Abstract

Single colloidal CdSe/ZnS nanocrystals are deposited at various distances from a gold film in order to improve their performance as single-photon sources. Photon antibunching is demonstrated and the experimental curves are accurately fitted by theoretical equations. Emission lifetime and intensity are measured and found in excellent agreement with theoretical values. The various effects of a neighbouring gold film are discussed : interferences of the excitation beam, interferences of the fluorescence light, opening of plasmon and lossy-surface-wave modes, modification of the radiation pattern leading to a modified objective collection efficiency. At 80 nm from the gold film, when using an objective with 0.75 numerical aperture, about a 2.4-fold increase of the detected intensity is evidenced.

© 2010 Optical Society of America

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  1. B. C. Buchler, T. Kalkbrenner, C. Hettich, and V. Sandoghdar, “Measuring the quantum efficiency of the optical emission of single radiating dipoles using a scanning mirror,” Phys. Rev. Lett. 95, 063003 (2005).
    [Crossref] [PubMed]
  2. X. Brokmann, L. Coolen, M. Dahan, and J.-P. Hermier, “Measurement of the radiative and nonradiative decay rates of single CdSe nanocrystals through a controlled modification of their spontaneous emission,” Phys. Rev. Lett. 93, 107403 (2004).
    [Crossref] [PubMed]
  3. M. D. Leistikow, J. Johansen, A. J. Kettelarij, P. Lodahl, and W. L. Vos, “Size-dependent oscillator strength and quantum efficiency of CdSe quantum dots controlled via the local density of states,” Phys. Rev. B 79, 045301 (2009).
    [Crossref]
  4. W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45, 661 (1998).
    [Crossref]
  5. 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, 023005 (2005).
    [Crossref] [PubMed]
  6. K. Ray, R. Badugu, and J. R. Lakowicz, “Metal-enhanced fluorescence from CdTe nanocrystals : A single-molecule fluorescence study,” J.Am. Chem. Soc. 128, 8998 (2006).
    [Crossref] [PubMed]
  7. X. Wu, Y. Sun, and M. Pelton, “Recombination rates for single colloidal quantum dots near a smooth metal film,” Phys. Chem. Chem. Phys. 11, 5867 (2009).
    [Crossref] [PubMed]
  8. E. Fort and S. Gresillon, “Surface enhanced fluorescence,” J. Phys. D 41, 013001 (2008).
    [Crossref]
  9. K. T. Shimizu, W. K. Woo, B. R. Fisher, H. J. Eisler, and M. G. Bawendi, “Surface-enhanced emission from single semiconductor nanocrystals,” Phys. Rev. Lett. 89, 117401 (2002).
    [Crossref] [PubMed]
  10. Ito Yuichi, Matsuda Kazunari, and Kanemitsu Yoshihiko, “Mechanism of photoluminescence enhancement in single semiconductor nanocrystals on metal surfaces,” Phys. Rev. B 75, 033309 (2007).
    [Crossref]
  11. E. D. Palik, Handbook of Optical Constants of Solids, (Academic Press, 2005).
  12. B. Lounis, H. A. Bechtel, D. Gerion, P. Alivisatos, and W. E. Moerner, “Photon antibunching in single cdse/zns quantum dot fluorescence,” Chem. Phys. Lett. 329, 399 (2000).
    [Crossref]
  13. P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282 (2000).
    [Crossref] [PubMed]
  14. G. Messin, J. P. Hermier, E. Giacobino, P. Desbiolles, and M. Dahan, “Bunching and antibunching in the fluorescence of semiconductor nanocrystals,” Opt. Lett. 26, 1891 (2001).
    [Crossref]
  15. R. Hanbury-Brown and R. Q. Twiss, “The Question of Correlation between Photons in Coherent Light Rays,” Nature 178, 1447–1448 (1956).
    [Crossref]
  16. M Nirmal, B. O. Dabbousi, M. G. Bawendi, J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, “Fluorescence intermittency in single cadmium selenide nanocrystals,” Nature 383, 802 (1996).
    [Crossref]
  17. B. R. Fisher, H.-J. Eisler, N. E. Stott, and M. G. Bawendi, “Emission Intensity Dependence and Single-Exponential Behavior In Single Colloidal Quantum Dot Fluorescence Lifetimes,” J. Phys. Chem. B 108, 143–148 (2004).
    [Crossref]
  18. X. Brokmann, E. Giacobino, M. Dahan, and J. P. Hermier, “Highly efficient triggered emission of single photons by colloidal cdse/zns nanocrystals,” Appl. Phys. Lett. 85, 712 (2004).
    [Crossref]
  19. S. A. Empedocles, R. Neuhauser, and M. G. Bawendi, “Three-dimensional orientation measurements of symmetric single chromophores using polarization microscopy,” Nature 399, 126 (1999).
    [Crossref]
  20. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. i. total radiated power,” J. Opt. Soc. Am. 67, 1607 (1977).
    [Crossref]
  21. R. R. Chance, A. Prock¡, and R. Silbey, “Lifetime of an emitting molecule near a partially reflecting surface,” J. Chem. Phys. 60, 2744 (1974).
    [Crossref]
  22. G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195 (1984).
    [Crossref]
  23. W. H. Weber and C. F. Eagen, “Energy transfer from an excited dye molecule to the surface plasmons of an adjacent metal,” Opt. Lett. 4, 236 (1979).
    [Crossref] [PubMed]
  24. K. Ray, H. Szmacinski, J. Enderlein, and J.R. Lakowicz, “Distance dependence of surface plasmon-coupled emission observed using Langmuir-Blodgett films,” Appl. Phys. Lett. 90, 251116 (2007).
    [Crossref] [PubMed]
  25. J. Enderlein, “Single-molecule fluorescence near a metal layer,” Chem. Phys. 247, 1 (1999).
    [Crossref]
  26. C. A. Leatherdale, W.-K. Woo, F. V. Mikulec, and M. G. Bawendi, “On the absorption cross section of CdSe nanocrystal quantum dots,” J. Phys. Chem. B 106, 7619 (2002).
    [Crossref]
  27. W. Lukosz, “Theory of optical-environment-dependent spontaneous emission rates for emitters in thin layers,” Phys. Rev. B 22, 3030 (1980).
    [Crossref]
  28. R. T. Holm, S. W. McKnight, E. D. Palik, and W. Lukosz, “Interference effects in luminescence studies of thin films,” Appl. Opt. 21, 2512 (1982).
    [Crossref] [PubMed]

