Abstract

We study the spontaneous emission rate of emitter in a periodically patterned metal or dielectric membrane in the picture of a multimode field of damped Bloch states. For Bloch states in dielectric structures, the approach fully describes the Purcell effect in photonic crystal or spatially coupled cavities with losses. For a metal membrane, the Purcell factor depends on resistive loss at the resonant frequency of surface plasmon polariton (SPP). Analysis of an InP-Au-InP structure indicates that the SPP’s Purcell effect can exceed a value of 50 in the ultraviolet. For a plasmonic crystal, we find a position-dependent Purcell enhancement with a mean value similar to the unpatterned membrane.

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  30. P. D. Drummond and M. Hillery, “Quantum theory of dispersive electromagnetic modes,” Phys. Rev. A 59(1), 691–707 (1999).
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    [CrossRef]
  36. Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61(3), 033807 (2000).
    [CrossRef]
  37. Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17(3), 387–400 (2000).
    [CrossRef]
  38. J. K. S. Poon and A. Yariv, “Active coupled-resonator optical waveguides. I. Gain enhancement and noise,” J. Opt. Soc. Am. B 24(9), 2378–2388 (2007).
    [CrossRef]
  39. S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996).
    [CrossRef] [PubMed]
  40. H. Iwase, D. Englund, and J. Vučković, “Spontaneous emission control in high-extraction efficiency plasmonic crystals,” Opt. Express 16(1), 426–434 (2008).
    [CrossRef] [PubMed]
  41. S. D. Liu, M. T. Cheng, Z. J. Yang, and Q. Q. Wang, “Surface plasmon propagation in a pair of metal nanowires coupled to a nanosized optical emitter,” Opt. Lett. 33(8), 851–853 (2008).
    [CrossRef] [PubMed]

2008

H. Iwase, D. Englund, and J. Vučković, “Spontaneous emission control in high-extraction efficiency plasmonic crystals,” Opt. Express 16(1), 426–434 (2008).
[CrossRef] [PubMed]

S. D. Liu, M. T. Cheng, Z. J. Yang, and Q. Q. Wang, “Surface plasmon propagation in a pair of metal nanowires coupled to a nanosized optical emitter,” Opt. Lett. 33(8), 851–853 (2008).
[CrossRef] [PubMed]

2007

J. K. S. Poon and A. Yariv, “Active coupled-resonator optical waveguides. I. Gain enhancement and noise,” J. Opt. Soc. Am. B 24(9), 2378–2388 (2007).
[CrossRef]

M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express 15(10), 5948–5965 (2007).
[CrossRef] [PubMed]

Y. Gong and J. Vučković, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90(3), 033113 (2007).
[CrossRef]

G. Lecamp, P. Lalanne, and J. P. Hugonin, “Very large spontaneous-emission β factors in photonic-crystal waveguides,” Phys. Rev. Lett. 99(2), 023902 (2007).
[CrossRef] [PubMed]

V. S. C. Manga Tao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75(20), 205437 (2007).
[CrossRef]

2006

S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14(5), 1957–1964 (2006).
[CrossRef] [PubMed]

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96(11), 113002 (2006).
[CrossRef] [PubMed]

2005

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6(1), 3–9 (2005).
[CrossRef]

2004

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

2002

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(7), 077402 (2002).
[CrossRef] [PubMed]

A. Neogi, C. Lee, H. O. Everitt, T. Kuroda, A. Tackeuchi, and E. Yablonovitch, “Enhancement of spontaneous recombination rate in a quantum well by resonant surface plasmon coupling,” Phys. Rev. B 66(15), 153305 (2002).
[CrossRef]

2000

J. Vučković, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quantum Electron. 36(10), 1131–1144 (2000).
[CrossRef]

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61(3), 033807 (2000).
[CrossRef]

Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17(3), 387–400 (2000).
[CrossRef]

R. K. Lee, Y. Xu, and A. Yariv, “Modified spontaneous emission from a two-dimensional photonic bandgap crystal slab,” J. Opt. Soc. Am. B 17(8), 1438–1442 (2000).
[CrossRef]

1999

P. D. Drummond and M. Hillery, “Quantum theory of dispersive electromagnetic modes,” Phys. Rev. A 59(1), 691–707 (1999).
[CrossRef]

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

W. L. Barnes, “Electromagnetic Crystals for Surface Plasmon Polaritons and the Extraction of Light from Emissive Devices,” J. Lightwave Technol. 17(11), 2170–2182 (1999).
[CrossRef]

