On the use of gauzes in electron optics

ON THE USE OF GAUZES

IN ELECTRON OPTIes

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAP AAN DE TECHNISCHE HOGESCHOOL TE DELFT, OP GEZAG VAN DE RECTOR MAGNI-FICUS Ir. H. J. DE WIJS, HOOGLERAAR IN DE AFDELING DER MIJNBOUWKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VER-DEDIGEN OP WOENSDAG 30 OCTOBER 1963

DES NA MIDDAGS TE 4 UUR

DOOR

JAN LOUIS

VERSTER

GEBOREN TE AMSTERDAM

BIBLIOTHEEK

D6R

TECHNISCHE HOGESCHOOL

DELFT

Aan de nagedachtenis van mijn moeder

Aan mijn vader

INTRODUCTION

It was discovered in the late twenties, that electrons, emerging from an object, could be made to give an image, after having traversed a properly shaped electromagnetic field. Hence it was discovered that such a field acts upon electrons as a lens acts upon rays of light. It was realized very soon that imaging by means of electrons and the focussing of electron beams opened up new possibilities in many branches of science and biology, to mention the elec-tron microscope and the cathode ray oscilloscope. Therefore a new branch of applied physics, electron optics, came into being. Electron optics studies the trajectories of electrons in electromagnetic fields in order to describe the optical properties of such fields.

Electron optics can be treated in the same way as light optics, at least theo-retically, once the electron optical index of refraction is introduced. This follows from the facts that rays of light satisfy Fermat's principle, in which the index of refraction appears, and th at the trajectories of electrons satisfy Maupertuis' principle of least action, both principles being variational equations with the same conditions. If the electron optical index of refraction is defined, an im-portant chapter of light optics, such as the theory of geometrical optics, can be applied to electron optics.

It might appear that electron optics is only a special kind of light optics and therefore cannot lead a life of its own. That this is not true wiU become clear if electron optics is considered closer.

An electron optical lens consists of an electromagnetic field, whose electro-static part is brought about by electrodes and whose magnetic part is brought about by pole pieces, by coils or by both.

If it is required that the region where the electrons move is free from elec-trodes and pole pieces, then the field in this region can only be controlled by electrodes etc. at the boundary. Therefore the most obvious difference with light optica I lenses is that, in order to knoweither the field or the electron optical index of refraction, a boundary value problem must be solved.

Since the electrons move in a field, their trajectories will in general be curved. In order to obtain the trajectories, certain differential equations, whose coef-ficients depend on the field, must be integrated.

Consequently, with regard to the calculation of the rays, electron optics is completely different from light optics.

A third important difference between electron optical and light opticallenses consists of the fact that it is very difficult to correct the former for third order spherical aberration, whereas designing light optical lenses that are free from this aberration presents no problem.

capable of solving Laplace's equation with boundary conditions. An electro-lytic tank wiU be described in Chapter 3.

The manner by which the second problem will be solved depends to a large extent upon the kind of rays th at are required. If, in order to determine the paraxial or Gaussian properties of the lens, paraxial rays must be calculated, it suffices to determine the axial potential distribution fr om which the rays can be calculated rather easily, using the so-called paraxial ray equation. The calcula-tion of rays that are no more paraxial is, however, very laborious, if they are to be calculated from the field. In order to obtain such rays analogue compu-ters have been built that trace the trajeetory, described by the electron. Compu-ters of this kind are connected with an electrolytic tank from which they derive the data, necessary for tracing the ray. In Chapter 3 a detailed description of such an analogue computer will be given.

With regard to the third difference between electron optical and light optical lenses, it is weIl known that the quality of the former can be improved by the use of a foil. Chapter 1 is devoted to a comparison of th ree kinds of lenses two of which have one foil. It will be shown that the one-foil lenses have better optical qualities. If, however, the scattering by a gauze-shaped foil, which is treated in Chapter 2, is taken into account, it appears that the kind of applica-tion determines which lens is to be preferred.

In Chapter 4 the properties of a special kind of television piek-up tube wil! be dealt with. Since it was necessary, for the determination of its properties, to know the trajectories of electrons that were so far off-axis that they were just reflected or just transmitted by a grid-shaped structure, the rays were traeed by the ray tracer.

CONTENTS

Chapter 1 ELECTRON OPTICAL PROPERTIES OF TWO-ELEC-TRODE LENSES WITHOUT AND wnH A FOIL. . 1 1.1 Introduction . . . . . 1 1.2 The three kinds of lenses considered and their axial potential

dis-tribution . . . 2 1.3 Ray equations and ray constants . . . . 8

1.31 Variational equation for the motion 8

1.32 Variational equation for the ray 11

1.33 The Lagrangian invariant 13

1.34 The paraxial ray equation . . . 15

1.4 Gaussian properties of the lenses considered 17 1.41 Definition of the cardinal points and the power 17 1.42 Locations of the cardinal points and the power expressed in

terms of matrix T. . . 18 1.43 Approximation of Tin terms of C/>(z) . . . 20 1.44 Approximations of P, ZHl, ZH2, ZF1 and ZF2 24 1.45 Application to the lenses considered . . . . 26 1.5 Third order properties of the lenses considered . 35 1.51 Relations between f1 and f2 and between the coefficients mtj 37

l.52 Third order aberrations 39

1.53 Spherical aberration . 41

1.54 Coma . . . 42

1.55 The principal surfaces 42

1.56 The sine condition 43

1.57 Results of the measurements of the coefficients mtj 45 1.6 Possible applications of one-foil lenses . . . 54 References . . . 56 Chapter 2 POTENTIAL DISTRIBUTION IN FLAT SYSTEMS WITH

A GAUZE AND THE SCATTERING OF ELECTRONS BYGAUZES . . •. . .

2.1 Introduction

57 57 2.2 Potential distribution in a planar triode with a gauze 57 2.3 Results of the measurements of Da and D' . . . . 63 2.4 Potential distribution in the neighbourhood of a gauze of thin wires 67

2.5 The scattering of electrons by gauzes 70

References . . . 78

Chapter 3 THE RA Y TRACER 79

3.12 Mechanical and electrical design 82

3.13 Accuracy . . . 85

3.14 Practical possibilities 86

3.2 The electrolytic trough . . 86

3.21 The three tanks. . . 86

3.22 The generation of the electrostatic field in the trough 88 3.23 Measuring potential distributions with the aid of the trough 91 3.24 Accuracy of potential-distribution measurements . . . . 91 3.3 Measurement and calculation of En and cp • . . . • . • 94

3.31 The influence of the induced dipole moment of the probes 95

3.32 En and cp expressed in terms of the probe voltages 97

3.33 The probe holder and the probes . 100

3.34 The cathode followers for the pro bes 102 3.35 The E-computer . . . 105 3.36 The cp-computer . . . . . 111 3.4 The trolley, the steering-motor amplifier and the mechanical

trans-mission of the trolley motion to the probes 114

3.41 The trolley . . . 114 3.411 Construction of the trolley . . . 115 3.412 The potentiometer circuit mounted on the trolley 116 3.413 Determination of the wheel base hand the distance g 119 3.42 The steering-motor amplifier . . . 120 3.43 Mechanical transmission of trolley motion to the probes 122 3.5 A test problem . . . 124 References . . . 125 Chapter 4 MEASUREMENTS ON' A NBW TELEVISION PICK-UP

