Abstract

A rigorous surface integral equation approach is proposed to study the spontaneous emission of a quantum emitter embedded in a multilayered plasmonic structure with the presence of arbitrarily shaped metallic nanoscatterers. With the aid of the Fermi’s golden rule, the spontaneous emission of the emitter can be calculated from the local density of states, which can be further expressed by the imaginary part of the dyadic Green’s function of the whole electromagnetic system. To obtain this Green’s function numerically, a surface integral equation is established taking into account the scattering from the metallic nanoscatterers. Particularly, the modeling of the planar multilayered structure is simplified by applying the layered medium Green’s function to reduce the computational domain and hence the memory requirement. Regarding the evaluation of Sommerfeld integrals in the layered medium Green’s function, the discrete complex image method is adopted to accelerate the evaluation process. This work offers an accurate and efficient simulation tool for analyzing complex multilayered plasmonic system, which is commonly encountered in the design of optical elements and devices.

© 2012 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, “Design of plasmonic nanoantennae for enhancing spontaneous emission,” Opt. Lett.32, 1623–1625 (2007).
    [CrossRef] [PubMed]
  2. M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett.35, 1863–1865 (2010).
    [CrossRef] [PubMed]
  3. P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman1, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature430, 654–657 (2004).
    [CrossRef] [PubMed]
  4. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev.69, 681 (1946).
  5. C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University Press, 2005).
  6. L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2006).
    [CrossRef]
  7. 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, 601–605 (2004).
    [CrossRef] [PubMed]
  8. J.-J. Greffet, “Nanoantennas for light emission,” Science308, 1561–1563 (2005).
    [CrossRef] [PubMed]
  9. A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
    [CrossRef] [PubMed]
  10. K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photo.5, 166–169 (2011).
    [CrossRef]
  11. J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B76, 245403 (2007).
    [CrossRef]
  12. X. W. Chen, W. C. H. Choy, S. He, and P.C. Chui, “Comprehensive analysis and optimal design of top-emitting organic light emitting devices,” J. Appl. Phys.101, 113107 (2007).
    [CrossRef]
  13. X. W. Chen, W. C. H. Choy, and S. He, “Efficient and rigorous modeling of light emission in planar multilayer organic light-emitting diodes,” IEEE/OSA J. Display Technol.3, 110–117 (2007).
    [CrossRef]
  14. W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, Norwood, 2001).
  15. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag.14, 302–307 (1966).
    [CrossRef]
  16. P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University Press, 2003).
    [CrossRef] [PubMed]
  17. W. C. Chew, M. S. Tong, and B. Hu, Integral Equations for Electromagnetic and Elastic Waves (Morgan & Claypool Publishers, 2009).
  18. A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A26, 732–740 (2009).
    [CrossRef]
  19. B. Gallinet, A. M. Kern, and O. J. F. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A27, 2261–2271 (2010).
    [CrossRef]
  20. J. M. Taboada, J. Rivero, F. Obelleiro, M. G. Araújo, and L. Landesa, “Method-of-moments formulation for the analysis of plasmonic nano-optical antennas,” J. Opt. Soc. Am. A28, 1341–1348 (2011).
    [CrossRef]
  21. M. G. Araújo, J. M. Taboada, D. M. Solís, J. Rivero, L. Landesa, and F. Obelleiro, “Comparison of surface integral equation formulations for electromagnetic analysis of plasmonic nanoscatterers,” Opt. Express20, 9161–9171 (2012).
    [CrossRef] [PubMed]
  22. K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propagat.45, 508–519 (1997).
    [CrossRef]
  23. Y. P. Chen, W. C. Chew, and L. Jiang, “A new Green’s function formulation for modeling homogeneous objects in layered medium,” IEEE Trans. Antennas Propagat. accepted for publication.
  24. D. G. Fang, J. J. Yang, and G. Y. Delisle, “Discrete image theory for horizontal electric dipoles in a multilayered medium,” Proc. Inst. Elect. Eng.135, 297–303 (1988).
  25. E. N. Economou, Green’s Functions in Quantum Physics (Springer, Berlin, 2006).
    [PubMed]
  26. R. Carminati, J.-J. Greffet, C. Henkel, and J.M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Comm.261, 368–375, 2006.
    [CrossRef]
  27. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990; IEEE Press, 1995).
  28. X. W. Chen, M. Agio, and V. Sandoghdar, “Metallodielectric hybrid antennas for ultrastrong enhancement of spontaneous emission,” Phys. Rev. Lett.108, 233001 (2012).
    [CrossRef]
  29. A. J. Poggio and E. K. Miller, “Integral equation solutions of three dimensional scattering problems,” in Computer Techniques for Electromagnetics (Permagon, 1973).
  30. Y. Chang and R. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag.25, 789–795 (1977).
    [CrossRef]
  31. T.-K. Wu and L. L. Tsai, “Scattering from arbitrarilyshaped lossy dielectric bodies of revolution,” Radio Sci.12, 709–718 (1977).
    [CrossRef]
  32. S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surface of arbitrary shape,” IEEE Trans. Antennas Propagat.30, 409–418 (1982).
    [CrossRef]
  33. W. C. Chew, J. L. Xiong, and M. A. Saville, “A matrix-friendly formulation of layered medium Green’s function,” IEEE Antennas Wireless Propagat. Lett.5, 490–494 (2006).
    [CrossRef]
  34. T. K. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” lEEE Antennas Propagat. Magazine37, 48–55 (1995).
    [CrossRef]
  35. A. Alparslan, M. I. Aksun, and K. A. Michalski, “Closed-form Green’s functions in planar layered media for all ranges and materials,” IEEE Trans. Microw. Theory Tech.58, 602–613 (2010).
    [CrossRef]
  36. Y. P. Chen, W. C. Chew, and L. Jiang, “A novel implementation of discrete complex image method for layered medium Green’s function,” IEEE Antennas Wireless Propagat. Lett.10, 419–422 (2011).
    [CrossRef]
  37. A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt.37, 5271–5283 (1998).
    [CrossRef]
  38. G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys.25, 377–445 (1908).
    [CrossRef]
  39. A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photo.3, 654–657 (2009).
    [CrossRef]

