Abstract

An axially symmetric three-dimensional finite element method model is applied to investigate the electromagnetic field distribution in the vicinity of a silver coated glass tip. Under radially polarized illumination, a strongly enhanced field located at the apex of the tip is found due to the constructive interference of surface plasmon propagating at the air/silver interface. The enhancement factor and surface plasmon resonance excitation are analyzed systematically. The optimal condition for field enhancement is investigated through the exploration of different taper angles of the tip and the illumination geometry. The numerical studies show that a significantly enhanced localized electromagnetic field with a full-width-half-maximum of 10 nm is obtainable with 632.8 nm optical excitation.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. A. Downes, D. Salter, and A. Elfick, “Simulations of atomic resolution tip-enhanced optical microscopy,” Opt. Express 14,11324–11329 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2007 (1)

W. Chen and Q. Zhan, “Optimal plasmonic focusing with radial polarization,” Proc. SPIE, 6450,64500D (2007).
[CrossRef]

2006 (4)

H. Frey, C. Bolwien, A. Brandenburg, R. Ros, and D. Anselmetti, “Optimized apertureless optical near-field probes with 15 nm optical resolution,” Nanotechnology 17,3105–3110 (2006).
[CrossRef]

A. Bouhelier, “Field-enhanced scanning near-field optical microscopy,” Microsc. Res. Tech. 69,563–579 (2006).
[CrossRef] [PubMed]

S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14,1957–1964 (2006).
[CrossRef] [PubMed]

A. Downes, D. Salter, and A. Elfick, “Simulations of atomic resolution tip-enhanced optical microscopy,” Opt. Express 14,11324–11329 (2006).
[CrossRef] [PubMed]

2005 (3)

2004 (2)

T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. kawata, “Application of tip-enhanced microscopy for nonlinear Raman spectroscopy,” Appl. Phys. Lett. 84,1768–1770 (2004).
[CrossRef]

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12,3377–3382 (2004).
[CrossRef] [PubMed]

2003 (6)

C. Sun and C. Liu, “Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation,” Opt. Lett. 28,99–101 (2003).
[CrossRef] [PubMed]

R. Dorn, S Qubis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91,233901 (2003).
[CrossRef] [PubMed]

A. Bouhelier, J. Renger, M. R. Beversluis, and L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210,220–224 (2003).
[CrossRef] [PubMed]

L. Vaccaro, L. Aeschimann, U. Staufer, H. P. Herzig, and R. Dändliker, “Propagation of the electromagnetic field in fully coated near-field optical probes,” Appl. Phys. Lett. 83,584–586 (2003).
[CrossRef]

R. Fikri, D. Barchiesi, F. H′Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2D finite element method,” Opt. Commun. 221,13–22 (2003).
[CrossRef]

R. Bachelot, F. H′Dhili, and D. Barchiesi et. al. “Apertureless near-field optical microscopy: A study of the local tip field enhancement using photosensitive azobenzene-containing films,” J. Appl. Phys. 94,2060–2072 (2003).
[CrossRef]

2002 (3)

A. Tarun, M. Daza, N. Hayazawa, Y. Inouye, and S. Kawata, “Apertureless optical near-field fabrication using an atomic force microscope on photoresists,” Appl. Phys. Lett. 80,3400–3402 (2002).
[CrossRef]

A. Hartschuh, N. Anderson, and L. Novotny, “Near-field Raman spectroscopy using a sharp metal tip,” J. Microsc. 210,234–240 (2002).
[CrossRef]

Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10,324–331 (2002).
[PubMed]

2000 (2)

T. J. Yang, G. A. Lessard, and S. R. Quake, “An apertureless near-field microscope for fluorescence imaging,” Appl. Phys. Lett. 76,378–380 (2000).
[CrossRef]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7,77–87 (2000).
[CrossRef] [PubMed]

1999 (1)

F. Keilmann, “Surface-polariton propagation for scanning near-field optical microscopy application,” J. Microsc. 194,567–570 (1999).
[CrossRef]

1993 (1)

E. Betzig and R. J. Chichester, “Single molecules observed by near-field scanning optical microscopy,” Science 262,1422–1425 (1993).
[CrossRef] [PubMed]

1991 (1)

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

Aeschimann, L.

