Abstract

A perturbation theory based on a single-scattering approximation is developed from the rigorous differential theory of diffraction in cylindrical coordinates. It results in analytical formulas in the inverse space for the field amplitudes providing results that are in quantitative agreement with the results of the rigorous method, in both the near- and far-field regions, when a proper correction to the incident field inside the aperture is made by using the renormalized Born approximation. When working in reflection by a screen having permittivity high in modulus, the method proposes an equivalence with the simple model consisting of the emission by a single magnetic dipole excited inside the pierced layer, emission that is then transferred back into the cladding following the Fresnel’s coefficients of transmission from the layer into the cladding. The theory predicts a directivity of the radiation pattern that increases for smaller values of modulus of permittivity, both for dielectrics and metals, thus independently of the possibility of plasmon surface wave excitation along the interface. The theory can take into account such surface wave resonances, as well as the waveguide supported by a dielectric slab, but cannot implicitly recognize the modes carried out by the cylindrical waveguide corresponding to the aperture. This fact limits its domain of validity when used in transmission, although the far- and near-field maps can be reconstructed sufficiently well within a multiplicative factor corresponding to the enhanced transmission due to the excitation of these modes.

© 2007 Optical Society of America

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    [Crossref]
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    [Crossref]
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2006 (2)

2005 (3)

2004 (4)

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, "Surface-topography-induced enhanced transmission and directivity of microwave radiation through a subwavelength circular metal aperture," Appl. Phys. Lett. 84, 2040-2042 (2004).
[Crossref]

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Commun. 239, 61-66 (2004).
[Crossref]

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, "Surface plasmons at single nanoholes in Au films," Appl. Phys. Lett. 85, 467-469 (2004).
[Crossref]

R. Zakharian, M. Mansuripur, and J. V. Moloney "Transmission of light through small elliptical apertures," Opt. Express 12, 2631-2648 (2004).
[Crossref] [PubMed]

2003 (4)

M. J. Levene, J. Kotach, S. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, "Zero-mode waveguide for single-molecule analysis at high concentrations," Science 209, 682-686 (2003).
[Crossref]

A. Moreau, G. Granet, F. I. Baida, and D. Van Labeke, "Light transmission by subwavelength square coaxial aperture arrays in metallic films," Opt. Express 11, 1131-1136 (2003).
[Crossref] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, and T. W. Ebbesen, "Focusing light with a single subwavelength aperture flanked by surface corrugations," Appl. Phys. Lett. 83, 4500-4502 (2003).
[Crossref]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003).
[Crossref] [PubMed]

2002 (2)

S. Enoch, E. Popov, M. Nevière, and R. Reinisch, "Enhanced light transmission by hole arrays," J. Opt. A, Pure Appl. Opt. 4, S83-S87 (2002).
[Crossref]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[Crossref] [PubMed]

2001 (4)

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[Crossref] [PubMed]

L. Salomon, F. Grillot, A. Zayats, and F. de Fornel, "Near-field distribution of optical transmission of periodic subwavelength holes in a metal film," Phys. Rev. Lett. 86, 1110-1113 (2001).
[Crossref] [PubMed]

A. Krishman, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolf, J. A. Pendry, L. Martin-Moreno, and F., J. Garcia-Vidal, "Evanescently-coupled surface resonance in surface plasmon enhanced transmission," Opt. Commun. 200, 1-7 (2001).
[Crossref]

R. Wanneracher, "Plasmon-supported transmission of light through nanometric holes in metallic thin films," Opt. Commun. 195, 107-118 (2001).
[Crossref]

2000 (1)

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[Crossref]

1998 (2)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature (London) 391, 667-669 (1998).
[Crossref]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through sub-wavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[Crossref]

1987 (1)

1980 (1)

A Yaghjian, "Electric dyadic Green's function in the source region," Proc. IEEE 68, 248-263(1980).
[Crossref]

1954 (1)

C. J. Bouwkamp, "Diffraction theory," Rep. Prog. Phys. 17, 35-100 (1954).
[Crossref]

1944 (1)

H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
[Crossref]

Abramovitz, M.

M. Abramovitz and I. A. Stegun, eds. Handbook of Mathematical Functions [9th printing (1970)] National Bureau of Standards Applied Mathematics Series 55 (US GPD, 1964).

Baida, F. I.

Bethe, H. A.

H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
[Crossref]

Bonod, N.

Bouwkamp, C. J.

C. J. Bouwkamp, "Diffraction theory," Rep. Prog. Phys. 17, 35-100 (1954).
[Crossref]

Brown, D. B.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, "Surface plasmons at single nanoholes in Au films," Appl. Phys. Lett. 85, 467-469 (2004).
[Crossref]

Chang, S. H.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, "Surface plasmons at single nanoholes in Au films," Appl. Phys. Lett. 85, 467-469 (2004).
[Crossref]

Chaumet, P.

Craighead, H. G.

M. J. Levene, J. Kotach, S. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, "Zero-mode waveguide for single-molecule analysis at high concentrations," Science 209, 682-686 (2003).
[Crossref]

de Fornel, F.

L. Salomon, F. Grillot, A. Zayats, and F. de Fornel, "Near-field distribution of optical transmission of periodic subwavelength holes in a metal film," Phys. Rev. Lett. 86, 1110-1113 (2001).
[Crossref] [PubMed]

Degiron, A.

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Commun. 239, 61-66 (2004).
[Crossref]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003).
[Crossref] [PubMed]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[Crossref] [PubMed]

Devaux, E.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[Crossref] [PubMed]

Ebbesen, T. W.

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Commun. 239, 61-66 (2004).
[Crossref]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003).
[Crossref] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, and T. W. Ebbesen, "Focusing light with a single subwavelength aperture flanked by surface corrugations," Appl. Phys. Lett. 83, 4500-4502 (2003).
[Crossref]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[Crossref] [PubMed]

A. Krishman, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolf, J. A. Pendry, L. Martin-Moreno, and F., J. Garcia-Vidal, "Evanescently-coupled surface resonance in surface plasmon enhanced transmission," Opt. Commun. 200, 1-7 (2001).
[Crossref]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[Crossref] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature (London) 391, 667-669 (1998).
[Crossref]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through sub-wavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[Crossref]

Enoch, S.

