Optical Design Using Stop-Shift Theory

Dave Shafer

• The use of first and 3rd order stop shift theory can lead to new types of designs and a better understanding of existing designs.

• No computations are necessary to benefit from stop-shift theory – it just involves a few basic principles and some temporary changes in aperture stop position.

• Experiments can be carried out in your head. Computer calculations only happen after you are done with the conceptual work.

The view of Copernicus, that the sun is the center of the solar system, is widely considered to be the correct view and the very complicated system of Ptolemy, with epicycles and with the earth the center of the solar system, is considered wrong. But neither is right or wrong, if they correctly predict the apparent motions of the planets. One system is much simpler and easier to understand. Stop shift, especially temporary shift, helps understanding in optical design through simplicity – just like Copernicus.

CopernicusPtolemy system

Let’s start out with 1st order stop-shift theory, which relates lateral and axial color.

1) If a system has axial color then lateral color is linear with stop position. That means that there must be a stop position that makes primary (1st-order) lateral color be zero.

2) If a system is corrected for primary axial color, then primary lateral color is independent of stop position.

3) A thin lens with the stop in contact has no lateral color.

4) A thin lens at a focus has no axial or lateral color.

Field lens

3 silica elements and a spherical mirror gives a deep UV high NA objective.

Design with broad spectral range

What is the aberration theory behind this very simple design?

Answer – it involves stop-shift theory

Schupmann design with virtual focus

Both lenses are same glass type

Axial color is linear with lens power, quadratic with beam diameter, so color here cancels between the lenses

Offner improvement – a field lens at the intermediate focus The field lens images the other two lenses onto each other

Field lens

Field lens

1) Put stop on first lens, then choose power of field lens to image it onto the lens/mirror element. Stop is then effectively at both places.2) Then neither of those elements has lateral color. Power of lens/mirror element corrects axial color.3) Field lens imaging and only one glass type corrects for secondary axial color too (Offner theory).4) Then can put stop anywhere.

Low-order theory of design

• A key point – the aperture stop was only temporarily located at a place where the theory is simple to understand and the aberration correction method becomes obvious.

• Then later the stop is moved to where it needs to be – like in order to have a telecentric system.

• Once the aberrations are well-corrected they do not change (at the lower-order levels) when the stop is moved.

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• Lateral color depends on aperture stop position, since axial color is not corrected.

• Move the stop around and find out what position makes lateral color be zero.

• Then correct axial color at that location. Let’s try using a diffractive surface.

Lateral color for front stop position

All same glass type

Telecentric design

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Aperture stop position that corrects lateral color

We move the stop position back and forth until we get lateral color = zero

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Aperture stop position that corrects lateral color

If we correct axial color here, with a diffractive surface, then both axial and lateral color will be corrected. Then we can move the stop back to where we want it, and both color types will still be corrected.

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• This same design method indicates where to add lenses for color correction

• It minimizes the number of extra lenses needed for color correction

• But it may indicate adding color correcting lenses where we don’t want them, because of space constraints

• Then we rely on conventional color correcting techniques

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Diffraction-limited monochromatic f/1.0 design with 5.0 mm field diameter

Telecentric image

Aperture stop

Long working distance design

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Axial color not corrected

Aperture stop position for no lateral color

Aperture stop position for no lateral color may not be in a desirable, place - as in this long working distance design. We don’t want to put axial color correcting lenses there, in the long working distance space.

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Cemented triplet Cemented doublets

In these cases you have to use two separated groups of color correcting lenses, instead of just one, for axial and lateral color correction.

Correcting secondary lateral color

Stop position for best performance

SF2

SK16

Design is corrected for primary axial and lateral color, has secondary axial and secondary lateral color.

• Suppose primary axial and lateral color are corrected.

• If a design has secondary axial color then secondary lateral color is linear with stop position.

• So there must be a stop position then that corrects for secondary lateral color.

• If you fix secondary axial color at that stop position, then both secondary axial and lateral color will be corrected.

• Then you can put the stop anywhere with no effect.

Stop position for best monochromatic performance

Stop position for no secondary lateral color

Semiconducter wafer metrology inspection design

• KLA-Tencor in 2005 wanted a “perfect” .80 NA design for .488u - .720u

• Requires correction of primary, secondary, and tertiary axial color to get .999 polychromatic Strehl over that spectral range.

