Abstract

A set of new surface integral equations (Fredholm equations of the second kind) has been systematically derived from the Stratton–Chu formulation of Maxwell’s equations for a two-dimensional TM mode to investigate the interactions of an incident electromagnetic wave with nanostructures, especially metals. With these equations, the surface components (the tangential magnetic field, the normal displacement, and the tangential electric field) on the boundary are solved simultaneously by the boundary-element method numerically. For nanometer-sized structures (e.g., dimension of 10nm), our numerical results show that surface plasmon resonance causes a strong near-field enhancement of the electric field within a shallow region close to the interface of metal and dielectric. In addition, the corresponding pattern of the far-field scattering cross section is like a dipole. For the submicrometer-sized cases (dimension of several hundreds of nanometers), the numerical results indicate the existence of a standing wave on the backside surface of metals. This phenomenon could be caused by two surface plasmon waves that creep along the contour of metals clockwise and counterclockwise, respectively, and interfere with each other.

© 2006 Optical Society of America

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    [PubMed]
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  6. G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher,A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu,B. Monacelli, and G. Boreman, "Plasmon dispersion relation of Au and Ag nanowires," Phys. Rev. B 68, 155427 (2003).
    [CrossRef]
  7. J.-C. Weeber, A. Dereux, C. Girard, G. C. des Francs, J. R. Krenn, and J.-P. Goudonnet, "Optical addressing at the subwavelength scale," Phys. Rev. E 62, 7381-7388 (2000).
    [CrossRef]
  8. J. W. Liaw and J. K. Wang, "Dispersion relation of plasmon wave in metallic nanowires," Scanning Microsc. 26, 106-108 (2004).
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    [CrossRef]
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2005

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang,J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405(R) (2005).
[CrossRef]

2004

L. B. Blanco and F. J. Garcia de Abajo, "Spontaneous emission enhancement near nanoparticles," J. Quant. Spectrosc. Radiat. Transf. 89, 37-42 (2004).
[CrossRef]

J. W. Liaw and J. K. Wang, "Dispersion relation of plasmon wave in metallic nanowires," Scanning Microsc. 26, 106-108 (2004).

L. Rogobete and C. Henkel, "Spontaneous emission in a subwavelength environment characterized by boundary integral equations," Phys. Rev. A 70, 063815 (2004).
[CrossRef]

2003

S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205402 (2003).
[CrossRef]

G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher,A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu,B. Monacelli, and G. Boreman, "Plasmon dispersion relation of Au and Ag nanowires," Phys. Rev. B 68, 155427 (2003).
[CrossRef]

C. I. Valencia, E. R. Mendez, and B. S. Mendoza, "Second-harmonic generation in the scattering of light by two-dimensional particles," J. Opt. Soc. Am. B 20, 2150-2161 (2003).
[CrossRef]

C. Rockstuhl, M. G. Salt, and H. P. Herzig, "Application of the boundary-element method to the interaction of light with single and coupled metallic nanoparticles," J. Opt. Soc. Am. A 20, 1969-1973 (2003).
[CrossRef]

2002

E. Moreno, D. Erni, C. Hafner, and R. Vahldieck, "Multiple multipole method with automatic multipole setting applied to the simulation of surface plasmons in metallic nanostructures," J. Opt. Soc. Am. A 19, 101-111 (2002).
[CrossRef]

F. J. Garcia de Abajo, "Light transmission through a single cylindrical hole in a metallic film," Opt. Express 10, 1475-1484 (2002).
[PubMed]

V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, "Plasmon modes in metal nanowires and left-handed materials," J. Nonlinear Opt. Phys. Mater. 11, 65-74 (2002).
[CrossRef]

F. J. Garcia de Abajo and A. Howie, "Retarded field calculation of electron energy loss in inhomogeneous dielectrics," Phys. Rev. B 65, 115418 (2002).
[CrossRef]

2001

2000

J. P. Kottmann, J. F. Martin, D. R. Smith, and S. Schultz, "Spectral response of plasmon resonant nanoparticles with a non-regular shape," Opt. Express 6, 213-219 (2000).
[CrossRef] [PubMed]

J.-C. Weeber, A. Dereux, C. Girard, G. C. des Francs, J. R. Krenn, and J.-P. Goudonnet, "Optical addressing at the subwavelength scale," Phys. Rev. E 62, 7381-7388 (2000).
[CrossRef]

