=================================================================================
Under the isoplanatic approximation as well as Taylor expansion, the wave aberration function, W, using the notation provided by Krivanek [1], can be concisely expanded to sixth order in k about the origin of zero scattering angle in terms of the coherent aberration coefficients, given by,
 [3740a]
The first subscript refers to the order of the coefficient in terms of real space displacement, while the second subscript describes the angular symmetry. If the vector k is replaced by a complex number ω = λk_{x} + iλk_{y}, the wave aberration function can be rewriten by,
 [3740b]
The wave aberration function can also be expanded in polar notation. The following equation shows the terms to fourth order,
 [3740c]
The other way of presenting the wave aberration function at the back focal plane of a lens is based on polar angular coordinates θ and φ, given by,
 [3740d]
The sum over m is taken from 0 (or 1) to n+1 for each order n with the additional constraint that m+n is odd.
Table 3740a lists the aberration coefficient nomenclature, their order (of both the ray deviation (N) and the wavefront), and radial (azimuthal) symmetry. This table also compares both commonly used notations [1, 2]. The spherical aberration (highlighted in orange in Table 3740a) is always real and are usually the most significant among all the aberrations, while the offaxial coma (highlighted in green in Table 3740a) is complex in the presence of an axial magnetic field. Table 3740b also lists more different methods of axial aberration notations presented in different publications.
Table 3740a. Aberration Coefficient Nomenclature. The aberration coefficients have two
main types of notations, namely Krivanek notation, and Typke and Dierksen notation.
Krivanek notation 
Haider[8] 
Radial Order 
Value 
Azimuthal Symmetry 
Wave aberration 
Nomenclature 
Ray 
Wave (k) 
C_{0,1} 

0 
1 
Complex 
1 

Beam/image Shift 
C_{1,2} 
A_{1} 
1 
2 
Complex 
2 

Twofold axial astigmatism (or axial astigmatism of the 1st order) 
C_{1,0} 
C_{1} 
1 
2 
Real 
0, ∞ 

Defocus (overfocus positive, or spherical aberration of the 1st order; Real numbers and describing rotationally symmetric contributions to the wave aberration) (alt: Δf) 
C_{2,3} 
A_{2} 
2 
3 
Complex 
3 

Threefold axial astigmatism (or axial astigmatism of the 2nd order) 
C_{2,1} 
B_{2} 
2 
3 
Complex 
1 

Secondorder axial coma 
C_{3,4} 
A_{3} 

4 
Complex 
4 

Fourfold axial astigmatism or axial astigmatism of the 3rd order C_{s} 
C_{3,2} 
S_{3} 

4 
Complex 
2 

Twofold astigmatism of C_{s} (or Third order twofold astigmatism, or Axial star aberration of the 3rd order) 
C_{3,0} 
C_{3} 

4 
Real 
0, ∞ 

Thirdorder spherical aberration (always positive for round lenses [3]; Real numbers and describing rotationally symmetric contributions to the wave aberration) (alt: C_{s} ) 
C_{4,5} 
A_{4} 

5 
Complex 
5 

Fivefold axial astigmatism or axial astigmatism of the 4th order 
C_{4,1} 
B_{4} 

5 
Complex 
1 

Fourthorder axial coma 
C_{4,3} 
D_{4} 
4 
5 
Complex 
3 

Fourth order threefold astigmatism (or Three lobe aberration) 
C_{5,4} 
R_{5} 
5 
6 
Complex 
4 

Fourfold astigmatism of C_{5 }(or Fifth order rosette aberration) 
C_{5,2} 
S_{5} 
5 
6 
Complex 
2 

Twofold astigmatism of C_{5} (or Fifthorder axial star aberration) 
C_{5,0} 
C_{5} 

6 
Real 
0, ∞ 

Fifthorder spherical aberration 
C_{5,6} 
A_{5} 

6 
Complex 
6 

Sixfold axial astigmatism or sixfold axial astigmatism of the 5th order 
C_{6,1} 
B_{6} 


Complex 
1 

Sixth order axial coma 
C_{6,3} 
D_{6} 


Complex 
3 

Sixth order threelobe aberration 
C_{6,5} 
F_{6} 


Complex 
5 

Sixth order pentacle aberration 
C_{6,7} 
A_{6} 


Complex 
7 

Sevenfold astigmatism 
C_{7,0} 
C_{7} 


Real 
0 

Seventhorder spherical aberration 
C_{7,2} 
S_{7} 


Complex 
2 

Seventhorder star aberration 
C_{7,4} 
R_{7} 


Complex 
4 

Seventhorder rosette aberration 
C_{7,6} 
G_{7} 


Complex 
6 

Seventhorder chaplet aberration 
C_{7,8} 
A_{7} 


Complex 
8 

Eightfold astigmatism 
The indices n and k denote the order and azimuthal symmetry of the aberration C_{n,k}. In the equations above, Factors 1/(n+1) are used for all contributions to the nth order. [4] The aberration coefficients C_{n,k} are complex
numbers denoting the two Cartesian components except
C_{1,0}(C_{1}), C_{3,0}(C_{3}), and C_{5,0} that are real numbers. The aberration coefficients C_{1} and C_{3} describe rotationally symmetric contributions to the wave aberration. The complex coefficients involve both the strength and direction of the aberration. In the list in Table 3740a only the spherical aberration, C_{3}, is completely unavoidable. C_{1} is almost always present since it is used to reduce the effects of the spherical aberration by slightly underfocusing the objective lens (corresponding to a negative C_{1}). Without aberration corrections, the other aberrations usually are left out since they are much smaller then the two main aberrations C_{1} and C_{3}. The wave aberration function can be separated into a symmetric part W_{s} with W_{s} = W_{s}, e.g. containing the terms in A_{1}, C_{1} and C_{3} and an antisymmetric part W_{a} with W_{a} = W_{a}, e.g. containing the terms in A_{2} and B_{2}.
Table 3740b. Different methods of axial aberration notations presented in different publications.
Krivanek[1] 
Typke and Dierksen[2] 
Sawada [5]* 
Zemlin [6] 
Isizuka[7] 
Haider[8] 
Pöhner and Rose 
Thust[9] 
C_{0,1} 
A_{0} 




