## Aperture and far-field distributions expressed by the Debye integral representation of focused fields

JOSA, Vol. 72, Issue 8, pp. 1076-1083 (1982)

http://dx.doi.org/10.1364/JOSA.72.001076

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### 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

**Citation**

George C. Sherman and W. C. Chew, "Aperture and far-field distributions expressed by the Debye integral representation of focused fields," J. Opt. Soc. Am. **72**, 1076-1083 (1982)

http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-72-8-1076

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### References

- M. Born and W. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, England, 1980), Sec. 8.8.
- P. Debye, "Das Verhalten von Lichtwellen in der Nähe eines Brennpuktes oder iener Brennlinie," Ann. Phys. 30, 755–776 (1909).
- 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.
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- A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), Sec. 2.8–2.9.
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- N. Chako, "Asymptotic expansions of double and multiple integrals occurring in diffraction theory," J. Inst. Math. Appl. 1, 372–422 (1965), Sec. 3.
- This is done for the critical points on the corners in Ref. 8. The same arguments can be applied to the other critical points.
- D. S. Jones and M. Kline, "Asymptotic expansion of multiple integrals and the method of stationary phase," J. Math. Phys. 37, 1–28 (1958).
- 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).

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