2009 (2)

M. D. Leistikow, J. Johansen, A. J. Kettelarij, P. Lodahl, and W. L. Vos, “Size-dependent oscillator strength and quantum efficiency of CdSe quantum dots controlled via the local density of states,” Phys. Rev. B 79, 045301 (2009).
[Crossref]

X. Wu, Y. Sun, and M. Pelton, “Recombination rates for single colloidal quantum dots near a smooth metal film,” Phys. Chem. Chem. Phys. 11, 5867 (2009).
[Crossref] [PubMed]

2008 (1)

E. Fort and S. Gresillon, “Surface enhanced fluorescence,” J. Phys. D 41, 013001 (2008).
[Crossref]

2007 (2)

Ito Yuichi, Matsuda Kazunari, and Kanemitsu Yoshihiko, “Mechanism of photoluminescence enhancement in single semiconductor nanocrystals on metal surfaces,” Phys. Rev. B 75, 033309 (2007).
[Crossref]

K. Ray, H. Szmacinski, J. Enderlein, and J.R. Lakowicz, “Distance dependence of surface plasmon-coupled emission observed using Langmuir-Blodgett films,” Appl. Phys. Lett. 90, 251116 (2007).
[Crossref] [PubMed]

2006 (1)

K. Ray, R. Badugu, and J. R. Lakowicz, “Metal-enhanced fluorescence from CdTe nanocrystals : A single-molecule fluorescence study,” J.Am. Chem. Soc. 128, 8998 (2006).
[Crossref] [PubMed]

2005 (2)

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, 023005 (2005).
[Crossref] [PubMed]

B. C. Buchler, T. Kalkbrenner, C. Hettich, and V. Sandoghdar, “Measuring the quantum efficiency of the optical emission of single radiating dipoles using a scanning mirror,” Phys. Rev. Lett. 95, 063003 (2005).
[Crossref] [PubMed]

2004 (3)

X. Brokmann, L. Coolen, M. Dahan, and J.-P. Hermier, “Measurement of the radiative and nonradiative decay rates of single CdSe nanocrystals through a controlled modification of their spontaneous emission,” Phys. Rev. Lett. 93, 107403 (2004).
[Crossref] [PubMed]

B. R. Fisher, H.-J. Eisler, N. E. Stott, and M. G. Bawendi, “Emission Intensity Dependence and Single-Exponential Behavior In Single Colloidal Quantum Dot Fluorescence Lifetimes,” J. Phys. Chem. B 108, 143–148 (2004).
[Crossref]

X. Brokmann, E. Giacobino, M. Dahan, and J. P. Hermier, “Highly efficient triggered emission of single photons by colloidal cdse/zns nanocrystals,” Appl. Phys. Lett. 85, 712 (2004).
[Crossref]

2002 (2)

K. T. Shimizu, W. K. Woo, B. R. Fisher, H. J. Eisler, and M. G. Bawendi, “Surface-enhanced emission from single semiconductor nanocrystals,” Phys. Rev. Lett. 89, 117401 (2002).
[Crossref] [PubMed]

C. A. Leatherdale, W.-K. Woo, F. V. Mikulec, and M. G. Bawendi, “On the absorption cross section of CdSe nanocrystal quantum dots,” J. Phys. Chem. B 106, 7619 (2002).
[Crossref]

2001 (1)

2000 (2)

B. Lounis, H. A. Bechtel, D. Gerion, P. Alivisatos, and W. E. Moerner, “Photon antibunching in single cdse/zns quantum dot fluorescence,” Chem. Phys. Lett. 329, 399 (2000).
[Crossref]

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282 (2000).
[Crossref] [PubMed]

1999 (2)

S. A. Empedocles, R. Neuhauser, and M. G. Bawendi, “Three-dimensional orientation measurements of symmetric single chromophores using polarization microscopy,” Nature 399, 126 (1999).
[Crossref]

J. Enderlein, “Single-molecule fluorescence near a metal layer,” Chem. Phys. 247, 1 (1999).
[Crossref]

1998 (1)

W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45, 661 (1998).
[Crossref]

1996 (1)

M Nirmal, B. O. Dabbousi, M. G. Bawendi, J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, “Fluorescence intermittency in single cadmium selenide nanocrystals,” Nature 383, 802 (1996).
[Crossref]

1984 (1)

G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195 (1984).
[Crossref]

1982 (1)

1980 (1)

W. Lukosz, “Theory of optical-environment-dependent spontaneous emission rates for emitters in thin layers,” Phys. Rev. B 22, 3030 (1980).
[Crossref]

1979 (1)

1977 (1)

1974 (1)

R. R. Chance, A. Prock¡, and R. Silbey, “Lifetime of an emitting molecule near a partially reflecting surface,” J. Chem. Phys. 60, 2744 (1974).
[Crossref]

1956 (1)

R. Hanbury-Brown and R. Q. Twiss, “The Question of Correlation between Photons in Coherent Light Rays,” Nature 178, 1447–1448 (1956).
[Crossref]

Alivisatos, P.

