Abstract

Previous asymptotic expansions of the diffraction integral are valid for only certain regions in the observation plane. This paper gives a method for calculating semiuniform expansions that are valid in the intermediate regions. They are especially useful and simple for the near-axis field of a circularly symmetric aperture. The case for circularly symmetric ρ<sup>2v.</sup> aberrations is presented.

© 1980 Optical Society of America

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  1. R. Ingarden, "A generalization of the Young–Rubinowicz principle in the theory of diffraction," Acta Phys. Pol. 14, 77–91 (1955).
  2. J. Keller, "Diffraction by an aperture," J. Appl. Phys. 28, 426–444 (1957).
  3. J. Keller, "Geometrical theory of diffraction," J. Opt. Soc. Am. 52, 116–130 (1962).
  4. N. Chako, "Développement asymptotique d’intégrales doubles que l’on rencontre dans la théorie de la diffraction," C. R. Acad. Sci. Paris 247, 436–438 (1953); "Application de la méthode de la phase stationnaire dans la théorie de la diffraction des images optiques," ibid. 247, 580–582 (1958).
  5. N. Van Kampen, "An asymptotic treatment of diffraction problems," Physica (Utrecht) 14, 575–589 (1949).
  6. D. Jones and M. Kline, "Asymptotic expansion of multiple integrals and the method of stationary phase," J. Math. Phys. 37, 1–28 (1958).
  7. M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley-Interscience, New York, 1965), Chap. XII.
  8. J. Focke, "Asymptotische Entwicklungen Mittels der Methode der Stationären Phase," Ber. Sächs. Geo. (Akad.) Wiss. 101 (1954), Heft. 3.
  9. N. Bleistein and R. Handelsman, Asymptotic Expansion of Integrals (Holt, Rinehart and Winston, New York, 1975).
  10. W. Thomson, "On the waves produced by a single impulse in water of any depth, or in a dispersive medium," Proc. R. Soc. London A 42, 80–83 (1887).
  11. G. Watson, "The limits of applicability of the principle of stationary phase," Proc. Cambridge Philos. Soc. 19, 49–55 (1918).
  12. J. van der Corput, "On the method of critical points," K. Ned. Akad. Wet. Indag. Math. 10, 201–209 (1948).
  13. A. Erdélyi, "Asymptotic representations of Fourier integrals and the method of stationary phase," J. Soc. Ind. Appl. Math. 3, 17–27 (1955).
  14. I. Gradshteyn and I. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), Formula 6.631, Number 6.

1962 (1)

1958 (1)

D. Jones and M. Kline, "Asymptotic expansion of multiple integrals and the method of stationary phase," J. Math. Phys. 37, 1–28 (1958).

1957 (1)

J. Keller, "Diffraction by an aperture," J. Appl. Phys. 28, 426–444 (1957).

1955 (2)

R. Ingarden, "A generalization of the Young–Rubinowicz principle in the theory of diffraction," Acta Phys. Pol. 14, 77–91 (1955).

A. Erdélyi, "Asymptotic representations of Fourier integrals and the method of stationary phase," J. Soc. Ind. Appl. Math. 3, 17–27 (1955).

1954 (1)

J. Focke, "Asymptotische Entwicklungen Mittels der Methode der Stationären Phase," Ber. Sächs. Geo. (Akad.) Wiss. 101 (1954), Heft. 3.

1953 (1)

N. Chako, "Développement asymptotique d’intégrales doubles que l’on rencontre dans la théorie de la diffraction," C. R. Acad. Sci. Paris 247, 436–438 (1953); "Application de la méthode de la phase stationnaire dans la théorie de la diffraction des images optiques," ibid. 247, 580–582 (1958).

1949 (1)

N. Van Kampen, "An asymptotic treatment of diffraction problems," Physica (Utrecht) 14, 575–589 (1949).

1948 (1)

J. van der Corput, "On the method of critical points," K. Ned. Akad. Wet. Indag. Math. 10, 201–209 (1948).

1918 (1)

G. Watson, "The limits of applicability of the principle of stationary phase," Proc. Cambridge Philos. Soc. 19, 49–55 (1918).

1887 (1)

W. Thomson, "On the waves produced by a single impulse in water of any depth, or in a dispersive medium," Proc. R. Soc. London A 42, 80–83 (1887).

Bleistein, N.

N. Bleistein and R. Handelsman, Asymptotic Expansion of Integrals (Holt, Rinehart and Winston, New York, 1975).

Chako, N.

N. Chako, "Développement asymptotique d’intégrales doubles que l’on rencontre dans la théorie de la diffraction," C. R. Acad. Sci. Paris 247, 436–438 (1953); "Application de la méthode de la phase stationnaire dans la théorie de la diffraction des images optiques," ibid. 247, 580–582 (1958).

