Abstract

We present a quantitative quasi-analytical model to predict and analyze signals on layered samples measured by infrared scattering-type scanning near-field optical microscopy. Our model predictions are compared to experimental data and to fully retarded calculations based on a point dipole approximation of the tip. The model is used to study the influence of the tip vibration amplitude and of the tip radius on the near-field contrasts of samples with particularly small variations in the layer thickness. Additionally the influence of a dielectric capping layer on the tip–substrate coupling is analyzed. When inversely applied, our calculation opens the possibility to extract the local layer thickness of thin films or the dielectric functions that allow one to draw conclusions on the material composition, conductivity or crystal structure on the nanoscale.

© 2012 OSA

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  47. T. Taubner, F. Keilmann, and R. Hillenbrand, “Nanomechanical resonance tuning and phase effects in optical near-field interaction,” Nano Lett. 4, 1669–1672 (2004).
    [CrossRef]
  48. J. A. Porto, P. Johansson, S. P. Apell, and T. López-Ríos, “Resonance shift effects in apertureless scanning near-field optical microscopy,” Phys. Rev. B 67, 085409 (2003).
    [CrossRef]

2012 (2)

L. M. Zhang, G. O. Andreev, Z. Fei, A. S. McLeod, G. Dominguez, M. Thiemens, A. H. Castro-Neto, D. N. Basov, and M. M. Fogler, “Near-field spectroscopy of silicon dioxide thin films,” Phys. Rev. B 85, 075419 (2012).
[CrossRef]

R. Krutokhvostov, A. A. Govyadinov, J. M. Stiegler, F. Huth, A. Chuvilin, P. S. Carney, and R. Hillenbrand, “Enhanced resolution in subsurface near-field optical microscopy,” Opt. Express 20, 593–600 (2012).
[CrossRef] [PubMed]

2011 (5)

K. Moon, E. Jung, M. Lim, Y. Do, and H. Han, “Quantitative analysis and measurements of near-field interactions in terahertz microscopes,” Opt. Express 19, 11539–11544 (2011).
[CrossRef] [PubMed]

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene–SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
[CrossRef] [PubMed]

J. M. Stiegler, Y. Abate, A. Cvitković, Y. E. Romanyuk, A. J. Huber, S. R. Leone, and R. Hillenband, “Nanoscale infrared absorption spectroscopy of individual nanoparticles enabled by scattering-type near-field microscopy,” ACS Nano 5, 6494–6499 (2011).
[CrossRef] [PubMed]

F. Huth, M. Schnell, J. Wittborn, N. Ocelić, and R. Hillenband, “Infrared-spectroscopic nanoimaging with a thermal source,” Nature Mat. 10, 352–356 (2011).
[CrossRef]

S. Amarie and F. Keilmann, “Broadband-infrared assessment of phonon resonance in scattering-type near-field microscopy,” Phys. Rev. B 83, 045404 (2011).
[CrossRef]

2010 (1)

J. M. Stiegler, A. J. Huber, S. L. Diedenhofen, J. G. Rivas, R. E. Algra, E. P. A. M. Bakkers, and R. Hillenband, “Nanoscale free-carrier profiling of individual semiconductor nanowires by infrared near-field nanoscopy,” Nano Lett. 10, 1387–1392 (2010).
[CrossRef] [PubMed]

2009 (4)

A. J. Huber, A. Ziegler, T. Köck, and R. Hillenband, “Infrared nanoscopy of strained semiconductors,” Nature Nanotech. 4, 153–157 (2009).
[CrossRef]

R. Esteban, R. Vogelgesang, and K. Kern, “Full simulations of the apertureless scanning near field optical microscopy signal: achievable resolution and contrast,” Opt. Express 17, 2518–2529 (2009).
[CrossRef] [PubMed]

J. Sun, J. C. Schotland, R. Hillenbrand, and P. S. Carney, “Nanoscale optical tomography using volume-scanning near-field microscopy,” Appl. Phys. Lett. 95, 121108 (2009).
[CrossRef]

A. A. Govyadinov, G. Y. Panasyuk, and J. C. Schotland, “Phaseless three-dimensional optical nanoimaging,” Phys. Rev. Lett. 103, 213901 (2009).
[CrossRef]

2008 (5)

2007 (2)

I. Kopf, J.-S. Samson, G. Wollny, C. Grunwald, E. Brüdermann, and M. Havenith, “Chemical imaging of microstructured self-assembled monolayers with nanometer resolution,” J. Phys. Chem. C 111, 8166–8171 (2007).
[CrossRef]

A. Cvitković, N. Ocelić, and R. Hillenbrand, “Analytical model for quantitative prediction of material contrasts in scattering-type near-field optical microscopy,” Opt. Express 16, 8550–8565 (2007).
[CrossRef]

2006 (4)

A. Cvitković, N. Ocelić, J. Aizpurua, R. Guckenberger, and R. Hillenbrand, “Infrared imaging of single nanoparticles via strong field enhancement in a scanning nanogap,” Phys. Rev. Lett. 97, 060801 (2006).
[CrossRef]

M. Brehm, T. Taubner, R. Hillenband, and F. Keilmann, “Infrared spectroscopic mapping of single nanoparticles and viruses at nanoscale resolution,” Nano Lett. 6, 1307–1310 (2006).
[CrossRef] [PubMed]

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenband, “Near-field microscopy through a SiC superlens,” Science 313, 1595–1595 (2006).
[CrossRef] [PubMed]

A. Huber, N. Ocelić, T. Taubner, and R. Hillenband, “Nanoscale resolved infrared probing of crystal structure and of plasmon–phonon coupling,” Nano Lett. 6, 774–778 (2006).
[CrossRef] [PubMed]