2012

2011

J. M. Taboada, J. Rivero, F. Obelleiro, M. G. Araújo, and L. Landesa, “Method-of-moments formulation for the analysis of plasmonic nano-optical antennas,” J. Opt. Soc. Am. A28, 1341–1348 (2011).
[CrossRef]

Y. P. Chen, W. C. Chew, and L. Jiang, “A novel implementation of discrete complex image method for layered medium Green’s function,” IEEE Antennas Wireless Propagat. Lett.10, 419–422 (2011).
[CrossRef]

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photo.5, 166–169 (2011).
[CrossRef]

2010

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

A. Alparslan, M. I. Aksun, and K. A. Michalski, “Closed-form Green’s functions in planar layered media for all ranges and materials,” IEEE Trans. Microw. Theory Tech.58, 602–613 (2010).
[CrossRef]

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett.35, 1863–1865 (2010).
[CrossRef] [PubMed]

B. Gallinet, A. M. Kern, and O. J. F. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A27, 2261–2271 (2010).
[CrossRef]

2009

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photo.3, 654–657 (2009).
[CrossRef]

A. M. Kern and O. J. F. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A26, 732–740 (2009).
[CrossRef]

2007

L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, “Design of plasmonic nanoantennae for enhancing spontaneous emission,” Opt. Lett.32, 1623–1625 (2007).
[CrossRef] [PubMed]

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B76, 245403 (2007).
[CrossRef]

X. W. Chen, W. C. H. Choy, S. He, and P.C. Chui, “Comprehensive analysis and optimal design of top-emitting organic light emitting devices,” J. Appl. Phys.101, 113107 (2007).
[CrossRef]

X. W. Chen, W. C. H. Choy, and S. He, “Efficient and rigorous modeling of light emission in planar multilayer organic light-emitting diodes,” IEEE/OSA J. Display Technol.3, 110–117 (2007).
[CrossRef]

2006

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

W. C. Chew, J. L. Xiong, and M. A. Saville, “A matrix-friendly formulation of layered medium Green’s function,” IEEE Antennas Wireless Propagat. Lett.5, 490–494 (2006).
[CrossRef]

2005

J.-J. Greffet, “Nanoantennas for light emission,” Science308, 1561–1563 (2005).
[CrossRef] [PubMed]

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, 601–605 (2004).
[CrossRef] [PubMed]

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman1, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature430, 654–657 (2004).
[CrossRef] [PubMed]

1998

1997

K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propagat.45, 508–519 (1997).
[CrossRef]

1995

T. K. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” lEEE Antennas Propagat. Magazine37, 48–55 (1995).
[CrossRef]

1988

D. G. Fang, J. J. Yang, and G. Y. Delisle, “Discrete image theory for horizontal electric dipoles in a multilayered medium,” Proc. Inst. Elect. Eng.135, 297–303 (1988).

