Abstract

On the basis of a rigorous, nonperturbative, purely numerical solution of the corresponding reduced Rayleigh equation for the scattering amplitudes we have studied the scattering of a surface plasmon polariton by a two dimensional dielectric defect on a planar metal surface. The profile of the defect is assumed to be an arbitrary single-valued function of the coordinates in the plane of the metal surface, and to be differentiable with respect to those coordinates. When the defect is circularly symmetric and the dependence of the scattering amplitudes on the azimuthal angle is expressed by a rotational expansion, the reduced Rayleigh equation is transformed into a pair of one-dimensional integral equations for each value of the rotational quantum number. This approach is applied to a defect in the form of an isotropic Gaussian function. The differential cross sections for the scattering of the incident surface plasmon polariton into volume electromagnetic waves in the vacuum above the surface and into other surface plasmon polaritons are calculated, as well as the intensity of the field near the surface. These results differ significantly from the corresponding results for a metallic defect on a metallic substrate.

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    [CrossRef]
  2. Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5, 1726–1729 (2005).
    [CrossRef] [PubMed]
  3. I. I. Smolyaninov, C. C. Davis, and A. V. Zayats, “Image formation in surface plasmon polariton mirrors: applications in high resolution optical microscopy,” New. J. Phys.7, 175 (2005).
    [CrossRef]
  4. E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science311, 189–193 (2006).
    [CrossRef] [PubMed]
  5. I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B76, 205424, (2007).
    [CrossRef]
  6. H. Kim, J. Hahn, and B. Lee, “Focusing properties of surface plasmon polariton floating dielectric lenses,” Opt. Express16, 3049–3057 (2008).
    [CrossRef] [PubMed]
  7. J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
    [CrossRef]
  8. I. Chremmos, “Magnetic field integral equation analysis of surface plasmon scattering by rectangular dielectric channel discontinuities,” J. Opt. Soc. Am. A27, 85–94 (2010).
    [CrossRef]
  9. G. Brucoli and L. Martín-Moreno, “Comparative study of surface plasmon polariton scattering by shallow ridges and grooves,” Phys. Rev. B83, 045422 (2011).
    [CrossRef]
  10. G. Brucoli and L. Martín-Moreno, “Effect of defect depth on surface plasmon scattering by subwavelength surface defects,” Phys. Rev. B83, 075433 (2011).
    [CrossRef]
  11. A. V. Shchegrov, I. V. Novikov, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a circularly symmetric surface defect,” Phys. Rev. Lett.78, 4269–4272 (1997). Erratum: Phys. Rev. Lett. 79, 2597 (1997).
    [CrossRef]
  12. E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Z. NaturforschA23, 2135–2136 (1963).
  13. M. Paulus and O. J. F. Martin, “Light propagation and scattering in stratified media: a Green’s tensor approach,” J. Opt. Soc. Am. A18, 854–861 (2001).
    [CrossRef]
  14. B. Baumeier, F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a localized dielectric surface defect studied using an effective boundary condition,” Phys. Rev.A84, 013810 (2011).
  15. I. I. Smolyaninov, J. E. Elliott, A.V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett.94, 057401 (2005).
    [CrossRef] [PubMed]
  16. T. Nordam, P. A. Letnes, I. Simonsen, and A. A. Maradudin, “Satellite peaks in the scattering of light from the two-dimensional randomly rough surface of a dielectric film on a planar metal surface,” Opt. Express20, 11336– 11350 (2012).
    [CrossRef] [PubMed]
  17. T. A. Leskova and A. A. Maradudin (unpublished work).
  18. A. A. Maradudin and W. M. Visscher, “Electrostatic and electromagnetic surface shape resonances,” Z. Phys.B60, 215–230 (1985).
    [CrossRef]
  19. D. L. Mills, “Attenuation of surface polaritons by surface roughness,” Phys. Rev. B12, 4036–4046 (1975). Erratum: Phys. Rev. B14, 5539 (1976).
    [CrossRef]
  20. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
    [CrossRef]
  21. A. A. Maradudin and D. L. Mills, “Attenuation of Rayleigh surface waves by surface roughness,” Ann. Phys. (N.Y.)100262–309 (1976), Ref. 9.
    [CrossRef]
  22. M. Kretschmann, T. A. Leskova, and A.A. Maradudin, “Excitation of surface plasmon polaritons by the scattering of a volume electromagnetic beam from a circularly symmetric defect on a planar metal surface,” Proc. SPIE4447, 24–33 (2001).
    [CrossRef]

2012 (1)

2011 (3)

G. Brucoli and L. Martín-Moreno, “Comparative study of surface plasmon polariton scattering by shallow ridges and grooves,” Phys. Rev. B83, 045422 (2011).
[CrossRef]

G. Brucoli and L. Martín-Moreno, “Effect of defect depth on surface plasmon scattering by subwavelength surface defects,” Phys. Rev. B83, 075433 (2011).
[CrossRef]

B. Baumeier, F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a localized dielectric surface defect studied using an effective boundary condition,” Phys. Rev.A84, 013810 (2011).

2010 (1)

2009 (1)

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
[CrossRef]

2008 (1)

2007 (1)

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B76, 205424, (2007).
[CrossRef]

2006 (1)

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science311, 189–193 (2006).
[CrossRef] [PubMed]

2005 (3)

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5, 1726–1729 (2005).
[CrossRef] [PubMed]

I. I. Smolyaninov, C. C. Davis, and A. V. Zayats, “Image formation in surface plasmon polariton mirrors: applications in high resolution optical microscopy,” New. J. Phys.7, 175 (2005).
[CrossRef]

I. I. Smolyaninov, J. E. Elliott, A.V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett.94, 057401 (2005).
[CrossRef] [PubMed]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London)424, 824–830 (2003).
[CrossRef]

2001 (2)

M. Paulus and O. J. F. Martin, “Light propagation and scattering in stratified media: a Green’s tensor approach,” J. Opt. Soc. Am. A18, 854–861 (2001).
[CrossRef]

M. Kretschmann, T. A. Leskova, and A.A. Maradudin, “Excitation of surface plasmon polaritons by the scattering of a volume electromagnetic beam from a circularly symmetric defect on a planar metal surface,” Proc. SPIE4447, 24–33 (2001).
[CrossRef]

1997 (1)

A. V. Shchegrov, I. V. Novikov, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a circularly symmetric surface defect,” Phys. Rev. Lett.78, 4269–4272 (1997). Erratum: Phys. Rev. Lett. 79, 2597 (1997).
[CrossRef]

1985 (1)

A. A. Maradudin and W. M. Visscher, “Electrostatic and electromagnetic surface shape resonances,” Z. Phys.B60, 215–230 (1985).
[CrossRef]

1976 (1)

A. A. Maradudin and D. L. Mills, “Attenuation of Rayleigh surface waves by surface roughness,” Ann. Phys. (N.Y.)100262–309 (1976), Ref. 9.
[CrossRef]

1975 (1)

D. L. Mills, “Attenuation of surface polaritons by surface roughness,” Phys. Rev. B12, 4036–4046 (1975). Erratum: Phys. Rev. B14, 5539 (1976).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

1963 (1)

E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Z. NaturforschA23, 2135–2136 (1963).

