Abstract

We simulate the remarkable changes that occur to the decay rates of a fluorescent molecule as a conical metal tip approaches. A new and simple model is developed to reveal and quantify which decay channels are responsible. Our analysis, which is independent of the method of molecular excitation, shows some universal characteristics. As the tip-apex enters the molecule’s near-field, the creation of surface plasmon polaritons can become extraordinarily efficient, leading to an increase in the nonradiative rate and, by proportional radiative-damping, in the radiative rate. Enhancements reaching 3 orders of magnitude have been found, which can improve the apparent brightness of a molecule. At distances less than ~5nm, short-ranged energy transfer to the nano-scale apex quickly becomes dominant and is entirely nonradiative.

© 2007 Optical Society of America

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References

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  1. R. R. Chance, A. Prock, and R. Silbey, "Molecular fluorescence and energy transfer near interfaces," Adv. Chem. Phys. 37, 1-65 (1978).
    [CrossRef]
  2. G. W. Ford and W. H. Weber, "Electromagnetic-interactions of molecules with metal-surfaces," Phys. Rep. 113, 195-287 (1984).
    [CrossRef]
  3. W. L. Barnes, "Fluorescence near interfaces: the role of photonic mode density," J. Mod. Opt. 45, 661-699 (1998).
    [CrossRef]
  4. R. X. Bian, R. C. Dunn, and X. S. Xie, "Single molecule emission characteristics in near-field microscopy," Phys. Rev. Lett. 75, 4772-4775 (1995).
    [CrossRef] [PubMed]
  5. N. Hayazawa, Y. Inouye, and S. Kawata, "Evanescent field excitation and measurement of dye fluorescence in a metallic probe near-field scanning optical microscope," J. Microsc. 194, 472-476 (1999).
    [CrossRef]
  6. E. J. Sanchez, L. Novotny, and X. S. Xie, "Near-field fluorescence microscopy based on two-photon excitation with metal tips," Phys. Rev. Lett. 82, 4014-4017 (1999).
    [CrossRef]
  7. T. J. Yang, G. A. Lessard, and S. R. Quake, "An apertureless near-field microscope for fluorescence imaging," Appl. Phys. Lett. 76, 378-380 (2000).
    [CrossRef]
  8. A. Kramer, W. Trabesinger, B. Hecht, and U. P. Wild, "Optical near-field enhancement at a metal tip probed by a single fluorophore," Appl. Phys. Lett. 80, 1652-1654 (2002).
    [CrossRef]
  9. H. G. Frey, S. Witt, K. Felderer, and R. Guckenberger, "High-resolution imaging of single fluorescent molecules with the optical near-field of a metal tip," Phys. Rev. Lett. 93, 200801 (2004).
    [CrossRef] [PubMed]
  10. J. N. Farahani, D. W. Pohl, H. J. Eisler, and B. Hecht, "Single quantum dot coupled to a scanning optical antenna: A tunable superemitter," Phys. Rev. Lett. 95, (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
  12. H. Gersen, M. F. Garcia-Parajo, L. Novotny, J. A. Veerman, L. Kuipers, and N. F. Van Hulst, "Near-field effects in single molecule emission," J. Microsc. 202, 374-378 (2001).
    [CrossRef] [PubMed]
  13. P. Anger, P. Bharadwaj, and L. Novotny, "Enhancement and quenching of single-molecule fluorescence," Phys. Rev. Lett. 96, 4 (2006).
    [CrossRef]
  14. S. Kuhn, U. Hakanson, L. Rogobete, and V. Sandoghdar, "Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna," Phys. Rev. Lett. 97, (2006).
    [CrossRef] [PubMed]
  15. F. Cannone, G. Chirico, A. R. Bizzarri, and S. Cannistraro, "Quenching and blinking of fluorescence of a single dye molecule bound to gold nanoparticles," J. Phys. Chem. B 110, 16491-16498 (2006).
    [CrossRef] [PubMed]
  16. F. D. Stefani, K. Vasilev, N. Bocchio, F. Gaul, A. Pomozzi, and M. Kreiter, "Photonic mode density effects on single-molecule fluorescence blinking," New J. Phys. 9, 21 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. J. T. KrugII, E. J. Sanchez, and X. S. Xie, "Fluorescence quenching in tip-enhanced nonlinear optical microscopy," Appl. Phys. Lett. 86, (2005).
  23. F. M. Huang and D. Richards, "Fluorescence enhancement and energy transfer in apertureless scanning near-field optical microscopy," J. Opt. A 8, S234-S238 (2006).
    [CrossRef]
  24. C. Girard, O. J. F. Martin, and A. Dereux, "Molecular lifetime changes induced by nanometer-scale optical-fields," Phys. Rev. Lett. 75, 3098-3101 (1995).
    [CrossRef] [PubMed]
  25. L. Novotny, "Single molecule fluorescence in inhomogeneous environments," Appl. Phys. Lett. 69, 3806-3808 (1996).
    [CrossRef]
  26. A. Rahmani, P. C. Chaumet, and F. de Fornel, "Environment-induced modification of spontaneous emission: Single-molecule near-field probe," Phys. Rev. A 63, 023819 (2001).
    [CrossRef]
  27. A. Downes, D. Salter, and A. Elfick, "Finite element simulations of tip-enhanced Raman and fluorescence spectroscopy," J. Phys. Chem. B 110, 6692-6698 (2006).
    [CrossRef] [PubMed]
  28. N. A. Issa and R. Guckenberger, "Optical nanofocusig on tapered metallic waveguides," Plasmonics 2, 31-37 (2007).
    [CrossRef]
  29. P. M. Whitmore, H. J. Robota, and C. B. Harris, "Mechanisms for electronic-energy transfer between molecules and metal-surfaces - a comparison of silver and nickel," J. Chem. Phys. 77, 1560-1568 (1982).
    [CrossRef]
  30. P. Avouris and B. N. J. Persson, "Excited-states at metal-surfaces and their nonradiative relaxation," J. Phys. Chem. 88, 837-848 (1984).
    [CrossRef]
  31. P. B. Johnson and R. W. Chirsty, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
    [CrossRef]
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    [CrossRef] [PubMed]
  35. R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux, "Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle," Opt. Commun. 261, 368-375 (2006).
    [CrossRef]
  36. L. Novotny and C. Hafner, "Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function," Phys. Rev. E 50, 4094-4106 (1994).
    [CrossRef]
  37. A. W. Snyder, and J. D. Love, Optical waveguide theory (Chapman and Hall, New York, 1983).

