Abstract

Recent advances in nano-optics elicit a growing need for more efficient numerical treatments of light-matter interaction in stratified backgrounds. While being known for its many favorable properties, usage of the surface integral approach is hindered by numerical difficulties associated with layered-medium Green’s functions. We present an efficient and robust implementation of this approach, addressing the limiting issues. The singularity extraction method is generalized to account for the occurring secondary-term singularities. The resulting scheme thus allows for arbitrary positioning of the scatterers. Further, the laborious matrix-filling process is dramatically accelerated through a simple and robustly-devised, spatial interpolation scheme, completely devoid of integral evaluations. The accuracy and versatility of the method are demonstrated by treating several representative plasmonic problems.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach

Benjamin Gallinet, Andreas M. Kern, and Olivier J. F. Martin
J. Opt. Soc. Am. A 27(10) 2261-2271 (2010)

Light propagation and scattering in stratified media: a Green’s tensor approach

Michael Paulus and Oliver J. F. Martin
J. Opt. Soc. Am. A 18(4) 854-861 (2001)

Efficient evaluation of Sommerfeld integrals for the optical simulation of many scattering particles in planarly layered media

Amos Egel, Siegfried W. Kettlitz, and Uli Lemmer
J. Opt. Soc. Am. A 33(4) 698-706 (2016)

References

  • View by:
  • |
  • |
  • |

  1. R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromag. Waves Appl. 3, 1–15 (1989).
    [Crossref]
  2. R. F. Harrington, Field Computation by Moment Methods (R. E. Krieger, 1968).
  3. 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]
  4. 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]
  5. 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 28, 1341–1348 (2011).
    [Crossref]
  6. M. G. Araújo, J. M. Taboada, D. M. Sols, 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]
  7. D. M. Solís, J. M. Taboada, F. Obelleiro, L. M. Liz-Marzán, and F. J. García de Abajo, “Toward ultimate nanoplas-monics modeling,” ACS Nano 8, 7559–7570 (2014).
    [Crossref]
  8. R. Rodríguez-Oliveros and J. A. Sánchez-Gil, “Localized surface-plasmon resonances on single and coupled nanoparticles through surface integral equations for flexible surfaces,” Opt. Express 19, 12208–12219 (2011).
    [Crossref] [PubMed]
  9. W. C. Chew, M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves (Morgan & Claypool, 2009).
  10. M. Paulus and O. J. F. Martin, “Light propagation and scattering in stratified media: a Green’s tensor approach,” J. Opt. Soc. Am. A 18, 854–861 (2001).
    [Crossref]
  11. J. Alegret, P. Johansson, and M. Käll, “Green’s tensor calculations of plasmon resonances of single holes and hole pairs in thin gold films,” New J. Phys. 10, 105004 (2008).
    [Crossref]
  12. K. A. Michalski and J. R. Mosig, “Multilayered media green’s functions in integral equation formulations,” IEEE Trans. Antennas Propag. 45, 508–519 (1997).
    [Crossref]
  13. 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 Propag. 60, 4766–4776 (2012).
    [Crossref]
  14. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, 1990).
  15. Y. P. Chen, W. E. I. Sha, W. C. H. Choy, L. Jiang, and W. C. 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).
    [Crossref] [PubMed]
  16. Y. P. Chen, W. E. I. Sha, L. Jiang, and J. Hu, “Graphene plasmonics for tuning photon decay rate near metallic split-ring resonator in a multilayered substrate,” Opt. Express 23, 2798–2807 (2015).
    [Crossref] [PubMed]
  17. 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 Wirel. Propag. Lett. 10, 419–422 (2011).
    [Crossref]
  18. 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. Microwave Theory Tech. 58, 602–613 (2010).
    [Crossref]
  19. E. P. Karabulut, A. T. Erdogan, and M. I. Aksun, “Discrete complex image method with spatial error criterion,” IEEE Trans. Microwave Theory Tech. 59, 793–802 (2011).
    [Crossref]
  20. 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]
  21. P. Yla-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]
  22. I. Hanninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Electromag. Res. 63, 243–278 (2006).
    [Crossref]
  23. J. Mosig, “The weighted averages algorithm revisited,” IEEE Trans. Antennas Propag. 60, 2011–2018 (2012).
    [Crossref]
  24. L. N. Medgyesi-Mitschang, J. M. Putnam, and M. B. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Soc. Am. A 11, 1383–1398 (1994).
    [Crossref]
  25. S. M. Rao, D. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
    [Crossref]
  26. W. C. Chew, J. L. Xiong, and M. A. Saville, “A matrix-friendly formulation of layered medium Green’s function,” IEEE Antennas Wireless Propag. Lett. 5, 490–494 (2006).
    [Crossref]
  27. W. C. Chew and S.-Y. Chen, “Response of a point source embedded in a layered medium,” IEEE Antennas Wireless Propag. Lett. 2, 254–258 (2003).
  28. A. Sommerfeld, Partial Differential Equations in Physics (Academic, 1949).
  29. M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
    [Crossref]
  30. R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack : A Subroutine Package for Automatic Integration (Springer-Verlag, 1983).
    [Crossref]
  31. R. Golubovic, A. G. Polimeridis, and J. R. Mosig, “Efficient algorithms for computing Sommerfeld integral tails,” IEEE Trans. Antennas Propag. 60, 2409–2417 (2012).
    [Crossref]
  32. P. R. Atkins and W. C. Chew, “Fast computation of the dyadic Green’s function for layered media via interpolation,” IEEE Antennas Wireless Propag. Lett. 9, 493–496 (2010).
    [Crossref]
  33. C. W. Ueberhuber, Numerical Computation 1: Methods, Software, and Analysis (Springer-Verlag, 1997).
  34. J.-P. Berrut and L. N. Trefethen, “Barycentric Lagrange interpolation,” SIAM Review 46, 501–517 (2004).
    [Crossref]
  35. N. J. Higham, “The numerical stability of barycentric Lagrange interpolation,” IMA J. Numer. Anal. 24, 547–556 (2004).
    [Crossref]
  36. J.-M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett. 109, 023901 (2012).
    [Crossref] [PubMed]
  37. F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” New J. Phys. 10, 105017 (2008).
    [Crossref]
  38. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [Crossref]
  39. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
    [Crossref]
  40. I. Chremmos, “Magnetic field integral equation analysis of surface plasmon scattering by rectangular dielectric channel discontinuities,” J. Opt. Soc. Am. A 27, 85–94 (2010).
    [Crossref]
  41. L. Lafone, T. P. H. Sidiropoulos, and R. F. Oulton, “Silicon-based metal-loaded plasmonic waveguides for low-loss nanofocusing,” Opt. Lett. 39, 4356–4359 (2014).
    [Crossref] [PubMed]
  42. K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2006).
  43. F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112, 17983–17987 (2008).
    [Crossref]
  44. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier Academic, 2007).
  45. L. Zhang, T. Cui, and H. Liu, “A set of symmetric quadrature rules on triangles and tetrahedra,” J. Comput. Math. 27, 89–96 (2009).