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London)424, 824–830 (2003).
[CrossRef]

Baumeier, B.

B. Baumeier, F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a localized dielectric surface defect studied using an effective boundary condition,” Phys. Rev.A84, 013810 (2011).

Bose, R.

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
[CrossRef]

Brucoli, G.

G. Brucoli and L. Martín-Moreno, “Comparative study of surface plasmon polariton scattering by shallow ridges and grooves,” Phys. Rev. B83, 045422 (2011).
[CrossRef]

G. Brucoli and L. Martín-Moreno, “Effect of defect depth on surface plasmon scattering by subwavelength surface defects,” Phys. Rev. B83, 075433 (2011).
[CrossRef]

Chremmos, I.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Davis, C. C.

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B76, 205424, (2007).
[CrossRef]

I. I. Smolyaninov, C. C. Davis, and A. V. Zayats, “Image formation in surface plasmon polariton mirrors: applications in high resolution optical microscopy,” New. J. Phys.7, 175 (2005).
[CrossRef]

I. I. Smolyaninov, J. E. Elliott, A.V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett.94, 057401 (2005).
[CrossRef] [PubMed]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London)424, 824–830 (2003).
[CrossRef]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London)424, 824–830 (2003).
[CrossRef]

Elliott, J. E.

I. I. Smolyaninov, J. E. Elliott, A.V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett.94, 057401 (2005).
[CrossRef] [PubMed]

Hahn, J.

Hong, B. H.

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
[CrossRef]

Huerkamp, F.

B. Baumeier, F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a localized dielectric surface defect studied using an effective boundary condition,” Phys. Rev.A84, 013810 (2011).

Hung, Y.-J.

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B76, 205424, (2007).
[CrossRef]

Hwang, I.-C.

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Jouravlev, M. V.

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
[CrossRef]

Kaufman, L. J.

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
[CrossRef]

Kim, H.

Kim, K. S.

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
[CrossRef]

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
[CrossRef]

Kim, P.

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
[CrossRef]

Kim, W. Y.

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
[CrossRef]

Kim, Y.

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
[CrossRef]

Kretschmann, E.

E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Z. NaturforschA23, 2135–2136 (1963).

Kretschmann, M.

M. Kretschmann, T. A. Leskova, and A.A. Maradudin, “Excitation of surface plasmon polaritons by the scattering of a volume electromagnetic beam from a circularly symmetric defect on a planar metal surface,” Proc. SPIE4447, 24–33 (2001).
[CrossRef]

Lee, B.

Lee, J. Y.

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
[CrossRef]

Leskova, T. A.

B. Baumeier, F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a localized dielectric surface defect studied using an effective boundary condition,” Phys. Rev.A84, 013810 (2011).

M. Kretschmann, T. A. Leskova, and A.A. Maradudin, “Excitation of surface plasmon polaritons by the scattering of a volume electromagnetic beam from a circularly symmetric defect on a planar metal surface,” Proc. SPIE4447, 24–33 (2001).
[CrossRef]

T. A. Leskova and A. A. Maradudin (unpublished work).

Letnes, P. A.

Liu, Z.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5, 1726–1729 (2005).
[CrossRef] [PubMed]

Maradudin, A. A.

T. Nordam, P. A. Letnes, I. Simonsen, and A. A. Maradudin, “Satellite peaks in the scattering of light from the two-dimensional randomly rough surface of a dielectric film on a planar metal surface,” Opt. Express20, 11336– 11350 (2012).
[CrossRef] [PubMed]

B. Baumeier, F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a localized dielectric surface defect studied using an effective boundary condition,” Phys. Rev.A84, 013810 (2011).

A. V. Shchegrov, I. V. Novikov, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a circularly symmetric surface defect,” Phys. Rev. Lett.78, 4269–4272 (1997). Erratum: Phys. Rev. Lett. 79, 2597 (1997).
[CrossRef]

A. A. Maradudin and W. M. Visscher, “Electrostatic and electromagnetic surface shape resonances,” Z. Phys.B60, 215–230 (1985).
[CrossRef]

A. A. Maradudin and D. L. Mills, “Attenuation of Rayleigh surface waves by surface roughness,” Ann. Phys. (N.Y.)100262–309 (1976), Ref. 9.
[CrossRef]

T. A. Leskova and A. A. Maradudin (unpublished work).

Maradudin, A.A.

M. Kretschmann, T. A. Leskova, and A.A. Maradudin, “Excitation of surface plasmon polaritons by the scattering of a volume electromagnetic beam from a circularly symmetric defect on a planar metal surface,” Proc. SPIE4447, 24–33 (2001).
[CrossRef]

Martin, O. J. F.

Martín-Moreno, L.

G. Brucoli and L. Martín-Moreno, “Effect of defect depth on surface plasmon scattering by subwavelength surface defects,” Phys. Rev. B83, 075433 (2011).
[CrossRef]

G. Brucoli and L. Martín-Moreno, “Comparative study of surface plasmon polariton scattering by shallow ridges and grooves,” Phys. Rev. B83, 045422 (2011).
[CrossRef]

Mills, D. L.

A. A. Maradudin and D. L. Mills, “Attenuation of Rayleigh surface waves by surface roughness,” Ann. Phys. (N.Y.)100262–309 (1976), Ref. 9.
[CrossRef]

D. L. Mills, “Attenuation of surface polaritons by surface roughness,” Phys. Rev. B12, 4036–4046 (1975). Erratum: Phys. Rev. B14, 5539 (1976).
[CrossRef]

Min, S. K.

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
[CrossRef]

Nordam, T.

Novikov, I. V.

A. V. Shchegrov, I. V. Novikov, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a circularly symmetric surface defect,” Phys. Rev. Lett.78, 4269–4272 (1997). Erratum: Phys. Rev. Lett. 79, 2597 (1997).
[CrossRef]

Ozbay, E.

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science311, 189–193 (2006).
[CrossRef] [PubMed]

Paulus, M.

Pikus, Y.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5, 1726–1729 (2005).
[CrossRef] [PubMed]

Raether, H.

E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Z. NaturforschA23, 2135–2136 (1963).

Shchegrov, A. V.

A. V. Shchegrov, I. V. Novikov, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a circularly symmetric surface defect,” Phys. Rev. Lett.78, 4269–4272 (1997). Erratum: Phys. Rev. Lett. 79, 2597 (1997).
[CrossRef]

Simonsen, I.

Smolyaninov, I. I.

I. I. Smolyaninov, Y.-J. Hung, and C. C. Davis, “Imaging and focusing properties of plasmonic metamaterial devices,” Phys. Rev. B76, 205424, (2007).
[CrossRef]

I. I. Smolyaninov, C. C. Davis, and A. V. Zayats, “Image formation in surface plasmon polariton mirrors: applications in high resolution optical microscopy,” New. J. Phys.7, 175 (2005).
[CrossRef]

I. I. Smolyaninov, J. E. Elliott, A.V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett.94, 057401 (2005).
[CrossRef] [PubMed]

Srituravanich, W.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5, 1726–1729 (2005).
[CrossRef] [PubMed]

Steele, J. M.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5, 1726–1729 (2005).
[CrossRef] [PubMed]

Sun, C.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5, 1726–1729 (2005).
[CrossRef] [PubMed]

Visscher, W. M.

