A study of magnetic-deflection errors

Citation for published version (APA):Kaashoek, J. (1968). A study of magnetic-deflection errors. Technische Hogeschool Eindhoven.https://doi.org/10.6100/IR147461

DOI:10.6100/IR147461

Document status and date:Published: 01/01/1968

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https://doi.org/10.6100/IR147461https://doi.org/10.6100/IR147461https://research.tue.nl/en/publications/a-study-of-magneticdeflection-errors(b725e812-eea4-4edc-8ca0-2cf55669620c).html

A STUDY OF

MAGNETIC-DEFLECTION ERRORS

J. KAASHOEK

A STUDY OF

MAGNETIC-DEFLECTION ERRORS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL TE EINDHOVEN OP GEZAG VAN DE RECTOR MAGNIFICUS, DR. K. POSTHUMUS, HOOGLERAAR IN DE AFDELING DER SCHEIKUNDIGE TECHNOLOGIE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP DINSDAG 2 JULl 1968, DES

NAMIDDAGS TE 4 UUR

DOOR

JOHANNES KAASHOEK ELEKTROTECHNISCH INGENIEUR

GEBOREN TE HAZERSWOUDE

DIT PROEFSCHRlFT IS GOEDGEKEURD

DOOR DE PROMOTOR PROF. DR. H. GROENDIJK

Aan Annie, Hans, Erik en Marianne

CONTENTS

I. INTRODUCTION . . . . 1

1.1. General considerations 1 1.2. Historical note . . . . 2 1.3. General outline of this study . 3

2. ABERRATION CALCULATION AS A PERTURBATION PROBLEM . . . . . . . . . . . . . . . . . . . 5

2.1. Introduction and general electron-path equations . 5 2.2. Field distributions and Lagrange functions . 6

2.2.1. Introduction . . . . . . . . 6 2.2.2. The twofold-symmetric set-up 7 2.2.3. Intrinsic asymmetry . 10 2.2.4. Extrinsic asymmetry . . 11 2.2.5. Diagonal symmetry . . 11

2.3. Solution of the path equations 12 2.3.1. Introduction . . . . . 12 2.3.2. The ideal or unperturbed trajectory 13 2.3.3. First-order perturbations . . 15 2.3.4. Second-order perturbations . . . . . 17

3. THIRD-ORDER ERRORS OF TWOFOLD-SYMMETRIC SYS-TEMS: QUALITATIVE CONSIDERATIONS 22

3.1. Introduction . . . . . . . . . 22 3.2. The ideal or gaussian deflection . 22 3.3. Third-order aberrations . . . . 23

3.3.1. Definition of the error coefficients 23 3.3.2. Influence of the deflection magnitude and the ray param-

eters . . . . . . . . . . . . . 25 3.3.3. Influence of the field configuration 27

Appendix I.

Appendix II

Appendix III

3.3.3.1. Introduction . . . . . . 27 3.3.3.2. The shape of H 0(z) . . . 28 3.3.3.3. The deflection-plane-to-screen distance and the

effective field length . . . 32 3.3.3.4. The field parameter H 2(z) 34

38

40

42

4. THIRD-ORDER ERRORS: CALCULATIONS AND MEASURE-MENTS. . . . . 44

4.1. Introduction . 44 4.2. Calculation of the field distribution . 44

4.2.1. Introduction . . . . . . . . 44 4.2.2. The field distribution of arbitrarily shaped single-winding

air coils . . . . . . . . . . . . . . . 45 4.2.3. Application to simplified coil geometries . 49

4.3. Measurement of the field distribution 54 4.3.1. General remarks 54 4.3.2. Practical set-up . . . 55

4.4. The computer program . . 57 4.4.1. The computer input . 58 4.4.2. The computer output 59

4.5. Error-coefficient diagrams . 59 4.6. Experiments . . . . . . . 63

4.6.1. Construction of deflection units 64 4.6.1.1. The saddle coil . 65 4.6.1.2. The toroidal coil. . . . 66

4.6.2. Measurements . . . . . . . . 67 4.6.2.1. A planar-screen measuring device 67 4.6.2.2. The rotating-beam method . . . 69

5. FIFTH-ORDER ABERRATIONS 72

5.1. Introduction . . . . . . . . 72 5.2. Fifth-order-aberration coefficients. 72 5.3. The character of the fifth-order errors . 74

5.3.1. Influence of the field parameters 75 5.4. Computation of the fifth-order errors . 75

5.4.1. Introduction . . . . . . . . . 75 5.4.2. Calculation of the fifth-order coefficients and related errors:

program II . . . . . . . . . . . . . . . . . . . 76 5.4.3. The fifth-order error as a function of z: program III 77 5.4.4. Integration of the path equations: program IV 78

5.5. Aberration-coefficient diagrams. 80 5.5.1. The influence of H 4 • • • 81

5.6. Preliminary experimental results 84

Appendix IV . . . . . . . . . . . 85

6. THE DEFLECTION PROBLEM OF THE 90° SHADOW-MASK TUBE. . . . . . 90

6.1. Introduction . 90 6.2. Basic principles of the shadow-mask tube and deflection require-

ments .....•... 6.3. Registration . . . . . . .

6.3.1. Convergence errors . 6.3.2. Dynamic convergence

6.3.2.1. Spherical screen 6.3.3. Calculation results . .

6.4. Choice of the deflection unit . 6.4.1. Registration . 6.4.2. Colour purity 6.4.3. Coma errors . 6.4.4. Raster distortion

6.5. Asymmetry errors . . 6.5.1. Introduction . . 6.5.2. Intrinsic asymmetry . 6.5.3. Extrinsic asymmetry . 6.5.4. Compensation of anisotropic astigmatism

Appendix V .

Appendix VI .

REFERENCES.

Dankbetuiging

Samenvatting .

Levensbericht .

90 91 91 93 95 96

100 100 102 102 104 106 106 106 108 110

111

111

113

115

116

119

1-

1. INTRODUCTION

1.1. General considerations

One of the operations required for the reproduction of information on the phosphor screen of a cathode-ray tube is the deflection of electron beams. The electrons within the beam are concentrated on the phosphor to form one or more constant or intensity-modulated light-emitting electron spots. The de-flection causes the spots to scan the screen, thus forming a one- or two-dimen-sional visible image, meant to display optically perceptible information.

The amount of information which can be maximally displayed at one time and the fidelity of reproduction depend on a number of factors. Among the most important are those related to the phosphor screen and those related to the moving spots.

As for the screen, the phosphor coverage may be either almost homogeneous, as is usual in monochrome cathode-ray tubes, or may show a certain more or less regular structure as in colour-television-picture tubes. The graininess, after-glow, and saturation effects of the phosphor, and the number of elements on a structure-type screen may influence the information-handling capacity of the display.

The influence of the spot is related to its dimensions and its position on the screen. The dimensions of the spot are primarily determined by the electron-emission system and the focusing mechanism. The electron-emission system consists of one or more electron guns. Unless otherwise stated in the following the electrons are presumed to emanate from one gun. The electron beam is more or less diverging, which necessitates a focusing mechanism, generally situated within a short distance from the gun. Either electric or magnetic fields are used for focusing. Electron-beam generation and focusing is the subject of a vast and still growing wealth of literature which is referred to in the various hand-books on electron optics 1 - 11 •74). Essentially the research in this field is aimed at increasing current density and decreasing spot dimensions or at achieving more light output at a higher resolution.

Apart from some exceptions, it is common practice to determine the position of the spot on the screen by means of two electric or magnetic deflecting fields which are generally situated between the focus system and the screen, producing independent displacements of the spot in two perpendicular directions.

An ideal deflection system is defined as a system which produces a displace-ment of the spot on the screen proportional to the deflecting-field strength whilst leaving the dimensions of the spot unchanged. It is an experimental fact that it is very difficult or impossible to fulfil these requirements in the case of displacements which are not negligibly small with regard to the deflection-system-to-screen distance. Practical deflecting fields introduce unwanted effects

-2-

which appear as a disproportionality between the wanted and the obtained spot position and as an increase in dimensions of the deflected spot. These effects undeniably lead to the distortion and loss of the displayed information and are called deflection errors.

1.2. Historical note

The study of deflection errors is relatively young as it started only about thirty years ago A firm basis for this branch of electron optics was provided by the theoretical work of Busch 15) and Glaser 16•17). Special mention must be made of the introduction by the latter of the electron-optical index of refrac-tion, based on the essential analogy of light optics and electron optics. The so-called third-order errors of deflection were derived along more or less dif-ferent ways by Glaser 18•19), Wendt 20- 23), Picht and Himpan 24•25 ) and Hut-ter 26 •27). Early experimental work aiming at a check of the validity of the theoretical results and the finding of means to decrease specific errors have been reported by Wendt 20 •28 •29), Hutter 26 •30), Schlesinger 31) and others 32 •33).

The relations between the coil geometry and the field configuration, and be-tween the field distribution and the magnitude of the aberrations were investi-gated from a qualitative point of view by Haantjes and Lubben 34•35). This has led to a better understanding of the factors affecting the usefulness of deflection units and to very valuable design information. However, a numerical application of the theory was seriously hampered by the limited possibilities of calculating the large series of cumbersome integrals which constitute the various aberra-tions. During the last fifteen years a great number of papers have been published describing deflectors, the design of which has been based on trial and error 36•37) or empirical data 38 - 43). In the most favourable cases some qualitative reason-ing 44 -48), approximate formulae based on the idealized case of a local uniform field 48) or other simplifying assumptions 49) are used.

Up till now very little has been published about quantitative calculations of deflection errors and the use of electronic computers in designing deflection units 50 •51 •59). In this thesis it will be shown that this approach opens up new perspectives regarding the evaluation of the deflection-error theory and the degree of validity of the various approximations as well as in the matter of distinguishing between essential and non-essential design parameters. Interesting relations are revealed between the parameters describing the magnetic-flux dis-tribution and the deflection-error coefficients which are representative for the various types of error. This in turn gives information about the field distribution and the beam geometry which is required to avoid specific errors and leads to an improved design procedure. This is illustrated by giving examples of applica-tions to deflector problems.

