Abstract

The contributions of homogeneous and evanescent waves to two-dimensional near-field diffraction patterns of scalar optical fields are examined in detail. The total plane-integrated intensities of the two contributions are introduced as convenient measures of their relative importance. As an example, the diffraction of a plane wave by a slit is considered.

© 1995 Optical Society of America

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References

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  1. E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
    [CrossRef] [PubMed]
  2. U. Dürig, D. W. Pohl, F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59, 3318–3327 (1986).
    [CrossRef]
  3. D. W. Pohl, “Scanning near-field optical microscopy,” in Advances in Optical and Electron Microscopy (Academic, New York, 1991), Vol. 12, pp. 242–312.
  4. E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. C. Finn, M. H. Kryder, C.-H. Chang, “Near-field magneto-optics and high-density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
    [CrossRef]
  5. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
    [CrossRef]
  6. G. A. Massey, “Microscopy and pattern generation with scanned evanescent waves,” Appl. Opt. 23, 658–660 (1984).
    [CrossRef] [PubMed]
  7. W. H. Carter, “Band-limited angular-spectrum approximation to a scalar dipole field,” Opt. Commun. 2, 142–148 (1970).
    [CrossRef]
  8. W. H. Carter, “Band-limited angular-spectrum approximation to a spherical scalar wave field,” J. Opt. Soc. Am. 65, 1054–1058 (1975).
    [CrossRef]
  9. See, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 560–561, or D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1984), Vol. XXI, pp. 1–67.
    [CrossRef]
  10. J. W Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–54.
  11. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, in press), Sec. 3.2.
  12. D. C. Bertilone, “The contributions of homogeneous and evanescent plane waves to the scalar optical field: exact diffraction formulae,” J. Mod. Opt. 38, 865–875 (1991).
    [CrossRef]
  13. D. C. Bertilone, “Wave theory for a converging spherical incident wave in an infinite-aperture system,” J. Mod. Opt. 38, 1531–1536 (1991).
    [CrossRef]
  14. G. C. Sherman, J. J. Stamnes, A. J. Devaney, É. Lalor, “Contribution of the inhomogeneous waves in angular spectrum representations,” Opt. Commun. 8, 271–274 (1973).
    [CrossRef]
  15. G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
    [CrossRef]
  16. See, for example, A. Papoulis, Systems and Transformations with Applications in Optics (McGraw-Hill, New York, 1968), pp. 81–83.
  17. J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), p. 43.
  18. D. Slepian, “Some comments on Fourier analysis, uncertainty and modeling,” SIAM Rev. 25, 379–393 (1983).
    [CrossRef]
  19. B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1971), Vol. IX, pp. 311–407.
    [CrossRef]
  20. G. Toraldo di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799–804 (1969).
    [CrossRef] [PubMed]
  21. E. Wolf, T. Habashy, “Reconstruction of scattering potentials from incomplete data,” J. Mod. Opt. 41, 1679–1685 (1994).
    [CrossRef]
  22. A. W. Lohmann, “Three-dimensional properties of wave fields,” Optik 51, 105–117 (1978).
  23. G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1969).
    [CrossRef]
  24. M. W. Kowarz, E. Wolf, “Conservation laws for partially coherent free fields,” J. Opt. Soc. Am. A 10, 88–94 (1993).
    [CrossRef]
  25. M. W. Kowarz, “Conservation laws for free electromagnetic fields,” J. Mod. Opt. 42, 109–115 (1995).
    [CrossRef]
  26. M. W. Kowarz, “Energy constraints in optimum apodization problems,” Opt. Commun. 110, 274–278 (1994).
    [CrossRef]
  27. E. Wolf, E. W. Marchand, “Comparison of the Kirchhoff and the Rayleigh–Sommerfeld theories of diffraction at an aperture,” J. Opt. Soc. Am. 54, 587–594 (1964).
    [CrossRef]
  28. For the case of a circular aperture a comparison of the approximate theories with the exact results may be found in C. J. Bouwkamp, “Theoretical and numerical treatment of diffraction through a circular aperture,” IEEE Trans. Antennas Propagat. AP-18, 152–176 (1970), and a comparison with experimental results may be found in E. W. Marchand, E. Wolf, “Consistent formulation of Kirchhoff’s diffraction theory,” J. Opt. Soc. Am. 56, 1712–1722 (1966). In Bouwkamp’s paper the two Rayleigh–Sommerfeld theories are referred to as improved Kirchhoff theories.
    [CrossRef]
  29. The decay of the total evanescent intensity Itot(i)(z) with z is much more rapid for the cases d = 1λ and d = 5λ, shown in Fig. 8. We may explain this behavior by performing an asymptotic expansion of Eq. (4.4b) for large kz. One then finds that, when the slit width is an integer number of wavelengths, the first term in the asymptotic expansion vanishes and a higher-order term, which decays more rapidly with z, becomes important.
  30. H. Levine, J. Schwinger, “On the theory of diffraction by an aperture in an infinite plane screen. I,” Phys. Rev. 74, 958–974 (1948), Eq. (2.24).
    [CrossRef]
  31. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 241–243.
  32. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 135–137.
  33. The electromagnetic analog of these quantities is discussed in G. V. Borgiotti, “Radiation and reactive energy of aperture antennas,” IEEE Trans. Antennas Propagat. AP-11, 94–95 (1963).
    [CrossRef]

