A variety of useful optical "tricks" or tools, based on aberration theory

1. Some Optical Design Tricks
Dave Shafer

2. Monocentric designs
A monocentric design is one where all the surfaces share
the same center of curvature. In such a system any line
through that common center of curvature is just as legitimate
an optical “axis” as any other line. In fact, there is no
unique optical axis for such a system. There is also no way
to distinguish going through the system forwards or going
through it backwards, since these terms loose their meaning
and cannot be defined for a monocentric design.

3. Monocentric systems
In the 1960s Charles Wynne showed with a simple
geometric diagram that it makes no difference to the
aberrations in a monocentric system in what order the
surfaces are seen by the rays. These two monocentric
Bouwers designs here are exactly equivalent to all orders
of aberrations.

4. Concentric lens in front
of aperture stop has an
exact concentric
equivalent behind the
aperture stop.
Exactly the same
aberrations to all orders,
but one lens version is
very much smaller than
the other one.

5. A concentric lens in double-pass has several exact equivalents

6. The three monocentric designs on
the left are exactly equivalent to the
one above here. Same radii, same
correction, just a different order of
the radii.
Has a well-corrected
virtual image

7. Nearly concentric meniscus lenses

8. A nearly concentric meniscus can often be flipped over to the other
side of its centers of curvature to give a new design version. Here
is a design corrected for all five 3rd-order aberrations. The two
outer lenses are not very close to being concentric. But we will try
flipping them over in the opposite direction, one at a time.

9. The three designs on the left are all
corrected for all the 3rd order aberrations.
Each design was the result of flipping one
or both of the outer lenses in the design
above over to the other side of its
average center of curvature and then
reoptimizing. Lens thickness is important
for the three designs on the left but not for
the one at the top here.

10. Monochromatic design, .30 NA, 15 degrees full field, no vignetting

11. .054
waves r.m.s. at edge of field at .55u

12. Flip nearly concentric
lens over to other side of its
centers of curvature to get
an alternate design. Then
reoptimize. Of the two
designs one will be better
than the other. There is no
way to tell in advance
which will be better.
Not yet reoptimized

13. Reoptimized design

14. .054 waves r.m.s. at edge
of field
.038 waves r.m.s. at edge
of field

15. Parabolic mirror has no
spherical aberration but has
coma and also astigmatism
(if stop is in contact). It is
equivalent to a spherical
mirror + aspheric.
Nearly concentric lens acts
like an aspheric Schmidt
plate located near its centers
of curvature.
So combine the two to get
a parabola simulator with a
spherical mirror.

16. It takes a double pass through the lens to get enough spherical aberration
to correct for the spherical mirror. This design has the same 3rd order coma
and astigmatism as a parabolic mirror. It makes a parabolic mirror
simulator. 5th order spherical aberration is also corrected in this design
here. Axial color can also be self-corrected in this single lens.

17. It is surprising that even
in this very simple design
there are two separate
solutions. The one on the
bottom is also a 3rd order
parabolic mirror simulator
but it cannot be corrected
for 5th-order spherical
aberration.
This same design type can
also be used to simulate the
spherical aberration, coma,
and astigmatism of an
elliptical or hyperbolic
mirror.

18. Gabor telescope
This is often confused with the Maksutov telescope and is only
correctly described in a 1941 patent by Gabor. It is not clear if
Gabor himself understood this design – probably not based on the
patent text. British patent #544,694

19. Bouwers design = all
surfaces concentric about
front aperture stop,
including image surface.
Performance is very
dependent on lens thickness
Gabor design = first
surface concentric about
aperture stop, second lens
surface is aplanatic for axial
rays. Mirror is concentric
about shifted pupil (due to
second lens surface). Lens
thickness has no effect

20. Bouwers monocentric
design. 100 mm F.L.,
.40 NA, .60 waves
r.m.s. over any field
angle on a curved
image, with BK7 glass
Gabor design. 100 mm F.L.,
.40 NA, .11 waves r.m.s.
over a 5 degree field on a
curved image, with BK7
glass. .18 waves r.m.s. over
10 degrees.
.05 waves over 10 degrees
if ray optimized.