TUBE. . . 126

4.1 Introduction 126

4.2 The potential distribution in the vicinity of the anode and the photo-emitter . . . 128 4.3 Results of the measurements of f3(Vp ) • • • • 133 4.4 Influence of thermal velocities of the electrons 135 References . . . 139

Chapter 1. ELECTRON OPTICAL PROPERTIES OF TWO-ELECTRODE LENSES WITHOUT AND WITH A FOIL

1.1 Introduction

The question of to which extent the properties of electron optical lenses can be improved by the use of a foil, has been answered önly incidentally. Symmetrical three-electrode electrostatic lenses, whose middle electrode con-sists of a gauze, have received much attention from French authors. Such a lens has been designed by Cartan as early as 1937 and will therefore be called a Cartan lens. Grivet 1), in his text-book, has dealt with the Gaussian properties of Cartan lenses and with the scattering of the electrons by the gauze. These problems have been attacked originally by Bernard and by Bertein. Bernard 2)

has derived a first order approximation of the focal distance of Cartan lenses. Further he has derived an expression for the spherical aberration, taking the discontinuity of the field strength into account, and has observed th at the sphe-rical aberration rnight become negative if the potentialof the foil is lower than the potentialof the outer electrodes 3). Bertein 4) has dealt with the scattering of the electrons by the gauze-shaped electrode. Gianola 5) has proposed to correct the spherical aberration by means of a uniform electrostaiic field be-tween two foils.

In order to compare the properties of one-foil lenses with those of lenses without a foil, we shall deal with three kinds of two-electrode lenses. The

V, 11! V, V2

1_.:

d

oil i.---._z

0,

t

i ___ _

~

_d

'-

'-'-'

-i

₀

'

---

'

-'_Z

t

lens of the 1st kind lens of the 2nd kind lens of the 3rd kind

Fig. l.I. The three kinds of lens es, whose properties will be dealt with. The gap between the electrodes is at z = O. The foil intersects the axis at z = Z2. The foil of the lens of the second kind is flat and therefore Z2(2) = O. The foil of the lens of the third kind is part of a sphere with radius 5/6 d and centre at z = 2/3 d, therefore Z2(3) = - 1/6 d.

electrodes of the lens of the first kind consist of two tubes of equal diameter, in the second and the third kind of lens, one of the tubes is replaced by a flat and a spherical foil respectively. Especially the second kind of lens is basic,

since two such lenses placed back to back yield a Cartan lens if the potentials ofthe outer electrodes are equal, and yield a lens ofthe fust kind ifthe potential of the foil is midway between the potentials of the outer electrodes.

Concerning the Gaussian properties, we shall present the actual values of the power and the locations of the cardinal points and we shall derive approxi-mate expressions for these quantities. The approximations will be simple func-tions of the ratio of the potentials of the electrodes and some integrals that depend on the geometry of the lens. The approximations will prove to be Father accurate and are more convenient to use than the conventional ones.

The third order aberrations will be computed fr om rays, traced by the ray tracer. They will be expressed by means of coefficients th at are independent ofthe locations of the object and the aperture. The said coefficients resembie the coefficients that are dealt with by van Heel 6) in his text-book on light optics and which are used extensively by Brouwer 7).

1.2 The three kinds of lenses considered and their axial potential distribution As mentioned in the introduction to this chapter, we shall compare the elec-tron optical properties of some kinds of rotationally symmetrical electrostatic lenses with two electrodes, see Fig. 1.1. The left electrode of these lenses is a circular cylinder with an inside diameter of d, that extends from z

=

0 to z

=

-~. In the first kind of lens the right electrode has the same form and inside diameter as the first one and extends from z

=

0 to z = ~; in the second kind of lens the right electrode is a flat foil, and in the third kind of lens it is a spherical foil with a radius of curvature of

t

d.

The gap between the e1ectrodes is in all cases much smaller than d; their potentials are VI and V2 respectively. Because the e1ectrodes have rotational symmetry we use cylindrical co-ordinates rand z in this section.

For the calculation of the paraxial rays it suffices to know cJ>(z)

=

</>(O,z)

(sèe sub section 1.3). General rays are obtained from the ray tracer (see chapter 3) for the use of which no calculations of the potential-distribution are neces-sary. We must only know the potential in off-axis points in order to determine the shape of an equipotential surface, and for this the potential distribution calculated by Bertram 8) is sufficiently accurate. Hence it suffices to determine

cJ>(z).

The following treatment becomes shorter and clearer if we introduce the function B(z). It is defined as the potential along the axis of the normalized lens of the fust kind, for which the inner radius of the e1ectrodes equals 1, and their potentials equal -! and ! respectively.

For the axial potentialof the lens of the first kind we find, in view of the definition of B(z),

cJ>(I)(Z)

=

(V2-VI) B(z/td)

+

!(VI

+

V2). (1.1) The subscript in 0 win indicate the kind oflens to which the quantity pertains. In order to express the axial potentialof the lens of the second kind in terms ot B(z), we observe that in the lens of the first kind z = 0 is an equipotential

3

-surface on which cj>(r,O) equals t(VI

+

V2). Hence if the first electrode of the lens of the first kind is at VI, and the second one is at 2 V2 - VI, cj>(r,O)

=

V2.

Therefore the axial potentialof the lens of the second kind is given by (1.2)

The potential distribution in the lens of the third kind, see Fig. 1.1, is ob-tained approximately from the potential distribution in the lens of the first kind because on the sphere with radius of curvature td, which passes through the gap, the potential proves to be very nearly constant and equal to

0·294(V2-VI)

+

Vl. Hence if the potentials of the electrodes of the lens of the fust kind are VI and 3·40 V2- 2·40 VI respectively, the potential on the sphere very nearly equals V2. Therefore the axial potentialof the lens of the third kind is obtained approximately by replacing in (1.1) V2 by 3·40 V2 - 2·40 Vl. This results in

n.. ( ) _ 3·40 (V2- VI) B(zJ-td)

+

1·70 V2 - 0·70 VI if z ~ -id

'V(3) z - V 'f '- -l-d

2 l Z :;::::'- 6 · (1.3)

In general we may say that, when the potential, cj>(td,z) on a circular cylinder with radius

-td

is given, the potential along the axis of the cylinder may be expressed in terms of cj>(td,z) and B(zl-td). We have tried to make c1ear, with the examples given above, that B(z) is a very useful function. This being so, we shall now give several expressions for B(z) from which it can be calculated'

very accurately, and also a table of B(z). We have chosen this high accuracy in order to make it possible to find exact values for d<P(z)/dz, which is neces-sary in the accurate calculation ofparaxial rays. The results of these calculations will be presented in section 1.4. The high accuracy makes it also possible to calculate higher derivatives, which occur in expressions for the aberrations, but which are not used here. In order to obtain an expression for B(z), we seek the solution, b(r,z), of Laplace's equation, which obeys the boundary condition :

b(l,z) =

t

sgn z, (1.4)

where sgn z = 1 if z

>

0 and sgn z = -1 if z

<

O. Using cylindrical co-ordi-nates, Laplace's equation reads

(1.5) for rotationally symmetrical potential distributions.

