An exploration of many multiple solutions to a design problem (null lens design for optical testing) with very few variables to work with.

1. Null lens design techniques
David Shafer
There are a surprising number of discrete, well-correctednull lens designs with just two elements for
testing a large, fast-speed parabolic mirror from its center of curvature. Some design techniques are
discussed for generating these solutions, which may be applicableto the design of null lenses for other
mirror shapes. Without examining these multiple solutions, which differgreatly in performance, there is
no wayto knowif the optimum configuration has beenfound.
Keywords: Nulllens, parabolic mirrors, aberrations.
Introduction
There are some interesting aspects to the design of
null lenses that are not fully covered in the Offner
chapter on the subject in Malacara's Optical Shop
Testing book.' Here we show how high-order aberra-
tion correction can be obtained for a sample design
case and the surprisingly large number of discrete
solutions that are possible with just a two-element
null lens design that is corrected for third-, fifth-,
seventh-, and ninth-order spherical aberrations.
Figure 1 shows the conventional Offner null lens
configuration for our test case. We assume that we
want to test a parabolic mirror from its center of
curvature. Here we look at a 1-m aperture f1.5
parabola that is f3 from its center of curvature and
has a large amount of spherical aberration when used
at that conjugate. The Offner null lens consists of a
front focusing lens and a second field lens near the
intermediate image. The shape of the front lens
generates the desired amount ofthird-order spherical
aberration (in double pass) to correct for the mirror
while the field lens power controls the amount of
fifth-order aberration.
Here we assume a collimated input beam and
arbitrarily choose a beam diameter of 40 mm. The
power of the front lens is then set at f/3 to match the
mirror when seen from its center of curvature. The
lens bending is chosen to achieve the desired third-
order spherical aberration. The power, not the bend-
ing, of the field lens permits correction of the fifth-
order spherical aberration (for reasons that are
The author is with David Shafer Optical Design,56 Drake Lane,
Fairfield, Connecticut 06430.
Received 25 January 1991.
0003-6935/92/132184-04$05.00/0.
© 1992 Optical Society ofAmerica.
explained in the Offner article) with essentially no
effect on the third-order aberration. In the Fig. 1
design the fieldlens is positioned at the paraxial focus
of the front lens. Then computer optimization can
take this starting design and balance the controllable
third- and fifth-order spherical aberrations against
the uncontrollable seventh and higher orders.
Seventh-Order Correction
So far all this is standard procedure as described by
Offner. Suppose now that we want to try to correct for
the seventh- and ninth-order spherical aberrations as
well but without adding any additional lenses. This
can be doneby moving the Offner fieldlens away from
the focus region and bending it. When it is a thin lens
and essentially right at the focus its bending has no
effect-only its power counts. Away from the focus
and with an appreciable thickness the bending ofthis
lens gives us the additional variables that are needed
to correct for the seventh- and ninth-order spherical
aberrations also.
Finding this higher-order solution can be quite
difficult. It involves much stronger curves on the field
lens than the conventional fifth-order solution and is
located in a highly nonlinear solution space. It is just
too far from the starting point for the computer to
find on its own: The designer must help it along. We
start by decreasing the amount of seventh-order
spherical aberration in several small steps. While
holding the third and fifth order to zero,we target the
seventh order to be 80%-90% of the starting value.
Then we vary both of the field lens curvatures and
also their separation from the front lens (and of
course the two front lens curvatures). If this process
converges, then the seventh order can be worked
downslowly to zero in this manner.
2184 APPLIED OPTICS / Vol. 31, No. 13 / 1 May 1992

2. Fig. 1. Conventional Offner null lens with third- and fifth-order
correction.
Ninth-Order Correction
Figure 2 shows one such higher-order solution. The
fieldlens has moved quite a ways from the intermedi-
ate image. Now it is time to get a little tricky. This
design was generated by assuming a particular input
beam diameter. Let us now take this Fig. 2 design
with corrected third-, fifth-, and seventh-order spher-
ical aberrations and slightly change the input beam
diameter by 10%-20%. Then we reoptimize while
keeping the f/3 cone angle match.
Note that the real marginal ray must hit the rim of
the parabolic mirror even though the paraxial height
may be quite different. A new solution is found that is
somewhat different from the Fig. 2 design. It has
either more ninth-order aberration or less aberration
than the Fig. 2 design as determined by the marginal
ray behavior. Changing the input beam diameter
changes the size ofthe null lens relative to the mirror.
Here enlarging the input beam causes the reopti-
mized solution to have less ninth order. This is
continued a fewtimes, in small steps, until the ninth
order changes sign. In the process the fieldlens moves
back toward the region of the focus and the final
result is shown in Fig. 3. It is corrected for third-,
fifth-, seventh-, and ninth-order spherical aberra-
tions.
