The application of spiral lenses in electron guns for cathode
ray tubes
Citation for published version (APA):
Spanjer, T. G. (1989). The application of spiral lenses in electron guns for cathode ray tubes. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR298472
DOI:
10.6100/IR298472
Document status and date: Published: 01/01/1989 Document Version:
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The application of spiral lenses in
electron guns for cathode ray tubes
PROEFSCHRIFT
TER VERKRIJGlNG VAN DE GRAAD VAN DOCTOR AAN DE TECHNISCHE UNIVERSITEIT EINDHOVEN,
OP GEZAG VAN DE RECTOR MAGNIFICUS PROF. IR. M. TELS, VOOR EEN COMMISIE AANGEWEZEN DOOR HET COLLEGE V AN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP
VRIJDAG 17 FEBRUARI 1989 TE 14.00 UUR
door
TJERK GERRIT SPANJER
Dit proefschrift is goedgekeurd door de promotoren: Prof.Dr.Ir. H,L. Hagedoorn en Prof.Odr. K.D. van der Mast Copromotor; Dr. W.M. van Alphen
The work described in this thesis has been carried out at the Philips Re~ search Laboratories Eindhoven as part of the Philips Research programme.
dla mojej iony Ewy
Contents
1 Intl;oduction 1.1 Electron optics
1.2 Lens design . . . . . . . 1.3 Scope of this thesis . . . __
2 A genel;alized compadson of sphedcal aberration of magnetic and 1 I 2 5 electrostatic lenses .••.•••••••••.••••••.•••...•.... 9 2.1 Introduction . . . 9 2.2 Theoretical background .... ___ . . . . . . . . . . . . . . . . . . . . .. 10 2.3 Results . . . 14 2.4 Discussion and conclusions . . . __ . _ . . . . . . . . . . . . . . . .. 17
3 Optimization of the configuration of spiral lenses for minimum
spherical aberration .••...•...•..••..•••.••....••.•• 18 3.1 Introduction . . . __ . _ . _ . . . 18
3.2 Geometry . . . 21
33 The potential distribution within the spiral lens ., ____ . __ . .. 24 3.4 Ray tracing and calculation of the spherical aberration .... _ _ _ 30 3.4.1 Ray tracing in rotationally-symmetric electrical fields .... 30 3.4.2 Spherical aberration . . . 32 3.5 The optimization procedure . . . _ . ___ .. 35 3.5.1 Formulation of the optimization problem . . . 35 3.5.2 Optimization algorithms . . . 36 3.6 Results . . . _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40 3.6.1 Results of the segment configuration optimization . . . 40
3.6.2 Analysis of the optimum ~egment configurations and axial po-tential distributions . . . ___ . . . ___ . . . . . . .. 44
3.7 Discus~ion _____ . . . __ . . . ____ . . . . .. 50
Appendix: De~cription of the "Stirling Engine" optimization algo-rithm. . . . ___ . . . __ . . . _____ . .. 55
4 Construction and manufacturing process of the glass gun and spiral lens . . . • . . . • . . . • . • . . . • . . . • . . . 4.1 Introduction
4.2 The glass~gun construction . . . _____ . . . -4.3 The spiral lens _ . . . ___ . . . - - . . . . 4.4 Tube processing __ . . . ___ . . . ____ ..
5 Design and evaluation of the glass gun with spiral lens ill a cathode ray lube for projection television • . . . 5_1 Introduction . . . ___ . . . __ . . . . 5.2 Design of the spiral lens . . . __ . . . - . . . . 5_3 Design of the beam-forming part ... ____ . . . _
5.3.1 Introduction . . . ___ . . . - - - . . . .
5.3_2 Electron-optical design __ . . . ___ . . . . 5.3.3 Electrical gun properties . . . ___ . . . ___ .. 5.3.4 Trajectory and phase-space calculations __ . . . __ _ 5.3.5 Simulation of the complete electron gun _ . . . __ 5.4 Experimental ... _____ . . . ____ . . . .
5.4_1 Introduction ... _______ . . . ______ . . . .
5.4.2 Experimental set-up
5A.3 Electrical measurements __ . . . ___ . . . . 5.4.4 Projected beam protlles and divergence angles . . . __ _
5.4.5 Spot sizes . . . _____ . . . _____ .. . 5.4.6 Experiments with simplified gun designs _ . . . . 5.5 Discussion _. _ . _ . . . ____ . . . _____ .
6 A model to determine the electron-spot size of rotationally
symmet-62 62 65 68
74
76 76 78 80 80 82 83 85 90 92 92 94 98 99 103 108 III6.1 Introduction 6.2 The triode section
6.3 The prefocusing and main lens section . . . . 6.4 The spot growth due to space-charge effects in the drift region 6.5 Application of the model to two types of electron guns ... .
6.5.1 Introduction . . . . 116 119 129 138 142 142
6.5.2 The spot sizes of the electron gun for projection television . 143 6.5.3 The spot sizes of the electron gun for a color CRT ... 149 6.6 Discussion and conclusions . . . 156
Summary l60
Samenvatting 163
Nawoord . . . , . . . 167
1 Introduction
1.1 Electron optics
Although electrons and photons differ in properties such as mass, charge and velocity, many laws that govern light rays can also be applied to electron beams. Therefore the tenn "electron optics" is used when we deal with the behaviour of electrons in electric and magnetic fields. The tenn "ion optics" is mostly used to describe -the behaviour of other charged particle beams. If we introduce the electron-optical index of re-fraction by relating it to the velocity of the electrons [IJ, we can treat electron optics in the same way as light optics. This is due to the fact that light rays satisfy Fennat's principle of least action, which is defined as the line integral of the refractive index along the trajectory of the light ray, while the electron trajectories obey Maupertuis' principle of least action, which is defined as the line integral of the momentum of the electron along its trajectory.
Light optics has been studied for about 4 centuries, while the study of electron optics dates back only about 50 years. Therefore, many terms used in electron optics have been taken directly from the older field of light optics. It might even appear that electron optics is only a special kind of light optics. It will become clear that this is not true if electron optics is examined more closely, There exist a few fundamental limita-tions to the analogy between the two fields and these must be considered when we try to apply the laws of light optics to electron optics.
One of the main differences is the way in which photons and electrons are forced to follow a certain trajectory: a photon trajectory is influenced by the index of optical refraction which is a material property, while the refractive index in electron optics is determined by electric and magnetic
2 Chapter I
fields. Therefore, the photon trajectory changes abruptly at the boundary between two materials of different indices of refraction, while the changes in electron optics are continuous. Some exceptions to this general rule can also occur: e.g. in graded index optical fibers and the human eye the refractive index varies continuously and, with the help of conducting foil lenses at different potentials, abrupt changes of the electron optical re-fraction index can be obtained. Another major difference is that the refractive index for light is usually between 1 and 2.5, while the refractive index in electron optics can vary enormously. Values as high as 1000 are often used. A third difference is the fact that it is very difficult to correct electron-optical lenses for third-order spherical aberration, whereas it is well known that aberrations in light optics can be avoided by the super-posilion of various lenses or the application of aspherical lens elements. Finally, the repulsive Coulomb interaction between charged particles has nO analogy in light optics.