M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol. 17(11), 2096–2112 (1999).
[CrossRef]

1998

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

Y. Jiang and M. Liu, “Electromagnetic force in dispersive and transparent media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(5), 6685–6694 (1998).
[CrossRef]

S. Nojima, “Enhancement of Optical Gain in Two-dimensional Photonic Crystals with Active Lattice Points,” Jpn. J. Appl. Phys. 37(Part 2, No. 5B), L565–L567 (1998).
[CrossRef]

1996

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996).
[CrossRef] [PubMed]

1994

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994).
[CrossRef]

1991

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43(1), 467–491 (1991).
[CrossRef] [PubMed]

1988

T. A. B. Kennedy and E. M. Wright, “Quantization and phase-space methods for slowly varying optical fields in a dispersive nonlinear medium,” Phys. Rev. A 38(1), 212–221 (1988).
[CrossRef] [PubMed]

1981

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981).
[CrossRef]

1978

R. R. Chance, A. Prock, and R. Silbey, “Molecular Fluorescence and Energy Transfer near Interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

1974

R. R. Chance, A. Prock, and R. Silbey, “Lifetime of an emitting molecule near partially reflecting surface,” J. Chem. Phys. 60(7), 2744–2748 (1974).
[CrossRef]

1970

H. Kuhn, “Classical aspects of energy transfer in molecular systems,” J. Chem. Phys. 53(1), 101–108 (1970).
[CrossRef]

1969

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. 182(2), 539–554 (1969).
[CrossRef]

1946

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

Anger, P.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96(11), 113002 (2006).
[CrossRef] [PubMed]

Asano, T.

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6(1), 3–9 (2005).
[CrossRef]

Bahriz, M.

M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express 15(10), 5948–5965 (2007).
[CrossRef] [PubMed]

Barnes, W. L.

W. L. Barnes, “Electromagnetic Crystals for Surface Plasmon Polaritons and the Extraction of Light from Emissive Devices,” J. Lightwave Technol. 17(11), 2170–2182 (1999).
[CrossRef]

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

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996).
[CrossRef] [PubMed]

Bharadwaj, P.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96(11), 113002 (2006).
[CrossRef] [PubMed]

Bhat, R.

M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol. 17(11), 2096–2112 (1999).
[CrossRef]

Bloemer, M. J.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994).
[CrossRef]

Boroditsky, M.

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol. 17(11), 2096–2112 (1999).
[CrossRef]

Bowden, C. M.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994).
[CrossRef]

Chance, R. R.

R. R. Chance, A. Prock, and R. Silbey, “Molecular Fluorescence and Energy Transfer near Interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

R. R. Chance, A. Prock, and R. Silbey, “Lifetime of an emitting molecule near partially reflecting surface,” J. Chem. Phys. 60(7), 2744–2748 (1974).
[CrossRef]

Cheng, M. T.

S. D. Liu, M. T. Cheng, Z. J. Yang, and Q. Q. Wang, “Surface plasmon propagation in a pair of metal nanowires coupled to a nanosized optical emitter,” Opt. Lett. 33(8), 851–853 (2008).
[CrossRef] [PubMed]

Chutinan, A.

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6(1), 3–9 (2005).
[CrossRef]

Coccioli, R.

M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol. 17(11), 2096–2112 (1999).
[CrossRef]

Colombelli, R.

M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express 15(10), 5948–5965 (2007).
[CrossRef] [PubMed]

Crisafulli, O.

M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express 15(10), 5948–5965 (2007).
[CrossRef] [PubMed]

DenBaars, S. P.

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

Dowling, J. P.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994).
[CrossRef]

Drummond, P. D.

P. D. Drummond and M. Hillery, “Quantum theory of dispersive electromagnetic modes,” Phys. Rev. A 59(1), 691–707 (1999).
[CrossRef]

Economou, E. N.

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. 182(2), 539–554 (1969).
[CrossRef]

Englund, D.

H. Iwase, D. Englund, and J. Vučković, “Spontaneous emission control in high-extraction efficiency plasmonic crystals,” Opt. Express 16(1), 426–434 (2008).
[CrossRef] [PubMed]

Everitt, H. O.

