Applied Charged Particle Optics www.com Helmut Liebl Applied Charged Particle Optics With 124 Figures 123 www. Helmut Liebl Hartstr. 17 85386 Eching Germany Library of Congress Control Number: 2007932728 ISBN 978-3-540-71924-3 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broad- casting, reproduction on microfilm or in any other way, and storage in data banks.
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Typesetting: Data prepared by the Author and by SPI Kolam Cover: eStudio Calamar Steinen Printed on acid-free paper SPIN 11903109 57/3180/SPI 543210 www.com To my dear wife Elfie, and our children Bernhard, Wolfgang, Regina, Christina, Martin, and our grandchildren.com Preface This booklet is essentially an extended English version of a course I taught at the Max Planck Institute for Plasma Physics in Garching/ Munich for physicists and graduate students working at the Institute and for the nearby Physics Department of the Technical University. It covers mostly applications of particle optics which I have designed, built and worked with myself during my career, such as mass spectrom- etry, focusing of ion beams, emission microscopy, ion and electron beam systems, in an energy range of less than 20 keV. It is intended to help physicists who have to design their own appa- ratus or to help them to better understand instruments they have to work with. Some of the subjects described date back quite some time, the oldest references as far back as the thirties in the last century.
And I am old enough to have met some of those authors personally. But the booklet also contains some material from my own file which has not been published previously. I should like to thank Dr. Dietmar Wagner for his invaluable help with the manuscript.
Eching, August 2007 Helmut Liebl www.com Contents 1 Lenses: Basic Optics .1 Simple Transfer Matrices .2 Passage of Charged Particles Through a Uniform Electrostatic Field .3 Transfer Matrix of the Uniform Field .4 Acceleration of Charged Particles Emitted from a Planar Surface .5 Transfer Matrix of Electrostatic Field Between Spherical Concentric Equipotential Surfaces .6 Acceleration of Charged Particles Emitted from a Spherical Surface .7 Passage of Charged Particles Through an Electrode with Round Aperture .9 Passage of Charged Particles Through an Electrode with Slotted Aperture. 39 2 Electrostatic Deflection .1 Parallel Plate Condenser. 59 3 Magnetic Deflection .1 Small Deflection Angles .2 Magnetic Sector Fields .3 Axial Focusing with Uniform Magnetic Sector Field .4 Non-Uniform Magnetic Sector Fields .2 General Toroidal Condenser .5 Uniform Magnetic Sector Fields .6 Non-Uniform Magnetic Sector Fields. 101 5 Fringe Field Confinement .1 Emission Lens Combined with Optical Mirror Objective Lens .2 Combined Objective and Emission Lens .3 Dynamic Emittance Matching .4 Energy Analyzer for Parallel Beam with Coinciding Entrance and Exit Axes .5 Elimination of Transverse Image Aberrations of Sector Fields .6 Energy-Focusing Mass Spectrometers .com 1 Lenses: Basic Optics Summary.
Basic optical formulae are derived, the transfer matrix method is ex- plained, the lens action of apertures is shown, and emission, immersion and einzel lenses are treated. A lens is characterized by the property that it imparts to a ray (particle trajectory) passing through it a deflection (∆r ) which is proportional to the distance r1 from the axis, at which the ray passes, but which is independent of the original slope r1. For thin lenses this deflection may be assumed to be a sharp kink, occurring at the single principal plane P. If the entrance side – left of P – is designated by the index 1, and the exit side – right of P – by the index 2, one can write that the exit distance r2 equals the entrance distance r1 (Fig.1) and the exit slope r2 equals the entrance slope r1 plus the (negative) change of slope ∆r : r2 = r1 + ∆r .2) As stated above, −∆r = cr1 , where c is the proportionality constant.
It can be derived from the special case that the exit ray is parallel to the axis: r2 = 0, r1 = −∆r = cr1 , (1.4) r1 f1 In this case (Fig.2) the entrance ray crosses the axis at the distance f1 from P ; f1 is the entrance focal length of the lens, F1 the entrance focal plane.2) can now be written as r1 r2 = r1 − .com 2 1 Lenses: Basic Optics Fig. Principle of a lens: A trajectory crossing the lens at distance r1 from the z-axis is deflected by an angle ∆r which is proportional to r1 Fig. Trajectories starting from the axis point F – the focal point – leave the lens parallel to the z-axis. The distance of the focal plane F1 to the lens plane P is the focal length f1 Equations (1.5) can be written in matrix form r 1 0 r r = = ML , (1.6) r 2 − 1 f1 1 r 1 r 1 ML is called the transfer matrix of the lens.com 1 Lenses: Basic Optics 3 The transfer matrix is a11 a12 ML = a21 a22 with the coefficients a11 = 1, a12 = 0, a21 = −1/f1 , a22 = 1.
