Abstract

This study constructs interference-based model of the apertureless scanning near-field optical microscopy (A-SNOM) heterodyne detection signal which takes account of both the tip enhancement phenomena and the tip reflective background electric field. The analytical model not only provides a meaningful explanation of the image artifacts and errors, but also suggests methods for reducing these effects. It is shown that the detection signal obtained in the heterodyne A-SNOM method has a significantly higher signal-to-background (S/B) ratio than in the homodyne method. It is also shown that the S/B ratio increases as the wavelength of the illuminating light source is increased or the incident angle is reduced. Finally, an inspection reveals two fundamental phenomena which may potentially be exploited to obtain further significant improvements, namely (1) the modulation depth parameter has certain specific values greater than 1; and (2) the AFM tip apparatus using a ramp function.

© 2008 Optical Society of America

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  1. E. Betzing and M. Isaacson, "Collection mode near-field scanning optical microscopy," Appl. Phys. Lett. 51, 2088-2090 (1987).
    [CrossRef]
  2. E. Betzing, 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]
  3. S. Patane, G. G. Gucciardi, M. Labardi, and M. Allegrini, "Apertureless near-field optical microscopy," Rivita Del Nuovo Cimento 27, 1-46 (2004).
  4. J. Wessel, "Surface-enhanced optical microscopy," J. Opt. Soc. Am. 2, 1538-1540 (1985).
    [CrossRef]
  5. H. K. Wickramasinghe and C. C. Williams, "Apertureless near field optical microscope," US Patent 4 947 034 (1990).
  6. Y. Inouye and S. Kawata, "Near-field scanning optical microscope with a metallic probe tip," Opt. Lett. 19, 159-161 (1994).
    [CrossRef] [PubMed]
  7. R. Hillenbrand and F. Keilmann, "Complex optical constants on a subwavelength scale," Phys. Rev. Lett. 85, 3029-3032 (2000).
    [CrossRef] [PubMed]
  8. R. Hillenbrand, B. Knoll, and F. Keilmann, "Pure optical contrast in scattering-type scanning near-field microscopy," J. Microsc. 202, 77-83 (2000).
    [CrossRef]
  9. B. Knoll and F. Keilmann, "Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy," Opt. Commun. 182, 321-328 (2000).
    [CrossRef]
  10. S. Hudlet, S. Aubert, A. Bruyant, R. Bachelot, P. M. Adam, J. L. Bijeon, G. Lerondel, P. Royer, and A. A. Stashkevich, "Apertureless near field optical microscopy: a contribution to the understanding of the signal detected in the presence of background field," Opt. Commun. 230, 245-251 (2004).
    [CrossRef]
  11. F. Formanek, Y. D. Wilde, and L. Aigouy, "Analysis of the measured signals in apertureless near-field optical microscopy," Ultramicroscopy 103, 133-139 (2005).
    [CrossRef] [PubMed]
  12. P. G. Gucciardi, G. Bachelier, and M. Allegrini, "Far-field background suppression in tip-modulated apertureless near-field optical microscopy," J. Appl. Phys. 99, Art. No. 124309 (2006).
    [CrossRef]
  13. P. G. Gucciardi, G. Bachelier, M. Allegrini, J. Ahn, M. Hong, S. Chang, W. Jhe, S. C. Hong, and S. H. Baek, "Artifacts identification in apertureless near-field optical microscopy," J. Appl. Phys. 101, Art. No. 064303 (2007).
    [CrossRef]
  14. L. Billot, M. Lamy de la Chapelle, D. Barchiesi, S. H. Chang, S. K. Gray, J. A. Rogers, A. Bouhelier, P. M. Adam, J. L Bijeon, G. P. Wiederrecht, R. Bachelot, and P. Royer, "Error signal artifact in apertureless scanning near-field optical microscopy," Appl. Phys. Lett. 89, Art. No. 023105 (2006).
  15. C. H. Chuang and Y. L. Lo, "Analytical Analysis of Modulated Signal in Apertureless Scattering Near-field Optical Microscopy," Opt. Express 15, 15782-15796 (2007).
    [CrossRef] [PubMed]
  16. L. Gomez, R. Bachelot, A. Bouhelier, G. P. Wiederrecht, S. Chang, S. K. Gary, F. Hua, S. Jeon, J. A. Rogers, M. E. Castro, S. Blaize, I. Stefanon, G. Lerondel, 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]
  17. I. Stefanon, S. Blaize, A. Bruyant, S Aubert, G. Lerondel, R. Bachelot, and P. Royer, "Heterodyne detection of guided waves using a scattering-type scanning near-field optical microscope," Opt. Express 13, 5553-5564 (2005).
    [CrossRef] [PubMed]
  18. J. N. Walford, J. A. Porto, R. Carminati, J. J. Greffet, P. M. Adam, S. Hudlet, J. L. Bijeon, A. Stashkevich, and P. Royer, "Influence of tip modulation on image formation in scanning near-field optical microscopy," J. Appl. Phys. 89, 5159-5169 (2001).
    [CrossRef]
  19. F. Keilmann and R. Hillenbrand, "Near-field microscopy by elastic light scattering from a tip," Phil. Trans. R. Soc. Lond. A. 362, 787-805 (2004).
    [CrossRef]
  20. M. Micic, N. Klymyshyn, Y. D. Sun, and H. P. Lu, "Finite element method simulation of the field distribution for AFM tip-enhanced surface Raman Scanning Microscopy," J. Phys. Chem. B. 107, 1574-1584 (2003).
    [CrossRef]

2007 (2)

C. H. Chuang and Y. L. Lo, "Analytical Analysis of Modulated Signal in Apertureless Scattering Near-field Optical Microscopy," Opt. Express 15, 15782-15796 (2007).
[CrossRef] [PubMed]

P. G. Gucciardi, G. Bachelier, M. Allegrini, J. Ahn, M. Hong, S. Chang, W. Jhe, S. C. Hong, and S. H. Baek, "Artifacts identification in apertureless near-field optical microscopy," J. Appl. Phys. 101, Art. No. 064303 (2007).
[CrossRef]

2006 (2)

L. Billot, M. Lamy de la Chapelle, D. Barchiesi, S. H. Chang, S. K. Gray, J. A. Rogers, A. Bouhelier, P. M. Adam, J. L Bijeon, G. P. Wiederrecht, R. Bachelot, and P. Royer, "Error signal artifact in apertureless scanning near-field optical microscopy," Appl. Phys. Lett. 89, Art. No. 023105 (2006).

