Table Of ContentHANDBOOK OF
VACUUM PHYSICS
VOLUME 2
PHYSICAL ELECTRONICS
EDITED BY
A. H. BECK
Engineering Laboratory, Cambridge
Part 4
E. A. ASH—Electron Optics
Part 5
A. H. BECK—Heavy Current Electron Beams
Part 6
A. H. BECK—Gridded Valves. Theory and Design Principles
Φ
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PART 4
ELECTRON OPTICS
E. A. ASH
Department of Electrical Engineering,
University College, London
PART 4
ELECTRON OPTICS
E. A. ASH
Department of Electrical Engineering,
University College, London
1. INTRODUCTION
The operation of an important class of electronic devices can be
described in terms which are very similar to those used in classical
light optics. The analogy is deep, in that both sciences flow from a
common fundamental principle, and at the same time is conceptually
extremely useful. There are naturally limits to how far the parallel
can be drawn; one of these limits is set by the need to take the mutual
repulsion of the electrons into account, when the current density
is sufficiently large. Although, even in this event, the picture could, if
desired, be stretched—one would have to talk about a current de
pendent refractive index—it is of very dubious value to do so. At the
same time it so happens that most devices in which this effect would
be important are distinguished also from the low current devices by
the nature of the applications. The discussion of high current beams
is presented in a companion article.
Another limit to the parallel with classical optics is encountered
if one wishes to treat time-varying fields by a generalization of
electron optics(1). Although mathematically the two subjects are
contiguous, the devices to which they relate are sharply differentiated
in function; the present article will be confined to time-independent
situations.
Although the science of electron optics has spread in so many
directions that it is not now easy to delineate its frontiers, its origin
is clearly defined, and is to be found in a single paper—that by
Busch(2,3), which first explained the action of "concentrating" coils
in cathode ray oscilloscopes. Even before this time there was a great
deal of existing art and theory of electron dynamics. However, the
399
400 HANDBOOK OF VACUUM PHYSICS
concept of an electron lens is, it is generally agreed, to be traced to this
paper. The concept at once stimulated further experiments which in
only six years led to an electron microscope which surpassed the
performance of the best optical microscope then available. The
fascinating early history of the electron microscope has been re
corded by Gabor(4) and the reader with a taste for the history of the
subject is referred to this article.
Electron optics shares with light optics the characteristics that it is
amenable to analytical treatment, without the need to make drastic
assumptions limiting the validity of the results. Most of the impor
tant properties of electron optical systems can be analysed starting
with only three equations—Newton's and Laplace's equations and
the Lorentz force law—and the applied mathematician need not
burden himself with any further knowledge of physics. This circum
stance has attracted a great deal of analytical effort; in a brief survey
it is therefore very pertinent to ask to what extent has this theoretical
attack contributed to the performance of electron optical instruments.
The answer, to those who have not succeeded in eliminating a utilitarian
streak from their natures, must be depressing. Indeed, in considering
the electron microscope one might speculate as to whether any other
invention has produced such a high ratio of theoretical expressions
propounded to expressions used in grim earnest. This is not to deny
the elegance and beauty of the mature theory of electron optics,
nor the use which might be made of it in the future, possibly in an
entirely different context. However, for the purpose of this survey,
as far as possible we will adopt the rule of giving only such analysis
as is very directly related to the design or understanding of electron
optical devices.
At this point we must raise another issue, and one which will
encourage us to adopt a slightly different point of view from that
taken by previous and longer treatments. How does the existence of
the high speed computer affect the kind of analysis which one needs
in order to obtain results by the quickest route? It is one of the
stranger doctrines of applied mathematics that once a differential
equation has been transformed to a definite integral the problem
has been "solved".
The "solution" is regarded as only a little tarnished if the integral
contains parameters which represent the solution of auxiliary equa
tions. On the other hand, the user has to decide whether it is quicker
to compute the "solution", or to effect a direct integration of the
ELECTRON OPTICS 401
starting equations. To be useful, an analytical solution must satisfy
one of three possible requirements:
(i) It should be of use in proving general theorems.
(ii) It should require fewer arithmetical operations in order to
attain a numerical result.
(iii) It should be so simple as to be meaningful and suggestive by
inspection.
It is our impression that this set of criteria means that in many cases
we are better advised to hand the raw material to the computer
rather than to attempt any prior cooking, f
In a short survey it is clear that we can only include the most basic
results. However, it is hoped that enough can be included to enable
the reader to gain a general understanding of the principles under
lying electron optical devices and their design. The equations have
been written in the non-relativistic form. This approximation is ade
quate for most of the applications discussed, but is not usable for
some special electron microscopes such as those which use voltages
up to 500 kV(5). In most cases the modifications required are simple
to effect. For a detailed treatment the reader is referred to L. de Brog-
lie'sbook(6).
In most of the discussion the electron will be regarded as a particle,
the wave description being referred to only where we are directly
concerned with wave phenomena, notably in discussing diffraction
effects in the electron microscope. The reason is simply that most of
the results required are adequately reflected in the corpuscular
theory.
It will be assumed that the reader is familiar with the electrostatic
and magnetostatic theory and that in some context he has encount
ered the Lorentz force F, which in the M.K.S. units, used throughout,
is
F = e(E+vxB). (1.1)
t The advent of the computer has, of course, no effect on which of two formu
lations leads more rapidly to a numerical result. The fact that, in the pre-com-
puter era, it had in many cases not been noticed that the analytical result was no
nearer the solution than the starting point suggests that neither had ever been
computed.
