Abstract

The Stratton–Chu theory of electromagnetic (EM) scattering is used to develop a Kirchhoff formalism of the diffraction of EM waves by an aperture. The theory is applied to the study of the diffraction of a polarized EM wave by small-Fresnel-number systems. It is demonstrated through sample numerical calculations that the vectorial aspects of the EM waves are important, especially for fields in the microwave regime. The focal shift is then calculated with the vectorial aspects taken into account.

© 1994 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1983), Chap. 8.
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  3. R. K. Lunenberg, Mathematical Theory of Optics (University of California, Berkeley, Calif., 1964), Chap. 6.
  4. M. Klein, I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, New York, 1965).
  5. S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, San Diego, Calif., 1986).
  6. B. Baker, E. Copson, The Mathematical Theory of Huygen’s Principle (Oxford U. Press, New York, 1953).
  7. D. J. Schroeder, Astronomical Optics (Academic, San Diego, Calif., 1987).
  8. T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, San Diego, Calif., 1984).
  9. M. Pluto, Advanced Light Microscopy (Elsevier, Amsterdam, 1984), Vols. 1 and 2.
  10. J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  11. A. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982).
  12. J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986).
  13. H. Kogelnik, “Modes in optical resonators,” in Advances in Lasers, A. K. Levine, ed. (Decker, New York, 1966), Vol. 1, pp. 295–347.
  14. L. A. Vaynshteyn, Open Resonators and Open Waveguides (Golem, Boulder, Colo., 1969).
  15. P. K. Das, Optical Signal Processing (Springer-Verlag, New York, 1991).
    [CrossRef]
  16. K. Strehl, Theorie des Fernrohrs (Barth, Leipzig, 1894), Vol. 1.
  17. H. H. Hopkins, “The Airy disc formula for systems of higher relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
    [CrossRef]
  18. H. Osterberg, “Diffraction theory of phase microscopy with Kohler illumination,” in Phase Microscopy, A. Bennett, H. Jupnik, H. Osterberg, O. Richards, eds. (Wiley, New York, 1951), Chap. 7.
  19. R. Barakat, D. Lev, “Transfer functions and total illuminance of high numerical aperture systems obeying the sine condition,”J. Opt. Soc. Am. 53, 324–332 (1963).
    [CrossRef]
  20. R. Barakat, “The intensity distribution and total illuminance of aberration-free diffraction images,” in Progress in Optics I, E. Wolf, ed. (North-Holland, Amsterdam, 1961).
    [CrossRef]
  21. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  22. 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]
  23. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  24. Y. Li, “Dependence of the focal shift on Fresnel number and fnumber,”J. Opt. Soc. Am. 72, 770–774 (1982).
    [CrossRef]
  25. Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 64, 207–228 (1983).
  26. Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1626 (1983).
    [CrossRef]
  27. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
    [CrossRef]
  28. D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric study of apertured focused Gaussian beams,” Appl. Opt. 11, 565–574 (1972).
    [CrossRef] [PubMed]
  29. G. C. Sherman, W. C. Chew, “Aperture and far-field distributions expressed by the Debye integral representation of focused fields,”J. Opt. Soc. Am. 72, 1076–1083 (1982).
    [CrossRef]
  30. E. Wolf, “The focal region in diffraction-limited systems,” Kiman 5, 257–259 (1983).
  31. V. S. Ignatovsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 4 (1919).
  32. V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 5 (1920).
  33. C. J. Bowkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
    [CrossRef]
  34. E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
    [CrossRef]
  35. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  36. A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
    [CrossRef]
  37. A. Carswell, “Measurements of the longitudinal component of the electromagnetic field at the focus of a coherent beam,” Phys. Rev. Lett. 15, 647–649 (1965).
    [CrossRef]
  38. R. Barakat, “Diffracted electromagnetic fields in the neighborhood of the focus of a paraboloidal mirror having a central obscuration,” Appl. Opt. 26, 3790–3795 (1987).
    [CrossRef] [PubMed]
  39. C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987); C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London Ser. A 379, 145–158 (1982); C. J. R. Sheppard, A. Choudhury, J. Gannaway, “Electromagnetic field near the focus of wide-angular lens and mirror systems,” Microwaves Opt. Acoust. 1, 129–132 (1977).
    [CrossRef]
  40. R. Barakat, “The numerical evaluation of diffraction integrals,” in Computer Application in Optical Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980).
  41. A. H. Shafer, “Hamilton’s mixed and angle characteristic functions and diffraction aberration theory,”J. Opt. Soc. Am. 57, 630–639 (1967).
    [CrossRef]
  42. J. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 8.
  43. M. S. Zhdanov, Integral Transforms in Geophysics (Springer-Verlag, New York, 1988), Chap. 8.
    [CrossRef]
  44. S. K. Cho, Electromagnetic Scattering (Springer-Verlag, New York, 1990).
    [CrossRef]
  45. W. Franz, Theorie der beugung elektromagnetischer Wellen (Springer-Verlag, Berlin, 1957).
    [CrossRef]
  46. M. P. Bachynski, G. Bekefi, “Study of optical diffraction images at microwave frequencies,”J. Opt. Soc. Am. 47, 428–438 (1957).
    [CrossRef]
  47. G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–784 (1957).
    [CrossRef]
  48. G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
    [CrossRef]