A_{0} 

C_{1,2} 
A_{1} 
A_{2} 
a_{2} 
a_{2} 
A_{1} 
A_{1} 
C_{22} 
C_{1,0} 
C_{1} 
O_{2} 
Δ 
z 
C_{1} 
C_{1} 
C_{20} 
C_{2,3} 
A_{2} 
A_{3} 
a_{3} 
a_{3} 
A_{2} 
A_{2} 
C_{33} 
(1/3)C_{2,1} 
B_{2} 
P_{3} 
b 
b 
B_{2} 
(1/3)B_{2} 
C_{31} 
C_{3,4} 
A_{3} 
A_{4} 


A_{3} 

C_{44} 
(1/4)C_{3,2} 
B_{3} 
Q_{4} 


S_{3} 
(1/4)B_{3} 
C_{42} 
C_{3,0} 
C_{3} 
O_{4} 
C_{s} 
C_{s} 
C_{3} 
C_{3} 
C_{40} 
C_{4,5} 
A_{4} 
A_{5} 


A_{4} 

C_{55} 
(1/4)C_{4,1} 
B_{4} 
P_{5} 


B_{4} 

C_{51} 
(1/4)C_{4,3} 
D_{4} 
R_{5} 


D_{4} 
(1/6)D_{5} 
C_{53} 
(1/6)C_{5,4} 
R_{5} 



R_{5} 


(1/6)C_{5,2} 
S_{5} 



S_{5} 
(1/6)B_{5} 

C_{5,0} 
C_{5} 
O_{6} 


C_{5} 
C_{5} 
C_{60} 
C_{5,6} 
A_{5} 
A_{6} 


A_{5} 
A_{5} 
C_{66} 

D_{5} 






(1/7)C_{6,1} 




B_{6} 


(1/7)C_{6,3} 




D_{6} 


(1/7)C_{6,5} 




F_{6} 


C_{6,7} 




A_{6} 


C_{7,0} 




C_{7} 


(1/8)C_{7,2} 




S_{7} 


(1/8)C_{7,4} 




R_{7} 


(1/8)C_{7,6} 




G_{7} 


C_{7,8} 




A_{7} 


* "A" denotes the coefficient of the primary aberration, which is introduced by simple multipole field; "O" denotes round symmetry; "P" denotes onefold symmetry; "Q" denotes towfold symmetry; "R" denotes threefold symmetry; "S" denotes fourfold symmetry. The suffix number of the aberration coefficient corresponds to the order of the wave aberrations.
Figure 3740 shows the aberration coefficients from 0^{th} to 5^{th} order. The vertical lines indicate the groups of different angular symmetries, while the colors of the schematic illustrations in the same rows represent the order of the aberration coefficients. Corresponding to Equation 3740d, in Figure 3740 all the schematic illustrations represent the distortions of the electron wavefronts induced by axial geometric aberrations. The n^{th}order aberrationinduced wavefront distortion on the diffraction plane increases with the distance (δ'_{r}) from the optical axis by (n+1)^{th} power (namely (δ'_{r})^{n+1}) or increases with the angle (θ) by (n+1)^{th} power (namely θ^{n+1}) in Equation 3740d, and the corresponding ray is displaced by n^{th} power with distance δr (See Figure 3752a) on the image plane (namely δ^{n}_{r}). The m index indicates the wavefront distortion repeats the maxima and mina m times for C_{n,m} when the coordinate system rotates for 360 degrees. The subscripts a and b represent that for the cases m > 0, the azimuthal variation of the aberrations have two orthogonal components, rotated by π/2m between C_{n,m,a} and C_{n,m,b}.
Figure 3740 shows the aberration coefficients from 0^{th} to 5^{th} order.
[1] Krivanek, O. L., Dellby, N., and Lupini, A. R. (1999). Towards subÅ electron beams.
Ultramicroscopy 78, 1.
[2] Typke, D., and Dierksen, K. (1995). Determination of image aberrations in highresolution
electron microscopy using diffractogram and cross correlation methods. Optik 99, 155.
[3] O. Scherzer, J. Appl. Phys. 20 (1949) 20.
[4] Saxton,W.O. (2000). A new way of measuring microscope aberrations. Ultramicroscopy 81, 41–45.
[5] H. Sawada, F. Hosokawa, T. Kaneyama, T. Tomita, Y. Kondo, T. Tanaka, Y.
Oshima, Y. Tanishiro, K. Takayanagi, Proc. Microsc. Microanal. 13 (2007) 880.
[6] F. Zemlin, K. Weiss, P. Schiske,W. Kunath, K.H. Herremann, Ultramicroscopy 3
(1978) 46.
[7] K. Ishizuka, Ultramicroscopy 55 (1994) 407.
[8] M. Haider, S. Uhlemann, J. Zach, Ultramicroscopy 81 (2000) 163.
[9] A. Thust, J. Barthel, L. Houben, C.L. Jia, M. Lentzen, K. Tillmann, K. Urban, Proc.
Microsc. Microanal. 11 (2005) 58.
