# Intensity loss due to vignetting

I was trying to get an expression for the loss of intensity due to vignetting in a simple optical system, and got a fairly complex integral. I was wondering if there's an easier way, or any book that does this calculation. I guess not, because basic books as Hetch or Born & Wolf don't give the expression, but it doesn't hurt to ask.

The system is simply an extended light source at the focal plane of a converging lens, a second converging lens at a certain distance, and a detector at the focal plane of the second lens. Everything is circular, both lenses are ideal and have the same diameter, and the detector is big enough. I'm interested in an isotropic light source with a Gaussian profile (i.e., a fluorescence emission from a surface when illuminated by a focused laser), but you can assume a different angular or radial profile if that makes things simpler.

If this is not usually done by hand, but you use an optical design program to get the result for each case, please do tell me which do you use.

In short, the questions are:

1. What is the an analytical expression for this?
2. Is there any book that explicitly makes the calculation?
3. If this is usually done with a specialized program, which one do you recommend?

Thanks in advance

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It is going to be an ugly integral. You are basically calculating the area of overlap between two circles as a function of field angle. It's gross, but not actually too difficult to derive. A far as books, I wouldn't expect Hecht or B&W to have something so"applied". Maybe look in one of the books by Warren Smith (which you had better own if you want to do anything related to optics!) For software: codeV does this, I think the other lens design tools do too. You could also model this easily in something like Matlab. – Colin K Nov 24 '11 at 17:25
I'd make a decent answer out of that comment but it's thanksgiving and I've only got a smartphone. – Colin K Nov 24 '11 at 17:26
@ColinK Thanks. Please do make it an answer when you have access to a keyboard ;) By the way, which Warren Smith? I found several books of him. – Arnoques Nov 25 '11 at 3:09

## 2 Answers

There's a nice quantitative discussion of natural, optical and mechanical vignetting in "Applied Photographic Optics" by Sidney F. Ray. It covers peripheral illuminance fall-off in detail.

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Since @ColinK didn't make an answer from his comment, I'll copy it here:

It is going to be an ugly integral. You are basically calculating the area of overlap between two circles as a function of field angle. It's gross, but not actually too difficult to derive. A far as books, I wouldn't expect Hecht or B&W to have something so "applied". Maybe look in one of the books by Warren Smith (which you had better own if you want to do anything related to optics!) For software: codeV does this, I think the other lens design tools do too. You could also model this easily in something like Matlab.

Thanks for the answer!

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Sorry! I totally forgot. I'll try to get to this today. – Colin K Dec 5 '11 at 18:18