Abstract

We present an approach to the problem of electromagnetic scattering by a subwavelength circular hole in a perfect metal plate of finite thickness. The matched asymptotic expansion is employed to solve the scattered fields in the inner and outer regions with respect to the hole position. By use of the dual potentials and Fabrikant’s theory, the solutions are expressed as a combination of spherical functions. In particular, the scattering behavior is illustrated with the near-field quasistatic potentials as well as the far-field radiation patterns. The dipole strengths associated with the subwavelength hole are scaled by the angle of incidence in terms of the vertical electric field and/or horizontal magnetic field components. The effect of the plate thickness is manifest on the attenuation of the dipole strengths associated with the subwavelength hole. For a large thickness, the dipole strengths approach the values for an infinitely deep hole, while for a very small thickness, the results coincide with the Bethe theory.

© 2010 Optical Society of America

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References

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  1. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
    [CrossRef] [PubMed]
  2. M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
    [CrossRef] [PubMed]
  3. H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P. F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
    [CrossRef] [PubMed]
  4. H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
    [CrossRef]
  5. A. F. Stevenson, “Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatterer)/wavelength,” J. Appl. Phys. 24, 1134–1142 (1953).
    [CrossRef]
  6. A. F. Stevenson, “Electromagnetic scattering by an ellipsoid in the third approximation,” J. Appl. Phys. 24, 1143–1151 (1953).
    [CrossRef]
  7. J. Bazer and L. Rubenfeld, “Diffraction of electromagnetic waves by a circular aperture in an infinitely conducting plane screen,” SIAM J. Appl. Math. 13, 558–585 (1965).
    [CrossRef]
  8. C. Huang, R. D. Kodis, and H. Levine, “Diffraction by apertures,” J. Appl. Phys. 26, 151–165 (1955).
    [CrossRef]
  9. C. J. Bouwkamp, “Theoretical and numerical treatment of diffraction through a circular aperture,” IEEE Trans. Antennas Propag. 18, 152–176 (1970).
    [CrossRef]
  10. C. Butler, Y. Rahmat-Samii, and R. Mittra, “Electromagnetic penetration through apertures in conducting surfaces,” IEEE Trans. Antennas Propag. 26, 82–93 (1978).
    [CrossRef]
  11. A. Roberts, “Electromagnetic theory of diffraction by a circular aperture in a thick, perfectly conducting screen,” J. Opt. Soc. Am. A 4, 1970–1984 (1987).
    [CrossRef]
  12. F. J. García de Abajo, “Light transmission through a single cylindrical hole in a metallic film,” Opt. Express 10, 1475–1484 (2002).
    [PubMed]
  13. C. W. Chang, A. K. Sarychev, and V. M. Shalaev, “Light diffraction by a subwavelength circular aperture,” Laser Phys. Lett. 2, 351–355 (2005).
    [CrossRef]
  14. A. Roberts, “Near-zone fields behind circular apertures in thick, perfectly conducting screens,” J. Appl. Phys. 65, 2896–2899 (1989).
    [CrossRef]
  15. J. H. Lee and H. J. Eom, “Electrostatic potential through a circular aperture in a thick conducting plane,” IEEE Trans. Microwave Theory Tech. 44, 341–343 (1996).
    [CrossRef]
  16. J. G. Lee and H. J. Eom, “Magnetostatic potential distribution through a circular aperture in a thick conducting plane,” IEEE Trans. Electromagn. Compat. 40, 97–99 (1998).
    [CrossRef]
  17. R. L. Gluckstern and J. A. Diamond, “Penetration of fields through a circular hole in a wall of finite thickness,” IEEE Trans. Microwave Theory Tech. 39, 274–279 (1991).
    [CrossRef]
  18. B. Radak and R. L. Gluckstern, “Penetration of electromagnetic fields through an elliptical hole in a wall of finite thickness,” IEEE Trans. Microwave Theory Tech. 43, 194–204 (1995).
    [CrossRef]
  19. D. G. Crighton, A. P. Dowling, J. E. Ffowcs Williams, M. Heckl, and F. G. Leppington, Modern Methods in Analytical Acoustics: Lecture Notes (Springer, 1992).
    [CrossRef]
  20. C. Kuo, R. L. Chern, and C. C. Chang, “Sound scattering by a compact circular pore,” J. Sound Vibrat. 319, 622–645 (2009).
    [CrossRef]
  21. V. I. Fabrikant, Applications of Potential Theory in Mechanics: A Selection of New Results (Kluwer, 1989).
  22. F. J. García de Abajo and J. J. Sáenz, “Electromagnetic surface modes in structured perfect-conductor surfaces,” Phys. Rev. Lett. 95, 233901 (2005).
    [CrossRef]
  23. D. M. Pozar, Microwave Engineering, 3rd ed. (Wiley, 2005).
  24. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
  25. D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, 1999).
  26. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
    [CrossRef]

2009

C. Kuo, R. L. Chern, and C. C. Chang, “Sound scattering by a compact circular pore,” J. Sound Vibrat. 319, 622–645 (2009).
[CrossRef]

2007

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
[CrossRef] [PubMed]

2005

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P. F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

C. W. Chang, A. K. Sarychev, and V. M. Shalaev, “Light diffraction by a subwavelength circular aperture,” Laser Phys. Lett. 2, 351–355 (2005).
[CrossRef]

F. J. García de Abajo and J. J. Sáenz, “Electromagnetic surface modes in structured perfect-conductor surfaces,” Phys. Rev. Lett. 95, 233901 (2005).
[CrossRef]

2003

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[CrossRef] [PubMed]

2002

1998

J. G. Lee and H. J. Eom, “Magnetostatic potential distribution through a circular aperture in a thick conducting plane,” IEEE Trans. Electromagn. Compat. 40, 97–99 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

1996

J. H. Lee and H. J. Eom, “Electrostatic potential through a circular aperture in a thick conducting plane,” IEEE Trans. Microwave Theory Tech. 44, 341–343 (1996).
[CrossRef]

1995

B. Radak and R. L. Gluckstern, “Penetration of electromagnetic fields through an elliptical hole in a wall of finite thickness,” IEEE Trans. Microwave Theory Tech. 43, 194–204 (1995).
[CrossRef]