1982

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surface of arbitrary shape,” IEEE Trans. Antennas Propagat.30, 409–418 (1982).
[CrossRef]

1977

Y. Chang and R. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag.25, 789–795 (1977).
[CrossRef]

T.-K. Wu and L. L. Tsai, “Scattering from arbitrarilyshaped lossy dielectric bodies of revolution,” Radio Sci.12, 709–718 (1977).
[CrossRef]

1966

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag.14, 302–307 (1966).
[CrossRef]

1946

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

1908

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys.25, 377–445 (1908).
[CrossRef]

Agio, M.

X. W. Chen, M. Agio, and V. Sandoghdar, “Metallodielectric hybrid antennas for ultrastrong enhancement of spontaneous emission,” Phys. Rev. Lett.108, 233001 (2012).
[CrossRef]

L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, “Design of plasmonic nanoantennae for enhancing spontaneous emission,” Opt. Lett.32, 1623–1625 (2007).
[CrossRef] [PubMed]

Aksun, M. I.

A. Alparslan, M. I. Aksun, and K. A. Michalski, “Closed-form Green’s functions in planar layered media for all ranges and materials,” IEEE Trans. Microw. Theory Tech.58, 602–613 (2010).
[CrossRef]

Alparslan, A.

A. Alparslan, M. I. Aksun, and K. A. Michalski, “Closed-form Green’s functions in planar layered media for all ranges and materials,” IEEE Trans. Microw. Theory Tech.58, 602–613 (2010).
[CrossRef]

Araújo, M. G.

Avlasevich, Y.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photo.3, 654–657 (2009).
[CrossRef]

Barnakov, Yu. A.

Bonner, C. E.

Carminati, R.

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

Chang, Y.

Y. Chang and R. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag.25, 789–795 (1977).
[CrossRef]

Chen, X. W.

X. W. Chen, M. Agio, and V. Sandoghdar, “Metallodielectric hybrid antennas for ultrastrong enhancement of spontaneous emission,” Phys. Rev. Lett.108, 233001 (2012).
[CrossRef]

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photo.5, 166–169 (2011).
[CrossRef]

X. W. Chen, W. C. H. Choy, S. He, and P.C. Chui, “Comprehensive analysis and optimal design of top-emitting organic light emitting devices,” J. Appl. Phys.101, 113107 (2007).
[CrossRef]

X. W. Chen, W. C. H. Choy, and S. He, “Efficient and rigorous modeling of light emission in planar multilayer organic light-emitting diodes,” IEEE/OSA J. Display Technol.3, 110–117 (2007).
[CrossRef]

Chen, Y. P.

Y. P. Chen, W. C. Chew, and L. Jiang, “A novel implementation of discrete complex image method for layered medium Green’s function,” IEEE Antennas Wireless Propagat. Lett.10, 419–422 (2011).
[CrossRef]

Y. P. Chen, W. C. Chew, and L. Jiang, “A new Green’s function formulation for modeling homogeneous objects in layered medium,” IEEE Trans. Antennas Propagat. accepted for publication.

Chew, W. C.

Y. P. Chen, W. C. Chew, and L. Jiang, “A novel implementation of discrete complex image method for layered medium Green’s function,” IEEE Antennas Wireless Propagat. Lett.10, 419–422 (2011).
[CrossRef]

W. C. Chew, J. L. Xiong, and M. A. Saville, “A matrix-friendly formulation of layered medium Green’s function,” IEEE Antennas Wireless Propagat. Lett.5, 490–494 (2006).
[CrossRef]

Y. P. Chen, W. C. Chew, and L. Jiang, “A new Green’s function formulation for modeling homogeneous objects in layered medium,” IEEE Trans. Antennas Propagat. accepted for publication.

W. C. Chew, M. S. Tong, and B. Hu, Integral Equations for Electromagnetic and Elastic Waves (Morgan & Claypool Publishers, 2009).

W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, Norwood, 2001).

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990; IEEE Press, 1995).

Choy, W. C. H.

X. W. Chen, W. C. H. Choy, and S. He, “Efficient and rigorous modeling of light emission in planar multilayer organic light-emitting diodes,” IEEE/OSA J. Display Technol.3, 110–117 (2007).
[CrossRef]

X. W. Chen, W. C. H. Choy, S. He, and P.C. Chui, “Comprehensive analysis and optimal design of top-emitting organic light emitting devices,” J. Appl. Phys.101, 113107 (2007).
[CrossRef]

Chui, P.C.

