Abstract

We present a surface integral equation (SIE) to model the electromagnetic behavior of metallic objects at optical frequencies. The electric and magnetic current combined field integral equation considering both tangential and normal equations is applied. The SIE is solved by using a method-of-moments (MoM) formulation. The SIE-MoM approach is applied only on the material boundary surfaces and interfaces, avoiding the cumbersome volumetric discretization of the objects and the surrounding space required in differential-equation formulations. Some canonical examples have been analyzed, and the results have been compared with analytical reference solutions in order to prove the accuracy of the proposed method. Finally, two plasmonic Yagi–Uda nanoantennas have been analyzed, illustrating the applicability of the method to the solution of real plasmonic problems.

© 2011 Optical Society of America

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    [CrossRef]
  2. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
    [CrossRef] [PubMed]
  3. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  4. J.-J. Greffet, “Nanoantennas for light emission,” Science 308, 1561–1563 (2005).
    [CrossRef] [PubMed]
  5. M. L. Brongersma, “Plasmonics: engineering optical nanoantennas,” Nat. Photon. 2, 270–273 (2008).
    [CrossRef]
  6. P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1, 438–483 (2009).
    [CrossRef]
  7. T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “λ/4 Resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. 7, 28–33(2007).
    [CrossRef] [PubMed]
  8. P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15, 14266–14274 (2007).
    [CrossRef] [PubMed]
  9. F. Tam, G. P. Goodrich, B. R. Johnson, and N. J. Halas, “Plasmonic enhancement of molecular fluorescence,” Nano Lett. 7, 496–501 (2007).
    [CrossRef] [PubMed]
  10. K. Sendur and W. Challener, “Near-field radiation of bow-tie antennas and apertures at optical frequencies,” J. Microsc. 210, 279–283 (2003).
    [CrossRef] [PubMed]
  11. D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nanoantennas resonant in the visible,” Nano Lett. 4, 957–961 (2004).
    [CrossRef]
  12. P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1608(2005).
    [CrossRef] [PubMed]
  13. J. Aizpurua, G. W. Bryant, L. J. Richter, and F. J. García de Abajo, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71, 235420 (2005).
    [CrossRef]
  14. A. Sundaramurthy, P. J. Schuck, N. R. Conley, D. P. Fromm, G. S. Kino, and W. E. Moerner, “Toward nanometer-scale optical photolithography: utilizing the near-field of bowtie optical nanoantennas,” Nano Lett. 6, 355–360 (2006).
    [CrossRef] [PubMed]
  15. O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Optical scattering resonances of single and coupled dimer plasmonic nanoantennas,” Opt. Express 15, 17736–17746(2007).
    [CrossRef] [PubMed]
  16. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Single emitters coupled to plasmonic nano-antennas: angular emission and collection efficiency,” New J. Phys. 10, 105005 (2008).
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  19. T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi–Uda antenna,” Opt. Express 16, 10858–10866(2008).
    [CrossRef] [PubMed]
  20. T. Kosako, Y. Kadoya, and H. F. Hofmann, “Directional control of light by a nano-optical Yagi–Uda antenna,” Nat. Photon. 4, 312–315 (2010).
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  21. 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 329, 930–933 (2010).
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  28. Y. Chang and R. F. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag. 25, 789–795 (1977).
    [CrossRef]
  29. T. K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709–718 (1977).
    [CrossRef]
  30. 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]
  31. 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 27, 2261–2271 (2010).
    [CrossRef]
  32. M. S. Yeung, “Single integral equation for electromagnetic scattering by three-dimensional dielectric objects,” IEEE Trans. Antennas Propag. 47, 1615–1622 (1999).
    [CrossRef]
  33. P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40, RS6002 (2005).
    [CrossRef]
  34. S. M. Rao and D. R. Wilton, “E-field, H-field, and combined field solution for arbitrarily shaped three-dimensional dielectric bodies,” Electromagnetics 10, 407–421 (1990).
    [CrossRef]
  35. K. C. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003).
    [CrossRef]
  36. P. Ylä-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag. 53, 1168–1173 (2005).
    [CrossRef]
  37. P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general integral equation method for general composite metallic and dielectric structures with junctions,” PIER 52, 81–108 (2005).
    [CrossRef]
  38. Ö. Ergül and L. Gürel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 57, 176–187 (2009).
    [CrossRef]
  39. J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express 18, 15876–15886(2010).
    [CrossRef] [PubMed]
  40. J. M. Taboada, L. Landesa, F. Obelleiro, J. L. Rodriguez, J. M. Bertolo, M. G. Araujo, J. C. Mouriño, and A. Gomez, “High scalability FMM-FFT electromagnetic solver for supercomputer systems,” IEEE Antennas Propag. Mag. 51 (6), 20–28 (2009).
    [CrossRef]
  41. M. G. Araújo, J. M. Taboada, F. Obelleiro, J. M. Bértolo, L. Landesa, J. Rivero, and J. L. Rodríguez, “Supercomputer aware approach for the solution of challenging electromagnetic problems,” PIER 101, 241–256 (2010).
    [CrossRef]
  42. J. M. Taboada, M. G. Araújo, J. M. Bértolo, L. Landesa, F. Obelleiro, and J. L. Rodríguez, “MLFMA-FFT parallel algorithm for the solution of large-scale problems in electromagnetics,” PIER 105, 15–30 (2010), invited paper.
    [CrossRef]
  43. S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
    [CrossRef]
  44. D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281(1984).
    [CrossRef]
  45. R. E. Hodges and Y. Rahmat-Samii, “The evaluation of MFIE integrals with the use of vector triangle basis functions,” Microw. Opt. Technol. Lett. 14, 9–14 (1997).
    [CrossRef]
  46. R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green’s function or its gradient on a plane triangle,” IEEE Trans. Antennas Propag. 41, 1448–1455(1993).
    [CrossRef]
  47. P. Ylä-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n^×RWG functions,” IEEE Trans. Antennas Propag. 51, 1837–1846 (2003).
    [CrossRef]
  48. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  49. C. A. Balanis, Antenna Theory: Analysis and Design (Wiley & Sons, 1982).
  50. A. Koenderink, “Plasmon nanoparticle array waveguides for single photon and single plasmon sources,” Nano Lett. 9, 4228–4233 (2009).
    [CrossRef] [PubMed]
  51. B. Stout, A. Devilez, B. Rolly, and N. Bonod, “Multipole methods for nano-antennas design: applications to Yagi–Uda configurations,” J. Opt. Soc. Am. B 28, 1213–1223 (2011).
    [CrossRef]
  52. A. Devilez, N. Bonod, and B. Stout, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano 4, 3390–3396 (2010).
    [CrossRef] [PubMed]