1995 (1)

M. W. Kowarz, “Conservation laws for free electromagnetic fields,” J. Mod. Opt. 42, 109–115 (1995).
[CrossRef]

1994 (2)

M. W. Kowarz, “Energy constraints in optimum apodization problems,” Opt. Commun. 110, 274–278 (1994).
[CrossRef]

E. Wolf, T. Habashy, “Reconstruction of scattering potentials from incomplete data,” J. Mod. Opt. 41, 1679–1685 (1994).
[CrossRef]

1993 (1)

1992 (2)

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. C. Finn, M. H. Kryder, C.-H. Chang, “Near-field magneto-optics and high-density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

1991 (2)

D. C. Bertilone, “The contributions of homogeneous and evanescent plane waves to the scalar optical field: exact diffraction formulae,” J. Mod. Opt. 38, 865–875 (1991).
[CrossRef]

D. C. Bertilone, “Wave theory for a converging spherical incident wave in an infinite-aperture system,” J. Mod. Opt. 38, 1531–1536 (1991).
[CrossRef]

1986 (1)

U. Dürig, D. W. Pohl, F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59, 3318–3327 (1986).
[CrossRef]

1984 (1)

1983 (1)

D. Slepian, “Some comments on Fourier analysis, uncertainty and modeling,” SIAM Rev. 25, 379–393 (1983).
[CrossRef]

1979 (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

1978 (1)

A. W. Lohmann, “Three-dimensional properties of wave fields,” Optik 51, 105–117 (1978).

1976 (1)

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

1975 (1)

1973 (1)

G. C. Sherman, J. J. Stamnes, A. J. Devaney, É. Lalor, “Contribution of the inhomogeneous waves in angular spectrum representations,” Opt. Commun. 8, 271–274 (1973).
[CrossRef]

1970 (2)

W. H. Carter, “Band-limited angular-spectrum approximation to a scalar dipole field,” Opt. Commun. 2, 142–148 (1970).
[CrossRef]

For the case of a circular aperture a comparison of the approximate theories with the exact results may be found in C. J. Bouwkamp, “Theoretical and numerical treatment of diffraction through a circular aperture,” IEEE Trans. Antennas Propagat. AP-18, 152–176 (1970), and a comparison with experimental results may be found in E. W. Marchand, E. Wolf, “Consistent formulation of Kirchhoff’s diffraction theory,” J. Opt. Soc. Am. 56, 1712–1722 (1966). In Bouwkamp’s paper the two Rayleigh–Sommerfeld theories are referred to as improved Kirchhoff theories.
[CrossRef]

1969 (2)

1964 (1)

1963 (1)

The electromagnetic analog of these quantities is discussed in G. V. Borgiotti, “Radiation and reactive energy of aperture antennas,” IEEE Trans. Antennas Propagat. AP-11, 94–95 (1963).
[CrossRef]

1948 (1)

H. Levine, J. Schwinger, “On the theory of diffraction by an aperture in an infinite plane screen. I,” Phys. Rev. 74, 958–974 (1948), Eq. (2.24).
[CrossRef]

Bertilone, D. C.

