International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
Projector Properties of the Magnetic Lens
Depending on Some Physical and Geometrical
Parameters
Mohammed Jawad Yaseen
Department of Physics, College of Education, the University of Mustansiriyah, Baghdad, Iraq.
ABSTRACT
A target function of conventional parameters has been suggested throughout this work to study symmetrical double polepiece
magnetic projector lens. These optimization parameters are respectively the excitation of magnetic lens, the effective axial length
of lens, the bore diameter width of lens and the air gap width of lens. The effect of each of these parameters is investigated
concerning the field parameters, projector focal properties and the reconstructed polepieces.
Keywords: Electron Optics, Projector Magnetic Lenses, Distortion, Synthesis
1. INTRODUCTION
The approximation of the axial magnetic field along certain interval plays an important role to investigate the properties
of imaging magnetic fields. Consequently, several models had been suggested since the invention of charged particle
optics. Most of these models use independent optimization parameters describe by the physical properties of the imaging
field see for example [1]. However, a few of them are consisting of parameters that describe the geometrical profile of the
imaging magnetic fields, see for example [2]-[5].
Indeed, the using of magnetic field that has geometrical parameters is more suitable to make an objective judgment of the
inverse design problem. Thus, the present work is mainly concern with adopting such like model. It should be mentioned
that according to our knowledge, this is the first time that the adopted model is used to synthesizing of a magnetic lenses.
2. MATHEMATICAL PROCEDURE
The approach by means this work is accomplished can be divided into mainly three steps. In the first one the axial
magnetic field is setup by using the new mathematical expression. While the second one aims at finding and evaluating
the projector focal properties of the magnetic field distribution determined in the first step. The final step, however, is
usually concern with finding the polepieces that have the ability to produce the imaging field handled in the first step.
3. MAGNTIC FILED DETERMINTION
The calculation of the magnetic flux density distribution along the optical axis has been calculated from paraxial-ray
equation [1].
r  
η
B z2 (z)r  0
8Vr
(1)
Where r″ is the second derivative of the electron beam trajectory, r is the electron beam trajectory,  is the charge -tomass quotient of the electron, Vr is the relativistically-corrected accelerating voltage and the primes are signs of
differentiation with respect to z. The axial magnetic field distribution has been approximated with the aid of the following
formula [6].
μ o NI
 2.636s 
sinh

s
 D 
B z (z) 
(2)

 2.636s 
 5.272z  
 cosh
  cosh
 
 D 
 D 

Where μo is the space permeability ( 4πx10-7 H.m-1), NI is the imaging field excitation, D and s are the bore diameter
width and the air gap width of the proposed lens. It is seen that the magnetic field approximated by equation 2 mainly
depends on the air gap width and the bore diameter width of the lens. In other word, the s and D are the optimization
parameters that have the most contribution to define the shape of the magnetic field. In fact the algorithem for analyting a
magnetic lens by solving Laplace's equation.
So as to calculate the scalar magnetic potential, axial magnetic field, reconstructed polepiece and the correspondence
focal properties have been writen in Fortran power station 90 language program. Therefore, figure 1 represents flowchart
to find single or double imaging field.
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Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
Figure 1 A flowchart used to find single or double imaging field [4]
4. ABREATION DETERMINATION
In the magnetic lens theory, there are two important results. The first one is that the focal length of the lens decreases
with an increase of magnetic flux density, the second one, the electron beam rotates in the magnetic field. This rotation is
simply proportional to the integral of the field strength and does not depend on the field distribution [7]. The present
work aims at investigating optimization by synthesis of a symmetrical double polepiece projector magnetic lens. As it is
well known that the image of this lens suffers from two important geometrical defects which are the radial and spiral
distortion. However, when these two defects are minimized as small as possible or eliminated the resulted image is of
high quality. Electrostatic, magnetic and light lenses suffer from radial distortion, while the spiral distortion occurs in the
magnetic lenses only as a result to the rotation of the image along the bore radius of the lens and it is proportional to the
excitation of the lens. Thus, to enhance the final image of the projector lens, the spiral distortion must be minimized or
eliminated. One of the methods that are used to eliminate this distortion in the magnetic lenses, one can use a double airgap lens with rotation-free image [8]-[11].
The radial Dr and spiral Ds distortion coefficients can be determined by using the following integrals [12].
 η
Dr  
 128Vr
z

