Differential algebraic description for third and fifth order
aberrations of electromagnetic lenses
Zhixiong Liu*
Department of Electronics, Peking University，Beijing 100871, China
Abstract
In this paper the modern map method has been applied to the third and fifth order
aberration analysis of electromagnetic lenses, focusing on the high order. It has also
been emphasized that there exist three types of differential algebraic (DA) descriptions
for electron optical aberrations. The numerical results of third and fifth order
aberrations of a given electromagnetic lens computed through the DA technique are in
excellent agreement with those evaluated by using the electron optical aberration
integrals. In conclusion, the DA technique is concise, efficient, and precise for electron
optical aberration analysis and COSY INFINITY has been proved to be an excellent
computer code for such purposes.
PACS: 41.85.-p; 41.85.Gy; 41.85.NE
Keywords: Differential algebra; Map method; Electromagnetic lens; aberration
Corresponding author: E-mail: zxliu@pku.edu.cn
Fax: +86-10-6275-1762
1
1. Introduction
In the previous work [1] it has been pointed out that there exist three types of DA
descriptions for aberration analysis of electron optical rotationally symmetric systems.
The main reason for this is that a rotating coordinate frame is always used for such
systems in order to simplify the analytical expression of the paraxial trajectory
equation and, as a result, the aberration is expanded in rotating coordinates and must be
corrected by the object magnetic immersion (OMI) effect when the object is immersed
in the lens magnetic field. Although the way developed in that work is suitable for
aberration analysis through arbitrary order, the DA description in rotating coordinates
for aberrations higher than third order becomes quite complicated. Therefore, in the
present work the general electron trajectory equation expressed in rotating coordinates
has been derived and the transfer map in rotating coordinates obtained by tracking this
equation will be simplified considerably.
The numerical calculation of third and fifth order aberrations of a given combined
electromagnetic lens was carried out by using the two completely different methods,
the DA technique and aberration integrals in electron optics, in order to make a
cross-check with each other. The results were extremely consistent. It comes to
conclusion that the DA descriptions for electron optical aberration analysis is very
concise, efficient, and precise and that COSY INFINITY [2,3] is an excellent computer
code for such purposes. In addition, the DA description in rotating coordinates derived
in this work is superior to the previous one [1] in simplicity with high precision
remaining unchanged.
2. General electron trajectory equation in rotating coordinates
In electron optics the general electron trajectory equation is usually expressed in a
fixed coordinate frame [4,5],
X  
Y  

 2 
(
2 X
 2 
(
2 Y
 X
Y
  2
)
( BY  Y Bt ) ,
Z

  2
)
( BX  X Bt ) ,
Z

(1)
e
1
,   1  X 2  Y 2 , Bt  ( BZ  X BX  Y BY ) ,
2m

where the uppercase letters, X , Y , and Z , indicate the fixed coordinates in this
context. By using Eq. (1) and the rotating transform of coordinates [4,5] in electron
2
optics, we can switch Eq. (1) to another form of the trajectory equation,
x  2 y   2 x   y 
 2 

