This document summarizes the design process for a Double-Gauss lens using aberration theory. It begins with the historical basis of two Gauss doublets back-to-back, then walks through building up a design from first principles using aberration theory. Key steps include: 1) Adding concentric surfaces to cancel astigmatism; 2) Adding an aplanatic/aplanatic shell to introduce Petzval curvature; 3) Adding a concentric/concentric shell to push the system to a telecentric exit pupil. This allows removing the final lens element far from the image. The result is a corrected Double-Gauss design arrived at through theoretical understanding rather than trial-and-error optimization.

1. Special invited Paper
Aberration theory - a spectrum of design techniques for the perplexed
David Shafer
David Shafer Optical Design, Inc., 56 Drake Lane, Fairfield, Connecticut 06430
(Telephone 203-259-4929)
Introduction
The early medieval scholar Maimonides wrote a famous book called "Guide for the
Perplexed", which explained various thorny philosophical and religious questions for the
benefit of the puzzled novice. I wish I had had such a person to guide me when I first
started a career in lens design. There the novice is often struck by how much of an
"art" this endeavor is. The best bet, for a beginner with no experience, should be to
turn to optical aberration theory - which, in principle, should explain much of what
goes into designing an optical system.
Unfortunately, this subject is usually presented in the form of proofs and derivations,
with little time spent on the practical implications of aberration theory. Furthermore,
a new generation of lens designers, who grew up with the computer, often consider aberra-
tion theory as an unnecessary relic from the past.
My career, by contrast, is based on the conviction that using the results of aberration
theory is the only intelligent way to design optical systems. Computers are an invaluable
aide, but we must, ultimately, bite the bullet and think. Along these lines, I have
given several papers over the last few years which deal directly with the philosophy of
lens design; the kind of guides for the perplexed that I wished I had had from the start.
These papers include: "Lens design on a desert island - A simple method of optical
design", "A modular method of optical design", "Optical design with air lenses", "Optical
design with 'phantom' aspherics", "Optical design methods: your head as a personal
computer", "Aberration theory and the meaning of life", and a paper at Innsbruck - "Some
interesting correspondences in aberration theory".
In all cases, the emphasis is on using your head to think, and the computer to help
you out with the numerical work and the "fine-tuning" of a design. To hope that the
computer will do the thinking for you is folly. Solutions gained by this route rarely
equal the results of an experienced and/or intelligent human thinker. Other writers who
have emphasized this same theme before me, and whom I highly recommend, are Bob Hopkins,
Jan Hoogland, and Erhard Glatzel.
This paper will not attempt to review the many different design methods set forth in
the papers just listed. It will, however, "walk through" the thinking behind just one
optical design in a way which will show the enormous power and versatility of aberration
theory as a design tool. It is my strong conviction that just about everything worth
saying about the results and implications of aberration theory (which is highly mathema-
tical) can be stated in ordinary English, so there will be no mathematics in this paper.
Design Examples
The design to be considered is the Double-Gauss lens. This is a very interesting
design for many reasons. Considering how common a design it is, with applications in
every conceivable situation, and how it has been'described in print many times, I find
it amazing that it is as little understood as it is. Figure 1 shows a typical modern
Double-Gauss design. It is well known that this design form can be easily corrected for
chromatic variation of spherical aberration. As Hoogland points out, it can be made
positive, negative, or zero.
Why is that? Find me a place where it explains this very valuable feature of the
Double-Gauss. The historical development of the Double-Gauss, as many authors have
related, is that it started out as two identical Gauss telescope doublets used back to
back. Figure 2 shows a Gauss doublet, which is corrected for spherical aberration, color,
and chromatic variation of spherical aberration (although it is not aplanatic). Again,
why is that - when the more common Fraunhofer doublet (shown in Figure 3) has lots of
chromatic variation of spherical aberration?
