OSA's Digital Library

Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Editor: Joseph N. Mait
  • Vol. 51, Iss. 18 — Jun. 20, 2012
  • pp: 4087–4091

Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils

Virendra N. Mahajan  »View Author Affiliations


Applied Optics, Vol. 51, Issue 18, pp. 4087-4091 (2012)
http://dx.doi.org/10.1364/AO.51.004087


View Full Text Article

Enhanced HTML    Acrobat PDF (113 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

In a recent paper, we considered the classical aberrations of an anamorphic optical imaging system with a rectangular pupil, representing the terms of a power series expansion of its aberration function. These aberrations are inherently separable in the Cartesian coordinates (x,y) of a point on the pupil. Accordingly, there is x-defocus and x-coma, y-defocus and y-coma, and so on. We showed that the aberration polynomials orthonormal over the pupil and representing balanced aberrations for such a system are represented by the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point; for example, Ll(x)Lm(y), where l and m are positive integers (including zero) and Ll(x), for example, represents an orthonormal Legendre polynomial of degree l in x. The compound two-dimensional (2D) Legendre polynomials, like the classical aberrations, are thus also inherently separable in the Cartesian coordinates of the pupil point. Moreover, for every orthonormal polynomial Ll(x)Lm(y), there is a corresponding orthonormal polynomial Ll(y)Lm(x) obtained by interchanging x and y. These polynomials are different from the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil. In this paper, we show that the orthonormal aberration polynomials for an anamorphic system with a circular pupil, obtained by the Gram–Schmidt orthogonalization of the 2D Legendre polynomials, are not separable in the two coordinates. Moreover, for a given polynomial in x and y, there is no corresponding polynomial obtained by interchanging x and y. For example, there are polynomials representing x-defocus, balanced x-coma, and balanced x-spherical aberration, but no corresponding y-aberration polynomials. The missing y-aberration terms are contained in other polynomials. We emphasize that the Zernike circle polynomials, although orthogonal over a circular pupil, are not suitable for an anamorphic system as they do not represent balanced aberrations for such a system.

© 2012 Optical Society of America

OCIS Codes
(010.7350) Atmospheric and oceanic optics : Wave-front sensing
(110.0110) Imaging systems : Imaging systems
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(220.0220) Optical design and fabrication : Optical design and fabrication
(220.1010) Optical design and fabrication : Aberrations (global)

ToC Category:
Imaging Systems

History
Original Manuscript: April 2, 2012
Manuscript Accepted: April 16, 2012
Published: June 14, 2012

Citation
Virendra N. Mahajan, "Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils," Appl. Opt. 51, 4087-4091 (2012)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-51-18-4087


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (SPIE, 2011).
  2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  3. V. N. Mahajan, Optical Imaging and Aberrations, Part I: Ray Geometrical Optics (SPIE, 2004).
  4. V. N. Mahajan, “Orthonormal aberration polynomials for anamorphic optical imaging systems with rectangular pupils,” Appl. Opt. 49, 6924–6929 (2010). [CrossRef]
  5. J. C. Burfoot, “Third-order aberrations of ‘doubly symmetric’ systems,” Proc. Phys. Soc. B 67, 523–528 (1954). [CrossRef]
  6. C. G. Wynne, “The primary aberrations of anamorphotic lens systems,” Proc. Phys. Soc. B 67, 529–537 (1954). [CrossRef]
  7. V. N. Mahajan, “Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils,” presented at the annual meeting of OSA, San Jose, Calif., 16Oct.2011.
  8. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).
  9. V. N. Mahajan and G.-M. Dai, “Orthonormal polynomials in wavefront analysis: analytical solution,” J. Opt. Soc. Am. A 24, 2994–3016 (2007). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited