/. Electron Microsc, Vol. 27, No. 4, 259-265, 1978
Characteristics of the Foil Lens for the Correction of the Spherical Aberration
of the Strongly Excited Magnetic Lens
Michio
HIBINO,
Setsuko
SUGIYAMA,
Takaaki
HANAI
and Susumu
MARUSE
Department of Electronics, Faculty of Engineering,
Nagoya University, Nagoya, 464 Japan
(Received July 6, 1978; accepted August 15, 1978)
INTRODUCTION
The spherical aberration of electron lenses
is an important factor which limits the performance of electron beam instruments such
as conventional transmission electron microscopes and various kinds of electron probe
instruments including scanning electron microscopes. This is the reason why considerable
efforts have been devoted to the reduction or
the correction of the spherical aberration of
lenses for the last 40 years. The lens of the
present high-level performance is attributable
to the efforts directed to the reduction of the
aberration. Examples are the determination
of the geometrical parameters of the lens which
give the smallest possible spherical aberration 1 '
and the reduction of the spherical aberration
by operating the lens at high excitaion so as
to be the condenser-objective2' or the second
zone lens,3' to name a few. It does not seem
likely, however, that further remarkable reduction of the spherical aberration can be
expected by the appropriate choice of design
and operating condition of the lens.
It was pointed out by Scherzer,4' on the
other hand, that the correction of the spherical
aberration of electron lenses is possible by
removing at least one of the conditions of the
rotational symmetry of the optical system, the
time invariance of the potential, no space
charge and the continuity of the potential and
its derivative. According to this suggestion,
various ideas were proposed and some of
them were studied experimentally.5' Furthermore the correction of the spherical aberration by means of the holographic reconstruction technique6' and the compensation of the
wave aberration with the zone plate71 were
conceived. None of these, however, have
been put to practical use because of technical
and practical difficulties due to the complexity of the structure and the alignment
required to the corrector system.
For a practical corrector of the spherical
aberration, we have been developing a foil
lens8' which has an important advantage of
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The foil lens, in which the voltage is applied between an aperture diaphragm
and a conducting foil, acts as a concave lens and produces the negative
spherical aberration. This negative spherical aberration can be utilized
for the correction of the positive spherical aberration of a magnetic lens.
The foil lens has been applied to the prefield lens formed by the magnetic
field before the center of the strongly excited objective of a standard electron
microscope. The spherical aberration of the lens system composed of the
foil lens and the prefield lens has been studied experimentally, and it has
been found that the correction is possible, at the voltage of 260 volts applied
in the foil lens, up to 1.0 x 10 ' rad in terms of the angle of the beam focusing
at the center of the objective. It has been shown as well that the experimental result agrees satisfactorily with numerical evaluations calculated
from the third and the fifth order spherical aberrations of the lens system.
260
M. HIBINO, S. SUGIYAMA, T. HANAI and S. MARUSE
EXPERIMENT
The foil lens was placed in front of the
objective, as is shown in Fig. 1. The arrangement of the foil lens is fundamentally the
same as the one reported previously9' except
that some of the dimensions were changed.
An aperture diaphragm of thickness of
t=0.1 mm and with an opening of diameter
of 2R=0.5 mm was separated from a conducting foil by an insulating film of thickness
of d=0.03 mm. As the conducting foil was
used a carbon thin foil supported on a 1,000
mesh square grid for the quantitative measurement of the spherical aberration. The
foil lens was positioned 14 mm before the
center of the objective which has a bore of
Z)=7 mm in diameter and a gap of S = 4 mm.
Magnetic Lens
Fig. 1. Schematic diagram of the experimental arrangement.
When the positive voltage is applied to the
aperture diaphragm and to the cylinder connected to it as is shown in Fig. 1, the convex
equipotential surfaces are produced around
the aperture. Due to this potential distribution, the electron beam incident nearly in
parallel with the optical axis diverges and
therefore the foil lens acts as a concave or a
negative lens. This concave lens produces the
negative spherical aberration and can be
utilized to correct the positive spherical aberration of the magnetic lens.