A. Neogi, C. Lee, H. O. Everitt, T. Kuroda, A. Tackeuchi, and E. Yablonovitch, “Enhancement of spontaneous recombination rate in a quantum well by resonant surface plasmon coupling,” Phys. Rev. B 66(15), 153305 (2002).
[CrossRef]

Feldmann, J.

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(7), 077402 (2002).
[CrossRef] [PubMed]

Franzl, T.

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(7), 077402 (2002).
[CrossRef] [PubMed]

Fujita, M.

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6(1), 3–9 (2005).
[CrossRef]

Glauber, R. J.

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43(1), 467–491 (1991).
[CrossRef] [PubMed]

Gong, Y.

Y. Gong and J. Vučković, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90(3), 033113 (2007).
[CrossRef]

Gontijo, I.

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

Hillery, M.

P. D. Drummond and M. Hillery, “Quantum theory of dispersive electromagnetic modes,” Phys. Rev. A 59(1), 691–707 (1999).
[CrossRef]

Hughes, S.

V. S. C. Manga Tao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75(20), 205437 (2007).
[CrossRef]

Hugonin, J. P.

G. Lecamp, P. Lalanne, and J. P. Hugonin, “Very large spontaneous-emission β factors in photonic-crystal waveguides,” Phys. Rev. Lett. 99(2), 023902 (2007).
[CrossRef] [PubMed]

Ishihara, K.

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6(1), 3–9 (2005).
[CrossRef]

Iwase, H.

H. Iwase, D. Englund, and J. Vučković, “Spontaneous emission control in high-extraction efficiency plasmonic crystals,” Opt. Express 16(1), 426–434 (2008).
[CrossRef] [PubMed]

Jiang, Y.

Y. Jiang and M. Liu, “Electromagnetic force in dispersive and transparent media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(5), 6685–6694 (1998).
[CrossRef]

Keller, S.

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

Kennedy, T. A. B.

T. A. B. Kennedy and E. M. Wright, “Quantization and phase-space methods for slowly varying optical fields in a dispersive nonlinear medium,” Phys. Rev. A 38(1), 212–221 (1988).
[CrossRef] [PubMed]

Kitson, S. C.

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996).
[CrossRef] [PubMed]

Krauss, T. F.

M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol. 17(11), 2096–2112 (1999).
[CrossRef]

Kuhn, H.

H. Kuhn, “Classical aspects of energy transfer in molecular systems,” J. Chem. Phys. 53(1), 101–108 (1970).
[CrossRef]

Kuroda, T.

A. Neogi, C. Lee, H. O. Everitt, T. Kuroda, A. Tackeuchi, and E. Yablonovitch, “Enhancement of spontaneous recombination rate in a quantum well by resonant surface plasmon coupling,” Phys. Rev. B 66(15), 153305 (2002).
[CrossRef]

Lalanne, P.

G. Lecamp, P. Lalanne, and J. P. Hugonin, “Very large spontaneous-emission β factors in photonic-crystal waveguides,” Phys. Rev. Lett. 99(2), 023902 (2007).
[CrossRef] [PubMed]

Lecamp, G.

G. Lecamp, P. Lalanne, and J. P. Hugonin, “Very large spontaneous-emission β factors in photonic-crystal waveguides,” Phys. Rev. Lett. 99(2), 023902 (2007).
[CrossRef] [PubMed]

Lee, C.

A. Neogi, C. Lee, H. O. Everitt, T. Kuroda, A. Tackeuchi, and E. Yablonovitch, “Enhancement of spontaneous recombination rate in a quantum well by resonant surface plasmon coupling,” Phys. Rev. B 66(15), 153305 (2002).
[CrossRef]

Lee, R. K.

R. K. Lee, Y. Xu, and A. Yariv, “Modified spontaneous emission from a two-dimensional photonic bandgap crystal slab,” J. Opt. Soc. Am. B 17(8), 1438–1442 (2000).
[CrossRef]

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61(3), 033807 (2000).
[CrossRef]

Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17(3), 387–400 (2000).
[CrossRef]

Lewenstein, M.

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43(1), 467–491 (1991).
[CrossRef] [PubMed]

Liu, M.

Y. Jiang and M. Liu, “Electromagnetic force in dispersive and transparent media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(5), 6685–6694 (1998).
[CrossRef]

Liu, S. D.

S. D. Liu, M. T. Cheng, Z. J. Yang, and Q. Q. Wang, “Surface plasmon propagation in a pair of metal nanowires coupled to a nanosized optical emitter,” Opt. Lett. 33(8), 851–853 (2008).
[CrossRef] [PubMed]

Loncar, M.

