Abstract

We propose an analytical theory which predicts that Converging Plasmon Resonance (CPR) at conical nanotips exhibits a red-shifted and continuous band of resonant frequencies and suggests potential application of conical nanotips in various fields, such as plasmonic solar cells, photothermal therapy, tip-enhanced Raman and other spectroscopies. The CPR modes exhibit superior confinement and ten times broader scattering bandwidth over the entire solar spectrum than smooth nano-structures. The theory also explicitly connects the optimal angles and resonant optical frequencies to the material permittivities, with a specific optimum half angle that depends only on the real permittivity for high-permittivity and low-loss materials.

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  1. V. Ferry, J. Munday, and H. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater.22, 4794–4808 (2010).
    [CrossRef] [PubMed]
  2. X. Lu, M. Rycenga, S. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem.60, 167–192 (2009).
    [CrossRef]
  3. C. Lee, S. Bae, S. Mobasser, and H. Manohara, “A novel silicon nanotips antireflection surface for the micro sun sensor,” Nano Lett.5, 2438–2442 (2005).
    [CrossRef] [PubMed]
  4. A. R. Parker and H. E. Townley, “Biomimetics of photonic nanostructures,” Nat. Nanotechnol.2, 347–353 (2007).
    [CrossRef]
  5. A. V. Goncharenko, H. C. Chang, and J. K. Wang, “Electric near-field enhancing properties of a finite-size metal conical nano-tip,” Ultramicroscopy107, 151–157 (2007).
    [CrossRef]
  6. X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc128, 2115–2120 (2006).
    [CrossRef] [PubMed]
  7. R. Rodriguez-Oliveros and J. A. Sanchez-Gil, “Gold nanostars as thermoplasmonic nanoparticles for optical heating,” Opt. Express20, 621–626 (2012).
    [CrossRef] [PubMed]
  8. A. Mohammadi, F. Kaminski, V. Sandoghdar, and M. Agio, “Fluorescence enhancement with the optical (bi-) conical antenna,” J. Phys. Chem. C114, 7372–7377 (2010).
    [CrossRef]
  9. H. Shen, N. Guillot, J. Rouxel, M. de la Chapelle, and T. Toury, “Optimized plasmonic nanostructures for improved sensing activities,” Opt. Express20, 21278–21290 (2012).
    [CrossRef] [PubMed]
  10. A. F. Stevenson, “Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatter)/wavelength,” Appl. Phys. Lett.24, 1134–1141 (1953).
  11. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett.93, 137404 (2004).
    [CrossRef] [PubMed]
  12. A. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric near-field enhancement of a sharp semi-infinite conical probe: Material and cone angle dependence,” Phys.Rev.B74, 235442 (2006).
    [CrossRef]
  13. J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys70, 1–87 (2007).
    [CrossRef]
  14. L. Novotny and S. J. Stranick, “Near-field optical microscopy and spectroscopy with pointed probes,” Annu. Rev. Phys. Chem.57, 303–331 (2006).
    [CrossRef]
  15. Y. Kawata, C. Xu, and W. Denk, “Feasibility of molecular-resolution fluorescence near-field microscopy using multi-photon absorption and field enhancement near a sharp tip,” J. Appl. Phys.85, 1294–1301 (1999).
    [CrossRef]
  16. P. B. Johnson and R. W. Christy, “Optical constants of the nobel metals,” Phys. Rev. B6, 4370–4379 (1972).
    [CrossRef]
  17. F. Wang and Y. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett.97, 206806 (2006).
    [CrossRef] [PubMed]
  18. N. A. Issa and R. Guckenberger, “Optical nanofocusing on tapered metallic waveguides,” Plasmonics2, 31–37 (2007).
    [CrossRef]

2012

2010

A. Mohammadi, F. Kaminski, V. Sandoghdar, and M. Agio, “Fluorescence enhancement with the optical (bi-) conical antenna,” J. Phys. Chem. C114, 7372–7377 (2010).
[CrossRef]