D. C. Bertilone, “The contributions of homogeneous and evanescent plane waves to the scalar optical field: exact diffraction formulae,” J. Mod. Opt. 38, 865–875 (1991).
[CrossRef]

D. C. Bertilone, “Wave theory for a converging spherical incident wave in an infinite-aperture system,” J. Mod. Opt. 38, 1531–1536 (1991).
[CrossRef]

Betzig, E.

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. C. Finn, M. H. Kryder, C.-H. Chang, “Near-field magneto-optics and high-density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Borgiotti, G. V.

The electromagnetic analog of these quantities is discussed in G. V. Borgiotti, “Radiation and reactive energy of aperture antennas,” IEEE Trans. Antennas Propagat. AP-11, 94–95 (1963).
[CrossRef]

Born, M.

See, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 560–561, or D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1984), Vol. XXI, pp. 1–67.
[CrossRef]

Bouwkamp, C. J.

For the case of a circular aperture a comparison of the approximate theories with the exact results may be found in C. J. Bouwkamp, “Theoretical and numerical treatment of diffraction through a circular aperture,” IEEE Trans. Antennas Propagat. AP-18, 152–176 (1970), and a comparison with experimental results may be found in E. W. Marchand, E. Wolf, “Consistent formulation of Kirchhoff’s diffraction theory,” J. Opt. Soc. Am. 56, 1712–1722 (1966). In Bouwkamp’s paper the two Rayleigh–Sommerfeld theories are referred to as improved Kirchhoff theories.
[CrossRef]

Carter, W. H.

W. H. Carter, “Band-limited angular-spectrum approximation to a spherical scalar wave field,” J. Opt. Soc. Am. 65, 1054–1058 (1975).
[CrossRef]

W. H. Carter, “Band-limited angular-spectrum approximation to a scalar dipole field,” Opt. Commun. 2, 142–148 (1970).
[CrossRef]

Chang, C.-H.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. C. Finn, M. H. Kryder, C.-H. Chang, “Near-field magneto-optics and high-density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Devaney, A. J.

G. C. Sherman, J. J. Stamnes, A. J. Devaney, É. Lalor, “Contribution of the inhomogeneous waves in angular spectrum representations,” Opt. Commun. 8, 271–274 (1973).
[CrossRef]

Dürig, U.

U. Dürig, D. W. Pohl, F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59, 3318–3327 (1986).
[CrossRef]

Finn, P. C.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. C. Finn, M. H. Kryder, C.-H. Chang, “Near-field magneto-optics and high-density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Frieden, B. R.

B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1971), Vol. IX, pp. 311–407.
[CrossRef]

Goodman, J. W

J. W Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–54.

Gyorgy, E. M.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. C. Finn, M. H. Kryder, C.-H. Chang, “Near-field magneto-optics and high-density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Habashy, T.

E. Wolf, T. Habashy, “Reconstruction of scattering potentials from incomplete data,” J. Mod. Opt. 41, 1679–1685 (1994).
[CrossRef]

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 241–243.

Kowarz, M. W.

M. W. Kowarz, “Conservation laws for free electromagnetic fields,” J. Mod. Opt. 42, 109–115 (1995).
[CrossRef]

M. W. Kowarz, “Energy constraints in optimum apodization problems,” Opt. Commun. 110, 274–278 (1994).
[CrossRef]

M. W. Kowarz, E. Wolf, “Conservation laws for partially coherent free fields,” J. Opt. Soc. Am. A 10, 88–94 (1993).
[CrossRef]

Kryder, M. H.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. C. Finn, M. H. Kryder, C.-H. Chang, “Near-field magneto-optics and high-density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Lalor, É.