S. Enoch, E. Popov, M. Nevière, and R. Reinisch, "Enhanced light transmission by hole arrays," J. Opt. A, Pure Appl. Opt. 4, S83-S87 (2002).
[Crossref]

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[Crossref]

Fehrembach, A.-L.

Foquet, M.

M. J. Levene, J. Kotach, S. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, "Zero-mode waveguide for single-molecule analysis at high concentrations," Science 209, 682-686 (2003).
[Crossref]

Garcia-Vidal, F. J.

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, and T. W. Ebbesen, "Focusing light with a single subwavelength aperture flanked by surface corrugations," Appl. Phys. Lett. 83, 4500-4502 (2003).
[Crossref]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003).
[Crossref] [PubMed]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[Crossref] [PubMed]

A. Krishman, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolf, J. A. Pendry, L. Martin-Moreno, and F., J. Garcia-Vidal, "Evanescently-coupled surface resonance in surface plasmon enhanced transmission," Opt. Commun. 200, 1-7 (2001).
[Crossref]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[Crossref] [PubMed]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature (London) 391, 667-669 (1998).
[Crossref]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through sub-wavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[Crossref]

Granet, G.

Gray, S. K.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, "Surface plasmons at single nanoholes in Au films," Appl. Phys. Lett. 85, 467-469 (2004).
[Crossref]

Grillot, F.

L. Salomon, F. Grillot, A. Zayats, and F. de Fornel, "Near-field distribution of optical transmission of periodic subwavelength holes in a metal film," Phys. Rev. Lett. 86, 1110-1113 (2001).
[Crossref] [PubMed]

Grupp, D. E.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through sub-wavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[Crossref]

Guérin, Ch.-A.

Hibbins, A. P.

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, "Surface-topography-induced enhanced transmission and directivity of microwave radiation through a subwavelength circular metal aperture," Appl. Phys. Lett. 84, 2040-2042 (2004).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

Kim, T. J.

A. Krishman, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolf, J. A. Pendry, L. Martin-Moreno, and F., J. Garcia-Vidal, "Evanescently-coupled surface resonance in surface plasmon enhanced transmission," Opt. Commun. 200, 1-7 (2001).
[Crossref]

Kimball, C. W.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, "Surface plasmons at single nanoholes in Au films," Appl. Phys. Lett. 85, 467-469 (2004).
[Crossref]

Kotach, J.

M. J. Levene, J. Kotach, S. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, "Zero-mode waveguide for single-molecule analysis at high concentrations," Science 209, 682-686 (2003).
[Crossref]

Krishman, A.

A. Krishman, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolf, J. A. Pendry, L. Martin-Moreno, and F., J. Garcia-Vidal, "Evanescently-coupled surface resonance in surface plasmon enhanced transmission," Opt. Commun. 200, 1-7 (2001).
[Crossref]

Lenne, P.-F.

Levene, M. J.

M. J. Levene, J. Kotach, S. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, "Zero-mode waveguide for single-molecule analysis at high concentrations," Science 209, 682-686 (2003).
[Crossref]

Lezec, H. J.

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Commun. 239, 61-66 (2004).
[Crossref]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003).
[Crossref] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, and T. W. Ebbesen, "Focusing light with a single subwavelength aperture flanked by surface corrugations," Appl. Phys. Lett. 83, 4500-4502 (2003).
[Crossref]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[Crossref] [PubMed]

A. Krishman, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolf, J. A. Pendry, L. Martin-Moreno, and F., J. Garcia-Vidal, "Evanescently-coupled surface resonance in surface plasmon enhanced transmission," Opt. Commun. 200, 1-7 (2001).
[Crossref]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[Crossref] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature (London) 391, 667-669 (1998).
[Crossref]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through sub-wavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[Crossref]

Linke, R. A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[Crossref] [PubMed]

Lockyear, M. J.

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, "Surface-topography-induced enhanced transmission and directivity of microwave radiation through a subwavelength circular metal aperture," Appl. Phys. Lett. 84, 2040-2042 (2004).
[Crossref]

Love, J.

A. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Mallet, P.

Mansuripur, M.

Martin-Moreno, L.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003).
[Crossref] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, and T. W. Ebbesen, "Focusing light with a single subwavelength aperture flanked by surface corrugations," Appl. Phys. Lett. 83, 4500-4502 (2003).
[Crossref]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[Crossref] [PubMed]

A. Krishman, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolf, J. A. Pendry, L. Martin-Moreno, and F., J. Garcia-Vidal, "Evanescently-coupled surface resonance in surface plasmon enhanced transmission," Opt. Commun. 200, 1-7 (2001).
[Crossref]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[Crossref] [PubMed]

Moloney, J. V.

Moreau, A.

Nevière, M.

Pearson, J.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, "Surface plasmons at single nanoholes in Au films," Appl. Phys. Lett. 85, 467-469 (2004).
[Crossref]

Pellerin, K. M.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[Crossref] [PubMed]

Pendry, J. A.

A. Krishman, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolf, J. A. Pendry, L. Martin-Moreno, and F., J. Garcia-Vidal, "Evanescently-coupled surface resonance in surface plasmon enhanced transmission," Opt. Commun. 200, 1-7 (2001).
[Crossref]

Pendry, J. B.

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[Crossref] [PubMed]

Popov, E.

Reinisch, R.

S. Enoch, E. Popov, M. Nevière, and R. Reinisch, "Enhanced light transmission by hole arrays," J. Opt. A, Pure Appl. Opt. 4, S83-S87 (2002).
[Crossref]

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[Crossref]

Rigneault, H.

Roberts, A.

Rydh, A.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, "Surface plasmons at single nanoholes in Au films," Appl. Phys. Lett. 85, 467-469 (2004).
[Crossref]

Salomon, L.

L. Salomon, F. Grillot, A. Zayats, and F. de Fornel, "Near-field distribution of optical transmission of periodic subwavelength holes in a metal film," Phys. Rev. Lett. 86, 1110-1113 (2001).
[Crossref] [PubMed]

Sambles, J. R.

