Abstract

We make a theoretical analysis of the interaction of the field transmitted by a subwavelength tip and a two-dimensional subwavelength lattice. Such a model provides a new insight into the resolution achievable by near-field microscopy and confirms the experimental results obtained recently. In the present model the probe, characterized by its electric dipolar susceptibility, is assumed to be locally spherical, and the representation of the sample is based on a discrete description of the matter. This permits separation of the electric field detected by the probe after reflection into two different parts that describe both the continuum character and the corrugation of the surface. Numerical results performed on a two-dimensional lattice are similar to those obtained by atomic force microscopy and exhibit specific behavior such as a strong dependence on the polarization of the incident field.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968) pp. 49–53.
  2. E. Wolf and M. Nieto-Vesperinas, “Analyticity of the angular spectrum amplitude of scattered fields and some of its consequences,” J. Opt. Soc. Am. 2, 886 (1985).
    [CrossRef]
  3. D. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
    [CrossRef]
  4. E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, and E. Kratschmer, “Near-field scanning optical microscopy (NSOM) development and biological applications,” Biophys. J. 49, 269 (1986).
  5. R. Reddick, R. Warmack, and T. Ferrell, “New form of scanning optical microscopy,” Phys. Rev. B 39, 767 (1989).
    [CrossRef]
  6. D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23 (1989).
    [CrossRef]
  7. E. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature (London) 237, 510 (1972).
    [CrossRef]
  8. G. Massey, “Microscopy and pattern generation with scanned evanescent waves,” Appl. Opt. 23, 658 (1984).
    [CrossRef] [PubMed]
  9. M. Spajer, D. Courjon, K. Sarayeddine, A. Jalocha, and J. M. Vigoureux, “Microscopie en champ proche par réflexion,” Rev. Phys. Appl. (to be published).
  10. D. Courjon, Scanning Near-Field Optical Microscopy (IBM Europe Institute, Garmisch-Partenkirchen, Federal Republic of Germany, 1989).
  11. H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163 (1944).
    [CrossRef]
  12. C. J. Bouwkamp, “On the diffraction of electromagnetic waves by small circular disks and holes,” Philips Res. Rep. 5, 401 (1950).
  13. E. W. Marchand and E. Wolf, “Diffraction at small apertures in black screens,” J. Opt. Soc. Am. 59, 79 (1969).
    [CrossRef]
  14. W. A. Steele, in The Interaction of Gases with Solid Surfaces, D. H. Everett, ed. (Pergamon, Oxford, 1974), p. 24.
  15. H. Hoinkes, Rev. Mod. Phys. 52, 933 (1980).
    [CrossRef]
  16. V. Celli, D. Eichenauer, A. Kaufhold, and J. P. Toennies, J. Chem. Phys. 83, 2504 (1985).
    [CrossRef]
  17. C. Girard and C. Girardet, J. Chem. Phys. 86, 6531 (1987).
    [CrossRef]
  18. C. Girard and C. Girardet, Phys. Rev. B 38, 1987 (1987).
  19. C. Girard and F. Hache, Chem. Phys. 118, 249 (1987).
    [CrossRef]
  20. F. Hache, D. Ricard, and C. Girard, Phys. Rev. B 38, p. 7990.
  21. C. Girard, S. Maghezzi, and F. Hache, J. Chem. Phys. 91, 5509 (1989).
    [CrossRef]
  22. C. J. F. Böttcher, in Theory of Electric Polarization, O. C. Van Belle, P. Bordewijk, and A. Rip, eds. (Elsevier, Amsterdam, 1973), Vol. 1, p. 55.
  23. M. Abramowitz and I. A. Stegun, in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (Dover, New York, 1968), p. 480.
  24. G. Binning, C. F. Quate, and Ch. Gerber, Phys. Rev. Lett. 56, 930 (1986).
    [CrossRef]
  25. W. Ekardt, Phys. Rev. B 29, 1558 (1984); Surf. Sci. 152, 180 (1985).
    [CrossRef]
  26. F. Hache, D. Ricard, and C. Flytzannis, J. Opt. Soc. Am. B 3, 1647 (1986).
    [CrossRef]
  27. D. Van Labeke, B. Labani, and C. Girard, Chem. Phys. Lett. 162, 399 (1989).
    [CrossRef]
  28. D. Van Labeke, S. Maghezzi, C. Girard, and J. M. Vigoureux, presented at the Institute of Physics Electron Microscopy and Analysis Group and Royal Microscopical Society Conference, London, 1989.
  29. C. Girard, D. Van Labeke, and J. M. Vigoureux, Phys. Rev. B 40, 2133 (1989).
    [CrossRef]