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

G. C. Sherman, J. J. Stamnes, A. J. Devaney, É. Lalor, “Contribution of the inhomogeneous waves in angular spectrum representations,” Opt. Commun. 8, 271–274 (1973).
[CrossRef]

Levine, H.

H. Levine, J. Schwinger, “On the theory of diffraction by an aperture in an infinite plane screen. I,” Phys. Rev. 74, 958–974 (1948), Eq. (2.24).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, “Three-dimensional properties of wave fields,” Optik 51, 105–117 (1978).

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, in press), Sec. 3.2.

Marchand, E. W.

Massey, G. A.

Papoulis, A.

See, for example, A. Papoulis, Systems and Transformations with Applications in Optics (McGraw-Hill, New York, 1968), pp. 81–83.

Pohl, D. W.

U. Dürig, D. W. Pohl, F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59, 3318–3327 (1986).
[CrossRef]

D. W. Pohl, “Scanning near-field optical microscopy,” in Advances in Optical and Electron Microscopy (Academic, New York, 1991), Vol. 12, pp. 242–312.

Rohner, F.

U. Dürig, D. W. Pohl, F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59, 3318–3327 (1986).
[CrossRef]

Schwinger, J.

H. Levine, J. Schwinger, “On the theory of diffraction by an aperture in an infinite plane screen. I,” Phys. Rev. 74, 958–974 (1948), Eq. (2.24).
[CrossRef]

Sherman, G. C.

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

G. C. Sherman, J. J. Stamnes, A. J. Devaney, É. Lalor, “Contribution of the inhomogeneous waves in angular spectrum representations,” Opt. Commun. 8, 271–274 (1973).
[CrossRef]

G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1969).
[CrossRef]

Slepian, D.

D. Slepian, “Some comments on Fourier analysis, uncertainty and modeling,” SIAM Rev. 25, 379–393 (1983).
[CrossRef]

Stamnes, J. J.

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

G. C. Sherman, J. J. Stamnes, A. J. Devaney, É. Lalor, “Contribution of the inhomogeneous waves in angular spectrum representations,” Opt. Commun. 8, 271–274 (1973).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), p. 43.

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 135–137.

Toraldo di Francia, G.

Trautman, J. K.

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. C. Finn, M. H. Kryder, C.-H. Chang, “Near-field magneto-optics and high-density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Wolf, E.

E. Wolf, T. Habashy, “Reconstruction of scattering potentials from incomplete data,” J. Mod. Opt. 41, 1679–1685 (1994).
[CrossRef]

M. W. Kowarz, E. Wolf, “Conservation laws for partially coherent free fields,” J. Opt. Soc. Am. A 10, 88–94 (1993).
[CrossRef]

E. Wolf, E. W. Marchand, “Comparison of the Kirchhoff and the Rayleigh–Sommerfeld theories of diffraction at an aperture,” J. Opt. Soc. Am. 54, 587–594 (1964).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, in press), Sec. 3.2.

See, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 560–561, or D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1984), Vol. XXI, pp. 1–67.
[CrossRef]

Wolfe, R.

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. C. Finn, M. H. Kryder, C.-H. Chang, “Near-field magneto-optics and high-density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

Am. J. Phys. (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

E. Betzig, J. K. Trautman, R. Wolfe, E. M. Gyorgy, P. C. Finn, M. H. Kryder, C.-H. Chang, “Near-field magneto-optics and high-density data storage,” Appl. Phys. Lett. 61, 142–144 (1992).
[CrossRef]

IEEE Trans. Antennas Propagat. (2)

For the case of a circular aperture a comparison of the approximate theories with the exact results may be found in C. J. Bouwkamp, “Theoretical and numerical treatment of diffraction through a circular aperture,” IEEE Trans. Antennas Propagat. AP-18, 152–176 (1970), and a comparison with experimental results may be found in E. W. Marchand, E. Wolf, “Consistent formulation of Kirchhoff’s diffraction theory,” J. Opt. Soc. Am. 56, 1712–1722 (1966). In Bouwkamp’s paper the two Rayleigh–Sommerfeld theories are referred to as improved Kirchhoff theories.
[CrossRef]