2005 (2)

T. Taubner, F. Keilmann, and R. Hillenband, “Nanoscale-resolved subsurface imaging by scattering-type near-field optical microscopy,” Opt. Express 13, 8893–8899 (2005).
[CrossRef] [PubMed]

J. Renger, S. Grafström, L. M. Eng, and R. Hillenbrand, “Resonant light scattering by near-field-induced phonon polaritons,” Phys. Rev. B 71, 075410 (2005).
[CrossRef]

2004 (3)

T. Taubner, R. Hillenbrand, and F. Keilmann, “Nanoscale polymer recognition by spectral signature in scattering infrared near-field microscopy,” Appl. Phys. Lett. 85, 5064–5066 (2004).
[CrossRef]

N. Ocelić and R. Hillenband, “Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation,” Nature Mat. 3, 606–609 (2004).
[CrossRef]

T. Taubner, F. Keilmann, and R. Hillenbrand, “Nanomechanical resonance tuning and phase effects in optical near-field interaction,” Nano Lett. 4, 1669–1672 (2004).
[CrossRef]

2003 (4)

J. A. Porto, P. Johansson, S. P. Apell, and T. López-Ríos, “Resonance shift effects in apertureless scanning near-field optical microscopy,” Phys. Rev. B 67, 085409 (2003).
[CrossRef]

T. Taubner, R. Hillenbrand, and F. Keilmann, “Performance of visible and mid-infrared scattering-type near-field optical microscopes,” J. Microscopy 210, 311–314 (2003).
[CrossRef]

M. B. Raschke and C. Lienau, “Apertureless near-field optical microscopy: Tip–sample coupling in elastic light scattering,” Appl. Phys. Lett. 83, 5089–5091 (2003).
[CrossRef]

B. Wang and C. H. Woo, “Atomic force microscopy-induced electric field in ferroelectric thin films,” J. Appl. Phys. 94, 4053–4059 (2003).
[CrossRef]

2002 (2)

B. B. Akhremitchev, Y. Sun, L. Stebounova, and G. C. Walker, “Monolayer-sensitive infrared imaging of DNA stripes using apertureless near-field optical microscopy,” Langmuir 18, 5325–5328 (2002).
[CrossRef]

R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature (London) 418, 159–162 (2002).
[CrossRef]

2000 (1)

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

1998 (2)

L. Novotny, B. Hecht, and D. W. Pohl, “Implications of high resolution to near-field optical microscopy,” Ultra-microscopy 71, 341–344 (1998).

F. Demming, J. Jersch, K. Dickmann, and P. I. Geshev, “Calculation of the field enhancement on laser-illuminated scanning probe tips by the boundary element method,” Appl. Phys. B 66, 593–598 (1998).
[CrossRef]

1995 (1)

F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, “Scanning interferometric apertureless microscopy: Optical imaging at 10 angstrom resolution,” Science 269, 1083–1085 (1995).
[CrossRef] [PubMed]

1994 (1)

1991 (1)

M. Nonnenmacher, M. P. O’Boyle, and H. K. Wickramasinghe, “Kelvin probe force microscopy,” Appl. Phys. Lett. 58, 2921–2923 (1991).
[CrossRef]

Abate, Y.

J. M. Stiegler, Y. Abate, A. Cvitković, Y. E. Romanyuk, A. J. Huber, S. R. Leone, and R. Hillenband, “Nanoscale infrared absorption spectroscopy of individual nanoparticles enabled by scattering-type near-field microscopy,” ACS Nano 5, 6494–6499 (2011).
[CrossRef] [PubMed]

Aizpurua, J.

J. Aizpurua, T. Taubner, F. J. García de Abajo, M. Brehm, and R. Hillenband, “Substrate-enhanced infrared near-field spectroscopy,” Opt. Express 16, 1529–1545 (2008).
[CrossRef] [PubMed]

A. Cvitković, N. Ocelić, J. Aizpurua, R. Guckenberger, and R. Hillenbrand, “Infrared imaging of single nanoparticles via strong field enhancement in a scanning nanogap,” Phys. Rev. Lett. 97, 060801 (2006).
[CrossRef]

Akhremitchev, B. B.

B. B. Akhremitchev, Y. Sun, L. Stebounova, and G. C. Walker, “Monolayer-sensitive infrared imaging of DNA stripes using apertureless near-field optical microscopy,” Langmuir 18, 5325–5328 (2002).
[CrossRef]

Algra, R. E.

J. M. Stiegler, A. J. Huber, S. L. Diedenhofen, J. G. Rivas, R. E. Algra, E. P. A. M. Bakkers, and R. Hillenband, “Nanoscale free-carrier profiling of individual semiconductor nanowires by infrared near-field nanoscopy,” Nano Lett. 10, 1387–1392 (2010).
[CrossRef] [PubMed]

Amarie, S.

S. Amarie and F. Keilmann, “Broadband-infrared assessment of phonon resonance in scattering-type near-field microscopy,” Phys. Rev. B 83, 045404 (2011).
[CrossRef]

Andreev, G. O.

L. M. Zhang, G. O. Andreev, Z. Fei, A. S. McLeod, G. Dominguez, M. Thiemens, A. H. Castro-Neto, D. N. Basov, and M. M. Fogler, “Near-field spectroscopy of silicon dioxide thin films,” Phys. Rev. B 85, 075419 (2012).
[CrossRef]

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene–SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
[CrossRef] [PubMed]

Apell, S. P.

