[ATM] Houghton corrector radii

[vladimir sacek]

Thought it would serve purpose to post equations for 
needed lens element radii of curvature for the Houghton 
corrector in Newtonian configuration. The primary is 
spherical. First comes an aplantic (third-order) form, 
and then a plano-lens form that cancels spherical 
aberration and significantly reduces coma of the primary mirror.

The aplantic corrector consists from a bi-convex (positive) 
front lens followed by bi-concave (negative) rear lens, 
the two being in near-contact. The radii of the first/third 
and second/fourth surface are nominally identical and, 
for the glass refrative index "n", given by:

R1 = 2(n-1)f/(1+q)

R2 = 2(n-1)f/(1-q)

q= [n(n-1)(f/R)^3]/(n+1) 

f = -2(1-s)R

where 
R1 is the absolute value of the first/third surface radius 
(positive and negative, respectively), 
R2 is the absolute value for the second/fourth surface radius 
(negative and positive, respectively), 
"f" is the absolute value of the focal length for the two lens elements, and 
"s" is the corrector to mirror separation in units of the mirror radius of curvature R
(which is of negative sign).

Note that "q" is the lens shape factor, also given by (R2+R1)/(R2-R1);
it is normally negative, which in the above relation for "q" results from the (f/R)^3 factor.

The formulae are somewhat simpler than those obtained from 
the Rutten/Vennroij book (given on Rick Scott's site).

The plano-lens corrector form consists from the plano-convex 
front lens followed by (near-contact) plano-concave rear lens, both facing 
incoming light with their curved side. The two radii are of 
identical absolute value (front positive, rear negative), given by:

R1,3 = R(n-1)[(n+1)/n(n-1)]^(1/3)

with "R" being the mirror ROC, and "n" the glass refractive. This corrector 
form leaves some residual coma, which is in a typical configuration about 
1/6 of the mirror coma (about half compared to a Mak-Newt, and about 
a third of the Schmidt-Newtonian coma). Astigmatism is negligible in all three.

Vlad