a general lens design method, with a photographic lens example

A General Design Method, With Photographic Lens Examples

Dave Shafer

A General Design Method

1) Always do a monochromatic design first, even if starting from an existing color-corrected design. See if color-correcting surfaces can be removed with no loss in monochromatic correction. Goal is simple starting design with single glass type.

1) Find the simplest design that meets the required monochromatic performance

2) Use aspherics during the monochromatic design evolution but remove them later

3) Find locations to add color correcting surfaces that require the least change in the monochromatic design and the least change in the monochromatic performance

4) Minimize amount of color inside the design. Use no more than 3 glass types.

5) If run into problems, always go back to an earlier monochromatic design to solve them

Design method shown with a photographic lens design example

• Design a 50 mm focal length f/1.7 camera lens for a digital sensor with a +/- 15.7 mm diagonal field

• Length to image < 170 mm

• Spectral weights quite small below .4861u, about equal from .4861u to .6563u

• Distortion < 1%, vignetting 50% at edge of field

• Focus down to 500 mm object to image, with no performance loss

• Very high MTF performance required

Need very high monochromatic performance in order to be able to reach desired polychromatic MTF, and even then need very good color correction for this 50 mm f/1.7 lens.

Need about .14 waves r.m.s. or better at .55u on axis and .22 waves r.m.s. or better at .55u at edge of field, with 50% vignetting

Typical monochromatic MTF that is needed Required polychromatic MTF

Axis spec

Edge of field spec

Polychromatic MTF specs

Monochromatic MTF

Design starting point – monochromatic, no aspheres, all BK7. Double-Gauss plus front lens, gives long BFL

On-axis = .26 waves r.m.s., edge of 50% vignetted field = .46 waves r.m.s.

Need 2X improvement in monochromatic correction

Low glass index designs

• In the beginning stages of a design it might not be clear how important secondary color will be

• I always start out with a low index monochromatic design. If secondary color is important, then reducing it is easiest in low-index designs – shown here later

• If it is not important then you can always raise the glass index later

Splitting front or back meniscus lens does little to help performance. Use aspheres to find out where design needs new correction means.


Split lens

Use “temporary” aspherics to

1) ease transition between one solution region and another nearby one.

2) Identify where in the design you need to split lenses or add elements.

3) Predict what performance will be of design once lenses have been added and the aspherics have been removed

1) Add an aspheric at front of design, at back of design, and one or two in the middle of the design

2) Use only 4th and 6th order terms

3) Optimize design

4) Remove aspherics, one at a time.

5) Some can be removed with little effect, others require replacing an aspheric lens with a doublet lens


Four Aspheres Design

This exceeds the monochromatic MTF goals

Lens added so can have an aspheric near the stop


Two of the four aspherics were removed without much effect, since they did very little to help the correction

Has acceptable monochromatic correction

2 aspherics remain

Strong lens with aspheric was split in two, with no aspheric

Only one aspheric left

On axis = .13 waves r.m.s., edge of 50% vignetted field = .25 waves r.m.s.

Almost meets monochromatic wavefront, MTF goals

New shape

Sometimes an extra lens, here used to help replace an aspheric single lens, can also help the design move into a different solution region – just as aspherics can do that

Once the design is in the new solution region, the extra lens might not be needed anymore, although it was needed to make the transition . Here this lens will be gone in the final design,just as we removed aspherics that were only temporarily in the design.

How to replace an aspheric surface with a doublet lens

1) Add zero-thickness flat lens next to aspheric

2) Write down system 3rd-order values

3) Remove aspheric terms Remove all system variables. Remove merit function

4) Vary only flat plate radii and aspheric surface radius

5) Correct spherical aberration, coma, and Petzval to values from step 2)

(A systematic method)

Aspheric

Zero power, zero thickness lens

Simplest case, where there is no other lens right next to aspheric surface

Case where there is already another lens surface next to aspheric

• 2 new surfaces plus aspheric surface radius = 3 radii variables

• Correct 3rd-order spherical aberration, coma, and Petzval = 3 aberrations

• Petzval correction makes power of these 3 surfaces be the same as original power of aspheric surface. Can be done by other means too.

• Solutions are found the easiest, due to non-linearities, if aperture stop is temporarily shifted to be at the aspheric surface

• With stop at aspheric, aspheric has astigmatism and Petzval linked together,

• So only need to correct for spherical aberration, coma, and Petzval

• Then insert the new doublet without an aspheric in the original system in place of the aspheric lens, and reoptimize with all system variables

1) When a lens is right next to an aspheric lens, we can use the radius of that lens that is closest to the aspheric as another variable, in addition to the two new radii added in of the flat plate. You just have to keep the sum of the curvatures fixed. If aspheric element is thin than can use both of its radii as variables. But best results happen if all radii variables are in direct contact.

2) Multiple solutions are best found by starting out with radii made to have right net power and be +/- doublet, -/+ doublet, or with one lens being a negative meniscus. Try all of these.

New solution region Aspheric replacement

No aspherics

Last aspheric

Is now weaker

Design with no aspherics meets monochromatic wavefront and MTF goals

Removing last aspheric was quite difficult. Several doublets were tried before this one was found with good higher-order match to aspheric it replaced

Monochromatic MTF

Summary

Starting point = very optimized monochromatic design with no aspherics

3 or 4 aspherics added to get best performance

One at a time, aspherics are removed or replaced with doublet. This can be tricky and take a lot of work. Result usually has same performance as aspheric design

Color correction plan

1) Minimize the amount of color inside the design. Use low dispersion crown glass like FK51.

2) Use glasses with good partial dispersion match. This requires strong curves, so have to compromise some in glass choice or design complexity

3) Don’t use positive lens flints to help with lateral color correction. It hurts secondary color. Might be hard to avoid in wide angle lenses.

4) Try to correct color with the smallest changes to the monochromatic design first-order

1) Minimize the amount of color inside the design. Use very low dispersion glass like FK51 for positive lenses.

2) Use glasses with good partial dispersion match.

Demonstration of first two color correction design principles

Ultra low dispersion glass

Relative partial dispersion

Herzberger secondary color correction methodConnect 3 glasses to give a triangle with largest possible area, to minimize lens powers

Extreme example = FK51, SF57, and KZFS1 - all three are anomalous dispersion glasses

Relative partialdispersion

Three anomalous glasses gives very small residual color. Very dense flint in front has very little power.

Herzberger method gives good results but requires 3 glasses. If you avoid extreme crowns and flints then result is very high lens powers.

SF57-FK51-KZFS1

100 mm focal length

Herzberger secondary color correction methodConnect 3 glasses to give a triangle with largest possible area, to minimize lens powers

As base of triangle becomes more horizontal , the power of the 3rd glass gets weaker and weaker. That 3rd glass disappears when triangle base is horizontal.

Want two glasses to be on horizontal line for super-achromat. FK51 and BK7 are a good pair. Glass pairs on “normal” glass line giveordinary secondary color, much larger than superachromatism.

BK7 and F2 fall on “normal’ glass line

Glass pairs with the same partial dispersion have relatively small dispersion difference, so strong lens powers are needed for a given focal length, or multiple doublets stacked up in a row.

BK7-F2 achromat FK51-BK7 superachromat

10X smaller scale than other graph on left

Quadratic color Cubic color

Crown/flint glass pair Crown/crown glass pair

• When partial dispersions match, want largest possible dispersion difference to reduce required lens powers for achromatism. FK51 and the BK glasses are therefore the optimum glass pair by this criterion.

• SSKN8 and KZFSN4, for example, have good match for partial dispersion but very small dispersion difference - so requires very strong lens curves in a doublet.

• There are significant differences within the BK glasses, when matched to FK51. BK1 is better than BK7 for residual secondary color.

Calcium Fluoride and Silica FK51 and BK1

Has small reverse secondary color Very flat over most of spectrum

High index of LAK8 makes for weaker curves, better aberrations and chromatic variation of aberrations, but it also increases the Petzval of the doublet.

FK51- LAK8 doublet

Relative glass priceF2 - 1.6BK7 - 1FK51 - 16LLF1 - 2.5KZFSN4 - 11LAK8 - 3

• Result of all this is good secondary color correction with just two glasses

• These two glasses, FK51 and a BK glass have very high transmission in the blue region

• The resulting design is a very low index design.

• If the blue wavelengths are not too important than can use a more dispersive flint than BK glasses. Result is reduced (but not corrected) secondary color and weaker lens curves.

• All of these results are for thin lenses in contact. In a real design with substantial lens separations, like a Double-Gauss or Distagon, the optimum glass pairs may shift some on the glass chart.

• But this is a relatively small effect. Try some glasses near the thin-lens optimum choice to see what gives the best result.

• The importance of all this depends on the spectral weighting. If deep blue wavelengths are important then secondary color can be very important.

