Abstract

The first- and second-order derivative matrices of the ray (i.e., ∂R¯i/∂X¯0 and ∂2i/∂X¯02) and optical path length (i.e., ∂OPLi/∂X¯0 and ∂2OPLi/∂X¯02) were derived with respect to the variable vector X¯0 of the source ray in an optical system by our previous papers. Using the first and second fundamental forms of the wavefront, these four matrices are used to investigate the local principal curvatures of the wavefront at each boundary surface encountered by a ray traveling through the optical system. The proposed method not only yields the data needed to compute the irradiance of the wavefront but also provides the information required to determine the caustics. Importantly, the proposed methodology is applicable to both axisymmetric and nonaxisymmetric optical systems.

© 2012 Optical Society of America

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2012 (1)

2011 (4)

2010 (1)

2008 (2)

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 91, 621–628 (2008).
[CrossRef]

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]

2007 (1)

1997 (1)

1988 (1)

A. M. Kassim and D. L. Shealy, “Wave front equation, caustics, and wave aberration function of simple lenses and mirrors,” Appl. Opt. 21, 516–522 (1988).
[CrossRef]

1985 (1)

1982 (1)

1981 (2)

1976 (1)

1973 (2)

1968 (1)

1964 (1)

1957 (1)

D. P. Feder, “Calculation of an optical merit function and its derivatives with respect to the system parameters,” J. Opt. Soc. Am. A 47, 913–925 (1957).
[CrossRef]

1906 (1)

A. Gullstrand, Svenska Vetensk. Handl. 41, 1 (1906).

Andersen, T. B.

Burke, W. L.

W. L. Burke, Applied Differential Geometry (Cambridge University, 1985).

Burkhard, D. G.

Chen, Y. B.

Feder, D. P.

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]

D. P. Feder, “Calculation of an optical merit function and its derivatives with respect to the system parameters,” J. Opt. Soc. Am. A 47, 913–925 (1957).
[CrossRef]

Gullstrand, A.

A. Gullstrand, Svenska Vetensk. Handl. 41, 1 (1906).

Hanrahan, P.

D. P. Mitchell and P. Hanrahan, “Illumination from curved reflectors,” in Proceedings of SIGGRAPH (1992), pp. 283–291.

Hoffnagle, J. A.

Kassim, A. M.

A. M. Kassim and D. L. Shealy, “Wave front equation, caustics, and wave aberration function of simple lenses and mirrors,” Appl. Opt. 21, 516–522 (1988).
[CrossRef]

Kneisly, J. A.

Laikin, M.

M. Laikin, Lens Design (Dekker, 1995), pp. 71–72.

Leveque, R. J.

R. J. Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady-State and Time-Dependent Problems (SIAM, 2007), pp. 3–4.

Lin, P. D.

Liu, C. S.

Mitchell, D. P.

D. P. Mitchell and P. Hanrahan, “Illumination from curved reflectors,” in Proceedings of SIGGRAPH (1992), pp. 283–291.

Pressley, A.

A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series (Springer, 2001), p. 123.

Shealy, D. L.

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), p. 372.

Spivak, M.

M. Spivak, A Comprehensive Introduction to DifferentialGeometry (Publish or Perish, 1999).

Stavroudis, O. N.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972).

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley-VCH, 2006).

Stone, B. D.

Tsai, C. Y.

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 91, 621–628 (2008).
[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]

Wu, W.

Appl. Opt. (9)

A. M. Kassim and D. L. Shealy, “Wave front equation, caustics, and wave aberration function of simple lenses and mirrors,” Appl. Opt. 21, 516–522 (1988).
[CrossRef]

D. L. Shealy and D. G. Burkhard, “Caustic surfaces and irradiance for reflection and refraction from an ellipsoid, elliptic parabolid and elliptic cone,” Appl. Opt. 12, 2955–2959 (1973).
[CrossRef]

D. L. Shealy, “Analytical illuminance and caustic surface calculations in geometrical optics,” Appl. Opt. 15, 2588–2596 (1976).
[CrossRef]

D. G. Burkhard and D. L. Shealy, “Simplified formula for the illuminance in an optical system,” Appl. Opt. 20, 897–909 (1981).
[CrossRef]

T. B. Andersen, “Optical aberration functions: computation of caustic surfaces and illuminance in symmetrical systems,” Appl. Opt. 20, 3723–3728 (1981).
[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]

C. S. Liu and P. D. Lin, “Computational method for deriving the geometrical point spread function of an optical system,” Appl. Opt. 49, 126–136 (2010).
[CrossRef]

Y. B. Chen and P. D. Lin, “Second-order derivatives of optical path length of ray with respect to variable vector of source ray,” Appl. Opt. 51, 5552–5562 (2012).
[CrossRef]

Appl. Phys. B (1)

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 91, 621–628 (2008).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (8)

Svenska Vetensk. Handl. (1)

A. Gullstrand, Svenska Vetensk. Handl. 41, 1 (1906).

Other (11)

R. J. Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady-State and Time-Dependent Problems (SIAM, 2007), pp. 3–4.

M. Laikin, Lens Design (Dekker, 1995), pp. 71–72.

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001), p. 372.

D. P. Mitchell and P. Hanrahan, “Illumination from curved reflectors,” in Proceedings of SIGGRAPH (1992), pp. 283–291.

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972).

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics (Wiley-VCH, 2006).

A. Pressley, Elementary Differential Geometry, Springer Undergraduate Mathematics Series (Springer, 2001), p. 123.

W. L. Burke, Applied Differential Geometry (Cambridge University, 1985).

M. Spivak, A Comprehensive Introduction to DifferentialGeometry (Publish or Perish, 1999).

T. Shifrin, “Differential geometry: a first course in curves and surfaces,” http://www.math.uga.edu/∼shifrin/ShifrinDiffGeo.pdf .

W. Rossmann, “Lectures on differential geometry,” http://mysite.science.uottawa.ca/rossmann/Differential%20Geometry%20book_files/Diffgeo.pdf .

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