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Chapter 10 Aberration Correction 725

3.3.4 Mirror Correctors

Several schemes for compensating the aberrations of round lenses by

introducing an electron mirror (see Recknagel, 1937, for early work on

electron mirrors) into the optical system have been proposed (Zworykin

et al., 1945; Ramberg, 1949; Kasper, 1968/9). Aberration correction has

now reached the stage at which simultaneous correction of all the aber-

rations that are liable to impair the performance of the instrument in

question must be envisaged: correction of individual aberrations without

considering the effect of the remainder is no longer sufﬁ cient. For this

reason, we describe here only the scheme devised by Preikszas and Rose

(1997) and surveyed by Hartel et al. (2002), intended for the spectromi-

croscope for all relevant techniques (SMART) project at BESSY II and

also adopted for PEEM3 at the Lawrence Berkeley National Laboratory

(Rose et al., 2004; Feng et al., 2005; Schmid et al., 2005). This differs from

earlier schemes, notably that of Rempfer and Mauck (1992) and Rempfer

et al. (1997), in that the beam splitter, the role of which is to separate the

beam incident on the mirror from the beam emerging from it, is nondis-

persive (Figure 10–15); however, for the application for which it was

designed, a low-energy emission microscope, the corrector shown in

Figure 10–15 is effective. A four-electrode mirror, such as that shown in

Figure 10–16, offers enough degrees of freedom to adjust the focal length

and spherical and chromatic aberration coefﬁ cients satisfactorily. As an

example, Preikszas and Rose (1997) show that the spherical and chro-

matic aberration coefﬁ cients can be chosen anywhere inside the shaded

region in Figure 10–17 for a ﬁ xed position of the Gaussian image plane.

Such a mirror could be combined with a dispersion-free magnetic beam

splitter as shown in Figure 10–18a. Figure 10–18b shows the device incor-

porated in the SMART.

Interface

Lens

Electron

Mirror

(Separating)

Magnet

Aberration Corrector

Magnet

Objective

Lens

Fine

Mesh

Final

Image

Shadow

Pattern

Electron

Source

Condenser

Lens

Magnet

Beam Separator

Image

of Source

Overcorrected

Image

Figure 10–15. The optical system described by Rempfer et al. (1997) for correcting chromatic and

spherical aberration with the aid of an electron mirror. T

and T

are lens triplets, which act as transfer

lenses. The shadow thrown by the ﬁ ne mesh gives a measure of the aberration coefﬁ cients. (After

Rempfer et al., 1997, courtesy of the authors and the Microscopy Society of America.)

726 P.W. Hawkes

3.4 All-Purpose Correctors

In his keynote lecture on “High-Performance Electron Microscopes of

the Future,” delivered at the twelfth European Congress on Electron

Microscopy in Brno, 2000, Rose described a “hexapole planator” capable

of correcting the ﬁ eld curvature and (third-order) astigmatism. Such

correction is essential for projection lithography systems. This pro-

posal was explored in detail by Munro et al. (2001) who, in an impor-

tant paper, ﬁ rst examined the present limitations of projection systems

for charged particles and then analyzed an electrostatic and a magnetic

conﬁ guration free of all third-order geometric aberrations, inspired by

Rose’s hexapole planator.

Subsequently, two very complex correctors have been proposed by

Rose, which are capable of correcting the spherical and chromatic aber-

rations as well as other primary aberrations that could be harmful once

the axial aberrations have been brought under control (Rose, 2003, 2004,

2005). These are the superaplanator and the ultracorrector. The ﬁ rst of

these is suitable for a transmission electron microscope: it consists of

two symmetrical quadrupole quintuplets and three (or more) octopoles

and corrects spherical and chromatic aberration. Very high stability of

the excitations is essential (Rose, 2005), of the order of a few parts in 10

but this is attainable today. Symmetry is exploited to make the device

as easy to use as possible. Thus the quadupole ﬁ elds are symmetric with

respect to the center plane of each quintuplet; conversely, the whole

optic axis

3.6 mm

Figure 10–16. Section of a tetrode mirror. The potentials applied to the elec-

trons determine the focal length and the chromatic and spherical aberration

coefﬁ cients. (After Preikszas and Rose, 1997, courtesy of the authors and

Oxford University Press.)