Erdélyi, A.

A. Erdélyi, "Asymptotic representations of Fourier integrals and the method of stationary phase," J. Soc. Ind. Appl. Math. 3, 17–27 (1955).

Focke, J.

J. Focke, "Asymptotische Entwicklungen Mittels der Methode der Stationären Phase," Ber. Sächs. Geo. (Akad.) Wiss. 101 (1954), Heft. 3.

Gradshteyn, I.

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), Formula 6.631, Number 6.

Handelsman, R.

N. Bleistein and R. Handelsman, Asymptotic Expansion of Integrals (Holt, Rinehart and Winston, New York, 1975).

Ingarden, R.

R. Ingarden, "A generalization of the Young–Rubinowicz principle in the theory of diffraction," Acta Phys. Pol. 14, 77–91 (1955).

Jones, D.

D. Jones and M. Kline, "Asymptotic expansion of multiple integrals and the method of stationary phase," J. Math. Phys. 37, 1–28 (1958).

Kay, I.

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley-Interscience, New York, 1965), Chap. XII.

Keller, J.

J. Keller, "Geometrical theory of diffraction," J. Opt. Soc. Am. 52, 116–130 (1962).

J. Keller, "Diffraction by an aperture," J. Appl. Phys. 28, 426–444 (1957).

Kline, M.

D. Jones and M. Kline, "Asymptotic expansion of multiple integrals and the method of stationary phase," J. Math. Phys. 37, 1–28 (1958).

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley-Interscience, New York, 1965), Chap. XII.

Ryzhik, I.

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), Formula 6.631, Number 6.

Thomson, W.

W. Thomson, "On the waves produced by a single impulse in water of any depth, or in a dispersive medium," Proc. R. Soc. London A 42, 80–83 (1887).

van der Corput, J.

J. van der Corput, "On the method of critical points," K. Ned. Akad. Wet. Indag. Math. 10, 201–209 (1948).

VanKampen, N.

N. Van Kampen, "An asymptotic treatment of diffraction problems," Physica (Utrecht) 14, 575–589 (1949).

Watson, G.

G. Watson, "The limits of applicability of the principle of stationary phase," Proc. Cambridge Philos. Soc. 19, 49–55 (1918).

Acta Phys. Pol. (1)

R. Ingarden, "A generalization of the Young–Rubinowicz principle in the theory of diffraction," Acta Phys. Pol. 14, 77–91 (1955).

Ber. Sächs. Geo. (1)

J. Focke, "Asymptotische Entwicklungen Mittels der Methode der Stationären Phase," Ber. Sächs. Geo. (Akad.) Wiss. 101 (1954), Heft. 3.

C. R. Acad. Sci. Paris (1)

N. Chako, "Développement asymptotique d’intégrales doubles que l’on rencontre dans la théorie de la diffraction," C. R. Acad. Sci. Paris 247, 436–438 (1953); "Application de la méthode de la phase stationnaire dans la théorie de la diffraction des images optiques," ibid. 247, 580–582 (1958).

J. Appl. Phys. (1)

J. Keller, "Diffraction by an aperture," J. Appl. Phys. 28, 426–444 (1957).

J. Math. Phys. (1)

D. Jones and M. Kline, "Asymptotic expansion of multiple integrals and the method of stationary phase," J. Math. Phys. 37, 1–28 (1958).

J. Opt. Soc. Am. (1)

J. Soc. Ind. Appl. Math. (1)

A. Erdélyi, "Asymptotic representations of Fourier integrals and the method of stationary phase," J. Soc. Ind. Appl. Math. 3, 17–27 (1955).

K. Ned. Akad. Wet. Indag. Math. (1)

J. van der Corput, "On the method of critical points," K. Ned. Akad. Wet. Indag. Math. 10, 201–209 (1948).

Physica (1)

N. Van Kampen, "An asymptotic treatment of diffraction problems," Physica (Utrecht) 14, 575–589 (1949).

Proc. Cambridge Philos. Soc. (1)

G. Watson, "The limits of applicability of the principle of stationary phase," Proc. Cambridge Philos. Soc. 19, 49–55 (1918).

Proc. R. Soc. London A (1)

W. Thomson, "On the waves produced by a single impulse in water of any depth, or in a dispersive medium," Proc. R. Soc. London A 42, 80–83 (1887).

Other (3)

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), Formula 6.631, Number 6.

M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley-Interscience, New York, 1965), Chap. XII.

N. Bleistein and R. Handelsman, Asymptotic Expansion of Integrals (Holt, Rinehart and Winston, New York, 1975).

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