21. 3rd order spherical
aberration for these surfaces
is, in arbitrary units, -1.23,
+.60, +.69 and 5th order is .19, +.06, +.06
3rd order spherical
aberration for these
surfaces is, in arbitrary
units, -.60, 0.0, +.60 and
5th order is -.06, 0.0, +.05
Total higher-order is
much less than Bouwers.

22. Both designs can be
achromatized. Gabor
design has more color due
to stronger lens power.
Bouwers design
Gabor design
Two mirror version of design
Both mirrors are concentric
about shifted pupil. Curved
image design.

23. Both designs suffer if the
aperture stop is moved to be
at the front lens but system
is then shorter. Gabor
design has better higherorder so it is less affected by
moving the aperture stop.

24. The point of showing the Gabor design is
1) Try to be aware of what has already been done, by
reading the optical design literature and patents
2) Use aberration theory to understand designs – why
they work well
3) Use this understanding to see how to change existing
designs to get new designs. Bouwers
Gabor

25. .50 NA,
very good
correction
Bouwers + Gabor combination
Second lens is concentric, with mirror, about shifted
pupil. Corrects 5th order spherical aberration

26. Air lenses

27. Achromatic aplanatic
doublet, SK2 and F2
Alternate design, F2 and SK2

28. Achromatic aplanat
Air lens
1) Combine first and
last lens into one lens
2) Change nearly
concentric air lens into
nearly concentric glass
lens
Glass lens
3) Re-correct for
spherical aberration and
coma
4) No longer have color
correction

29. Alternate design conversion
Air lens
1) Combine first and last
lens into one lens
2) Change nearly concentric
air lens into nearly
concentric glass lens
Glass lens
3) Re-correct for spherical
aberration and coma
4) No longer have color
correction

30. Centers of curvature
A nearly concentric lens acts as if it is located near the two centers of
curvature – both with respect to first order optics and also aberrations.
It acts like a weak power (at the first order level) aspheric Schmidt plate
located near the centers of curvature, with considerable spherical
aberration.

31. Air lens converted to glass
lens can give anastigmatic
doublet. Meniscus lens glass
thickness is a key variable
3rd-order spherical
aberration, coma,
and astigmatism =
0.0
Aplanat, no
color correction
Anastigmat, no
color correction
Stronger meniscus radii in
anastigmat design

32. Because of the way a nearly
concentric lens acts this design
is equivalent to two widely
separated lenses, which is why
it can be corrected for
astigmatism.
It is quite surprising that
there are actually two different
3rd-order anastigmatic
solutions.
There is another one where
the front mensicus lens is quite
thin and is very strongly
curved, giving terrible fifthorder spherical aberration. If
you try to set up and find this
good solution here you might
start out within the capture
range of the terrible solution
and you will not find the good
one.

33. Aspheric
plate
Located at center of curvature of the mirror
Schmidt telescope with aspheric plate and spherical mirror

34. Spherical mirror
Same glass
Houghton design – zero power doublet simulates aspheric Schmidt plate

35. Houghton doublet lens is a
nearly concentric outer
menisicus lens with a nearly
concentric inner air lens
inside.
Air lens becomes
glass lens
Pull out the air lens and
convert design to two nearly
concentric glass lenses

36. Same glass
Corrected for spherical aberration, coma,
astigmatism and axial and lateral color
Both lenses act as if they
are far from their actual
location - near their
centers of curvature

37. Notice that bottom design
is shorter than Houghton one
– nearly one focal length
long instead of two focal
lengths long. Yet the front
lens in the bottom design,
since it is nearly concentric,
acts as if it was located
further out in front. The
physical length of the design
is about ½ as long as the
“optical length”

38. Extreme wavelength range design

39. BK7
lenses
Corrected from .365u to 1.50u
My design from 1979 that can be corrected for spherical aberration,
coma, astigmatism, Petzval, primary and secondary axial and lateral
color, and chromatic variation of spherical aberration, coma, astigmatism
and Petzval. All spherical surfaces. 5 degree full field and f/2.5

40. Polychromatic wavefront from .365u to 1.5u is .035 waves r.m.s.

41. The “air lens” between
the 2nd and 3rd lenses was
“released” and turned into
glass. The best result has
the air lens first turned in
the opposite direction and
then converted into glass
to give this design here.
The same extreme
chromatic performance
but much looser decenter
tolerances and a shorter
design.

