Volume 2, Issue 10, October 2013
Page 25
ABSTRACT
A new analytical target function to start the synthesis approach has been introduced in the present work. The proposed mathematical expression has been carried out to approximate the axial magnetic scalar potential distribution of asymmetrical double pole piece magnetic lenses. Thereby, the influence of the main effective optimization parameter on the reconstructed pole piece profile and the objective optical focal properties for lenses operate under zero and infinite magnification modes has been investigated. The results have shown that the behavior of most objective properties evaluated in sense of synthesis procedure is in an excellent correspondence with that of analysis approach except the spherical aberration coefficient. Also, the results have points out that several design considerations must be followed in sense of the adopted target function.
Keywords: Electron Optics, Magnetic Lenses, Simulink, Modeling, Optimization, Charged Particle
1. INTRODUCTION
Examination of objects on a very fine scale in electron microscope can be achieved by using a high-energetic electron beams. However, aberrations have been introduced in the obtained images when these beam passing throughout the electron lenses of the electron optical instruments such as electron microscope. Minimization of electron optical aberrations has significantly improved the performance of scanning electron microscopes, transmission electron microscopes and scanning transmission microscopes. For decades most effort on charged particle optics was in improving the optical quality of such systems from micrometer down to sub-nanometer resolution. The major limiting factor for resolution is aberration (e.g. spherical or chromatic aberration). During image formation the magnetic field has to be in steady state, which implies that aberrations are only of importance during the image acquisition. As a result the design of the microscope is highly optimized for static use [1]. Improvement of the resolution of electron microscope is limited by the aberrations of the objective lens. Spherical and chromatic aberrations are the most effective ones. Unfortunately, both spherical and chromatic aberrations cannot eliminate completely. But they can be minimized to improve the resolution of the electron microscope. Traditionally, effort in improving electron lenses has concentrated on minimizing these aberrations and searching for field distributions with minimum or zero spherical aberration [2]. According to Scherzer theorem, it is impossible to correct spherical and chromatic aberrations of a charged particle lens just by a combination of ordinary lenses as it can be achieved for light-optical instruments. Therefore, aberration correction can be achieved when one has to break one of the Scherzer postulates that the electromagnetic focusing fields are free of space charge, time-independent and cylinder-symmetric and the optic axis has no point of reversal [3].The main component in any electron optical system is the electron lens that suffers from many different types of defects through the image formation process. However, due to imperfections of electron lenses and electron sources, aberration are introduced to the electron wave, thus reducing the spatial resolution of microscope. The spherical aberration, defocus, axial astigmatism and chromatic aberration are the most important aberrations for conventional transmission electron microscope [4]. Therefore, since 60 years many different efforts have been concentrated for minimization or correction these defects of the electron lens. The desire to produce electromagnetic lenses, with prescribed first order properties and minimized third order aberrations, is the main task of both researchers and designers since the invention of charged particle optics. There are two optimization procedures to be followed in the design of electron lenses, namely; analysis and synthesis. The synthesis approach which is followed in the present article called in some times the inverse design procedure. There are different ways may be followed to perform such approach. The designers or researchers may start with the trajectory of the charged particle beam to be the target function. Accordingly, the charged particle beam trajectory is represented by an appropriate mathematical expression such as the polynomial [5]-[8]; or like an n-time differentiable function [9]. However, with determining beam trajectory along the optical axis, the paraxial ray equation can be solved to deduce the axial field distribution capable of producing such a trajectory. However, the axial magnetic or electrostatic fields can be approximated by an appropriate analytical mathematical target function [10]-[11]. Also, the electrostatic potential may be represented by a mathematical expression to be the target function [12]-[15], or it being the axial magnetic scalar potential distribution [16]. Thus, by solving the paraxial ray equation the path of the charged particle beam will be determined along the optical axis.
Modeling and Design For Objective Charged
Particle Lens
Ali Hadi Hassan Al-Batat
Volume 2, Issue 10, October 2013
Page 26
Present article, however, compromise on using a new target function in the field of electron optics to approximate the axial magnetic scalar potential distribution of asymmetrical double pole piece charged particle lens. The Simulink in MATLAB environment is proposed to be a powerful tool for carrying out the present approach.
