This document summarizes the properties of multi-element electrostatic lenses. It discusses how the focal properties, zoom lens curves, and aberrations of three-element, four-element, and five-element lenses are affected by varying the voltage ratios between elements and the aperture to diameter ratio. Increasing the number of elements and final accelerating potential ratio improves focusing power and reduces aberrations. A four-element lens with an aperture to diameter ratio of 1 provides good overall performance with focusing power at low voltage.

1. Dr. Omer Sise
Afyon Kocatepe University, TURKEY
omersise@aku.edu.tr
Lecture 3.
Computation of Parameters of Some Electrostatic L
enses
Charged Particle Optics:
Theory & Simulation
My Current Adress:
Suleyman Demirel University, TURKEY
omersise@sdu.edu.tr
omersise.com

2. THREE-ELEMENT LENSES
Focal properties
Zoom-lens curves
Figures of merits
Spherical and Chromatic Aberrations

3. Focal lengths (A/D=0.5)
As V2/V1 is increased from 0.1 to 10, the focal and midfocal lengths pass through a maximum value and
then decrease to a minimum value. This behavior is repeated when V3/V1 is varied from unity (< 1 or> 1),
but f and F tend to become shorter and less dependent on V2/V1. These results are almost identical to
those of aperture lenses. The overall effect on the focal properties is not large.

4. The lens with A/D=0.5 has focal points slightly different
from the lens with A/D=1.
Focal lengths (A/D=1.0)

5. For a given voltage ratio cylinder lenses are a little stronger and have smaller aberration
coefficients. Cylinder lenses tend to be slightly superior to the aperture lenses.
= Cs0/f2, = Cs4F2
2
/f1
3

6. P
Object
mage
Q
Reference
Plane
D
A
G
V
1
V
2
V
3
1
10
1
0,01
0,1
1
10
A/D=0.5 (a)M
V 3
/V 1V
2 /V
1 +1
0,1
1
10
1
0,1
1
10
(b)A/D=1M
V 3
/V 1V
2 /V
1
0,1
1
10
1
0,01
0,1
1
10
(c)A/D=2M
V 3
/V 1V
2 /V
1
The properties of a zoom lens depend on both lens geometry and
voltage ratios. Especially, the different lengths of the center element
A/D affects the properties of the lens
The properties of a zoom lens depend on both lens geometry and
voltage ratios. Especially, the different lengths of the center element
A/D affects the properties of the lens
Zoom Lens Curves
The range of magnification of
the lens with A/D=2 increases
substantially for fixed P and Q
values while the lens with A/D=
0.5 has the range 0.2–2.2.
The range of magnification of
the lens with A/D=2 increases
substantially for fixed P and Q
values while the lens with A/D=
0.5 has the range 0.2–2.2.

7. On each zoom-lens curve for V3/V1=1 two
points correspond to an einzel lens and
for V2/V1=1 or V2/V1=V3/V1, four points
correspond to a two-element lens.
On each zoom-lens curve for V3/V1=1 two
points correspond to an einzel lens and
for V2/V1=1 or V2/V1=V3/V1, four points
correspond to a two-element lens.

8. Aperture lenses are ‘weaker’ requiring a higher V2/V1 ratio for a
given object and image distance for a fixed V3/V1. Since the
aberrations are usually lower for the higher values of V2/V1, these
lenses are expected to have lower aberration coefficients than
cylinder lenses.

9. Acceleration & Deceleration
0,1
1
10
1
10
100
0,1
1
10
(a)
CC
/D
CS
/D Q/D=2.8
V 2
/V 1V
3 /V
1
0,1
1
10
1
10
100
0,1
1
10
CC
/D
CS
/D Q/D=4
V 2
/V 1V
3 /V
1
0,1
1
10
1
10
100
0,1
1
CC
/D
CS
/D
(c)Q/D=10
V 2
/V 1V
3 /V
1
Decelerating lensDecelerating lens
Accelerating lensAccelerating lens