2
[  ( x   y ) ] 
[ by  ( y   x)bt ] ,
2 x
z

y  2 x   2 y   x 
 2 

2
[  ( y   x) ] 
[ bx  ( x   y )bt ] ,
2 y
z

  1  ( x   y ) 2  ( y   x) 2 , bt 
1

(2)
[bz  ( x   y )bx  ( y   x)by ] .
where the lowercase letters, x, y, and z imply the corresponding rotating coordinates,
 and b are the potential and magnetic induction depending on x, y, and z , and  is
the rotating angle of the rotating coordinate frame. Although Eq. (2) becomes more
complicated than Eq. (1), it will clearly be seen later that it does take an important role
in simplifying the DA description in rotating coordinates.
3. Three types of DA description for third and fifth order aberrations
The DA description in fixed coordinates is produced by tracking Eq. (1) through a
DA integrator from the object plane zo to the image plane zi . For third and fifth
order aberrations we have
X 3 i 
Y3i 
X 5 i 
Y5i 
k  l  m  n 3
M
(1, klmn) X ok X o'lYomYo'n ,
f
k ,l , m , n  0 ,1, 2 , 3
k  l  m  n 3
M
f
k ,l , m , n  0 ,1, 2 , 3
(3, klmn) X ok X o'lYomYo'n ,
(3)
k  l  m  n 5
M
f
k ,l , m , n  0 ,1,..., 5
k
o
'l m 'n
o o o
(1, klmn) X X Y Y ,
k  l  m  n 5
M
f
k ,l , m , n  0 ,1,..., 5
(3, klmn) X ok X o'lYomYo'n ,
while the DA description in rotating coordinates is obtained by tracking Eq. (2) and
hve the form of
x3i 
y3i 
x5i 
y5i 
k  l  m  n 3
M
r
k ,l , m , n  0 ,1, 2 , 3
(1, klmn) xok xo'l yom yo'n ,
k  l  m  n 3
M
r
k ,l , m , n  0 ,1, 2 , 3
(3, klmn) xok xo'l yom yo'n ,
(4)
k  l  m  n 5
M
r
k ,l , m , n  0 ,1,..., 5
k
o
'l
o
m
o
'n
o
(1, klmn) x x y y ,
k  l  m  n 5
M
r
k ,l , m , n  0 ,1,..., 5
(3, klmn) xok xo'l yom yo'n .
3
In Eqs. (3) and (4) M f (i, klmn) and M r (i, klmn) for i  1 or 3 are the third and
fifth order transfer map elements in fixed and rotating coordinates, respectively, the
number of order depending on the sum of k , l , m, and n .
Since the expansion of the third order aberration is known in the textbooks on
electron optics, we only write the expansion of the fifth order one below,

 


 

*
r5i  (ro  ro) 2 ( A5 ro  a5 ro* )  (ro  ro) 2 ( B51ro  b51ro ) 
   


   

*
(ro  ro)( ro  ro )( B52ro  b52ro* )  (ro  ro)( ro  ro )(C51ro  c51ro ) 
   


 


(ro  ro)( ro  ro )(C52ro  c52ro* )  (ro  ro ) 2 (C53ro  c53ro* ) 
   

*
 

*
(ro  ro)( ro  ro )( D51ro  d 51ro )  (ro  ro ) 2 ( D52ro  d 52ro ) 
   


   

*
(ro  ro )( ro  ro )( D53ro  d 53ro* )  (ro  ro )( ro  ro )( E51ro  e51ro ) 
 


 