X. W. Chen, W. C. H. Choy, S. He, and P.C. Chui, “Comprehensive analysis and optimal design of top-emitting organic light emitting devices,” J. Appl. Phys.101, 113107 (2007).
[CrossRef]

Curto, A. G.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

Delisle, G. Y.

D. G. Fang, J. J. Yang, and G. Y. Delisle, “Discrete image theory for horizontal electric dipoles in a multilayered medium,” Proc. Inst. Elect. Eng.135, 297–303 (1988).

Djurisic, A. B.

Dryden, D.

Economou, E. N.

E. N. Economou, Green’s Functions in Quantum Physics (Springer, Berlin, 2006).
[PubMed]

Eghlidi, H.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photo.5, 166–169 (2011).
[CrossRef]

Elazar, J. M.

Engheta, N.

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B76, 245403 (2007).
[CrossRef]

Fan, S.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photo.3, 654–657 (2009).
[CrossRef]

Fang, D. G.

D. G. Fang, J. J. Yang, and G. Y. Delisle, “Discrete image theory for horizontal electric dipoles in a multilayered medium,” Proc. Inst. Elect. Eng.135, 297–303 (1988).

Gallinet, B.

Gerry, C.

C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University Press, 2005).

Glisson, A. W.

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surface of arbitrary shape,” IEEE Trans. Antennas Propagat.30, 409–418 (1982).
[CrossRef]

Götzinger, S.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photo.5, 166–169 (2011).
[CrossRef]

Greffet, J.-J.

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

J.-J. Greffet, “Nanoantennas for light emission,” Science308, 1561–1563 (2005).
[CrossRef] [PubMed]

Harrington, R.

Y. Chang and R. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag.25, 789–795 (1977).
[CrossRef]

He, S.

X. W. Chen, W. C. H. Choy, and S. He, “Efficient and rigorous modeling of light emission in planar multilayer organic light-emitting diodes,” IEEE/OSA J. Display Technol.3, 110–117 (2007).
[CrossRef]

X. W. Chen, W. C. H. Choy, S. He, and P.C. Chui, “Comprehensive analysis and optimal design of top-emitting organic light emitting devices,” J. Appl. Phys.101, 113107 (2007).
[CrossRef]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2006).
[CrossRef]

Henkel, C.

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

Hu, B.

W. C. Chew, M. S. Tong, and B. Hu, Integral Equations for Electromagnetic and Elastic Waves (Morgan & Claypool Publishers, 2009).

Irman1, A.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman1, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature430, 654–657 (2004).
[CrossRef] [PubMed]

Jacob, Z.

Jiang, L.

Y. P. Chen, W. C. Chew, and L. Jiang, “A novel implementation of discrete complex image method for layered medium Green’s function,” IEEE Antennas Wireless Propagat. Lett.10, 419–422 (2011).
[CrossRef]

Y. P. Chen, W. C. Chew, and L. Jiang, “A new Green’s function formulation for modeling homogeneous objects in layered medium,” IEEE Trans. Antennas Propagat. accepted for publication.

Jin, J. M.

W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, Norwood, 2001).

Kaminski, F.

Kern, A. M.

Kinkhabwala, A.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photo.3, 654–657 (2009).
[CrossRef]

Knight, P.

C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University Press, 2005).

Kreuzer, M. P.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

Kukura, P.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photo.5, 166–169 (2011).
[CrossRef]

Landesa, L.

Lee, K. G.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photo.5, 166–169 (2011).
[CrossRef]

Lettow, R.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photo.5, 166–169 (2011).
[CrossRef]

Li, H.

Li, J.

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B76, 245403 (2007).
[CrossRef]

Lodahl, P.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman1, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature430, 654–657 (2004).
[CrossRef] [PubMed]

Majewski, M. L.

Martin, O. J. F.

Mayy, M.

Michalski, K. A.

A. Alparslan, M. I. Aksun, and K. A. Michalski, “Closed-form Green’s functions in planar layered media for all ranges and materials,” IEEE Trans. Microw. Theory Tech.58, 602–613 (2010).
[CrossRef]

K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propagat.45, 508–519 (1997).
[CrossRef]

Michielssen, E.

W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, Norwood, 2001).

Mie, G.

