I want to discuss several ways of designing optical systems from scratch

and then optimizing them. Of the 5 completely different design methods I

will be covering, 4 are semi-automatic or are completely so and not much

thinking or experience is required on the part of the designer.

One of the methods, however, requires a lot from the designer – a solid

understanding of aberration theory, close attention to what has been

published in journals and books, especially from more than 20 years ago, a

lot of creativity, and finally – a certain playfulness and delight in doing

optical design. For me this is the most fun and productive way of doing

optical design and I will start with a few examples, then cover the other four

design methods and then return at the end to this, my main personal design

method.

Unlike the other design methods that I will be discussing it is hard to

reduce this one to a set of rules or procedures, so it is best illustrated by

simply showing several design examples. The basic idea here is to file away

in your memory any interesting or unusual designs or principles of aberration

theory that you run across and then, maybe many years later, to pull

that out and use it as the basis for some idea about a new type of design. An

equally important component of this design method is to

constantly look for and then question hidden assumptions that you or other

people have brought to the design problem you are working on.

My first design example is exceedingly simple. When I was 15 years old

I bought the book “Mirrors, Lenses, and Prisms” by Southall. This was first

published in 1918 and has many things in it that have long since

disappeared from optics texts. It is full of very complicated geometric

constructions and proofs, like this one. One simple picture stuck in my mind

and it is shown here. When a prism is being rotated to find the minimum

deviation angle in order to measure its index of refraction, the refracted ray

deviation turns around at some point and reverses direction. It is hard to

decide exactly when that happens since it is a stationary point with no first

derivative. This drawing shows how a parallel input ray reflected from the

outside base of the prism, a ray moving very fast as the prism is rotated, will

become parallel with the refracted ray at the minimum deviation setting.

This seemed very clever to me and I filed it away in my memory as a

non-obvious way to use a simple prism. Later, when I was an optics student

at the Univeristy of Rochester, one of my professors was MVRK Murty. He

was and is an extremely creative person. He devised some new ways to use

simple prisms and found some non-obvious ray paths through prisms. This

further alerted me to the fact that simple prisms might still have some

unexplored potentials. Now let us jump forward from back then to 1980.

The Spanish surrealist artist Salvador Dali had recently mastered the art of

making stereo paintings, of imaginary scenes. He wanted a new and unusual

kind of stereo viewer to have for viewing these stereo painting pairs. I met

with him for about one hour and discussed this project, which would involve

a limited edition printing of 2000 lithographs of one set of his stereo

paintings. Later I came up with a prism viewer that is shown here. The line

of sight is reflected off the base of the prism, and this ray path is equivalent

to going through a thick tilted parallel plate, with no color effects or

distortion. The two prisms are joined together with a stiff but flexible hinge

and so the angle between them can be adjusted. That then gives the option

of being at a variety of different distances from the stereo paintings. An

especially nice feature of this design is that the two prisms can then be

folded together when not being used, as shown here. Salvador Dali was

thinking of decorating the outside faces of this folded up unit with some of

his unusual images. 2000 prism pairs were made, using old war surplus

optics.

What I like about this design, aside from its extreme simplicity, is that

there already existed some prism viewers, but with prism wedges that

introduce a lot of color and keystone distortion. I had to question the

assumption that a prism viewer would necessarily have those problems.

What is different about my prism viewer is the complete lack of any color

effects, due to the different ray path through the prism. It is a different way

to think of using prisms. It also turned out that there were some advantages

to the field of view of the prism to using this alternate arrangement of the

lines of sight instead of the obvious one of this way.

Before leaving this prism example, I just want to show you how a simple

right angle prism can be used to display several different pictures in a novel

fashion. The top drawing should be self-explanatory, and gives a rapid

switch between the two pictures as you change slightly your viewing angle

and go from transmission to total internal reflection off the base. In the

bottom drawing, which I use to various images, view #1 is of a picture on the

outside of a two-picture sandwich. View #2 is a view of the back side of the

two-picture sandwich, after two internal reflections. View #3 is a view of a

picture on the base after one internal reflection and view #4 is a direct view

of the picture on the vertical side of the prism. These views seem to be in the

same location and change quickly from one to the other as you rotate the

prism in your hand or move your head a little.

