Abstract

The second-order derivative of a scalar function with respect to a variable vector is known as the Hessian matrix. We present a computational scheme based on the principles of differential geometry for determining the Hessian matrix of a skew ray as it travels through a prism system. A comparison of the proposed method and the conventional finite difference (FD) method is made at last. It is shown that the proposed method has a greater inherent accuracy than FD methods based on ray-tracing data. The proposed method not only provides a convenient means of investigating the wavefront shape within complex prism systems, but it also provides a potential basis for determining the higher order derivatives of a ray by further taking higher order differentiations.

© 2011 Optical Society of America

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  1. R. J. Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady-State and Time-Dependent Problems (SIAM, 2007), pp 3–4.
  2. B. D. Stone, “Determination of initial ray configurations for asymmetric systems,” J. Opt. Soc. Am. A 14, 3415–3429 (1997).
    [CrossRef]
  3. R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE , 3737, 149–160 (1999).
    [CrossRef]
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    [CrossRef]
  5. T. B. Andersen, “Optical aberration functions: derivatives with respect to surface parameters for symmetrical systems,” Appl. Opt. 24, 1122–1129 (1985).
    [CrossRef]
  6. 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]
  7. 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]
  8. O. Stavroudis, “A simpler derivation of the formulas for generalized ray tracing,” J. Opt. Soc. Am. 66, 1330–1333 (1976).
    [CrossRef]
  9. J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–18(1988).
  10. W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26(1988).
  11. W. Mandler, “Uber die Berechnung Einfacher GauB-Objektive,” Optik 55, 219–240 (1980).
  12. B. D. Stone and G. W. Forbes, “Differential ray tracing in inhomogeneous media,” J. Opt. Soc. Am. A 14, 2824–2836(1997).
    [CrossRef]
  13. 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]
  14. 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]
  15. R. P. Paul, Robot Manipulators—Mathematics, Programming and Control (MIT Press, 1982).
  16. G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed. (Oxford Univ. Press, 1985).

2008 (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]

2007 (1)

1999 (1)

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE , 3737, 149–160 (1999).
[CrossRef]

1997 (2)

1988 (2)

J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–18(1988).

W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26(1988).

1985 (1)

1982 (1)

1980 (1)

W. Mandler, “Uber die Berechnung Einfacher GauB-Objektive,” Optik 55, 219–240 (1980).

1976 (1)

1968 (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]

Andersen, T. 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]

Forbes, G. W.

Kross, J.

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE , 3737, 149–160 (1999).
[CrossRef]

J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–18(1988).

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.

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]

Mandler, W.

W. Mandler, “Uber die Berechnung Einfacher GauB-Objektive,” Optik 55, 219–240 (1980).

Oertmann, W.

W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26(1988).

Paul, R. P.

R. P. Paul, Robot Manipulators—Mathematics, Programming and Control (MIT Press, 1982).

Shi, R.

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE , 3737, 149–160 (1999).
[CrossRef]

Smith, G. D.

G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed. (Oxford Univ. Press, 1985).

Stavroudis, O.

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]

Appl. Opt. (2)

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. (2)

J. Opt. Soc. Am. A (4)

Optik (1)

W. Mandler, “Uber die Berechnung Einfacher GauB-Objektive,” Optik 55, 219–240 (1980).

Proc. SPIE (3)

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE , 3737, 149–160 (1999).
[CrossRef]

J. Kross, “Differential ray tracing formulae for optical calculations: principles and applications,” Proc. SPIE 1013, 10–18(1988).

W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26(1988).

Other (3)

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

R. P. Paul, Robot Manipulators—Mathematics, Programming and Control (MIT Press, 1982).

G. D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, 3rd ed. (Oxford Univ. Press, 1985).

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