2015 (1)

2014 (2)

D. M. Solís, J. M. Taboada, F. Obelleiro, L. M. Liz-Marzán, and F. J. García de Abajo, “Toward ultimate nanoplas-monics modeling,” ACS Nano 8, 7559–7570 (2014).
[Crossref]

L. Lafone, T. P. H. Sidiropoulos, and R. F. Oulton, “Silicon-based metal-loaded plasmonic waveguides for low-loss nanofocusing,” Opt. Lett. 39, 4356–4359 (2014).
[Crossref] [PubMed]

2012 (6)

J.-M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett. 109, 023901 (2012).
[Crossref] [PubMed]

J. Mosig, “The weighted averages algorithm revisited,” IEEE Trans. Antennas Propag. 60, 2011–2018 (2012).
[Crossref]

R. Golubovic, A. G. Polimeridis, and J. R. Mosig, “Efficient algorithms for computing Sommerfeld integral tails,” IEEE Trans. Antennas Propag. 60, 2409–2417 (2012).
[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 Propag. 60, 4766–4776 (2012).
[Crossref]

Y. P. Chen, W. E. I. Sha, W. C. H. Choy, L. Jiang, and W. C. 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).
[Crossref] [PubMed]

M. G. Araújo, J. M. Taboada, D. M. Sols, 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]

2011 (4)

E. P. Karabulut, A. T. Erdogan, and M. I. Aksun, “Discrete complex image method with spatial error criterion,” IEEE Trans. Microwave Theory Tech. 59, 793–802 (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 Wirel. Propag. Lett. 10, 419–422 (2011).
[Crossref]

R. Rodríguez-Oliveros and J. A. Sánchez-Gil, “Localized surface-plasmon resonances on single and coupled nanoparticles through surface integral equations for flexible surfaces,” Opt. Express 19, 12208–12219 (2011).
[Crossref] [PubMed]

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 28, 1341–1348 (2011).
[Crossref]

2010 (4)

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]

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. Microwave Theory Tech. 58, 602–613 (2010).
[Crossref]

P. R. Atkins and W. C. Chew, “Fast computation of the dyadic Green’s function for layered media via interpolation,” IEEE Antennas Wireless Propag. Lett. 9, 493–496 (2010).
[Crossref]

I. Chremmos, “Magnetic field integral equation analysis of surface plasmon scattering by rectangular dielectric channel discontinuities,” J. Opt. Soc. Am. A 27, 85–94 (2010).
[Crossref]

2009 (2)