L. Gomez, R. Bachelot, A. Bouhelier, G. P. Wiederrecht, S. Chang, S. K. Gary, F. Hua, S. Jeon, J. A. Rogers, M. E. Castro, S. Blaize, I. Stefanon, G. Lerondel, 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]

2005 (2)

2004 (3)

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

S. Patane, G. G. Gucciardi, M. Labardi, and M. Allegrini, "Apertureless near-field optical microscopy," Rivita Del Nuovo Cimento 27, 1-46 (2004).

S. Hudlet, S. Aubert, A. Bruyant, R. Bachelot, P. M. Adam, J. L. Bijeon, G. Lerondel, P. Royer, and A. A. Stashkevich, "Apertureless near field optical microscopy: a contribution to the understanding of the signal detected in the presence of background field," Opt. Commun. 230, 245-251 (2004).
[CrossRef]

2003 (1)

M. Micic, N. Klymyshyn, Y. D. Sun, and H. P. Lu, "Finite element method simulation of the field distribution for AFM tip-enhanced surface Raman Scanning Microscopy," J. Phys. Chem. B. 107, 1574-1584 (2003).
[CrossRef]

2001 (1)

J. N. Walford, J. A. Porto, R. Carminati, J. J. Greffet, P. M. Adam, S. Hudlet, J. L. Bijeon, A. Stashkevich, and P. Royer, "Influence of tip modulation on image formation in scanning near-field optical microscopy," J. Appl. Phys. 89, 5159-5169 (2001).
[CrossRef]

2000 (3)

R. Hillenbrand and F. Keilmann, "Complex optical constants on a subwavelength scale," Phys. Rev. Lett. 85, 3029-3032 (2000).
[CrossRef] [PubMed]

R. Hillenbrand, B. Knoll, and F. Keilmann, "Pure optical contrast in scattering-type scanning near-field microscopy," J. Microsc. 202, 77-83 (2000).
[CrossRef]

B. Knoll and F. Keilmann, "Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy," Opt. Commun. 182, 321-328 (2000).
[CrossRef]

1994 (1)

1991 (1)

E. Betzing, 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]

1987 (1)

E. Betzing and M. Isaacson, "Collection mode near-field scanning optical microscopy," Appl. Phys. Lett. 51, 2088-2090 (1987).
[CrossRef]

1985 (1)

J. Wessel, "Surface-enhanced optical microscopy," J. Opt. Soc. Am. 2, 1538-1540 (1985).
[CrossRef]

Adam, P. M.

L. Billot, M. Lamy de la Chapelle, D. Barchiesi, S. H. Chang, S. K. Gray, J. A. Rogers, A. Bouhelier, P. M. Adam, J. L Bijeon, G. P. Wiederrecht, R. Bachelot, and P. Royer, "Error signal artifact in apertureless scanning near-field optical microscopy," Appl. Phys. Lett. 89, Art. No. 023105 (2006).

S. Hudlet, S. Aubert, A. Bruyant, R. Bachelot, P. M. Adam, J. L. Bijeon, G. Lerondel, P. Royer, and A. A. Stashkevich, "Apertureless near field optical microscopy: a contribution to the understanding of the signal detected in the presence of background field," Opt. Commun. 230, 245-251 (2004).
[CrossRef]

J. N. Walford, J. A. Porto, R. Carminati, J. J. Greffet, P. M. Adam, S. Hudlet, J. L. Bijeon, A. Stashkevich, and P. Royer, "Influence of tip modulation on image formation in scanning near-field optical microscopy," J. Appl. Phys. 89, 5159-5169 (2001).
[CrossRef]

Ahn, J.

P. G. Gucciardi, G. Bachelier, M. Allegrini, J. Ahn, M. Hong, S. Chang, W. Jhe, S. C. Hong, and S. H. Baek, "Artifacts identification in apertureless near-field optical microscopy," J. Appl. Phys. 101, Art. No. 064303 (2007).
[CrossRef]

Aigouy, L.

F. Formanek, Y. D. Wilde, and L. Aigouy, "Analysis of the measured signals in apertureless near-field optical microscopy," Ultramicroscopy 103, 133-139 (2005).
[CrossRef] [PubMed]

Allegrini, M.

P. G. Gucciardi, G. Bachelier, M. Allegrini, J. Ahn, M. Hong, S. Chang, W. Jhe, S. C. Hong, and S. H. Baek, "Artifacts identification in apertureless near-field optical microscopy," J. Appl. Phys. 101, Art. No. 064303 (2007).
[CrossRef]

S. Patane, G. G. Gucciardi, M. Labardi, and M. Allegrini, "Apertureless near-field optical microscopy," Rivita Del Nuovo Cimento 27, 1-46 (2004).

Aubert, S

Aubert, S.

S. Hudlet, S. Aubert, A. Bruyant, R. Bachelot, P. M. Adam, J. L. Bijeon, G. Lerondel, P. Royer, and A. A. Stashkevich, "Apertureless near field optical microscopy: a contribution to the understanding of the signal detected in the presence of background field," Opt. Commun. 230, 245-251 (2004).
[CrossRef]

Bachelier, G.