B. Lounis, H. A. Bechtel, D. Gerion, P. Alivisatos, and W. E. Moerner, “Photon antibunching in single cdse/zns quantum dot fluorescence,” Chem. Phys. Lett. 329, 399 (2000).
[Crossref]

Badugu, R.

K. Ray, R. Badugu, and J. R. Lakowicz, “Metal-enhanced fluorescence from CdTe nanocrystals : A single-molecule fluorescence study,” J.Am. Chem. Soc. 128, 8998 (2006).
[Crossref] [PubMed]

Barnes, W. L.

W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45, 661 (1998).
[Crossref]

Bawendi, M. G.

B. R. Fisher, H.-J. Eisler, N. E. Stott, and M. G. Bawendi, “Emission Intensity Dependence and Single-Exponential Behavior In Single Colloidal Quantum Dot Fluorescence Lifetimes,” J. Phys. Chem. B 108, 143–148 (2004).
[Crossref]

K. T. Shimizu, W. K. Woo, B. R. Fisher, H. J. Eisler, and M. G. Bawendi, “Surface-enhanced emission from single semiconductor nanocrystals,” Phys. Rev. Lett. 89, 117401 (2002).
[Crossref] [PubMed]

C. A. Leatherdale, W.-K. Woo, F. V. Mikulec, and M. G. Bawendi, “On the absorption cross section of CdSe nanocrystal quantum dots,” J. Phys. Chem. B 106, 7619 (2002).
[Crossref]

S. A. Empedocles, R. Neuhauser, and M. G. Bawendi, “Three-dimensional orientation measurements of symmetric single chromophores using polarization microscopy,” Nature 399, 126 (1999).
[Crossref]

M Nirmal, B. O. Dabbousi, M. G. Bawendi, J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, “Fluorescence intermittency in single cadmium selenide nanocrystals,” Nature 383, 802 (1996).
[Crossref]

Becher, C.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282 (2000).
[Crossref] [PubMed]

Bechtel, H. A.

B. Lounis, H. A. Bechtel, D. Gerion, P. Alivisatos, and W. E. Moerner, “Photon antibunching in single cdse/zns quantum dot fluorescence,” Chem. Phys. Lett. 329, 399 (2000).
[Crossref]

Bocchio, N.

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, 023005 (2005).
[Crossref] [PubMed]

Brokmann, X.

X. Brokmann, L. Coolen, M. Dahan, and J.-P. Hermier, “Measurement of the radiative and nonradiative decay rates of single CdSe nanocrystals through a controlled modification of their spontaneous emission,” Phys. Rev. Lett. 93, 107403 (2004).
[Crossref] [PubMed]

X. Brokmann, E. Giacobino, M. Dahan, and J. P. Hermier, “Highly efficient triggered emission of single photons by colloidal cdse/zns nanocrystals,” Appl. Phys. Lett. 85, 712 (2004).
[Crossref]

Brus, L. E.

M Nirmal, B. O. Dabbousi, M. G. Bawendi, J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, “Fluorescence intermittency in single cadmium selenide nanocrystals,” Nature 383, 802 (1996).
[Crossref]

Buchler, B. C.

B. C. Buchler, T. Kalkbrenner, C. Hettich, and V. Sandoghdar, “Measuring the quantum efficiency of the optical emission of single radiating dipoles using a scanning mirror,” Phys. Rev. Lett. 95, 063003 (2005).
[Crossref] [PubMed]

Chance, R. R.

R. R. Chance, A. Prock¡, and R. Silbey, “Lifetime of an emitting molecule near a partially reflecting surface,” J. Chem. Phys. 60, 2744 (1974).
[Crossref]

Coolen, L.

X. Brokmann, L. Coolen, M. Dahan, and J.-P. Hermier, “Measurement of the radiative and nonradiative decay rates of single CdSe nanocrystals through a controlled modification of their spontaneous emission,” Phys. Rev. Lett. 93, 107403 (2004).
[Crossref] [PubMed]

Dabbousi, B. O.

M Nirmal, B. O. Dabbousi, M. G. Bawendi, J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, “Fluorescence intermittency in single cadmium selenide nanocrystals,” Nature 383, 802 (1996).
[Crossref]

Dahan, M.

X. Brokmann, E. Giacobino, M. Dahan, and J. P. Hermier, “Highly efficient triggered emission of single photons by colloidal cdse/zns nanocrystals,” Appl. Phys. Lett. 85, 712 (2004).
[Crossref]

X. Brokmann, L. Coolen, M. Dahan, and J.-P. Hermier, “Measurement of the radiative and nonradiative decay rates of single CdSe nanocrystals through a controlled modification of their spontaneous emission,” Phys. Rev. Lett. 93, 107403 (2004).
[Crossref] [PubMed]

G. Messin, J. P. Hermier, E. Giacobino, P. Desbiolles, and M. Dahan, “Bunching and antibunching in the fluorescence of semiconductor nanocrystals,” Opt. Lett. 26, 1891 (2001).
[Crossref]

Desbiolles, P.