10. FOUR ELEMENT LENSES
Focal properties
Zoom-lens curves
Figures of merits
Spherical and Chromatic Aberrations

11. Since the potential of the
final element V4/V1 takes
high values, the lens
parameters are
independent of the voltage
fluctuations of V3/V1.
Since the potential of the
final element V4/V1 takes
high values, the lens
parameters are
independent of the voltage
fluctuations of V3/V1.
0,1
1
10
1
10
0
1
2
3
4
5
(a)V4
/V1
=1
A/D=0.5
F1
/D
V 3
/V 1
V
2 /V
1
0,1
1
10
0,1
1
10
0
1
2
3
4
5
V4
/V1
=1
A/D=1
(b)F1
/D
V 3
/V 1
V
2 /V
1
0,1
1
10
1
0
1
2
3
4
5
V4
/V1
=10
A/D=0.5
(g)F1
/D
V 3
/V 1V
2 /V
1
0,1
1
10
0,1
1
0
1
2
3
4
5
V4
/V1
=10
A/D=1
(h)F1
/D
V 3
/V 1V
2 /V
1
The mid-focal lengths, F1/D
and F2/D, have the same
values when the values of
V3/V1 and V4/V1 are near
1.0, but as V3/V1 diverges
from unity, the four-
element lens becomes
stronger.
The mid-focal lengths, F1/D
and F2/D, have the same
values when the values of
V3/V1 and V4/V1 are near
1.0, but as V3/V1 diverges
from unity, the four-
element lens becomes
stronger.
Focal lengths

14. The overall effect on the focal properties is not large, but the lower aberration of the cylinder
lens is quite noticeable. An increase in the accelerating potential ratio V4/V1 greatly reduces
the lens aberrations. So it is clearly advantageous to use a four-element lens as a focusing
lens.

15. Zoom Lens Curves
The four-aperture
lens has three
degrees of freedom
(three potential
ratios). One degree
of freedom is used
to set the ratio of
final-to-initial
energy, the second
to fix the focal
points. The third
degree of freedom
may be used to
control the
magnification
(linear or angular).
The four-aperture
lens has three
degrees of freedom
(three potential
ratios). One degree
of freedom is used
to set the ratio of
final-to-initial
energy, the second
to fix the focal
points. The third
degree of freedom
may be used to
control the
magnification
(linear or angular).
The relationship between V2/V1 and V3/V1, for fixed values of P, and V4/V1=5, for either Q is
constant, or M is constant. It is possible to find a value of V3-V2 which gives a constant
magnification and fixed image distance for a given voltage ratio V4/V1 by using the crossing
points on the curves.

17. The ranges of values of V4/V1 are considerably wider than those available with the four-cylinder lens.The ranges of values of V4/V1 are considerably wider than those available with the four-cylinder lens.
Calculated values
of the mid-focal
length, F1, as a
function of V4/V1
for each of the six
sets. If F1 is
known for a
constant (P,Q,M)
set then the
remaining lens
parameters (f1, f2,
F2) can be
obtained from
the lens equation:
Calculated values
of the mid-focal
length, F1, as a
function of V4/V1
for each of the six
sets. If F1 is
known for a
constant (P,Q,M)
set then the
remaining lens
parameters (f1, f2,
F2) can be
obtained from
the lens equation:
2
2
1
1
f
FQ
FP
f
M
−
−=
−
−=

18. P
Object
mage
Q
Reference
Plane
D
A
G
V
1
V
2
V
3
V
4
A
Increasing V4/V1 has the effect
of reducing the aberration
coefficients of the lens, and
theminimumsof coefficients
are significantly smaller than
those of the three-element
lens. In general, lens with A/D
= 1 appears to provide a good
overall design, giving
focussing power at a low
voltage.
Increasing V4/V1 has the effect
of reducing the aberration
coefficients of the lens, and
theminimumsof coefficients
are significantly smaller than
those of the three-element
lens. In general, lens with A/D
= 1 appears to provide a good
overall design, giving
focussing power at a low
voltage.