I'm glad you asked. We will digress briefly from the main point of this paper in
order to follow up this train of thought. The reason is that the spherical aberration
contribution of a surface used near its aplanatic conjugates is changing very rapidly
Proc. of SPIE Vol. 0655, Optical System Design, Analysis, and Production for Advanced Technology Systems, ed. Fischer, Rogers (Apr 1986) Copyright
SPIE
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2. with conjugate and/or index of refraction (which have equivalent effects). Figure 4,
from Kingslake's book, points this out quite dramatically. On either side of the
aplanatic point, in conjugate or index (changed due to wavelength), the amount of
spherical aberration passes through zero and changes sign.
The front lens of the Gauss doublet has no surfaces anywhere near their aplanatic
conjugates. The back surface is very nearly concentric about the marginal ray. This
front lens has a certain amount of spherochromatism, all coming from the front surface.
It has more net undercorrected spherical aberration towards the blue, due to the higher
index of refraction (more power).
The back lens, however, has its front surface working very close to its aplanatic
conjugates. The changing index, with wavelength, of the front surface of the back lens
makes its front input conjugate seem more inside the aplanatic conjugates as you go
towards shorter wavelengths. Thisgives (see Figure 4) more overcorrected spherical
aberration towards the blue, and is a small effect intrinsic to the surface - not enough
to cancel the spherochromatism of the first lens.
The input color, from the first lens, also makes the input conjugate to the third
surface of the doublet seem more inside its aplanatic conjugates, towards the blue, and
gives more overcorrected spherical aberration, which is induced. By proper design, these
two sources of spherochromatism in the back lens, which are the same sign, can cancel that
of the front lens and give the Gauss doublet. The back surface of the second lens plays
essentially no role in this, for reasons I am still confused about.
Such correction for spherochromatism is also possible with the Fraunhofer doublet, butonly
if it is airspaced by a considerable amount, as shown in Figure 5. Here the aplanatic
conjugate factor, just given, plays no part. It is simply a matter of color giving more
intrinsic undercorrected spherical aberration to the front lens, towards the blue, and
more overcorrected to the back lens.
These are not equal, however, due to the dispersion differences of the crown and flint,
and so are fixed up by separating them enough so that the blue ray drops enough on the
second lens compared to the red ray (an induced effect). This is explained nicely, in a
different context, in an early article by Rosin in the Journal of the Optical Society of
America.
What is lacking here, but which plays a big role in the Gauss doublet, is any signifi-
cant effect due to the changing input conjugate (as opposed to ray height) at the second
lens due to input color from the first lens. On the other hand, ray height plays no role
in the Gauss doublet since it is two thin lenses in contact. It has essentially all the
spherical aberration occuring at the first surface and being corrected at the last surface,
while all the spherochromatism occurs at the first surface and is corrected by the third
surface.
Two of these Gauss objectives, used back-to-back, are the historical basis of the
Double-Gauss lens, and explain why it has correction for spherochromatism. On the other
hand, such a lens is only suitable for slow speeds and has thin negative lenses such as
in the design worked out by Kingslake in his book on optical design.
Having said all of this, it then turns out that the actual Double-Gauss lens of Figure
1 is not based on this explanation at all. The thick shells in the modern Double-Gauss
always have achromatizing surfaces in the middle of the shells. These may be cemented
or airspaced. The point is that these achromatizing surfaces are spaced far enough away
from the front lens and the front surface of the shell (taking just the first half of
the system, for simplicity) that the mechanism is really different.
It is that axial color coming into the achromatizing surface has dropped the ray down
(in the blue) enough to correct for chromatic variation of spherical aberration, even
if the previously explained principle (involving aplanatic conjugates) were not at work
at all - although it still is to some extent. The dominant mechanism, then, in the
modern Double-Gauss is the same as that in the air-spaced Fraunhofer doublet of Figure 5.
To that extent, "Double-Gauss" is a misnomer for this modern lens, although not so for
the old slow-speed version with thin shells.
This situation involves some subtle effects and I don't understand it completely myself.