-70 -10-20-30-50-60

-5

-10

-15

-20

-25

-40

[km]

[m]

Figure 10–17. Behavior of the

aberration coefﬁ cients of the

tetrode mirror of Figure 10–16.

For a ﬁ xed image plane, the

spherical and chromatic aberra-

tion coefﬁ cients fall within the

shaded zone. (After Preikszas

and Rose, 1997, courtesy of the

authors and Oxford University

Press.)

Chapter 10 Aberration Correction 727

Figure 10–18. Incorporation of the tetrode mirror of Figure 10–16 into a complete system. The beam

separator is free of dispersion. (a) Basic conﬁ guration. (b) The entire layout of the SMART (spectro-

microscope for all relevant techniques). [(a) After Preikszas and Rose, 1997, courtesy of the authors

and Oxford University Press. (b) After Hartel et al., 2002, courtesy of the authors and Elsevier.]

beam

separator

optic axis

specimen and

objective lens

projective

system

tetrode

mirror

optic

axis

electron source

deflector D1

transfer lens T1

field aperture

electric-magnetic

multipole deflector

elements

electron

mirror

beam

separator

Mirror corrector

T3 D5

600 mm

500

400300200

1000

energy selection slit

projector

field

lens L1

objective

lens

Projector/Detector

axial ray

dispersive ray

Energy filterObjective Transfer optics

electric/

magnetic

deflector

specimen

x-ray

mirror

x-ray

illumination

energy selection

slit for x-ray

illumination

quadrupoledipole P1

hexapole H1

apertures

dodecapole

camera system

field ray

coils

electrode surfaces

polepieces

condenser

728 P.W. Hawkes

(double-quintuplet) unit exhibits antisymmetry about its mid-plane

(the plane midway between the two quintuplets). An octopole is placed

in this mid-plane and at the center of each quintuplet (Figure 10–19).

The superaplanator is adequate for the transmission electron

microscope, where no particular attention need be paid to third-

order astigmatism and ﬁ eld curvature, the zone to be imaged being

so small. In electron projection lithography, however, it is desirable to

correct all the primary chromatic and geometric aberrations. This can

in principle be achieved with the conﬁ guration known as the ultra-

corrector. Here, two identical multipole multiplets are used. Each of

these consists of seven quadrupoles and seven octopoles, themselves

disposed symmetrically about the center plane of the multiplet. Once

again, the multiplet ﬁ elds are antisymmetric with respect to the plane

midway between two multiplets in which an additional octopole is

situated (Figure 10–20). Detailed description of the modes of action of

these complex structures is to be found in the work of Rose already

cited. The suitability of these correctors for the Trans mission Electron

Aberration-corrected Microscope (TEAM) project is currently (2005)

being examined (O’Keefe, 2003, 2004; Kabius et al., 2004).

4 Concluding Remarks

If any lesson is to be learned from past attempts to correct spherical

and chromatic aberration, it is that complex systems require a rapid

and efﬁ cient means of adjustment and that beyond a certain degree of

complexity, only computer control offers the necessary speed and ﬂ ex-

ibility. Nevertheless, it remains important to simplify the task as far as

possible and this is achieved by exploiting the symmetry properties.

The intrinsic simpliﬁ cation of the optics provided by the Russian qua-

druplet (or multiplet) has long been known and, more recently, very

a.u.

–1

–2

–3

02 4681012

Figure 10–19. The superaplanator. (After Rose, 2005, courtesy of the author

and Elsevier.)

Chapter 10 Aberration Correction 729

ingenious blends of symmetry and antisymmetry have been beneﬁ cial,

or indeed indispensable, in the design of Ω and related types of ﬁ lters,

such as the mandoline ﬁ lter (see Rose and Krahl, 1995, for a thorough

account and Tsuno, 2001, for a schematic appraisal). The symmetry and

antisymmetry on which the multicomponent structures of the supera-

planator and the untracorrector are based not only help us to under-

stand how they work but also render the adjustment more systematic

and less liable to instability.