Now I am going to break away from this design method and cover four

semi-automatic design methods that do not require so much thought from

the designer, and then will return to this later. The first one we will look at

now is the

Brixner parallel plate method

This design method could not be simpler. It was described in several

journal articles by Berlyn Brixner about 30 years ago. You basically just

stack up a bunch of parallel plates in a row, tell the design program what you

want for focal length, speed, track length, back focus, and whatever else you

want to constrain, and then let the design program crank away until it spits

out something. You don’t have to know anything at all about much of

anything to use this design method and you certainly don’t have to know

anything about lens design, or aberration theory. For these reasons it would

be to us professional lens designers completely beneath contempt except for

one thing – it works surprisingly well, in some circumstances.

Unfortunately it is extremely sensitive to the initial conditions, as I will see.

The first picture shows my starting point. We are going to first look at a

very simple system of five lenses and a monochromatic design. I show the

first lens having the power that we want for the design, but you can also start

with all parallel plates. The aperture stop is allowed to move around during

optimization. We are going to optimize this system for a 20 degree full field

at f/2.0, with no vignetting. My merit function consists of r.m.s. wavefront

optimization at three field points, plus targets for the system focal length and

for distortion correction. I also have a constraint that the system length not

be more than 1.5X the focal length, as well as lower boundaries for all the

lens rim thicknesses and lens edge to edge clearances. I only used a thinly

populated ray grid over the pupil because I am just looking for different types

of solutions and not a completely optimized design.

Now let us do an optimization run on this system. You see the result is a

design where the last lens is curved towards the image. Next I slightly

changed the spacing of the parallel plates in the starting point, as shown

here. Now we do another optimization run and the result is a different

design. Finally I changed the plate spacing again in the starting point and

begin with this system here. Now we do an optimization run and get a new

design, which turns out to be a very good design with better performance

than the other two designs already generated. A comparison of these results

is shown next.

As you can imagine, a more complicated design with more parallel plates in

the starting lineup will generate many different designs just by making very

small changes in the starting point, because this design method is very

sensitive to the initial conditions. Some years ago I got a call from another

designer, Jan Hoogland, who told me that he has just discovered a new five

element design that had better performance than a 6 lens Double-Gauss, and

which could work well without the vignetting that is usually necessary in a

Double Gauss just to get the off-axis light through. He also said that it could

be corrected for distortion, as well as axial and lateral color, with just five

lenses. But he did not want to tell me what the five lens configuration

looked like. I thought about this and decided that if the new design had very

good performance and also could avoid the vignetting of a Double-Gauss

design, then it must be because the design had relatively weak curves and

was therefore a “relaxed” design.

Now the Brixner design method was ideally suited for finding this new

design, because it starts out with flat plates – so the first solution that it

comes to will be one with relatively long radii and therefore small higher-

order aberrations. So I set up the five element parallel plate system that we

just looked at, put a very high weight on distortion, and immediately found

the new design, which is the last one shown here. I called Jan Hoogland

back about one hour later and he verified that I had just found his new

design.

Now I see that I was simply lucky that day, because even a very small

change in my starting point and I would not have found the good design but

instead one of the other designs that is not nearly as good. These other

designs will also have long radii, but that just by itself is not enough to give

a good design. The Hoogland design is really quite remarkable. Here are

some optimized examples that show that it can cover both fast speeds and

wide fields at the same time, and with no vignetting. These are color

corrected designs. The six lens version really holds up well at very extreme

aperture and field combinations, like this f/2, 60 degrees and this f/1.25, 35

degrees color corrected design, both with no vignetting.

Finally, here is an example where I have 8 lenses and have imposed a

constraint on the working distance, to be half the focal length.