1991

R. L. Gluckstern and J. A. Diamond, “Penetration of fields through a circular hole in a wall of finite thickness,” IEEE Trans. Microwave Theory Tech. 39, 274–279 (1991).
[CrossRef]

1989

A. Roberts, “Near-zone fields behind circular apertures in thick, perfectly conducting screens,” J. Appl. Phys. 65, 2896–2899 (1989).
[CrossRef]

1987

1978

C. Butler, Y. Rahmat-Samii, and R. Mittra, “Electromagnetic penetration through apertures in conducting surfaces,” IEEE Trans. Antennas Propag. 26, 82–93 (1978).
[CrossRef]

1970

C. J. Bouwkamp, “Theoretical and numerical treatment of diffraction through a circular aperture,” IEEE Trans. Antennas Propag. 18, 152–176 (1970).
[CrossRef]

1965

J. Bazer and L. Rubenfeld, “Diffraction of electromagnetic waves by a circular aperture in an infinitely conducting plane screen,” SIAM J. Appl. Math. 13, 558–585 (1965).
[CrossRef]

1955

C. Huang, R. D. Kodis, and H. Levine, “Diffraction by apertures,” J. Appl. Phys. 26, 151–165 (1955).
[CrossRef]

1953

A. F. Stevenson, “Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatterer)/wavelength,” J. Appl. Phys. 24, 1134–1142 (1953).
[CrossRef]

A. F. Stevenson, “Electromagnetic scattering by an ellipsoid in the third approximation,” J. Appl. Phys. 24, 1143–1151 (1953).
[CrossRef]

1944

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
[CrossRef]

Bazer, J.

J. Bazer and L. Rubenfeld, “Diffraction of electromagnetic waves by a circular aperture in an infinitely conducting plane screen,” SIAM J. Appl. Math. 13, 558–585 (1965).
[CrossRef]

Bethe, H. A.

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
[CrossRef]

Bonod, N.

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P. F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

Bouwkamp, C. J.

C. J. Bouwkamp, “Theoretical and numerical treatment of diffraction through a circular aperture,” IEEE Trans. Antennas Propag. 18, 152–176 (1970).
[CrossRef]

Butler, C.

C. Butler, Y. Rahmat-Samii, and R. Mittra, “Electromagnetic penetration through apertures in conducting surfaces,” IEEE Trans. Antennas Propag. 26, 82–93 (1978).
[CrossRef]

Capoulade, J.

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P. F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

Chang, C. C.

C. Kuo, R. L. Chern, and C. C. Chang, “Sound scattering by a compact circular pore,” J. Sound Vibrat. 319, 622–645 (2009).
[CrossRef]

Chang, C. W.

C. W. Chang, A. K. Sarychev, and V. M. Shalaev, “Light diffraction by a subwavelength circular aperture,” Laser Phys. Lett. 2, 351–355 (2005).
[CrossRef]

Chern, R. L.

C. Kuo, R. L. Chern, and C. C. Chang, “Sound scattering by a compact circular pore,” J. Sound Vibrat. 319, 622–645 (2009).
[CrossRef]

Craighead, H. G.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[CrossRef] [PubMed]

Crighton, D. G.

D. G. Crighton, A. P. Dowling, J. E. Ffowcs Williams, M. Heckl, and F. G. Leppington, Modern Methods in Analytical Acoustics: Lecture Notes (Springer, 1992).
[CrossRef]

Diamond, J. A.

R. L. Gluckstern and J. A. Diamond, “Penetration of fields through a circular hole in a wall of finite thickness,” IEEE Trans. Microwave Theory Tech. 39, 274–279 (1991).
[CrossRef]

Dintinger, J.

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P. F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

Dowling, A. P.

D. G. Crighton, A. P. Dowling, J. E. Ffowcs Williams, M. Heckl, and F. G. Leppington, Modern Methods in Analytical Acoustics: Lecture Notes (Springer, 1992).
[CrossRef]

Ebbesen, T. W.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
[CrossRef] [PubMed]

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P. F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Eom, H. J.

J. G. Lee and H. J. Eom, “Magnetostatic potential distribution through a circular aperture in a thick conducting plane,” IEEE Trans. Electromagn. Compat. 40, 97–99 (1998).
[CrossRef]

J. H. Lee and H. J. Eom, “Electrostatic potential through a circular aperture in a thick conducting plane,” IEEE Trans. Microwave Theory Tech. 44, 341–343 (1996).
[CrossRef]

Fabrikant, V. I.

V. I. Fabrikant, Applications of Potential Theory in Mechanics: A Selection of New Results (Kluwer, 1989).

Ffowcs Williams, J. E.

D. G. Crighton, A. P. Dowling, J. E. Ffowcs Williams, M. Heckl, and F. G. Leppington, Modern Methods in Analytical Acoustics: Lecture Notes (Springer, 1992).
[CrossRef]

Foquet, M.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[CrossRef] [PubMed]

García de Abajo, F. J.

F. J. García de Abajo and J. J. Sáenz, “Electromagnetic surface modes in structured perfect-conductor surfaces,” Phys. Rev. Lett. 95, 233901 (2005).
[CrossRef]

F. J. García de Abajo, “Light transmission through a single cylindrical hole in a metallic film,” Opt. Express 10, 1475–1484 (2002).
[PubMed]

Genet, C.

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
[CrossRef] [PubMed]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Gluckstern, R. L.

B. Radak and R. L. Gluckstern, “Penetration of electromagnetic fields through an elliptical hole in a wall of finite thickness,” IEEE Trans. Microwave Theory Tech. 43, 194–204 (1995).
[CrossRef]

R. L. Gluckstern and J. A. Diamond, “Penetration of fields through a circular hole in a wall of finite thickness,” IEEE Trans. Microwave Theory Tech. 39, 274–279 (1991).
[CrossRef]

Griffiths, D. J.

D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, 1999).

Heckl, M.

D. G. Crighton, A. P. Dowling, J. E. Ffowcs Williams, M. Heckl, and F. G. Leppington, Modern Methods in Analytical Acoustics: Lecture Notes (Springer, 1992).
[CrossRef]

Huang, C.

