A lens in a Box

Dave Shafer

Design reference point – a monochromatic triplet, f/3, 15 degrees total field, 100 mm focal length, no vignetting, distortion corrected, .70 waves r.m.s.

BK7 lenses

Any new designs must have same length and BFL as this triplet

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75% of focal length

47.5% of focal length

Lens in a box problem – replace triplet with a high performance design in the same space

f/3, 15 degrees total field, worst r.m.s. = .70 waves at .5876u

A local minimum,but no better than the triplet

A 4 lens design

With the thin lens triplet lens there is more than one 3rd-order solution. The “right hand” and “left hand” solution are shown here. But in a monochromatic design where lens thickness is a variable there are a large number of 3rd-order solutions with just 3 lenses and most of them look completely different from each other. With a monochromatic 4 lens design there are even more solutions. The Double-Gauss type shown on the last slide (but not the example that is shown) is probably the best type but that is really not known for sure, for this small box size problem. It is a lot of work to find the best solution.

Two examples of monochromatic triplets with all 3rd-order aberrations = 0,where lens thickness is a very important parameter. With thickness as a variable there are many such 3rd order solutions. With thin lenses there are just a few. With four lenses there are a very large number of distinct 3rd-order solutions.We want designs with the best 5th and higher orderaberrations and then there are not so many good ones.

f/3, 15 degrees total field, worst r.m.s = .29 waves r.m.s. at .5876u

Stronger curves than other Double-Gauss solution already shown. The Double-Gauss form has more than one local minimum.

Monochromatic design Distortion corrected

Best local minimum

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f/3, 15 degrees total field, worst r.m.s = .25 waves at .5876u

Best r.m.s. wavefront so far but meniscus lens rims exceed box length, as well as too large lens diameters and an aperture stop outside of the box. Not acceptable.

One of very many 4 lens local minima

The very tight space constraints in this problem kill off almost all of the very many multiple solutions (local minima) that are possible with multi-element designs.The triplet starting point is a very compact design.

Not quite as good as the four lens Double-Gauss, .31 waves r.m.s.

Meniscus lens flipped over to other side is not as good as this solution

Constraints on back focus distance and lens length, .29 waves r.m.s.

No space constraints, .075 waves r.m.s.

Floating stop = a paraxial pupil, not a real stop

Double-Gauss works best with no space constraints

Local minimum, .17 waves r.m.s.

Local minimum, .12 waves r.m.s.

Best five-lens design found so far

Five–Lens designsDifferent power and bending

Very different local minimum, .15 waves r.m.s.

Better local minimum, .12 waves r.m.s.

Different power and bending Alternate 5 lens designs

Both 5-lens designs have same performance = .12 waves r.m.s. But bottom design has strong curves, tighter tolerances. This shows the value of looking for multiple solutions.

A simple splitting of the middle section does very little, .095 waves r.m.s. We need a different distribution of power inside the lens

Six-lens design

Good six-lens design, .065 waves r.m.s.

Another good six-lens design, .065 waves r.m.s.

Good seven-lens design, .036 waves r.m.s.

Better seven-lens design, .029 waves r.m.s.

Sum of absolute value of curvatures = .428, sum for starting triplet = .129

Three lenses, .70 waves r.m.s.

Four lenses, .29 waves r.m.s.

Five lenses, .12 waves r.m.s.

Six lenses, .065 waves r.m.s.

About 2X improvement for each extra lens

(Two different solutions)

(Two different solutions)

Seven lenses = .029 waves r.m.s.

So for this very space-constrained example, every lens added to the starting triplet doubles the performance, but

1) that is only true for the best of the multiple solutions

2) Eventually there will be no room for more lenses, and ever thinner lens edges and center thicknesses are needed

3) Improvements start to get smaller with many more lenses – there is an ultimate limit to what can be done within these space constraints.

4) It takes more and more time to find the best n-lens solution, as n increases and there are many local minima

2) Adding some aspherics indicates that there might be one last 2X improvement possible over the best seven-lens design (.029 waves r.m.s.), maybe in a nine-lens design without aspherics.

1) In my camera lens talk, 6 months ago, I showed how aspherics in a design can be used to predict the ultimate performance level without aspherics and also can show where in a design to add a lens to improve performance.

3) But strongly curved surfaces have aberrations that cannot be duplicated by higher-order aspherics on weakly curved surfaces.So points 1) and 2) above have large but also limited value when there are some strong surfaces in a design.

Now we will look at some of the difficulties involved in finding multiple local minima in a design

Most designs have multiple local minima. There can be tipping points in a design starting point where a very small change in one radius, for example, will cause the design to branch off in different directions to different local minima.

Goes to

Starting 1st radius = 24.003

For a fixed focal length there are always two lens bending that give the same amount of spherical aberration

Starting design, .31 units of spherical aberration, but want .50

These two designs have the same amount of spherical aberration, .50 units of it

Goes to

Starting 1st radius = 24.002

For values of the last radius > -39.41 the exact solution is found, with no “kicks” involved. - 39.41 is right on the edge of a tipping point. For the last radius between -39.42 and -39.54, a non-exact local minimum solution is found. -39.54 is right at the edge of another tipping point. For the last radius < -39.55 a different local minimum is found. There is nothing magic about the last radius and any of the other radii probably has some tipping point value or values. Here the last radius has two tipping points (that I have found close to each other, there may be more).

Exact solution

Local minimum

Local minimum

Starting point, last radius = -39.41

Starting point, last radius = -39.42 to 39.54

Starting point, last radius < -39.55

All 3rd-order = 0

Usually there will be fewer local minimum with a merit function that is only real rays than if the merit function is only 3rd order or 3rd and 5th order.

So the examples just shown are a special case. A design tipping point will not be anywhere near as sharply defined when only rays are optimized.

All of the multi-element designs shown earlier were found by real ray optimization using the OSLO ASA program, a simulated annealing design method. ASA gives the best chance of finding multiple good solutions.

Some very simple optical designs can show extreme non-linearities during optimization. This makes it hard to find good solutions and very hard to find multiple solutions. Here is an example with just 2 surfaces!

Two spherical mirrors

For a given object distance there are only three variables – two radii and the mirror separation. It is possible to make 3rd, 5th, and 7th order spherical aberration exactly 0.0 but it is very hard to find this solution!

Design has bad coma

Starting point = looks very much like exact solution

But can’t get there from here. Starting point needs to be even closer to final variable values.

Exact solution

Starting point

But a purely ray or wavefront optimization has much less trouble getting to the good solution

3rd, 5th,7th order spherical aberration = 0.0

This is a very non-linear system

aberration

Starting point

When starting point is not close enough to the desired exact solution the computer heads straight for one of two alternate exact solutions that we don’t want.

1) It makes both mirrors concentric about the object point so neither mirror has any spherical aberration of any order

2) Or it makes one mirror be concentric about the object point and puts the other mirror at its focus.

The exact solution that we want has large amounts of aberration cancelling out exactly between the two mirrors. In the starting point it doesn’t cancel exactly. The computer sees that the cancellation gets better if both mirrors are given smaller aberration so it heads for one of these alternate exact solutions.

Conclusions

Even very simple designs can have multiple good solutions

Non-linearities can make these alternate solutions very hard to find

A tight space constraint makes for difficult design challenges. The usual design forms might not work well with a short lengthand long back focal length

The End