Abstract

The mathematical process of combining the known characteristic functions of a number of optical systems to obtain the characteristic function of the system created by adjoining these component systems is shown to permit a significant simplification: If the intermediate variables are solved as power series to a given order in the external variables, then it is possible to determine the power series for the resulting characteristic function to at least twice this order. This doubling of order is shown to be a direct result of the extremal property of the rays. By way of illustration, the point characteristic of a refracting plane is determined explicitly as a power series accurate to within terms of degree 16.

© 1982 Optical Society of America

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  1. A. W. Conway and J. L. Synge, eds., The Mathematical Papers of Sir William Rowan Hamilton (Cambridge U. Press, Cambridge, 1931), Vol. 1, pp. 218, 492–494, 512–517.
  2. H. Bruns, "Das Eikonal," Abh. K. Sächs. Ges. Wiss. Leipzig Math. Phys. K1. 21, 325–436 (1895), pp. 366–368 and Sec. XV.
  3. J. L. Synge, Geometrical Optics, An Introduction to Hamilton's Method (Cambridge U. Press, Cambridge, 1931), pp. 61–68.
  4. M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), Chap. 31.
  5. R. K. Luneburg, Mathematical Theory of Optics, (U. California Press, Berkeley, Calif., 1964), Sec. 26.
  6. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970), Sec. 124.
  7. M. Andrews, "The concatenation of characteristic functions in Hamiltonian optics," J. Opt. Soc. Am. (to be published).
  8. Ref. 1, p. 408, p. 507.
  9. Ref. 3, p. 67.
  10. Ref. 6, p. 288.
  11. Ref. 7, Sec. 4, Eq. (4.2).
  12. If a characteristic function is specified as a power series to order N in its arguments, then the equations for the initial and final behavior of the rays are known to order N-1 (since the configuration of the rays is determined by differentiating this function). Consequently, the characteristic function is said to be determined to order N-1. For example, the expansion of V(y, y″) given in Eq. (3.11) is correct to order 14 or, using the above convention, determined to "order" 13. To avoid confusion, this convention will always be distinguished with the use of quotation marks.
  13. H. A. Buchdahl, "Hamiltonian optics: the point characteristic of a refracting plane," J. Opt. Soc. Am. 60, 997–1000 (1970).
  14. Ref. 6, pp. 15–17.
  15. H. A. Buchdahl, "Hamiltonian optics. V. On the point characteristic of a spherical refracting surface," Optik 42, 135–146 (1975).

1975

H. A. Buchdahl, "Hamiltonian optics. V. On the point characteristic of a spherical refracting surface," Optik 42, 135–146 (1975).

1970

1895

H. Bruns, "Das Eikonal," Abh. K. Sächs. Ges. Wiss. Leipzig Math. Phys. K1. 21, 325–436 (1895), pp. 366–368 and Sec. XV.

Andrews, M.

M. Andrews, "The concatenation of characteristic functions in Hamiltonian optics," J. Opt. Soc. Am. (to be published).

Bruns, H.

H. Bruns, "Das Eikonal," Abh. K. Sächs. Ges. Wiss. Leipzig Math. Phys. K1. 21, 325–436 (1895), pp. 366–368 and Sec. XV.

Buchdahl, H. A.

H. A. Buchdahl, "Hamiltonian optics. V. On the point characteristic of a spherical refracting surface," Optik 42, 135–146 (1975).

H. A. Buchdahl, "Hamiltonian optics: the point characteristic of a refracting plane," J. Opt. Soc. Am. 60, 997–1000 (1970).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970), Sec. 124.

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), Chap. 31.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics, (U. California Press, Berkeley, Calif., 1964), Sec. 26.

Synge, J. L.

J. L. Synge, Geometrical Optics, An Introduction to Hamilton's Method (Cambridge U. Press, Cambridge, 1931), pp. 61–68.

Abh. K. Sächs. Ges. Wiss. Leipzig Math. Phys. K1.

H. Bruns, "Das Eikonal," Abh. K. Sächs. Ges. Wiss. Leipzig Math. Phys. K1. 21, 325–436 (1895), pp. 366–368 and Sec. XV.

J. Opt. Soc. Am.

Optik

H. A. Buchdahl, "Hamiltonian optics. V. On the point characteristic of a spherical refracting surface," Optik 42, 135–146 (1975).

Other

A. W. Conway and J. L. Synge, eds., The Mathematical Papers of Sir William Rowan Hamilton (Cambridge U. Press, Cambridge, 1931), Vol. 1, pp. 218, 492–494, 512–517.

J. L. Synge, Geometrical Optics, An Introduction to Hamilton's Method (Cambridge U. Press, Cambridge, 1931), pp. 61–68.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), Chap. 31.

R. K. Luneburg, Mathematical Theory of Optics, (U. California Press, Berkeley, Calif., 1964), Sec. 26.

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970), Sec. 124.

M. Andrews, "The concatenation of characteristic functions in Hamiltonian optics," J. Opt. Soc. Am. (to be published).

Ref. 1, p. 408, p. 507.

Ref. 3, p. 67.

Ref. 6, p. 288.

Ref. 7, Sec. 4, Eq. (4.2).

If a characteristic function is specified as a power series to order N in its arguments, then the equations for the initial and final behavior of the rays are known to order N-1 (since the configuration of the rays is determined by differentiating this function). Consequently, the characteristic function is said to be determined to order N-1. For example, the expansion of V(y, y″) given in Eq. (3.11) is correct to order 14 or, using the above convention, determined to "order" 13. To avoid confusion, this convention will always be distinguished with the use of quotation marks.

Ref. 6, pp. 15–17.

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