C. Huang, R. D. Kodis, and H. Levine, “Diffraction by apertures,” J. Appl. Phys. 26, 151–165 (1955).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

Kodis, R. D.

C. Huang, R. D. Kodis, and H. Levine, “Diffraction by apertures,” J. Appl. Phys. 26, 151–165 (1955).
[CrossRef]

Korlach, J.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[CrossRef] [PubMed]

Kuo, C.

C. Kuo, R. L. Chern, and C. C. Chang, “Sound scattering by a compact circular pore,” J. Sound Vibrat. 319, 622–645 (2009).
[CrossRef]

Lee, J. G.

J. G. Lee and H. J. Eom, “Magnetostatic potential distribution through a circular aperture in a thick conducting plane,” IEEE Trans. Electromagn. Compat. 40, 97–99 (1998).
[CrossRef]

Lee, J. H.

J. H. Lee and H. J. Eom, “Electrostatic potential through a circular aperture in a thick conducting plane,” IEEE Trans. Microwave Theory Tech. 44, 341–343 (1996).
[CrossRef]

Lenne, P. F.

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P. F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

Leppington, F. G.

D. G. Crighton, A. P. Dowling, J. E. Ffowcs Williams, M. Heckl, and F. G. Leppington, Modern Methods in Analytical Acoustics: Lecture Notes (Springer, 1992).
[CrossRef]

Levene, M. J.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[CrossRef] [PubMed]

Levine, H.

C. Huang, R. D. Kodis, and H. Levine, “Diffraction by apertures,” J. Appl. Phys. 26, 151–165 (1955).
[CrossRef]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Mittra, R.

C. Butler, Y. Rahmat-Samii, and R. Mittra, “Electromagnetic penetration through apertures in conducting surfaces,” IEEE Trans. Antennas Propag. 26, 82–93 (1978).
[CrossRef]

Popov, E.

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P. F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

Pozar, D. M.

D. M. Pozar, Microwave Engineering, 3rd ed. (Wiley, 2005).

Radak, B.

B. Radak and R. L. Gluckstern, “Penetration of electromagnetic fields through an elliptical hole in a wall of finite thickness,” IEEE Trans. Microwave Theory Tech. 43, 194–204 (1995).
[CrossRef]

Rahmat-Samii, Y.

C. Butler, Y. Rahmat-Samii, and R. Mittra, “Electromagnetic penetration through apertures in conducting surfaces,” IEEE Trans. Antennas Propag. 26, 82–93 (1978).
[CrossRef]

Rigneault, H.

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P. F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

Roberts, A.

A. Roberts, “Near-zone fields behind circular apertures in thick, perfectly conducting screens,” J. Appl. Phys. 65, 2896–2899 (1989).
[CrossRef]

A. Roberts, “Electromagnetic theory of diffraction by a circular aperture in a thick, perfectly conducting screen,” J. Opt. Soc. Am. A 4, 1970–1984 (1987).
[CrossRef]

Rubenfeld, L.

J. Bazer and L. Rubenfeld, “Diffraction of electromagnetic waves by a circular aperture in an infinitely conducting plane screen,” SIAM J. Appl. Math. 13, 558–585 (1965).
[CrossRef]

Sáenz, J. J.

F. J. García de Abajo and J. J. Sáenz, “Electromagnetic surface modes in structured perfect-conductor surfaces,” Phys. Rev. Lett. 95, 233901 (2005).
[CrossRef]

Sarychev, A. K.

C. W. Chang, A. K. Sarychev, and V. M. Shalaev, “Light diffraction by a subwavelength circular aperture,” Laser Phys. Lett. 2, 351–355 (2005).
[CrossRef]

Shalaev, V. M.

C. W. Chang, A. K. Sarychev, and V. M. Shalaev, “Light diffraction by a subwavelength circular aperture,” Laser Phys. Lett. 2, 351–355 (2005).
[CrossRef]

Stevenson, A. F.

A. F. Stevenson, “Electromagnetic scattering by an ellipsoid in the third approximation,” J. Appl. Phys. 24, 1143–1151 (1953).
[CrossRef]

A. F. Stevenson, “Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatterer)/wavelength,” J. Appl. Phys. 24, 1134–1142 (1953).
[CrossRef]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Turner, S. W.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[CrossRef] [PubMed]

Webb, W. W.

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[CrossRef] [PubMed]

Wenger, J.

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P. F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

Wolff, P.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

IEEE Trans. Antennas Propag.

C. J. Bouwkamp, “Theoretical and numerical treatment of diffraction through a circular aperture,” IEEE Trans. Antennas Propag. 18, 152–176 (1970).
[CrossRef]

C. Butler, Y. Rahmat-Samii, and R. Mittra, “Electromagnetic penetration through apertures in conducting surfaces,” IEEE Trans. Antennas Propag. 26, 82–93 (1978).
[CrossRef]

IEEE Trans. Electromagn. Compat.

J. G. Lee and H. J. Eom, “Magnetostatic potential distribution through a circular aperture in a thick conducting plane,” IEEE Trans. Electromagn. Compat. 40, 97–99 (1998).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

R. L. Gluckstern and J. A. Diamond, “Penetration of fields through a circular hole in a wall of finite thickness,” IEEE Trans. Microwave Theory Tech. 39, 274–279 (1991).
[CrossRef]

B. Radak and R. L. Gluckstern, “Penetration of electromagnetic fields through an elliptical hole in a wall of finite thickness,” IEEE Trans. Microwave Theory Tech. 43, 194–204 (1995).
[CrossRef]

J. H. Lee and H. J. Eom, “Electrostatic potential through a circular aperture in a thick conducting plane,” IEEE Trans. Microwave Theory Tech. 44, 341–343 (1996).
[CrossRef]

J. Appl. Phys.

C. Huang, R. D. Kodis, and H. Levine, “Diffraction by apertures,” J. Appl. Phys. 26, 151–165 (1955).
[CrossRef]

A. Roberts, “Near-zone fields behind circular apertures in thick, perfectly conducting screens,” J. Appl. Phys. 65, 2896–2899 (1989).
[CrossRef]

A. F. Stevenson, “Solution of electromagnetic scattering problems as power series in the ratio (dimension of scatterer)/wavelength,” J. Appl. Phys. 24, 1134–1142 (1953).
[CrossRef]

A. F. Stevenson, “Electromagnetic scattering by an ellipsoid in the third approximation,” J. Appl. Phys. 24, 1143–1151 (1953).
[CrossRef]

J. Opt. Soc. Am. A

J. Sound Vibrat.

C. Kuo, R. L. Chern, and C. C. Chang, “Sound scattering by a compact circular pore,” J. Sound Vibrat. 319, 622–645 (2009).
[CrossRef]

Laser Phys. Lett.