42. Thicker lenses allow for a
shorter design with a small
change in performance.
The very weak middle lens
is required for this design
to work well.
This simpler design shown
earlier does not have the same
extreme chromatic performance
and has a curved image.

43. Alignment insensitive designs

44. • The astigmatism of a thin aplanatic element or elements is
independent of stop position.
• The astigmatism is proportional to the net power of the thin
element or elements in contact.
• The Petzval of the thin element or elements in contact is
proportional to their net power
• If two such thin aplanatic systems have equal and opposite
power then the combination will be corrected for astigmatism
and Petzval.
• Each subsystem will be alignment insensitive since tilt or
decenter will not introduce coma.

45. Alignment insensitive design
Aspheric aplanatic lenses, equal and opposite powers
Astigmatism and Petzval cancel. Each lens can be
tilted and decentered without introducing coma.

46. Each doublet can be tilted or decentered without introducing coma
Alignment insensitive system
All spherical – two cemented achromatic aplanatic doublets.
Corrected for spherical aberration, coma, astigmatism, Petzval,
and axial and lateral color

47. Alignment insensitive system
Achromatic aplanatic cemented doublet lens/mirror elements
Corrected for spherical aberration, coma, astigmatism, and Petzval

48. Lens used backwards

49. Petzval is the most difficult
aberration to correct and
results in most of a design’s
complexity. But once it is
corrected it stays corrected
for any conjugate and also if
the design is turned around
and used backwards.
Monochromatic design
New designs can be
generated by this method.
Here the design is reversed
and then reoptimized. Now
it wants a front aperture stop
and can be made telecentric.
The performance is the same
in both designs. This idea
came from Jan Hoogland, a
Monochromatic design great designer.

50. This lens is turned around
and reoptimized backwards.
When it is reversed it may be
necessary to move the stop
and temporarily reduce the
NA, in order to get the rays
through the design. Then it
is optimized and the NA
slowly increased back to the
original value.
Reversed design reoptimized
Notice the distant external stop

51. Replacing an aspheric singlet with two or three spherical lenses

52. Monochromatic design
A strong aspheric
f/1.0, 20 degree full field,
.03 waves r.m.s.

53. Monochromatic design
What is limiting the performance – higher-order aperture
aberrations or higher-order field aberrations?

54. Monochromatic design
Make image surface be aspheric and vary the lowest order
aspheric term, but not the curvature. If the performance improves
a lot with optimization then it is higher-order field aberrations that
are limiting performance, probably higher-order Petzval.

55. Paraxial pupil
Aspheric
Monochromatic design
Aspheric image surface helped a lot so we go back to the original
design and reoptimize with a flat image (no image surface aspheric)
and an extra field lens to control higher-order Petzval. Result is 2X
better performance. Then we slowly reduce the strength of the lens
aspheric until the performance starts to suffer.

56. Paraxial pupil
Monochromatic design
Aspheric
Front lens has become very weak so we remove it and reoptimize
= no change in performance = .015 waves r.m.s. at edge of field.

57. Paraxial pupil
Monochromatic design
Aspheric
Extra lens allows aspheric to be weakened a lot, even
though it does not want to have much power

58. Paraxial pupil
Aspheric
Monochromatic design
Aspheric was moved to a different lens, with low incidence
angles. Result is very small 6th order aspheric term, but +/- 20u
aspheric deformation from best fit sphere = a strong aspheric.

59. Aspherics are usually easiest
to remove if the higher-order
terms are small. This is
often the case if the aspheric
incidence angles are small.
So it is a good idea to
optimize the design with a
preferred location of the
aspheric. Here instead of
here. The locations are quite
close to each other but one
has low incidence angles and
that aspheric has much
smaller higher-order terms
than in the other location.