Eagen, C. F.

Eisler, H. J.

K. T. Shimizu, W. K. Woo, B. R. Fisher, H. J. Eisler, and M. G. Bawendi, “Surface-enhanced emission from single semiconductor nanocrystals,” Phys. Rev. Lett. 89, 117401 (2002).
[Crossref] [PubMed]

Eisler, H.-J.

B. R. Fisher, H.-J. Eisler, N. E. Stott, and M. G. Bawendi, “Emission Intensity Dependence and Single-Exponential Behavior In Single Colloidal Quantum Dot Fluorescence Lifetimes,” J. Phys. Chem. B 108, 143–148 (2004).
[Crossref]

Empedocles, S. A.

S. A. Empedocles, R. Neuhauser, and M. G. Bawendi, “Three-dimensional orientation measurements of symmetric single chromophores using polarization microscopy,” Nature 399, 126 (1999).
[Crossref]

Enderlein, J.

K. Ray, H. Szmacinski, J. Enderlein, and J.R. Lakowicz, “Distance dependence of surface plasmon-coupled emission observed using Langmuir-Blodgett films,” Appl. Phys. Lett. 90, 251116 (2007).
[Crossref] [PubMed]

J. Enderlein, “Single-molecule fluorescence near a metal layer,” Chem. Phys. 247, 1 (1999).
[Crossref]

Fisher, B. R.

B. R. Fisher, H.-J. Eisler, N. E. Stott, and M. G. Bawendi, “Emission Intensity Dependence and Single-Exponential Behavior In Single Colloidal Quantum Dot Fluorescence Lifetimes,” J. Phys. Chem. B 108, 143–148 (2004).
[Crossref]

K. T. Shimizu, W. K. Woo, B. R. Fisher, H. J. Eisler, and M. G. Bawendi, “Surface-enhanced emission from single semiconductor nanocrystals,” Phys. Rev. Lett. 89, 117401 (2002).
[Crossref] [PubMed]

Ford, G. W.

G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195 (1984).
[Crossref]

Fort, E.

E. Fort and S. Gresillon, “Surface enhanced fluorescence,” J. Phys. D 41, 013001 (2008).
[Crossref]

Gerion, D.

B. Lounis, H. A. Bechtel, D. Gerion, P. Alivisatos, and W. E. Moerner, “Photon antibunching in single cdse/zns quantum dot fluorescence,” Chem. Phys. Lett. 329, 399 (2000).
[Crossref]

Giacobino, E.

X. Brokmann, E. Giacobino, M. Dahan, and J. P. Hermier, “Highly efficient triggered emission of single photons by colloidal cdse/zns nanocrystals,” Appl. Phys. Lett. 85, 712 (2004).
[Crossref]

G. Messin, J. P. Hermier, E. Giacobino, P. Desbiolles, and M. Dahan, “Bunching and antibunching in the fluorescence of semiconductor nanocrystals,” Opt. Lett. 26, 1891 (2001).
[Crossref]

Gresillon, S.

E. Fort and S. Gresillon, “Surface enhanced fluorescence,” J. Phys. D 41, 013001 (2008).
[Crossref]

Hanbury-Brown, R.

R. Hanbury-Brown and R. Q. Twiss, “The Question of Correlation between Photons in Coherent Light Rays,” Nature 178, 1447–1448 (1956).
[Crossref]

Harris, T. D.

M Nirmal, B. O. Dabbousi, M. G. Bawendi, J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, “Fluorescence intermittency in single cadmium selenide nanocrystals,” Nature 383, 802 (1996).
[Crossref]

Hermier, J. P.

X. Brokmann, E. Giacobino, M. Dahan, and J. P. Hermier, “Highly efficient triggered emission of single photons by colloidal cdse/zns nanocrystals,” Appl. Phys. Lett. 85, 712 (2004).
[Crossref]

G. Messin, J. P. Hermier, E. Giacobino, P. Desbiolles, and M. Dahan, “Bunching and antibunching in the fluorescence of semiconductor nanocrystals,” Opt. Lett. 26, 1891 (2001).
[Crossref]

Hermier, J.-P.

X. Brokmann, L. Coolen, M. Dahan, and J.-P. Hermier, “Measurement of the radiative and nonradiative decay rates of single CdSe nanocrystals through a controlled modification of their spontaneous emission,” Phys. Rev. Lett. 93, 107403 (2004).
[Crossref] [PubMed]

Hettich, C.

B. C. Buchler, T. Kalkbrenner, C. Hettich, and V. Sandoghdar, “Measuring the quantum efficiency of the optical emission of single radiating dipoles using a scanning mirror,” Phys. Rev. Lett. 95, 063003 (2005).
[Crossref] [PubMed]

Holm, R. T.

Hu, E.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282 (2000).
[Crossref] [PubMed]

Imamoglu, A.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282 (2000).
[Crossref] [PubMed]

Johansen, J.

M. D. Leistikow, J. Johansen, A. J. Kettelarij, P. Lodahl, and W. L. Vos, “Size-dependent oscillator strength and quantum efficiency of CdSe quantum dots controlled via the local density of states,” Phys. Rev. B 79, 045301 (2009).
[Crossref]

Kalkbrenner, T.

