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

1. Ultra-broad spectrum color correction
Dave Shafer
David Shafer Optical Design

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

3. 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.

4. 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?

5. 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 .

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

14. 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

15. •
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.

16. 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

17. 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

18. 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.

19. 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.

20. 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

21. 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.

22. 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.

23. 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.

24. 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.

25. 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

26. 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

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

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

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

30. 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).

31. 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.

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

33. 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.

34. .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.

35. 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.

36. 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

37. 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.

38. 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.

39. 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.

40. 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.

41. 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.

42. 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

43. 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.

44. 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.

45. 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.

46. 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

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

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