Abstract

Heterodyne detection for apertureless near-field scanning optical microscopy was used to study periodic gold nanowell arrays. Optical near-field amplitude and phase signals were obtained simultaneously with the topography of the gold nanowells and with different polarizations. Theoretical calculations of the near-fields were consistent with the experiments; in particular, the calculated amplitudes were in especially good agreement. The heterodyne method is shown to be particularly effective for these types of periodic photonic structures and other highly scattering media, which can overwhelm the near-field scattered signal when conventional apertureless near-field scanning optical microscopy is used.

© 2007 Optical Society of America

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2006 (3)

L. Gomez, R. Bachelot, A. Bouhelier, G. P. Wiederrecht, S. H. Chang, S. K. Gray, G. Lerondel, F. Hua, S. Jeon, J. Rogers, S. Blaize, I. Stefanon, and P. Royer, “Apertureless scanning Near-field Optical Microscopy: a comparison between homodyne and heterodyne approaches,” J. Opt. Soc. Am. B 23,823–833 (2006).
[Crossref]

L. Aigouy, V. Mathet, and P. Beauvillainu, “Electromagnetic field distribution on a rough gold thin film: An experimental study as a function of the gold thickness,” Opt. Commun. 262,263–269 (2006).
[Crossref]

M. E. Stewart, N. H. Mack, V. Malyarchuk, J. A. N. T. Soares, T.-W. Lee, S. K. Gray, R. G. Nuzzo, and J. A. Rogers, “Quantitative multispectral biosensing and 1D imaging using quasi-3D plasmonic crystals,” PNAS 103,17143–17148 (2006).
[Crossref] [PubMed]

2005 (2)

2004 (6)

E. Hutter and J. Fendler, “Exploitation of Localized Surface Plasmon Resonance,” Adv. Mater. 16,1685–1706 (2004).
[Crossref]

J. Kim, J. H. Kim, and K. H. Park, “Local excitation of surface plasmon in sturctured Au films by atomic force anodic oxidation,” J. Vac. Sci. Technol. B 22,212–215 (2004).
[Crossref]

R. Quidant, G. Badenes, S. Cheylan, R. Alcubilla, J. C. Weeber, and C. Girard, “Sub-wavelength patterning of the optical near-field,” Opt. Express 12,282–7 (2004).
[Crossref] [PubMed]

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, “Surface plasmons at single nanoholes in Au films,” Appl. Phys. Lett. 85,467–469 (2004).
[Crossref]

R. Hillenbrand and F. Keilmann, “Near-field microscopy by elastic light scattering from a tip,” Phil. Trans. R. Soc. Lond. A 362,787–805 (2004).
[Crossref]

F. Hua, Y. Sun, A. Gaur, M. A. Meitl, L. Bilhaut, L. Rotkina, J. Wang, P. Geil, M. Shim, and J. A. Rogers, “Polymer Imprint Lithography with Molecular-Scale Resolution,” Nano Lett. 4,2467–2471 (2004).
[Crossref]

2003 (3)

R. Hillenbrand, F. Keilmann, P. Hanarp, D. S. Sutherland, and J. Aizpurua, “Coherent imaging of nanoscale plasmon patterns with a carbon nanotube probe,” Appl. Phys. Lett. 83,368–70 (2003).
[Crossref]

G. A. Wurtz, J. S. Im, S. K. Gray, and G. P. Wiederrecht, “Optical scattering from isolated metal nanoparticles and arrays,” J. Phys. Chem. B 107,14191 (2003).
[Crossref]

J. Seidel, S. Grafstrom, L. Eng, and L. Bischoff, “Surface plasmon transmission across narrow grooves in thin silver films,” Appl. Phys. Lett. 82,1368–70 (2003).
[Crossref]

2001 (1)

R. Hillenbrand and F. Keilmann, “Optical oscillation modes of plasmon particles observed in direct space by phase-contrast near-field microscopy,” Appl. Phys. B 73,239–243 (2001).
[Crossref]

