A modification of the Double-Gauss design with two diffractive surfaces is described with very enhanced performance. The key is an interaction between the aberrations of the two diffractive surfaces and the aberrations of a curved substrate lens.

1. Dave Shafer
David Shafer Optical Design
Fairfield, CT 06824
203-259-1431
shaferlens@sbcglobal.net
Practical refractive/diffractive hybrid optical designs

2. By using a combination of refractive power and aspherics,
and diffractive power and diffractive asphericity, some
amazing performance can be achieved with just 2 or 3
elements. The 10 mm focal length 3 lens design above is
diffraction-limited at .55u over a 60 degree field at f/1.25
while the 2 lens designs shown here, for 20 mm focal
length, are diffraction-limited at .55u over a 60 degree
field at f/1.0 or a 30 degree field at f/.70 There is no
vignetting over the field.

3. These very simple designs differ in how many aspherics and diffractive
surfaces they use. They have all of the aberration correction and almost of
the focusing power provided by very strong aspherics and strong diffractive
power. Because of that they would be very difficult to make and are not
practical designs. They just show how aspheric and diffractive surfaces are
extremely powerful design tools. But there is another reason why they are
not practical. The strong diffractive surface(s) cause an enormous amount of
color and because of that these designs have a useful spectral bandwidth
that is extremely small. They would be confined to use with a laser.
By going to more complex designs, with 4 or 5 lens elements, most of the
aberration correction and focusing power can be provided by conventional
means. Then aspheric and diffractive surfaces can be used to improve
performance instead of doing all of the hard work. Color is then much less
of a problem as well.

4. The Double-Gauss design is a
very common type of camera lens
design that combines very good
performance with simplicity.
Here is an example with no
aspherics or diffractive surfaces
and with no color correction. It is
just 4 lenses and is f/2.0 with a 20
degree field with no vignetting.
For this 50 mm focal length design the amount of color between .45u and .65u is about
90 waves, with an axial focus shift of 1.25 mm. Usually the Double-Gauss design has one
or two additional lenses for color correction but next we will do that with a single flat
diffractive surface.
All same glass type = SK2

5. The flat plate in the middle of
the design has a diffractive
surface on it and its diffractive
power corrects the color of
the rest of the lenses. But
there is then residual
secondary color due to the
nature of diffractive surfaces.
The uncorrected 90 waves of
primary color of the 4 lens
design is reduced by the
diffractive surface to 10 waves
of residual color over the .45u
to .65u range.
It is an odd feature of the Double-Gauss design
that adding aspherics to all the lenses does very
little to improve performance. This design, with a
diffractive surface, is dominated by secondary
color.
Diffractive
surface

6. A dispersion engineered metasurface (a special kind of diffractive
surface) could reduce that residual 10 waves of secondary color to
essentially zero. But this kind of metasurface is very hard to make.
If there are errors of 10% in the controlled dispersion values (that
have to deal with 90 waves of uncorrected color from the lenses)
that would still leave significant amounts of secondary color left.
Fortunately there is a better way. Instead of using a diffractive
surface to do all of the color correction of this single glass type
design we add a lens of a different glass type that largely corrects
the design for color. Then the diffractive surface has much less
work to do. That is one type of hybrid design. A conventional
diffractive surface then will still leave some residual secondary color
but it will be very much less than the 10 waves of the previous
design example.

7. 2.5 waves of secondary color over .45u to .65u range
This shows how adding a negative flint glass lens to the 4 crown glass lenses corrects the 90
waves of uncorrected color and brings it down to a small residual of 2.5 waves of secondary
color. That is much better than the 10 waves of secondary color that results when all the color
correction is done by a diffractive surface. Next we will add a weak diffractive surface to
further improve the residual color situation.
Flint glass
Perfect lens
Effect of secondary color

8. By using both the extra glass type, with an additional element, and the anomalous
properties of the dispersion of a diffractive surface, it is possible to essentially eliminate
the residual secondary color on-axis from .45u to .65u. But residual lateral color and
chromatic variation of aberrations then limits the performance. This kind of hybrid
refractive/diffractive design is already well-known. Next we will see something new.
Diffractive surface

9. It turns out that there is a significant performance improvement if the flat plate has a
diffractive surface on both sides. The best results happen if the flat plate is allowed to be curved,
and then very good performance is possible. The two separated diffractive surfaces have an
aberration interaction that produces results not achievable with a single diffractive surface. And
if there are no diffractive surfaces, but the same curved base lens in the middle of the design,
there is almost no performance improvement due to that lens being in the design. It is that
interaction with that lens and the diffractive surface(s) that is key to the great performance.
Diffractive on both sides Diffractive on both sides

10. This is an interesting phenomenon – the design
with the curved base lens and diffractive surfaces
on both sides has much better performance than
one or two diffractive surfaces on a flat plate. The
two diffractive surfaces for the curved lens are
moderately strong and have diffractive power of
opposite signs, and they add up to the same net
value as the diffractive power of the weak single
diffractive surface flat plate design. So the color
correction is the same in both designs. When the
two diffractive surfaces are on both sides of a flat
plate the good effect is not nearly as good. So the
diffractive monochromatic aberrations interact in
some way with the curved base lens aberrations in
a way that helps the performance.
Flat plate diffractive design
Curved lens diffractive design

11. Designs
with two
diffractive
surfaces.
curved
flat

12. The surprise is that if the design has the diffractive surfaces removed then the presence of the curved
lens in the middle of the design does not help the monochromatic performance at all, or the color
correction either. By separating the two diffractive surfaces from the curved base lens and putting
them on separate flat plates we find that the good performance does not change. So it is not due to a
curved shape to the diffractive surfaces. It must be some interaction between the diffractive
aberrations and the aberrations of the curved lens that accounts for the good results. Adding aspherics
to the curved lens does almost nothing to improve performance since the diffractive surfaces already
have diffractive asphericity as well as power.
As the very first slide shows, when strong aspherics
and strong diffractive power and diffractive asphericity
are combined then there is aberration interaction that
can give amazing correction potential, but with large
amounts of color. But for these color corrected designs
here the diffractive surfaces are not all that strong and
so there is little benefit to adding aspherics - since the
interaction is then quite weak.
Conclusion – a new type of design is shown in slide
#11 that is practical to make and gives very high color
corrected performance. There are no aspherics. There
are two conventional diffractive surfaces, with a curved
lens between them. That is key to the great results.
Separate flat plates for the diffractive surfaces
The front diffractive surface has more
beneficial effects than the rear one.