Abstract

The behavior along the axis of the intensity arising from the diffraction of a uniform, converging spherical wave at a circular aperture is studied on the basis of the theory of the boundary-diffraction wave. The results are used to determine the location of the principal intensity maximum and to elucidate the dependence of the focal shift both on the Fresnel number and on the ƒ number of the focusing geometry. Analytic as well as numerical results are obtained. Comparison with microwave experiments of Farnell [Can. J. Phys. 36, 935 (1958)] is also made.

© 1982 Optical Society of America

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  1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.
  2. M. P. Bachynski and G. Bekefi, "Study of optical diffraction images at microwave frequencies," J. Opt. Soc. Am. 47, 428–438 (1957):
  3. G. W. Farnell, "Calculated intensity and phase distribution in the image space of a microwave lens," Can. J. Phys. 35, 777–783 (1957).
  4. G. W. Farnell, "Measured phase distribution in the image space of a microwave lens," Can. J. Phys. 36, 935–943 (1958).
  5. G. W. Farnell, "On the axial phase anomaly for microwave lenses," J. Opt. Soc. Am. 48, 643–647 (1958).
  6. H. Osterberg and L. W. Smith, "Closed solutions of Rayleigh's diffraction integral for axial points," J. Opt. Soc. Am. 51, 1050–1054 (1961).
  7. G. Goubau, "Optical relations for coherent wave beams," in Electromagnetic Theory and Antennas, Part 2, E. C. Jordan, ed. (Macmillan, New York, 1963), pp. 907–918.
  8. A. G. Van Nie, "Rigorous calculation of the electromagnetic field for wave beams," Philips Res. Rep. 19, 378–394 (1964).
  9. H. Kogelnik, "Imaging of optical modes—resonators with internal lenses," Bell Syst. Tech. J. 44, 455–494 (1965).
  10. D. A. Holmes, J. E. Korka, and P. V. Avizonis, "Parametric study of apertured focused Gaussian beams," Appl. Opt. 11, 565–574 (1972).
  11. A. Arimoto, "Intensity distribution of aberration-free diffraction patterns due to circular apertures in large ƒ-number optical systems," Opt. Acta 23, 245–250 (1976).
  12. M. A. Gusinow, M. E. Riley, and M. A. Palmer, "Focusing in a large ƒ-number optical system," Opt. Quantum Electron. 9, 465–471 (1977).
  13. J. J. Stamnes, "Focusing of two-dimensional waves," J. Opt. Soc. Am. 71, 15–31 (1981).
  14. J. H. Erkkila and M. E. Rogers, "Diffracted fields in the focal volume of a converging wave," J. Opt. Soc. Am. 71, 904–905 (1981).
  15. Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211–215 (1981).
  16. E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39, 205–210 (1981).
  17. J. J. Stamnes and B. Spjelkavik, "Focusing at small angular apertures in the Debye and Kirchhoff approximations," Opt. Commun. 40, 81–85 (1981).
  18. K. Miyamoto and E. Wolf, "Generalization of the Maggi—Rubinowicz theory of the boundary diffraction wave," Part I, J. Opt. Soc. Am. 52, 615–625 (1962); Part II, J. Opt. Soc. Am. 52, 626–637 (1962).
  19. B. Richard and E. Wolf, "Electromagnetic diffraction in optical systems-II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253, 385–379 (1959).

1981

Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211–215 (1981).

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39, 205–210 (1981).

J. J. Stamnes and B. Spjelkavik, "Focusing at small angular apertures in the Debye and Kirchhoff approximations," Opt. Commun. 40, 81–85 (1981).

J. J. Stamnes, "Focusing of two-dimensional waves," J. Opt. Soc. Am. 71, 15–31 (1981).

J. H. Erkkila and M. E. Rogers, "Diffracted fields in the focal volume of a converging wave," J. Opt. Soc. Am. 71, 904–905 (1981).

1977

M. A. Gusinow, M. E. Riley, and M. A. Palmer, "Focusing in a large ƒ-number optical system," Opt. Quantum Electron. 9, 465–471 (1977).

1976

A. Arimoto, "Intensity distribution of aberration-free diffraction patterns due to circular apertures in large ƒ-number optical systems," Opt. Acta 23, 245–250 (1976).

1972

1965

H. Kogelnik, "Imaging of optical modes—resonators with internal lenses," Bell Syst. Tech. J. 44, 455–494 (1965).

1964

A. G. Van Nie, "Rigorous calculation of the electromagnetic field for wave beams," Philips Res. Rep. 19, 378–394 (1964).

1962

1961

1959

B. Richard and E. Wolf, "Electromagnetic diffraction in optical systems-II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253, 385–379 (1959).

1958

G. W. Farnell, "On the axial phase anomaly for microwave lenses," J. Opt. Soc. Am. 48, 643–647 (1958).

G. W. Farnell, "Measured phase distribution in the image space of a microwave lens," Can. J. Phys. 36, 935–943 (1958).

1957

G. W. Farnell, "Calculated intensity and phase distribution in the image space of a microwave lens," Can. J. Phys. 35, 777–783 (1957).

M. P. Bachynski and G. Bekefi, "Study of optical diffraction images at microwave frequencies," J. Opt. Soc. Am. 47, 428–438 (1957):

Arimoto, A.