2000 (2)

J. M. Brockman, B. P. Nelson, and R. M. Corn, “Surface plasmon resonance imaging measurements of Ultrathin Organic Films,” Annu. Rev. Phys. Chem. 51,41–63 (2000).
[Crossref] [PubMed]

R. Hillenbrand and F. Keilmann, “Complex Optical Constants on a subwavelength scale,” Phys. Rev. Lett. 85,3029–3032 (2000).
[Crossref] [PubMed]

1999 (1)

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54,3–15 (1999).
[Crossref]

1995 (1)

1994 (2)

Y. Inouye and S. Kawata, “Near-field scanning optical microscope with a metallic probe tip,” Opt. Lett. 19,159–161 (1994).
[Crossref] [PubMed]

F. Zenhausern, M. P. O’Boyle, and H. K. Wickramasinghe, “Apertureless near-field optical microscope,” App. Phys. Lett. 65,1623–1625 (1994).
[Crossref]

1985 (1)

Aigouy, L.

L. Aigouy, V. Mathet, and P. Beauvillainu, “Electromagnetic field distribution on a rough gold thin film: An experimental study as a function of the gold thickness,” Opt. Commun. 262,263–269 (2006).
[Crossref]

Aizpurua, J.

R. Hillenbrand, F. Keilmann, P. Hanarp, D. S. Sutherland, and J. Aizpurua, “Coherent imaging of nanoscale plasmon patterns with a carbon nanotube probe,” Appl. Phys. Lett. 83,368–70 (2003).
[Crossref]

Alcubilla, R.

Bachelot, R.

Badenes, G.

Beauvillainu, P.

L. Aigouy, V. Mathet, and P. Beauvillainu, “Electromagnetic field distribution on a rough gold thin film: An experimental study as a function of the gold thickness,” Opt. Commun. 262,263–269 (2006).
[Crossref]

Bilhaut, L.

F. Hua, Y. Sun, A. Gaur, M. A. Meitl, L. Bilhaut, L. Rotkina, J. Wang, P. Geil, M. Shim, and J. A. Rogers, “Polymer Imprint Lithography with Molecular-Scale Resolution,” Nano Lett. 4,2467–2471 (2004).
[Crossref]

Bischoff, L.

J. Seidel, S. Grafstrom, L. Eng, and L. Bischoff, “Surface plasmon transmission across narrow grooves in thin silver films,” Appl. Phys. Lett. 82,1368–70 (2003).
[Crossref]

Blaize, S.

Boccara, A. C.

Bouhelier, A.

Brockman, J. M.

J. M. Brockman, B. P. Nelson, and R. M. Corn, “Surface plasmon resonance imaging measurements of Ultrathin Organic Films,” Annu. Rev. Phys. Chem. 51,41–63 (2000).
[Crossref] [PubMed]

Brown, D. B.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, “Surface plasmons at single nanoholes in Au films,” Appl. Phys. Lett. 85,467–469 (2004).
[Crossref]

Chang, S. H.

Cheylan, S.

Corn, R. M.

J. M. Brockman, B. P. Nelson, and R. M. Corn, “Surface plasmon resonance imaging measurements of Ultrathin Organic Films,” Annu. Rev. Phys. Chem. 51,41–63 (2000).
[Crossref] [PubMed]

Eng, L.

J. Seidel, S. Grafstrom, L. Eng, and L. Bischoff, “Surface plasmon transmission across narrow grooves in thin silver films,” Appl. Phys. Lett. 82,1368–70 (2003).
[Crossref]

Fendler, J.

E. Hutter and J. Fendler, “Exploitation of Localized Surface Plasmon Resonance,” Adv. Mater. 16,1685–1706 (2004).
[Crossref]

Gauglitz, G.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54,3–15 (1999).
[Crossref]

Gaur, A.