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, "Surface-topography-induced enhanced transmission and directivity of microwave radiation through a subwavelength circular metal aperture," Appl. Phys. Lett. 84, 2040-2042 (2004).
[Crossref]

Schatz, G. C.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, "Surface plasmons at single nanoholes in Au films," Appl. Phys. Lett. 85, 467-469 (2004).
[Crossref]

Sentenac, A.

Snyder, A.

A. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

Stegun, I. A.

M. Abramovitz and I. A. Stegun, eds. Handbook of Mathematical Functions [9th printing (1970)] National Bureau of Standards Applied Mathematics Series 55 (US GPD, 1964).

Thio, T.

A. Krishman, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolf, J. A. Pendry, L. Martin-Moreno, and F., J. Garcia-Vidal, "Evanescently-coupled surface resonance in surface plasmon enhanced transmission," Opt. Commun. 200, 1-7 (2001).
[Crossref]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[Crossref] [PubMed]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through sub-wavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[Crossref]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature (London) 391, 667-669 (1998).
[Crossref]

Turner, S.

M. J. Levene, J. Kotach, S. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, "Zero-mode waveguide for single-molecule analysis at high concentrations," Science 209, 682-686 (2003).
[Crossref]

Van Labeke, D.

Vlasko-Vlasov, V. K.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, "Surface plasmons at single nanoholes in Au films," Appl. Phys. Lett. 85, 467-469 (2004).
[Crossref]

Wanneracher, R.

R. Wanneracher, "Plasmon-supported transmission of light through nanometric holes in metallic thin films," Opt. Commun. 195, 107-118 (2001).
[Crossref]

Webb, W. W.

M. J. Levene, J. Kotach, S. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, "Zero-mode waveguide for single-molecule analysis at high concentrations," Science 209, 682-686 (2003).
[Crossref]

Welp, U.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, "Surface plasmons at single nanoholes in Au films," Appl. Phys. Lett. 85, 467-469 (2004).
[Crossref]

Wenger, J.

Wolf, P. A.

A. Krishman, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolf, J. A. Pendry, L. Martin-Moreno, and F., J. Garcia-Vidal, "Evanescently-coupled surface resonance in surface plasmon enhanced transmission," Opt. Commun. 200, 1-7 (2001).
[Crossref]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature (London) 391, 667-669 (1998).
[Crossref]

Yaghjian, A

A Yaghjian, "Electric dyadic Green's function in the source region," Proc. IEEE 68, 248-263(1980).
[Crossref]

Yamamoto, N.

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Commun. 239, 61-66 (2004).
[Crossref]

Yin, L.

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, "Surface plasmons at single nanoholes in Au films," Appl. Phys. Lett. 85, 467-469 (2004).
[Crossref]

Zakharian, R.

Zayats, A.

L. Salomon, F. Grillot, A. Zayats, and F. de Fornel, "Near-field distribution of optical transmission of periodic subwavelength holes in a metal film," Phys. Rev. Lett. 86, 1110-1113 (2001).
[Crossref] [PubMed]

Appl. Opt. (2)

Appl. Phys. Lett. (3)

F. J. Garcia-Vidal, L. Martin-Moreno, H. J. Lezec, and T. W. Ebbesen, "Focusing light with a single subwavelength aperture flanked by surface corrugations," Appl. Phys. Lett. 83, 4500-4502 (2003).
[Crossref]

M. J. Lockyear, A. P. Hibbins, and J. R. Sambles, "Surface-topography-induced enhanced transmission and directivity of microwave radiation through a subwavelength circular metal aperture," Appl. Phys. Lett. 84, 2040-2042 (2004).
[Crossref]

L. Yin, V. K. Vlasko-Vlasov, A. Rydh, J. Pearson, U. Welp, S. H. Chang, S. K. Gray, G. C. Schatz, D. B. Brown, and C. W. Kimball, "Surface plasmons at single nanoholes in Au films," Appl. Phys. Lett. 85, 467-469 (2004).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

S. Enoch, E. Popov, M. Nevière, and R. Reinisch, "Enhanced light transmission by hole arrays," J. Opt. A, Pure Appl. Opt. 4, S83-S87 (2002).
[Crossref]

J. Opt. Soc. Am. A (4)

Nature (London) (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through subwavelength hole arrays," Nature (London) 391, 667-669 (1998).
[Crossref]

Opt. Commun. (3)

A. Krishman, T. Thio, T. J. Kim, H. J. Lezec, T. W. Ebbesen, P. A. Wolf, J. A. Pendry, L. Martin-Moreno, and F., J. Garcia-Vidal, "Evanescently-coupled surface resonance in surface plasmon enhanced transmission," Opt. Commun. 200, 1-7 (2001).
[Crossref]

R. Wanneracher, "Plasmon-supported transmission of light through nanometric holes in metallic thin films," Opt. Commun. 195, 107-118 (2001).
[Crossref]

A. Degiron, H. J. Lezec, N. Yamamoto, and T. W. Ebbesen, "Optical transmission properties of a single subwavelength aperture in a real metal," Opt. Commun. 239, 61-66 (2004).
[Crossref]

Opt. Express (2)

Phys. Rev. (1)

H. A. Bethe, "Theory of diffraction by small holes," Phys. Rev. 66, 163-182 (1944).
[Crossref]

Phys. Rev. B (2)

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, and H. J. Lezec, "Surface plasmons enhance optical transmission through sub-wavelength holes," Phys. Rev. B 58, 6779-6782 (1998).
[Crossref]

E. Popov, M. Nevière, S. Enoch, and R. Reinisch, "Theory of light transmission through subwavelength periodic hole arrays," Phys. Rev. B 62, 16100-16108 (2000).
[Crossref]

Phys. Rev. Lett. (3)

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, "Theory of extraordinary optical transmission through subwavelength hole arrays," Phys. Rev. Lett. 86, 1114-1117 (2001).
[Crossref] [PubMed]

L. Salomon, F. Grillot, A. Zayats, and F. de Fornel, "Near-field distribution of optical transmission of periodic subwavelength holes in a metal film," Phys. Rev. Lett. 86, 1110-1113 (2001).
[Crossref] [PubMed]

L. Martin-Moreno, F. J. Garcia-Vidal, H. J. Lezec, A. Degiron, and T. W. Ebbesen, "Theory of highly directional emission from a single subwavelength aperture surrounded by surface corrugations," Phys. Rev. Lett. 90, 167401 (2003).
[Crossref] [PubMed]