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys.25, 377–445 (1908).
[CrossRef]

Miller, E. K.

A. J. Poggio and E. K. Miller, “Integral equation solutions of three dimensional scattering problems,” in Computer Techniques for Electromagnetics (Permagon, 1973).

Moerner, W. E.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photo.3, 654–657 (2009).
[CrossRef]

Monk, P.

P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University Press, 2003).
[CrossRef] [PubMed]

Mosig, J. R.

K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propagat.45, 508–519 (1997).
[CrossRef]

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, 601–605 (2004).
[CrossRef] [PubMed]

Müllen, K.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photo.3, 654–657 (2009).
[CrossRef]

Narimanov, E. E.

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, 601–605 (2004).
[CrossRef] [PubMed]

Nataraj, G.

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, 601–605 (2004).
[CrossRef] [PubMed]

Nikolaev, I. S.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman1, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature430, 654–657 (2004).
[CrossRef] [PubMed]

Noginov, M. A.

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2006).
[CrossRef]

Obelleiro, F.

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, 601–605 (2004).
[CrossRef] [PubMed]

Overgaag, K.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman1, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature430, 654–657 (2004).
[CrossRef] [PubMed]

Pereira, O.

T. K. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” lEEE Antennas Propagat. Magazine37, 48–55 (1995).
[CrossRef]

Poggio, A. J.

A. J. Poggio and E. K. Miller, “Integral equation solutions of three dimensional scattering problems,” in Computer Techniques for Electromagnetics (Permagon, 1973).

Purcell, E. M.

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

Quidant, R.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

Rakic, A. D.

Rao, S. M.

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surface of arbitrary shape,” IEEE Trans. Antennas Propagat.30, 409–418 (1982).
[CrossRef]

Renn, A.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photo.5, 166–169 (2011).
[CrossRef]

Rivero, J.

Rogobete, L.

Salandrino, A.

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B76, 245403 (2007).
[CrossRef]

Sandoghdar, V.

X. W. Chen, M. Agio, and V. Sandoghdar, “Metallodielectric hybrid antennas for ultrastrong enhancement of spontaneous emission,” Phys. Rev. Lett.108, 233001 (2012).
[CrossRef]

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photo.5, 166–169 (2011).
[CrossRef]

L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, “Design of plasmonic nanoantennae for enhancing spontaneous emission,” Opt. Lett.32, 1623–1625 (2007).
[CrossRef] [PubMed]

Sarkar, T. K.

T. K. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” lEEE Antennas Propagat. Magazine37, 48–55 (1995).
[CrossRef]

Saville, M. A.

W. C. Chew, J. L. Xiong, and M. A. Saville, “A matrix-friendly formulation of layered medium Green’s function,” IEEE Antennas Wireless Propagat. Lett.5, 490–494 (2006).
[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, 601–605 (2004).
[CrossRef] [PubMed]

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, 601–605 (2004).
[CrossRef] [PubMed]

Solís, D. M.

Song, J. M.

W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, Norwood, 2001).

Taboada, J. M.

Taminiau, T. H.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

Tong, M. S.

W. C. Chew, M. S. Tong, and B. Hu, Integral Equations for Electromagnetic and Elastic Waves (Morgan & Claypool Publishers, 2009).

Tsai, L. L.

T.-K. Wu and L. L. Tsai, “Scattering from arbitrarilyshaped lossy dielectric bodies of revolution,” Radio Sci.12, 709–718 (1977).
[CrossRef]

van Driel, A. F.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman1, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature430, 654–657 (2004).
[CrossRef] [PubMed]

van Hulst, N. F.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

Vanmaekelbergh, D.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman1, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature430, 654–657 (2004).
[CrossRef] [PubMed]

Vigoureux, J.M.

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

Volpe, G.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

Vos, W. L.

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman1, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature430, 654–657 (2004).
[CrossRef] [PubMed]

Wilton, D. R.

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surface of arbitrary shape,” IEEE Trans. Antennas Propagat.30, 409–418 (1982).
[CrossRef]

Wu, T.-K.

T.-K. Wu and L. L. Tsai, “Scattering from arbitrarilyshaped lossy dielectric bodies of revolution,” Radio Sci.12, 709–718 (1977).
[CrossRef]

Xiong, J. L.

W. C. Chew, J. L. Xiong, and M. A. Saville, “A matrix-friendly formulation of layered medium Green’s function,” IEEE Antennas Wireless Propagat. Lett.5, 490–494 (2006).
[CrossRef]

Yang, J. J.