19. FIVE ELEMENT LENSES
P
Object
mage
Q
Reference
Plane
D
A
G
V
1
V
2
V
3
V
4
A
V
5
Zoom-lens Curves
Spherical and Chromatic Aberrations
Afocal Lenses
T
V1
D
A A
V2 V3
Equipotential
Contours
Image
Point
3D
R
A AL
V4 V5
Object
Point

20. Zoom-lens Curves
There are four
possible
configurations of
V2/V1 and V4/V1
because the middle
electrodes are
usually either higher
or lower in voltage
to satisfy the zoom-
lens condition.
There are four
possible
configurations of
V2/V1 and V4/V1
because the middle
electrodes are
usually either higher
or lower in voltage
to satisfy the zoom-
lens condition.
If V4=V1 and V2=5.2 V1 -> M=3
If V2=V1 and V4=5.2 V1 -> M=1/3
The crossing point
on curves shows the
required voltage
ratios for afocal
lenses, where we
mark the points
corresponding to
the afocal lens.
The crossing point
on curves shows the
required voltage
ratios for afocal
lenses, where we
mark the points
corresponding to
the afocal lens.
P
Object
mage
Q
Reference
Plane
D
A
G
V
1
V
2
V
3
V
4
A
V
5

21. Spherical and Chromatic Aberration Coefficients
0,1
1
1
10
100
1000
0,1
1
V5
/V1
=1
Long
Cc
/D
Cs
/D
V 2
/V 1V
4 /V
1
1
10
1
10
100
1000
1
10
V5
/V1
=1
Short
Cc
/D
Cs
/D
V 2
/V 1
+2V
4 /V
1 +2
0,1
1
10
100
1000
0,1
1
V5
/V1
=0.1
Long
Cc
/D
Cs
/D
V 2
/V 1V
4 /V
1
1
10
1
10
100
1000
1
10
V5
/V1
=10
Long
Cc
/D
Cs
/D
V 2
/V 1V
4 /V
1
1
10
100
1000
1
V5
/V1
=0.1
Short
Cc
/D
Cs
/D
V 2
/V 1
+2V
4 /V
1 +2
1
10
1
10
100
1000
1
10
V5
/V1
=10
Short
Cc
/D
Cs
/D
V 2
/V 1
+2V
4 /V
1 +2
V3/V1=0.32 V3/V1=1.00 V3/V1=3.16
V3/V1=0.32 (+2) V3/V1=1.00 (+2) V3/V1=3.16 (+2)
P
Object
mage
Q
Reference
Plane
D
A
G
V
1
V
2
V
3
V
4
A
V
5

22. Cs and Cc versus M
AB: V2/V1 > 1 and V3/V1 > 1
BC: V2/V1 > 1 and V3/V1 < 1
CD: V2/V1 < 1 and V3/V1 < 1
DA: V2/V1 < 1 and V3/V1 > 1
AB: V2/V1 > 1 and V3/V1 > 1
BC: V2/V1 > 1 and V3/V1 < 1
CD: V2/V1 < 1 and V3/V1 < 1
DA: V2/V1 < 1 and V3/V1 > 1
AA
BB
CC
DD
If a high positive voltage with respect to the
first element’s potential is applied to the lens
elements, the aberrations as well as the
minimum beam divergence can be reduced.
The reason, obtained from numerical
simulation, is that a positive voltage increases
the electron velocity, shortening the electron
drift time across the region with aberrant field.

23. P
Object
mage
Q
Reference
Plane
D
A
G
V
1
V
2
V
3
V
4
A
V
5
Design Curves
This representation allows one to read
off values for the required voltage
ratios to satisfy constant magnification
at an overall voltage ratio. In this way,
one can design both a lens having a
fixed magnification overall voltage
ratio and one having the true-zoom
lens with variable magnification at a
fixed voltage ratio.
This representation allows one to read
off values for the required voltage
ratios to satisfy constant magnification
at an overall voltage ratio. In this way,
one can design both a lens having a
fixed magnification overall voltage
ratio and one having the true-zoom
lens with variable magnification at a
fixed voltage ratio.

24. Summary
• Aperture lenses can be made compact and light.
• Cylinder lenses have the advantage that in
comparison with aperture lenses the spherical
and chromatic aberration are smaller for a given
lens diameter. This is the consequence of the
smoother axial potential distribution.
• The lens having A/D = 1 offers the advantage of a
slightly wider range of magnifications and small
aberrations compared with A/D = 0.5 for the
same object and image distance.

25. Cappadocia, Turkey