Proc. IEEE (1)

A Yaghjian, "Electric dyadic Green's function in the source region," Proc. IEEE 68, 248-263(1980).
[Crossref]

Rep. Prog. Phys. (1)

C. J. Bouwkamp, "Diffraction theory," Rep. Prog. Phys. 17, 35-100 (1954).
[Crossref]

Science (2)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, "Beaming light from a subwavelength aperture," Science 297, 820-822 (2002).
[Crossref] [PubMed]

M. J. Levene, J. Kotach, S. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, "Zero-mode waveguide for single-molecule analysis at high concentrations," Science 209, 682-686 (2003).
[Crossref]

Other (3)

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

A. Snyder and J. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

M. Abramovitz and I. A. Stegun, eds. Handbook of Mathematical Functions [9th printing (1970)] National Bureau of Standards Applied Mathematics Series 55 (US GPD, 1964).

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Figures (12)

Fig. 1
Fig. 1

Schematic representation of a screen with an aperture.

Fig. 2
Fig. 2

Spectral amplitudes M m E , R k m and M m H , R k m in reflection as function of k m for an aluminum infinitely thick screen, air as cladding and inside the aperture, R = 10 nm , wavelength λ = 500 nm . Open triangles, rigorous results; open circles, analytical results; solid squares, renormalized analytical results.

Fig. 3
Fig. 3

Amplitude of the diffracted electric field calculated along the x axis at a height z = 1 nm . Low-conductivity screen material. The other parameters are as in Fig. 2.

Fig. 4
Fig. 4

Electric field intensity just over the center of the opening of the aperture ( z = 1 nm ) as a function of the aperture radius R. Infinitely thick silver screen, air as cladding and inside the aperture, λ = 500 nm . Open triangles, rigorous results; open circles, analytical results; solid squares, renormalized analytical results; solid circles, analytical results by taking into account terms proportional to both R 2 and R 4 .

Fig. 5
Fig. 5

(a) Electric field intensity after the screen in the center of the aperture ( z = 201 nm ) for a silver screen with thickness t = 200 nm . Open circles, analytical results; open triangles rigorous results; dotted line R 7 . (b) Comparison between the rigorous and the analytical (in a.u.) results of the electric field amplitude along the x axis 1 nm below the aperture for R = 30 nm . (c) Comparison between the analytical results and the Hankel function H 1 + ( k p x ) , representing the plasmon field traveling away from the aperture.

Fig. 6
Fig. 6

Spectral amplitude M m E , R k m for k m k 0 = 0.05 calculated in reflection using the rigorous (open triangles), the analytical (open circles), the renormalized analytical (solid squares), and the “extremely truncated” (solid circles) rigorous method, as a function of the real part of permittivity of the screen for R = 20 nm and λ = 500 nm . (a) Metals; (b) dielectrics, with a zoom (bottom panel) close to the origin.

Fig. 7
Fig. 7

Polar angle distribution of the power radiated in reflection in a radial direction ρ (see Fig. 1) for three different values of ρ lying in the plane of incident polarization θ = 0 . Infinitely thick silver screen, R = 10 nm , and λ = 500 nm .

Fig. 8
Fig. 8

Variation of the power radiated in reflection with the polar angle ψ in the two planes θ = 0 , 90 ° for two different metals and relatively wide aperture, R = 100 nm , and λ = 500 nm . For θ = 0 and very highly conducting material ( ε 2 = 100 ε A 1 ) : open circles, analytical results; half-filled circles, Jackson’s formula; open triangles, rigorous results. Open squares, real metal (aluminum) and θ = 0 . Solid triangles, in the plane θ = 90 ° for both aluminum and very high conductivity.

Fig. 9
Fig. 9

Similar to Fig. 8 but for much smaller aperture, R = 10 nm , and screen materials (metals) with values of permittivity increasing in modulus as indicated at bottom of graph. Symbols, θ = 0 ; solid curve θ = 90 ° .

Fig. 10
Fig. 10

Variation of P ρ with polar angle ψ in the plane θ = 0 for different dielectric screens with gradually increasing permittivity as given in the inset. R = 10 nm , λ = 500 nm .

Fig. 11
Fig. 11

Same as Fig. 10 but in the θ = 90 ° plane, perpendicular to the incident wave polarization.

Fig. 12
Fig. 12

Variation of the spectral amplitude P m E , R k m as a function of k m for (a) metals, (b) dielectrics for different permittivity of the screen as shown in the insets. R = 10 nm , λ = 500 nm .

Equations (171)