2007 (3)

F. D. Stefani, K. Vasilev, N. Bocchio, F. Gaul, A. Pomozzi, and M. Kreiter, "Photonic mode density effects on single-molecule fluorescence blinking," New J. Phys. 9, 21 (2007).
[CrossRef]

Y. X. Zhang, K. Aslan, M. J. R. Previte, and C. D. Geddes, "Metal-enhanced fluorescence: Surface plasmons can radiate a fluorophore's structured emission," Appl. Phys. Lett. 90, (2007).

N. A. Issa and R. Guckenberger, "Optical nanofocusig on tapered metallic waveguides," Plasmonics 2, 31-37 (2007).
[CrossRef]

2006 (8)

R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux, "Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle," Opt. Commun. 261, 368-375 (2006).
[CrossRef]

G. Winter and W. L. Barnes, "Emission of light through thin silver films via near-field coupling to surface plasmon polaritons," Appl. Phys. Lett. 88, (2006).
[CrossRef]

F. M. Huang and D. Richards, "Fluorescence enhancement and energy transfer in apertureless scanning near-field optical microscopy," J. Opt. A 8, S234-S238 (2006).
[CrossRef]

A. Downes, D. Salter, and A. Elfick, "Finite element simulations of tip-enhanced Raman and fluorescence spectroscopy," J. Phys. Chem. B 110, 6692-6698 (2006).
[CrossRef] [PubMed]

J. R. Lakowicz, "Plasmonics in biology and plasmon-controlled fluorescence," Plasmonics 1, 5-33 (2006).
[CrossRef] [PubMed]

P. Anger, P. Bharadwaj, and L. Novotny, "Enhancement and quenching of single-molecule fluorescence," Phys. Rev. Lett. 96, 4 (2006).
[CrossRef]

