Abstract

The Abbe transform is used to derive rigorous algebraic formulas for the wave function describing Fraunhofer diffraction at apertures in the form of arbitrary polygons. Two remarks bring the Abbe transform and the Abbe theorem into the context of the history of diffraction.

© 1982 Optical Society of America

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  1. W. Sillitto, "Fraunhofer diffraction at straight-edged apertures," J. Opt. Soc. Am. 69, 765–770 (1979).
  2. R. M. Sillitto and W. Sillitto, "A simple Fourier approach to Fraunhofer diffraction by triangular apertures," Opt. Acta 22, 999–1010 (1975).
  3. I. E. Golberg, B. E. Kinber, and V. B. Tzeitlin, "Diffraction by the bodies with broken edges," Radiotech. Elektron. 20, 1582–1591 (1975).
  4. J. Komrska, "Fraunhofer diffraction at apertures in the form of regular polygons. I," Opt. Acta 19, 807–816 (1972).
  5. J. Komrska, "Fraunhofer diffraction at apertures in the form of regular polygons. II," Opt. Acta 20, 549–563 (1973).
  6. R. C. Smith and J. S. March, "Diffraction patterns of simple apertures," J. Opt. Soc. Am. 64, 798–803 (1974).
  7. W. Witz, "Fraunhofer diffraction pattern of an aperture," J. Opt. Soc. Am. 65, 1077–1078 (1975).
  8. E. Abbe, Gesammelte Abhandlungen (Verlag Gustav Fischer, Jena, Germany, 1904), Vols. 1–3.
  9. R. Straubel, Über die Berechnung der Fraunhoferschen Beugungserscheinungen durch Randintegrale mit besonderer Berück-Sichtigung der Theorie der Beugung in Heliometer (Frommansche Buchdruckerei, Jena, Germany, 1888).
  10. M. v. Laue, "Bemerkung über Fraunhofersche Beugung," Sitzungsber. Preuss. Akad. Wiss. Phys. Math. KI. 89–91 (1936).
  11. M. v. Laue, "Die äussere Form der Kristalle in ihrem Einfluss auf die Interferenzerscheinungen an Raumgittern," Ann. Phys. (Leipzig) 26, 55–68 (1936).
  12. R. W. James, The Optical Principles of the Diffraction of X-Rays (Bell, London, 1967), pp. 548–555.
  13. A. Rubinowicz, "Die Rolle der Beugungswelle in der Fraunhoferschen Beugungserscheinungen," Acta Phys. Pol. 13, 3–13 (1954).
  14. A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung, 2nd ed. (Springer-Verlag, Berlin, 1966), pp. 214–225.
  15. R. Straubel, "Zwei allgemeine Sätze über Fraunhofersche Beugungserscheinungen," Ann. Phys. Chemie, Neue Folge 56, 746–761 (1895); cf. pp. 757–758.
  16. J. Komrska, "Abbe transform and Abbe theorem in the theory of Fraunhofer diffraction," Cesk. Cas. Fys. (A) 27, 579–586 (1977).
  17. F. M. Schwerd, Die Beugungserscheinungen aus den Fundamehtalgesetzen der,Undulationstheorie analytisch entwickelt und in Bildern dargestellt (Schwan und Goetz'sche Hofbuchhandlung, Mannheim, Germany, 1835), Sec. 182, p. 117.
  18. Ref. 2, Fig. 4(a).
  19. J. Larmor, "The stellate appendages of telescopic and entoptic diffraction," Proc. Cambridge Philos. Soc. 21, 410–413 (1923).
  20. A. Sommerfeld, Optics (Academic, New York, 1954), pp. 233–237.

1979

1977

J. Komrska, "Abbe transform and Abbe theorem in the theory of Fraunhofer diffraction," Cesk. Cas. Fys. (A) 27, 579–586 (1977).

1975

R. M. Sillitto and W. Sillitto, "A simple Fourier approach to Fraunhofer diffraction by triangular apertures," Opt. Acta 22, 999–1010 (1975).

