1

2π-Wien filters with negative aberrations

K. Tsuno*, D. Ioanoviciu** and G. Martinez***

* JEOL Ltd., 1-2, Musashino 3-chome, Akishima, Tokyo 196-8558, Japan

** Institute of Isotopic and Molecular Technology, P.O. Box 700, R-3400, Cluj-Napoca,

Rumania

*** Dept. Fisica Aplicada III, Facide Fisica, Univ. Complutense, E-28040, Madrid, Spain

Abstract

Aberrations of 2π-(double focus) Wien filters are analyzed. There are conditions of multipole field

components giving negative chromatic and negative aperture aberrations at the same time. Under

these conditions, chromatic aberration does not depend on the distance from the central axis.

Aperture aberration has positional dependence and the beam size increases with increasing from the

axis, but roundness of the beam remains even in the off axis region and no distortions take place.

The 2π-Wien filter is suitable for an aberration corrector in direct imaging electron microscopes at

low accelerating voltages.

1. Introduction

Wien filters have significant properties of straight optical axis and round lens focus,

although they consist of non-axially symmetric multipole fields. As is well known, any

axially symmetric field lens cannot create negative spherical and negative chromatic

aberrations. The Wien filters have a round lens focus, although the electric and magnetic

field distributions have non-axial symmetry. Therefore, it has a possibility of having

negative spherical and negative chromatic aberrations under round focus. In recent

aberration correctors both of 4-8 poles and 6-poles, a beam shape is elliptic or

triangular.

Rose [1] proposed a double focus Wien filter (in the following, we call this filter as

2π-Wien filter) as an axial chromatic aberration corrector. He pointed out that the

corrector must be (i) non dispersive, (ii) free from second order geometrical aberrations

and (iii) round negative chromatic aberration. Stigmatic focus 2π-Wien filter

automatically gives properties (i) and (ii). Those properties are the same with the

requirements of a monochromator. He derived a condition for the hexapole electric and

magnetic fields to satisfy the condition (iii). Mentink et al. [2] improved the 2π-Wien

2

filter to correct both of spherical and chromatic aberrations by combining two filters

with opposite hexapole field components.

According to our analysis of the third order aberrations of the 2π-Wien filter [3], the

beam shape is distorted by the third order aberrations even if all of the above three

conditions are satisfied. We have further found [4] that there are regions of the round

beam with negative aperture and negative chromatic aberrations. The chromatic

aberrations do not depend on the distance from the axis. Aperture aberrations increase in

its size according to increase the distance from the center. However, no distortions take

place and the beam keeps its roundness even at the off axial regions.

From the above properties, the 2π-Wien filter can be used as an aberration corrector.

The corrector has advantages in microscopes operated at low voltages, such as scanning

electron microscopes (SEM), low voltage reflection electron microscope (LEEM) and

transmission electron microscopes (TEM).

2. The Wien condition

A famous straight optical axis condition, called the Wien

condition, is written as

E1 = vB1. (1)

Here, E1 and B1 are the electric and magnetic dipole

components and v the velocity of electrons derived from

the following equation of motion.

mv2 / 2 = eUo, (2)

where, e, m and Uo are the charge, mass and pass energy of electrons.

In usual Wien filters, electrodes were inserted in a gap of magnet pole-pieces. In order

to keep the field homogeneity, an electrode distance is smaller than the gap of magnet.

In such a case, the electric field decreases sharply while the magnetic field decreases

gradually at the fringing regions. Then, electrons feel only magnetic field when it

approaches the filter. Because the Wien condition is not fulfilled, the electron beam

must be bent in such a fringing region and the filter cannot have a straight optical axis.

Figure 1 shows an example of a Wien filter, in which the gap of electrodes and magnetic

pole-pieces has the same distance to keep the fringing region of magnetic and electric

field equal. However, such a filter has another problem of field homogeneity. A large

Fig. 1. Wien filter with the

same gap length.

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hexapole field component E3 appears due to the

inhomogeneity of electric field. The problem of

fulfilling the Wien condition at the fringing region has

been discussed by Kohashi et al.[5]. If we use a

multipole filter such as shown in Fig. 2, which is made

of metallic magnetic material, there is no problem in

obtaining the Wien condition at the fringing regions.

Effect of the direct magnetic field from the coils can be

neglected.

3. Stigmatic focus conditions

It is well known that the homogeneous electric field E1 and

magnetic field B1 have only lens action on the electric field

direction (x). There is no focusing effect in the magnetic

field direction (y) such as shown in Fig. 3.

In order to make the y-axis focus, it is necessary to excite a

quadrupole electric (E2) or magnetic (B2) field. When E2 is

applied, the beam focuses in both directions as shown in Fig.

4. The quadrupole field acts as a focusing lens on the

y-direction but as a diverging lens on the x-direction, so that

the length necessary to focus the beam for the same

excitations is (√2) times

longer than that of the

homogeneous field filter.