*
(ro  ro ) 2 ( E52ro  e52ro* )  (ro  ro ) 2 ( F5 ro  f 5 ro ) ,


*

ro  ( xo , yo ) , ro  ( xo , yo ) , ro  ( yo , xo ) , ro*  ( yo , xo ) ,
(5)
In comparison of Eq. (4) with these expansions of third and fifth order aberrations,
the relationships between the map elements and aberration coefficients can be
established. The total number of relationships is 2  20  40 for the third order
aberration and 2  56  112 for the fifth order one. Among them the simplest set of
relationships has been chosen. Third order aberration coefficients in Glaser’s notation
are
B  M r (1,0300) ,
F  M r (1,1002) ,
C  M r (1,1011) / 2 ,
D  M r (1,0120) ,
E  M r (1,3000) ,
f  M r (3,1002) / 3 ,
c  M r (3,1011) / 2 , e  M r (3,3000) ;
(6)
and the fifth order aberration coefficients in Eq. (5) are
A5  M r (1,0500) ,
B51  M r (1,1004) ,
B52  M r (1,0311) ,
C51  M r (1,1013) ,
C52  M r (1,0320) ,
C53  M r (1,0122)  C52 ,
D51  M r (1,3002) , D52  M r (1,1022)  D51 , D53  M r (1,0131) ,
E51  M r (1,3011) ,
E52  M r (1,0140) ,
F5  M r (1,5000) ,
a5  M r (3,0500) ,
b51  M r (3,1004) ,
b52  M r (3,0311) ,
c51  M r (3,1013) ,
c52  M r (3,0320) ,
c53  M r (3,0122)  c52 ,
d 51  M r (3,3002) ,
d 52  M r (3,1022)  d 51 ,
d 53  M r (3,0131) ,
e51  M r (3,3011) ,
e52  M r (3,0140) ,
f 5  M r (3,5000) .
4
(7)
The DA description in hybrid coordinates takes the form of
x3mi 
y3mi 
x5 mi 
y5 mi 
k  l  m  n 3
M
h
k,l,m,n  0 ,1, 2 , 3
k  l  m  n 3
M
h
k,l,m,n  0 ,1, 2 , 3
k  l  m  n 5
M
h
k,l,m,n  0 ,1,..., 3
k  l  m  n 5
M
h
k,l,m,n  0 ,1,..., 3
(1, klmn) X ok X olYomYon ,
(3, klmn) X ok X olYomYon ,
（8）
(1, klmn) X X o Y Y  ,
k
o
l
m
n
o o
(3, klmn) X ok X olYomYon ,
where suffix " m " in aberration coefficients signifies that the aberrations have been
corrected by the OMI effect [1,6], M h (1,klmn) and M h (3,klmn) are the transfer map
elements in hybrid coordinates, which can be derived from M f (1,klmn) and
M f (3,klmn) . As a result, the simplest set of relationships between the zmap elements
in hybrid coordinates and the third and fifth order aberration coefficients with the
inclusion of the OMI effect has been found.
Third order isotropic aberration coefficients:
Bm  M h (1,0300)  M f (1,0300) cos i  M f (3,0300) sin i ,
Fm  M h (1,1002)  M f (1,1002) cos i  M f (3,1002) sin i ,
Cm  M h (1,1011)  [ M f (1,1011) cos i  M f (3,1011) sin i ]/2 ,
(9)
Dm  M h (1,0120)  M f (1,0120) cos i  M f (3,0120) sin i ,
Em  M h (1,3000)  M f (1,3000) cos i  M f (3,3000) sin i ;
Third order anisotropic aberration coefficients:
f m  M h (3,1002)  [ M f (3,1002) cos i  M f (1,1002) sin i ]/3 ,
cm  M h (3,1011)  [ M f (3,1011) cos i  M f (1,1011) sin i ]/2 ,
(10)
em  M h (3,3000)  M f (3,3000) cos i  M f (1,3000) sin i ;
Fifth order isotropic aberration coefficients:
A5 m  M h (1,0500)  M f (1,0500)co s i  M f (3,0500)si n i ,
B51m  M h (1,1004)  M f (1,1004)co s i  M f (3,1004)si n i ,
B52m  M h (1,0311)  M f (1,0311)co s i  M f (3,0311)si n i ,
C51m  M h (1,1013)  M f (1,1013)co s i  M f (3,1013)si n i ,
C52m  M h (1,0320)  M f (1,0320)co s i  M f (3,0320)si n i ,
C53m  M h (1,0122) - C52m  M f (1,0122)co s i  M f (3,0122)si n i  C52m ,
5
(11)
D51m  M h ( 1 , 3 0 02M) f ( 1 , 3 0 0 2s) icoM f ( 3 , 3 0 0 2n)i s, i
D52m  M h ( 1 , 1 0 2- 2D51
) m  M f ( 1 , 1 0 2 2s) icoM f ( 3 , 1 0 2 2n)i s- iD51m ,
D53m  M h ( 1 , 0 1 31M) f ( 1 , 0 1 3 1s) icoM f ( 3 , 0 1 3 1n)i s, i
(12)
