1) The document describes several simple mirror systems that have unusual optical characteristics despite using few elements. 2) Many of the designs use multiple reflections off of spherical or aspheric surfaces to correct aberrations like astigmatism. 3) Unexpected solutions are found, such as designs that correct third-order spherical aberration using a single reflective surface.

1. Simple mirror systems with unusual characteristics
Dave Shafer
Example: a non-reversing mirror

2. Single mirror examples

3. Sagittal rays are collimated
between the two reflections,
while tangential rays form an
intermediate image. No
astigmatism for narrow ring field.
Single
spherical
mirror, 2
reflections

4. tangential
No astigmatism
sagittal
Higher order astigmatism is
opposite sign to lower order
Design has Petzval and astigmatism from two reflections on
concave mirror, but higher-order astigmatism allows for one
good astigmatism-free field point. Sagittal field is always exactly
flat for object at center of curvature of a spherical mirror.

5. Non-reversing mirror. Concave in horizontal
direction, convex in vertical direction
Mirror sends rays across the front of mirror and gives two reflections off
of the same concave surface. Only one reflection in vertical direction
The convex vertical curvature is to keep the image from being very tall and skinny in the vertical direction.
A cylinder mirror with two reflections is non-reversing but would give a tall and skinny image

6. My door handle – is
concave in one
direction and convex
in other. Towards
the base it becomes
convex in both
directions. Print
reflection is nonreversed in middle
of handle but then
becomes reversed
near the base.

7. Three reflections, no
astigmatism for very narrow
ring field. Petzval keeps
adding with more reflections,
but sagittal field is always flat.
2 intermediate images for tangential
rays, but only one for sagittal rays
Sagittal rays
focus here

8. Four reflections, no astigmatism
for very narrow ring field.
This is a “whispering gallery”
phenomenon. N reflections are
possible with a single surface.
As go towards top of
sphere, get more reflections

9. Field Point A
Two other four-reflection no-astigmatism
solutions. Even very simple systems can have
more than one solution to a given condition
(here it is no astigmatism). For “n” reflections
there are n-1 separate field heights with no
astigmatism.
If this spherical mirror is replaced by a glass
sphere, then TIR keeps the “Whispering Gallery”
rays going around and around forever with little
attenuation.
Field Point B
Field point B
Astigmatism
curves
Field point A

10. Even very simple systems can have more
than one solution to a particular problem.
Here there are multiple field heights where
N reflections gives no astigmatism, yet it is
just a single spherical mirror. Always look
for alternate solutions in any situation.

11. Single reflective surface NA=1.0 aplanat
Focal length = radius
of mirror, due to
negative diffractive
power or effect of
Fresnel surface.
No spherical aberration
or coma of any order
Spherical
mirror with
diffractive
surface, or
reflective
Fresnel lens.
Simple diffractive
power – no diffractive
or reflective asphericity

12. 1) A thin (zero thickness) system can be corrected for 3rd
order spherical aberration for all conjugates if it satisfies
certain conditions
2) These conditions require certain values of Petzval
and pupil aberration, and a system thickness of zero.
3) This is very counter-intuitive!
4) A single surface can meet these conditions, and
that is very surprising!

13. An aspheric Mangin mirror can meet the required Petzval
condition by the right combination of lens power and mirror power.
But it is not zero thickness

14. No aspheric is required if separate lens from mirror and then bend the lens.
But still is not zero thickness

15. Diffractive
mirror
Diffractive mirror has zero thickness, can be given required Petzval.
This is corrected for 3rd-order spherical aberration for all conjugates.
Negative diffractive power, positive mirror power
Petzval of diffractive power is always zero

16. Possible use of this idea
2X to 10X
zoom beam
expander
Reflective
diffractive
element works
over a range of
conjugates

17. Pre-correction
mirror
Post - correction
mirror
If need both images to exact
same scale, then use sandwich
beam splitter and pre-correction
Beamsplitter in
converging light
puts in several
different types of
aberrations, in
conventional view,
but if shift axis it is
only a small offcenter piece of axial
pupil and spherical
aberration. Can
then be corrected
with a weak power
spherical mirror.

18. Two-mirror designs

19. Offner concentric 2 mirror relay versions
Three reflections. Working distance
= concave radius/2
Five reflections. Working distance =
2/3 concave radius
Notice the 10X
scale difference

20. Two spherical mirrors, 5 reflections, plus fold mirrors = thin
package in this plane, narrow width out of plane. Correction for
spherical aberration, coma, astigmatism, Petzval and distortion.