Against the
cylinder at the positive potential, another
cylinder connected to the ground potential
was placed for the formation of a simple
two-cylinder lens field, otherwise a complicated field, whose effect can not be estimated
easily, would be produced between the cylinder
and the column of the electron microscope.
The effect of this two-cylinder lens is very
weak and is completely negligible within a
range of the applied voltages used in the
experiment.
The objective was excited at 5,460 ampereturns at the accelerating voltage of 50 kV so
as to be the condenser-objective and to focus
the beam incident in parallel with the axis at
the center of the objective. In this operation,
the prefield lens formed by the magnetic field
before the lens center is the same as the
postfield lens formed by the field after the
lens center and has a focal length of 2.1 mm.
The variation of the spherical aberration was
investigated for a combined lens system composed of the foil lens and the prefield lens.
This lens system is similar to the probeforming system of the electron probe instruments.
The operating condition of the lens system
is schematically illustrated in Fig. 2. As is
shown by the broken lines, the condenser is
first adjusted so as to make the beam parallel
to the axis when the voltage is not applied in
the foil lens. The objective is then adjusted so
that the prefield lens forms the gaussian focus
at the center of the objective and the postfield
lens forms its image at the object plane of the
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the simple structure and therefore the simple
alignment. It was already shown that the foil
lens corrects the spherical aberration of a
relatively weak magnetic lens with large dimensions and that the behavior of the foil
lens agrees with the numerical evaluation
calculated from the third and the fifth order
spherical aberrations. 9 ' This paper is concerned with the application of the foil lens to
the objective of a standard electron microscope. Experimental studies and numerical
evaluations will be described on the characteristics of the foil lens for the correction of
the spherical aberration of the prefield lens
of the strongly excited objective.
Foil Lens for the Correction of the Spherical Aberration
Images of the aberration figures at various foil lens voltages,
showing the deformed images of a 25.4 /im spacing square grid
used for supporting the foil in the foil lens. The spherical aberration due to the magnifying lens system is included in these
figures in addition to the aberration formed at the Gaussian
focal plane of the prefield lens.
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V
Fig. 2. Ray trajectories showing the electron optical
conditions. The broken lines show the rays
when the foil lens is not operated. The
solid lines show the rays when the foil lens
is operated and the spherical aberration of
the prefield lens is corrected.
intermediate lens. In the Gaussian focal plane
at the center of the objective is formed the
aberration figure due to the spherical aberration of the prefield lens, which appears as a
deformed image of the grid used for supporting the foil, instead of a point focus.
Then the foil lens voltage is applied between
the aperture diaphragm and the foil. This
shifts the Gaussian focal plane in consequence
of the concave lens action of the foil lens. In
order to maintain the same Gaussian fecal
plane position, the condenser is varied so as to
make the paraxial beam parallel to the axis
after passing the foil lens, while the objective
is kept at the same excitation. The solid lines
in Fig. 2 illustrate the ray trajectories at this
operating condition. The aberration figure
262
M. HIBINO, S. SUGIYAMA, T. HANAI and S. MARUSE
12
8
4
0
4
8
12
xlO" 2 (rad)
12
8
4
(a)
0
4
8 12
x10' ! (rad)
(b)
5r((jm)
T 0.5
x*>oo
» V
0
12/ 8
4
8 \12
xio-!(rad)\
12
8
4
^>-C
V
+-as
/ = 200vo1t
-L-1.0
(C)
12
I
8
iP-
4
0
x10"'(rad)
4
8 12
12
-1—2.0
6r(|jm)
(e)
(f)
Fig. 4. Amount of the spherical aberration measured
and calculated from the aberration figures
of Fig. 3 (O and x), as a function of the
angle at which the beam focuses. The solid
and broken curves show the calculations
obtained from the third and the fifth order
spherical aberrations for rf=0.03 mm and
0.05 mm respectively.
