A study shows how the number of diffraction-limited pixels that can be produced by an optical design correlates with the number of lens elements.

1. Diffraction-limited pixels versus number of lens elements
David Shafer
David Shafer Optical Design
Fairfield, CT 06824
#203-259-1431
shaferlens@sbcglobal.net

2. For any optical design with a given
focal length there is some product
of the pupil diameter D and the
full field angle A for which he
design can be made diffraction-
limited over the whole field.
1) For a 100 mm focal length design what is the maximum number of diffraction-limited
pixels possible with a one-element design, a two-element design, a three element
design, etc. D is the pupil diameter in mm and A is the full field angle in degrees
2) What is the ratio of the maximum number of pixels divided by the number of
elements? And how does that change?

3. 100 mm focal length
Monochromatic at .55u
Only BK7 glass lenses
No aspherics
Real image on flat surface
.07 waves r.m.s. or better over field
Distortion not relevant
Non-zero center and edge thickness
and edge gaps to next lens.
Real stop can be inside glass
No vignetting
Design Problem Groundrules

4. For any design where the product of D and A gives
diffraction-limited performance over the whole field
both D and A can usually be changed by +/- 20% or
sometimes a lot more, to give the same D x A product,
and the reoptimized performance will still stay
diffraction-limited. There is a “sweet spot” within that
range where the D x A product is maximized.
There are two good single element designs, one
with a front aperture stop and one with a rear stop.
The sweet spot for the D x A product is near 190 for
the rear stop design and that is better than the 140
value of the front stop design. The first has a sweet
spot that does better with a larger D value while the
front stop does better with a large field A.
Not to same scale
D x A = 140
D x A = 190

5. D x A = 430
D x A = 415
With two lenses there are again two solutions, now
according to if the negative power is in front of or in
back of the aperture stop. Both designs have nearly
the same sweet spot combination of the optimum
choice for the pupil diameter D and the full field
angle A (in degrees), and they have nearly the same
value for the D X A product.
The best single lens value for D x A is 190. Here
with two lenses the value of D x A divided by the
number of lenses is 430/2 = 215, which is very close
to the single lens value. The number of pixels is an
area number and goes with the square of D x A, but
we will first just be looking at how D x A changes with
the number of lenses, not the squared value.
The two plots here are to the same scale and you
can see the design lengths differ.

6. D x A = 680
D x A = 528
Not to same scale
With three lenses there are again two best
solutions. The lesser of the two is much longer and
has a front external aperture stop. Now here is the
data so far.
Number of lenses Best D x A value Best value/ number of lenses
1 190 190
2 430 215
3 680 227

7. D x A = 1010
D x A = 770
Not to same scale
D x A = 700
With 4 lenses we know from the Monochromatic
Quartet IODC contest that the best design form is the
one shown on the upper left here and the 2nd best
form is shown below that. With 4 lenses and the
important extra variables from the glass thickness
there are several discrete good solution regions. With
5 and 6 lenses there are a large number of local
minima and it is hard to know if the best has been
found. Here D x A divided by 4 lenses gives 252
Front exterior stop design

8. D x A = 1400
For 5 lenses I have several quite different
configurations, one shown below, and there
are probably a few more good ones, but this
one here on the left is the best so far.
What is most interesting is the trend here
of D x A divided by the number of lenses.
Clearly that number has to eventually turn
around and decline for some value, not yet
reached, of the number of lenses. And D x A
also has to also slow down in its growth.
Number of lenses Best D x A value Best value/ number of lenses
1 190 190
2 430 215
3 680 227
4 1010 252
5 1550 310
D x A = 1550

9. D x A = 1800
It is interesting that this 6 lens design here
shows that, like the previous designs, the
highest “sweet spot” value for the D x A
product occurs when the field angle A is
increased but the D value does not change
much. This design here and the best 5 lens
design sort of look like a “ball lens” with a
field flattening lens at the image. It has a 90
degree field for A. Further improvements in
D x A will probably need a larger aperture D.
Number of lenses Best D x A value Best value/ number of lenses
1 190 190
2 430 215
3 680 227
4 1010 252
5 1550 310
6 1800 300
There is a lot of work involved in trying to find the
best 6 lens design and I can’t tell for sure if this one
here is it. If it is, then the numbers here are starting
to maybe turn around in value. But finding the best
7 lens design has not been done yet to show that
trend, if it is true. So I tried that next.

