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A Study on Scaling Symmetrical Condenser-Objective Lenses

A. Alamir

To cite this version:

A. Alamir. A Study on Scaling Symmetrical Condenser-Objective Lenses. Journal de Physique III,

EDP Sciences, 1995, 5 (5), pp.641-645. �10.1051/jp3:1995151�. �jpa-00249336�

Classification

Physics

Abstracts 41.80D

A Study

_on

Scaling Synunetrical Condenser-Objective Lenses

A.SA. Alamir

Physics Department, Faculty

of science, Assuit University, Assuit,

Egypt (Received

12

September

1994,

accepted

2

February 1995)

Abstract.

Scaling

^{down the dimensions} of _a conventional

magnetic objective

lens to reduce aberrations is studied in detail. The value of other aberration coefficients for

optimum

lens size

corresponding

to desired minimum aberration

are estimated.

1. Introduction

In

light optics,

it is well known

that, disregarding

the diffraction

limit,

a

glass

lens _can

always

be

improved by reducing

its dimensions. If the _same conclusion _can be drawn for

magnetic lens,

the

magnetic

circuit _can be made smaller

(conserved similarity).

In

magnetic lenses,

for _a constant flux

density

at the

poleface

and constant excitation parameter

NI/I~~/~,

_at different

accelerating voltages,

the dimensions and excitation NI of the lens _must be

multiplied by

a

scaling

factor _n. The

objective

focal

length f~,

the

spherical

and chromatic aberration coefficients

Cs

and

Cc

will also be scaled

by

the _same factor

ill.

If the

relativistically

corrected

accelerating voltage

of the

original

lens _is

('

and that to be

applied

to the scaled lens is

("

the

scaling

factor _n is

given by:

~"l/2

~

v'l/2

The factor _n is useful in

calculating

the electron

optical

_parameters at different

accelerating voltages.

The

relativistically

corrected

accelerating voltage I~

is calculated from the

following

equation:

I~

=

V(I

₊ o.978 _x

10~~ V)

where V is the

accelerating voltage

in volts.

Kamminga

_[2] has discussed in detail the variation of the aberration coefficients and other

optical quantities

when _a lens is scaled

by

_a factor _n.

In the absence of saturation

effects,

the aberrations of _a

magnetic

^lens _may in

principle

be reduced

indefinitely simply by making

it smaller _at the _cost of

increasing

current

density

in the coil.

©

Les Editions de

Physique

1995

642 JOURNAL DE

PHYSIQUE

III N°3

Mulvey

_[3] has shown that

scaling

down the dimensions of _a conventional

magnetic objective

lens increases the

peak gap-flux density

and reduces the field half-width but does not

necessarily

reduce the lens aberrations.

The purpose of the

present

discussion is to

study

the

advantages

^and

disadvantages

of

scaling

the

magnetic

^{lens dimensions} so far _as the lens aberrations _are concerned.

2.

Double-Polepiece Magnetic

Lens

A test lens

[3],

^{shown in}

Figure I,

of the Riecke-Ruska type

(condenser-objective)

_was devised and checked

by computation

to _ensure that _no saturation effects _{occur at any}

part

of the

polepieces, consisting

of two

spherical pole-caps joined

to the main

magnetic

circuit

by

_a

pair

of truncated _cones. The axial field distribution of such lens is calculated

using

_a standard finite

element _program AMAG

[4].

^This program is well

adapted

to the calculation of lenses with

large

extremes of flux

density.

2.I. THE AXIAL FIELD DISTRIBUTION AND THE FOCAL PROPERTIES. The axial field

distribution of

Figure

I is shown _{as a} function of

scaling

^factor n in

Figure

2. Curves

1,

² indicate that the lens

operates

almost

linearly. Scaling

_up

(n

₌

2)

increases the half-width

remarkably

and decreases the

peak

value.

Scaling

down

(n

=

1/2),

as indicated

by

Curve

3,

^the field start to widen and the

parasitic

field appears. This _means that the lens reaches saturation.

On the other hand the

peak

rises and the half-width decreases. In Curve 4

(n

=

1/3),

the

peak

value is increased but the half-width is reduced

slightly,

and the

parasitic

^field

plays

an

important

role. The overall result is that the focal

properties

_{are worse} in

spite

of

reducing

lens

size, I-e-, increasing

current

density,

_as shown in

Figure

3. This

figure shows,

also that for _a

given S/D

ratio there is a

particular

lens size for _an

optimum Cs

or

Cc. Figure

4 shows the variation of the coefficient

Cs

and

Cc

with

S/D

ratio for the

original

lenses. It _seems that

there is _a certain size of each lens _to reach the best condition.

