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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 12, Iss. 9 — Sep. 1, 1995
  • pp: 1947–1953

Structure of focused fields in systems with large Fresnel numbers

Weijian Wang, Ari T. Friberg, and Emil Wolf  »View Author Affiliations

JOSA A, Vol. 12, Issue 9, pp. 1947-1953 (1995)

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A new and simple formula valid within the framework of the Debye theory is derived for determining the structure of focused fields in diffraction-limited systems. It is first applied to study the field behavior in the focal region, and the results are compared with those of the classic theory of Lommel. The field distribution in the intermediate zone between the focal region and the far zone is then studied, and the changes of the field with increasing distance from the geometrical focus are examined. An estimate is obtained for the distance from focus at which the field behaves as a cutoff portion of a uniform spherical wave.

© 1995 Optical Society of America

Original Manuscript: January 31, 1995
Manuscript Accepted: April 12, 1995
Published: September 1, 1995

Weijian Wang, Ari T. Friberg, and Emil Wolf, "Structure of focused fields in systems with large Fresnel numbers," J. Opt. Soc. Am. A 12, 1947-1953 (1995)

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  1. For comprehensive reviews, see E. Wolf, “The diffraction theory of aberrations,” Rep. Progr. Phys. (The Physical Society, London) 14, 95–120 (1951), Secs. 3.1 and 3.2, and J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Sec. 1.5. [CrossRef]
  2. G. B. Airy, “On the diffraction of an object-glass with circular aperture,” Trans. Cambridge Philos. Soc. 5, 283–291 (1835).
  3. E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisnunden Schirmchens,” Abh. Bayer Akad. Math. Naturwiss. 15, 229–328 (1885). An account of Lommel’s theory in English is given in Ref. 17 below, Sec. 8.8.
  4. M. Berek, “Über Kohärenz und Konsonanz des Lichtes,” Z. Phys. 40, 420–450 (1927). [CrossRef]
  5. P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Physik 30, 755–776 (1909). An account of Debye’s theory in English is given in A. Sommerfeld, Optics (Academic, New York, 1954), Sec. 45. [CrossRef]
  6. J. Picht, Optische Abbildung (Vieweg, Braunschweig, 1931). [CrossRef]
  7. F. Zernike, B. R. A. Nijboer, “Théorie de la diffraction des aberrations,” in La Théorie des Images Optiques, P. Fleury, A. Maréchal, C. Anglade, eds. (Editions de la Revue d’Optique, Paris, 1949), p. 227.
  8. E. H. Linfoot, E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B 66, 145–149 (1953). [CrossRef]
  9. E. H. Linfoot, E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. B 69, 823–832 (1956). [CrossRef]
  10. A. G. Van Nie, “Rigorous calculation of the electromagnetic field for wave beams,” Philips Res. Rep. 19, 378–394 (1964).
  11. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Technol. J. 44, 455–494 (1965).
  12. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981). [CrossRef]
  13. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981). [CrossRef]
  14. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984). [CrossRef]
  15. W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel numbers,” Opt. Commun. (to be published).
  16. For more general symmetry properties of focused fields, see E. Collett, E. Wolf, “Symmetry properties of focused fields,” Opt. Lett. 5, 264–266 (1980). [CrossRef] [PubMed]
  17. See, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980).
  18. Strictly speaking, this behavior does not apply to field points on, or sufficiently close to, the zaxis or to points located on the geometrical shadow boundary or in the immediate vicinity of it. However, these two regions of anomalous behavior of the Debye integral become negligibly small when the distance from the aperture is sufficiently large [see Sherman, Chew, “Aperture and far-field distributions expressed by the Debye integral representation of focused fields,” J. Opt. Soc. Am. 72, 1076–1083 (1982)]. [CrossRef]

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