1989 (5)

R. Reddick, R. Warmack, and T. Ferrell, “New form of scanning optical microscopy,” Phys. Rev. B 39, 767 (1989).
[CrossRef]

D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23 (1989).
[CrossRef]

C. Girard, S. Maghezzi, and F. Hache, J. Chem. Phys. 91, 5509 (1989).
[CrossRef]

D. Van Labeke, B. Labani, and C. Girard, Chem. Phys. Lett. 162, 399 (1989).
[CrossRef]

C. Girard, D. Van Labeke, and J. M. Vigoureux, Phys. Rev. B 40, 2133 (1989).
[CrossRef]

1987 (3)

C. Girard and C. Girardet, J. Chem. Phys. 86, 6531 (1987).
[CrossRef]

C. Girard and C. Girardet, Phys. Rev. B 38, 1987 (1987).

C. Girard and F. Hache, Chem. Phys. 118, 249 (1987).
[CrossRef]

1986 (3)

G. Binning, C. F. Quate, and Ch. Gerber, Phys. Rev. Lett. 56, 930 (1986).
[CrossRef]

F. Hache, D. Ricard, and C. Flytzannis, J. Opt. Soc. Am. B 3, 1647 (1986).
[CrossRef]

E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, and E. Kratschmer, “Near-field scanning optical microscopy (NSOM) development and biological applications,” Biophys. J. 49, 269 (1986).

1985 (2)

E. Wolf and M. Nieto-Vesperinas, “Analyticity of the angular spectrum amplitude of scattered fields and some of its consequences,” J. Opt. Soc. Am. 2, 886 (1985).
[CrossRef]

V. Celli, D. Eichenauer, A. Kaufhold, and J. P. Toennies, J. Chem. Phys. 83, 2504 (1985).
[CrossRef]

1984 (3)

W. Ekardt, Phys. Rev. B 29, 1558 (1984); Surf. Sci. 152, 180 (1985).
[CrossRef]

D. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

G. Massey, “Microscopy and pattern generation with scanned evanescent waves,” Appl. Opt. 23, 658 (1984).
[CrossRef] [PubMed]

1980 (1)

H. Hoinkes, Rev. Mod. Phys. 52, 933 (1980).
[CrossRef]

1972 (1)

E. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature (London) 237, 510 (1972).
[CrossRef]

1969 (1)

1950 (1)

C. J. Bouwkamp, “On the diffraction of electromagnetic waves by small circular disks and holes,” Philips Res. Rep. 5, 401 (1950).

1944 (1)

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163 (1944).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (Dover, New York, 1968), p. 480.

Ash, E.

E. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature (London) 237, 510 (1972).
[CrossRef]

Bethe, H. A.

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163 (1944).
[CrossRef]

Betzig, E.

E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, and E. Kratschmer, “Near-field scanning optical microscopy (NSOM) development and biological applications,” Biophys. J. 49, 269 (1986).

Binning, G.

G. Binning, C. F. Quate, and Ch. Gerber, Phys. Rev. Lett. 56, 930 (1986).
[CrossRef]

Böttcher, C. J. F.

C. J. F. Böttcher, in Theory of Electric Polarization, O. C. Van Belle, P. Bordewijk, and A. Rip, eds. (Elsevier, Amsterdam, 1973), Vol. 1, p. 55.

Bouwkamp, C. J.

C. J. Bouwkamp, “On the diffraction of electromagnetic waves by small circular disks and holes,” Philips Res. Rep. 5, 401 (1950).

Celli, V.

V. Celli, D. Eichenauer, A. Kaufhold, and J. P. Toennies, J. Chem. Phys. 83, 2504 (1985).
[CrossRef]

Courjon, D.

