Some optical design tricks

Some Optical Design Tricks

Dave Shafer

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.

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.

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.

A concentric lens in double-pass has several exact equivalents

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

Nearly concentric meniscus lenses

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.

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 itsaverage 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.

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

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

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

Reoptimized design

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

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

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.

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.

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.

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

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

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.

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, +.05Total higher-order is much less than Bouwers.

Both designs can be achromatized. Gabor design has more color due to stronger lens power.

Both mirrors are concentric about shifted pupil. Curved image design.

Bouwers design

Gabor design

Two mirror version of design

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

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

Second lens is concentric, with mirror, about shifted pupil. Corrects 5th order spherical aberration

.50 NA, very good correction

Bouwers + Gabor combination

Air lenses

Achromatic aplanatic doublet, SK2 and F2

Alternate design, F2 and SK2

Air lens

Glass lens

Achromatic aplanat

1) Combine first and last lens into one lens

2) Change nearly concentric air lens into nearly concentric glass lens

3) Re-correct for spherical aberration and coma

4) No longer have color correction

Air lens

Glass lens

Alternate design conversion

1) Combine first and last lens into one lens

2) Change nearly concentric air lens into nearly concentric glass lens

3) Re-correct for spherical aberration and coma

4) No longer have color correction

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.

Centers of curvature

Air lens converted to glass lens can give anastigmatic doublet. Meniscus lens glass thickness is a key variable

Anastigmat, no color correction

Aplanat, no color correction

Stronger meniscus radii in anastigmat design

3rd-order spherical aberration, coma, and astigmatism = 0.0

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 fifth-order 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.

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.

Schmidt telescope with aspheric plate and spherical mirror

Located at center of curvature of the mirror

Aspheric plate

Houghton design – zero power doublet simulates aspheric Schmidt plate

Same glassSpherical mirror

Houghton doublet lens is a nearly concentric outer menisicus lens with a nearly concentric inner air lens inside.

Pull out the air lens and convert design to two nearly concentric glass lenses

Air lens becomes glass lens

Corrected for spherical aberration, coma, astigmatism and axial and lateral color

Same glass

Both lenses act as if they are far from their actual location - near their centers of curvature

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”

Extreme wavelength range design

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

BK7 lenses

Corrected from .365u to 1.50u

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

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.

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.

Alignment insensitive designs

• 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.

Aspheric aplanatic lenses, equal and opposite powers

Astigmatism and Petzval cancel. Each lens can be tilted and decentered without introducing coma.

Alignment insensitive design

All spherical – two cemented achromatic aplanatic doublets.Corrected for spherical aberration, coma, astigmatism, Petzval, and axial and lateral color

Alignment insensitive system

Each doublet can be tilted or decentered without introducing coma

Achromatic aplanatic cemented doublet lens/mirror elements

Corrected for spherical aberration, coma, astigmatism, and Petzval

Alignment insensitive system

Lens used backwards

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. 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 great designer.

Monochromatic design


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

Replacing an aspheric singlet with two or three spherical lenses

A strong asphericf/1.0, 20 degree full field, .03 waves r.m.s.


What is limiting the performance – higher-order aperture aberrations or higher-order field aberrations?


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.


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.


Paraxial pupil

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.


Paraxial pupil

Aspheric

Extra lens allows aspheric to be weakened a lot, even though it does not want to have much power


Paraxial pupil

Aspheric

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.


Paraxial pupilAspheric

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.

*SEIDEL ABERRATIONS SRF SA3 CMA3 AST3 PTZ3 DIS3 1 -0.186965 0.122397 -0.080127 -0.050523 0.085530 2 -1.008444 -0.075329 -0.005627 -0.022490 -0.002100 3 0.527528 0.124226 0.029253 -0.000935 0.006668 4 0.752953 -0.135018 0.024211 0.036744 -0.010930 5 0.173777 0.046864 0.012638 0.018152 0.008303 6 -0.582068 0.001699 0.003497 -0.007389 -0.009706Aspheric lens

Aspheric removed from lens

*SEIDEL ABERRATIONS SRF SA3 CMA3 AST3 PTZ3 DIS3 1 -0.186965 0.122397 -0.080127 -0.050523 0.085530 2 -1.008444 -0.075329 -0.005627 -0.022490 -0.002100 3 0.527528 0.124226 0.029253 -0.000935 0.006668 4 0.752953 -0.135018 0.024211 0.036744 -0.010930 5 0.173777 0.046864 0.012638 0.018152 0.008303 6 0.000562 0.001402 0.003497 -0.007389 -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.

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.

If the aspheric lensis 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 plates right against the aspheric surface.

The goal is to replace the aspheric with spherical surfaces that are as close as possible to that location. Then it is easy to insert the resulting solution, of some 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.

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

Replaces aspheric lens

A different aspheric design example

.50 NA, 10 degrees full field, telecentric, monochromatic design

aspheric

Long working distance design

f/1.0, 10 degrees full field monochromatic design

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.

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.

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.

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.

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

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

aspheric

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Some optical design tricks

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