21. 5X, anastigmat
Curved image
Concentric spheres
5X, no 3rd, 5th
spherical aberration
More obscuration
Bad coma
Not concentric
If magnification is used as a
variable then there is this
3.73X solution where the 3rd,
5th, and 7th order spherical
aberration = 0. Bad coma
Obscuration = 60% diameter.
Not concentric

22. Aplanatic
Only spherical surfaces
1.0X relay, bad coma cancels by symmetry
No 3rd, 5th, or 7th order spherical aberration

23. Stray light problem
Blue shows
outer rays of
light cone.
4 reflection stray
light path
Rays hit
area
unused
by main
ray path
Main image
Small
unused
area of
mirror
around
hole
Red shows
inner rays of
obscuration
Small unused area around
hole in concave mirror allows
for a four reflection light
path to get through the
system. This can be stopped
by sizing the hole to be
larger.

24. The 4 reflection stray light path, an
unexpected phenomenon, is not just a
problem. It is also an opportunity to
explore new designs that are based on this
phenomenon.
Let us see what can be done with
multiple reflections between two spherical
mirrors.

25. Concentric spheres anastigmats
Obscuration = 45% diameter,
Concave mirror area (ignore hole) =
22X effective area of obscured pupil.
Obscuration = 70% diameter
Concave mirror area (ignore hole) =
22 X effective area of obscured pupil
For a given effective area of the obscured pupil, you need the
same amount of large mirror area (ignoring the hole) in both
designs. But the 2 reflection design requires a 30% larger
diameter concave mirror than the 4 reflection design. Both
designs are anastigmats.

26. If we drop the concentric arrangement,
what can be done to correct for Petzval as
well as the other aberrations, to get a flat
image anastigmat? There are only two
surfaces and both are spheres. Is it
possible? I’m glad you asked.

27. Flat Image Anastigmat - 3.3X Relay
Magnification is an important variable
and 3.3X is needed for this solution
2 spheres, 4 reflections,
corrected for 3rd-order
spherical aberration, coma,
astigmatism and Petzval.
Mirrors have same radius

28. Move field off-axis until system
becomes unobscured. Then the 4
reflections are on 4 separate mirrors.
Then we can independently vary 4 radii
instead of just 2. But keep them spheres.
Result is unobscured flat image
anastigmat. Next slide shows infinite
conjugate example but finite conjugate
examples work well too.

29. 4 spherical
mirrors – all
nearly the
same radius
Finite
conjugate
versions
are also
possible
Flat image anastigmatic telescope. Best used for ring field or strip field.

30. What else can be done with mirrors the
same radius?
We started with concentric mirrors and 2
reflections, then added reflections, then
dropped concentricity.
Now let us back up a little and start over
again with just two spherical mirrors and
only two reflections. The mirrors are not
concentric and have the same radius.

31. Bad coma
Small obscuration
Spherical mirrors, same radius, corrected for 3rd order spherical aberration

32. Pupil position for no astigmatism
Two symmetrical systems make coma cancel, give a 1.0X magnification aplanat
Each half has a stop position which eliminates
astigmatism, since each half has coma. But
pupil can’t be in both places at the same time.

33. Astigmatism-correcting pupil positions are imaged onto each other by
positive power field lens.
System is then corrected for spherical aberration, coma,
and astigmatism, but there is Petzval from field lens.

34. Thick meniscus field lens pair has positive power but no Petzval or axial or lateral color
Result is corrected for all 5 Seidel aberrations, plus axial and
lateral color. This shows how a simple building block of two
spherical mirrors was turned into something quite useful.

35. Equal radii (R) spherical mirror pair
2 reflection separation = .866 R, 4 reflections = .588 R, 6 reflections = .434 R
There is always a mirror separation where after any number of
even reflections the object and image are at the mirror vertex
locations. Then 3rd –order spherical aberration is always
corrected. Why is that? A big mystery! Only true for equal
radii on mirrors. Use as a long path cell for gas absorption?

36. Two spheres, equal and
opposite radii R, and
separated by R/2 . This
6 reflection system is --1.0X, afocal, and is
corrected for 3rd order
spherical aberration,
coma, astigmatism,
Petzval, and distortion
for all conjugates
Two spheres, six reflections
Different mirror separation
from previous slide examples

37. Alternate solution – same
mirrors but different spacing,
of .866 R instead of R/2
This is +1.0X afocal and
every point is imaged back
onto itself after 6 reflections,
with no 3rd –order
aberrations.
Two spheres, six reflections
The lesson here is that even
very simple systems can have
more than one solution region.