1987 (2)

1984 (1)

1983 (3)

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 64, 207–228 (1983).

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1626 (1983).
[CrossRef]

E. Wolf, “The focal region in diffraction-limited systems,” Kiman 5, 257–259 (1983).

1982 (2)

1981 (3)

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

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

1972 (1)

1967 (1)

1965 (2)

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

A. Carswell, “Measurements of the longitudinal component of the electromagnetic field at the focus of a coherent beam,” Phys. Rev. Lett. 15, 647–649 (1965).
[CrossRef]

1963 (1)

1959 (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

1958 (1)

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

1957 (2)

G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–784 (1957).
[CrossRef]

M. P. Bachynski, G. Bekefi, “Study of optical diffraction images at microwave frequencies,”J. Opt. Soc. Am. 47, 428–438 (1957).
[CrossRef]

1954 (1)

C. J. Bowkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[CrossRef]

1943 (1)

H. H. Hopkins, “The Airy disc formula for systems of higher relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
[CrossRef]

1920 (1)

V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 5 (1920).

1919 (1)

V. S. Ignatovsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 4 (1919).

Avizonis, P. V.

Bachynski, M. P.

Baker, B.

B. Baker, E. Copson, The Mathematical Theory of Huygen’s Principle (Oxford U. Press, New York, 1953).

Barakat, R.

R. Barakat, “Diffracted electromagnetic fields in the neighborhood of the focus of a paraboloidal mirror having a central obscuration,” Appl. Opt. 26, 3790–3795 (1987).
[CrossRef] [PubMed]

R. Barakat, D. Lev, “Transfer functions and total illuminance of high numerical aperture systems obeying the sine condition,”J. Opt. Soc. Am. 53, 324–332 (1963).
[CrossRef]

R. Barakat, “The numerical evaluation of diffraction integrals,” in Computer Application in Optical Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980).

R. Barakat, “The intensity distribution and total illuminance of aberration-free diffraction images,” in Progress in Optics I, E. Wolf, ed. (North-Holland, Amsterdam, 1961).
[CrossRef]

Bekefi, G.

Boivin, A.

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1983), Chap. 8.

Bowkamp, C. J.

C. J. Bowkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[CrossRef]

Carswell, A.

A. Carswell, “Measurements of the longitudinal component of the electromagnetic field at the focus of a coherent beam,” Phys. Rev. Lett. 15, 647–649 (1965).
[CrossRef]

Chew, W. C.

Cho, S. K.

S. K. Cho, Electromagnetic Scattering (Springer-Verlag, New York, 1990).
[CrossRef]

Copson, E.

B. Baker, E. Copson, The Mathematical Theory of Huygen’s Principle (Oxford U. Press, New York, 1953).

Crosignani, B.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, San Diego, Calif., 1986).