A. A. Maradudin and W. M. Visscher, “Electrostatic and electromagnetic surface shape resonances,” Z. Phys.B60, 215–230 (1985).
[CrossRef]

Wong, C. W.

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
[CrossRef]

Zayats, A. V.

I. I. Smolyaninov, C. C. Davis, and A. V. Zayats, “Image formation in surface plasmon polariton mirrors: applications in high resolution optical microscopy,” New. J. Phys.7, 175 (2005).
[CrossRef]

Zayats, A.V.

I. I. Smolyaninov, J. E. Elliott, A.V. Zayats, and C. C. Davis, “Far-field optical microscopy with a nanometer-scale resolution based on the in-plane image magnification by surface plasmon polaritons,” Phys. Rev. Lett.94, 057401 (2005).
[CrossRef] [PubMed]

Zhang, X.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5, 1726–1729 (2005).
[CrossRef] [PubMed]

Ann. Phys. (N.Y.) (1)

A. A. Maradudin and D. L. Mills, “Attenuation of Rayleigh surface waves by surface roughness,” Ann. Phys. (N.Y.)100262–309 (1976), Ref. 9.
[CrossRef]

J. Opt. Soc. Am. A (2)

Nano Lett. (1)

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with a plasmonic lens,” Nano Lett.5, 1726–1729 (2005).
[CrossRef] [PubMed]

Nature (1)

J. Y. Lee, B. H. Hong, W. Y. Kim, S. K. Min, Y. Kim, M. V. Jouravlev, R. Bose, K. S. Kim, I.-C. Hwang, L. J. Kaufman, C. W. Wong, P. Kim, and K. S. Kim, “Near-field focusing and magnification through self-assembled nanoscale spherical lenses,” Nature460, 498–501 (2009).
[CrossRef]

Nature (London) (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London)424, 824–830 (2003).
[CrossRef]

New. J. Phys. (1)

I. I. Smolyaninov, C. C. Davis, and A. V. Zayats, “Image formation in surface plasmon polariton mirrors: applications in high resolution optical microscopy,” New. J. Phys.7, 175 (2005).
[CrossRef]

Opt. Express (2)

Phys. Rev. (1)

B. Baumeier, F. Huerkamp, T. A. Leskova, and A. A. Maradudin, “Scattering of surface plasmon polaritons by a localized dielectric surface defect studied using an effective boundary condition,” Phys. Rev.A84, 013810 (2011).

Phys. Rev. B (5)

D. L. Mills, “Attenuation of surface polaritons by surface roughness,” Phys. Rev. B12, 4036–4046 (1975). Erratum: Phys. Rev. B14, 5539 (1976).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

A contour plot of σvac(θx, ϕx). The concentric dashed circles are the lines of constant θx, with θx = 0° at the center, and θx = 90° at the boundary. The azimuthal angle ϕx varies from 0° to 360°. ε2 = 2.69;A/R = 0.1 (a), 0.2 (b), 0.3 (c).

Fig. 2
Fig. 2

The same as Fig. 1, but with ε2 = 5.0.

Fig. 3
Fig. 3

Plots of σspp(ϕx). ε2 = 2.69;A/R = 0.1 (a), 0.2 (b), 0.3 (c).

Fig. 4
Fig. 4

The same as Fig. 3, but with ε2 = 5.0.

Fig. 5
Fig. 5

The field intensity |E(x|ω)|2 as a function of x at x3 = ζ (x)+5 nm. ε2 = 2.69; A/R = 0.1 (a), 0.2 (b), 0.3 (c).

Fig. 6
Fig. 6

The same as Fig. 5, but with ε2 = 5.0.

Equations (105)