J. A. Porto, P. Johansson, S. P. Apell, and T. López-Ríos, “Resonance shift effects in apertureless scanning near-field optical microscopy,” Phys. Rev. B 67, 085409 (2003).
[CrossRef]

Arsov, Z.

Bakkers, E. P. A. M.

J. M. Stiegler, A. J. Huber, S. L. Diedenhofen, J. G. Rivas, R. E. Algra, E. P. A. M. Bakkers, and R. Hillenband, “Nanoscale free-carrier profiling of individual semiconductor nanowires by infrared near-field nanoscopy,” Nano Lett. 10, 1387–1392 (2010).
[CrossRef] [PubMed]

Bao, W.

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene–SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
[CrossRef] [PubMed]

Basov, D. N.

L. M. Zhang, G. O. Andreev, Z. Fei, A. S. McLeod, G. Dominguez, M. Thiemens, A. H. Castro-Neto, D. N. Basov, and M. M. Fogler, “Near-field spectroscopy of silicon dioxide thin films,” Phys. Rev. B 85, 075419 (2012).
[CrossRef]

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene–SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
[CrossRef] [PubMed]

Behr, N.

N. Behr and M. Raschke, “Optical antenna properties of scanning probe tips: Plasmonic light scattering, tip–sample coupling, and near-field enhancement,” J. Phys. Chem. C 112, 3766–3773 (2008).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Section 8.6.

Brehm, M.

J. Aizpurua, T. Taubner, F. J. García de Abajo, M. Brehm, and R. Hillenband, “Substrate-enhanced infrared near-field spectroscopy,” Opt. Express 16, 1529–1545 (2008).
[CrossRef] [PubMed]

M. Brehm, A. Schliesser, F. Čajko, I. Tsukerman, and F. Keilmann, “Antenna-mediated back-scattering efficiency in infrared near-field microscopy,” Opt. Express 16, 11203–11215 (2008).
[CrossRef] [PubMed]

M. Brehm, T. Taubner, R. Hillenband, and F. Keilmann, “Infrared spectroscopic mapping of single nanoparticles and viruses at nanoscale resolution,” Nano Lett. 6, 1307–1310 (2006).
[CrossRef] [PubMed]

M. Brehm, “Infrarot-Mikroskopie mit einem Nahfeldmikroskop,” Dissertation, TU München (2006), Chap. 4.

Brüdermann, E.

G. Wollny, E. Brüdermann, Z. Arsov, L. Quaroni, and M. Havenith, “Nanoscale depth resolution in scanning near-field infrared microscopy,” Opt. Express 16, 7453–7459 (2008).
[CrossRef] [PubMed]

I. Kopf, J.-S. Samson, G. Wollny, C. Grunwald, E. Brüdermann, and M. Havenith, “Chemical imaging of microstructured self-assembled monolayers with nanometer resolution,” J. Phys. Chem. C 111, 8166–8171 (2007).
[CrossRef]

Cajko, F.

Carney, P. S.

R. Krutokhvostov, A. A. Govyadinov, J. M. Stiegler, F. Huth, A. Chuvilin, P. S. Carney, and R. Hillenbrand, “Enhanced resolution in subsurface near-field optical microscopy,” Opt. Express 20, 593–600 (2012).
[CrossRef] [PubMed]

J. Sun, J. C. Schotland, R. Hillenbrand, and P. S. Carney, “Nanoscale optical tomography using volume-scanning near-field microscopy,” Appl. Phys. Lett. 95, 121108 (2009).
[CrossRef]

Castro-Neto, A. H.

L. M. Zhang, G. O. Andreev, Z. Fei, A. S. McLeod, G. Dominguez, M. Thiemens, A. H. Castro-Neto, D. N. Basov, and M. M. Fogler, “Near-field spectroscopy of silicon dioxide thin films,” Phys. Rev. B 85, 075419 (2012).
[CrossRef]

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene–SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
[CrossRef] [PubMed]

Chuvilin, A.

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A. Cvitković, N. Ocelić, and R. Hillenbrand, “Material-specific infrared recognition of single sub 10 nm particles by substrate-enhanced scattering-type near-field microscopy,” Opt. Express 16, 7453–7459 (2008).

A. Cvitković, N. Ocelić, and R. Hillenbrand, “Analytical model for quantitative prediction of material contrasts in scattering-type near-field optical microscopy,” Opt. Express 16, 8550–8565 (2007).
[CrossRef]

A. Cvitković, N. Ocelić, J. Aizpurua, R. Guckenberger, and R. Hillenbrand, “Infrared imaging of single nanoparticles via strong field enhancement in a scanning nanogap,” Phys. Rev. Lett. 97, 060801 (2006).
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[CrossRef]

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Dominguez, G.

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[CrossRef]

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene–SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
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J. Renger, S. Grafström, L. M. Eng, and R. Hillenbrand, “Resonant light scattering by near-field-induced phonon polaritons,” Phys. Rev. B 71, 075410 (2005).
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Fei, Z.

