Abstract

Hyperbolic metamaterials can enhance spontaneous emission, but the radiation-matter coupling is not optimized if the light source is placed outside such media. We demonstrate a 3-fold improvement of the Purcell factor over its outer value and a significant enlargement in bandwidth by including the emitter within a Si/Ag periodic multilayer metamaterial. To extract the plasmonic modes of the structure into the far field we implement two types of 1D grating with triangular and rectangular profile, obtaining a 10-fold radiative enhancement at visible frequencies.

© 2014 Optical Society of America

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References

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  1. A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Mullen, W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).
  2. K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Mater. 3(9), 601–605 (2004).
    [PubMed]
  3. I. De Leon, P. Berini, “Theory of noise in high-gain surface plasmon-polariton amplifiers incorporating dipolar gain media,” Opt. Express 19(21), 20506–20517 (2011).
    [PubMed]
  4. A. G. Brolo, “Plasmonics for future biosensors,” Nat. Photonics 6(11), 709–713 (2012).
  5. H. G. Frey, S. Witt, K. Felderer, R. Guckenberger, “High-resolution imaging of single fluorescent molecules with the optical near-field of a metal tip,” Phys. Rev. Lett. 93(20), 200801 (2004).
    [PubMed]
  6. M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299(5607), 682–686 (2003).
    [PubMed]
  7. G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
    [PubMed]
  8. L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University, 2012)
  9. G. W. Ford, W. H. Weber, “Electromagnetic-interactions of molecules with metal-surfaces,” Phys. Rep. 113(4), 195–287 (1984).
  10. W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45(4), 661–699 (1998).
  11. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
  12. M. Noginov, M. Lapine, V. Podolskiy, Y. Kivshar, “Focus issue: hyperbolic metamaterials,” Opt. Express 21(12), 14895–14897 (2013).
    [PubMed]
  13. Z. Jacob, I. I. Smolyaninov, E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
  14. D. Lu, J. J. Kan, E. E. Fullerton, Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
    [PubMed]
  15. P. Yeh, Optical Waves in Layered Media (Wiley, 1988).
  16. P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
  17. D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27(2), 985–1009 (1983).
  18. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  19. P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photonics 1(3), 484–588 (2009).

2014 (1)

D. Lu, J. J. Kan, E. E. Fullerton, Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
[PubMed]

2013 (1)

2012 (2)

Z. Jacob, I. I. Smolyaninov, E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).

A. G. Brolo, “Plasmonics for future biosensors,” Nat. Photonics 6(11), 709–713 (2012).

2011 (2)

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[PubMed]

I. De Leon, P. Berini, “Theory of noise in high-gain surface plasmon-polariton amplifiers incorporating dipolar gain media,” Opt. Express 19(21), 20506–20517 (2011).
[PubMed]

2009 (2)

P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photonics 1(3), 484–588 (2009).

A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Mullen, W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).

2004 (2)

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Mater. 3(9), 601–605 (2004).
[PubMed]

H. G. Frey, S. Witt, K. Felderer, R. Guckenberger, “High-resolution imaging of single fluorescent molecules with the optical near-field of a metal tip,” Phys. Rev. Lett. 93(20), 200801 (2004).
[PubMed]

2003 (1)

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299(5607), 682–686 (2003).
[PubMed]

1998 (1)

W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45(4), 661–699 (1998).

1984 (1)

G. W. Ford, W. H. Weber, “Electromagnetic-interactions of molecules with metal-surfaces,” Phys. Rep. 113(4), 195–287 (1984).

1983 (1)

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27(2), 985–1009 (1983).

1972 (1)

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Aspnes, D. E.

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27(2), 985–1009 (1983).

Avlasevich, Y.

A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Mullen, W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).

Barnes, W. L.

W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45(4), 661–699 (1998).

Berini, P.

Brolo, A. G.

A. G. Brolo, “Plasmonics for future biosensors,” Nat. Photonics 6(11), 709–713 (2012).

Christy, R. W.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).

Craighead, H. G.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299(5607), 682–686 (2003).
[PubMed]

De Leon, I.

Ellis, B.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[PubMed]

Fan, S. H.

A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Mullen, W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).

Felderer, K.