F. Hua, Y. Sun, A. Gaur, M. A. Meitl, L. Bilhaut, L. Rotkina, J. Wang, P. Geil, M. Shim, and J. A. Rogers, “Polymer Imprint Lithography with Molecular-Scale Resolution,” Nano Lett. 4,2467–2471 (2004).
[Crossref]

Geil, P.

F. Hua, Y. Sun, A. Gaur, M. A. Meitl, L. Bilhaut, L. Rotkina, J. Wang, P. Geil, M. Shim, and J. A. Rogers, “Polymer Imprint Lithography with Molecular-Scale Resolution,” Nano Lett. 4,2467–2471 (2004).
[Crossref]

Girard, C.

Gleyzes, P.

Gomez, L.

Grafstrom, S.

J. Seidel, S. Grafstrom, L. Eng, and L. Bischoff, “Surface plasmon transmission across narrow grooves in thin silver films,” Appl. Phys. Lett. 82,1368–70 (2003).
[Crossref]

Gray, S. K.

M. E. Stewart, N. H. Mack, V. Malyarchuk, J. A. N. T. Soares, T.-W. Lee, S. K. Gray, R. G. Nuzzo, and J. A. Rogers, “Quantitative multispectral biosensing and 1D imaging using quasi-3D plasmonic crystals,” PNAS 103,17143–17148 (2006).
[Crossref] [PubMed]

L. Gomez, R. Bachelot, A. Bouhelier, G. P. Wiederrecht, S. H. Chang, S. K. Gray, G. Lerondel, F. Hua, S. Jeon, J. Rogers, S. Blaize, I. Stefanon, and P. Royer, “Apertureless scanning Near-field Optical Microscopy: a comparison between homodyne and heterodyne approaches,” J. Opt. Soc. Am. B 23,823–833 (2006).
[Crossref]

S. H. Chang, S. K. Gray, and G. C. Schatz, “Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films,” Opt. Express 13,3150–1365 (2005).
[Crossref] [PubMed]

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, “Surface plasmons at single nanoholes in Au films,” Appl. Phys. Lett. 85,467–469 (2004).
[Crossref]

G. A. Wurtz, J. S. Im, S. K. Gray, and G. P. Wiederrecht, “Optical scattering from isolated metal nanoparticles and arrays,” J. Phys. Chem. B 107,14191 (2003).
[Crossref]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 2000).

Hanarp, P.

R. Hillenbrand, F. Keilmann, P. Hanarp, D. S. Sutherland, and J. Aizpurua, “Coherent imaging of nanoscale plasmon patterns with a carbon nanotube probe,” Appl. Phys. Lett. 83,368–70 (2003).
[Crossref]

Hillenbrand, R.

R. Hillenbrand and F. Keilmann, “Near-field microscopy by elastic light scattering from a tip,” Phil. Trans. R. Soc. Lond. A 362,787–805 (2004).
[Crossref]

R. Hillenbrand, F. Keilmann, P. Hanarp, D. S. Sutherland, and J. Aizpurua, “Coherent imaging of nanoscale plasmon patterns with a carbon nanotube probe,” Appl. Phys. Lett. 83,368–70 (2003).
[Crossref]

R. Hillenbrand and F. Keilmann, “Optical oscillation modes of plasmon particles observed in direct space by phase-contrast near-field microscopy,” Appl. Phys. B 73,239–243 (2001).
[Crossref]

R. Hillenbrand and F. Keilmann, “Complex Optical Constants on a subwavelength scale,” Phys. Rev. Lett. 85,3029–3032 (2000).
[Crossref] [PubMed]

Homola, J.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B 54,3–15 (1999).
[Crossref]

Hua, F.

Hutter, E.

E. Hutter and J. Fendler, “Exploitation of Localized Surface Plasmon Resonance,” Adv. Mater. 16,1685–1706 (2004).
[Crossref]

Im, J. S.

