation characteristic functions. A proof of the equivalence of the aberration formulae obtained by Melkich, using the variation of parameters method, and those obtained below is offered in an appendix.

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Third-order aberration of soft X-ray optical systems with orthogonal and coplanar arrangement of the main planes of elements

Journal of Synchrotron Radiation ◽

10.1107/s1600577520010474 ◽

2020 ◽

Vol 27 (6) ◽

pp. 1477-1484

Author(s):

Yiqing Cao ◽

Zhijuan Shen ◽

Haihe Xie

Keyword(s):

Optical System ◽

Optical Systems ◽

Analysis Method ◽

Third Order ◽

Optical Elements ◽

Analytical Analysis ◽

X Ray ◽

Plane Symmetric ◽

Design And Optimization ◽

Aberration Theory

A third-order aberration analytical analysis method of soft X-ray optical systems with orthogonal and coplanar arrangement of the main planes of elements is proposed. Firstly, the transfer equations of the aperture ray and the principle ray are derived; then, based on the third-order aberration theory with the aperture-ray coordinates on the reference exit wavefront of a plane-symmetric optical system, the aberration expressions contributed by the wave aberration and defocus of this kind of optical system are studied in detail. Finally, the derived aberration calculation expressions are applied to calculate the aberration of two design examples of such types of optical systems; the images are compared with ray-tracing results obtained using the Shadow software to validate the aberration expressions. The study shows that the accuracy of the aberration expressions is satisfactory. The analytical analysis method of aberration is helpful in the design and optimization of the soft X-ray optical systems with orthogonal and coplanar arrangement of the main planes of optical elements.

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Atomic Resolution in Crystal Aperture Stem

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100179932 ◽

1990 ◽

Vol 48 (1) ◽

pp. 236-237

Author(s):

J T Fourie

Keyword(s):

Optical System ◽

Spherical Aberration ◽

Three Dimensional ◽

Transmission Function ◽

Optical Systems ◽

Bragg Angle ◽

Electron Optical System ◽

Intimate Contact ◽

Diffraction Effects ◽

Design Aspect

The attempts at improvement of electron optical systems to date, have largely been directed towards the design aspect of magnetic lenses and towards the establishment of ideal lens combinations. In the present work the emphasis has been placed on the utilization of a unique three-dimensional crystal objective aperture within a standard electron optical system with the aim to reduce the spherical aberration without introducing diffraction effects. A brief summary of this work together with a description of results obtained recently, will be given.The concept of utilizing a crystal as aperture in an electron optical system was introduced by Fourie who employed a {111} crystal foil as a collector aperture, by mounting the sample directly on top of the foil and in intimate contact with the foil. In the present work the sample was mounted on the bottom of the foil so that the crystal would function as an objective or probe forming aperture. The transmission function of such a crystal aperture depends on the thickness, t, and the orientation of the foil. The expression for calculating the transmission function was derived by Hashimoto, Howie and Whelan on the basis of the electron equivalent of the Borrmann anomalous absorption effect in crystals. In Fig. 1 the functions for a g220 diffraction vector and t = 0.53 and 1.0 μm are shown. Here n= Θ‒ΘB, where Θ is the angle between the incident ray and the (hkl) planes, and ΘB is the Bragg angle.

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The axion-baryon coupling in SU(3) heavy baryon chiral perturbation theory

Journal of High Energy Physics ◽

10.1007/jhep08(2021)024 ◽

2021 ◽

Vol 2021 (8) ◽

Author(s):

Thomas Vonk ◽

Feng-Kun Guo ◽

Ulf-G. Meißner

Keyword(s):

Perturbation Theory ◽

Chiral Perturbation Theory ◽

Heavy Baryon ◽

Power Counting ◽

Flavor Symmetry ◽

Chiral Perturbation ◽

Third Order ◽

Stellar Objects ◽

The Past ◽

Dense Nuclear Matter

Abstract In the past, the axion-nucleon coupling has been calculated in the framework of SU(2) heavy baryon chiral perturbation theory up to third order in the chiral power counting. Here, we extend these earlier studies to the case of heavy baryon chiral perturbation theory with SU(3) flavor symmetry and derive the axion coupling to the full SU(3) baryon octet, showing that the axion also significantly couples to hyperons. As studies on dense nuclear matter suggest the possible existence of hyperons in stellar objects such as neutron stars, our results should have phenomenological implications related to the so-called axion window.