What the Brixner design method will probably not produce is a Double-

Gauss type of design, which has steep curves. The main appeal of the

Brixner design method is also its greatest weakness. Since it starts with

parallel plates it will stop at the first local minimum that its finds, and that

will tend to be one with long radii, like the 8 element picture above. This

gives very relaxed designs with small higher-order aberration and loose

tolerances. But it will miss finding those designs, like the Double-Gauss,

where steep curves might give better performance design. Like any design

method, lots of experience with the Brixner method will give better results

than the first few times you try it. It is important to remember that the

results you get are extremely dependent on the initial conditions, and

multiple solutions of quite different types can be generated just by slightly

changing the initial parallel plate setup. Next up is the

ASA design method

This design method uses adaptive simulated annealing to do a random

search through all of parameter space, and it generates a large number of

local minima, most of which are of poor quality. It has been described in

several journal articles years ago by Greg Forbes and Brian Stone. It

simulates the way molten glass, for example, is cooled slowly so that the

material can move into lower energy stages. This corresponds to local

minima in lens design. Since it is completely automated it is possible to have

it run for several hours, or overnight, or over the weekend. The final merit

function of each solution that is found is given in a long list by the program

and then it is simple to select out the 1 in 10 or so that have pretty good

performance and work on them further with conventional optimization.

This design method is the opposite of the Brixner method, for here the

starting point does not make the least difference to the progress of the

simulated annealing. The program immediately makes random changes in

all the parameters of the starting point. What is quite important, however, is

the boundary conditions you set on the design parameters. If you do not

constrain the radii and thicknesses and airspaces sufficiently then the

program will spend most of its running time wandering around in very bad

regions of design space. On the other hand, if you restrict the parameter

space with boundaries that are too tight you may prevent the program from

finding interesting new solutions.

I am going to just briefly describe what I have found to be very useful

guidelines for using this type of program.

Typical ASA user imposed variable constraints and optimization weights

Radii - longer than about +/- 20% of system focal length

Airspaces and thicknesses - less than 50% of system focal length

Floating stop position - less than 50% of focal length inside or outside system

System length to image - less than 1.5X to 2.0X system focal length, unless

looking for inverse telephoto configurations

Very heavy weights on focal length, system length, distortion (if important) and

Petzval sum.

Put boundaries on edge thicknesses and lens edge interferences for edge of field

rim rays.

First I restrict the range of curvatures to be no stronger than about +/- 20% of

the system focal length. That covers the most likely design types and is a

little stronger than the strongest radii of Double Gauss designs – which have

quite steep radii on the meniscus shells. Then I restrict the lens thicknesses to

be no more than 50% of the system focal length. That is a little arbitrary, but

there can be very good designs that have one quite thick lens in them and we

don’t want to prevent that. The airspaces I restrict by the same amount. Then

I constrain the total length to be no more than twice the system focal length,

assuming it has parallel input light. This is quite important. It has to be

increased if you are looking for, or expect, an inverse telephoto design but

for many other good designs the system length tends to be under twice the

value of the focal length.

Finally, I put an extremely heavy weight on the focal length and on

Petzval curvature. The purpose of the very strong weight on Petzval

curvature is to try to keep the automated design process from exploring

designs with intermediate images – assuming that we don’t want that.

Designs with intermediate images will usually have very bad Petzval sum

and give poor performance. The running time of ASA is very much

affected by the number of rays that are in the merit function, so I use a

pretty thin grid of rays over the pupil. The best use of the program is to

discover new solution regions with pretty good performance, and not many

rays are needed for that. Then conventional optimization with many rays

can be used to push these good solutions towards better performance.

Simulated Annealing

Best one found, diffraction limited at

.5u over field, distortion corrected. 6th

lens can be removed with little effect,

after more regular optimization

ASA starting point

– 8 BK7 elements

Simulated Annealing

Almost as good as best one

found. Simulated annealing

+ regular optimization

Brixner method

Almost as good as best one found.

Only regular optimization used.

Most “relaxed” design, with weak curves.

It is interesting to see that the Brixner design method gave almost as good

results as the best ASA design that was found, and has weaker curves. Of

course we never know with either design method that we have found the best

possible design – as there are just too many local minima to ever find them

all.