I. E. Golberg, B. E. Kinber, and V. B. Tzeitlin, "Diffraction by the bodies with broken edges," Radiotech. Elektron. 20, 1582–1591 (1975).

W. Witz, "Fraunhofer diffraction pattern of an aperture," J. Opt. Soc. Am. 65, 1077–1078 (1975).

1974

1973

J. Komrska, "Fraunhofer diffraction at apertures in the form of regular polygons. II," Opt. Acta 20, 549–563 (1973).

1972

J. Komrska, "Fraunhofer diffraction at apertures in the form of regular polygons. I," Opt. Acta 19, 807–816 (1972).

1954

A. Rubinowicz, "Die Rolle der Beugungswelle in der Fraunhoferschen Beugungserscheinungen," Acta Phys. Pol. 13, 3–13 (1954).

1936

M. v. Laue, "Die äussere Form der Kristalle in ihrem Einfluss auf die Interferenzerscheinungen an Raumgittern," Ann. Phys. (Leipzig) 26, 55–68 (1936).

1923

J. Larmor, "The stellate appendages of telescopic and entoptic diffraction," Proc. Cambridge Philos. Soc. 21, 410–413 (1923).

1895

R. Straubel, "Zwei allgemeine Sätze über Fraunhofersche Beugungserscheinungen," Ann. Phys. Chemie, Neue Folge 56, 746–761 (1895); cf. pp. 757–758.

Abbe, E.

E. Abbe, Gesammelte Abhandlungen (Verlag Gustav Fischer, Jena, Germany, 1904), Vols. 1–3.

Golberg, I. E.

I. E. Golberg, B. E. Kinber, and V. B. Tzeitlin, "Diffraction by the bodies with broken edges," Radiotech. Elektron. 20, 1582–1591 (1975).

James, R. W.

R. W. James, The Optical Principles of the Diffraction of X-Rays (Bell, London, 1967), pp. 548–555.

Kinber, B. E.

I. E. Golberg, B. E. Kinber, and V. B. Tzeitlin, "Diffraction by the bodies with broken edges," Radiotech. Elektron. 20, 1582–1591 (1975).

Komrska, J.

J. Komrska, "Abbe transform and Abbe theorem in the theory of Fraunhofer diffraction," Cesk. Cas. Fys. (A) 27, 579–586 (1977).

J. Komrska, "Fraunhofer diffraction at apertures in the form of regular polygons. II," Opt. Acta 20, 549–563 (1973).

J. Komrska, "Fraunhofer diffraction at apertures in the form of regular polygons. I," Opt. Acta 19, 807–816 (1972).

Larmor, J.

J. Larmor, "The stellate appendages of telescopic and entoptic diffraction," Proc. Cambridge Philos. Soc. 21, 410–413 (1923).

Laue, M. v.

M. v. Laue, "Die äussere Form der Kristalle in ihrem Einfluss auf die Interferenzerscheinungen an Raumgittern," Ann. Phys. (Leipzig) 26, 55–68 (1936).

M. v. Laue, "Bemerkung über Fraunhofersche Beugung," Sitzungsber. Preuss. Akad. Wiss. Phys. Math. KI. 89–91 (1936).

March, J. S.

Rubinowicz, A.

A. Rubinowicz, "Die Rolle der Beugungswelle in der Fraunhoferschen Beugungserscheinungen," Acta Phys. Pol. 13, 3–13 (1954).

A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung, 2nd ed. (Springer-Verlag, Berlin, 1966), pp. 214–225.

Schwerd, F. M.

F. M. Schwerd, Die Beugungserscheinungen aus den Fundamehtalgesetzen der,Undulationstheorie analytisch entwickelt und in Bildern dargestellt (Schwan und Goetz'sche Hofbuchhandlung, Mannheim, Germany, 1835), Sec. 182, p. 117.

Sillitto, R. M.

R. M. Sillitto and W. Sillitto, "A simple Fourier approach to Fraunhofer diffraction by triangular apertures," Opt. Acta 22, 999–1010 (1975).

Sillitto, W.