2. MATHEMETICAL TREATMENT
2.1 THE TARGET FUNCTION
A new target function has been introduced to approximate the axial magnetic scalar potential distribution of asymmetrical electron charged particle lens given by.
1
a z V
z
V( ) o exp(tan 1 ) (1)
Where Vo and a are an optimization parameters that controlled the distribution of the magnetic field and hence the
electron optical properties of the lens. With aid of the formula
dz z dV z
Bz o
) ( )
( the axial magnetic field distribution corresponding to the potential given in equation (1) can be deduced as follows.
) exp(tan ) ) ( ( ) ( a z a z 1 a V z B 1 2 o o z
(2)
Where μo=4π x 10-7H/m is the space permeability. The first derivative of the axial magnetic field distribution B'z(z) given
by the formula.
2 1 2 2 o o z a z 2 a 1 a z a z 1 a V z
B exp(tan ) )
) ( ( )
( (3)
The final task of any synthesis optimization procedure is the desire to construct the shape of the pole piece (equipotential surface) that can met equation(2) with aid of the formula[17].
2 1 P P (z) V V V(z) 2 (z) R
(4)
Indeed, Rp(z) represents the radial height of the pole piece, i.e. the distance between the material of the pole piece and the
optical axis.
V
(z)
is the second derivative of V(z), and Vp represents the axial potential value at any of the terminals ofthe lens optical axis. For a symmetrical magnetic lenses such a value being equivalent to half of the lens excitation (NI). However, in the case of asymmetrical charged particle lens the values of potential at terminals are not equivalent; thus, each potential value can be evaluated by using Ampere's circuital law along the axial interval on which the magnetic field is distributed.
2.2 ELECTRON LENS DEFECTS
The electron lenses suffer from many different types of imperfections through the image formation process. These defects are different according to the function of the charged particle lens. However, the objective electron lenses suffer from two main important defects namely; spherical and chromatic aberrations. The spherical aberration is caused by the in-homogeneity of the lens field affecting the off-axis rays, which reduces the focal length for electron rays passing the outer zones of the lens. As a result, electrons from a point object crossing the optical axis at different angles is imaged as a disc
with a minimum diameter of s Cs 3
2 1
d . Chromatic aberration is caused by spread of the energy ΔE of the electron source, variations of accelerating voltage ΔV and the lens current ΔL , resulting in a variation of the focal length, the diameter of disc confusion can be expressed as[4].
2 2 2 c c I I 2 V V eV E C d
(5)
Where Cs and Cc are the spherical and chromatic aberration coefficients, E, V, and I are the electron energy, accelerating
voltage and the lens current respectively. Actually, any task in the field of the electron and ion optics aims to determine the aberrations of an optical instrument. To evaluating these aberrations, several axial functions must be determined. The geometrical and chromatic aberration coefficients for any charged particle lens are expressed as a definite integral of the form [18].
V z V z V z B z B z B z r z r z r z r z
dzF C 2 1 z z
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Where z1 and z2 are the limits of the field distribution on the optical axis for the projector system, while these limits
represent the object and image coordinates for the objective lens. The symbols ' and '' refer to the first and second derivatives with respect to z respectively V(z), B(z) are the axial distribution of the electrostatic potential and the magnetic field respectively. In the case of magnetic lens, rα(z) and rγ(z) are two independent solutions of the following
well known paraxial ray equation[18].
0 (z) (z)r B 8mV
e
dz r d
α
2 z r 2
α
2
(7)
Where e and m are the electric charge and the rest mass of the electrons, usually e/m denoted by the symbol that known
as the electron to mass quotient and is the relativistically corrected accelerating voltage. Since the present work concerns with the synthesis optimization procedure of magnetic lenses, thus, the most two important spherical and chromatic aberration coefficients of any objective magnetic field may be represented by the following integral.
B z B z B z r z r z
dz FC
i
o z
z
( ), ( ), ( ), ( ), ( ) (8)
In last integral zo and zi are the object and image coordinates respectively. The coefficients of spherical Cs and chromatic
Cc aberration for the magnetic round lenses can be abbreviated in the following two integrals [18].