V. Ferry, J. Munday, and H. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater.22, 4794–4808 (2010).
[CrossRef] [PubMed]

2009

X. Lu, M. Rycenga, S. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem.60, 167–192 (2009).
[CrossRef]

2007

A. R. Parker and H. E. Townley, “Biomimetics of photonic nanostructures,” Nat. Nanotechnol.2, 347–353 (2007).
[CrossRef]

A. V. Goncharenko, H. C. Chang, and J. K. Wang, “Electric near-field enhancing properties of a finite-size metal conical nano-tip,” Ultramicroscopy107, 151–157 (2007).
[CrossRef]

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys70, 1–87 (2007).
[CrossRef]

N. A. Issa and R. Guckenberger, “Optical nanofocusing on tapered metallic waveguides,” Plasmonics2, 31–37 (2007).
[CrossRef]

2006

F. Wang and Y. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett.97, 206806 (2006).
[CrossRef] [PubMed]

A. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric near-field enhancement of a sharp semi-infinite conical probe: Material and cone angle dependence,” Phys.Rev.B74, 235442 (2006).
[CrossRef]

L. Novotny and S. J. Stranick, “Near-field optical microscopy and spectroscopy with pointed probes,” Annu. Rev. Phys. Chem.57, 303–331 (2006).
[CrossRef]

X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc128, 2115–2120 (2006).
[CrossRef] [PubMed]

2005

C. Lee, S. Bae, S. Mobasser, and H. Manohara, “A novel silicon nanotips antireflection surface for the micro sun sensor,” Nano Lett.5, 2438–2442 (2005).
[CrossRef] [PubMed]

2004

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett.93, 137404 (2004).
[CrossRef] [PubMed]

1999

Y. Kawata, C. Xu, and W. Denk, “Feasibility of molecular-resolution fluorescence near-field microscopy using multi-photon absorption and field enhancement near a sharp tip,” J. Appl. Phys.85, 1294–1301 (1999).
[CrossRef]

1972

P. B. Johnson and R. W. Christy, “Optical constants of the nobel metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

1953

A. F. Stevenson, “Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatter)/wavelength,” Appl. Phys. Lett.24, 1134–1141 (1953).

Agio, M.

A. Mohammadi, F. Kaminski, V. Sandoghdar, and M. Agio, “Fluorescence enhancement with the optical (bi-) conical antenna,” J. Phys. Chem. C114, 7372–7377 (2010).
[CrossRef]

Atwater, H.

V. Ferry, J. Munday, and H. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater.22, 4794–4808 (2010).
[CrossRef] [PubMed]

Bae, S.

C. Lee, S. Bae, S. Mobasser, and H. Manohara, “A novel silicon nanotips antireflection surface for the micro sun sensor,” Nano Lett.5, 2438–2442 (2005).
[CrossRef] [PubMed]

Chang, H. C.

A. V. Goncharenko, H. C. Chang, and J. K. Wang, “Electric near-field enhancing properties of a finite-size metal conical nano-tip,” Ultramicroscopy107, 151–157 (2007).
[CrossRef]

Chang, Y. C.

A. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric near-field enhancement of a sharp semi-infinite conical probe: Material and cone angle dependence,” Phys.Rev.B74, 235442 (2006).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the nobel metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Chulkov, E. V.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys70, 1–87 (2007).
[CrossRef]

de la Chapelle, M.

Denk, W.

Y. Kawata, C. Xu, and W. Denk, “Feasibility of molecular-resolution fluorescence near-field microscopy using multi-photon absorption and field enhancement near a sharp tip,” J. Appl. Phys.85, 1294–1301 (1999).
[CrossRef]

Echenique, P. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys70, 1–87 (2007).
[CrossRef]

El-Sayed, I. H.

X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc128, 2115–2120 (2006).
[CrossRef] [PubMed]

El-Sayed, M. A.

X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc128, 2115–2120 (2006).
[CrossRef] [PubMed]

Ferry, V.