3

1.3. General outline of this study

In this thesis the theory of the deflection aberrations will first of all be con-sidered. It will be shown that the error expressions can be obtained from very simple mathematical manipulations with the so-called Lagrange function of the deflection field. This function being written as a power-series expansion, the second-degree part of it gives the so-called ideal deflection. The higher-order parts give rise to deflection aberrations which may be arranged in classes, the order of which depends on the power of the relevant part of the Lagrange function. First of all the third-order aberrations will be considered. After a qualitative study which reveals a number of interesting properties of the various error coefficients, the results of numerical calculations will be given and com-pared with measurements. For large deflection angles a description which is limited to third-order errors does not fully cover the phenomena and higher-order errors must be taken into account. As an extension of the classical analytic approach the fifth-order errors will be considered. The results obtained by these means will be compared with those resulting from a completely different ap-proach made possible by the computer, namely a direct integration of the elec-tron-path equations.

The magnetic-field distribution can be calculated from expressions which are derived for arbitrarily shaped air coils. The design parameters that are the most important and the range of realizable field configurations are revealed from computations on simplified coil configurations. From this knowledge combined with the relations between the field-distribution parameters and the error coef-ficients conclusions can be drawn as to what extent deflection errors can be reduced or avoided.

A practical deflection unit may show unwanted effects, which, amongst other things, are due to asymmetries and the presence of the earth's magnetic field. An analysis of some characteristic situations will reveal the nature of the various allied errors and some possibilities of avoiding or reducing them. It will be shown that situations may occur in which a reduction of the over-all error is obtained by appropriately introducing a certain asymmetry.

Finally, details of the design of a deflector for the shadow-mask colour-tele-vision display tube will be given to illustrate the practical application of the qualitative and quantitative results of the deflection-error theory.

Throughout this study the Cartesian system of coordinates will be used. The z axis is chosen coincident with the axis of the electron-optical system under consideration. If undeflected, the electrons of the beam are supposed to travel in the z direction. A deflection in the x direction is also defined as horizontal or line deflection, while a deflection in the y direction may be indicated as vertical or frame deflection.

4-

As it is the aim of this investigation to study the effect on electron beams of deflection fields and more particularly of magnetic deflection fields, several simplifications are introduced. Firstly, the undeflected beam or beams are assumed to be rectilinear flows of electrons, homogeneous in velocity and un-affected by space-charge action. Moreover, the electron velocity is presumed to be small compared with the velocity of light. Secondly, the deflection area is supposed to be free from accelerating or focusing fields. Thirdly, the relatively weak magnetic field which originates from the electron stream is neglected. Although in practice some cross-talk between focusing and deflection may be encountered and for instance space-charge action may give additional changes of the electron spot 52•53), a study of a combination of the effects mentioned only makes sense if the individual effects are adequately known.

-5

2. ABERRATION CALCULATION AS A PERTURBATION PROBLEM

2.1. Introduction and general electron-path equations

Geometrical electron optics is derived from Maupertuis' principle of least action, a variational equation which is formally identical with Fermat's prin-ciple of light optics. This principle states that in a combination of electrostatic and magnetostatic fields the trajectory of an electron, moving from a point A to a point B, is such that, compared with all possible paths between A and B, the characteristic function

B

S Jnds A

measured along the actual trajectory has an extreme value, or

B

~S = ~ j n ds = o. (2.1) A

In this ds is an element of path between A and B and n is the electron-optical index of refraction. As long as the square of the electron velocity is small compared with the square of the velocity of light, the index of refraction may be written as

n {2 em (s)}112 - e(A • s), (2.2)

in which s is the unit vector in the direction of the electron path, (s) is the electrostatic potential referred to the potential of the cathode which has emitted the electrons, A is the vector potential of the magnetic field and -e and m are charge and mass of the electron. If in a rectangular system of coordinates z is introduced as the independent variable, the electron trajectory may be written as

with slopes

Then one may write

x'

x = x(z);

bx(z)

bz

B ZB

y y(z),

by(z) y' = .

bz

~ Jn ds = ~ j F(x,y, x',y', z)dz = 0, A ZA

with

(2.3)

(2.4)

(2.5)

F {2 em (x, y, z) }1' 2 (1 + x' 2 + y'2 ) 112 - e (Ax x' + Ay y' + Az). (2.6)

-6

This expression is called the Lagrange function. The necessary and sufficient conditions required for F to satisfy (2.5) are the well-known Euler-Lagrange differential equations

d ?JF ?JF ---- = 0, dz ?Jx' ?Jx

d ?JF ?JF (2.7)

0. dz by' ?Jy

On the other hand, the solution of eqs (2.7) after the appropriate F function has been substituted, gives the path of the electron in the electromagnetic field under study.

2.2. Field distributions and Lagrange functions

2.2.1. Introduction

Before proceeding to the calculation of the trajectory coordinates from the ray equations, a survey will be given of the Lagrange functions which are representative of deflection fields to be considered in this study.

A description of field distributions and the associated Lagrange functions which is best suited to the current methods of solving the ray equations, is the power-series expansion of (x, y, z), A(x, y, z) and F(x, y, z) in powers of x and y. A general procedure for carrying out this series expansion and series expressions for the general electrostatic and magnetic field have been given by Glaser 4 ).

In the ideal case of complete separation between accelerating or focusing fields and the deflection field, the electrostatic potential in the deflection area is constant and equal to the screen potential U., so

(x, y, z) = u. (2.8)

and the only field to be considered is that due to the deflector. As multiplication ofF with a constant factor does not influence the ray equations, the Lagrange function (2.6) can be simplified with eq. (2.8). We then obtain:

F* = F(2 em u.)- 112 (I + x'2 + y' 2 ) 112 k(Ax x' + Ay y' + A.), (2.9)

with k (e/2 m U.)1 12 .

As long as (2.8) holds, eq. (2.9) will be used as the Lagrange function and in the following equations the asterisk will be omitted.

In general the deflection fields are generated by means of electric currents in two pairs of coils situated around the neck of the cathode-ray tube. A schematic view of the horizontal and vertical deflection coils and their nominal

y

t

7-

Vertical deflection co its

Fig. 2.1. Schematic representation of a deflection unit. The currents i H produce a deflection field in the positive y direction; the currents iv give a field in the positive x direction.

position with respect to the coordinate system is given in fig. 2.1. In order to accomplish a symmetrical deflection in two perpendicular directions each pair of coils should show a certain symmetry in shape as well as in position and the two coils forming a pair should be fed with the same current. Moreover, the positional mirror planes through the axis of the system should be perpendicular to each other. One symmetry plane is parallel to the deflection field and the other plane is perpendicular to it. This set-up gives the twofold-symmetric deflector which is mostly used in practice. The greater part of this study is concerned with this type of deflection systems.

With practical winding techniques an exact symmetry cannot be realized and a more or less serious departure from the ideal set-up is often encountered, giving rise to certain asymmetries and inherent additional components in the field distribution. This may be due to asymmetry in the shape or position of the two coils forming a pair or due to unequal currents through the coils of one pair. The asymmetry which occurs if the field configuration is not sym-metrical with respect to the symmetry plane parallel to the field is called intrinsic asymmetry (see sees 2.2.3 and 6.5.2). An asymmetry with respect to the plane perpendicular to the deflection field is called extrinsic asymmetry (see sees 2.2.4 and 6.5.3).

Another source of errors may be that, due to positional or shape aberrations, the planes of symmetry of the horizontal and vertical deflection field do not coincide.

Finally the assumption that the deflection area is free from additional fields does not always hold. The presence of the earth's magnetic field or the fringe area of adjacent focusing fields may produce unwanted effects.

2.2.2. The twofold-symmetric set-up

First the case of twofold symmetry of both the horizontal and the vertical

8-

deflection fields will be considered. The two perpendJcular symmetry planes are chosen as xz and yz planes, respectively.

The symmetry conditions are expressed by two sets of equations: for the horizontal-deflection field-strength components:

Hx(x, y, z) = -Hx(-x, y, z) -Hx(x, -y, z), Hy(x, y, z) = Hy{-x, y, z) = Hy(x, -y, z), (2.10) H,:(x, y, z) Hz(-x, y, z) = -Hz(x, -y, z);

for the vertical-deflection field-strength components:

Hx(X, y, z) = Hx(-x, y, z) Hx(X, -y, z), Hy(X, y, z) = -HY(-x, y, z) -Hy{x, -y, z), (2.11) Hz( X, y, z) = -Hz(-x, y, z) = Hz(x, -y, z).

The power-series expressions for the complete deflection field are found to be 21,26):

Hx Vo- (V2 .,. Vo")x2 2 H2 x y + V2 y2 + (V4 + i V2" 2\ V0"")x4 + 4 H4 x 3y + (6 V4 + t V2")x2Y2 - ( 4 H4 + t Hz")x y 3 + V4 Y4 + ... ,

Hy = H0 (Hz +! Ho")yz + 2 Vz x y +Hz xz + (H4 + i Hz" + 2\ Ho'"')y4 + 4 v4 X y 3 (2.12)

(6 H4 + t H2")x2yz (4 V4 + t V2")x3y H4 x4 + ... , Hz = Vo' X + Ho' Y (t V2' + i Vo'")x3 + Hz' x 2y +

Vz' xy2 - H H2' + i Ho"')y3 + ... ,

in which the field parameters H 0 , V0 , Hz, Vz etc., are mutually independent functions of z. The H parameters belong to the horizontal deflection field and are defined as follows:

Ho = (Hy)x=y=O• (2.13a)

Hz=~! (~:;Y )x=y=O' (2.13b)

H4 :! c:; )x=y=O· (2.13c)

The corresponding expressions for the V parameters of the vertical field are obtained by replacing H by v and by interchanging X andy in (2.13). For details about the relations between these functions and the coil geometry see sec. 4.2. The primes denote differentiation with respect to z.

The magnetic-vector-potential components may be computed from 54)

-9-

z y

Ax flo J H_. dz-! flo J Hz(x, y, z1)dy,

(2.14)

Az = 0, with

flo= 4n. 10- 7 H/m.