]dz
2
r
2
r
2
z
B
8
4
r
2
z
B
8
4
r
4
z
B
i
z
o
z
V
r
3
η
[
r
128V
η
s
C
(9)dz
r
B
8V
η
C
2z 2z
z r c
i
o
(10)For a certain magnetic field distribution of the lens these coefficients can be evaluated provided the electron beam trajectory rα(z) and its derivative are known along the lens length. However, trajectory of the electron beam can be
obtained by solving equation (7) numerically using the fourth-order Rang-Kutta method.
3. SIMULINK MODEL FOR THE ORIGINAL TARGET FUNCTION
Figure1 shows a simulink model for determining the axial magnetic scalar potential distribution V(z), the axial magnetic field of the charged particle lens Bz(z), and the first derivative of the field B'z(z) with aid of equations (1-3). The output
results of the simulink model that the axial potential target function, the axial magnetic field, and the first derivative of the magnetic field are plotted in figure 2 at a=1mm, V0=200A-t, and L=60mm. From figure 2b, it seen that the resulted
field is asymmetrical about the plane z=0.5mm. However, to be the field asymmetrical about the common symmetry plane (z=0), it can be shifted depending on the position of the inflection point of the field curve as shown in figure 3. The values of the two main important properties of the field distribution that the maximum flux density value Bmax and the
half width W are 0.3197 Tesla and 2.268mm respectively, where the half width of the left field side equals to 0.8503mm, while that for the right field side is 1.4177mm. Also, similar steps have been used to manipulate the curve of the departure of the magnetic field B'z(z) to be asymmetrical distribution about the common symmetry plane. It is worthwhile
to find the correspondence pole piece shape with aid of a simulink model by using equation (4). Thus, figure 4 shows the reconstructed above part of the pole piece shape that met the asymmetrical field plotted in figure 3. It is noted that the bore radius of the right pole piece is greater than that of the left side one. This result is agreement with the behavior of the field distribution plotted in figure 3, where the slowly right field is corresponding to the pole piece of large bore radius via the deeply decreasing field on the left side.
Figure1 The built up simulink model for the magnetic scalar potential V(z) and its corresponding axial functions Bz(z)
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-30 -20 -10 0 10 20 30
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
z(mm)
B
z
(T
)
a=1mm V0=200A-t L=60mm
-30 -20 -10 0 10 20 30
-200 -100 0 100 200 300 400 500 600 700 800
z(mm)
V(
z)
(A
-t
)
a=1mm V0=200 A-t L=60mm
(a) (b)
-30 -20 -10 0 10 20 30
-150 -100 -50 0 50 100 150 200 250 300
z(mm)
T
h
e
F
ir
s
t
F
ie
ld
D
e
riv
a
ti
v
e
a=1mm V0=200A-t L=60mm
(c)
Figure2 The output results of the simulink model for (a) the axial potential distribution (b) the axial magnetic field, and (c)the first derivative of Bz(Tesla) at a=1mm and V0=200 (ampere-turn)
-30 -20 -10 0 10 20 30 0
0.05 0.1 0.15 0.2 0.25 0.3 0.35
z(mm)
B
z
(T
)
a=1mm V0=200A-t L=60mm
Figure3 The asymmetrical magnetic field distributed about the common symmetry plane (z=0)
-30 -20 -10 0 10 20 30
0 2 4 6 8 10 12 14 16 18 20
z(mm)
Rp
(m
m
)
a=1mm V0=200 L=60mm
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4. RESULTS AND DISCUSSION
Obviously, the new suggested target function that represents the distribution of the axial magnetic potential consists of three optimization parameters namely the potential parameter denoted by V0, the length of the lens L=ZF-Zi, and the more
effective parameter a. In the present work the effect of the latter parameter has been investigated when the other two parameters are kept constants concerning image field parameters, objective focal properties for zero and infinite magnification modes, and the reconstructed pole piece shape with aid of block diagram shown in figure5.