1.2 Lens design
The development of electron optics started after the construction of the first magnetic electron lenses by H. Busch in 1926 [2]. The first electrostatic lenses consisting of two cylinders at different potentials were built by Davisson and Calbick in 1931 [3[. The new optics developed rapidly, considerably aided by the application in electron microscopes and radio valves. Well-known classical applications include cathode-ray tubes, tranSmiSSIOn and scanning-electron microscopes, electron spectrometers, microwave generators and amplifier tubes. More recent applications are electron and ion microprobes and ion implanters.
One of the most important goals in electron optics is the quest for the aberration-free lens. As early as 1936 it was shown by Scherzer [ 4] that all ordinary rotationally-symmetric lenses are affected by positive spheri-cal aberration, which means that rays going through the outer zones of the lens are refracted more strongly than paraxial rays- Since then, the search has continued to find ways to circumvent the proof of Scherzer's theorem, for instance by introducing gauze lenses [5, 6J, or by the use of complicated non-rotationally symmetric field distributions [7]. An
over-Introduction 3
view of all these attempts can be found in Septier's work: "The struggle to overcome spherical aberration in electron opticsN
[8].
An attempt has also been made to calculate the field distribution necessary to give zero spherical aberration for a magnetic lens [9], but, unfortunately, the resulting field was not strong enough to form a real image. It was also shown that aberration-free electrostatic lenses do not form a real image either [10]. Therefore, it seems better to concentrate the efforts on the minimization rather than the elimination of aberrations. Tretner was the first to apply the calculus of variations to determine the lower limits of the spherical and chromatic aberration coefficients for both magnetic and electrostatic lenses [IIJ. He does not give a general solution for the field distribution needed to achieve these limits. He managed to simplify the mathematical problem by a series of variable transformations for special cases only.
The calculus of variations is in principle a suitable approach to min-imize aberrations, because all aberration coefficients can be expressed as definite integrals. The difficulty is that the integrand is a complicated function of the unknown axial electrostatic and magnetic field distrib-ution and their first and second derivatives and of the two linearly inde-pendent solutions of the paraxial ray equation. This equation is a second-order differential equation with coefficients depending upon the same field distributions and their derivatives. Constraints on the field distribution and the trajectories of the paraxial rays can be added by us-ing Lagrange multipliers [12]. This leads to a system of three high-order, non linear, differential equations. In the case of magnetic lenses, these equations are of the second order, and they have been solved by Moses. In this way he designed a coma-free magnetic lens with low spherical ab· erration [13].
Unfortunately, the situation is more complicated for electrostatic lenses, where a similar procedure leads to coupled fourth-order differen-tial equations with complicated inidifferen-tial conditions [14]. Until now the solution of such a system has been intractable. If one succeeds in solving the complex system of differential equations, the final result is the opti-mum axial electrostatic Or magnetic scalar potential distribution, which still has to be translated into the electrode Or pole piece configuration that will produce this optimized distribution. The optimum axial potential
4 Chapter 1
distribution is given in the form of a set of numerical values. In order to extend this distribution off axis to obtain equipotential surfaces for the electrodes, the axial potential has to be approximated by a continuous,
2(n - 1) times differentiable function, where n is the number of terms used
in the Taylor power expansion. Unfortunately, the value of n needed for a good convergence is high, while at the same time the values of the higher derivatives as produced by numerical methods will be inaccurate. Szilagyi [12, 14] has circumvented this difficulty by replacing the axial field distribution by a continuous curve constructed of cubic splines- The Taylor power expansion is formally not valid for an axial function with discontinuous higher derivatives, but it should be noted that the focusing properties of the lens and the third order aberrations do not depend on Lhese derivatives either. Szilagyi used the technique of dynamic pro-gramming to optimize the field distribution of electrostatic lenses in fo-cused ion-beam systems_ In this way it was found to be possible to make the reconstruction process of the electrodes an integral part of the opti-mization procedure. The electrode configuration can also be recon-structed using the charge density method [15], but this method requires excessive computer time in order to obtain a reasonable accuracy [16].
Another method developed by Chu and Munro [17] for designing combined focusing and dual-channel deflection systems for electron beam lithography minimizes the aberrations by meanS of an optimization pro-gram. The program uses the damped least squares method [18] for mini-mizing a weighted sum of squares of aberrations, subject to specified physical constraints, as a function of a set of variables. These variables represent the position, size, and strength of each element present in the optical system. In each optimization cycle, the field is calculated using the charge-density method, the paraxial ray equation is solved and the aberration integrals are evaluated. This method, however, does not opti-mize the shape of each of the electron-optical elements,
Most of the work on electron lenses was carried out in connection with the development of the electron microscope. It appeared that, within the limits set by the electron microscope, such as the short working dis-tance and the large magnification, magnetic lenses were affected by less spherical aberration than electrostatic Ones_ As will be shown in this thesis, this superiority of magnetic lenses need not necessarily be true in
Introduction 5
the field of cathode ray tubes (CRTs) with a magnification varying be-tween 1 and
to.
1.3 Scope of this thesis
This thesis describes the optimization and application of a new kind of electrostatic main lens in electron guns for CRTs: the spiral lens. This lens comprises a helical structure, machined in the resistance layer which is applied on the inside of a tubular glass envelope. The use of these lenses has been made possible by the development by G.A.H.M. Vrijssen at Philips Research Laboratories Of a stable, homogeneous, high-resistivity layer which can withstand the tube processing at a temperature of about 400°C. The interruptions between the windings can withstand electrical fields up to 20 k V Imm.
By varying the configuration of the helix:, it is possible to improve the axial potential of the spiral lens for lower spherical aberration. The spiral lens construction has been combined with a novel technique for mounting the electrodes of the beam-forming part of the gun in the same tubular glass envelope without the use of multiform rods. This technique allows the manufacture of accurate, well aligned, reproducible and compact electron guns intended for high brightness and high resolution.
The electron guns in which the spiral lens has been applied belong to the class of triode electron guns [19]. In this type of gun shown in Fig. t -1, the electron beam is extracted from the cathode and converged into a crossover by the electric field in front of the cathode, which acts as a positive lens. The size of the crossover is determined by the spherical aberration of this cathode lens and the initial velocities of the electrons at the cathode. After passing through the crossover, the electrons diverge towards the main lens, which images the crossover upon the screen of the CRT.