J. Vučković, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quantum Electron. 36(10), 1131–1144 (2000).
[CrossRef]

Maier, S. A.

S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14(5), 1957–1964 (2006).
[CrossRef] [PubMed]

Manga Tao, V. S. C.

V. S. C. Manga Tao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75(20), 205437 (2007).
[CrossRef]

Mishra, U. K.

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

Moreau, V.

M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express 15(10), 5948–5965 (2007).
[CrossRef] [PubMed]

Mukai, T.

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

Mulvaney, P.

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(7), 077402 (2002).
[CrossRef] [PubMed]

Narukawa, Y.

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

Neogi, A.

A. Neogi, C. Lee, H. O. Everitt, T. Kuroda, A. Tackeuchi, and E. Yablonovitch, “Enhancement of spontaneous recombination rate in a quantum well by resonant surface plasmon coupling,” Phys. Rev. B 66(15), 153305 (2002).
[CrossRef]

Niki, I.

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

Noda, S.

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6(1), 3–9 (2005).
[CrossRef]

Nojima, S.

S. Nojima, “Enhancement of Optical Gain in Two-dimensional Photonic Crystals with Active Lattice Points,” Jpn. J. Appl. Phys. 37(Part 2, No. 5B), L565–L567 (1998).
[CrossRef]

Novotny, L.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96(11), 113002 (2006).
[CrossRef] [PubMed]

Okamoto, K.

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

Painter, O.

M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express 15(10), 5948–5965 (2007).
[CrossRef] [PubMed]

Poon, J. K. S.

J. K. S. Poon and A. Yariv, “Active coupled-resonator optical waveguides. I. Gain enhancement and noise,” J. Opt. Soc. Am. B 24(9), 2378–2388 (2007).
[CrossRef]

Prock, A.

R. R. Chance, A. Prock, and R. Silbey, “Molecular Fluorescence and Energy Transfer near Interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

R. R. Chance, A. Prock, and R. Silbey, “Lifetime of an emitting molecule near partially reflecting surface,” J. Chem. Phys. 60(7), 2744–2748 (1974).
[CrossRef]

Purcell, E. M.

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

Sambles, J. R.

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996).
[CrossRef] [PubMed]

Sarid, D.

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981).
[CrossRef]

Scalora, M.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994).
[CrossRef]

Scherer, A.

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

J. Vučković, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quantum Electron. 36(10), 1131–1144 (2000).
[CrossRef]

Shvartser, A.

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

Silbey, R.

R. R. Chance, A. Prock, and R. Silbey, “Molecular Fluorescence and Energy Transfer near Interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

R. R. Chance, A. Prock, and R. Silbey, “Lifetime of an emitting molecule near partially reflecting surface,” J. Chem. Phys. 60(7), 2744–2748 (1974).
[CrossRef]

Sönnichsen, C.

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(7), 077402 (2002).
[CrossRef] [PubMed]

Tackeuchi, A.

A. Neogi, C. Lee, H. O. Everitt, T. Kuroda, A. Tackeuchi, and E. Yablonovitch, “Enhancement of spontaneous recombination rate in a quantum well by resonant surface plasmon coupling,” Phys. Rev. B 66(15), 153305 (2002).
[CrossRef]

von Plessen, G.

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(7), 077402 (2002).
[CrossRef] [PubMed]

Vrijen, R.

M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol. 17(11), 2096–2112 (1999).
[CrossRef]

Vuckovic, J.

H. Iwase, D. Englund, and J. Vučković, “Spontaneous emission control in high-extraction efficiency plasmonic crystals,” Opt. Express 16(1), 426–434 (2008).
[CrossRef] [PubMed]

Y. Gong and J. Vučković, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90(3), 033113 (2007).
[CrossRef]

J. Vučković, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quantum Electron. 36(10), 1131–1144 (2000).
[CrossRef]

Wang, Q. Q.

S. D. Liu, M. T. Cheng, Z. J. Yang, and Q. Q. Wang, “Surface plasmon propagation in a pair of metal nanowires coupled to a nanosized optical emitter,” Opt. Lett. 33(8), 851–853 (2008).
[CrossRef] [PubMed]

Wilk, T.

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(7), 077402 (2002).
[CrossRef] [PubMed]

Wilson, O.

