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Information about Ultra- broad spectrum color correction in optical design

Optical design techniques for extremely broad spectrum color correction, by the optimum selection of glass types.

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Design Goals – Correct secondary color to a very high level. - Avoid strong lens powers - Minimize use of expensive special glasses

Prisms with equal vertex angle (= light deviation power) and same glass type (= equal dispersion) can exactly cancel out color that is between them. The color of a positive lens can be cancelled by an equal power negative lens of the same glass, but then the focal length of the lens pair would be zero, if they were in contact. Instead we want the negative lens to be a more dispersive glass than the positive lens, so that a weaker power negative lens can still cancel out the color and give a total power of the lens pair that is not zero. When the red and blue light rays come to the same focus primary color has been corrected. In a typical contact doublet the negative lens glass is about 1.5X to 2X more dispersive than the positive lens glass.

In a typical cemented doublet lens of BK7 and F2 glass – both very inexpensive glasses – the positive lens power is about 1.7X larger than the negative lens power. So for a 100 mm focal length doublet the positive lens focal length is +41.2 mm and the negative one is -70 mm. If we plot axial focus position versus wavelength in such a color corrected doublet (= an achromat) we get a curve like this one. When the red and blue light rays are brought to the same focus the green light rays are out of focus (= secondary color). Why is that?

The dispersion of a glass is the incremental change in its refractive index for a small change in wavelength. But that changes across the spectrum. For long wavelengths the index changes more slowly than for short wavelengths. For a positive/negative doublet lens with color correction we want the ratio of the two lens powers to not change with wavelength (so that the doublet focal length will not change with wavelength) and this will happen if the ratio of the two glass dispersions does not change with wavelength .

The very non-linear change of a glass dispersion (slope of these curves) with wavelength is not by itself a problem for color correction. There could be two glass types that had the same dispersion ratio across the whole spectrum, as long as they were both non-linear with a fixed relationship to each other. Unfortunately that almost never happens. There are some glass pairs where their dispersion ratio might be 2X in the red wavelength region but 3X or more in the deep blue region. So then the color cancellation in the positive/negative doublet will not be uniform across the spectrum.

A typical glass chart shows glasses as points on a graph with glass dispersion and index of refraction as the two axes. The dispersion can be thought of as the local slope of the index versus wavelength curve (not this one here), which was shown in the previous slide. The local rate of change of that slope (the second derivative) is called the partial dispersion. It is almost the same for all glass types but the small differences result in secondary color. If these differences are plotted versus glass dispersion each glass is a point on this chart.

Small differences in partial dispersion Dispersion Most glasses lie on or very close to what is called the normal glass line. The result is that the amount of secondary color in an achromatized design (with thin lenses in contact) that only uses glasses on the normal glass line will be about the same, regardless of which glasses are used or how many lenses. But if any two glasses have the same partial dispersion in this chart they will lie on a horizontal line. Then the secondary color for that glass pair can be nearly zero.

Normal doublet achromat There are some unusual glass types that can part of a doublet lens and give either very reduced secondary color or nearly complete correction of focus shift over a broad spectrum . But the best design results happen when 3 or 4 glass types are used, instead of just two. At least one of these must be an unusual glass type. Then it is possible to bring 3, 4, or even 5 wavelengths to the same focus and have greatly reduced residual color. We will see now how to do this.

Reference design = BK7 and F2 glass. Both are “normal” glasses. 100 mm focal length. Spectral range is .365u to 1.0u Focus shift over .365u to 1.0u range is .75 mm. If two wavelengths come to the same focus (primary color is corrected) then this residual focus shift is called secondary color.

Typical doublet pair = BK7 and F2 glass BK7 F2 Any two glasses that lie on or close to the “normal” line of partial dispersion versus dispersion will give about the same amount of secondary color – which is related to the slope of the line connecting the two glasses. We want the slope to be zero. Let’s see what can be done with just two glasses.

FK51 and BK7 lie on a nearly horizontal line, with slope about zero = good. But dispersion difference is small = bad = leads to strong lens powers to correct primary color. FK51 is a “special glass” since it is well off the “normal” glass line.