S. Kuhn, U. Hakanson, L. Rogobete, and V. Sandoghdar, "Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna," Phys. Rev. Lett. 97, (2006).
[CrossRef] [PubMed]

F. Cannone, G. Chirico, A. R. Bizzarri, and S. Cannistraro, "Quenching and blinking of fluorescence of a single dye molecule bound to gold nanoparticles," J. Phys. Chem. B 110, 16491-16498 (2006).
[CrossRef] [PubMed]

2005 (5)

J. N. Farahani, D. W. Pohl, H. J. Eisler, and B. Hecht, "Single quantum dot coupled to a scanning optical antenna: A tunable superemitter," Phys. Rev. Lett. 95, (2005).
[CrossRef] [PubMed]

F. M. Huang, F. Festy, and D. Richards, "Tip-enhanced fluorescence imaging of quantum dots," Appl. Phys. Lett. 87, (2005).
[CrossRef]

F. D. Stefani, K. Vasilev, N. Bocchio, N. Stoyanova, and M. Kreiter, "Surface-plasmon-mediated single-molecule fluorescence through a thin metallic film," Phys. Rev. Lett. 94, (2005).
[CrossRef] [PubMed]

J. T. KrugII, E. J. Sanchez, and X. S. Xie, "Fluorescence quenching in tip-enhanced nonlinear optical microscopy," Appl. Phys. Lett. 86, (2005).

E. Dulkeith, M. Ringler, T. A. Klar, J. Feldmann, A. M. Javier, and W. J. Parak, "Gold nanoparticles quench fluorescence by phase induced radiative rate suppression," Nano Lett. 5, 585-589 (2005).
[CrossRef] [PubMed]

2004 (2)

M. Thomas, J. J. Greffet, R. Carminati, and J. R. Arias-Gonzalez, "Single-molecule spontaneous emission close to absorbing nanostructures," Appl. Phys. Lett. 85, 3863-3865 (2004).
[CrossRef]

H. G. Frey, S. Witt, K. Felderer, and R. Guckenberger, "High-resolution imaging of single fluorescent molecules with the optical near-field of a metal tip," Phys. Rev. Lett. 93, 200801 (2004).
[CrossRef] [PubMed]

2002 (1)

A. Kramer, W. Trabesinger, B. Hecht, and U. P. Wild, "Optical near-field enhancement at a metal tip probed by a single fluorophore," Appl. Phys. Lett. 80, 1652-1654 (2002).
[CrossRef]

2001 (2)

H. Gersen, M. F. Garcia-Parajo, L. Novotny, J. A. Veerman, L. Kuipers, and N. F. Van Hulst, "Near-field effects in single molecule emission," J. Microsc. 202, 374-378 (2001).
[CrossRef] [PubMed]

A. Rahmani, P. C. Chaumet, and F. de Fornel, "Environment-induced modification of spontaneous emission: Single-molecule near-field probe," Phys. Rev. A 63, 023819 (2001).
[CrossRef]

2000 (1)

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

1999 (2)

N. Hayazawa, Y. Inouye, and S. Kawata, "Evanescent field excitation and measurement of dye fluorescence in a metallic probe near-field scanning optical microscope," J. Microsc. 194, 472-476 (1999).
[CrossRef]

E. J. Sanchez, L. Novotny, and X. S. Xie, "Near-field fluorescence microscopy based on two-photon excitation with metal tips," Phys. Rev. Lett. 82, 4014-4017 (1999).
[CrossRef]

1998 (1)

W. L. Barnes, "Fluorescence near interfaces: the role of photonic mode density," J. Mod. Opt. 45, 661-699 (1998).
[CrossRef]

1997 (1)

1996 (1)

L. Novotny, "Single molecule fluorescence in inhomogeneous environments," Appl. Phys. Lett. 69, 3806-3808 (1996).
[CrossRef]

1995 (2)

C. Girard, O. J. F. Martin, and A. Dereux, "Molecular lifetime changes induced by nanometer-scale optical-fields," Phys. Rev. Lett. 75, 3098-3101 (1995).
[CrossRef] [PubMed]

