Aspheric and diffractive optics extend monochromatic imaging limits 1999

Aspheric and diffractive optics extend monochromatic imaging limits
David Shafer
David Shafer Optical Design
56 Drake Lane, Fairfield, CT. 06430
phone 203-259-4929, e-mail: Shaferlens@worldnet.att.net
Abstract
The highest possible performance in an optical
design occurs when the aberrations of each
element are as decoupled as possible from each
other. This is achieved when each optical surface
is aspheric and each element also has diffractive
power in addition to refractive or reflective
power. Some very simple design examples are
shown with amazing performance.
Key Words
Aspheric, diffractive
Introduction
In all but very simple optical systems there are
generally enough design variables to correct all of
the 3rd
-order image aberrations. Performance is
then determined by higher-order aberrations and
chromatic aberrations. The higher-order
aberrations which limit performance, in a well-
corrected design, are usually higher-order field
curvature and oblique spherical aberration. Both
of these higher-order aberrations are usually
present at every optical surface in a design, but
partially cancel out to give a small net sum for the
whole system.
The higher-order contributions of each optical
surface can be broken into two components:
aberrations which are intrinsic to the surface, and
aberrations, called induced aberrations, which are
the result of aberrations in the light coming into
the surface. Figure 1 here illustrates this
difference. It shows a triplet lens that has been
specifically designed to have all the 3rd
-order
aberrations be exactly zero. Nobody would want
such a lens, because best performance occurs
when the unavoidable higher-order aberrations
that occur in such a simple design are partially
balanced out by deliberately introducing non-zero
values of the lower-order (the 3rd
-order)
aberrations.
Figure 1. Triplet with 3rd
-order aberrations = 0.0
The reason I am showing such an undesirable
type of aberration correction will be clear in a
moment. Now this design has astigmatic field
curves shown in Figure 2. The tangential rays
Figure 2. Field curves for triplet
focus considerably short of the image plane, and
the amount quickly grows with larger field angles.
This is higher-order astigmatism, and we can see
the large focus error at the edge of the field on the
ray plot in the lens drawing. The tangential rays
are the ones that lie in the plane of the page here.
Normally this higher-order astigmatism would be
balanced out by non-zero 3rd
-order astigmatism, to
give a best average focus over the field. By
making the 3rd
-order aberrations zero here, we
have deliberately highlighted the higher-order
problem.
© 1999 OSA/FTA 1999

Now in this design we have done a little
experiment. The middle lens was made the
aperture stop. It turns out that in most high
performance systems it is not the higher-order
aberrations that are intrinsic to the surfaces that
are most important, but rather those that are
induced in the surface by aberrations coming into
the surfaces. I have shown in Figure 3 what
Figure 3. Chief ray behavior
happens when you aim a series of chief rays, for
different field angles, at the paraxial entrance
pupil. 3rd
-order aberration theory assumes the
chief ray goes through the paraxial entrance pupil
for any field angle. But look what actually
happens. There is aberration of the chief ray, and
it doesn’t hit the middle lens at the center except
for small field angles. The ray aberration also
bends the chief ray around to be a steeper angle
than what is assumed by 3rd
-order theory. The
discrepancy between the actual ray path and that
assumed by 3rd
-order theory diverges even more
by the time we reach the last lens. Is it any
wonder, then that the aberrations of the middle
lens and especially the last lens are different from
what is assumed by 3rd
-order theory - and the
discrepancy increases with larger field angles.
But this is the nature of induced higher-order
aberrations.
The last two lenses do have some higher-order
aberrations, intrinsic ones, that occur even if the
ray paths were somehow made to be just what 3rd
-
order theory assumes. But in this example, the
main thing of importance is rather the effect
induced on the lenses by pupil aberration (chief
ray aberration) coming into the surfaces - not by
any higher-order aberrations that are intrinsic to
the surfaces.
One more point before we move on to some
new designs. Look at the rays in Figure 1 on the
middle lens for the on-axis beam and the off-axis
beam. Clearly the beam diameter at that lens for
the edge of the field is substantially less than it is
on-axis. Figures 4 and 5 show the ray traces and
the field curves for just the first lens by itself.
The astigmatism and Petzval curvature of this
single lens makes both the tangential and sagittal
Figure 4. Field curves of first lens alone
Figure 5. Rays for first lens alone
rays focus short, compared to the on-axis focus
position. This causes the beam diameter in both
the tangential and sagittal directions to be
reduced, at the middle negative lens, when
compared to the on-axis beam.
Figures 6 and 7 show the actual beam
footprints on the middle lens, when seen head-on
for the on-axis case and also at the edge of the
field. Now this middle lens is the only lens of the
three triplet components which puts in
overcorrected spherical aberration - to correct for
the undercorrected spherical aberration from the
two outer positive lenses (most of it coming from
the first lens). Since the beam diameter is less on
that middle lens off-axis, especially for the
tangential rays, we should expect to see that the
negative lens puts in an insufficient amount of
spherical aberration correction for the off-axis
field points. That is exactly what is observed, in
fact, and is called oblique spherical aberration - a
© 1999 OSA/FTA 1999