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E j ( r , θ , z ) = n = + E j , n ( r , z ) exp ( i n θ ) ,
H j ( r , θ , z ) = n = + H j , n ( r , z ) exp ( i n θ ) , j = r , θ , z ,
E θ , n z = i n r E z , n i ω μ 0 H r , n , H θ , n z = i n r H z , n + i ω ε 0 ε E r , n ,
E r , n z = E z , n r + i ω μ 0 H θ , n , H r , n z = H z , n r i ω ε 0 ε E θ , n ,
i ω μ 0 H z , n = E θ , n r + E θ , n r i n r E r , n , i ω ε 0 ε E z , n = H θ , n r + H θ , n r i n r H r , n ,
E θ , n ( r , z ) = m = 0 [ b n , m E ( z ) J n + 1 ( k m r ) + c n , m E ( z ) J n 1 ( k m r ) ] Δ m ,
E r , n ( r , z ) = i m [ b n , m E ( z ) J n + 1 ( k m r ) c n , m E ( z ) J n 1 ( k m r ) ] Δ m ,
k 0 2 ε E z , n ( r , z ) = i m [ b n , m H ( z ) c n , m H ( z ) ] J n ( k m r ) k m Δ m ,
ω μ 0 H θ , n ( r , z ) = m [ b n , m H ( z ) J n + 1 ( k m r ) + c n , m H ( z ) J n 1 ( k m r ) ] Δ m ,
ω μ 0 H r , n ( r , z ) = i [ b n , m H ( z ) J n + 1 ( k m r ) c n , m H ( z ) J n 1 ( k m r ) ] Δ m ,
ω μ 0 H z , n ( r , z ) = i m [ b n , m E ( z ) c n , m E ( z ) ] J n ( k m r ) k m Δ m ,
Δ m = k m + 1 k m .
d d z b n , m E = b n , m H k m 2 k 0 2 ε 2 ( b n , m H c n , m H ) m m ( ε 1 ) m , m n , n k m k m 2 k 0 2 ( b n , m H c n , m H ) ,
d d z c n , m E = c n , m H k m 2 k 0 2 ε 2 ( b n , m H c n , m H ) m m ( ε 1 ) m , m n , n k m k m 2 k 0 2 ( b n , m H c n , m H ) ,
d d z b n , m H = k m 2 2 ( b n , m E c n , m E ) k 0 2 ε 2 b n , m E k 0 2 m m ( ε ) m , m n + 1 , n + 1 b n , m E ,
d d z c n , m H = k m 2 2 ( b n , m E c n , m E ) + k 0 2 ε 2 c n , m E + k 0 2 m m ( ε ) m , m n 1 , n 1 c n , m E ,
( ε ) m , m n , n = k m Δ m 0 ε ( r ) J n ( k m r ) J n ( k m r ) r d r .
0 J n ( k m r ) J n ( k m r ) r d r = δ ( k m k m ) k m ,
( ε ) m , m n , n = ε 2 δ m , m + ( ε d ε 2 ) k m Δ m 0 R J n ( k m r ) J n ( k m r ) r d r = ε 2 δ m , m + ( ε d ε 2 ) k m Δ m R k m 2 k m 2 [ J n + 1 ( k m R ) J n ( k m R ) k m J n ( k m R ) J n + 1 ( k m R ) k m ] .
( ε ) m , m n , n k m Δ m R 2 ( n + 1 ) , m m .
( ε d ε 2 ) k m Δ m R k m 2 k m 2 [ J 1 ( k m R ) J 0 ( k m R ) k m J 0 ( k m R ) J 1 ( k m R ) k m ] .
( ε d ε 2 ) Δ i R J 1 ( k m R ) ,
d d z b 1 , m E = b 1 , m H k m 2 2 k 0 2 ε 2 ( b 1 , m H c 1 , m H ) ,
d d z c 1 , m E = c 1 , m H k m 2 2 k 0 2 ε 2 ( b 1 , m H c 1 , m H ) ,
d d z b 1 , m H = k m 2 2 ( b 1 , m E c 1 , m E ) k 0 2 ε 2 b 1 , m E ,
d d z c 1 , m H = k m 2 2 ( b 1 , m E c 1 , m E ) + k 0 2 ε 2 c 1 , m E + k 0 2 m m ( ε d ε 2 ) Δ m R J 1 ( k m R ) c 1 , m E .
d 2 d z 2 c 1 , m E = k m z 2 c 1 , m E + R J 1 ( k m R ) m η m m c 1 , m E ,
k m z 2 = k 0 2 ε 2 k m 2
η m m = ( 1 k m 2 2 k 0 2 ε 2 ) k 0 2 ( ε d ε 2 ) Δ m .
M m E = b 1 , m E c 1 , m E , P m E = b 1 , m E + c 1 , m E ,
M m H = b 1 , m H c 1 , m H , P m H = b 1 , m H + c 1 , m H .
( M m E ) = P m H ,
( P m H ) = k m z 2 M m E R J 1 ( k m R ) Δ ε c ̂ i E ,
( M m H ) = k 0 2 ε 2 P m E + R J 1 ( k m R ) Δ ε c ̂ i E ,
( P m E ) = k m z 2 k 0 2 ε 2 M m H ,
Δ ε = k 0 2 ( ε d ε 2 ) ,
c ̂ i E = Δ i c 1 , i E .
( M i E ) = P i H , ( P i H ) = k i z 2 M i E ,
( M i H ) = k 0 2 ε 2 P i E , ( P i E ) = k i z 2 k 0 2 ε 2 M i H .
M i E ± = M i E ± exp ( ± i k i z z ) ,
P i E ± = P i E ± exp ( ± i k i z z ) ,
( M m E ) = k m z 2 M m E R J 1 ( k m R ) Δ ε c ̂ i E
M m E = M m E ± exp ( ± i k m z z ) R J 1 ( k m R ) Δ ε k m z 2 γ i 2 C ̂ i ± exp ( ± i γ i z ) ,
C ̂ i ± = C i ± Δ i = Δ i ( P i E ± M i E ± ) 2 .
M m H = M m H ± exp ( ± i k m z z ) i R J 1 ( k m R ) Δ ε γ i k m z 2 γ i 2 C ̂ i ± exp ( ± i γ i z ) ,
