Abstract

We study the anomalous asymptotic behavior of the Debye integral far from focus that occurs in the vicinities of the axis of the focusing system and the boundary of the geometrical-optics shadow. The first terms in the asymptotic power series of the far field valid on the axis, on the shadow boundary, in the shadow, and in the geometrical illuminated region off axis are obtained to show how they change discontinuously as the field point passes from one region to another. We obtain the second-order term in the asymptotic power series valid in the last-named region to show how it grows without limit as the field point approaches the axis or the shadow boundary. We then derive an approximation valid far from focus that remains continuous as the field point approaches the axis and the shadow boundary. This approximation agrees with the asymptotic power-series results where they are valid. The continuous approximation is applied to determine the sizes of the regions where the field does not approximate the geometrical-optics field.

© 1982 Optical Society of America

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  1. M. Born and W. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, England, 1980), Sec. 8.8.
  2. P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpuktes oder iener Brennlinie," Ann. Phys. 30, 755–776 (1909).
  3. E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39, 204–210 (1981), and references therein.
  4. A. Sommerfeld, Partial Differential Equations in Physics, Vol. VI of Lectures on Theoretical Physics (Academic, New York, 1964), Sec. 28.
  5. A. Sommerfeld, Optics, Vol. IV of Lectures on Theoretical Physics (Academic, New York, 1964), Sec. 45.
  6. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Sec 46.
  7. E. Collett and E. Wolf, "Symmetry properties of focused fields," Opt. Lett., 5, 264–266 (1980).
  8. G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976).
  9. A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), Sec. 2.8–2.9.
  10. H. Bremermann, Distributions, Complex Variables and Fourier Transforms (Addison-Wesley, Reading, Mass., 1965), Sec. 3.3–3.4.
  11. K. Miyamoto and W. Wolf, "Generalization of the Maggi—Rubinowicz theory of the boundary diffraction wave—Part I," J. Opt. Soc. Am. 52, 615–625 (1962). See also Ref. 8.
  12. N. Chako, "Asymptotic expansions of double and multiple integrals occurring in diffraction theory," J. Inst. Math. Appl. 1, 372–422 (1965), Sec. 3.
  13. This is done for the critical points on the corners in Ref. 8. The same arguments can be applied to the other critical points.
  14. D. S. Jones and M. Kline, "Asymptotic expansion of multiple integrals and the method of stationary phase," J. Math. Phys. 37, 1–28 (1958).
  15. G. Tyras, Radiation and Propagation of Electromagnetic Waves (Academic, New York, 1969). Our analysis here parallels the analysis of a similar integral presented in Sec. 5.1.1 of this reference.
  16. Ref. 15, Eq. (5.35b).

1981 (1)

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39, 204–210 (1981), and references therein.

1980 (1)

1976 (1)

G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976).

1965 (1)

N. Chako, "Asymptotic expansions of double and multiple integrals occurring in diffraction theory," J. Inst. Math. Appl. 1, 372–422 (1965), Sec. 3.

1962 (1)

1958 (1)

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

1909 (1)

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpuktes oder iener Brennlinie," Ann. Phys. 30, 755–776 (1909).

Born, M.

M. Born and W. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, England, 1980), Sec. 8.8.

Bremermann, H.

H. Bremermann, Distributions, Complex Variables and Fourier Transforms (Addison-Wesley, Reading, Mass., 1965), Sec. 3.3–3.4.

Chako, N.

N. Chako, "Asymptotic expansions of double and multiple integrals occurring in diffraction theory," J. Inst. Math. Appl. 1, 372–422 (1965), Sec. 3.

Collett, E.

Debye, P.

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpuktes oder iener Brennlinie," Ann. Phys. 30, 755–776 (1909).

Erdélyi, A.

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), Sec. 2.8–2.9.

Jones, D. S.

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

Kline, M.

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

Lalor, E.

G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976).

Li, Y.

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39, 204–210 (1981), and references therein.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Sec 46.

Miyamoto, K.

Sherman, G. C.

G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976).

Sommerfeld, A.

A. Sommerfeld, Optics, Vol. IV of Lectures on Theoretical Physics (Academic, New York, 1964), Sec. 45.

A. Sommerfeld, Partial Differential Equations in Physics, Vol. VI of Lectures on Theoretical Physics (Academic, New York, 1964), Sec. 28.

Stamnes, J. J.

G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976).

Tyras, G.

G. Tyras, Radiation and Propagation of Electromagnetic Waves (Academic, New York, 1969). Our analysis here parallels the analysis of a similar integral presented in Sec. 5.1.1 of this reference.

Wolf, E.

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39, 204–210 (1981), and references therein.

E. Collett and E. Wolf, "Symmetry properties of focused fields," Opt. Lett., 5, 264–266 (1980).

Wolf, W.

Ann. Phys. (1)

P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpuktes oder iener Brennlinie," Ann. Phys. 30, 755–776 (1909).

J. Inst. Math. Appl. (1)

N. Chako, "Asymptotic expansions of double and multiple integrals occurring in diffraction theory," J. Inst. Math. Appl. 1, 372–422 (1965), Sec. 3.

J. Math. Phys. (2)

G. C. Sherman, J. J. Stamnes, and E. Lalor, "Asymptotic approximations to angular-spectrum representations," J. Math. Phys. 17, 760–776 (1976).

D. S. 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)

Opt. Commun. (1)

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39, 204–210 (1981), and references therein.

Opt. Lett. (1)

Other (9)

M. Born and W. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, England, 1980), Sec. 8.8.

G. Tyras, Radiation and Propagation of Electromagnetic Waves (Academic, New York, 1969). Our analysis here parallels the analysis of a similar integral presented in Sec. 5.1.1 of this reference.

Ref. 15, Eq. (5.35b).

A. Sommerfeld, Partial Differential Equations in Physics, Vol. VI of Lectures on Theoretical Physics (Academic, New York, 1964), Sec. 28.

A. Sommerfeld, Optics, Vol. IV of Lectures on Theoretical Physics (Academic, New York, 1964), Sec. 45.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Sec 46.

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), Sec. 2.8–2.9.

H. Bremermann, Distributions, Complex Variables and Fourier Transforms (Addison-Wesley, Reading, Mass., 1965), Sec. 3.3–3.4.

This is done for the critical points on the corners in Ref. 8. The same arguments can be applied to the other critical points.

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