V. Ferry, J. Munday, and H. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater.22, 4794–4808 (2010).
[CrossRef] [PubMed]

Goncharenko, A.

A. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric near-field enhancement of a sharp semi-infinite conical probe: Material and cone angle dependence,” Phys.Rev.B74, 235442 (2006).
[CrossRef]

Goncharenko, A. V.

A. V. Goncharenko, H. C. Chang, and J. K. Wang, “Electric near-field enhancing properties of a finite-size metal conical nano-tip,” Ultramicroscopy107, 151–157 (2007).
[CrossRef]

Guckenberger, R.

N. A. Issa and R. Guckenberger, “Optical nanofocusing on tapered metallic waveguides,” Plasmonics2, 31–37 (2007).
[CrossRef]

Guillot, N.

Huang, X.

X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc128, 2115–2120 (2006).
[CrossRef] [PubMed]

Issa, N. A.

N. A. Issa and R. Guckenberger, “Optical nanofocusing on tapered metallic waveguides,” Plasmonics2, 31–37 (2007).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the nobel metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Kaminski, F.

A. Mohammadi, F. Kaminski, V. Sandoghdar, and M. Agio, “Fluorescence enhancement with the optical (bi-) conical antenna,” J. Phys. Chem. C114, 7372–7377 (2010).
[CrossRef]

Kawata, Y.

Y. Kawata, C. Xu, and W. Denk, “Feasibility of molecular-resolution fluorescence near-field microscopy using multi-photon absorption and field enhancement near a sharp tip,” J. Appl. Phys.85, 1294–1301 (1999).
[CrossRef]

Lee, C.

C. Lee, S. Bae, S. Mobasser, and H. Manohara, “A novel silicon nanotips antireflection surface for the micro sun sensor,” Nano Lett.5, 2438–2442 (2005).
[CrossRef] [PubMed]

Lu, X.

X. Lu, M. Rycenga, S. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem.60, 167–192 (2009).
[CrossRef]

Manohara, H.

C. Lee, S. Bae, S. Mobasser, and H. Manohara, “A novel silicon nanotips antireflection surface for the micro sun sensor,” Nano Lett.5, 2438–2442 (2005).
[CrossRef] [PubMed]

Mobasser, S.

C. Lee, S. Bae, S. Mobasser, and H. Manohara, “A novel silicon nanotips antireflection surface for the micro sun sensor,” Nano Lett.5, 2438–2442 (2005).
[CrossRef] [PubMed]

Mohammadi, A.

A. Mohammadi, F. Kaminski, V. Sandoghdar, and M. Agio, “Fluorescence enhancement with the optical (bi-) conical antenna,” J. Phys. Chem. C114, 7372–7377 (2010).
[CrossRef]

Munday, J.

V. Ferry, J. Munday, and H. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater.22, 4794–4808 (2010).
[CrossRef] [PubMed]

Novotny, L.

L. Novotny and S. J. Stranick, “Near-field optical microscopy and spectroscopy with pointed probes,” Annu. Rev. Phys. Chem.57, 303–331 (2006).
[CrossRef]

Parker, A. R.

A. R. Parker and H. E. Townley, “Biomimetics of photonic nanostructures,” Nat. Nanotechnol.2, 347–353 (2007).
[CrossRef]

Pitarke, J. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys70, 1–87 (2007).
[CrossRef]

Qian, W.

X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc128, 2115–2120 (2006).
[CrossRef] [PubMed]

Rodriguez-Oliveros, R.

Rouxel, J.

Rycenga, M.

X. Lu, M. Rycenga, S. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem.60, 167–192 (2009).
[CrossRef]

Sanchez-Gil, J. A.

Sandoghdar, V.

A. Mohammadi, F. Kaminski, V. Sandoghdar, and M. Agio, “Fluorescence enhancement with the optical (bi-) conical antenna,” J. Phys. Chem. C114, 7372–7377 (2010).
[CrossRef]

Shen, H.

Shen, Y.