A solution of (1.5) is obtained by separating the variables, i.e. by putting cj>(r,z)

=

F(r) X G(z). Substituting for cj> in (1.5) and deviding by F(r) X G(z) yields:

Since the fust term depends on r onIy, the second term depends on z only and because rand z may vary independently, both terms must be constants. Therefore we put:

F"IF

+

F'lFr

=

k2 _{and G"IG}

₌

_{- k}2 where k is a constant.

Since b(r,z) is an odd function of zand remains finite for r

=

0, F(r)

=

lo(kr)

=

Jo(ikr) and G(z)

=

sin (kz).

A more generaI soIution of ,1cp

=

°

is obtained by multiplying the particular soIution by eek) and integrating with respect to k; i.e.:

0:>

cp(r,z) =

I

eek) sin (kz) lo(kr) dk. (1.6)

- 0:>

If we put r

=

land the integral thus obtained is compared with the identity 0:>

I

{sin (kz) I k} dk

=

7T sgn Z, - 0:>

we see that cp(r,z), given by (1,6) meets the boundary condition (lA) if eek)

=

1/{27Tklo(k)}.

Consequently,

and hence

0:>

b(r,z)

=

~f{Sin

(kz)/o(kr) I kloek)} dk, 27T

0:>

B(z) = b(O,z)

=

~f{Sin

(kz) I kloek)} dk. 27T

(1.7)

(1.8)

Since the integrand of (1.8) is an even function of k, we may at the same time replace the factor 1/27T by 1/7T and replace the lower limit of integration by 0. Expression (1.8) can be found in the paper by Bertram 8) and in the books by e.g. RusterhoIz 9) and E. Weber 10). It is shown by the authors just mentioned, that the expression for B(z) can be worked out into

0:>

B(z)

=

-t-

~ exp(-/Lnz)l/Lnh(/Ln) (1.9) n=l

where /Ln is the nth root of JO(/L) = 0. The series even converges for z = 0, because also then the terms decrease in magnitude with limit zero and have alternating signs. Because B(z) is an odd and continuous function of z it vanishes if z

=

0, and hence if z

=

0, the series equals 1/2.

5

-Expression (1.9) is very useful if z is not too smalI, z

>

0·3 say. If z

<

0·3 the convergence of (1.9) is too slow to be useful for the calculation of B(z).

In order to compute B(z) for small values of Izl, Bremmer 11) replaces in (1.8) sin (kz) by its power series and then integrates term by term with respect to k.

His result, if written in our notation, reads

0> 0> B(z)

=

~

1

dk/lo(k) z - _1_

1

{k2_{/Io(k)} dk z3}

+ ...

₌

~ 3!~ o 0

=

0·663114 z - 0-4156 z3

+

0·3366 Z5 - ... , (1.10) 0>

where we have made use of

f

dk/lo(k) = 2·083233, as ca1culated by Bouwkamp o

and de Bruyn 12).

The convergence ofthe last series is rather slow for say z

>

0'2, but the main difficulty is that the ca1culation of the coefficients of the later terms becomes very laborious. For th is reasons it is better to calculate B(z) for small values of z by a method that is derived from the relation

1

{cos P,k)/lo(k)}

dk

=

fim~o

cos {\8(m +!)}/Io{fi(m +.i)} o

0>

+

2~ ~ cosh (A/1-n)/h(/1-n) {I

+

exp (2~/1-n/fi)}, (1.11)

11=1

where À, fi, and k are real, fi

>

0 and filAI

< 2~,

which relation has been derived by Bouwkamp and de Bruyn 12). If the left-hand side of (1.11) is integrated under the integral sign with respect to A from 0 to zand is divided by ~, we obtain

0> 0> Z

1 f'

1

11 1

-; j [

{cos (Ak)/lo(k)}dk] dA = -; [ {COS(Ak)/Io(k)}dA] dk =

o 0 0 0

=

~

.r

{sin (zk)/k1o(k)} dk = B(z).

o

Consequently, we obtain an expression for B(z) by applying the same opera-tion as above to the right-hand side of (1.11). If summaopera-tion and integraopera-tion are interchanged one obtains

0>

B(z)

=

~ sin{zfi(m+!)}/(m+!)lo{fi(m+!)} m=O

0>

+

~ sinh (/1-nZ)/ /1-nh(/1-n) {I

+

exp (2~/1-n/fi)}. (1.12)

n=l

by substituting

f3

=

21T/Z. If

f3

is so chosen the first series vanishes and the second one becomes

ex>

~ {exp (!-'nz) - exp (-!-'nZ)}/ !-'nh(!-'n) {I

+

exp (!-'nZ)}.

n~l

If the numerator of the nth term is written as

1

+

exp (!-'nz) - {I

+

exp (!-'nZ)} exp (-!-'nZ) one easily finds:

ex> cx> B(z)

=

~ I/!-'nh(!-'n) - ~ exp (-!-'nz)/!-'nh(!-'n)

=

n~O n~O ex>

t

-

~ exp(-!-'nz)/!-'nh(!-'n). n=O

For the calculation of B(z), if z

=

0·05,0·10, ... 0·30, we have used (1.12)

with f3

=

3. The first 7 terms ofthe fust series and the first 3 terms ofthe second one have been calculated with the aid of a desk calculator to 10 decimal places. The other terms of both series are all

<

10-10 • If the same result should have been obtained with (1.10) about the same number of 10 terms should have been necessary. However the calculation of the coefficients should have taken up much more time than the whole calculation with the aid of (1.12). The cal-culation of B(z) for z

=

0·30, 0·35, ... 5·00, has been carried out by Mr. H. G. Kaper, to which our thanks are due, at the Pbilips Computer Cent re with the

aid of an I.B.M. 650 using (1.8). Table I gives B(z) for z

=

0, 0·05, 0.10, ... 5·00.

Let us finally compare B(z) with

t

tanh (wz), which function is often used instead of B(z) to express the axial potentialof our lenses of the first, second and third kind. The factor w is chosen so that both functions have the same

slope for z

=

0, i.e.:

ex>

w

=

2B'(0)

=

~

f

dk/lo(k)

=

1·3262275 .... (1.13)

o

With tbis value of w we obtain the series expansion:

t

tanh (wz)

=

0·663114 z - 0·38878 z3

+

0·273528 Z5 - •••

In order to give an idea of the different behaviour of both functions, table 11 gives B(z) and tanh (wz) - B(z) as a function of z.