Suppose the point (x 1 , Yt. z 1 ) coincides with the origin of the coordinate system and the magnetic-field-strength distribution is such that in the xy plane its z component is negligibly small. Then eq. (2.14) simplifies into

Ax= flo J Hy dz, 0

z

A_. = -flo J Hx dz, 0

(2.15)

Substitution of (2.12) into (2.15) yields expressions for the vector-potential components which contain a number of integrals. These integrals must be eliminated in order to obtain simple Lagrange-function expressions and to avoid repeated integrals in the aberration coefficients. This elimination can easily be carried out using the relation

rot A rot (A + grad 1p), (2.16) in which 'If is an arbitrary scalar field. Now 1p can be chosen so that the com-ponents of grad 1p are equal and opposite to the algebraic sum of the integrals of the relevant vector-potential components 54). The resulting magnetic vector potential, for which we use again the symbol A, turns out to be

A, = flo{-! Y2 Ho' + Y4 (1; Hz' + 2\ Ho"')-! X 2Y2 H 2 ' + - t x 3y V2 ' + ... },

Ay = flo{! x 2 Vo'- X4(! Vz' 2\ Vo"') + t x 2y 2 Vz' + + t xy3 H 2 ' + ... },

Az = flo{-xHo + yVo- t x 3H2- x 2yVz + xy2Hz + t y 3 V2 - l x 5 H4 x4y V4 + 2 x 3y 2 H4 - 2 x 2y 3 V4 - xy4 H4 +iysV4 ... }.

(2.17)

The series expansions (2.17) of the components of A may be substituted into

10-

(2.9). After writing the first part of F also as a series expansion, all the terms ofF can be arranged with regard to powers of x', x, y' andy. In this arrange-ment the dimension of the field parameters and their derivatives may be assumed equal to that of the ray coordinates 23). Then F appears to contain only terms of even order:

... ' (2.18) with

Fo = 1,

F2 = !(x'2 y'2) t-tok(xHo yVo),

F4 -t(x' 2 + y'2 ) 2 t-tokH x'y2 Ho'- t X 2y'Vo' xy2H2 + + -1 x3 H2 +- X 2Y V2 t Y3 Vz), (2.19)

F6 = 1\(x'2 y' 2) 3 t-tok(! x 5 H4 + ! x4y' V2' + 2\ X 4y' Vo"' +

- x4y V4 t x'x3y V/- 2 x 3y 2 H 4 +! x'x2y2H 2 ' ! x 2y'y 2 V2 ' 2 x 2y 3 V4 t xy'y~ Hz' + xy4 H4

- i x'y4 H2'- 2\ x'y4 Ho"' -1; Y5 V4).

2.2.3. Intrinsic asymmetry

A deflection system which shows intrinsic asymmetry as defined in sec. 2.2.1 is characterized by the fact that the vertical or yz plane is not a mirror-sym-metry plane for the horizontal deflection field and/or the xz plane is not a symmetry plane for the vertical field. Then the symmetry conditions reduce to

H,;(x, y, z) = -Hx(x, -y, z), Hy{x, y, z) = Hy(x, -y, z), H.(x, y, z) = -H.(x, -y, z)

for the line-field components, and

Hx(x, y, z) Hx(-x, y, z), Hv(x, y, z) -Hy{-x, y, z), H.(x, y, z) = -H.(-x, y, z)

for the vertical-field components.

(2.20)

(2.21)

Besides the field distribution given in (2.12) the intrinsic asymmetry, if present in both coils at one time, gives rise to the following additional field-strength components:

Hx (Hu Vu)Y + ... , Hv = (V11 + H 11)x H. 0+ ....

(2.22)

In (2.22) the parameter H 11 represents the intrinsic asymmetry of the line-deflection field and V11 is the vertical-field equivalent. Only the lowest-degree part is given, as higher-order parts will not be used in the error calculations

11

in sec. 6.5. The field components (2.22) contribute to a third-power part of the Lagrange function:

(2.23)

2.2.4. Extrinsic asymmetry

A line-deflection field shows extrinsic asymmetry if the xz plane is not a symmetry plane and an extrinsically asymmetric vertical deflection field has no vertical symmetry plane. Now the symmetry relations are

Hx(x, y, z) = -Hx(-x, y, z), Hy(x, y, z) = Hy(-x, y, z), Hz(x, y, z) = Hz(-x, y, z)

for the horizontal-deflection field, and

Hx(x, y, z) Hx(x, -y, z), Hy(X, y, z) = -Hy(X, -y, z), Hz(x, y, z) = Hz(x, -y, z)

for the vertical-field components.

(2.24)

(2.25)

Compared with the field configuration (2.12) of the twofold-symmetric sys-tem, the extrinsic asymmetry gives rise to additional components as follows:

(H12 V12 Zv')x ... , (V12 H12 ZH')y + ... , (2.26) ZH+Zv ... ,

in which H 12 and Z1h being functions of z, describe the extrinsic asymmetry of the horizontal field and V12 and Zv are the asymmetry parameters of the vertical field. The corresponding part of the Lagrange function is found to be

(2.27)

2.2.5. Diagonal symmetry

Besides the field configurations considered so far, a situation which may arise in practice is that the field distribution has no symmetry plane at all, but shows only a so-called diagonal symmetry. This means that the magnitude of the field strength in an arbitrary point (x, y, z) is equal to that in the point (-x, -y, z).

This leads to the following symmetry relations:

Hx(x, y, z) Hy{x, y, z) Hz(X, y, z)

Hx(-x, -y, z), Hy(-x, -y, z),

-Hz(:-X, -y, z). (2.28)

It can easily be seen that this field can be supposed to consist of the super-

-12-

position of two field configurations which each have the xz plane and the yz plane as symmetry planes. So in principle this set-up is equivalent to the configuration dealt with in sec. 2.2.2. For the horizontal deflection field the power-series expansions for the field components turn out to be

H, = Ho1 - (Hzt t Hol")x2 + 2 HzXY + H21Y2 + ... , Hy = H 0 (H2 t Ho")y2 2 HztXY + H2 X2 ••• , (2.29)

The parameters Ho~> H 21 , etc. are functions of z and determine the extent of "obliquity" of the field. As these functions originate from the same current as H 0 , they are proportional to H 0 , although the proportionality constants may be functions of z.

The expressions for the vector-potential components and the Lagrange function are equivalent in form to those given in sec. 2.2.2, and will not be reproduced here. An example of the errors due to H01 and H 21 is given in sec. 6.5.3.

As a general conclusion from this survey it may be stated that the power-series expansion of the Lagrange function contains only terms of even power for the twofold-symmetric deflection field. Terms of third- and higher-order odd powers are introduced by intrinsic and extrinsic asymmetry. The second-degree part of F, being the lowest-degree part to be taken into consideration (see next section), can be regarded as the description of the ideal or unperturbed system. In this conception the higher-order parts of F are considered as per-turbations of the ideal system.

2.3. Solution of the path equations

2.3.1. Introduction

In the literature two essentially different methods are given for calculating from the Euler-Lagrange equations the trajectory coordinates of an electron moving in electrostatic and magnetic fields. The first method is called the "path method" and is based on the concept of electrons being small charged particles which move through electric and magnetic force fields. The field distributions 4> 4>(x, y, z) and A A(x, y, z) are substituted in (2.7), using eq. (2.6). This results in a set of two partial differential equations which are basically equiv-alent to Newton's equations for the motion of charged particles. A solution of these equations, yielding the electron trajectory, may be obtained using the methods of successive approximation 20) or numerical integration 4 ). Haantjes and Lubben 34) simplified the differential equations of the trajectory such that the magnetic-field strength H can be used instead of the vector potential A.

The second method makes use of certain properties of Hamilton's point-characteristic function or point iconal W 55). Therefore this approach is called

-13-

the "iconal method". The geometrical aberrations, being interpreted as devia-tions from the ideal behaviour of the system, are considered to originate from perturbations of the originally unperturbed system. Consequently the calcula-tion of electron trajectories is a perturbation problem. The expressions for the aberrations are functions of the perturbation of the point-characteristic function or of the Lagrange function. In this thesis the method of perturbations will be used. It will be shown that in the case of flux distributions which are typical for deflecting fields, this method yields expressions which reduce the process of deriving the various aberrations to some simple mathematical manipulations.

2.3.2. The ideal or unperturbed trajectory

Suppose the field distributions f(x, y, z) and A(x, y, z) and the Lagrange function (2.6) are known. Written as a power-series expansion F may be

F F0 + F 2 -+- Fr + Fn ... , (2.30) in which F0 = 1, F 2 is the second-order part ofF, Fr is the next-higher-order part, and so on. The zero-order part F0 does not contribute to the ray equa-tions (2.7), so it may be omitted. As long as the electron moves relatively close to the axis and the slope of its path is small, the higher-order parts Fr. F11 , etc. may be considered as negligibly small perturbations of the second-order part and the Lagrange function may with little error be replaced by F 2 • Now the ideal, unperturbed or paraxial trajectory is defined as given by (2.7) after the substitution of F2 , so

d ?:IF2 ?:IFz ----=0, dz ?:lx' ?:lx

d ?:IF2 ?:!Fz 0.

(2.31)

dz ?:ly' ?:ly

These equations give a set of two second-order partial differential equations. In general they can be rearranged into two identical, mutually independent, linear second-order differential equations with or without a right part:

x" a(z) x' + b(z) x nx(z), y" + a(z) y' + b(z) y ny(z), (2.32)

in which a, band n are functions of z. Suppose g(z) and h(z) are two arbitrary, mutually independent solutions of

the homogeneous part of (2.32) andpx(z) andpy(z), respectively, are particular integrals of the complete equations, then the general solution of (2.32) being called the ideal or gaussian trajectory is

x 11 = c1g(z) c2h(z) Px(z), Y11 = c3g(z) -+- c4h(z) py(z),

(2.33)

in which c1 ••• c4 are arbitrary integration constants.