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E ( x | ω ) = c ω [ i β 1 ( k | | ) k ^ | | + k | | x ^ 3 ] E o p ( k | | ) exp [ i k | | x | | β 1 ( k | | ) x 3 ] + d 2 q | | ( 2 π ) 2 { c ω [ i β 1 ( q | | ) q ^ | | q | | x ^ 3 ] a p ( q | | ) ε 3 β 1 ( q | | ) + ε 1 β 3 ( q | | ) + ( x ^ 3 + q ^ | | ) a s ( q | | ) β 1 ( q | | ) + β 3 ( q | | ) } exp [ i q | | x | | β 1 ( q | | ) x 3 ] .
k | | = k | | ( cos ϕ 0 , sin ϕ 0 , 0 ) ,
k | | = ω c ( ε 1 ε 3 ε 1 + ε 3 ) 1 2
β j ( q | | ) = [ q | | 2 ε j ( ω / c ) 2 ] 1 2 , Re β j ( q | | ) > 0 , Im β j ( q | | ) < 0 .
a p ( p | | ) ε 2 ε 1 2 ε 2 β 2 ( p | | ) d 2 q | | ( 2 π 2 ) { ( ε 2 β 3 ( p | | ) + ε 3 β 2 ( p | | ) ) × [ p | | q | | β 2 ( p | | ) ( p ^ | | q ^ | | ) β 1 ( q | | ) ] J ( + ) ( β 2 ( p | | ) β 1 ( q | | ) | p | | q | | ) β 2 ( p | | ) β 1 ( q | | ) + ( ε 2 β 3 ( p | | ) ε 3 β 2 ( p | | ) ) [ p | | q | | + β 2 ( p | | ) ( p ^ | | q ^ | | ) β 1 ( q | | ) ] × J ( ) ( β 2 ( p | | ) + β 1 ( q | | ) | p | | q | | ) β 2 ( p | | ) + β 1 ( q | | ) } a p ( q | | ) ε 3 β 1 ( q | | ) + ε 1 β 3 ( q | | ) ε 2 ε 1 2 ε 2 β 2 ( p | | ) d 2 q | | ( 2 π ) 2 { i ω c β 2 ( p | | ) ( p ^ | | × q ^ | | ) 3 × [ ( ε 2 β 3 ( p | | ) + ε 3 β 2 ( p | | ) ) J ( + ) ( β 2 ( p | | ) β 1 ( q | | ) | p | | q | | ) β 2 ( p | | ) β 1 ( q | | ) ( ε 2 β 3 ( p | | ) ε 3 β 2 ( p | | ) ) J ( ) ( β 2 ( p | | ) + β 1 ( q | | ) | p | | q | | ) β 2 ( p | | ) + β 1 ( q | | ) ] × a s ( q | | ) β 3 ( q | | ) + β 1 ( q | | ) = ε 2 ε 1 2 ε 2 β 2 ( p | | ) { ( ε 2 β 3 ( p | | ) + ε 3 β 2 ( p | | ) ) [ p | | k | | β 2 ( p | | ) ( p ^ | | k ^ | | ) β 1 ( k | | ) ] × J ( + ) ( β 2 ( p | | ) β 1 ( k | | ) | p | | k | | ) β 2 ( p | | ) β 1 ( k | | ) + ( ε 2 β 3 ( p | | ) ε 3 β 2 ( p | | ) ) [ p | | k | | + β 2 ( p | | ) ( p ^ | | k ^ | | ) β 1 ( k | | ) ] × J ( ) ( β 2 ( p | | ) + β 1 ( k | | ) | p | | k | | ) β 2 ( p | | ) + β 1 ( k | | ) } E o p ( k | | )
a s ( p | | ) ( ω c ) 2 ε 2 ε 1 2 β 2 ( p | | ) d 2 q | | ( 2 π ) 2 i c ω ( p ^ | | × q ^ | | ) β 1 ( q | | ) × [ ( β 3 ( p | | ) + β 2 ( p | | ) ) J ( + ) ( β 2 ( p | | ) β 1 ( q | | ) | p | | q | | ) β 2 ( p | | ) β 1 ( q | | ) + ( β 3 ( p | | ) β 2 ( p | | ) ) J ( ) ( β 2 ( p | | ) + β 1 ( q | | ) | p | | q | | ) β 2 ( p | | ) + β 1 ( q | | ) ] × a p ( q | | ) ε 3 β 1 ( q | | ) + ε 1 β 3 ( q | | ) ( ω c ) 2 ε 2 ε 1 2 β 2 ( p | | ) d 2 q | | ( 2 π ) 2 ( p ^ | | q ^ | | ) × [ ( β 3 ( p | | ) + β 2 ( p | | ) ) J ( + ) ( β 2 ( p | | ) β 1 ( q | | ) | p | | q | | ) β 2 ( p | | ) β 1 ( q | | ) + ( β 3 ( p | | ) β 2 ( p | | ) ) J ( ) ( β 2 ( p | | ) + β 1 ( q | | ) | p | | q | | ) β 2 ( p | | ) + β 1 ( q | | ) ] × a s ( q | | ) β 3 ( q | | ) + β 1 ( q | | ) = ( ω c ) 2 ε 2 ε 1 2 β 2 ( p | | ) i ( c ω ) ( p ^ | | × k ^ | | ) 3 β 1 ( k | | ) × [ ( β 3 ( p | | ) + β 2 ( p | | ) ) J ( + ) ( β 2 ( p | | ) β 1 ( k | | ) | p | | k | | ) β 2 ( p | | ) β 1 ( k | | ) + ( β 3 ( p | | ) β 2 ( p | | ) ) J ( ) ( β 2 ( p | | ) + β 1 ( k | | ) | p | | k | | ) β 2 ( p | | ) + β 1 ( k | | ) ] E o p ( k | | ) ,
J ( ± ) ( β 2 ( p | | ) β 1 ( q | | ) | p | | q | | ) = d 2 x | | exp [ i ( p | | q | | ) x | | ] { exp [ ( ± β 2 ( p | | ) β 1 ( q | | ) ) ζ ( x | | ) ] 1 } .
a p , s ( p | | ) = k = a k ( p , s ) ( p | | ) exp ( i k ϕ p ) ,
J ( + ) ( β 2 ( p | | ) β 1 ( q | | ) | p | | q | | ) = k = c k ( + ) ( p | | | q | | ) exp [ i k ( ϕ p ϕ q ) ]
J ( ) ( β 2 ( p | | ) + β 1 ( q | | ) | p | | q | | ) = k = c k ( ) ( p | | | q | | ) exp [ i k ( ϕ p ϕ q ) ]
c k ( + ) ( p | | | q | | ) = 2 π n = 1 ( β 2 ( p | | ) β 1 ( q | | ) ) n n ! × 0 d x | | x | | ζ n ( x | | ) J k ( p | | x | | ) J k ( q | | x | | )
c k ( ) ( p | | | q | | ) = 2 π n = 1 ( 1 ) n ( β 2 ( p | | ) + β 1 ( q | | ) ) n n ! × 0 d x | | x | | ζ n ( x | | ) J k ( p | | x | | ) J k ( q | | x | | ) ,
