Electron Beams, Lenses, and Optics. Volume 2

325 Pages·1970·3.921 MB·English

by  A. B. El-Kareh and J. C. J. El-Kareh (Auth.)

ELECTRON BEAMS, LENSES,
AND OPTICS
Volume 2
A. B. EL-KAREH
DEPARTMENT OF ELECTRICAL ENGINEERING
SYRACUSE UNIVERSITY
SYRACUSE, NEW YORK
J. C J. EL-KAREH
SYRACUSE, NEW YORK
1970
ACADEMIC PRESS New York and London
COPYRIGHT © 1970, BY ACADEMIC PRESS, INC.
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List of Symbols
all subscripts a refer to aperture plane
all subscripts i refer to image side
all subscripts o refer to object side
all primes are derivatives with respect to z
all dots are derivatives with respect to time
A magnetic vector potential
A a factor
a acceleration
a parameter occurring in the formula for the Glaser bell-shaped field;
in this text a = d is the half width of the field
B magnetic flux density; also a factor
B maximum flux density along the axis of a magnetic lens; in the sym-
0
metrical case, B is the flux density at the center of the lens
0
B magnetic flux density in the core of the magnetic lens
p
C coefficient of chromatic aberration for object or image at infinity
c
C coefficient of spherical aberration for object or image at infinity
s
C coefficient of spherical aberration for finite object and image distances
s
C coefficient of spherical aberration, image side, object at infinity
si
C coefficient of spherical aberration, image side, object not at infinity
si
C coefficient of spherical aberration object side, image at infinity
so
C coefficient of spherical aberration, object side, image not at infinity
so
c velocity of light
D diameter of tube or aperture of an electrostatic lens ; also bore dia-
meter of a magnetic lens
D diameter of central electrode of an einzel lens
l
ix
X List of Symbols
d half width of central electrode in an einzel lens; also half width of
magnetic field
E electric field
e charge of the electron
F position of focal point; also a function
F reduced focal length
/ focal length
^obj objective focal length
F projective focal length
proj
G coil form factor; also center of gravity of an electrostatic lens
H magnetic field intensity
h Planck's constant
/ electric current; also a factor (Chapter VI)
/ current density; also a factor (Chapter VI)
K2 Glaser's factor equal to eB 2a2/SmV (Chapter VII); also a constant
0
(Chapter XIII)
k2 factor equal to eB 2R2ßm V (Chapter VIII)
0
k Boltzmann's constant
/ distance from the geometrical center of an einzel lens to the outer
electrode
log logarithm to base e
M magnification ; also a factor (Chapter VI)
m mass of the electron
TV factor (Chapter VI)
NI ampere turns
n index of refraction
P position of principal plane
p momentum
R reduced height of ray from axis in electrostatic lenses; also radius of
tube or aperture of an electrostatic lens; also bore radius of a magnetic
lens
Ri radius of central electrode of an einzel lens
R radius of outer electrode of an einzel lens
2
R mean radius of a torus (Chapter VIII)
m
r height of electron from axis in polar coordinates
r minimum radius of beam
m
r height of the first paraxial ray in the aberration expressions
a
r ' slope of the first paraxial ray in the aberration expressions
a
r height of the second paraxial ray in the aberration expressions
y
r' slope of the second paraxial ray in the aberration expressions
y
S action integral, ray function ; also gap width of a magnetic lens
List of Symbols XI
s spacing between electrodes in electrostatic lens
T temperature in degrees Kelvin
/ time
u ratio of potentials at the center of the central and outer electrodes of
an einzel lens
u image distance
{
u object distance
0
V fixed voltage of electrode in electrostatic lenses; also accelerating
potential in magnetic lenses
V accelerating potential in magnetic lenses corrected for relativity
r
v velocity of the electron
W energy
x distance from image to image focal point
{
x distance from object to object focal point
0
z coordinate of axis of rotational symmetry
z midfocal length
m
z distance from principal plane to plane of reference
p
a angle of incidence; also half angle of arrival or departure of electron
trajectory; also a factor (Chapter VI)
ß angle; also a factor (Chapter VI); also a factor equal to k2VJ(NI)2 of
a magnetic lens
Δ/ longitudinal spherical aberration
Θ half angle of arrival or departure of electron trajectory; also angle of
rotation of the path in a magnetic lens
Ô Larmor angular velocity
0
K wave number
λ wavelength ; also a factor (Chapter VI)
μ magnetic permeability in vacuum
0
v frequency
p radius of curvature; also radius in cylindrical coordinates
σ ratio of voltage (V /V ) in electrostatic immersion lens
2 1
τ period
φ variable voltage in electrostatic lenses
φ voltage corresponding to initial velocity of emitted electrons
{
φ magnetic scalar potential
IX
Geometrical Aberrations of Lenses
In general, when we speak of aberration in electron optics, we refer to the
case in which rays emanating from one and the same object point converge to
different image points. So far we have confined our analysis to first-order
theory. We have shown that electron paths, which leave points of the object
close to the axis at small inclinations with respect to the axis, intersect the
image plane in points forming a geometrically similar pattern. This ideal
image is known as the Gaussian image, and the plane in which it is formed as
the Gaussian image plane. If an electron path leaving an object point a finite
distance from the axis with a particular direction and electron velocity inter-
sects the Gaussian image plane at a point displaced from the Gaussian image
point, this displacement is defined as the aberration.
