Abstract

We present a Green’s-function/Green’s-theorem integral equation approach to numerically modeling two-dimensional, s-polarized, wave propagation problems effectively for a variety of geometries. The model accurately calculates both near fields and far fields because of the minimal assumptions made on the behavior of the scattered radiation. The method was applied to modeling light emission from a near-field scanning optical microscope fiber tip. Several convergence and energy tests were used to give confidence in the results. The behavior of intensity and power near the tip was investigated. The effects of changing the dielectric constant of a sample material located below the tip were also examined.

© 2001 Optical Society of America

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References

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  1. U. Durig, D. W. Pohl, F. Rohner, “Near-field optical-scanning microscopy,” J. Appl. Phys. 59, 3318–3327 (1986).
    [CrossRef]
  2. E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1468–1470 (1991).
    [CrossRef] [PubMed]
  3. G. A. Valaskovic, M. Holton, G. H. Morrison, “Parameter control, characterization, and optimization in the fabrication of optical fiber near-field probes,” Appl. Opt. 34, 1215–1228 (1995).
    [CrossRef] [PubMed]
  4. K. Fukuzawa, H. Kuwano, “Conversion of evanescent into propagating light in near-field scanning optical microscopy,” J. Appl. Phys. 79, 8174–8178 (1996).
    [CrossRef]
  5. E. Wolf, J. T. Foley, “Do evanescent waves contribute to the far field?” Opt. Lett. 23, 16–18 (1998).
    [CrossRef]
  6. L. Novotny, B. Hecht, D. W. Pohl, “Implications of high resolution to near-field optical microscopy,” Ultramicroscopy 71, 341–343 (1998).
    [CrossRef]
  7. H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
    [CrossRef]
  8. C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Philips Res. Rep. 5, 321–332 (1950).
  9. G. W. Bryant, “Probing quantum nanostructures with near-field optical microscopy and vice versa,” Appl. Phys. Lett. 72, 768–770 (1998).
    [CrossRef]
  10. G. W. Bryant, E. L. Shirley, L. S. Goldner, E. B. McDaniel, J. W. P. Hsu, R. J. Tonucci, “Theory of probing photonic crystal with transmission near-field optical microscopy,” Phys. Rev. B 58, 2131–2141 (1998).
    [CrossRef]
  11. O. J. F. Martin, C. Girard, A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
    [CrossRef] [PubMed]
  12. P. J. Valle, R. Carminati, J-J. Greffet, “Contrast mechanisms in illumination-mode SNOM,” Ultramicroscopy 71, 39–48 (1998).
    [CrossRef]
  13. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, Mass., 1995).
  14. D. A. Christensen, “Analysis of near field tip patterns including object interaction using finite-difference time-domain,” Ultramicroscopy 57, 189–195 (1995).
    [CrossRef]
  15. H. Furukawa, S. Kawata, “Analysis of image formation in a near-field scanning optical microscope: effects of multiple scattering,” Opt. Commun. 132, 170–178 (1996).
    [CrossRef]
  16. R. Muller, C. Lienau, “Propagation of femtosecond optical pulses through uncoated and metal-coated near-field fiber probes,” Appl. Phys. Lett. 76, 3367–3369 (2000).
    [CrossRef]
  17. E. Vasilyeva, A. Taflove, “Three-dimensional modeling of amplitude-object imaging in scanning near-field optical microscopy,” Opt. Lett. 23, 1155–1157 (1998).
    [CrossRef]
  18. L. Novotny, D. W. Pohl, P. Regali, “Light propagation through nanometer-sized structures: the two-dimensional-aperture scanning near-field optical microscope,” J. Opt. Soc. Am. A 11, 1768–1779 (1994).
    [CrossRef]
  19. L. Novotny, D. W. Pohl, B. Hecht, “Light confinement in scanning near-field optical microscopy,” Ultramicroscopy 61, 1–9 (1995).
    [CrossRef]
  20. D. W. Pohl, D. Courjon, eds., Near Field Optics, NATO ASI Series E (Kluwer, Academic, Dordrecht, The Netherlands, 1993), Vol. 242.
  21. J. A. DeSanto, Scalar Wave Theory: Green’s Functions and Applications (Springer, Heidelberg, 1992).
  22. D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Wiley, New York, 1983).
  23. J. A. DeSanto, P. A. Martin, “On angular-spectrum representations for scattering by infinite rough surfaces,” J. Wave Motion 24, 421–433 (1996).
    [CrossRef]
  24. J. A. DeSanto, P. A. Martin, “On the derivation of boundary integral equations for scattering by an infinite one-dimensional rough surface,” J. Acoust. Soc. Am. 102, 67–77 (1997).
    [CrossRef]
  25. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  26. D. S. Jones, Methods in Electromagnetic Wave Propagation (Oxford U. Press, Oxford, UK, 1994).