C. W. Chang, A. K. Sarychev, and V. M. Shalaev, “Light diffraction by a subwavelength circular aperture,” Laser Phys. Lett. 2, 351–355 (2005).
[CrossRef]

Nature

C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Opt. Express

Phys. Rev.

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
[CrossRef]

Phys. Rev. Lett.

H. Rigneault, J. Capoulade, J. Dintinger, J. Wenger, N. Bonod, E. Popov, T. W. Ebbesen, and P. F. Lenne, “Enhancement of single-molecule fluorescence detection in subwavelength apertures,” Phys. Rev. Lett. 95, 117401 (2005).
[CrossRef] [PubMed]

F. J. García de Abajo and J. J. Sáenz, “Electromagnetic surface modes in structured perfect-conductor surfaces,” Phys. Rev. Lett. 95, 233901 (2005).
[CrossRef]

Science

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, “Zero-mode waveguides for single-molecule analysis at high concentrations,” Science 299, 682–686 (2003).
[CrossRef] [PubMed]

SIAM J. Appl. Math.

J. Bazer and L. Rubenfeld, “Diffraction of electromagnetic waves by a circular aperture in an infinitely conducting plane screen,” SIAM J. Appl. Math. 13, 558–585 (1965).
[CrossRef]

Other

V. I. Fabrikant, Applications of Potential Theory in Mechanics: A Selection of New Results (Kluwer, 1989).

D. G. Crighton, A. P. Dowling, J. E. Ffowcs Williams, M. Heckl, and F. G. Leppington, Modern Methods in Analytical Acoustics: Lecture Notes (Springer, 1992).
[CrossRef]

D. M. Pozar, Microwave Engineering, 3rd ed. (Wiley, 2005).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice-Hall, 1999).

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Figures (6)

Fig. 1
Fig. 1

Schematics of (a) a subwavelength hole in a perfect metal plate of finite thickness and (b) TM and TE incidences.

Fig. 2
Fig. 2

Schematic of the inner and outer regions with respect to the hole position.

Fig. 3
Fig. 3

Near-field potential contours for (a) ψ e ( 0 ) / k x , (b) ψ e , sc ( 0 ) / k x , (c) ψ m ( 0 ) , and (d) ψ m , sc ( 0 ) . The shaded region corresponds to the metal plate with thickness 2 l = 1.5 a . In (d), the hole region is blanked.

Fig. 4
Fig. 4

Near-field potential contours for (a) ψ e ( 1 ) / i k x 2 , and (b) ψ e , sc ( 1 ) / i k x 2 , (c) ψ m ( 1 ) / i k x , and (d) ψ m , sc ( 1 ) / i k x . The hole geometry is the same as in Fig. 3. In (d), the hole region is blanked.

Fig. 5
Fig. 5

Strengths of the normalized effective (a) electric and (b) magnetic dipole moments associated with the radiation fields for a subwavelength circular hole. The red (dark gray) and green (light gray) curves denote the strenghts of the upper and lower dipoles, respectively. The black (dark) dotted lines denote the analytical results of Bethe’s theory [4]: | p ± | = 4 / 3 and | m ± | = 8 / 3 . The red (gray) dotted lines denote the numerical results for an infinitely deep hole [22]: | p + | 1.138 and | m + | 1.876 .

Fig. 6
Fig. 6

Schematic of the effective dipole moments associated with the radiation fields for a subwavelength hole in (a) TM incidence and (b) TE incidence.

Equations (117)