These twoseparate optimization sequencesof slowly
working down the seventh order and of slowlychang-
ing the input beam diameter movethe design in small
enough steps so that the localsolution space does not
depart too far from linearity. In the process described,
the field lens moved quite far away from focus and
then moved back again for the final solution. Mean-
while, the field lens power and bending changed a
Fig. 2. Additional seventh-order correction with the field lens
moved away from the focus region.
gooddeal. The final design in Fig. 3had to gothrough
this circuitous route sothat we did not lose control of
it. Even then many iterations and a fewoptimization
tricks were required to reach a solution.
In an ideal world one would take the Offner third-
and fifth-order starting points and just ask the com-
puter to find the seventh- and ninth-order (as indi-
cated by the marginal ray) solutions. Since existing
optimization programs cannot do this because of the
extreme nonlinearity of the problem, we are forced to
manage the optimization progress closely.
It is interesting to note that, once such a ninth-
order solution has been obtained and the marginal
ray is being corrected as part of the merit function
(alongwith the third-, fifth-, and seventh-order coeffi-
cients), then the design often can be worked back
down to the original input beam diameter. The slow
change of the input beam diameter is a tool for
driving the design toward a ninth-order solution,
which is difficult to find. Once the solution found,
however, it is possible to keep control over it while
slowlyreversing the path just taken.
Multiple Solutions
Now comes the big surprise. The correction for
seventh- and ninth-order spherical aberrations in
such a simple design was unexpected. Much more
surprising is the fact that there are at least seven
discrete solutions with just these two lenses. Sasian2
has described how he found one of these solutions in a
similar type of design study. Let us see how these
multiple solutions can arise.
Figure 4 shows the next one that was found. It is
well known that there are always two bendings of a
lens that give the same third-order spherical aberra-
tion. Figure 4 represents this alternative bending for
the front lens. Rather than going through the slow
optimization sequence described above, we substi-
tuted this alternative bending for the front lens was
substituted into the Fig. 3 design and that was used
as a starting point. This was then close enough to the
final Fig. 4 solution so that the seventh and ninth
orders could be brought in rather quickly. The hard
part in going from the conventional Offner solution to
the Figs. 3 and 4 designs is getting close to the final
shape, power, and position of the fieldlens. Once they
have been established, other design solutions are
much easier to find.
Figure 5 showsthe next solution that was found. It
is always a good bet, anytime there is a nearly
zero-power meniscus lens in any design, to try flip-
ping the lens over and reoptimizing. There is often an
alternative solution that can be found by this tech-
Fig.3. Third-, fifth-,seventh-, and ninth-order corrections. There
isstrong bending of the fieldlens. Fig. 4. Alternative bending solution for the front lens.
1 May 1992 / Vol. 31, No. 13 / APPLIED OPTICS 2185

3. Fig. 5. Alternative bending solution for the fieldlens.
nique. Once this worked, and the Fig. 5 design was
found with third-, fifth-, seventh-, and ninth-order
corrections, the same thing was done with the Fig. 4
design. The resulting new solution is shown in Fig. 6.
Alternative Configuration
At this point a completely different line of thought
was investigated. Suppose that we place a nearly
zero-power (i.e., afocal) meniscus lens in front of the
main focusing lens and omit the field lens from the
design. Figure 7 shows one example. The front afocal
meniscus lens has a certain amount of third-order
spherical aberration. The net third order of the two
lenses then does not depend significantly on their
separation. The fifth-order aberration ofthe pair does
depend on this separation and can be used as a means
of correction.
There are many such possible designs with third-
and fifth-order correction, depending on how strong
and thick the meniscus lens is. They differ in the
amounts of seventh-order aberration, which can be
slowly worked down to zero in small steps. Then the
input beam size can be changed in small steps, and
the direction that eases the design slowly toward
ninth-order correction can be found. During this
process the meniscus lens is allowed to depart freely
from the initial afocal condition.
Figure 8 shows the result of the same line of
thought as applied to the alternative third-order
bending solution for the main focusinglens. As before
it was easy to reach these new third-, fifth-, seventh-,
and ninth-order solutions once the approximate form
of the meniscus lens had been determined from the
Fig. 7 design. Figure 9 shows the next solution that
results when the meniscus lens of Fig. 7 is flipped
over and the system is reoptimized. Figure 10 shows
Fig. 8. Meniscuslens with alternative focusinglens bending.
what should have been the last design, as based on
the Fig. 8 system. There does not seem to be a
ninth-order solution here, however, there is correc-
tion onlythrough the seventh order.