2011 (1)

2010 (9)

A. Devilez, N. Bonod, and B. Stout, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano 4, 3390–3396 (2010).
[CrossRef] [PubMed]

J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express 18, 15876–15886(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. A 27, 2261–2271 (2010).
[CrossRef]

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef] [PubMed]

T. Kosako, Y. Kadoya, and H. F. Hofmann, “Directional control of light by a nano-optical Yagi–Uda antenna,” Nat. Photon. 4, 312–315 (2010).
[CrossRef]

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 329, 930–933 (2010).
[CrossRef] [PubMed]

J. Dorfmüller, R. Vogelgesang, W. Khunsin, C. Rockstuhl, C. Etrich, and K. Kern, “Plasmonic nanowire antennas: experiment, simulation, and theory,” Nano Lett. 10, 3596–3603 (2010).
[CrossRef] [PubMed]

M. G. Araújo, J. M. Taboada, F. Obelleiro, J. M. Bértolo, L. Landesa, J. Rivero, and J. L. Rodríguez, “Supercomputer aware approach for the solution of challenging electromagnetic problems,” PIER 101, 241–256 (2010).
[CrossRef]

J. M. Taboada, M. G. Araújo, J. M. Bértolo, L. Landesa, F. Obelleiro, and J. L. Rodríguez, “MLFMA-FFT parallel algorithm for the solution of large-scale problems in electromagnetics,” PIER 105, 15–30 (2010), invited paper.
[CrossRef]

2009 (5)

Ö. Ergül and L. Gürel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 57, 176–187 (2009).
[CrossRef]

J. M. Taboada, L. Landesa, F. Obelleiro, J. L. Rodriguez, J. M. Bertolo, M. G. Araujo, J. C. Mouriño, and A. Gomez, “High scalability FMM-FFT electromagnetic solver for supercomputer systems,” IEEE Antennas Propag. Mag. 51 (6), 20–28 (2009).
[CrossRef]

A. Koenderink, “Plasmon nanoparticle array waveguides for single photon and single plasmon sources,” Nano Lett. 9, 4228–4233 (2009).
[CrossRef] [PubMed]