G. A. Wurtz, J. S. Im, S. K. Gray, and G. P. Wiederrecht, “Optical scattering from isolated metal nanoparticles and arrays,” J. Phys. Chem. B 107,14191 (2003).
[Crossref]

Inouye, Y.

Jeon, S.

Kawata, S.

Keilmann, F.

R. Hillenbrand and F. Keilmann, “Near-field microscopy by elastic light scattering from a tip,” Phil. Trans. R. Soc. Lond. A 362,787–805 (2004).
[Crossref]

R. Hillenbrand, F. Keilmann, P. Hanarp, D. S. Sutherland, and J. Aizpurua, “Coherent imaging of nanoscale plasmon patterns with a carbon nanotube probe,” Appl. Phys. Lett. 83,368–70 (2003).
[Crossref]

R. Hillenbrand and F. Keilmann, “Optical oscillation modes of plasmon particles observed in direct space by phase-contrast near-field microscopy,” Appl. Phys. B 73,239–243 (2001).
[Crossref]

R. Hillenbrand and F. Keilmann, “Complex Optical Constants on a subwavelength scale,” Phys. Rev. Lett. 85,3029–3032 (2000).
[Crossref] [PubMed]

Kim, J.

J. Kim, J. H. Kim, and K. H. Park, “Local excitation of surface plasmon in sturctured Au films by atomic force anodic oxidation,” J. Vac. Sci. Technol. B 22,212–215 (2004).
[Crossref]

Kim, J. H.

J. Kim, J. H. Kim, and K. H. Park, “Local excitation of surface plasmon in sturctured Au films by atomic force anodic oxidation,” J. Vac. Sci. Technol. B 22,212–215 (2004).
[Crossref]

Kimball, C. W.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, “Surface plasmons at single nanoholes in Au films,” Appl. Phys. Lett. 85,467–469 (2004).
[Crossref]

Lee, T.-W.

M. E. Stewart, N. H. Mack, V. Malyarchuk, J. A. N. T. Soares, T.-W. Lee, S. K. Gray, R. G. Nuzzo, and J. A. Rogers, “Quantitative multispectral biosensing and 1D imaging using quasi-3D plasmonic crystals,” PNAS 103,17143–17148 (2006).
[Crossref] [PubMed]

Lerondel, G.

Mack, N. H.

M. E. Stewart, N. H. Mack, V. Malyarchuk, J. A. N. T. Soares, T.-W. Lee, S. K. Gray, R. G. Nuzzo, and J. A. Rogers, “Quantitative multispectral biosensing and 1D imaging using quasi-3D plasmonic crystals,” PNAS 103,17143–17148 (2006).
[Crossref] [PubMed]

V. Malyarchuk, F. Hua, N. H. Mack, V. T. Velesquez, J. O. White, R. G. Nuzzo, and J. A. Rogers, “A High performance plasmonic crystal sensor formed by soft nanoimprint lithography,” Opt. Express 13,5669–5675 (2005).
[Crossref] [PubMed]

Malyarchuk, V.

M. E. Stewart, N. H. Mack, V. Malyarchuk, J. A. N. T. Soares, T.-W. Lee, S. K. Gray, R. G. Nuzzo, and J. A. Rogers, “Quantitative multispectral biosensing and 1D imaging using quasi-3D plasmonic crystals,” PNAS 103,17143–17148 (2006).
[Crossref] [PubMed]

V. Malyarchuk, F. Hua, N. H. Mack, V. T. Velesquez, J. O. White, R. G. Nuzzo, and J. A. Rogers, “A High performance plasmonic crystal sensor formed by soft nanoimprint lithography,” Opt. Express 13,5669–5675 (2005).
[Crossref] [PubMed]

Mathet, V.

L. Aigouy, V. Mathet, and P. Beauvillainu, “Electromagnetic field distribution on a rough gold thin film: An experimental study as a function of the gold thickness,” Opt. Commun. 262,263–269 (2006).
[Crossref]

Meitl, M. A.