R. X. Bian, R. C. Dunn, and X. S. Xie, "Single molecule emission characteristics in near-field microscopy," Phys. Rev. Lett. 75, 4772-4775 (1995).
[CrossRef] [PubMed]

1994 (1)

L. Novotny and C. Hafner, "Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function," Phys. Rev. E 50, 4094-4106 (1994).
[CrossRef]

1984 (2)

P. Avouris and B. N. J. Persson, "Excited-states at metal-surfaces and their nonradiative relaxation," J. Phys. Chem. 88, 837-848 (1984).
[CrossRef]

G. W. Ford and W. H. Weber, "Electromagnetic-interactions of molecules with metal-surfaces," Phys. Rep. 113, 195-287 (1984).
[CrossRef]

1982 (1)

P. M. Whitmore, H. J. Robota, and C. B. Harris, "Mechanisms for electronic-energy transfer between molecules and metal-surfaces - a comparison of silver and nickel," J. Chem. Phys. 77, 1560-1568 (1982).
[CrossRef]

1978 (1)

R. R. Chance, A. Prock, and R. Silbey, "Molecular fluorescence and energy transfer near interfaces," Adv. Chem. Phys. 37, 1-65 (1978).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Chirsty, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Adv. Chem. Phys. (1)

R. R. Chance, A. Prock, and R. Silbey, "Molecular fluorescence and energy transfer near interfaces," Adv. Chem. Phys. 37, 1-65 (1978).
[CrossRef]

Appl. Phys. Lett. (8)

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

A. Kramer, W. Trabesinger, B. Hecht, and U. P. Wild, "Optical near-field enhancement at a metal tip probed by a single fluorophore," Appl. Phys. Lett. 80, 1652-1654 (2002).
[CrossRef]

F. M. Huang, F. Festy, and D. Richards, "Tip-enhanced fluorescence imaging of quantum dots," Appl. Phys. Lett. 87, (2005).
[CrossRef]

Y. X. Zhang, K. Aslan, M. J. R. Previte, and C. D. Geddes, "Metal-enhanced fluorescence: Surface plasmons can radiate a fluorophore's structured emission," Appl. Phys. Lett. 90, (2007).

G. Winter and W. L. Barnes, "Emission of light through thin silver films via near-field coupling to surface plasmon polaritons," Appl. Phys. Lett. 88, (2006).
[CrossRef]

M. Thomas, J. J. Greffet, R. Carminati, and J. R. Arias-Gonzalez, "Single-molecule spontaneous emission close to absorbing nanostructures," Appl. Phys. Lett. 85, 3863-3865 (2004).
[CrossRef]

J. T. KrugII, E. J. Sanchez, and X. S. Xie, "Fluorescence quenching in tip-enhanced nonlinear optical microscopy," Appl. Phys. Lett. 86, (2005).

L. Novotny, "Single molecule fluorescence in inhomogeneous environments," Appl. Phys. Lett. 69, 3806-3808 (1996).
[CrossRef]

J. Chem. Phys. (1)

P. M. Whitmore, H. J. Robota, and C. B. Harris, "Mechanisms for electronic-energy transfer between molecules and metal-surfaces - a comparison of silver and nickel," J. Chem. Phys. 77, 1560-1568 (1982).
[CrossRef]

J. Microsc. (2)

N. Hayazawa, Y. Inouye, and S. Kawata, "Evanescent field excitation and measurement of dye fluorescence in a metallic probe near-field scanning optical microscope," J. Microsc. 194, 472-476 (1999).
[CrossRef]

H. Gersen, M. F. Garcia-Parajo, L. Novotny, J. A. Veerman, L. Kuipers, and N. F. Van Hulst, "Near-field effects in single molecule emission," J. Microsc. 202, 374-378 (2001).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

W. L. Barnes, "Fluorescence near interfaces: the role of photonic mode density," J. Mod. Opt. 45, 661-699 (1998).
[CrossRef]

J. Opt. A (1)

F. M. Huang and D. Richards, "Fluorescence enhancement and energy transfer in apertureless scanning near-field optical microscopy," J. Opt. A 8, S234-S238 (2006).
[CrossRef]

