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A Study on Scaling Symmetrical Condenser-Objective Lenses
A. Alamir
To cite this version:
A. Alamir. A Study on Scaling Symmetrical Condenser-Objective Lenses. Journal de Physique III, EDP Sciences, 1995, 5 (5), pp.641-645. �10.1051/jp3:1995151�. �jpa-00249336�
Classification
Physics Abstracts 41.80D
A
Study
onScaling Synunetrical Condenser-Objective
LensesA.SA. Alamir
Physics Department, Faculty of science, Assuit University, Assuit, Egypt (Received 12 September 1994, accepted 2 February 1995)
Abstract. Scaling down the dimensions of a conventional magnetic objective lens to reduce aberrations is studied in detail. The value of other aberration coefficients for optimum lens size
corresponding to desired minimum aberration
are estimated.
1. Introduction
In light optics, it is well known that, disregarding the diffraction limit, a glass lens can always be improved by reducing its dimensions. If the same conclusion can be drawn for magnetic lens, the magnetic circuit can be made smaller (conserved similarity). In magnetic lenses, for a constant flux density at the poleface and constant excitation parameter NI/I~~/~, at different
accelerating voltages, the dimensions and excitation NI of the lens must be multiplied by
a scaling factor n. The objective focal length f~, the spherical and chromatic aberration coefficients Cs and Cc will also be scaled by the same factor ill. If the relativistically corrected
accelerating voltage of the original lens is (' and that to be applied to the scaled lens is ("
the scaling factor n is given by:
~"l/2
~ v'l/2
The factor n is useful in calculating the electron optical parameters at different accelerating voltages. The relativistically corrected accelerating voltage I~ is calculated from the following
equation:
I~ = V(I + o.978 x 10~~ V)
where V is the accelerating voltage in volts.
Kamminga [2] has discussed in detail the variation of the aberration coefficients and other
optical quantities when a lens is scaled by a factor n.
In the absence of saturation effects, the aberrations of a magnetic lens may in principle be reduced indefinitely simply by making it smaller at the cost of increasing current density in the coil.
© Les Editions de Physique 1995
642 JOURNAL DE PHYSIQUE III N°3
Mulvey [3] has shown that scaling down the dimensions of a conventional magnetic objective
lens increases the peak gap-flux density and reduces the field half-width but does not necessarily
reduce the lens aberrations.
The purpose of the present discussion is to study the advantages and disadvantages of scaling
the magnetic lens dimensions so far as the lens aberrations are concerned.
2. Double-Polepiece Magnetic Lens
A test lens [3], shown in Figure I, of the Riecke-Ruska type (condenser-objective) was devised and checked by computation to ensure that no saturation effects occur at any part of the
polepieces, consisting of two spherical pole-caps joined to the main magnetic circuit by a pair of truncated cones. The axial field distribution of such lens is calculated using a standard finite
element program AMAG [4]. This program is well adapted to the calculation of lenses with
large extremes of flux density.
2.I. THE AXIAL FIELD DISTRIBUTION AND THE FOCAL PROPERTIES. The axial field
distribution of Figure I is shown as a function of scaling factor n in Figure 2. Curves 1, 2 indicate that the lens operates almost linearly. Scaling up (n = 2) increases the half-width
remarkably and decreases the peak value. Scaling down (n
= 1/2), as indicated by Curve 3, the field start to widen and the parasitic field appears. This means that the lens reaches saturation.
On the other hand the peak rises and the half-width decreases. In Curve 4 (n
= 1/3), the
peak value is increased but the half-width is reduced slightly, and the parasitic field plays an
important role. The overall result is that the focal properties are worse in spite of reducing
lens size, I-e-, increasing current density, as shown in Figure 3. This figure shows, also that for a given S/D ratio there is a particular lens size for an optimum Cs or Cc. Figure 4 shows the variation of the coefficient Cs and Cc with S/D ratio for the original lenses. It seems that
there is a certain size of each lens to reach the best condition.
IRON CIRCUIT
cot
Fig. 1. Mulvey [3] test lens (Riecke-Ruska type). Bore diameter D, gap length S, object in mid- plane of gap. Lens is scaled down to increase gap field.
I
z = o-o
V = TOO KV
' lln=1
2. n =2
' 31n =)
4,n=J
M 2 3
m
4
,,
3 ',
'
o 5 io
Zimml
Fig. 2. The axial field distribution for lens of Figure 1 (S/D
= 4). The specimen is focused at
mid-plane of gap at 100 kV. Lens scaled down by factor of 2, 1, 1/2, 1/3.
~,Cs
B ,"'
~
," °
~
,"
u- 6 ,"
j
~,/"4 "
u
z
O 2 3 4 5 6
n
Fig. 3. The focal properties of scaled lenses as a function of scaling factors.
3. Minimum Spherical Aberration
Figure 5 shows the variation of (Cs)ruin of the optimum scaled lenses with the associated chromatic aberration coefficient. From the figure it is clear that the lens of S/D
= 2 has the
lowest value of Cs(Cs = o.38 mm), but the associated Cc
= o.9 mm is about 50Sl higher than that of the original lens. The resolution is 6
=
2.61 (6 = 1.1 1at I~
= I MV).
644 JOURNAL DE PHYSIQUE III N°3
,~
Cc /"
I'
U I'
U /~~
~
l'
U ',
",
o
lo s/D
Fig. 4. Spherical and chromatic aberration coefficients of original lenses as a function of S/D.
s/D n
O 64
2 o e7
4 O 46
e o 2e
", Cc
~ ', ,,'
fi '"-
_,"
~~~~-~~
d Cs
'
s/D
Fig. 5. Minimum spherical aberration coefficient of scaled lenses together with the resulting coef- ficient of chromatic aberration Cc as a function of S/D.
4. Minimum Chromatic Aberration
The minimum chromatic aberration coefficient (Cc)m;n of optimum scaled lenses with the associated spherical aberration coefficient is shown in Figure 6. A lens of S/D = 4 has the lowest value of Cc(Cc
= o.74 mm). The associated value of Cs is o.93 mm (25Sl higher than that of the original lens).
S/0 n
O 94
2 0 57
4 O 74
e o 3e
° Cs
c
Cc __-~
---~~~
Uu
o
jo s/D
Fig. 6. Minimum chromatic aberration coefficient of scaled lens together with associated spherical
aberration coefficient of scaled lenses as a function of S/D.
5. Conclusion
In double polepiece condenser-objective lenses, one cannot always take advantage of making
the lenses smaller (n < I). As the lens reaches full saturation, there exist some optimum size and excitation which should be taken into consideration when the lens is designed.
Reducing the lens size, improves one of the lens aberration coefficients at the cost of increas-
ing the other.
Acknowledgments
The author would like to thank Professor T. Mulvey of Asten University in Birmingham, U-K- for useful and stimulating discussions.
References
ill Juma S-M-, Kaliq M.A. and Antar F-H-, J. Phys. E: Sm. Instrum. 16 (1983) 1063-1068.
[2] Kamminga W., Optik 45 (1976) 39-54.
[3] Mulvey T. and Yin H-C-, Inst. Phys. Cant. Ser. 1 (1988) 109-10, paper presented at EUREM 88, York, U-K-
[4] Lenc M. and Lencova B., SEMB6 3 (SEM Inc., Chicago, 1986) p.897.
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