B. C. Buchler, T. Kalkbrenner, C. Hettich, and V. Sandoghdar, “Measuring the quantum efficiency of the optical emission of single radiating dipoles using a scanning mirror,” Phys. Rev. Lett. 95, 063003 (2005).
[Crossref] [PubMed]

Kazunari, Matsuda

Ito Yuichi, Matsuda Kazunari, and Kanemitsu Yoshihiko, “Mechanism of photoluminescence enhancement in single semiconductor nanocrystals on metal surfaces,” Phys. Rev. B 75, 033309 (2007).
[Crossref]

Kettelarij, A. J.

M. D. Leistikow, J. Johansen, A. J. Kettelarij, P. Lodahl, and W. L. Vos, “Size-dependent oscillator strength and quantum efficiency of CdSe quantum dots controlled via the local density of states,” Phys. Rev. B 79, 045301 (2009).
[Crossref]

Kiraz, A.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282 (2000).
[Crossref] [PubMed]

Kreiter, M.

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, 023005 (2005).
[Crossref] [PubMed]

Kunz, R. E.

Lakowicz, J. R.

K. Ray, R. Badugu, and J. R. Lakowicz, “Metal-enhanced fluorescence from CdTe nanocrystals : A single-molecule fluorescence study,” J.Am. Chem. Soc. 128, 8998 (2006).
[Crossref] [PubMed]

Lakowicz, J.R.

K. Ray, H. Szmacinski, J. Enderlein, and J.R. Lakowicz, “Distance dependence of surface plasmon-coupled emission observed using Langmuir-Blodgett films,” Appl. Phys. Lett. 90, 251116 (2007).
[Crossref] [PubMed]

Leatherdale, C. A.

C. A. Leatherdale, W.-K. Woo, F. V. Mikulec, and M. G. Bawendi, “On the absorption cross section of CdSe nanocrystal quantum dots,” J. Phys. Chem. B 106, 7619 (2002).
[Crossref]

Leistikow, M. D.

M. D. Leistikow, J. Johansen, A. J. Kettelarij, P. Lodahl, and W. L. Vos, “Size-dependent oscillator strength and quantum efficiency of CdSe quantum dots controlled via the local density of states,” Phys. Rev. B 79, 045301 (2009).
[Crossref]

Lodahl, P.

M. D. Leistikow, J. Johansen, A. J. Kettelarij, P. Lodahl, and W. L. Vos, “Size-dependent oscillator strength and quantum efficiency of CdSe quantum dots controlled via the local density of states,” Phys. Rev. B 79, 045301 (2009).
[Crossref]

Lounis, B.

B. Lounis, H. A. Bechtel, D. Gerion, P. Alivisatos, and W. E. Moerner, “Photon antibunching in single cdse/zns quantum dot fluorescence,” Chem. Phys. Lett. 329, 399 (2000).
[Crossref]

Lukosz, W.

Macklin, J. J.

M Nirmal, B. O. Dabbousi, M. G. Bawendi, J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, “Fluorescence intermittency in single cadmium selenide nanocrystals,” Nature 383, 802 (1996).
[Crossref]

McKnight, S. W.

Messin, G.

Michler, P.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282 (2000).
[Crossref] [PubMed]

Mikulec, F. V.

C. A. Leatherdale, W.-K. Woo, F. V. Mikulec, and M. G. Bawendi, “On the absorption cross section of CdSe nanocrystal quantum dots,” J. Phys. Chem. B 106, 7619 (2002).
[Crossref]

Moerner, W. E.

B. Lounis, H. A. Bechtel, D. Gerion, P. Alivisatos, and W. E. Moerner, “Photon antibunching in single cdse/zns quantum dot fluorescence,” Chem. Phys. Lett. 329, 399 (2000).
[Crossref]

Neuhauser, R.

S. A. Empedocles, R. Neuhauser, and M. G. Bawendi, “Three-dimensional orientation measurements of symmetric single chromophores using polarization microscopy,” Nature 399, 126 (1999).
[Crossref]

Nirmal, M

M Nirmal, B. O. Dabbousi, M. G. Bawendi, J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, “Fluorescence intermittency in single cadmium selenide nanocrystals,” Nature 383, 802 (1996).
[Crossref]

Palik, E. D.

Pelton, M.

X. Wu, Y. Sun, and M. Pelton, “Recombination rates for single colloidal quantum dots near a smooth metal film,” Phys. Chem. Chem. Phys. 11, 5867 (2009).
[Crossref] [PubMed]

Petroff, P. M.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282 (2000).
[Crossref] [PubMed]

Prock¡, A.

R. R. Chance, A. Prock¡, and R. Silbey, “Lifetime of an emitting molecule near a partially reflecting surface,” J. Chem. Phys. 60, 2744 (1974).
[Crossref]

Ray, K.

K. Ray, H. Szmacinski, J. Enderlein, and J.R. Lakowicz, “Distance dependence of surface plasmon-coupled emission observed using Langmuir-Blodgett films,” Appl. Phys. Lett. 90, 251116 (2007).
[Crossref] [PubMed]

K. Ray, R. Badugu, and J. R. Lakowicz, “Metal-enhanced fluorescence from CdTe nanocrystals : A single-molecule fluorescence study,” J.Am. Chem. Soc. 128, 8998 (2006).
[Crossref] [PubMed]

Sandoghdar, V.

B. C. Buchler, T. Kalkbrenner, C. Hettich, and V. Sandoghdar, “Measuring the quantum efficiency of the optical emission of single radiating dipoles using a scanning mirror,” Phys. Rev. Lett. 95, 063003 (2005).
[Crossref] [PubMed]

Schoenfeld, W. V.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282 (2000).
[Crossref] [PubMed]

Shimizu, K. T.

K. T. Shimizu, W. K. Woo, B. R. Fisher, H. J. Eisler, and M. G. Bawendi, “Surface-enhanced emission from single semiconductor nanocrystals,” Phys. Rev. Lett. 89, 117401 (2002).
[Crossref] [PubMed]

Silbey, R.

R. R. Chance, A. Prock¡, and R. Silbey, “Lifetime of an emitting molecule near a partially reflecting surface,” J. Chem. Phys. 60, 2744 (1974).
[Crossref]

Stefani, F. D.

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, 023005 (2005).
[Crossref] [PubMed]

Stott, N. E.

B. R. Fisher, H.-J. Eisler, N. E. Stott, and M. G. Bawendi, “Emission Intensity Dependence and Single-Exponential Behavior In Single Colloidal Quantum Dot Fluorescence Lifetimes,” J. Phys. Chem. B 108, 143–148 (2004).
[Crossref]

Stoyanova, N.

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, 023005 (2005).
[Crossref] [PubMed]

Sun, Y.

X. Wu, Y. Sun, and M. Pelton, “Recombination rates for single colloidal quantum dots near a smooth metal film,” Phys. Chem. Chem. Phys. 11, 5867 (2009).
[Crossref] [PubMed]

Szmacinski, H.

K. Ray, H. Szmacinski, J. Enderlein, and J.R. Lakowicz, “Distance dependence of surface plasmon-coupled emission observed using Langmuir-Blodgett films,” Appl. Phys. Lett. 90, 251116 (2007).
[Crossref] [PubMed]

Trautman, J. K.

M Nirmal, B. O. Dabbousi, M. G. Bawendi, J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, “Fluorescence intermittency in single cadmium selenide nanocrystals,” Nature 383, 802 (1996).
[Crossref]

Twiss, R. Q.

R. Hanbury-Brown and R. Q. Twiss, “The Question of Correlation between Photons in Coherent Light Rays,” Nature 178, 1447–1448 (1956).
[Crossref]

Vasilev, K.

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, 023005 (2005).
[Crossref] [PubMed]

Vos, W. L.

M. D. Leistikow, J. Johansen, A. J. Kettelarij, P. Lodahl, and W. L. Vos, “Size-dependent oscillator strength and quantum efficiency of CdSe quantum dots controlled via the local density of states,” Phys. Rev. B 79, 045301 (2009).
[Crossref]

Weber, W. H.

G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195 (1984).
[Crossref]

W. H. Weber and C. F. Eagen, “Energy transfer from an excited dye molecule to the surface plasmons of an adjacent metal,” Opt. Lett. 4, 236 (1979).
[Crossref] [PubMed]

Woo, W. K.

K. T. Shimizu, W. K. Woo, B. R. Fisher, H. J. Eisler, and M. G. Bawendi, “Surface-enhanced emission from single semiconductor nanocrystals,” Phys. Rev. Lett. 89, 117401 (2002).
[Crossref] [PubMed]

Woo, W.-K.

C. A. Leatherdale, W.-K. Woo, F. V. Mikulec, and M. G. Bawendi, “On the absorption cross section of CdSe nanocrystal quantum dots,” J. Phys. Chem. B 106, 7619 (2002).
[Crossref]

Wu, X.

X. Wu, Y. Sun, and M. Pelton, “Recombination rates for single colloidal quantum dots near a smooth metal film,” Phys. Chem. Chem. Phys. 11, 5867 (2009).
[Crossref] [PubMed]

Yoshihiko, Kanemitsu

Ito Yuichi, Matsuda Kazunari, and Kanemitsu Yoshihiko, “Mechanism of photoluminescence enhancement in single semiconductor nanocrystals on metal surfaces,” Phys. Rev. B 75, 033309 (2007).
[Crossref]

Yuichi, Ito

Ito Yuichi, Matsuda Kazunari, and Kanemitsu Yoshihiko, “Mechanism of photoluminescence enhancement in single semiconductor nanocrystals on metal surfaces,” Phys. Rev. B 75, 033309 (2007).
[Crossref]

Zhang, L.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282 (2000).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

K. Ray, H. Szmacinski, J. Enderlein, and J.R. Lakowicz, “Distance dependence of surface plasmon-coupled emission observed using Langmuir-Blodgett films,” Appl. Phys. Lett. 90, 251116 (2007).
[Crossref] [PubMed]

X. Brokmann, E. Giacobino, M. Dahan, and J. P. Hermier, “Highly efficient triggered emission of single photons by colloidal cdse/zns nanocrystals,” Appl. Phys. Lett. 85, 712 (2004).
[Crossref]

Chem. Phys. (1)

J. Enderlein, “Single-molecule fluorescence near a metal layer,” Chem. Phys. 247, 1 (1999).
[Crossref]

Chem. Phys. Lett. (1)

B. Lounis, H. A. Bechtel, D. Gerion, P. Alivisatos, and W. E. Moerner, “Photon antibunching in single cdse/zns quantum dot fluorescence,” Chem. Phys. Lett. 329, 399 (2000).
[Crossref]

J. Chem. Phys. (1)

R. R. Chance, A. Prock¡, and R. Silbey, “Lifetime of an emitting molecule near a partially reflecting surface,” J. Chem. Phys. 60, 2744 (1974).
[Crossref]