L. M. Zhang, G. O. Andreev, Z. Fei, A. S. McLeod, G. Dominguez, M. Thiemens, A. H. Castro-Neto, D. N. Basov, and M. M. Fogler, “Near-field spectroscopy of silicon dioxide thin films,” Phys. Rev. B 85, 075419 (2012).
[CrossRef]

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene–SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
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[CrossRef]

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene–SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
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A. Cvitković, N. Ocelić, J. Aizpurua, R. Guckenberger, and R. Hillenbrand, “Infrared imaging of single nanoparticles via strong field enhancement in a scanning nanogap,” Phys. Rev. Lett. 97, 060801 (2006).
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F. Huth, M. Schnell, J. Wittborn, N. Ocelić, and R. Hillenband, “Infrared-spectroscopic nanoimaging with a thermal source,” Nature Mat. 10, 352–356 (2011).
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J. M. Stiegler, A. J. Huber, S. L. Diedenhofen, J. G. Rivas, R. E. Algra, E. P. A. M. Bakkers, and R. Hillenband, “Nanoscale free-carrier profiling of individual semiconductor nanowires by infrared near-field nanoscopy,” Nano Lett. 10, 1387–1392 (2010).
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A. J. Huber, A. Ziegler, T. Köck, and R. Hillenband, “Infrared nanoscopy of strained semiconductors,” Nature Nanotech. 4, 153–157 (2009).
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T. Taubner, F. Keilmann, and R. Hillenband, “Nanoscale-resolved subsurface imaging by scattering-type near-field optical microscopy,” Opt. Express 13, 8893–8899 (2005).
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A. Cvitković, N. Ocelić, and R. Hillenbrand, “Material-specific infrared recognition of single sub 10 nm particles by substrate-enhanced scattering-type near-field microscopy,” Opt. Express 16, 7453–7459 (2008).

A. Cvitković, N. Ocelić, and R. Hillenbrand, “Analytical model for quantitative prediction of material contrasts in scattering-type near-field optical microscopy,” Opt. Express 16, 8550–8565 (2007).
[CrossRef]

A. Cvitković, N. Ocelić, J. Aizpurua, R. Guckenberger, and R. Hillenbrand, “Infrared imaging of single nanoparticles via strong field enhancement in a scanning nanogap,” Phys. Rev. Lett. 97, 060801 (2006).
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T. Taubner, R. Hillenbrand, and F. Keilmann, “Nanoscale polymer recognition by spectral signature in scattering infrared near-field microscopy,” Appl. Phys. Lett. 85, 5064–5066 (2004).
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T. Taubner, F. Keilmann, and R. Hillenbrand, “Nanomechanical resonance tuning and phase effects in optical near-field interaction,” Nano Lett. 4, 1669–1672 (2004).
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T. Taubner, R. Hillenbrand, and F. Keilmann, “Performance of visible and mid-infrared scattering-type near-field optical microscopes,” J. Microscopy 210, 311–314 (2003).
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A. Huber, N. Ocelić, T. Taubner, and R. Hillenband, “Nanoscale resolved infrared probing of crystal structure and of plasmon–phonon coupling,” Nano Lett. 6, 774–778 (2006).
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J. M. Stiegler, Y. Abate, A. Cvitković, Y. E. Romanyuk, A. J. Huber, S. R. Leone, and R. Hillenband, “Nanoscale infrared absorption spectroscopy of individual nanoparticles enabled by scattering-type near-field microscopy,” ACS Nano 5, 6494–6499 (2011).
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J. A. Porto, P. Johansson, S. P. Apell, and T. López-Ríos, “Resonance shift effects in apertureless scanning near-field optical microscopy,” Phys. Rev. B 67, 085409 (2003).
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M. Brehm, T. Taubner, R. Hillenband, and F. Keilmann, “Infrared spectroscopic mapping of single nanoparticles and viruses at nanoscale resolution,” Nano Lett. 6, 1307–1310 (2006).
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T. Taubner, F. Keilmann, and R. Hillenband, “Nanoscale-resolved subsurface imaging by scattering-type near-field optical microscopy,” Opt. Express 13, 8893–8899 (2005).
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T. Taubner, F. Keilmann, and R. Hillenbrand, “Nanomechanical resonance tuning and phase effects in optical near-field interaction,” Nano Lett. 4, 1669–1672 (2004).
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T. Taubner, R. Hillenbrand, and F. Keilmann, “Nanoscale polymer recognition by spectral signature in scattering infrared near-field microscopy,” Appl. Phys. Lett. 85, 5064–5066 (2004).
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T. Taubner, R. Hillenbrand, and F. Keilmann, “Performance of visible and mid-infrared scattering-type near-field optical microscopes,” J. Microscopy 210, 311–314 (2003).
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R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature (London) 418, 159–162 (2002).
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J. M. Stiegler, Y. Abate, A. Cvitković, Y. E. Romanyuk, A. J. Huber, S. R. Leone, and R. Hillenband, “Nanoscale infrared absorption spectroscopy of individual nanoparticles enabled by scattering-type near-field microscopy,” ACS Nano 5, 6494–6499 (2011).
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J. A. Porto, P. Johansson, S. P. Apell, and T. López-Ríos, “Resonance shift effects in apertureless scanning near-field optical microscopy,” Phys. Rev. B 67, 085409 (2003).
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J. M. Stiegler, Y. Abate, A. Cvitković, Y. E. Romanyuk, A. J. Huber, S. R. Leone, and R. Hillenband, “Nanoscale infrared absorption spectroscopy of individual nanoparticles enabled by scattering-type near-field microscopy,” ACS Nano 5, 6494–6499 (2011).
[CrossRef] [PubMed]

J. M. Stiegler, A. J. Huber, S. L. Diedenhofen, J. G. Rivas, R. E. Algra, E. P. A. M. Bakkers, and R. Hillenband, “Nanoscale free-carrier profiling of individual semiconductor nanowires by infrared near-field nanoscopy,” Nano Lett. 10, 1387–1392 (2010).
[CrossRef] [PubMed]

Sun, J.

J. Sun, J. C. Schotland, R. Hillenbrand, and P. S. Carney, “Nanoscale optical tomography using volume-scanning near-field microscopy,” Appl. Phys. Lett. 95, 121108 (2009).
[CrossRef]

Sun, Y.

