## Study on spontaneous emission in complex multilayered plasmonic system via surface integral equation approach with layered medium Green’s function |

Optics Express, Vol. 20, Issue 18, pp. 20210-20221 (2012)

http://dx.doi.org/10.1364/OE.20.020210

Acrobat PDF (2893 KB)

### 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

## 1. Introduction

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]

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. B **76**, 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]

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]

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. A **26**, 732–740 (2009). [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. Express **20**, 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]

## 2. Theoretical principles

### 2.1. Green’s function approach in spontaneous emission

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]

### 2.2. Surface integral equation with layered medium Green’s function

**E**/

**H**) generated by equivalent electric/magnetic currents (

**J**/

**M**) [17] where We only discuss the electric-type LMGF

**G**̄

*in Eq. (11) and Eq. (12) here since the magnetic-type LMGF*

_{e}**G**̄

*in Eq. (13) and Eq. (14) can be easily obtained via the duality principle [27]. The*

_{m}**G**̄

*is defined as where*

_{e}*m*and

*n*are the layer indices of the source and observation points, and Here

*F*

^{TE/TM}(

*k*,

_{ρ}*z*,

*z*′) is the propagation factor in the layered medium [27],

*J*

_{0}(

*k*) is the zeroth order Bessel function.

_{ρ}ρ28. X. W. Chen, M. Agio, and V. Sandoghdar, “Metallodielectric hybrid antennas for ultrastrong enhancement of spontaneous emission,” Phys. Rev. Lett. **108**, 233001 (2012). [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]

*ℒ*

_{E}is

**f**

*= -*

_{s}*z*̂ ×

*z*̂ ×

**f**is the horizontal projection of the basis function; the inner product is defined as 〈

**f**(

**r**) ·

**g**(

**r**)〉 = ∫

*d*

**rf**(

**r**) ·

**g**(

**r**), and Similarly the matrix representation of

*𝒦*

_{H}operator is

*ϕ*= arctan[(

*y*-

*y*′)/(

*x*-

*x*′)], and

*J*

_{1}(

*k*) is the first order Bessel function.

_{ρ}ρ*α*-polarized Hertzian dipole to excite the structure, and calculate the scattered field at the same location by using the SIE with LMGF. Finally the Green’s function (of the whole hybrid structure) can be deduced as The first term of the right-hand side is from the scattered field, the second term is from the secondary term of the incident field (due to the reflection and transmission of the layered medium), and the third term is from the primary term of the incident field (the field is singular but the imaginary part of Green’s function is regular and has analytic solution).

### 2.3. Evaluation of Sommerfeld integrals via discrete complex image method

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. Magazine **37**, 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]

## 3. Simulation results and discussion

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]

*λ*= 510 nm, the configuration is shown in Fig 2(a), and the near field along the observation line is shown in Fig. 2(b) and Fig. 2(c), compared with the one from MIE series [38

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

*z*-polarized dipole, as shown in Fig. 3(a). In this case, no analytic solution is available. The accuracy is validated by an approximated model, where the substrate is truncated with a dimension of 600 × 600 nm, and the SIE based on FSGF is applied. The real part of the

*z*-component of electric field Re{

*E*} is calculated, which is proportional to the imaginary part of the Green’s function Im{

_{z}*G*}. The agreement of the two results is reasonably good, as shown in Fig. 3(b). After the validation we then calculate the SER of the configuration as shown in Fig. 3(a). The normalized SER with different polarizations of emitters are shown in Fig. 4(a) (

_{zz}*x*-polarized), and Fig. 4(b) (

*z*-polarized), respectively. To clearly demonstrate the effect of the substrate, the results of the nanosphere in air and the substrate (the LMGF itself) are also calculated. It is observed that the vertically polarized dipole excites stronger field than the one from horizontally polarized dipole. This is because strong near-field evanescent wave coupling between sphere and substrate can be expected for

*z*-polarized dipole imposed between the two metallic nanostructures. The localized plasmon from the nanosphere will strongly interact with the surface plasmon from the plate substrate resulting in a strong confinement and large spontaneous decay rate. Particularly, constructive or coherent interferences by evanescent wave coupling in the case of

*z*-polarized dipole makes the normalized SER of the hybrid system stronger than the summation of the nano sphere and the substrate. In Fig. 4(c), the normalized SER versus the distance between the nano particle and the metallic substrate is shown, where the SER decreases monotonically when the distance increases. This suggests that the near-field surface waves from plasmonic resonances contribute to the boosted SE in comparison with leaky waves in the middle or far field regions, which have weaker effects on SE. The near field distribution is also calculated at

*λ*= 510 nm with

*z*-polarized dipole as shown in Fig. 5. In Fig. 5(a), the mesh of the nanosphere is shown; in Fig. 5(b) and Fig. 5(c), the near scattered field of the nanosphere (in logarithmic scale) with and without the substrate is shown for comparison. The singular primary field of the dipole is subtracted for better demonstration. A strong E-field is concentrated at the gap between the nanosphere and substrate.