H. G. Frey, S. Witt, K. Felderer, R. Guckenberger, “High-resolution imaging of single fluorescent molecules with the optical near-field of a metal tip,” Phys. Rev. Lett. 93(20), 200801 (2004).
[PubMed]

Foquet, M.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299(5607), 682–686 (2003).
[PubMed]

Ford, G. W.

G. W. Ford, W. H. Weber, “Electromagnetic-interactions of molecules with metal-surfaces,” Phys. Rep. 113(4), 195–287 (1984).

Frey, H. G.

H. G. Frey, S. Witt, K. Felderer, R. Guckenberger, “High-resolution imaging of single fluorescent molecules with the optical near-field of a metal tip,” Phys. Rev. Lett. 93(20), 200801 (2004).
[PubMed]

Fullerton, E. E.

D. Lu, J. J. Kan, E. E. Fullerton, Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
[PubMed]

Guckenberger, R.

H. G. Frey, S. Witt, K. Felderer, R. Guckenberger, “High-resolution imaging of single fluorescent molecules with the optical near-field of a metal tip,” Phys. Rev. Lett. 93(20), 200801 (2004).
[PubMed]

Haller, E. E.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[PubMed]

Harris, J.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[PubMed]

Jacob, Z.

Z. Jacob, I. I. Smolyaninov, E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).

Johnson, P. B.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).

Kan, J. J.

D. Lu, J. J. Kan, E. E. Fullerton, Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
[PubMed]

Kinkhabwala, A.

A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Mullen, W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).

Kivshar, Y.

Korlach, J.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299(5607), 682–686 (2003).
[PubMed]

Lapine, M.

Levene, M. J.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299(5607), 682–686 (2003).
[PubMed]

Liu, Z.

D. Lu, J. J. Kan, E. E. Fullerton, Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
[PubMed]

Lu, D.

D. Lu, J. J. Kan, E. E. Fullerton, Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
[PubMed]

Majumdar, A.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[PubMed]

Mayer, M. A.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[PubMed]

Moerner, W. E.

A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Mullen, W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).

Mukai, T.

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Mater. 3(9), 601–605 (2004).
[PubMed]

Mullen, K.

A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Mullen, W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).

Narimanov, E. E.

Z. Jacob, I. I. Smolyaninov, E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).

Narukawa, Y.

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Mater. 3(9), 601–605 (2004).
[PubMed]

Niki, I.

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Mater. 3(9), 601–605 (2004).
[PubMed]

Noginov, M.

Okamoto, K.

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Mater. 3(9), 601–605 (2004).
[PubMed]

Petykiewicz, J.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[PubMed]

Podolskiy, V.

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Sarmiento, T.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[PubMed]

Scherer, A.

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Mater. 3(9), 601–605 (2004).
[PubMed]

Shambat, G.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[PubMed]

Shvartser, A.

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Mater. 3(9), 601–605 (2004).
[PubMed]

Smolyaninov, I. I.

Z. Jacob, I. I. Smolyaninov, E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).

Studna, A. A.

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27(2), 985–1009 (1983).

Turner, S. W.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299(5607), 682–686 (2003).
[PubMed]

Vuckovic, J.

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[PubMed]

Webb, W. W.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299(5607), 682–686 (2003).
[PubMed]

Weber, W. H.

G. W. Ford, W. H. Weber, “Electromagnetic-interactions of molecules with metal-surfaces,” Phys. Rep. 113(4), 195–287 (1984).

Witt, S.

H. G. Frey, S. Witt, K. Felderer, R. Guckenberger, “High-resolution imaging of single fluorescent molecules with the optical near-field of a metal tip,” Phys. Rev. Lett. 93(20), 200801 (2004).
[PubMed]

Yu, Z. F.

A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Mullen, W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).

Adv. Opt. Photonics (1)

P. Berini, “Long-range surface plasmon polaritons,” Adv. Opt. Photonics 1(3), 484–588 (2009).

Appl. Phys. Lett. (1)

Z. Jacob, I. I. Smolyaninov, E. E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).

J. Mod. Opt. (1)

W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45(4), 661–699 (1998).