B. B. Akhremitchev, Y. Sun, L. Stebounova, and G. C. Walker, “Monolayer-sensitive infrared imaging of DNA stripes using apertureless near-field optical microscopy,” Langmuir 18, 5325–5328 (2002).
[CrossRef]

Tauber, M. J.

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene–SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
[CrossRef] [PubMed]

Taubner, T.

J. Aizpurua, T. Taubner, F. J. García de Abajo, M. Brehm, and R. Hillenband, “Substrate-enhanced infrared near-field spectroscopy,” Opt. Express 16, 1529–1545 (2008).
[CrossRef] [PubMed]

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenband, “Near-field microscopy through a SiC superlens,” Science 313, 1595–1595 (2006).
[CrossRef] [PubMed]

M. Brehm, T. Taubner, R. Hillenband, and F. Keilmann, “Infrared spectroscopic mapping of single nanoparticles and viruses at nanoscale resolution,” Nano Lett. 6, 1307–1310 (2006).
[CrossRef] [PubMed]

A. Huber, N. Ocelić, T. Taubner, and R. Hillenband, “Nanoscale resolved infrared probing of crystal structure and of plasmon–phonon coupling,” Nano Lett. 6, 774–778 (2006).
[CrossRef] [PubMed]

T. Taubner, F. Keilmann, and R. Hillenband, “Nanoscale-resolved subsurface imaging by scattering-type near-field optical microscopy,” Opt. Express 13, 8893–8899 (2005).
[CrossRef] [PubMed]

T. Taubner, F. Keilmann, and R. Hillenbrand, “Nanomechanical resonance tuning and phase effects in optical near-field interaction,” Nano Lett. 4, 1669–1672 (2004).
[CrossRef]

T. Taubner, R. Hillenbrand, and F. Keilmann, “Nanoscale polymer recognition by spectral signature in scattering infrared near-field microscopy,” Appl. Phys. Lett. 85, 5064–5066 (2004).
[CrossRef]

T. Taubner, R. Hillenbrand, and F. Keilmann, “Performance of visible and mid-infrared scattering-type near-field optical microscopes,” J. Microscopy 210, 311–314 (2003).
[CrossRef]

R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature (London) 418, 159–162 (2002).
[CrossRef]

Thiemens, M.

L. M. Zhang, G. O. Andreev, Z. Fei, A. S. McLeod, G. Dominguez, M. Thiemens, A. H. Castro-Neto, D. N. Basov, and M. M. Fogler, “Near-field spectroscopy of silicon dioxide thin films,” Phys. Rev. B 85, 075419 (2012).
[CrossRef]

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene–SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
[CrossRef] [PubMed]

Tsukerman, I.

Urzhumov, Y.

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenband, “Near-field microscopy through a SiC superlens,” Science 313, 1595–1595 (2006).
[CrossRef] [PubMed]

Vogelgesang, R.

Walker, G. C.

B. B. Akhremitchev, Y. Sun, L. Stebounova, and G. C. Walker, “Monolayer-sensitive infrared imaging of DNA stripes using apertureless near-field optical microscopy,” Langmuir 18, 5325–5328 (2002).
[CrossRef]

Wang, B.

B. Wang and C. H. Woo, “Atomic force microscopy-induced electric field in ferroelectric thin films,” J. Appl. Phys. 94, 4053–4059 (2003).
[CrossRef]

Wang, C.

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene–SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
[CrossRef] [PubMed]

Wickramasinghe, H. K.

F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, “Scanning interferometric apertureless microscopy: Optical imaging at 10 angstrom resolution,” Science 269, 1083–1085 (1995).
[CrossRef] [PubMed]

M. Nonnenmacher, M. P. O’Boyle, and H. K. Wickramasinghe, “Kelvin probe force microscopy,” Appl. Phys. Lett. 58, 2921–2923 (1991).
[CrossRef]

Wittborn, J.

F. Huth, M. Schnell, J. Wittborn, N. Ocelić, and R. Hillenband, “Infrared-spectroscopic nanoimaging with a thermal source,” Nature Mat. 10, 352–356 (2011).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Section 8.6.

Wollny, G.

G. Wollny, E. Brüdermann, Z. Arsov, L. Quaroni, and M. Havenith, “Nanoscale depth resolution in scanning near-field infrared microscopy,” Opt. Express 16, 7453–7459 (2008).
[CrossRef] [PubMed]

I. Kopf, J.-S. Samson, G. Wollny, C. Grunwald, E. Brüdermann, and M. Havenith, “Chemical imaging of microstructured self-assembled monolayers with nanometer resolution,” J. Phys. Chem. C 111, 8166–8171 (2007).
[CrossRef]

Woo, C. H.

B. Wang and C. H. Woo, “Atomic force microscopy-induced electric field in ferroelectric thin films,” J. Appl. Phys. 94, 4053–4059 (2003).
[CrossRef]

Zenhausern, F.

F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, “Scanning interferometric apertureless microscopy: Optical imaging at 10 angstrom resolution,” Science 269, 1083–1085 (1995).
[CrossRef] [PubMed]

Zhang, L. M.

L. M. Zhang, G. O. Andreev, Z. Fei, A. S. McLeod, G. Dominguez, M. Thiemens, A. H. Castro-Neto, D. N. Basov, and M. M. Fogler, “Near-field spectroscopy of silicon dioxide thin films,” Phys. Rev. B 85, 075419 (2012).
[CrossRef]

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene–SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
[CrossRef] [PubMed]

Zhao, Z.

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene–SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
[CrossRef] [PubMed]

Ziegler, A.