Optimum Design
Figure 3 has the smallest amount of eleventh-order
aberration. This design was given a wave-front error
optimization and the result, in double pass at 0.6328
with the 1.0-m aperture f/1.5 parabola, was so phe-
nomenally good that the design was redone for a
1.0-m aperture f.0 parabola. Then the wave-front
error was 0.003 waves peak to peak. Design data
are given in Table I and a picture of the ray paths is
shown in Fig. 11. The lenses are 50 mm or less in
diameter. This wave-front optimized solution is quite
similar to the one that Sasian featured as his best
result. 2
Note that in this optimized solution, given in Fig.
11, the correction of third-, fifth-, seventh-, and
ninth-order spherical aberrations to zero is aban-
doned. Instead computer optimization ofthe rms spot
size balances these coefficients, including defocus,
Fig. 9. Alternative meniscus lens bending solution.
_,~~~~/
Fig. 10. The two alternative bendings with no ninth-order solu-
tion.
Fig. 6. Alternative bendings for both the field lens and the front
lens.
Fig. 7. Nearly afocalmeniscus lens approach.
TableI. DesignData
Surface Radius Thickness Aperture Glass
1 - - 25.00000 Air
2 -205.50673 13.00000 30.00000 BK7
3 -45.55029 127.76558 30.00000 Air
4 -1212.07451 5.00000 25.00000 BK7
5 -131.58100 1980.59334 25.00000 Air
6 -2000.00000 -1980.59334 500.00000 Reflect
7 -131.58100 -5.00000 25.00000 BK7
8 -1212.07451 -127.76558 25.00000 Air
9 -45.55029 -13.00000 30.00000 BK7
10 -205.50657 -10.00000 30.00000 Air
11 - - - Air
2186 APPLIED OPTICS / Vol. 31, No. 13 / 1 May 1992

4. Fig. 11. Wave-front optimizedresult for the 1.0-mf/l.0 parabolic
mirror.
against uncorrectable eleventh- and higher-order
spherical aberrations to givethe best result.
Discussion
A conic mirror tends to have a relatively small
higher-order content of total spherical aberration
when tested from its center of curvature. Even so the
designs discussed here still need to have control at
least as far as the ninth order if one is to get good
performance when the conic speed is fast. In design-
ing null lenses for more general aspheric mirrors, it
may be more difficult to achieve such a high-level
correction with just two lenses. This is especiallytrue
if the mirror shape has a large amount of higher-
order content when compared with a conic surface.
Furthermore the number of discrete solutions may be
quite different.
An interesting aspect of the seven solutions dis-
cussed here concerns their eleventh- and higher-
order spherical aberrations. The Fig. 3 design has
eleventh order that is - 10times smaller than most of
the other solutions, mainly because of its weaker
curves. All seven designs givegoodresults when used
with thef/ 1.5speed parabola, although some (such as
the Fig. 3 design) are much better than others. With
an f/1.0 parabola, however, several of the designs
give total internal reflection for the outer rays or
some rays miss a surface entirely. The designs are
still corrected through the ninth order but they have
horrendous eleventh and higher orders.
The conventional Offner design of Fig. 1, by con-
trast, is corrected only through the fifth order. When
used with an f/1.0 parabola though, its net higher-
order aberration (ofall orders) is much lessthan that
of most of the designs discussed here. There are no
ray failures or missed surfaces. What counts, then, is
not how many orders of spherical aberration can be
corrected but rather how small the net higher order
can be made. This depends on the particular situa-
tion. It is possible to have a design that is much better
than the conventional Offner design at one speed
such as f 1.5but much worse at a faster speed such as
fI 1.0.Each case needs to be examinedwithout precon-
ceptions. Tolerances will be looser on designs with
shallower curves.
When the Fig. 3 design was optimized for a wave
front with an f/1.0 parabola it moved quite a bit away
from the starting point with respect to the shape of
the fieldlens. This reduced the net higher order while
giving the design weaker curves. This particular
design is so good that it is probably better than a
conventional Offner design at any speed. When the
weak radius on one side of the fieldlens was made flat
the reoptimized design gave a much worse perfor-
mance. This design needs all the available variables to
get such a high performance.
References
1. A. Offner, in Optical Shop Testing, D. Malacara, ed. (Wiley, New
York, 1978), Chap. 14, pp. 439-457.
2. J. M. Sasian, "Design of null correctors for testing of astronom-
icaloptics," Opt. Eng. 27, 1051-1056 (1988).
1 May 1992 / Vol. 31, No. 13 / APPLIED OPTICS 2187