Das, P. K.

P. K. Das, Optical Signal Processing (Springer-Verlag, New York, 1991).
[CrossRef]

Di Porto, P.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, San Diego, Calif., 1986).

Farnell, G. W.

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

G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–784 (1957).
[CrossRef]

Franz, W.

W. Franz, Theorie der beugung elektromagnetischer Wellen (Springer-Verlag, Berlin, 1957).
[CrossRef]

Gaskill, J.

J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Goodman, J. W.

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

Holmes, D. A.

Hopkins, H. H.

H. H. Hopkins, “The Airy disc formula for systems of higher relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
[CrossRef]

Ignatovsky, V. S.

V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 5 (1920).

V. S. Ignatovsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 4 (1919).

Kay, I.

M. Klein, I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, New York, 1965).

Klein, M.

M. Klein, I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, New York, 1965).

Kogelnik, H.

H. Kogelnik, “Modes in optical resonators,” in Advances in Lasers, A. K. Levine, ed. (Decker, New York, 1966), Vol. 1, pp. 295–347.

Korka, J. E.

Lev, D.

Li, Y.

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]

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1626 (1983).
[CrossRef]

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 64, 207–228 (1983).

Y. Li, “Dependence of the focal shift on Fresnel number and fnumber,”J. Opt. Soc. Am. 72, 770–774 (1982).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Lunenberg, R. K.

R. K. Lunenberg, Mathematical Theory of Optics (University of California, Berkeley, Calif., 1964), Chap. 6.

Marathay, A.

A. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982).

Matthews, H. J.

Osterberg, H.

H. Osterberg, “Diffraction theory of phase microscopy with Kohler illumination,” in Phase Microscopy, A. Bennett, H. Jupnik, H. Osterberg, O. Richards, eds. (Wiley, New York, 1951), Chap. 7.

Platzer, H.

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1626 (1983).
[CrossRef]

Pluto, M.

M. Pluto, Advanced Light Microscopy (Elsevier, Amsterdam, 1984), Vols. 1 and 2.

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics (Academic, San Diego, Calif., 1987).

Shafer, A. H.

Sheppard, C.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, San Diego, Calif., 1984).

Sheppard, C. J. R.

Sherman, G. C.

Solimeno, S.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, San Diego, Calif., 1986).

Spjelkavik, B.

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

Stamnes, J.

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

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986).

Stratton, J.

J. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 8.

Strehl, K.

K. Strehl, Theorie des Fernrohrs (Barth, Leipzig, 1894), Vol. 1.

Vaynshteyn, L. A.

L. A. Vaynshteyn, Open Resonators and Open Waveguides (Golem, Boulder, Colo., 1969).

Wilson, T.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, San Diego, Calif., 1984).

Wolf, E.

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]

E. Wolf, “The focal region in diffraction-limited systems,” Kiman 5, 257–259 (1983).

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1983), Chap. 8.

Zhdanov, M. S.

M. S. Zhdanov, Integral Transforms in Geophysics (Springer-Verlag, New York, 1988), Chap. 8.
[CrossRef]

Appl. Opt. (2)

Can. J. Phys. (2)

G. W. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–784 (1957).
[CrossRef]

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

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (2)

Kiman (1)

E. Wolf, “The focal region in diffraction-limited systems,” Kiman 5, 257–259 (1983).

Opt. Acta (1)

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1626 (1983).
[CrossRef]

Opt. Commun. (3)

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

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

Optik (1)

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 64, 207–228 (1983).

Phys. Rev. B (1)

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

Phys. Rev. Lett. (1)

A. Carswell, “Measurements of the longitudinal component of the electromagnetic field at the focus of a coherent beam,” Phys. Rev. Lett. 15, 647–649 (1965).
[CrossRef]

Proc. Phys. Soc. London (1)

H. H. Hopkins, “The Airy disc formula for systems of higher relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
[CrossRef]

Proc. R. Soc. London Ser. A (2)

E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London Ser. A 253, 349–357 (1959).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Rep. Prog. Phys. (1)

C. J. Bowkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[CrossRef]

Trans. Opt. Inst. Petrograd (2)

V. S. Ignatovsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 4 (1919).

V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 5 (1920).

