Lens design

[alanrockwood]

Today, just for fun, I designed a lens using WinLens 3D Basic. The lens design uses six elements in 4 groups, and all Seidel aberrations are essentially zero, although there was an extremely small amount (negligible really) of Petzval curvature and astigmatism. Chromatic aberrations are zero, except for a very small amount of secondary color.

I designed this lens to cover a 7mm radius at 1:1 magification, and the spot diagram fits well inside of the diffraction limit circle over the full field. This is for an f/4 aperture. It can even go to f/3.5. The on-center MTF is at the diffraction limit clear down to a spatial frequency of 200/mm at f/4, which is basically where the contrast at the diffraction limit bottoms out to zero. The basic version of the WinLens won't report the field dependent MTF functions, so I can't report those numbers, but they should also be pretty close to the diffraction limit for this design.

The design uses three kinds of glass, and I don't know if they are expensive types or not, though my guess is that they are.

The design is not compact, being rather long.

I think this lens could be useful for reproduction at around 1:1 (or even ratios significantly different from this), or as a lens in a scanner of certain designs.

 

[Nodda Duma]

Awesome! Here are a couple of questions that I can help you with:

What are the glass types? I can tell you if they are cheap, expensive, etc., and give you Ohara equivalents (where applicable) which is a better supplier than Schott. Keep in mind that for prototype numbers, cost will be $1000-$1500 per element even for the cheapest glass (BK-7).

Coming up with a prescription is about 1/3 of the design effort. You should also examine tolerances to ensure it can be made. Producing perfect designs that cannot be made is a common pitfall. Dffraction-limited performance on-axis at f/4 plus being long tells me that fabrication and alignment tolerances will be a challenge, but the small image circle may counterbalance that. I'm not familiar enough with winlens to know how/if they support tolerancing. If they don't, a basic approach is to adjust the position of an element with respect to the others by 0.001" to 0.005" and see if refocusing corrects the resultant degradation of performance. That approach can be misleading, but it will give you a feel for whether to continue with real tolerancing or whether to abandon all hope.

Once you start designing a barrel, evaluate stray light for ghosting, flare, etc, and then there are various mitigation techniques to address any issues that crop up.

Cheers,
Jason

 

[alanrockwood]

...What are the glass types?...

...You should also examine tolerances to ensure it can be made.... I'm not familiar enough with winlens to know how/if they support tolerancing...

Cheers,
Jason

Thanks Jason. WinLens has a companion application for tolerancing, but the tolerancer does not come with the free version of WinLens.

This is actually a bit of a sore spot with me because a few years ago I ordered the full version of WinLens. Not long afterwards (I think it was a few months after I paid for the full version, but it may have been a little longer) unbeknownst to me they released a later version of WinLens. A few years later my employer updgraded my computer, and my copy of WinLens would not run on my new computer. I contacted Qioptiq/Linos (the distributer for WinLens) about this, and they said I would have to pay again if I wanted the version which would run on my new computer. Nowadays I don't have funding to buy the full version, so I have reverted to using the free version which does not include the tolerancer. Since I am just a hobbiest and am currently unemployed it is hard to justify paying for the full version.

Anyway, I will try some of the strategies you mentioned. A quick check showed the design quite robust against spacing of the elements.

Regarding the glass types, it will be interesting to see if they are expensive glasses. I suspect they are because they have dispersion values that tend to be at the extreme high and extreme low end of the range for their refractive indexes.

By the way, I started designing a monochromatic version using BK7 but gradually reverted to other glasses as I went along, so I could make the system monochromatic and also to get the Petzval curvature and astigmatism down to residual levels.

Regarding tolerancing, there is at least one aspect of the design where tolerances are probably fairly forgiving. There are two cemented doublets in the system. They are identical. The two glass types in each cemented doublet have virtually the same refractive index, but differ in dispersion. This means that if the radius of the cemented surface changes it may change the color correction a bit, but not the monochromatic aberrations. Also, the system is symmetrical, so I think that might tend to make the design a little more robust. Also, the color correction is split between multiple negative lenses, so if one is a little off at least it won't take all of the color correction along with it.

By the way, I mentioned that two of the glasses have virtually the same refractive index. The refractive index of the third is very close to the other two as well. I don't know if this is good design practice, but it is more or less an artifact of my seat-of-the-pants approach to this design, which was to start the design with a single glass type.

I have a question about Petzval curvature. (Assume for sake of discussion that we are talking about a design using a single refractive index.) I had read that if the curvatures of the glass surfaces (taking into account the sign conventions for the front and back surfaces) add up to zero then the Petzval curvature will be zero. However, as I spaced the elements further apart this turned out to not be true. Is the theorem about Petzval curvature just an approximation?

Also, I found that astigmatism was extremely hard to get rid of. Are there simple rules of thumb for reducing astigmatism?

 

[Nodda Duma]

Establishing a monochromatic design and then correcting for color is the right approach. Doesn't always work but it keeps you out of trouble.

Perturbations of radius of curvature will impact field curvature as well as spherical aberration and axial color. It also affects focus, but that's usually a non-issue.

Tolerances on airgaps (element spacing) is usually fairly benign. You'll also want to decenter each element 50 to 75 microns (0.002" - 0.003") and perhaps shift one element in one direction and another in the other direction to see how the MTF responds. 50 to 75 micron decenter is achievable without having to do anything "special". By special I mean driving your costs way up. Also evaluate 3 arcminute wedge in each element.