IRON CIRCUIT

cot

Fig.

1.

Mulvey

[3] test lens

(Riecke-Ruska type).

Bore diameter

D,

_gap

length S, object

in mid-

plane

of _gap. Lens is scaled down to increase gap field.

I

z ₌ o-o

V = TOO KV

' lln=1

2. _n =2

' 31n

=)

4,n=J

M 2 ³

m

4

,,

3 ^',

'

o 5 io

Zimml

Fig.

2. The axial field distribution for lens of

Figure

1

(S/D

=

4).

The

specimen

is focused at

mid-plane

of gap ^at 100 kV. Lens scaled down

by

factor of 2, 1,

1/2, 1/3.

~,Cs

B

,"'

~

," ^°

~

,"

u- 6 ,"

j _~,/"

4 ^"

u

z

O 2 3 4 5 6

n

Fig.

3. The focal

properties

of scaled lenses _{as a} function of

scaling

factors.

3. Minimum

Spherical

Aberration

Figure

5 shows the variation of

(Cs)ruin

of the

optimum

^scaled lenses with the associated chromatic aberration coefficient. From the

figure

it is clear that the lens of

S/D

= 2 has the

lowest value of

Cs(Cs

₌ o.38

mm),

but the associated

Cc

= o.9 _mm is about

50Sl higher

than that of the

original

lens. The resolution is 6

=

2.61 (6

₌ 1.1

1at _I~

= I

MV).

644 JOURNAL DE

PHYSIQUE

III N°3

,~

Cc /"

I'

U I'

U /~~

~

l'

U ',

",

o

lo s/D

Fig.

4.

Spherical

and chromatic aberration coefficients of

original

lenses _{as a} function of

S/D.

s/D _n

O 64

2 o e7

4 O 46

e o 2e

", Cc

~ ', ,,'

fi '"-

_,"

~~~~-~~

d _Cs

'

s/D

Fig.

5. Minimum

spherical

aberration coefficient of scaled lenses

together

with the

resulting

coef- ficient of chromatic aberration Cc _{as a} function of

S/D.

4. Minimum Chromatic Aberration

The minimum chromatic aberration coefficient

(Cc)m;n

of

optimum

scaled lenses with the associated

spherical

aberration coefficient is shown in

Figure

6. A lens of

S/D

₌ 4 has the lowest value of

Cc(Cc

= o.74

mm).

The associated value of

Cs

is o.93 _mm

(25Sl higher

than that of the

original lens).

S/0 _n

O 94

2 0 57

4 O 74

e o 3e

° _Cs

c

Cc __-~

---~~~

Uu

o

jo s/D

Fig.

6. Minimum chromatic aberration coefficient of scaled lens

together

with associated

spherical

aberration coefficient of scaled lenses _{as a} function of

S/D.

5. Conclusion

In double

polepiece condenser-objective lenses,

_one cannot

always

take

advantage

of

making

the lenses smaller

(n

_<

I).

As the lens reaches full

saturation,

there exist _some

optimum

size and excitation which should be taken into consideration when the lens is

designed.

Reducing

the lens

size, improves

_one of the lens aberration coefficients at the _cost of increas-

ing

the other.

Acknowledgments

The author would like to thank Professor T.

Mulvey

of Asten

University

in

Birmingham,

U-K- for useful and

stimulating

discussions.

References

ill

Juma

S-M-, Kaliq

M.A. and Antar

F-H-,

J.

Phys.

E: Sm. Instrum. 16

(1983)

1063-1068.

[2]

Kamminga

W.,

Optik

45

(1976)

39-54.

[3]

Mulvey

T. and Yin

H-C-,

Inst.

Phys. Cant.

Ser. 1

(1988)

109-10, _paper

presented

at EUREM 88,

York,

U-K-

[4] ^{Lenc M. and Lencova} B., SEMB6 3

(SEM

Inc.,

Chicago, 1986)

p.897.

Commission paitaire N° 57920

©

Lea Editions de Physique 1995 Dilectdce de la Publication : Jeanne BERGER

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