1. Dr. Omer Sise
Afyon Kocatepe University, TURKEY
omersise@aku.edu.tr
Lecture 2. Focal and Zoom Lens Properties
Charged Particle Optics:
Theory & Simulation
1
My Current Adress:
Suleyman Demirel University, TURKEY
omersise@sdu.edu.tr
omersise.com
2. So far, we covered:
the bits of optics we need to know (ray diagrams,
optical elements, lens equation, magnification,
focus)
why we need electrostatic lenses & how we get it
how they work, how electrons travel through
them
This presentation covers the focal and zoom-
lens properties of electrostatic lenses
I will also mention the (primary) aberrations.
Introduction
2
3. We use conventional terminology, from light optics, to describe many similar features
here!
Photon Optics and Charged
Particle Optics are Equivalent
Photons or ions can be trapped in
a resonator
R R
L
V V
Ek, q
V>Ek/q
V1 V2
V1<V2
Physical Principle
Photons or ions can be energy
analyzed usin a filter
3
4. α
Convex Lens
Optic axis
PointObject
PointImage
Excluded
Rays
• Fraction of rays from the object gathered by the lens is defined by
the semiangle, α
• The lens forms a magnified or de-magnified image of an object
Photon Optics:
Imaging with a simple lens
4
5. All parallel rays (whether parallel to the optic axis or not) are brought to a
focus in a plane at a position depending on their angle to the axis
Parallel Rays
Brought to Focus
Convex
Lens
Optic axis
ObjectPlane
ImagePlane
FocalPlane
Image formed after each lens is
rotated by 180o
with respect to the
object
Photon Optics:
Image Formation
5
7. defocusingfocusingdefocusing
0 V +V 0 V
Charged Particle Optics:
Electrostatic Lenses
Lenses are used to focus the beam and adjust the current .
Purpose: control the size and shape of the final beam spot. 7
9. Cardinal Points, Principal Planes,
And etc.
The representation of the four cardinal points, and the focal and mid-focal
lengths of the einzel lens is also shown. 9
10. • Electrostatic lenses are fairly simple static field
systems, composed of a few electrodes held at
possibly different potentials.
• They are widely used in electron and ion beam
instruments, both in sources and for extraction and
imaging.
• Electrodes of lenses are often made of coaxial
• cylinders or
• thin apertures
(O. Sise, Master Thesis, July 2005)
(N. Okumus, BSC Thesis, July 2007)
Types of Electrostatic Lenses
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12. The two-electrode
lens is often used
when the image and
object are required to
be in space of
different potential.
The second electrode
is placed at a higher
or lower potential,
thus providing
acceleration or
deceleration of the
beam.
Two-element lens
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13. The electron-optical
properties of a two-
element lens can be
presented in a diagram,
showing the image
position corresponding to
a given object distance,
with the acceleration ratio
(V2/V1>1) as a parameter
(the corresponding data
for retarding lenses
(V2/V1<1) can be obtained
from the same diagram).
Magnification lines are
also indicated.
Two-element lens:
P-Q Diagrams
13
14. In order to focus a beam without changing its
final energy, a lens with at least three
electrodes is necessary.
Three-element lens
14
15. In an acceleration or
deceleration lens, it is
very often desirable to
be able to keep the
image of a given
object fixed when the
acceleration /
deceleration ratio and
magnification are
changed. A lens
operated in this way is
usually referred to as
a zoom lens.
Three-element lens:
Zoom Lenses
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16. Calculation of zoom-lens
curves using simplex
optimization which is used
to identify locations of some
representative set of
minimums (dots) for a
three-aperture lens with
A/D = 1, P/D = 5 and Q/D =
5. One can roughly see from
the plots where the surface
minimums are, but it does
not precisely identify the
locations of the minimums.
Therefore, an optimization
is necessary.
Three-element lens:
Zoom Lenses
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17. The paraxial approximation applies whenever the angle
and distance between the system’s optical axis and the
ray of interest are small.
This allows the use of the small angle approximations
(sin(α) ≈ α, tan(α) ≈ α and cos(α) ≈ 1) when tracing the
path of the ray through an optical system (such as a
lens).
However, when the electrons are not moving close to
the axis, then the basic approximation begins to fail and
aberrations start to form.
As in light optics, the optics of charged particles suffers
from a number of image aberrations and distortions. Of
primary concern are spherical and chromatic
aberrations.
Paraxial Approximation & Aberrations
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18. Spherical aberration is most important as
compared with other lens aberrations. It appears
in the cubic dependence of the angle of
trajectory refraction on its radial position.
Aberrations:
Spherical Aberration
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19. y (µm)
-4 -2 0 2 4
z(µm)
-4
-2
0
2
4
y (µm)
-4 -2 0 2 4
z(µm)
-4
-2
0
2
4
Analogously, in charged
particle optics, spherical
aberration refers to the
variation in the focal
properties of the lens with
distance (or angle) from the
optical axis
Aberrations:
Spherical Aberration
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21. The spherical aberration coefficients are obtained from
Δr = −MCsα0
3
.
Aberrations:
Spherical Aberration
Cs is a fourth-order polynomial in 1/M:
Cs(M) = Cs0 + Cs1/M + Cs2/M2
+ Cs3/M3
+ Cs4/M4
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22. The particles in a beam will vary some in velocity, so
particles with slightly different energies (δE) get focused
at different image planes, and the focal point becomes
blurred.
Always some spread
in initial electron
energies as leave
cathode
W ~ 2 eV
LaB6 ~ 1 eV
FE ~ 0.2 to 0.5 eV
Minimize by decrease
in α
Aberrations:
Chromatic Aberration
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23. The chromatic aberration coefficient in the image plane is
described by δr = −MCcα0 δE/E0.
The same reasoning can be applied to find the chromatic
aberration coefficients Ci.
Cc = chromatic aberration
coefficient
α = convergence angle
Directly related to focal
length
Much less significant at high
E0
Aberrations:
Chromatic Aberration
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25. It is not yet clear the extent to which type of lenses is more
desirable over the other.This largely depends on the
specific application and the overall length of the lens
system employed.
Nevertheless, computer optimization of electrostatic
lenses is certainly important.This will save time and
resources in design of future charged particle optical
systems.
To provide more flexibility, both in terms of energy range
and for optimization with respect to optical properties,
more independent lens elements have to be added.
If more degrees of freedom are desirable, it is probably
better to use some combination of such basic units
separated by field-free regions than to design a lens where
more electrodes are closely spaced.
Summary
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