The main way in which I use this adaptive simulated annealing program is

to look for designs which are corrected for all the 3rd

and 5th

order

aberrations, which is the topic of one of the other design methods. Then I

only trace two real rays during optimization, to control lens edge thicknesses

and clearances, and these runs go very fast. Now we will look at the

next design method, which I call

Design with phantom aspherics

This design method uses aspherics in the design process, but they are

completely gone in the final design, where they have been replaced by extra

lenses added to the design.. The use of aspherics during optimization has

two big benefits, other than the very obvious one of improving the system

performance. One benefit is that aspherics make it easier for a design to

move between two different solution regions. If there is a bad performance

region of the optimization merit function, between two local minima that are

near each other, it is difficult for some design programs to move through the

bad region. You have to climb uphill to go from one valley to the next one,

and damped least squares optimization is not good for that. But adding

aspherics to the design can make the bad regions be smoothed out or even

eliminated. Then it is much easier for the design program to move between

different merit function minima.

The other benefit, which I will discuss now, is that using aspherics in the

design allows you to quickly and easily find out where in the design

something new is needed to improve performance. I have made up a design

example that we can use to show how this works. This is a 2.0X

magnification relay design that is telecentric on both ends. It is a

monochromatic design. The wavelength is the .6328u laser and the lenses

are all the same glass, which is SK2. The speed is .40 NA on the left hand

side and .20 NA on the right hand side. The object on the left is 21 mm in

diameter. The design is corrected for distortion. There are no aspherics. It

is a very thoroughly optimized design and it has a wavefront that is .042

waves r.m.s. over the field. It is better than the .07 waves r.m.s. diffraction-

limited quality that we will pretend that we need, but there is not much

excess performance here to allow for a good tolerance budget. We want

better performance but it is certainly not obvious where to add a lens to the

design to improve it.

Now the higher-order aberrations of a design are often largely determined

by the first - order optics configuration of the design. It might be that this

particular design configuration has already had most of the possible

performance squeezed out of it and that simply adding a lens or splitting an

existing one will do very little to improve the design. If that were true then

what would be needed to get better performance is a different configuration,

a different first-order distribution of power inside the design.

The great benefit of using aspherics during the design process is that it

allows you to quickly and easily answer this question, as to whether or not

there is still performance potential left in a design. We take the design and

add aspherics to the front and back of the design and also one or two inside

locations. The aspherics are shown in red here. Then we optimize and see

how much performance improvement is possible. There are some well-

optimized designs where adding aspherics does very little to improve

performance. This design here, however, benefits a lot. The r.m.s.

wavefront quality drops from .042 waves down to .013 waves r.m.s., or a

little more than three times better. The design, therefore, clearly has a lot of

improvement potential left in it.

Now I have to explain that when you do this you must only use the

two lowest order aspheric terms, the 4th

and 6th

power terms. The reason is

that our ultimate goal is to not have aspherics in the design, since what we

want to do is to improve it by adding lenses to it instead. I have found that

aspherics that only use the two lowest terms can often be replaced by a

airspaced lens doublet, which then gives similar results. So we only want to

use aspherics in this little exercise which we think we will be able to replace

later on, with additional lens equivalents.

Now here is the really neat part of all of this. It turns out that usually

almost all of the performance improvement that you get, when you

distribute 3 or 4 aspherics through the design the way I have done here, is

due to just one of the aspherics and the others have little effect. So we

then try just a single aspheric in each of my four locations to find out which

one is the important one. It turns out that the aspherics at the front and back

of this particular design have very little effect and a single aspheric in either

location does almost nothing of value. A single aspheric at either of the two

middle locations gives most of the benefits, with the one shown here being

the best location. The design performance with just this one aspheric is then

.016 waves r.m.s., or only a little worse than the .013 waves you get with all

four aspherics.

So, to back up a little, we first used several aspherics just to quickly see if

the design has much further performance potential without any major

changes in the first order configuration. Once we got a very positive answer

to this question then we determined which one particular aspheric location it

is that usually gives most of the benefits. There was no point in doing this

unless we first got a positive answer with the four aspherics. Now we have

to replace this aspheric with some extra lenses.

Now as I said before, an aspheric surface on a lens, or at least an

aspheric with only the two lowest order terms, can often be replaced by

splitting that lens or adding a lens next to it. This can be quite tricky to do.

Here I show five possible lens pairs that might turn out to be equivalent to

Original aspheric

lens

a single aspheric lens. Each of these will have different degrees of matching

the higher-order aberrations of the aspheric surface. The airspace can be a

very important variable, as well as the thickness of a meniscus lens. There is

not time here for me to discuss a simple and quick method that I use to find

out which of these choices is the best match to the aspheric.