H. Xu, J. Aizpurua, M. Kall, and P. Apell, "Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering," Phys. Rev. E 62, 4318-4324 (2000).
[CrossRef]

1999

K. Tanaka, M. Tanaka, and K. Katayama, "Simulations of two-dimensional photon scanning tunneling microscope by boundary integral equation method: p-polarization," Opt. Rev. 6, 249-256 (1999).
[CrossRef]

J. W. Liaw, S. L. Chu, C. S. Yeh, and M. K. Kuo, "Analysis of eddy current in a bar containing an embedded defect," NDT & E Int. 32, 293-303 (1999).
[CrossRef]

1998

D. W. Prather, J. N. Mait, M. S. Mirotznik, and J. P. Collins, "Vector-based synthesis of finite aperiodic subwavelength diffractive optical elements," J. Opt. Soc. Am. A 15, 1599-1607 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, andP. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Aizpurua, J.

H. Xu, J. Aizpurua, M. Kall, and P. Apell, "Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering," Phys. Rev. E 62, 4318-4324 (2000).
[CrossRef]

Apell, P.

H. Xu, J. Aizpurua, M. Kall, and P. Apell, "Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering," Phys. Rev. E 62, 4318-4324 (2000).
[CrossRef]

Atwater, H. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205402 (2003).
[CrossRef]

Aussenegg, F. R.

G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher,A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu,B. Monacelli, and G. Boreman, "Plasmon dispersion relation of Au and Ag nanowires," Phys. Rev. B 68, 155427 (2003).
[CrossRef]

Blanco, L. B.

L. B. Blanco and F. J. Garcia de Abajo, "Spontaneous emission enhancement near nanoparticles," J. Quant. Spectrosc. Radiat. Transf. 89, 37-42 (2004).
[CrossRef]

Boreman, G.

G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher,A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu,B. Monacelli, and G. Boreman, "Plasmon dispersion relation of Au and Ag nanowires," Phys. Rev. B 68, 155427 (2003).
[CrossRef]

Chang, Y. C.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang,J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405(R) (2005).
[CrossRef]

Chen, Y. C.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang,J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405(R) (2005).
[CrossRef]

Choi, M. K.

Chu, S. L.

J. W. Liaw, S. L. Chu, C. S. Yeh, and M. K. Kuo, "Analysis of eddy current in a bar containing an embedded defect," NDT & E Int. 32, 293-303 (1999).
[CrossRef]

Collins, J. P.

Dereux, A.

J.-C. Weeber, A. Dereux, C. Girard, G. C. des Francs, J. R. Krenn, and J.-P. Goudonnet, "Optical addressing at the subwavelength scale," Phys. Rev. E 62, 7381-7388 (2000).
[CrossRef]

des Francs, G. C.

J.-C. Weeber, A. Dereux, C. Girard, G. C. des Francs, J. R. Krenn, and J.-P. Goudonnet, "Optical addressing at the subwavelength scale," Phys. Rev. E 62, 7381-7388 (2000).
[CrossRef]

Ditlbacher, H.

G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher,A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu,B. Monacelli, and G. Boreman, "Plasmon dispersion relation of Au and Ag nanowires," Phys. Rev. B 68, 155427 (2003).
[CrossRef]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, andP. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Erni, D.

Garcia de Abajo, F. J.

L. B. Blanco and F. J. Garcia de Abajo, "Spontaneous emission enhancement near nanoparticles," J. Quant. Spectrosc. Radiat. Transf. 89, 37-42 (2004).
[CrossRef]

F. J. Garcia de Abajo, "Light transmission through a single cylindrical hole in a metallic film," Opt. Express 10, 1475-1484 (2002).
[PubMed]

F. J. Garcia de Abajo and A. Howie, "Retarded field calculation of electron energy loss in inhomogeneous dielectrics," Phys. Rev. B 65, 115418 (2002).
[CrossRef]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, andP. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Girard, C.

J.-C. Weeber, A. Dereux, C. Girard, G. C. des Francs, J. R. Krenn, and J.-P. Goudonnet, "Optical addressing at the subwavelength scale," Phys. Rev. E 62, 7381-7388 (2000).
[CrossRef]

Goudonnet, J.-P.

J.-C. Weeber, A. Dereux, C. Girard, G. C. des Francs, J. R. Krenn, and J.-P. Goudonnet, "Optical addressing at the subwavelength scale," Phys. Rev. E 62, 7381-7388 (2000).
[CrossRef]

Hafner, C.

Henkel, C.