• Needs correction of primary lateral color, and secondary lateral color is a very big problem – doesn’t hurt image quality but gives wafer measurement error.

• Olympus, Tropel, and ORA all worked on this and could not get any better than 10 to 20X more secondary lateral color than was acceptable (needed = 1.0 nanometer over a 80u field diameter).

• I tried a design and also was about 10X too much lateral color

• The solution – use stop-shift theory

• I corrected primary axial and lateral color and partially corrected secondary axial color.

• I found out where the stop position is then where secondary lateral color is zero

• I corrected the remaining secondary axial color at that stop position

• Then I moved the stop back to its telecentric position

Telecentric stop position

Stop position for no secondary lateral color, when secondary axial color is partly uncorrected. Here I add a very low power “dense flint” lens (SF6 glass) with anomalous dispersion and fixed secondary and tertiary axial color. Result is a design with <1.0 nanometer of lateral color, but with telecentric stop in very different stop position from this lens location.

.80 NA microscope objective, 80u field, .999 polychromatic Strehl from .488u-.720u, lateral color <1.0 nanometer.

Aperture stop for telecentric design

Semiconducter wafer metrology inspection design

A 1.0X catadioptric relay system developed using stop shift theory

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

Bad comaSmall obscuration

• If a design has spherical aberration then coma is linear with stop position and astigmatism is quadratic with stop position

• If spherical aberration is corrected then coma is constant with stop position and astigmatism is linear with stop position. Then, for non-zero coma, there is always a stop position that corrects for astigmatism.

• If both spherical aberration and coma are corrected then astigmatism is a constant

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. Plus, how stop shift theory is useful for thinking of a new design.

Aft-Schmidt Design

• If spherical aberration is uncorrected then coma is linear with stop position and astigmatism is quadratic with stop position.

• So then, for non-zero spherical aberration, there is always a stop position that corrects for coma and either 2 or none that correct for astigmatism.

• In some cases (like the Schmidt telescope) the stop position which corrects coma also corrects astigmatism.

Aperture stop at center of curvature of M1

Three spherical mirrors with decentered pupil

Field mirror

Pupil at center of curvature of M3, due to field mirror power

Much spherical aberration

Field mirror images M1 center of curvature onto M3 center of curvature

Aspheric plate

Because of field mirror power the aspheric acts like it is in both the aperture stop and the exit pupil, at the centers of curvature of M1 and M3

Exit pupil

Design is good for rectangular strip fields

Not there

Aspheric plate

Smaller aspheric but more higher-order aberrations

Aspheric acts like it is at the centers of curvature of both M1 and M3, due to power of field mirror

Aspheric mirror and aperture stop

All-reflective - 3 spheres and one asphere

In all of these designs the image is curved

After the system is given good correction, with the Schmidt aspheric, the aperture stop can be moved if that is wanted, maybe to minimize the size of M1. Higher-order aberrations will be affected and the best stop position is at the centers of curvatures of M1 and M3

For afocal case, Petzval is zero

2 X afocal pupil relay

Aspheric plate at either pupil or a concentric Bouwers lens in either place does the spherical aberration correction

Best for rectangular fields, with long direction in X field direction.

Can be a building block in other designs

Afocal version of system

Infrared Target Simulator Design

A system from 1984 – customer wanted an infrared target simulator to test missile heat seeking heads. Requires a distant external pupil. Goals – all-reflective, inexpensive, 8 X 8 degree square field, f/4.5, 200 mm aperture, unobscured, .05 to .10 millirad spot on a flat image

External pupil of simulator matches internal pupil of missile head

Part of the solution – two aspheric mirrors with same radius. Corrected for spherical aberration, coma, astigmatism and Petzval. One of Schwarzschild’s designs from the 1890’s

Two oblate spheroid mirrors

Field is all set to one side of axis. Stop could be on either mirror. Here it is on the larger mirror to minimize its size due to field size. Now how do we get an external pupil?

Reed patent – images one pupil to another. Offner independently invented this system but with finite conjugates, imaging an object to an image, not pupils – which is done here.

Center of curvature of monocentric Reed system is imaged by convex Schwarzschild mirror onto concave Schwarzschild mirror

Reed 1X afocal pupil relay

Also a pupil

Schmidt aspheric needed to correct Reed system could be placed either at first pupil or at second one.

