Estimation of the Most Probable Values for the OFAD

Theory should accurately predict the shape of the distortion curve, yet the measured values of d5 were consistently approximately 100,000 units smaller than expected. According to the Monte Carlo calculations, d5 is unlikely to have decreased as a result of manufacturing.

A difference of 100,000 in values of d5 causes a maximum difference in image positions of 17micron (.3 arcsec) at the edge of the field. When the theoretical and empirical models are adjusted to coincide as well as possible, larger values of theoretical d5 produce the best fit with slightly lower, compensating values of f and d3, reducing the residual errors to about rms 4micron, roughly the same as the intrinsic errors of the measuring process. Thus, the difference between the experimental and theoretical values of d5 is not significant here and can be safely ignored for the present, but it is unclear why this discrepancy exists. The most likely explanation is that it is some kind of systematic difference in how positions are predicted with a computer and measured photographically.

The MC calculations indicate that due to fabrication tolerances, the measured focal length might vary by be as much as 5mm from the predicted values. As previously mentioned, the best agreement occurs when the focal lengths are shortened by 1.8mm. The fact that a correction of this degree is sufficient to minimize the difference between experiment and theory strongly suggests that the corrector was assembled within specifications.

Adjusting the focal length by reduces the rms difference between the predicted and measured image positions to less than 4micron (.08 arcsec), which is comprable to the experimental error in the positions predicted by the measured OFAD. Once this adjustment has been made, the measured and predicted values tabulated for the OFAD of the PF camera in Table 2 are essentially indistinguishable.

This focal length adjustment can be put into further perspective by noting that the theoretical focal length agrees almost perfectly with the plate scale derived from the M68 plates while it differs by 3mm from the scale on the LP543 plates. This provides some support to the supposition that the M68 scale is more likely to be correct and that an error may have been made correcting for refraction on the LP543 plates.

Summarizing, the theoretical OFAD coefficients are probably the more reliable, certainly for determining how d3 and d5 vary with wavelength. The photographic modeling gives us assurance that the true image scale is within the expected range. However the errors in fabrication appear to have been smaller than those made in the measurement of the OFAD. The theoretical values appear to be the best predictors we have of the corrector's behavior until we can obtain another, more accurate measurement of the paraxial focal length.

Table 3: OFAD Coefficients for Nominal PFCCD

Band f.l (mm) d3 d5
U 11466.5 359.5 900000
B 11468.3 355.9 875000
V 11469.3 353.8 840000
R 11469.8 351.3 830000
I 11470.3 350.2 810000
With 4mm filter and 6mm window
d3 and d5 are dimensionless
Errors are mainly in focal length: see text

Zemax can now be used to calculate the OFAD to use for the PFCCD and Argus. The OFAD for the nominal PFCCD with a 4mm BK7 filter and 6mm fused silica window are given in Table 3. Argus should be focused through a blue filter and the B band OFAD used, i.e. f=11466.5mm, d3 = 357.9 and d5 =835000.

The focal lengths for the nominal PFCCD are approximately 1.5mm longer than for the PF Camera while d3 and d5 are negligibly different. The focal length difference is because the PFCCD is .3mm from the nominal position while the PF Camera is within .1mm of the best location. It is also interesting to note that there is a slightly greater variation of focal length with color for the PF camera. As previously mentioned, this comes from a small amount of chromatic aberration in Argus and the photographic camera caused by the incorrect thickness of the filters.