a k ( p ) ( p | | ) ε 2 ε 1 2 ε 2 β 2 ( p | | ) 0 d q | | 2 π q | | { ε 2 β 3 ( p | | ) + ε 3 β 2 ( p | | ) β 2 ( p | | ) β 1 ( q | | ) × [ p | | q | | c k ( + ) ( p | | | q | | ) 1 2 β 2 ( p | | ) β 1 ( q | | ) ( c k 1 ( + ) ( p | | | q | | ) + c k + 1 ( + ) ( p | | | q | | ) ) + ε 2 β 3 ( p | | ) ε 3 β 2 ( p | | ) β 2 ( p | | ) + β 1 ( q | | ) × [ p | | q | | c k ( ) ( p | | | q | | ) + 1 2 β 2 ( p | | ) β 1 ( q | | ) ( c k 1 ( ) ( p | | | q | | ) + c k + 1 ( ) ( p | | | q | | ) ) ] } a k ( p ) ( q | | ) ε 3 β 1 ( q | | ) + ε 1 β 3 ( q | | ) ε 2 ε 1 4 ε 2 ω c 0 d q | | 2 π q | | { ε 2 β 3 ( p | | ) + ε 3 β 3 ( p | | ) β 2 ( p | | ) β 1 ( q | | ) × [ c k 1 ( + ) ( p | | | q | | ) c k + 1 ( + ) ( p | | | q | | ) ] ε 2 β 3 ( p | | ) ε 3 β 2 ( p | | ) ) β 2 ( p | | ) + β 1 ( q | | ) [ c k 1 ( ) ( p | | | q | | ) c k + 1 ( ) ( p | | | q | | ) ] } a k ( s ) ( q | | ) β 3 ( q | | ) + β 1 ( q | | ) = ( ε 2 ε 1 ) 2 ε 2 β 2 ( p | | ) { ε 2 β 3 ( p | | ) + ε 3 β 2 ( p | | ) β 2 ( p | | ) β 1 ( k | | ) [ p | | k | | c k ( + ) ( p | | | k | | ) 1 2 β 2 ( p | | ) β 1 ( k | | ) ( c k 1 ( + ) ( p | | | k | | ) + c k + 1 ( + ) ( p | | | k | | ) ) ] + ε 2 β 3 ( p | | ) ε 3 β 2 ( p | | ) β 2 ( p | | ) + β 1 ( k | | ) [ p | | k | | c k ( ) ( p | | | k | | ) + 1 2 β 2 ( p | | ) β 1 ( k | | ) ( c k 1 ( ) ( p | | | k | | ) + c k + 1 ( ) ( p | | | k | | ) ) ] } × exp ( i k ϕ 0 ) E o p ( k | | ) .
a k ( s ) ( p | | ) + ω c ( ε 2 ε 1 ) 4 β 2 ( p | | ) 0 d q | | 2 π q | | { β 3 ( p | | ) + β 2 ( p | | ) β 2 ( p | | ) β 1 ( q | | ) × [ c k 1 ( + ) ( p | | | q | | ) c k + 1 ( + ) ( p | | | q | | ) ] + β 3 ( p | | ) β 2 ( p | | ) β 2 ( p | | ) + β 1 ( q | | ) × [ c k 1 ( ) ( p | | | q | | ) c k + 1 ( ) ( p | | | q | | ) ] } β 1 ( q | | ) a k ( p ) ( q | | ) ε 3 β 1 ( q | | ) + ε 1 β 3 ( q | | ) ω 2 c 2 ε 2 ε 1 4 β 2 ( p | | ) 0 d q | | ( 2 π ) q | | { β 3 ( p | | ) + β 2 ( p | | ) β 2 ( p | | ) β 1 ( q | | ) × [ c k 1 ( + ) ( p | | | q | | ) + c k + 1 ( + ) ( p | | | q | | ) ] + β 3 ( p | | ) β 2 ( p | | ) β 2 ( p | | ) + β 1 ( q | | ) × [ c k 1 ( ) ( p | | | q | | ) + c k + 1 ( ) ( p | | | q | | ) ] } a k ( s ) ( q | | ) β 3 ( q | | ) + β 1 ( q | | ) = ω c ε 2 c 1 4 β 2 ( p | | ) { β 3 ( p | | ) + β 2 ( p | | ) β 2 ( p | | ) β 1 ( k | | ) [ c k 1 ( + ) ( p | | | k | | ) c k + 1 ( + ) ( p | | | k | | ) ] + β 3 ( p | | ) β 2 ( p | | ) β 2 ( p | | ) + β 1 ( k | | ) [ c k 1 ( ) ( p | | | k | | ) c k + 1 ( ) ( p | | | k | | ) ] } × β 1 ( k | | ) exp [ i k ϕ 0 ] E o p ( k | | ) .
1 f p ( q | | ) = P ( 1 f p ( q | | ) ) + i π ε 1 β 3 ( k | | ) ( ε 1 2 ε 3 2 ) k | | δ ( q | | k | | ) ,
f p ( q | | ) ε 3 β 1 ( q | | ) + ε 1 β 3 ( q | | ) ,
E vac ( s c ) ( x | ω ) = = i ε 1 ω cos θ x 2 π c e i ε 1 ( ω / c ) x x × [ ε 1 e ^ p A p ( ε 1 x ^ | | ( ω c ) sin θ x ) + e ^ s A s ( e 1 x ^ | | ( ω c ) sin θ x ) ] , ε 1 ( ω / c ) x 1.
e ^ p = ( cos θ x cos ϕ x , cos θ x sin ϕ x , sin θ x )
e ^ s = ( sin ϕ x , cos ϕ x , 0 ) ,
cos θ x = x 3 x , sin θ x = x | | x
cos ϕ x = x 1 x | | , sin ϕ x = x 2 x | | ,
A p ( q | | ) = a p ( q | | ) ε 3 β 1 ( q | | ) + ε 1 β 3 ( q | | )
A s ( q | | ) = a s ( q | | ) β 1 ( q | | ) + β 3 ( q | | ) .
E spp ( s c ) ( x | ω ) = i e i k | | x | | β 1 ( k | | ) x 3 i π 4 ( 2 π k | | x | | ) 1 2 × c ε 1 β 3 ( k | | ) ω i x ^ | | β 1 ( k | | ) x ^ 3 k | | ε 3 2 ε 1 2 a p ( x ^ | | k | | ) , k | | x | | 1.
σ vac ( θ x , ϕ x ) = P vac ( θ x , ϕ x ) P inc
σ spp ( ϕ x ) = P spp ( ϕ x ) P inc ,
P vac ( θ x , ϕ x ) = c 8 π ε 1 ( ω 2 π c ) 2 cos 2 θ x × { ε 1 | A p ( ε 1 x ^ | | ( ω c ) sin θ x ) | 2 + | A s ( ε 1 x ^ | | ( ω c ) sin θ x ) | 2 }
P spp ( ϕ x ) = 1 2 ε 1 3 ( c 4 π ) 2 β 3 2 ( k | | ) ω β 1 ( k | | ) | a p ( x ^ | | k | | ) | 2 ( ε 3 2 ε 1 2 ) 2
P inc = ε 1 c 2 8 π ω k | | 2 β 1 ( k | | ) ( 1 1 ε 3 2 ( ω ) ) | E o p ( k | | ) | 2 .
ζ ( x | | ) = A exp ( x | | 2 / R 2 ) ,
c k ( ± ) ( p | | | q | | ) = π R 2 n = 1 [ ± A ( β 2 ( p | | ) β 1 ( q | | ) ) ] n n n ! × I k ( R 2 p | | q | | 2 n ) exp [ R 2 ( p | | 2 + q | | 2 ) 4 n ] ,
a p ( q | | ) = ε 2 ε 1 ε 2 ζ ^ ( q | | k | | ) [ ε 3 q | | k | | ε 2 β 3 ( q | | ) ( q ^ | | k ^ | | ) β 1 ( k | | ) ] E 0 p ( k | | )
a s ( q | | ) = i ( ε 2 ε 1 ) ζ ^ ( q | | k | | ) ω c ( q ^ | | × k ^ | | ) 3 β 1 ( k | | ) E 0 p ( k | | ) ,
ζ ^ ( Q | | ) = d 2 x | | ζ ( x | | ) exp ( i Q | | x | | ) = π A R 2 exp ( R 2 Q | | 2 / 4 ) .
