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[ATM] Thin Lens Approximations
I've been working against the thin lens approximation:
(n1-1)*c1 + (n2-1)*c2 + ... = 1/f
where n1,n2,... are the refractive indices for each lens and c1,c2,...
is the net curvature (i.e.: 1/R1 - 1/R2 for the ROCs R1 and R2 in a lens).
Another way to put this is in terms of power:
p1 + p2 + ... = 1/f
For achromatism, the original is differentiated w.r.t. some convenient
measure of wavelength:
n1'*c1 + n2'*c2 + ... = 0
or:
(n1'/(n1-1))*p1 + (n2'/(n2-1))*p2 + ... = 0
The inverse of the coefficients here ( (ni-1)/ni' ) is very much akin to
the Abbe number. My version of the GSUM calculator, in fact, uses a
constant multiple of the derivative w.r.t. (1/lambda)^2 for its calculation
of Abbe number (multiplication of this equation by a non-zero constant
doesn't change its validity).
However, I found some curious things about this variant of Abbe number:
a) When plotted against wavelength, it appears to be unimodal,
with the mode in the visible range.
b) The peak location correlates strongly with the peak value.
The peak for flint glasses is at a longer wavelength than it
is for crown glasses.
For K7, for example, this Abbe number variant can be
calculated as:
60.5054901 - 130.2547804*w^2 + 137.3255789*w^3
- 197.6323419*w^4 + 198.6569122*w^5
where w = lambda - 0.492848286988, is in microns.
For F2, this Abbe number variant can be calculated as:
36.5867060 - 58.8969612*w^2 + 76.4565084*w^3
- 264.7729207*w^4 + 395.6434458*w^5
where w = lambda - 0.630243796250, again in microns.
The fit of these polynomials is on the order of 0.01 to 0.02
over the 0.365 to 1.015 micron band.
I surmise that, if only the first two terms of these are considered,
the result would indicate the quadratic performance of ordinary
achromats.
--
Rick S.
http://users.rcn.com/rflrs
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