Often an extra lens, the prefocusing lens, is present betwe~n the crossover and the main lens to adapt the aperture angle of the beam emerging from the triode to the main lens. In the drift regions between crossover and main lens, and between main lens and screen, the beam is influenced by space-charge effects. The space-charge force acts as a negative lens, which can introduce extra aberrations. In convergent beams it sets a lower limit
6 Chapter I screen
---~~~~--~~~--- -==-==:::---co/hode IMS prrdocusi()g IMs ---mal/) lensFig. 1.1 Schematic drawing of an ekctmn gun with spiral lens for a cathode ray tube. to the spot size which can be obtained at the screen. The spherical aber-ration of the main lens often contributes significantly to the total spot size at the screen. This thesis shows how the performance of these lenses and thus of the electron gun can be improved significantly by the use of spiral lenses in a resistance layer.
In chapter 2 a criterion is worked out to compare the spherical aber-ration of electron lenses, which is especially useful in the field of CR Ts. This criterion can be applied when a small object of finite size is imaged on a target with a magnification between 1 and to. It is proportional to the minimum obtainable spot size due to the combined effect of magnified object size and spherical aberration of the main lens. Results are pre-sented for a magnetic lens and for two kinds of conventional electrostatic lenses.
This criterion is used in chapter 3 to improve the configuration of the spiral lens for lower spherical aberration. The optimization has been car-ried out for an electron gun to be used in a CRT for projection television. Graphs are presented of the resulting axial potential, and the low spheri-cal aberration of this lens is demonstrated by the use of acceptance dia-grams.
Chapter 4 deals with the manufacturing process of the electron gun, which consists of an accurately preformed tubular glass envelope in which
Introduction 7
the electrodes of the beam forming part of the gun are mounted. The technology of the resistance layer with the helical structure which is ap-plied On the inside of the tubular glass envelope is also described.
Chapter 5 is devoted to the design and evaluation of a triode electron gun with a spiral lens for a projection CRT. Calculated phase-space dia-grams of the beam emerging from the triode section are presented. Graphs are given of the calculated and measured spot sizes at the screen and of the measured aperture angles.
In chapter 6 a model is presented to determine the contributions to the spot size due to thermal velocities at the cathode, cathode- and main lens aberrations, and defocusing and aberrations caused by space charge in the drift region towards the screen. The model is partly based upon theories derived from simulations and experiments on the behaviour of the triode and prefocusing sections of electron guns carried out by van den Broek and van Gorkum, two former members of the Display tubes group at Philips Research Laboratories, The model is applied to an electron gun for projection television, where space-charge effects are rel-atively unimportant, and to a gun for a large color CRT, where space charge has a significant effect on the spot size. The model is used to esti-mate the consequences of replacing a conventional electrostatic lens by a spiral lens with low spherical aberration, both for guns which are equipped with a cathode with a low current density and for guns equipped with a high current-density cathode.
Chapter 2, in a slightly different form, has already been published in Optik, 72, 4, (1986), 134. Most of the other chapters were also written in a form suitable for future publication. Consequently, there will be some overlap between parts of these chapters. On the other hand, this choice has the advantage that these parts can be read independently.
References
[I] kR EI-Kareh, J.c' EI-Kareh. Electron Beams, Lenses and Optics, Academic Press, New York and London (l970) vol I.
[2] H. Busch, Ann. Physik 81 (1926) 974.
[3] e.l. Davisson, C.l. Calbick, Phys. Rev. 38 (1931). [4] O. Scherzer, Z. Phys. 101 (1936) 593.
[5] O. Scherzer, Optik 2 (1947) 114.
[6] J.L. Verster, Philips Res. Repts. 18 (1965) 465. [7] J.e. Burfoot, Proc. Phys. Soc. Lond. 668 (1953) 775.
8 Chapter 1
[8] A. Septier, in Advllnces in Optical and Electron Microscopy, edited by R. Barer lind V.E. Cosslett, Academic Press, London and New York (1966) 1 204. [9] W. Glaser, Z. Phys. 116 (1940) 19.
[10] A. Recknagel, Z. Phys, 117 (1940) 67. [11] W_ Tretner, Optik 16 1 (1959) 155.
[12] R. Courant, D. Hilbert, Methods of Mathematical Physics, Interscience Pub-lishers Inc., New York and London (1953) 1 164.
[\3J R.W. Moses, in "Image Processing and Computer-Aided Design in Electron Optics·, edited by P.W. Hawkes, Academic Press, London lind New York
(1973) 250_
[14] M_ Szilagyi, in "Electron Optical Systems' by U. Hren e.a., Eds. AMP O'Hare, II: SCllnning Electron Microscopy, Inc. (1984) 75.
[15] P_W_ Hawkes, J. Phys. E: Sci. lnstrum. 14 (1981) 1353. [16] M. Szilagyi; Proc. of the lEE, 733 (1985) 412.
[17] H. Chu, E. Munro, J. Vac- ScL Techno!. 194 (1981) 1053. [l8] K. Levenbe,g, AppL Math. 2 (1944) 164.
[19] H_ Moss, Narrow Angle Electron Guns and Cathode Ray Tubes. Adv, Elec-tronics and Electron Physics, Supp!.3, Academic Press, London and New York (1968).
2 A generalized comparison of spherical aberration of
magnetic
and electrostatic lenses
Abstract
9
A general comparative basis for the spherical aberration of lenses is given. The criterion is applicable to both magnetic and electrostatic lenses of various types. Re-sults for a magnetic and for two electrostatic lenses show that the quality of these lenses as applied in CRTs is nearly equal. Simple approximation equations in terms of geometrical parameters are derived.
2.t Introduction
Both magnetic and electrostatic lenses may be used to focus electron beams at the screen of cathode ray tubes (eRTs). The main defect of these lenses is spherical aberration, causing rays that pass through the outer zoneS of the lens to be refracted more strongly than paraxial rays
[I
J.
This defect contributes to the spot size on the screen,The history of the development of electron microscopes [2] has shown that magnetic lenses are superior to electrostatic lenses in the sense that they introduce less spherical aberration. It should, however, be pointed out that this is true within the boundaries set by the electron microscope. The superiority of magnetic lenses does not necessarily apply to the field of CRTs as has often been presumed [3, 4]. In CRTs the value of the magnification lies typically between 1 and 10. The calculations by Dosse [5J that are often referred to, were performed for infinite magnification and were based on simple approximation equations for the magnetic and electrostatic fields in the lenses.
In this chapter a better suited criterion is gjven to compare lenses as applied in CRTs. It enables a comparison of a wide variety of magnetic and electrostatic lenses. The new criterion is explained in section 2.2.
IO Chapter 2
Results for a magnetic lens and for two types of electrostatic lenses (bipotential and unipotential) are given in section 2.3. The results are discussed in section 2.4.
2.2 Theoretical background
In order to develop a comparative basis for the spherical aberration in different types of lenses, three starting points are taken for the analysis:
I the object to be imaged at the screen is of a fixed brightness
2 the spherical aberration is calculated for the optimized aperture angle of the beam (optimum beam diameter in the lens)
3 the spherical aberration is expressed in terms of geometrical parame-ters.
The first point is stated in more detail as follows: the brightness of the object is inversely proportional to the so-called Helmholtz-Lagrange product . This product has been derived in many text books on electron optics [4, 6J and is equal to:
(2.1 )
where
ro
is the radius of the (virtual) object,IX the maximum aperture angle of the object rays (this angle can be varied with the help of a prefocusing lens, which is positioned between the triode and the main lens),
V" the potential of the object space.
The quantity H
1..0
sets a lower limit to the spot size that can be obtained in an electron optical system. In practice this lower limit is not attained due to aberrations of the lenses and space-charge effects.As to the second point: the magnification of the object size and the spherical aberration of the main lens both contribute to the spot size in the minimum cross section of the beam, which is located near the paraxial image plane of the lens. The contribution M( ro of the object magnification
A generalized comparison of spherical aberration of lenses II
can be found from the Helmholtz-Lagrange product H~ at the object side of the main lens as
(2.2)
where M[ is the linear magnification of the lens,
P
is the maximum aper-ture angle of the rays at the image side of the lens and Vi is the potential at the image side of the lens.The contribution rsa at the minimum cross section due to the spherical aberration of the main lens is usually expressed as [ 7]
M[Cs 3
'sa
=
~~4-(X , (2.3)where Cs is the third order spherical aberration coefficient of the lens. It can be shown that C .• is a fourth order polynomial in I! M/ [8]. The factor 4 in Eq. (2.3) arises from the fact that the minimum croSS section of the beam is not located at the paraxial image plane, but at a small distance in front of this plane [9]. The radius of the aberration circle at the mini-mum cross section is a quarter of the radius of the aberration circle at the paraxial image plane. We can also relate rsa to the beam aperture angle
p
at the image side of the lens, yielding(2.4)
where C is the spberical aberration constant with respect to the image side of the lens. It is related to the spherical aberration constant Cs by
(2.5)
where Vo is the potential at the object side of the lens. The angles (X and j3 are coupled to the linear magnification via the equation
fJiV:
12 Chapter 2
Since the total spot size at the image plane consists of one term which is inversely proportional to the aperture angle {3 and One term propor-tional to {33, it can be minimized with respect to
fl.
If we assume that rS<1can be added linearly to the magnified object size M/ ro, the result for the minimum obtainable spot size is
(
HLo
)3/4 1/4
rs min
=
krFt
C , (2.7) where kr=
(4/3)3/4=
1.24. The optimum aperture angle flmin is given by. = (
4 HLa )
1/4f3
mm ! T ? "3
c.",
Vi(2.8)
Linear addition is justified when ro is predominated by spherical ab-erration of the cathode lens and an ordinary 'integral' prefocusing lens is used to adapt the aperture angle of the beam as emerging from the triode part of the gun to the main lens" In this case ro and rS<1 are both
deter-mined by the same outer ray of the electron beam.
This is no longer true when the integral pre focusing lens is replaced by a selective prefocusing lens [9], which acts close to the crOSsover formed in the triode section and refracts the rays selectively" Hence one should not calculate ro and r$<1 independently and add them afterwards"
Nevertheless, it has been shown [ 9] that in this case the minimum ob-tainable spot size can also be written in the form of Eg. (2"7), only with a drastically reduced proportionality constant k, of 0.667.
Linear addition is not justified either when the object size '0 is mainly determined by thermal velocities of the electrons at the cathode, but there are indications that linear addition still yields a reasonable approximation of the total spot size [101. However, one can easily derive that even when M/ '0 and rsa are added quadratically, the minimum obtainable spot size
can still be written in the form of Eq. (2.7) , now with a value for kr of 0.937.
In a CRT, the thermal velocities of the electrons at the cathode, ab-errations of the triode section of the gun and of the main lens, and space-charge effects in the drift region between lens and screen all
con-A generalized comparison of spherical aberration of lenses 13
tribute to the spot size_ An overview of the dependency of these con-tributions On the aperture angle will be given in Chapter 6.
Expression [2.7] shows that
ct/
4 is directly proportional to the mini-mum obtainable spot size. Consequently CI/4 can be used as a parameter for the comparison of the spherical aberration of electron lenses.Fig. 2.1 A schematic overview of the geometrical parameters PI> Qt, and Dt used to compare different types of lenses in a CRT.
As to the third point: the lenses to be compared are used to image an object at a screen. A schematic drawing of the relevant geometric pa-rameters is given in Fig. 2.1. The distance QI between the midplane of the main lens of the electron gun and the screen is dependent on the screen size and the deflection angle. It is a fixed distance for a particular type of tube with given screen size and deflection angle.
The length of the gun, being approximately equal to the distance PI
between the object (the crossover Or cathode) and the midplane of the lens, contributes to the length of the tube, and is consequently a critical parameter.
It should be noted that these distances PI and Q; need not necessarily correspond to the electron optical object and image distances, especially not for bipotential lenses.
14 Chapter 2
The diameter D[ of the lens is restricted in practice by constructional considerations such as the neck diameter of the tube.
The quantities PI' Q[, and DI are chosen as geometrical parameters for the comparison of the spherical aberration, because they correspond to practical boundary conditions in the design of lenses for CR Ts.
2.3 Results
The spherical aberration parameter
ct
l4 was calculated for a magnetic lens as shown in Fig. 2-2a: a coil lens is shielded with iron, leaving a gap of width OJ D/ between two cylinders of equal diameter. It has been shown [6] , that the magnetic flux density along the axis can be approximated by an analytical expression under two assumptions:~
---
..
~-.-.-.-.-.-.-.-.---.-.-=-:---'="'"~JfJa
Clxis ~ mogneflc Q ---=---=-===-: - -- -- -_. -- - -_. - . QipofenflQI..
... b .-.-.-.~-".u;.-.-"-.-.-.-.--~. _ _ _ ~___ vnrpQt(:nfl()t D~is +----~---~I.~'~---~~---~ ~Fig. 2.2 Magnetic (a), e1ect.-ostatic bipotcutia.l (b) and electrostatic unipotential (0)
lenses for which the spherical aberration has been evaluated. The gaps are all O.ID[ in width. In this regime the results are insensitive to the choice of the width.