The electromagnetic analog of these quantities is discussed in G. V. Borgiotti, “Radiation and reactive energy of aperture antennas,” IEEE Trans. Antennas Propagat. AP-11, 94–95 (1963).
[CrossRef]

J. Appl. Phys. (1)

U. Dürig, D. W. Pohl, F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59, 3318–3327 (1986).
[CrossRef]

J. Math. Phys. (1)

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

J. Mod. Opt. (4)

D. C. Bertilone, “The contributions of homogeneous and evanescent plane waves to the scalar optical field: exact diffraction formulae,” J. Mod. Opt. 38, 865–875 (1991).
[CrossRef]

D. C. Bertilone, “Wave theory for a converging spherical incident wave in an infinite-aperture system,” J. Mod. Opt. 38, 1531–1536 (1991).
[CrossRef]

E. Wolf, T. Habashy, “Reconstruction of scattering potentials from incomplete data,” J. Mod. Opt. 41, 1679–1685 (1994).
[CrossRef]

M. W. Kowarz, “Conservation laws for free electromagnetic fields,” J. Mod. Opt. 42, 109–115 (1995).
[CrossRef]

J. Opt. Soc. Am. (4)

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

Opt. Commun. (3)

M. W. Kowarz, “Energy constraints in optimum apodization problems,” Opt. Commun. 110, 274–278 (1994).
[CrossRef]

W. H. Carter, “Band-limited angular-spectrum approximation to a scalar dipole field,” Opt. Commun. 2, 142–148 (1970).
[CrossRef]

G. C. Sherman, J. J. Stamnes, A. J. Devaney, É. Lalor, “Contribution of the inhomogeneous waves in angular spectrum representations,” Opt. Commun. 8, 271–274 (1973).
[CrossRef]

Optik (1)

A. W. Lohmann, “Three-dimensional properties of wave fields,” Optik 51, 105–117 (1978).

Phys. Rev. (1)

H. Levine, J. Schwinger, “On the theory of diffraction by an aperture in an infinite plane screen. I,” Phys. Rev. 74, 958–974 (1948), Eq. (2.24).
[CrossRef]

Science (1)

E. Betzig, J. K. Trautman, “Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit,” Science 257, 189–195 (1992).
[CrossRef] [PubMed]

SIAM Rev. (1)

D. Slepian, “Some comments on Fourier analysis, uncertainty and modeling,” SIAM Rev. 25, 379–393 (1983).
[CrossRef]

Other (10)

B. R. Frieden, “Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1971), Vol. IX, pp. 311–407.
[CrossRef]

See, for example, A. Papoulis, Systems and Transformations with Applications in Optics (McGraw-Hill, New York, 1968), pp. 81–83.

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986), p. 43.

See, for example, M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), pp. 560–561, or D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, E. Wolf, ed. (North-Holland, New York, 1984), Vol. XXI, pp. 1–67.
[CrossRef]

J. W Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–54.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, in press), Sec. 3.2.

D. W. Pohl, “Scanning near-field optical microscopy,” in Advances in Optical and Electron Microscopy (Academic, New York, 1991), Vol. 12, pp. 242–312.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 241–243.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), pp. 135–137.

The decay of the total evanescent intensity Itot(i)(z) with z is much more rapid for the cases d = 1λ and d = 5λ, shown in Fig. 8. We may explain this behavior by performing an asymptotic expansion of Eq. (4.4b) for large kz. One then finds that, when the slit width is an integer number of wavelengths, the first term in the asymptotic expansion vanishes and a higher-order term, which decays more rapidly with z, becomes important.

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Figures (8)

Fig. 1
Fig. 1

Diagrams of (a) a conventional optical microscope and (b) a collection-mode near-field scanning optical microscope.

Fig. 2
Fig. 2

Slit of width d in an opaque screen, illuminated by a normally incident plane wave.