_{2}substrate. SE or Raman scattering from the molecular material can be amplified by the metallic structure. The dimension of the prism and the substrate is shown in Fig. 6(a), where the refractive indices are:

*n*= 1.49 (PMMA),

*n*= 2 (ITO), and

*n*= 1.47 (SiO

_{2}) [39

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]

*z*direction. The normalized SER (with respect to the one in a homogeneous and unbounded PMMA space) for two different locations of the emitter is shown in Fig. 6(b). Location

*a*is above the center of the prism (0, 0, 10), and location

*b*is around one corner of the prism (-35, -17.32, 0). It is observed that the SER is strongly location dependent, which should be taken into account in the design of the nanoantenna. For location

*a*, there are two resonant peaks while for location

*b*there is only one. The scattered near field distribution at the two resonance frequencies for the case of the location

*a*is shown in Fig. 6(c) and Fig. 6(d), respectively. The primary field of the dipole is also subtracted and again logarithmic scale is adopted. As shown in Fig. 6(c), the near-field of the metallic prism is concentrated at the center region of its cross section at the resonance peak of 520 nm. The supported photonic-like mode contributes the boosted SE. Furthermore, the whole prism is fully excited at the second resonance peak of 680 nm as depicted in Fig. 6(d). The sharp E-field hotspots are focused on the tips of the prism induced by the plasmonic and lightning-rod effects. The dipole, with different positions and orientations, enables distinguished eigenstates of the nanoantenna system excited involving plasmonic and photonic-like modes. As a result, the enhanced SER shows position and polarization dependent features. All the numerical simulations except for the approximated validation model mentioned above are run on a personal computer with 2.66 GHz processor and 2 Gb memory, without the necessity to invoke super computers.

## 4. Conclusion

## Acknowledgments

## References and links

1. | L. Rogobete, F. Kaminski, M. Agio, and V. Sandoghdar, “Design of plasmonic nanoantennae for enhancing spontaneous emission,” Opt. Lett. |

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. |

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,” Nature |

4. | E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. |

5. | C. Gerry and P. Knight, |

6. | L. Novotny and B. Hecht, |

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. |

8. | J.-J. Greffet, “Nanoantennas for light emission,” Science |

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,” Science |

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. |

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. B |

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. |

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. |

14. | W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, |

15. | K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. |

16. | P. Monk, |

17. | W. C. Chew, M. S. Tong, and B. Hu, |

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. A |

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. A |

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. A |

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. Express |

22. | K. A. Michalski and J. R. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propagat. |

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. |

25. | E. N. Economou, |

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. |

27. | W. C. Chew, |

28. | X. W. Chen, M. Agio, and V. Sandoghdar, “Metallodielectric hybrid antennas for ultrastrong enhancement of spontaneous emission,” Phys. Rev. Lett. |

29. | A. J. Poggio and E. K. Miller, “Integral equation solutions of three dimensional scattering problems,” in |

30. | Y. Chang and R. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag. |

31. | T.-K. Wu and L. L. Tsai, “Scattering from arbitrarilyshaped lossy dielectric bodies of revolution,” Radio Sci. |

32. | S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surface of arbitrary shape,” IEEE Trans. Antennas Propagat. |

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. |

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. Magazine |

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. |

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. |

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. |

38. | G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys. |

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. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(260.2110) Physical optics : Electromagnetic optics

(050.1755) Diffraction and gratings : Computational electromagnetic methods

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: July 5, 2012

Revised Manuscript: August 10, 2012

Manuscript Accepted: August 11, 2012

Published: August 20, 2012

**Virtual Issues**

Vol. 7, Iss. 10 *Virtual Journal for Biomedical Optics*

**Citation**

Yongpin P. Chen, Wei E. I. Sha, Wallace C. H. Choy, Lijun Jiang, and Weng Cho Chew, "Study on spontaneous emission in complex multilayered plasmonic system via surface integral equation approach with layered medium Green’s function," Opt. Express **20**, 20210-20221 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-18-20210

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### References

- 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]
- 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]
- 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]
- E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev.69, 681 (1946).
- 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]
- 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]
- 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]
- 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]
- 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]
- W. C. Chew, J. M. Jin, E. Michielssen, and J. M. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, Norwood, 2001).
- 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]
- 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. 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]
- 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]
- 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]
- 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]
- 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]
- 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.
- 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).
- E. N. Economou, Green’s Functions in Quantum Physics (Springer, Berlin, 2006). [PubMed]
- 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, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990; IEEE Press, 1995).
- X. W. Chen, M. Agio, and V. Sandoghdar, “Metallodielectric hybrid antennas for ultrastrong enhancement of spontaneous emission,” Phys. Rev. Lett.108, 233001 (2012). [CrossRef]
- A. J. Poggio and E. K. Miller, “Integral equation solutions of three dimensional scattering problems,” in Computer Techniques for Electromagnetics (Permagon, 1973).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys.25, 377–445 (1908). [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]

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