NUMERICAL EVALUATION OF THE
SPHERICAL ABERRATION AND THE
COMPARISON WITH THE EXPERIMENT
The coefficients of the third and the fifth
order spherical aberrations of the superposed
electric and magnetic fields were already
derived91 for evaluations of the spherical
aberration of a lens system composed of the
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formed at the Gaussian focal plane of the
prefield lens is, in practice, observed after the
subsequent magnification by the postfield lens
of the objective, the intermediate lens and the
projector. Typical aberration figures observed
for various foil lens voltages are shown in
Fig. 3. It is seen that the aberration figure
decreases as the foil lens voltage is increased.
The spherical aberration of the lens system
composed of the foil lens and the prefield lens
of the objective can be calculated from the
aberration figures in Fig. 3. For the calculation it must be noticed that the aberration
figures in Fig. 3 are subject to the influence
of the additional spherical aberration of the
lens system used for magnifying the aberration figure produced by the prefield lens.
Among the magnifying lenses the postfield
lens has the dominant effect and the influence
of other lenses can be neglected. The postfield
lens produces the same amount of the spherical
aberration as the prefield lens does, because
of the identity of the postfield lens and the
prefield lens. The contribution of the magnifying lens system to the final aberration figure
image is therefore a half of the final image
observed at the foil lens voltage of K=0 volt.
Accordingly the spherical aberration of the
combined system of the foil lens and the
prefield lens is obtained by subtracting a half
of the aberration in the final image at K=0
volt from the aberration in the final image at
any foil lens voltage. The amount of the
spherical aberration was calculated for the
mesh points in two directions perpendicular
each other of the aberration figure of Fig. 3.
The results are plotted in Fig. 4 with o and *,
as a function of the angle at which the beam
focuses. The spherical aberration reduces to
about half at V= 100 volts and reduces further
at 200 volts. At 260 volts the spherical aberration is corrected up to the beam angle of
l.Ox 10"1 rad and is negative at higher angles.
At 320 volts the aberration is negative or
overcorrected over all beam angles and increases further in the negative sign at the higher
voltage.
Foil Lens for the Correction of the Spherical Aberration
foil lens and the magnetic lens. The formulae
were modified into a rather simplified form
by the integration by parts. The modified
formulae of the coefficients C,3 and C, 5 ' for
the third and the fifth order spherical aberrations, defined with respect to the beam angle
at the image plane, are given respectively by
G33(h) ^ ^ - ^ p j h S h - ^
(3)
where
(4)
9e
dition of /ii=0, / j j ' = l , hx=\ and ^ ' = 0 at the
object plane, and C733(//i) and G33(hi) in Eqs. 1
and 5 are given by substituting fa and \
respectively in place of h in G33{h) of Eq. 3.
Knowledge of axial distributions of the potential of the foil lens and the magnetic field
of the objective are indispensable for numerical
evaluations of the spherical aberration. The potential distribution given by Eq. 18 of Ref. 9),
which was obtained by summing contributions
of the surface charges on the aperture diaphragm and their image charges with respect
to the foil, was used since this potential was
already proved to give a good agreement
with the experiment.9' The axial magnetic
field of the objective was measured with a
Hall probe gaussmeter and the measured
values are plotted in Fig. 5. As is seen from
the curves illustrated in the figure, the measured values are well approximated by the
generalized bell-shaped distribution of y=oo
or the Gaussian distribution, of the half width
of 2a=6.9 mm.
With these field distributions, the coefficients C,3' and C, 5 ' were calculated for the
spacing between the aperture diaphragm and
the foil of 0.03 mm (which corresponds to the
thickness of the insulating film) and for the
larger spacing of 0.05 mm, and are plotted in
Fig. 6 as a function of the foil lens voltage.
