xmpl2d17, 17th 'example' data file for CPO2D
A mirror with negative aberrations.
The equipotential surface at which the electrons are reflected has a curvature that decreases as the distance from the axis increases, which gives rise to a negative third order spherical aberration. The chromatic aberration is also negative. The mirror is therefore useful for correcting the aberrations of electron microscope lenses.
From D Preikszas, PhD thesis, Technical University of Darmstadt (1995) and H Rose and D Preikszas, Optik, vol 92, 3144 (1992).
This simulation is a severe test of the CPO2D program (or any other program).
The 'mesh' method of ray tracing is used for a quick calculation (but see below for further comments), with a mesh size of 0.2mm. A mesh size of 0.1mm gives almost the same results, but more slowly.
The properties of this mirror are very sensitive to the details of the field, so the 'improve nearaxis radial field' option is used.
The 'uniform' beam option is used, with a 'window' (that is, a sizedefining aperture) at z = 35 mm and a point axial 'pupil' (that is, an angledefining aperture) at z = 120 mm, so it would seem that the electrons are being directed towards the desired position if the object (z = 120 mm) instead of away from it, but the time direction is made negative. 12 rays are taken, with initial angles proportional to the square roots of 1, 3, 5, ... 23.
By running the program with different values of the total number of segments, N (166 in the present example), it is found that the final parameters depend linearly on 1/N**2. For example, the dependencies of the final angle alpha_i and the final extrapolated axis crossing of the image, z_i, on the initial object angle alpha_o, for the outermost ray (alpha_o = 0.009789) treated in the above file are:
N = 
166 
332 
799 
therefore infinity 
alpha_i*1000 = 
4.9362 
4.9160 
4.9094 
4.9086 
z_i(mm) = 
235.48 
236.43 
236.75 
236.79 
The extrapolated values (usually deduced from the results at N = 166 and 799) are used for all the results quoted below. The 't' option (for 'total' energy) has been used in obtaining these results (but this option has not been included in the lines of the given data file because of ease of misuse).
It is found that the value of z becomes more negative as the object angle increases, which implies that the spherical aberration is negative (and which also implies that this mirror can be used to correct the spherical aberration of accompanying lenses).
Delta_r/alpha_o**3 has a linear dependence on alpha_o**2 for a gaussian image position 119.95(20)mm (where the number in parenthesis gives the estimated error in the last digits), with an intercept of 528(6)m and a slope of 9.4(4)E5m. The angular magnification is found by plotting (alpha_o/alpha_i) against alpha_o**2, and is 1.00045(50). The third order spherical aberration coefficient of the lens is therefore Cs3 = 528(6)m, and the fifth order coefficient Cs5 is 9.4(4)E5m. These and other results are summarised in the table at the end of these comments.
To investigate the chromatic aberration, the energy has been changed to 10.01eV. The resulting change in delta_r is divided by alpha_o and then plotted against alpha_o**2. The slope of the graph gives 9.63(6)m for the chromatic aberration coefficient Cc (see the relevant 2D note).
To obtain a more accurate, 'benchmark' result, inaccuracy levels of 0.0000001 (not available in the commercial version of CPO2D) and the 'direct' method of ray tracing have been used. The image position is found to be 119.994(2)mm, the intercept and slope of the graph of delta_r/alpha_o**3 versus alpha_o**2 are 532.37(6)m and 8.90(3)E5m respectively, and the angular magnification is 1.00007(2), giving Cs3 = 532.37(6)m and Cs5 = 8.90(3)E5m. Changing the energy to 10.01 it is found that Cc = 9.623(4)m.
The accurate values of the gaussian image position, Cs3, Cs5 and Cc compare well with those deduced by Preikszas (who used a form of Boundary Element Method with 3000 segments), 120.0mm, 532.9m, 8.743E5m and 9.62m respectively (the comparisons are summarised below).
This mirror is surprisingly sensitive to relativistic corrections. For example, the following results (using the present data file, with N extrapolated to infinity) show how alpha_i and z_i change (for alpha_o*1000 = 5.4006) when the energy is scaled up by a factor of 1000 (to give the energy that is intended to be used for the mirror):
energy = 
10eV 
10keV 
alpha_i*1000 = 
4.6587 
4.0919 
z_i(mm) = 
138.72 
157.60 
These changes are very much larger than for a twocylinder lens of voltage ratio 10 (see xmpl2d13.dat). The change in z_i in the above table is 18.88mm, which is close to the value 19.00mm found by Preikszas under the same conditions. Preikszas finds that to obtain a gaussian image point at 120mm when the energy is 10keV requires that a reflecting voltage of 3182.085.
Summary of some of the results, and comparisons with the results of other calculations (all nonrelativistic, and all distances in m, and the numbers in parenthesis give the estimated error in the last digits):
Authors 
Voltage 
image z 
Cs3 
Cs5 
Cc 
CPO 'benchmark' 
3.198593 
0.119994(2) 
532.37(6) 
8.90(3)E5 
9.623(4) 
D Preikszas 
3.198593 
0.120000 
532.9 
8.743E5 
9.62 
Present results 
3.198593 
0.11995(20) 
528(6) 
9.4(4)E5 
9.63(6) 
Comparison of the results of Preikszas with those obtained by authors who have used the Finite Difference Method:
Authors 
Voltage 
image z 
Cs3 
Cs5 
Cc 
D Preikszas 
3.1936 
0.11646 
531.8 
8.747E5 
9.62 
B Lencova 
3.1936 
0.12008 
495 
1.18E6 
9.60 
B Lencova and J Zlamal 
3.1936 

487 
1.29E6 

E Munro et al 
3.1936 

667 
28.7E6 
8.40 
(Note that, as mentioned above, Preikszas used a form of Boundary Element Method.)
References:
CPO 'benchmark', using inaccuracies of 0.0000001, the direct method of ray tracing and extrapolation of the number of segments to infinity.
D Preikszas, Doctoral thesis, Technical University of Darmstadt (1995), and private communication and H Rose and D Preikszas, Optik, vol 92, 3144 (1992).
Present results, using the above file, as explained.
B Lencova, private communication (fit with MapleV.3 to alpha_o up to 10 mrad).
B Lencova and J Zlamal, Proceedings of the Symposium on Recent Advances in Charged Particle Optics and Surface Science Instrumentation, Skalsky Dvur, Czech Republic (1996).
E Munro, X Zhu and J Rouse, J of Microscopy, vol 179, 161 (1995).