D. G. Fang, J. J. Yang, and G. Y. Delisle, “Discrete image theory for horizontal electric dipoles in a multilayered medium,” Proc. Inst. Elect. Eng.135, 297–303 (1988).

Yee, K.

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag.14, 302–307 (1966).
[CrossRef]

Yu, Z.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photo.3, 654–657 (2009).
[CrossRef]

Zhu, G.

Ann. Phys.

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys.25, 377–445 (1908).
[CrossRef]

Appl. Opt.

IEEE Antennas Wireless Propagat. Lett.

W. C. Chew, J. L. Xiong, and M. A. Saville, “A matrix-friendly formulation of layered medium Green’s function,” IEEE Antennas Wireless Propagat. Lett.5, 490–494 (2006).
[CrossRef]

Y. P. Chen, W. C. Chew, and L. Jiang, “A novel implementation of discrete complex image method for layered medium Green’s function,” IEEE Antennas Wireless Propagat. Lett.10, 419–422 (2011).
[CrossRef]

IEEE Trans. Antennas Propag.

Y. Chang and R. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag.25, 789–795 (1977).
[CrossRef]

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag.14, 302–307 (1966).
[CrossRef]

IEEE Trans. Antennas Propagat.

K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propagat.45, 508–519 (1997).
[CrossRef]

S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surface of arbitrary shape,” IEEE Trans. Antennas Propagat.30, 409–418 (1982).
[CrossRef]

IEEE Trans. Microw. Theory Tech.

A. Alparslan, M. I. Aksun, and K. A. Michalski, “Closed-form Green’s functions in planar layered media for all ranges and materials,” IEEE Trans. Microw. Theory Tech.58, 602–613 (2010).
[CrossRef]

IEEE/OSA J. Display Technol.

X. W. Chen, W. C. H. Choy, and S. He, “Efficient and rigorous modeling of light emission in planar multilayer organic light-emitting diodes,” IEEE/OSA J. Display Technol.3, 110–117 (2007).
[CrossRef]

J. Appl. Phys.

X. W. Chen, W. C. H. Choy, S. He, and P.C. Chui, “Comprehensive analysis and optimal design of top-emitting organic light emitting devices,” J. Appl. Phys.101, 113107 (2007).
[CrossRef]

J. Opt. Soc. Am. A

lEEE Antennas Propagat. Magazine

T. K. Sarkar and O. Pereira, “Using the matrix pencil method to estimate the parameters of a sum of complex exponentials,” lEEE Antennas Propagat. Magazine37, 48–55 (1995).
[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, 601–605 (2004).
[CrossRef] [PubMed]

Nat. Photo.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Götzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photo.5, 166–169 (2011).
[CrossRef]

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photo.3, 654–657 (2009).
[CrossRef]

Nature

P. Lodahl, A. F. van Driel, I. S. Nikolaev, A. Irman1, K. Overgaag, D. Vanmaekelbergh, and W. L. Vos, “Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals,” Nature430, 654–657 (2004).
[CrossRef] [PubMed]

Opt. Comm.

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

Opt. Express

Opt. Lett.

Phys. Rev.

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

Phys. Rev. B

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Uda nanoantenna in the optical domain,” Phys. Rev. B76, 245403 (2007).
[CrossRef]

Phys. Rev. Lett.

X. W. Chen, M. Agio, and V. Sandoghdar, “Metallodielectric hybrid antennas for ultrastrong enhancement of spontaneous emission,” Phys. Rev. Lett.108, 233001 (2012).
[CrossRef]

Proc. Inst. Elect. Eng.

D. G. Fang, J. J. Yang, and G. Y. Delisle, “Discrete image theory for horizontal electric dipoles in a multilayered medium,” Proc. Inst. Elect. Eng.135, 297–303 (1988).

Radio Sci.

T.-K. Wu and L. L. Tsai, “Scattering from arbitrarilyshaped lossy dielectric bodies of revolution,” Radio Sci.12, 709–718 (1977).
[CrossRef]

Science

J.-J. Greffet, “Nanoantennas for light emission,” Science308, 1561–1563 (2005).
[CrossRef] [PubMed]

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

Other

C. Gerry and P. Knight, Introductory Quantum Optics (Cambridge University Press, 2005).

L. Novotny and B. Hecht, Principles of Nano-optics (Cambridge University Press, 2006).
[CrossRef]

E. N. Economou, Green’s Functions in Quantum Physics (Springer, Berlin, 2006).
[PubMed]

Y. P. Chen, W. C. Chew, and L. Jiang, “A new Green’s function formulation for modeling homogeneous objects in layered medium,” IEEE Trans. Antennas Propagat. accepted for publication.