P. G. Gucciardi, G. Bachelier, M. Allegrini, J. Ahn, M. Hong, S. Chang, W. Jhe, S. C. Hong, and S. H. Baek, "Artifacts identification in apertureless near-field optical microscopy," J. Appl. Phys. 101, Art. No. 064303 (2007).
[CrossRef]

Bachelot, R.

L. Billot, M. Lamy de la Chapelle, D. Barchiesi, S. H. Chang, S. K. Gray, J. A. Rogers, A. Bouhelier, P. M. Adam, J. L Bijeon, G. P. Wiederrecht, R. Bachelot, and P. Royer, "Error signal artifact in apertureless scanning near-field optical microscopy," Appl. Phys. Lett. 89, Art. No. 023105 (2006).

L. Gomez, R. Bachelot, A. Bouhelier, G. P. Wiederrecht, S. Chang, S. K. Gary, F. Hua, S. Jeon, J. A. Rogers, M. E. Castro, S. Blaize, I. Stefanon, G. Lerondel, 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]

I. Stefanon, S. Blaize, A. Bruyant, S Aubert, G. Lerondel, R. Bachelot, and P. Royer, "Heterodyne detection of guided waves using a scattering-type scanning near-field optical microscope," Opt. Express 13, 5553-5564 (2005).
[CrossRef] [PubMed]

S. Hudlet, S. Aubert, A. Bruyant, R. Bachelot, P. M. Adam, J. L. Bijeon, G. Lerondel, P. Royer, and A. A. Stashkevich, "Apertureless near field optical microscopy: a contribution to the understanding of the signal detected in the presence of background field," Opt. Commun. 230, 245-251 (2004).
[CrossRef]

Baek, S. H.

P. G. Gucciardi, G. Bachelier, M. Allegrini, J. Ahn, M. Hong, S. Chang, W. Jhe, S. C. Hong, and S. H. Baek, "Artifacts identification in apertureless near-field optical microscopy," J. Appl. Phys. 101, Art. No. 064303 (2007).
[CrossRef]

Barchies, D.

L. Billot, M. Lamy de la Chapelle, D. Barchiesi, S. H. Chang, S. K. Gray, J. A. Rogers, A. Bouhelier, P. M. Adam, J. L Bijeon, G. P. Wiederrecht, R. Bachelot, and P. Royer, "Error signal artifact in apertureless scanning near-field optical microscopy," Appl. Phys. Lett. 89, Art. No. 023105 (2006).

Betzing, E.

E. Betzing, 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]

E. Betzing and M. Isaacson, "Collection mode near-field scanning optical microscopy," Appl. Phys. Lett. 51, 2088-2090 (1987).
[CrossRef]

Bijeon, J. L.

L. Billot, M. Lamy de la Chapelle, D. Barchiesi, S. H. Chang, S. K. Gray, J. A. Rogers, A. Bouhelier, P. M. Adam, J. L Bijeon, G. P. Wiederrecht, R. Bachelot, and P. Royer, "Error signal artifact in apertureless scanning near-field optical microscopy," Appl. Phys. Lett. 89, Art. No. 023105 (2006).

S. Hudlet, S. Aubert, A. Bruyant, R. Bachelot, P. M. Adam, J. L. Bijeon, G. Lerondel, P. Royer, and A. A. Stashkevich, "Apertureless near field optical microscopy: a contribution to the understanding of the signal detected in the presence of background field," Opt. Commun. 230, 245-251 (2004).
[CrossRef]

J. N. Walford, J. A. Porto, R. Carminati, J. J. Greffet, P. M. Adam, S. Hudlet, J. L. Bijeon, A. Stashkevich, and P. Royer, "Influence of tip modulation on image formation in scanning near-field optical microscopy," J. Appl. Phys. 89, 5159-5169 (2001).
[CrossRef]

Billot, L.

L. Billot, M. Lamy de la Chapelle, D. Barchiesi, S. H. Chang, S. K. Gray, J. A. Rogers, A. Bouhelier, P. M. Adam, J. L Bijeon, G. P. Wiederrecht, R. Bachelot, and P. Royer, "Error signal artifact in apertureless scanning near-field optical microscopy," Appl. Phys. Lett. 89, Art. No. 023105 (2006).

Blaize, S.

Bouhelier, A.

L. Gomez, R. Bachelot, A. Bouhelier, G. P. Wiederrecht, S. Chang, S. K. Gary, F. Hua, S. Jeon, J. A. Rogers, M. E. Castro, S. Blaize, I. Stefanon, G. Lerondel, 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. Billot, M. Lamy de la Chapelle, D. Barchiesi, S. H. Chang, S. K. Gray, J. A. Rogers, A. Bouhelier, P. M. Adam, J. L Bijeon, G. P. Wiederrecht, R. Bachelot, and P. Royer, "Error signal artifact in apertureless scanning near-field optical microscopy," Appl. Phys. Lett. 89, Art. No. 023105 (2006).

Bruyant, A.

I. Stefanon, S. Blaize, A. Bruyant, S Aubert, G. Lerondel, R. Bachelot, and P. Royer, "Heterodyne detection of guided waves using a scattering-type scanning near-field optical microscope," Opt. Express 13, 5553-5564 (2005).
[CrossRef] [PubMed]

S. Hudlet, S. Aubert, A. Bruyant, R. Bachelot, P. M. Adam, J. L. Bijeon, G. Lerondel, P. Royer, and A. A. Stashkevich, "Apertureless near field optical microscopy: a contribution to the understanding of the signal detected in the presence of background field," Opt. Commun. 230, 245-251 (2004).
[CrossRef]

Carminati, R.

