[ATM] stalking the wild achromat: what price broadband correction?

["dave w" <daze39@earthlink.net>]

Well, you know what they say about idle hands + stuff... here I was
noodling around the Schott optical glass web pages and I found they
had a spreadsheet with a bunch of different parameters for all 105
flavors - including the type-specific "Sellmeier" coefficients for
an empirical curve-fit of refractive index vs. wavelength... and,
of course as luck would have it I had just acquired a rather fast
new linux computer (core2 2.2Gz), so with the power of raw C at my
command, unhampered by the bloat of Vista (or even XP), it seemed 
only natural to code up some search loops to try to come up with 
really precise achromatic combinations over various design wavelength 
ranges.

There's a good chance I'll have things to the point where I might
have some code and sample outputs to post within maybe a few weeks -
but some preliminary comments seemed to be worth posting at this time.

General scheme is that:
0. I have the glass types and coefficients from the spreadsheet 
in a text file that gets read in at program startup.
1. For any given glass type I can calculate the refractive index
at any given wavelength. If I know the curvatures of the positive
and negative lenses in a doublet or triplet, I can then calculate 
the net power of the combination.
2. The procedure for evaluating a given combination consists 
of calculating the net power for each a set of wavelengths 
and determining the maximum, minimum, and average of these 
values. The relative deviation is calculated by finding the 
difference between the maximum and minimum of these values
and dividing it by their average. I then take the reciprocal
of that to get the "achromatic quality" - a large number means
that the overall power (i.e., inverse focal length) of the
combination under consideration is very consistent over the
wavelength range calculated (which would be specified as e.g.
350 to 950 nm in steps of 50 or 480 to 660 nm in steps of 30,
etc. as desired). So a factor of 1000 would mean consistent 
over one part in 1000...

So I write some code to loop through possible pairs and triplets
making this sort of calculation - the holy grail would be a perfect
"photovisual" combination with minimum diff across that 350-950 nm
range (so everything from B+W film that picks up especially well 
in the deep blue-violet and near-UV, to CCD's with their natural 
extended red/near-IR sensitivity, would image sharply at the point 
of optimum visual focus... but dang, there sure ain't no free lunch,
is there? It's that last 50-100 nm on the blue end that seems to be
the hardest to match... setting the short wavelength end of the search
range to 400 or 450 nm lets a few combinations optimize a lot better 
than they do if the far blue end is included in the calculation.

Some general observations...
1. It's easy to see why fluorite has the reputation it does:
Lithotec-CaF2 (pure optical fluorite) and FK-51A (a "fluorite
crown" seem to show up in a lot of the best pairs.
2. The best thing to do with BK-7 in a doublet (as far as sheer
achromatic accuracy over the 350-950 nm range) seems to be to
use it as the -negative- element with an FK-51A positive...
however, the tradeoff is that while this combination makes an 
-accurate- achromat, it's sure not a very "strong" one: the 
ratio between the relative dispersions, and the refractive 
indices of both glasses, are all relatively low - which means 
that strong individual element curvatures would be needed for 
a given net power, and it could be hard to lay out such a lens 
with good geometrical (spherical / coma etc.) correction, especially
at a "fast" focal ratio where the advantage of accurate color
correction would be most desirable.

This seems to be something of a general pattern (with doublets
at least) - there are really "strong" achromats (which use things
like high-dispersion "dense flint" glass types), where color 
correction between one end of the range and the other is achieved
with a comparatively low-power negative lens, and there are very
"accurate" achromats, with very small focal shift across a range
of wavelengths... and it seems to be a bit of a "pick one or the
other" situation for the most part (especially for the broadband
"photovisual" case!).

There do seem to be a few cases with triplets that will
call for a deeper look: a few glass combinations where
there's a "sweet spot", where dividing the "compensation"
between two negative lenses of different glass types gives 
a noticeably better result than using a single negative of
either type... alas, these effects seem to be much more
pronounced with a narrower wavelength range. A combination
that seems to have really good consistency over the 480-660
nm range is frequently nothing special when calculated over
the 350-950 nm range.

-dave w

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