 2  3η 2

   Bz  8Bz2 rα rγ3  4Bz2 (rγ2 rα rγ  rγ rα2rα ) dz
z V


 1  r

z2 
3  η

Ds   


z1 128  Vr

3/2

1
 rα2 Bz3 

16

 η

V
 r
1/2


 rα2 Bz  dz




(3)
(4)
Where rα and rγ are two linearly independent solutions of the paraxial-ray equation(1). The limits of integration are the
two terminals points' z1 and z2 of the magnetic field.
5. MAGNETIC SCALAR POTENTIALDETERMENTION
The axial magnetic scalar potential is essential parameter by means the polepiece profile can reconstructed. Thus, the
determination of such an axial distribution must be achieved first before one can talk about the polepieces reconstruction.
So, to calculate the magnetic scalar distribution corresponds to the input Bz using equation [13].
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Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
n1
i
V z(i1)  0.5  Ei   E j
i 1
j 1
(5)
Thereby the axial magnetic scalar potential Vz along the lens interval z1≤ z ≤ z2 can be computed.
6. POLEPEICE RECONSTRUCTION
The final task of any synthesis procedure is to find the polepieces that can produce the wanted field distribution that
determined in last section. In this work the technique of Laplace’s equation solution has been followed to determine the
polepieces profile. However, the following equations are used to reconstruct the lens polepieces [1], [14].
1/2
V z  V p 
(6)
R(z)  2 