W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, Norwood, 2001).

P. Monk, Finite Element Methods for Maxwell’s Equations (Oxford University Press, 2003).
[CrossRef] [PubMed]

W. C. Chew, M. S. Tong, and B. Hu, Integral Equations for Electromagnetic and Elastic Waves (Morgan & Claypool Publishers, 2009).

A. J. Poggio and E. K. Miller, “Integral equation solutions of three dimensional scattering problems,” in Computer Techniques for Electromagnetics (Permagon, 1973).

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990; IEEE Press, 1995).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Configuration profile: arbitrarily shaped nano particles embedded in a multilayered structure. Surface integral equation can be set up by invoking the surface equivalence principle. With the help of LMGF, only the surfaces of the scatterers excluding the substrate need discretization.

Fig. 2
Fig. 2

(a) Configuration profile: a gold sphere is located in the air, illuminated by a normal incident plane wave, r = 20 nm, the observation line is shown by a dash line. (b) Near electric field, validated by MIE series. (c) Near magnetic field, validated by MIE series.

Fig. 3
Fig. 3

(a) Configuration profile: a gold sphere is located in an air-metal-air substrate, excited by a z-polarized dipole, where r = 20 nm, d = 20 nm, h = 30 nm; (b) the real part of z-component of the electric field is calculated by our scheme (corresponding to the Im{Gzz}), validated by an approximated model, where the substrate is truncated into a finite domain for discretization, and the SIE with FSGF is adopted.

Fig. 4
Fig. 4

(a) Normalized SER of a x-polarized emitter versus wavelength; (b) Normalized SER of a z-polarized emitter versus wavelength; (c) Normalized SER of a z-polarized emitter at λ = 510 nm versus distance.

Fig. 5
Fig. 5

(a) Mesh of the nano sphere; (b) Scattered near field distribution of the sphere with the absence of substrate (in logarithmic scale); (c) Scattered near field distribution of the sphere with the presence of substrate (in logarithmic scale).

Fig. 6
Fig. 6

(a) Configuration profile of the multilayered structure and mesh of the nano scatterer; (b) Normalized SER for different locations of z-polarized emitters: a: above the center of the prism (0, 0, 10), b: around one corner of the prism (-35, -17.32, 0); (c) Scattered near field distribution at the resonance peak of 520 nm; (d) Scattered near field distribution at the second resonance peak of 680 nm.

Equations (35)