D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23 (1989).
[CrossRef]

M. Spajer, D. Courjon, K. Sarayeddine, A. Jalocha, and J. M. Vigoureux, “Microscopie en champ proche par réflexion,” Rev. Phys. Appl. (to be published).

D. Courjon, Scanning Near-Field Optical Microscopy (IBM Europe Institute, Garmisch-Partenkirchen, Federal Republic of Germany, 1989).

Denk, W.

D. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

Eichenauer, D.

V. Celli, D. Eichenauer, A. Kaufhold, and J. P. Toennies, J. Chem. Phys. 83, 2504 (1985).
[CrossRef]

Ekardt, W.

W. Ekardt, Phys. Rev. B 29, 1558 (1984); Surf. Sci. 152, 180 (1985).
[CrossRef]

Ferrell, T.

R. Reddick, R. Warmack, and T. Ferrell, “New form of scanning optical microscopy,” Phys. Rev. B 39, 767 (1989).
[CrossRef]

Flytzannis, C.

Gerber, Ch.

G. Binning, C. F. Quate, and Ch. Gerber, Phys. Rev. Lett. 56, 930 (1986).
[CrossRef]

Girard, C.

D. Van Labeke, B. Labani, and C. Girard, Chem. Phys. Lett. 162, 399 (1989).
[CrossRef]

C. Girard, D. Van Labeke, and J. M. Vigoureux, Phys. Rev. B 40, 2133 (1989).
[CrossRef]

C. Girard, S. Maghezzi, and F. Hache, J. Chem. Phys. 91, 5509 (1989).
[CrossRef]

C. Girard and C. Girardet, Phys. Rev. B 38, 1987 (1987).

C. Girard and C. Girardet, J. Chem. Phys. 86, 6531 (1987).
[CrossRef]

C. Girard and F. Hache, Chem. Phys. 118, 249 (1987).
[CrossRef]

F. Hache, D. Ricard, and C. Girard, Phys. Rev. B 38, p. 7990.

D. Van Labeke, S. Maghezzi, C. Girard, and J. M. Vigoureux, presented at the Institute of Physics Electron Microscopy and Analysis Group and Royal Microscopical Society Conference, London, 1989.

Girardet, C.

C. Girard and C. Girardet, Phys. Rev. B 38, 1987 (1987).

C. Girard and C. Girardet, J. Chem. Phys. 86, 6531 (1987).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968) pp. 49–53.

Hache, F.

C. Girard, S. Maghezzi, and F. Hache, J. Chem. Phys. 91, 5509 (1989).
[CrossRef]

C. Girard and F. Hache, Chem. Phys. 118, 249 (1987).
[CrossRef]

F. Hache, D. Ricard, and C. Flytzannis, J. Opt. Soc. Am. B 3, 1647 (1986).
[CrossRef]

F. Hache, D. Ricard, and C. Girard, Phys. Rev. B 38, p. 7990.

Harootunian, A.

E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, and E. Kratschmer, “Near-field scanning optical microscopy (NSOM) development and biological applications,” Biophys. J. 49, 269 (1986).

Hoinkes, H.

H. Hoinkes, Rev. Mod. Phys. 52, 933 (1980).
[CrossRef]

Isaacson, M.

E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, and E. Kratschmer, “Near-field scanning optical microscopy (NSOM) development and biological applications,” Biophys. J. 49, 269 (1986).

Jalocha, A.

M. Spajer, D. Courjon, K. Sarayeddine, A. Jalocha, and J. M. Vigoureux, “Microscopie en champ proche par réflexion,” Rev. Phys. Appl. (to be published).

Kaufhold, A.

V. Celli, D. Eichenauer, A. Kaufhold, and J. P. Toennies, J. Chem. Phys. 83, 2504 (1985).
[CrossRef]

Kratschmer, E.

E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, and E. Kratschmer, “Near-field scanning optical microscopy (NSOM) development and biological applications,” Biophys. J. 49, 269 (1986).

Labani, B.

D. Van Labeke, B. Labani, and C. Girard, Chem. Phys. Lett. 162, 399 (1989).
[CrossRef]

Lanz, M.

D. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

Lewis, A.

E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, and E. Kratschmer, “Near-field scanning optical microscopy (NSOM) development and biological applications,” Biophys. J. 49, 269 (1986).

Maghezzi, S.

C. Girard, S. Maghezzi, and F. Hache, J. Chem. Phys. 91, 5509 (1989).
[CrossRef]

D. Van Labeke, S. Maghezzi, C. Girard, and J. M. Vigoureux, presented at the Institute of Physics Electron Microscopy and Analysis Group and Royal Microscopical Society Conference, London, 1989.

Marchand, E. W.

Massey, G.

Nicholls, G.

E. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature (London) 237, 510 (1972).
[CrossRef]

Nieto-Vesperinas, M.

E. Wolf and M. Nieto-Vesperinas, “Analyticity of the angular spectrum amplitude of scattered fields and some of its consequences,” J. Opt. Soc. Am. 2, 886 (1985).
[CrossRef]

Pohl, D.

D. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

Quate, C. F.

G. Binning, C. F. Quate, and Ch. Gerber, Phys. Rev. Lett. 56, 930 (1986).
[CrossRef]

Reddick, R.

R. Reddick, R. Warmack, and T. Ferrell, “New form of scanning optical microscopy,” Phys. Rev. B 39, 767 (1989).
[CrossRef]

Ricard, D.

F. Hache, D. Ricard, and C. Flytzannis, J. Opt. Soc. Am. B 3, 1647 (1986).
[CrossRef]

F. Hache, D. Ricard, and C. Girard, Phys. Rev. B 38, p. 7990.

Sarayeddine, K.

D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23 (1989).
[CrossRef]

M. Spajer, D. Courjon, K. Sarayeddine, A. Jalocha, and J. M. Vigoureux, “Microscopie en champ proche par réflexion,” Rev. Phys. Appl. (to be published).

Spajer, M.

D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23 (1989).
[CrossRef]

M. Spajer, D. Courjon, K. Sarayeddine, A. Jalocha, and J. M. Vigoureux, “Microscopie en champ proche par réflexion,” Rev. Phys. Appl. (to be published).

Steele, W. A.

W. A. Steele, in The Interaction of Gases with Solid Surfaces, D. H. Everett, ed. (Pergamon, Oxford, 1974), p. 24.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (Dover, New York, 1968), p. 480.

Toennies, J. P.

V. Celli, D. Eichenauer, A. Kaufhold, and J. P. Toennies, J. Chem. Phys. 83, 2504 (1985).
[CrossRef]

Van Labeke, D.

D. Van Labeke, B. Labani, and C. Girard, Chem. Phys. Lett. 162, 399 (1989).
[CrossRef]

C. Girard, D. Van Labeke, and J. M. Vigoureux, Phys. Rev. B 40, 2133 (1989).
[CrossRef]

D. Van Labeke, S. Maghezzi, C. Girard, and J. M. Vigoureux, presented at the Institute of Physics Electron Microscopy and Analysis Group and Royal Microscopical Society Conference, London, 1989.

Vigoureux, J. M.

C. Girard, D. Van Labeke, and J. M. Vigoureux, Phys. Rev. B 40, 2133 (1989).
[CrossRef]

D. Van Labeke, S. Maghezzi, C. Girard, and J. M. Vigoureux, presented at the Institute of Physics Electron Microscopy and Analysis Group and Royal Microscopical Society Conference, London, 1989.

M. Spajer, D. Courjon, K. Sarayeddine, A. Jalocha, and J. M. Vigoureux, “Microscopie en champ proche par réflexion,” Rev. Phys. Appl. (to be published).

Warmack, R.

R. Reddick, R. Warmack, and T. Ferrell, “New form of scanning optical microscopy,” Phys. Rev. B 39, 767 (1989).
[CrossRef]

Wolf, E.