With aid of simulink model in figure 1 for determining the distributions of magnetic scalar potential, axial magnetic field, and the reconstructed pole piece shape, figure6 shows these axial functions distributed along the z-optical axis for different values of the parameter a(i.e., 0.5,1,1.5,2, and 2.5mm) when the other parameters are maintained fixed at the values: V0=200(ampere-turn) and L=60mm. It is noted that the increasing of the parameter a causes the refractive power
to extending on a wide optical axial interval while its strength being decreases. However, it can be seen that as the parameter a increases the half width W increases while the maximum flux density value Bmax decreases as shown in figure
7. Accordingly, the excitation NI of the lens decreases with increasing a.
As it shown in figure 6a and b, it is noted that the distribution of the axial potential and consequently the axial magnetic field are asymmetric about the common symmetry plane (z=0). However, the right hand side distribution, i.e., the field that distributed along the positive z-optical axis interval is more affected by varying the parameter a, where the area under the field distribution is greater than that of the left hand side field. Therefore, the refractive power of the right hand side field is greater than that of the left hand side field. Indeed, one can conclude that as the parameter a decreases the axial magnetic field becomes more localized in the air gap region.
Actually, one can expect that the right hand side reconstructed pole pieces are of large bore radii from those of the left hand side ones as shown in figure 6c. This result can be explained according to the field distribution, where, the reconstructed magnetic pole piece of large bore radius is corresponding to slowly decreasing field and via versa.
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-30 -20 -10 0 10 20 30
-200 -100 0 100 200 300 400 500 600 700 800
z(mm)
V
(z
)
A
-t
a=0.5mm =1 =1.5 =2 =2.5
-20 -15 -10 -5 0 5 10 15 20 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
z(mm)
B
z
(T
)
a=0.5mm =1 =1.5 =2 =2.5
(a) (b)
-30 -20 -10 0 10 20 30
0 5 10 15 20 25 30
z(mm)
R
p
(m
m
)
a=0.5mm =1 =1.5 =2 =2.5
(c)
Figure 6 The axial distributions of a) V(z), b) Bz(z), and c) Rp along the optical axis for different values of the parameter
a
0 0.2 0.4 0.6 0.8 1
0 1 2 3
The parameter a(mm)
B
m
ax (
Te
sl
a)
, N
I
x1
0
3 (
a
m
p
er
e-tu
rn
),
W
x1
0(
m
m
)
Bmax(T) NI(A-t) W(mm)
Figure7 The field parameters Bmax, NI, and W as a function of a at V0 =200(ampere-turn) and L=60mm
Since the axial field distribution of the charged particle lens is asymmetric, therefore, it is expected that the first order properties and third-order aberrations for the object and image spaces are different. However, figure 8 shows the variation of the objective focal length fo, spherical Cs and chromatic Cc aberration coefficients with optimization parameter a at
NI/Vr1/2=20 for zero and infinite magnification modes. Obviously, the behavior of fo and Cc evaluated in present work
concerning the synthesis procedure is similar to that of conventional one, where these optical properties are of small values for the more localized field distribution that is distributed in the object space. Therefore, the charged particle lens that operates under zero magnification mode concerning considered target function is of better fo and Cc from that of
infinite magnification mode. This means that these parameters for the condenser field distribution are better than those of the objective field one. On the other hand, the behavior of spherical aberration Cs with the parameter a for the two sides
operations is not in agreement with the conventional behavior. However, it is seen that the values of Cs for the lens
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as shown in table1. This means that the trajectory of electron beam under infinite magnification mode focusing near the maximum flux density value, i.e., in the more effective lens region.
Table1: Field values and intersection points at object and image planes a(mm) Zero intersection axial
point (mm)
Infinite intersection axial point (mm)
Field value at image space (Tesla)
Field value at object space (Tesla)
0.5 0.14 -0.07 0.4560 0.6282
1.0 0.80 -0.13 0.2320 0.3145
1.5 1.20 -0.15 0.1574 0.2111
2.0 1.50 -0.13 0.1200 0.1593
2.5 1.80 -0.09 0.0976 0.1277
(a)
(b)
(c)
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5. CONCLUSIONS
It appears that the results have shown the validity of the magnetostatic reduction problem using optimization by synthesis. Also, the computations of the present work proved the efficiency magnetostatic reduction problem. Several remarks could record in accordance the results shown above. The most important ones are that, for all values of the parameter a, lenses operate under infinite magnification mode have small spherical aberration values compared with those operate under zero magnification mode. Actually, this result is not a conventional one as in the analysis approach. Indeed, the results showed that computations of synthesis approach with aid of simulink for more focal properties are in agreement with the conventional design of charged particle lenses.
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