J. Phys. Chem. (1)

P. Avouris and B. N. J. Persson, "Excited-states at metal-surfaces and their nonradiative relaxation," J. Phys. Chem. 88, 837-848 (1984).
[CrossRef]

J. Phys. Chem. B (2)

F. Cannone, G. Chirico, A. R. Bizzarri, and S. Cannistraro, "Quenching and blinking of fluorescence of a single dye molecule bound to gold nanoparticles," J. Phys. Chem. B 110, 16491-16498 (2006).
[CrossRef] [PubMed]

A. Downes, D. Salter, and A. Elfick, "Finite element simulations of tip-enhanced Raman and fluorescence spectroscopy," J. Phys. Chem. B 110, 6692-6698 (2006).
[CrossRef] [PubMed]

Nano Lett. (1)

E. Dulkeith, M. Ringler, T. A. Klar, J. Feldmann, A. M. Javier, and W. J. Parak, "Gold nanoparticles quench fluorescence by phase induced radiative rate suppression," Nano Lett. 5, 585-589 (2005).
[CrossRef] [PubMed]

New J. Phys. (1)

F. D. Stefani, K. Vasilev, N. Bocchio, F. Gaul, A. Pomozzi, and M. Kreiter, "Photonic mode density effects on single-molecule fluorescence blinking," New J. Phys. 9, 21 (2007).
[CrossRef]

Opt. Commun. (1)

R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux, "Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle," Opt. Commun. 261, 368-375 (2006).
[CrossRef]

Opt. Lett. (1)

Phys. Rep. (1)

G. W. Ford and W. H. Weber, "Electromagnetic-interactions of molecules with metal-surfaces," Phys. Rep. 113, 195-287 (1984).
[CrossRef]

Phys. Rev. A (1)

A. Rahmani, P. C. Chaumet, and F. de Fornel, "Environment-induced modification of spontaneous emission: Single-molecule near-field probe," Phys. Rev. A 63, 023819 (2001).
[CrossRef]

Phys. Rev. B (1)

P. B. Johnson and R. W. Chirsty, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Phys. Rev. E (1)

L. Novotny and C. Hafner, "Light propagation in a cylindrical waveguide with a complex, metallic, dielectric function," Phys. Rev. E 50, 4094-4106 (1994).
[CrossRef]

Phys. Rev. Lett. (8)

E. J. Sanchez, L. Novotny, and X. S. Xie, "Near-field fluorescence microscopy based on two-photon excitation with metal tips," Phys. Rev. Lett. 82, 4014-4017 (1999).
[CrossRef]

C. Girard, O. J. F. Martin, and A. Dereux, "Molecular lifetime changes induced by nanometer-scale optical-fields," Phys. Rev. Lett. 75, 3098-3101 (1995).
[CrossRef] [PubMed]

R. X. Bian, R. C. Dunn, and X. S. Xie, "Single molecule emission characteristics in near-field microscopy," Phys. Rev. Lett. 75, 4772-4775 (1995).
[CrossRef] [PubMed]

H. G. Frey, S. Witt, K. Felderer, and R. Guckenberger, "High-resolution imaging of single fluorescent molecules with the optical near-field of a metal tip," Phys. Rev. Lett. 93, 200801 (2004).
[CrossRef] [PubMed]

J. N. Farahani, D. W. Pohl, H. J. Eisler, and B. Hecht, "Single quantum dot coupled to a scanning optical antenna: A tunable superemitter," Phys. Rev. Lett. 95, (2005).
[CrossRef] [PubMed]

F. D. Stefani, K. Vasilev, N. Bocchio, N. Stoyanova, and M. Kreiter, "Surface-plasmon-mediated single-molecule fluorescence through a thin metallic film," Phys. Rev. Lett. 94, (2005).
[CrossRef] [PubMed]

P. Anger, P. Bharadwaj, and L. Novotny, "Enhancement and quenching of single-molecule fluorescence," Phys. Rev. Lett. 96, 4 (2006).
[CrossRef]

S. Kuhn, U. Hakanson, L. Rogobete, and V. Sandoghdar, "Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna," Phys. Rev. Lett. 97, (2006).
[CrossRef] [PubMed]

Plasmonics (2)

J. R. Lakowicz, "Plasmonics in biology and plasmon-controlled fluorescence," Plasmonics 1, 5-33 (2006).
[CrossRef] [PubMed]

N. A. Issa and R. Guckenberger, "Optical nanofocusig on tapered metallic waveguides," Plasmonics 2, 31-37 (2007).
[CrossRef]

Other (2)

A. W. Snyder, and J. D. Love, Optical waveguide theory (Chapman and Hall, New York, 1983).

D. R. Lide, ed. CRC handbook of chemistry and physics (CRC press, London, 1996).

Supplementary Material (1)

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Figures (9)

Fig. 1.
Fig. 1.