2000 (1)

R. Muller, C. Lienau, “Propagation of femtosecond optical pulses through uncoated and metal-coated near-field fiber probes,” Appl. Phys. Lett. 76, 3367–3369 (2000).
[CrossRef]

1998 (6)

E. Vasilyeva, A. Taflove, “Three-dimensional modeling of amplitude-object imaging in scanning near-field optical microscopy,” Opt. Lett. 23, 1155–1157 (1998).
[CrossRef]

P. J. Valle, R. Carminati, J-J. Greffet, “Contrast mechanisms in illumination-mode SNOM,” Ultramicroscopy 71, 39–48 (1998).
[CrossRef]

E. Wolf, J. T. Foley, “Do evanescent waves contribute to the far field?” Opt. Lett. 23, 16–18 (1998).
[CrossRef]

L. Novotny, B. Hecht, D. W. Pohl, “Implications of high resolution to near-field optical microscopy,” Ultramicroscopy 71, 341–343 (1998).
[CrossRef]

G. W. Bryant, “Probing quantum nanostructures with near-field optical microscopy and vice versa,” Appl. Phys. Lett. 72, 768–770 (1998).
[CrossRef]

G. W. Bryant, E. L. Shirley, L. S. Goldner, E. B. McDaniel, J. W. P. Hsu, R. J. Tonucci, “Theory of probing photonic crystal with transmission near-field optical microscopy,” Phys. Rev. B 58, 2131–2141 (1998).
[CrossRef]

1997 (1)

J. A. DeSanto, P. A. Martin, “On the derivation of boundary integral equations for scattering by an infinite one-dimensional rough surface,” J. Acoust. Soc. Am. 102, 67–77 (1997).
[CrossRef]

1996 (3)

K. Fukuzawa, H. Kuwano, “Conversion of evanescent into propagating light in near-field scanning optical microscopy,” J. Appl. Phys. 79, 8174–8178 (1996).
[CrossRef]

H. Furukawa, S. Kawata, “Analysis of image formation in a near-field scanning optical microscope: effects of multiple scattering,” Opt. Commun. 132, 170–178 (1996).
[CrossRef]

J. A. DeSanto, P. A. Martin, “On angular-spectrum representations for scattering by infinite rough surfaces,” J. Wave Motion 24, 421–433 (1996).
[CrossRef]

1995 (4)

L. Novotny, D. W. Pohl, B. Hecht, “Light confinement in scanning near-field optical microscopy,” Ultramicroscopy 61, 1–9 (1995).
[CrossRef]

D. A. Christensen, “Analysis of near field tip patterns including object interaction using finite-difference time-domain,” Ultramicroscopy 57, 189–195 (1995).
[CrossRef]

G. A. Valaskovic, M. Holton, G. H. Morrison, “Parameter control, characterization, and optimization in the fabrication of optical fiber near-field probes,” Appl. Opt. 34, 1215–1228 (1995).
[CrossRef] [PubMed]

O. J. F. Martin, C. Girard, A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[CrossRef] [PubMed]

1994 (1)

1991 (1)

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1468–1470 (1991).
[CrossRef] [PubMed]

1986 (1)

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

1950 (1)

C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Philips Res. Rep. 5, 321–332 (1950).

