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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MAGNETIC-DEFLECTION ERRORS
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
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 H0(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 H2(z) 34
38 40 42
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 H4 • • • 81
5.6. Preliminary experimental results 84
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 1191. 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-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
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.
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.
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
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 prinprin-ciple 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 <f>(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 <f>(x, y, z) }1'2 (1+
x'2+
y'2 )112- e (Ax x'+
Ay y'+
Az). (2.6)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 multiplicationofF 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.)112 .
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
Verticaldeflection 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
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 H 2 x y
+
V2 y 2+
(V4+
i
V2" 2\ V0"")x4+
4 H4 x3y+
(6 V4+
t
V2")x2Y2 - ( 4 H4+
t Hz")x
y3+
V4 Y4+ ... ,
Hy=
H0 (Hz+!
Ho")yz+
2 Vz x y +Hz xz+
(H4+
i
Hz"+
2\ Ho'"')y4+
4 v4 X y3 (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' x2y+
Vz' xy2
-
H
H2'+
i
Ho"')y3+ ... ,
in which the field parameters H0 , 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). Fordetails about the relations between these functions and the coil geometry see
sec. 4.2. The primes denote differentiation with respect to z.
z y
Ax flo
J
H_. dz-! floJ
Hz(x, y, z1)dy,(2.14)
Az = 0, with
flo= 4n. 10-7 H/m.
Suppose the point (x1 , Yt. z1 ) 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_. = -floJ
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"')-! X2Y2 H2 '+
- t
x3y V 2 '+ ... },
Ay = flo{! x2 Vo'-X4(! Vz' 2\ Vo"')+
t x
2y2 Vz'+
+
t xy
3 H 2 '+ ... },
Az
=
flo{-xHo+
yVo-t
x3H2- x2yVz+
xy2Hz+
t
y3V2- l
x5 H 4 x4y V4+
2 x3y 2 H4 - 2 x2y3 V 4 - xy4 H 4 +iysV4 ... }. (2.17)(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
X2y'Vo' xy2H2+
+
-1
x3 H2+-
X2Y V2t
Y3 Vz), (2.19) F6=
1\(x'2 y'2)3 t-tok(! x5 H4+
!
x4y' V2'+
2\ X4y' Vo"'+
- x4y V 4t
x'x3y V/-2 x3y2 H4+!
x'x2y 2H 2 '!
x2y'y2 V2 ' 2 x 2 y3 V4t
xy'y~ Hz'+
xy4 H4- i
x'y4 H2'- 2\ x'y4 Ho"'-1;
Y5 V4). 2.2.3. Intrinsic asymmetryA 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+
H11)xH.
0+ ....
(2.22)
In (2.22) the parameter H11 represents the intrinsic asymmetry of the
line-deflection field and V11 is the vertical-field equivalent. Only the lowest-degree
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 H12 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)
super-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 - (Hztt
Hol")x2+
2 HzXY+
H21Y2+ ... ,
Hy = H0 (H2t
Ho")y22 HztXY
+
H2X2 ••• , (2.29)The parameters Ho~> H21 , etc. are functions of z and determine the extent of
"obliquity" of the field. As these functions originate from the same current
as H0 , they are proportional to H0 , 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 H21 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
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 perturbacalcula-tion 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, F2 is the second-order part ofF, Fr is the next-higher-orderpart, 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 F2 • 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
x11
=
c1g(z) c2h(z) Px(z), Y11=
c3g(z)-+-
c4h(z) py(z),(2.33)
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 , y0 , z0 ) of the trajectory. We choose (x0 , y0 , z0 ) at the gun side of the
de-flection area, at such a distance from that area that the dede-flection-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'
andys'
isg(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 parametersx., y., xs'
andys'
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
Xg x.
+
x,'(z z,)+
J
(z C) nx(C) dC, zoz (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
f
z g(C) r X 1 = Xg+
h(z) (fi)11 d._, D(C) zo zh(C) g(z)J --
(h)0 dC. D(C) zo (2.43)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
- [<z
l;) (bF:) Jz .
(2.46)i:lx 9 zo
Since (F1) 9 is found by substituting the gaussian path (2.39) into FI> 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)
(zz.)
-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:
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.
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)Application of the Taylor series expansion to (2.60) yields Zs L1XI
+
L1Xuf
(z.f
=s{(jpl
bx' zo zo z) -{+ -- +
L1u1+
L1u/- - dz+
bF1 bFn ( () () )bF
1} 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
bF
1} 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
f
zs{ bFnbFu}
( z . - z ) - - - , dz bx bx 9 zo [ bF 11J
(zs-zo) - ,+
CJX g;z=zo bF1bF1}]
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)gJ .
(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)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
oF
1/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.f
zs{
bxs()
(0~)
bu 9 OX0
5(0~)}
0 u 9zo (2.68)
The vertical component of the second-order perturbation is obtained from (2.68) after replacing X5 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 zoo
(oF
1 ) } L1u/ - , dz+
OX8 bu 9 (2.69)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).
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' JtokJ
H0dC
=X', zo z (3.2)Llyg'
=
-JtokJ
VodC
=
Y'. zoy
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)
X9 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.=
-tt0kJ
(zs- z) V0 dz, (3.4)zo zo
and the change in slope is
zs
Xs' ttok
J
H0 dz; Ys'=
-ttokJ
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)x 9' dC-t
(x/2+
y.' 2)xs'(z zo z zo z+
ttokzJ
(3.6) zoand 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 - ~ X9 Y9 X9 - ~ X5 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' b311 Y,xs'y,
+
a3tsX.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 aand 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
that the given coefficients are functions of
z.
Expressions fora,
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 disappearat 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)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 n304-306 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 21.29
) 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.
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 H2 and those not containing H2 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 H0 as such and those which only contain integrals of H0 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
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
