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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 16, Iss. 3 — Mar. 1, 1999
  • pp: 596–601

General method for the determination of matrix coefficients for high-order optical system modeling

José B. Almeida  »View Author Affiliations


JOSA A, Vol. 16, Issue 3, pp. 596-601 (1999)
http://dx.doi.org/10.1364/JOSAA.16.000596


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Abstract

The nonlinear transformations incurred by the rays in an optical system can be suitably described by matrices to any desired order of approximation. In systems composed of uniform refractive-index elements each individual ray refraction or translation has an associated matrix, and a succession of transformations corresponds to the product of the respective matrices. A general method is described to find the matrix coefficients for translation and surface refraction, irrespective of the surface shape or the order of approximation. The choice of coordinates is unusual, as the orientation of the ray is characterized by the direction cosines rather than by the slopes; this is shown to greatly simplify and generalize coefficient calculation. Two examples are shown in order to demonstrate the power of the method: The first is the determination of seventh-order coefficients for spherical surfaces, and the second is the determination of third-order coefficients for a toroidal surface.

© 1999 Optical Society of America

OCIS Codes
(080.2730) Geometric optics : Matrix methods in paraxial optics
(220.1010) Optical design and fabrication : Aberrations (global)
(220.2740) Optical design and fabrication : Geometric optical design

History
Original Manuscript: July 14, 1998
Revised Manuscript: November 16, 1998
Manuscript Accepted: October 30, 1998
Published: March 1, 1999

Citation
José B. Almeida, "General method for the determination of matrix coefficients for high-order optical system modeling," J. Opt. Soc. Am. A 16, 596-601 (1999)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-16-3-596


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References

  1. A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Dover, New York, 1994).
  2. M. Born, E. Wolf, Principles of Optics (Pergamon Press, New York, 1980).
  3. G. G. Slyusarev, Aberration and Optical Design Theory (Hilger, Bristol, UK, 1984).
  4. M. Kondo, Y. Takeuchi, “Matrix method for nonlinear transformation and its application to an optical lens system,” J. Opt. Soc. Am. A 13, 71–89 (1996). [CrossRef]
  5. V. Lakshminarayanam, S. Varadharajan, “Expressions for aberration coefficients using nonlinear transforms,” Optom. Vis. Sci. 74, 676–686 (1997). [CrossRef]
  6. J. B. Almeida, “Use of matrices for third-order modeling of optical systems,” in International Optical Design Conference, K. P. Thompson, L. R. Gardner, eds., Proc. SPIE3482, 917–925 (1998). [CrossRef]
  7. D. S. Goodman, in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1995), Vol. 1, p. 93.

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