A generalized comparison of spherical aberration of lenses
The pole pieces are unsaturated; Le., the core material has a high sat-uration flux density.
2 The magnetic scalar potential varies Jjnearly between the pole pieces. This assumption is well fulfilled in cases where the gap width does not exceed 2D,.
The equations of motion were solved with a third order Runge Kutta method [11] to obtain the paraxial trajectory for a ray starting in the object plane with zero initial radius. The spherical aberration parameter
cl/
4was derived using Eq. (2.5) and the standard integral for the spherical aberration coefficient Cs, which is a function of the axial flux~density
distribution, its derivatives and the paraxial trajectory.
For the electrostatic lenses two types of lenses were chosen. The first type, as drawn in Fig. 2.2b, is an immersion or bipotentiallens consisting of two cylinders of equal diameter at different potentials and with a gap of width O.lD,_ The second type, shown in Fig. 2.2c, is the einzel or unipotential lens consisting of a central electrode with length D/, and two outer cylinders of equal diameter at the same potential and with gap widths of O.ID,. (For values below 0.5D1 the gap widths are of minor in-fluence). The spherical aberration parameters CI/4 were calculated using
the data of C.s given by Harting and Read [8].
The value of
d/
4 was calculated for values of PI between 50 androo
mm and for Q/ between ISO and 210 mm typical for a projection type CRT. The diameter D, was chosen between 12 and 30 mm for the electrostatic lenses, and between 18 and 36 mm for the magnetic lens.Fig. 2.3 shows some of the results that were obtained. The value of CI/4 is plotted as a function of PI for values of QI equal to 150 and 210 mm. The diameter of all three lenses was 30 moo. The figure shows that that there are only small differences in the spherical aberration quality of the three types of lenses. Moreover PI has only a slight effect on the quality, whereas the influence of {1 is relatively strong.
The data for the three types of lenses in the mentioned range can be summarized in the following empirical equation:
16 5~---~----. {m'I'J 3 2 1.0 '::,-",
"'<::::-.-.
- _ -.--.. uni ''---. bi mog '::--'.
... :::::--.----. __ uni - - - - bio(
=150mm 60 80F/---mag 100 /mmJ Chapter 2
Fig. 2.3 Thl;: ~pherical aberration parameter C'14 for the three types of lenses with a diameter D, = 30 mm, a distance Q, between midplane of the lens and screen of 150 mm (lower three curves) and 210 mm (upper three curves) as a function of the
di~tance PI between object and lens.
Moreover, kc is neatly equal for all three lenses:
kc
=
[1.54±
0.10] bipotential kc=
[L59±
0.06J unipotentialkc = [1.45
±
0.06J magnetic.(2.9)
A generalized comparison of spherical aberration of lenses 17
The variation in kc for one type of lens is mainly due to the fact that the exponents in Eg. (2.9) are not exactly equal for the three types of lenses as can easily be seen from Fig. 2.3.
2.4 Discussion and conclusions
The results of the previous section show that only marginal differ-ences exist between the three types of lenses for equal geometrical pa-rameters PI, Q" and D, in the ranges investigated. The magnetic lens, normally applied on the outside of the neck of the tube of a CRT, will only exhibit a superior performance due to its larger diameter (Eq. (2.9)). The simple approximation given in Eq. (2.9) shows that the minimum obtainable spot size is proportional to the distance Q, between lens and screen, decreases with the diameter of the lens (DI-1/2) and only slightly decreases with the distance from object to lens (PI-I/4).
The spherical aberration parameter Cl/4will also be used in the next chapter to investigate and optimize the configuration of the spiral lens.
References
[I] W. Glaser, Grundlagen der Elektronenoptik, Springer Verlag, Wien (1952). [2] E. Ruska, The early development of electron lenses and electron microscopes.
Hirzel, Stuttgart (1980).
[3] D.L. Say, Information Display, May 1970,29
[4] P. Grivet, Electron Optics. Pergamon Press, Oxford (1972). [5] J. Dosse, Z. Physik 117 (1941) 722.
[6] A.B. El-Kareh, J.e. EI-Kareh, Electron Beams, Lenses and Optics, Academic Press, New York and London (970) vol L
[7] O. Klemperer, M.E. Barnett, Electron Optics, Cambridge University Press, Cambridge (l971).
[8] E. Harting, F.H. Read, Electrostatic Lenses, Elsevier Scientific Pub!. Comp., Amsterdam (1976).
[9] A,A, van Gorkum, M,H.L,M, van den Broek, j, App!. Phys, 58, 8 (1985) 2902.
[10] J. Hasker, IEEE Trans. Electron Dev. ED1S (1971) 703.
18
3 Optimization of the configuration of spiral lenses for
minimum spherical aberration
Abstract
The configuration of segmented spiral lenses with constant pih:h has been opti-mized to obtain lenses with low 8pherical aberration for application within electron guns for CRTs, The method presented in the previous chapter has been used to yield an objective function for minimization of the spherical aberration, The results are presented as a function of the number of helical segments in tbe spiral lens, the object and image distance, and the lens length and lens radius. General conclusions are drawn for the axial pOLential distribution and its derivatives. It appears that the re· suiting spiral lenses have a quality unacbievabk by conventional electrostatic lenses presently used in CRTs.
3.1 Introduction
Electrostatic lenses for the focusing of charged particle beams have been studied extensively in the literature [1, 2, 3]. Conventional electrostatic lenses in cathode ray tubes (CRTs) usually consist of simple electrodes (combinations of apertures and tubes), which are easy to manufacture and align. The only major problems in manufacturing are electrical breakdown and charging of the insulating surfaces between the electrodes.
However, it remains difficult to evaluate the relation between the properties of such lenses and the large number of characteristic parame-ters. The number of possible electrode combinations is unlimited, and there exists a wide variety of feasible sets of electrode parameters, all leading to different lenses. Even the properties of a simple lens consisting of a pair of rotationally-symmetric tubes depend on a large number of
Optimization of the configuration of spiral lenses 19
parameters, like the voltage ratio of the electrodes, the lengths and the radii of the tubes and the distance between them.
Lens data for a wide range of simple lens geometries consisting of two and three cylinders or apertures have been presented by Harting and Read [3J. These data, comprising the focal lengths, positions of the principal planes and the spherical and chromatic aberration coefficients, were obtained by determining the potential field with the charge density method, and subsequent numerical integration of the appropriate equations of motion.
Szilagyi et a1. have presented a thorough investigation of a class of axially-symmetric two-electrode bipotential [ 4J and three-electrode unipotential lenses [5J. The study is based on the analysis of the axial potential distribution, which has been approximated by a cubic spline. They have examined the dependency of the spherical and chromatic ab-erration coefficients for two voltage ratios on the central and outer electrodes.