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(7), 077402 (2002).
[CrossRef] [PubMed]

Wright, E. M.

T. A. B. Kennedy and E. M. Wright, “Quantization and phase-space methods for slowly varying optical fields in a dispersive nonlinear medium,” Phys. Rev. A 38(1), 212–221 (1988).
[CrossRef] [PubMed]

Xu, Y.

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61(3), 033807 (2000).
[CrossRef]

Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17(3), 387–400 (2000).
[CrossRef]

R. K. Lee, Y. Xu, and A. Yariv, “Modified spontaneous emission from a two-dimensional photonic bandgap crystal slab,” J. Opt. Soc. Am. B 17(8), 1438–1442 (2000).
[CrossRef]

Yablonovitch, E.

A. Neogi, C. Lee, H. O. Everitt, T. Kuroda, A. Tackeuchi, and E. Yablonovitch, “Enhancement of spontaneous recombination rate in a quantum well by resonant surface plasmon coupling,” Phys. Rev. B 66(15), 153305 (2002).
[CrossRef]

M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol. 17(11), 2096–2112 (1999).
[CrossRef]

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

Yang, Z. J.

S. D. Liu, M. T. Cheng, Z. J. Yang, and Q. Q. Wang, “Surface plasmon propagation in a pair of metal nanowires coupled to a nanosized optical emitter,” Opt. Lett. 33(8), 851–853 (2008).
[CrossRef] [PubMed]

Yariv, A.

J. K. S. Poon and A. Yariv, “Active coupled-resonator optical waveguides. I. Gain enhancement and noise,” J. Opt. Soc. Am. B 24(9), 2378–2388 (2007).
[CrossRef]

Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17(3), 387–400 (2000).
[CrossRef]

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61(3), 033807 (2000).
[CrossRef]

R. K. Lee, Y. Xu, and A. Yariv, “Modified spontaneous emission from a two-dimensional photonic bandgap crystal slab,” J. Opt. Soc. Am. B 17(8), 1438–1442 (2000).
[CrossRef]

Adv. Chem. Phys.

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.

Y. Gong and J. Vučković, “Design of plasmon cavities for solid-state cavity quantum electrodynamics applications,” Appl. Phys. Lett. 90(3), 033113 (2007).
[CrossRef]

IEEE J. Quantum Electron.

J. Vučković, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quantum Electron. 36(10), 1131–1144 (2000).
[CrossRef]

J. Appl. Phys.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: A new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994).
[CrossRef]

J. Chem. Phys.

H. Kuhn, “Classical aspects of energy transfer in molecular systems,” J. Chem. Phys. 53(1), 101–108 (1970).
[CrossRef]

R. R. Chance, A. Prock, and R. Silbey, “Lifetime of an emitting molecule near partially reflecting surface,” J. Chem. Phys. 60(7), 2744–2748 (1974).
[CrossRef]

J. Lightwave Technol.

W. L. Barnes, “Electromagnetic Crystals for Surface Plasmon Polaritons and the Extraction of Light from Emissive Devices,” J. Lightwave Technol. 17(11), 2170–2182 (1999).
[CrossRef]

M. Boroditsky, R. Vrijen, T. F. Krauss, R. Coccioli, R. Bhat, and E. Yablonovitch, “Spontaneous emission extraction and Purcell enhancement from thin-film 2-D photonic crystals,” J. Lightwave Technol. 17(11), 2096–2112 (1999).
[CrossRef]

J. Mod. Opt.

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

J. Opt. Soc. Am. B

R. K. Lee, Y. Xu, and A. Yariv, “Modified spontaneous emission from a two-dimensional photonic bandgap crystal slab,” J. Opt. Soc. Am. B 17(8), 1438–1442 (2000).
[CrossRef]

Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17(3), 387–400 (2000).
[CrossRef]

J. K. S. Poon and A. Yariv, “Active coupled-resonator optical waveguides. I. Gain enhancement and noise,” J. Opt. Soc. Am. B 24(9), 2378–2388 (2007).
[CrossRef]

Jpn. J. Appl. Phys.

S. Nojima, “Enhancement of Optical Gain in Two-dimensional Photonic Crystals with Active Lattice Points,” Jpn. J. Appl. Phys. 37(Part 2, No. 5B), L565–L567 (1998).
[CrossRef]

Nat. Mater.

