## Second-order derivatives of optical path length of ray with respect to variable vector of source ray |

Applied Optics, Vol. 51, Issue 22, pp. 5552-5562 (2012)

http://dx.doi.org/10.1364/AO.51.005552

Enhanced HTML Acrobat PDF (766 KB)

### Abstract

A method is proposed for determining the second-order derivatives (i.e., the Hessian matrix) of the optical path length of a ray with respect to the variable vector of the source ray in an optical system comprising both flat and spherical boundary surfaces. Several wavefront aberration problems are investigated using the Hessian matrix proposed in this study and the Jacobian (first-order derivatives) matrix presented in the literature. It is found that when using the Hessian matrix the precision of wavefront aberration is significantly improved when evaluated up to the quadratic term of the Taylor series expansion. The methodology proposed in this study not only provides the means to investigate the principal curvatures of the wavefront along a ray, but also yields the information required to determine the irradiance and caustics of both axisymmetric and nonaxisymmetric optical systems.

© 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

**History**

Original Manuscript: May 24, 2012

Revised Manuscript: July 9, 2012

Manuscript Accepted: July 9, 2012

Published: July 30, 2012

**Citation**

Yu-Bin Chen and Psang Dain Lin, "Second-order derivatives of optical path length of ray with respect to variable vector of source ray," Appl. Opt. **51**, 5552-5562 (2012)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-51-22-5552

Sort: Year | Journal | Reset

### References

- R. J. Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady-State and Time-Dependent Problems (SIAM, 2007), pp. 3–4.
- B. D. Stone, “Determination of initial ray configurations for asymmetric systems,” J. Opt. Soc. Am. A 14, 3415–3429 (1997). [CrossRef]
- T. B. Andersen, “Optical aberration functions: derivatives with respect to axial distances for symmetrical systems,” Appl. Opt. 21, 1817–1823 (1982). [CrossRef]
- T. B. Andersen, “Optical aberration functions: derivatives with respect to surface parameters for symmetrical systems,” Appl. Opt. 24, 1122–1129 (1985). [CrossRef]
- D. P. Feder, “Calculation of an optical merit function and its derivatives with respect to the system parameters,” J. Opt. Soc. Am. 47, 913–925 (1957). [CrossRef]
- D. P. Feder, “Differentiation of ray-tracing equations with respect to constructional parameters of rotationally symmetric systems,” J. Opt. Soc. Am. 58, 1494–1505 (1968). [CrossRef]
- P. D. Lin and C. Y. Tsai, “First order gradients of skew rays of axis-symmetrical optical systems,” J. Opt. Soc. Am. A 24, 776–784 (2007). [CrossRef]
- P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew-rays of optical systems with non-coplanar optical axes,” Appl. Phys. B: Lasers Opt. 91, 621–628 (2008). [CrossRef]
- 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: Lasers Opt. 98, 471–479 (2010). [CrossRef]
- P. D. Lin and W. Wu, “Determination of second-order derivatives of a skew-ray with respect to the variables of its source ray in optical prism systems,” J. Opt. Soc. Am. A 28, 1600–1609 (2011). [CrossRef]
- P. D. Lin, “Second-order derivatives of a ray with respect to the variables of its source ray in optical systems containing spherical boundary surfaces,” J. Opt. Soc. Am. A 28, 1995–2005 (2011). [CrossRef]
- O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley, 2006).
- O. N. Stavroudis, The Optics Of Rays, Wavefronts, and Caustics (Academic, 1972).
- D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surace,” J. Opt. Soc. Am. A 25, 2370–2382 (2008). [CrossRef]
- J. A. Hoffnagle and D. L. Shealy, “Refracting the k-function: Stavroudis’s solution to the eikonal equation for multi-element optical systems,” J. Opt. Soc. Am. A 28, 1312–1321 (2011). [CrossRef]
- R. P. Paul, Robot Manipulators-Mathematics, Programming and Control (MIT, 1982).
- A. Mikŝ, “Dependence of the wavefront aberration on the radius of the reference sphere,” J. Opt. Soc. Am. A 19, 1187–1190 (2002). [CrossRef]
- J. Meiron, “The use of merit functions based on wavefront aberrations in automatic lens design,” Appl. Opt. 7, 667–672(1968). [CrossRef]
- M. Laikin, Lens Design (Marcel, 1995).

## 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.