- 7 -T ABLE 1. B(z) as a function of z ;t B(x) %

B(z)

z

B(z)

0.00 0.00000000

1.70 0.48661255

3.40 0.49977476

0.05 0.03310382

1.75 0.48812358

3.45 0.49980028

0.10 0.06589891

1.80 0.48946478

3.50 0.49982290

0.15 0.09808882

1.85 0.49065508

3.55 0.49984297

0.20 0.12940050

1.90 0.49171131

3.60 0.49986076

0.25 0.15959357

1.95 0.49264847

3.65 0.49987653

0.30 0.18846689

2.00 0.4934-7992

3.70 0.49989052

0.35 0.21586233

2.05 0.49421751

3.75 0.49990292

0.40 0.24166572

2.10 0.49487180

3.80 0.49991392

0.45 0.26580528

2.15 0.49545217

3.85 0.49992367

0.50 0.28824827

2.20 0.49596693

3.90 0.49993232

0.55 0.30899619

2.25 0.49642349

3.95 0.49993999

0.60 0.32807928

2.30 0.49682841

4.00 0.49994679

0.65 0.34555085

2.35 0.49718752

4.05 0.49995282

0.70 0.36148170

2.40 0.49750600

4.10 0.49995816

0.75 0.37595512

2.45 0.49778843

4.15 0.49996290

0.80 0.38906240

2.50 0.49803890

4.20 0.49996710

0.85 0.40089911

2~55

0.49826101

4.25 0.49997083

0.90 0.41156208

2.60 0.49845797

4.30 0.49997413

0.95 0.42114703

2.65 0.49863264

4.35 0.49997707

1.00 0.42974680

2.70 0.49878752

4.40 0.49997966

1.05 0.43745008

2.75 0.49892486

4.45 0.49998197

1.10 0.44434060

2.80 0.49904665

4.50

o

.499984()1

1.15 0.45049657

2.85 0.49915465

4.55 0.49998582

1.20 0.45599050

2.90 0.49925041

4.60 0.49998.743

1.25 0.46088911

2.95 0.49933533

4.65 0.49998885

1.30 0.46525349

3.00 0.49941062

4.70 0.49999012

1.35 0.46913927

3.05 0.49947739

_{4.75 0.49999124}

1.40 0.47259692

3.10 0.49953660

4.80 0.49999223

1.45 0.47567208

3.15 0.49958909

4.85 0.49999311

1.50 0.47840588

3.20 0.49963565

_{4.90 0.49999389}

1.55 0.48083533

3.25 0.49967692

4.95 0.49999458

1.60 0.48299361

3.30 0.49971352

_{5.00 0.49999520}

1.65 0.48491048

3.35 0.49974598

TABLE II. B(z) and

t

tanh (wz) - B(z) as a function of z. z B(z)

t

tanh (wz) - B(z) 0 0 0 0·5 0·28825 0·00197

}·o

0·42975 0·00442 }'5 0·47841 0·00323 2·0 0·49348 0·00158 2·5 0·49804 0·00064 3·0 0·49941 0·00024 3·5 0·49982 0·00008

Table II shows that

t

tanh (wz) is good approximation of B(z)

1.3 Ray equations and ray constants

In this section we shall derive variational equations to characterize the motion

of an electron and the ray traced by it. The variational equation for the ray

will anab1e us to define the electron optical index of refraction. By varying

not only the path of integration but also the end points we shall be able to derive the Lagrangian invariant. The paraxial ray equation for an electron moving in a rotationally symmetrical field will be derived from the variational equation for the ray. The paraxial ray equation will be used in section 1.4, dealing with the Gaussian properties of lenses, and in section 2.4 dealing with the scattering of electrons by gauzes. The Lagrangian invariant wiJl be used in section 1.5, dealing with the third order properties of lenses.

1.31 Variational equation for the motion

The motion of the electron can be described by means of the variational

equation, known as Hamilton's principle. It states th at the motion, given by

the displacement vector q(t), makes

02.12

W

=

f

L(q,v,t) dt, (1.14)

01./1

stationary with respect to weak variations, oq(t), of q(t), provided all motions

go through the same terminal points, Ql,tl and Q2,l2, in space time. L(q,v,t)

denotes the Lagrangian function of the electron, it depends explicitly on

dis-placement q, velocity v

=

dq/dt and time t.

The variation of W, 0 W, brought about by varying the terminal points and

- 9

-QZ+~qZ.I,+~/, QZ.IZ

.3W =

I

L(q+.3q, v+ .3v,t) dt -

I

L(q, v, t) dt.

Q, +~q,./, +~/, Q,./,

Since L(q+.3q, v+.3v,t) = L(q,v,t) + bL/bq . .3q+.3L/ov . .3v, where bL/bq and

bL/bv denote grad L, considering q and v respectively as the independent

varia-bie while keeping the other variables, v,t and q,t respectively, constant, and

because .3v = .3dq/dt = d.3q/dt, we obtain

L(q+.3q, v+ .3v,t) = L(q,v,t) + bL/bq . .3q+bL/bv.d.3q/dt.

Substituting for L(q+ .3q, v+.3v,t) yields

I, I, Iz+~/, .3 W

=

I

Ldt

+

I

{bLjbq . .3q

+

bLjbv.d.3qjdt}dt

+

I

L dt.

11+ 6(1 /1 /2

Integration by parts of the second term of the second integral and replacing

the first and the last integral by - L(ql,Vl,tI).3tl and by L(q2,v2,t2).3t2

respec-tively gives:

I,

.3 W

=

[bL/bv . .3q]:~

+

[Ht]:~

+

f

{bL/bv - (djdt)bLjbv} . .3qdl.

I,

.3q(t= tI) and .3q(t= t2) in the first term stands for the variations of the

displace-ment vectors at t = tI and t = t2 respectively. We want however to express

.3 W in terms of .3ql and .3q2, which are the variations of the terminal points of

the varied line in space time with respect to the terminal points of the original

one. Since .3ql,2 = .3q(t= tl,2) + Vl,2.3tl,2, we have .3q(t=tl,2) = .3ql,2

-Vl,2.3tl,2. Substitution in the expression for .3 W yield

Q,.IZ

.3 W = [bL/bv . .3q]g~ +[(L-v.bLjbv).3t]:~ +

I

{bL/bq - (d/dt)bL/bv} . .3qdt. (1.15)

Q,./,

Since it is assumed in Hamilton's principle that the terminal points in space

time are fixed, it states th at the integral of the right side vanishes for arbitrary

.3q. Hence the motion of the electron must satisfy

bL d bL

- - - - =0.

bq dt bv ( 1.16)

Equation (1.16) represents three so caUed Euler-Lagrange equations. If we

caU the actualline in space time described by the electron, the world line, (1.16)

is the equation of motion of the electron or of its world line.

We choose L(q,v,t) so that (1.16) is equivalent to the Lorentz equation of motion of an electron in an electromagnetic field. The latter equation reads:

(d/dt) (mv;Vl- v2/c2) = - eE - eB /\ v, (1.17)

of light, E is the electric field strength, B is the magnetic induction and B /\ v denotes the vector product of Band v.

The Lorentz equation of motion will now be derived from (1.16), using the

Lagrangian

L(q,v,t)

=

- mc2_{Vl - v2/C}2

+

_{e4> - eA.v,}

where cp is the electric potential and A is the magnetic vector potential.

It follows from (1.18) and the definition of oL/ov that

oL/ov

=

mv/Vl - v2_{/C2 - eA,}

and hence that

(d/dt) (mv;Vl - V 2/C2)

=

(d/dt) (oL/ov - eA).

If one uses (1.16) and the relation

dA/dt

=

(v grad) A

+

oA/ot,

one finds

(d/dt) (mv;Vl - v2/C2)

=

oL/oq

+

e(v grad) A

+

e oA/ot.

From (1.18) and the definition of oL/dq we derive

oL/oq

=

e grad cp - e(v grad) A - e(curl A) /\ v,

which together with (1.20) yields

(d/dl) (mv;Vl - v2/C2)

=

e grad cp

+

e oA/ot - e(curl A) /\ v.