W. Sillitto, "Fraunhofer diffraction at straight-edged apertures," J. Opt. Soc. Am. 69, 765–770 (1979).

R. M. Sillitto and W. Sillitto, "A simple Fourier approach to Fraunhofer diffraction by triangular apertures," Opt. Acta 22, 999–1010 (1975).

Smith, R. C.

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1954), pp. 233–237.

Straubel, R.

R. Straubel, "Zwei allgemeine Sätze über Fraunhofersche Beugungserscheinungen," Ann. Phys. Chemie, Neue Folge 56, 746–761 (1895); cf. pp. 757–758.

R. Straubel, Über die Berechnung der Fraunhoferschen Beugungserscheinungen durch Randintegrale mit besonderer Berück-Sichtigung der Theorie der Beugung in Heliometer (Frommansche Buchdruckerei, Jena, Germany, 1888).

Tzeitlin, V. B.

I. E. Golberg, B. E. Kinber, and V. B. Tzeitlin, "Diffraction by the bodies with broken edges," Radiotech. Elektron. 20, 1582–1591 (1975).

Witz, W.

Acta Phys. Pol.

A. Rubinowicz, "Die Rolle der Beugungswelle in der Fraunhoferschen Beugungserscheinungen," Acta Phys. Pol. 13, 3–13 (1954).

Ann. Phys.

M. v. Laue, "Die äussere Form der Kristalle in ihrem Einfluss auf die Interferenzerscheinungen an Raumgittern," Ann. Phys. (Leipzig) 26, 55–68 (1936).

Ann. Phys. Chemie, Neue Folge

R. Straubel, "Zwei allgemeine Sätze über Fraunhofersche Beugungserscheinungen," Ann. Phys. Chemie, Neue Folge 56, 746–761 (1895); cf. pp. 757–758.

Cesk. Cas. Fys.

J. Komrska, "Abbe transform and Abbe theorem in the theory of Fraunhofer diffraction," Cesk. Cas. Fys. (A) 27, 579–586 (1977).

J. Opt. Soc. Am.

Opt. Acta

R. M. Sillitto and W. Sillitto, "A simple Fourier approach to Fraunhofer diffraction by triangular apertures," Opt. Acta 22, 999–1010 (1975).

J. Komrska, "Fraunhofer diffraction at apertures in the form of regular polygons. I," Opt. Acta 19, 807–816 (1972).

J. Komrska, "Fraunhofer diffraction at apertures in the form of regular polygons. II," Opt. Acta 20, 549–563 (1973).

Proc. Cambridge Philos. Soc.

J. Larmor, "The stellate appendages of telescopic and entoptic diffraction," Proc. Cambridge Philos. Soc. 21, 410–413 (1923).

Radiotech. Elektron.

I. E. Golberg, B. E. Kinber, and V. B. Tzeitlin, "Diffraction by the bodies with broken edges," Radiotech. Elektron. 20, 1582–1591 (1975).

Other

E. Abbe, Gesammelte Abhandlungen (Verlag Gustav Fischer, Jena, Germany, 1904), Vols. 1–3.

R. Straubel, Über die Berechnung der Fraunhoferschen Beugungserscheinungen durch Randintegrale mit besonderer Berück-Sichtigung der Theorie der Beugung in Heliometer (Frommansche Buchdruckerei, Jena, Germany, 1888).

M. v. Laue, "Bemerkung über Fraunhofersche Beugung," Sitzungsber. Preuss. Akad. Wiss. Phys. Math. KI. 89–91 (1936).

F. M. Schwerd, Die Beugungserscheinungen aus den Fundamehtalgesetzen der,Undulationstheorie analytisch entwickelt und in Bildern dargestellt (Schwan und Goetz'sche Hofbuchhandlung, Mannheim, Germany, 1835), Sec. 182, p. 117.

Ref. 2, Fig. 4(a).

R. W. James, The Optical Principles of the Diffraction of X-Rays (Bell, London, 1967), pp. 548–555.

A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung, 2nd ed. (Springer-Verlag, Berlin, 1966), pp. 214–225.

A. Sommerfeld, Optics (Academic, New York, 1954), pp. 233–237.

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