M i E = M i E , I exp ( i α i z z ) + M i E , R exp ( i α i z z ) ,
M i E , I + M i E , R = M i E ,
α i z M i E , I + α i z M i E , R = k i z M i E ;
M i E = T i TE M i E , I , T i TE = 2 α i z α i z + k i z ,
M i E , R = R i TE M i E , I , R i TE = α i z k i z α i z + k i z .
M i H = T i TM M i H , I , T i TM = 2 α i z ε 1 α i z ε 1 + k i z ε 2 ,
M i H , R = R i TM M i H , I , R i TM = 2 α i z ε 1 k i z ε 2 α i z ε 1 + k i z ε 2 .
M m E , R = M m E R J 1 ( k m R ) Δ ε k m z 2 γ i 2 C ̂ i ,
α m z M m E , R = k m z M m E + R J 1 ( k m R ) Δ ε γ i k m z 2 γ i 2 C ̂ i ,
M m E , R = R J 1 ( k m R ) Δ ε C ̂ i ( α m z + k m z ) ( k m z + γ i ) ,
P m H , R = i α m z M m E , R = i α m z R J 1 ( k m R ) Δ ε C ̂ i ( α m z + k m z ) ( k m z + γ i ) .
M m H , R = M m H i R J 1 ( k m R ) Δ ε γ i k m z 2 γ i 2 C ̂ i ,
α m z ε 1 M m H , R = k m z ε 2 M m H + i k m z 2 ε 2 R J 1 ( k m R ) Δ ε k m z 2 γ i 2 C ̂ i ,
M m H , R = i k m z ε 2 R J 1 ( k m R ) Δ ε C ̂ i ( α m z ε 1 + k m z ε 2 ) ( k m z + γ i ) ,
P m E , R = i α m z k 0 2 ε 1 M m H , R = α m z k 0 2 ε 1 k m z ε 2 R J 1 ( k m R ) Δ ε C ̂ i ( α m z ε 1 + k m z ε 2 ) ( k m z + γ i ) .
b n , m E = c n , m E ,
b n , m H = c n , m H .
[ 1 1 α i z α i z ] ( M i E , R M i E , I ) = [ 1 1 k i z k i z ] [ exp ( i k i z t ) 0 0 exp ( i k i z t ) ] ( M i E + M i E ) ,
( 1 β i z ) M i E , T = [ 1 1 k i z k i z ] ( M i E + M i E ) ,
T i TE = [ cos ( k i z t ) i k i z sin ( k i z t ) i k i z sin ( k i z t ) cos ( k i z t ) ] ,
[ 1 1 α i z α i z ] ( M i E , R M i E , I ) = T i TE ( 1 β i z ) M i E , T .
( M i E , R M i E , T ) = [ 1 α i z T i TE ( 1 β i z ) ] 1 ( 1 α i z ) M i E , I ,
T i TM = [ cos ( k i z t ) i ε 2 k i z sin ( k i z t ) i k i z ε 2 sin ( k i z t ) cos ( k i z t ) ] ,
( M i H , R M i H , T ) = [ 1 α i z ε 1 T i TM ( 1 β i z ε 3 ) ] 1 ( 1 α i z ε 3 ) M i H , I .
( 1 β i z ε 3 ) M i H , T = [ 1 1 k i z ε 2 k i z ε 2 ] ( M i H + M i H ) .
( 1 α m z ) M m E , R = ( 1 1 k m z k m z ) ( exp ( i k m z t ) 0 0 exp ( i k m z t ) ) ( M m E + M m E ) + R J 1 ( k m R ) Δ ε k m z 2 γ i 2 ( 1 1 γ i z γ i z ) ( exp ( i γ i z t ) 0 0 exp ( i γ i z t ) ) ( C ̂ i + C ̂ i )
( 1 β m z ) M m E , T = ( 1 1 k m z k m z ) ( M m E + M m E ) + R J 1 ( k m R ) Δ ε k m z 2 γ i 2 ( 1 1 γ i z γ i z ) ( C ̂ i + C ̂ i ) .
( M m E , R M m E , T ) = R J 1 ( k m R ) Δ ε k m z 2 γ i 2 [ 1 α m z T m TE ( 1 β m z ) ] 1 × { [ 1 1 γ i z γ i z ] [ exp ( i γ i z t ) 0 0 exp ( i γ i z t ) ] T m TE [ 1 1 γ i z γ i z ] } ( C ̂ i + C ̂ i ) .
( 1 α m z ε 1 ) M m H , R = [ 1 1 k m z ε 2 k m z ε 2 ] [ exp ( i k m z t ) 0 0 exp ( i k m z t ) ] ( M m H + M m H ) + i R J 1 ( k m R ) Δ ε k m z 2 γ i 2 [ γ i z γ i z k m z 2 ε 2 k m z 2 ε 2 ] [ exp ( i γ i z t ) 0 0 exp ( i γ i z t ) ] ( C ̂ i + C ̂ i ) ,
( 1 β m z ε 3 ) M m H , T = [ 1 1 k m z ε 2 k m z ε 2 ] ( M m H + M m H ) + i R J 1 ( k m R ) Δ ε k m z 2 γ i 2 [ γ i z γ i z k m z 2 ε 2 k m z 2 ε 2 ] ( C ̂ i + C ̂ i ) ,
( M m H , R M m H , T ) = i R J 1 ( k m R ) Δ ε k m z 2 γ i 2 [ 1 α m z ε 1 T m TM ( 1 β m z ε 3 ) ] 1 × { [ γ i z γ i z k m z 2 ε 2 k m z 2 ε 2 ] [ exp ( i γ i z t ) 0 0 exp ( i γ i z t ) ] T m TM [ γ i z γ i z k m z 2 ε 2 k m z 2 ε 2 ] } ( C ̂ i + C ̂ i ) .
( ε d ε 2 ) Δ i R J 1 ( k m R )
( ε d ε 2 ) k m Δ m R k m 2 k m 2 [ J 1 ( k m R ) J 0 ( k m R ) k m J 0 ( k m R ) J 1 ( k m R ) k m ] ,
( ε ) m , m 0 , 0 = ε 2 δ m m + ( ε d ε 2 ) k m Δ m R k m 2 k m 2 [ J 1 ( k m R ) × J 0 ( k m R ) k m J 0 ( k m R ) J 1 ( k m R ) k m ] ,
( ε 1 ) m , m 0 , 0 = 1 ε 2 δ m m ( ε d ε 2 ) ε 2 2 k m Δ m R k m 2 k m 2 [ J 1 ( k m R ) × J 0 ( k m R ) k m J 0 ( k m R ) J 1 ( k m R ) k m ] ,