L. Zhang, T. Cui, and H. Liu, “A set of symmetric quadrature rules on triangles and tetrahedra,” J. Comput. Math. 27, 89–96 (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. A 26, 732–740 (2009).
[Crossref]

2008 (3)

J. Alegret, P. Johansson, and M. Käll, “Green’s tensor calculations of plasmon resonances of single holes and hole pairs in thin gold films,” New J. Phys. 10, 105004 (2008).
[Crossref]

F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112, 17983–17987 (2008).
[Crossref]

F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” New J. Phys. 10, 105017 (2008).
[Crossref]

2006 (2)

I. Hanninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Electromag. Res. 63, 243–278 (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 Propag. Lett. 5, 490–494 (2006).
[Crossref]

2004 (2)

J.-P. Berrut and L. N. Trefethen, “Barycentric Lagrange interpolation,” SIAM Review 46, 501–517 (2004).
[Crossref]

N. J. Higham, “The numerical stability of barycentric Lagrange interpolation,” IMA J. Numer. Anal. 24, 547–556 (2004).
[Crossref]

2003 (2)

W. C. Chew and S.-Y. Chen, “Response of a point source embedded in a layered medium,” IEEE Antennas Wireless Propag. Lett. 2, 254–258 (2003).

P. Yla-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]

2001 (1)

2000 (1)

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[Crossref]

1997 (1)

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

1994 (1)

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]

1989 (1)

R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromag. Waves Appl. 3, 1–15 (1989).
[Crossref]

1982 (1)

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

1972 (1)

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

Aksun, M. I.

E. P. Karabulut, A. T. Erdogan, and M. I. Aksun, “Discrete complex image method with spatial error criterion,” IEEE Trans. Microwave Theory Tech. 59, 793–802 (2011).
[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. Microwave Theory Tech. 58, 602–613 (2010).
[Crossref]

Alegret, J.

J.-M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett. 109, 023901 (2012).
[Crossref] [PubMed]

J. Alegret, P. Johansson, and M. Käll, “Green’s tensor calculations of plasmon resonances of single holes and hole pairs in thin gold films,” New J. Phys. 10, 105004 (2008).
[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. Microwave Theory Tech. 58, 602–613 (2010).
[Crossref]

Araújo, M. G.

Atkins, P. R.

P. R. Atkins and W. C. Chew, “Fast computation of the dyadic Green’s function for layered media via interpolation,” IEEE Antennas Wireless Propag. Lett. 9, 493–496 (2010).
[Crossref]

Berrut, J.-P.

J.-P. Berrut and L. N. Trefethen, “Barycentric Lagrange interpolation,” SIAM Review 46, 501–517 (2004).
[Crossref]

Brucoli, G.

F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” New J. Phys. 10, 105017 (2008).
[Crossref]

Chen, S.-Y.

W. C. Chew and S.-Y. Chen, “Response of a point source embedded in a layered medium,” IEEE Antennas Wireless Propag. Lett. 2, 254–258 (2003).

Chen, Y. P.

Y. P. Chen, W. E. I. Sha, L. Jiang, and J. Hu, “Graphene plasmonics for tuning photon decay rate near metallic split-ring resonator in a multilayered substrate,” Opt. Express 23, 2798–2807 (2015).
[Crossref] [PubMed]

Y. P. Chen, W. E. I. Sha, W. C. H. Choy, L. Jiang, and W. C. 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).
[Crossref] [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 Propag. 60, 4766–4776 (2012).
[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 Wirel. Propag. Lett. 10, 419–422 (2011).
[Crossref]

Chew, W. C.

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 Propag. 60, 4766–4776 (2012).
[Crossref]

Y. P. Chen, W. E. I. Sha, W. C. H. Choy, L. Jiang, and W. C. 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).
[Crossref] [PubMed]

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 Wirel. Propag. Lett. 10, 419–422 (2011).
[Crossref]

P. R. Atkins and W. C. Chew, “Fast computation of the dyadic Green’s function for layered media via interpolation,” IEEE Antennas Wireless Propag. Lett. 9, 493–496 (2010).
[Crossref]

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

W. C. Chew and S.-Y. Chen, “Response of a point source embedded in a layered medium,” IEEE Antennas Wireless Propag. Lett. 2, 254–258 (2003).

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

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

Choy, W. C. H.

Chremmos, I.

Christy, R. W.

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

Cuche, A.

J.-M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett. 109, 023901 (2012).
[Crossref] [PubMed]

Cui, T.

L. Zhang, T. Cui, and H. Liu, “A set of symmetric quadrature rules on triangles and tetrahedra,” J. Comput. Math. 27, 89–96 (2009).

de Doncker-Kapenga, E.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack : A Subroutine Package for Automatic Integration (Springer-Verlag, 1983).
[Crossref]

de León-Pérez, F.

J.-M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett. 109, 023901 (2012).
[Crossref] [PubMed]

F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” New J. Phys. 10, 105017 (2008).
[Crossref]

Degiron, A.