J. Mod. Opt. (1)

W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45, 661 (1998).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. Chem. B (2)

C. A. Leatherdale, W.-K. Woo, F. V. Mikulec, and M. G. Bawendi, “On the absorption cross section of CdSe nanocrystal quantum dots,” J. Phys. Chem. B 106, 7619 (2002).
[Crossref]

B. R. Fisher, H.-J. Eisler, N. E. Stott, and M. G. Bawendi, “Emission Intensity Dependence and Single-Exponential Behavior In Single Colloidal Quantum Dot Fluorescence Lifetimes,” J. Phys. Chem. B 108, 143–148 (2004).
[Crossref]

J. Phys. D (1)

E. Fort and S. Gresillon, “Surface enhanced fluorescence,” J. Phys. D 41, 013001 (2008).
[Crossref]

J.Am. Chem. Soc. (1)

K. Ray, R. Badugu, and J. R. Lakowicz, “Metal-enhanced fluorescence from CdTe nanocrystals : A single-molecule fluorescence study,” J.Am. Chem. Soc. 128, 8998 (2006).
[Crossref] [PubMed]

Nature (3)

R. Hanbury-Brown and R. Q. Twiss, “The Question of Correlation between Photons in Coherent Light Rays,” Nature 178, 1447–1448 (1956).
[Crossref]

M Nirmal, B. O. Dabbousi, M. G. Bawendi, J. J. Macklin, J. K. Trautman, T. D. Harris, and L. E. Brus, “Fluorescence intermittency in single cadmium selenide nanocrystals,” Nature 383, 802 (1996).
[Crossref]

S. A. Empedocles, R. Neuhauser, and M. G. Bawendi, “Three-dimensional orientation measurements of symmetric single chromophores using polarization microscopy,” Nature 399, 126 (1999).
[Crossref]

Opt. Lett. (2)

Phys. Chem. Chem. Phys. (1)

X. Wu, Y. Sun, and M. Pelton, “Recombination rates for single colloidal quantum dots near a smooth metal film,” Phys. Chem. Chem. Phys. 11, 5867 (2009).
[Crossref] [PubMed]

Phys. Rep. (1)

G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195 (1984).
[Crossref]

Phys. Rev. B (3)

W. Lukosz, “Theory of optical-environment-dependent spontaneous emission rates for emitters in thin layers,” Phys. Rev. B 22, 3030 (1980).
[Crossref]

M. D. Leistikow, J. Johansen, A. J. Kettelarij, P. Lodahl, and W. L. Vos, “Size-dependent oscillator strength and quantum efficiency of CdSe quantum dots controlled via the local density of states,” Phys. Rev. B 79, 045301 (2009).
[Crossref]

Ito Yuichi, Matsuda Kazunari, and Kanemitsu Yoshihiko, “Mechanism of photoluminescence enhancement in single semiconductor nanocrystals on metal surfaces,” Phys. Rev. B 75, 033309 (2007).
[Crossref]

Phys. Rev. Lett. (4)

K. T. Shimizu, W. K. Woo, B. R. Fisher, H. J. Eisler, and M. G. Bawendi, “Surface-enhanced emission from single semiconductor nanocrystals,” Phys. Rev. Lett. 89, 117401 (2002).
[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, 023005 (2005).
[Crossref] [PubMed]

B. C. Buchler, T. Kalkbrenner, C. Hettich, and V. Sandoghdar, “Measuring the quantum efficiency of the optical emission of single radiating dipoles using a scanning mirror,” Phys. Rev. Lett. 95, 063003 (2005).
[Crossref] [PubMed]

X. Brokmann, L. Coolen, M. Dahan, and J.-P. Hermier, “Measurement of the radiative and nonradiative decay rates of single CdSe nanocrystals through a controlled modification of their spontaneous emission,” Phys. Rev. Lett. 93, 107403 (2004).
[Crossref] [PubMed]

Science (1)

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. Imamoglu, “A quantum dot single-photon turnstile device,” Science 290, 2282 (2000).
[Crossref] [PubMed]

Other (1)

E. D. Palik, Handbook of Optical Constants of Solids, (Academic Press, 2005).

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Figures (8)

Figure 1.
Figure 1.

Schematic of the sample and the observation configuration.

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Figure 2.
Figure 2.

Histogram of the time intervals τ between one photon detection on a photodiode and the next detection on the other photodiode for a nanocrystal at distance d = (a) 256 nm, (b) 18 nm from a gold/silica interface. The red lines are theoretical curves obtained as explained in the text and Appendix.

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Figure 3.
Figure 3.

Schematic representation of (a) a linear dipole (arrow), oriented at an angle δ from the normal to the sample plane, and (b) a nanocrystal with crystalline c-axis oriented at an angle θ from the sample plane : the 2D-degenerate emission can be decomposed into two dipoles (indicated by arrows) perpendicular to the c axis.

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Figure 4.
Figure 4.

Theoretical decay rates into each channel : far field radiation Γ rad (blue), surface plasmon mode Γ SP (red), lossy surface waves Γ LSW (green), and total decay rate Γ (black) (normalized to Γ(d = ∞)) for a nanocrystal with c-axis (a) normal and (b) parallel to the interface. (c) Dots : measured lifetimes 1/Γ of single CdSe/ZnS nanocrystals, normalized to the lifetime in a homogeneous medium of index 1.5 (18 ns), as a function of the distance d to the gold film. Lines : Calculated theoretical decay rates for a nanocrystal with c-axis normal (green line) and parallel (blue line) to the gold/silica interface. Inset: measured decay curves of three nanocrystals at 18 nm (red), 150 nm (black) and 256 nm (blue) from the gold film. The corresponding fitted decay rates are respectively 3.5, 15.9 and 21.3 ns.