-14

The particular integrals are found by applying the method of variation of parameters to (2.32) and using the solutions g(z) and h(z). This yields

z

p,(z) h(z) J D- 1g(C) nx(C) dC- g(z) J D- 1h(C) nx(C) dC, (2.34) with

D g(C) h'(C) g'(C) h(C), (2.35)

and a similar expression for py(z). The integration starts at an arbitrary point (x0 , y 0 , z0) of the trajectory. We choose (x0 , y 0 , z0) at the gun side of the de-flection area, at such a distance from that area that the deflection-field strength in this point can be neglected.

In the study of focus-lens aberrations it is common practice to specify the integration constants by means of two points of the paraxial trajectory: an object point at one side and an image point at the other side of the focus-field area. For focusing fields the trajectory equations generally are such that it is impossible to find analytical expressions for g(z) and h(z). Approximate solu-tions yielding the paraxial trajectory (2.33) and (2.34) may then be obtained by using one of the methods of numerical integration.

When calculating the effect of deflecting fields it is more convenient to define the integration constants by using one point of the undeflected trajectory and the slope of the trajectory in this point. It is common practice to choose the point of intersection (x., y., zs) of the undefi.ected ray with the screen and the slope of the ray in this point given by xs', ys'.

The homogeneous part of the paraxial-ray differential equation found after substituting F2 of (2.19) into (2.31) is

x" O· '

y" 0. (2.36)

From this, g(z) and h(z) are found to be linear functions of z. It will be shown that this is of great importance as it opens up the possibility of simplifying the procedure of calculating the paraxial path and the ray perturbations. The set of solutions which is obtained with the integration constants x., y., xs' and ys' is

g(z) = 1; h(z) z- z., (2.37)

giving the undeflected trajectory

x = x. x.'(z- z.); y = Ys +- y/(z- Z5 ). (2.38)

The parameters x., y., xs' and ys' will be called ray parameters. Applying (2.37) to (2.33) and (2.34) and combining the result with (2.38)

yields the ideally deflected trajectory

15-

Xg x. + x,'(z z,) + J (z C) nx(C) dC, zo

z (2.39)

Yg = Ys + y,'(z- z,) J (z- C) ny(C) dC. zo

The slope of this trajectory is given by

z

(2.40) zo zo

As a matter of fact, the part due to nx(z) and ny{z) in eqs (2.39) and (2.40) can be considered as a perturbation of the originally rectilinear electron path. It can easily be verified that nx(z) and ny{z) originate from the part of F2 which contains the field parameters. This in turn can be considered as a perturbation of F2 for the field-free case. The mathematical treatment of this perturbation might be chosen analogous to that of the first-order perturbation which is dealt with in the next section.

2.3.3. First-order perturbations

If the next-higher-order part F1 of the Lagrange function (2.30) is taken into account, eq. (2. 7) yields for the x coordinate:

dz ()x' ()x ()x dz bx' (2.41)

In this and in the next section the expressions for the corresponding y coordinate will not be given explicitly as they can be obtained from the x expression by simply interchanging the symbols x andy.

Substitution of the paraxial or gaussian trajectory (2.39) into F1 yields a known function of z for the right-hand member of (2.41), which we write as

( bF1 d ()FI)

(fi)g = ~~ dz ()x' g. (2.42)

Now (2.41) can again be written as a linear second-order differential equation with a known right part. As long as Xg, y0 , Xg' and y0' are sufficiently small, the difference between the solution of this equation and the exact trajectory will be negligible.

Analogous to (2.33) and (2.34) the solution is

fz g(C) r

X1 = Xg + h(z) (fi)11 d._, D(C)

zo

zh(C) g(z) J -- (h)0 dC.

D(C) zo

(2.43)

-16

We write

(2.44)

in which Ax1 is called the first~order perturbation. Replacing (fi)9 by the ap~ propriate part (2.42) of the Euler~Lagrange equation and applying partial integration gives the first~order perturbation of the trajectory as a function of the first~order perturbation of the Lagrange function. Materially identical ex~ pressions have been given by Glaser 57) and Sturrock 58).

In our particular case with g(z) 1, h(z) z- Zs and D I the pertur-bation expression can be greatly simplified. The part Ax1 of (2.43) reduces to

z

(2.45) zo

Substituting (2.42) and working this out gives

- [ it is a function of the ray parameters x., x;, y., y;. So Ax1 can also be expressed in derivatives of (F1) 9 with respect to the ray parameters. It is easily seen that

i:l(FI)g = ( i:lfj_) i:lX3 {)x 9

(2.47)

and i:l(FI)9 (bFI) ({)FI). (2.48) (z-z.) -bx,' i:lx g i:lx' g

Using (2.47) and (2.48), (2.46) finally gives a general expression for the deflection aberrations as a next-higher-order correction to the paraxial trajectory:

(2.49)

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and at the screen, using the notation L1x1(z.) = L1X1 :

(2.50)

The first-order perturbation of the slope of the electron trajectory is found by differentiating (2.49). The result is

(2.51)

and at the screen

With the relations (2.49) to (2.52), the calculation of deflection aberrations has been reduced to some simple mathematical manipulations with the approp-riate part of the Lagrange function. In a completely different way, Glaser has derived an expression for the deflection errots at z = z. which, however, is not the same as our equation (2.50), since Glaser's formula only gives the first part of it 19). Application of that formula requires the omission of the part ofF, that is not related to the field distribution. It may be noted that eqs (2.49) to (2.52) can also be used to calculate the part Px(z) of the paraxial trajectory (2.33). It can easily be shown that this requires substitution of (2.38) in the part of F2 which gives rise to the right member in (2.32) and substitution of the result instead of (F1)g into (2.49) to (2.52).

2.3.4. Second-order perturbations

The accuracy of electron-trajectory calculations can be increased if, besides F" the next-higher-order part Fn of the Lagrange function is also taken into con-sideration.

In this section expressions will be derived which are an extension of eqs (2.49) to (2.52) and which enable the second-order perturbations to be calculated directly from the appropriate part of the Lagrange function.

-18

The Lagrange function (2.30) up to F11 substituted in (2.7) yields

d bF2 bFz bF, d bF1 bFn d bF11 ---- ---,

dz bx' bx bx dz bx' bx dz bx'

d bF2 bFz bF, d bF1 bFn d bFn (2.55)

--- + dz by' by by dz by' dy dz by'

This gives a set of linear second-order differential equations with a known right part if the sum of the paraxial trajectory and the first-order perturbation is substituted in F1 and F11 • This case will be worked out only for the x coordinate; the corresponding y coordinate is obtained from x by interchanging x andy. We define

and

d bF1) dz bx' 1

(2.56)

(2.57)

in which the suffix I outside the brackets means a substitution in F1 and Fn of the trajectory which is obtained when applying the first-order perturbation.

Under the conditions (2.37) and on the analogy of (2.45) the total perturba-tion is found to be

(2.58)

and at the screen: Zs

LlX1 · 1-- LlXn J (z, z) (fi -+- fn), dz. (2.59) zo

These expressions can again be rearranged giving the perturbations directly as functions of F1 and F11 • This will be done for eq. (2.59). Substitution of (2.56) and (2.57) and partial integration gives

lt(F, ():' Fnl dz zo

(z.- z0) • [b(F1 + F11)]

bx' I;z=zo (2.60)

-19-

Application of the Taylor series expansion to (2.60) yields

Zs

L1XI + L1Xu f (z.

f=s{(jpl

bx' zo

zo

z) - + -- + L1u1 + L1u/- - dz + { bF1 bFn ( () () ) bF1} bx bx bu bu' bx 9

higher-order terms, (2.61)

in which u has to be successively replaced by x and y and the results must be added up. The first-order perturbations L1u1 and L1u/ are given by (2.49) and (2.51 ), respectively. Substituting z = z. into (2.46) yields

?!F1 bF1} z)--- dz

?!x bx' 9 [ ()FIJ zo) - . bx' g;z=zo (2.62)

Subtracting (2.62) from (2.61) and assuming that the higher-order terms are negligibly small, we find

fzs{ bFn bFu} (z.-z)---, dz bx bx 9 zo [ bF11 J (zs-zo) -, + CJX g;z=zo

bF1 bF1}] z) -- dz. bx bx' 9

(2.63)

The part containing Fn is similar to (2.62). The contribution of Fn to t1Xu can be found from (2.50) after replacing F1 by F11, so

?!(Fu)g + (zs- zo) ?!(Fu)g J . (2.64) ?!xs' bx, z=zo

Changing the sequence of differentiation the second part of (2.63) gives

L1u/{(zs- z) ~ bx

() } bF1] --- dz bx' ?!u' 9

(2.65)

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On the analogy of (2.47) and (2.48) the derivatives of F1 can be written as

(2.66)

and

( 0 bF1)

ox' ou g - + (z5 -Z)- - . 0 (oF1) 0 (oF1)

oxs' bu g OXs ou g (2.67)

Equivalent expressions are found for oF1/bu'. By substituting this in (2.65) and adding (2.64) the total second-orde1 per-

turbation is finally found to be

+ (z. z0 )

L1u1 , -- + L1u1' --, -, dz. fzs{ () (0~) 0 (0~)} bxs bu 9 OX5 0 u 9 zo (2.68)

The vertical component of the second-order perturbation is obtained from (2.68) after replacing X 5 by Ys and xs' by y,'.

The second-order perturbation of the slope of the trajectory may be found using an analogous procedure. The x component of this perturbation at z Zs turns out to be

Zs o(F ) L1Xn' = J __ n_g dz

OX8 zo

z, 0 (oF) + J{L1ul- _I OX5 OU g zo

o (oF1)} L1u/ -, dz + OX8 bu 9

(2.69)

Replacement of Xs by Ys and xs' by ys' again produces they component.

21-

With (2.68) and (2.69), combined with (2.49) and (2.51) for Llu1 and ;1u/, respectively, the calculation of second-order perturbations is reduced to a simple and straightforward mathematical procedure (see sec. 5.2).

It may be remarked here that general expressions for second-order ray per-turbations have been given by Sturrock in 1951 58). However, they are not very suitable for direct application to deflection-aberration computations. At the expense of several pages of tedious mathematics they can be transformed into eq. (2.68).