L. Vaccaro, L. Aeschimann, U. Staufer, H. P. Herzig, and R. Dändliker, “Propagation of the electromagnetic field in fully coated near-field optical probes,” Appl. Phys. Lett. 83,584–586 (2003).
[CrossRef]

Ait-Ameur, K.

Anderson, N.

A. Hartschuh, N. Anderson, and L. Novotny, “Near-field Raman spectroscopy using a sharp metal tip,” J. Microsc. 210,234–240 (2002).
[CrossRef]

Anselmetti, D.

H. Frey, C. Bolwien, A. Brandenburg, R. Ros, and D. Anselmetti, “Optimized apertureless optical near-field probes with 15 nm optical resolution,” Nanotechnology 17,3105–3110 (2006).
[CrossRef]

Bachelot, R.

R. Bachelot, F. H′Dhili, and D. Barchiesi et. al. “Apertureless near-field optical microscopy: A study of the local tip field enhancement using photosensitive azobenzene-containing films,” J. Appl. Phys. 94,2060–2072 (2003).
[CrossRef]

R. Fikri, D. Barchiesi, F. H′Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2D finite element method,” Opt. Commun. 221,13–22 (2003).
[CrossRef]

Baghdasaryan, K. S.

N. A. Janunts, K. S. Baghdasaryan, Kh.V. Nerkararyan, and B. Hecht, “Excitation and superfocusing of surface plasmon polaritons on a silver-coated optical fiber tip,” Opt. Commun. 253,118–124 (2005).
[CrossRef]

Barchiesi, D.

R. Bachelot, F. H′Dhili, and D. Barchiesi et. al. “Apertureless near-field optical microscopy: A study of the local tip field enhancement using photosensitive azobenzene-containing films,” J. Appl. Phys. 94,2060–2072 (2003).
[CrossRef]

R. Fikri, D. Barchiesi, F. H′Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2D finite element method,” Opt. Commun. 221,13–22 (2003).
[CrossRef]

Betzig, E.

E. Betzig and R. J. Chichester, “Single molecules observed by near-field scanning optical microscopy,” Science 262,1422–1425 (1993).
[CrossRef] [PubMed]

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

Beversluis, M. R.

A. Bouhelier, J. Renger, M. R. Beversluis, and L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210,220–224 (2003).
[CrossRef] [PubMed]

Bolwien, C.

H. Frey, C. Bolwien, A. Brandenburg, R. Ros, and D. Anselmetti, “Optimized apertureless optical near-field probes with 15 nm optical resolution,” Nanotechnology 17,3105–3110 (2006).
[CrossRef]

Bouhelier, A.

A. Bouhelier, “Field-enhanced scanning near-field optical microscopy,” Microsc. Res. Tech. 69,563–579 (2006).
[CrossRef] [PubMed]

A. Bouhelier, J. Renger, M. R. Beversluis, and L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210,220–224 (2003).
[CrossRef] [PubMed]

Brandenburg, A.

H. Frey, C. Bolwien, A. Brandenburg, R. Ros, and D. Anselmetti, “Optimized apertureless optical near-field probes with 15 nm optical resolution,” Nanotechnology 17,3105–3110 (2006).
[CrossRef]

Brown, T. G.

Chen, W.

W. Chen and Q. Zhan, “Optimal plasmonic focusing with radial polarization,” Proc. SPIE, 6450,64500D (2007).
[CrossRef]

Chichester, R. J.

E. Betzig and R. J. Chichester, “Single molecules observed by near-field scanning optical microscopy,” Science 262,1422–1425 (1993).
[CrossRef] [PubMed]

Dändliker, R.

L. Vaccaro, L. Aeschimann, U. Staufer, H. P. Herzig, and R. Dändliker, “Propagation of the electromagnetic field in fully coated near-field optical probes,” Appl. Phys. Lett. 83,584–586 (2003).
[CrossRef]

Daza, M.

A. Tarun, M. Daza, N. Hayazawa, Y. Inouye, and S. Kawata, “Apertureless optical near-field fabrication using an atomic force microscope on photoresists,” Appl. Phys. Lett. 80,3400–3402 (2002).
[CrossRef]

Denis, R.

