Abstract

The existing analyses on extraordinary optical transmission through apertures on a metal screen have been carried out assuming perfect conductivity or by examining arrays of closely spaced holes with subwavelength dimensions. We present an electromagnetic analysis of a single hole (modeled by use of an array of distant holes) in a finitely conducting metal membrane, applying no approximations. We demonstrate that finite conductivity is not of remarkable importance with small hole-diameter-to-wavelength ratios in the absence of strong resonances. However, if the angle of incidence of a plane wave is such that surface plasmons are excited, substantial enhancement of the transmittance can be observed, and the effect of finite conductivity will no longer be negligible. Our analysis also reveals that transmission of small apertures in highly conducting membranes can be described by approximate analytical formulas if surface waves are not excited, but with poor conductors the full electromagnetic analysis should be applied.

© 2004 Optical Society of America

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2003 (3)

2002 (2)

Q. Cao, P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 0574031–0574034 (2002).
[CrossRef]

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

2001 (4)

E. Silberstein, P. Lalanne, J.-P. Hugonin, Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[CrossRef]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[CrossRef]

T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-discontinuity approach for non-paraxial diffractive optics,” J. Mod. Opt. 48, 1195–1210 (2001).
[CrossRef]

L. Martı́n-Moreno, F. J. Garcı́a-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

2000 (4)

E. Popov, M. Nevière, S. Enoch, R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000).
[CrossRef]

S. Astilean, P. Lalanne, M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265–273 (2000).
[CrossRef]

T. Thio, H. J. Lezec, T. W. Ebbesen, “Strongly enhanced optical transmission through subwavelength holes in metal films,” Physica B 279, 90–93 (2000).
[CrossRef]

I. Avrutsky, Y. Zhao, V. Kochergin, “Surface-plasmon-assisted resonant tunneling of light through a periodically corrugated thin metal film,” Opt. Lett. 25, 595–597 (2000).
[CrossRef]

1999 (3)

T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, T. W. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B 16, 1743–1748 (1999).
[CrossRef]

M. M. J. Treacy, “Dynamical diffraction in metallic optical gratings,” Appl. Phys. Lett. 75, 606–608 (1999).
[CrossRef]

J. A. Porto, F. J. Garcı́a-Vidal, J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
[CrossRef]

1998 (6)

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

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998).
[CrossRef]

U. Schröter, D. Heitmann, “Surface-plasmon-enhanced transmission through metallic gratings,” Phys. Rev. B 58, 15419–15421 (1998).
[CrossRef]

J. R. Sambles, “More than transparent,” Nature 391, 641–642 (1998).
[CrossRef]

P. Dansas, N. Paraire, “Fast modeling of photonic bandgap structures by use of a diffraction-grating approach,” J. Opt. Soc. Am. A 15, 1586–1598 (1998).
[CrossRef]

V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, “Spectral filtering with finitely conducting inductive grids,” J. Opt. Soc. Am. A 15, 2783–2785 (1998).
[CrossRef]

1997 (2)

1996 (4)

1994 (1)

1993 (1)

J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[CrossRef]

1987 (1)

1978 (1)

1954 (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 18, 35–100 (1954).
[CrossRef]

1944 (1)

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

1908 (1)

G. Mie, “Beitrage zur optik truber median, speziell kolloidaler metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

1907 (1)

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London Ser. A 79, 399–416 (1907).
[CrossRef]

1896 (1)

A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47, 317–374 (1896).
[CrossRef]

Astilean, S.

S. Astilean, P. Lalanne, M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265–273 (2000).
[CrossRef]

Avrutsky, I.

Baida, F. I.

Bethe, H. A.

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

Bonod, N.

Botten, L. C.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.

Bouwkamp, C. J.

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 18, 35–100 (1954).
[CrossRef]

Cao, Q.

Q. Cao, P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 0574031–0574034 (2002).
[CrossRef]

E. Silberstein, P. Lalanne, J.-P. Hugonin, Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[CrossRef]

Chavel, P.

P. Lalanne, C. Sauvan, J.-P. Hugonin, J. C. Rodier, P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68, 125404 (2003).
[CrossRef]

Dansas, P.

Derrick, G. H.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.

Durnin, J.

Ebbesen, T. W.