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E = 0 ,     × E = i ω B ,
B = 0 ,     × B = i ω c 2 E ,
B = × A m ,     E = ψ e + i ω A m .
E = × A e ,     B = ψ m + i ω c 2 A e .
2 ψ e ( m ) + k 2 ψ e ( m ) = 0 ,     2 A m ( e ) + k 2 A m ( e ) = 0
A m + i ω c 2 ψ e = 0 ,     A e + i ω ψ m = 0.
E = i ψ e + i ε A m i × A e ,
B = i ψ m + i ε A e + i × A m ,
( i 2 + ε 2 ) ψ e ( m ) = 0 ,     ( i 2 + ε 2 ) A m ( e ) = 0 ,
i A m ( e ) + i ε ψ e ( m ) = 0 ,
E = ε o ψ e + i ε A m ε o × A e ,
B = ε o ψ m + i ε A e + ε o × A m ,
( o 2 + 1 ) ψ e ( m ) = 0 ,     ( o 2 + 1 ) A m ( e ) = 0 ,
o A m ( e ) + i ψ e ( m ) = 0 ,
ψ e ( m ) = ψ e ( m ) ( 0 ) + ε ψ e ( m ) ( 1 ) + ε 2 ψ e ( m ) ( 2 ) + ,
A m ( e ) = A m ( e ) ( 0 ) + ε A m ( e ) ( 1 ) + ε 2 A m ( e ) ( 2 ) + .
E = i ψ e ( 0 ) + ε ( i ψ e ( 1 ) i × A e ( 1 ) ) + ,
B = i ψ m ( 0 ) + ε ( i ψ m ( 1 ) + ε i × A m ( 1 ) ) + ,
i 2 ψ e ( m ) ( 0 , 1 ) = 0 ,     i 2 A m ( e ) ( 0 , 1 ) = 0 ,
i A m ( e ) ( 1 ) + i ψ e ( m ) ( 0 ) = 0 ,     i A m ( e ) ( 0 ) = 0.
E = ε ( o ψ e ( 0 ) + i A m ( 0 ) o × A e ( 0 ) ) + ,
B = ε ( o ψ m ( 0 ) + i A e ( 0 ) + o × A m ( 0 ) ) + ,
( o 2 + 1 ) ψ e ( m ) ( 0 ) = 0 ,     ( o 2 + 1 ) A m ( e ) ( 0 ) = 0 ,
o A m ( e ) ( 0 ) + i ψ e ( m ) ( 0 ) = 0.
B ext , + = [ 0 1 0 ] e i ε k x x   cos ( ε k z ( z l ) ) ,    
E ext , + = i [ k z   sin ( ε k z ( z l ) ) 0 i k x   cos ( ε k z ( z l ) ) ] e i ε k x x ,
B ext , + = [ 0 1 0 ] ( 1 + i ε k x x ) + O ( ε 2 ) ,    
E ext , + = k x [ 0 0 1 ] + i ε [ k z 2 ( z l ) 0 k x 2 x ] + O ( ε 2 ) ,
B ext , + = i ( ρ   sin   ϕ ) + i ε k x 2 i × ( ρ ( z l ) ρ ̂ ) + ε i ( i k x 4 ρ 2   sin ( 2 ϕ ) ) + O ( ε 2 ) ,
E ext , + = k x i ( z l ) + i ε i × ( ρ ( z l ) sin   ϕ z ̂ ) + ε i ( i k x 2 ( z l ) ρ   cos   ϕ ) + O ( ε 2 ) .
ψ e , ext , + ( 0 ) = k x ( z l ) ,     ψ m , ext , + ( 0 ) = ρ   sin   ϕ ,
ψ e , ext , + ( 1 ) = i k x 2 ( z l ) ρ   cos   ϕ ,     ψ m , ext , + ( 1 ) = i k x 4 ρ 2   sin ( 2 ϕ ) .
ψ e , sc , ± ( 0 , 1 ) = ± 0 2 π d ϕ 0 0 1 ρ 0 d ρ 0 z l 2 π σ 3 ψ e , sc , ± ( 0 , 1 ) < ,
ψ m , sc , ± ( 0 , 1 ) = 0 2 π d ϕ 0 0 1 ρ 0 d ρ 0 1 2 π σ | ψ m , sc , ± ( 0 , 1 ) z | < ,
ψ e , sc , ± ( 0 , 1 ) < = 2 π ρ 1 d x x 2 ρ 2 0 x ρ 0 d ρ 0 x 2 ρ 0 2 L ( ρ ρ 0 x 2 ) | ψ e , sc , ± ( 0 , 1 ) z | < ,
± π t 2 | ψ m , sc , ± ( 0 , 1 ) z | < = L ( t ) d d t t 1 r d r r 2 t 2 L ( 1 r 2 ) d d r 0 r ρ d ρ r 2 ρ 2 L ( ρ ) ψ m , sc , ± ( 0 , 1 ) < ,
L ( λ ) f ( r , ϕ ) = 1 2 π 0 2 π ( 1 λ 2 ) f ( r , ϕ 0 ) 1 + λ 2 2 λ   cos ( ϕ ϕ 0 ) d ϕ 0 .
ψ e , hole ( 0 ) = n = 1 k x j 0 n [ a e , n ( 0 ) e j 0 n ( z l ) + b e , n ( 0 ) e j 0 n ( z + l ) ] J 0 ( j 0 n ρ ) ,
ψ m , hole ( 0 ) = n = 1 [ a m , n ( 0 ) e j 1 n ( z l ) + b m , n ( 0 ) e j 1 n ( z + l ) ] J 1 ( j 1 n ρ ) sin   ϕ ,
ψ e , hole ( 1 ) = n = 1 i k x 2 j 1 n [ a e , n ( 1 ) e j 1 n ( z l ) + b e , n ( 1 ) e j 1 n ( z + l ) ] J 1 ( j 1 n ρ ) cos   ϕ ,
ψ m , hole ( 1 ) = n = 1 i k x [ a m , n ( 1 ) e j 2 n ( z l ) + b m , n ( 1 ) e j 2 n ( z + l ) ] J 2 ( j 2 n ρ ) sin ( 2 ϕ ) ,
ψ e ( m ) , hole ( 0 , 1 ) < = ψ e ( m ) , ext , ± ( 0 , 1 ) < + ψ e ( m ) , sc , ± ( 0 , 1 ) < ,
| ψ e ( m ) , hole ( 0 , 1 ) z | < = | ψ e ( m ) , ext , ± ( 0 , 1 ) z | < + | ψ e ( m ) , sc , ± ( 0 , 1 ) z | < .