If one substitutes

E

= -

grad 4> - ~A/ot and B

=

curl A

in the last equation, the Lorentz equation of motion is obtained.

We define the generalized momenturn, p, by means of

p

=

oL/ov, (1.18) (1.19) (1.20) (1.21 ) (1.22)

as is commonly done in electronoptics. This way of defining p has the advantage

that p has the same èlirection as v if A

=

O. It must be emphasized that p from

(1.22) differs in general from p as defined in classical mechanics, i.e. by Pi

=

oL/o(dqt/dt), qi being the ith generalized co-ordinate, we have retained however

the expression 'generalized momentum' since p differs from mv/Vl - V 2/C 2

if A of=. O.

From (1.19) we get

p = mv;Vl - v2/C 2 - eÁ. (1.23)

The second term of the right side of (1.15) may be calculated from (1.18) and (1.19), to yield

1 1

-E stands for the total energy, since (1.24) clearly represents the sum of the kinetic energy, the energy at rest and the potential energy.

If bL(öv

=

pand v.bL/bv - L

=

E are introduced in (1.15) one finds

02.12

8W= P2. 8q2- pl. 8ql-E28t2+EI 8tl

+

f

{bL/bq- (d/dt)bL/bv}.8qdt. (1.25) OJ,II

1.32. Variational equation for the ray

In static electron optics, electron rays are studied only in systems, where the electromagnetic field is independent of time. For the rays in such fields we shall derive a variational equation in which time does not appear.

Along any line in space time we have:

dE/dt

=

(d/dt) (v.bL/bv - L)

=

dv/dt.bL/bv

+

v.(d/dt) bLjbv

-(bLjbq.dqjdt

+

bLjbv.dv/dt

+

bLjbt)

=

v.{(djdt) bLjbv - bLjbq} - bLjbt.

It follows fr om (1.16) and the last identity that

dEjdt

=

- bLjbt

along any world line of an electron.

The fields, in which the motions of electrons are considered now, are inde-pendent of time and hence the same holds for the potentials. Consequently

bLjM

=

0 and hence E

=

constant along the world lines of the electrons th at are considered in static electron optics.

In such fields it is possible to decribe the world line bya variational equation with the conditions 8ql

=

8q2

=

8E

=

0 instead of the conditions made in Hamilton's principle.

If E is constant and only world lines are considered

8W

=

P2 . 8q2 - P18ql - E. 8(t2 - fI),

where E is chosen so that the time of ftight of the electron from Ql to Q2

equals t2-tl. Therefore, the function

has the variation

8S

=

8 W

+

8E.(t2- tt)

+

E.8(t2- tt)

=

P2 . 8q2 - PI . 8ql

+

8E.(t2- tl). (1.26) Since v dt

=

sv dt

=

s ds, where s stands for the arc length and s is a vector of unit length along the tangent of the ray and which points in the direction of the motion

02 02 02

S

=

f

(L+E) dt

=

f

p. v dt

=

f

p. s ds. (l.27)

Ol QI . Ol

ft and t2 are omitted from the integration limits since the integrals are in-dependent of hand Eis chosen so that the time of ftight equals t2-ft.

We fix the total energy so that 4> vanishes where v vanishes. Equation (1.24) shows that tbis amounts to

E

=

mc2• (1.28)

Since we shaII consider only e1ectrons that start with zero velocity from cathodes th at have the same potential, wruch must be equated to zero, oE

=

O. The lines in space or the trajectories described be the electrons wil! be called rays. Because only rays with equal energy are considered

oS

=

P2 . dq2 - Pi . dql. (1.29)

The rays themselves are deterrnined by the variational equation

Q2

oS

=

0

f

p.S ds

=

O. (1.30)

Q,

for weak variations of the motion from Ql to Q2.

Equation (1.30) is known as the principle of stationary action. If S is not only stationary but shows a minimum for the actual motion, it may be called the principle of least action, as it was caIIed by its discoverer Maupertuis. It wiII be clear th at if the motion of the electron satisfies Harnilton's principle and the energy is a constant of the motion, the principle of stationary action is satisfied too. If, on the other hand, the principle of stationiuy action is satis-fied and the energy of the electron is so chosen th at it takes t2 - ti to go from Ql to Q2 th en oS = 0, oE = 0 and Ot2 = otl, and hence, by virtue of (1.26), oW

=

O. Therefore both principles define the same motion or world line, pro-vided the field is time independent and the energy, when using the principle of stationary action, is properly chosen.

An equation in which only quantities appear that depend on location and direction will be derived from (1.30). Jt follows from v

=

v S and (1.23), that

p

=

m vsjVI- v2jc2- eA.

v may be elirninated from the expression for p by expressing the former in terms of m,4>,e and c. This is achieved by equating E given by (1.24) and given by (1.28); i.e. by solving

mc2jVl - v2jc2 - e4>

=

mc2

for v. If v obtained thus is substituted in the expression for p, one gets:

p

=

V2em4>

+

e24>2jmc2 s - eÁ. (1.31) Substituting in (1.30) for p yields the variational equation :

Q2

oS = 0

f

(V2em4>

+

e24>2jmc2 _{- eA.s) ds}

₌

_{0 (1.32)}

0,

1 3

-Several remarks must be made about (1.32). Firstly, the potentials 4> and A

are not uniquely defined by the electromagnetic field. It follows from (l.21)

th at any constant may be added to 4> and any vector field who se curl vanishes

may be added to A without altering the electromagnetic field. Now 4> is fixed

by putting E

=

mc2 , but A may be replaced by A

+

grad X(q) without changing

B. This means th at p contains the same arbitrary vector grad X(q). However,

(l.32) is not altered by adding grad X(q) to A because this amounts to

subtract-ing the constant quantity e{X(Q2)- X(Ql)} from the integraI, which constant

drops out when taking the variation.

Secondly, one might expect from (l.32) as it stands th at if Ql and Q2 are

interchanged, the same ray will be determined by it. It follows however from

the definition of s that interchanging Ql and Q2 causes s to change sign.

Conse-quently, if A

=

°

along the ray, an electron when running from Q2 to Ql

des-cri bes the same ray as when running from Ql to Q2 and if A -=1=

°

the ray

des-cribed by an electron moving from Q2 to Ql will differ from the original ray.

If, however, the magnetic field is reversed before the motion starts, the ray,

described by an electron moving from Q2 to Ql, will coincide with the original

ray.

Thirdly, from comparing (1.32) with the principle of Fermat of light optics,

Q2

a

f

n ds

=

0,

0,

where n denotes the index of refraction, it follows that the light ray that

passes through Ql and Q2 will coincide with the electron ray through Ql and

Q2 if

(1.33) The integrand of (1.32) is therefore often called the anisotropic electron optical

index of refraction. It must be observed however that the anisotropic index of

refraction of several kinds of crystals cannot be represented by (1.33) because the latter quantity does not change if the direction of the light ray is reversed

whereas the integrand of (1.32) gene rally does. Obviously the index of

refrac-ti on of a crystal is an even funcrefrac-tion of the direcrefrac-tion cosines, Sa;, Sy and Sz, of the

light ray.