σ vac ( θ x , ϕ x ) = 1 2 R ( A R ) 2 ( ω R c ) 5 ε 1 5 / 2 ε 2 2 | ε 2 | 1 2 ( ε 2 ε 1 ) 2 ( ε 3 2 ε 1 2 ) cos 2 θ x × exp [ R 2 2 ( k | | 2 2 ε 1 ω c k | | sin θ x cos ϕ x + ε 1 ω 2 c 2 sin 2 θ x ) ] × a 0 ( θ x ) + a 1 ( θ x ) cos ϕ x + a 2 ( θ x ) cos 2 ϕ x [ | ε 3 | ( | ε 3 | ε 1 ) sin 2 θ x ] ,
a 0 ( θ x ) = ε 2 2 | ε 3 | cos 2 θ x + ( | ε 3 | 3 + ε 2 2 ε 1 ) sin 2 θ x
a 1 ( θ x ) = 2 ε 2 | ε 3 | 3 / 2 sin θ x ( | ε 3 | + ε 1 sin 2 θ x ) 1 2
a 2 ( θ x ) = ε 2 2 | ε 3 | sin 2 θ x .
σ spp ( θ x ) = π 2 R ( A R ) ( ω R c ) 5 | ε 3 | 7 / 2 ε 1 7 / 2 ε 2 2 ( ε 2 ε 1 ) 2 ( | ε 2 | ε 1 ) 9 / 2 ( | ε 2 | + ε 1 ) 2 × exp ( 2 R 2 k | | 2 sin 2 1 2 ϕ x ) ( | ε 3 | + ε 2 cos ϕ x ) 2 .
E > ( x | ω ) = E 0 ( k | | ) exp ( i Q 0 x ) + d 2 q | | ( 2 π ) 2 A ( q | | ) exp ( i Q 1 x )
E ( f ) ( x | ω ) = d 2 q | | ( 2 π ) 2 [ F + ( q | | ) exp ( i Q + x ) + F ( q | | ) exp ( i Q x ) ]
E < ( x | ω ) = d 2 q | | ( 2 π ) 2 B ( q | | ) exp ( i Q 2 x )
Q 0 ( k | | ) = k | | α 1 ( k | | ) x ^ 3
Q 1 ( q | | ) = q | | + α 1 ( q | | ) x ^ 3
Q ± ( q | | ) = q | | ± α 2 ( q | | ) x ^ 3
Q 2 ( q | | ) = q | | α 3 ( q | | ) x ^ 3 ,
α j ( q | | ) = [ ε j ( ω / c ) 2 q | | 2 ] 1 2 Re α j ( q | | ) > 0 , Im α j ( q | | ) > 0 .
n × E > ( x | ω ) = n × E ( f ) ( x | ω )
n × ( × E > ( x | ω ) ) = n × ( × E ( f ) ( x | ω ) ) )
ε 1 n E > ( x | ω ) = ε 2 n E ( f ) ( x | ω ) ,
n = ( ζ ( x | | ) x 1 , ζ ( x | | ) x 2 , 1 ) .
( ε 2 ε 1 ) e i α 1 ( k | | ) d I ( α 2 ( p | | ) + α 1 ( k | | ) | p | | k | | ) α 2 ( p | | ) + α 1 ( k | | ) [ P + ( p | | ) × ( P | | + ( p | | ) × E 0 ( k | | ) ) ] + ( ε 2 ε 1 ) d 2 q | | ( 2 π ) 2 e i α 1 ( q | | ) d I ( α 2 ( p | | ) α 1 ( q | | ) | p | | q | | ) α 2 ( p | | ) α 1 ( q | | ) [ P + ( p | | ) × ( P + ( p | | ) × A ( q | | ) ) ] = 2 ε 2 α 2 ( p | | ) e i α 2 ( p | | ) d F + ( p | | ) ,
I ( γ | Q | | ) = d 2 x | | e i Q | | x | | e i γ ζ ( x | | ) .
( ε 2 ε 1 ) e i α 1 ( k | | ) d I ( ( α 2 ( p | | ) α 1 ( k | | ) ) | p | | k | | | ) α 2 ( p | | ) α 1 ( k | | ) [ P ( p | | ) × ( P ( p | | ) × E 0 ( k | | ) ) ] ( ε 2 ε 1 ) d 2 q | | ( 2 π ) 2 e i α 1 ( q | | ) d I ( ( α 2 ( p | | ) + α 1 ( q | | ) ) | p | | q | | | ) α 2 ( p | | ) + α 1 ( q | | ) [ P ( p | | ) × ( P ( p | | ) × A ( q | | ) ) ] = 2 ε 2 α 2 ( p | | ) e i α 2 ( p | | ) d F ( p | | ) .
x ^ 3 × E ( f ) ( x | ω ) = x ^ 3 × E < ( x | ω ) )
x ^ 3 × ( × E ( f ) ( x | ω ) ) ) = x ^ 3 × ( × E < ( x | ω ) )
ε 2 x ^ 3 E ( f ) ( x | ω ) = ε 3 x ^ 3 E < ( x | ω ) .
x ^ 3 × F + ( q | | ) + x ^ 3 × F ( q | | ) = x ^ 3 × B ( q | | )
i x ^ 3 × [ Q + ( q | | ) × F + ( q | | ) ] + i x ^ 3 × [ Q ( q | | ) × F ( q | | ) ] = i x ^ 3 × [ Q 2 ( q | | ) × B ( q | | ) ]
ε 2 x ^ 3 [ F + ( q | | ) + F ( q | | ) ] = ε 3 x ^ 3 B ( q | | ) .
d p ( q | | ) [ q | | F + ( q | | ) ] Δ p ( q | | ) [ q | | F ( q | | ) ] = 0
d s ( q | | ) [ x ^ 3 ( q | | × F + ( q | | ) ) ] + Δ s ( q | | ) [ x ^ 3 ( q | | × F ( q | | ) ) ] = 0 ,
d p ( q | | ) = ε 2 α 3 ( q | | ) + ε 3 α 2 ( q | | )
Δ p ( q | | ) = ε 2 α 3 ( q | | ) ε 3 α 2 ( q | | )
d s ( q | | ) = α 3 ( q | | ) + α 2 ( q | | )
Δ s ( q | | ) = α 3 ( q | | ) α 2 ( q | | ) .
p | | { d p ( p | | ) e i [ α 2 ( p | | ) + α 1 ( k | | ) ] d [ P + ( p | | ) × ( P + ( p | | ) × E 0 ( k | | ) ) ] × I ( α 2 ( p | | ) + α 1 ( k | | ) | p | | k | | ) α 2 ( p | | ) + α 1 ( k | | ) Δ p ( p | | ) e i [ α 2 ( p | | ) α 1 ( k | | ) ] d × [ P ( p | | ) × ( P ( p | | ) × E 0 ( k | | ) ) ] I ( ( α 2 ( p | | ) α 1 ( k | | ) ) | p | | k | | ) α 2 ( p | | ) α 1 ( k | | ) } + p | | d 2 q | | ( 2 π ) 2 { d p ( p | | ) e i [ α 2 ( p | | ) α 1 ( q | | ) ] d [ P + ( p | | ) × ( P + ( p | | ) × A ( q | | ) ) ] × I ( α 2 ( p | | ) α 1 ( q | | ) | p | | q | | ) α 2 ( p | | ) α 1 ( q | | ) Δ p ( p | | ) e i [ α 2 ( p | | ) + α 1 ( q | | ) ] d × [ P ( p | | ) × ( P ( p | | ) × A ( q | | ) ) ] I ( ( α 2 ( p | | ) + α 1 ( q | | ) ) | p | | q | | ) α 2 ( p | | ) + α 1 ( q | | ) } = 0 .
