Aberration coefficients.


Some of this information is also give in the note on aberration formulas.



Most of the aberration coefficients given below are output by the cpo2d program when the printing level is sufficiently high.  They are not output by the cpo3d program, but an alternative is given by the lens coefficients option.


Aberration coefficients that are ‘referred to the object’.


The third order 'spherical aberration coefficient' Cs is defined by

dr = M*Cs*alpha**3,

where dr is the value of r at the gaussian image plane for an axial point object that emits a ray at an angle alpha and where ** signifies ‘to the power of’. This coefficient can be expanded in terms of the linear magnification M -see E Harting and F H Read, Electrostatic Lenses, Elsevier Publ. Co., 1976, or P W Hawkes and E Kasper, Principles of Electron Optics (Academic Press, 1989):

Cs = Cs0 + Cs1/M + Cs2/M**2 + Cs3/M**3 + Cs4/M**4,

where M is negative when the image is real (and no cross-overs inside the lens). The quantities Csn are given by the program.


One further quantity is needed to provide a complete specification of all the third-order aberrations of an electrostatic lens. This is the Petzval integral (see Hawkes and Kasper) and is labelled 'pzv' in the program output. The general aberration ceofficients i1 to i6, defined by Hawkes and Kasper) are related to the coefficients Csn and the Petzval integral as follows:

i1 = u*Cs4, i2 = -(u/8)*((f1**3/f2)+2*Cs3),

i3+i4 = u*Cs2/2, i5 = -(u/8)*(f1+2*Cs1),

i6 = u*Cs0, i4-2*i3 = ptz/f1**2,

where u = (1/f1)**4.


The 'chromatic aberration coefficient' Cc is defined by

dr = -M*Cc*alpha*(dphi/phi),

where dphi/phi is the fractional spread in the initial energy of the electrons, and dr is added to the dr given above for spherical aberration. For relativistic motion phi is replaced by the 'acceleration potential'

phi_star = phi*(1 + epsilon*phi),


epsilon = e/(2*m0*c**2) = 9.7847E-7,

and similarly for dphi. This aberration coefficient can also be expanded in terms of M (see L A Baranova et al, Sov Phys Tech Phys, 34, 1409, 1989, or Hawkes and Kasper):

Cc = Cc0 + Cc1/M + Cc2/M**2.


Another relevant coefficient is the 'chromatic aberration of magnification' (see Hawkes and Kasper) defined by

dr = -M*Cd*r1*(dphi/phi),

where r1 is the initial off-axis distance. Expanding in terms on M:

Cd = Cd0 + Cd1/M,


Cd0 = -Cc1/(2*f1) + Ce,

Cd1 = -Cc2/f1,

Ce = 0.25*phi_star1*(gamma2/phi_star2-gamma1/phi_star1),

gamma = 1 + 2*epsilon*phi,

and labels 1 and 2 refer to object and image space respectively. It is therefore straightforward to evaluate Cd0 and Cd1, which are not given by the program.


Examples of these aberration coefficients are given in the files xmpl2d01.dat, xmpl2d02.dat and xmpl2d13.dat.



At the highest printing level CPO2D also outputs the three ‘figures of merit’ g1, g2 and g3, defined by Harting and Read (see publications). For the definitions, see lens aberrations.



The aberration coefficients that are ‘referred to the image’:


We start by re-writing the first equation above as

dr = M*Cso*alphao**3,

where the spherical aberration coefficient Cso is referred to the object and alpha0 is the angle at the object. We then write

dr = Csi*alphai**3,

where Csi is referred to the image and alphai is the angle at the image. Using the Helmholtz⌀Lagrange relationship for small angles

alpha0/alphai = M*sqrt(Ei/Eo),

where Eo and Ei are the particle energies at the object and image respectively, we obtain

Csi = Cso*M**4*(Ei/Eo)**(3/2).

Similarly for the chromatic aberration coefficient

Cci = Cco*M**2*(Ei/Eo)**(3/2).



Non-axial aberration coefficients.


We use the coefficients B, C, D and F defined by P. Grivet, Electron Optics (Pergamon Press, 1965), which are the coefficients of astigmatism, field curvature, coma and distortion respectively. It can be shown that these are related to the expansion coefficient Cs0 to Cs4 defined above, as follows (see E. Harting and F. H. Read, Electrostatic lenses, Elsevier, 1976).

B + C = 0.5*M**4*f1**(-3)*(6*Cs0 + 3*Cs1/M + Cs2/M**2 1.5*f1/M).

D = -(M**4/12*f1**3)*(12*Cs0 + 9*Cs1/M + 6*Cs2/M**2 + 3*Cs3/M**3 1.5*f1/M + 1.5*f1**3/(f2**2*M**3)).

F = -(M**4/4*f1**3)*(4*Cs0 + Cs1/M 1.5*f1/M).

Grivet’s coefficient A is related to Cs by

A = Cs*M**4/f1**3


Further information that can be output.


For the purposes of the paper published in the CPO9 proceedings in 2015, further data are printed, the trigger for which is P < -1E10:

Q,M and object-related Cs, Cc, Cs4/f1^3 and Cc2/f1.



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