[ 2 k x π 1 ρ 2 0 ] ± k x n = 1 2 π j 0 n [ a e , n ( 0 ) b e , n ( 0 ) e 2 j 0 n l a e , n ( 0 ) e 2 j 0 n l b e , n ( 0 ) ] ρ 1 x J 1 / 2 ( j 0 n x ) x 2 ρ 2 d x = n = 1 k x j 0 n [ a e , n ( 0 ) + b e , n ( 0 ) e 2 j 0 n l a e , n ( 0 ) e 2 j 0 n l + b e , n ( 0 ) ] J 0 ( j 0 n ρ ) .
[ N e ( 0 ) S e ( 0 ) S e ( 0 ) N e ( 0 ) ] [ a e ( 0 ) b e ( 0 ) ] = [ u e ( 0 ) 0 ] .
n = 1 t 2 π j 1 n 2 [ a m , n ( 0 ) + b m , n ( 0 ) e 2 j 1 n l a m , n ( 0 ) e 2 j 1 n l + b m , n ( 0 ) ] ( J 1 / 2 ( j 1 n ) 1 t 2 + j 1 n t 1 J 3 / 2 ( j 1 n r ) r ( r 2 t 2 ) d r ) = [ 2 t 2 1 t 2 0 ] ± π t 2 n = 1 [ a m , n ( 0 ) b m , n ( 0 ) e 2 j 1 n l a m , n ( 0 ) e 2 j 1 n l b m , n ( 0 ) ] J 1 ( j 1 n t ) .
[ 3 4 i k x 2 π ρ 1 ρ 2 0 ] ± i k x 2 n = 1 2 π j 1 n [ a e , n ( 1 ) b e , n ( 1 ) e 2 j 1 n l a e , n ( 1 ) e 2 j 1 n l b e , n ( 1 ) ] ρ ρ 1 x J 3 / 2 ( j 1 n x ) x 2 ρ 2 d x , = n = 1 i k x 2 j 1 n [ a e , n ( 1 ) + b e , n ( 1 ) e 2 j 1 n l a e , n ( 1 ) e 2 j 1 n l + b e , n ( 1 ) ] J 1 ( j 1 n ρ ) ,
i k x n = 1 π j 2 n 2 [ a m , n ( 1 ) + b m , n ( 1 ) e 2 j 2 n l a m , n ( 1 ) e 2 j 2 n l + b m , n ( 1 ) ] ( t 3 J 3 / 2 ( j 2 n ) 1 t 2 + j 2 n t 3 t 1 J 5 / 2 ( j 2 n r ) r 3 ( r 2 t 2 ) d r ) , = [ 2 3 i k x t 3 1 t 2 0 ] ± i k x π t 2 n = 1 j 2 n [ a m , n ( 1 ) b m , n ( 1 ) e 2 j 2 n l a m , n ( 1 ) e 2 j 2 n l b m , n ( 1 ) ] J 2 ( j 2 n t ) ,
A m z , hole ( 1 ) = k x n = 1 i j 0 n 2 [ a e , n ( 0 ) e j 0 n ( z l ) b e , n ( 0 ) e j 0 n ( z + l ) ] J 0 ( j 0 n ρ ) ,
A e ρ , hole ( 1 ) = n = 1 i 2 j 1 n [ a m , n ( 0 ) e j 1 n ( z l ) + b m , n ( 0 ) e j 1 n ( z + l ) ] J 0 ( j 1 n ρ ) sin   ϕ + n = 1 i 2 j 1 n [ a m , n ( 0 ) e j 1 n ( z l ) + b m , n ( 0 ) e j 1 n ( z + l ) ] J 2 ( j 1 n ρ ) sin   ϕ ,
A e ϕ , hole ( 1 ) = n = 1 i 2 j 1 n [ a m , n ( 0 ) e j 1 n ( z l ) + b m , n ( 0 ) e j 1 n ( z + l ) ] J 0 ( j 1 n ρ ) cos   ϕ n = 1 i 2 j 1 n [ a m , n ( 0 ) e j 1 n ( z l ) + b m , n ( 0 ) e j 1 n ( z + l ) ] J 2 ( j 1 n ρ ) cos   ϕ .
ψ e , sc , ± ( 0 , 1 ) = ± 0 2 π d ϕ 0 0 1 ρ 0 d ρ 0 ψ e , sc , ± ( 0 , 1 ) < s   cos   θ 2 π s 3 ( 1 2 ρ 0   sin   θ s cos   ( ϕ ϕ 0 ) + ρ 0 2 s 2 ) 3 / 2 ,
ψ m , sc , ± ( 0 , 1 ) = 0 2 π d ϕ 0 0 1 ρ 0 d ρ 0 | ψ m , sc , ± ( 0 , 1 ) z | < 1 2 π s ( 1 2 ρ 0   sin   θ s cos   ( ϕ ϕ 0 ) + ρ 0 2 s 2 ) 1 / 2 ,
ψ e , sc , ± ( 0 ) k x   cos   θ s 2 n = 1 1 j 0 n 2 f e , n , ± ( 0 ) J 1 ( j 0 n ) ± 3 k x   cos   θ 2 s 4 ( 5 2 cos 2 θ 3 2 ) n = 1 1 j 0 n 2 f e , n , ± ( 0 ) ( J 1 ( j 0 n ) 2 j 0 n J 2 ( j 0 n ) ) ,
ψ e , sc , ± ( 1 ) 3 i k x 2 2 s 3 cos   θ   sin   θ   cos   ϕ n = 1 1 j 1 n 2 f e , n ( 1 ) J 2 ( j 1 n ) ,
ψ m , sc , ± ( 0 ) 1 2 s 2 sin   θ   sin   ϕ n = 1 f m , n , ± ( 0 ) J 2 ( j 1 n ) ± 3   sin   θ 16 s 4 ( 5 cos 2 θ 1 ) sin   ϕ n = 1 f m , n , ± ( 0 ) ( J 2 ( j 1 n ) 2 j 1 n J 3 ( j 1 n ) ) ,
ψ m , sc , ± ( 1 ) 3 i k x 8 s 3 sin 2 θ   sin ( 2 ϕ ) n = 1 f m , n ( 1 ) J 3 ( j 2 n ) ,
A m , sc , ± ( 1 ) k x i s n = 1 1 j 0 n 2 f e , n , ± ( 0 ) J 1 ( j 0 n ) z ̂ ,
A e , sc , ± ( 1 ) i s n = 1 1 2 j 1 n f m , n , ± ( 0 ) J 1 ( j 1 n ) y ̂ ,
ψ e ( m ) , sc , ± ( S , ϕ , θ ) ε 2 ψ e ( m ) , sc , ± ( 2 ) ( S , ϕ , θ ) + ε 4 ψ e ( m ) , sc , ± ( 4 ) ( S , ϕ , θ ) ,