x ^ 3 p | | × { d s ( p | | ) e i [ α 2 ( p | | ) + α 1 ( k | | ) ] d [ P + ( p | | ) × P + ( p | | ) × E 0 ( k | | ) ) ] × I ( α 2 ( p | | ) + α 1 ( k | | ) | p | | k | | ) α 2 ( p | | ) + α 1 ( k | | ) ] + Δ s ( p | | ) e i [ α 2 ( p | | ) α 1 ( k | | ) ] d × [ P ( p | | ) × P ( p | | ) × E 0 ( k | | ) ) ] I ( ( α 2 ( p | | ) α 1 ( k | | ) ) | p | | k | | ) α 2 ( p | | ) α 1 ( k | | ) } + x ^ 3 p | | × d 2 q | | ( 2 π ) 2 { d s ( p | | ) e i [ α 2 ( p | | ) α 1 ( q | | ) ] d × [ P + ( p | | ) × ( P + ( p | | ) × A ( q | | ) ) ] I ( α 2 ( p | | ) α 1 ( q | | ) | p | | q | | ) α 2 ( p | | ) α 1 ( q | | ) + Δ s ( p | | ) e i [ α 2 ( p | | ) + α 1 ( q | | ) ] d [ P ( p | | ) × P ( p | | ) × A ( q | | ) ) ] × I ( ( α 2 ( p | | ) + α 1 ( q | | ) ) | p | | q | | ) α 2 ( p | | ) + α 1 ( q | | ) } = 0 ,
E 0 ( k | | ) = c ω [ α 1 ( k | | ) k ^ | | + k | | x ^ 3 ] E o p ( k | | ) + ( x ^ 3 × k ^ | | ) E 0 s ( k | | )
A ( q | | ) = c ω [ α 1 ( q | | ) q ^ | | q | | x ^ 3 ] A p ( q | | ) + ( x ^ 3 × q ^ | | ) A s ( q | | ) .
d 2 q | | ( 2 π ) 2 [ M p p ( p | | | q | | ) A p ( q | | ) + M p s ( p | | | q | | ) A s ( q | | ) ] = [ N p p ( p | | | k | | ) E 0 p ( k | | ) + N p s ( p | | | k | | ) E 0 s ( k | | ) ]
d 2 q | | ( 2 π ) 2 [ M s p ( p | | | q | | ) A p ( q | | ) + M s s ( p | | | q | | ) A s ( q | | ) ] = [ N s p ( p | | | k | | ) E 0 p ( k | | ) + N s s ( p | | | k | | ) E 0 s ( k | | ) ] ,
M p p ( p | | | q | | ) = d p ( p | | ) e i [ α 2 ( p | | ) α 1 ( q | | ) ] d × [ α 2 ( p | | ) ( p ^ | | q ^ | | ) α 1 ( q | | ) + p | | q | | ] I ( α 2 ( p | | ) α 1 ( q | | ) | p | | q | | ) α 2 ( p | | ) α 1 ( q | | ) Δ p ( p | | ) e i [ α 2 ( p | | ) + α 1 ( q | | ) ] d × [ α 2 ( p | | ) ( p ^ | | q ^ | | ) α 1 ( q | | ) p | | q | | ] I ( ( α 2 ( p | | ) + α 1 ( q | | ) ) | p | | q | | ) α 2 ( p | | ) + α 1 ( q | | )
M p s ( p | | | q | | ) = ω c α 2 ( p | | ) ( p ^ | | × q ^ | | ) 3 × { d p ( p | | ) e i [ α 2 ( p | | ) α 2 ( q | | ) ] d I ( α 2 ( p | | ) α 1 ( q | | ) | p | | q | | ) α 2 ( p | | ) α 1 ( q | | ) Δ p ( p | | ) e i [ α 2 ( p | | ) + α 1 ( q | | ) ] d I ( ( α 2 ( p | | ) + α 1 ( q | | ) ) | p | | q | | ) α 2 ( p | | ) + α 1 ( q | | ) }
M s p ( p | | | q | | ) = c ω α 1 ( q | | ) ( p ^ | | × q ^ | | ) 3 × { d s ( p | | ) e i [ α 2 ( p | | ) α 1 ( q | | ) ] d I ( α 2 ( p | | ) α 1 ( q | | ) | p | | q | | ) α 2 ( p | | ) α 1 ( q | | ) + Δ s ( p | | ) e i [ α 2 ( p | | ) + α 1 ( q | | ) ] d I ( ( α 2 ( p | | ) + α 1 ( q | | ) ) | p | | q | | ) α 2 ( p | | ) + α 1 ( q | | ) }
M s s ( p | | | q | | ) = ( p ^ | | q ^ | | ) × { d s ( p | | ) e i [ α 2 ( p | | ) α 1 ( q | | ) ] d I ( α 2 ( p | | ) α 1 ( q | | ) | p | | q | | ) α 2 ( p | | ) α 1 ( q | | ) + Δ s ( p | | ) e i [ α 2 ( p | | ) + α 1 ( q | | ) ] d I ( ( α 2 ( p | | ) + α 1 ( q | | ) ) | p | | q | | ) α 2 ( p | | ) + α 1 ( q | | ) }
N p p ( p | | | k | | ) = d p ( p | | ) e i [ α 2 ( p | | ) + α 1 ( k | | ) ] d × [ α 2 ( p | | ) ( p ^ | | k ^ | | ) α 1 ( k | | ) p | | k | | ] I ( α 2 ( p | | ) + α 1 ( k | | ) | p | | k | | ) α 2 ( p | | ) + α 1 ( k | | ) Δ p ( p | | ) e i [ α 2 ( p | | ) α 1 ( k | | ) ] d × [ α 2 ( p | | ) ( p ^ | | k ^ | | ) α 1 ( k | | ) + p | | k | | ] I ( ( α 2 ( p | | ) α 1 ( k | | ) ) | p | | k | | ) α 2 ( p | | ) α 1 ( k | | )
N p s ( p | | | k | | ) = ω c α 2 ( p | | ) ( p ^ | | × k ^ | | ) 3 × { d p ( p | | ) e i [ α 2 ( p | | ) + α 1 ( k | | ) ] d I ( α 2 ( p | | ) + α 1 ( k | | ) | p | | k | | ) α 2 ( p | | ) + α 1 ( k | | ) Δ p ( p | | ) e i [ α 2 ( p | | ) α 1 ( k | | ) ] d I ( ( α 2 ( p | | ) α 1 ( k | | ) ) | p | | k | | ) α 2 ( p | | ) α 1 ( k | | )
N s p ( p | | | k | | ) = c ω α 1 ( k | | ) ( p ^ | | × k ^ | | ) 3 × { d s ( p | | ) e i [ α 2 ( p | | ) + α 1 ( k | | ) ] d I ( α 2 ( p | | ) + α 1 ( k | | ) | p | | k | | ) α 2 ( p | | ) + α 1 ( k | | ) + Δ s ( p | | ) e i [ α 2 ( p | | ) α 1 ( k | | ) ] d I ( ( α 2 ( p | | ) α 1 ( k | | ) ) | p | | k | | ) α 2 ( p | | ) α 1 ( k | | )
N s s ( p | | | k | | ) = ( p ^ | | k ^ | | ) { d s ( p | | ) e i [ α 2 ( p | | ) + 1 ( k | | ) ] d I ( α 2 ( p | | ) + α 1 ( k | | ) | p | | k | | ) α 2 ( p | | ) + α 1 ( k | | ) + Δ s ( p | | ) e i [ α 2 ( p | | ) α 1 ( k | | ) ] d I ( ( α 2 ( p | | ) α 1 ( k | | ) ) | p | | k | | ) α 2 ( p | | ) α 1 ( k | | ) } .