M m E = b 0 , m E c 0 , m E , P m E = b 0 , m E + c 0 , m E ,
M m H = b 0 , m H c 0 , m H , P m H = b 0 , m H + c 0 , m H .
( M m E ) = P m H ,
( P m H ) = k m z 2 M m E ,
( M m H ) = k 0 2 ε 2 P m E ,
( P m E ) = k m z 2 k 0 2 ε 2 M m H + R G m , i ( R ) M i H ,
G m , i ( R ) = ( ε d ε 2 ) ε 2 2 k m Δ i k m 2 k i 2 [ J 1 ( k m R ) J 0 ( k i R ) k m J 0 ( k m R ) J 1 ( k i R ) k i ] .
( M m H ) = k m z 2 M m H k 0 2 ε 2 R G m , i ( R ) M i H ,
M m H = M m H ± exp ( ± i k m z z ) k 0 2 ε 2 R G m , i ( R ) k m z 2 γ i 2 M i H ± exp ( ± i γ i z ) .
C ̂ i 2 ε 2 ε 2 + ε d C ̂ i ,
E ( x ) H 1 + ( k p x ) ,
P ρ = 1 2 ρ 2 ( E × H ¯ ) ρ ̂ ,
P p J 1 ( sin 2 π λ R sin ψ ) sin 2 π λ R sin ψ 2 ( cos 2 ψ + sin 2 ψ cos 2 θ ) ,
( cos 2 ψ + sin 2 ψ cos 2 θ ) = { 1 , θ = 0 cos 2 ψ , θ = 90 ° .
M m E , R = T m TE , inv M m E , D M ,
M m H , R = T m TM , inv M m H , D M ,
T m TE , inv = 2 k m z α m z + k m z ,
T m TM , inv = 2 k m z ε 2 α m z ε 1 + k m z ε 2 ,
M m E , D M = R J 1 ( k m R ) Δ ε C ̂ i 2 k m z ( k m z + γ i ) ,
M m H , D M = i R J 1 ( k m R ) Δ ε C ̂ i 2 ( k m z + γ i ) ,
J 1 ( k m R ) k m R 2 ,
M m E , D M k m ε 2 8 ε 2 R 2 C ̂ 1 = k m k 2 Z 2 D M 8 π ,
D M = π R 2 ε 2 ε 2 C ̂ i k 2 Z 2 .
M m H , D M i k m k 0 2 ε 2 k m z k 2 Z 2 D M 8 π
T m TE , inv 2 ,
T m TM , inv 2 ε 1 α m z k m z ε 2 ,
M m E , R k m 4 R 2 ε 2 C ̂ i ε 2 , P m H , R i α m z k m 4 R 2 ε 2 C ̂ i ε 2 ,
M m H , R i k 0 2 ε 1 α m z k m 4 R 2 ε 2 C ̂ i ε 2 , P m E , R k m 4 R 2 ε 2 C ̂ i ε 2 .
M m E , R R J 1 ( k m R ) ε 2 C ̂ i 2 ε 2 , P m H , R i α m z R J 1 ( k m R ) ε 2 C ̂ i 2 ε 2 ,
M m H , R i k 0 2 ε 1 α m z R J 1 ( k m R ) ε 2 C ̂ i 2 ε 2 , P m E , R R J 1 ( k m R ) ε 2 C ̂ i 2 ε 2 ,
Δ E θ , n E θ , n r 2 + 2 i n r E r , n + k 2 E θ , n = 0 ,
Δ E r , n E r , n r 2 2 i n r E θ , n + k 2 E r , n = 0 ,
Δ = 2 r 2 + 1 r r + 1 r 2 2 θ 2 + 2 z 2 .
E , n = E θ , n ± i E r , n ,
[ 2 r 2 + 1 r r ( n ± 1 ) 2 r 2 + k 2 + 2 z 2 ] E ± , n ( r , z ) = 0 .
E , n = 0 c ̂ n E ( k r , z ) J n 1 ( k r r ) k r d k r = m = 0 c n , m E ( z ) J n 1 ( k m r ) Δ m ,
E + , n = 0 b ̂ n E ( k r , z ) J n + 1 ( k r r ) k r d k r = m = 0 b n , m E ( z ) J n + 1 ( k m r ) Δ m ,
( d d r + n + 1 r ) J n + 1 ( k m r ) = k m J n ( k m r ) ,
( d d r n 1 r ) J n 1 ( k m r ) = k m J n ( k m r ) ,
k 0 2 ε ( r ) E z , n = k 0 2 ε ( r ) m e n , m ( z ) J n ( k m r ) Δ m = Eq . ( 5 ) i m [ b n , m H ( z ) c n , m H ( z ) ] J n ( k m r ) k m Δ m .
k 0 2 m ( ε ) m , m n , n e n , m ( z ) = i ω k m [ b n , m H ( z ) c n , m H ( z ) ] ,
e n , m ( z ) = i k 0 2 m ( ε 1 ) m , m n n k m [ b n , m H ( z ) c n , m H ( z ) ] .
h n , m ( z ) = 1 i ω μ 0 [ b n , m H ( z ) c n , m H ( z ) ] .
d d r J n ( k m r ) = k m 2 [ J n 1 ( k m r ) J n + 1 ( k m r ) ] ,
z m c n , m E ( z ) J n 1 ( k m r ) Δ m = m c n , m H ( z ) J n 1 ( k m r ) Δ m k m 2 k 0 2 m , m ( ε n , n ) m , m 1 k m [ b n , m H ( z ) c n , m H ( z ) ] J n 1 ( k m r ) ,
z m b n , m E ( z ) J n + 1 ( k m r ) Δ m = m b n , m H ( z ) J n + 1 ( k m r ) Δ m k m 2 k 0 2 m , m ( ε n , n ) m , m 1 k m [ b n , m H ( z ) c n , m H ( z ) ] J n + 1 ( k m r ) .
z m c n , m H ( z ) J n 1 ( k m r ) Δ m = m k m 2 2 [ b n , m E ( z ) c n , m E ( z ) ] J n 1 ( k m r ) Δ m + k 0 2 ε ( r ) m c n , m E ( z ) J n 1 ( k m r ) Δ m ,
z m b n , m H ( z ) J n + 1 ( k m r ) Δ m = m k m 2 2 [ b n , m E ( z ) c n , m E ( z ) ] J n + 1 ( k m r ) Δ m k 0 2 ε ( r ) m b n , m E ( z ) J n + 1 ( k m r ) Δ m .
c m ( z ) = k m z 2 c m ( z ) + R J 1 ( k m R ) n η m n c n ( z ) .