The error particularly dominant in electron guns for CRTs is spheri-cal aberration, which is the only error also present for objects on the op-tical axis. In the previous chapter a performance parameter has been introduced to compare the spherical aberration of electron lenses, which is especially useful for lenses in CRTs having a magnification between I and 10. This parameter is directly proportional to the minimum obtaina-ble spot size due to object magnification and spherical aberration of the main lens. It has been shown that for conventional electrostatic lenses which are most often used in CRTs (i.e. the bipotential and unipotential equidiameter cylinder lenses), this parameter can be approximated by a very simple expression only depending on the lens diameter, and the ge-ometrical object and image distances.
The same parameter will be used in this chapter fOr the optimization of a new kind of electrostatic lens: the spiral lens. This lens, developed in the group Display Tubes and Cathodes of Philips Research Laboratories, consists of a helical structure machined in a high-ohmic resistance layer on the inside of a tubular glass envelope. The spiral acts simultaneously as a lens electrode and as a voltage divider. By varying the distribution of the windings, the axial potential of the lens can be optimized to obtain minimum spherical aberration.
20 Chapter 3
There are several approaches which may be used to design lenses with minimum spherical aberration.
- One approach is the trial and error method, in which the designer starts with a given set of electrodes and analyzes the paraxial properties and aberrations of the system. Next he tries to improve the design by vary-ing the geometrical dimensions and electrical parameters. Normally this process is extremely slow because of the vast amount of possible electrode configurations and the considerable amount of computer time needed for the calculation of the potential distribution.
- A second approach is based On the fact that the optical properties and aberrations of any lens field are tOlally determined by the axial potential distribution and its derivatives. Thus, instead of analyzing an enormous amount of different electrode configurations, the designer tries to de-termine the optimum axial potential distribution, which yields a mini-mum spherical aberration coefficient and at the same time satisfies some additional constraints, for instance with respect to the maximum field strength and the focal length of the lens. Tretner 16'1 and Moses [7J have applied the calculus of variations to solve this minimization problem for magnetic lenses, but until now this method has not been successfully applied to the case of electroSLati<.: lenses. Moreover, one is still left with thc mathematically ill-posed problem of how to translate the optimized axial potential distribution into an electrode configuration which would produce this distribution. Szilagyi [4, 5] has circumvented this problem by replacing the axial potential distribution by a cubic spline,
In the .:;ase of a spiral lens, we have the ability to prescribe the po-tential along a cylinder with a fixed radius, by varying the pitch of the spiral. The potential distribution completely determines the axial poten-lial, and through the axial potential also the optical lens properties and the spherical aberration coefficient. We have therefore decided to express the potential along the cylinder as a function of a limited number of pa-rameters which can be easily adjusted during the manufacturing of the spiral in the resistance layer. The optimum lens configuration has been determined by optimizing these parameters to obtain a lens with mini-mUm spherical aberration.
Optimization of the configuration of spiral lenses 21
The optimization of the spiral lens described in this chapter has been carried out for a lens to be used in an electron gun for projection CRTs (see chapter 5). Scaling of its dimensions allows one to translate its properties to lenses to be used in various other electron guns. The ge-ometry of the lens and the parameters used for the optimization of the spiral configuration are described in the next section. In section 33, we will derive how the axial potential distribution depends on the parameters which detennine the configuration of the spiral. The ray-tracing proce-dure and the calculation of the spherical aberration coefficient are ex-plained in section 3_4, and the optimization procedure used is described
in section 3_5. The results of the optimization for various lens configura-tions are presented in section 3.6. The quality of these lenses will be demonstrated by acceptance CUrves. We will also study the dependence of the lens quality on several geometrical parameters. The results will be analyzed and discussed in section 3.7.
3.2 Geometry
The configuration of the spiral lens to be optimized is drawn in Fig. 3.1a. The lens consists of a cylinder with length Ls and radius Rs' As long as R,4.L., the length of the lens field will be mainly determined by L,_ Since the definition of the geometrical centre of the spiral lens in the axial direction is not so convenient as in the case of the conventional, tubular, bipotential and unipotential lenses described in the previous chapter, we have defined the paraxial object and image distances Pc and Q., with respect to the end of the spiral. To enable a fair comparison be-tween the spiral lenses, we have kept these parameters, which are imposed by practical and technical requirements, constant during the optimization procedure_
For technological reasons we have chosen the resistive layer to be provided with a constant square resistance (typically in the order of 3 MQ ). Consequently, the potential distribution along the spiral can only be varied by varying the distribution of the windings along the helix. The spiral has been split up in a number of helical segments alternated with intermediate segments of the same resistance layer without helix. The lengths of the helical segments is.i and intermediate segments It.i are all
22 Chapter 3
I
, '"
~
Po'r"~
0,2R4
L~
.
.::::::l=--~-_=----L=--::._.:-=_-=
___.= __ --. __
-=_ -=--::::!-;"":---==--::::...-=--=-::."""--=----~. ==~_...J.
---·~·---IL
5.1
1
~I
L - -:';---' _ _ - ' - - 'o
z _ ... "... L,fig. 3.1 (a) Geometry of the segmeoted spi,al lens with radius R, and length L.,. Also indicated are the object distance Pe, image distance Q. and the potentials applied to (he spiraL All voltages are normaiiz.ed with rcspect to the cnd potential of the lens. (b) The potential distribution V.(;) along the spiral for the lens of Fig. 3.1(a).
variable. All helical segments are employed with constant and identical pitch. This way of segmenting the lens has the advantage that the axial potential is related to the parameters
i",i
and It,i through a simple summa-tion instead of a convolusumma-tion integral. Furthermore the segmentasumma-tion links up nicely with the way in which the helix is fabricated (see next chapter).In all optimizations we have added a prefocusing part to the spiral lens to obtain a changeover from the end potential Vo of the beam-forming section of the gun to the focusing potential Vjo In this way we can
also include the aberrations introduced by the prefocusing part, which allows for a fair comparison of spiral lenses with a different focusing voltage. We normalized all potentials applied to the spiral to the end potential of the spiral, which is usually equal to the screen potential.
Optimization of the configuration of spiral lenses 23
Since the helical segments act as a potential divider, the potential of the
/h
intennediate segment is given by(3.1)
where Ns denotes the total number of spiral segments, and M~ the number of helical segments in the prefocusing part of the lens (usually just ooe). Hence, if we neglect the deviations from the rotational symmetry due to the helical structure, the potential distribution Viz) along the lens is given by
where
i-I
for z:s; z),
for Zi
+
is,i :-:;;; Z :s; zi+),for Zj :5:; Z :s;; Zi
+
("i' for Z ;;::; z\+
Ls, Zj = ZI+
I(lsJ+
It) j-d (3.2) (3.3)denotes the z-coordinate at the beginning of the jlh helical segment. The potential distribution Viz) along the spiral for the lens of Fig. 3.1a is shown in Fig. 3.1b.