It also follows from comparing (1.32) with the principle of Fermat that S

may be identified with the point characteristic function V( Ql, Q2) of Hamilton.

1.33 The Lagrangian invariant

In this subsection we shall derive an invariant relation, which holds between three rays that are infinitely near one another.

In the subsequent derivation, the fact will be used that rays are completely

to consider only rays that belong to a narrow bundie that contains the terminal points. Since arelation between three rays that are infinitely near one another will be derived, tbis is no restriction of the validity of the invariant.

Since any ray of the bundIe is completely deterrnined by its terminal points, the momenta at this points an~ the action are determined by them too. If the terminal points QI and Q2 are sbifted by the vectors 8qI and 8q2 respectively, the second order approximation of 8S is given by

8S = bS/bql.8ql + bS/bq2.8q2 + t{(8ql gradI) (bS/bq!) + + (8q2 grad2)(bS/8ql)} . 8qI + t{(ql gradI) (bS/bq2) + + (8q2 grad2) (8S/bq2)}. 8q2,

where the index 1 or 2 of grad indicates whether the gradient is taken with respect to ql or q2 respectively.

Tt follows from comparing this expression with (l.29) th at

PI = - bS/bql and P2 = bS/bq2. (l.34) Hence the shifting of the terminal points causes PI and P2 to change by 8PI = ( 8qI grad l) PI + (8q2 grad2)pI = - (8ql gradI)(bS/bql) - (8q2 grad 2)(bS/bql) and

8P2=(8ql gradI)p2 + (8q2 grad2)P2 = (8ql gradI)(bS/bq2) +( 8q2 grad2)(bS/bq2). By virtue of (l.34) and the expressions for 8PI and 8P2, the second order ap-proximation of 8S may be expressed by

8S = (P2 + -t8p2). 8q2 - (PI + -t8pI). 8ql. (l.35)

If the terminal points are shifted from QI + 8qI and Q2 + 8q2 to QI + 8qI + dql and Q2 + 8q2 + dq2 respectively, (1.35) shows that Sis changed by

dS = (P2+ 8P2 + -tdp2).dq2 - (PI + 8PI +-tdpI).dql.

Sbifting the terminal points from Ql and Q2 to Ql + 8qI + dql and Q2

+

8q2 + dq2 respectively, causes S to change by

d8S = {p2 + t( 8P2 + dp2)}. (8q2 + dp2) - {PI + -t(8pl + dpI)}. (8ql + dqI). Since S depends only on the terminal points

d8S = 8S + dS. (l.36)

The Lagrangian invariant is obtained from substituting the second order approximations of d8S, 8S and dS, derived in tbis subsection, in (1.36), i.e.:

8P2 . dq2 - dP2 . 8q2 = 8PI . dql - dPl . 8ql (l.37) Because any point on the ray through QI and Q2 may be taken as terminal point, it follows from (l.37) that 8p. dq - dp . 8q = constant for three rays that infinitely near one another.

1 5

-lt must be observed that the derivation of the Lagrangian invariant by Sturrock 13) is not correct, since he does not use a correct second order ap-proximation of oS .For example, if the terminal points are shifted to Q1

+

Oq1 and Q2 = OQ2 and then back to Q1 and Q2 and the resulting changes of S are added, Sturrock's method will show th at OP2 • OQ2 - OP1 . OQ1

=

0, which is not generally true.

1.34 The paraxial ray equation

In trus subsection we shaU derive the paraxial ray eqation in fields that are rotationally symmetrical. lt will be assumed that the region where the electrons move is free from charges and currents.

There are several methods to derive the paraxial ray equation, notably the one given by Francken 14) and the more conventional one, for which we refer to Glaser 15).

Francken uses cylindrical co-ordinates r,

e

and z. He first derives an equation for rand one for

e

,

both equations being valid for arbitrary rays. From these equations he derives equations that are only valid for paraxial rays. The equa-tions for r, he has derived so far, are non-linear to such an extent that they will be very hard to solve. He finally derives an equation for the projection of the paraxial ray upon a screw-like meridional surface and an equation for this surface. The equation for the projection is the paraxial ray equation in its weU known form.

Because the conventional method is shorter and because it starts fr om the variational equation for the ray obtained already, we shall use this method.

The fust step will be to translate the variational equation into an equatión in rectangular co-ordinates x, y and z, where z is the independent variabie. The components of a vector in the x, y and z directions will be indicated by the subscripts x, y and z respectively. Total ditferentiation with respect to z will be indicated by a dash.

The components Sz, Sy and Sz of vector s must satisfy Sz : Sy : Sz = x' : y' : 1 and sz2

+

Sy2

+

sz2 = 1. Therefore we find:

Sz

=

x';VI

+

X'2

+

y'2, Sy

=

y';VI + X'2

+

y'2, Sz

=

I

/V

I +

X'2

+

y'2 and ds = (I/sz) dz =

V

I +

x'2

+

y'2 dz.

Because the last of these expressions only makes sense when Sz remains finite, for which it is necessary that x' and y' remain finite, we have excluded from our further considerations reflection of the electrons.

Substituting in (1.32) for Sz, Sy, Sz and ds and dividing by V2em yields

02

o

J

{V.p

+

e.p2/2mc2

VI

+

X'2

+

y'2 - Ve/2m (Azx'

+

Ayy'

+

Az)}dz, (l.38) 01

The second step depends on which kind of rays are considered. If general rays are considered, equations for such rays consist of the Euler-Langrange equations th at follow from (1.38), i.e.

oL d oL - - - - = 0 ox dz ox' and (1.39) oL d oL - - - - =0, oy dz oy' where L denotes the integrand of (1.38).

If, on the other hand, only paraxial rays are considered, the paraxial ray

equa-tions are obtained in the following way. Firstly, cp and A are expressed in terms

of a series of ascending powers of x and y, of which the coefficients are func-tions of z only. Secondly, one writes

L

=

Lo

+

L2

+

L4

+ ..

.

,

where Lo is independent of x, y, x' and y' and L2 and L4 are second and fourth degree polynomials respectively of x, y, x' and y'. Thirdly, one derives

the paraxial ray equation from the Euler-Lagrange equations, where L is

replaced by L2 thus neglecting L4, L6 etc., Lo being left out since oL%x to

oL% l' vanish.

If the fields, where the paraxial rays are considered, are e.g. non-electric, non-magnetic or cp is so small that

lvi

«

c, the derivation of the paraxial ray equation is simplified by introducing the assumptions about the field in L, Since we shall only consider paraxial rays in electrostatic fields and we shall assume that

l

v

i

«

c, the second step consists ofneglecting ecp2/2mc with respect to rp, equating A to zero, which is allo wed since B

=

0 in the region where the electrons move and hence X(x,y,z) can be so chosen that A

+

grad X(x,y,z)

=

o.

and expressing rp in terms of a power series in x and y.

Using cylindrical co-ordinates, the assumptions made about the field and about the region where the electrons move yield

6.rp

=

o2rp/or2

+

(ocp/or)/r

+

02rp/OZ2

=

o.

(1.5) It can be verified by substitution in (1.5) that rp may be expressed by

rp(r,z)

=

<P(z) -

t

r2<P"(z)

+

114; r4<P""(z) - ... , where <P(z)

=

rp(O,z) is considered as a function of z only.

