Test2d15, 15th 'benchmark test' data file for CPO2D
Field at a circular hole in an infinite sheet
The problem is simulated by putting a hole of diameter 1 in an electrode of diameter 5, and enclosing the regions on both sides of the electrode with cylinders of length 10. With 50 segments the errors are less than 0.0006 for the potential and field on the axis, but go up to 0.003 for the more difficult region near the aperture edge (where the fields become infinitely large at the edge itself). The values extrapolated to an infinite number of segments are much more accurate.
The problem is simulated by putting a hole of diameter 1 in an electrode of diameter 5, and enclosing the regions on both sides of the electrode with cylinders of length 10. The potentials and field components are sampled at 3 points on the axis and 3 points near to the edge of the aperture (at a distance of 0.1 from the edge).
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.
Taking E1 and E2 as the asymptotic fields for z<0 and z>0 respectively, then for a hole of diameter R the exact expression for the potential along the axis is
V(0,z) = (E2-E1)*(z/pi)*(arctan(z/R)+(R/z)) + (E2+E1)*(z/2)
In the present example, E1 = 0, E2 = -1, R = 1.
The equation for the off-axis potential is given by P W Hawkes and E Kasper, Principles of Electron Optics. Further details and an algorithm are given in equation details.
The table below shows the comparison between the results obtained with the present simulation and the exact values of the potentials and field components. The table also gives values labelled 'extrapolated', which have been obtained by doubling the size of the shielding cylinders and extrapolating the number of segments N to infinity. This extrapolation is carried with N = 100, 200 and 400, using the observation that the potentials and field components depend linearly on (1/N)^2.
z= -1.0 0.0 1.0 -0.1 0.0 0.1
r= 0.0 0.0 0.0 1.0 0.9 1.0
potl: present 0.06777 0.31831 1.06820 0.05846 0.13925 0.15850
extrapolated 0.06812 0.31815 1.06813 0.05814 0.13874 0.15874
exact 0.06831 0.31831 1.06831 0.05815 0.13878 0.15815
ez: present -0.09149 -0.50021 -0.90898 0.11214 -0.50020 -1.11254
extrapolated -0.09088 -0.50000 -0.90912 0.11512 -0.50000 -1.11513
exact -0.09085 -0.50000 -0.90915 0.11519 -0.49991 -1.11519
er: present 0.0 0.0 0.0 0.44551 0.65371 0.44551
extrapolated 0.0 0.0 0.0 0.44358 0.65716 0.44358
exact 0.0 0.0 0.0 0.44362 0.65723 0.44362
It can be seen from the table that the present simulation with N = 50 gives errors that are less than 0.0006 for the potential and field on the axis, but go up to 0.003 for the more difficult region near the aperture edge (where the fields become infinitely large at the edge itself). The 'extrapolated' values are much more accurate, having maximum errors of 0.0002 and 0.0006 respectively (and these errors are also partly due to the finite size of the shielding cylinders).