 V Z 
Where Vz is the magnetic scalar potential and Vp is the value of V(z) at any of the terminal points of the optical axis. It is
important to mention that V(z) and its second derivative V″(z) have been calculated using the cubic-spline integration
technique, for more details see [13]. Figure 2 represents a flowchart to find projector properties and pole shape.
Figure 2 A flowchart used to find projector properties and pole shape [4]
7. RESULTS AND DISCUSSION
The influence of varying each of D, s and NI on the projector focal properties and hence the reconstructed polepieces have
been carried out when the other are maintained fixed.
7.1 THE INFLUNCE OF THE BORE DIAMETER WIDTH
Five values of the bore diameter width have been chosen, namely (1, 2, 3, 4 and 5) (in unit of millimeter), in order to
reveal its influence on the magnetic field distribution profile and its properties. However, the values of s, NI and the
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Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
length of lens are maintained constant during these calculations at 1mm, 100A.t and 20mm, to satisfy higher objectivity.
The distributions of the axial magnetic field correspond to each value of D are plotted in figure 3.
0.12
D=1mm
D=2mm
D=3mm
D=4mm
D=5mm
0.1
Bz(T)
0.08
0.06
0.04
0.02
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
Z(mm)
Figure 3 The Bz(Tesla) distributions at different values of D(mm)
It can be seen that as D vary the distribution parameters become totally different. In other word, the halfwidth of the
magnetic field distribution w increases and the maximum field value Bmax decreases whenever, D increases and vice
versa, see figure 4. In actual fact, this behavior in agreement with conventional results concerning the variation of the
lens bore dimension, see for example [15].
12
w
Bmax
10
Bmaxx10-2(T), w(mm)
8
6
4
2
0
0
1
2
3
4
5
6
D(mm)
Figure 4 The magnetic field parameters Bmax(Tesla) and w(mm) as a function of D(mm)
Figure 5 shows the distribution of the magnetic scalar potential Vz(z) at the chosen values of D. It can be seen that the
influence of increasing D leads to making Vz(z) curve departure toward the optical axis as a result of extending of the
action interval. However, this is not a strange result as long as Bmax decreases and w increases when D increases.
60
D=1mm
D=2mm
D=3mm
D=4mm
40
D=5mm
Vz(A.t)
20
0
-20
-40
-60
-10
-8
-6
-4
-2
0
2
4
6
8
10
Z(mm)
Figure 5 The magnetic scalar potential distributions Vz(z)(ampere-turn) at different values of D(mm)
The total reconstructed polepieces for each value of D are shown in figure 6. Obviously, one may note that the increasing
of D will significantly change the polepiece profile. The coincidence between this synthesis result and the well known
conventional counterparts is that both of them announce the increasing of the bore diameter width will redacting the
ability of the polepieces in localizing and confining the magnetic flux lines. This consequence can be forced with aid of
figure 7, which represents the behavior of electron beam trajectory along the optical axis at various values of D for the
case of infinite operating mode that dependent throughout this work. Furthermore, these R(z) curves are plotted at an
excitation parameter NI/Vr1/2=20.
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10
D=1mm
D=2mm
8
D=3mm
D=4mm
D=5mm
6
4
Rp(mm)
2
0
-2
-4
-6
-8
-10
-10
-8
-6
-4
-2
0
2
4
6
8
10
Z(mm)
Figure 6 The total reconstructed polepieces for each value of D(mm)
0.006
D=1mm
D=2mm
0.004
D=3mm
D=4mm
D=5mm
0.002
0
R(mm)
-0.002
-0.004
-0.006
-0.008
-0.01
-0.012
-0.014
-10
-8
-6
-4
-2
0
2
4
6
8
10
Z(mm)
Figure 7 The electron beam trajectory along the optical axis at different values of D(mm)
Variation of the radial Dr, the spiral Ds distortion coefficients and the minimum projector focal length (Fp)min with the optimization
parameter D is shown in figure 8, these values calculate the excitation parameter of which the minimum projector focal length is
occurred. One can see that as D increases the minimum projector focal length (Fp)min increases as a result to the broadening of the
magnetic field distribution with increasing the parameter D. However, it should be noted that the distortion of the magnetic lens decreases
with increasing D. Specially in the values of D that less than 4mm, while for the values of the parameter D approximately greater than
4mm, the magnetic lens has very small amount of these distributions and approximately are not equal to each other.
4
(Fp)min
Dr
Ds
(Fp)min(mm), (Dr, Ds)(mm-2)
3
2
1
0
0
1
2
3
4
5
6
D(mm)
Figure 8 The radial Dr(mm-2), spiral distortion Ds(mm-2) coefficients at the minimum projector focal length and the
minimum projector focal length(Fp)min(mm) as a function of D(mm)
7.2 THE INFLUNCE OF THE AIR GAP WIDTH
Figure 9 shows various Bz distributions corresponding to different values of the air gap s(1, 2, 3, 4 and 5) (in unit of
millimeter). However, the bore diameter width and the lens excitation are kept constantan, the following values; D=1mm,
NI=100A.t and the length of lens 20mm respectively. It is seen that as the value of s increases, the peak value of Bz
distribution decreases and the corresponding halfwidth w increase as shown in figure 10. This result, however, is in
agreement with the corresponding one that may be obtained from the analysis procedure. Furthermore, the values of w