F. Wang and Y. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett.97, 206806 (2006).
[CrossRef] [PubMed]

Silkin, V. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys70, 1–87 (2007).
[CrossRef]

Skrabalak, S.

X. Lu, M. Rycenga, S. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem.60, 167–192 (2009).
[CrossRef]

Stevenson, A. F.

A. F. Stevenson, “Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatter)/wavelength,” Appl. Phys. Lett.24, 1134–1141 (1953).

Stockman, M. I.

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett.93, 137404 (2004).
[CrossRef] [PubMed]

Stranick, S. J.

L. Novotny and S. J. Stranick, “Near-field optical microscopy and spectroscopy with pointed probes,” Annu. Rev. Phys. Chem.57, 303–331 (2006).
[CrossRef]

Toury, T.

Townley, H. E.

A. R. Parker and H. E. Townley, “Biomimetics of photonic nanostructures,” Nat. Nanotechnol.2, 347–353 (2007).
[CrossRef]

Wang, F.

F. Wang and Y. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett.97, 206806 (2006).
[CrossRef] [PubMed]

Wang, J. K.

A. V. Goncharenko, H. C. Chang, and J. K. Wang, “Electric near-field enhancing properties of a finite-size metal conical nano-tip,” Ultramicroscopy107, 151–157 (2007).
[CrossRef]

A. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric near-field enhancement of a sharp semi-infinite conical probe: Material and cone angle dependence,” Phys.Rev.B74, 235442 (2006).
[CrossRef]

Wiley, B.

X. Lu, M. Rycenga, S. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem.60, 167–192 (2009).
[CrossRef]

Xia, Y.

X. Lu, M. Rycenga, S. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem.60, 167–192 (2009).
[CrossRef]

Xu, C.

Y. Kawata, C. Xu, and W. Denk, “Feasibility of molecular-resolution fluorescence near-field microscopy using multi-photon absorption and field enhancement near a sharp tip,” J. Appl. Phys.85, 1294–1301 (1999).
[CrossRef]

Adv. Mater.

V. Ferry, J. Munday, and H. Atwater, “Design considerations for plasmonic photovoltaics,” Adv. Mater.22, 4794–4808 (2010).
[CrossRef] [PubMed]

Annu. Rev. Phys. Chem.

X. Lu, M. Rycenga, S. Skrabalak, B. Wiley, and Y. Xia, “Chemical synthesis of novel plasmonic nanoparticles,” Annu. Rev. Phys. Chem.60, 167–192 (2009).
[CrossRef]

L. Novotny and S. J. Stranick, “Near-field optical microscopy and spectroscopy with pointed probes,” Annu. Rev. Phys. Chem.57, 303–331 (2006).
[CrossRef]

Appl. Phys. Lett.

A. F. Stevenson, “Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatter)/wavelength,” Appl. Phys. Lett.24, 1134–1141 (1953).

J. Am. Chem. Soc

X. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc128, 2115–2120 (2006).
[CrossRef] [PubMed]

J. Appl. Phys.

Y. Kawata, C. Xu, and W. Denk, “Feasibility of molecular-resolution fluorescence near-field microscopy using multi-photon absorption and field enhancement near a sharp tip,” J. Appl. Phys.85, 1294–1301 (1999).
[CrossRef]

J. Phys. Chem. C

A. Mohammadi, F. Kaminski, V. Sandoghdar, and M. Agio, “Fluorescence enhancement with the optical (bi-) conical antenna,” J. Phys. Chem. C114, 7372–7377 (2010).
[CrossRef]

Nano Lett.

C. Lee, S. Bae, S. Mobasser, and H. Manohara, “A novel silicon nanotips antireflection surface for the micro sun sensor,” Nano Lett.5, 2438–2442 (2005).
[CrossRef] [PubMed]

Nat. Nanotechnol.

A. R. Parker and H. E. Townley, “Biomimetics of photonic nanostructures,” Nat. Nanotechnol.2, 347–353 (2007).
[CrossRef]

Opt. Express

Phys. Rev. B

P. B. Johnson and R. W. Christy, “Optical constants of the nobel metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Phys. Rev. Lett.