A. J. Huber, A. Ziegler, T. Köck, and R. Hillenband, “Infrared nanoscopy of strained semiconductors,” Nature Nanotech. 4, 153–157 (2009).
[CrossRef]

ACS Nano (1)

J. M. Stiegler, Y. Abate, A. Cvitković, Y. E. Romanyuk, A. J. Huber, S. R. Leone, and R. Hillenband, “Nanoscale infrared absorption spectroscopy of individual nanoparticles enabled by scattering-type near-field microscopy,” ACS Nano 5, 6494–6499 (2011).
[CrossRef] [PubMed]

Appl. Phys. B (1)

F. Demming, J. Jersch, K. Dickmann, and P. I. Geshev, “Calculation of the field enhancement on laser-illuminated scanning probe tips by the boundary element method,” Appl. Phys. B 66, 593–598 (1998).
[CrossRef]

Appl. Phys. Lett. (4)

T. Taubner, R. Hillenbrand, and F. Keilmann, “Nanoscale polymer recognition by spectral signature in scattering infrared near-field microscopy,” Appl. Phys. Lett. 85, 5064–5066 (2004).
[CrossRef]

M. B. Raschke and C. Lienau, “Apertureless near-field optical microscopy: Tip–sample coupling in elastic light scattering,” Appl. Phys. Lett. 83, 5089–5091 (2003).
[CrossRef]

J. Sun, J. C. Schotland, R. Hillenbrand, and P. S. Carney, “Nanoscale optical tomography using volume-scanning near-field microscopy,” Appl. Phys. Lett. 95, 121108 (2009).
[CrossRef]

M. Nonnenmacher, M. P. O’Boyle, and H. K. Wickramasinghe, “Kelvin probe force microscopy,” Appl. Phys. Lett. 58, 2921–2923 (1991).
[CrossRef]

J. Appl. Phys. (1)

B. Wang and C. H. Woo, “Atomic force microscopy-induced electric field in ferroelectric thin films,” J. Appl. Phys. 94, 4053–4059 (2003).
[CrossRef]

J. Microscopy (1)

T. Taubner, R. Hillenbrand, and F. Keilmann, “Performance of visible and mid-infrared scattering-type near-field optical microscopes,” J. Microscopy 210, 311–314 (2003).
[CrossRef]

J. Phys. Chem. C (2)

I. Kopf, J.-S. Samson, G. Wollny, C. Grunwald, E. Brüdermann, and M. Havenith, “Chemical imaging of microstructured self-assembled monolayers with nanometer resolution,” J. Phys. Chem. C 111, 8166–8171 (2007).
[CrossRef]

N. Behr and M. Raschke, “Optical antenna properties of scanning probe tips: Plasmonic light scattering, tip–sample coupling, and near-field enhancement,” J. Phys. Chem. C 112, 3766–3773 (2008).
[CrossRef]

Langmuir (1)

B. B. Akhremitchev, Y. Sun, L. Stebounova, and G. C. Walker, “Monolayer-sensitive infrared imaging of DNA stripes using apertureless near-field optical microscopy,” Langmuir 18, 5325–5328 (2002).
[CrossRef]

Nano Lett. (5)

Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Infrared nanoscopy of Dirac plasmons at the graphene–SiO2 interface,” Nano Lett. 11, 4701–4705 (2011).
[CrossRef] [PubMed]

A. Huber, N. Ocelić, T. Taubner, and R. Hillenband, “Nanoscale resolved infrared probing of crystal structure and of plasmon–phonon coupling,” Nano Lett. 6, 774–778 (2006).
[CrossRef] [PubMed]

M. Brehm, T. Taubner, R. Hillenband, and F. Keilmann, “Infrared spectroscopic mapping of single nanoparticles and viruses at nanoscale resolution,” Nano Lett. 6, 1307–1310 (2006).
[CrossRef] [PubMed]

J. M. Stiegler, A. J. Huber, S. L. Diedenhofen, J. G. Rivas, R. E. Algra, E. P. A. M. Bakkers, and R. Hillenband, “Nanoscale free-carrier profiling of individual semiconductor nanowires by infrared near-field nanoscopy,” Nano Lett. 10, 1387–1392 (2010).
[CrossRef] [PubMed]

T. Taubner, F. Keilmann, and R. Hillenbrand, “Nanomechanical resonance tuning and phase effects in optical near-field interaction,” Nano Lett. 4, 1669–1672 (2004).
[CrossRef]

Nature (London) (1)

R. Hillenbrand, T. Taubner, and F. Keilmann, “Phonon-enhanced light-matter interaction at the nanometre scale,” Nature (London) 418, 159–162 (2002).
[CrossRef]

Nature Mat. (2)

F. Huth, M. Schnell, J. Wittborn, N. Ocelić, and R. Hillenband, “Infrared-spectroscopic nanoimaging with a thermal source,” Nature Mat. 10, 352–356 (2011).
[CrossRef]

N. Ocelić and R. Hillenband, “Subwavelength-scale tailoring of surface phonon polaritons by focused ion-beam implantation,” Nature Mat. 3, 606–609 (2004).
[CrossRef]

Nature Nanotech. (1)

A. J. Huber, A. Ziegler, T. Köck, and R. Hillenband, “Infrared nanoscopy of strained semiconductors,” Nature Nanotech. 4, 153–157 (2009).
[CrossRef]

Opt. Commun. (1)

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

Opt. Express (9)

R. Krutokhvostov, A. A. Govyadinov, J. M. Stiegler, F. Huth, A. Chuvilin, P. S. Carney, and R. Hillenbrand, “Enhanced resolution in subsurface near-field optical microscopy,” Opt. Express 20, 593–600 (2012).
[CrossRef] [PubMed]

T. Taubner, F. Keilmann, and R. Hillenband, “Nanoscale-resolved subsurface imaging by scattering-type near-field optical microscopy,” Opt. Express 13, 8893–8899 (2005).
[CrossRef] [PubMed]

J. Aizpurua, T. Taubner, F. J. García de Abajo, M. Brehm, and R. Hillenband, “Substrate-enhanced infrared near-field spectroscopy,” Opt. Express 16, 1529–1545 (2008).
[CrossRef] [PubMed]

A. Cvitković, N. Ocelić, and R. Hillenbrand, “Material-specific infrared recognition of single sub 10 nm particles by substrate-enhanced scattering-type near-field microscopy,” Opt. Express 16, 7453–7459 (2008).