Thicker lenses (or higher index) and more compact layouts reduce astigmatism in symmetrical designs.

Cost vs. glass map is all over the place... there's no rhyme or reason except with newer very high index / low dispersion. You can find cost info in Schott's catalog. 95% of the time you can sub out expensive glasses and re-optimize the prescription with cheaper glass. The optimizer algorithm in winlens is separating the dispersion values because that's an effective way to correct color in a doublet.

Re: Petzval curvature. Adding up the radii of curvature for an element doesn't do anything for you except give you practice at addition. You need to include the effect of refractive index of the air and glass (or glass and glass in a doublet). See the equation on the Petzval field curvature wikipedia page for the governing equation.

 

[alanrockwood]

...Re: Petzval curvature. Adding up the radii of curvature for an element doesn't do anything for you except give you practice at addition. You need to include the effect of refractive index of the air and glass (or glass and glass in a doublet). See the equation on the Petzval field curvature wikipedia page for the governing equation.

Thanks for the comments.

I am aware that Petzval curvature depends on refractive indexes, but in the special case that all of the lens elements have the same refractive index, and if adding the curvatures gives a zero result, then the Petzval curvature will be zero as well.

 

[alanrockwood]

...Thicker lenses (or higher index) and more compact layouts reduce astigmatism in symmetrical designs...

That's good to know.

In my design the air gaps are big. I did that so I could get positive focusing while still keeping the Petzval sum nearly equal to zero. Otherwise the positive and negative lens elements would nearly cancel out. I put the positive elements on outside with large air gaps before getting to the negative elements on the inside. This results in a physically long lens. Of course, this approach will limit the field of view, but I figure at 1:1 magnification the field of view going to be fairly narrow anyway, so I think this is OK.

In my design both the Petzval curvature and astigmatism were really small. I think I could probably tweek the design a little more to zero them out altogether, but I haven't gone quite that far, and it seems unnecessary.

One thing I have wondered about is whether one can find two glasses that can be combined to result in both a zero Petzval sum and an achromatic condition in a cemented doublet, or even an air-spaced doublet. I haven't tried to analyze this.

 

[Nodda Duma]

The other thing to do is correct for sagittal field curvature and balance out astigmatism with tangential. That's a common balancing act and results in acceptable performance.

Thanks for the comments.

I am aware that Petzval curvature depends on refractive indexes, but in the special case that all of the lens elements have the same refractive index, and if adding the curvatures gives a zero result, then the Petzval curvature will be zero as well.

True, but that is impractical for anything but a single wavelength... like a laser.

Zen wisdom: Lens design is all about balance and symmetry ....

 

[Sirius Glass]

Assuming the design is mature and ready for production, how do you plan to have it built?

 

[alanrockwood]

Assuming the design is mature and ready for production, how do you plan to have it built?

Not thinking that far ahead yet... for now I am just working on a paper design.

 

[Dan Fromm]

I designed this lens to cover a 7mm radius at 1:1 magification, and the spot diagram fits well inside of the diffraction limit circle over the full field.

I understand the joy of tinkering, also that prescriptions scale up/down as needed. What's the design focal length and, if you're thinking about making a lens of that focal length, what's your intended application? A 7mm image circle seems small even for a subminiature camera. Cell phone, perhaps?

 

[alanrockwood]

Here's more information. This is a spot diagram for the current version of the lens I am tinkering with.

This is for a f-number of 6.3 and an image circle of radius 13.5mm (which I think covers Dan's question). This is about the size of the diagonal of a Canon Digital Rebel or similar camera. The circle on the plots is the Airy circle, which basically represents one version of a diffraction limit. By the way, This version of the lens covers a wider circle, but the aperture is less than my earlier version. The plots to look at are the two plots in the center of the figure. They represent the image center and the image corner.

Here is a snapshot of a raytrace:

Another detail, I dropped the number of glass types from three to two.

Here's the aberration report:

SphAbn: 0.00107

Coma: 0

Astig: 0

PtzCv: 0

Distn: 0

CI: -0.000075

CII : -7.5E-05

Secndry Spec: -0.000311

There are a lot of zeros, which is a good thing. I was able to make all of the listed aberrations zero (except for the secondary spectrum), but I found I got better spot sizes by under correcting spherical aberration, probably by balancing it against higher order spherical aberration. Also, slightly over-correcting chromatic aberration made a better balance between the spot sizes, considering all three colors.

 

[Sirius Glass]

Interesting software. It looks like fun to play with it.

 

[alanrockwood]

Interesting software. It looks like fun to play with it.

Yes, WinLens fun to play with, and the basic version of the software is free. It's actually quite powerful, even the free version.

There is a bit of a learning curve, but not too bad.

By the way, here is what the spot size looks like at a -0.1mm focal plane shift for green light. This is the optimum focal point for green.

As you can see, this is somewhat better than plot given in an earlier post, but the red and blue spot sizes are correspondingly worse. Here is the blue spot size at a focal plane shift of -0.1mm:

The spot size for red at this shift in focal plane is pretty similar to the one for blue.

On the other hand, I can also pick a better focus point for the red and blue, which track each other pretty well in this design, but the green gets worse. This difference is a result of the secondary color aberration.