Figure 6. On-axis beam on middle lens
Figure 7. Off-axis beam on middle lens
higher-order aberration.Clearly it is mainly an
induced aberration here. It is not so much due to
any intrinsic properties of the middle lens, but
rather due to the effect of the aberrations, mostly
astigmatism here, in the light coming into the lens
from the previous lens. A close study of Figure 7
also shows the presence of some coma of the off-
axis pupil, causing the bottom of the ray footprint
to be flatter than the top, which is an effect due to
the previous lens. The result will naturally be
some odd looking imbalance in the aberration
correction. What we want to do is to somehow get
control over these aberrations inside the design
and thereby get control over these induced
aberrations that limit the performance of the
system. What we would ideally like to do is to
have the aberrations of each optical element be
made to be whatever we choose. We would like,
for example, to be able to make the pupil
aberrations, like spherical aberration of the chief
ray that we saw in Figure 3, be made for each
element to be independent of the image
aberrations of that same element. It turns out,
unfortunately, to be theoretically impossible to do
that.
The main reason, and this involves a very
obscure point about aberration theory, is that for
conventional lenses the image aberrations and the
pupil aberrations are inter-related in a way that
involves both the Petzval curvature of the element
as well as its power. Unfortunately, the Petzval
curvature of an element is normally very closely
related to its power. Aspherics have no effect on
this. What we need is some way to make the
power and Petzval curvature of an optical element
be completely independent. Then we could obtain
the maximum amount of control that is
theoretically possible over induced higher-order
aberrations, and greatly improve design
performance.
It turns out that there is just such a way of
decoupling the power of an optical element from
its Petzval curvature, and that is to put a
diffractive surface on the element. By choosing
the right ratio, for a lens, of refractive power to
diffractive power we can make the Petzval
curvature be anything we want, including having
it be either positive or negative, while not
changing the total power. The catch, and it is a
pretty big one, is that this normally puts in a lot of
chromatic aberration and may limit the resulting
designs to nearly monochromatic use.
New Designs
When diffractive surfaces are added to a design -
and specifically used in the manner I have
indicated - great performance improvements are
possible. The design configurations which
achieve the highest levels will not generally be
anything like the triplet of Figure 1, which starts
off bad, but rather those lens configurations which
already have good performance. I described a
three element design at the 1994 International
Optical Design Conference. By using asphericity
on the surfaces as well as diffractive power, the
most performance possible can be squeezed out of
this simple design, without going to more
elements. This design is diffraction-limited, at
.6328u, over a 70 degree diameter flat field at f/.9,
for a focal length of 10 mm. There is no
vignetting, and distortion is well-corrected. Every
surface is aspheric, and each of the three
elements has an additional superimposed
diffractive overlay. It turns out, however, that
most of the performance can be achieved with just
one diffractive surface in the system, on the front
of the last element. It would take a great many
additional elements to eliminate the aspherics
from this design, while keeping the performance,
but another design was also shown where the
diffractive surfaces were eliminated. It simulates
the effect of the diffractive surfaces by combining
glasses with a very low index of refraction with
© 1999 OSA/FTA 1999