L. Rogobete and C. Henkel, "Spontaneous emission in a subwavelength environment characterized by boundary integral equations," Phys. Rev. A 70, 063815 (2004).
[CrossRef]

Herzig, H. P.

Hohenau, A.

G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher,A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu,B. Monacelli, and G. Boreman, "Plasmon dispersion relation of Au and Ag nanowires," Phys. Rev. B 68, 155427 (2003).
[CrossRef]

Howie, A.

F. J. Garcia de Abajo and A. Howie, "Retarded field calculation of electron energy loss in inhomogeneous dielectrics," Phys. Rev. B 65, 115418 (2002).
[CrossRef]

Huang, K. T.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang,J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405(R) (2005).
[CrossRef]

Kall, M.

H. Xu, J. Aizpurua, M. Kall, and P. Apell, "Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering," Phys. Rev. E 62, 4318-4324 (2000).
[CrossRef]

Katayama, K.

K. Tanaka, M. Tanaka, and K. Katayama, "Simulations of two-dimensional photon scanning tunneling microscope by boundary integral equation method: p-polarization," Opt. Rev. 6, 249-256 (1999).
[CrossRef]

Kik, P. G.

S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205402 (2003).
[CrossRef]

Kottmann, J. P.

Krenn, J. R.

G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher,A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu,B. Monacelli, and G. Boreman, "Plasmon dispersion relation of Au and Ag nanowires," Phys. Rev. B 68, 155427 (2003).
[CrossRef]

J.-C. Weeber, A. Dereux, C. Girard, G. C. des Francs, J. R. Krenn, and J.-P. Goudonnet, "Optical addressing at the subwavelength scale," Phys. Rev. E 62, 7381-7388 (2000).
[CrossRef]

Kuo, M. K.

J. W. Liaw, S. L. Chu, C. S. Yeh, and M. K. Kuo, "Analysis of eddy current in a bar containing an embedded defect," NDT & E Int. 32, 293-303 (1999).
[CrossRef]

Lee, C. K.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang,J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405(R) (2005).
[CrossRef]

Leitner, A.

G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher,A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu,B. Monacelli, and G. Boreman, "Plasmon dispersion relation of Au and Ag nanowires," Phys. Rev. B 68, 155427 (2003).
[CrossRef]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, andP. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Liaw, J. W.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang,J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405(R) (2005).
[CrossRef]

J. W. Liaw and J. K. Wang, "Dispersion relation of plasmon wave in metallic nanowires," Scanning Microsc. 26, 106-108 (2004).

J. W. Liaw, S. L. Chu, C. S. Yeh, and M. K. Kuo, "Analysis of eddy current in a bar containing an embedded defect," NDT & E Int. 32, 293-303 (1999).
[CrossRef]

Lin, D. Z.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang,J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405(R) (2005).
[CrossRef]

Liu, J. M.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang,J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405(R) (2005).
[CrossRef]

Maier, S. A.

S. A. Maier, P. G. Kik, and H. A. Atwater, "Optical pulse propagation in metal nanoparticle chain waveguides," Phys. Rev. B 67, 205402 (2003).
[CrossRef]

Mait, J. N.

Martin, J. F.

Mendez, E. R.

Mendoza, B. S.

Mirotznik, M. S.

Monacelli, B.

G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher,A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu,B. Monacelli, and G. Boreman, "Plasmon dispersion relation of Au and Ag nanowires," Phys. Rev. B 68, 155427 (2003).
[CrossRef]

Moreno, E.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

Podolskiy, V. A.

V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, "Plasmon modes in metal nanowires and left-handed materials," J. Nonlinear Opt. Phys. Mater. 11, 65-74 (2002).
[CrossRef]

Prather, D. W.

Puscasu, I.

G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher,A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu,B. Monacelli, and G. Boreman, "Plasmon dispersion relation of Au and Ag nanowires," Phys. Rev. B 68, 155427 (2003).
[CrossRef]

Rockstuhl, C.

Rogobete, L.

L. Rogobete and C. Henkel, "Spontaneous emission in a subwavelength environment characterized by boundary integral equations," Phys. Rev. A 70, 063815 (2004).
[CrossRef]

Salt, M. G.

Sarychev, A. K.

V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, "Plasmon modes in metal nanowires and left-handed materials," J. Nonlinear Opt. Phys. Mater. 11, 65-74 (2002).
[CrossRef]

Schaich, W. L.