ψ e , sc , ± ( 2 ) ( S , ϕ , θ ) i k x B e , 10 , ± ( 2 ) h 1 ( 1 ) ( S ) cos   θ ,
ψ e , sc , ± ( 4 ) ( S , ϕ , θ ) i k x B e , 30 , ± ( 4 ) h 3 ( 1 ) ( S ) cos   θ ( 5 cos 2 θ 3 ) + k x 2 B e , 21 , ± ( 4 ) h 2 ( 1 ) ( S ) cos   θ   sin   θ   cos   ϕ i k x B e , 10 , ± ( 4 ) h 1 ( 1 ) ( S ) cos   θ ,
ψ m , sc , ± ( 2 ) ( S , ϕ , θ ) i B m , 11 , ± ( 2 ) h 1 ( 1 ) ( S ) sin   θ   sin   ϕ ,
ψ m , sc , ± ( 4 ) ( S , ϕ , θ ) i B m , 31 , ± ( 4 ) h 3 ( 1 ) ( S ) sin   θ ( 5 cos 2 θ 1 ) sin   ϕ + k x B m , 22 , ± ( 4 ) h 2 ( 1 ) ( S ) sin 2 θ   sin ( 2 ϕ ) i B m , 11 , ± ( 4 ) h 1 ( 1 ) ( S ) sin   θ   sin   ϕ ,
h 0 ( 1 ) ( S ) = i S + 1 + O ( S ) ,     h 1 ( 1 ) ( S ) = i S 2 i 2 + O ( S ) ,
h 2 ( 1 ) ( S ) = 3 i S 3 i 2 S + O ( 1 ) ,     h 3 ( 1 ) ( S ) = 15 i S 4 3 i 2 S 2 + O ( 1 ) .
B e , 10 , ± ( 2 ) = ± n = 1 1 j 0 n 2 f e , n , ± ( 0 ) J 1 ( j 0 n ) ,
B e , 30 , ± ( 4 ) = ± 1 20 n = 1 1 j 0 n 2 f e , n , ± ( 0 ) ( J 1 ( j 0 n ) 2 j 0 n J 2 ( j 0 n ) ) ,    
B e , 21 , ± ( 4 ) = ± 1 2 n = 1 1 j 1 n 2 f e , n , ± ( 1 ) J 2 ( j 1 n ) ,
B e , 10 , ± ( 4 ) = ± 1 15 n = 1 1 j 0 n 2 f e , n , ± ( 0 ) ( J 1 ( j 0 n ) 2 j 0 n J 2 ( j 0 n ) ) .
B m , 11 , ± ( 2 ) = ± 1 2 n = 1 f m , n , ± ( 0 ) J 2 ( j 1 n ) ,
B m , 31 , ± ( 4 ) = ± 1 80 n = 1 f m , n , ± ( 0 ) ( J 2 ( j 1 n ) 2 j 1 n J 3 ( j 1 n ) ) ,    
B m , 22 , ± ( 4 ) = ± 1 8 n f m , n , ± ( 1 ) J 3 ( j 2 n ) ,
B m , 11 , ± ( 4 ) = ± 1 30 n = 1 1 j 1 n f m , n , ± ( 0 ) ( 2 J 3 ( j 1 n ) 1 j 1 n J 2 ( j 1 n ) ) .
A m , sc , ± ( 2 ) k x B e , 10 , ± ( 2 ) h 0 ( 1 ) ( S ) z ̂ ,     A e , sc , ± ( 2 ) B e , 11 , ± ( 2 ) h 0 ( 1 ) ( S ) y ̂ ,
E ε 3 ( o ψ e ( 2 ) + i A m ( 2 ) ) + ε 5 ( o ψ e ( 4 ) + i A m ( 4 ) ) ,
B ε 3 ( o × A m ( 2 ) ) + ε 5 ( o × A m ( 4 ) ) .
h 0 ( 1 ) ( S ) = i e i S 1 S + O ( 1 ) ,     h 1 ( 1 ) ( S ) = e i S 1 S + O ( 1 ) ,
h 2 ( 1 ) ( S ) = i e i S 1 S + O ( 1 ) ,     h 3 ( 1 ) ( S ) = e i S 1 S + O ( 1 ) ,
E rad , ± ε 3 k x e i S S n = 1 1 j 0 n 2 f e , n , ± ( 0 ) J 1 ( j 0 n ) sin   θ θ ̂ ,
B rad , ± ε 3 k x e i S S n = 1 1 j 0 n 2 f e , n , ± ( 0 ) J 1 ( j 0 n ) sin   θ ϕ ̂ ,
E rad , ± 2 E inc k x ω 2 a 3 e i k s c 2 ( sin   θ s ) n = 1 1 j 0 n 2 f e , n , ± ( 0 ) J 1 ( j 0 n ) θ ̂ ,
p ± ε 0 E 0 a 3   sin   φ p ± = ± 4 π n = 1 1 j 0 n 2 f e , n , ± ( 0 ) J 1 ( j 0 n ) z ̂ ,
p ± = ± 4 3 z ̂ .
B ε 3 ( o ψ m ( 2 ) + i A e ( 2 ) ) + ε 5 ( o ψ m ( 4 ) + i A e ( 4 ) ) ,
E ε 3 ( o × A e ( 2 ) ) ε 5 ( o × A e ( 4 ) ) .
B rad , ± ± ε 3 e i S 2 S n = 1 1 j 1 n f m , n , ± ( 0 ) J 1 ( j 1 n ) ( sin   ϕ   cos   θ θ ̂ + cos   ϕ ϕ ̂ ) ,
E rad , ± ± ε 3 e i S 2 S n = 1 1 j 1 n f m , n , ± ( 0 ) J 1 ( j 1 n ) ( cos   ϕ θ ̂ cos   θ   sin   ϕ ϕ ̂ ) ,
B rad ± 2 E inc ω 2 a 3 e i k s 2 c 3 s n = 1 1 j 1 n f m , n , ± ( 0 ) J 1 ( j 1 n ) ( sin   ϕ   cos   θ θ ̂ + cos   ϕ ϕ ̂ ) .
m ± 1 μ 0 B 0 a 3 m ± = ± 2 π n = 1 1 j 1 n f m , n ( 0 ) J 1 ( j 1 n ) y ̂ ,
m ± = ± 8 3 y ̂ .
E ext , + = i [ 0 1 0 ] e i ε k x x   sin ( ε k z ( z l ) ) ,    
B ext , + = i [ i k z   cos ( ε k z ( z l ) ) 0 k x   sin ( ε k z ( z l ) ) ] e i ε k x x .
E ext , + = i ε k z [ 0 z l 0 ] + O ( ε 2 ) ,    
B ext , + = k z [ 1 0 0 ] + i ε k x k z [ x 0 ( z l ) ] + O ( ε 2 ) ,
E ext , + = i ε k z i × ( ρ ( z l ) cos   ϕ z ̂ ) + O ( ε 2 ) ,
