xmpl2d04, 4th 'example' data file for CPO2D

A double rectangular slit lens


This example is the analogue in planar symmetry of the double aperture lens dealt with in file xmpl2d02.dat. Each slit is composed of 2 plates separated by a distance of 1, and the slits are situated at z=+/-0.25. The lens is enclosed by plates at x=+/-1, capped at their far ends. The plates between the 2 slits have the appropriate voltage gradient. Only one value of the voltage ratio v2/v1 is chosen in this example, namely 6. 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 voltage gradient between the 2 slits prevents the z=0 plane being used as a plane of voltage antisymmetry in the present program (for technical reasons). The x=0 plane is a plane of voltage symmetry. The charge density on the slit plates is highest at their edges, so the segments into which they are subdivided are made the smallest there (by using 'type 1' of subdivision). A total of 49 segments is used.

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 -2.5 and +2.5, 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 500 steps. The requested value of the voltage ratio V2/V1 (which is essentially E2/E1) is 6.


The table shows some representative results for V2/V1=6 (1) obtained with the above conditions, (2) obtained with 224 segments, (3) obtained with 408 segments, inaccuracies of 0.00001 and 4000 tracing steps, and (4) given by E. Harting and F. H. Read, Electrostatic Lenses (Elsevier, Amsterdam, 1976) -copies of the text (non-data) pages of this book are available in the ‘document’ folder of the CPO package.

The f's and F's are focal and mid-focal lengths respectively.


































(*Harting and Read)

It can be seen that the average inaccuracy of the first line of data is 0.5%, that the inaccuracies in the second line are consistent with the requested inaccuracy of 0.1%, and that the data of Harting and Read is considerably less accurate than their claimed inaccuracy of 1% It should be noted that it is exceptionally difficult to obtain accurate values for the parameters of lenses that consist of infinitely thin apertures, because of the singularity of the fields at the edges. As far as we know, this can only be attempted by using the Boundary Charge Method. It can also be remarked that the calculated lens properties are sensitive to the density of segments at the aperture edges (which was not fully appreciated by Harting and Read). A point arising from this is that the technique of iterative subdivision (also called adaptive segmentation) will not necessarily produce the most accurate lens parameters, because this procedure tends to optimise the complete field, rather than the part of the field that dominates the lens properties.


The next table shows some representative results for aberration coefficients, again for v2/v1=10. '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).







































It can be seen that the average inaccuracies in the first line of data are 0.4% and 0.2% respectively, and that the data of Harting and Read is even more inaccurate (but see the comments above).


To check the computed aberration coefficients, the program has also been asked to trace some specific rays through the lens. The voltages on the slits are -2.5V and +2.5V, and the initial kinetic energy of the electrons is 1 eV, so the voltage ratio v2/v1 is effectively 6. The rays all start at z1 = -4.0, and so in the absence of aberrations should cross the axis at z2 = 4.608, 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 48.3 (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 =




computed rf (N=49) =




computed rf (N=224) =




'calculated' rf =





The discrepancy for alpha=0.03 is mainly due to the inherent difficulties that exist in accurately tracing rays that are very near to the axis (and perhaps also to an inaccurate value of z2). The discrepancies for the higher valued of alpha are 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 0.00541 in the value of rf. It can be deduced therefore that the chromatic aberration coefficient is 3.26 for this position of the object. The coefficient becomes 3.50 when 224 segments are used. At present there are no known values with which this can be compared.


The total computing time is 27 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.)