E51m  M h ( 1 , 3 0 11M) f ( 1 , 3 0 1 1s) icoM f ( 3 , 3 0 1 1n)i s, i
E52m  M h ( 1 , 0 1 40M) f ( 1 , 0 1 4 0s) icoM f ( 3 , 0 1 4 0n)i s, i
F5 m  M h ( 1 , 5 0 00M) f ( 1 , 5 0 0 0s) icoM f ( 3 , 5 0 0 0n)i s; i
Fifth order anisotropic aberration coefficients:
a5 m  M h (3,0500)  M f (3,0500)co s i  M f (1,0500)si n i ,
b51m  M h (3,1004)  M f (3,1004)co s i  M f (1,1004)si n i ,
b52m  M h (3,0311)  M f (3,0311)co s i  M f (1,0311)si n i ,
(13)
c51m  M h (3,1013)  M f (3,1013)co s i  M f (1,1013)si n i ,
c52m  M h (3,0320)  M f (3,0320)co s i  M f (1,0320)si n i ,
c53m  M h (3,0122) - c52m  M f (3,0122)cos i  M f (1,0122)si n i  c52m ,
d 51m  M h (3,3002)  M f (3,0300) cos i  M f (1,3002) sin i ,
d 52m  M h (3,1022)  d 51m  M f (3,1022) cos i  M f (1,1022) sin i - d 51m ,
d 53m  M h (3,0131)  M f (3,0131) cos i  M f (1,0131) sin i ,
(14)
e51m  M h (3,3011)  M f (3,3011) cos i  M f (1,3011) sin i ,
e52m  M h (3,0140)  M f (3,0140) cos i  M f (1,0140) sin i ,
f 5 m  M h (3,5000)  M f (3,5000) cos i  M f (1,5000) sin i ;
4. Fifth order aberrations of electromagnetic lenses in electron optics
The basic idea of the derivation of the fifth order aberration coefficients is the same
as that in references [7,8]. In the present work, however, the forms of the aberration
integrals are somewhat different. According to Eq. (5) there are 12 isotropic (uppercase
letters) and 12 anisotropic (lowercase letters) aberration coefficients:
(spherical aberration),
aberration),
A5 , a5
B51 , B52 , b51 , b52 (coma), C51 , C52 , C53 , c51 , c52 , c53 (peanut
D51 , D52 , D53 , d51 , d52 , d53
(elliptical
coma),
E51 , E52 , e51 , e52
(astigmatism and field curvature), and F5 , f 5 (distortion) [9], each of which includes
both intrinsic and combined components. In order to save space we will not list all fifth
order aberration integrals here, but they are available at request. What should be
emphasized is that they have been proved to be correct because the aberration
coefficients calculated through these analytical expressions have been cross-checked
with those computed through the DA technique (see the next section). Nevertheless,
6
following the guidelines in reference [6], the OMI correction formulas of fifth order
aberration coefficients are given here,
A5 m  A5 ,
B51m  B51  5 o a5 ,
B52m  B52  4 o a5 ,
C51m  C51   o (4b51  3b52 )  4 o A5 ,
2
C52m  C52  4 ob51  1 0 o2 A5 ,
C53m  C53  2 ob52  4 o2 A5 ,
D51m  D51  3 oc52  6 o2 B51  1 0 o3 a5 ,
D52m  D52   o (2c51  c53 )  2 o2 (2 B51  B52 ) - 4 o3 a5 ,
（15）
D53m  D53  2 o (c51  c53 )  3 o2 B52 - 4 o3 a5 ,
E51m  E51   o (2d 51  d 53 )   o2 (C51  2C52 )   o3 (4b51  b52 )  4 o4 A5 ,
E52m  E52  2 o d 51  3 o2C52  4 o3b51  5 o4 A5 ,
F5 m  F5   oe52   o2 D51   o3c52   o4 B51   o5 a5 ;
a5 m  a5 ,
b51m  b51  5 o A5 ,
b52m  b52  4 o A5 ,
c51m  c51   o (4 B51  3B52 )  4 o a5 ,
2
c52m  c52  4 o B51  10 o2 a5 ,
c53m  c53  2 o B52  4 o2 a5 ,
d 51m  d 51  3 oC52  6 o2b51  10 o3 A5 ,
d 52m  d 52   o (2C51  C53 )  2 o2 (2b51  b52 )  4 o3 A5 ,
(16)
d 53m  d 53  2 o (C51  C52 )  3 o2b52  4 o3 A5 ,
e51m  e51   o (2 D51  D53 )   o2 (c51  2c52 )   o3 (4 B51  B52 )  4 o4 a5 ,
e52m  e52  2 o D51  3 o2 c52  4 o3 B51  5 o4 a5 ,
f 5 m  f 5   o E52   o2 d 51   o3C52   o4b51   o5 A5 .
5. Computational illustration
The analytical model of the combined bell-shaped electromagnetic lens [10] was