Going back to our design we find that the single aspheric lens can be

replaced by the lens plus the meniscus shell lens that is shown. The

resulting design then has an r.m.s. wavefront that is .014 waves over the

field – a little better than the single aspheric design that it replaces and

almost exactly the same performance that I had gotten with four aspherics.

Before we leave this design method I want to point out that now we can

do the same procedure all over again with this new design. The single

aspheric or its meniscus lens replacement, corrected whatever was the

limiting aberration in the original design, and gave a 3X improvement in

performance. But now there is a new limiting aberration, whatever it is, and

maybe it is possible to fix that too. So we put in the four aspherics again,

and again only use the two lowest order terms, and find that after

optimization the .014 waves r.m.s. has dropped down to .007 waves, or 2X

better. Then we try the different locations for a single aspheric and find that

now the best place for a single aspheric is right near the aperture stop. That

then gives .008 waves r.m.s, or almost exactly the same as what I got with

four aspherics.

Now, unfortunately, we have a problem. The limiting higher-order

aberration that is being fixed here requires the 4th

and 6th

order on that

aspheric to be of opposite signs and of such relative magnitudes that I have

not been able to find a double or even a triplet lens to replace it. It must be

that a rather steep surface or two is needed. The best that I could do is

shown here and it is .010 waves r.m.s., with two extra meniscus lenses.

Summary of design progress

1) Starting design, no aspherics > .042 waves r.m.s. over field

2) Add four 4th

and 6th

order aspherics > .013 waves r.m.s.

3) Just one aspheric, in best location > .016 waves r.m.s.

4) Add lens and remove aspheric > .014 waves r.m.s.

So one extra lens = 3X improvement in performance

Second round of process

5) Add four 4th

and 6th

order aspherics > .007 waves r.m.s.

6) Just one aspheric, in best location > .008 waves r.m.s.

7) add two lenses and remove aspheric > .010 waves r.m.s.

Diminishing returns in this series – two extra lenses gives only 1.4X improvement.

There is one final other temporary use of aspherics in lens design that I

want to describe. In any very well-corrected high performance design the

limiting aberrations are usually higher-order Petzval curvature and higher-

order oblique spherical aberration. Often you cannot tell just by looking at

ray traces and field curves which of these two aberrations is the main

limiting aberration for any particular design. Sometimes higher-order

Petzval is not controllable and the design has introduced oblique spherical

aberration to partially focus it out over the field, while other times the

reverse is true and the design has put in higher-order Petzval curvature to

partially focus out uncontrollable oblique spherical aberration residuals. Yet

the ray traces and field curves might not allow you to see the difference

between these two situations.

What I do with a very well-corrected design is to temporarily make the

image have an aspheric surface, with 3 or 4 variable terms in it, but no base

curvature. Then I reoptimize the design using this aspheric image surface to

see if the performance improves significantly. If it does not improve much

then it means that higher-order Petzval curvature is already controllable and

the design is being limited by higher order oblique spherical aberration. On

the other hand, if the performance does improve quite a bit it then that means

that higher-order Petzval is the limiting aberration. Once I know what the

limiting aberration is, by this little trick of having an aspheric image as a

diagnostic tool, then I can go back to the original design and know if I

should be concentrating on making changes to lenses far from the stop or

close to the stop.

1) Limiting aberrations are usually higher‐order Petzval curvature and oblique spherical

aberration

2)

3) Make image surface aspheric, flat base curvature, and reoptimize

4) If performance improves quite a bit, then higher‐order Petzval is main problem

5) So then work on lenses far from aperture stop

6) If performance does not improve much with aspheric image surface, then oblique

spherical aberration is main problem

7) So then work on lenses close to aperture stop.

Only a temporary diagnostic use of an aspheric

When this procedure was done on the various versions of the 2X relay

design that was just discussed, it was found that an aspheric image surface

did very little to improve performance. This then tells me that the design has

good control of higher-order Petzval curvature – almost certainly due to the

two last lenses on the right hand side of the design. Efforts to improve the

design, therefore, should be made fairly close to the aperture stop.