Other (23)

H. Osterberg, “Diffraction theory of phase microscopy with Kohler illumination,” in Phase Microscopy, A. Bennett, H. Jupnik, H. Osterberg, O. Richards, eds. (Wiley, New York, 1951), Chap. 7.

R. Barakat, “The intensity distribution and total illuminance of aberration-free diffraction images,” in Progress in Optics I, E. Wolf, ed. (North-Holland, Amsterdam, 1961).
[CrossRef]

J. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. 8.

M. S. Zhdanov, Integral Transforms in Geophysics (Springer-Verlag, New York, 1988), Chap. 8.
[CrossRef]

S. K. Cho, Electromagnetic Scattering (Springer-Verlag, New York, 1990).
[CrossRef]

W. Franz, Theorie der beugung elektromagnetischer Wellen (Springer-Verlag, Berlin, 1957).
[CrossRef]

R. Barakat, “The numerical evaluation of diffraction integrals,” in Computer Application in Optical Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1983), Chap. 8.

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

R. K. Lunenberg, Mathematical Theory of Optics (University of California, Berkeley, Calif., 1964), Chap. 6.

M. Klein, I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, New York, 1965).

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, San Diego, Calif., 1986).

B. Baker, E. Copson, The Mathematical Theory of Huygen’s Principle (Oxford U. Press, New York, 1953).

D. J. Schroeder, Astronomical Optics (Academic, San Diego, Calif., 1987).

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, San Diego, Calif., 1984).

M. Pluto, Advanced Light Microscopy (Elsevier, Amsterdam, 1984), Vols. 1 and 2.

J. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

A. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982).

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986).

H. Kogelnik, “Modes in optical resonators,” in Advances in Lasers, A. K. Levine, ed. (Decker, New York, 1966), Vol. 1, pp. 295–347.

L. A. Vaynshteyn, Open Resonators and Open Waveguides (Golem, Boulder, Colo., 1969).

P. K. Das, Optical Signal Processing (Springer-Verlag, New York, 1991).
[CrossRef]

K. Strehl, Theorie des Fernrohrs (Barth, Leipzig, 1894), Vol. 1.

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Figures (6)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Total energy density of the diffracted field along the optical axis for N = 5, a = 31 cm, λ = 3.22 cm, f = 59.7 cm.

Fig. 3
Fig. 3

Total energy density of diffracted fields for data in Fig. 2 in three receiving planes: –, z = 0; – · –, z = +5 cm; - - - - -, z = −5 cm.

Fig. 4
Fig. 4

Contours of constant electric energy density for data in Fig. 2 in the receiving plane z = 0.

Fig. 5
Fig. 5

Contours of constant electric energy density for data in Fig. 2 in the receiving plane z = +5 cm.

Fig. 6
Fig. 6

Contours of constant electric energy density for data in Fig. 2 in the receiving plane z = −5 cm.

Equations (48)