and s become very close for the high values of s. It should be mentioned that, throughout this work, the lens length is
chosen to be 20mm.
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0.12
s=1mm
s=2mm
s=3mm
s=4mm
0.1
s=5mm
Bz(T)
0.08
0.06
0.04
0.02
0
-10
-8
-6
-4
-2
0
2
4
6
8
10
Z(mm)
Figure 9 The Bz(Tesla) distributions at different values of s(mm)
6
w
Bmax
5
Bmaxx10-1(T),w(mm)
4
3
2
1
0
0
1
2
3
4
5
6
s(mm)
Figure 10 The magnetic field parameters Bmax(Tesla) and w(mm) as a function of s(mm)
The axial magnetic scalar potential distribution Vz(z) and the corresponding reconstructed magnetic total polepieces for
different values of the air gap width at D=1mm, NI=100A.t and length of lens 20mm are shown in figures 11 and 12
respectively. In fact, each of the reconstructed polepiece represents the total lens. So, the measured distance from the
center of the concave profile of each polepiece to the symmetry plane (z=0) represents half of the air gap width (i.e. s/2).
Therefore, when this distance is multiplied by the factor 2, the resulting distance is equal to the s value that is supplied to
the mathematical expression shown in equation (2). Similarly, the distance from the center of concave profile of the
polepiece to the optical axis z represents half of the bore diameter (i.e. D/2); this is in agreement with the input bore
diameter value after multiplying it by the factor 2.
60
s=1mm
s=2mm
s=3mm
40
s=4mm
s=5mm
Vz(A.t)
20
0
-20
-40
-60
-10
-8
-6
-4
-2
0
2
4
6
8
10
Z(mm)
Figure 11 The magnetic scalar potential distributions Vz(z)(ampere-turn) at different values of s(mm)
10
s=1mm
s=2mm
8
s=3mm
s=4mm
6
s=5mm
4
Rp(mm)
2
0
-2
-4
-6
-8
-10
-10
-8
-6
-4
-2
0
2
4
6
8
10
Z(mm)
Figure 12 The total reconstructed polepieces for each value of s(mm)
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Consequently, one can suggest that the position from which the air gap width and the bore diameter width are measured
in the centre of the concave profile of the reconstructed polepiece. It is important to mention now that the closeness
between the values of s and D determined from figure 12 and the corresponding one used to initialize the calculation
proves the validity of present approach, i.e. justifies the procedure by which the magnetostatic problem has been reduced
to that of electrostatic [7].
Variation of the radial Dr, the spiral Ds distortion coefficients and the minimum projector focal length (Fp)min with the optimization
parameter s is shown in figure 13, these values are calculated at the excitation parameter of which the minimum projector focal length is
occurred. One can see that as s increases the minimum projector focal length (Fp)min increases as a result to the broadening of the
magnetic field distribution with increasing the parameter s. However, it should be noted that the distortion of the magnetic lens decreases
with increasing s specially in the values of s that less than 4mm, while for the values of the parameter s approximately greater than 4mm,
the magnetic lens has very small amount of these distributions and approximately are equal to each other.
4
(Fp)min
Dr
Ds
(Fp) min(mm), (Dr, Ds)(mm-2)
3
2
1
0
-1
0
1
2
3
4
5
6
s(mm)
Figure 13 The radial Dr(mm-2), spiral distortion Ds(mm-2) coefficients at the minimum projector focal length and the
minimum projector focal length (Fp)mim(mm) as a function of s(mm)
7.3 THE INFLUNCE OF THE LENS EXCITION
Five values of the parameter NI have been chosen (NI=100, 200, 300, 400 and 500) (in unit of ampere-turn) when other parameters D
and s are kept constant at 1mm, while the length of lens 20mm. The investigation shows that the most important parameters D and s
remain constant when the excitation of the lens varied. Accordingly, the field and the potential distributions as well as the polepiece
profile are unaffected with varying NI as shown in figure 14. It is well known that when the halfwidth of the field is unaffected with the
optimization parameter under consideration, the focal properties are unaffected with that parameter as shown in figure 15.
300
0.6
NI=100 A.t
NI=100A.t
NI=200 A.t
NI=200A.t
NI=300 A.t
NI=300A.t
0.5
200
NI=400A.t
NI=400 A.t
NI=500 A.t
NI=500A.t
100
Vz(A.t)
B z(T)
0.4
0.3
0
-100
0.2
-200
0.1
-300
0
-10
-8
-6
-4
-2
0
2
4
6
8
-10
10
-8
-6
-4
-2
0
2
4
6
8
10
Z(mm)
Z(mm)
a
b
10
8
6
4
RP(mm)
2
0
-2
-4
-6
-8
-1 0
-10
-8
-6
-4
-2
0
2
4
6
8
10
Z(mm)
c
Figure14 a) The axial magnetic field distribution Bz(Tesla) b) The axial potential distribution Vz(z)( ampere-turn) and c) The total
polepiece profile for different NI values (ampere-turn)
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Volume 2, Issue 9, September 2013
ISSN 2319 - 4847
7
(Fp)min
Dr
Ds
(Fp)minx10-1(mm), (Drx10-1, Ds)(mm-2)
6
5
4
3
2
1
0
0
100
200
300
400
500
600
NI(A.t)
Figure 15 The radial Dr(mm-2), spiral distortion Ds(mm-2) coefficients at the minimum projector focal length and the
minimum projector focal length (FP)min (mm) as a function of NI(ampere-turn)
8. CONCLUTIONS
According to the previous results, several remarks can be stated. The most important of them is that, the mathematical
nature of a model used to approximate such a magnetic field plays an important role to specify its optical properties
(objective and projector). In this sense one may replace the conventional investigation of a magnetic lens by a careful
choice of mathematical model to approximate the imaging field distribution.
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