E. Wolf and M. Nieto-Vesperinas, “Analyticity of the angular spectrum amplitude of scattered fields and some of its consequences,” J. Opt. Soc. Am. 2, 886 (1985).
[CrossRef]

E. W. Marchand and E. Wolf, “Diffraction at small apertures in black screens,” J. Opt. Soc. Am. 59, 79 (1969).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

D. Pohl, W. Denk, and M. Lanz, “Optical stethoscopy: image recording with resolution λ/20,” Appl. Phys. Lett. 44, 651 (1984).
[CrossRef]

Biophys. J. (1)

E. Betzig, A. Lewis, A. Harootunian, M. Isaacson, and E. Kratschmer, “Near-field scanning optical microscopy (NSOM) development and biological applications,” Biophys. J. 49, 269 (1986).

Chem. Phys. (1)

C. Girard and F. Hache, Chem. Phys. 118, 249 (1987).
[CrossRef]

Chem. Phys. Lett. (1)

D. Van Labeke, B. Labani, and C. Girard, Chem. Phys. Lett. 162, 399 (1989).
[CrossRef]

J. Chem. Phys. (3)

C. Girard, S. Maghezzi, and F. Hache, J. Chem. Phys. 91, 5509 (1989).
[CrossRef]

V. Celli, D. Eichenauer, A. Kaufhold, and J. P. Toennies, J. Chem. Phys. 83, 2504 (1985).
[CrossRef]

C. Girard and C. Girardet, J. Chem. Phys. 86, 6531 (1987).
[CrossRef]

J. Opt. Soc. Am. (2)

E. W. Marchand and E. Wolf, “Diffraction at small apertures in black screens,” J. Opt. Soc. Am. 59, 79 (1969).
[CrossRef]

E. Wolf and M. Nieto-Vesperinas, “Analyticity of the angular spectrum amplitude of scattered fields and some of its consequences,” J. Opt. Soc. Am. 2, 886 (1985).
[CrossRef]

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

Nature (London) (1)

E. Ash and G. Nicholls, “Super-resolution aperture scanning microscope,” Nature (London) 237, 510 (1972).
[CrossRef]

Opt. Commun. (1)

D. Courjon, K. Sarayeddine, and M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23 (1989).
[CrossRef]

Philips Res. Rep. (1)

C. J. Bouwkamp, “On the diffraction of electromagnetic waves by small circular disks and holes,” Philips Res. Rep. 5, 401 (1950).

Phys. Rev. (1)

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163 (1944).
[CrossRef]

Phys. Rev. B (5)

C. Girard and C. Girardet, Phys. Rev. B 38, 1987 (1987).

F. Hache, D. Ricard, and C. Girard, Phys. Rev. B 38, p. 7990.

R. Reddick, R. Warmack, and T. Ferrell, “New form of scanning optical microscopy,” Phys. Rev. B 39, 767 (1989).
[CrossRef]

W. Ekardt, Phys. Rev. B 29, 1558 (1984); Surf. Sci. 152, 180 (1985).
[CrossRef]

C. Girard, D. Van Labeke, and J. M. Vigoureux, Phys. Rev. B 40, 2133 (1989).
[CrossRef]

Phys. Rev. Lett. (1)

G. Binning, C. F. Quate, and Ch. Gerber, Phys. Rev. Lett. 56, 930 (1986).
[CrossRef]

Rev. Mod. Phys. (1)

H. Hoinkes, Rev. Mod. Phys. 52, 933 (1980).
[CrossRef]

Other (7)

C. J. F. Böttcher, in Theory of Electric Polarization, O. C. Van Belle, P. Bordewijk, and A. Rip, eds. (Elsevier, Amsterdam, 1973), Vol. 1, p. 55.

M. Abramowitz and I. A. Stegun, in Handbook of Mathematical Functions, M. Abramowitz and I. A. Stegun, eds. (Dover, New York, 1968), p. 480.

D. Van Labeke, S. Maghezzi, C. Girard, and J. M. Vigoureux, presented at the Institute of Physics Electron Microscopy and Analysis Group and Royal Microscopical Society Conference, London, 1989.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968) pp. 49–53.

M. Spajer, D. Courjon, K. Sarayeddine, A. Jalocha, and J. M. Vigoureux, “Microscopie en champ proche par réflexion,” Rev. Phys. Appl. (to be published).