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

Opt. Express

S. A. Maier, “Plasmonic field enhancement and SERS in the effective mode volume picture,” Opt. Express 14(5), 1957–1964 (2006).
[CrossRef] [PubMed]

M. Bahriz, V. Moreau, R. Colombelli, O. Crisafulli, and O. Painter, “Design of mid-IR and THz quantum cascade laser cavities with complete TM photonic bandgap,” Opt. Express 15(10), 5948–5965 (2007).
[CrossRef] [PubMed]

H. Iwase, D. Englund, and J. Vučković, “Spontaneous emission control in high-extraction efficiency plasmonic crystals,” Opt. Express 16(1), 426–434 (2008).
[CrossRef] [PubMed]

Opt. Lett.

S. D. Liu, M. T. Cheng, Z. J. Yang, and Q. Q. Wang, “Surface plasmon propagation in a pair of metal nanowires coupled to a nanosized optical emitter,” Opt. Lett. 33(8), 851–853 (2008).
[CrossRef] [PubMed]

Org. Electron.

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6(1), 3–9 (2005).
[CrossRef]

Phys. Rev.

E. N. Economou, “Surface Plasmons in Thin Films,” Phys. Rev. 182(2), 539–554 (1969).
[CrossRef]

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

Phys. Rev. A

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61(3), 033807 (2000).
[CrossRef]

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43(1), 467–491 (1991).
[CrossRef] [PubMed]

T. A. B. Kennedy and E. M. Wright, “Quantization and phase-space methods for slowly varying optical fields in a dispersive nonlinear medium,” Phys. Rev. A 38(1), 212–221 (1988).
[CrossRef] [PubMed]

P. D. Drummond and M. Hillery, “Quantum theory of dispersive electromagnetic modes,” Phys. Rev. A 59(1), 691–707 (1999).
[CrossRef]

Phys. Rev. B

V. S. C. Manga Tao and S. Hughes, “Single quantum-dot Purcell factor and β factor in a photonic crystal waveguide,” Phys. Rev. B 75(20), 205437 (2007).
[CrossRef]

A. Neogi, C. Lee, H. O. Everitt, T. Kuroda, A. Tackeuchi, and E. Yablonovitch, “Enhancement of spontaneous recombination rate in a quantum well by resonant surface plasmon coupling,” Phys. Rev. B 66(15), 153305 (2002).
[CrossRef]

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics

Y. Jiang and M. Liu, “Electromagnetic force in dispersive and transparent media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(5), 6685–6694 (1998).
[CrossRef]

Phys. Rev. Lett.

D. Sarid, “Long-range surface-plasma waves on very thin metal films,” Phys. Rev. Lett. 47(26), 1927–1930 (1981).
[CrossRef]

G. Lecamp, P. Lalanne, and J. P. Hugonin, “Very large spontaneous-emission β factors in photonic-crystal waveguides,” Phys. Rev. Lett. 99(2), 023902 (2007).
[CrossRef] [PubMed]

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96(11), 113002 (2006).
[CrossRef] [PubMed]

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(7), 077402 (2002).
[CrossRef] [PubMed]

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996).
[CrossRef] [PubMed]

Other

M. O. Scully, and M. S. Zubairy, Quantum Optics (Cambridge University Press,1997), Chap. 9.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, Berlin, 1988).

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, New Jersey, 1984).

L. A. Coldren, and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, New York, 1995).

L. D. Landau, and E. M. Lifshitz, Electrodynamics of Continuum Media (Pergamon, New York, 1984).

E. A. Hinds, “Perturbative cavity quantum electrodynamics,” in Cavity Quantum Electrodynamics, P. R. Berman, ed. (Academic, New York, 1994).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton University Press, New Jersey, 1995).

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Figures (10)

Fig. 1
Fig. 1

(a) A metal membrane with uniform metal-dielectric interfaces and the electrical field cross-section of a SPP mode. (b) A plasmonic crystal fabricated on top of a metal membrane.

Fig. 2
Fig. 2

(a) SPP’s electromagnetic field with a momentum k at a uniform metal (z ≤ 0) -dielectric (z > 0) boundary with an area l 2. E k z is an out-of-plane component of the electric field, and E k / / and H k are in-plane components of the electric and magnetic fields. E k z and E k / / are parallel to z-direction and k, respectively. An emitter with an electric dipole μ lies at the distance of zA from the metal surface (positioned at z = 0). (b) Time-evolution of E k z and E k / / oscillating in the z-k plane. ek is a unit vector composed of the amplitudes of E k z and E k / / .

