Lens aberrations.

(Taken from the note on lens properties).

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

See also the copies of the text pages of Harting and Read in the ‘document’ folder of the CPO package.

The quantities f1 and f2 are the focal lengths (for the object and image spaces respectively) and F1 and F2 are the mid-focal lengths, which are the distances of the principal focal points from the reference plane z = 0 (which is usually the mid-point of the lens). F1 is positive if the first principal focus has z < 0:

(1) The object distance P (which is taken as positive if the object is at z < 0) and the image distance Q (which is taken as positive if the object is at z > 0) are related by

(P - F1)*(Q - F2) = f1*f2.

(2) The linear magnification is

M = -f1/(P - F1).

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. 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 (copies of the text (non-data) pages of this book are available in the ‘document’ folder of the CPO package), 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 for a real image and no cross-overs inside the lens. The quantities Csn are given by the program.

A 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 coefficients 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),

where

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,

where

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.

Figures of Merit

At the highest printing level CPO2D also outputs the three ‘figures of merit’ g1, g2 and g3, defined by Harting and Read (see publications or see the copies of the text (non-data) pages of this book in the ‘document’ folder of the CPO package). We give here a summary of the definitions.

The three figures of merit g1, g2 and g3 apply to M = 0, 1 and infinity respectively, where M is the linear magnification. The figures of merit are dimensionless and are therefore independent of the actual size of the lens. In general the best geometry of lens is that for which the appropriate figure of merit is smallest.

Figure of merit g1, for M = 0:

If a parallel beam of diameter d is focussed at a distance L from a lens, then the radius of the spherical aberration disc at the gaussian image position is g1*d**3/(8*L**2).

Figure of merit g2, for M = 1:

If a beam emanates from a point axial source with a half-angle a and is brought to a focus at a distance L from the source, with unit linear magnification, then the radius of the spherical aberration disc is g2*L*a**3.

Figure of merit g3, for M = infinity:

If a beam emanates from a point axial source with a half-angle a and is brought to a focus at a very large distance L from the source, then the radius of the spherical aberration disc is g3*L*a**3.

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.

For the coefficients of astigmatism, field curvature, coma and distortion see note on aberrations).