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3. THIRD-ORDER ERRORS OF TWOFOLD-SYMMETRIC SYSTEMS: QUALITATIVE CONSIDERATIONS

3.1. Introduction

It was stated in sec. 2.2.1 that the magnetic-deflection systems we consider are those which nominally show a twofold symmetry. In the following the electron trajectory including third-order aberrations in this type of field con-figuration will be considered. The Lagrange function of twofold-symmetric de-flectors has only terms of even power in x, y, x' andy', leading to electron-path expressions containing only terms of odd power. The first-power part is called the ideal or gaussian deflection, the third-power part can be subdivided into a number of third-order aberrations, the fifth-power part gives the fifth-order errors, etc.

The various aberrations are characterized by means of so-called aberration or error coefficients. The dependence of the third-order coefficients on a number of variables such as the field distribution given by the field parameters, the position of the screen with respect to the deflection area, and the axial length of the magnetic-field region, are qualitatively discussed in this chapter.

3.2. Ideal or gaussian deflection

The deflection field of twofold-symmetric systems is given by (2.12). The ideal or gaussian deflection is obtained by applying F2 of (2.19) to the path equations. The resulting trajectory is given by (2.39) in which nx(z) has to be put equal to flo k H0 and ny(z) equal to -Jt0 k V0 (see (2.31) and (2.32)). The part Jtok(xH0 - y V0 ) of F2 can also be seen as a perturbation of the field-free case. The influence of this perturbation on the trajectory is found from (2.49) to (2.52) after replacing (F1)g by the second part of F2 and substituting the recti-linear path (2.38). The result is

z

Llxg = Jtok f (z- C)H0 dC = X, zo

z (3.1)

zo and

z

Llxg' Jtok J H0 dC =X', zo

z (3.2)

Llyg' = -Jtok J Vo dC = Y'. zo

This deflection apparently is proportional to the field intensity and independent

-23-

y

i

Ys Xs

Fig. 3.1. Ray parameters, gaussian deflection and third-order errors.

of the ray parameters and is generally called the ideal or gaussian deflection. The complete gaussian trajectory is (see fig. 3.1)

X 9 Xs + xs'(z- Z5) +X, Y9 Ys + Ys'(z- Z5) + Y. (3.3)

The displacement of the point of intersection of the electron ray with the screen is

Zs Zs

X.= ttok J (zs z)H0 dz; Y. = -tt0k J (zs- z) V0 dz, (3.4) zo zo

and the change in slope is zs

Xs' ttok J H0 dz; Ys' = -ttok J Vo dz. (3.5) ZQ

3.3. Third-order aberrations

3.3.1. Definition of the error coefficients

ln the preceding section it was found that F2 leads to a differential equation (2.32), the right part of which is independent of the ray parameters. The cal-culation of the third-order aberrations due to F4 of (2.19) can therefore be considered as a first-order-perturbation problem to be solved with the formulae of sec. 2.3.3.

Substituting (3.3) in F4 of (2.19) which gives (F4 ) 9 and putting (F1)" = (F4 ) 9 in (2.49) or using (2.46) we find

z

Llx3 = t f (x9' 2 + yg' 2 )x9' dC- t (x/ 2 + y.' 2)xs'(z zo

z

zo z

+ ttokz J (3.6) zo

-24-

and a similar expression for Ay3 • Here and in the following expressions the y component of the aberration can be found from the x component by inter-changing the x andy symbols and replacing H field components by the corre-sponding-V components and V components by the corresponding-H com-ponents (see (3.1 )). It may be remarked that the present method of calculating aberrations produces error expressions in which the number of integrations to be successively carried out is reduced by one compared with the path meth-od 2o,34).

From (2.52) or from a differentiation with respect to z of (3.6) the third-order slope error is found to be

A I _ ~1( 12 + 12) I 1( 12 + 12) I 1 k 2H, I + LJX3 - ~ X 9 Y9 X 9 - ~ X 5 Ys Xs -2 flo Y9 o

The aberrations (3.6) and (3.7) can be worked out, the terms can be arranged with regard to powers of the ray parameters, and by applying partial integration the derivatives of H0 and V0 are eliminated. With an arbitrary gaussian de-flection X.,Y. at z z., the total third-order aberration as a function of z is found to be

Ax3 = Ax3o1 Ax3o2 + Ax3u + Ax312 + Ax313 + Ax314 + Ax321 + Ax323, (3.8)

with

Ax3ot = a3otX/,

Ax3o2 = (a3o2 + b3o3)X.Y/, Ax3ll = (a3o4X/ b3osY/)xs',

Ax312 (a3o6 b3o6 b3o6c)X.Y,y.',

Ax3t3 = (a3o9X/ + b31oY/)x., Ax314 = (a312 b3u)X,Y.y., Ax321 = a3o1X,xs'2 + a3oaX.ys'2 2(b3os b3osc)Y.x,'y,', Ax322 = a3t3Xsx/ + a3t4XsY/ + b3tsY,x,y., Ax323 = a316X.x.xs' b311Y,xs'y, + a3 tsX.y,ys' +

+ (b318 + b3tac)Y,x.y,',

(3.9a)

(3.9b)

(3.9c)

(3.9d)

(3.9e)

(3.9f)

(3.9g)

(3.9h)

(3.9i)

and a similar expression for AYJ, obtained after interchanging the symbols a and b, X and Y, and x andy. The coefficients an and bn defined in this way are called the third-order-error coefficients. Small letters a and b are used to indicate

-25-

that the given coefficients are functions of z. Expressions for a, are given in appendix I (eq. (3.10)).

Since the error Llx3 ' is found from (3.6) after a differentiation with respect to z, the same procedure applied to a, and b, gives the third-order-slope-error coefficients a,' and b,'.

A substitution of z = z. in (3.6) to (3.10) and in the corresponding expres-sions for Lly3 , bm LlYJ', a,' and b,' gives the third-order position and slope error and error coefficients at the flat screen z z •.

The error coefficients for z = z. will be indicated as A, and Bn with n 301-318. They can be found in the literature but in order to facilitate the dis-cussion in the next section they are reproduced in appendix II, eq. (3.11), in a form which makes them directly applicable for numerical calculations. Appendix III gives the coefficients An' (eq. (3.12)).

It must be noted that besides the coefficients ranging from n = 301 to n = 318, additional or correction coefficients with n = 306c, 308c and 318c are introduced in (3.9), giving a total of twenty-one error coefficients. The notation given here is used in order to keep the nomenclature in agreement with the literature, in which only eighteen coefficients are produced. It has been common practice to define the error coefficients at z = z. and there the corrections given in (3.9) turn out to be zero. Inspection of An' and B,' shows that the correction coefficients A306c', B306/, A308/, B308/ do not disappear at z = z •. If the screen is positioned within the deflection field this is also the case for A318/ and/or B31sc'·

In principle a calculation of the electron trajectory as long as the third-order approximation holds can be carried out using (3.3) and (3.8) and the associated expression for Lly3 . However, instead of (3.8) generally the more direct expression (3.6) or other methods will be preferred (see sec. 5.4.4), and (3.8) and (3.9) only find application when the fifth-order errors have to be calculated.

The great value of a knowledge of the error coefficients lies in the fact that they give a clear insight into the way in which the deflection-field distribution, the screen position, the wanted deflection and the ray parameters influence the position of the point of intersection of the ray with the tube screen. A discussion of the error coefficients as given in the next sections may therefore be limited to the behaviour of these coefficients at z z •.

3.3.2. Influence of the deflection magnitude and the ray parameters

A primary subdivision of the error coefficients is the one which regards the gaussian-deflection magnitude and the ray parameters.

Some of the error contributions given in (3.9) are independent of the ray parameters. These are LlX301 and LlX302 being the values at z = z. of (3.9a) and (3.9b), respectively, and the corresponding errors LlY301 and LlY302 • They

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are proportional to the third power of the gaussian deflection and are called geometry aberrations. They give rise to positional errors of the deflected spot which are recognized as a distortion of the displayed information. On a flat screen LIX301 and Ll Y301 produce a linearity error and LIX302 and Ll Y302 lead to the so-called pin-cushion or barrel distortion of the scanned raster 21).

The remaining error contributions depend both on the deflection and the ray parameters. In the literature the discussion of these errors is limited in general to the case x" Ys 0. This is based on the assumption of an infinites-imally small undeflected spot impinging on the screen at the point of inter-section of the deflection-field axis with this screen. However, the majority of practical set-ups do not meet these requirements and then the error coefficients related to x, and Ys play an important role.

The errors LlXm and LlYm with m = 311-314 due to A. and B. with n 304-306 and 309-312 (eqs (3.9c)-(3.9f)), being proportional to the first power of the ray parameters and to the square of the gaussian deflection, are called curvature of the image field, astigmatism and astigmatism-like errors. The nature of the aberrations due to curvature of the image field and isotropic astigmatism (LIX311 and LIY311) and due to anisotropic astigmatism (LIX312 and LIY312) is amply discussed in the literature 21 •34•35). As far as the rays within an electron beam have a common x, and y., the errors (3.9e) and (3.9f) have the character of a raster distortion 60). When different rays within a beam show a difference in x.- and Ys·parameter values, which is the case if the spot is not infinitesimally small, LIX313 and Ll Y3 13 give isotropic astigmatism and LIX314 and LIY314 produce anisotropic astigmatism.

Finally there are various eHor contributions that are proportional to the gaussian deflection and to the square of the ray parameters. These are called coma and coma-like aberrations. The nature of coma represented by A. and B. with n = 307 and 308 (LIX321 and LIY321 of (3.9g)), has been discussed by Wendt 2 1. 29) and Haantjes and Lubben 34 •35). The error contributions LIX323 and Ll Y323 appear when the centre of the undeflected spot does not coincide with the point (0, 0, z.). They cause dimensional changes of the deflected spot that have the character of coma. The errors LIX322 and Ll Y322 can be divided into two parts which are found as follows. Let us write

X,= Xm +X,

Ys = Ym +Yr.

in which Xm and Ym are the coordinates of .the centre of the spot and x, and y, are the horizontal and vertical distance between the point of intersection of the undeflected ray and the spot centre. Substitution of these Xs and Ys into (3.9h) and working this out gives a number of terms. Those with Xm 2 , XmYm and Ym 2 give an additional geometry error, while the remaining part of (3.9h) is a coma-like aberration.