Strong powers due to small dispersion difference requires slower speed design Reverse secondary color compared to BK7-F2 design. FK51 and BK7 gives greatly reduced secondary color. If the focus shift is optimized over a spectral range of .40u to .65u then the curve is a cubic, shown above. But if it is corrected to minimize the total shift over .365u to 1.0u then we get the curve on the left, which is sort of quadratic.

Glasses very close to BK7, like BK1 and BK3, give quite different amounts of secondary spectrum when combined with FK51 glass. BK1, above here, gives 2X smaller secondary color than BK7 and BK3, at left here, gives 1.8X larger secondary spectrum than BK7

• Glasses very close to BK7, like BK1 and BK3, give quite different amounts of secondary spectrum when combined with FK51 glass, assuming primary color has been corrected. Large amounts of color of opposite sign are nearly cancelling. So a very small change in one large number can affect the degree of color cancellation and the amount of residual color. • With just two glasses there are only two variables – the total lens power for one glass type and the total lens power for the other glass type. If the focal length and primary color are corrected then there are no degrees of freedom left and secondary color is fixed, for those two glass types. • With three or more glass types there are extra degrees of freedom and the same amount of secondary color can result for a variety of different glass type combinations.

BK7 – F2 doublet FK51 – BK1 doublet Only BK7 – a single lens Only FK51 – a single lens Scales are all different BK7 power = 2.1X larger than F2 power BK7 power = 1.7X larger than F2 power FK51 power = 1.36X larger than BK1 power FK51 power = 1.34X larger than BK1 power

Plot scales are all different FK51 and BK7 glass, above, has 3X smaller secondary color than FK51 and BK1 glass, below, when both are optimized for .45u - .55u But when design is optimized over .365u – 1.0u, the FK51 and BK7 design, above, has 2X larger color than the FK51 and BK1 design, below

Conclusion • Glasses that give the best color correction for a very broad spectrum may be different from the glasses that are best for a more limited spectral range. • Glasses very close to each other, like the BK glasses, may give considerably different residual color in a two or three glass type design. • These results do not depend on the number of lenses – just on the number of different glass types.

A line with a slope of exactly zero is not possible with FK51 and the BK glasses. As we have just seen, small differences in the BK glasses give sizable changes in the secondary color. Maybe there is an Ohara glass that is like the BK7 glasses but which makes a perfect match to FK51. Let’s find out.

Best Schott match to FK51 glass = BK1 (n = 1.517, v = 64) or PSK3 glass (n = 1.552, v = 63.5) Scales are different Best Ohara match to FK51 glass = Ohara BAL50 (n = 1.560, v = 61) About 70% of FK51-BK1 secondary color Secondary color is about 12X smaller than BK7-F2 combination

It is not physically or chemically impossible for there to be a glass which exactly matches the partial dispersion of a different glass, over a wide spectral region, to give a perfect two-glass super-apochromat. But actual glasses pairs all depart some from that perfect match, especially over a very broad spectral region. By using a third or fourth glass type as well, these departures can be made to almost completely cancel each other out, since there are both + and – departures from a perfect match. That cannot be done with only two glass types. It is hard to discover the best combination of 3 and 4 glass types, as we shall soon see.

Conclusion is that mixing sources of glass, like Schott and Ohara, may give the best possible match to FK51 glass in a two glass design. Let’s look at a match not quite as good – FK51 and KZFSN2, but with a larger dispersion difference = weaker lens powers compared to FK51-BK7 design but worse residual color.

Larger color than FK51BK1 design, but weaker lens powers The larger dispersion difference of KZFSN2 compared to FK51 makes the lens powers considerably weaker than a FK51 –BK1 doublet. Weaker curves means smaller aberrations = good. Both glasses in this FK51-KZFSN2 design are off the normal glasses line. Now let us look at the other end of the glass chart, for glasses, like SF11 - not on the normal glass line.

The usual glass chart is very misleading because a small horizontal change in the glass position on the right hand end, in the dense flint glasses, gives a much larger % change in dispersion than the same shift of a glass on the left hand side of the chart. Let us look at the very dense flints now, which are off the normal glass line.