1944 (1)

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
[CrossRef]

Bethe, H. A.

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
[CrossRef]

Betzig, E.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1468–1470 (1991).
[CrossRef] [PubMed]

Bouwkamp, C. J.

C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Philips Res. Rep. 5, 321–332 (1950).

Bryant, G. W.

G. W. Bryant, “Probing quantum nanostructures with near-field optical microscopy and vice versa,” Appl. Phys. Lett. 72, 768–770 (1998).
[CrossRef]

G. W. Bryant, E. L. Shirley, L. S. Goldner, E. B. McDaniel, J. W. P. Hsu, R. J. Tonucci, “Theory of probing photonic crystal with transmission near-field optical microscopy,” Phys. Rev. B 58, 2131–2141 (1998).
[CrossRef]

Carminati, R.

P. J. Valle, R. Carminati, J-J. Greffet, “Contrast mechanisms in illumination-mode SNOM,” Ultramicroscopy 71, 39–48 (1998).
[CrossRef]

Christensen, D. A.

D. A. Christensen, “Analysis of near field tip patterns including object interaction using finite-difference time-domain,” Ultramicroscopy 57, 189–195 (1995).
[CrossRef]

Colton, D.

D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Wiley, New York, 1983).

Dereux, A.

O. J. F. Martin, C. Girard, A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[CrossRef] [PubMed]

DeSanto, J. A.

J. A. DeSanto, P. A. Martin, “On the derivation of boundary integral equations for scattering by an infinite one-dimensional rough surface,” J. Acoust. Soc. Am. 102, 67–77 (1997).
[CrossRef]

J. A. DeSanto, P. A. Martin, “On angular-spectrum representations for scattering by infinite rough surfaces,” J. Wave Motion 24, 421–433 (1996).
[CrossRef]

J. A. DeSanto, Scalar Wave Theory: Green’s Functions and Applications (Springer, Heidelberg, 1992).

Durig, U.

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

Foley, J. T.

Fukuzawa, K.

K. Fukuzawa, H. Kuwano, “Conversion of evanescent into propagating light in near-field scanning optical microscopy,” J. Appl. Phys. 79, 8174–8178 (1996).
[CrossRef]

Furukawa, H.

H. Furukawa, S. Kawata, “Analysis of image formation in a near-field scanning optical microscope: effects of multiple scattering,” Opt. Commun. 132, 170–178 (1996).
[CrossRef]

Girard, C.

O. J. F. Martin, C. Girard, A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[CrossRef] [PubMed]

Goldner, L. S.

G. W. Bryant, E. L. Shirley, L. S. Goldner, E. B. McDaniel, J. W. P. Hsu, R. J. Tonucci, “Theory of probing photonic crystal with transmission near-field optical microscopy,” Phys. Rev. B 58, 2131–2141 (1998).
[CrossRef]

Greffet, J-J.

P. J. Valle, R. Carminati, J-J. Greffet, “Contrast mechanisms in illumination-mode SNOM,” Ultramicroscopy 71, 39–48 (1998).
[CrossRef]

Harris, T. D.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1468–1470 (1991).
[CrossRef] [PubMed]

Hecht, B.

L. Novotny, B. Hecht, D. W. Pohl, “Implications of high resolution to near-field optical microscopy,” Ultramicroscopy 71, 341–343 (1998).
[CrossRef]

L. Novotny, D. W. Pohl, B. Hecht, “Light confinement in scanning near-field optical microscopy,” Ultramicroscopy 61, 1–9 (1995).
[CrossRef]

Holton, M.

Hsu, J. W. P.

G. W. Bryant, E. L. Shirley, L. S. Goldner, E. B. McDaniel, J. W. P. Hsu, R. J. Tonucci, “Theory of probing photonic crystal with transmission near-field optical microscopy,” Phys. Rev. B 58, 2131–2141 (1998).
[CrossRef]

Jones, D. S.