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3.3.3. Influence of the deflection-field configurations

3.3.3.1. Introduction

In a few words the general deflection problem is to realize an electron-spot displacement which approximates ideal deflection as well as possible. Avoidance of all or some of the aberrations means that the relevant error coefficients have to be made zero. In this section the possibilities of changing the value of these coefficients will be studied qualitatively.

A general conclusion from eqs (3.10)-(3.12) (appendices I-III) is that the qualitative and quantitative behaviour of the third-order-error coefficients de-pend on the field parameters H0 (z), Hiz), V0 (z) and Vz(z) and on the position of the screen with respect to the field area. The following discussion will be restricted to the coefficients An of (3.11) but the results bear equally upon B, and to a certain extent upon A,' and Bn' as well.

The integrals that constitute the error coefficients can be divided into two classes: those containing H 2 and those not containing H 2 in the integrand. A further subdivision of the first class is possible with regard to powers of X and Y in the integrand. The second class splits up into integrals which con-tain H 0 as such and those which only contain integrals of H 0 and V0 , such as X', X, Y' and Y.

The best way of analyzing the influence of the field parameters on the numerical value of the error coefficients at z zs is to consider three aspects separately: the shape of the field curves as functions of z, the distance between the field area and the screen, and the effective axial length of the field area.

The field-area-to-screen distance is defined as the distance between the screen and the gaussian-deflection plane of the deflector. This deflection plane is the plane perpendicular to the z axis through the points of intersection of un-deflected rays and the tangents at z = zs to the corresponding paraxially de-flected rays, assuming that at z Zs the deflection-field strength is zero.

The deflection-plane-to-screen distance is

LH = Xs/Xs'

for the horizontal-deflecting field, and

(3.13)

(3.14)

for the vertical-deflecting field, the deflection in these fields being given by (3.4) and (3.5).

The effective length of a deflection field is defined as follows. The gaussian deflection of an arbitrary field is compared with the deflection produced by a homogeneous field positioned in such a way that the deflection planes of the two fields coincide. The field strength within the latter field is chosen equal to the maximum value of the field strength in the former field. The length of the

-28-

homogeneous field required to give the same gaussian deflection as the arbitrary field on a screen outside the field area is called the effective field length

-ro

Starting from the given subdivision of the integrals and considering the mentioned aspects, in the next sections a number of properties of the error coefficients will be revealed. Although idealized field configurations will be used, the conclusions are shown to be of general importance.

3.3.3.2. The shape of H0 (z)

Since for practical deflection units the field parameters H0 and V0 cannot be exactly described with simple mathematical expressions, a computation of the error coefficients requires the application of numerical integration. ln order to answer the question how the coefficients depend on the shape of the func-tions H0 (z) and V0 (z), the results of some calculations on idealized field con-figurations will be considered. Before going into details the assumption of equally shaped horizontal and vertical deflection fields is introduced. This assumption does not stretch reality too much and leads to more transparent error-coefficient expressions. By substituting v0 = -h0 ; Y/ =X/; Y, = X, and v2 -h2 in (3.11) (appendix II), and denoting the part which does not contain h2 by En and the part that contains h2 by Fn, the error coefficients are simplified to

A3o1 = _E 1 -l-A3oz = Ez A3o3 = £3

A3o4 £4

A3os Es + A3o6 = E6 -A307 = I A3os =-! A3o9 = A31o E1o-

A3u

A31z

Ft.

Ft. 2F1,

F4, F4,

F4, F7, F7,

. Fg,

Fg, Fg, Fg,

A313 = F13, A3t4 =- A313,

A31s = -2A3t3• A3t6 Ft6•

A317 - A3t6•

A31s E1s + F16•

(3.15)

-29-

with z,

E1 -! J X/ 3 dz, zo

Zs

E2 -! Xrs' 2 .1 J X ' 3 dz 4 r • zo

Zs Zs

E3 t X,/ 2 -{2 f X/ 3 dz J ho 2 X,(zs z)dz, zo zo

z,

~ J X/ 2 dz, zo

(3.16)

zo zo

Zs

E6 X,s'- f X/ 2 dz, zo

Zs

-! X,,' 2 - J h0 2(z.- z) dz, zo

in which X,s' X,'(z = z.). If H2 is assumed to be zero the functions Fn are zero. We now introduce the field configurations given in fig. 3.2. Curve a represents

a homogeneous field of restricted area, which, notwithstanding its marked difference from actual field configurations, is frequently used for approximate calculations of deflection errors. Curve b represents one period of a cosine

H 0

i Ham ;

Screen I \ l-ora

I \ I \

I \

I \~b I

I / '

12o 0~ I -z Zs

2l L

Fig. 3.2. Idealized representations of H 0 (z).

-30-

function; it is in acceptable agreement with the bell-shaped curves as measured on actual deflection units, and is still feasible for exact computations. The two curves have the same maximum value, they are positioned with regard to each other in such a manner that the deflection planes coincide, and the length of the homogeneous field is equal to the effective length of field b. The origin of the coordinate system is chosen in the gaussian deflection plane, which gives z. = L, where L is the deflection-plane-to-screen distance. The two field curves are

Hao Hom for -1 ::::;;; z ::::;;; I, Hao 0 for lzl I,

and

Hbo ! Hom { 1 +cos ( ~ z )} for -21::::;;; z 21,

The gaussian deflections are

Xa' f-to k Hom (z + /) Xa' 2 f-to k Hom I Xa tto k Hom H z2 -7- I z Xa = 2 f-to k Hom I Z

and

for lzl > 21.

for -/::::;;; z::::;;; l, for z > l;

! F) for -l ::::;;; z ::::;;; l, for z >I,

Xb'=ttt0 kHom{z+21+:sin(;,z)} for -21 z:::;;;2l,

Xb' 2ttok Homl for z > 21;

Xb = t tto k Hom {t z2 + 2/z -+- 2J2 4J2 4f2 cos ( ;n; z)} ;n;2 21

for -2/ ::::;;; z ::::;;; 2/, Xb 2 tto k H0 m1z for z > 21.

In both cases

X8 = 2 tto k Hom I L.

(3.17)

(3.18)

(3.19)

(3.20)

(3.21)

Within the region 21 < x ::::;;; z. the gaussian trajectories are equal. The func-tions LX,' and (L/2/)X, within the deflection area are given in fig. 3.3. This diagram shows clearly that differences in the shape of X' are far less pronounced than differences in the shape of H0 , while the diffetences between the gaussian trajectories Xa and Xb are almost negligible. The conclusion from this is that changes in the shape of H 0 (z) which do not alter the deflection-plane position and the effective length of the deflection region, hardly influence the numerical values of integrals of (3.1 0) which only contain X' or X in the integrand. How-

-31-

LX~ (L/2l)Xr

L I

LXra

(L/2l)Xra

2l -z

Fig. 3.3. Normalized gaussian trajectories and trajectory slopes for the fields of fig. 3.2.

ever, the integrals that contain H0(z) as such will show some change. This is illustrated in table 3-1, which gives the error coefficients (3.16) for the field distributions of fig. 3.2. Taking into consideration that in practice l/L is smaller than unity, it can be noted that A301 , A302 and A304 contain a large shape-indifferent part and a small shape-dependent part, so that in total the relative changes of these coefficients will be smalL The coefficient A306 forms an excep-tion as it so happens that the constant part contained in the integral is cancelled by Xs'/Xs. Due to the presence of the integrals containing H0 2 , the coefficients A303 , A305 and A 310 show a pronounced dependence on the shape of H 0 •

TABLE 3-1

I 0·035; rela-

curve b L 0·28 tive

curve a (m) change

a b (%)

E1 L- 2(0·50 0·25/fL) L- 2(0·50- 0·31//L) 5·98 5·88 1·7 E2 L- 2 (0·25 + 0·125//L) L- 2 (0·25 + 0·16) 1/L) 3·39 3·43 1·2 E3 L- 2(-0·25 + 0·3751/L) L- 2(-0·22 T 0·421/L) -2·59 -2·1" 16 E4 L - 1(1·50- 0·50 1/L) L- 1(1·50-0·621/L) 5·13 5·08 1 Es t- 1{0·50 -0·50 1/L t- 1 {0·38- 0·50 ljL

0·33(l/L)2 } 0·33(1/L)2 } 12·65 9·0" 28

E6 1- 1 {0·33(l/L)2 } /- 1 {0·41(l/L)2 } 0·15 0·19 27

E1o (lL)- 1(-0·50 + (/L)- 1(0·38 + 0·50 lfL) 0·50 1/L) --44·64 -31·89 29

E12 0·5 L- 2 0·5L- 2 6·38 6·38 0 E1s L-1 L-1 3·58 3·58 0

-32-

The remaining coefficients are independent of the shape of H 0 • The numerical values of the error coefficients and the relative changes of these coefficients as given in table 3-I are computed for the arbitrarily chosen but practical substi-tutions L 0·28 m and I 0·035 m. These results confirm the conclusions given in this paragraph. It is an experimental fact that the relative difference in shape between the H 0 curves of actual deflection units and curve b of fig. 3.2 is smaller than that between the curves a and b. Furthermore the relative differences between H0 (z) and V0 (z) are small as well. Therefore it may be stated that the conclusions regarding the error coefficients based upon calculations using the two hypothetical curves fully apply to practical situations.

Summarizing the above considerations, we can say that the possibility of changing the numerical value of the third-order-error coefficients by changing the z dependence of the main field component in such a way that the effective field length and the deflection-plane-to-screen distance remain constant, is limited. The realizable change is smaller than about one per cent for most of the coefficients; some of them may be varied up to about ten per cent.