J. N. Walford, J. A. Porto, R. Carminati, J. J. Greffet, P. M. Adam, S. Hudlet, J. L. Bijeon, A. Stashkevich, and P. Royer, "Influence of tip modulation on image formation in scanning near-field optical microscopy," J. Appl. Phys. 89, 5159-5169 (2001).
[CrossRef]

Castro, M. E.

Chang, S.

Chang, S. H.

L. Billot, M. Lamy de la Chapelle, D. Barchiesi, S. H. Chang, S. K. Gray, J. A. Rogers, A. Bouhelier, P. M. Adam, J. L Bijeon, G. P. Wiederrecht, R. Bachelot, and P. Royer, "Error signal artifact in apertureless scanning near-field optical microscopy," Appl. Phys. Lett. 89, Art. No. 023105 (2006).

Chuang, C. H.

Formanek, F.

F. Formanek, Y. D. Wilde, and L. Aigouy, "Analysis of the measured signals in apertureless near-field optical microscopy," Ultramicroscopy 103, 133-139 (2005).
[CrossRef] [PubMed]

Gary, S. K.

Gomez, L.

Gray, S. K.

L. Billot, M. Lamy de la Chapelle, D. Barchiesi, S. H. Chang, S. K. Gray, J. A. Rogers, A. Bouhelier, P. M. Adam, J. L Bijeon, G. P. Wiederrecht, R. Bachelot, and P. Royer, "Error signal artifact in apertureless scanning near-field optical microscopy," Appl. Phys. Lett. 89, Art. No. 023105 (2006).

Greffet, J. J.

J. N. Walford, J. A. Porto, R. Carminati, J. J. Greffet, P. M. Adam, S. Hudlet, J. L. Bijeon, A. Stashkevich, and P. Royer, "Influence of tip modulation on image formation in scanning near-field optical microscopy," J. Appl. Phys. 89, 5159-5169 (2001).
[CrossRef]

Gucciardi, G. G.

S. Patane, G. G. Gucciardi, M. Labardi, and M. Allegrini, "Apertureless near-field optical microscopy," Rivita Del Nuovo Cimento 27, 1-46 (2004).

Gucciardi, P. G.

P. G. Gucciardi, G. Bachelier, M. Allegrini, J. Ahn, M. Hong, S. Chang, W. Jhe, S. C. Hong, and S. H. Baek, "Artifacts identification in apertureless near-field optical microscopy," J. Appl. Phys. 101, Art. No. 064303 (2007).
[CrossRef]

Harris, T. D.

E. Betzing, 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]

Hillenbrand, R.

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

R. Hillenbrand and F. Keilmann, "Complex optical constants on a subwavelength scale," Phys. Rev. Lett. 85, 3029-3032 (2000).
[CrossRef] [PubMed]

R. Hillenbrand, B. Knoll, and F. Keilmann, "Pure optical contrast in scattering-type scanning near-field microscopy," J. Microsc. 202, 77-83 (2000).
[CrossRef]

Hong, M.

P. G. Gucciardi, G. Bachelier, M. Allegrini, J. Ahn, M. Hong, S. Chang, W. Jhe, S. C. Hong, and S. H. Baek, "Artifacts identification in apertureless near-field optical microscopy," J. Appl. Phys. 101, Art. No. 064303 (2007).
[CrossRef]

Hong, S. C.

P. G. Gucciardi, G. Bachelier, M. Allegrini, J. Ahn, M. Hong, S. Chang, W. Jhe, S. C. Hong, and S. H. Baek, "Artifacts identification in apertureless near-field optical microscopy," J. Appl. Phys. 101, Art. No. 064303 (2007).
[CrossRef]

Hua, F.

Hudlet, S.

S. Hudlet, S. Aubert, A. Bruyant, R. Bachelot, P. M. Adam, J. L. Bijeon, G. Lerondel, P. Royer, and A. A. Stashkevich, "Apertureless near field optical microscopy: a contribution to the understanding of the signal detected in the presence of background field," Opt. Commun. 230, 245-251 (2004).
[CrossRef]

J. N. Walford, J. A. Porto, R. Carminati, J. J. Greffet, P. M. Adam, S. Hudlet, J. L. Bijeon, A. Stashkevich, and P. Royer, "Influence of tip modulation on image formation in scanning near-field optical microscopy," J. Appl. Phys. 89, 5159-5169 (2001).
[CrossRef]

Inouye, Y.

Isaacson, M.

E. Betzing and M. Isaacson, "Collection mode near-field scanning optical microscopy," Appl. Phys. Lett. 51, 2088-2090 (1987).
[CrossRef]

Jeon, S.

Jhe, W.

P. G. Gucciardi, G. Bachelier, M. Allegrini, J. Ahn, M. Hong, S. Chang, W. Jhe, S. C. Hong, and S. H. Baek, "Artifacts identification in apertureless near-field optical microscopy," J. Appl. Phys. 101, Art. No. 064303 (2007).
[CrossRef]

Kawata, S.

Keilmann, F.

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

B. Knoll and F. Keilmann, "Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy," Opt. Commun. 182, 321-328 (2000).
[CrossRef]

R. Hillenbrand, B. Knoll, and F. Keilmann, "Pure optical contrast in scattering-type scanning near-field microscopy," J. Microsc. 202, 77-83 (2000).
[CrossRef]

R. Hillenbrand and F. Keilmann, "Complex optical constants on a subwavelength scale," Phys. Rev. Lett. 85, 3029-3032 (2000).
[CrossRef] [PubMed]

Klymyshyn, N.

M. Micic, N. Klymyshyn, Y. D. Sun, and H. P. Lu, "Finite element method simulation of the field distribution for AFM tip-enhanced surface Raman Scanning Microscopy," J. Phys. Chem. B. 107, 1574-1584 (2003).
[CrossRef]

Knoll, B.