In explicit form one has r2 = a11 r1 + a12 r1 , r2 = a21 r1 + a22 r1. This, with the above coefficients for a lens, yields (1. Another way of describing the action of a lens is in the form of the exit equation of the ray in the z–r coordinate system (Fig.3): r1 r = ar1 + zr2 = ar1 + z r1 − , f1 where a is the distance of the object point A from P. With r1 = ar1 we have a r = a+z 1− r1.
The object point A is imaged to a virtual image point B if a < f1 www.com 4 1 Lenses: Basic Optics Fig. Imaging of an extended object through a lens to a real image The distance of the image point B – the point where the exit ray crosses the axis – from P, i. the image distance b, is obtained from (1. f1 By dividing it by ab, this yields the familiar lens equation 1 1 1 + = .8) a b f1 For a < f1 , as in Fig.3, the image distance b is negative, i.
the image is virtual. For lenses, where the particle energy is the same on both the en- trance and exit side, the focal lengths are also the same on both sides: f2 = f1 = f.9) For the imaging of an extended object (Fig.4) the same rules apply as in light optics: s2 b lateral magnification: M = = , (1.10) s1 a r 1 angular magnification: 2 = .11) r1 M Electrostatic lenses are generally thick lenses which have two princi- pal planes P1 and P2 , and these are usually interchanged as shown in Fig.1 Simple Transfer Matrices The transfer matrix of a lens has been introduced above. Transfer ma- trices become very useful when composite optical systems consisting www.1 Simple Transfer Matrices 5 Fig. Schematic of electrostatic lens with interchanged principal planes P1 and P2 Fig.
Drift space without deflection of several elements in tandem are to be treated. Frequently, just one property of the composite system is of interest, e. the magnification, and therefore only one of the matrix coefficients needs to be calculated, which can often be done very quickly with the aid of a hand calculator. The simplest transfer matrix is that of a drift space (Fig.
From the figure one can see immediately that r2 = r1 + L r1 , (1.13) or in transfer matrix form r 1L r r = = MD .14) r 2 01 r 1 r 1 When two optical elements are combined in tandem, their respective transfer matrices have to be multiplied. For the combination of a lens www.com 6 1 Lenses: Basic Optics Fig. Combination of einzel lens with drift space and a drift space (Fig.7) one has therefore to multiply the transfer matrices of the lens and the drift space: r b11 b12 r b11 b12 a11 a12 r = = , r 3 b21 b22 r 2 b21 b22 a21 a22 r 1 drift space drift space lens r c11 c12 r = .15) r 3 c21 c22 r 1 The coefficients cik are found according to the scheme 2 cik = bis ask , i, k = 1, 2, s=1 or explicitely c11 = b11 a11 + b12 a21 , c12 = b11 a12 + b12 a22 , c21 = b21 a11 + b22 a21 , c22 = b21 a12 + b22 a22. In our example, we have with (1.2 Passage of Charged Particles Through a Uniform Electrostatic Field 7 With these and the coefficients of the lens transfer matrix, one then obtains L c11 = 1 − , c12 = L, f 1 c21 = − , c22 = 1.
f Explicitly, this reads r1 r2 = r1 + L r1 − , f r1 r2 = r1 −. f When the sequence of the two elements is reversed, i. drift space followed by lens, the aik s and bik s are interchanged.2 Passage of Charged Particles Through a Uniform Electrostatic Field Figure 1.8 shows the important case where a charged particle is acceler- ated through a uniform field between the equipotential planes denoted P1 and P2. The spaces to the left of P1 and right of P2 are field free and have the potentials V1 and V2.
The potentials are counted from where the charged particles have zero energy so that their kinetic energy in flight direction at any point in space with the potential Vi is eVi. In this simple case the differential equations of motion can be straightforwardly integrated and yield the motion of the particle in the z–r-coordinate system as a function of time t: m r̈ = 0, mz̈ = eE, z1 = r1 = 0, V1 ż1 = v1 cos α1 , ṙ1 = v1 sin α1 , v1 = 2e m www.com 8 1 Lenses: Basic Optics Fig. Acceleration of charged particle through a uniform field (v1 is the velocity of the particle at energy eV1 , m its mass). eE eE 2 ż = t + v1 cos α1 , z = t + v1 cos α1 · t, (1.17) By eliminating the time t one obtains the trajectory: eE 2 z t + cos α1 · t − = 0, 2mv1 v1 mv1 2eE t= cos2 α1 + z − cos α1 , eE mv12 2V1 E r= sin α1 z + cos2 α1 − cos α1 .18) E V1 This is the equation of the trajectory within the field.