F. Hua, Y. Sun, A. Gaur, M. A. Meitl, L. Bilhaut, L. Rotkina, J. Wang, P. Geil, M. Shim, and J. A. Rogers, “Polymer Imprint Lithography with Molecular-Scale Resolution,” Nano Lett. 4,2467–2471 (2004).
[Crossref]

Nelson, B. P.

J. M. Brockman, B. P. Nelson, and R. M. Corn, “Surface plasmon resonance imaging measurements of Ultrathin Organic Films,” Annu. Rev. Phys. Chem. 51,41–63 (2000).
[Crossref] [PubMed]

Nuzzo, R. G.

M. E. Stewart, N. H. Mack, V. Malyarchuk, J. A. N. T. Soares, T.-W. Lee, S. K. Gray, R. G. Nuzzo, and J. A. Rogers, “Quantitative multispectral biosensing and 1D imaging using quasi-3D plasmonic crystals,” PNAS 103,17143–17148 (2006).
[Crossref] [PubMed]

V. Malyarchuk, F. Hua, N. H. Mack, V. T. Velesquez, J. O. White, R. G. Nuzzo, and J. A. Rogers, “A High performance plasmonic crystal sensor formed by soft nanoimprint lithography,” Opt. Express 13,5669–5675 (2005).
[Crossref] [PubMed]

O’Boyle, M. P.

F. Zenhausern, M. P. O’Boyle, and H. K. Wickramasinghe, “Apertureless near-field optical microscope,” App. Phys. Lett. 65,1623–1625 (1994).
[Crossref]

Park, K. H.

J. Kim, J. H. Kim, and K. H. Park, “Local excitation of surface plasmon in sturctured Au films by atomic force anodic oxidation,” J. Vac. Sci. Technol. B 22,212–215 (2004).
[Crossref]

Pearson, J.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, “Surface plasmons at single nanoholes in Au films,” Appl. Phys. Lett. 85,467–469 (2004).
[Crossref]

Quidant, R.

Rogers, J.

Rogers, J. A.

M. E. Stewart, N. H. Mack, V. Malyarchuk, J. A. N. T. Soares, T.-W. Lee, S. K. Gray, R. G. Nuzzo, and J. A. Rogers, “Quantitative multispectral biosensing and 1D imaging using quasi-3D plasmonic crystals,” PNAS 103,17143–17148 (2006).
[Crossref] [PubMed]

V. Malyarchuk, F. Hua, N. H. Mack, V. T. Velesquez, J. O. White, R. G. Nuzzo, and J. A. Rogers, “A High performance plasmonic crystal sensor formed by soft nanoimprint lithography,” Opt. Express 13,5669–5675 (2005).
[Crossref] [PubMed]

F. Hua, Y. Sun, A. Gaur, M. A. Meitl, L. Bilhaut, L. Rotkina, J. Wang, P. Geil, M. Shim, and J. A. Rogers, “Polymer Imprint Lithography with Molecular-Scale Resolution,” Nano Lett. 4,2467–2471 (2004).
[Crossref]

Rotkina, L.

F. Hua, Y. Sun, A. Gaur, M. A. Meitl, L. Bilhaut, L. Rotkina, J. Wang, P. Geil, M. Shim, and J. A. Rogers, “Polymer Imprint Lithography with Molecular-Scale Resolution,” Nano Lett. 4,2467–2471 (2004).
[Crossref]

Royer, P.

Rydh, A.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, “Surface plasmons at single nanoholes in Au films,” Appl. Phys. Lett. 85,467–469 (2004).
[Crossref]

Schatz, G. C.

S. H. Chang, S. K. Gray, and G. C. Schatz, “Surface plasmon generation and light transmission by isolated nanoholes and arrays of nanoholes in thin metal films,” Opt. Express 13,3150–1365 (2005).
[Crossref] [PubMed]

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, “Surface plasmons at single nanoholes in Au films,” Appl. Phys. Lett. 85,467–469 (2004).
[Crossref]

Seidel, J.