R. Hillenbrand, B. Knoll, and F. Keilmann, "Pure optical contrast in scattering-type scanning near-field microscopy," J. Microsc. 202, 77-83 (2000).
[CrossRef]

B. Knoll and F. Keilmann, "Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy," Opt. Commun. 182, 321-328 (2000).
[CrossRef]

Kostelak, R. L.

E. Betzing, 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]

Labardi, M.

S. Patane, G. G. Gucciardi, M. Labardi, and M. Allegrini, "Apertureless near-field optical microscopy," Rivita Del Nuovo Cimento 27, 1-46 (2004).

Lamy de la Chapelle, M.

L. Billot, M. Lamy de la Chapelle, D. Barchiesi, S. H. Chang, S. K. Gray, J. A. Rogers, A. Bouhelier, P. M. Adam, J. L Bijeon, G. P. Wiederrecht, R. Bachelot, and P. Royer, "Error signal artifact in apertureless scanning near-field optical microscopy," Appl. Phys. Lett. 89, Art. No. 023105 (2006).

Lerondel, G.

Lo, Y. L.

Lu, H. P.

M. Micic, N. Klymyshyn, Y. D. Sun, and H. P. Lu, "Finite element method simulation of the field distribution for AFM tip-enhanced surface Raman Scanning Microscopy," J. Phys. Chem. B. 107, 1574-1584 (2003).
[CrossRef]

Micic, M.

M. Micic, N. Klymyshyn, Y. D. Sun, and H. P. Lu, "Finite element method simulation of the field distribution for AFM tip-enhanced surface Raman Scanning Microscopy," J. Phys. Chem. B. 107, 1574-1584 (2003).
[CrossRef]

Patane, S.

S. Patane, G. G. Gucciardi, M. Labardi, and M. Allegrini, "Apertureless near-field optical microscopy," Rivita Del Nuovo Cimento 27, 1-46 (2004).

Porto, J. A.

J. N. Walford, J. A. Porto, R. Carminati, J. J. Greffet, P. M. Adam, S. Hudlet, J. L. Bijeon, A. Stashkevich, and P. Royer, "Influence of tip modulation on image formation in scanning near-field optical microscopy," J. Appl. Phys. 89, 5159-5169 (2001).
[CrossRef]

Rogers, J. A.

L. Billot, M. Lamy de la Chapelle, D. Barchiesi, S. H. Chang, S. K. Gray, J. A. Rogers, A. Bouhelier, P. M. Adam, J. L Bijeon, G. P. Wiederrecht, R. Bachelot, and P. Royer, "Error signal artifact in apertureless scanning near-field optical microscopy," Appl. Phys. Lett. 89, Art. No. 023105 (2006).

L. Gomez, R. Bachelot, A. Bouhelier, G. P. Wiederrecht, S. Chang, S. K. Gary, F. Hua, S. Jeon, J. A. Rogers, M. E. Castro, S. Blaize, I. Stefanon, G. Lerondel, 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]

Royer, P.

L. Gomez, R. Bachelot, A. Bouhelier, G. P. Wiederrecht, S. Chang, S. K. Gary, F. Hua, S. Jeon, J. A. Rogers, M. E. Castro, S. Blaize, I. Stefanon, G. Lerondel, 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. Billot, M. Lamy de la Chapelle, D. Barchiesi, S. H. Chang, S. K. Gray, J. A. Rogers, A. Bouhelier, P. M. Adam, J. L Bijeon, G. P. Wiederrecht, R. Bachelot, and P. Royer, "Error signal artifact in apertureless scanning near-field optical microscopy," Appl. Phys. Lett. 89, Art. No. 023105 (2006).

I. Stefanon, S. Blaize, A. Bruyant, S Aubert, G. Lerondel, R. Bachelot, and P. Royer, "Heterodyne detection of guided waves using a scattering-type scanning near-field optical microscope," Opt. Express 13, 5553-5564 (2005).
[CrossRef] [PubMed]

S. Hudlet, S. Aubert, A. Bruyant, R. Bachelot, P. M. Adam, J. L. Bijeon, G. Lerondel, P. Royer, and A. A. Stashkevich, "Apertureless near field optical microscopy: a contribution to the understanding of the signal detected in the presence of background field," Opt. Commun. 230, 245-251 (2004).
[CrossRef]

J. N. Walford, J. A. Porto, R. Carminati, J. J. Greffet, P. M. Adam, S. Hudlet, J. L. Bijeon, A. Stashkevich, and P. Royer, "Influence of tip modulation on image formation in scanning near-field optical microscopy," J. Appl. Phys. 89, 5159-5169 (2001).
[CrossRef]

Stashkevich, A.

J. N. Walford, J. A. Porto, R. Carminati, J. J. Greffet, P. M. Adam, S. Hudlet, J. L. Bijeon, A. Stashkevich, and P. Royer, "Influence of tip modulation on image formation in scanning near-field optical microscopy," J. Appl. Phys. 89, 5159-5169 (2001).
[CrossRef]

Stashkevich, A. A.

S. Hudlet, S. Aubert, A. Bruyant, R. Bachelot, P. M. Adam, J. L. Bijeon, G. Lerondel, P. Royer, and A. A. Stashkevich, "Apertureless near field optical microscopy: a contribution to the understanding of the signal detected in the presence of background field," Opt. Commun. 230, 245-251 (2004).
[CrossRef]

Stefanon, I.

Sun, Y. D.

M. Micic, N. Klymyshyn, Y. D. Sun, and H. P. Lu, "Finite element method simulation of the field distribution for AFM tip-enhanced surface Raman Scanning Microscopy," J. Phys. Chem. B. 107, 1574-1584 (2003).
[CrossRef]

Trautman, J. K.