J.-M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett. 109, 023901 (2012).
[Crossref] [PubMed]

Devaux, E.

J.-M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett. 109, 023901 (2012).
[Crossref] [PubMed]

Ebbesen, T. W.

J.-M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett. 109, 023901 (2012).
[Crossref] [PubMed]

Erdogan, A. T.

E. P. Karabulut, A. T. Erdogan, and M. I. Aksun, “Discrete complex image method with spatial error criterion,” IEEE Trans. Microwave Theory Tech. 59, 793–802 (2011).
[Crossref]

Gallinet, B.

García de Abajo, F. J.

D. M. Solís, J. M. Taboada, F. Obelleiro, L. M. Liz-Marzán, and F. J. García de Abajo, “Toward ultimate nanoplas-monics modeling,” ACS Nano 8, 7559–7570 (2014).
[Crossref]

F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112, 17983–17987 (2008).
[Crossref]

García-Vidal, F. J.

F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” New J. Phys. 10, 105017 (2008).
[Crossref]

Gay-Balmaz, P.

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[Crossref]

Gedera, M. B.

Genet, C.

J.-M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett. 109, 023901 (2012).
[Crossref] [PubMed]

Glisson, A. W.

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

Golubovic, R.

R. Golubovic, A. G. Polimeridis, and J. R. Mosig, “Efficient algorithms for computing Sommerfeld integral tails,” IEEE Trans. Antennas Propag. 60, 2409–2417 (2012).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier Academic, 2007).

Graglia, R. D.

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]

Hanninen, I.

I. Hanninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Electromag. Res. 63, 243–278 (2006).
[Crossref]

Harrington, R. F.

R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromag. Waves Appl. 3, 1–15 (1989).
[Crossref]

R. F. Harrington, Field Computation by Moment Methods (R. E. Krieger, 1968).

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

Higham, N. J.

N. J. Higham, “The numerical stability of barycentric Lagrange interpolation,” IMA J. Numer. Anal. 24, 547–556 (2004).
[Crossref]

Hu, B.

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

Hu, J.

Jiang, L.

Y. P. Chen, W. E. I. Sha, L. Jiang, and J. Hu, “Graphene plasmonics for tuning photon decay rate near metallic split-ring resonator in a multilayered substrate,” Opt. Express 23, 2798–2807 (2015).
[Crossref] [PubMed]

Y. P. Chen, W. E. I. Sha, W. C. H. Choy, L. Jiang, and W. C. 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).
[Crossref] [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 Propag. 60, 4766–4776 (2012).
[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 Wirel. Propag. Lett. 10, 419–422 (2011).
[Crossref]

Johansson, P.

J. Alegret, P. Johansson, and M. Käll, “Green’s tensor calculations of plasmon resonances of single holes and hole pairs in thin gold films,” New J. Phys. 10, 105004 (2008).
[Crossref]

Johnson, P. B.

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

Kahaner, D. K.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack : A Subroutine Package for Automatic Integration (Springer-Verlag, 1983).
[Crossref]

Käll, M.

J. Alegret, P. Johansson, and M. Käll, “Green’s tensor calculations of plasmon resonances of single holes and hole pairs in thin gold films,” New J. Phys. 10, 105004 (2008).
[Crossref]

Karabulut, E. P.

E. P. Karabulut, A. T. Erdogan, and M. I. Aksun, “Discrete complex image method with spatial error criterion,” IEEE Trans. Microwave Theory Tech. 59, 793–802 (2011).
[Crossref]

Kern, A. M.

Lafone, L.

Landesa, L.

Laux, E.

J.-M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett. 109, 023901 (2012).
[Crossref] [PubMed]

Liu, H.

L. Zhang, T. Cui, and H. Liu, “A set of symmetric quadrature rules on triangles and tetrahedra,” J. Comput. Math. 27, 89–96 (2009).

Liz-Marzán, L. M.

D. M. Solís, J. M. Taboada, F. Obelleiro, L. M. Liz-Marzán, and F. J. García de Abajo, “Toward ultimate nanoplas-monics modeling,” ACS Nano 8, 7559–7570 (2014).
[Crossref]

Martin, O. J. F.

Martín-Moreno, L.

J.-M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett. 109, 023901 (2012).
[Crossref] [PubMed]

F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” New J. Phys. 10, 105017 (2008).
[Crossref]

Medgyesi-Mitschang, L. N.

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. Microwave 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 Propag. 45, 508–519 (1997).
[Crossref]

Mosig, J.

J. Mosig, “The weighted averages algorithm revisited,” IEEE Trans. Antennas Propag. 60, 2011–2018 (2012).
[Crossref]

Mosig, J. R.