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]

P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon. 1, 438–483 (2009).
[CrossRef]

2008 (3)

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi–Uda antenna,” Opt. Express 16, 10858–10866(2008).
[CrossRef] [PubMed]

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Single emitters coupled to plasmonic nano-antennas: angular emission and collection efficiency,” New J. Phys. 10, 105005 (2008).
[CrossRef]

M. L. Brongersma, “Plasmonics: engineering optical nanoantennas,” Nat. Photon. 2, 270–273 (2008).
[CrossRef]

2007 (6)

F. Tam, G. P. Goodrich, B. R. Johnson, and N. J. Halas, “Plasmonic enhancement of molecular fluorescence,” Nano Lett. 7, 496–501 (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. B 76, 245403 (2007).
[CrossRef]

H. F. Hofmann, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical Yagi–Uda antenna,” New J. Phys. 9, 217(2007).
[CrossRef]

T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “λ/4 Resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. 7, 28–33(2007).
[CrossRef] [PubMed]

P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express 15, 14266–14274 (2007).
[CrossRef] [PubMed]

O. L. Muskens, V. Giannini, J. A. Sánchez-Gil, and J. Gómez Rivas, “Optical scattering resonances of single and coupled dimer plasmonic nanoantennas,” Opt. Express 15, 17736–17746(2007).
[CrossRef] [PubMed]

2006 (1)

A. Sundaramurthy, P. J. Schuck, N. R. Conley, D. P. Fromm, G. S. Kino, and W. E. Moerner, “Toward nanometer-scale optical photolithography: utilizing the near-field of bowtie optical nanoantennas,” Nano Lett. 6, 355–360 (2006).
[CrossRef] [PubMed]

2005 (7)

J.-J. Greffet, “Nanoantennas for light emission,” Science 308, 1561–1563 (2005).
[CrossRef] [PubMed]

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1608(2005).
[CrossRef] [PubMed]

J. Aizpurua, G. W. Bryant, L. J. Richter, and F. J. García de Abajo, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71, 235420 (2005).
[CrossRef]

P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40, RS6002 (2005).
[CrossRef]

P. Ylä-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag. 53, 1168–1173 (2005).
[CrossRef]

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general integral equation method for general composite metallic and dielectric structures with junctions,” PIER 52, 81–108 (2005).
[CrossRef]

2004 (1)

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nanoantennas resonant in the visible,” Nano Lett. 4, 957–961 (2004).
[CrossRef]

2003 (3)

K. Sendur and W. Challener, “Near-field radiation of bow-tie antennas and apertures at optical frequencies,” J. Microsc. 210, 279–283 (2003).
[CrossRef] [PubMed]

K. C. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003).
[CrossRef]

P. Ylä-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n^×RWG functions,” IEEE Trans. Antennas Propag. 51, 1837–1846 (2003).
[CrossRef]

1999 (1)

M. S. Yeung, “Single integral equation for electromagnetic scattering by three-dimensional dielectric objects,” IEEE Trans. Antennas Propag. 47, 1615–1622 (1999).
[CrossRef]

1997 (1)

R. E. Hodges and Y. Rahmat-Samii, “The evaluation of MFIE integrals with the use of vector triangle basis functions,” Microw. Opt. Technol. Lett. 14, 9–14 (1997).
[CrossRef]

1993 (1)

R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green’s function or its gradient on a plane triangle,” IEEE Trans. Antennas Propag. 41, 1448–1455(1993).
[CrossRef]

1990 (1)

S. M. Rao and D. R. Wilton, “E-field, H-field, and combined field solution for arbitrarily shaped three-dimensional dielectric bodies,” Electromagnetics 10, 407–421 (1990).
[CrossRef]

1984 (1)

D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281(1984).
[CrossRef]

1982 (1)

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

1977 (3)

T. Weiland, “A discretization method for the solution of Maxwell’s equations for six-component fields,” AEU Arch. Elektron. Übertragungstech. 31, 116–120 (1977).

Y. Chang and R. F. 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 arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci. 12, 709–718 (1977).
[CrossRef]

1975 (1)

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
[CrossRef]

1972 (1)

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

Aizpurua, J.

J. Aizpurua, G. W. Bryant, L. J. Richter, and F. J. García de Abajo, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71, 235420 (2005).
[CrossRef]

Al-Bundak, O. M.