J. Seidel, S. Grafstrom, L. Eng, and L. Bischoff, “Surface plasmon transmission across narrow grooves in thin silver films,” Appl. Phys. Lett. 82,1368–70 (2003).
[Crossref]

Shim, M.

F. Hua, Y. Sun, A. Gaur, M. A. Meitl, L. Bilhaut, L. Rotkina, J. Wang, P. Geil, M. Shim, and J. A. Rogers, “Polymer Imprint Lithography with Molecular-Scale Resolution,” Nano Lett. 4,2467–2471 (2004).
[Crossref]

Soares, J. A. N. T.

M. E. Stewart, N. H. Mack, V. Malyarchuk, J. A. N. T. Soares, T.-W. Lee, S. K. Gray, R. G. Nuzzo, and J. A. Rogers, “Quantitative multispectral biosensing and 1D imaging using quasi-3D plasmonic crystals,” PNAS 103,17143–17148 (2006).
[Crossref] [PubMed]

Stefanon, I.

Stewart, M. E.

M. E. Stewart, N. H. Mack, V. Malyarchuk, J. A. N. T. Soares, T.-W. Lee, S. K. Gray, R. G. Nuzzo, and J. A. Rogers, “Quantitative multispectral biosensing and 1D imaging using quasi-3D plasmonic crystals,” PNAS 103,17143–17148 (2006).
[Crossref] [PubMed]

Sun, Y.

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L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, “Surface plasmons at single nanoholes in Au films,” Appl. Phys. Lett. 85,467–469 (2004).
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L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, “Surface plasmons at single nanoholes in Au films,” Appl. Phys. Lett. 85,467–469 (2004).
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J. Opt. Soc. Am. B (2)

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F. Hua, Y. Sun, A. Gaur, M. A. Meitl, L. Bilhaut, L. Rotkina, J. Wang, P. Geil, M. Shim, and J. A. Rogers, “Polymer Imprint Lithography with Molecular-Scale Resolution,” Nano Lett. 4,2467–2471 (2004).
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Opt. Commun. (1)

L. Aigouy, V. Mathet, and P. Beauvillainu, “Electromagnetic field distribution on a rough gold thin film: An experimental study as a function of the gold thickness,” Opt. Commun. 262,263–269 (2006).
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Other (1)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 2000).

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Figures (4)

Fig. 1.
Fig. 1.

(a). Schematic of homodyne detection method. IL is the incident light, Eb is the background light, Es near-field signal from the tip-sample interaction, LD lock-in detector system. (b) Schematic of heterodyne detection method. AOM is the acousto-optical modulators, Er is the reference beam or Δω, BS is a beam splitter, O is the focusing/collecting objective.

Fig. 2.
Fig. 2.

NSOM images of gold nanowells. (a) Topography, (b) homodyne detected optical near-field amplitude taken at tapping mode frequency f, (c) heterodyne detected optical near-field amplitude (d) phase image taken at 2f-Δω.[10] The incident light travels from the bottom of the images

Fig. 3.
Fig. 3.

Theoretical FDTD calculations and experimental NSOM images of the near-field amplitude signal of gold nanowell films. Each left-side panel is a theoretical result for the specified incident/observed polarization state, with the corresponding experimental result to its right. The incident light travels from the bottom of the images.

Fig. 4.
Fig. 4.

Theoretical FDTD calculations and experimental NSOM images of the near-phase signal [-π, π] of gold nanowell films. Each left-side panel is a theoretical result for the specified incident/observed polarization state, with the corresponding experimental result to its right. The incident light travels from the bottom of the images

Equations (2)

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E s 2 + 2 E s E b cos ( ϕ b ϕ s ) ,
2 E s E r cos ( Δ ω t + ϕ r ϕ s ) ,

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