OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 28, Iss. 6 — Jun. 1, 2011
  • pp: 1312–1321

Refracting the k-function: Stavroudis’s solution to the eikonal equation for multielement optical systems

John A. Hoffnagle and David L. Shealy  »View Author Affiliations

JOSA A, Vol. 28, Issue 6, pp. 1312-1321 (2011)

View Full Text Article

Enhanced HTML    Acrobat PDF (591 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The k-function of Stavroudis describes a solution of the eikonal equation in a region of constant refractive index. Given the k-function describing the optical field in one region of space, and given a prescribed refractive or reflective boundary, we construct the k-function for the refracted or reflected field. This procedure, which Stavroudis calls refracting the k-function, can be repeated any number of times, and therefore extends the usefulness of the k-function formalism to multielement optical systems. As examples, we present an analytic solution for the k-function, wavefronts, and caustics generated by a biconvex thick lens illuminated by a plane wave propaga ting parallel to the symmetry axis, and numerical results for off-axis plane-wave illumination of a two-mirror telescope.

© 2011 Optical Society of America

OCIS Codes
(080.0080) Geometric optics : Geometric optics
(080.2720) Geometric optics : Mathematical methods (general)
(080.2740) Geometric optics : Geometric optical design

Original Manuscript: March 14, 2011
Manuscript Accepted: March 31, 2011
Published: May 31, 2011

John A. Hoffnagle and David L. Shealy, "Refracting the k-function: Stavroudis’s solution to the eikonal equation for multielement optical systems," J. Opt. Soc. Am. A 28, 1312-1321 (2011)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), pp. 109–142.
  2. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, 3rd ed. (Pergamon, 1971), pp. 130–136.
  3. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley, 2006), pp. 67–146. [CrossRef]
  4. K. Schwarzschild, “Untersuchungen zur geometrischen Optik. I. Einleitung in die Fehlertheorie optischer Instrumente auf Grund des Eikonalbegriffs,” in Astronomische Mittheilungen der Königlichen Sternwarte zu Göttingen (Dieterich’schen University, 1905), pp. 1–28.
  5. R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964), pp. 82–116.
  6. A. Walther, The Ray and Wave Theory of Lenses (Cambridge University, 1995), pp. 59–68. [CrossRef]
  7. D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surface,” J. Opt. Soc. Am. A 25, 2370–2382 (2008). [CrossRef]
  8. J. A. Hoffnagle and D. L. Shealy, “Wavefront generated by reflection of a plane wave from a conic section,” Proc. SPIE 7060, 70600V-1–70600V-9 (2008). [CrossRef]
  9. E. Román-Hernández, J. G. Santiago-Santigo, G. Silva-Ortigoza, and R. Silva-Ortigoza, “Wavefronts and caustics of a spherical wave reflected by an arbitrary smooth surface,” J. Opt. Soc. Am. A 26, 2295–2305 (2009). [CrossRef]
  10. A. Aveñdana-Alejo, L. Castañedo, and I. Moreno, “Properties of caustics produced by a positive lens: meridional rays,” J. Opt. Soc. Am. A 27, 2252–2260 (2010). [CrossRef]
  11. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972), p. 83
  12. O. N. Stavroudis and R. C. Fronczek, “Caustic surfaces and the structure of the geometrical image,” J. Opt. Soc. Am. 66, 795–800 (1976). [CrossRef]
  13. O. N. Stavroudis, “The k-function in geometrical optics and its relationship to the archetypal wavefront and the caustic surface,” J. Opt. Soc. Am. A 12, 1010–1016 (1995). [CrossRef]
  14. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flanney, Numerical Recipes in Fortran 77: The Art of Scientific Computing, 2nd ed., (Cambridge University, 2003), p. 181, Eq. (5.7.7).
  15. A. Cordero-Dávila and J. Castro-Ramos, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror,” Appl. Opt. 37, 6774–6778 (1998). [CrossRef]
  16. G. Silva-Ortigoza, J. Castro-Ramos, and A. Cordero-Dávila, “Exact calculation of the circle of least confusion of a rotationally symmetric mirror. II,” Appl. Opt. 40, 1021–1028(2001). [CrossRef]
  17. R. W. Hosken, “Circle of least confusion of a spherical reflector,” Appl. Opt. 46, 3107–3117 (2007). [CrossRef] [PubMed]
  18. J. A. Hoffnagle and D. L. Shealy, “Caustic surfaces of a Keplerian two-lens beam shaper,” Proc. SPIE 6663, 666304-1–666304-9 (2007). [CrossRef]
  19. J. J. Stamnes, Waves in Focal Region: Propagation, Diffraction and Focusing of Light, Sound and Water Waves (Adam Hilger, 1986), pp. 163–200.
  20. D. J. Schroeder, Astronomical Optics (Academic, 1987), pp. 15–19.
  21. R. W. Wood, Physical Optics (Optical Society of America, 1988), 3rd ed.,, p. 50.
  22. M.Born and E.Wolf, Principles of Optics, 7th ed., (Cambridge University, 1999), pp. 203–207.

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