The last design method we will look at, before going back to my

personal favorite that we started with, is that of

Correcting all 3rd

and 5th

order aberrations to zero

This design method consists of looking for designs which have the

ability to be corrected, to zero, for all five 3rd

order aberrations as well as all

12 of Buchdahl’s 5th

order aberrations. Since only 9 of the 12 Buchdahl 5th

order aberrations are independent of each other, this then consists of

correcting the 5th

order aberrations of spherical aberration, coma,

astigmatism, Petzval curvature, distortion, sagital and tangential elliptical

coma and sagital and tangential oblique spherical aberration. Now the point

is not to make these aberrations all be zero, because in any real design they

must be balanced out against 7th

and higher order aberrations. And any final

design will have real ray or wavefront optimization. The point is to establish

whether or not you actually can make all these 3rd

and 5th

order aberrations be

zero, because if you can’t then it means that there are some aberrations that

you can’t control and they will be the ones which limit the performance of

the design.

It is interesting to learn, for example, that a Double-Gauss design

cannot be corrected for 5th

order oblique spherical aberration, even with

aspherics on several surfaces. Yet a design with the fewer lenses, shown

next to it here, can be corrected for all the 3rd

and 5th

order aberrations

and with no aspherics.

I have found a very large number of 6 element designs that can be

corrected for all the 3rd

and 5th

order aberrations and they have very different

configurations. All of them seem to need a rather short back focus distance

– no larger than about 25% of the focal length. At first glance this one

looks like the triplet form, but the powers in the second half are much

stronger than in a triplet.

The best that I have been able to do with an inverse telephoto

configuration, with a long back focus, cannot correct for all the 3rd

and 5th-

order but if I drop 5th

order distortion then I can do it – but with some very

strong curves. Maybe there is some theoretical limitation to having a long

back focus and correction for all the 3rd

and 5th-order aberrations. So the

purpose of this design method is to find new design forms with high

performance levels and also to find out if any particular design is being

limited at the 5th-order level and what particular aberrations are then causing

the problem. That can then indicate if you should be making changes close

to or far from the aperture stop.

A good example of the use of this design method is that of lithographic

stepper lenses. In addition to incredibly tight specs on wavefront quality and

distortion, there is also a tight spec on the variation of telecentricity over the

field. Here is a typical lithographic 4X stepper lens design, from 2004. It is

.80 NA, 1000mm long, has 27 lenses and 3 aspherics. The 27 mm field

diameter on the fast speed end has distortion of about 1.0 nanometer,

telecentricity of about 2 milliradians, and better than .005 waves r.m.s. over

the field at .248u. More modern designs have more aspherics and fewer

lenses.

If you are given a thoroughly optimized design like this and asked to

improve the telecentricity variation over the field, you will find that doing

this is very difficult without screwing up the image quality or the distortion.

You want to improve spherical aberration of the chief ray. But it turns out

that higher-order Petzval curvature is related to the aberrations of the chief

ray inside the design, and distortion is also affected by what happens to the

spherical aberration of the chief ray. The result is that there are almost too

many constraints on the behavior of the chief ray, both inside and outside the

system. Using aspherics on lenses near the object and image helps, but partly

the chief ray behavior is sort of built in to the design by this late stage of

the design evolution. It may be necessary to go back and start over at an

earlier stage of the design and gain control of the chief ray behavior much

earlier in the design evolution.

This is where the 3rd

and 5th-order design method is very handy. We want

to know if some major changes are necessary in the design, or not. So we

drop all the ray and wavefront corrections and only optimize the 3rd

and 5th

order aberrations, plus 3rd

and 5th

order spherical aberration of the telecentric

chief ray. If these can all be corrected to zero, then further design work on

the original design can probably eventually give you the telecentricity

variation improvement you are looking for. But if you can’t make these all

zero then you need to go back to an earlier stage in the design evolution,

possibly much earlier, and find out when you can establish the chief ray

control that you need.

Now let us return to the design method I started out with, which require a

lot from the designer. Let us review its features here with this chart.

My main design method

1) Remember interesting and/or unusual designs, facts, and design principles – for

later use

2) Read the literature – journals, patents, books

3) Understand the implications of aberration

theory, like stop-shift theory

4) Identify and question hidden assumptions

5) try to think outside the box

Abe Offner was a pioneer in the use of a field lens to correct secondary

color, which is a very counterintuitive idea. I learned about this about 40

years ago and I filed away this odd idea in my memory. 30 years later I

pulled it out to design a series of extremely broad spectral band high

numerical aperture microscope objectives for lithographic wafer inspection.

Only one glass material is used, which is silica. This design is based on the

idea of producing a very good virtual image, and then relaying that to a real

image.

We start off with a conventional doublet that I will use as a comparison,

to show how secondary color can be enormously improved by using a

different type of design. This doublet has about 26u of focus shift between

the mercury i line and the g line. Next we look at the Schupmann design,

from over 100 years ago. It is two lenses of the same glass type and it

produces a virtual image, which is corrected for primary axial color. At the

same focal length as our comparison doublet this virtual image design has

7.5 times smaller secondary color.

Now let us put an Offner field lens right at the intermediate image. By

choosing the power of this lens correctly we can correct the secondary color

and are then left with tertiary color. As you see from the graph, this cubic

curve only has .01u of focus shift over that particular spectral range. So now

let us reoptimize the system for a much broader range, that goes from the

.193u laser line up to the mercury i line, with all three lenses being made of

silica, for its deep UV transmission. Now the focus shift is about +/- 10u

over this very broad deep UV spectral range. The last step is to make the

field lens be an achromatic doublet, of silica and calcium fluoride. Then the

theory indicates that the tertiary color can also be corrected and what you

then get is the 4th-order curve shown here, where the focus shift has been

reduced from 10u to +/- .25u. That is all fine and dandy, you might say, but

it is a virtual image. The last step is to use a concave spherical mirror to

relay that virtual image to a real one, while also speeding up the numerical

aperture by a factor of two. There are some further design variables, like the

position of the field lens doublet relative to the intermediate focus, which

allow a further improvement in focus shift and keep the field lens surfaces

from being right at a focus.

There are some more complicated version of this type of design where

only silica is used and yet it is well corrected from the very deep UV right

through to the near infrared. One example is shown here, with a numerical

aperture of .90, using only silica.

Another design example that starts out with a virtual image is based on a

curious feature of monocentric systems. A long time ago Charles Wynne

proved by means of a simple geometric construction that the surface order in

which light goes through a monocentric system makes absolutely no

difference to the aberrations. A monocentric system when used backwards

has exactly the same aberrations to all orders as when it is used frontwards.

I read that a long time ago and filed it away in my memory as a curious fact.

Some time later I considered the two concentric spherical mirror system

shown here, which is well corrected for aberrations on a curved image, and

turned it around and used it backwards to give the virtual image system that

is shown. By the way, although this design is always attributed to

Schwarschild, he did not seem to realize that his equations describing all

possible anastigmatic two mirror systems has this non-aspheric solution with

two spherical mirrors. As far as I can tell, Burch discovered that much later.

It is too bad that we can’t do something with this well-corrected virtual

image. Then I got the idea of using the Offner unit magnification relay

system, another monocentric system, to relay the virtual image to

become a real image. The two separated monocentric systems when

combined then define a unique optical axis. By offsetting the entrance pupil

and field you can then get an unobscured design. Furthermore you can then

change all the variables and it is then possible to correct the system for

Petzval curvature, to give a flat well-corrected image. It is still a single axis

design but all the centers of curvature are spread out along that axis.

This telescope was on the Cassini spacecraft that went to Saturn. A

grating was put on the convex Offner mirror, which is an aperture stop, and

the resulting spectrum has very high image quality. The same telescope

design is also on a mission to land on a comet and another one is due to

fly by an asteroid. It has only spherical mirrors and also has loose

tolerances.

My next to last example concerns color correction. Let us take a

monochromatic design, like the one shown here, and correct it for axial and

lateral color. It is for use in the deep UV. Suppose we want to consider

using diffractive surfaces to give color correction. We know that two

separated diffractive surfaces are sufficient to correct for both axial and

lateral color. But two diffractive surfaces would have more scattering and

expense than we would like.

