Abstract

We present a formulation of the scalar diffraction of spherical waves through plane apertures that takes advantage of the importance of the critical points and the semiperiodic zones that are defined in the plane of the aperture. Our approach overcomes the intrinsic limitations of stationary-phase methods (when the critical points coalesce) and allows a direct implementation in terms of numerical algorithms for computations of any predefined accuracy. The method is illustrated with examples corresponding to a rectangular aperture.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. H. Hopkins and M. J. Yzuel, Opt. Acta 17, 157 (1970).
    [CrossRef]
  2. L. A. D’Arcio, J. J. M. Braat, and H. J. Frankena, J. Opt. Soc. Am. A 11, 2664 (1994).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. VIII.
  4. M. Born and E. Wolf, eds., Principles of Optics (Pergamon, Oxford, 1975), Appendix III, Sec. 3.
  5. J. B. Keller, J. Opt. Soc. Am. 52, 116 (1962).
    [PubMed]
  6. J. J. Stamnes, J. Opt. Soc. Am. 73, 96 (1983).
  7. J. Ferré-Borrull, S. Bosch, and S. Vallmitjana, J. Mod. Opt. 45, 555 (1998).
    [CrossRef]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  9. J. Ferré-Borrull and S. Bosch, “On the integration of highly oscillatory functions: algorithms based on semiperiodic zones and critical points in scalar diffraction,” J. Comput. Phys. (to be published).

1998 (1)

J. Ferré-Borrull, S. Bosch, and S. Vallmitjana, J. Mod. Opt. 45, 555 (1998).
[CrossRef]

1994 (1)

1983 (1)

1970 (1)

H. H. Hopkins and M. J. Yzuel, Opt. Acta 17, 157 (1970).
[CrossRef]

1962 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. VIII.

Bosch, S.

J. Ferré-Borrull, S. Bosch, and S. Vallmitjana, J. Mod. Opt. 45, 555 (1998).
[CrossRef]

J. Ferré-Borrull and S. Bosch, “On the integration of highly oscillatory functions: algorithms based on semiperiodic zones and critical points in scalar diffraction,” J. Comput. Phys. (to be published).

Braat, J. J. M.

D’Arcio, L. A.

Ferré-Borrull, J.

J. Ferré-Borrull, S. Bosch, and S. Vallmitjana, J. Mod. Opt. 45, 555 (1998).
[CrossRef]

J. Ferré-Borrull and S. Bosch, “On the integration of highly oscillatory functions: algorithms based on semiperiodic zones and critical points in scalar diffraction,” J. Comput. Phys. (to be published).

Frankena, H. J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hopkins, H. H.

H. H. Hopkins and M. J. Yzuel, Opt. Acta 17, 157 (1970).
[CrossRef]

Keller, J. B.

Stamnes, J. J.

Vallmitjana, S.

J. Ferré-Borrull, S. Bosch, and S. Vallmitjana, J. Mod. Opt. 45, 555 (1998).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. VIII.

Yzuel, M. J.

H. H. Hopkins and M. J. Yzuel, Opt. Acta 17, 157 (1970).
[CrossRef]

J. Mod. Opt. (1)

J. Ferré-Borrull, S. Bosch, and S. Vallmitjana, J. Mod. Opt. 45, 555 (1998).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

H. H. Hopkins and M. J. Yzuel, Opt. Acta 17, 157 (1970).
[CrossRef]

Other (4)

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. VIII.

M. Born and E. Wolf, eds., Principles of Optics (Pergamon, Oxford, 1975), Appendix III, Sec. 3.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. Ferré-Borrull and S. Bosch, “On the integration of highly oscillatory functions: algorithms based on semiperiodic zones and critical points in scalar diffraction,” J. Comput. Phys. (to be published).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Geometry for the case of a diverging wave.

Fig. 2
Fig. 2

Geometry for the case of a converging wave.

Fig. 3
Fig. 3

Profiles for the modulus of a diffracted diverging wave. Thick solid curve, exact results; ×, method 2; , method 1. The abcissas are the distance (from the z axis) along the projection of a diagonal of the diffracting rectangle (see text for full details about the configuration).

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

UP=ΣUx,yexpikrPrP1rP-ikcosn,rPdxdy,
rS+rP=r0+nλ2, n=1,2,,
x-x0n2an2+y2bn2=1,
x0n=zS+zP2bn2bn2 sin2 γ+an2 cos2 γtan γ,
bn2=bn21-zS+zP221an2 cos2 γ+bn2 sin2 γ,
an2=bn2an2an2 cos2 γ+bn2 sin2 γ,
an=12r0+nλ2,
bn=12nλ22r0+nλ21/2,
-rS+rP=r0-nλ2, n=0,1,2,nmax.

Metrics