Herzberger secondary color correction method Connect 3 glasses to give a triangle with largest possible area, to minimize lens powers Extreme example = FK51, SF57, and KZFSN4 - all three are anomalous dispersion glasses Relative partial dispersion

Design Goals – Correct secondary color to a very high level. - Avoid strong lens powers - Minimize use of expensive special glasses Herzberger method of correcting secondary color avoids strong powers but does not give the smallest secondary color

FK51-SF57 – KZFSN4 make a triangle with large area = minimum lens powers, but not the best residual color (tertiary color)

FK51 – SF57 – KZFSN4 does not have the best residual color for a three-glass type design.

Red triangle is FK51 – SF57 – KF3, smaller area than green triangle of FK51- SF57 – KZFSN4, so stronger powers. But much better color.

Best three glass result New Scale FK51-SF57-KF3 design has 3X smaller color than FK51 –SF57 – KZFSN4 design and can bring 4 wavelengths to the same focus, but has stronger powers. Color here is 7X better than best two glass design (Schott glasses).

Conclusions Herzberger glass selection method can minimize lens powers, but will not give the best residual color. If the two extreme glasses on the glass chart triangle are fixed, it can take a lot of experiments to find the best 3rd glass for minimizing color. FK51 – KF3 – SF57 gives the best result but FK51 – LAFn24 – SF57 gives almost the same result, yet KF3 and LAFn24 are very different glasses Two glasses that are extremely close to each other on the partial dispersion glass chart, like LAFn21 and LAFn24 can still give quite different amounts of residual color.

Fixed glasses – FK51 and SF57 Best glass match for third glass is KF3 or LAFN24 Yet these are very different glasses.

Best three glass designs for minimal color Design #1 - FK51, KF3, SF57 Design #2 – FK51, LAFn24, SF57 Relative lens power Relative lens power Radius Radius FK51 = +1.00 +13.8 mm FK51 = +1.00 +17.1 mm KF3 = -.77 -19.0 mm LAFn24 = -.73 - 36.5 mm SF57 = +.055 +450 mm SF57 = + .084 +364 mm Both designs are for 100 mm focal length and .365u-1.0u and both have almost exactly the same amount of residual color. Note how Design #2 has weaker radii and also that the SF57 lens has very little power in both designs.

.20 NA 100 mm focal length, .365u – 1.0u, small field Axial correction Only 5 lenses are needed to give diffraction-limited correction onaxis over the whole .365u – 1.0u spectrum. Design on the left has FK51-KF3- FK51-KF3-SF57. Design on right has LAFn24 instead of KF3. Almost same performance but weaker curves due to more dispersion from LAFn24.

Focus shift over .365u to 1.0u limits performance of these designs. To have diffraction-limited correction at higher NA values (which have a smaller depth of focus) we need to further reduce the residual color. This requires a fourth glass type and it allows a 10X reduction in residual color!! The focus then only shifts by +/-0.5u over the .365u to 1.0u range, in a 100 mm focal length design. There are 5 wavelengths brought to the same focus. But the lens powers are stronger than the best 3 glass design. It also comes with a surprise: The best possible 4 glass design seems to consist of FK51– BAK1- SF1- SF57, with two high index flint glasses, in a positive/negative pair. Of course in a paraxial design with thin lenses in contact it makes no difference what order the lenses are in. This is a really amazing level of color correction, over that very broad spectrum, and includes (in the next design) spherochromatism correction to the same high level as the focus shift correction. But is it real??? What is effect of a very small glass index measurement error at several wavelengths? This needs more study.

Brings 5 wavelengths to the same focus Best 4 glass type design. Focus shift for 100 mm focal length is +/- .50u over .365u to 1.0u spectral range

4 glass types, 7 lenses, NA = .25 100 mm focal length, .365u-1.0u FK51, BAK4, FK51, BAK4, SF57, SF1 Because beam diameter becomes smaller as the rays go through the design, the optimum glass choice changes a little from when all lenses are assumed to be thin lenses in contact. So the thin lens optimum of FK51, BAK1, SF1 and SF57 changes a little – the BAK1 glass wants to become BAK4, which we did here.

In any fairly simple design there will be a competition between correcting paraxial focus shift and correcting the spherochromatism and higherorder spherical aberration. The top design has FK51, BAK4, FK51, BAK4, SF57, SF1 and has the best focus shift but not the optimum spherochromatism. The bottom design has LAFn28 instead of BAK4 and has better spherochromatism and higherorder spherical, due to weaker curves, but not quite optimum focus shift. The combined performance is about the same for both designs.