Dorn, R.

R. Dorn, S Qubis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91,233901 (2003).
[CrossRef] [PubMed]

Downes, A.

Elfick, A.

Fikri, R.

R. Fikri, D. Barchiesi, F. H′Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2D finite element method,” Opt. Commun. 221,13–22 (2003).
[CrossRef]

Frey, H.

H. Frey, C. Bolwien, A. Brandenburg, R. Ros, and D. Anselmetti, “Optimized apertureless optical near-field probes with 15 nm optical resolution,” Nanotechnology 17,3105–3110 (2006).
[CrossRef]

Gan, X.

Gu, M.

Harris, T. D.

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

Hartschuh, A.

A. Hartschuh, N. Anderson, and L. Novotny, “Near-field Raman spectroscopy using a sharp metal tip,” J. Microsc. 210,234–240 (2002).
[CrossRef]

Hashimoto, M.

T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. kawata, “Application of tip-enhanced microscopy for nonlinear Raman spectroscopy,” Appl. Phys. Lett. 84,1768–1770 (2004).
[CrossRef]

Hayazawa, N.

T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. kawata, “Application of tip-enhanced microscopy for nonlinear Raman spectroscopy,” Appl. Phys. Lett. 84,1768–1770 (2004).
[CrossRef]

A. Tarun, M. Daza, N. Hayazawa, Y. Inouye, and S. Kawata, “Apertureless optical near-field fabrication using an atomic force microscope on photoresists,” Appl. Phys. Lett. 80,3400–3402 (2002).
[CrossRef]

H'Dhili, F.

R. Fikri, D. Barchiesi, F. H′Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2D finite element method,” Opt. Commun. 221,13–22 (2003).
[CrossRef]

R. Bachelot, F. H′Dhili, and D. Barchiesi et. al. “Apertureless near-field optical microscopy: A study of the local tip field enhancement using photosensitive azobenzene-containing films,” J. Appl. Phys. 94,2060–2072 (2003).
[CrossRef]

Hecht, B.

N. A. Janunts, K. S. Baghdasaryan, Kh.V. Nerkararyan, and B. Hecht, “Excitation and superfocusing of surface plasmon polaritons on a silver-coated optical fiber tip,” Opt. Commun. 253,118–124 (2005).
[CrossRef]

Herzig, H. P.

L. Vaccaro, L. Aeschimann, U. Staufer, H. P. Herzig, and R. Dändliker, “Propagation of the electromagnetic field in fully coated near-field optical probes,” Appl. Phys. Lett. 83,584–586 (2003).
[CrossRef]

Hierle, R.

Ichimura, T.

T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. kawata, “Application of tip-enhanced microscopy for nonlinear Raman spectroscopy,” Appl. Phys. Lett. 84,1768–1770 (2004).
[CrossRef]

Inouye, Y.

T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. kawata, “Application of tip-enhanced microscopy for nonlinear Raman spectroscopy,” Appl. Phys. Lett. 84,1768–1770 (2004).
[CrossRef]

A. Tarun, M. Daza, N. Hayazawa, Y. Inouye, and S. Kawata, “Apertureless optical near-field fabrication using an atomic force microscope on photoresists,” Appl. Phys. Lett. 80,3400–3402 (2002).
[CrossRef]

Janunts, N. A.

N. A. Janunts, K. S. Baghdasaryan, Kh.V. Nerkararyan, and B. Hecht, “Excitation and superfocusing of surface plasmon polaritons on a silver-coated optical fiber tip,” Opt. Commun. 253,118–124 (2005).
[CrossRef]

Jia, B.

kawata, S.

T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. kawata, “Application of tip-enhanced microscopy for nonlinear Raman spectroscopy,” Appl. Phys. Lett. 84,1768–1770 (2004).
[CrossRef]

A. Tarun, M. Daza, N. Hayazawa, Y. Inouye, and S. Kawata, “Apertureless optical near-field fabrication using an atomic force microscope on photoresists,” Appl. Phys. Lett. 80,3400–3402 (2002).
[CrossRef]

Keilmann, F.