24 Chapter 3
3.3 The potential distribution within the spiral lens
In this section we derive the dependence of the axial potential dis-tribution upon the configuration of the segments along the spiral lens, and the potentials applied to the lens. Let us consider the spiral lens as an infinitely long cylinder of radius R; with a given potential distribution Viz), whereby Vs(z ... ~ 00) = Vo and Vs(z ---'> =) = I , In this
configura-tion, the derivative of Vs(z) will vanish for z --->
±
(Xl. The Laplaceequation V2¢
=
0 reduces in a rotationally-symmetric system to(3.4)
Using the method of separation of variables, we can express the potential ¢(r,z) as
¢ = R(r).2(z). (3.5)
Substitution of Eq. (3.4) in Eq. (3.5) leads to the general solution [ 2J
(3.6)
where a and bare arbritrary constants and Io(kr) and Ko(kr) represent the zero-order, modified Bessel functions of respectively first and second kind. The summation extends over all possible real as well as imaginary values of k.
Since for r ""- 0 the potential should still be finite for all values of z and
Ko (0) --> - 00, we must choose b(k) = 0 for all values of k. Moreover,
since (/J(z,r) should remain finite for all values of z, we must restrict our-selves to real values of k only. Letting k change rather continuously, we can write 1>(r,z) as 1 k
f
oo ¢(r,z)=
211: a(k) Io(kr)d
zdk. -<Xl (3.7)Optimization of the configuration of spirallenses 25
The coefficients a(k) represent the Fourier coefficients of the axial po-tential distribution, which can be written as
f
<Xl I 'k ~(z)=
Tn""
a(k) r! zdk. -co (3.8)If we substitute the boundary condition for r
=
Rs, we obtain(3.9)
The values of the coefficients a(k) can be obtained by applying Fourier's Integral Theorem (FIT) [8J to Eq. (3.9). To circumvent mathematical difficulties when integrating Vs(z) towards infinity, we will apply FIT to the derivative of V.(z), which can be written as
V;Cz) -
2~ f~
a(k)jk I"CkR,) d"dk. - 0 0Application of FIT to Eq. (3.10) leads to
Hence the derivative of the potential 4J(r,z) can be expressed as
o</l(r
,z)oz
(3.10)
(3.11 )
26 Cbapter 3
The derivative of the axial potential with r = 0 equals
(3.13)
The imaginary part of Eq. (3.13) will vanish since Jo(kR.,) is an even function of k. Therefore 4l'(z) can also be expressed as
iI>'(z)
=
~1
fOO
V/(O
B'(
z -()d('
210 Rs Rs - 0 0 (3.14) where B'(z) = _1_
foo
cos k:z dk 210 IO(k) - 0 0 (3.15)represents the derivative of the electron optical B-function, which has been defined by Verster [9J as
B (z)
=
_1_foe
sin kz dk. 211: k Io(k)-00
(3.16)
The function B(z) is often approximated by tanh OJZ , where the fac" tor 0) is chosen so that both functions have the samlslope for z
=
0, i.c.:w = 2B'(O) = - }
fOC
I:(~)
=
1.3262275.- 0 0
(3.17)
This is a good approximation for B(z) and its first derivative as is illus-trated in Fig. 3.2a, but as shown in Fig. 3.2b it is not valid anymore for
OptimizatioIl of the configuration of spiral lenses 27
the third derivative which will be used later on. Here the deviations can amount to as much as 10%. g 1.0 . . . - - - , 05 0 0 -1 -2
o
/
/i /, 0.5 0.5 tonhlw;l:) 2 -B(z) ~ tarl>' (WZ)"-~
,-",...L
8'(z) 1.0 1.5 2.0 25 z _ 1,0 1.5 2.0 2.5Fig. 3.2 A comparison between the electron optical B-function as given in Eq. (3.16) and the function tanh(wz)/2 with w = 1.3262275. The first three derivatives of the two functions are also drawn.
The axial potential can be obtained by integrating Eq. (3.14) towards z, yielding
28 Chapter 3
<I>(z) - V,
+
f~
v/((){
B(
z ;;,( ) - B( - =)}d(,
(3.18:-00
which can be worked out into
<D(z)
~
1-
(vo+
I)+
f=
V/(()B(
Z ;;,()d(
(3,19)-00
Tn our case the potential distribution V,(z) along the spiral is given by Eq. (3.2) Hence we obtain for its derivative
V/(z) = 0
(Vj-
Vol
V'(z) = V ' = for Zj ~ Z ~ Zj
+
Is,i with i.:::; MpS - M,
\ /
. LS,1j=! (3.20)
(I - VI)
V/(z)
=
V+'=
for zi':::; z .$ zi+
Is,1 with Ms < i .:::; N.~.N,
I
Is.1i""M,+i
If we substitute this expression into Eq. (3.19), the convolution integral reduces to a simple summation
Optimization of the configuration of spirallenses 29 1
~[
(z - z·
- I . )(Z Z)]
«l>(z)=T(VO+ 1)- V_'RsL
A ~s S,I -A ~s i+
i=! N,
(3.21)+
V;R,
I
H
Z~~~l',j )~A( Z~Zj)
1
i= M,+!where the function A(z) is the primitive of the electron optical B-function. The derivatives of the axial potential are given by
The last two expressions Eqs. (3.21) and (3.22) relate the axial potential distribution and its derivatives to the configuration of the spiral lens seg-ments determined by IS.i and
'"i
and to the voltages Vo and VI that are applied to the lens. They will be used in the next sections to solve the paraxial ray equation and to determine the spherical aberration coeffi-cient.30 Chapter 3
3.4 Ray tracing and calculation of the spherical aberration 3.4.1 Ray tracing in rotationally-symmetric electrical fields
When the initial direction of an electron trajectory is in a plane through the z-axis, the trajectory will not leave this plane, since in a rotationally-symmetric (RS) electric field there is no variation in the
po-tential as one rotates around the z-axis. The motion of such a meridional electron with a negative charge of e = 1.6 10-19 C and mass m,. = 9_11 10-31 kg can be described by the following equations:
(3.23)
and
N' ow 0/, - -A.
81
an - - can e expresse as a power senes d 8¢ b d . . -JO r, Sinceor oz 2 4 _ q,(r,z)
=
<ll(z) -T
<ll"(z)+
~4 (JlIV(Z)+ ._-,
(3.25) iJ¢(r,z)= _
~
1P"(z)+
r63 <llif(Z)+ ... ,
(3.26)or
2 I DA.(r z) 2 -'1'-,--:,-' --'---=
<ll'(z) - ~ W"'(z)+ ,...