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Figure 5.
Figure 5.

Image measured by CCD camera of a 13×13 μm portion of (a) the reference sample of nanocrystals on glass covered by PMMA and (b) the sample with nanocrystals at d = 80 nm from a gold film. (c) Histogram of the detected nanocrystal fluorescence intensities D (in arbitrary units) from the reference sample (red) and the d = 80 nm sample (blue). The lines are fits obtained as explained later in the text.

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Figure 6.
Figure 6.

Theoretical influence of the silica spacer thickness d on (a) the normalized excitation surface power β, (b) the quantum yield Y = Γ rad. /Γ and (c) the collection efficiency C. The curves of (b) and (c) are plotted for a nanocrystal parallel (blue line) and normal (green line) to the sample plane. The values for the reference sample are indicated as dotted lines.

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Figure 7.
Figure 7.

Theoretical emission diagrams (in arbitrary units) of a nanocrystal with vertical c axis (θ = 0), (a) in a homogeneous medium, (b) in the reference sample, and (c) in three gold samples with different values of d (and no immersion oil). The green lines indicate the solid angle which is collected by an air objective with 0.75 numerical aperture.

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Figure 8.
Figure 8.

Theoretical plots of the factors (a) βYC, (b) Γ radC (with Γ rad normalized to the decay rate in a homogeneous medium of index 1.5 : 1/18 ns) and (c)YC, which can be used to characterize the quality of a single-photon source (see text).

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Equations (23)

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I NC ( t ) = I ˜ NC ( t ) ∑ n Γ T e − Γ ( t − nT ) H ( t − nT )
B ( t ) = B 0 + B P ∑ n Γ P T e − Γ P ( t − nT ) H ( t − nT )
γ = r Γ rad / ( r + Γ )
r = σP h ¯ ω exc
D ( d , θ ) = r ( d ) Y ( d , θ ) C ( d , θ ) η
d 2 N ( θ , ϕ ) = N tot sin ⁡ θdθdϕ 2 π
D ( θ ) = AβY ( θ ) C ( θ )
G ( A ) = 1 2 π w exp ⁡ ( − ( A − A 0 ) 2 2 w 2 )
dN ( D ) = ∫ θ = 0 π / 2 ∫ ϕ = 0 2 π d 2 N ( θ , ϕ ) G ( D βY ( θ ) C ( θ ) ) dD βY ( θ ) C ( θ )
= N tot dD ∫ θ = 0 π / 2 G ( D βY ( θ ) C ( θ ) ) sin ⁡ θ βY ( θ ) C ( θ ) dθ
c ( τ ) = ∫ t = 0 Θ I ( t ) dtI ( t + τ ) δτ = 〈 I ( t ) I ( t + τ ) 〉 . Θ . δτ
I NC ( t ) = I ˜ NC ( t ) ∑ n = 0 Θ / T Γ T e − Γ ( t − nT ) H ( t − nT )
B ( t ) = B 0 + B P ∑ n = 0 Θ / T Γ P T e − Γ P ( t − nT ) H ( t − nT )
B 0 2 + 2 B 0 B P + ( B P Γ P T ) 2 Θ ∑ n = 0 Θ / T ∑ m = 0 Θ / T ∫ 0 Θ f ( t ) dt
f ( t ) = e − Γ P ( t − nT ) e − Γ P ( t + τ − mT ) H ( t + τ − mT ) H ( t − nT )
∫ t = 0 Θ f ( t ) dt = ∫ mT − τ Θ e − 2 Γ P t e − Γ P τ e Γ P ( n + m ) T dt = 1 2 Γ P e − Γ P ( ( m − n ) T − τ )
∫ t = 0 Θ f ( t ) dt = ∫ nT Θ e − 2 Γ P t e − Γ P τ e Γ P ( n + m ) T dt = 1 2 Γ P e − Γ P ( − ( m − n ) T + τ )
B 0 2 + 2 B 0 B P + ( B P T ) 2 Γ P 2 Θ ∑ n = 0 Θ / T ∑ m = 0 Θ / T e − Γ P ∣ τ − lT ∣
B 0 2 + 2 B 0 B P + ( B P T ) 2 Γ P 2 Θ ∑ l = − Θ / T Θ / T ∑ k = − ( Θ / T − l ) / 2 ( Θ / T − l ) / 2 e − Γ P ∣ τ − lT ∣
B 0 2 + 2 B 0 B P + B P 2 T Γ P 2 ∑ l = − 2 2 e − Γ P ∣ τ − lT ∣
〈 B ( t ) I NC ( t + τ ) + I NC ( t ) B ( t + τ ) 〉 = 2 B 0 〈 I ˜ NC 〉
+ B P 〈 I ˜ NC 〉 T 1 / Γ + 1 / Γ P ∑ l ( e − Γ P ∣ τ − lT ∣ + e − Γ ∣ τ − lT ∣ )
〈 I NC ( t ) I NC ( t + τ ) 〉 = 〈 I ˜ NC 2 〉 T Γ 2 ∑ l ≠ 0 e − Γ ∣ τ − lT ∣

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