Fig. 3
Fig. 3

Near-field diffraction patterns showing |U(x, z)|, |Uh(x, z)|, and Ui(x, z) for a slit of width d = 0.2λ with the choice K = λ/d = 5.

Fig. 4
Fig. 4

Same as Fig. 3 but for width d = 1λ and K = λ/d = 1.

Fig. 5
Fig. 5

Same as Fig. 3 but for width d = 5λ and K = λ/d = 0.2.

Fig. 6
Fig. 6

Total homogeneous intensity I tot ( h ) (which is independent of z) and total evanescent intensity I tot ( i ) ( 0 ) in the plane z = 0 as functions of the slit width d. These curves were computed from Eqs. (4.5) and (4.8).

Fig. 7
Fig. 7

Total intensity I tot ( z ) = I tot ( h ) + I tot ( i ) ( z ) computed from Eqs. (4.4b) and (4.5) as a function of the distance z for various slit widths.

Fig. 8
Fig. 8

Total evanescent intensity I tot ( i ) ( z ) computed from Eq. (4.4b) as a function of the distance z for various slit widths.

Equations (54)

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( 2 + k 2 ) U ( x , z ) = 0 ,
U ( x , z ) = U h ( x , z ) + U i ( x , z ) .
U h ( x , z ) = | u x | 1 a ( u x ) exp ( i k u x x ) exp [ i k z ( 1 u x 2 ) 1 / 2 ] d u x ,
U i ( x , z ) = | u x | > 1 a ( u x ) exp ( i k u x x ) exp [ k z ( u x 2 1 ) 1 / 2 ] d u x .
a ( u x ) = k 2 π U ( x , 0 ) exp ( i k u x x ) d x .
U h ( x , z ) = H h ( x x , z ) U ( x , 0 ) d x ,
U i ( x , z ) = H i ( x x , z ) U ( x , 0 ) d x ,
H h ( x , z ) k π 0 1 cos ( k u x x ) exp [ i k z ( 1 u x 2 ) 1 / 2 ] d u x
H i ( x , z ) k π 1 cos ( k u x x ) exp [ k z ( u x 2 1 ) 1 / 2 ] d u x .
H ( x , z ) H h ( x , z ) + H i ( x , z ) = i k z 2 ( x 2 + z 2 ) 1 / 2 H 1 ( 1 ) [ k ( x 2 + z 2 ) 1 / 2 ] ,
H h ( x , 0 ) = 1 π sin ( k x ) x ,
H i ( x , 0 ) = δ ( x ) 1 π sin ( k x ) x ,
U h ( x , 0 ) = 1 π sin [ k ( x x ) ] ( x x ) U ( x , 0 ) d x ,
U i ( x , 0 ) = U ( x , 0 ) 1 π sin [ k ( x x ) ] ( x x ) U ( x , 0 ) d x .
U h ( x , z ) = 1 π sin [ k ( x x ) ] ( x x ) U ( x , z ) d x ,
U i ( x , z ) = U ( x , z ) 1 π sin [ k ( x x ) ] ( x x ) U ( x , z ) d x .
I ( x , z ) = I h ( x , z ) + I i ( x , z ) + I h i ( x , z ) ,
I h ( x , z ) | U h ( x , z ) | 2 ,
I i ( x , z ) | U i ( x , z ) | 2 ,
I h i ( x , z ) U h * ( x , z ) U i ( x , z ) + U h ( x , z ) U i * ( x , z )
I tot ( z ) I ( x , z ) d x ,
I tot ( h ) ( z ) I h ( x , z ) d x ,
I tot ( i ) ( z ) I i ( x , z ) d x ,
I tot ( z ) = I tot ( h ) ( z ) + I tot ( i ) ( z ) ,