D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281(1984).
[CrossRef]

Araujo, M. G.

J. M. Taboada, L. Landesa, F. Obelleiro, J. L. Rodriguez, J. M. Bertolo, M. G. Araujo, J. C. Mouriño, and A. Gomez, “High scalability FMM-FFT electromagnetic solver for supercomputer systems,” IEEE Antennas Propag. Mag. 51 (6), 20–28 (2009).
[CrossRef]

Araújo, M. G.

M. G. Araújo, J. M. Taboada, F. Obelleiro, J. M. Bértolo, L. Landesa, J. Rivero, and J. L. Rodríguez, “Supercomputer aware approach for the solution of challenging electromagnetic problems,” PIER 101, 241–256 (2010).
[CrossRef]

J. M. Taboada, M. G. Araújo, J. M. Bértolo, L. Landesa, F. Obelleiro, and J. L. Rodríguez, “MLFMA-FFT parallel algorithm for the solution of large-scale problems in electromagnetics,” PIER 105, 15–30 (2010), invited paper.
[CrossRef]

Atwater, H. A.

S. A. Maier and H. A. Atwater, “Plasmonics: localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Antenna Theory: Analysis and Design (Wiley & Sons, 1982).

Barnard, E. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef] [PubMed]

Bertolo, J. M.

J. M. Taboada, L. Landesa, F. Obelleiro, J. L. Rodriguez, J. M. Bertolo, M. G. Araujo, J. C. Mouriño, and A. Gomez, “High scalability FMM-FFT electromagnetic solver for supercomputer systems,” IEEE Antennas Propag. Mag. 51 (6), 20–28 (2009).
[CrossRef]

Bértolo, J. M.

M. G. Araújo, J. M. Taboada, F. Obelleiro, J. M. Bértolo, L. Landesa, J. Rivero, and J. L. Rodríguez, “Supercomputer aware approach for the solution of challenging electromagnetic problems,” PIER 101, 241–256 (2010).
[CrossRef]

J. M. Taboada, M. G. Araújo, J. M. Bértolo, L. Landesa, F. Obelleiro, and J. L. Rodríguez, “MLFMA-FFT parallel algorithm for the solution of large-scale problems in electromagnetics,” PIER 105, 15–30 (2010), invited paper.
[CrossRef]

Bharadwaj, P.

Bonod, N.

B. Stout, A. Devilez, B. Rolly, and N. Bonod, “Multipole methods for nano-antennas design: applications to Yagi–Uda configurations,” J. Opt. Soc. Am. B 28, 1213–1223 (2011).
[CrossRef]

A. Devilez, N. Bonod, and B. Stout, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano 4, 3390–3396 (2010).
[CrossRef] [PubMed]

Brodwin, M. E.

A. Taflove and M. E. Brodwin, “Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell’s equations,” IEEE Trans. Microwave Theory Tech. 23, 623–630 (1975).
[CrossRef]

Brongersma, M. L.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef] [PubMed]

M. L. Brongersma, “Plasmonics: engineering optical nanoantennas,” Nat. Photon. 2, 270–273 (2008).
[CrossRef]

Bryant, G. W.

J. Aizpurua, G. W. Bryant, L. J. Richter, and F. J. García de Abajo, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71, 235420 (2005).
[CrossRef]

Butler, C. M.

D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281(1984).
[CrossRef]

Cai, W.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef] [PubMed]

Challener, W.

