OSA's Digital Library

Applied Optics

Applied Optics


  • Editor: Joseph N. Mait
  • Vol. 51, Iss. 4 — Feb. 1, 2012
  • pp: 486–493

Determination of first-order derivative matrix of wavefront aberration with respect to system variables

Psang Dain Lin  »View Author Affiliations

Applied Optics, Vol. 51, Issue 4, pp. 486-493 (2012)

View Full Text Article

Enhanced HTML    Acrobat PDF (697 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The first-order derivative matrix of a function with respect to a variable vector is referred to as the Jacobian matrix in mathematics. Current commercial software packages for the analysis and design of optical systems use a finite difference (FD) approximation methodology to estimate the Jacobian matrix of the wavefront aberration with respect to all of the independent system variables in a single raytracing pass such that the change of the wavefront aberration can be determined simply by computing the product of the developed Jacobian matrix and the corresponding changes in the system variables. The proposed method provides an ideal basis for automatic optical system design applications in which the merit function is defined in terms of wavefront aberration. The validity of the proposed approach is demonstrated by means of two illustrative examples. It is shown that the proposed method requires fewer iterations than the traditional FD approach and yields a more reliable and precise optimization performance. However, the proposed method incurs an additional CPU overhead in computing the Jacobian matrix of the merit function. As a result, the CPU time required to complete the optimization process is longer than that required by the FD method.

© 2012 Optical Society of America

OCIS Codes
(080.0080) Geometric optics : Geometric optics
(080.1010) Geometric optics : Aberrations (global)
(080.2740) Geometric optics : Geometric optical design
(080.1753) Geometric optics : Computation methods

Original Manuscript: July 11, 2011
Revised Manuscript: August 30, 2011
Manuscript Accepted: September 3, 2011
Published: January 27, 2012

Psang Dain Lin, "Determination of first-order derivative matrix of wavefront aberration with respect to system variables," Appl. Opt. 51, 486-493 (2012)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. H. H. Hopkins, Wave Theory of Aberrations (OxfordU. Press, 1950), Chap. 1.
  2. T. Suzuki and I. Uwoki, “Differential method for adjusting the wave-front aberrations of a lens system,” J. Opt. Soc. Am. A 49, 402–404 (1958).
  3. J. A. Kneisly, “Local curvature of wavefronts in an optical system,” J. Opt. Soc. Am. 54, 229–235 (1964). [CrossRef]
  4. J. Meiron, “The use of merit functions on wave-front aberrations in automatic lens design,” Appl. Opt. 7, 667–672 (1968). [CrossRef]
  5. W. T. Welford, “A new total aberration Formula,” Opt. Acta 19, 719–727 (1972). [CrossRef]
  6. D. L. Shealy and D. G. Burkhard, “Caustic surface merit functions in optical design,” J. Opt. Soc. Am. 66, 1122 (1976). (Program of the 1976 Annual Meeting of the Optical Society of America, Tucson, Ariz., 18–22October1976). [CrossRef]
  7. A. M. Kassim, D. L. Shealy, and D. G. Burkhard, “Caustic merit function for optical design,” Appl. Opt. 28, 601–606 (1989). [CrossRef]
  8. J. Braat, “Analytical expressions for the wave-front aberration coefficients of a tilted plane-parallel plate,” Appl. Opt. 36, 8459–8466 (1997). [CrossRef]
  9. A. Mikŝ, “Dependence of the wave-front aberration on the radius of the reference sphere,” J. Opt. Soc. Am. 19, 1187–1190 (2002). [CrossRef]
  10. D. P. Feder, “Automatic optical design,” Appl. Opt. 2, 1209–1226 (1963). [CrossRef]
  11. C. G. Wynne and P. Wormell, “Lens design by computer,” Appl. Opt. 2, 1233–1238 (1963). [CrossRef]
  12. D. S. Grey, “The inclusion of tolerance sensitivities in the merit function for lens optimization,” SPIE Rev. 147, 63–65 (1978).
  13. D. P. Feder, “Calculation of an optical merit function and its derivatives with respect to the system parameters,” J. Opt. Soc. Am. 47, 913 (1957). [CrossRef]
  14. D. P. Feder, “Differentiation of ray-tracing equations with respect to construction parameters of rotationally symmetric optics,” J. Opt. Soc. Am. 58, 1494–1495 (1968). [CrossRef]
  15. A. Maréchal, “Etude des effets combines de la diffraction et des aberrations geometriques sur l’image d’un point lumineux,” Revue d'Optique, Theorique et Instrumentale 26, 257–277 (1947).
  16. R. Barakat and A. Houston, “Transfer function of an optical system in the presence of off-axis aberrations,” J. Opt. Soc. Am. 55, 1142–1148 (1965). [CrossRef]
  17. H. H. Hopkins, “The use of diffraction-based criteria of image quality in automatic optical design,” Opt. Acta 13, 343–369 (1966). [CrossRef]
  18. C. C. Hsueh and P. D. Lin, “Gradient matrix of optical path length for evaluating the effects of design variable changes in a prism,” Appl. Phys. B 98, 471–479 (2010). [CrossRef]
  19. C. C. Hsueh and P. D. Lin, “Computationally efficient gradient matrix of optical path length in axisymmetric optical systems,” Appl. Opt. 48, 893–902 (2009). [CrossRef]
  20. S. K. Gupta and R. Hradaynath, “Angular tolerance on Dove prisms,” Appl. Opt. 22, 3146–3147 (1983). [CrossRef]
  21. R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT Press, 1982).
  22. J. S. Arora, Introduction to Optimum Design (McGraw-Hill, 1989), p. 313.

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