3) Don’t use positive lens flints in front of aperture stop to help with lateral color correction. That increases the amount of color inside the design and hurts secondary color. Might be hard to avoid in a wide angle lens.

Demonstration of 3rd part of color correction method

Use temporary stop shift to help in correcting lateral color

• If axial color is uncorrected then there is always an aperture stop position that corrects for lateral color

• If we achromatize at that stop position, the design is then corrected for both axial and lateral color

• It stays that way then, regardless of stop position

• In an ideal world we could then correct both axial and lateral color with a single cemented surface between two glasses.

Temporary stop shift

10X smaller scale

Lateral color in all BK7 design

Move aperture stop to find out what position corrects for lateral color, then achromatize at that location, then move stop back to original position.

Original stop location

Temporary stop location

= lateral color corrected stop position

All same glass typeActual stop position

More dispersive positive power to left of stop moves lateral color corrected stop position to the left. So does more dispersive negative power to right of stop.

1) Achromatize at stop position that corrects for lateral color

2) May require thickening up a lens there to give enough room for a strong cemented surface

Thicker lens

Too thin for strong cemented surface

3) There might be a small monochromatic correction penalty

Another example – a more inverse front end shape.

All same glass type – SK16

= stop position for no lateral color

To achromatize at this stop location requires much too strong a cemented surface of F2 glass – not practical

F2 flint glass here

Shifts lateral color corrected stop position to here

Then achromatize here

Stronger flint here All SK16 except

Shifts achromatizing position further to left

Then achromatize here

Color Correction Summary

• Goal is to correct color with smallest change to the good monochromatic correction

• Temporary stop shift shows where to add color correcting surfaces

• Glass choice can minimize color inside the design, giving good chromatic variation in aberrations and low secondary color

• Result has few flint lenses and very few glass types

Chromatic variation in aberrations is mostly induced by color coming into certain surfaces, and is not mainly an intrinsic aberration.

All lenses are FK51 glass

Largest amount of spherical aberration in design

Proof –

1) set index of that lens to be same for all wavelengths. Result is almost no change in chromatic variation of spherical aberration. It is not intrinsic to that surface

2)) set index of lenses before that surface to be same for all wavelengths, so no color coming into that surface. Result = almost zero chromatic variation in spherical aberration

3) This shows why we want to minimize the amount of color inside the design

Design is not good enough – too much secondary color. Need different glasses. But keep the very good monochromatic correction

All BK7 except F5

BK7 and F5 design, no aspherics

Almost meets polychromatic MTF spec, but is slightly low

All FK51 except LF5 50 mm f/1.7, no aspheres, just two glass types

Doesn’t quite meet the on-axis MTF goal. Very constant over the whole field

50 mm, f/1.7 design with no aspheric and just two glasses

All FK51 except LLF6

Extra lens added

A low index design – highest index is n = 1.53!

Once get a good design, look for other versions with same number of lenses

Very slightly short of MTF specs

Meets MTF specs

This alternate solution has a stronger negative lens, so making it a flint glass shifts the lateral color corrected stop position further to the left = a good thing

This doublet replacement for the last aspheric lens has very little negative power, so making that lens a flint glass has little effect on lateral color

Try to correct axial and lateral color with the smallest change to a good monochromatic design, and adding the least number of new lenses

• If blue spectrum is more important than in this example, then secondary color must be better corrected

• Then flint glasses must be better partial dispersion match to FK51 crowns, such as BK glasses

• This requires more lenses to keep the negative lens powers from being too strong.

Moves as a pair

Focusing from infinity down to 500 mm object to image distance.

Back end of design, after aperture stop, might be common to several different front ends – for different focal lengths

stop

stop Correction should not change, throughout focusing range - very hard to do

And you thought that you were all done!

Best plan is to go back to monochromatic design and temporary aspherics and build in good focusing correction at an early stage of the design evolution. Be willing to start over !Difficult tasks should not be left to the end of the design process, but should be solved much earlier.

A General Design Method - Review

1) Always do a monochromatic design first, even if starting from an existing color-corrected design. See if color-correcting surfaces can be removed with no loss in monochromatic correction. Goal is simple starting design with single glass type.

1) Find the simplest design that meets the required monochromatic performance

2) Use aspherics during the monochromatic design evolution but remove them later

3) Find locations to add color correcting surfaces that require the least change in the monochromatic design and the least change in the monochromatic performance

4) Minimize amount of color inside the design. Use no more than 3 glass types.

5) If run into problems, always go back to an earlier monochromatic design to solve them

My time is finished - any questions?