9.1 Types of Aberrations
Aberrations in electron optical devices arise from various sources. If the
velocity of the electron is kept constant and if the electron optical system is
perfectly aligned, any deviation from the ideal image is known as geometrical
aberration.
If the electron optical system is not perfectly executed, some asymmetries
will exist and a deviation in the Gaussian image plane occurs even if we re-
strict ourselves to paraxial rays. In that case, we refer to these as aberrations
due to asymmetry.
Because of the thermal velocity spread of electrons and the interaction of
the beam with matter, such as in an electron microscope, the velocity of the
electron is not constant, and this leads to a deviation referred to as chromatic
1
2 IX GEOMETRICAL ABERRATIONS OF LENSES
aberration—a notation analogous to that of light optics. Just as in light optics
in which a change in wavelength or color of the light alters the index of re-
fraction, a change in velocity of the electrons can be regarded as a change in
refractive index of the medium through which the electrons travel, thus re-
sulting in a change in optical behavior. Chromatic aberration also results
when the accelerating potential is not constant.
In the presence of an appreciable concentration of electrons, the repulsion
that electrons exercise on each other causes a deviation of the electron path
from the one that was considered in the completely isolated medium. This
type of aberration is known as a space charge effect.
We showed in a previous chapter how the paths of electrons in electro-
static fields remain unaltered if all the potentials in a system are scaled up or
down in proportion. However, when some of the potentials attain values
exceeding, say 25 kV, this proportionality is no longer valid because of rela-
tivistic effects. We thus refer to this distortion as a relativistic aberration.
In what follows we shall deal with geometrical aberrations. Subsequent
chapters will be devoted to the other types of aberrations.
9.2 Third-Order Theory
If we consider the expansion for sin Θ
sine = θ -- + --■■■ (9.1)
and we limit the angle that the electron trajectory makes with the optical
axis to less than π/20, the error incurred if we make the approximation
sin θ « Θ, as we have done in all the previous chapters, is of the order of 0.1 %.
In many electron optical devices it is necessary to let a large percentage of
the electrons, emanating from the cathode, reach the target. In order to
achieve this, a large aperture must be used at the entrance of the lens. We are
thus forced to consider angles for which sin Θ is no longer closely approxi-
mated by the first term in the series.
In light optics, since most of the aberrations can be compensated for, Θ
can have large values. In electron optics, because of the aberrations involved,
the angle Θ cannot be increased too much. However, even if we take an extreme
value and consider the case in which θ = π/4, the error incurred by taking
only the first two terms of the series is less than 0.5%. Therefore, when
dealing with cases in which there is a departure from the paraxial condition,
we can investigate the imaging properties of the lenses to a good degree of
accuracy by considering terms up to and including the third-order terms. We
9.3 Geometrical Aberrations 3
refer to this study as the third-order theory as compared with the first-order
theory we have treated so far.
9.3 Geometrical Aberrations
We shall start by deriving the general form of the aberration equations
and relating them to the aberration figures. Then, we shall derive explicit
formulas in which each aberration coefficient is evaluated in terms of the
known parameters of the optical system. There are numerous methods
available to treat this problem. We are following one developed by Rogowski
and Ramberg.
Let us consider an electron optical system of rotational symmetry as shown
in Fig. 9.1. The z axis is the axis of symmetry. Normal to this axis, we con-
0 A I
y0 y0 Yi
/*'
//Xo Aa /xi
ιΑ'<
°0 cT7 y z axis
/
Fig. 9.1. The coordinate system of the trajectories. Initially the planes y * , y» * , and
0 0 a
yi Xi are coplanar. The plane y 'Xo' is rotated by a given angle for simplification (object plane,
0
O; aperture plane, A; image plane, /).
sider an object plane O of coordinates x , y , and further along the axis an
0 Q
image plane / perpendicular to the axis, whose position is yet to be fixed, of
coordinates Xj, y. We assume the space between O and / to be free from
{
matter in the vicinity of the axis so that Laplace's equation holds for both the
electric and the magnetic fields. We also assume that the space surrounding
the image area / is field free. We now introduce between O and /, also normal
to the axis, an aperture plane A with coordinates x , y close to / so that the
a a
entire space between A and / may be regarded as field free.