D. S. Jones, Methods in Electromagnetic Wave Propagation (Oxford U. Press, Oxford, UK, 1994).

Kawata, S.

H. Furukawa, S. Kawata, “Analysis of image formation in a near-field scanning optical microscope: effects of multiple scattering,” Opt. Commun. 132, 170–178 (1996).
[CrossRef]

Kostelak, R. L.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1468–1470 (1991).
[CrossRef] [PubMed]

Kress, R.

D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Wiley, New York, 1983).

Kuwano, H.

K. Fukuzawa, H. Kuwano, “Conversion of evanescent into propagating light in near-field scanning optical microscopy,” J. Appl. Phys. 79, 8174–8178 (1996).
[CrossRef]

Lienau, C.

R. Muller, C. Lienau, “Propagation of femtosecond optical pulses through uncoated and metal-coated near-field fiber probes,” Appl. Phys. Lett. 76, 3367–3369 (2000).
[CrossRef]

Martin, O. J. F.

O. J. F. Martin, C. Girard, A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[CrossRef] [PubMed]

Martin, P. A.

J. A. DeSanto, P. A. Martin, “On the derivation of boundary integral equations for scattering by an infinite one-dimensional rough surface,” J. Acoust. Soc. Am. 102, 67–77 (1997).
[CrossRef]

J. A. DeSanto, P. A. Martin, “On angular-spectrum representations for scattering by infinite rough surfaces,” J. Wave Motion 24, 421–433 (1996).
[CrossRef]

McDaniel, E. B.

G. W. Bryant, E. L. Shirley, L. S. Goldner, E. B. McDaniel, J. W. P. Hsu, R. J. Tonucci, “Theory of probing photonic crystal with transmission near-field optical microscopy,” Phys. Rev. B 58, 2131–2141 (1998).
[CrossRef]

Morrison, G. H.

Muller, R.

R. Muller, C. Lienau, “Propagation of femtosecond optical pulses through uncoated and metal-coated near-field fiber probes,” Appl. Phys. Lett. 76, 3367–3369 (2000).
[CrossRef]

Novotny, L.

L. Novotny, B. Hecht, D. W. Pohl, “Implications of high resolution to near-field optical microscopy,” Ultramicroscopy 71, 341–343 (1998).
[CrossRef]

L. Novotny, D. W. Pohl, B. Hecht, “Light confinement in scanning near-field optical microscopy,” Ultramicroscopy 61, 1–9 (1995).
[CrossRef]

L. Novotny, D. W. Pohl, P. Regali, “Light propagation through nanometer-sized structures: the two-dimensional-aperture scanning near-field optical microscope,” J. Opt. Soc. Am. A 11, 1768–1779 (1994).
[CrossRef]

Pohl, D. W.

L. Novotny, B. Hecht, D. W. Pohl, “Implications of high resolution to near-field optical microscopy,” Ultramicroscopy 71, 341–343 (1998).
[CrossRef]

L. Novotny, D. W. Pohl, B. Hecht, “Light confinement in scanning near-field optical microscopy,” Ultramicroscopy 61, 1–9 (1995).
[CrossRef]

L. Novotny, D. W. Pohl, P. Regali, “Light propagation through nanometer-sized structures: the two-dimensional-aperture scanning near-field optical microscope,” J. Opt. Soc. Am. A 11, 1768–1779 (1994).
[CrossRef]

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

Regali, P.

Rohner, F.

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

Shirley, E. L.

G. W. Bryant, E. L. Shirley, L. S. Goldner, E. B. McDaniel, J. W. P. Hsu, R. J. Tonucci, “Theory of probing photonic crystal with transmission near-field optical microscopy,” Phys. Rev. B 58, 2131–2141 (1998).
[CrossRef]

Taflove, A.