The stop position is chosen to give a telecentric design. The graph shows

lateral color for this stop position. Now let us temporarily move the stop

around and find the positon which makes lateral color be zero. We know

there has to be such a position because if axial color is uncorrected then

lateral color is linear with stop position and that means it must go to zero

somewhere. We find that the stop position that makes lateral color be zero is

further back in the design, towards the image side. Now let us correct axial

color at this location, with one diffractive surface. Now that both axial and

lateral color are corrected, we can then move the stop position back to where

it needs to be – to give a telecentric system – and the axial and lateral color

will stay corrected. We only moved the stop temporarily as a conceptual

maneuver. Here is another example. The lateral color at the telecentric stop

position is shown next and then the stop position which eliminates lateral

color is then found by moving the stop back and forth until we find that a

surface position much further back is needed. So we achromatize there –

with a diffractive surface. So we avoid using two diffractive surfaces to

correct axial and lateral color and get away with just one.

My final design example comes from the very early days of research into

laser fusion – around 1976. KMS Fusion, in Michigan, was doing very high

power irradiation of a tiny target pellet, using what was known back then as a

“clamshell” optical system, which is shown here.

Two extremely high power laser beams came into the system from

opposite directions. Each was focused to a point by an extremely aspheric

high numerical aperture single lens. That focus was then located at one of

the focii of an elliptical conic mirror. The mirror then formed another point

image, with almost a hemisphere of light, at the other focus of the ellipse –

where the target pellet was located at the center of the system. It was

important to provide almost complete 360 degree illumination of the target

pellet. Everything was in a vacuum, because air would break down and

explode at the focus of the lenses.

This design had two main problems, aside from the extremely high cost of

the aspheric lens. Ghost images from single and double surface

reflections would tend to form inside the lens and it was hard to design the

lens to avoid that. Several of these very expensive lenses blew up because of

the power levels of the internal ghost images. The other problem is the high

power levels would heat up the lens due to glass absorbtion and that would

change the glass index of refraction locally inside the lens. The result would

be a spoiled point image and poor results at the target focus.

I inherited this design from someone else and was given the task of

somehow fixing these problems. Splitting the lens into two weaker ones

had been already been considered but it added a lot to the multiple reflection

ghost image problems. I stared at this lens drawing for a long time one

sleepy autumn afternoon. Suddenly I questioned a hidden assumption that I

had been making and that was the key to finding a very good new solution.

In optics text books you will often see a picture that looks like this,

Which shows light coming to a focus and then stopping. Usually we bring

light to a focus because we want to put something there, like film, a CCD

array, or a laser fusion target pellet. What you will not see nearly as often

in books is a picture that looks like this one, where the light keeps on going

after it comes to a focus. I suddenly wondered what would happen to the

light in the clamshell optical system if I removed the target pellet. If you

look at the picture of the system you will see that the rays then keep on

going through the target focus, hit the other mirror, go through the other

lens, and exit out the system as a collimated beam. That by itself is hardly a

useful insight but it gave me the very important idea of hitting both mirrors.

I then came up with what I called the double-bounce clamshell design,

shown next.

Only one of the two light paths is shown, to keep the picture from being too

confusing. By reflecting off both mirrors before coming to the target pellet

focus, more of the work of bending the rays around into the near hemisphere

angle is done by the mirrors and less is required of the lens. The next picture

shows a comparison and you can see that the lens speed required is

considerably reduced in this new design.

The result of that is that it is easy to make the high power ghost images fall

outside the lens, the glass heating is much less because of smaller glass

volume, and the cost of the aspheric lens is dramatically less. In the final

system that was built, a three reflection clamshell design (still just two

mirrors) was used. That further reduces the speed required of the lens and it

was replaced with an off-axis piece of a parabolic mirror, with a fold flat, for

an all reflective system with no ghosts at all or lens heating. The clamshell

mirrors cannot be pure ellipses in these multibounce designs but the

advantages of the new designs far outweigh that factor.

In summary, there are many possible methods for designing optical

systems and I have just talked about a few that I use myself. A whole other

topic, for some other time, is ways of using the optimization damping factor

to improve optimization results and to find new solution regions. Since I

have written and talked about that on other occasions I did not include it

here.