B ext , + = i ( k z ρ   cos   ϕ ) + ε i ( 1 2 i k x k z ( ρ 2 cos 2 ϕ ( z l ) 2 ) ) + O ( ε 2 ) .
ψ m , ext , + ( 0 ) = k z ρ   cos   ϕ ,     ψ e , ext , + ( 0 ) = 0 ,
ψ m , ext , + ( 1 ) = 1 2 i k x k z ( ρ 2 cos 2 ϕ ( z l ) 2 ) ,     ψ e , ext , + ( 1 ) = 0.
ψ m , hole ( 0 ) = k z n = 1 [ a m , n ( 0 ) e j 1 n ( z l ) + b m , n ( 0 ) e j 1 n ( z + l ) ] J 1 ( j 1 n ρ ) cos   ϕ ,
ψ m , hole ( 1 ) = i k x k z { a 0 ( 1 ) + a 1 ( 1 ) z + n = 1 [ a m , n ( 1 ) e j 2 n ( z l ) + b m , n ( 1 ) e j 2 n ( z + l ) ] J 2 ( j 2 n ρ ) cos   ( 2 ϕ ) } .
B rad , ± i k z ε 3 e i S 2 S n = 1 1 j 1 n f m , 1 n ( 0 ) J 1 ( j 1 n ) ( cos   ϕ   cos   θ θ ̂ sin   ϕ ϕ ̂ ) ,
E rad , ± i k z ε 3 e i S 2 S n = 1 1 j 1 n f m , n ( 0 ) J 1 ( j 1 n ) ( sin   ϕ θ ̂ cos   θ   cos   ϕ ϕ ̂ ) .
B rad i k z 2 E inc ω 2 a 3 e i k s 2 c 3 s n = 1 1 j 1 n f m , n , ± ( 0 ) J 1 ( j 1 n ) ( cos   ϕ   cos   θ θ ̂ sin   ϕ ϕ ̂ ) .
m ± 1 μ 0 B 0 a 3   cos   φ m ± = i 2 π n = 1 1 j 1 n f m , n ( 0 ) J 1 ( j 1 n ) x ̂ ,
[ N e ( 0 ) ] p n = { α e , p n ( 0 ) + β e , p n ( 0 ) , n = p α e , p n ( 0 ) , n p , }     [ S e ( 0 ) ] p n = { e 2 j 0 n l ( α e , p n ( 0 ) β e , p n ( 0 ) ) , n = p e 2 j 0 n l α e , p n ( 0 ) , n p , }
α e , p n ( 0 ) = { 1 2 j 0 n J 1 ( j 0 n ) J 1 ( j 0 n ) , n = p ( j 0 n j 0 p ) 1 / 2 j 0 n 2 j 0 p 2 ( j 0 p J 1 / 2 ( j 0 n ) J 1 / 2 ( j 0 p ) j 0 n J 1 / 2 ( j 0 n ) J 1 / 2 ( j 0 p ) ) , n p , }
β e , p n ( 0 ) = 1 2 j 0 n ( J 1 / 2 2 ( j 0 n ) J 1 / 2 ( j 0 n ) J 3 / 2 ( j 0 n ) ) ,
[ N m ( 0 ) ] p n = { α m , p n ( 0 ) + β m , p n ( 0 ) , n = p α m , p n ( 0 ) , n p , }     [ S m ( 0 ) ] p n = { e 2 j 1 n l ( α m , p n ( 0 ) β m , p n ( 0 ) ) , n = p e 2 j 1 n l α m , p n ( 0 ) , n p , }
α m , p n ( 0 ) = { π 2 { j 1 n 2 [ J 3 / 2 2 ( j 1 n ) J 1 / 2 ( j 1 n ) J 5 / 2 ( j 1 n ) ] + J 3 / 2 ( j 1 n ) J 1 / 2 ( j 1 n ) } , n = p π 2 ( j 1 n j 1 p ) 1 / 2 j 1 n 2 j 1 p 2 { j 1 n J 3 / 2 ( j 1 n ) J 1 / 2 ( j 1 p ) j 1 p J 1 / 2 ( j 1 n ) J 3 / 2 ( j 1 p ) } , n p , }
β m , p n ( 0 ) = π 4 j 1 n ( 1 1 j 1 n 2 ) J 1 2 ( j 1 n ) ,
[ N e ( 1 ) ] p n = { α e , p n ( 1 ) + β e , p n ( 1 ) , n = p α e , p n ( 1 ) , n p , }     [ S e ( 1 ) ] p n = { e 2 j 1 n l ( α e , p n ( 1 ) β e , p n ( 1 ) ) , n = p e 2 j 1 n l α e , p n ( 1 ) , n p , }
α e , p n ( 1 ) = { 1 2 j 1 n [ J 3 / 2 2 ( j 1 n ) J 1 / 2 ( j 1 n ) J 5 / 2 ( j 1 n ) ] , n = p ( j 1 n j 1 p ) 1 / 2 j 1 n 2 j 1 p 2 { j 1 p J 3 / 2 ( j 1 n ) J 1 / 2 ( j 1 p ) j 1 n J 1 / 2 ( j 1 n ) J 3 / 2 ( j 1 p ) } , n p , }
β e , p n ( 1 ) = 1 2 j 1 n J 0 ( j 1 n ) J 2 ( j 1 n ) ,
[ N m ( 1 ) ] p n = { α m , p n ( 1 ) + β m , p n ( 1 ) , n = p α m , p n ( 1 ) , n p , }     [ S m ( 1 ) ] p n = { e 2 j 2 n l ( α m , p n ( 1 ) β m , p n ( 1 ) ) , n = p e 2 j 2 n l α m , p n ( 1 ) , n p , }
α m , p n ( 1 ) = { π 2 ( j 2 n 2 ( J 5 / 2 2 ( j 2 n ) J 3 / 2 ( j 2 n ) J 7 / 2 ( j 2 n ) ) + J 3 / 2 ( j 2 n ) J 5 / 2 ( j 2 n ) ) , n = p π 2 ( j 2 n j 2 p ) 1 / 2 j 2 n 2 j 2 p 2 ( j 2 n J 5 / 2 ( j 2 n ) J 3 / 2 ( j 2 p ) j 2 p J 3 / 2 ( j 2 n ) J 5 / 2 ( j 2 p ) ) , n p , }
β m , p n ( 1 ) = π 4 j 2 n ( 1 4 j 2 n 2 ) J 2 2 ( j 2 n ) ,

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