In the following, we use dimensionless multipole

field components for electric and magnetic

quadrupole, hexapole and octupole field

components E2, B2, E3, B3, E4 and B4, respectively,

as follows,

e2 = E2R/E1, e3 = E3R2 / E1, e4 =E4R

3 / E1,

b2 = B2R/B1, b3 = B3R2 / B1, b4 = B4R

3 / B1, (3)

where, R is the cyclotron radius of electrons,

R = 2Uo / E1. (4)

Then the stigmatic focus condition is written as

Fig.3. Trajectory in the

homogeneous field

-30 -20 -10 0 10 20 30

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

(b)

y (

mm

)

z (mm)

-30 -20 -10 0 10 20 30

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

(a)

object plane image plane

x (

mm

)

z (mm)

Fig.4. Trajectory with

electric quadrupole

Fig. 2. Cross section of

12-pole Wien filter

X

Y

4

follows.

e2 – b2 = -1/4. (5)

At the image plane of the filter, there is a

large second order geometrical aberration as

seen at the end of the X-directional beam

(180°) in Fig. 4. The origin of this

aberration is the change of the axial velocity

of the beam. Electrons directed to the

positive electrode are accelerated and

those directed to the negative electrode

are decelerated as shown in Fig. 5 [6].

This change of the velocity creates

aberrations but the velocity change

vanishes at the exit of the filter. Because

the initial and final energy does not

change, the generated aberrations are not

the chromatic aberrations but the

geometrical aberrations.

4. Cancellation of the dispersion and the geometrical second order aberrations

The dispersion and the second order

geometrical aberrations are cancelled out

at the second focus of the 2π-Wien filter as

shown in Fig. 6. When a Wien filter is

used as a monochromator, the slit is

inserted at the first focus. It is important to

reduce aberrations at the first focus to get

higher energy resolution.

5. Conditions for round beam or zero second order chromatic aberrations

The condition to get the round beam of the second order chromatic aberrations is,

Fig. 5. The velocity change in Wien

filter.

Fig. 6. Ray trajectories in x- (upper) and

y-direction (lower) of the 2π-Wien filter.

Fig. 7. Round second order chromatic

aberration condition.

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e3 – b3 = m/16. (6)

Here, we introduced a parameter m, which connects all the multipole components as

shown in Table I. Up to here, only e2,b2 and (e3-b3) are determined but in Table I, all the

other components are also listed,

which will be described in the

following.

In Fig. 7, the second order

aberration figures are shown by

changing e3 – b3 and m. The first

thing we should point out is that

all the second order geometrical

aberrations are zero. All the

aberrations shown in Fig. 7 are

chromatic aberrations. When (e3 –

b3) = m/16 is satisfied, all the aberration figures become round independent of m. On

the other hand, when m=0 or 2, all the chromatic aberrations becomes round

independent of (e3 – b3). A significant result is that the chromatic aberration becomes

zero when m=2 under e3 – b3 = m/16.

6. Conditions for round third order aberrations and zero aperture aberrations

The round beam conditions of the third order

aberrations are as follows:

b3 = 5m2/144, (7)

e4-b4 = - 29m2/1152 (8)

Fig. 8 shows aberration figures with the change

of both b3 and e4-b4. Only when both of those

conditions are satisfied, the third order

aberration figure becomes round both of the

aperture aberrations (blue lines) and chromatic

aberrations (red lines).

Under above conditions, the third order aberration can be written as [4],

uf = u0 + A2αδα0δ + A2uδδ(u0δ2 – δ

3) + A2uuα{2α0 (α0

2 +β0

2)+α0(u0

2 + v0

2)

– 2α0u0δ) } + A2αδδα0δ2 (9)

Fig. 8. Condition of the third order

round beam.

m e2 b2 e3 b3 e4 - b4

-2 0 2/8 1/72 5/36 -29/288

-1 -1/8 1/8 -2/72 5/144 -29/1152

0 -2/8 0 0 0 0

1 -3/8 -1/8 7/72 5/144 -29/1152

1.101 -0.3876 -0.1376 0.1109 0.0421 -0.03051

2 -4/8 -2/8 19/72 5/36 -29/288

4 -6/8 -4/8 7/18 5/9 -29/72

6 -8/8 -6/8 37/72 5/4 -29/32

Table I. Relation between multipole components

and the parameter m.

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vf = v0 + A2αδβ0δ + A2uδδv0δ2 +A2uuα{2β0(α0

2+β0

2) +β0(u0

2+v0

2) - 2β0 u0δ}

+ A2αδδβ0δ2 (10)

Here, ufR and vfR are aberrations for x- and y-directions. Coefficients are written as,

A2αδ= π(1-m/2)/k , (11)

A2uδδ=-π2(1-m/2)

2/2, (12)

A2uuα= -π(2m2 – 3m/8 + 3)/(6k), (13)

A2αδδ= -π(m2 / 2 – 5m / 2 +1)/(4k), (14)

where, k=√(1/2).