K. Sendur and W. Challener, “Near-field radiation of bow-tie antennas and apertures at optical frequencies,” J. Microsc. 210, 279–283 (2003).
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K. C. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003).
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J. M. Taboada, L. Landesa, F. Obelleiro, J. L. Rodriguez, J. M. Bertolo, M. G. Araujo, J. C. Mouriño, and A. Gomez, “High scalability FMM-FFT electromagnetic solver for supercomputer systems,” IEEE Antennas Propag. Mag. 51 (6), 20–28 (2009).
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F. Tam, G. P. Goodrich, B. R. Johnson, and N. J. Halas, “Plasmonic enhancement of molecular fluorescence,” Nano Lett. 7, 496–501 (2007).
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F. Tam, G. P. Goodrich, B. R. Johnson, and N. J. Halas, “Plasmonic enhancement of molecular fluorescence,” Nano Lett. 7, 496–501 (2007).
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K. C. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003).
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F. Tam, G. P. Goodrich, B. R. Johnson, and N. J. Halas, “Plasmonic enhancement of molecular fluorescence,” Nano Lett. 7, 496–501 (2007).
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P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
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J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
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H. F. Hofmann, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical Yagi–Uda antenna,” New J. Phys. 9, 217(2007).
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J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express 18, 15876–15886(2010).
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J. M. Taboada, L. Landesa, F. Obelleiro, J. L. Rodriguez, J. M. Bertolo, M. G. Araujo, J. C. Mouriño, and A. Gomez, “High scalability FMM-FFT electromagnetic solver for supercomputer systems,” IEEE Antennas Propag. Mag. 51 (6), 20–28 (2009).
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A. Sundaramurthy, P. J. Schuck, N. R. Conley, D. P. Fromm, G. S. Kino, and W. E. Moerner, “Toward nanometer-scale optical photolithography: utilizing the near-field of bowtie optical nanoantennas,” Nano Lett. 6, 355–360 (2006).
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D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nanoantennas resonant in the visible,” Nano Lett. 4, 957–961 (2004).
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P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1608(2005).
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Novotny, L.

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M. G. Araújo, J. M. Taboada, F. Obelleiro, J. M. Bértolo, L. Landesa, J. Rivero, and J. L. Rodríguez, “Supercomputer aware approach for the solution of challenging electromagnetic problems,” PIER 101, 241–256 (2010).
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J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express 18, 15876–15886(2010).
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J. M. Taboada, M. G. Araújo, J. M. Bértolo, L. Landesa, F. Obelleiro, and J. L. Rodríguez, “MLFMA-FFT parallel algorithm for the solution of large-scale problems in electromagnetics,” PIER 105, 15–30 (2010), invited paper.
[CrossRef]

J. M. Taboada, L. Landesa, F. Obelleiro, J. L. Rodriguez, J. M. Bertolo, M. G. Araujo, J. C. Mouriño, and A. Gomez, “High scalability FMM-FFT electromagnetic solver for supercomputer systems,” IEEE Antennas Propag. Mag. 51 (6), 20–28 (2009).
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A. J. Poggio and E. K. Miller, Computer Techniques for Electromagnetics (Pergamon, 1973).

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P. Mühlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1608(2005).
[CrossRef] [PubMed]

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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 329, 930–933 (2010).
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R. E. Hodges and Y. Rahmat-Samii, “The evaluation of MFIE integrals with the use of vector triangle basis functions,” Microw. Opt. Technol. Lett. 14, 9–14 (1997).
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S. M. Rao and D. R. Wilton, “E-field, H-field, and combined field solution for arbitrarily shaped three-dimensional dielectric bodies,” Electromagnetics 10, 407–421 (1990).
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D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281(1984).
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S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
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J. Aizpurua, G. W. Bryant, L. J. Richter, and F. J. García de Abajo, “Optical properties of coupled metallic nanorods for field-enhanced spectroscopy,” Phys. Rev. B 71, 235420 (2005).
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Rivas, J. Gómez

Rivero, J.

M. G. Araújo, J. M. Taboada, F. Obelleiro, J. M. Bértolo, L. Landesa, J. Rivero, and J. L. Rodríguez, “Supercomputer aware approach for the solution of challenging electromagnetic problems,” PIER 101, 241–256 (2010).
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J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express 18, 15876–15886(2010).
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Rockstuhl, C.

J. Dorfmüller, R. Vogelgesang, W. Khunsin, C. Rockstuhl, C. Etrich, and K. Kern, “Plasmonic nanowire antennas: experiment, simulation, and theory,” Nano Lett. 10, 3596–3603 (2010).
[CrossRef] [PubMed]

Rodriguez, J. L.

J. M. Taboada, L. Landesa, F. Obelleiro, J. L. Rodriguez, J. M. Bertolo, M. G. Araujo, J. C. Mouriño, and A. Gomez, “High scalability FMM-FFT electromagnetic solver for supercomputer systems,” IEEE Antennas Propag. Mag. 51 (6), 20–28 (2009).
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Rodríguez, J. L.

