xmpl2d13, 13th 'example' data file for CPO2D
A relativistic two-tube lens.
This file is almost identical to the 1st example file xmpl2d01.dat, except that the energies are relativistic
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.
This file is identical to the 1st example file xmpl2d01.dat, except that (1) more segments are used, (2) the lens parameters are asked for at a range of initial energies (but always with a voltage ratio of 10), and (3) rays are found for V1=100000, V2=1000000 (and also the test plane is the gaussian image plane for these voltages).
The following lens parameters are obtained (see the output file temp13a.dat):
|
v1=1.000E+00 |
v2=1.000E+01 |
|
|
|
|
|
f1,2, F1,2 = |
7.99736E-01 |
2.52900E+00 |
1.61639E+00 |
1.17959E+00 |
|
|
cs0,1,2,3,4= |
3.67480E+00 |
-7.49808E+00 |
5.95693E+00 |
-2.19336E+00 |
3.18477E-01 |
|
cc0,1,2,ptz= |
8.28703E-01 |
-1.00366E+00 |
3.16363E-01 |
4.53177E-01 |
|
|
v1=1.000E+01 |
v2=1.000E+02 |
|
|
|
|
|
f1,2, F1,2 = |
7.99735E-01 |
2.52910E+00 |
1.61642E+00 |
1.17962E+00 |
|
|
cs0,1,2,3,4= |
3.67501E+00 |
-7.49838E+00 |
5.95706E+00 |
-2.19337E+00 |
3.18471E-01 |
|
cc0,1,2,ptz= |
8.28742E-01 |
-1.00369E+00 |
3.16363E-01 |
4.53181E-01 |
|
|
v1=1.000E+02 |
v2=1.000E+03 |
|
|
|
|
|
f1,2, F1,2 = |
7.99727E-01 |
2.53007E+00 |
1.61672E+00 |
1.17987E+00 |
|
|
cs0,1,2,3,4= |
3.67713E+00 |
-7.50142E+00 |
5.95838E+00 |
-2.19342E+00 |
3.18409E-01 |
|
cc0,1,2,ptz= |
8.29128E-01 |
-1.00393E+00 |
3.16365E-01 |
4.53222E-01 |
|
|
v1=1.000E+03 |
v2=1.000E+04 |
|
|
|
|
|
f1,2, F1,2 = |
7.99642E-01 |
2.53979E+00 |
1.61968E+00 |
1.18233E+00 |
|
|
cs0,1,2,3,4= |
3.69810E+00 |
-7.53145E+00 |
5.97131E+00 |
-2.19384E+00 |
3.17779E-01 |
|
cc0,1,2,ptz= |
8.32958E-01 |
-1.00637E+00 |
3.16379E-01 |
4.53636E-01 |
|
|
v1=1.000E+04 |
v2=1.000E+05 |
|
|
|
|
|
f1,2, F1,2 = |
7.98244E-01 |
2.63204E+00 |
1.64669E+00 |
1.20517E+00 |
|
|
cs0,1,2,3,4= |
3.89210E+00 |
-7.79967E+00 |
6.07772E+00 |
-2.19172E+00 |
3.11067E-01 |
|
cc0,1,2,ptz= |
8.68299E-01 |
-1.02784E+00 |
3.16044E-01 |
4.57934E-01 |
|
|
v1=1.000E+05 |
v2=1.000E+06 |
|
|
|
|
|
f1,2, F1,2 = |
7.55588E-01 |
3.20760E+00 |
1.75668E+00 |
1.31290E+00 |
|
|
cs0,1,2,3,4= |
4.66543E+00 |
-8.35628E+00 |
5.79704E+00 |
-1.85303E+00 |
2.31752E-01 |
|
cc0,1,2,ptz= |
1.03898E+00 |
-1.08301E+00 |
2.91493E-01 |
5.01960E-01 |
|
|
v1=1.000E+06 |
v2=1.000E+07 |
|
|
|
|
|
f1,2, F1,2 = |
4.93507E-01 |
3.64362E+00 |
1.51689E+00 |
1.11369E+00 |
|
|
cs0,1,2,3,4= |
2.34421E+00 |
-3.17360E+00 |
1.68356E+00 |
-4.17681E-01 |
4.14294E-02 |
|
cc0,1,2,ptz= |
1.37075E+00 |
-1.08975E+00 |
2.27081E-01 |
6.53795E-01 |
|
Using the results obtained for v1=100000 and v2=1000000, the gaussian image is at z=2.3933 (which is therefore used for the test plane), and M = -0.3368, Cs*M = -49.54, Cc*M = -2.298, Cd*M = -0.4566.
The rays are intended to provide a check on these aberration coefficients. The following results are obtained for the values of r at the test plane:
r1 = -0.00125, r2 = -0.01141, r3 = -0.04301,
r4-r2 = 0.00614,
r6-r5 = 0.00242.
(and they change by less than 1% if a very small step length and inaccuracy parameter are used). Using the values given above for the aberration coefficients, the r's should be (ignoring higher order aberrations):
r1 = -0.00134, r2 = -0.01070, r3 = -0.03611,
r4-r2 = 0.00690,
r6-r5 = 0.00228.
The discrepancies seem to be caused by the presence of higher order aberrations, which are therefore more significant than they are at non-relativistic energies (see the first example file xmpl2d01.dat). To investigate this we have reduced the initial angles to .01, .02 and .03, the fractional energy change to 2.5% and the off-axis position to 0.05, and have also used the best possible ray inaccuracy, obtaining:
r1 = -0.00062, r2 = -0.00052, r3 = -0.00125,
r4-r2 = 0.00101,
r6-r5 = 0.00061.
These should be compared with the expected (lowest order) values:
r1 = -0.00005, r2 = -0.00040, r3 = -0.00134,
r4-r2 = 0.00115,
r6-r5 = 0.00057.
This comparison is more convincing (allowing for the numerical errors that are always present when tracing paraxial rays).