By putting Schmidt aspheric onto this pupil an oblate spheroid becomes a sphere!!!

Fold flat is made a very long radius sphere.

Only one asphere and that is a centered one, not an off-axis one

New idea for design – get almost constant astigmatism over field and then correct with weak sphere on tilted fold flat mirror

Fold flat is made a very long radius sphere.

Only one asphere and that is a centered one, not an off-axis one

This gives a 3X improvement in performance.

Two-Axis Asphere Design

Schmidt aspheric is sum of what corrects the spherical aberration of the primary mirror + what corrects for the secondary mirror

Hard to baffle image

Separate part of aspheric for primary mirror from that for secondary mirror, and place on opposite sides of aspheric plate. Then tilt secondary mirror and decenter its aspheric to follow secondary’s center of curvature.

Easy to baffleTwo-Axis Aspheric Design

Instead of two rotationally symmetric aspherics on opposite sides of the Schmidt plate, with decentered axis, combine aspherics into a single non-rotationally symmetric aspheric.

Early warning missile defense system

Work I did in 1972, 40 years ago.

If a missile comes over the rim of the earth it will be seen here by a satellite against a black sky, but it will be very close to an extremely bright earth, which gives an unwanted signal that vastly exceeds the missile’s heat signal. But that is the easy case. Much worse is when the satellite is on the night side and the missile is seen against a sun-lit earth’s limb.

With the sun behind the horizon, the earth’s limb is 1.0 e+10 times brighter than the missile signal.

Rim of aperture stop is source of diffracted light

Light from earth limb

Second aperture stop is smaller than image of first stop, blocks out-of-field diffracted light from earth limb.

Lyot stop principle

Two confocal parabolic mirrors give well-corrected imagery

(Mersenne design)

Aperture stop

Lyot stop

Add M3, a spherical mirror with M2 at center of curvature

Put Schmidt aspheric for M3 onto M2, then M2 parabola becomes a hyperbola

Image from M3 is not accessible

M1

M2

M3

parabola sphere

Image of M1 by M2, at center of curvature of M3

Accessible image with conventional aspheres, but a long system

Alternate design, with Schmidt aspheric added to M1 instead of M2

Parabola + Schmidt aspheric = hyperbola

parabola

sphere2-axis aspheric

Well-corrected image in an accessible location

Image is curved because of Petzval

Parabola + decentered Schmidt aspheric = 2-axis aspheric

High NA laser beam expander

Aplanatic surface

Surface radius chosen to correct spherical aberration of first surface

Surface at focus of first surface

(There are two different values that do this, on either side of the perpendicular incidence condition. One speeds up the divergence, and we choose that, while the other one slows down the beam divergence.)

Put stop at center of curvature of first surface

Choose curvature of surface at the focus to make the chief ray go through the center of curvature of the 4th surface

1st surface has no coma or astigmatism. 2nd surface is at an image, 3rd surface is aplanatic, so no coma or astigmatism, 4th surface has no coma or astigmatism because of where pupil is. Spherical aberration cancels between 1st and 4th surface

Stop can be placed anywhere, once aberrations are corrected. Then computer optimize the design

So system is insensitive to tilt of entering collimated beam

Cascaded Conic Mirrors

• A conic mirror with the aperture stop at either of its focii has no astigmatism of any order.

• This can be proven mathematically with the Coddington equations.

• Some interesting designs are possible using this fact.

Eye pupilellipse

hyperbola hyperbola

Collimated pupil

Part of a fundus camera to look at the eye’s retina

Corrected for astigmatism and Petzval

No common axisof mirrors

OSLO can’t draw this partof surface

ellipse

hyperbola hyperbola

Corrected for astigmatism and Petzval

No common axisof mirrors

Each conic mirror shares one of its focii with the next mirror 2.2X afocal pupil relay

Hand drawn part

pupil

pupil

• Spherical aberration and coma are uncorrected in this design but the pupil size is very small so they don’t matter very much

• But still this means that the aperture stop and pupils cannot be moved from the mirror focii without hurting the zero astigmatism situation of the system

Conclusion

• Stop shift theory gives insight into the aberration theory of a design and also suggests new design possibilities

• Temporary stop shift is a powerful design tool and does not usually require changing the actual final position of the stop, which may be set by the telecentric condition or other constraints