E. Betzing, 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]

Walford, J. N.

J. N. Walford, J. A. Porto, R. Carminati, J. J. Greffet, P. M. Adam, S. Hudlet, J. L. Bijeon, A. Stashkevich, and P. Royer, "Influence of tip modulation on image formation in scanning near-field optical microscopy," J. Appl. Phys. 89, 5159-5169 (2001).
[CrossRef]

Weiner, J. S.

E. Betzing, 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]

Wessel, J.

J. Wessel, "Surface-enhanced optical microscopy," J. Opt. Soc. Am. 2, 1538-1540 (1985).
[CrossRef]

Wiederrecht, G. P.

L. Billot, M. Lamy de la Chapelle, D. Barchiesi, S. H. Chang, S. K. Gray, J. A. Rogers, A. Bouhelier, P. M. Adam, J. L Bijeon, G. P. Wiederrecht, R. Bachelot, and P. Royer, "Error signal artifact in apertureless scanning near-field optical microscopy," Appl. Phys. Lett. 89, Art. No. 023105 (2006).

L. Gomez, R. Bachelot, A. Bouhelier, G. P. Wiederrecht, S. Chang, S. K. Gary, F. Hua, S. Jeon, J. A. Rogers, M. E. Castro, S. Blaize, I. Stefanon, G. Lerondel, 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]

Wilde, Y. D.

F. Formanek, Y. D. Wilde, and L. Aigouy, "Analysis of the measured signals in apertureless near-field optical microscopy," Ultramicroscopy 103, 133-139 (2005).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

E. Betzing and M. Isaacson, "Collection mode near-field scanning optical microscopy," Appl. Phys. Lett. 51, 2088-2090 (1987).
[CrossRef]

L. Billot, M. Lamy de la Chapelle, D. Barchiesi, S. H. Chang, S. K. Gray, J. A. Rogers, A. Bouhelier, P. M. Adam, J. L Bijeon, G. P. Wiederrecht, R. Bachelot, and P. Royer, "Error signal artifact in apertureless scanning near-field optical microscopy," Appl. Phys. Lett. 89, Art. No. 023105 (2006).

J. Appl. Phys. (2)

P. G. Gucciardi, G. Bachelier, M. Allegrini, J. Ahn, M. Hong, S. Chang, W. Jhe, S. C. Hong, and S. H. Baek, "Artifacts identification in apertureless near-field optical microscopy," J. Appl. Phys. 101, Art. No. 064303 (2007).
[CrossRef]

J. N. Walford, J. A. Porto, R. Carminati, J. J. Greffet, P. M. Adam, S. Hudlet, J. L. Bijeon, A. Stashkevich, and P. Royer, "Influence of tip modulation on image formation in scanning near-field optical microscopy," J. Appl. Phys. 89, 5159-5169 (2001).
[CrossRef]

J. Microsc. (1)

R. Hillenbrand, B. Knoll, and F. Keilmann, "Pure optical contrast in scattering-type scanning near-field microscopy," J. Microsc. 202, 77-83 (2000).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Wessel, "Surface-enhanced optical microscopy," J. Opt. Soc. Am. 2, 1538-1540 (1985).
[CrossRef]

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

J. Phys. Chem. B. (1)

M. Micic, N. Klymyshyn, Y. D. Sun, and H. P. Lu, "Finite element method simulation of the field distribution for AFM tip-enhanced surface Raman Scanning Microscopy," J. Phys. Chem. B. 107, 1574-1584 (2003).
[CrossRef]

Opt. Commun. (2)

B. Knoll and F. Keilmann, "Enhanced dielectric contrast in scattering-type scanning near-field optical microscopy," Opt. Commun. 182, 321-328 (2000).
[CrossRef]

S. Hudlet, S. Aubert, A. Bruyant, R. Bachelot, P. M. Adam, J. L. Bijeon, G. Lerondel, P. Royer, and A. A. Stashkevich, "Apertureless near field optical microscopy: a contribution to the understanding of the signal detected in the presence of background field," Opt. Commun. 230, 245-251 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phil. Trans. R. Soc. Lond. A. (1)

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

Phys. Rev. Lett. (1)

R. Hillenbrand and F. Keilmann, "Complex optical constants on a subwavelength scale," Phys. Rev. Lett. 85, 3029-3032 (2000).
[CrossRef] [PubMed]

Rivita Del Nuovo Cimento (1)

S. Patane, G. G. Gucciardi, M. Labardi, and M. Allegrini, "Apertureless near-field optical microscopy," Rivita Del Nuovo Cimento 27, 1-46 (2004).

Science (1)

E. Betzing, 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]

Ultramicroscopy (1)

F. Formanek, Y. D. Wilde, and L. Aigouy, "Analysis of the measured signals in apertureless near-field optical microscopy," Ultramicroscopy 103, 133-139 (2005).
[CrossRef] [PubMed]

Other (2)

P. G. Gucciardi, G. Bachelier, and M. Allegrini, "Far-field background suppression in tip-modulated apertureless near-field optical microscopy," J. Appl. Phys. 99, Art. No. 124309 (2006).
[CrossRef]

H. K. Wickramasinghe and C. C. Williams, "Apertureless near field optical microscope," US Patent 4 947 034 (1990).

Cited By

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Figures (13)

Fig. 1.
Fig. 1.

Schematic illustration of heterodyne A-SNOM.

Fig. 2.
Fig. 2.

Detailed view of near-field region in A-SNOM.

Fig. 3.
Fig. 3.

Simulation results showing variation of amplitude of E T S l ω 0 component of near-field interaction electric field with tip-to-sample distance Z 0 (nm).

Fig. 4.
Fig. 4.

