ISSN 1063-7397, Russian Microelectronics, 2006, Vol. 35, No. 6, pp. 372–381. © Pleiades Publishing, Inc., 2006.
Original Russian Text © V.A. Zhukov, A.V. Zav’yalova, 2006, published in Mikroelektronika, 2006, Vol. 35, No. 6, pp. 434–444.
The axial aberrations—whether spherical or chro-
matic—of electromagnetic lenses have been a major
issue of axially symmetric charged-particle optics ever
since the very beginning [1, 2]. Finding ways to reduce
them is the central concern in electron and ion micros-
copy . Scherzer’s theorem of 1936 states that the
sign of these aberrations cannot be changed . Later,
he discovered three exceptions :
(i) There is a nonzero current or charge density on
(ii) A time-dependent focusing ﬁeld is used.
(iii) The system is modiﬁed to include optical ele-
ments of lower symmetry.
Ramberg  added a fourth one: electron-optical
systems with one or more electrostatic mirrors.
These exceptions might seem to offer four respec-
tive avenues for suppressing the axial aberrations.
However, the ﬁrst one is hopelessly impractical, and the
use of time-dependent focusing ﬁelds is still little more
than a theoretical possibility because, with existing
technologies, it is limited to particles of very low
energy, less than a few electronvolts for electrons or a
few tens of electronvolts for ions .
The third approach has been studied in the context
of electron microscopy. Seeliger  was the ﬁrst to
design and fabricate a correction system with quadru-
pole and octupole lenses. Archard  showed that the
potential on any of the lenses must be proportional to
the aberration coefﬁcient being corrected (whether
spherical or chromatic). It follows that for a magnetic
objective lens these potentials must be 2–16 kV, the
aberration coefﬁcients being of order 1 mm . Such
correction systems are obviously unsuitable for the
electrostatic objective lenses of focused-ion-beam
microscopes and ion-projection-lithography machines,
whose chromatic aberration coefﬁcients are in excess
of 1 m. Later variants were not especially stable during
operation as they contained about 40 lower symmetry
elements requiring individual alignment, mechanical or
electrical [9–13]. It was not long ago that the problem
of stable alignment was solved . Note also that cor-
rection with lower symmetry elements increases chro-
matic aberration when reducing spherical one.
Electrostatic-mirror correction, like the ﬁrst two
approaches, is examined only theoretically [5, 15–18].
With negative axial aberrations, the ranges of adjust-
ment of the electrode potentials turn out to be very nar-
row due to high sensitivity of the aberration coefﬁcients
to variations in the potentials; this relationship is repre-
sented by a polynomial of degree 5 or greater [16–18].
It is therefore extremely difﬁcult to match the negative
aberration coefﬁcients of correction mirrors to the pos-
itive aberration coefﬁcients of an objective lens to be
corrected. We know of no such experiment.
At the same time, we ﬁnd it worthwhile to extend
the mirror-correction concept to composite mirrors
Axially Symmetric Composite Electromagnetic Mirror
for Perfect Axial-Aberration Correction
V. A. Zhukov
and A. V. Zav’yalova
St. Petersburg Institute for Informatics and Automation, Russian Academy of Sciences, St. Petersburg, Russia
Vavilov State Optical Institute, St. Petersburg, Russia
Received October 10, 2005
—A new element of charged-particle optics is proposed and examined: an axially symmetric compos-
ite electromagnetic mirror employing a focusing magnetic ﬁeld as well as an electrostatic one, the two ﬁelds
overlapping. It is shown that the chromatic and spherical aberration coefﬁcients of the mirror are opposite in
sign to those found in conventional axially symmetric optics; their magnitude is twice the length of the paraxial
particle trajectory as projected on the mirror axis of symmetry. As a result, the negative axial aberration coefﬁ-
cients can be varied over the wide range from a few millimeters to one meter with the mirror focal length and
the particle energy being ﬁxed. This property offers a way to fully compensate for the axial aberration of opti-
mized ion or electron objective lenses in current use. A correction system incorporating the mirror is proposed.
As compared with the sole correction arrangement used in electron optics, the system considered has about 1/5
times as many alignment degrees of freedom (mechanical and electrical) and should therefore display far
greater stability; furthermore, it can work with ions as well as electrons.
PACS numbers: 74.25.Nf