From: "Richard F.L.R. Snashall" 
To: ATM List 
Reply-To: "Richard F.L.R. Snashall" 


Hypothesis:
    What you can do twice, you often can do thrice;-)

My apologies ahead of time if this sounds somewhat like a math book...
that's my training kicking in, I guess.


A restricted set of triplets are those which consist of a doublet/singlet
or singlet/doublet combination.  By introducing a dummy parameter, B, such
that:

         B = ( Pa - Pc ) / ( Pa + Pc )

where Pa and Pc are the powers of the first and third lenses, respectively,
the achromatism equations can be reduced to the equivalent of those for a
simple doublet and the additional equation:

        (B-1) * Pa - (B+1) * Pc = 0

The three lens powers and curvatures can then be calculated.

The Gee and Wyld solutions to the doublet solved the combination of linear
and quadratic relations that approximated a nulling of LA and OSC, in terms
of the two curvatures that separated the two lenses.

In the restricted set of triplets, the same action could be performed, as
the other inner curvature can be determined in terms of the computable
curvatures by:

        C2a = C2b + Cb

in the case of a doublet/singlet combination, and:

        C1c = C1b - Cb

in the case of a singlet/doublet combination.  (Cb is the net curvature of
the middle lens, C1 and C2 are the curvatures of the first and second
surfaces of a given lens.)

The G-sums for the full triplet are then calculable as linear (OSC) and
quadratic (LA) functions of the two curvatures that separate the two lens
groups.

Now, hopefully, the parameter B would not be too much of a difficulty to
handle.  I think mostly that the value would lie between -1 and 1,
indicating that the first and third lenses are either both positive
(Flint/Crown/Flint) or both negative (Crown/Flint/Crown).

Questions:

Would this be a useful tool?

My guess is that, unless the glass choice is predetermined, we expect
somewhat more out of a triplet than just solving the same set of
requirements.  What measures of performance would be useful?  (e.g.:
secondary color, provided I can come up with an equivalent variable band
form of it)

        Rick S.

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