The quest for aberration correctors has always been fuelled by the

desire for ever higher resolution but, now that correction has been

achieved, microscopists are beginning to realise that such devices will

be useful in many other areas of microscopy. In-situ microscopy

requires lenses with ample space between the polepieces and the coef-

ﬁ cients of spherical and chromatic aberration of such lenses are large.

By correcting these aberrations, in-situ microscopy should be capable

of providing much ﬁ ner information. In the life sciences, chromatic

correction will make it possible to work with thicker specimens. And

no doubt this is only the beginning. The new century is witnessing a

new era in electron microscopy.

5 Appendix I

5.1 Power Series Expansions of Electrostatic Potential and

Vector Potential

The power series expansions used in the foregoing text are identical with

those adopted in Hawkes and Kasper (1989) and are reproduced here

for the reader’s convenience. However, since the work of Rose is very

frequently referred to, the relation between the notation used in his

chapter in Ernst and Rühle (2003) and that employed here is also given.

Figure 10–20. The ultracorrector. (After Rose, 2005, courtesy of the author and

Elsevier.)

730 P.W. Hawkes

5.1.1 Expansions for the Electrostatic Potential

In Hawkes and Kasper (1989), the electrostatic potential is written as

Φ xyz z x y z x y xFz yF z,, () () () ()

(

)

=− +

()

′′

()

−−

φφ

22 22

xxyxFyF xypz xyqz

xyx

222

()

′′

(

)

+−

()

−+

()

−

() ()

yyp x yxyq

xxypz yxyq

()

′′

−+

()

′′

−−

()

+−

()

() (zz

xxyypz xyxyqz

)

() ()+−+

()

+−

()

4224

Rose (2003c) uses the expansion

ϕ(,) ()

!( )!

()

[]

lm l

=−

(

)

∞

∑∑

in which

w = x + iy

and [2l] signiﬁ es 2l-fold differentiation with respect to z. An overbar

indicates the complex conjugate.

This leads to the following correspondences:

φ() ()

() , () , ( )

()

() ()

(

Fz Fz F iF

=− =− =− +

ΦΦΦ

112

rri

qz p iq

pz qz

)()

() ()

,() , ( )

() , () ,

222

==+

=− =−

ΦΦ

ΦΦΦ

333

444

24 24

=− +

===+

()

() , () , ( )

() ()

piq

pz qz p iq

and we have written

ΦΦ Φ

i=+

() ()

5.1.2 Expansions for the Vector Potential

In Hawkes and Kasper, the components of the vector potential are

written as

BxyB xyB

xyxyB

=− − +

′′

(

)

+−

′

−−+

′′

−

22 22

2222

() ()

()()

xyB xy x y B

xxyQ yxyP

′

′′′

−−

′

−−

′

()

()()

++−+

′

−−

′

4224

()()xxyyQ xyxyP

Chapter 10 Aberration Correction 731

BxyB xyB

xyxyB

=− +

′′

(

)

+−

′

−−+

′′′

22 22

2222

() ()

()()

xyB xy x y B

xxyP yxyQ

′

−+

′′′

−−

′

+−

′

()

()()

++−+

′

+−

′

4224

()()xxyyP xyxyQ

AxBzyBz xyxB yB

xyQzx

=− − + +

′′

−

′′

+− −

21 1

() () ( )( )

()()yyP z x y x y Q

xyxyP x xy

2222

() ( )( )

() ( )

−+−

′′

−−

QQz

yxyPz xxyyQz

xyxyP

4224

()

()()()()

()

−− + − +

−−

()z

In Rose, the same gauge is adopted and the components of the vector

potential are given by

AA iA

lm l

=+ =

−

(

)

∞

∑∑

() !

!( )!

()

[]

214

121

lm l

=−

(

)

∞

∑∑

Im ( )

!( )!

()

[]

The correspondence is now

112

222

333

444

=− +

()

BiB

PiQ

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