Simulation results showing variation of amplitude and phase of (Δω+1ω 0)- order component of near-field interaction electric field with tip-to-sample distance Z 0 (nm).

Fig. 5.
Fig. 5.

Simulation results for variation of homodyne S/B ratio with wavelength of illuminating light source at various radian frequency orders, nω0.

Fig. 6.
Fig. 6.

Simulation results showing variation of homodyne intensity with wavelength of illuminating light source at various radian frequency orders, nω0.

Fig. 7.
Fig. 7.

Simulation results showing variation of homodyne signal contrast with wavelength of illuminating light source at various radian frequency orders, nω0.

Fig. 8.
Fig. 8.

Simulation results showing variation of heterodyne signal intensity with wavelength of illuminating light source at various radian frequency orders, nω0.

Fig. 9.
Fig. 9.

Simulation results showing variation of heterodyne S/B ratio with wavelength of illuminating light source at various radian frequency orders,Δω+nω0.

Fig. 10.
Fig. 10.

Simulation results showing variation of heterodyne signal contrast with wavelength of illuminating light source at various radian frequency orders, Δω+nω0.

Fig. 11.
Fig. 11.

Simulation results showing variation of heterodyne S/B ratio with modulation depth ψ3 in range ψ3 < 1 at various radian frequency orders, IΔω+nω0.

Fig. 12.
Fig. 12.

Simulation results showing variation of heterodyne S/B ratio with incident angle θ at various radian frequency orders, IΔω+nω0.

Fig. 13.
Fig. 13.

Simulation results obtained for variation of heterodyne S/B ratio with modulation depth ψ3 in range ψ3 >1 at various values of radian frequency order, IΔω+nω0.

Equations (70)

