[ATM] Design Tool (longish)

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

The other week, someone (can't remember their first name;-) asked
if I had ever considered extending the Gee-Wyld doublet tool to design
triplets.  Actually I had, but it was more or less just hacked together.
That tool was lost (or not, depending on your opinion) when my previous
machine died. Some time ago, on another list, Roland Christen suggested
that he thought of a triplet design as being more of a doublet
with an additional singlet.  My brain has been going over these to
see if I could come up with a reasonable design tool concept.

There are two design components that need to be considered:
   a) Initial Design
   b) Design Iteration

My references here are mostly the Gee and Wyld articles in the
ATM series, Vol 2 (in the later editions -- in much earlier
editions, I think they were in Vol 3).

Initial Design

The Gee-Wyld tool computes the lens powers in a preliminary design
using the standard linear equations for achromatism:

       P1    + P2       = 1
       P1/v1 + P2/v2    = 0

where the P's are powers and the v's are Abbe Numbers.  Even an
apochromat satisfies this criterion, albeit at a wavelength further
from the center of the band of interest. (Yes, the Gee-Wyld tool
calculates everything based on a unit power lens, converting to
real focal length scaling only for the display function.)

My initial thoughts about a triplet is that it is normally thought
of as being a balanced design; either a convex center element
surrounded by concave outer elements or a concave center element
surrounded by convex outer elements.  In this case, the corresponding
powers of the outer elements could be described with a balance factor
relating them to the net combined power:

     P1 = (0.5+fbal)*Pouter
     P3 = (0.5-fbal)*Pouter

The balance factor could be allowed over a wide range,
say [-1e20,1e20].  The achromatism equations are then:

    (0.5+fbal)*Pouter    + Pinner     + (0.5-fbal)*Pouter     = 1
    (0.5+fbal)*Pouter/v1 + Pinner/v2  + (0.5-fbal)*Pouter/v3  = 0

Thus, given this additional user input, the inner and outer
powers would be easily computable.  That would make each of
the lens powers also easily computable.

 >From the powers, the net curvatures of the elements may
are computed from:

     Ci = Pi / (ni - 1)

The other major component of the initial design is the
bending of the elements.  For this, I thought that the
extension from the doublet case would be the freezing
of the two inner surfaces.  In the current Gee-Wyld tool,
there is an option to hold the relationship between the
last surface of the first lens and the first surface of
the second lens.  For the initial design, this was taken
to imply that the two curvatures would be equal.  For the
triplet, I think I can see a three-option extension of
this: hold the relationship of the first air-gap RoCs,
hold the relationship of the second air-gap RoCs, or
hold both.  When the relationship is held in the initial
design, there is a linear relationship between the
bending of middle lens and the other lens.  Letting
Bi, i = 1..3 represent the bending of each lens, i.e.:
Bi is twice the mean curvature of the two surfaces,
then:

        B1 = B2 + ( C1 + C2 )
        B3 = B2 - ( C2 + C3 )

as appropriate.

However, if the fixed curvature relationship is not held, the
coma sum is linear in the bending.  The net effect is to end
up with an equation of the form:

        a1*B1+a2*B2+a3*B3 = a4

where the a's are functions of the refractive indices
and net lens curvatures.  As I mentioned above, the
triplet extension of the doublet design tool would force one
of the bendings fixed.  The fixed relationship equation above
would then be substituted back into the coma sum relationship,
eliminating either B1 or B3.  The remaining equation
would then easily be solved for the other outer lens bending
as a linear function of the inner lens bending, say:

       B3 = a23 * B2 + b23

The full set of equations for this would then be either:

       B1 =   1 * B2 + ( C1 + C2 )
       B3 = a23 * B2 +     b23

or:

       B1 = a21 * B2 +     b21
       B3 =   1 * B2 - ( C2 + C3 )

Now for the messy part.  The spherical aberration sum
is a quadratic in the bending of the three lenses.  However,
if the two linear relationships determined above are
substituted in, the result is a simple quadratic in the
bending of the inner element; the recombination is
performed algebraicly, though.  The resulting equation
can be solved for the inner element bending, and the linear
relationships above can be reused to solve for the outer
lens bendings.  Finally, given both the bending and the
net curvatures of each lens, the curvatures of both sides
of each lens may be calculated, i.e.:  (Bi +/- Ci)/2.  In
the Gee-Wyld tool, these curvatures are then used to establish
the thicknesses of the lenses and air-spaces; this operation
would not change (except, of course, that now three
lenses must be computed).

Design Iteration

The Gee-Wyld tool calculates an achromatic update of the
last surface radius using the path difference method in
the Wyld article.  The path difference for the first two
lenses will, however, be computed using the calculated value
of dispersion, rather than substitution back into the
equation for the Abbe number.  In all, I think it should be
reasonably extendable to three lenses, using the sum of the
path differences of the first two lenses in the update
equations for the third lens.

Looking at the sums developed in the Gee article, I'm
not sure exactly what the extension would be for three
lenses.  However, reverting back to the old stand-by,
Born&Wolf, the equations can be simply evaluated
(section 5.6) for the thin lens case by recognizing
that, for the entrance pupil coincident with the
lens:

      scrK = -( P/2 + SUM( P's ) )

i.e.: the value of "script K" is the negative of the
sum of the previous lens powers and half the current lens
power -- totally computable from the lens powers.

When the air-space is variable, I use the two empirical
update equations given by Wyld:

       dLA'/dt   = 80 / f#^2
       dOSC'/dBi = y^2 / (1.16*f')

For fixed air-space, I use the G values to generate
updates to the lens bending, as given in that article
(my own extension):

     dLA'/dBi = y^2*lb'^2*Mi

where

     Mi = 2*G4*Ci*Cia - G2*Ci^2 - G5*Ci/Li

In this, Li is the distance to the object as seen by that
lens and Cia is the first surface curvature ( (Bi + Ci)/2 ).
The other equation is:

     dOSC/dBi = 0.25*y^2*G5*Ci

Note: Exactly where I got this second equation from, I
      do not remember..... it's not in the Wyld article,
      I don't think.

As in the current Gee-Wyld tool, the goal amount of LA or
OSC change will still be dependent on the user desired
relationship to the 70% values.

The point here is that I think, provided the limitations
presented (only one air-space and one internal bending
being variable) are enforced, the triplet design may be
just a simple extension of the double design.  The onus
on the user here is that they must:

   a) shift the wavelength of minimal focus as desired
      to achieve some level (maybe none, for all normal
      glasses) of apochromatism.
   b) decide the balance of the outer elements.
      I have no idea how much iterative work this will entail for
      the user.  The default value for the balance factor, though,
      will be zero, implying that the two outer elements will be
      of equal strength.
   c) choose amongst the new option extesions.
      Instead of a "fix t2" option (yes/no), for example,
      there will be a "fix t2"/"fix t4"/"fix t2 and t4"
      choice.

Watta ya say? (OTHER than that you think I'm mad!;-)
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