(3.27) oz 4Next we shall eliminate the time variable I by regarding r as a fUnction of z and z as a function of t. Hence we can write
r
= r"il+
r'i. (3.28)where the fluxion indicates differentiation with respect to t and the superscript' indicates differentiation to z. Since the kinetic energy of the electron equals e times the potential of the electron related to the cathode, we can express i2 as
Optimi~tion of the configuration of spiral lenses
i2 2e ¢(r,z)
=
me 1+
r,2 .Substitution of Eqs. (3.23), (3.24) and (3.29) into Eq. (3.28) leads to
r"
=
-=.!.. { -
!!t
+ "
~
}{l
+
r'2}.2<b
or
oz
31
(3.29)
(3.30)
This expression is generally valid for the motion of meridional charged particles in a rotationally symmetric electric field <b(r,z).
We can obtain the paraxial ray equation by substituting Eqs. (3.25). (3.26) and (3.27), neglecting all tenns containing r2 , rJ, etc. and assuming that r'<O. Hence Eq. (3.30) reduces to
<D' cD" r"+~-r'+~-r=O
2cD 4$ , (3.31 )
which is a linear homogeneous second-order differential equation for the distance of the electron to the axis in an RS electric field.
It can be simplified Significantly by introducing a new variable
R
=
r $1/4 which finally yields the well known Picht equationwhere the function I/I(z) is defined as
cD'(z)
I/I(z)
=
Il>(z) .(3.32)
(3.33)
The paraxial ray is obtained by solving the Picht equation numerically with the help of a third order Runge-Kutta method, after Weber [10J . Only two function evaluations are required per step. The paraxial object plane with x-coordinate zp is located at a distance of Pe from the end of
the spiral lens. The initial conditions for the fundamental ray which starts on the axis with an initial angle of 45° are
R(zp)
=
0,32 Chapter :3
The ray is traced through the lens-field and drifted through equipotential space towards the image plane at z-coordinate Zq' which is located at a
distance
Q"
from the end of the spiral lens. The focusing voltage Vr is adjusted iteratively, until the ray is actually focused at the image plane, which is expressed by the condition(3.35) The value of R'(zq) for this ray yields the angular magnification MIX' be-cause we have normalized all potentials with respect to the end potential of the lens. We can also derive the linear magnification M, froro R'(zq) , with the help of the Helmholtz-Lagrange relation [ I1J which reads in our case
(3.36)
The result is
(3.37)
The data of the fundamental ray R(z) are also needed for the com-putation of the spherical aberration coefficient, that will be treated in the next sub-section.
3.4.2 Spherical aberration
Spherical aberration, which is illustrated in Fig. 3.3 is one of the third order geometrical aberrations, that arise from terms in ,3, which we have ignored so far in the paraxial approximation. It is the most important defect of electron optical lenses used in triode electron guns for CRTs and the only aberration also present for rays starting from a point object on the lens axis.
Paraxial rays fonn a focus at the Gaussian image plane, whereas non-paraxial rays, which go through the outer zones of the lens are re-fracted more strongly, and cross the axis in front of this plane. The ab-erration tlgure is a circle in the Gaussian image plane of radius
co-Optimization of the configuration of spiral lenses disc of least confusion Gaussian image plane ~ ______ -4 ______ ~ ________________ ~~~=l~o 33
Fig. 3.3 lmaging of a point axial object by a lens with spherical aberration. The ra-dius of the image in the Gaussian image plane is riU ' Also indicated is the "disc ofIeast confusion" which has a radius of r,.a/4.
efficient defined at the object side of the lens and a the maximum half angle of the rays emanating from the object plane. It can be seen from Fig. 3.3 that the smallest size of the image disc does not occur at the Gaussian image plane, but at a distance proportional to Cs
ri
in front of this plane. The smallest disc, usually called the Hdisc of least confusionlthas a radius of fsa
I
4.It can be shown [9J that CI1(M/), which has the dimension of length is
a fourth order polynomial in 1/M/. Harting and Read have tabulated the values of Cs for a variety of conventional electrostatic lenses [3J.
The integral for the spherical aberration coefficient whose integrand in-volves a complex function of the paraxial ray, the axial potential distrib-ution and its derivatives, has been derived in many text books on electron optics [1, 2, 12]. It is often expressed as
34 Chapter 3
If necessary, it is possible to eliminate the third derivative of ell in Eq. (3.38) by integrating by parts, but then we will obtain extra terms in-volving r'. We have used the integral expressed in tenns of Rand
1/1,
which is written as(3.39)
where the integrand
rs
equals(3.40)
The integration of Eq. (3.40) has been carried out numerically. Since
rs
vanishes in field free space, the limits of integration may also be re-placed by the beginning and end coordinates of the lens field if the field does not overlap with the object and image plane. In the computations we have assumed that the lens field extends over a length of 5 times the lens radius beyond the beginning and end coordinate of the spiral lens.In the previous chapter we have shown that a perfonnance parameter which is well adapted to compare all kinds of electron lenses applied in
eRn
with respect to spherical aberration js given by(3.41 )
where the voltage ratio VN across the lens is in our case equal to IjVo (see section 3.2), and C represents the spherical aberration constant with re-spect to the image side of the lens. The factor C1/4 is directly proportional to the minimum obtainable spot size due to object magnification and
Optimization of the configuration of spiral lenses 35
spherical aberration of the main lens. With the help of Eqs. (3.37) and (3.39) it is easily derived that the factor
c1/
4 can be written as(3.42)
The quantity C1/4 has been used as minimization criterion for the opti-mization of the segment configuration to be described in the next section.
3.5 The optimization procedure
3.5.1 Formulation of the optimization problem
On the basis of the treatment given in the previous sections we can formulate the optimization problem of the segment configuration in the spiral lens in a way that closely reflects the technological possibilities. The objective function, which has to be minimized as a function of the design variables is the factor
d/
4 as given in Eq. (3.42). The computations will be superposed with some noise due to numerical inaccuracies. The inac-curacies are predominated by the tolerances during the ray-tracing with the Runge-Kutta method and the evaluation of the spherical aberration integraL Usually they are in the order of 10-2 - 10-3 relative tod/
4•The design variables are the normalized focusing potential VI and the lengths of the spiral and intermediate segments normalized to the total length of the spiral lens
is,i Xs,i=Y' s
'Ii
xt,;=t-· s (3.43)The parameters that we have kept constant during one optimization cycle are the spiral length Ls, the paraxial object and image distances with re-spect to the end of the spiral P" and