F. Keilmann, “Surface-polariton propagation for scanning near-field optical microscopy application,” J. Microsc. 194,567–570 (1999).
[CrossRef]

Kostelak, R. L.

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

Leger, J. R.

Lessard, G. A.

T. J. Yang, G. A. Lessard, and S. R. Quake, “An apertureless near-field microscope for fluorescence imaging,” Appl. Phys. Lett. 76,378–380 (2000).
[CrossRef]

Leuchs, G.

R. Dorn, S Qubis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91,233901 (2003).
[CrossRef] [PubMed]

Liu, C.

Maier, S. A.

Nerkararyan, Kh.V.

N. A. Janunts, K. S. Baghdasaryan, Kh.V. Nerkararyan, and B. Hecht, “Excitation and superfocusing of surface plasmon polaritons on a silver-coated optical fiber tip,” Opt. Commun. 253,118–124 (2005).
[CrossRef]

Novotny, L.

A. Bouhelier, J. Renger, M. R. Beversluis, and L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210,220–224 (2003).
[CrossRef] [PubMed]

A. Hartschuh, N. Anderson, and L. Novotny, “Near-field Raman spectroscopy using a sharp metal tip,” J. Microsc. 210,234–240 (2002).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids, page 356 (Academic Press, 1998).

Passilly, N.

Quake, S. R.

T. J. Yang, G. A. Lessard, and S. R. Quake, “An apertureless near-field microscope for fluorescence imaging,” Appl. Phys. Lett. 76,378–380 (2000).
[CrossRef]

Qubis, S

R. Dorn, S Qubis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91,233901 (2003).
[CrossRef] [PubMed]

Renger, J.

A. Bouhelier, J. Renger, M. R. Beversluis, and L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210,220–224 (2003).
[CrossRef] [PubMed]

Roch, J.

Ros, R.

H. Frey, C. Bolwien, A. Brandenburg, R. Ros, and D. Anselmetti, “Optimized apertureless optical near-field probes with 15 nm optical resolution,” Nanotechnology 17,3105–3110 (2006).
[CrossRef]

Royer, P.

R. Fikri, D. Barchiesi, F. H′Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2D finite element method,” Opt. Commun. 221,13–22 (2003).
[CrossRef]

Salter, D.

Staufer, U.

L. Vaccaro, L. Aeschimann, U. Staufer, H. P. Herzig, and R. Dändliker, “Propagation of the electromagnetic field in fully coated near-field optical probes,” Appl. Phys. Lett. 83,584–586 (2003).
[CrossRef]

Sun, C.

Tarun, A.

A. Tarun, M. Daza, N. Hayazawa, Y. Inouye, and S. Kawata, “Apertureless optical near-field fabrication using an atomic force microscope on photoresists,” Appl. Phys. Lett. 80,3400–3402 (2002).
[CrossRef]

Trautman, J. K.

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

Treussart, F.

Vaccaro, L.

L. Vaccaro, L. Aeschimann, U. Staufer, H. P. Herzig, and R. Dändliker, “Propagation of the electromagnetic field in fully coated near-field optical probes,” Appl. Phys. Lett. 83,584–586 (2003).
[CrossRef]

Vial, A.

R. Fikri, D. Barchiesi, F. H′Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2D finite element method,” Opt. Commun. 221,13–22 (2003).
[CrossRef]

Weiner, J. S.

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

Yang, T. J.

T. J. Yang, G. A. Lessard, and S. R. Quake, “An apertureless near-field microscope for fluorescence imaging,” Appl. Phys. Lett. 76,378–380 (2000).
[CrossRef]

Youngworth, K. S.

Zhan, Q.