3.3.3.3. The deflection-plane-to-screen distance and the effective field length

A consequence of the conclusion at the end of the preceding section is that the manner in which the error coefficients are influenced by the other parameters mentioned in sec. 3.3.3.1 can be studied by using the simple H 0 configuration given by curve a in fig. 3.2. In addition the influence of H 2 is considered here using an identical curve H 2 H 2 rn within-/ z ~ I and H:z = 0 for jzj > I. Again we suppose that the horizontal and the vertical fields are equal. In order to avoid interpretation difficulties the following remark must be made. A com-parison of deflection errors at varying screen positions makes sense only if the deflection angle is kept constant. This means that the gaussian deflection as defined in (3.3) must be adapted to the screen position. This is easily done by writing x. =LX,' (eq. (3.13)). For this deflection the error contributions of the various coefficients are calculated for unity beam parameters. The sub-coefficients En for curve a in table 3-1 give

E1Xs3 -! X/ 3 (L t /), EzXs 3 ! X/ 3 (L t /), E3Xs3 !X/3 (-L + i /), E4Xs2 = t X/ 2 (3 L -I), EsX/ = t X/ 2 (£2/l L t /), E6X. 2 = t X 5 ' 2 I, E10Xs 2 fX/ 2 (-L/1 1), E12X/ t X/ 2 , E18X, X,'.

(3.22)

-33-

The contributions of the sub-coefficients Fn are, with h2 H 2m/H0 ,

F1Xs3 F4Xsz F7Xs F9X,Z

=! X/ 3h2P (L- t f), '2h2 l (£2 -IL + g P),

Xs'h2 L (U + P), ~- Xs' 2h21 (L t /),

F13X, = Xs'hzL, F1oXs =2Xs'hz(£2 !P).

(3.23)

Outside the field area the gaussian trajectory and the trajectory including the deflection errors are straight lines, so the error itself is a linear function of L. However, this is not the case for all of the individual contributions, as (3.22) and (3.23) show that the parts due to E 5 , E6 , E 12 , E18, F4 , F7 and F16 are not proportional to L. This is linked up with the fact that the ray coordinates used to define the error coefficients are chosen in the plane of the screen. If the screen position changes, the slope parameters x.' andy,' of a fixed electron ray remain constant but the position parameters Xs and Ys vary linearly with L (see fig. 3.1 ). It can be shown that the sum of the contributions of coefficients which by nature belong together, such as A 305 and A 310, A 3 o6 and A312, A 304 and A 309, A307 with A313 and A316 , is indeed a linear function of L.

On the other hand this property of the error coefficients means that with constant ray parameters the relative contributions of the error coefficients change with changing screen position. The relative contributions of E 5 , F4 ,

and F16 increase and that of E6 decreases with increasing L. Consequently picture tubes with equal maximum deflection angle and equivalent beam geometry but different screen dimensions require differently designed deflection units. For instance, suppose we have a deflection unit giving anastigmatic deflection on a screen in a certain position L'. Now it can be derived from (3.22) and (3.23) that the consequence of a shift of the screen away from the deflector is the appearance of astigmatic errors. This is due to the fact that the radius of curvature of the sagittal image plane 34) l!s = 1/2A305 decreases and the radius of curvature of the meridional image plane l!m 1/2A304 increases with increasing L - L'.

The influence of the field length 2/ on the various errors is somewhat complex. The most important conclusions from (3.22) and (3.23) are the following. (1) A field distribution with a negative H 2 causes pin-cushion raster distortion

due to the positive value of the sum of A302 and A303, a situation which frequently occurs in practice. This distortion decreases with decreasing field length.

(2) In general anisotropic astigmatism decreases as well with decreasing coil length. On the other hand astigmatism due to A305 and A310 increases with decreasing /.

(3) In the literature 35) it has been shown that the radius of curvature of the

-34-

image field is about equal to 21. From this point of view a large 1 is favour-able.

(4) In practice H 2 is a very valuable design parameter. Some of the important sub-coefficients, viz. Fh F4 and F9 , increase with increasing l. So a relatively small H 2 , which means a relatively small field inhomogeneity, already leads to considerable contributions from these coefficients if a long coil is used. Furthermore the realization of intricate H 2 functions, which are required in practice (see next section), is only feasible if the coil is not too short.

Finally some aspects regarding the coil-to-screen distance and the coil length must be mentioned which are not related to electron optics. Frequently the deflection-plane-to-screen distance is fixed a priori, being given by the dimen-sions of the display tube. In many cases the coil length is dictated by such con-siderations as deflection sensitivity (power economics), impedance (circuit simplicity), and cost price. In those situations the only parameter which is available for changing the third-order-error coefficients is the field parameter H 2 • In the next section the influence of the value and of the shape of the H 2 function will be considered.

3.3.3.4. The field parameter H 2

From eqs (3.15) and (3.23) it is clear that the error coefficients can be in-fluenced by changing the value of h2 rn. For some of the coefficients Haantjes and Lubben 35) have discussed how certain types of error can be avoided. They found that when H 2 is constant within the coil only one kind of error can be eliminated at a time. One of the most promising situations that could be sug-gested at a first glance from (3.15) and (3.11) (appendix II) is h2 m = 0, which means that the deflecting field is homogeneous. However, due to the contribu-tion of H0 some coefficients do not disappear. This is in contradiction with the still wide-spread misunderstanding that a homogeneous field should give error-free deflection.

An interesting point to which we shall return later on, but which is worth mentioning here, is the fact revealed from (3.15) that regarding the dependence on H 2 the error coefficients can be divided into six groups. For a constant H 0 and a changing H 2 , in value as well as in shape, the error coefficients of one group can be written as linear functions of one of the coefficients of that group. It must be recalled that (3.15) is obtained by putting XfXs and Y/Ys equal. In practice the relative differences between these functions prove to be very small, and therefore the given subdivision can be maintained. A practical consequence of these interrelations is an increased transparency of the field-design procedure. Further details are given in sec. 4.5.

We now come to the influence of the shape of Hz(z). A very useful approach, which has been indicated in the quoted article 35), is to choose more intricate functions for H 2 • A qualitative consideration of the integral expressions led to

35-

l 1 .I -z Screen

-z

Fig. 3.4. Idealized field-parameter curve for studying the influence of H 2 (z).

the conclusion that this opens the possibility of avoiding two types of error at the same time.

A clear insight into the role of the shape of H 2(z) can be obtained from ap-plying the field distribution of fig. 3.4 to the aberration expressions. Again H 0 is constant within the coil. For H 2 we choose a small section of a constant value H 2m which can be shifted along the z axis. The length of the section is given by al with 0 a ~ 2 and the position by the parameter t with

+ a) t ~ 1. Substitution of this field configuration into the sub-coefficients Fn of (3.15) gives

F1 lhzm(~Y{:{0 ts+it4+-!t3 ·!tz+it1+ - ~ (i4 t6 + i ls + i !4 + t !3 + t tz)},

F4 = -! hzm I {t t3 + -! lz + 1 t1 ~ (! t4 t !3 + 1 lz) + +(~Y({0 ts+!t4 !t3)},

F1=-!hzmL2 {t1-~ ~ tz+(~Y !3-t(~Y t4}, (3.24) F9 -! hzm ~ {t t3 + t lz +-! t1- ~ (-§- !4 + t !3 + i tz+ F13 -!hzm(t1-!~t2). F16 = hzmL{t1- ~ lz + t(~ r t3},

with h2m Hom/Hzm and tn = rn (t- a)n.

36-

Suppose a O·l, which means a section length of one twentieth of the field length, and h2 m 400, which means that at a point 5 mm from the axis in the deflection direction the field strength is 1 per cent higher than on the axis. For the arbitrarily chosen values I 0·04 m and L 0·35 m the functions Fn are given in fig. 3.5. This diagram is typical in so far as it does not change materially

102 ·····-~ ~·

10

~ m -·· m ~ .l --+- r: Co~ r-r-,_ :._ - ~

= -~~ ~r- - r-- '--

Fn

i 9·

/ --1-1-k:: v v F7 1 Asti! rrn:Jtism L

-r-· - - r- vm ~ 1- ·-t

- r-···· 1/ /n -! v 1 q - ~ r-- ...

. .. - 1-.

_.--""'/ I ! I Ditlrtit 10-2 - '/-4. -1-2 - (}8 -(}4. 0 0·4. (}8 1-:2, /-4.

-t Fig. 3.5. The error-coefficient contributions of an H 2 section as a function of the section position.

if l and L are changed or if a is decreased. The possibilities of changing the error coefficients by changing the shape of H 2 can easily be derived from it. Bearing in mind that F 1 refers to raster-distortion coefficients, F4 and F9 to isotropic-and anisotropic-astigmatism errors and F7 , F13 and F16 to coma coefficients, the following remarks can be made.

Firstly, a certain section of H 2 in a certain position contributes differently to the different types of error. This is due to the presence of different powers of X/X, Y/Y. and z. z in the coefficients. Within the deflection field X/X. and Y/Y, are much smaller than unity, and in practical set-ups Zs z is also smaller than unity. Consequently F1 is relatively small and F13 is relatively large, the other coefficients lying somewhere in between. A certain change of the geometry errors therefore requires a relatively large change of the value of H 2 while the other types of error, particularly coma, are already affected by small changes of H 2 •

-37-

A second interesting point is the dependence of the functions Fn on the posi-tion of the H 2 section. In fig. 3.5 roughly three parts can be discerned. In the first part at the gun side of the field area the gaussian deflection is still negligibly small, so the integrals containing X and Y can be neglected. Only the integrals of the coma coefficients represented by F7 , F13 and F16 give a noticeable con-tribution in this part of the field. In the second, central part of the field, although the deflection is still small, integrals containing X or Y to the first power are no longer negligible. Consequently F4 and F9 start to play a role and changes of H 2 in this part of the field influence the astigmatism errors. Due to the presence of powers of zs- z in the coma integrals the relative influence of H 2 on this type of error decreases with increasing z. Finally fig. 3.5 shows that in the third part of the field at the screen side of the coil the function F 1 related to the geometry errors at last acquires an appreciable value.