Nat. Commun. (1)

G. Shambat, B. Ellis, A. Majumdar, J. Petykiewicz, M. A. Mayer, T. Sarmiento, J. Harris, E. E. Haller, J. Vučković, “Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,” Nat. Commun. 2, 539 (2011).
[PubMed]

Nat. Mater. (1)

K. Okamoto, I. Niki, A. Shvartser, Y. Narukawa, T. Mukai, A. Scherer, “Surface-plasmon-enhanced light emitters based on InGaN quantum wells,” Nat. Mater. 3(9), 601–605 (2004).
[PubMed]

Nat. Nanotechnol. (1)

D. Lu, J. J. Kan, E. E. Fullerton, Z. Liu, “Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials,” Nat. Nanotechnol. 9(1), 48–53 (2014).
[PubMed]

Nat. Photonics (2)

A. G. Brolo, “Plasmonics for future biosensors,” Nat. Photonics 6(11), 709–713 (2012).

A. Kinkhabwala, Z. F. Yu, S. H. Fan, Y. Avlasevich, K. Mullen, W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photonics 3(11), 654–657 (2009).

Opt. Express (2)

Phys. Rep. (1)

G. W. Ford, W. H. Weber, “Electromagnetic-interactions of molecules with metal-surfaces,” Phys. Rep. 113(4), 195–287 (1984).

Phys. Rev. (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Phys. Rev. B (2)

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).

D. E. Aspnes, A. A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27(2), 985–1009 (1983).

Phys. Rev. Lett. (1)

H. G. Frey, S. Witt, K. Felderer, R. Guckenberger, “High-resolution imaging of single fluorescent molecules with the optical near-field of a metal tip,” Phys. Rev. Lett. 93(20), 200801 (2004).
[PubMed]

Science (1)

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299(5607), 682–686 (2003).
[PubMed]

Other (3)

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University, 2012)

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

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Figures (4)

Fig. 1
Fig. 1

(a) Uniform Ag/Si ML, (b) triangular and (c) rectangular grating.

Fig. 2
Fig. 2

(a) Purcell factor FP for an isotropic dipole in glass 10 nm above the Si/Ag uniform ML. The Ag volume fraction ρ is varied from 1 (in which case the Si capping layer is excluded) to 0.2, as indicated above each curve. The tunability is finer in the yellow-green region of the visible range, highlighted by the colored background. (b) Power density spectrum (logarithmic scale) of an s- dipole for ρ = 1, (c) 0.6 and (d) 0.3. (e) Comparison between the Purcell factors of an s-, p- and isotropic dipole with ρ = 0.6, obtained with analytical calculations (continuous curves) and FDTD simulations (open circles).

Fig. 3
Fig. 3

(a) Positioning of a dipole (indicated by a red dot) outside and inside selected Si layers (ochre stripes) of the uniform ML. (b) Purcell factor FP of a parallel or (c) orthogonal dipole for the outer (dashed line) and inner positions (solid lines). The insets show the coupling of the dipole field, directed from the tail to the tip of the triangles and proportional in magnitude to their size, to the Ag (grey)/Si (yellow) interfaces. (d) Purcell factor of an isotropic emitter for the positions considered in (b) and (c).

Fig. 4
Fig. 4

(a) Isotropic Purcell factor for the triangular grating (blue curve, triangular markers), the rectangular grating (green curve, square markers) and an unpatterned structure with an emitting layer of 20 nm (red curve, circular markers). (b) Radiative enhancement at different angles for an isotropic dipole, provided by the triangular grating, and (c) by the rectangular grating, sketched in the insets.

Equations (8)

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

F || =1 η 0 ( 1 3 4 Re 0 d k || D || ( k || ) ).
F =1 η 0 ( 1 3 2 Re 0 d k || D ( k || ) ).
D || = 1 k z k || ε 1 k 0 [ 1+ r S e 2i k z d + ( k z ε 1 k 0 ) 2 ( 1 r P e 2i k z d ) ].
D = 1 k z ( k || ε 1 k 0 ) 3 ( 1+ r P e 2i k z d ).
F iso = 1 3 ( F +2 F || ).
F ||, = P ||, ML P 0 = P ||, r + P ||, nr P 0 .
RE(θ)= P r (θ) P 0 (θ) .
F iso = 1 3 ( F x + F y + F z ),R E iso = 1 3 (R E x +R E y +R E z ).

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