xmpl2d08, 8th 'example' data file for CPO2D

Lens coefficients of double-cylinder lens.

This file is identical to the 1st example file, xmpl2d01.dat, except that various 'lens coefficients' are asked for, including the linear and angular magnifications and the spherical and chromatic aberrations. For the analogous 3D example see xmpl3d32.dat.

Initially the following lens coefficients are asked for:

coefficients, aberration - definition of rays

0 -4 0 1 1 r,z,vr,vz,eV,I of median ray

0.05 0 0 0.01 0.05 Increments for initial parameters

1 0 0 0 0 Choice of coefficients

The first line defines the starting conditions of the median ray. It starts at z = -4 and is directed along the z axis with an initial energy of 1 eV. The next line defines some increments (not all of which are necessarily used, depending on the next line). The initial distance r from the axis is

incremented or stepped by 0.05. The the angle a_z to the z axis is incremented by 0.01 radians and the energy is incremented by 0.05.

The first 'test plane' is z=2.0274, which is the gaussian image plane. This value was obtained in a preliminary run, using a set of rays that start at (r,z) = (0,-4) with vz = 1 and vr = 0.01, 0.02 and 0.03, and with the test plane at r = 0. The output results are then the z values, zc say, of the ray

as it crosses the axis in image space. Then the 'Origin' package is used to plot zc versus vr^2 and to extrapolate linearly to vr = 0, which gives the required z of the gaussian image plane, z = 2.0274. It is important to follow this procedure to obtain an accurate value. Otherwise the value of Cs obtained below would be inaccurate.

The 'choice of coefficients' has given an order of 1 for the initial distance r from the axis, and so the program will find the variation of each of 4 final parameters (r, z, a_r, a_z) at the test plane, with respect to r_initial. In other words the program will give the coefficient c(r,r) in the expansion

r_final - r_final_median = c(r,r)*(r_initial - r_initial_median) +...

and the coefficient c(r,z) in the expansion

z_final - z_final_median = c(z,r)*(r_initial - r_initial_median) +...

and the analogous coefficients for the other 2 final parameters. For example, c(r,r) is the differential of r_final with respect to r_initial, which is the linear magnification M of the lens.

For another explanation of the meaning of the coefficients see the note on coefficients.

The output lines are:

Initial Coefficients for:

parameters: r z angle_r angle_z

r -3.378E-01 0.000E+00 3.965E-01 -3.965E-01

Therefore M = -3.378E-01, which is close to the value -0.336 derived from the lens parameters produced by xmpl2d01.dat.

Another 'choice of coefficients' is:

coefficients, aberration - definition of rays

0 -4 0 1 1 r,z,vr,vz,eV,I of median ray

0.05 0 0 0.01 0.05 Increments for initial parameters

0 0 0 3 0 Choice of coefficients

Here there is an order of 3 for the angle a_z, and so the program will find the variation of each of 4 final parameters (r, z, a_r, a_z) at the test plane, with respect to a_z cubed. That is, it will give the coefficient c(r,a_z*a_z*a_z) in the expansions

r - r_median = c(r,a_z*a_z*a_z)*a_z*a_z*a_z +...

etc. For example the coefficient c(r,a_z*a_z*a_z) is the third order differential of r_final with respect to a_z_initial, which is Cs*M, where Cs is the third order spherical aberration coefficient and M is the linear magnification of the lens. (Note that this is the aberration coefficient referred to the object -it can be converted to the coefficient referred to the image by using the usual formulae, given for example in E Harting and F H Read, Electrostatic Lenses, Elsevier 1976 -copies of the text (non-data) pages of this book are available in the ‘document’ folder of the CPO package.)

The output lines are:

Initial Coefficients for:

parameters: r z angle_r angle_z

a_z a_z a_z -5.490E+01 3.811E-04 1.072E+03 -1.072E+03

The first result given here is effectively Cs*M = -5.490E+01, therefore Cs = 162.4. This is close to the value 162.1 quoted in the footnotes of xmpl2d01.dat and obtained from the calculated coefficients Csn, n = 1 to 4. The other coefficients here are not useful.

Another 'choice of coefficients' is:

coefficients, aberration - definition of rays

0 -4 0.01 1 1 r,z,vr,vz,eV,I of median ray

0.05 0 0 0.01 0.05 Increments for initial parameters

0 0 0 0 1 Choice of coefficients

Here the median ray now starts at an angle a_z = 0.01 to the axis and there is an order of 1 for the energy en. Therefore the program will find the variation of each of 4 final parameters with respect to en. The coefficient c(r,en) is related to the chromatic aberration coefficient Cc.

The radial distance at the gaussian image plane is

dr = M*Cc*a_z*en/E

where E is the energy of the median ray. Therefore the coefficient obtained here is

c(r,en) = dr/en = M*Cc*a_z/E.

Therefore

Cc = c(r,en)*E/(M*a_z).

Note that by using the median ray specified above and varying only the energy, we remove any contribution to dr caused by spherical aberration.

The output lines are:

Initial Coefficients for:

parameters: r z angle_r angle_z

en 2.183E-02 0.000E+00 -6.963E-03 6.963E-03

The first result given here is effectively c(r,en) = 2.183E-02. Therefore Cc = 6.46. This is close to the value 6.63 quoted in xmpl2d01.dat.

The errors found above can be reduced by using higher charge and ray accuracies, but it is difficult to obtain very accurate lens coefficients due to the inherent errors in integrating paraxial rays. There is always a conflict between using small increments for the initial parameters, in which case numerical errors might be significant, and using large increments, when higher order effects might be present. The user should always carefully explore the dependence on these increments.