R. Golubovic, A. G. Polimeridis, and J. R. Mosig, “Efficient algorithms for computing Sommerfeld integral tails,” IEEE Trans. Antennas Propag. 60, 2409–2417 (2012).
[Crossref]

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

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

Obelleiro, F.

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2006).

Oulton, R. F.

Paulus, M.

M. Paulus and O. J. F. Martin, “Light propagation and scattering in stratified media: a Green’s tensor approach,” J. Opt. Soc. Am. A 18, 854–861 (2001).
[Crossref]

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[Crossref]

Piessens, R.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack : A Subroutine Package for Automatic Integration (Springer-Verlag, 1983).
[Crossref]

Polimeridis, A. G.

R. Golubovic, A. G. Polimeridis, and J. R. Mosig, “Efficient algorithms for computing Sommerfeld integral tails,” IEEE Trans. Antennas Propag. 60, 2409–2417 (2012).
[Crossref]

Putnam, J. M.

Rao, S. M.

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

Rivero, J.

Rodríguez-Oliveros, R.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier Academic, 2007).

Sánchez-Gil, J. A.

Sarvas, J.

I. Hanninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Electromag. Res. 63, 243–278 (2006).
[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 Propag. Lett. 5, 490–494 (2006).
[Crossref]

Sha, W. E. I.

Sidiropoulos, T. P. H.

Solís, D. M.

D. M. Solís, J. M. Taboada, F. Obelleiro, L. M. Liz-Marzán, and F. J. García de Abajo, “Toward ultimate nanoplas-monics modeling,” ACS Nano 8, 7559–7570 (2014).
[Crossref]

Sols, D. M.

Sommerfeld, A.

A. Sommerfeld, Partial Differential Equations in Physics (Academic, 1949).

Taboada, J. M.

Taskinen, M.

I. Hanninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Electromag. Res. 63, 243–278 (2006).
[Crossref]

P. Yla-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]

Tong, M. S.

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

Trefethen, L. N.

J.-P. Berrut and L. N. Trefethen, “Barycentric Lagrange interpolation,” SIAM Review 46, 501–517 (2004).
[Crossref]

Überhuber, C. W.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack : A Subroutine Package for Automatic Integration (Springer-Verlag, 1983).
[Crossref]

Ueberhuber, C. W.

C. W. Ueberhuber, Numerical Computation 1: Methods, Software, and Analysis (Springer-Verlag, 1997).

Wilton, D.

S. M. Rao, D. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[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 Propag. Lett. 5, 490–494 (2006).
[Crossref]

Yi, J.-M.

J.-M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett. 109, 023901 (2012).
[Crossref] [PubMed]

Yla-Oijala, P.

P. Yla-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]

Zhang, L.

L. Zhang, T. Cui, and H. Liu, “A set of symmetric quadrature rules on triangles and tetrahedra,” J. Comput. Math. 27, 89–96 (2009).

ACS Nano (1)

D. M. Solís, J. M. Taboada, F. Obelleiro, L. M. Liz-Marzán, and F. J. García de Abajo, “Toward ultimate nanoplas-monics modeling,” ACS Nano 8, 7559–7570 (2014).
[Crossref]

IEEE Antennas Wirel. Propag. Lett. (1)

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 Wirel. Propag. Lett. 10, 419–422 (2011).
[Crossref]

IEEE Antennas Wireless Propag. Lett. (3)

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

W. C. Chew and S.-Y. Chen, “Response of a point source embedded in a layered medium,” IEEE Antennas Wireless Propag. Lett. 2, 254–258 (2003).

P. R. Atkins and W. C. Chew, “Fast computation of the dyadic Green’s function for layered media via interpolation,” IEEE Antennas Wireless Propag. Lett. 9, 493–496 (2010).
[Crossref]

IEEE Trans. Antennas Propag. (7)

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

R. Golubovic, A. G. Polimeridis, and J. R. Mosig, “Efficient algorithms for computing Sommerfeld integral tails,” IEEE Trans. Antennas Propag. 60, 2409–2417 (2012).
[Crossref]

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]

P. Yla-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]

J. Mosig, “The weighted averages algorithm revisited,” IEEE Trans. Antennas Propag. 60, 2011–2018 (2012).
[Crossref]

K. A. Michalski and J. R. Mosig, “Multilayered media green’s functions in integral equation formulations,” IEEE Trans. Antennas Propag. 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 Propag. 60, 4766–4776 (2012).
[Crossref]

IEEE Trans. Microwave Theory Tech. (2)

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. Microwave Theory Tech. 58, 602–613 (2010).
[Crossref]

E. P. Karabulut, A. T. Erdogan, and M. I. Aksun, “Discrete complex image method with spatial error criterion,” IEEE Trans. Microwave Theory Tech. 59, 793–802 (2011).
[Crossref]

IMA J. Numer. Anal. (1)

N. J. Higham, “The numerical stability of barycentric Lagrange interpolation,” IMA J. Numer. Anal. 24, 547–556 (2004).
[Crossref]

J. Comput. Math. (1)

L. Zhang, T. Cui, and H. Liu, “A set of symmetric quadrature rules on triangles and tetrahedra,” J. Comput. Math. 27, 89–96 (2009).