D. Courjon, Scanning Near-Field Optical Microscopy (IBM Europe Institute, Garmisch-Partenkirchen, Federal Republic of Germany, 1989).

W. A. Steele, in The Interaction of Gases with Solid Surfaces, D. H. Everett, ed. (Pergamon, Oxford, 1974), p. 24.

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 (5)

Fig. 1
Fig. 1

Geometry of the tip and of the nanometric lattice. R = (0, 0, R) characterizes the approach distance of the spherical probe with respect to the object; ls describes the scanning in the plane (X, Y); (η1, η2) are the vectors of the primitive cell associated to the periodic lattice.

Fig. 2
Fig. 2

Representation of the intensity Iob(X, Y) detected by a spherical probe of radius a = 1.8 nm scanning several cells of a lattice formed of small spheres of radius b = 1 nm. |η1| = |η2| = 2.4 nm, η1η2 = 0, R0 = Ra = 1.5 nm. (a) θ = π/2. The excitation field E0(ω) is polarized along the OY axis. (b) θ = π/4. E 0 ( ω ) = ( 1 2 , 1 2 , 0 ) E 0 ( ω ). (c) θ = 0. The excitation field E0(ω) is polarized along the OX axis.

Fig. 3
Fig. 3

Ratio between corrugation and continuum parts of the detected intensity versus the approach distance R0.

Fig. 4
Fig. 4

Corrugation intensity ΔI = Iob(max) − Iob(min) detected by a spherical probe as a function of its radius a for different values of the approach distance R0 = Ra.

Fig. 5
Fig. 5

Corrugation intensity ΔI = Iob(max) − Iob(min) experienced by a spherical probe as a function of the approach distance R0 = Ra. (a) The radius of the probe is smaller than the period of the lattice (|η1| ≥ a). (b) The radius of the probe is greater than the period of the lattice (|η1| ≤ a).

Equations (54)