38. Is there any use for
this system, which
images the whole 3D
space between the
mirrors back onto
itself with good
image quality?
6 reflections gives +1.0X

39. These designs so far are almost all
with just spherical surfaces.
What can be done with simple
aspheric designs?

40. Two conics (oblate
spheroids) with same
radius and object and
image at mirror centers
gives correction for
spherical aberration,
coma, astigmatism, and
Petzval.
3.7X relay
With 2 spheres it is corrected only
for spherical aberration and Petzval

41. Schwarzschild two aspheric mirror design for collimated light
With just two mirrors the first order layout is an important design variable
Unobscured version
Schwarzschild flat image anastigmat with two oblate spheroids
Concave mirror must be 2.4X larger than convex mirror for collimated input

42. 2 aspheric
diffractive
mirrors
Or two aspheric
Fresnel mirrors
Corrected for spherical aberration, coma, astigmatism, and Petzval
Diffractive surface adds variables to mirror surface

43. Two conic mirrors, three
reflections.
Corrected for spherical
aberration, coma, and
astigmatism, but only for
this geometry configuration.
Alternate solution –
Another example of
multiple solutions in
a simple system

44. Three-mirror designs
There are many possible 3 mirror
designs. Here are just a few that
are more unusual than most.

45. Image derotator for system with an intermediate image
Intermediate image
Grazing intersection angle
can give huge size, and
limits possible f# of system

46. Fast f# solution – split wavefront
Derotator for system
with intermediate image

47. 5X, anastigmat
5X, no 3rd, 5th
spherical aberration
More obscuration
Bad coma
With just two spheres you cannot correct 3rd and 5th
order spherical aberration and also 3rd order coma – you
need more variables. If you stay with spheres then you
need another mirror. One unusual solution has a third
mirror that is almost flat and is three mirrors but four
reflections. It is sort of a folded version of the design on
the upper left here and it is shown next.

48. The nearly flat 3rd
mirror allows the
design to be corrected
for 3rd and 5th order
spherical aberration
and 3rd order coma and
astigmatism.
3 spherical mirrors, 4 reflections

49. Next are several afocal systems

50. Diffraction-limited at .6328 for 15 mm
output beam, in 3X expanded direction
Astigmatism between tilted spherical mirrors can give
intentional anamorphic effects.

51. Offner patent design. Anastigmat that can also be corrected for Petzval
Unobscured system requires three off-axis conics

52. Unobscured ring-field design corrected for spherical
aberration, coma, astigmatism and Petzval with a centered
aspheric. Very good higher-order aberration. First and last
mirrors are imaged onto each other by middle mirror.

53. Folded version of design
Best higher-order aberrations when both first and last
mirrors are centered parabolas.

54. A conic mirror with a pupil at either
focii has no astigmatism of any order
2 or 3 conic mirrors can have their focii coincide
Conic axes don’t have to be colinear
No astigmatism
No astigmatism
pupil
pupil
Co-linear ellipses
Crossed axis ellipses

55. Ellipse-hyperbola-hyperbola
pupil
2.2 X afocal wide
angle pupil relay
Astigmatism and Petzval corrected
pupil

56. Offner concentric design, 2 spheres with 3 reflections, used with collimated input
1.0X afocal pupil relay design
Pupils are at center of curvature.
Corrected for coma and astigmatism and
Petzval but not for spherical aberration
pupil
pupil
Afocal 3 spheres design, with magnification
2.0X afocal pupil relay design
Field mirror images pupils to be at centers
of curvature of both mirrors. For 2.0X or
any other afocal magnification this also
corrects for Petzval
pupil
pupil

57. Corrected monocentric 1.0X afocal pupil relay
10 degree field pupil
10 degree field pupil
Bouwers concentric lens corrects spherical aberration

58. Combined systems
This will show how two very simple
systems can be combined to give a new
design with very attractive characteristics

59. Same system used backwards
Concentric spheres
Real image anastigmat
Virtual image anastigmat
Any concentric system of spherical surfaces has exactly the
same aberrations, to all orders, when used backwards.
Very strange, but true!

60. Unobscured virtual image anastigmat
Offner 1.0X relay, also concentric
Combined
systems. Virtual
image is relayed
to a real image.
By dropping concentricity, can
correct Petzval and distortion too.

61. This telescope/spectrometer from the previous slide, with 5
spherical mirrors, was sent to Saturn on the Cassini spacecraft
and another one will arrive at the asteroid Vesta in July 2011.
This design was one of my first patents, back in 1975.

62. This is a lot of
material to remember,
but this is all available
as a Powerpoint file
that you can have.

63. Had enough?
The End

64. Any questions?