J. Electromag. Waves Appl. (1)

R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromag. Waves Appl. 3, 1–15 (1989).
[Crossref]

J. Opt. Soc. Am. A (6)

J. Phys. Chem. C (1)

F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C 112, 17983–17987 (2008).
[Crossref]

New J. Phys. (2)

F. de León-Pérez, G. Brucoli, F. J. García-Vidal, and L. Martín-Moreno, “Theory on the scattering of light and surface plasmon polaritons by arrays of holes and dimples in a metal film,” New J. Phys. 10, 105017 (2008).
[Crossref]

J. Alegret, P. Johansson, and M. Käll, “Green’s tensor calculations of plasmon resonances of single holes and hole pairs in thin gold films,” New J. Phys. 10, 105004 (2008).
[Crossref]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. B (1)

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

Phys. Rev. E (1)

M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000).
[Crossref]

Phys. Rev. Lett. (1)

J.-M. Yi, A. Cuche, F. de León-Pérez, A. Degiron, E. Laux, E. Devaux, C. Genet, J. Alegret, L. Martín-Moreno, and T. W. Ebbesen, “Diffraction regimes of single holes,” Phys. Rev. Lett. 109, 023901 (2012).
[Crossref] [PubMed]

Prog. Electromag. Res. (1)

I. Hanninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Electromag. Res. 63, 243–278 (2006).
[Crossref]

SIAM Review (1)

J.-P. Berrut and L. N. Trefethen, “Barycentric Lagrange interpolation,” SIAM Review 46, 501–517 (2004).
[Crossref]

Other (9)

K. Okamoto, Fundamentals of Optical Waveguides (Academic, 2006).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier Academic, 2007).

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack : A Subroutine Package for Automatic Integration (Springer-Verlag, 1983).
[Crossref]

A. Sommerfeld, Partial Differential Equations in Physics (Academic, 1949).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
[Crossref]

C. W. Ueberhuber, Numerical Computation 1: Methods, Software, and Analysis (Springer-Verlag, 1997).

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

R. F. Harrington, Field Computation by Moment Methods (R. E. Krieger, 1968).

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

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 (7)

Fig. 1
Fig. 1 (a) A homogeneous scatterer with EM properties εi and µi is embedded in a medium with stratified properties εo(r) and µo(r). (b) A schematic representation of an RWG function f(r) = ±L(rp±)/2A±, rT±. T+ and T denote the two triangles that form its supported area, outside of which f(r) vanishes. L is the length of the common edge and A± is the area of T±. Implied is the relation ∇ · f(r) = ±L/A± for rT±.
Fig. 2
Fig. 2 Deformed integration path in the complex kρ-plane. {ξn} denote a set of break points involved in evaluating the tail integral within the generalized weighted-averages scheme.
Fig. 3
Fig. 3 (a) A schematic representation of the single-hole geometry. (b) A side view showing the corresponding scatterer surface S and its relative position. (c) A 3D view of the triangulated S, shown with a reduced number of triangles for visibility.
Fig. 4
Fig. 4 Normalized intensities as functions of θ obtained from the CMM (gray lines) and the SIE-MoM (black lines) for ϕ = 0 () and ϕ = π/2 (⊥), for hole diameters (a) d = 200 nm, (b) 300 nm, and (c) 600 nm. The incident plane wave Epl(r) is specified by λ = 660 nm, k = | k | z ^ and |Epl(r)| = 1, linearly polarized in the x-direction. The corresponding spatial distributions of |E(r)| at z = 10 nm are shown in (d), (e), and (f). For reference and scaling, the holes are drawn as white circles in the color plots.
Fig. 5
Fig. 5 (a) Cross-sectional geometry of the hybrid plasmonic waveguide. (b) A 3D view of the tapered metal strip. The surface mesh is shown with a reduced number of triangles for visibility. (c) A top view of the apex region.
Fig. 6
Fig. 6 Spatial distribution of |E(r)| at z = 55 nm, i.e., 5 nm above the metal strip for the fundamental (a) TM and (b) TE incident waves satisfying max(|Einc(r)|) = 1, with λ = 1,550 nm. For reference and scaling, the lateral boundaries of the metal strip are drawn in white lines. For the TM case, the same quantity is plotted as a function of z at (x, y) = (xapex + Δ, yapex), with Δ = 5 nm, where (c) the lateral and (d) transverse components of the field are shown along with (e) the total field. For comparison, the corresponding components of the TM incident field are shown as gray lines. The layer interfaces are indicated by solid red lines, and the transverse extent of the strip by dashed black lines. In (d), εr(r) denotes the relative permittivity of the background medium.
Fig. 7
Fig. 7 A line segment bounded by p1 and p2, along which r′ is located and the integrand is evaluated.