From the nature of the error integrals it is clear that if more H 2 sections are present at the same time the value of the functions Fn is obtained from a linear addition of the contributions of the individual sections. This suggests the follow-ing procedure for the realization of prescribed values of the aberration coeffi-cients.

Suppose the avoidance of raster distortion requires a certain negative value of F1 • This can be obtained with a number of H 2 sections having appropriate values of H 2 at the screen side of the field area. Instead of giving the sections equal negative values of H 2 , which is difficult to realize in practice, a more or less continuous transition from section to section is built up (see fig. 3.6, part I).

H 2 rf r\

r I 1 I

~ I I Screen -I t I I I I I I ; I

I

t~ ~ 1 -z

I I

~1 III II .1. ~. 17 • Fig. 3.6. The H 2 curve schematically represented as the addition of a number of separate sections.

-38-

In the central part of the field the H2 sections can be chosen to produce, in collaboration with the front-part sections, values of F4 or F9 required to avoid a certain astigmatism error. Due to the negative contribution of part I the realization of e.g. a positive F4 requires a strongly positive H 2 in part II of fig. 3.6. Finally the gun part of the field can be used to compensate for the coma contributions of parts I and II and to realize prescribed values of the coma-error coefficients. If for instance H 2 in I and II is as given in fig. 3.6 and F7 has to be zero, negative H 2 sections are required in part IlL It will be clear that in practice H 2 is not built up of a number of separate sections but is a continuous curve the shape of which should be chosen in accordance with the outlined procedure (e.g. the dashed curve in fig. 3.6).

Still another property of some of the error coefficients is indicated by the functions Fn of fig. 3.5. From (3.11) (appendix II) and (3.15) we see that the coma coefficients are mainly determined by the values of the contributions of F7 , F 13 and F16. Now fig. 3.5 shows that these parameters as a function of t have roughly the same shape. So modelling the Hz(z) to avoid one of the coma coefficients has the consequence that all of the coma coefficients tend to be zero. Furthermore there is a strong resemblance between F4 and F9 , their ratio being about constant as a function of t. Combining this with what has been remarked in the second paragraph of this section means that with regard to changes of H 2(z) there is roughly a linear relationship between all of the astigmatism coefficients.

The final and important conclusion from the qualitative description of the field design based on the properties of the third-order-aberration coefficients given here is that the parameter H2 is the most important design parameter. In principle three different kinds of third-order errors can be avoided at the same time. The feasibility of this procedure depends primarily on the possibili-ties of adequately shaping the H 2 curve. This will be dealt with in the next chapter.

It must, however, be kept in mind that practical deflection fields cannot be free from curvature of the image field 35).

Appendix I

Equation (3.1 0)

The third-order-error coefficients an as a function of z, as introduced in (3.9), are as follows:

z

zo z zo

z zo

P,okz f H2Y2 dC-! p,0kH0 Y 2 }, •o

39-

z

X -zv 1 { f (~4 X'zy' a3o3 = s .~. s zo

z

z

zo z

z

z

z

a 3o6 =X. 1 Y. 1 [ J {t X'Y' + flokHoY- flokH0 Y'(z.- C) +

- 2fl0kH2Y(z.- C)C} dC +

2 flokz J HzY(z.- C) dC + flokH0 Y(z.- z)], z

a3o6c= flo k X. 1 Y.- 1 {-HoY(z.- z) (z.- z) J HoY' de}, zo

z z

z z

a X 1 [ J {1 X'+ u kH (z r)2r} dt uokz jH2 ( 7 5 308 = s 2 rO 2 s ~ ~ , r -

z

a3osc= 1 flo k X. - 1 {H0(z.- z)2 (z.- z) J H 0 de}, zo

z z

z

z

-40-

z z

z z

z

zo zo z

z z

zo zo

z

a31s = -a316 P,ok Xs 1 {Ho(Zs- z) + J Ho de}, zo

(3.10)

The corresponding coefficients bn are obtained from an by replacing H0 by-V0 , H 2 by -V2 and by interchanging X andY.

Appendix II

Equation (3.11)

Simplified expressions for the error coefficients are obtained with the intro-duction of the relative magnitudes

X'/X. =X/; X/Xs =X.; Y' /Y. Y,'; YJY. = Y.;

P.,o k Ho/Xs ho; p,0 k H 2/Xs = h 2 ; P.,o k V0 /Ys v0 ; fl-o k V2 /Ys = v 2 •

The resulting coefficients An are Zs Z,s

A3ol J-! X/3 dz + J hz X/(zs z) dz, zo ~ ~

A3o2 j Ct X/Y/2 + ho Y.'Y.) dz j h2 Y.2(zs z) dz, zo

zs

A 303 J {!X,' 2 Y,' -h0 2 Y.(zs z) h0 X,'Y,'(zs- z) + h0 X,'Y.} dz Zs

J 2 h2 X,Y,(zs- z) dz,

-41-

zo zo

zs

A3os fa X,' 2 h0 2(z. z)2 2 h0 X,'(z.- z)} dz +

Zs

z, Zs

f a X,'Y,' + h0 Y, h0 Y,'(z.- z)} dz + 2 f h2 Y,(z. z)2 dz, zo zo

Zs

A3o7 i + f hizs- z)3 dz,

Zs

A3o9 = 2 f h2X,(z,. z) dz,

Zs

A31o = f {hoX,'- ho2(z. z)} dz- A3o9,

z., A3tt = 2 f h2Yr(z.- z) dz,

Zs

A312 = f ho Y/ dz + A3 11 , zo

Zs

A313 = f h2(zs- z) dz, zo

zs

A3J6 = 2 f h2(z,. z)2 dz,

f ho dz + A3t7· (3.11)

The coefficients Bn again are obtained from An in the known way.

Appendix ill

Equation (3.12)

-42-

The third-order coefficients An' are

"• ! X,/3 + J h2 X, 2 dz,

zo Zs

A302' = i-X,/Y,/2 - j h2 Y,Z dz-lho.'Y,/, zo

Zs

j (h 0 X,'Y,' h0 2 Y, + 2 h2 X,Y,) dz,

Zs

A 304' = £ X,/2 - 2 J h2 X,Z(z,- z) dz, zo

Zs

l X,/ 2 + j {h0 2(Zs- z) h0 X,' zo

zs

A3o61 = l X,.'Y,,' 2 j h2 Y,(zs- z) dz, zo

zs

hosYrs- j ho Y/ dz, zo

zo Zs

A3osc' -l J ho dz, zo

Zs

A3o9 ' = 2 j h2 X, dz, zo

zs

hosXrs' J (ho 2 + 2 hzX,) dz, zo

zo

2 h2 X,(zs- z)} dz,

-43-

Zs

A3l61 =- 2 J h2(zs z) dz, zo

with Xrs' X/(z Z5), etc. (3.12)

The coefficients Bn' follow after the usual substitutions.

-44-

4. THIRD-ORDER ERRORS: CALCULATIONS AND MEASUREMENTS

4.1. Introbuction

In this chapter some quantitative aspects of the calculation of the third-order aberrations are dealt with. Firstly expressions are derived for the field distribu-tion of arbitarily shaped air coils. The influence of the dimensions and the shape of the coil on the field parameters H 0 , Hz and H 4 are investigated on a number of simplified deflection-coil configurations. The field distribution of intricately shaped practical coils must be found from measurements. A short description is given of a measuring method that permits a direct determination of H 0 and Hz, while H 4 can be derived from the difference between two differential measurements. Details are then given of the computer program used to cal-culate third-order-aberration coefficients and deflection errors. A clear survey of the results of the calculations is obtained by displaying them in error-coefficient diagrams. These diagrams are found to be of essential practical importance for designing deflection units. Finally, after consideration of some technological aspects of deflection coils, the methods of measuring deflection errors are described and some preliminary results are given.

4.2. Calculation of the field distribution

4.2.1. General remarks

A calculation of deflection aberrations requires a knowledge of the field con-figuration. Several authors have reported field-distribution calculations on rectangular coils 60•61 ), toroidal coils 44), and electron-optical devices of a more general nature 62). On the whole the given expressions are very complicat-ed and rearranging them into a field description of the form of (2.12) does not seem feasible. Special attention has been paid to the realization of uniform deflection fields 60•63 •64). A uniform field is obtained if the winding distribution of a saddle or toroidal coil is chosen to give a current density at any point on the circumference of the tube neck which is a cosine function of the angle between a symmetry plane used as reference plane and the plane through the considered point and the tube axis. Any deviation of the cosine distribution will cause a non-uniform field.

The field configuration of the idealized case of a pair of symmetrical coils, each consisting of one single winding, has been calculated by Haantjes and Lubben 35). The shape of the windings is as given in fig. 2.1. The field param-eters H 0 and are expressed in the radius of the coil surface and in the length and the circumference angle of the coil. Assuming a relatively large length-to-diameter ratio a very simple expression for Hz is obtained, containing as most important design parameter the circumference angle 0. If (} 120°,

-45

Hz is about zero. A value of () exceeding 120° results in a mainly positive Hz while a negative Hz is obtained when ()is smaller than 120°.

A very interesting and useful approach indicated by the same authors 35) is to choose a different () in different parts of the coil, thus giving a varying magnitude and sign of Hz along the coil axis. In this way intricate H2 functions such as given in fig. 3.6 may possibly be realized.

Practical coil geometries differ considerably from the schematic represen-tation in fig. 2.1. In general the length-to-diameter ratio is relatively small, the parameter () is generally not constant along the coil and moreover the coils mostly show a certain flare (see sec. 4.6.1). In the next section expressions are derived for calculating the field distribution of air coils of arbitrary shape 75). Again the importance of the parameter will become evident.

4.2.2. The field distribution of arbitrarily shaped single-winding air coils

Assume a current-carrying wire element ds positioned at the point Q (fig. 4.1). The direction of ds is given by the direction cosines or, which is more con-venient here, by the parameters a1 , a2 and a3 , with

dx0 cos et 1 -=---; az dz0 cos et3

dy0 COS Ct2 -=---; a3 dz0 cos et3

l. (4.1)

Here a 1 , rx 2 and a3 are the angles between the w