Suppose now that you are only looking at a monochromatic design. Then the bottom of
Figure 1 shows the basic Double-Gauss without color correction, as described by Mandler.
Figure 6 shows my version of this, which I call a 1 l/2 Gauss design. Figure 7 shows how
it can be achromatized. It is corrected for all the third-order aberrations. There is
no magic, then, about the double-shell aspect of the Double-Gauss lens.
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3. This 1 l/2 Gauss lens is an obvious idea, once the theory of thick nearly concentric
shells has been studied. Let us, however, back-track some and build up the Double-Gauss
from a completely different point of view. This will illustrate the enormous power of
aberration theory to help one "force" a design into existence just by thinking about it,
which is the main point of this paper.
Figure 8 shows the starting point. An aplanatic doublet lens, with the aperture stop
in contact, has astigmatism and Petzval curvature. Neither of these depends on stop
position, but we will want the stop here later, for other reasons. The thick rear lens
has its front surface concentric about the focus of the front doublet. This surface has
no spherical aberration or coma, but does have astigmatism and Petzval curvature. We
move the concentric surface back and forth along the axis until a position is found where
its astigmatism cancels that of the front lens.
Aberration theory, or experience, tells us that such a concentric surface has opposite
astigmatism from the front aplanatic doublet. The back surface of the thick lens goes
almost right up to the image. This back surface can be curved, concave, to give a flat
image to the design. This results in a steep back surface, as shown, to put in enough
compensating Petzval curvature, and results in a lot of distortion and no working distance
to speak of.
If the back surface is moved away from the image, to give more working distance, it
introduces some spherical aberration, coma, and astigmatism. The first two can be compen-
sated for by redesigning the front doublet, while astigmatism can be fixed by changing
the front concentric radius of the thick lens a little bit. The result is a three element
lens with only distortion at the third-order monochromatic level.
The front doublet will be updated several more times (actually, we don't really do it,
but just assume it has been done, for reasons that will soon be clear). Since it is
assumed to be a thin lens at the aperture stop, such updating will have no effect on its
astigmatism, as long as its net power remains the same and it stays at the stop.
Figure 9 shows the next step. We follow the front doublet with a thick shell which
has both surfaces working at their aplanatic conjugates. This has no net effect on the
cone angle of the light beam. The first aplanatic surface speeds up the light by a factor
of n, its refractive index, and the back aplanatic surface slows it down by just the same
ratio. Neither surface puts in spherical aberration, coma, or astigmatism. The only
point of adding this lens is that it puts in a lot of Petzval curvature of the kind we
want to flatten the image.
It turns out that, assuming all the same index glass (n = 1.6 here), this aplanatic/
aplanatic shell will just flatten the image if it is extended all the way up to the front
of the back thick lens (with its front surface concentric about the image), as shown
here. This means the back surface of the last lens can now be flat. It must still be
right at, or very close to, the final image, however.
The next step is to consider the chief ray path. The stop is on the front doublet
lens, which is assumed to be a thin lens, for now. The chief ray is bent up by the nega-
tive power of the thick aplanatic shell, mainly by its back surface. The apparent stop
position, going into the last thick lens, is therefore somewhere in the middle of the
thick aplanatic/aplanatic lens.
Let us now add one more lens to the system, which is a shell with both surfaces concen-
tric about this chief ray, as shown in Figure 10. Ignore, for the moment, other changes
that have happened in this picture, such as the change of the front doublet into a single
lens. The concentric shell has no coma or astigmatism, due to the chief ray position
coming into it, but it does have spherical aberration and Petzval curvature. It is the
latter that we want.
The negative power of this shell pushes the last lens back further. Its (the last
lens) astigmatism correcting radius does not change, however. The thick aplanatic/
aplanatic lens has to have its thickness reduced enough to allow the thick concentric/
concentric (about the chief ray) lens to fit in without having the two strong inner facing
surfaces bump into each other.