Substituting r2

=

x 2

+

y2 in (1.40) yields

(1.40)

rp(x,y,z)

=

<P(z) -

t

(x2+y2)<P"(z)

+

ll-:dx2+y2)2<P""(z) - .... (l.41)

It follows from the definition of L2 and (1.41) that L2

=

-t

(X2+y2)<P"

/

0P

+!

(X'2+l'2)

vq;.

1 7

-Substitution of L

=

L2 in the Euler-Lagrange equations, (1.39), and multiplying by

-

2W>

yields

<1>r"

+

-t<1>' r'

+

1;<1>" r = O. (1.42) where r stands for x and for y.

Because the paraxial rays are defined by two second order Unear differential equations, any paraxial ray may be considered as a linear combination of four linearly independent paraxjal rays. For the latter rays we may chose paraxial rays that lie in meridional planes. Such rays exist because the equations for x and y admit solutions where y

=

0 and solutions where x

=

O. Hence any paraxial ray may be considered as a linear combination of two linearly inde-pendent paraxial rays in a meridional plane and two linearly indeinde-pendent par-axial rays in another meridional plane. In order to obtain these rays it suffices

to know two linearly independent solutions of (1.65). 1.4. The Gaussian properties of the lenses considered

In this section approximations of the power, to be defined, and of the loca-tions of the principal planes and the foci will be derived. The approximaloca-tions will be expressed as simple functions of

h

l

VI, where certain integrals depending on the axial potential if - h

=

V2

=

t

occur as parameters. Approximations of this kind offer the advantage that, once the integrals just mentioned have been calculated the power and the locations of the cardinal points become functions of V21VI only.

It will be assumed th at the field in the lenses considered decreases so fa st if z either increases or decreases, that the integrals just mentioned exist, or what amounts to the same, that the rays have straight line asymptotes. In order to check the approximations just mentioned, they will be applied to our lenses of the first, second and third kind, of which the Gaussian elements are computed

numerically.

The foci and the principal planes are defined as usual; i.e. in terms of two rays, indicated by the subscribts I and IT, that satisfy

rI'(~) = 0 and "II'(-~)

=

O.

The asymptote for z

=

-~ to a ray will be called the incident ray and the asymptote for z

=

~ to a ray will be called the emerging ray. In this section they will be indicated by the subscribts in and em respectively. Therefore we may write

rIem'

=

"I'(~)

=

0 and rIIin'

=

"II'(-~)

=

0 and hence

rIem

=

constant and rmn

=

constant. 1.41 Definition of rhe cardinal points and rhe power

V, electrodes 1/.2 (=10 \/', J. ---~---~~~/~~ _{f ï i}

.

'ir in O/~~---=:...::..:...---..c~~c..;-;;:: - - - _ _ L -rIl ".. : I rIem rIin2 : 'irem 2

---r-

J -Z2

Fig. 1.2. Gaussian rays land II in the lens of the second kind, where V2/ VI = 10. The

inci-dent ray and the emerging ray denote the asymptotes for z = - 00 and for z = 00 respectively

to the ray_ The emergent ray land the incident ray n are parallel to the axis. The foei are

indicated by Fl and F2, the principal planes are indicated by Hl and H2.

of an incident ray I. The first principal plane, Hl, is defined as the geometrie locus of the intersections of the incident and emerging rays I. Fl is located at

Z = ZF!, r = 0, the equation for Hl reads Z = ZHl. The second focus, F2,

and the second principal plane, H2 are defined in a analogous manner in terms of ray II. The nodal points, NI and N2, are defined as the intersections with the axis of an incident ray and an emergent ray respectively, provided both rays are parallel to each other. The focallengths,fl andf2, are defined by means

of fl

=

ZHl - ZFl and f2

=

ZF2 - ZH2. (1.43)

We introduce the power, P, of the lens, defined by

p

=

±

l/Vj;fi,

(1.44)

where the

+

sign is to be taken if fl and hare positive and the - sign if fl and f2 are negative.

1.42 Locations of the cardinal points and the power expressed in terms of matrix T

In one-foil lenses, we assume that the foil intersects the axis at Z

=

Z2,

that it has a potential V2 and that the region to the right of the foil, i.e. for Z ~ Z2 is field free. Hence the emergent rays are given by:

rem(Z)

=

(z-z2)rem'

+

rem2,

where rem' stands for the slope of the emerging ray and rem2

=

rem(Z2)-The incident ray is given by:

rtn(z)

=

(z-z2)rtn'

+

rfn2,

(1.45)

(1.46) where rtn' denotes the slope of the incident ray and rfn2

=

rtn(Z2); it must be remembered that the incident ray is defined as the asymptote for Z

=

-~ to the ray.

1 9

-Let us now intro duce matrix T by means of which rem' and r em2 can be

ex-pressed in terms of rin' and rin2, i.e.:

rem'

=

IUr1n'

+

h2r1n2

fem2 = 121rtn'

+

122f1n2. (1.47) lt is c1ear that the Gaussian elements, defined by means of the asymptotes

to the rays, are completely d~termined by T. We shall first express them in

terms of Tand in the next subsection derive the desired approximation of T.

In view of the definitions of ZF1 and ZH1 and because of (I.46) we have:

rHn(ZF1)

=

(ZF1- Z2)fHn'

+

fHn2

=

0 and rHn(ZH1)

=

(ZH1- z 2)rHn'

+

rHn2

=

rIem.

ZF1 and ZH1 are obtained if rHn' and rHn2 are expressed in terms of rIem by

means of (1.47), where it is to be remembered that rIem'

=

O. We thus find

ZF1

=

Z2

+

lu/112 and ZHl

=

Z2

+

(lu - ITI)/112.

(1.48) It follows from the definitions of ZF2 and ZH2 that,

rUem(ZF2)

=

(ZF2- z2)rUem'

+

fllem2

=

0 and fUem(ZH2) = (ZH2-z2)rUem'

+

rUem2 = rIlin·

We solve for ZF2 and ZH2 by expressing rllem' and rUem2 in terms of fmn by

means of (I.47) remembering th at rmn'

=

o.

This results In

ZF2 = Z2 - 122/112 and ZH2

=

Z2

+

(1 - t22)/112. The locations of the nodal points are found from

rem'

=

rin'

r1n(ZNI)

=

(ZN1-Z2)rin'

+

rin2

=

0 rem(ZN2)

=

(ZN2-Z2)rem'

+

rem2

=

O.

(1.49)

We solve for ZN1 and ZN2 by expressing rtn', rem' and rem2 in terms of rtn2

using (1.47) and taking into account that rem'

=

rin'. We find

ZN1

=

Z2 - (1-lll)/112 and

ZN2

=

Z2

+

(IT I - (22)/122.

From the definitions of

/1

and

/2,

(l.48) and (1.49) we find:

/1

=

- ITI/t12 and /2

=

- 1/112. The power of the lens follows fr om (1.44) and (1.51), i.e.:

P

=

-1I2/M.