A. Arimoto, "Intensity distribution of aberration-free diffraction patterns due to circular apertures in large ƒ-number optical systems," Opt. Acta 23, 245–250 (1976).

Avizonis, P. V.

Bachynski, M. P.

Bekefi, G.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.

Erkkila, J. H.

Farnell, G. W.

G. W. Farnell, "On the axial phase anomaly for microwave lenses," J. Opt. Soc. Am. 48, 643–647 (1958).

G. W. Farnell, "Measured phase distribution in the image space of a microwave lens," Can. J. Phys. 36, 935–943 (1958).

G. W. Farnell, "Calculated intensity and phase distribution in the image space of a microwave lens," Can. J. Phys. 35, 777–783 (1957).

Goubau, G.

G. Goubau, "Optical relations for coherent wave beams," in Electromagnetic Theory and Antennas, Part 2, E. C. Jordan, ed. (Macmillan, New York, 1963), pp. 907–918.

Gusinow, M. A.

M. A. Gusinow, M. E. Riley, and M. A. Palmer, "Focusing in a large ƒ-number optical system," Opt. Quantum Electron. 9, 465–471 (1977).

Holmes, D. A.

Kogelnik, H.

H. Kogelnik, "Imaging of optical modes—resonators with internal lenses," Bell Syst. Tech. J. 44, 455–494 (1965).

Korka, J. E.

Li, Y.

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39, 205–210 (1981).

Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211–215 (1981).

Miyamoto, K.

Osterberg, H.

Palmer, M. A.

M. A. Gusinow, M. E. Riley, and M. A. Palmer, "Focusing in a large ƒ-number optical system," Opt. Quantum Electron. 9, 465–471 (1977).

Richard, B.

B. Richard and E. Wolf, "Electromagnetic diffraction in optical systems-II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253, 385–379 (1959).

Riley, M. E.

M. A. Gusinow, M. E. Riley, and M. A. Palmer, "Focusing in a large ƒ-number optical system," Opt. Quantum Electron. 9, 465–471 (1977).

Rogers, M. E.

Smith, L. W.

Spjelkavik, B.

J. J. Stamnes and B. Spjelkavik, "Focusing at small angular apertures in the Debye and Kirchhoff approximations," Opt. Commun. 40, 81–85 (1981).

Stamnes, J. J.

J. J. Stamnes and B. Spjelkavik, "Focusing at small angular apertures in the Debye and Kirchhoff approximations," Opt. Commun. 40, 81–85 (1981).

J. J. Stamnes, "Focusing of two-dimensional waves," J. Opt. Soc. Am. 71, 15–31 (1981).

Van Nie, A. G.

A. G. Van Nie, "Rigorous calculation of the electromagnetic field for wave beams," Philips Res. Rep. 19, 378–394 (1964).

Wolf, E.

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39, 205–210 (1981).

Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211–215 (1981).

K. Miyamoto and E. Wolf, "Generalization of the Maggi—Rubinowicz theory of the boundary diffraction wave," Part I, J. Opt. Soc. Am. 52, 615–625 (1962); Part II, J. Opt. Soc. Am. 52, 626–637 (1962).

B. Richard and E. Wolf, "Electromagnetic diffraction in optical systems-II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253, 385–379 (1959).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, "Imaging of optical modes—resonators with internal lenses," Bell Syst. Tech. J. 44, 455–494 (1965).

Can. J. Phys.

G. W. Farnell, "Calculated intensity and phase distribution in the image space of a microwave lens," Can. J. Phys. 35, 777–783 (1957).

G. W. Farnell, "Measured phase distribution in the image space of a microwave lens," Can. J. Phys. 36, 935–943 (1958).

J. Opt. Soc. Am.

Opt. Acta

A. Arimoto, "Intensity distribution of aberration-free diffraction patterns due to circular apertures in large ƒ-number optical systems," Opt. Acta 23, 245–250 (1976).

Opt. Commun.

Y. Li and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39, 211–215 (1981).

E. Wolf and Y. Li, "Conditions for the validity of the Debye integral representation of focused fields," Opt. Commun. 39, 205–210 (1981).

J. J. Stamnes and B. Spjelkavik, "Focusing at small angular apertures in the Debye and Kirchhoff approximations," Opt. Commun. 40, 81–85 (1981).

Opt. Quantum Electron.

M. A. Gusinow, M. E. Riley, and M. A. Palmer, "Focusing in a large ƒ-number optical system," Opt. Quantum Electron. 9, 465–471 (1977).

Philips Res. Rep.

A. G. Van Nie, "Rigorous calculation of the electromagnetic field for wave beams," Philips Res. Rep. 19, 378–394 (1964).

Proc. R. Soc. London Ser. A

B. Richard and E. Wolf, "Electromagnetic diffraction in optical systems-II. Structure of the image field in an aplanatic system," Proc. R. Soc. London Ser. A 253, 385–379 (1959).

Other

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 8.8.

G. Goubau, "Optical relations for coherent wave beams," in Electromagnetic Theory and Antennas, Part 2, E. C. Jordan, ed. (Macmillan, New York, 1963), pp. 907–918.

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