(a). Example solution, Re(H ϕ), with diagram overlay of the model geometry. This cross-section view is symmetric about indicated axis of rotation. In this figure, the tip (silver) is at a distance D=100nm from the dipole and λ=550nm. The metal tip supports a propagating SPP that is clearly visible. The power in propagating SPPs remaining at the top of the tip is measured prior to the top PML, and is counted as nonradiative. (b) Example FEM meshing near the tip showing adaptation and fine mesh near nano-scale features: D=10nm. (c) Diagram illustrating different parts of the tip where the integrated resistive losses are attributed to SPP losses (volume 1) and local energy transfer (volume 2).

Fig. 2.
Fig. 2.

Comparison of the numerical solution of γ LET/γ̃o with the analytic solution [Eq. (6)] that neglects retardation effects. The numerical solution is for a silver tip using the geometry of Fig. 1(a) (without glass substrate) and λ=550nm. The analytic solution assumes the molecule is at distance D from a flat silver substrate (no tip). For comparison, the normalized total nonradiative rate γ nr/γ̃o is shown.

Fig. 3.
Fig. 3.

Numerical solution for a silver tip using the geometry of Fig. 1(a). (a). All calculated rates shown. γ LET approximates well Eq. (6) when D<~10nm. Γsc is defined in Section 6. (b). Validation of the tip portioning: γ prop/γ SPP≅ constant when D<~20nm and γ SPP/γ nr≠constant.

Fig. 4.
Fig. 4.

Relationship between the radiative rate and the nonradiative-SPP rate. A chance the emission spatial distribution is observed.

Fig. 5.
Fig. 5.

Comparison of efficiencies for two different initial quantum efficiencies. The curves are nearly identical for D<~10nm (see text).

Fig. 6.
Fig. 6.

A hypothetical silver tip without loss

Fig. 7.
Fig. 7.

Simulations for exploring the influence of metal dielectric constant. γ LET approximates well Eq. (6) when D<~10nm for both metals.

Fig. 8.
Fig. 8.

Animation showing the approach of a gold tip (λ=565nm). |Re(H)| is plotted, which has been normalized at each frame (excitation is distance independent). The value of D is stated above. The prevalence of the radiative, SPP and LET rates at different distances is distinct. File size: 3.7 Mb. [Media 1]

Fig. 9.
Fig. 9.

Spectra for gold tips in air and water surrounding medium, with fixed D=30nm.

Tables (1)

Tables Icon

Appendix B: Table of terms

Equations (14)

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

γ = γ r + γ i + γ nr .
γ nr = 1 ω tip 1 2 Re ( j * · E ) dV .
γ nr = γ SPP + γ LET .
q = γ r γ = γ r γ r + γ nr + ( 1 q o 1 ) γ o .
x LET < x < x SPP .
γ LET γ ˜ o 3 8 1 ε medium Im { ε metal ε medium ε metal + ε medium } c 3 ω 3 1 D 3 ,
P ( 2 ) = 1 2 ( 2 ) Re ( j * · E ) dV .
γ SPP = γ ( 1 ) + γ prop .
Γ sc = 1 2 ω tip Re ( E s × H s * ) · n ̂ dS γ prop .
σ = ( D 10 nm ) γ r γ SPP + γ r .
P ( 1 ) = 1 2 ( 1 ) Re ( j * · E ) dV .
a = 1 2 S ( E ¯ × H ) · z ̂ dr ,
P prop 2 π R o a 2 P ¯
P ¯ = 1 2 ( E ¯ × H ¯ * ) · z ̂ ' dr ' ,

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