Equations (55)

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

( D ^ i E ( r ) + D ^ o E ( r ) K ^ i H ( r ) K ^ o H ( r ) K ^ i E ( r ) + K ^ o E ( r ) D ^ i H ( r ) + D ^ o H ( r ) ) ( J M ) | tan = ( E i inc ( r ) E o inc ( r ) H i inc ( r ) H o inc ( r ) ) | tan ,
D ^ ν γ ( r ) F = i ω S p ν γ ( r ) G ¯ ν γ ( r , r ) F ( r ) d S , K ^ ν γ ( r ) F = 1 p ν γ ( r ) S p ν γ ( r ) [ × G ¯ ν γ ( r , r ) ] F ( r ) d S , with γ = E , H ν = i , o
( E ν ( r ) H ν ( r ) ) = ( E ν inc ( r ) H ν inc ( r ) ) ( D ^ ν E ( r ) K ^ ν H ( r ) K ^ ν E ( r ) D ^ ν H ( r ) ) ( J M ) ,
S i f i ( r ) D ^ o E ( r ) f j d S = i ω μ m { [ f i ( r ) ] g d d E ( r , r ) [ f j ( r ) ] + [ z ^ f i ( r ) ] g z d E ( r , r ) [ f j ( r ) ] + [ f j ( r ) ] g d z E ( r , r ) [ z ^ f i ( r ) ] + f i ( r ) [ x ^ g s s E ( r , r ) x ^ + y ^ g s s E ( r , r ) y ^ + z ^ g z z E ( r , r ) z ^ ] f j ( r ) } ,
g d d E ( r , r ) = k n m 2 z z g TM ( r , r ) g TE ( r , r ) ,
g z d E ( r , r ) = μ n μ m 1 z g TM ( r , r ) z g TE ( r , r ) ,
g d z E ( r , r ) = ε m ε n 1 z g TM ( r , r ) z g TE ( r , r ) ,
g s s E ( r , r ) = ( k n 2 + z 2 ) g TM ( r , r ) ,
g z z E ( r , r ) = k m n 2 g TM ( r , r ) z z g TE ( r , r ) .
g α ( r , r ) = i 4 π 0 d k ρ k ρ k m z J 0 ( k ρ r s ) F α ( k ρ , z , z ) , α = TM , TE
S i f i ( r ) K ^ o H ( r ) f j d S = ε m ε n 1 { z ^ [ f i ( r ) × ( r r ) ] r s 1 g s d H ( r , r ) [ f j ( r ) ] + [ f i ( r ) ] r s 1 g d s H ( r , r ) z ^ [ f j ( r ) × ( r r ) ] + z ^ [ f i ( r ) × ( r r ) ] r s 1 g s z H ( r , r ) [ z ^ f j ( r ) ] + [ z ^ f i ( r ) ] r s 1 g z s H ( r , r ) z ^ [ f j ( r ) × ( r r ) ] } ,
g s d H ( r , r ) = ε n ε m 1 r s z g TE ( r , r ) ,
g d s H ( r , r ) = r s z g TM ( r , r ) ,
g s z H ( r , r ) = k n m 2 r s g TE ( r , r ) ,
g z s H ( r , r ) = k n 2 r s g TM ( r , r ) .
g m α , k ( r s , Z k ) = i 4 π 0 d k ρ k ρ k m z U m α , k ( k ρ ) J 0 ( k ρ r s ) e i k m z Z k ( z , z ) , if r , r m ,
g m n α , k ( r s , Z m k , Z n k ) = i 4 π 0 d k ρ k ρ k m z U m n α , k ( k ρ ) J 0 ( k ρ r s ) e i [ k m z Z m k ( z ) + k n z Z n k ( z ) ] , if r n , r m ,
b = { min ( k 0 , 8 π r s ) : r s 0 k 0 : r s = 0
g m qs ( r s , Z ) = U ˜ m 0 d k ρ k ρ k m z 3 J 0 ( k ρ r s ) e i k m z Z
G ¯ 0 γ ( r , r ) = G ¯ pri γ ( r , r ) + G ¯ sec γ ( r , r ) ,
G ¯ pri E ( r , r ) = G ¯ pri H ( r , r ) = ( I ¯ + k m 2 ) e i k m | r r | 4 π | r r | ,
ω μ m ( U ˜ m TM [ f i ( r ) ] [ f i ( r ) ] T i ± I m S ( r eff ; T j ± ) d S + U ˜ m TE T i ± f i ( r ) K 2 m R ( r eff ; T j ± ) d S )