c m ( z ) = n C m n ± exp ( ± i γ n z ) ,
n C m n ± γ n 2 exp ( ± i γ n z ) = p , q ( k m z 2 δ m p R J 1 ( k m R ) η m p ) C p q ± exp ( ± i γ q z ) .
C m n ± γ n 2 = p ( k m z 2 δ m p R J 1 ( k m R ) η m p ) C p n ± .
γ m 2 k m z 2 R J 1 ( k m R ) η m m .
C m n ± R J 1 ( k m R ) k m z 2 γ n 2 p η m p C p n ± .
C m n ± C m m ± δ m n + O ( R 2 ) .
C m n ± = R J 1 ( k m R ) η m n k m z 2 γ n 2 C n n ± .
c m ( z ) = C m m ± exp ( ± i γ m z ) + R J 1 ( k m R ) n m η m n k m z 2 γ n 2 C n n ± exp ( ± i γ n z ) .
c i ( z ) = C i ± exp ( ± i γ i z ) ,
c m i ( z ) = R J 1 ( k m R ) η m i k m z 2 γ i 2 C i ± exp ( ± i γ i z ) ,
E i = E 0 n = i n J n ( k r r ) exp [ i n ( θ θ 0 ) ] exp ( i k z z ) .
E 0 = E 0 x x ̂ + E 0 y y ̂ + E 0 z z ̂ .
E 0 = E 0 x ( r ̂ cos θ θ ̂ sin θ ) + E 0 y ( r ̂ sin θ + θ ̂ cos θ ) + E 0 z z ̂ ,
E 0 = r ̂ [ ( E 0 x 2 + E 0 y 2 i ) exp ( i θ ) + ( E 0 x 2 E 0 y 2 i ) exp ( i θ ) ]
+ θ ̂ [ ( E 0 x 2 i + E 0 y 2 ) exp ( i θ ) + ( E 0 x 2 i + E 0 y 2 ) exp ( i θ ) ] + z ̂ E 0 z .
E r i = n = N N ( E 0 x 2 + E 0 y 2 i ) i n exp ( i n θ 0 ) J n ( k r r ) exp [ i ( n + 1 ) θ ] exp ( i k z z ) + n = N N ( E 0 x 2 E 0 y 2 i ) i n exp ( i n θ 0 ) J n ( k r r ) exp [ i ( n 1 ) θ ] exp ( i k z z ) .
E r i = n = N + 1 N + 1 i ( E 0 x 2 i E 0 y 2 ) i n 1 exp [ i ( n 1 ) θ 0 ] J n 1 ( k r r ) exp ( i n θ ) exp ( i k z z ) + n = N 1 N 1 i ( E 0 x 2 i + E 0 y 2 ) i n + 1 exp [ i ( n + 1 ) θ 0 ] J n + 1 ( k r r ) exp ( i n θ ) exp ( i k z z )
E θ i = n = N + 1 N + 1 ( E 0 x 2 i + E 0 y 2 ) i n 1 exp [ i ( n 1 ) θ 0 ] J n 1 ( k r r ) exp ( i n θ ) exp ( i k z z ) + n = N 1 N 1 ( E 0 x 2 i + E 0 y 2 ) i n + 1 exp [ i ( n + 1 ) θ 0 ] J n + 1 ( k r r ) exp ( i n θ ) exp ( i k z z ) .
b n , m E , I ( z = 0 ) = ( E 0 x 2 i + E 0 y 2 ) i n + 1 Δ m exp [ i ( n + 1 ) θ 0 ] ,
c n , m E , I ( z = 0 ) = ( E 0 x 2 i + E 0 y 2 ) i n 1 Δ m exp [ i ( n 1 ) θ 0 ] .
E ( ρ ) = E i ( ρ ) + G ( ρ ρ ) ( ε d ε 2 1 ) E ( ρ ) d 3 ρ ,
G ( ρ ρ ) = L δ ( ρ ρ ) + P v G ( ρ ρ ) ,
L = I ( 2 ) 2 ,
I ( 2 ) = [ 1 0 0 0 1 0 0 0 0 ] ,
E ( ρ ) E i ( ρ ) + L ( ε d ε 2 1 ) E ( ρ ) ,
E ( ρ ) [ I ( 3 ) L ( ε d ε 2 1 ) ] 1 E i ( ρ ) ,
[ I ( 3 ) L ε d ε 2 ε 2 ] 1 = [ 2 ε 2 ε d + ε 2 0 0 0 2 ε 2 ε d + ε 2 0 0 0 1 ] ,
E = Z k 2 4 π ρ ̂ × D M ( 1 1 i k ρ ) exp ( i k ρ ) ρ ,
H = 1 4 π { k 2 ( ρ ̂ × D M ) × ρ ̂ [ 3 ρ ̂ ( ρ ̂ D M ) D M ] ( 1 ρ 2 i k ρ ) } exp ( i k ρ ) ρ ,
D M = D M y ̂ ,
E z = Z D M k 2 8 π ( 1 1 i k r ) exp ( i k r ) r [ exp ( i θ ) + exp ( i θ ) ] .
Z D M k 4 8 π ( 1 1 i k r ) exp ( i k r ) r = i m [ b n , m H ( 0 ) c n , m H ( 0 ) ] J n ( k m r ) k m Δ m
M m H b 1 , m H c 1 , m H = i Z D M k 4 8 π 0 ( 1 1 i k r ) exp ( i k r ) r J 1 ( k m r ) r d r = i Z D M k 4 8 π k m k k m z ,
H θ , 1 i H r , 1 = 0 b ̂ 1 H ( k r , z ) j n + 1 ( k r r ) k r d k r = m = 0 b 1 , m H ( z ) J 2 ( k m r ) Δ m .
ρ ̂ × D M = D M cos θ z ̂ , ( ρ ̂ × D M ) × ρ ̂ = D M cos θ θ ̂ ,
ρ ̂ D M = D M sin θ ,
H r = 2 D M k 2 4 π ( 1 k 2 r 2 + 1 i k r ) exp ( i k r ) r 1 2 i [ exp ( i θ ) exp ( i θ ) ] ,
H θ = D M k 2 4 π ( 1 1 k 2 r 2 1 i k r ) exp ( i k r ) r 1 2 [ exp ( i θ ) + exp ( i θ ) ] .
b 1 , m H = ω μ 0 2 k m 0 ( H θ , 1 i H r , 1 ) J 2 ( k m r ) r d r = ω μ 0 2 k m D M k 2 8 π 0 ( 1 3 k 2 r 2 3 i k r ) exp ( i k r ) r J 2 ( k m r ) r d r = ω μ 0 2 k m D M k 2 8 π ( i k m 2 k 2 k m z ) ,
P m H = 2 b 1 , m H M m H = i Z D M 8 π k k m k m z .
M m E = 1 i k m z P m H = k m k Z D M 8 π ,
P m E = i k m z k 2 M m H = k m k Z D M 8 π .

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