10. Next there was a surprise. I found a
pretty good 7 lens design and after a lot
of work on it the design slowly moved
to a place where one lens could be
removed with no performance penalty.
The result was a better 6 lens design
than what is shown in the previous
slide. With more work on that it
became clear that another lens could
come out and the result is this new 5
lens design shown here, with better
performance than my best 5 lens
design shown earlier, and also shown at
the bottom left here. It is also a little
better than the earlier best 6 lens
design. The new 5 lens design is much
larger in size than the old one.
D x A = 1840
D x A = 1550
These are two
very different
solution regions,
especially for the
first lens shape

11. D x A = 2024
After a lot of searching I found what
seems to be the best 6 lens design and
glass path is a very important part of the
solution. There are several good solutions
and it is hard to move from one to another.
This is not a practical design, in this 100
mm focal length, because of all the glass.
It is a very “relaxed” design with gentle
bending of the rays. All of the designs
shown so far have distortion (not a lot) and
that helps performance.
Here is the revised data, with the new 5
and 6 element results. There is a big jump
going from 4 to 5 lenses and it looks like
maybe at six lenses the trend is starting to
slow down. The best D x A values still
occur with a slow speed and a large field,
which is 90 degrees here.
Number of lenses Best D x A value Best value/ number
of lenses
1 190 190
2 430 215
3 680 227
4 1010 252
5 1840 368
6 2024 337

12. D x A = 2230
The best 7 lens design I could
find, after a lot of work, is
basically the 6 lens design with
one split lens. It looks like the
entrance pupil keeps moving
closer to the front as the
number of lenses increases.
This lens is the end of the line
for this study. I can not find an
8 lens design that is better
except by just a little bit.
The design has very gentle
bending of the rays.
Very many attempts were made to add a lens to this design but
it looks like this particular configuration is pretty much exhausted
now, or that it is in a very deep local minimum where nothing can
improve it. That is common among very “relaxed” designs.

13. Number of lenses Best D x A value Best value/ number of lenses
1 190 190
2 430 215
3 680 227
4 1010 252
5 1840 368
6 2025 337
7 2230 319
8 ? ?
The D x A value is proportional to the
diameter of the image, measured in Airy
disks. The total number of Airy disks
would be the square of that and is a
measure of how much information the
designs can transmit. Usually the
performance drops off very quickly as
soon as you get even slightly outside of
the optimized field diameter so there is
very little useful information outside of
that. So the square of D x A does
accurately indicate the total amount of
information transfer by the design.
It looks like the effect of adding additional lenses
has started to decline, which it must eventually do.
But it might be that some new design configuration
can be found that breaks away from the designs
shown here of 4,5,6, and 7 lenses, which have a
similar type of configuration. But I have looked long
and hard for that and have not found something
new that is better than what is covered here.

14. Discussion
This is a very artificial design problem in several ways. But by limiting the scope of it the
difficulty of finding optimum designs is reduced to a reasonable level. No cemented lenses are
allowed and an index break across a cemented surface is a very powerful design tool, like in
monocentric ball lenses. But that would expand the scope of the problem too much. Having
all the lenses be low index BK7 is also a restriction which hurts potential performance. I have
limited the length to be 4X the focal length and all of these more complex designs, with close
to 90 degree fields, want to be at least that long. And not allowing any vignetting removes a
way to extend the field size while cutting off bad rays. But within these various constraints I
think that the results given here have some validity and predictive value.
Not requiring distortion correction makes a big difference in the wide angle performance
and allows for lens configurations that maximize the D x A product but have considerable
distortion at 90 degree fields. The D x A product is easiest to improve by increasing A, the field
angle, and not D, the aperture diameter. That is because higher-order aperture aberrations
are mostly a higher polynomial degree than higher-order field angle dependency and
therefore increase the fastest.

15. The optimum designs for the D x A product all have very gentle ray bendings
and are “relaxed” designs. It turns out that designs like that are very difficult to
optimize. The low-order derivatives of the parameters are very “weak” because
of the way the rays are bent gradually from surface to surface. That makes it hard
for an optimization program to work well. It is also very hard to find out how to
add a lens to a “relaxed” design to improve the performance.
In this study here I was not able to find an 8 lens design that is more than
slightly better than the best 7 lens design, and that 7 lens design is not as
improved as I would like compared to the 6 lens design. Going to longer lengths,
higher index glass, cemented lenses, Merte surfaces, etc. would change this but
without that it looks like I am stuck at 7 lenses. It is hard to see how the
configuration could be any different, for the best wide angle form, from what we
have here – an inverse form with front negative power, long length, and a
negative field flattener at the image.
More discussion

16. If anyone wants to take on the 8 lens design problem and gets something
good I would love to hear about it, at shaferlens@sbcglobal.net.