402 HANDBOOK OF VACUUM PHYSICS
2. PARAXIAL THEORY OF ELECTROSTATIC AND MAGNETIC
LENSES
2.1. General considerations
The principal concern of the theory of electron optics is the de
monstration and evaluation of the imaging properties of rotationally
symmetric electric and magnetic fields. We will see that the action of
such fields on electron streams is very closely analogous to that of
lens systems on beams of light. The analogy with light imaging is
sufficiently close to allow us to use much of the same terminology,
not only with regard to the Gaussian imaging but also with regard to
the aberrations.
The phenomenon of image formation by particles travelling in a
system having circular symmetry is quite general; it is by no means
FIG. 1. Two particular solutions of eq. (2.4).
confined to the case of charged particles in electrostatic or magneto-
static fields. Neutral particles travelling in a rotationally symmetric
gravitational field, or compressible fluids, represent other examples.
With reference to Fig. 1 consider a particle starting from the
symmetry axis at z = z moving through the force field, and inter
0
secting the symmetry axis for a second time at z = z.t Let the radial
f
f This implies a converging lens; the proof is readily extended to diverging
lenses. However, as will appear in section 2.5, divergent electron lenses in which
object and image are outside the force field do not exist.
ELECTRON OPTICS 403
force experienced by the particle be F (r, z). Suppose we now expand
r
F in powers of r:
r
F(r,z)= f G«W- (2-1)
r
/i = 0
The restraint imposed by circular symmetry immediately requires that
all even powers of r must vanish, and the first term in the expansion
is G (z)r. For sufficiently small values of r the higher terms can be
1
neglected, and we may write the radial equation of motion in the form,
mr = G (z)r (2.2)
1 9
where m is the mass of the particle. Let the particle velocity on the
axis be v(z). At a small distance r from the axis the velocity will
differ from v(z) on two counts: firstly, due to a difference in potential
energy arising from the radial force field, secondly because part of
the kinetic energy is associated with the radial and azimuthal motion.
Both these effects contribute terms in r2 to the velocity which can be
neglected for sufficiently small values of r. We may therefore write,
d d
dt=TzMz)>
^ = ^ ^ 2 ( z) + ^y(z)"'(z)' (2'3)
where the prime indicates differentiation with respect to z. Equation
(2.2) can then be brought into the form,
_,v'(z) G.jz)
" ^_J±J_£. . (2.4)
r + r v(z) mv2(z)r' = 0
This is a second order linear differential equation which will have two
linearly independent solutions. We will denote these by r R(z) and
Q x
r R (z), where r is a constant having the dimension of length and
0 2 0
R are non-dimensional functions f of z. The general solution of
12
eq. (2.4) therefore takes the form,
r-^-=c R (z) + c R (z) (2.5)
1 1 2 2 9
where c and c are constants to be determined by the boundary
i 2
t This notation is at variance with that more generally adopted; the motive
is to avoid a notation in which two functions differing only in subscript are
dimensionally different.
404 HANDBOOK OF VACUUM PHYSICS
conditions. We will now choose for our particular solutions two
functions which satisfy the following conditions at z = z :
0
*i(*o) = 0 Ä (*o) = l
2 (26)
Äi(z ) = l/r R' (z ) = 0.
0 0 2 0
We will assume that R becomes zero for a second time at z = z .
x x
With this representation, indicated in Fig. 1, eq. (2.5) becomes:
^ = r'(z ). R (z) + r-^-R (z). (2.7)
0 t 2
Does eq. (2.7) represent the behaviour of an imaging system ? To see
that it does, we can write down the form of the equation at z = z- \
v
r(z) = r(z ). R (z ). (2.8)
i 0 J 2 i
Thus the trajectory intercepts this plane at a radius which is directly
proportional to the radius at z = z , and is independent of the initial
0
slope, r'(z ). We can identify the constant of proportionality as the
0
lateral magnification, which we will denote by M:
M = R ( ). (2.9)
2 Zi
Equations (2.4) and (2.8) are the basis to the study of the behaviour
of any axial symmetric force field in the vicinity of the axis. We can
now proceed to examine the explicit forms which are relevant to the
case when the force field arises from distributions of electric and
magnetic fields.
2.2. The ray equation in electric fields
The electric field can be completely described by a potential function
in cylindrical polar coordinates Φ(Γ,Ο,Ζ). The electric fields which
determine the forces exerted on the electron are found by taking the
appropriate derivative. The ray equation is derived very directly from
the basic equations of motion for the electron, which in cylindrical
coordinates and for circular symmetry may be written in the form
r-ηψ. (2-10)
or
f - , 5. (2-11)
oz
where η is the ratio of charge to mass, and x = ax /at.
ELECTRON OPTICS 405
Equation (2.10) fully describes the radial motion of the electron
with which we are primarily concerned. In the above form of the
equations, the independent variable is the time. Now, in electron
optics it is the variation of the radial distance with position along the
axis with which we are primarily concerned. We must therefore
utilize eq. (2.11) in order to eliminate time. This is most conveniently
accomplished by using the energy integral.
Multiplying (2.10) by rdt, (2.11) by zdt, adding and integrating, we
have
Carrying out the integration we obtain the energy integral
f2 + z2 = 2*/Φ + const. (2.12)
Adopting the usual convention, the potential is taken as zero at the
cathode. The integration constant is then itself zero, provided that we
can neglect the initial thermal velocities. For most purposes this is
entirely justified but it is a point to which we will return in our dis
cussion of chromatic aberrations.
Noting that:
f dr
i dz
eq. (2.12) may be written in the form
. 2,/Φ
2 (2.13)
ί + r'2
The second derivative of r with respect to time may be transformed as
follows:
dz d fdr dz|
~ at 'dz jdz'dij
= z\ zr" + r' —
and inserting eq. (2.13), we find the following expression for r:
r ~ ΰ^ϊ {ϊΤ^+2Φ ^ +2Φ δ7|· (2'14)