chosen, whose axial potential and magnetic induction distributions have the form of
z
V ( z )  V0 e x pk(a r c t a n) ,
a
k
z
B0 e x p ( a r c t a n)
2
a .
B( z ) 
2
z
1 2
a
(17)
For numerical computation the lens parameters were V0  100 V, k  0.5 , a=0.05 m,
and B0  0.01 T, and the object was properly positioned so that magnification was
equal to -1000. The programs were written in COSY INFINITY [11]. The trajectory
7
equations, Eqs. (1) and (2), were tracked by using an eighth order Runge-Kutta
integrator [12] to produce the transfer map in fixed and rotating coordinates,
respectively. Then, all the third and fifth order aberration coefficients were calculated
in terms of Eqs. (6-7) (without the inclusion of the OMI effect) and Eqs. (9-14) (with
the inclusion of the OMI effect) for the given electromagnetic lens. A cross-check of
these coefficients was made with those evaluated by using the third and fifth order
aberration integrals and Mathematica [13] at the same condition. All the numerical
results are shown in Tables 1 for the third order aberration coefficients and Tables 2-5
for the fifth order aberration coefficients.
6. Discussion and conclusion
In section 2 it is very evident that the general electron trajectory equation in rotating
coordinates, Eq. (2), has not been made simple by the rotating transform of coordinates.
The equations are still coupled with one another, so it is out of use in electron optics.
The simple form of the paraxial trajectory equation can be derived directly from the
general trajectory equation in fixed coordinates [4,5]. However, Eq. (2) does become
most important in the DA technique because it makes the DA description in rotating
coordinates not only very simple, but easy to be extended to higher order aberrations
(compare Eqs. (6-7 with Eqs. (8-9) in reference [1]). Obviously, the DA description
developed in this paper is superior to that in the previous work [1]. As is shown in
references [14,15], the present work has also proved that the DA technique is very
concise, efficient, and precise for electron optical aberration analysis. Tables 1-5 have
demonstrated that the numerical results computed through the DA technique are in
excellent agreement with those calculated by using electron optical aberration integrals
for the third and fifth order aberrations. It is confident that the same precision can be
obtained for higher order aberration analysis and the DA technique will be extensively
applied to electron optics.
Acknowledgements
The author is very grateful for Professor M. Berz to provide COSY INFINITY 8.1
for this work.
8
References
[1] Z. Liu, The Seventh International Computational Accelerator Physics,
Michigan State University, East Lansing, MI 48824, USA, October 22-25, 2002.
[2] M. Berz, Nucl. Instr. and Meth. A 298 (1990) 473.
[3] K. Makino, M. Berz, Nucl. Instr. and Meth. A 427 (1999) 338.
[4] P.W. Hawkes, E. Kasper, Principles of Electron Optics, Academic Press,
London, 1989.
[5] J. Ximen, Aberration Theory in Electron and Ion Optics, Adv. in Electronics and
Electron Phys. Supplement 17, Academic Press, New York, 1986.
[6] J. Ximen, Z. Liu, Optik, 111 (2000) 355.
[7] M. Wu, Acta Phys. Sinica, 13 (1957) 181.
[8] P.W. Hawkes, Phil. Trans. A 257 (1965) 523.
[9] Z. Liu, Nucl. Instr. and Meth. A 488 (2002) 42.
[10] J. Ximen, Z. Liu, Optik, 111 (2000) 75.
[11] M. Berz. COSY INFINITY Version 8, User's Guide and Reference Manual,