G. Wollny, E. Brüdermann, Z. Arsov, L. Quaroni, and M. Havenith, “Nanoscale depth resolution in scanning near-field infrared microscopy,” Opt. Express 16, 7453–7459 (2008).
[CrossRef] [PubMed]

M. Brehm, A. Schliesser, F. Čajko, I. Tsukerman, and F. Keilmann, “Antenna-mediated back-scattering efficiency in infrared near-field microscopy,” Opt. Express 16, 11203–11215 (2008).
[CrossRef] [PubMed]

A. Cvitković, N. Ocelić, and R. Hillenbrand, “Analytical model for quantitative prediction of material contrasts in scattering-type near-field optical microscopy,” Opt. Express 16, 8550–8565 (2007).
[CrossRef]

R. Esteban, R. Vogelgesang, and K. Kern, “Full simulations of the apertureless scanning near field optical microscopy signal: achievable resolution and contrast,” Opt. Express 17, 2518–2529 (2009).
[CrossRef] [PubMed]

K. Moon, E. Jung, M. Lim, Y. Do, and H. Han, “Quantitative analysis and measurements of near-field interactions in terahertz microscopes,” Opt. Express 19, 11539–11544 (2011).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. B (4)

L. M. Zhang, G. O. Andreev, Z. Fei, A. S. McLeod, G. Dominguez, M. Thiemens, A. H. Castro-Neto, D. N. Basov, and M. M. Fogler, “Near-field spectroscopy of silicon dioxide thin films,” Phys. Rev. B 85, 075419 (2012).
[CrossRef]

J. A. Porto, P. Johansson, S. P. Apell, and T. López-Ríos, “Resonance shift effects in apertureless scanning near-field optical microscopy,” Phys. Rev. B 67, 085409 (2003).
[CrossRef]

J. Renger, S. Grafström, L. M. Eng, and R. Hillenbrand, “Resonant light scattering by near-field-induced phonon polaritons,” Phys. Rev. B 71, 075410 (2005).
[CrossRef]

S. Amarie and F. Keilmann, “Broadband-infrared assessment of phonon resonance in scattering-type near-field microscopy,” Phys. Rev. B 83, 045404 (2011).
[CrossRef]

Phys. Rev. Lett. (2)

A. A. Govyadinov, G. Y. Panasyuk, and J. C. Schotland, “Phaseless three-dimensional optical nanoimaging,” Phys. Rev. Lett. 103, 213901 (2009).
[CrossRef]

A. Cvitković, N. Ocelić, J. Aizpurua, R. Guckenberger, and R. Hillenbrand, “Infrared imaging of single nanoparticles via strong field enhancement in a scanning nanogap,” Phys. Rev. Lett. 97, 060801 (2006).
[CrossRef]

Science (2)

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenband, “Near-field microscopy through a SiC superlens,” Science 313, 1595–1595 (2006).
[CrossRef] [PubMed]

F. Zenhausern, Y. Martin, and H. K. Wickramasinghe, “Scanning interferometric apertureless microscopy: Optical imaging at 10 angstrom resolution,” Science 269, 1083–1085 (1995).
[CrossRef] [PubMed]

Ultra-microscopy (1)

L. Novotny, B. Hecht, and D. W. Pohl, “Implications of high resolution to near-field optical microscopy,” Ultra-microscopy 71, 341–344 (1998).

Other (8)

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, New York, 1985).

H. Kuzmany, Solid State Spectroscopy, 2nd ed. (Springer, Berlin, Heidelberg, 2009), Chap. 10.
[CrossRef]

F. Keilmann and R. Hillenbrand, “Near-field nanoscopy by elastic light scattering from a tip,” in A. Zayats and D. Richards, eds., Nano-optics and near-field optical microscopy (Artech House, Boston, London, 2009), Chap. 11, pp. 235–265.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Section 8.6.

G. Friedbacher and H. Bubert, eds., Surface and Thin Film Analysis, 2nd ed. (Wiley-VCH, Weinheim, 2011).
[CrossRef]

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Section 4.4.

M. Brehm, “Infrarot-Mikroskopie mit einem Nahfeldmikroskop,” Dissertation, TU München (2006), Chap. 4.

N. Ocelić, “Quantitative near-field phonon-polariton spectroscopy,” Dissertation, TU München (2007), Chapters 5 and 6.

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Figures (8)

Fig. 1
Fig. 1

Illustration of the FDM. A spheroid represents the tip. Its polarization in the external field Ein is approximated by the monopoles Q0 and −Q0 forming the dipole p0. The near-field interaction with the sample is treated electrostatically by introducing virtual image charges −βQ0 and −βQ1 in the sample. In the tip, the near-field induced charges Q1 and −Q1 form the dipole p1. Esc is the propagating field scattered from the system. The bars on the right mark the position of the respective charges and the height H of the tip above the sample. ε1 and ε2 are the dielectric functions of the surrounding medium and the sample, respectively. The tip vibrates at a frequency Ω and an amplitude A.