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

N a 2 / λ f ,
E ( r ) = - S [ i k ( n 0 × H ) G ( r , r ) + ( n 0 × E ) × G ( r , r ) + ( n 0 · E ) G ( r , r ) ] d S ,
H ( r ) = - S [ i k ( n 0 × E ) G ( r , r ) + ( n 0 × H ) × G ( r , r ) + ( n 0 · H ) G ( r , r ) ] d S .
G ( r , r ) = exp ( i k r - r ) 4 π r - r
E ( r ) = - A ¯ [ i k ( n 0 × H i ) G ( r , r ) + ( n 0 × E i ) × G ( r , r ) + ( n 0 · E i ) G ( r , r ) ] d S ,
H ( r ) = - A ¯ [ i k ( n 0 × E i ) G ( r , r ) + ( n 0 × H i ) × G ( r , r ) + ( n 0 · H i ) G ( r , r ) ] d S ,
E i ( θ , ϕ ) = E 0 ( θ , ϕ ) ( d Ω / d Ω ) 1 / 2 = E 0 ( θ , ϕ ) g ( θ ) ,
E i ( θ , ϕ ) = g ( θ ) ( ϕ ^ ϕ ^ + θ ^ θ ^ ) · E i ( θ , ϕ ) ,
E i ( θ , ϕ ) = E 0 ( h cos ϕ , h sin ϕ ) [ h sin θ ( d h d θ ) ] 1 / 2 = E 0 ( h cos ϕ , h sin ϕ ) f g 0 ( θ ) ,
E i ( θ , ϕ ) = g 0 ( θ ) [ ϕ ^ ϕ ^ + θ ^ ( z ^ × ϕ ^ ) ] · E 0 ,
E i ( θ , ϕ ) = g 0 ( θ ) E 0 [ ( cos θ cos 2 ϕ + sin 2 ϕ ) i - ( 1 - cos θ ) sin ϕ cos ϕ j + sin θ cos ϕ k ] ,
ϕ ^ = - sin ϕ i + cos ϕ j ,
θ ^ = - cos θ cos ϕ i - cos θ sin ϕ j - sin θ k .
H i = E i × n 0 .
E ( r ) = - i k A ¯ [ n 0 × ( E i × n 0 ) + ( n 0 × E i ) × R ^ + ( n 0 · E i ) R ^ ] exp ( i k R ) 4 π R d S ,
H ( r ) = - i k A ¯ [ n 0 × E i + ( n 0 × E i × n 0 ) × R ^ ] × exp ( i k R ) 4 π R d S .
x Q = f sin θ cos ϕ ,             y Q = f sin θ sin ϕ ,         z Q = - f cos θ .
x P = ρ cos γ ,             y P = ρ sin γ ,             z Q = z ,
r - r = [ f 2 + z 2 + ρ 2 + 2 f z - 2 f z ( 1 - cos θ ) - 2 f ρ sin θ cos ( γ - ϕ ) ] 1 / 2 f + z + 1 2 ( f + z ) [ ρ 2 - 2 f p sin θ cos ( γ - ϕ ) - 2 z f ( 1 - cos θ ) ] + ,
E = E 0 f 2 exp ( i Φ ) i λ ( z + f ) sin ϑ g 0 ( ϑ ) exp { - i k f z + f × [ ρ sin ϑ cos ( γ - ϕ ) + z ( 1 - cos ϑ ) ] } × [ ( cos ϑ cos 2 ϕ + sin 2 ϕ ) i + ( 1 - cos ϑ ) sin ϕ cos ϕ j + sin ϑ cos ϕ k ] d ϑ d ϕ ,
H = E 0 f 2 exp ( i Φ ) i λ ( z + f ) sin ϑ g 0 ( ϑ ) exp { - i k f z + f × [ ρ sin ϑ cos ( γ - ϕ ) + z ( 1 - cos ϑ ) ] } × { ( 1 - cos ϑ ) sin ϕ cos ϕ i + [ cos ϑ - ( 1 - cos ϑ ) sin 2 ϕ ] j + sin ϑ sin ϕ k } d ϑ d ϕ ,
Φ = k ( z + f ) + k ρ 2 2 ( z + f ) .
u N = 2 π N z z + f ,
ν N = 2 π N ( f a ) ρ z + f ,