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

T ( r 1 , r 2 ) = [ 3 ( r 1 r 2 ) ( r 1 r 2 ) I | r 1 r 2 | 2 ] | r 1 r 2 | 5 ,
S ( r , r , ω ) = n 1 , n 2 T ( r R n 1 , n 2 ) α b ( ω ) T ( R n 1 , n 2 r ) ,
R n 1 , n 2 = l s ( n 1 η 1 + n 2 η 2 ) ,
α b ( ω ) = b 3 [ 4 π χ b ( ω ) 3 + 4 π χ b ( ω ) ] I ,
S ( r , r , ω ) = b 3 [ 4 π χ b ( ω ) 3 + 4 π χ b ( ω ) ] n 1 , n 2 d r δ ( r R n 1 , n 2 ) × T ( r r ) T ( r r ) ,
n 1 , n 2 δ ( r R n 1 , n 2 ) = 1 A δ ( z ) g exp [ i g ( l + l s ) ] ,
S ( r , r , ω ) = b 3 A [ 4 π χ b ( ω ) 3 + 4 π χ b ( ω ) ] g exp ( i g · l ) s G ( r , r ) ,
G ( r , r , g ) = d l T ( r l ) T ( l r ) exp ( i g l ) .
μ 0 ( ω ) = a 3 [ 4 π χ p ( ω ) 3 + 4 π χ p ( ω ) ] E 0 ( ω ) ,
μ ( ω ) = μ 0 ( ω ) + μ ˜ ( ω ) ,
μ ˜ ( ω ) = f ( ω ) υ χ p ( ω ) E r ( r , ω ) d r ,
f ( ω ) = 3 3 + 4 π χ p ( ω ) .
E r ( r , ω ) = S ( r , R , ω ) μ ( ω ) ,
μ ( ω ) = μ 0 ( ω ) + M ( ω ) μ ( ω ) ,
M ( ω ) = f ( ω ) α b ( ω ) χ p ( ω ) A g exp ( i g l s ) F ( g , R ) ,
F α β ( g , R ) = d l exp ( i g l ) [ υ d r T α γ ( r l ) ] T γ β ( l R ) ,
μ ( ω ) = [ I M ( ω ) ] 1 μ 0 ( ω ) .
[ I M ( ω ) ] 1 = I + M ( ω ) + M ( ω ) M ( ω ) + .
μ ( ω ) = μ 0 ( ω ) + M ( ω ) μ 0 ( ω ) .
I ob | T ( R ob R ) M ( ω ) μ 0 ( ω ) | 2 ,
M ( ω ) = M ¯ ( ω ) + M ˜ ( ω ) ,
M ¯ ( ω ) = f ( ω ) α b ( ω ) χ p ( ω ) A F ( 0 , R )
M ˜ ( ω ) = f ( ω ) α b ( ω ) χ p ( ω ) A g 0 exp ( i g l s ) F ( g , R ) ,
I ob = I ¯ ob + I ˜ ob ,
I ¯ ob = | T ( R ob R ) M ¯ ( ω ) μ 0 ( ω ) | 2
I ˜ ob = 2 [ T ( R ob R ) M ¯ ( ω ) μ 0 ( ω ) ] [ T ( R ob R ) M ˜ ( ω ) μ 0 ( ω ) ] + | T ( R ob R ) M ˜ ( ω ) μ 0 ( ω ) | 2 .
μ ( ω ) = a 3 [ 4 π χ p ( ω ) 3 + 4 π χ p ( ω ) ] E 0 ( ω ) ( cos θ , sin θ , 0 ) ,
I ¯ ob = 144 π 6 a 8 d 6 R 4 A 2 ( 1 + a 2 3 R 2 ) 2 χ p 4 ( ω ) ( 3 + 4 π χ p ( ω ) 4 α b 2 ( ω ) E 0 2 ( ω ) ,
I ˜ ob = I ˜ ob ( 1 ) + I ˜ ob ( 2 ) ,
I ˜ ob ( 1 ) = 144 π 6 a 8 d 6 R 2 A 2 ( 1 + a 2 3 R 2 ) 4 χ p 4 ( ω ) [ 3 + 4 π χ p ( ω ) ] 4 × α b 2 ( ω ) E 0 2 ( ω ) P ( l s , R ) ,
P ( l s , R ) = m 1 > 0 m 2 0 cos g l s [ A ( R , θ ) + B ( R , θ ) + C ( R , θ ) ] ,
A ( R , θ ) = { 3 g x 2 K 0 ( g R ) + g R ( 1 + a 2 g x 2 2 + 3 g x 2 g 2 ) K 1 ( g R ) + [ a 2 g 2 6 R 2 ( 1 + g x 2 g 2 ) ] K 2 ( g R ) } cos 2 θ ,
B ( R , θ ) = { 3 g y 2 K 0 ( g R ) + g R ( 1 + a 2 g y 2 2 + 3 g y 2 g 2 ) K 1 ( g R ) + [ a 2 g 2 6 R 2 ( 1 + g y 2 g 2 ) ] K 2 ( g R ) } sin 2 θ ,
C ( R , θ ) = 2 g x g y [ 3 K 0 ( g R ) + ( 3 g R + a 2 g 6 R ) K 1 ( g R ) + a 2 2 R 2 K 2 ( g R ) ] sin θ cos θ ,