M. G. Araújo, J. M. Taboada, F. Obelleiro, J. M. Bértolo, L. Landesa, J. Rivero, and J. L. Rodríguez, “Supercomputer aware approach for the solution of challenging electromagnetic problems,” PIER 101, 241–256 (2010).
[CrossRef]

J. M. Taboada, M. G. Araújo, J. M. Bértolo, L. Landesa, F. Obelleiro, and J. L. Rodríguez, “MLFMA-FFT parallel algorithm for the solution of large-scale problems in electromagnetics,” PIER 105, 15–30 (2010), invited paper.
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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. B 76, 245403 (2007).
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D. R. Wilton, S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281(1984).
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A. Sundaramurthy, P. J. Schuck, N. R. Conley, D. P. Fromm, G. S. Kino, and W. E. Moerner, “Toward nanometer-scale optical photolithography: utilizing the near-field of bowtie optical nanoantennas,” Nano Lett. 6, 355–360 (2006).
[CrossRef] [PubMed]

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nanoantennas resonant in the visible,” Nano Lett. 4, 957–961 (2004).
[CrossRef]

Schuller, J. A.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9, 193–204 (2010).
[CrossRef] [PubMed]

Segerink, F. B.

T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “λ/4 Resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. 7, 28–33(2007).
[CrossRef] [PubMed]

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K. Sendur and W. Challener, “Near-field radiation of bow-tie antennas and apertures at optical frequencies,” J. Microsc. 210, 279–283 (2003).
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T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Single emitters coupled to plasmonic nano-antennas: angular emission and collection efficiency,” New J. Phys. 10, 105005 (2008).
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T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi–Uda antenna,” Opt. Express 16, 10858–10866(2008).
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B. Stout, A. Devilez, B. Rolly, and N. Bonod, “Multipole methods for nano-antennas design: applications to Yagi–Uda configurations,” J. Opt. Soc. Am. B 28, 1213–1223 (2011).
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A. Devilez, N. Bonod, and B. Stout, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano 4, 3390–3396 (2010).
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Sundaramurthy, A.

A. Sundaramurthy, P. J. Schuck, N. R. Conley, D. P. Fromm, G. S. Kino, and W. E. Moerner, “Toward nanometer-scale optical photolithography: utilizing the near-field of bowtie optical nanoantennas,” Nano Lett. 6, 355–360 (2006).
[CrossRef] [PubMed]

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nanoantennas resonant in the visible,” Nano Lett. 4, 957–961 (2004).
[CrossRef]

Taboada, J. M.

M. G. Araújo, J. M. Taboada, F. Obelleiro, J. M. Bértolo, L. Landesa, J. Rivero, and J. L. Rodríguez, “Supercomputer aware approach for the solution of challenging electromagnetic problems,” PIER 101, 241–256 (2010).
[CrossRef]

J. Rivero, J. M. Taboada, L. Landesa, F. Obelleiro, and I. García-Tuñón, “Surface integral equation formulation for the analysis of left-handed metamaterials,” Opt. Express 18, 15876–15886(2010).
[CrossRef] [PubMed]

J. M. Taboada, M. G. Araújo, J. M. Bértolo, L. Landesa, F. Obelleiro, and J. L. Rodríguez, “MLFMA-FFT parallel algorithm for the solution of large-scale problems in electromagnetics,” PIER 105, 15–30 (2010), invited paper.
[CrossRef]

J. M. Taboada, L. Landesa, F. Obelleiro, J. L. Rodriguez, J. M. Bertolo, M. G. Araujo, J. C. Mouriño, and A. Gomez, “High scalability FMM-FFT electromagnetic solver for supercomputer systems,” IEEE Antennas Propag. Mag. 51 (6), 20–28 (2009).
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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,” Science 329, 930–933 (2010).
[CrossRef] [PubMed]

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Single emitters coupled to plasmonic nano-antennas: angular emission and collection efficiency,” New J. Phys. 10, 105005 (2008).
[CrossRef]

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi–Uda antenna,” Opt. Express 16, 10858–10866(2008).
[CrossRef] [PubMed]

T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “λ/4 Resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. 7, 28–33(2007).
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P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40, RS6002 (2005).
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P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface integral equation method for general integral equation method for general composite metallic and dielectric structures with junctions,” PIER 52, 81–108 (2005).
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P. Ylä-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag. 53, 1168–1173 (2005).
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Supplementary Material (1)

» Media 1: AVI (2731 KB)     

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Figures (6)

Fig. 1
Fig. 1

Bistatic RCS of a gold nanosphere with a radius of 200 nm at a wavelength of 550 nm ( ε r = 8.0 1.66 j ).