E = E 0 f i λ ( 1 - u N 2 π N ) exp ( i Φ ) sin ϑ g 0 ( ϑ ) × exp ( - i f a [ ν N sin ϑ cos ( γ - ϕ ) - f a u N ( 1 - cos ϑ ) ] × { [ cos ϑ + ( 1 - cos ϑ ) sin 2 ϕ ] i + ( 1 - cos ϑ ) sin ϕ cos ϕ j + sin ϑ cos ϕ k } d ϑ d ϕ ) ,
H = E 0 f i λ ( 1 - u N 2 π N ) exp ( i Φ ) sin ϑ g 0 ( ϑ ) × exp ( - i f a [ ν N sin ϑ cos ( γ - ϕ ) - f a u N ( 1 - cos ϑ ) ] × { ( 1 - cos ϑ ) sin ϕ cos ϕ i + [ cos ϑ + ( 1 - cos ϑ ) sin 2 ϕ ] j + sin ϑ cos ϕ k } d ϑ d ϕ ) ,
Φ = 2 π N 2 π N - u N ( f a ) 2 + N 2 2 ( 2 π N - u N ) ν N 2 .
0 2 π cos n ϕ exp [ i ρ cos ( ϕ - γ ) ] d ϕ = 2 π i n j n ( ρ ) cos n γ , 0 2 π sin n ϕ exp [ i ρ cos ( ϕ - γ ) ] d ϕ = 2 π i n J n ( ρ ) sin n γ ,
E x = - i A ( L 0 + L 2 cos 2 γ ) , H x = - i A ( L 2 sin 2 γ ) , E y = - i A ( L 2 sin 2 γ ) , H y = - i A ( L 0 - L 2 cos 2 γ ) , E z = - A ( 2 L 1 cos γ ) , H z = - A ( 2 L 1 sin γ ) .
A = π E 0 f λ ( f z + f ) exp ( i Φ ) ,
L 0 0 θ 0 g 0 ( ϑ ) sin ϑ ( 1 + cos ϑ ) exp [ i f 2 a 2 ( 1 - cos ϑ ) u N ] × J 0 ( f a ν N sin ϑ ) d ϑ ,
L 1 0 θ 0 g 0 ( ϑ ) sin 2 ϑ exp [ i f 2 a 2 ( 1 - cos ϑ ) u N ] × J 1 ( f a ν N sin ϑ ) d ϑ ,
L 2 0 θ 0 g 0 ( ϑ ) sin ϑ ( 1 - cos ϑ ) exp [ i f 2 a 2 ( 1 - cos ϑ ) u N ] × J 2 ( f a ν N sin ϑ ) d ϑ .
E x ( - z , ρ , γ ) = - E x * ( z , ρ , γ ) , H x ( - z , ρ , γ ) = - H x * ( z , ρ , γ ) , E y ( - z , ρ , γ ) = - E y * ( z , ρ , γ ) , H y ( - z , ρ , γ ) = - H y * ( z , ρ , γ ) , E z ( - z , ρ , γ ) = E y * ( z , ρ , γ ) , H z ( - z , ρ , γ ) = H z * ( z , ρ , γ ) .
W e 1 16 π E · E * ,
W m 1 16 π H · H * .
W e A 2 16 π [ l 0 2 + L 2 2 + 4 L 1 2 cos 2 γ + 2 Re ( L 0 L 2 * ) cos ( 2 γ ) ] ,
W m A 2 16 π [ L 0 2 + L 2 2 + 4 L 1 2 sin 2 γ - 2 Re ( L 0 L 2 * ) cos ( 2 γ ) ] .
W = W e + W m = A 2 8 π ( L 0 2 + L 2 2 + 2 L 1 2 ) .
W e = W m = A 2 16 π L 0 2 ,
W = A 2 8 π L 0 2 ,
L 0 = 0 θ 0 g 0 ( ϑ ) sin ϑ ( 1 + cos ϑ ) exp [ i f 2 a 2 ( 1 - cos ϑ ) u N ] d ϑ .
sin ϑ ϑ ,             1 - cos ϑ ϑ 2 / 2 ,
L 0 = 0 θ 0 2 g 0 ( ϑ ) ϑ exp ( i f 2 2 a 2 u N ϑ 2 ) J 0 ( f a ν N ϑ ) d ϑ ,
L 1 = 0 θ 0 g 0 ( ϑ ) ϑ 2 exp ( i f 2 a 2 u N ϑ 2 ) J 1 ( f a ν N ϑ ) d ϑ ,
L 2 = 0 θ 0 g 0 ( ϑ ) ϑ 3 2 exp ( i f 2 2 a 2 u N ϑ 2 ) J 2 ( f a ν N ϑ ) d ϑ .
g 0 ( ϑ ) = cos ϑ ;
g 0 ( ϑ ) = 2 1 + cos ϑ .

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