xmpl2d02, 2nd 'example' data file for CPO2D

A double aperture lens

The number of segments used in the present example is small enough for the example to be run with the 'student' edition of CPO2D. Higher accuracy could of course be obtained with more segments, using the full edition of CPO2D.

The 2 apertures have a radius of 0.5 and are situated at z=+/-0.25. The apertures have a zero thickness in the present simulation, but this can easily be changed (but would give too many segments for the demonstration version). The lens is enclosed by cylinders of radius 2.5, which are themselves capped by discs at their far ends. The cylinder between the 2 apertures has the appropriate voltage gradient. The charge density on the aperture plates is highest at their edges, so the segments into which the plates are subdivided are made the smallest there (by using 'type -2' of subdivision, as explained in Help). A total of 146 segments is used.

The requested fractional inaccuracy of the boundary charges is 0.00001, and that of the potentials, fields and ray tracings is also 0.00001.

The voltages on the lens elements are specified in the data file as 1 and 10.

The region from z=-4.8 to +4.8 is specified for obtaining the lens parameters, and this length is divided into 400 steps.

The table shows some representative results for V2/V1=10 obtained with the above conditions and also obtained with more segments. The results given by E. Harting and F. H. Read, Electrostatic Lenses (Elsevier, Amsterdam, 1976) are also shown. The f's and F's are focal and mid-focal lengths respectively.

 N f1 f2 F1 F2 146 1.08003E+00 3.41382E+00 2.14389E+00 1.63856E+00 160 1.07969E+00 3.41257E+00 2.14313E+00 1.63808E+00 1800 1.07968E+00 3.41248E+00 2.14308E+00 1.63805E+00 HR* 1.01 3.20 2.01 1.53

The data of Harting and Read are less accurate than their claimed inaccuracy of 1%, but 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 iterative subdivision (adaptive segmentation, see Help or the note at the end of the 6th example file xmpl2d06.dat) 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. The cs's are the spherical aberration coefficients defined by Harting and Read, and the cc's are the chromatic aberration coefficients defined by L. A. Baranova et al, Sov. Phys. Tech. Phys. 34, 1409 (1989).

 N cs0 -cs1 cs2 -cs3 cs4 cc0 -cc1 cc2 ptz 146 6.0079 12.310 9.7075 3.4982 0.4884 1.0683 1.2782 0.3983 0.3223 600 6.0074 12.310 9.7083 3.4988 0.4885 1.0681 1.2779 0.3982 0.3224 1800 6.0073 12.310 9.7085 3.4989 0.4886 1.0681 1.2779 0.3982 0.3224 HR 5.51 11.3 8.93 3.23 0.453

To check the computed aberration coefficients, the program has also been asked to trace some specific rays through the lens. The rays all start at z1 = -4.0, and so in the absence of aberrations should cross the axis at z2 = 3.626, which has therefore been used as the position of the 'special 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 77.84 (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=1800) = 0.00174 0.01138 0.03943 'calculated' rf = 0.00122 0.00978 0.03302

The discrepancy for alpha=0.03 is due to the inherent difficulties that exist in accurately integrating rays that are very near to the axis. The discepancy for alpha=0.09 is almost certainly due to the presence of higher order aberrations.

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

(In the 'student' edition 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.)

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