D ( ω ) = χ p 2 ( ω ) α b 2 ( ω ) 3 + 4 π χ p ( ω ) 4
F α β ( g , R ) = d l exp ( i g l ) J α γ ( l ) T γ β ( l R ) ,
J α γ ( l ) = υ d r T α γ ( r l ) .
J ( l ) = 4 π a ( R 2 + l 2 ) 1 / 2 × [ 1 + a 2 3 ( R 2 + l 2 ) 0 0 0 1 + a 2 3 ( R 2 + l 2 ) 0 0 0 1 2 a 2 3 ( R 2 + l 2 ) ] .
T ( l R ) [ 3 x 2 ( R 2 + l 2 ) 3 x y 3 x R 3 x y 3 y 2 ( R 2 + l 2 ) 3 y R 3 x R 3 y R 2 R 2 l 2 ] 1 ( l 2 + R 2 ) 5 / 2 .
F ( g , R ) = 4 π a d l exp ( i g l ) × [ 1 ( R 2 + l 2 ) 2 + a 2 3 ( R 2 + l 2 ) 3 3 x y ( R 2 + l 2 ) 3 a 2 x y ( R 2 + l 2 ) 4 3 x R ( R 2 + l 2 ) 3 + a 2 x R ( R 2 + l 2 ) 4 3 x 2 ( R 2 + l 2 ) 3 a 2 x 2 ( R 2 + l 2 ) 4 1 ( R 2 + l 2 ) 2 + a 2 3 ( R 2 + l 2 ) 3 3 y l ( R 2 + l 2 ) 3 + a 2 y R ( R 2 + l 2 ) 4 3 x R ( R 2 + l 2 ) 3 2 a 2 x R ( R 2 + l 2 ) 4 3 y R ( R 2 + l 2 ) 3 2 a 2 y R ( R 2 + l 2 ) 4 2 R 2 l 2 ( R 2 + l 2 ) 3 + 2 a 2 ( 2 R 2 l 2 ) 3 ( R 2 + l 2 ) 4 ] .
P ( n ) = exp ( i g l ) ( l 2 + R 2 ) n d l = 2 π ( n 1 ) ! ( g 2 R ) n 1 K n 1 ( g R ) ,
F α α = π 2 a { 3 g α 2 K 0 ( g R ) + [ g R ( 1 + a 2 g α 2 6 + 3 g α 2 g 2 ) ] K 1 ( g R ) + [ 1 6 a 2 g 2 R 2 ( 1 + g α 2 g 2 ) ] K 2 ( g R ) } ,
F x y = F y x = π 2 a g x g y × [ 3 K 0 ( g R ) + ( 3 g R + a 2 g 6 R ) K 1 ( g R ) + a 2 6 R 2 K 2 ( g R ) ] ,
F α z = i π 2 a g α g [ 3 K 1 ( g R ) + a 2 g 6 R K 2 ( g r ) ] ,
F z α = i π 2 a g α g [ 3 K 1 ( g R ) a 2 g 3 R K 2 ( g R ) ] ,
F z z = π 2 a [ K 0 ( g R ) g 2 + ( 2 g R + 1 9 a 2 g 3 R 1 ) K 1 ( g R ) ( 2 g 2 + 1 9 a 2 g 2 R 2 ) K 2 ( g R ) + K 3 ( g R ) ( 2 9 a 2 g 3 R ) ] .
F x y ( 0 , R ) = F y x ( 0 , R ) = F y x ( 0 , R ) = F x , z ( 0 , R ) = F z x ( 0 , R ) = F z y ( 0 , R ) = 0
F y y ( 0 , R ) = F x x ( 0 , R ) = 4 π 2 a R 2 ( 1 4 + a 2 12 R 2 ) .
I ˜ ob ( 2 ) = 4 π 4 a 2 d 6 E 0 2 ( ω ) γ 4 ( ω ) m 1 > 0 m 2 0 m 1 > 0 m 2 0 [ β 1 ( g , R ) × β 1 ( g , R ) cos ( g l s ) cos ( g l s ) + β 2 ( g , R ) β 2 ( g , R ) cos ( g l s ) cos ( g l s ) + β 3 ( g , R ) β 3 ( g , R ) sin ( g l s ) sin ( g l s ) ] ,
γ ( ω ) = 3 a 3 4 π A [ 4 π χ p ( ω ) 3 + 4 π χ p ( ω ) ] 2 α b ( ω ) ,
β 1 ( g , R ) = 3 g y 2 K 0 ( g R ) + g / R ( 1 + a 2 g y 2 6 + 3 g y 2 g 2 ) K 1 ( g R ) + 1 6 a 2 g 2 R 2 ( 1 + g y 2 g 2 ) K 2 ( g R ) ,
β 2 ( g , R ) = g x g y [ 3 K 0 ( g R ) + ( 3 R g + a 2 g 6 R ) K 1 ( g R ) + a 2 R 2 K 2 ( g R ) ] ,
β 3 ( g , R ) = 6 g y g [ K 1 ( g R ) + 1 9 a 2 g R K 2 ( g R ) ] .
g = m 1 g 1 + m 2 g 2 , g = m 1 g 1 + m 2 g 2 .

Metrics