Technical Report MSUCL-1173, National Superconducting Cyclotron Laboratory,
Michigan State University, East Lansing, 2000.
[12] B. Hartmann, M. Berz. H. Wollnik, Nucl. Instr. and Meth. A 297 (1990) 343.
[13] S. Wolfram, The MATHEMATICA book, Third edition,
Cambridge University Press, Cambridge, 1996.
[14] L. Wang, T. Tang, B. Cheng, J. Cai, Optik, 110 (1999) 408.
[15] M. Cheng, T. Tang, Z. Yao, Optik, 112 (2001) 250.
9
Table 1
Third order aberration coefficients without and with the inclusion of the object
magnetic immersion effect for a combined bell-shaped electromagnetic lens with
B0  0.01 T, V0  100 V, a  0.05 m, and K  0.5 and under the magnification of
-1000. Note: The value in the second column is chosen as the criterion in the relative
error.
--------------------------------------------------------------------------------------------------------Aberr. coeff.
DA technique
Aberr. integral
Relative error
--------------------------------------------------------------------------------------------------------B (m)
2.85456035214 10 -1
2.85456035210 10 -1
1.235 10 11
F
8.25795388659
8.25795378007
1.290 10 8
C (m -1 )
2.37432784958 10 2
2.37432785020 10 2
-2.616 10 10
D (m -1 )
2.71994620417 10 2
2.71994620479 10 2
-2.286 10 10
E ( m -2 )
7.63653380978 10 3
7.63653380975 10 3
3.453 10 12
f
6.49587925428 10 -1
6.49587925425 10 -1
4.701 } 10 12
c (m -1 )
3.48851564964 101
3.48851564963 101
2.182 10 12
e ( m -2 )
5.95569993408 10 2
5.95569993406 10 2
3.803 10 12
B m (m)
2.85456035214 10 -1
2.85456035210 10 -1
1.235 10 11
Fm
8.25795388659
8.25795378007
1.290 10 8
C m (m -1 )
2.38466506135 10 2
2.38466506197 10 2
-2.609 10 10
D m (m -1 )
2.68893456886 10 2
2.68893456948 10 2
-2.301 10 10
E m ( m -2 )
7.60940292147 10 3
7.60940292134 10 3
1.772 10 11
fm
3.56205732871 10 -1
3.56205732872 10 -1
c m (m -1 )
1.79106573203 101
1.79106575391 101
e m ( m -2 )
3.17771580676 10 2
3.17771580610 10 2
10
-2.252 10 12
-1.222 10 8
2.068 10 10
Table 2
Fifth order isotropic aberration coefficients without the inclusion of the object
magnetic immersion effect for a combined bell-shaped electromagnetic lens under the
same conditions as those in Table 1.
--------------------------------------------------------------------------------------------------------Aberr. coeff.
DA technique
Aberr. integral
Relative error
--------------------------------------------------------------------------------------------------------2.213 10 8
A5 (m)
4.11524978819
4.11524969711
B51
1.17392977649 10 2
1.17392979463 10 2
-1.545 10 8
B52
5.12551884960 10 2
5.12551888575 10 2
-7.053 10 9
C51 (m -1 )
1.45629574927 10 4
1.45630343531 10 4
-5.278 10  6
C52 (m -1 )
8.29257364254 10 3
8.29261191267 10 3
-4.615 10  6
C53 (m -1 )
1.56399264890 104
1.56399266852 104
-1.254 10 8
D51 ( m -2 )
2.34517151844 10 5
2.34517151695 10 5
6.353 10 10
D52 ( m -2 )
4.42630983925 10 5
4.42630983684 10 5
5.445 10 10
D53 ( m -2 )
4.98640409322 10 5
4.98642487358 10 5
-4.167 10  6
E51 ( m -3 )
1.40423272443 107
1.40423214128 107
4.153 10 7
E52 ( m -3 )
3.91684999270 10 6
3.91684871207 10 6
F5 ( m -4 )
1.09746204513 108
1.09746204599 108
11
3.270 10 7
-7.836 10 10
Table 3
Fifth order anisotropic aberration coefficients without the inclusion of the object
magnetic immersion effect for a combined bell-shaped electromagnetic lens under the
same conditions as those in Table 1.