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

γ = π ω 0 h ¯ ε 0 | p | 2 k [ n p ( u k u k * ) n p ] δ ( ω k ω 0 )
× × u k ( r , ω k ) ω k 2 c 2 u k ( r , ω k ) = 0
× × G ¯ ( r , r , ω ) ω 2 c 2 G ¯ ( r , r , ω ) = I ¯ δ ( r r )
Im [ G ¯ ( r , r , ω ) ] = π c 2 2 ω k u k ( r , ω k ) u k * ( r , ω k ) δ ( ω ω k )
γ ( r 0 , ω 0 ) = 2 ω 0 2 h ¯ ε 0 c 2 | p | 2 { n p Im [ G ¯ ( r 0 , r 0 , ω 0 ) ] n p }
γ ( r 0 , ω 0 ) = 2 ω 0 2 3 h ¯ ε 0 c 2 Im { Tr [ G ¯ ( r 0 , r 0 , ω 0 ) ] }
ρ ( r 0 , ω 0 ) = k | u k | 2 δ ( ω k ω 0 ) = 2 ω 0 π c 2 Im { Tr [ G ¯ ( r 0 , r 0 , ω 0 ) ] }
γ γ 0 = ρ ( r 0 , ω 0 ) ρ 0 ( r 0 , ω 0 ) = Im { Tr [ G ¯ ( r 0 , r 0 , ω 0 ) ] } Im { Tr [ G 0 ( r 0 , r 0 , ω 0 ) ] }
E ( r ) = E ( r , r ) J ( r ) + 𝒦 E ( r , r ) M ( r )
H ( r ) = H ( r , r ) M ( r ) + 𝒦 H ( r , r ) J ( r )
E ( r , r ) = i ω d r G ¯ e ( r , r ) μ ( r ) .
𝒦 H ( r , r ) = μ 1 ( r ) d r × G ¯ e ( r , r ) μ ( r ) .
H ( r , r ) = i ω d r G ¯ m ( r , r ) ε ( r ) .
𝒦 E ( r , r ) = ε 1 ( r ) d r × G ¯ m ( r , r ) ε ( r ) .
G ¯ e ( r , r ) = G ¯ e TE ( r , r ) + 1 k n m 2 G ¯ e TM ( r , r )
G ¯ e TE ( r , r ) = ( × z ^ ) ( × z ^ ) i 4 π 0 d k ρ J 0 ( k ρ ρ ) F TE ( k ρ , z , z ) k m z k ρ
G ¯ e TM ( r , r ) = ( × × z ^ ) ( × × z ^ ) i 4 π 0 d k ρ J 0 ( k ρ ρ ) F TM ( k ρ , z , z ) k m z k ρ
[ E inc o H inc o ] | tan = [ ( E o + E i ) ( 𝒦 E o + 𝒦 E i ) ( 𝒦 H o + 𝒦 H i ) ( H o + H i ) ] [ J M ] | tan
f j ( r ) , E ( r , r ) , f i ( r ) = i ω μ m f j s ( r ) , g e , s s ( r , r ) , f i s ( r ) + i ω μ m z ^ f j ( r ) , g e , z z ( r , r ) , z ^ f i ( r ) + i ω μ m z ^ f j ( r ) , g e , z d ( r , r ) , f i ( r ) + i ω μ m f j ( r ) , g e , d z ( r , r ) , z ^ f i ( r ) + i ω μ m f j ( r ) , g e , d d ( r , r ) , f i ( r )
g e , s s = i 4 π 0 d k ρ J 0 ( k ρ ρ ) F TE k ρ k m z
g e , z z = i 4 π 0 d k ρ J 0 ( k ρ ρ ) ( z z F TE + k m n 2 F TM ) 1 k m z k ρ
g e , z d = i 4 π 0 d k ρ J 0 ( k ρ ρ ) ( z F TE μ n μ m z F TM ) 1 k m z k ρ
g e , d z = i 4 π 0 d k ρ J 0 ( k ρ ρ ) ( z F TE ε m ε n z F TM ) 1 k m z k ρ
g e , d d = i 4 π 0 d k ρ J 0 ( k ρ ρ ) ( F TE + z z k n m 2 F TM ) 1 k m z k ρ .
f j ( r ) , 𝒦 H ( r , r ) , f i ( r ) = μ m μ n f j ( r ) , g c e , d s ( r , r ) , f i s ( r ) + μ m μ n z ^ f j ( r ) , g c e , z s ( r , r ) , f i s ( r ) + μ m μ n f j s ( r ) , g c e , s d ( r , r ) , f i ( r ) + μ m μ n f j s ( r ) , g c e , s z ( r , r ) , z ^ f i ( r )
g c e , d s ( r , r ) = [ sin ϕ cos ϕ ] i 4 π 0 d k ρ J 1 ( k ρ ρ ) z F TE k m z
g c e , z s ( r , r ) = k n 2 [ sin ϕ cos ϕ ] i 4 π 0 d k ρ J 1 ( k ρ ρ ) F TE k m z
g c e , s d ( r , r ) = μ n μ m [ sin ϕ cos ϕ ] i 4 π 0 d k ρ J 1 ( k ρ ρ ) z F TM k m z
g c e , s z ( r , r ) = k m n 2 [ sin ϕ cos ϕ ] i 4 π 0 d k ρ J 1 ( k ρ ρ ) F TM k m z
Im [ G ¯ α α ( r 0 , r 0 ) ] = Im E sca α ( r 0 ) i ω μ ( r 0 ) + Im E inc , sec α ( r 0 ) i ω μ ( r 0 ) + k m 6 π
g ( ρ ) = i 4 π 0 d k ρ k ρ k m z J 0 ( k ρ ρ ) g ˜ ( k ρ ) .
g ˜ ( k ρ ) = i = 1 M a i e i k m z b i
e i k m r r = i 0 d k ρ k ρ k m z J 0 ( k ρ ρ ) e i k m z | z | , r = ρ 2 + z 2
g ( ρ ) = i = 1 M a i e i k m r i 4 π r i , r i = ρ 2 + b i 2 .
1 ρ [ e i k m | z | | z | e i k m r r ] = 0 d k ρ J 1 ( k ρ ρ ) e i k m z | z |

Metrics