E. Vasilyeva, A. Taflove, “Three-dimensional modeling of amplitude-object imaging in scanning near-field optical microscopy,” Opt. Lett. 23, 1155–1157 (1998).
[CrossRef]

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, Mass., 1995).

Tonucci, R. J.

G. W. Bryant, E. L. Shirley, L. S. Goldner, E. B. McDaniel, J. W. P. Hsu, R. J. Tonucci, “Theory of probing photonic crystal with transmission near-field optical microscopy,” Phys. Rev. B 58, 2131–2141 (1998).
[CrossRef]

Trautman, J. K.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1468–1470 (1991).
[CrossRef] [PubMed]

Valaskovic, G. A.

Valle, P. J.

P. J. Valle, R. Carminati, J-J. Greffet, “Contrast mechanisms in illumination-mode SNOM,” Ultramicroscopy 71, 39–48 (1998).
[CrossRef]

Vasilyeva, E.

Weiner, J. S.

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1468–1470 (1991).
[CrossRef] [PubMed]

Wolf, E.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

G. W. Bryant, “Probing quantum nanostructures with near-field optical microscopy and vice versa,” Appl. Phys. Lett. 72, 768–770 (1998).
[CrossRef]

R. Muller, C. Lienau, “Propagation of femtosecond optical pulses through uncoated and metal-coated near-field fiber probes,” Appl. Phys. Lett. 76, 3367–3369 (2000).
[CrossRef]

J. Acoust. Soc. Am. (1)

J. A. DeSanto, P. A. Martin, “On the derivation of boundary integral equations for scattering by an infinite one-dimensional rough surface,” J. Acoust. Soc. Am. 102, 67–77 (1997).
[CrossRef]

J. Appl. Phys. (2)

K. Fukuzawa, H. Kuwano, “Conversion of evanescent into propagating light in near-field scanning optical microscopy,” J. Appl. Phys. 79, 8174–8178 (1996).
[CrossRef]

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

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

J. Wave Motion (1)

J. A. DeSanto, P. A. Martin, “On angular-spectrum representations for scattering by infinite rough surfaces,” J. Wave Motion 24, 421–433 (1996).
[CrossRef]

Opt. Commun. (1)

H. Furukawa, S. Kawata, “Analysis of image formation in a near-field scanning optical microscope: effects of multiple scattering,” Opt. Commun. 132, 170–178 (1996).
[CrossRef]

Opt. Lett. (2)

Philips Res. Rep. (1)

C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Philips Res. Rep. 5, 321–332 (1950).

Phys. Rev. (1)

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
[CrossRef]

Phys. Rev. B (1)

G. W. Bryant, E. L. Shirley, L. S. Goldner, E. B. McDaniel, J. W. P. Hsu, R. J. Tonucci, “Theory of probing photonic crystal with transmission near-field optical microscopy,” Phys. Rev. B 58, 2131–2141 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

O. J. F. Martin, C. Girard, A. Dereux, “Generalized field propagator for electromagnetic scattering and light confinement,” Phys. Rev. Lett. 74, 526–529 (1995).
[CrossRef] [PubMed]

Science (1)

E. Betzig, J. K. Trautman, T. D. Harris, J. S. Weiner, R. L. Kostelak, “Breaking the diffraction barrier: optical microscopy on a nanometric scale,” Science 251, 1468–1470 (1991).
[CrossRef] [PubMed]

Ultramicroscopy (4)

L. Novotny, B. Hecht, D. W. Pohl, “Implications of high resolution to near-field optical microscopy,” Ultramicroscopy 71, 341–343 (1998).
[CrossRef]

P. J. Valle, R. Carminati, J-J. Greffet, “Contrast mechanisms in illumination-mode SNOM,” Ultramicroscopy 71, 39–48 (1998).
[CrossRef]

D. A. Christensen, “Analysis of near field tip patterns including object interaction using finite-difference time-domain,” Ultramicroscopy 57, 189–195 (1995).
[CrossRef]

L. Novotny, D. W. Pohl, B. Hecht, “Light confinement in scanning near-field optical microscopy,” Ultramicroscopy 61, 1–9 (1995).
[CrossRef]

Other (6)

D. W. Pohl, D. Courjon, eds., Near Field Optics, NATO ASI Series E (Kluwer, Academic, Dordrecht, The Netherlands, 1993), Vol. 242.