Fig. 3
Fig. 3

(a) Distributions of the SPP spontaneous emission enhancement spectrum i(υ,k) by classical electrodynamics analysis (left-side panel), following References [1316] and by quantum electrodynamics analysis, f(υ,k) (right-side panel). The electric dipole lies normal to the Au surface at a distance of 10 nm from it. ωp and τp are set 1.21 × 1016 sec−1 and 1.05 × 10−14 sec, respectively. (b) Spectra of f(υ,k) and i(υ,k) at k = 0.1 nm−1 along the dashed lines in Fig. 3(a), showing an excellent match between the two analyses. The Δυ k shows a spectral half-width. (c) Distributions of i(υ,k) above the light line. The distances of μ from the Au surface are set to 200 nm and 400 nm.

Fig. 4
Fig. 4

Purcell enhancement factors at various frequencies from quantum analysis - F sp (black lines) and classical analysis - I sp (red lines) for an exciton zA = 10 nm and 30 nm away from the Au/InP boundary, estimated by summing up f(υ,k) and i(υ,k) over k ≤ 0.3 nm−1, respectively. The values of f(υ,k) and i(υ,k) for zA = 10 nm are plotted in Fig. 3(a). Blue lines show the Purcell enhancement factor F non-ab sp for non-absorbing media. The electric dipole lies normal to the Au surface. Non-SPP modes were ignored in the calculation of these plots.

Fig. 5
Fig. 5

(a) 1D-array of high-Q resonators, composed of defects in 2D-photonic crystal with a periodic permittivity ε(r). The defects are aligned in x-direction with a lattice vector R e x . An exciton lies at the position r A in a defect. (b) A single defect in a 2D-photonic crystal with a permittivity ε 0(r).

Fig. 6
Fig. 6

(a) The plasmonic crystal consists of hexagonally arranged InP-filled holes in the Au membrane. The thickness of the Au layer is 20 nm, and the periodicity of the crystal a and radius r of the InP holes are determined by a = 450 nm and r/a = 0.2. (b) Dispersion diagrams of antisymmetric modes in the plasmonic crystal, obtained by FDTD simulation (dots). Dashed lines show the dispersion branches of the unpatterned InP/Au/InP structure.

Fig. 7
Fig. 7

(a) Field patterns on surface of Au layer in the plasmonic crystal, belonging to the upper edge (top figures) and the lower edge (bottom figures) of the plasmonic band gap shown in Fig. 6(b), obtained by FDTD simulation. The field of each band edge consists of three orthogonal modes (monopoles in the left figures and dipoles in the right figures). At the band edge, three dispersion branches overlap so close that their eigen-modes could not be separated by our FDTD simulation. (b) The sum of the density of photonic states j K D K , j ( υ ) for the plasmonic crystal (red line) and unpatterned structure (black line). For computation, we take the summation over the region K ≤ 0.0005 nm−1, considering the degeneracy of each branch. The Q-factor for the plasmonic crystal and unpatterned structure equals 96 and 98 by FDTD simulation, respectively. In the estimation, the upper branches were ignored because of the large leakage loss.

Fig. 8
Fig. 8

Values of 1 / ( 1 + Θ k ) , W E z / W , W E / / / W , and W H / W plotted for different k’s for a SPP mode at Au/InP interface, where W E / / 1 2 ( ω ε ) / ω | E k / / | 2 d r , W E z 1 2 ( ω ε ) / ω | E k z | 2 d r , W H 1 2 μ 0 | H k | 2 d r , and W is a k-mode’s total field energy. The lines show the values estimated by analytically solving Maxwell’s equations [21], and the marks by FDTD. The index and plasma frequency at Au/InP interface are set n = 3.2, and 2 π c / ω p = 156 nm, respectively.