glasses with a very high index of refraction. The
design is diffraction-limited at .6328u over a 60
degree diameter flat field at f/1.0 for a 25 mm
focal length.
Now for some new designs. Let us start out
with a conventional Double-in the left hand
design and is in the front doublet in the Gauss
design, shown in Figure 8. Here I show a 50 mm
Figure 8. Monochromatic Double-Gauss design
Figure 9. monochromatic MTF
focal length f/2.0 monochromatic design with a
full field angle of 45 degrees, with no vignetting.
All four lenses are BK7, a low index glass,
because that is what my new designs will be, and I
want a fair comparison. Distortion is held to less
than 2%, and the back focal length is made 25
mm - half the focal length. Performance is quite
poor, because normally vignetting would be
allowed that would make quite a difference. The
next step was to add a diffractive surface to both
the front lens pair and the rear lens pair. Since
that allowed the design to be easily corrected for
Petzval curvature - the most difficult aberration
to fix and the one with the most consequences for
the other aberrations - the reoptimized design
quickly evolved into a different configuration,
shown in Figure 9. The diffractive power has
allowed the design to move into its preferred
configuration, which is that of a Petzval lens. This
has inherently better performance because of the
more gentle way it bends the rays around. The
next step is to add aspherics to all the surfaces. It
then turns out the last lens pair can be combined
into a single lens, and the first lens pair can also
be combined into a single thick meniscus lens.
Figure 10. Diffractive surfaces lead to new design
Figure 11. Diffractive and aspheric surfaces
Figure 12. Almost perfect MTF for Figure 11
Figure 10 shows the result. It has the same
parameters of 50 mm focal length, f/2.0, 45
degree field, and no vignetting. This low-index
glass monochromatic design is diffraction-limited
across the field. The two surfaces facing the air-
gap both have diffractive power. All the surfaces
© 1999 OSA/FTA 1999

are aspheric, and there is diffractive asphericity in
addition to the glass substrate asphericity.
This is a very tricky design to optimize and it
takes a lot of special massaging and optimization
tricks to squeeze the most performance out of the
design. I used the Oslo design program and the
“Blow-up, settle-down” optimization method I
have described elsewhere. The spot size over the
field is about 50X better than the Figure 8
Double-Gauss design. You can see that the
external shape of this front lens is that of a
negative lens, yet it clearly has positive power.
That is because of the diffractive power on the
concave surface. The performance is so good that
another version, not diffraction-limited, is shown
in Figure 13. It is a 50 mm focal length, f/1.25
design with a 45 degree field, and a back focus of
25 mm.
Figure 13. F/1.25, 45 degree design
Figure 14. MTF for Figure 13 design
. It is not clear yet in this new design how much
its amazing performance would degrade if only
one diffractive surface were used instead of two,
or if some of the aspherics were removed. It
would be interesting to apply this design approach
to a laser scan lens, where the large amount of
color put in by the diffractive surface or surfaces
would not matter.
In the Figure 11 design, the diffractive fringe
spacing goes to about .65u at the worst location
on the apertures of the two diffractive surfaces.
So this is not really a practical design to make. It
mainly illustrates what can be done if the limits
are really pushed. Much more benign diffractive
surfaces result when the diffractive power or
asphericity is just used as a kind of touch-up on a
design which already has good performance.
Figure 15 shows what results when the two
diffractive surfaces in this design are replaced by
a cemented doublet of very high and very low
index glasses. Here I use Laf-N21 and fused
silica. This design is about 5 times worse in
performance than the Figure 16 design.. Most of
the power is in the rear doublet right-hand design.
The Figure 16 design is very close to diffraction-
limited over the whole field in this f/2.0
monochromatic design. Many field points must
be optimized to make sure there are no bad
quality field zones in the image.
Figure 15. No diffractive surfaces
Figure 16. Much better solution
© 1999 OSA/FTA 1999

Figure 17. MTF for Figure 16
The best result obtained, with a quick partial
optimization, is shown here. More work needs to
be done on this.
Figure 18. Color corrected design
Figure 19. MTF for Figure 18 design
The steep inflection region in the last of the two
“cemented surfaces” does not actually have to be
there. Once the best performance has been
obtained in one of these designs, the next step is
to then try to make the aspheric surfaces as beign
as possible, by controlling their departures from
the best-fit spheres. Often it turns out that large
amounts of asphericity, and especially sharp
infection areas, like in Figure 18, are not needed
and can be made to go away. A design of that
type was done, with much more reasonable
surface shapes in the second doublet, and
essentially the same performance, but is not
shown here.
Summary
Very high performance levels can be achieved by
using a combination of diffractive surfaces and
aspheric surfaces in quite simple lens designs.
The diffractive surfaces can be eliminated if a
combination of very high and very low index
glasses is used.
© 1999 OSA/FTA 1999

Aspheric and diffractive optics extend monochromatic imaging limits 1999

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Aspheric and diffractive optics extend monochromatic imaging limits 1999