Appl. Phys. Lett. (4)

A. Tarun, M. Daza, N. Hayazawa, Y. Inouye, and S. Kawata, “Apertureless optical near-field fabrication using an atomic force microscope on photoresists,” Appl. Phys. Lett. 80,3400–3402 (2002).
[CrossRef]

L. Vaccaro, L. Aeschimann, U. Staufer, H. P. Herzig, and R. Dändliker, “Propagation of the electromagnetic field in fully coated near-field optical probes,” Appl. Phys. Lett. 83,584–586 (2003).
[CrossRef]

T. Ichimura, N. Hayazawa, M. Hashimoto, Y. Inouye, and S. kawata, “Application of tip-enhanced microscopy for nonlinear Raman spectroscopy,” Appl. Phys. Lett. 84,1768–1770 (2004).
[CrossRef]

T. J. Yang, G. A. Lessard, and S. R. Quake, “An apertureless near-field microscope for fluorescence imaging,” Appl. Phys. Lett. 76,378–380 (2000).
[CrossRef]

J. Appl. Phys. (1)

R. Bachelot, F. H′Dhili, and D. Barchiesi et. al. “Apertureless near-field optical microscopy: A study of the local tip field enhancement using photosensitive azobenzene-containing films,” J. Appl. Phys. 94,2060–2072 (2003).
[CrossRef]

J. Microsc. (3)

A. Bouhelier, J. Renger, M. R. Beversluis, and L. Novotny, “Plasmon-coupled tip-enhanced near-field optical microscopy,” J. Microsc. 210,220–224 (2003).
[CrossRef] [PubMed]

F. Keilmann, “Surface-polariton propagation for scanning near-field optical microscopy application,” J. Microsc. 194,567–570 (1999).
[CrossRef]

A. Hartschuh, N. Anderson, and L. Novotny, “Near-field Raman spectroscopy using a sharp metal tip,” J. Microsc. 210,234–240 (2002).
[CrossRef]

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

Microsc. Res. Tech. (1)

A. Bouhelier, “Field-enhanced scanning near-field optical microscopy,” Microsc. Res. Tech. 69,563–579 (2006).
[CrossRef] [PubMed]

Nanotechnology (1)

H. Frey, C. Bolwien, A. Brandenburg, R. Ros, and D. Anselmetti, “Optimized apertureless optical near-field probes with 15 nm optical resolution,” Nanotechnology 17,3105–3110 (2006).
[CrossRef]

Opt. Commun. (2)

R. Fikri, D. Barchiesi, F. H′Dhili, R. Bachelot, A. Vial, and P. Royer, “Modeling recent experiments of apertureless near-field optical microscopy using 2D finite element method,” Opt. Commun. 221,13–22 (2003).
[CrossRef]

N. A. Janunts, K. S. Baghdasaryan, Kh.V. Nerkararyan, and B. Hecht, “Excitation and superfocusing of surface plasmon polaritons on a silver-coated optical fiber tip,” Opt. Commun. 253,118–124 (2005).
[CrossRef]

Opt. Express (6)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

R. Dorn, S Qubis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91,233901 (2003).
[CrossRef] [PubMed]

Proc. SPIE (1)

W. Chen and Q. Zhan, “Optimal plasmonic focusing with radial polarization,” Proc. SPIE, 6450,64500D (2007).
[CrossRef]

Science (2)

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

E. Betzig and R. J. Chichester, “Single molecules observed by near-field scanning optical microscopy,” Science 262,1422–1425 (1993).
[CrossRef] [PubMed]

Other (1)

E. D. Palik, Handbook of Optical Constants of Solids, page 356 (Academic Press, 1998).

Supplementary Material (2)

» Media 1: AVI (364 KB)     
» Media 2: AVI (384 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Diagram of an apertureless NSOM probe structure with radial polarization input.

Fig. 2.
Fig. 2.

(a) 2-D electric energy density distribution at the end of probe tip. (b) 3-D electric energy density distribution at the end of probe tip.

Fig. 3.
Fig. 3.

Movie showing the propagation of (a) longitudinal, and [Media 1] (b) radial, components of the electric field. Only half of the cone is shown in this figure. [Media 2]

Fig. 4.
Fig. 4.

Normalized electric energy density distribution at 0nm, 5nm, 10nm distances from the tip.

Fig. 5.
Fig. 5.

Electric field enhancement factor with respect to half taper angle of probe.

Fig. 6.
Fig. 6.

Electric field enhancement factor with respect to illumination spot size.

Equations (1)

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E ( r ) = r exp [ ( r w ) 2 ] e r

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