If the aperture plane is placed ahead of the lens such that the region be-
tween it and the image plane is no longer field free ( for example, it could
happen that the fields reach up to the image plane), one can always consider
4 IX GEOMETRICAL ABERRATIONS OF LENSES
a virtual aperture plane such that the virtual electron paths between it and the
image plane are tangents to the electron trajectories in the image plane.
In Chapter I, we saw that if one defines
5 = n ds (9.2)
and 5 is the ray function, then öS = 0. Starting from any point P in any
0
arbitrary direction, the locus of the points P for which Eq. (9.2) is a constant
x
is a surface known as the wave surface. This surface can have any shape. If
the index of refraction is constant between P and P , then the wave surface
0 x
is a sphere about P . Let us consider the case in which P and P are separated
0 0 x
by a lens. The initial wave surfaces are spheres. As the wave penetrates inside
the lens, the spheres become distorted. An ideal lens is one which will change
the incoming wave surfaces which are spheres of center P to new spheres of
Q
center P.
{
In practice, lenses change the incoming diverging, spherical wave surface
into one which approximates a sphere around the Gaussian point P , the
{
deviation increasing with increasing distance of P from the axis and with
0
increasing angular aperture of the imaging pencil.
y2 -jP2
Fig. 9.2. Rotation of the object and
yi ^|f?
V i / aperture planes by an angle φ.
A < P|
//y |
ö *2 Xl
The path of any electron leaving the point x , y in the object plane and
0 Q
passing through the point x , y of the aperture plane is uniquely determined,
a a
and accordingly also its intersection x , y with the image plane.
{ x
The optical distance S between the points of the object plane and those of
the aperture plane, being a function of initial and final points only, can be
written as power series in the coordinates JC , y , x , y .
0 0 a a
S = S + a^x + a y + a x + a y + bx 2 + b x y + · · · (9.3)
0 Q 2 Q 3 a 4 a x 0 2 0 0
or
S = S + S + S + S + S + · · · S (9.4)
0 t 2 3 4 n
9.3 Geometrical A berrations 5
with S denoting the part of the series containing terms of the nth order in
n
the coordinates. The series in Eqs. (9.3) and (9.4) can be considerably sim-
plified if we exploit the symmetry of the system.
Since the system is rotationally symmetrical, the optical distance S does
not change if the coordinates in both the object and aperture planes are
rotated by an identical angle.
Let us, for example, consider an object point P with coordinates x
1 l9
y and radial distance r as in Fig. 9.2.
u
x = r cos y and y = r sin y (9.5)
t l
Now if we rotate r by an angle φ, we have
x = r cos(y + φ) and y = r sin(y + φ) (9.6)
2 2
or
x = r cos y cos φ — r sin y sin φ (9.7)
2
and
y2 = r sin y cos φ + r cos y sin φ (9.8)
Hence
χ = X\ cos φ — y sin φ (9.9)
2 l
and
y = ;q sin φ + y cos φ (9.10)
2 x
Therefore, in general,
*o -> *ocos φ — y sm <p» * ~* * cos φ — y&sm ^>
0 a a (9.11)
j -> x sin φ + Jo cos φ, y -► x sin <p + j cos φ
0 0 a a a
If φ = π, all the coordinates change sign. Thus, all the odd terms in Eq.
(9.3) change sign, and therefore must vanish if S must remain invariant under
the transformation.
Hence
S = S + 5 + S + .·· (9.12)
0 2 4
The term S can now be expanded as
2
S = Mo2 + b x y + b y 2 + b x x + b x y + b y x
2 2 0 0 3 Q 4 Q a 5 Q a 6 Q a
+ b y y + b x 2 + Z> x y + b y 2 (9.13)
7 Q a s a 9 a a i0 a
Next, let us rotate by an angle φ = π/2. The optical distance should still
remain invariant. Reference to Eq. (9.11) gives
S = b y2 - b x y + b x2 + b y y - b y x - b x y + b x x
2 x 0 2 0 0 3 A Q a 5 Q a 6 0 a 7 0 a
+ by 2 - b x y + b x 2 (9.14)
s a 9 a a 10 a