Good news - there are several glass combinations that give very nearly the same smallest possible residual paraxial color. We here are not looking for just reduced secondary color - very many glass pairs or combinations can give that - but instead are trying to find the absolute smallest possible result – and there is more than one solution with about the same minimum. That is good – we have choices. The different best glass combinations for paraxial color will differ in ability to correct spherochromatism (depends on how many lens elements are used) and so a compromise merit function results. The combined total color merit function may be smallest when one or more of the glasses are shifted a little away from the paraxial study choices. Bad news - there does not seem to be any systematic way to find these optimum glass matches. Varying model glasses is no good. In addition to a model index, dispersion, and partial dispersion you would also need the change in the partials across a wide spectrum. Worse still –glasses with almost the same index, dispersion, and partial dispersion can give quite different residual color. And what about mixing Schott and Ohara? It is a nightmare – lots of trial and error. There is no way to be sure of getting the optimum choices.

Summary of results For the extremely broad spectrum studied here I have identified the best two glass pair for FK51: (FK51 and BK1 or PSK3) as well as the best three glass combinations: FK51, KF3 or LAFn24, and SF57, and four glass combinations: FK51, BAK1, SF1, and SF57. Residual paraxial color is about 10X better with the best three glasses compared to the best two glasses and a further 10X improvement comes with the best four glasses. In actual designs (not zero thickness lenses in contact) the axial ray height going through the design will change some and so the optimum glass choices may shift a little. Optimizing spherochromatism and higher-order spherical may require some compromise with paraxial focus shift and the best total color result may shift glasses a little away from the paraxial. There is no good way, by trial and errors, to find these best glass combinations. And the best glasses combinations change for different spectral band width situations. But there are some automated glass selection programs available and some seems to work quite well.

Zeiss, Olympus, and Nikon make very broad spectrum high NA microscope objectives, with excellent correction for secondary color. However, these are very small and very short focal length designs compared to the 100 mm focal length case we have been looking at here. If those designs are scaled up to be 100 mm focal length the residual secondary color would be much larger than the best 4 glass type design discussed here.

Some further thoughts The dense flint glass SF10 has pretty good transmission at .365u and should be used instead of the other very dense flints. The near UV spectral region is where almost all glasses stop transmitting. Only a few have good transmission at .365u and the very dense flints glasses like SF11 or SF57 do very badly. So unless the dense flint glass is quite thin then it is not a good choice for use in a broad spectral band design that includes .365u

The very dense flint design with SF57 has longer radii than a SF5 design but there is not a big difference, as this 3 glass type design shows, and there is no comparison to the transmission at .365u Design comparison for highest level color correction Glass Lens power Glass Lens power FK51 + 1.0 FK51 + 1.1 KF3 -.77 BAK4 -.94 SF57 +. 054 SF5 +.12 The focus shift versus wavelength is about the same for both designs and is essentially the best that is possible.

Design comparison for highest level color correction Glass Lens power Glass Lens power FK51 + 1.0 FK51 + 1.1 KF3 -.77 BAK4 -.94 SF57 +. 054 SF5 +.12 Total absolute power of lenses = 1.0 +.77+.054 = 1.824 = 1.1+.94+.12 = 2.16 The normal dense flint glass (SF5) design on the right has 18.5% higher total absolute lens powers than the design on the left with the anomalous dispersion very dense flint glass (SF57). But the highest power element in the design on the right is only 10% stronger than the other design.

Glass choices should therefore be more based on lens glass cost, transmission, and amount of residual secondary color – and not on total lens powers. Trying to get the largest possible chart area linking three glasses in the Herzberger color correction method may give the smallest absolute total lens powers but it may not make a big difference to the power of the strongest lens. It depends on the particular case.

Glasses are FK51, Lafn28, FK51, Lafn28, FK51, SF10, SF8 By trying several glasses a better 4 glass design was found, for higher NA use, and it can be pushed to .32 NA with diffraction-limited axial wavefront from .365u to 1.0u

Axial OPD curves for .32 NA design with 7 lenses and 4 glass types, for .365u through 1.0u.

Conclusion The optimum broad spectral band design needs to balance • Lens powers and aberrations • Transmission • Cost • Amount of residual focus shift

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