According to the definition of the coefficients,
the amount of the spherical aberration for the
*"'+?£,
and
measured distribution
(5)
Here 0 and B are the axial electric potential
and the flux density of the axial magnetic field
respectively, (J)o and @f are the potentials at
the object plane zo and at the image plane zt,
and primes denote the derivatives with respect
to the z-coordinate of the optical axis. The
rays hi and A, are the particular solutions of
the paraxial ray satisfying the boundary con-
Fig. 5. Measured distribution of the axial magnetic
field of the objective and the generalized
bell-shaped distributions.
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(2)
with
263
264
M. HIBINO, S. SUGIYAMA, T. HANAI and S. MARUSE
Cs'5 (cm)
o.io
-
6
0.05-
-0.10 -
-0.15
L
Fig. 6. Variation of the coefficients of the third and
the fifth order spherical aberrations, CSz and
Css', for </=0.03 mm and 0.05 mm, as a
function of the foil lens voltage.
ray which focuses at the beam angle of 6 is
calculated by
Sr=CB3'd3 + C'.5d5.
(6)
The calculation results are indicated in Fig. 4
with the solid curves for the spacing of 0.03
mm and with the broken curves for 0.05 mm.
A satisfactory agreement is seen between the
experiment and the numerical evaluation.
Better fitting of the curves evaluated for the
spacing of 0.05 mm may be attributable to the
possible situation that the actual spacing was
larger than the thickness of the insulating film.
The waving of the curves at K=200 volts is
caused by the positive C, 3 ' and the negative
C, 6 '.
The negative fifth order aberration,
which is negligibly small at small beam angles,
becomes effective at large angles and cancels
the positive third order aberration.
This
tendency is also seen in the experimental
result. At 260 volts, the third order aberration
is corrected but the negative C, 5 ' produces
the negative aberration at large angles. At
higher voltages both C,3' and C, 5 ' are negative and produce the negative aberration over
CONCLUSION
The foil lens, which has a feature of the
simple structure and the simple alignment,
has been applied to the objective of a standard
electron microscope and its characteristics of
correcting the spherical aberration of the
prefield lens of the strongly excited objective
have been studied. The experiment has revealed that the correction is possible up to the
large beam angle of 1.0xl0~ 1 rad and that
the characteristics of the foil lens are in a
reasonable agreement with the numerical
evaluation calculated from the third and the
fifth order spherical aberrations.
The lens system studied in this paper can
be applied to the practical electron probe
instruments as it stands. The improvement of
the performance of these instruments can be
expected by the formation of the electron
probe which is not affected by the spherical
aberration, although the effect of scattering
of the electron beam within the foil must be
investigated for detailed discussions of the gain
expected with the foil lens. It is also important to note that the characteristics of the
foil lens for the probe-forming system studied
here are applicable, in a quite similar way, to
the imaging system of conventional transmission electron microscopy.
Acknowledgement. One of the authors (M.H.) is
most grateful to Kazato Foundation for the grant to
cover a part of the cost of presenting the work at the
International Congress on Electron Microscopy,
Toronto. The work was partly supported by the
Grant-in-Aid for Scientific Research from the Ministry
of Education, Science and Culture of Japan.
REFERENCES
1) Liebmann, G.: Proc. Phys. Soc. (London), 64B,
972 (1951)
2) Riecke, W. D.: Proc. 5th Int. Congr. Electron
Microsc, Philadelphia, 1962, p. KK-5
3) Suzuki, S., Akashi, K. and Tochigi, H.: Proc.
26th EMSA Meeting, 1968, p. 320
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-0.05 -
all angles.
Foil Lens for the Correction of the Spherical Aberration
4) Scherzer, O.: Oplik, 2, 114 (1947)
5) For example, see the review paper by Septier, A.:
Advances in Optical and Electron Microscopy,
1, Academic Press, London and New York, 1966,
p. 204
6) Gabor, D.: Proc. Roy. Soc. (London), A197,
265
454 (1949)
7) Hoppe, W.: Optik, 20, 599 (1963)
8) Ichihashi, M. and Maruse, S.: /. Electron
Microsc, 20, 167 (1971)
9) Hibino, M. and Maruse, S.: J. Electron Microsc,
25, 229 (1976)
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