Test3d33, an example of an electrode defined by users equations.

A parabolic reflector.

The original data can be seen in the input data file, which is available as part of the CPO3D package.

This illustrates the option (added July 2012) for electrodes to be defined by user-supplied equations.

See also test2d27.

2 parabolas are defined, z = r^2 and z = r^2 - 0.05. They have the applied voltages 0 and -2 respectively, and so form a parabolic mirror for electrons of energy 1eV.

The same two variables r and theta are defined by the user for the two parabolas. The variable r is given 30 uniform subdivisions, from 0 to 1, while theta has 8, from 0 to 1.57 (the program recognises 1.57 as being pi/2 and so replaces it by the exact value).

The first parabola is defined by the equations

x = r*cos(theta)

x = r*sin(theta)

z = r^2.

The second parabola is defined similarly but with the addition of the fixed parameter b = 0.01 and with z defined as

z = r^2-b.

Electrons start from z = 0.75, with the energy 1eV.

The theoretical point of focus for the first parabola is at z = 0.25. The electrons are reflected midway between the the parabolas and so the expected focal point is at z = 0.225. In fact in this example the crossing points are in the range z = 0.223 +- 0.004. There seems to be some form of spherical aberration, perhaps caused by the gap between the parabolas.

Note that this 3D simulation is less accurate than the analogous 2D one, test2d27. In the 3D simulation we have started the rays away from the base plate, and used the 'total energy' option, to try to avoid errors arising from imperfections in the initial potential distribution. We have also used a larger separation betwen the parabolas, to avoid difficulties in accurately evaluating the field between them when their separation is less than the dimensions of the segments.

Higher accuracy would of course be obtained by using more segments.

See notes on the 'users equations' option.