F. Wang and Y. Shen, “General properties of local plasmons in metal nanostructures,” Phys. Rev. Lett.97, 206806 (2006).
[CrossRef] [PubMed]

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett.93, 137404 (2004).
[CrossRef] [PubMed]

Phys.Rev.B

A. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric near-field enhancement of a sharp semi-infinite conical probe: Material and cone angle dependence,” Phys.Rev.B74, 235442 (2006).
[CrossRef]

Plasmonics

N. A. Issa and R. Guckenberger, “Optical nanofocusing on tapered metallic waveguides,” Plasmonics2, 31–37 (2007).
[CrossRef]

Rep. Prog. Phys

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys70, 1–87 (2007).
[CrossRef]

Ultramicroscopy

A. V. Goncharenko, H. C. Chang, and J. K. Wang, “Electric near-field enhancing properties of a finite-size metal conical nano-tip,” Ultramicroscopy107, 151–157 (2007).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

(a) Conformal map in complex permittivities domain. Grey regions shows CPR sustainable region for a cone with half angle α = π/6 and the contour lines correspond to constant υr. The branch point dg(α, υ)/ = 0 of the conformal map is indicated as a solid dot and the limit point (υi → ∞) is indicated as an open circle on the negative εr axis. The bold circular contour is the image of υr = 1. The hollow circles are empirical permittivities for gold ranging from infrared to ultraviolet wavelength [16] and the dashed curve is the Drude permittivity model for gold [13]. The band of excited frequencies for the Drude model fall at the “resonance” band between the branch point and the limit point on the real axis. (b) Intensification exponents for the Drude gold cone with half angle 5, 10, 15 and 20 degrees are plotted as solid curves. The dotted line marks the resonance peak position of a single nanosphere for comparison [2]. The broad bandwidth extends beyond the visible range for the Drude cone and the resonance frequency is red-shifted compared to gold nanospheres.

Fig. 2
Fig. 2

Constant intensification exponent contours lines (1 −υr = 1.428, 1.4, 1.3, 1.1) for a gold cone for empirical permittivities from [16], with different angles and incident light wavelengths, where the dashed curve connects the local resonant conditions from large angles until the global optimal angle (circle). The inset (a) plots the intensification exponent for a gold cone with different cone angles (α = 5°, 10°, 15°, 20°, 30°) and shows the CPR spectrum broadens and exhibits a red shift away from the planar plasmon resonant wavelength λs = c/ωs. The inset (b) plots the ratio of the imaginary to real permittivity for gold as a function of wavelength [16], where maximum intensification occurs at λ = 892nm.

Fig. 3
Fig. 3

Optimum angles of a gold cone at different εr corresponding to different excitation wavelength. Simulation data from the solution of the Maxwell equation [18] are compared to estimates from the branch point, the critical point and the exact solution of the CPR dispersion relationship Eq. (1). The inset depicts the dominant CPR intensification exponent as a function of the half angle for five imaginary permittivities εi= 0.01, 0.5, 1, 3 and 10.0 at εr = −182. The optimum angle yielding the most negative υr is computed numerically for each εi and is marked as a solid circle on each root locus. Analytical estimate, with a εi/εr scaling, from an expansion about the critical point is shown as open triangles. At the limit of εi/εr → 0, the eigenvalue of this optimum angle approaches the critical point at υo = −1/2 +υioi.

Fig. 4
Fig. 4

(a) Comparison of theoretical intensification factor (υr − 1)log10(R1/R2) (solid curves) against the literature values of [18] (scattered points) for gold cones of different angles (R1=5 nm, R2=300 nm). (b) Comparison against literature data [18] for three different materials at excitation wavelength 488 nm with the same R1 and R2.

Equations (2)

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ε m ε o + f ( π α , υ ) f θ ( α , υ ) f ( α , υ ) f θ ( π α , υ ) = ε m ε o + g ( α , υ ) = 0
ε r = 8 α * 2 ln ( α * / 8 )

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