Aberration with explicit dependance on object coordinates

[frederic leroux]

TL;DR Summary

How to obtain the coefficients of the aberration expansion with explicit dependance on object coordinates for a single spherical or aspheric surface ?

Hello,
In order to get the coefficients of the aberration expansion with no explicit dependance on object coordinates I fit the optical path difference with the Zernike basis and convert with the paper of Robert K. Tyson "Conversion of Zernike aberration coefficients to Seidel and
higher-order power-series aberration coefficients". Now how could I obtain the coefficients with explicit difference on object coordinates ? Should I repeat with n objects and solve a n linear equations system to get the n coefficients of, let's say comma, h.r.cos(theta) ... h^(2n+1).r.cos(theta)?

Also I would like to know if they are already analytically solved for a single spherical or aspheric surface. For instance Mahajan solved analytically the Seidel aberration coefficients with a spherical and an aspheric surface. Is it solved analytically somewhere for 6th and 8th order wave aberration ? I read the book Optical Aberration Coefficients from Buchdahl would contain what I need but this book is unobtainable. I can't find what I need with Kidger, Kingslake and Welford. Which book as thorough as Buchdahl would you advise me? Also knowing the aberration at a single surface, how would you update it with a stop shift because the stop dang equation works only for Seidel ?

Thank you,
Frederic Leroux

 

[Andy Resnick]

TL;DR Summary: How to obtain the coefficients of the aberration expansion with explicit dependance on object coordinates for a single spherical or aspheric surface ?

[...]

Your question is a little confusing (to me)- I have a copy of Buchdahl, and there is a section "The dependence of aberration coefficients on the position of the object plane". Is that what you are hoping to learn? The book is very complete and detailed and largely beyond my ability to understand :)

 

[Tom.G]

A Google search shows several sources for the book in the USD$100 - $130 range:
https://www.google.com/search?&q=+Optical+Aberration+Coefficients+by+Buchdahl

Cheers,
Tom

 

[frederic leroux]

Your question is a little confusing (to me)- I have a copy of Buchdahl, and there is a section "The dependence of aberration coefficients on the position of the object plane". Is that what you are hoping to learn? The book is very complete and detailed and largely beyond my ability to understand :)

The dependance of aberration oefficients on the position of the object plane is more about conjugate shift. It is well known for Seidel terms but maybe Buchdahl wrote it explicitly for higher-order aberrations. It could be interesting.
Otherwise I know how to get aberration for coma for instance without explicit dependance on object. For coma I have something a_31*r^3*cos(theta). But this a_31 term is composed of 1_a_31*h+3_a_31*h^3+5_a_31*h^5+... I'm trying to get these aberration terms 1_a_31, 3_a_31, 5_a_31 and I don't know how to do.

 

[Andy Resnick]

[...]. I'm trying to get these aberration terms 1_a_31, 3_a_31, 5_a_31 and I don't know how to do.

Sorry for the delayed response- I was trying to come up with an answer other than "this is way beyond my understanding". Unfortunately, that's the only answer I have :)

Honestly, get a copy of Buchdahl. FWIW, the Dover edition has reprints of a series of his published papers, you may find what you want there (I assume you have access to the Journal of the Optical Society of America [JOSA] archive):

JOSA 46 (11) 941-943 (1951)
JOSA 48 (8) 563-567 (1958)
JOSA 48 (10) 747-756 (1958)
JOSA 48 (10) 757-759 (1958)
JOSA 49 (11) 1113-1121 (1959)
JOSA 50 (6) 534-539 (1960)
JOSA 50 (6) 539-544 (1960)
JOSA 50 (7) 678-683 (1960)