Fig. 9
Fig. 9

(a) An InP/Au/InP structure with an electric dipole lying on the dielectric side (top), and electric field components parallel to the Au surface of an antisymmetric and symmetric modes (bottom). The Au membrane is 10 nm thick. The electric dipole lies normal to the Au surface at a distance of 10 nm apart from it. (b) The distribution of i l a y e r ( υ , k ) estimated for the structure shown in Fig. 9(a). The υ v = 0 is a frequency with v g = 0 for the antisymmetric modes, and υ p ω p / ( 1 + n 2 ) 1 / 2 . (c) Dissipation spectrum i l a y e r ( υ , k ) at k = 0.1 nm−1 along the dashed line in Fig. 9(b). The i l a y e r ( υ , k ) below the light line is expressed by the sum of two Lorentzian spectra, i l a y e r ( υ , k ) = i a n t i ( υ , k ) + i s y m ( υ , k ) , corresponding to SE into the antisymmetric and symmetric modes. The Δυ k is a half-width of the spectrum.

Fig. 10
Fig. 10

(a) Quality factor υ k 0 / Δ υ k of symmetric and antisymmetric modes plotted with different k’s, estimated by fitting i l a y e r ( υ , k ) at RT (shown in Fig. 9(c)) and 77 K with Lorentzian spectra. The τp is set 1.05 × 10−14 sec at RT and 3.72 × 10−14 sec at 77 K, respectively, and ωp = 1.21 × 1016 sec−1. The circles show the quality factors, estimated by υ / 2 k v g for antisymmetric modes at RT [8]. (b) Purcell enhancements for the symmetric and anti-symmetric mode, I s y m and I a n t i , at RT and 77 K, estimated by summing up i s y m and i a n t i shown in Figs. 9(b) and 9(c) over k ≤ 0.3 nm−1, respectively.

Equations (23)

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E k wp ( r , t ) Δ k A Δ k 2 [ f k + Δ k ( r ) e i ω k t + f k + Δ k * ( r ) e i ω k t ]   e ω k t / 2 Q k .
F sp ( υ ) Γ sp n Γ 0 = k 2 π n Γ 0 ( υ ) | g k ( z A ) | 2 D k ( υ )   ,
D k ( υ ) = 1 π ω k / 2 Q k ( ω k υ ) 2 + ( ω k / 2 Q k ) 2   .
F dis ( υ , k ) 1 Δ k x Δ k y 2 π n Γ 0 ( υ ) | g k ( z A ) | 2 D k ( υ )  ,
F dis ( υ , k ) = 3 2 1 n 3 c 3 υ 2 1 1 + Θ k h k ( z A ) L k ( e k e μ ) 2 D k ( υ )  ,
f ( υ , k ) = 3 π n 3 c 3 υ 2 1 1 + Θ k ς   h k ( z A ) L k k D k ( υ )   .
f K , j ( r ) e i ω t = u K , j ( r ) e i K x e i ω t  ,
V K , j cell unit cell d r ( ε ω ) / ω | u K , j | 2 / max [ ( ε ω ) / ω | u K , j | 2 ]   .
F cry ( υ ) = Δ K x Δ K y j K F cry-dis ( υ ,   K , j ) + Δ K x Δ K y j K O K , j ( υ )  ,
F cry-dis ( υ ,   K , j ) = 3 2 1 n 3 c 3 υ 2 1 1 + Θ K , j S cell V K , j cell h K , j ( r A ) ( e μ e K , j ) 2 D K , j ( υ )  ,
h K , j ( r A ) = n 2 | u K , j ( υ , r A ) | 2 / max [ ( ε ω ) / ω | u K , j ( υ , r ) | 2 ]   .
F j ( υ ) = 2 π Δ K K 3 4 1 n 3 c 3 υ 2 R V K , j cell D K , j ( υ )  ,
E ( r , t ) = 1 2 [ D f k ( r ) η ( t ) + c . c . ]   ,
H ( r , t ) = 1 2 1 μ 0 [ D ω k × f k ( r ) χ ( t ) + c . c . ]   ,
η ( t ) = q ( t ) + i p ( t )  ,
χ ( t ) = p ( t ) i q ( t )  ,
W = 1 2 [ ( ε ω ) / ω E ( r , t ) 2 + μ 0 H ( r , t ) 2 ] d r
= 1 4 D 2 ( p 2 + q 2 ) + 1 4 D 2 Θ k ( p 2 + q 2 ) = 1 2 D 2 1 + Θ k 2 ( p 2 + q 2 )  ,
W q = ω k q = p ˙  ,
W p = ω k p = q ˙   .
H = ω k ( a + a + 1 2 )  ,
E ( r , t ) = i ω k 1 + Θ k f k ( r ) a + H . C .  ,
H ( r , t ) = ( 1 + Θ k ) ω k 1 μ 0 × f k ( r ) a + H . C .   .

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