G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher,A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu,B. Monacelli, and G. Boreman, "Plasmon dispersion relation of Au and Ag nanowires," Phys. Rev. B 68, 155427 (2003).
[CrossRef]

Schider, G.

G. Schider, J. R. Krenn, A. Hohenau, H. Ditlbacher,A. Leitner, F. R. Aussenegg, W. L. Schaich, I. Puscasu,B. Monacelli, and G. Boreman, "Plasmon dispersion relation of Au and Ag nanowires," Phys. Rev. B 68, 155427 (2003).
[CrossRef]

Schultz, S.

J. P. Kottmann, J. F. Martin, D. R. Smith, and S. Schultz, "Plasmon resonances of silver nanowires with a nonregular cross section," Phys. Rev. B 64, 235402 (2001).
[CrossRef]

J. P. Kottmann, J. F. Martin, D. R. Smith, and S. Schultz, "Spectral response of plasmon resonant nanoparticles with a non-regular shape," Opt. Express 6, 213-219 (2000).
[CrossRef] [PubMed]

Shalaev, V. M.

V. A. Podolskiy, A. K. Sarychev, and V. M. Shalaev, "Plasmon modes in metal nanowires and left-handed materials," J. Nonlinear Opt. Phys. Mater. 11, 65-74 (2002).
[CrossRef]

Smith, D. R.

J. P. Kottmann, J. F. Martin, D. R. Smith, and S. Schultz, "Plasmon resonances of silver nanowires with a nonregular cross section," Phys. Rev. B 64, 235402 (2001).
[CrossRef]

J. P. Kottmann, J. F. Martin, D. R. Smith, and S. Schultz, "Spectral response of plasmon resonant nanoparticles with a non-regular shape," Opt. Express 6, 213-219 (2000).
[CrossRef] [PubMed]

Stratton, A.

A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), pp. 464-467.

Tanaka, K.

K. Tanaka, M. Tanaka, and K. Katayama, "Simulations of two-dimensional photon scanning tunneling microscope by boundary integral equation method: p-polarization," Opt. Rev. 6, 249-256 (1999).
[CrossRef]

Tanaka, M.

K. Tanaka, M. Tanaka, and K. Katayama, "Simulations of two-dimensional photon scanning tunneling microscope by boundary integral equation method: p-polarization," Opt. Rev. 6, 249-256 (1999).
[CrossRef]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, andP. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Vahldieck, R.

Valencia, C. I.

Wang, J. K.

J. W. Liaw and J. K. Wang, "Dispersion relation of plasmon wave in metallic nanowires," Scanning Microsc. 26, 106-108 (2004).

Weeber, J.-C.

J.-C. Weeber, A. Dereux, C. Girard, G. C. des Francs, J. R. Krenn, and J.-P. Goudonnet, "Optical addressing at the subwavelength scale," Phys. Rev. E 62, 7381-7388 (2000).
[CrossRef]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, andP. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Xu, H.

H. Xu, J. Aizpurua, M. Kall, and P. Apell, "Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering," Phys. Rev. E 62, 4318-4324 (2000).
[CrossRef]

Yeh, C. S.

L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang,J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405(R) (2005).
[CrossRef]

J. W. Liaw, S. L. Chu, C. S. Yeh, and M. K. Kuo, "Analysis of eddy current in a bar containing an embedded defect," NDT & E Int. 32, 293-303 (1999).
[CrossRef]

Yeh, J. T.

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L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang,J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405(R) (2005).
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L. B. Yu, D. Z. Lin, Y. C. Chen, Y. C. Chang, K. T. Huang,J. W. Liaw, J. T. Yeh, J. M. Liu, C. S. Yeh, and C. K. Lee, "Physical origin of directional beaming emitted from a subwavelength slit," Phys. Rev. B 71, 041405(R) (2005).
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Figures (8)

Fig. 1
Fig. 1

Configuration of a plane p-polarized EM wave illuminating on a scatterer Ω 2 in an infinite domain Ω 1 with an incident angle θ i

Fig. 2
Fig. 2

Mean SCS, σ ¯ , versus the aspect ratio b a for elliptical silver ( a = 10 nm ) with ϵ 2 r = ( 4.42 , 0.73 ) , in air, ϵ 1 r = 1 , at ω = 4.558 × 10 15 rad s , ( 3 eV ) , θ i = 0 ° .