Fig. 2
Fig. 2

Surface mesh composed of 1620 flat triangular facets of the Yagi–Uda antenna embedded in glass ( ε r = 2.25 ) of [50]. The antenna is made of spherical silver nanoparticles with ε r = 20.09 0.45 j [48] at an operating wavelength of 650 nm . Note that only the metal– dielectric interfaces must be modeled.

Fig. 3
Fig. 3

Directivity of the Yagi–Uda antenna of [50] in dBi for a near-field coupled Hertzian dipole emitter: (a) H plane and (b) E plane.

Fig. 4
Fig. 4

Surface mesh composed of 3012 flat triangular facets for the analysis of the Yagi–Uda antenna of [19] with the proposed JMCFIE-MoM formulation. The antenna is made of aluminum, and it is optimized for an operating wavelength of 570 nm . The relative permittivity constant of aluminum at this wavelength is ε r = 38.0 10.9 j . Note that only the metal–dielectric interfaces must be modeled. Inset, detail of the 4 nm gap introduced to the feed element to analyze the usability of the antenna for field enhancement.

Fig. 5
Fig. 5

Directivity of the Yagi–Uda antenna of [19] in dBi for a near-field coupled Hertzian dipole emitter: (a) H plane and (b) E plane.

Fig. 6
Fig. 6

We show (Media 1, 1744 K ) the total near-electric-field distribution in the vertical and horizontal planes crossing the Yagi–Uda antenna of [19] versus the impinging direction, for an elevation angular sweep from θ inc = 0 ° to 180 ° ( ϕ inc = 0 ° ). Media 1 illustrates the gap confinement of light and the directional field enhancement provided by the Yagi–Uda antenna. The electric field distribution is shown both inside and outside the metallic nanoelements. Dimensions are in nanometers.

Equations (47)

