Simple mirror systems with unusual characteristics

Dave Shafer

Example: a non-reversing mirror

Single mirror examples

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

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.

Higher order astigmatism is opposite sign to lower order

No astigmatismtangential

sagittal

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

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

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

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

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 A

Field Point A

Field point B

Field Point B

Astigmatism curves

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.

Single reflective surface NA=1.0 aplanat

Spherical mirror with diffractive surface, or reflective Fresnel lens.

Focal length = radius of mirror, due to negative diffractive power or effect of Fresnel surface.

Simple diffractive power – no diffractive or reflective asphericity

No spherical aberration or coma of any order

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!

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

No aspheric is required if separate lens from mirror and then bend the lens.

But still is not zero thickness

Diffractive mirror has zero thickness, can be given required Petzval. This is corrected for 3rd-order spherical aberration for all conjugates.

Diffractive mirror

Negative diffractive power, positive mirror power

Petzval of diffractive power is always zero

2X to 10X zoom beam expander

Reflective diffractive element works over a range of conjugates

Possible use of this idea

Beamsplitter in converging light puts in several different types of aberrations, in conventional view, but if shift axis it is only a small off-center piece of axial pupil and spherical aberration. Can then be corrected with a weak power spherical mirror.

If need both images to exact same scale, then use sandwich beam splitter and pre-correction

Post - correction

Pre-correctionmirror

mirror

Two-mirror designs

Three reflections. Working distance = concave radius/2

Five reflections. Working distance = 2/3 concave radius

Offner concentric 2 mirror relay versions

Notice the 10X scale difference

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.

5X, anastigmat 5X, no 3rd, 5th spherical aberration

More obscurationBad coma

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 comaObscuration = 60% diameter.

Concentric spheres Not concentric

Not concentric

Curved image

1.0X relay, bad coma cancels by symmetryNo 3rd, 5th, or 7th order spherical aberration

Aplanatic Only spherical surfaces

Red shows inner rays of obscuration

Blue shows outer rays of light cone.

Small unused area of mirror around hole

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.

Stray light problem

Rays hit area unused by main ray path

4 reflection stray light path

Main image

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.

Obscuration = 45% diameter,Concave mirror area (ignore hole) = 22X effective area of obscured pupil.

Obscuration = 70% diameterConcave mirror area (ignore hole) = 22 X effective area of obscured pupil

Concentric spheres anastigmats

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.

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.

Flat Image Anastigmat - 3.3X Relay

2 spheres, 4 reflections, corrected for 3rd-order spherical aberration, coma, astigmatism and Petzval.

Mirrors have same radius

Magnification is an important variable and 3.3X is needed for this solution

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.

Flat image anastigmatic telescope. Best used for ring field or strip field.

4 spherical mirrors – all nearly the same radius

Finite conjugate versions are also possible

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.

Spherical mirrors, same radius, corrected for 3rd order spherical aberration

Bad comaSmall obscuration

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.

Pupil position for no astigmatism

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.

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.

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?

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 reflectionsDifferent mirror separation from previous slide examples

Two spheres, six reflections

Alternate solution – same mirrors but different spacing, of .866 R instead of R/2This is +1.0X afocal and every point is imaged back onto itself after 6 reflections, with no 3rd –order aberrations.

The lesson here is that even very simple systems can have more than one solution region.

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

These designs so far are almost all with just spherical surfaces.

What can be done with simple aspheric designs?

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

Schwarzschild two aspheric mirror design for collimated light

With just two mirrors the first order layout is an important design variable

Schwarzschild flat image anastigmat with two oblate spheroids

Concave mirror must be 2.4X larger than convex mirror for collimated input

Unobscured version

Corrected for spherical aberration, coma, astigmatism, and Petzval

2 aspheric diffractive mirrors

Or two aspheric Fresnel mirrors

Diffractive surface adds variables to mirror surface

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

Three-mirror designs

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

Image derotator for system with an intermediate image

Intermediate image

Grazing intersection angle can give huge size, and limits possible f# of system

Fast f# solution – split wavefront

Derotator for system with intermediate image

5X, anastigmat 5X, no 3rd, 5th spherical aberration

More obscurationBad 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.

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

Next are several afocal systems

Astigmatism between tilted spherical mirrors can give intentional anamorphic effects.

Diffraction-limited at .6328 for 15 mm output beam, in 3X expanded direction

Offner patent design. Anastigmat that can also be corrected for Petzval

Unobscured system requires three off-axis conics

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.

Best higher-order aberrations when both first and last mirrors are centered parabolas.

Folded version of design

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

Co-linear ellipses Crossed axis ellipses

No astigmatism

No astigmatism

pupilpupil

Astigmatism and Petzval corrected

Ellipse-hyperbola-hyperbola

2.2 X afocal wide angle pupil relay

pupil

pupil

Offner concentric design, 2 spheres with 3 reflections, used with collimated input

pupil

pupil

1.0X afocal pupil relay design

Pupils are at center of curvature. Corrected for coma and astigmatism and Petzval but not for spherical aberration

2.0X afocal pupil relay design pupil

pupil

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

Afocal 3 spheres design, with magnification

Corrected monocentric 1.0X afocal pupil relay

10 degree field pupil

10 degree field pupil

Bouwers concentric lens corrects spherical aberration

Combined systems

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

Concentric spheresSame system used backwards

Real image anastigmatVirtual image anastigmat

Any concentric system of spherical surfaces has exactly the same aberrations, to all orders, when used backwards. Very strange, but true!

Unobscured virtual image anastigmatOffner 1.0X relay, also concentric

Combined systems. Virtual image is relayed to a real image.

By dropping concentricity, can correct Petzval and distortion too.

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.

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

Had enough?

The End

Any questions?