Fig. 3
Fig. 3

(a) Fresnel reflection coefficient rp depending on the in-plane component of the wave vector q in the regime of evanescent waves for TM polarized light, evaluated for a vacuum wavelength of 10 μm. The curves represent bare silicon (red, solid), bare PMMA (red, dashed), and PMMA-covered silicon with PMMA film thicknesses of 20, 50, and 100 nm. Horizontal arrows mark the respective values of βeff for the tip being in contact with the sample. The values q*ρ−1 and are marked with the vertical gray lines. (b) Effective electrostatic reflection coefficients βeff evaluated for a tip radius of 30 nm depending on the height H of the tip above the sample for the same samples as in (a).

Fig. 4
Fig. 4

Approach curves for the scattering amplitude on silicon covered with 40 nm PMMA at a tip vibration amplitude of A = 27 nm (left column) and A = 50 nm (right column) and two demodulation orders (upper row: n = 2, lower row: n = 3). The curves are normalized to the signal on bare silicon for H0 = 0. Red line: experimental curve, black solid line: FDM calculation, black dashed line: point dipole approximation [14].

Fig. 5
Fig. 5

Film thickness variation: Measured dependence of the scattering amplitude on a silicon sample covered by PMMA layers with thicknesses of d = 0, 40, 54, 81, 115 nm for n = 3 and A = 50 nm as approach curves (red curves). Red dots: Signals in contact (H0 = 0). The bars mark the estimated error according to the uncertainty in the determination of H0. Black solid line: FDM calculation of the scattering amplitude in contact; black dotted line: point dipole approximation (PDM) [14]; black dashed lines: FDM calculation of the approach curves (H0 > 0).

Fig. 6
Fig. 6

(a) Demodulated scattering amplitude S2 on a silicon sample covered by a PMMA layer of thickness d for the tip vibration amplitudes A = 10 nm (black), A = 30 nm (red), and A = 60 nm (black, dashed). (b) Relative contrast change function [Eq. (14)]. The inset is a sketch for the corresponding derivation in the main text.

Fig. 7
Fig. 7

(a) Demodulated signal S2 on a silicon sample covered by a PMMA layer of the thickness d for the tip radii ρ = 10 nm (black), ρ = 30 nm (red), and ρ = 60 nm (black, dashed). The spheroid length is L = 300 nm in all cases. (b) Relative contrast change function [Eq. (14)].

Fig. 8
Fig. 8

Comparison of resonance curves. The demodulated scattering amplitude at n = 2 and A = 30 nm is shown depending on the real part of the substrate dielectric function. ε2 = 2, Im(ε3) = 1. Solid curves: d = 10 nm (black), d = 30 nm (red); dashed curves: H0 = 10 nm (black), H0 = 30 nm (red); dotted black curve: d = 0, H0 = 0.

Equations (24)

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

Q 1 = β ( f 0 Q 0 + f 1 Q 1 ) ,
f 0 , 1 = ( g ρ + 2 H + W 0 , 1 2 L ) ln 4 L ρ + 4 H + 2 W 0 , 1 ln 4 L ρ .
α eff p 1 p 0 + 1 = 1 2 β f 0 1 β f 1 + 1 .
H ( t ) = H 0 + A ( 1 + cos Ω t )
U = Q 4 π ɛ 0 ( 1 r 2 + z 2 + Φ ) .
Φ = 0 A ( k ) e k z J 0 ( k r ) d k ,
A ( k ) = e 2 k z 0 β 12 + β 23 e 2 k d 1 β 21 β 23 e 2 k d ,
β i j = ɛ i ɛ j ɛ i + ɛ j .
Φ | z = 0 = β X z 0 + X ,
Φ ( z ) z | z = 0 = Φ | z = 0 = β X ( z 0 + X ) 2 .
β X = Φ 2 Φ | z = 0 ,
X = Φ Φ | z = 0 z 0 .
f 0 , 1 = ( g ρ + H + X 0 , 1 2 L ) ln 4 L ρ + 2 H + 2 X 0 , 1 ln 4 L ρ ,
β eff r p ( q ¯ ) ,
S n , r : = S n B S n A = S n A + S n | A Δ S n A ,
S n , r 1 | S n S n | .
A ( k ) = e 2 k z 0 β 12 + β A e 2 k ( d 1 + d 2 + d 3 ) 1 + β B e 2 k ( d 1 + d 2 + d 3 ) ,
β A = β 12 β 23 β 34 e 2 k ( d 1 + d 3 ) + β 12 β 23 β 45 e 2 k d 1 + β 12 β 34 β 45 e 2 k ( d 1 + d 2 ) + β 23 e 2 k ( d 2 + d 3 ) + β 23 β 34 β 45 e 2 k d 2 + β 34 e 2 k d 3 + β 45
β B = β 12 β 23 e 2 k ( d 2 + d 3 ) + β 12 β 23 β 34 β 45 e 2 k d 2 + β 12 β 34 e 2 k d 3 + β 12 β 45 + β 23 β 34 e 2 k ( d 1 + d 3 ) + β 23 β 45 e 2 k d 1 + β 34 β 45 e 2 k ( d 1 + d 2 ) .
1 1 β 12 β 23 e 2 k d = m = 0 ( β 21 β 23 ) m e 2 m k d ,
0 e k | z | J 0 ( k r ) d k = 1 z 2 + r 2
Φ = β 12 1 2 z 0 z + γ 12 γ 21 m = 1 β 21 m 1 β 23 n 2 z 0 z + 2 m d
R 0 = β 12 Q ,
R m > 0 = γ 21 β 21 m 1 β 23 m γ 12 Q

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