I m S ( r ; T ) = 1 k m 2 I 1 S ( r ; T ) 1 2 I + 1 S ( r ; T ) ,
K 2 m R ( r ; T ± ) = [ x ^ K 2 m ( r ; T ± ) ] x ^ + [ y ^ K 2 m ( r ; T ± ) ] y ^ [ z ^ K 2 m ( r ; T ± ) ] z ^ ,
K 2 m ( r ; T ± ) = K 2 1 ( r ; T ± ) k m 2 2 K 2 + 1 ( r ; T ± ) .
I q S ( r ; T ) = T | r r | q d S ,
K 2 q ( r ; T ± ) = T ± | r r | q f ( r ) d S ,
i p z ^ ( U ˜ m TE [ f j ( r ) ] T i ± f i ( r ) × K 8 ( r eff ; T j ± ) dS + U ˜ m TM [ f i ( r ) ] T i ± K 9 ( r eff ; T j ± ) d S )
K 8 ( r ; T ) = T log ( | r r | ) d S ,
K 9 ( r ; T ± ) = T ± log ( | r r | ) × f ( r ) d S .
( D ^ ν E ( r ) K ^ ν H ( r ) K ^ ν E ( r ) D ^ ν H ( r ) ) ( J M ) = ± ( E ν inc ( r ) H ν inc ( r ) ) ,
β m α k m z 1 F n , m α ( z , d m ) = β m + 1 α k m + 1 1 F n , m + 1 α ( z , d m ) ,
k m z 1 z F n , m α ( z , z ) | z = d m = k m + 1 , z 1 z F n , m + 1 α ( z , z ) | z = d m ,
S i f i ( r ) O ^ o n , m ( r ) f j d S = S i f i ( r ) O ^ o n , m + 1 ( r ) f j d S ,
ε m ε n 1 T j d l f j ( r ) m j ± T i ± d S z ^ [ f i ( r ) × ( r r ) ] r s 1 g s d H ( r , r ) ,
ε m ε n 1 T i d l f i ( r ) m i ± T j ± d S z ^ [ f j ( r ) × ( r r ) ] r s 1 g s d H ( r , r ) ,
u n = ξ n 1 ξ n t ( k ρ ) d k ρ ,
I N * = n = 1 N w n I n n = 1 N w n ,
w n = ( 1 ) n + 1 exp ( γ ξ n ) ( N 1 n 1 ) ξ n N 2 q .
g ˜ m qs ( r s , Z ) = Z [ i I ˜ 1 m ( r s , Z ) + π H 0 ( 1 ) ( k m r s ) / 2 ] e i k m R ˜ / k m ,
Z g ˜ m qs ( r s , Z ) = i I ˜ 1 m ( r s , Z ) + π H 0 ( 1 ) ( k m r s ) / 2 ,
Z 2 g ˜ m qs ( r s , Z ) = i e i k m R ˜ / R ˜ ,
r s g ˜ m qs ( r s , Z ) = Z [ i I ˜ 2 m ( r s , Z ) π k m H 1 ( 1 ) ( k m r s ) / 2 ] i r s e i k m R ˜ / R ˜ ,
r s Z g ˜ m qs ( r s , Z ) = i I ˜ 2 m ( r s , Z ) π k m H 1 ( 1 ) ( k m r s ) / 2 ,
I ˜ 1 m ( r s , Z ) = ( 1 k m 2 r s 2 / 4 ) log [ ( Z + R ˜ ) / r s ] k m 2 Z R ˜ / 4 ,
I ˜ 2 m ( r s , Z ) = r s R ˜ ( Z + R ˜ ) 1 r s k m 2 r s 2 [ ( 1 k m 2 r s 2 8 ) log ( Z + R ˜ r s ) k m 2 8 Z R ˜ ] .
Z 2 g ˜ m n qs ( r s , Z ) = i 0 J 0 ( k ρ r s ) e k ρ Z d k ρ = i / R ˜ ,
r s g ˜ m n qs ( r s , Z ) = i 0 J 1 ( k ρ r s ) e k ρ Z d k ρ / k ρ = i ( Z R ˜ ) / r s ,
Z g ˜ m n qs ( r s , Z ) = i log ( Z + R ˜ ) ,
K 8 ( r ; T ) = k = 1 3 m k I log L ( r ; T k ) ,
K 8 n ( r ; T ) = n ( T ) h I 2 S ( r ; T ) .
I log L ( r ; Δ L ) = Δ L log ( | r r | ) d l .
I log L ( r ; Δ L ) = 1 2 s s + log ( R 0 2 + s 2 ) d s ,
{ s + log ( R + ) s log ( R ) Δ L + R 0 [ arctan ( s + R 0 ) arctan ( s R 0 ) ] : R 0 0 s + log ( | s + | ) s log ( | s | ) Δ L : R 0 = 0 .
K 9 ( r ; T ± ) = ± L 2 A ± K 8 ( r ; T ± ) × ( r p ± ) ,

Metrics