xmpl2d03, 3rd 'example' data file for CPO2D

A double rectangular tube lens

This example is the analogue in planar symmetry of the double cylinder lens dealt with in file xmpl2d01.dat. One pair of plates extends from z=-5.0 to z=-0.05, the other from z=0.05 to z=5.0, both with a spacing of 0.5, and both capped at their far ends. Only one value of the voltage ratio v2/v1 is chosen in this example, namely 4. The focal and mid-focal lengths and the third order spherical aberration coefficients are obtained, and are checked by tracing some specific rays through the lens.

The number of segments used in the present example is small enough for the example to be run with the ‘demo’ version of CPO2D (but see the comment at the end of this file). Higher accuracy could of course be obtained with more segments, using the standard or full versions of CPO2D.

Detailed description:

The following data were obtained when the memory and speed of PC's was much more limited than at present, so the available number of segments was small and the requested inaccuracies were fairly high to give a quick demonstration.

The voltages are made antisymmetric about the z=0 plane, and symmetric about the x=0 plane. The charge densities on the plates are highest at the ends near the gap, so the segments into which they are subdivided are made the smallest there (by using 'type 1' of subdivision). A total of 50 segments is used (before the 2 reflections).

The requested fractional inaccuracy of the potentials, fields and ray tracings is 0.001.

The voltages on the cylinders are specified in the data file as -1.5 and +1.5, but when the lens properties are calculated the program takes the electrode voltages specified by the user and then scales the original potential field and adds a constant potential to the whole field so that an electron which is at a point of potential V has a kinetic energy E = eV.

The region from z=-4.8 to +4.8 is specified for obtaining the lens parameters, and this length is divided into 1000 steps. The requested value of the voltage ratio V2/V1 (which is essentially E2/E1) is 4.

The table shows some representative results for V2/V1=4 (1) obtained with the above conditions, (2) obtained with 160 segments, (3) obtained with 320 segments, inaccuracies of 0.00001 and 4000 tracing steps, and (4) given by E. Harting and F. H. Read, Electrostatic Lenses (Elsevier, Amsterdam, 1976). The f's and F's are focal and mid-focal lengths respectively.

 N acc nz f1 f2 F1 F2 50 .001 1000 1.333 2.665 2.675 1.241 160 .001 1000 1.334 2.667 2.677 1.242 320 .00001 2000 1.334 2.668 2.678 1.242 *HR: 1.33 2.67 2.68 1.24

It can be seen that the inaccuracies of the first and second lines of data are consistent with the requested inaccuracy of 0.1%, and that the data of Harting and Read are consistent with their claimed maximum inaccuracy of 1%

The next table shows some representative results for aberration coefficients, again for v2/v1=4. 'N', 'acc' and 'nz' have the same meaning as in the first table. The cs's are the spherical aberration coefficients defined by Harting and Read (see the relevant note for further information).

 N acc nz cs0 cs1 cs2 cs3 cs4 50 .001 1000 17.79 -35.58 27.19 -9.460 1.265 160 .001 1000 17.81 -35.61 27.20 -9.461 1.265 320 .00001 2000 17.81 -35.61 27.21 -9.468 1.266 *HR: 17.9 -35.6 27.3 -9.50 1.27

It can be seen that the inaccuracies of the first and second lines of data are consistent with the requested inaccuracy of 0.1%, and that the data of Harting and Read are consistent with their claimed maximum inaccuracy of 1%

To check the computed aberration coefficients, the program has also been asked to trace some specific rays through the lens. The voltages on the plates are -1.5V and +1.5V, and the initial kinetic energy of the electrons is 1 eV, so the voltage ratio v2/v1 is effectively 4. The rays all start at z1 = -4.0, and so in the absence of aberrations should cross the axis at z2 = 3.931, which has therefore been used as the position of the 'test plane'. The initial angles to the axis are 0.03, 0.06 and 0.09. The table below shows the computed values of the distance rf from the axis at the test plane. The spherical aberration coefficient for z1 = -4.0 is 90.33 (as obtained from the calculated cs's), and this has been used to give the values of rf in the row labelled 'calculated' in the table.

 alpha = 0.03 0.06 0.09 computed rf (N=50) = 0.0021 0.02021 0.07668 computed rf (N=160) = 0.00249 0.02098 0.07739 'calculated' rf = 0.00246 0.01967 0.0664

The discrepancies for the higher values of alpha are presumably caused by the presence of higher order aberrations.

Finally, a further ray has been started with an energy that is 5% higher. It can be seen that this causes a change of 0.008229 in the value of rf. This corresponds to a chromatic aberration coefficient of 5.44 for this position of the object. The coefficient becomes 5.12 for 160 segments. At present there are no known values with which this can be compared.

The total computing time is 35 sec, using a 66MHz 486 PC.

To improve the calculation of the near-axis rays the 'improve near-axis radial field' option could be used.

(In the ‘demo’ version of the program the maximum numbers of axial and ray points are easily exceeded. The program automatically reduces the number of axial points, but to obtain all the ray information it might be necessary to use longer step lengths.)

The option to output lens parameters for a specific object position can be used.