Fig. 9 shows off axial aberrations

for the initial beam radius u0, v0 equal

to 0.01 for the left and 0.1 for the

right figures, respectively. The third

order aberration depending on α0(u02

+ v02) becomes larger in the off axial

region than the center. However, there

is no distortion of the beam. The

roundness of the beam is kept even in

the off axial region.

The zero aperture aberration

condition is written as,

m2 – 12m + 12 = 0. (15)

There are two solutions.

m = 10.8990 and 1.1010 (16)

Fig. 10 shows aberration figures including

the defocus around the aperture aberration

zero condition of b2 = 1.1010. It is easy to

see the zero aberration condition at the

center, where the spot size is zero.

When the defocus is considered, added

terms up to the third order are derived

from the second order aberrations. Then

the additional terms are written as,

(a) (b)

Fig.9. Comparison of the off axis aberrations

for u0=v0=0.01(left) and 0.1(right). Vertical

and horizontal axes are written in the

unit of u0,v0. The aberration figures are 100

times magnified to the positional dimension.

Fig.10. Aberration figures of under, just

and over focus states around the zero

aperture aberration conditions.

m=0.101

m=0.601

m=1.101

m=1.601

m=2.101

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defU = εα + (u0 +Δ) ε2 / 4 + π(m – 2)u0 Δε/ (4k) +π(m – 2)Δ

2ε/ (8k), (17)

defV =εβ+ v0 ε2 / 4 + π(m – 2) v0 Δε/ (4 k). (18)

Under the condition of aperture aberration zero, A2uuα becomes zero and the aberration

equations are written as,

uf = u0 + A2αδα0δ + A2uδδ(u0δ2 – δ

3) + A2αδδα0δ

2, (19)

vf = v0 + A2αδβ0δ + A2uδδv0δ2 + A2αδδβ0δ

2, (20)

where,

A2αδ =π(√6-2)/k, (21)

A2uδδ =π2(2√6-5)/k, (22)

A2αδδ =- π (7√6/4 - 4)/k. (23)

These contain only the chromatic aberrations. Here, A2αδ is the axial chromatic

aberration and the other

terms are the third order

chromatic aberrations.

Figs. 11 (a) and (b) show

aberration figures under

the condition of the zero

aperture aberration,

which means that

aberrations shown in

these figures are all

chromatic aberrations. There is no difference in

off axial aberrations with the axial one.

7. 2π-Wien corrector

The aberrations created by the multipole

corrector is written X = uf R and Y=vf R [see eqs.

(9) and (10)]. Here, the term RA2uuα (see

eq.(9)) include not only the spherical aberration

but also terms proportional to α0u02 and α0u0δ.

On the other hand, RA2αδ is the coefficient only

of the axial chromatic aberration. The x and y

include the third order chromatic aberration

Cc

Cs

Cc

Cs

Fig. 12. Aperture aberration (Cs)

and chromatic aberration (Cc) as

functions of the parameter m.

Fig. 11. Off axial aberrations for u0=0.01 and 0.1 under

zero aperture aberrations.

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RA2αδδα0δ2.

When those relations are kept in the 2π-Wien corrector, the beam shape through the

corrector is always round including the off axial regions. The residual aberrations

include aperture aberrations and the chromatic aberrations. A significant point appears at

m=0, in which it is only necessary the electric quaerupole. In the actual system, this

condition gives quite a simple corrector.

Fig.12 shows RA2uuα and RA2αδagainst the parameter m. The former corresponds to Cs

and the latter to Cc as follows,

Cs = RA2uuα, (24)

Cc = RA2αδ. (25)

8. Conclusion

Multipole field conditions are shown to get the round negative second and third order

aberration beam.

e2 – b2 = -1/4

e3 – b3 = m/16.

b3 = 5m2/144,

e4-b4 = - 29m2/1152.

The chromatic aberration becomes zero at

m=2

and aperture aberration becomes zero for

m2 – 12m + 12 = 0.

The Wien type multipole corrector under those conditions can be used as aberration

corrector of low voltage microscopes.

References

[1] Rose H Optik 84 (1990) 91-107.

[2] Mentink SA, Steffen T, Tiemeijer PC & Krijin MPC, Inst. Phys. Conf. Ser. 161 (1999) 83-86.

[3] Ioanoviciu D, Tsuno K & Martinez G, Rev. Sci. Instrum. 75 (2004) 4434-4441.

[4] Tsuno K, Ioanoviciu D & Martinez G., J. Microscopy 217 (2005) 205-215.

[5] Kohashi T, Konoto M & Koike K, Rev. Sci. Instrum. 75 (2004) 2003-2007.

[6]. Martinez G., Tsuno K, Ultramicroscopy 93 (2002) 253-261.