Fig. 3
Fig. 3

Nanometer-sized elliptical silver ( a = 10 nm , b = 38 nm ) with ϵ 2 r = ( 4.42 , 0.73 ) , in air, ϵ 1 r = 1 , at ω = 4.558 × 10 15 rad s ( 3 eV , λ 0 = 413 nm ), θ i = 0 ° . (a) The real parts of the surface components of the total field along the circumference. (b) The imaginary parts. ϕ is the angle of the polar coordinate. Solid curves H z H z i ; curves with circles, D n ϵ 1 E i ; curves with triangles, E t E i .

Fig. 4
Fig. 4

Field distribution of the total field of Fig. 2. (a) The total electric field distribution. (b) The total magnetic field distribution. (c) Far-field SCS. The length scale is in nanometers for the x and y-axes. σ ¯ = 176.4 nm .

Fig. 5
Fig. 5

Submicrometer-sized circular silver ( r = 400 nm ) with ϵ 2 r = ( 4.42 , 0.73 ) , in air, ϵ 1 r = 1 , at ω = 4.558 × 10 15 rad s ( 3 eV , λ 0 = 413 nm ) θ i = 0 ° . (a) The total electric field distribution. (b) The total magnetic field distribution. The length scale is in nanometers for the x and y axes. σ ¯ = 2136 nm .

Fig. 6
Fig. 6

Submicrometer-sized elliptical silver ( a = 400 nm , b = 100 nm ) with ϵ 2 r = ( 4.42 , 0.73 ) , in air, ϵ 1 r = 1 , at ω = 4.558 × 10 15 rad s ( 3 eV ) , θ i = 90 ° . (a) The total electric field distribution. (b) The total magnetic field distribution. The length scale is in nanometers for the x and y axes. σ ¯ = 1648 nm .

Fig. 7
Fig. 7

Total electric field distribution of an elliptical silver ( a = 400 nm , b = 100 nm ) in air, ϵ 1 r = 1 , at ω = 4.558 × 10 15 rad s ( 3 eV ) , θ i = 45 ° . The length scale is in nanometers for the x and y axes. σ ¯ = 599 nm .

Fig. 8
Fig. 8

Total electric field distribution of an elliptical silver ( a = 400 nm , b = 100 nm ) in water, ϵ 1 r = 1.777 , at ω = 4.558 × 10 15 rad s ( 3 eV ) , θ i = 90 ° . The length scale is in nanometers for the x and y axes. σ ¯ = 1608 nm .

Equations (42)