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

E Reference = E R e i ( ( ω + Δ ω ) t + ϕ R ) ,
E T S = α eff E i e i ( ω t + ϕ TS ) E T S e i ( ω t + ϕ TS ) ,
E Probe = E P e i ( ω t + ϕ P ) e i ( 2 K sin ( θ ) Z ( t ) ) ,
Z ( t ) = Z 0 + A cos ( ω 0 t ) .
E Probe _ Reflective = E P _ R e i ( ω t + ϕ P + 2 K sin ( θ ) Z ( t ) ) e i ( 2 K sin ( θ ) Z ( t ) )
= E P _ R e i ( ω t + ϕ P ) e i ( 4 K sin ( θ ) Z ( t ) ) .
E Sample = E S e i ( ω t + ϕ S ) ,
E total = E Reference + E T− S + E Probe + E Probe_ Reflective + E Sample .
I ( t ) = I hom ( t ) + I het ( t ) ,
I hom ( t ) = E T S 2 + E P 2 + E P _ R 2 + E S 2
+ 2 E T S E S cos ( ϕ TS ϕ S )
+ 2 E P E S cos [ ϕ T ϕ S + 2 K sin ( θ ) Z 0 + 2 K sin ( θ ) A cos ( ω 0 t ) ]
+ 2 E T S E P cos [ ϕ T ϕ TS + 2 K sin ( θ ) Z 0 + 2 K sin ( θ ) A cos ( ω 0 t ) ]
+ 2 E P _ R E S cos [ ϕ T ϕ S + 4 K sin ( θ ) Z 0 + 4 K sin ( θ ) A cos ( ω 0 t ) ]
+ 2 E T S E P _ R cos [ ϕ T ϕ TS + 4 K sin ( θ ) Z 0 + 4 K sin ( θ ) A cos ( ω 0 t ) ]
+ 2 E P E P _ R cos [ 2 K sin ( θ ) Z 0 + 2 K sin ( θ ) A cos ( ω 0 t ) ]
I het ( t ) = E R 2
+ 2 E T S E R cos ( Δ ω t + ϕ R ϕ TS )
+ 2 E S E R cos ( Δ ω t + ϕ R ϕ S )
+ 2 E R E P cos ( Δ ω t + ϕ R ϕ T 2 K sin ( θ ) Z 0 2 K sin ( θ ) A cos ( ω 0 t ) )
+ 2 E R E P _ R cos ( Δ ω t + ϕ R ϕ T 4 K sin ( θ ) Z 0 4 K sin ( θ ) A cos ( ω 0 t ) )
I het ( t ) = E R 2
+ 2 E T S E R cos ( ϕ R ϕ TS + Δω t )
+ 2 E S E R cos ( ϕ R ϕ S + Δ ω t )
+ 2 E R E P { [ J 0 ( ψ 3 ) + 2 j = 1 ( 1 ) j J 2 j ( ψ 3 ) cos ( 2 j ω 0 t ) ] cos ( Δ ω t + ψ 1 )
+ 2 j = 0 ( 1 ) j J 2 j + 1 ( ψ 3 ) cos [ ( 2 j + 1 ) ω 0 t ] sin ( Δ ω t + ψ 1 ) }
+ 2 E R E P _ R { [ J 0 ( 2 ψ 3 ) + 2 j = 1 ( 1 ) j J 2 j ( 2 ψ 3 ) cos ( 2 j ω 0 t ) ] cos ( Δ ω t + ψ 2 )
+ 2 j = 0 ( 1 ) j J 2 j + 1 ( 2 ψ 3 ) cos [ ( 2 j + 1 ) ω 0 t ] sin ( Δ ω t + ψ 2 ) }
E T S = E T S 0 ω 0 + E T S 1 ω 0 cos ( ω 0 t ) + E T S 2 ω 0 cos ( 2 ω 0 t ) + E T S 3 ω 0 + cos ( 3 ω 0 t ) +
I het ( t ) = E R 2
+ 2 n = 0 E T S n ω 0 cos ( n ω 0 t ) E R cos ( ϕ R ϕ TS + Δω t )
+ 2 E S E R cos ( ϕ R ϕ S + Δ ω t )
+ 2 E R E P { [ J 0 ( ψ 3 ) + 2 j = 1 ( 1 ) j J 2 j ( ψ 3 ) cos ( 2 j ω 0 t ) ] cos ( Δ ω t + ψ 1 )
+ 2 j = 0 ( 1 ) j J 2 j + 1 ( ψ 3 ) cos [ ( 2 j + 1 ) ω 0 t ] sin ( Δ ω t + ψ 1 ) }
+ 2 E R E P _ R { [ J 0 ( 2 ψ 3 ) + 2 j = 1 ( 1 ) j J 2 j ( 2 ψ 3 ) cos ( 2 j ω 0 t ) ] cos ( Δ ω t + ψ 2 )
+ 2 j = 0 ( 1 ) j J 2 j + 1 ( 2 ψ 3 ) cos [ ( 2 j + 1 ) ω 0 t ] sin ( Δ ω t + ψ 2 ) }
I het ( t ) = E R 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DC
+ 2 E R E P J 0 ( ψ 3 ) cos ( Δ ω t + ψ 1 ) + 2 E T S 0 ω 0 E R cos ( ϕ R ϕ TS + Δω t )
+ 2 E S E R cos ( ϕ R ϕ S + Δ ω t ) + 2 E R E P _ R J 0 ( 2 ψ 3 ) cos ( Δ ω t + ψ 2 ) . . . . . . . . . . . . . . . . . . Δ ω t
+ 4 E R E P J 1 ( ψ 3 ) cos ( ω 0 t ) sin ( Δ ω t + ψ 1 )
+ 4 E R E P _ R J 1 ( 2 ψ 3 ) cos ( ω 0 t ) sin ( Δ ω t + ψ 2 )
+ 2 E T S 1 ω 0 E R cos ( ω 0 t ) cos ( ϕ R ϕ TS + Δω t ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( Δ ω + 1 ω 0 ) t
4 E R E P J 2 ( ψ 3 ) cos ( 2 ω 0 t ) cos ( Δ ω t + ψ 1 )
4 E R E P _ R J 2 ( 2 ψ 3 ) cos ( 2 ω 0 t ) cos ( Δ ω t + ψ 2 )
+ 2 E T S 2 ω 0 E R cos ( 2 ω 0 t ) cos ( ϕ R ϕ TS + Δω t ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( Δ ω + 2 ω 0 ) t
4 E R E P J 3 ( ψ 3 ) cos ( 3 ω 0 t ) sin ( Δ ω t + ψ 1 )
4 E R E P _ R J 3 ( 2 ψ 3 ) cos ( 3 ω 0 t ) sin ( Δ ω t + ψ 2 )
+ 2 E T S 3 ω 0 E R cos ( 3 ω 0 t ) cos ( ϕ R ϕ TS + Δω t ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( Δ ω + 3 ω 0 ) t
+ 4 E R E P J 4 ( ψ 3 ) cos ( 4 ω 0 t ) cos ( Δ ω t + ψ 1 )
+ 4 E R E P _ R J 4 ( 2 ψ 3 ) cos ( 4 ω 0 t ) cos ( Δ ω t + ψ 2 )
+ 2 E T S 4 ω 0 E R cos ( 4 ω 0 t ) cos ( ϕ R ϕ TS + Δω t ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( Δ ω + 4 ω 0 ) t
+ Higher Order Heterodyne Modulation Radian Frequencym Terms
ψ 1 = ϕ R ϕ P 2 K sin ( θ ) Z 0 2 K sin ( θ ) sin ΔZ
ψ 2 = ϕ R ϕ P 4 K sin ( θ ) Z 0 2 K sin ( θ ) sin ΔZ
( S B ) Radian Frequency Order = Signal Intensity Background Intensity Radian Frequency Order
S 1 / S 2 Radian Frequency Order = I n ω 0 ( S 1 ) / I n ω 0 ( S 2 ) Radian Frequency Order
Z ( t ) = Z 0 + A T ( t m T )
I het ( t ) = E R 2
+ 2 E T S E R cos ( Δ ω t + ϕ R ϕ TS )
+ 2 E S E R cos ( Δ ω t + ϕ R ϕ S )
+ 2 E R E P cos ( Δ ω t + ψ 1 ψ 3 ω 0 t )
+ 2 E R E P R cos ( Δ ω t + ψ 2 2 ψ 3 ω 0 t )
I het ( t ) = E R 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DC
+ 2 E S E R cos ( Δ ω t + ϕ R ϕ S ) + 2 E T S ' 0 ω 0 E S cos ( Δ ω t + ϕ R ϕ TS ) . . . . . . . . . . . . . . . . . . . . . . . Δ ω t
+ 2 E R E P cos ( Δ ω t + ψ 1 1 ω 0 t )
+ 2 E T S ' 1 ω 0 E R cos ( 1 ω 0 ) cos ( Δ ω t + ϕ R ϕ TS ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( Δ ω ± 1 ω 0 ) t
+ 2 E R E P R cos ( Δ ω t + ψ 2 2 ω 0 t )
+ 2 E T S ' 2 ω 0 E R cos ( 2 ω 0 ) cos ( Δ ω t + ϕ R ϕ TS ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( Δ ω ± 2 ω 0 ) t
+ 2 E T S ' 3 ω 0 E R cos ( 3 ω 0 ) cos ( Δ ω t + ϕ R ϕ TS ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( Δ ω ± 3 ω 0 ) t
+ 2 E T S ' 4 ω 0 E R cos ( 4 ω 0 ) cos ( Δ ω t + ϕ R ϕ TS ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ( Δ ω ± 4 ω 0 ) t

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