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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 15, Iss. 5 — May. 1, 1998
  • pp: 1308–1319

Exact and approximate solutions for focusing of two-dimensional waves. III. Numerical comparisons between exact and Rayleigh–Sommerfeld theories

Hans A. Eide and Jakob J. Stamnes  »View Author Affiliations


JOSA A, Vol. 15, Issue 5, pp. 1308-1319 (1998)
http://dx.doi.org/10.1364/JOSAA.15.001308


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Abstract

Recently exact and approximate solutions were given for focusing of two-dimensional scalar and electromagnetic waves through a slit aperture in a perfectly reflecting screen. We present numerical comparisons, based on these solutions, between exact results and approximate Rayleigh–Sommerfeld results for a variety of different focusing geometries. These comparisons show that, at sufficiently low Fresnel numbers and large angular apertures, approximate solutions based on the first and the second Rayleigh–Sommerfeld diffraction integrals agree well with the corresponding exact solutions for hard and soft screens, respectively. Inasmuch as these results are contrary to what one would expect from considerations of the boundary conditions, we give an analytical explanation of them through comparisons between exact and approximate near-field solutions for the corresponding half-plane problems.

© 1998 Optical Society of America

OCIS Codes
(220.2560) Optical design and fabrication : Propagating methods
(350.7420) Other areas of optics : Waves

History
Original Manuscript: May 19, 1997
Revised Manuscript: October 3, 1997
Manuscript Accepted: December 12, 1997
Published: May 1, 1998

Citation
Hans A. Eide and Jakob J. Stamnes, "Exact and approximate solutions for focusing of two-dimensional waves. III. Numerical comparisons between exact and Rayleigh–Sommerfeld theories," J. Opt. Soc. Am. A 15, 1308-1319 (1998)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-15-5-1308


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References

  1. J. J. Stamnes, H. A. Eide, “Exact and approximate solutions for focusing of two-dimensional waves. I. Theory,” J. Opt. Soc. Am. A 15, 1285–1291 (1998). [CrossRef]
  2. H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. II. Numerical comparisons among exact, Debye, and Kirchhoff theories,” J. Opt. Soc. Am. A 15, 1292–1307 (1998). [CrossRef]
  3. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954). [CrossRef]
  4. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), p. 447.
  5. Ref. 4, Eqs. (8.19).
  6. Ref. 1, Eqs. (3.1a) and (3.1c).
  7. Ref. 4, lowest-order terms of Eqs. (8.8a) and (8.14a).
  8. Ref. 4, Sec. 4.4.

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