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A spread spectrum clock generator with a wide frequency and attenuation tuning range based on a third-order error-feedback delta-sigma modulator

Analog Integrated Circuits and Signal Processing ◽

10.1007/s10470-009-9428-1 ◽

2009 ◽

Vol 65 (1) ◽

pp. 115-121

Author(s):

Jia Peng Zheng ◽

Ruth Wei Li ◽

Peng Ren ◽

Yi Yang ◽

Chien Chun Shao ◽

...

Keyword(s):

Spread Spectrum ◽

Tuning Range ◽

Clock Generator ◽

Error Feedback ◽

Third Order ◽

Order Error ◽

Delta Sigma Modulator ◽

Delta Sigma

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Certain Mathematical Instruments for Graphically Indicating the Direction of Refracted and Reflected Light Rays

Proceedings of the Royal Society of Edinburgh ◽

10.1017/s0370164600016618 ◽

1906 ◽

Vol 25 (2) ◽

pp. 806-812

Author(s):

J.R. Milne

Keyword(s):

Optical System ◽

The Other ◽

Reflected Light ◽

Light Rays ◽

Image Position

The refraction equation sin i == μ sin r, though simple in itself, is apt to give rise, in problems connected with refraction, to formulæ too involved for arithmetical computation. In such cases it may be necessary to trace the course through the optical system in question of a certain number of arbitrarily chosen rays, and thence to find the course of the other rays by interpolation. Thelinkage about to be described affords a rapid and accurate means of determining the paths of the rays through any optical system.

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Origin of the residual line width under frequency-switched Lee–Goldburg decoupling in MAS solid-state NMR

Magnetic Resonance ◽

10.5194/mr-1-13-2020 ◽

2020 ◽

Vol 1 (1) ◽

pp. 13-25 ◽

Cited By ~ 1

Author(s):

Johannes Hellwagner ◽

Liam Grunwald ◽

Manuel Ochsner ◽

Daniel Zindel ◽

Beat H. Meier ◽

...

Keyword(s):

Solid State ◽

Line Width ◽

Floquet Theory ◽

Pulse Sequence ◽

Second Order ◽

Magic Angle ◽

Line Broadening ◽

Third Order ◽

Order Error ◽

Error Terms

Abstract. Homonuclear decoupling sequences in solid-state nuclear magnetic resonance (NMR) under magic-angle spinning (MAS) show experimentally significantly larger residual line width than expected from Floquet theory to second order. We present an in-depth theoretical and experimental analysis of the origin of the residual line width under decoupling based on frequency-switched Lee–Goldburg (FSLG) sequences. We analyze the effect of experimental pulse-shape errors (e.g., pulse transients and B1-field inhomogeneities) and use a Floquet-theory-based description of higher-order error terms that arise from the interference between the MAS rotation and the pulse sequence. It is shown that the magnitude of the third-order auto term of a single homo- or heteronuclear coupled spin pair is important and leads to significant line broadening under FSLG decoupling. Furthermore, we show the dependence of these third-order error terms on the angle of the effective field with the B0 field. An analysis of second-order cross terms is presented that shows that the influence of three-spin terms is small since they are averaged by the pulse sequence. The importance of the inhomogeneity of the radio-frequency (rf) field is discussed and shown to be the main source of residual line broadening while pulse transients do not seem to play an important role. Experimentally, the influence of the combination of these error terms is shown by using restricted samples and pulse-transient compensation. The results show that all terms are additive but the major contribution to the residual line width comes from the rf-field inhomogeneity for the standard implementation of FSLG sequences, which is significant even for samples with a restricted volume.

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The Static Aeroelasticity of Slender Configurations

Aeronautical Quarterly ◽

10.1017/s0001925900002675 ◽

1963 ◽

Vol 14 (1) ◽

pp. 75-104 ◽

Cited By ~ 7

Author(s):

G. J. Hancock

Keyword(s):

Static Stability ◽

Aerodynamic Forces ◽

Flexible Aircraft ◽

The Past ◽

Plate Models ◽

Non Linear ◽

The Future ◽

Static Aeroelasticity ◽

Definition Of ◽

Image Position

SummaryThe validity and applicability of the static margin (stick fixed) Kn,where as defined by Gates and Lyon is shown to be restricted to the conventional flexible aircraft. Alternative suggestions for the definition of static margin are put forward which can be equally applied to the conventional flexible aircraft of the past and the integrated flexible aircraft of the future. Calculations have been carried out on simple slender plate models with both linear and non-linear aerodynamic forces to assess their static stability characteristics.

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