# I 3rd Order Aberrations from Paraxial Rays?

1. May 12, 2018

### Twigg

Hey all,

In chapter 6 section 3 of Modern Optical Engineering, 4th edition, by Warren J. Smith, it claims you can calculate all the 3rd order aberrations by considering two paraxial rays. I'm trying to convince myself of this and not having much success. Unfortunately, the article that this book gets this info from is subscription-walled to me (Article: https://doi.org/10.1364/JOSA.41.000630).

My question is, how would you explain that you can get 3rd order aberrations from two paraxial rays, even though the whole problem with aberrations is that they are missing from the paraxial approximation? Is it just some algebraic coincidence, or is there an intuitive reason?

2. May 12, 2018

### sophiecentaur

Could it just be that the two paraxial rays will intersect the prime axis at different distances if there is aberration.? (The axial ray is 'assumed') But the actual order???

3. May 12, 2018

### Twigg

Yeah I should have specified: the two paraxial rays are one axial ray (starting from the optical axis at the object position and going through the edge of the entrance pupil) and one principal ray (starting at an off-axis point at the object position and going through the center of the entrance pupil).

I don't think this is the case. If you just traced these as paraxial rays (in my mind, I think of "tracing the paraxial rays" as using, say, ABCD matrices, maybe this is the wrong interpretation?), then they should focus at the same image point, no?

Unfortunately the author doesn't really explain, but just gives a laundry list of formulas based on the raytrace of these two rays. I'll try to paraphrase in another reply to this thread.

4. May 12, 2018

### Twigg

5. May 13, 2018

### Twigg

Friend of mine has an idea what is going in this book. He thinks they have analytical solutions for the aberrations that can be parametrized in terms of the paraxial raytrace data. Seems like that may be the case...

6. May 13, 2018

### sophiecentaur

It's too hard for me, I'm afraid. Specialised optics stuff.

7. May 13, 2018

### Twigg

No worries, I appreciate the effort :)

8. May 14, 2018

### Andy Resnick

As you have figured out so far, there are 2 special rays, the chief and marginal ray, which are used to parametrize imaging systems. Just to back up a bit, the aberrations you showed in the pages are called '3rd order' because of the Taylor series expansion of sin(u) = u - u^3/3! + u^5/5! -...; paraxial ray tracing uses the approximation sin(u) = u, so the first 'aberration' is 3rd order in u. There are also 5th order, 7th order...

The basic starting point, provided by Buchdahl, is the 'optical invariant', also known as the etendue. The derivations of the various formulas are incredibly opaque, especially the stop-shift formulas, so the final formulas are typically just provided 'fait accompli'. Buchdahl's book 'Optical Aberration Coefficients' is extremely thorough and has the 'original' derivations all the way through 7th order, including aspherical surfaces and chromatic aberrations (variations of the monochromatic aberrations with refractive index).

Does that help?