Xmpl3d32, 32nd 'example' data file for CPO3D

A double-cylinder lens, parameters and coefficients.

This is a 2-dimensional problem and so can be treated much more accurately using CPO2D -see xmpl2d08.

Here we concentrate on the use of the 'lens coefficients' option to obtain the lens parameters.

The segments are smallest near to the gap. The numbers of subdivisions are chosen so that all the segments near to the gap carry approximately the same charge.

USE OF 'LENS COEFFICIENTS' OPTION TO OBTAIN LENS PARAMETERS.

This file is the 3D analogue of xmpl2d08.dat (and the 2D simulation is of course more accurate).

Here the kinetic energy option must be used.

Also the 'set of single rays' option is replaced by the 'lens coefficients' option.

The first 'test plane' is z=2.0289, which is the gaussian image plane. This value was obtained in a preliminary run, using a set of rays that start at (x,y,z) = (0,0,-4) with vx = 0.01, 0.02 and 0.03 and vy = 0, vz = 1, and with the test plane at x = 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 vx^2 and to extrapolate linearly to vx = 0, which gives the required z of the gaussian image plane, z = 2.0289.

It is important to follow this procedure to obtain an accurate value. Otherwise the value of Cs obtained below would be inaccurate.

The first choice of parameters is:

coefficients, aberration - definition of rays

0 0 -4 0 0 1 1 x,y,z,vx,vy,vz,eV,I of median ray

0.03 0 0 0.03 0 0 0.03 increments for initial parameters

1 0 0 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 x from the axis is incremented or stepped by 0.03. The the angle a_x to the z axis is incremented by 0.03 radians and the energy is incremented by 0.03.

The 'choice of coefficients' has given an order of 1 for the initial distance x from the axis, and so the program will find the variation of each of 6 final parameters (x,y,z,a_x,a_y,a_z) at the test plane, with respect to x_initial. In other words the program will give the coefficient c(x,x) in the expansion

x_final - x_final_median = c(x,x)*(x_initial - x_initial_median) +...

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

z_final - z_final_median = c(x,r)*(x_initial - x_initial_median) +...

and the analogous coefficients for the other 4 final parameters. For example, c(x,x) is the differential of x_final with respect to x_initial, which is the linear magnification M of the lens in the x direction.

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

The output lines are:

Initial parameters (in digital and word forms),

followed by coefficients for final x, y, z, cosx, cosy, cosz:

1 0 0 0 0 0 0 (that is, x )

-3.364E-01 0.000E+00 0.000E+00 -3.959E-01 0.000E+00 -2.351E-03

Therefore M = -3.364E-01, which is close to the value -0.3355 derived from the lens parameters produced by xmpl2d01.dat.

Another 'choice of coefficients' is:

coefficients, aberration - definition of rays

0 0 -4 0 0 1 1 x,y,z,vx,vy,vz,eV,I of median ray

0.03 0 0 0.03 0 0 0.03 increments for initial parameters

0 0 0 3 0 0 0 choice of coefficients

Here there is an order of 3 for the angle a_x, and so the program will find the variation of each of 6 final parameters at the test plane, with respect to a_x cubed. That is, it will give the coefficient c(x,a_x*a_x*a_x) in the expansions

x - x_median = c(x,a_x*a_x*a_x)*a_x*a_x*a_x +...

etc. For example the coefficient c(x,a_x*a_x*a_x) is the third order differential of x_final with respect to a_x_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 parameters (in digital and word forms),

followed by coefficients for final x, y, z, cosx, cosy, cosz:

0 0 0 3 0 0 0 (that is, cosx cosx cosx)

-5.483E+01 0.000E+00 7.631E-04 -1.071E+03 0.000E+00 -1.550E+01

The first result given here is effectively Cs*M = -5.483E+01, therefore Cs = 163.0. 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 0 -4 0.03 0 1 1 x,y,z,vx,vy,vz,eV,I of median ray

0.03 0 0 0.03 0 0 0.03 increments for initial parameters

0 0 0 0 0 0 1 choice of coefficients

Here the median ray now starts at an angle a_x = 0.03 to the axis and there is an order of 1 for the energy en. Therefore the program will find the variation of each of 6 final parameters with respect to en.

The coefficient c(x,en) is related to the chromatic aberration coefficient Cc. The radial distance at the gaussian image plane is

dx = M*Cc*a_x*en/E

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

c(x,en) = dx/en = M*Cc*a_x/E.

Therefore

Cc = c(x,en)*E/(M*a_x).

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

The output lines are:

Initial parameters (in digital and word forms),

followed by coefficients for final x, y, z, cosx, cosy, cosz:

0 0 0 0 0 0 1 (that is, en )

6.604E-02 0.000E+00 0.000E+00 2.098E-02 0.000E+00 6.004E-04

The first result given here is c(x,en) = 6.604E-02. Therefore Cc = 6.54. This is close to the value 6.63 quoted in xmpl2d01.dat.

Finally we consider a 'mixed' spherical aberration coefficient, using the input lines:

coefficients, aberration - definition of rays

0 0 -4 0 0 1 1 x,y,z,vx,vy,vz,eV,I of median ray

0.03 0 0 0.02 0.02 0 0.03 increments for initial parameters

0 0 0 1 2 0 0 choice of coefficients

Here a_x has order 1 and a_y has order 2. The output is:

Initial parameters (in digital and word forms),

followed by coefficients for final x, y, z, cosx, cosy, cosz:

0 0 0 1 2 0 0 (that is, cosx cosy cosy)

-1.093E+02 -1.093E+02 1.797E-03 -2.404E+03 -2.404E+03 -4.625E+01

The first coefficient is

c(x,a_x*a_y*a_y) = dx/(a_x*a_y^2)

so

dx = -1.093E+02*a_x*a_y^2

In fact this can be slightly misleading, because there are two contributions to this dx, as can be seen in the usual formula (see P W Hawkes and E Kasper, Principles of Electron Optics, Acadmenic Press, 1989) for the spherical aberration in the xz plane:

dx = (M*Csx(x,x,x)*a_x + M*Csx(x,y,y)*a_y)*(a_x^2 + a_y^2)

Here Csx(x,x,x) is the aberration coefficient found above, 163.0.

We illustrate this by a specific example, with a_x = a_y = 0.02. Then:

-1.093E+02*a_x*a_y^2 = -8.744E-4,

M*Csx(x,x,x)*a_x^3 = -4.387E-4.

Therefore

M*Csx(x,y,y)*a_y*(a_x^2 + a_y^2) = -8.744E-3 - (-4.387E-3) = -4.357E-3.

Therefore

M*Csx(x,y,y) = -4.357E-3/0.02^3 = -5.446E2, Csx(x,y,y) = 161.9.

In fact for an axially symmetric system Csx(x,y,y) should be the same as Csx(x,x,x), so there is a small error here.

WARNING

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.