J. A. DeSanto, Scalar Wave Theory: Green’s Functions and Applications (Springer, Heidelberg, 1992).

D. Colton, R. Kress, Integral Equation Methods in Scattering Theory (Wiley, New York, 1983).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

D. S. Jones, Methods in Electromagnetic Wave Propagation (Oxford U. Press, Oxford, UK, 1994).

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, Mass., 1995).

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Figures (5)

Fig. 1
Fig. 1

Geometry used throughout this paper to model the NSOM tip. The upper aperture in region 1 is 850 nm. The lower aperture between regions 1 and 4 is 50 nm. The vertical distance between the upper and lower apertures is 1000 nm. The thinner aluminum strips along the taper of the tip are 70 nm thick. The thicker aluminum strips are 500 nm thick. The flux and the integrated intensity are computed along C1 and C2, as discussed in Section 3.

Fig. 2
Fig. 2

Integrated intensity along the curve C2 in Fig. 1 and integrated flux (power) through C2 for a geometry where the sample (region 5) has been removed. The lower line of curve C2 was shifted progressively further from the tip. The distance from the tip is a measure of how far this lower line was from the lower tip aperture. The integrated intensity is normalized to the incident integrated intensity along the upper aperture. The power is normalized to the incident power through the upper aperture.

Fig. 3
Fig. 3

Contour plot of log(|E|2) (logarithm of the electric field intensity) with glass (n=1.5) as the sample, including Poynting vectors. Each contour line differs by a factor of 100.6, with the lowest contour at 10-8.6 of the incident intensity. The inset is a linear plot of the decay of the percent intensity of the field (normalized to the incident intensity) along a vertical starting from the center of the lower tip aperture.

Fig. 4
Fig. 4

Far-field radiation intensity patterns on the same scale for three different cases: (a) vacuum sample (n=1), (b) glass sample (n=1.5), (c) GaAs sample (n=3.4). The contours differ by a factor of 100.6, where the lowest contour is 10-8.6 of the incident intensity. In (b) and (c), the dashed lines indicate the critical angles in the sample.

Fig. 5
Fig. 5

Plot of the integrated flux (power) into the sample as a function of the index of refraction of the sample material. The power is normalized to the incident power through the upper aperture.

Tables (1)

Tables Icon

Table 1 Percent Change (Δ) from Previous Value of the Field (u) and Its Normal Derivative (N) at Center of Lower Tip Aperture as a Function of Total Number of Boundary Points

Equations (15)

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

(2+ki2)ui(x)=0
Gi(x, x)=i4 H0(1)(ki|x-x|),
ui(x)=sids{ui(x)(nˆ  Gi)-Gi[nˆ  ui(x)]},
Ii+(x, y)=Rdx{u0(x)(nˆGi)-Gi[nˆu0(x)]},
Δθ2π ui(x)=uinc(x)+Ii+(x, y)+Ii-(-x, y)+sids[ui(nˆGi)-Gi(nˆui)],
Δθ2π up=uinc(xp, yp)+Ii+(xp, yp)+Ii-(-xp, yp)+q=1nΔlq(apquq-bpqNq).
f(l)=12 (fq+fq+1)+1Δlq (fq+1-fq)l,
μiui=μjuj,Ni=Nj.
Zb=uinc+I++I-.
zpq=apqΔlq-Δθ2π δpq-μiμjΔθ2π δ(p+m,q),qmbpqΔlq,q>m,
S=iμi*2ω (uiui*).
lRe(S)  n^ldl,
Ei=iωuiy xˆ-uix yˆ,
iui=juj,Ni=Nj.
S=ii*2ω (uiui*).

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