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T - EFIE i : j ( η i L i ( J i j ) K i ( M i j ) ) tan + 1 2 n ^ i j × M i j = ( E i inc ) tan ,
T - MFIE i : j ( K i ( J i j ) + 1 η i L i ( M i j ) ) tan 1 2 n ^ i j × J i j = ( H i inc ) tan ,
N - EFIE i : n ^ i j × j ( η i L i ( J i j ) K i ( M i j ) ) 1 2 M i j = n ^ i j × E i inc ,
N - MFIE i : n ^ i j × j ( K i ( J i j ) + 1 η i L i ( M i j ) ) + 1 2 J i j = n ^ i j × H i inc .
L i ( X i j ) = j k i [ S X i j ( r ) G i ( r , r ) d S + 1 k i 2 S · X i j G i ( r , r ) d S ] ,
K i ( X i j ) = S , PV X i j ( r ) × G i ( r , r ) d S ,
G i ( r , r ) = e j k i | r r | 4 π | r r | ,
J i j ( r ) = n ^ i j × H i ( r ) ,
M i j ( r ) = n ^ i j × E i ( r ) ,
JCFIE i = a i 1 η i T - EFIE i + b i N - MFIE i ,
MCFIE i = c i N - EFIE i + d i η i T - MFIE i ,
JCFIE i j : a i j ( L i ( J i j ) 1 η i K i ( M i j ) ) tan + a j j ( L j ( J j j ) 1 η j K j ( M j j ) ) tan + b i n ^ i j × j ( K i ( J i j ) + 1 η i L i ( M i j ) ) + b j n ^ j i × j ( K j ( J j j ) + 1 η j L j ( M j j ) ) + 1 2 ( a i η i n ^ i j × M i j + a j η j n ^ j i × M j i + b i J i j + b j J j i ) = a i ( 1 η i E i inc ) tan + a j ( 1 η j E j inc ) tan + b i n ^ i j × H i inc + b j n ^ j i × H j inc , r S i j .
MCFIE i j : c i n ^ i j × j ( η i L i ( J i j ) K i ( M i j ) ) c j n ^ j i × j ( η j L j ( J j j ) K j ( M j j ) ) + d i j ( η i K i ( J i j ) + L i ( M i j ) ) tan + d j j ( η j K j ( J j j ) + L j ( M j j ) ) tan + 1 2 ( c i M i j + c j M j i d i η i n ^ i j × J i j d j η j n ^ j i × J j i ) = c i n ^ i j × E i inc + c j n ^ j i × E j inc d i ( η i H i inc ) tan d j ( η j H j inc ) tan , r S i j .
J i j = n J n f n ; r S i j ,
M i j = n M n f n ; r S i j ,
a i j s i j n S i j ( A m n i J n 1 η i B m n i M n ) a j j s j j n S j j ( A m n j J n 1 η j B m n j M n ) + b i j s i j n S i j ( B m n i J n + 1 η i A m n i M n ) + b j j s j j n S j j ( B m n j J n + 1 η j A m n j M n ) + 1 2 n S i j [ ( a i η i a j η j ) I m n M n ] + 1 2 n S i j [ ( b i + b j ) I m n J n ] = E m i , j + H m i , j , m S i j ,
c i j s i j n S i j ( η i A m n i J n B m n i M n ) c j j s j j n S j j ( η j A m n j J n B m n j M n ) + d i j s i j n S i j ( η i B m n i J n + A m n i M n ) d j j s j j n S j j ( η j B m n j J n + A m n j M n ) + 1 2 n S i j [ ( c i + c j ) I m n M n ] 1 2 n S i j [ ( d i η i d j η j ) I m n J n ] = E m i , j + H m i , j , m S i j ,
A m n i = Δ m f m · L i ( f n ) d S ,
B m n i = Δ m f m · K i ( f n ) d S ,
A m n i = Δ m f m · n ^ m × L i ( f n ) d S ,
B m n i = Δ m f m · n ^ m × K i ( f n ) d S ,
E m i , j = Δ m f m · [ a i ( 1 η i E i inc ) tan a j ( 1 η j E j inc ) tan ] d S ,
H m i , j = Δ m f m · [ d i ( η i H i inc ) tan d j ( η j H j inc ) tan ] d S ,
H m i , j = Δ m f m · [ c i n ^ m × E i inc + c j n ^ m × E j inc ] d S ,
H m i , j = Δ m f m · [ b i n ^ m × H i inc + b j n ^ m × H j inc ] d S ,
I m n = Δ m f m · f n d S ,
I m n = Δ m f m · n ^ m × f n d S .
s i j = { + 1 , i < j 1 , i > j
[ Z ¯ i j , i j Z ¯ i j , k l Z ¯ i j , p q Z ¯ k l , i j Z ¯ k l , k l Z ¯ k l , p q Z ¯ p q , i j Z ¯ p q , k l Z ¯ p q , p q ] · [ I i j I k l I p q ] = [ V i j V k l V p q ] .
Z ¯ i j , i j = [ Z ¯ i j , i j 1 J Z ¯ i j , i j 1 M Z ¯ i j , i j 2 J Z ¯ i j , i j 2 M ] ,
Z ¯ i j , i j 1 J [ m , n ] = a i A m n i + a j A m n j + b i B m n i b j B m n j + 1 2 ( b i + b j ) I m n ,
Z ¯ i j , i j 1 M [ m , n ] = a i η i B m n i a j η j B m n j + b i η i A m n i b j η j A m n j + 1 2 ( a i η i a j η j ) I m n ,
Z ¯ i j , i j 2 J [ m , n ] = d i η i B m n i + d j η j B m n j c i η i A m n i + c j η j A m n j 1 2 ( d i η i d j η j ) I m n ,
Z ¯ i j , i j 2 M [ m , n ] = d i A m n i + d j A m n j + c i B m n i c j B m n j + c i + c j 2 I m n ,
Z ¯ i j , i j 1 J [ m , n ] = s i j a i A m n i + s i j b i B m n i ,
Z ¯ i j , i j 1 M [ m , n ] = s i j a i η i B m n i + s i j b i η i A m n i ,
Z ¯ i j , i j 2 J [ m , n ] = s i j d i η i B m n i s i j c i η i A m n i ,
Z ¯ i j , i j 2 M [ m , n ] = s i j d i A m n i + s i j c i B m n i ,
Z ¯ i j , j j 1 J [ m , n ] = s j j a j A m n j + s j j b j B m n j ,
Z ¯ i j , j j 1 M [ m , n ] = s j j a j η j B m n j + s j j b j η j A m n j ,
Z ¯ i j , j j 2 J [ m , n ] = s j j d j η j B m n j s i j c j η j A m n j ,
Z ¯ i j , j j 2 M [ m , n ] = s j j d j A m n j + s j j c j B m n j ,
I i j = [ J i j M i j ] ,
V i j = [ V i j 1 V i j 2 ] ,
V i j 1 [ m ] = E m i , j + H m i , j ,
V i j 2 [ m ] = E m i , j + H m i , j .
k i = ω μ i ε i ; η i = μ i ε i .

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