(1.50)

(1.51)

(1.52) Let us show now that the locations of the cardinal points can be expressed

that ZP1 = ZH1 - f1 and ZP2 = ZH2

+

f2. From comparing (1.48), (1.49), (1.50) and (1.51) we see that ZN1

=

ZH1 - f1

+

f2 and ZN2

=

ZH2 - f1

+

f2. Because of these relations and because of the definition of tbe power of the lens, we may say that the locations of the cardinal points can all be expressed in terms of ZHl, ZH2, f1 and f2. Now

ft

and f2 can be expressed in terms of V2jV1 and P because of the definition of P, (1.68), and

f21f1

=

VV2jV1, (1.53)

'>Yhere we have restricted ourselves to non-relativistic velocities.

Relation (1.53) follows from the facts that f1 and f2 are defined in terms of the asymptotes to the rays and tbat rI and rn obey the paraxial ray equation,

(1.66). Hence

rn(rr"cP+trr'cP' +i' rcP") - rI(ru"cP+tru'cP' +irncP")

=

(rurr" - rIru")cP + Ji(rurr' - rrru')cP'

=

(rurr'- rIrn')'cP + t(ru/,r'- /'Iru')cP'

=

o.

This ditferential equation in rurI' - rrru', the Wronskian, is solved by rurr' - rrru'

=

cP-1_/2 _constant.

Therefore,

The left member of this relation equals f21f1 because rI'(~)

=

rII'(-~)

=

0, f1

=

rI(~)jrr'(-~) and f2

=

-rIl(-~)jrII'(~), which proves (1.53).

If follows from (1.44) and (1.53) tbat f1

=

(hjVÜ-I

/4jP and f2

=

(V2j VÜ'/4jP. Hence we have proved tbe statement made just below (1.52).

Substitution in (1.53) for f1 and f2 from (1.51) yields

I

T

I

=

VV1jV2,

which re1ation will be used below as a check.

1.43 Approximation of T in terms of cP(z) If we introduce

(1.54) ,

(1.55)

(1.56) matrix T, and bence P, ZH1 and ZH2 can be expressed as functions of N, provided the geometry ofthe lens is given. If N = 1 there is no field in the lens and hence

P = O. Tberefore, P can be expressed as a series whose terms are O(N- l), O(N- l)2 etc., wbich we shall call for short a power series of N - 1.

Le Rütte 16) has shown tb at the power of a lens without a foil can be approx-imated to a given power of N- l by calculating tbe rays by means of tbe

well 2 1 well

-known equation of Picht, viz.:

Rif

+

Q(z)R

=

0, (1.57)

where R

=

,cJ>1/4 and Q(z)

= -t

\ {cJ>'(z)fcJ>(z)}2.

This equation is obtained from the paraxial ray equation, (1.42), by

substi-tuting r

=

RcJ>-1/4.

We shall show that the use, of R yields P, ZUl and Zll2 to a given power of

N - 1 also for one-foil lenses. 1t will be shown too, that the aproximations

thus obtained contain those for lenses without a foil as a special case. For

two-foil lenses, however, the use of R can lead to erroneous results. .

In order to express T as a power series of N - 1 we integrate (1.57) with the

initial conditions, i.e. the asymptote for z

=

-~, given by R(z)

=

Rin(Z)

=

(z - z2)Rin'

+

Rin2. The integration yields R'2 = R'(Z2 - 0) and R2 = R(Z2),

in terms of which rem/ and rent2 can be expressed. Although formally the

inte-gration yields R'(Z2+0) just as weU, 1"2 derived from R'(Z2+0) is incorrect.

The reason is that R = 1'(/)1/4 shows a bend when crossing a foil because cJ>'

is discontinuous there and I' shows no bend. Because this behaviour of R does

not follow from the Picht eq uation, the latter may be used only in regions where

cJ>/ is continuous.

The subsequent derivation will be simplified by the introduction of the

two-row vectors, ( I' ent' ) rem

=

,'em2 ( 1.58) and by writing, (1.59)

where X, Y and Z are transformation matrices, which have two rows and two

columns.

It follows from (1.58) and (1.59) that rem

=

ZYX rin and hence by virtue of

(1.47) we have:

T= Z Y

x.

(1.60)

The matrices X and Z are independent of the order of the approximation

but Y depends on it. X follows from R

=

,cJ>1/4 and from the fact that cJ>

=

V1

along the incident ray; i.e.:

X =

(V1~

/

4

°

)

=

_V11/4(1

V11/4 0 (1.61 )

Z may be derived from I' = RcJ>1/4 which yields 1"

=

R'cJ>1/4 - tRcJ>'cJ>-5/4;

hen ce '/2

=

R'2V2- tR2(/)/2V2-5/4 where cJ>/2

=

cJ>'(Z2- 0). Consequently,

Y is expressed by means of a power series of N - I, by solving (1.57) ap-proximately, where (CP' jCP)2 in its turn is approximated by a power series of

N - I. Integrating (1.57) once yields

z z R'(z) - Rtn'

=

J

R"(X)dx

= -

J

Q(x)R(x)dx -0:> - 0:> z z

=

-

J

Q(x)Rtn(x)dx -

J

Q(x){ R(x) - Rtn(x)} dx. (1.63) - 0:> - 0:> By integrating (1.63) we find z z y R(z) - Rtn(z) ~

J

{R'(x) - Rtn'(x)}dx

=

-

J [ J

Q(x)R(x)dx] dy. (1.64) - 0:> - co - co

In order to express Q(z) by means of a power series of N - 1, we introduce the function C(z), which is defined as the potential along the axis if VI

=

-t

and V2

=

l

It folIows from the definition of C(z) that

CP(z) = {(N- I) C(z)

+

(N+I)j2} VI (1.65)

and hence according to the definition of Q(z):

z _

3

(

(N- I) C'(z)

)

2

Q( ) - 16 (N+ I)j2

+

(N- I) C(z)

~ ~{(N-l)j(N+I)P C'2(Z) {l- 4C(z)(N- I)j(N+ I) + ... }.- (1.66)

This expansion is uniformly vaIid in z if N is positive and finite, because it is sufficient that 12C(z)(N- I)j(N+ 1)1

<

l.

For the sake of simplicity we restriet ourselves to the first terms of the power series in N - I.

The last integral of (1.63) may be dropped in this approximation because by virtue of (1.64) and (1.66) R(z) - Rtn(z)

=

O(N- I)2 and hence the integral just mentioned is O(N- I)4.

At this stage we can see that the use of R can lead to wrong results if applied to two-foil lenses. If the foils interseet the axis at z

=

ZI and at z

=

Z2 and the

distance, Z2 - ZI, between the foils is small compared with their diameters, CP'(z) ~ (V2- VI)j(Z2- ZI). Since R(z), Rin(Z) and R(z) - Rtn(z) may all have a constant sign along the path of integration and the latter has a length

Z2 - ZI, the integrals of (1.63) may tend to ~ if Z2 - ZI decreases to O.

There-fore, the second integral of the last member of (1.63) may only be dropped if Z2 - ZI is not smalI. If Z2 - ZI is small only the exact integration will yi~ld the

correct value of R'(Z2) - R'(zI). In the latter case it is even advantageous to use r instead of R.

After this digression we resume the caIculation of R'2 and R2. Substitution of z

=

Z2 in (1.63) and dropping of the last integral yield the third approxima-ti on i.e.: