[ATM] Littrow Solutions

["Richard F.L.R. Snashall" <rflrs@rcn.com>]

To me, the Littrow doublet is the refractive equivalent of the
"Newt".  From a construction viewpoint, it doesn't really matter
much if the radii non-flat surfaces change much (as long as they
are made equal) -- it just results in a different focal length.
I spent a bit of time trying to isolate searches for them.

In a Littrow doublet, to a first approximation (thin lens), the
derivative of the BFL with respect to wavelength, is zero at the
minimum focus wavelength.  For this to occur, the derivative of
the flint refractive index must be twice that of the crown.  If
we want the last surface to actually be a bit concave, it must
be just under twice.  Assuming the F-C dispersion to be proportional
to the derivative and using the search mechanism of the Schott
catalog, I found the following pairs (ratios: 1.95 to 2.00) in that
catalog:

    N-BK7/F5
    N-K5/N-LAF32,N-LASF44
    N-BAK2/N-BASF64,N-LAF33
    N-BK10/N-BAF51
    N-FK5/KZFSN4,N-KZFS4
    N-ZK7/KZFSN5,N-LAF2,N-LAF21
    N-BAK1/N-LASF43
    N-BAK4/N-LASF43
    N-SSK2/N-LASF45,N-SF15,N-SF64,SF15
    N-SK11/N-BASF2

(recall the first and the last??;-)

This lens generally has coma. However, by choosing the refractive
indices, spherical aberration can be removed.  Again, to a first
approximation, the G-Sum must be null.  If the crown refractive
index is known, this results in a quartic in the flint refractive
index.  Using the crown index as the initial guess of the flint
index for Newton-Raeffson iteration gives the following table:

   Crown      Flint

   1.45     1.546776
   1.50     1.598978
   1.55     1.652089
   1.60     1.706211
   1.65     1.761396
   1.70     1.817666
   1.75     1.875025

A cubic fits this within about 2e-5:

    0.056558x^3 - 0.063360x^2 + 0.861277x + 0.258722

If on one side of this line (I think on the high side, but I'm not
sure), residual spherical aberration can be removed with air-space.
However, I think this also makes coma and lateral-color worse.

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