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H z ( x ) = S H z ( x ) n G 2 d l + i ω S ϵ 2 E t ( x ) G 2 d l , x Ω 2 ,
E ( x ) = i ω S μ 2 G 2 H z d l + 1 ϵ 2 S D n G 2 d l S E t e z × G 2 d l , x Ω 2 ,
G 2 = i 4 H 0 ( 1 ) ( k 2 r ) .
2 G 2 + ω 2 ϵ 2 μ 2 G 2 = δ ( x x ) ,
H z s ( x ) = S H z s ( x ) n G 1 d l i ω S ϵ 1 E t s ( x ) G 1 d l , x Ω 1 ,
E s ( x ) = i ω S μ 1 G 1 H z s d l 1 ϵ 1 S D n s G 1 d l + S E t s e z × G 1 d l , x Ω 1 ,
G 1 = i 4 H 0 ( 1 ) ( k 1 r ) .
2 G 1 + ω 2 ϵ 1 μ 1 G 1 = δ ( x x ) .
H z i ( x ) = S H z i ( x ) n G 1 d l + i ω S ϵ 1 E t i ( x ) G 1 d l , x Ω 2 ,
E i ( x ) = i ω S μ 1 G 1 H z i d l + 1 ϵ 1 S D n i G 1 d l S E t i e z × G 1 d l , x Ω 2 .
1 2 H z ( x 0 ) = S H z ( x ) n G 2 d l + i ω S E t ( x ) ϵ 2 G 2 d l , x 0 S .
1 2 D n ( x 0 ) = i ω S ϵ 2 μ 2 G 2 H z n ( x 0 ) d l + S D n n ( x 0 ) G 2 d l S ϵ 2 E t n ( x 0 ) e z × G 2 d l , x 0 S .
1 2 E t ( x 0 ) = i ω S μ 2 G 2 H z t d l + 1 ϵ 2 S D n t G 2 d l S E t t e z × G 2 d l , x 0 S .
1 2 H z s ( x ) = S H z s ( x ) n G 1 d l i ω S E t s ( x ) ϵ 1 G 1 d l , x S ,
1 2 D n s = i ω S H z s ϵ 1 μ 1 G 1 n d l S D n s n G 1 d l + S ϵ 1 E t s n e z × G 1 d l , x S ,
1 2 E t s = i ω S μ 1 G 1 H z s t d l 1 ϵ 1 S D n s t G 1 d l + S E t s t e z × G 1 d l , x S .
1 2 H z i ( x ) = S H z i ( x ) n G 1 d l + i ω S E t i ( x ) ϵ 1 G 1 d l , x S ,
1 2 D n i = i ω S H z i ϵ 1 μ 1 G 1 n d l + S D n i n G 1 d l S ϵ 1 E t i n e z × G 1 d l , x S ,
1 2 E t i = i ω S μ 1 G 1 H z i t d l + 1 ϵ 1 S D n i t G 1 d l S E t i t e z × G 1 d l , x S .
1 2 H z ( x ) = H z i S H z ( x ) n G 1 d l i ω S E t ( x ) ϵ 1 G 1 d l , x S ,
1 2 D n = D n i i ω S H z ϵ 1 μ 1 G 1 n d l S D n n G 1 d l + S ϵ 1 E t n e z × G 1 d l , x S ,
1 2 E t = E t i i ω S μ 1 G 1 H z t d l 1 ϵ 1 S D n t G 1 d l + S E t t e z × G 1 d l , x S .
H z ( x ) = H z i ( x ) S H z ( x ) n [ G 1 G 2 ] d l i ω S E t ( x ) [ ϵ 1 G 1 ϵ 2 G 2 ] d l , x S ,
D n = D n i i ω S H z [ ϵ 1 μ 1 G 1 ϵ 2 μ 2 G 2 ] n d l S D n n [ G 1 G 2 ] d l + S E t n e z × [ ϵ 1 G 1 ϵ 2 G 2 ] d l ,
E t = E t i i ω S H z [ μ 1 G 1 μ 2 G 2 ] t d l S D n t [ G 1 ϵ 1 G 2 ϵ 2 ] d l + S E t t e z × [ G 1 G 2 ] d l .
0 = H z i ( x ) S H z ( x ) n [ G 1 + G 2 ] d l i ω S E t ( x ) [ ϵ 1 G 1 + ϵ 2 G 2 ] d l , x S ,
0 = D n i i ω S H z [ ϵ 1 μ 1 G 1 + ϵ 2 μ 2 G 2 ] n d l S D n n [ G 1 + G 2 ] d l + S E t n e z × [ ϵ 1 G 1 + ϵ 2 G 2 ] d l ,
0 = E t i i ω S H z [ μ 1 G 1 + μ 2 G 2 ] t d l S D n t [ G 1 ϵ 1 + G 2 ϵ 2 ] d l + S E t t e z × [ G 1 + G 2 ] d l .
σ ( θ ; θ i ) = lim x 2 π x E s × H ¯ s e r E i × H ¯ i = lim x 2 π x E s e r 2 E i 2 = lim x 2 π x H z s 2 H z i 2 ,
H n ( 1 ) ( z ) 2 π z exp [ i ( z n π 2 π 4 ) ] , as z 1 ;
G 1 ( k 1 r ) 1 8 π k 1 x exp [ i ( k 1 x + π 4 k 1 x e r ) ] , as x .
σ ( θ ; θ i ) = k 1 4 S [ n e r H z s H z i + E t s E i ] exp ( i k 1 x e r ) d l 2 .
σ ¯ ( θ i ) = 1 2 π 0 2 π σ d θ ,
x j ( ξ ) = i = 1 3 N i ( ξ ) x i j ,
F j ( ξ ) = i = 1 3 N i ( ξ ) F i j ,
N 1 ( ξ ) = 1 2 ξ ( ξ 1 ) ,
N 2 ( ξ ) = 1 ξ 2 ,
N 3 ( ξ ) = 1 2 ξ ( ξ + 1 ) ,
H z H z i , D n ϵ 1 E i , E t E i , E E i ,
E i = k 1 ω ϵ 1 H z i .
k sp = k 0 ϵ 1 r ϵ 2 r ϵ 1 r + ϵ 2 r ,
γ 1 2 = k 0 2 ϵ 1 r 2 ϵ 1 r + ϵ 2 r , γ 2 2 = k 0 2 ϵ 2 r 2 ϵ 1 r + ϵ 2 r .

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