I ( γ | Q | | ) = ( 2 π ) 2 δ ( Q | | ) + J ( γ | Q | | ) ,
J ( γ | Q | | ) = d 2 x | | e i Q | | x | | ( e i γ ζ ( x | | ) 1 ) .
M p p ( p | | | q | | ) = ( 2 π ) 2 δ ( p | | q | | ) 2 ε 2 α 2 ( p | | ) ε 2 ε 1 [ ε 1 α 3 ( p | | ) + ε 3 α 1 ( p | | ) ] + M ˜ p p ( p | | | q | | )
M p s ( p | | | q | | ) = M ˜ p s ( p | | | q | | )
M s p ( p | | | q | | ) = M ˜ s p ( p | | | q | | )
M s s ( p | | | q | | ) = ( 2 π ) 2 δ ( p | | | q | | ) c 2 ω 2 2 α 2 ( p | | ) ε 2 ε 1 [ α 3 ( p | | ) + α 1 ( p | | ) ] + M ˜ s s ( p | | | q | | )
N p p ( p | | | k | | ) = ( 2 π ) 2 δ ( p | | | k | | ) 2 ε 2 α 2 ( k | | ) ε 2 ε 1 [ ε 3 α 1 ( k | | ) ε 1 α 3 ( k | | ) ] + N ˜ p p ( p | | | k | | )
N s p ( p | | | k | | ) = N ˜ s p ( p | | | k | | ) ,
M ˜ p p ( p | | | q | | ) = d p ( p | | ) [ α 2 ( p | | ) ( p ^ | | q ^ | | ) α 1 ( q | | ) + p | | q | | ] × J ( α 2 ( p | | ) α 1 ( q | | ) | p | | q | | ) α 2 ( p | | ) α 1 ( q | | ) Δ p ( p | | ) [ α 2 ( p | | ) ( p ^ | | q ^ | | ) α 1 ( q | | ) p | | q | | ] × J ( ( α 2 ( p | | ) + α 1 ( q | | ) ) | p | | k | | ) α 2 ( p | | ) + α 1 ( q | | )
M ˜ p s ( p | | | q | | ) = ω c α 2 ( p | | ) ( p ^ | | × q ^ | | ) 3 × { d p ( p | | ) J ( α 2 ( p | | ) α 1 ( q | | ) | p | | q | | ) α 2 ( p | | ) α 1 ( q | | ) Δ p ( p | | ) J ( ( α 2 ( p | | ) + α 1 ( q | | ) ) | p | | q | | ) α 2 ( p | | ) + α 1 ( q | | ) }
M ˜ s p ( p | | | q | | ) = c ω ( p ^ | | × q ^ | | ) 3 α 1 ( q | | ) × { d s ( p | | ) J ( α 2 ( p | | ) α 1 ( q | | ) | p | | q | | ) α 2 ( p | | ) α 1 ( q | | ) + Δ s ( p | | ) J ( ( α 2 ( p | | ) + α 1 ( q | | ) ) | p | | q | | ) α 2 ( p | | ) + α 1 ( q | | ) }
M ˜ s s ( p | | | q | | ) = ( p ^ | | q ^ | | ) { d s ( p | | ) J ( α 2 ( p | | ) α 1 ( q | | ) | p | | q | | ) α 2 ( p | | ) α 1 ( q | | ) + Δ s ( p | | ) J ( ( α 2 ( p | | ) + α 1 ( q | | ) ) | p | | q | | ) α 2 ( p | | ) + α 1 ( q | | ) }
N ˜ p p ( p | | | k | | ) = d p ( p | | ) [ α 2 ( p | | ) ( p ^ | | k ^ | | ) α 1 ( k | | ) p | | k | | ] × J ( α 2 ( p | | ) + α 1 ( k | | ) | p | | k | | ) α 2 ( p | | ) + α 1 ( k | | ) Δ p ( p | | ) [ α 2 ( p | | ) ( p ^ | | k ^ | | ) α 1 ( k | | ) + p | | k | | ] × J ( ( α 2 ( p | | ) α 1 ( k | | ) ) | p | | k | | ) α 2 ( p | | ) α 1 ( k | | )
N ˜ s p ( p | | | k | | ) = c ω ( p ^ | | × k ^ | | ) 3 α 1 ( k | | ) × { d s ( p | | ) J ( α 2 ( p | | ) + α 1 ( k | | ) | p | | k | | ) α 2 ( p | | ) + α 1 ( k | | ) + Δ s ( p | | ) J ( ( α 2 ( p | | ) α 1 ( k | | ) ) | p | | k | | ) α 2 ( p | | ) α 1 ( k | | ) } .
[ ε 1 α 3 ( p | | ) + ε 3 α 1 ( p | | ) ] A p ( p | | ) + ε 2 ε 1 2 ε 2 α 2 ( p | | ) d 2 q | | ( 2 π ) 2 [ M ˜ p p ( p | | | q | | ) A p ( q | | ) + M ˜ p s ( p | | | q | | ) A s ( q | | ) ] = { ( 2 π ) 2 δ ( p | | k | | ) [ ε 3 α 1 ( k | | ) ε 1 α 3 ( k | | ) ] + ε 2 ε 1 2 ε 2 α 2 ( p | | ) N ˜ p p ( p | | | k | | ) } E 0 p ( k | | )
[ α 3 ( p | | ) + α 1 ( p | | ) ] A s ( p | | ) + ω 2 c 2 ε 2 ε 1 2 α 2 ( p | | ) d 2 q | | ( 2 π ) 2 [ M ˜ s p ( p | | | q | | ) A p ( q | | ) + M ˜ s s ( p | | | q | | ) A s ( q | | ) ] = ω 2 c 2 ε 2 ε 1 2 α 2 ( p | | ) N ˜ s p ( p | | | k | | ) E 0 p ( k | | ) .
α 1 ( k | | ) = i β 1 ( k | | )
α 1 ( q | | ) = i β 1 ( q | | )
α 2 ( q | | ) = i β 2 ( q | | )
α 3 ( q | | ) = i β 3 ( q | | ) ,
β j ( q | | ) = [ q | | 2 ε j ( ω / c ) 2 ] 1 2 , Re β j ( q | | ) > 0 , Im β j ( q | | ) < 0 .
J ( i ( β 2 ( p | | ) β 1 ( q | | ) ) | p | | q | | ) = d 2 x | | e i ( p | | q | | ) x | | ( e ( β 2 ( p | | ) β 1 ( q | | ) ) ζ ( x | | ) 1 ) = J ( + ) ( β 2 ( p | | ) β 1 ( q | | ) | p | | q | | )
J ( i ( β 2 ( p | | ) + β 1 ( q | | ) ) | p | | q | | ) = d 2 x | | e i ( p | | q | | ) x | | ( e ( β 2 ( p | | ) + β 1 ( q | | ) ) ζ ( x | | ) 1 ) = J ( ) ( β 2 ( p | | ) + β 1 ( q | | ) | p | | q | | )
A p ( q | | ) = a p ( q | | ) ε 3 β 1 ( q | | ) + ε 1 β 3 ( q | | )
A s ( q | | ) = a s ( q | | ) β 1 ( q | | ) + β 3 ( q | | ) ,

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