Now the total amount of Petzval correction needed to give a flat image to the system
can be shared between these two inner lenses in Figure 10. The aplanatic/aplanatic lens
does not change the focal length of the system as its thickness is changed. The concen-
tric/concentric (about the chief ray) lens does change the system focal length and
introduces spherical aberration as its thickness is changed.
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4. What we want is to have enough power in the third lens so that it pushes the last lens
back far enough so that the chief ray, after the seventh surface, is telecentric or nearly
so. If the computer program being used has aplanatic and concentric solve options in it,
then it is easy to play around with the thicknesses of the two shells until this condition
is achieved. I have found that it only takes a few trials to reach such a state. Of
course every time the second lens thickness is changed, it alters the apparent pupil
position going into the third shell. Even without automatic solves, however, it is easy
to zero in by trial and error.
The purpose of getting the chief ray telecentric going into the last surface is that
then the glass going all the way up to the image in Figure 9 can then be removed, as is
shown in Figure 10. The flat back surface will only introduce spherical aberration into
a telecentric beam as it is moved away from the image. This could not be done in Figure 9
because the exit pupil was very far from telecentric.
A side benefit of adding the concentric/concentric shell to the system is that when it
pushes the last lens further back (to the point of giving a telecentric exit pupil) the
front surface of this last lens almost corrects the distortion introduced by the back
surface of the second lens. The result is a system nearly corrected for distortion, and
with a reasonable working distance to the image.
Now all that is left to fix, at least monochromatically, is spherical aberration
introduced by the concentric/concentric shell and by moving the last flat surface away
from the final image. This is easily done by redesigning the front doublet again. The
result is always that the negative lens in the front doublet (remember, it is not doing
any color correction) essentially disappears. Its overcorrected spherical aberration is
now being put in by the concentric/concentric shell.
Furthermore, the amount of spherical aberration that this shell puts in can be changed
without changing its power, by changing both concentric (about the chief ray) radii and
its thickness. Then third and fifth-order spherical can be balanced against each other
without changing or affecting any other element, as long as its power is fixed.
The result, then, as shown in Figure 10, is that the front doublet is replaced by a
single lens. Its bending can change its coma, for a slight touch-up, but for an index
of n = 1.6, a plano-convex lens is fine. The spherical aberration of the system is tuned
up by changing the third concentric/concentric lens while keeping its power fixed.
At this point we give in and go to computer optimization, for the first time. Notice
that the front doublet was not actually ever designed until the very end, at which point
it was a single element. It was just assumed to have been designed, updated, and to have
certain properties. The whole process just described can be run through quickly in a few
minutes, with the aid of a computer (especially with the right kind of angle solves).
It is only at the very end, when the design is essentially already finished, that anything
approaching computer optimization is needed.
This is, of course, just one form of the Double-Gauss. Somewhat more conventional
looking forms result when the starting point for the aperture stop is not right at the
front lens in the above procedure.
The point here, clearly, is not to save a little money on computer costs. The point
is to understand what is going on in the system and to appreciate why the different lenses
are where they are and. what they are doing. What are we here for, anyway? Of course
we will optimize this design on the computer, we will move the stop (after the design is
well-corrected) to the center of the lens where it usually is, split the shells for color
correction, play with vignetting, etc. Everybody does that.
Understanding the basic theory of the system will make these activities more meaning-
ful and help one to think of other design versions, such as the 1 l/2 Gauss of Figure 6.
Lens design is an interesting and exciting profession. It is even more so when you try
to use aberration theory to understand designs and to help construct new ones. Although
aberration theory may be highly mathematical, the use of its results usually consists of
reasoning things through in ordinary English, as demonstrated here, with few (if any)
calculations.
This, then, is my guide for the perplexed: study the various design methods proposed
by myself and others, learn the basics of aberration theory, think a lot, and as Bob
Hopkins recently said at the International Lens Design Conference - "keep your hands off
the keyboard until you thoroughly understand the first and third-order problem."
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