I tot ( h ) ( z ) = 2 π k | u x | 1 | a ( u x ) | 2 d u x ,
I tot ( i ) ( z ) = 2 π k | u x | > 1 | a ( u x ) | 2 exp [ 2 k z ( u x 2 1 ) 1 / 2 ] d u x .
I tot ( h ) = 1 π d x d x sin [ k ( x x ) ] ( x x ) × U * ( x , 0 ) U ( x , 0 ) ,
I tot ( i ) ( z ) = d x d x H i ( x x , 2 z ) U * ( x , 0 ) U ( x , 0 ) ,
I tot ( h ) = 1 π d x d x sin [ k ( x x ) ] ( x x ) × U * ( x , z ) U ( x , z ) ,
I tot ( i ) ( z ) = | U ( x , z ) | 2 d x 1 π d x d x × sin [ k ( x x ) ] ( x x ) U * ( x , z ) U ( x , z ) .
I tot ( z ) I tot ( 0 ) , I tot ( h ) < I tot ( 0 ) , I tot ( i ) ( z ) < I tot ( 0 ) .
F tot ( h ) = 4 πωα | u x | 1 ( 1 u x 2 ) 1 / 2 | a ( u x ) | 2 d u x ,
F tot ( i ) ( z ) = 4 πωα | u x | > 1 ( u x 2 1 ) 1 / 2 | a ( u x ) | 2 × exp [ 2 k z ( u x 2 1 ) 1 / 2 ] d u x ,
U ( x , 0 ) = { U inc ( x , 0 ) for d / 2 x d / 2 0 otherwise ,
a ( u x ) = K π sin ( u x k d / 2 ) u x .
U h ( x , z ) = K π | u x | 1 sin ( u x k d / 2 ) u x exp ( i k u x x ) × exp [ i k z ( 1 u x 2 ) 1 / 2 ] d u x ,
U i ( x , z ) = K π | u x | > 1 sin ( u x k d / 2 ) u x exp ( i k u x x ) × exp [ k z ( u x 2 1 ) 1 / 2 ] d u x ,
I tot ( h ) = 4 I tot ( 0 ) π k d 0 1 sin 2 ( u x k d / 2 ) u x 2 d u x ,
I tot ( i ) ( z ) = 4 I tot ( 0 ) π k d 1 sin 2 ( u x k d / 2 ) u x 2 × exp [ 2 k z ( u x 2 1 ) 1 / 2 ] d u x .
I tot ( h ) = I tot ( 0 ) [ 2 π Si ( k d ) 4 π k d sin 2 ( k d / 2 ) ] ,
Si ( β ) 0 β sin t t d t .
U h ( x , 0 ) = K π { Si [ k ( x + d / 2 ) ] Si [ k ( x d / 2 ) ] } ,
U i ( x , 0 ) = { K π { π Si [ k ( x + d / 2 ) ] + Si [ k ( x d / 2 ) ] } for d / 2 x d / 2 K π { Si [ k ( x + d / 2 ) ] Si [ k ( x d / 2 ) ] } otherwise ,
I tot ( i ) ( 0 ) = I tot ( 0 ) [ 1 2 π Si ( k d ) + 4 π k d sin 2 ( k d / 2 ) ] .
F ( x , z ) 2 i ωα U ( x , z ) U * ( x , z ) ,
· F ( x , z ) + 2 i ω E ( x , z ) = 0 ,
E ( x , z ) α [ k 2 | U ( x , z ) | 2 | U ( x , z ) | 2 ] .
S F ( x , z ) · n ˆ d S + 2 i ω V E ( x , z ) d V = 0 ,
S Re [ F ( x , z ) · n ˆ ] d S = 0 ,
S Im [ F ( x , z ) · n ˆ ] d S + 2 ω V E ( x , z ) d V = 0 .
F tot ( h ) ( z ) Re [ F ( x , z ) · z ˆ ] d x ,
F tot ( i ) ( z ) Im [ F ( x , z ) · z ˆ ] d x ,
F tot ( h ) = 4 πωα | u x | 1 ( 1 u x 2 ) 1 / 2 | a ( u x ) | 2 d u x ,
F tot ( i ) ( z ) = 4 πωα | u x | > 1 ( u x 2 1 ) 1 / 2 | a ( u x ) | 2 × exp [ 2 k z ( u x 2 1 ) 1 / 2 ] d u x ,

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