--------------------------------------------------------------------------------------------------------Aberr. coeff.
DA technique
Aberr. integral
Relative error
---------------------------------------------------------------------------------------------------------
a5 (m)
1.85428793710 10 -1
1.85428793712 10 -1
-1.079 10 11
b51
4.32709988901 101
4.32709984297 101
1.064 10 8
b52
-1.02433684537 101
-1.02433681910 101
2.565 10 8
c51 (m -1 )
4.23204041645 10 3
4.23204611502 10 3
-1.347 10  6
c52 (m -1 )
-5.19729347955 10 2
-5.19729348120 10 2
-3.175 10 10
c53 (m -1 )
-1.29834906864 10 3
-1.29834909761 10 3
-2.231 10 8
d 51 ( m -2 )
5.36807062376 104
5.36808655466 104
-2.968 10  6
d 52 ( m -2 )
9.67804447458 104
9.67804536999 104
-9.252 10 8
d 53 ( m -2 )
-6.21228596368 104
-6.21228730784 104
-2.164 10 7
e51 ( m -3 )
2.26018791456 10 6
2.26018791478 10 6
-9.734 10 11
e52 ( m -3 )
-6.48270015136 10 5
-6.48270014473 10 5
1.023 10 9
f 5 ( m -4 )
1.17037727110 107
1.17037593832 107
1.139 10  6
12
Table 4
Fifth order isotropic aberration coefficients with the inclusion of the object magnetic
immersion effect for a combined bell-shaped electromagnetic lens under the same
conditions as those in Table 1.
--------------------------------------------------------------------------------------------------------Aberr coeff.
DA technique
Aberr integral
Relative error
--------------------------------------------------------------------------------------------------------2.213 10 8
A 5m (m)
4.11524978819
4.11524969711
B51m
1.18345865298 102
1.18345867112 102
-1.533 10 8
B52m
5.11789574840 102
5.11789578456 102
7.065 10 9
C51m ( m-1 )
1.46918762494 104
1.46919531091 104
C52m ( m-1 )
8.15815324433 10 3
8.15819151539 10 3
-4.691 10  6
C53m ( m-1 )
1.56435942500 104
1.56435944461 104
-1.253 10 8
D51m ( m -2 )
2.33660699683 10 5
2.33660699545 10 5
D52m ( m -2 )
4.48416046488 10 5
4.48416057916 10 5
D53m ( m -2 )
4.92633057337 10 5
4.92635123671 10 5
-4.194 10  6
E51m ( m -3 )
1.40868442909 107
1.40868387734 107
3.917 10 7
E52m ( m -3 )
3.83262100944 10 6
3.83261952262 10 6
3.879 10 7
F5 m ( m -4 )
1.09327222645 108
1.09327222732 108
13
-5.231 10  6
5.887 10 10
-2.548 10 8
-7.967 10 10
Table 5
Fifth order anisotropic aberration coefficients with the inclusion of the object magnetic
immersion effect for a combined bell-shaped electromagnetic lens under the same
conditions as those in Table 1.
--------------------------------------------------------------------------------------------------------Aberr coeff.
DA technique
Aberr integral
Relative error
--------------------------------------------------------------------------------------------------------am (m)
1.85428793710 10 -1
1.85428793712 10 -1
-1.079 10 11
b51m
2.21234164808 101
2.21234164886 101
-3.504 10 10
b52m
6.67469747375
6.67469736192
c51m ( m-1 )
2.16829539214 10 3
2.16830107210 10 3
-2.620 10  6
c52m ( m-1 )
-3.51603069621 101
-3.51602996698 101
2.074 10  7
c53m ( m-1 )
-2.45565079762 102
-2.45565101297 102
-8.770 10 8
d 51m ( m -2 )
2.83417818669 104
2.83418231757 104
-1.458 10  6
d 52m ( m -2 )
5.06282873816 104
5.06281381460 104
2.948 10  6
d 53m ( m -2 )
-1.51571440920 104
-1.51569208785 104
1.473 10 5
e51m ( m -3 )
1.27014537030 10 6
1.27014324111 10 6
1.676 10  6
1.675 10 8
e52m ( m -3 )
-1.67348360027 10 5
-1.67348359663 10 5
2.174 10 9
f 5m ( m -4 )
7.72590894110 10 6
7.72589705622 10 6
1.538 10  6
14