60. Aspheric lens
Aspheric removed
from lens
*SEIDEL ABERRATIONS
SRF SA3
CMA3
AST3
PTZ3
DIS3
1 -0.186965 0.122397 -0.080127 -0.050523
2 -1.008444 -0.075329 -0.005627 -0.022490
3 0.527528 0.124226 0.029253 -0.000935
4 0.752953 -0.135018 0.024211 0.036744
5 0.173777 0.046864 0.012638 0.018152
6 -0.582068 0.001699 0.003497 -0.007389
0.085530
-0.002100
0.006668
-0.010930
0.008303
-0.009706
*SEIDEL ABERRATIONS
SRF SA3
CMA3
AST3
PTZ3
DIS3
1 -0.186965 0.122397 -0.080127 -0.050523
2 -1.008444 -0.075329 -0.005627 -0.022490
3 0.527528 0.124226 0.029253 -0.000935
4 0.752953 -0.135018 0.024211 0.036744
5 0.173777 0.046864 0.012638 0.018152
6 0.000562 0.001402 0.003497 -0.007389
0.085530
-0.002100
0.006668
-0.010930
0.008303
-0.009706
Strong aspheric puts in enough overcorrected spherical aberration to cancel
about ½ of that from the very strong lens at surface 3 and 4.

61. To remove aspheric first add a zero
thickness parallel plate right against the
aspheric surface. Then remove the
aspheric from the surface. Then do an
optimization run where the only variables
are the curvatures of the parallel plate and
those of the lens that it is next to = 4
variables. The only aberrations to be
corrected are 3rd order spherical
aberration, 3rd order coma, the paraxial
focus, and 5th-order spherical aberration. It
may take several tries before you find a
good solution because there will be several
local minima, mostly bad ones. There will
probably not be an exact solution. Then
this change is put back into the original
design and the whole system reoptimized.

62. If the aspheric lens
is thin then both of
its curvatures
should be varied in
this very simple
optimization, along
with the parallel
plate curvatures. If
the aspheric lens is
thick then you need
to add two parallel
The goal is to replace the aspheric with spherical
plates right against
surfaces that are as close as possible to that location.
Then it is easy to insert the resulting solution, of some the aspheric surface.
surfaces without an aspheric, back into the original
design without disturbing the first-order optics. Then
everything is varied and optimized. Don’t forget to
remove the aspheric from the variable list.

63. Replaces aspheric lens
No aspheric, f/1.0, 20 degrees full field, .02 wave r.m.s. at edge of field

64. A different aspheric design example

65. Long working
distance design
aspheric
.50 NA, 10 degrees full field, telecentric, monochromatic design

66. f/1.0, 10 degrees full field monochromatic design

67. Add a positive lens to
reduce lens powers and
weaken aspheric. Then
slowly reduce lowest order
aspheric coefficient until
performance suffers.
The top aspheric completely
corrects the spherical
aberration of both the 3rd and
4th lenses = strong aspheric.
Aspheric deformation of top
design is +/- 80u while the
aspheric deformation of the
bottom design is +/-20u.
Any further reduction here
in asphericity starts to hurt
performance, unless we add
another lens.

68. Aspheric is removed from surface and thin parallel plate is inserted
next to it. These radii, and only these, are then varied and 3rd order
spherical aberration, coma, and paraxial focus are corrected to zero.

69. Here is the difficult part. The
aspheric surface radius and the
parallel plate radii may not be
enough to correct the 3rd order
spherical aberration, coma, and
focus when the aspheric is
removed. If the aspheric lens is
thin then vary its other radius as
well. Then there may be more than one solution. Add 5th order
spherical to what is being corrected and minimize. Look for several
alternate solutions. This may take some experimenting.

70. If the aspheric is not on a
thin lens then try to move it
to a nearby thin lens or
move it to a parallel plate
added in next to the thick
lens. You may need two
parallel plates then to get
the variables that you need
to be able to remove the
aspheric. This will give a
more complicated design
than if you don’t need to do
this.

71. Two alternate solutions to
replace the aspheric, with
very similar performance.

72. aspheric
Adding two lenses allowed
the strong aspheric to be
removed while keeping the
same performance.

73. Questions?