L. Martı́n-Moreno, F. J. Garcı́a-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[CrossRef]

T. Thio, H. J. Lezec, T. W. Ebbesen, “Strongly enhanced optical transmission through subwavelength holes in metal films,” Physica B 279, 90–93 (2000).
[CrossRef]

T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, T. W. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B 16, 1743–1748 (1999).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998).
[CrossRef]

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

Enoch, S.

N. Bonod, S. Enoch, L. Li, E. Popov, M. Nevière, “Resonant optical transmission through thin metallic films with and without holes,” Opt. Express 11, 482–490 (2003).
[CrossRef] [PubMed]

E. Popov, M. Nevière, S. Enoch, R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000).
[CrossRef]

Friberg, A. T.

J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[CrossRef]

Garci´a de Abajo, F. J.

Garci´a-Vidal, F. J.

L. Martı́n-Moreno, F. J. Garcı́a-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

J. A. Porto, F. J. Garcı́a-Vidal, J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
[CrossRef]

Ghaemi, H. F.

T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, T. W. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B 16, 1743–1748 (1999).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998).
[CrossRef]

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

Granet, G.

Grupp, D. E.

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998).
[CrossRef]

Heitmann, D.

U. Schröter, D. Heitmann, “Surface-plasmon-enhanced transmission through metallic gratings,” Phys. Rev. B 58, 15419–15421 (1998).
[CrossRef]

Hugonin, J.-P.

P. Lalanne, C. Sauvan, J.-P. Hugonin, J. C. Rodier, P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68, 125404 (2003).
[CrossRef]

E. Silberstein, P. Lalanne, J.-P. Hugonin, Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[CrossRef]

Kettunen, V.

T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-discontinuity approach for non-paraxial diffractive optics,” J. Mod. Opt. 48, 1195–1210 (2001).
[CrossRef]

V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, “Spectral filtering with finitely conducting inductive grids,” J. Opt. Soc. Am. A 15, 2783–2785 (1998).
[CrossRef]

Knop, K.

Kochergin, V.

Kuittinen, M.

T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-discontinuity approach for non-paraxial diffractive optics,” J. Mod. Opt. 48, 1195–1210 (2001).
[CrossRef]

V. Kettunen, M. Kuittinen, J. Turunen, P. Vahimaa, “Spectral filtering with finitely conducting inductive grids,” J. Opt. Soc. Am. A 15, 2783–2785 (1998).
[CrossRef]

Lalanne, P.

P. Lalanne, C. Sauvan, J.-P. Hugonin, J. C. Rodier, P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68, 125404 (2003).
[CrossRef]

Q. Cao, P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 0574031–0574034 (2002).
[CrossRef]

E. Silberstein, P. Lalanne, J.-P. Hugonin, Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001).
[CrossRef]

S. Astilean, P. Lalanne, M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265–273 (2000).
[CrossRef]

P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (1997).
[CrossRef]

P. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

Layet, B.

Lemercier-Lalanne, D.

P. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

Lezec, H. J.

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[CrossRef]

L. Martı́n-Moreno, F. J. Garcı́a-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

T. Thio, H. J. Lezec, T. W. Ebbesen, “Strongly enhanced optical transmission through subwavelength holes in metal films,” Physica B 279, 90–93 (2000).
[CrossRef]

T. Thio, H. F. Ghaemi, H. J. Lezec, P. A. Wolff, T. W. Ebbesen, “Surface-plasmon-enhanced transmission through hole arrays in Cr films,” J. Opt. Soc. Am. B 16, 1743–1748 (1999).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998).
[CrossRef]

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

Li, L.

Linke, R. A.

Mansuripur, M.

M. Mansuripur, The Physical Principles of Magneto-Optical Recording (Cambridge U. Press, Cambridge, 1995).

Marti´n-Moreno, L.

L. Martı́n-Moreno, F. J. Garcı́a-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

McPhedran, R. C.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.

Mie, G.

G. Mie, “Beitrage zur optik truber median, speziell kolloidaler metallosungen,” Ann. Phys. (Leipzig) 25, 377–445 (1908).
[CrossRef]

Moreau, A.

Nevière, M.

N. Bonod, S. Enoch, L. Li, E. Popov, M. Nevière, “Resonant optical transmission through thin metallic films with and without holes,” Opt. Express 11, 482–490 (2003).
[CrossRef] [PubMed]

E. Popov, M. Nevière, S. Enoch, R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100–16108 (2000).
[CrossRef]

Noponen, E.

Palamaru, M.

S. Astilean, P. Lalanne, M. Palamaru, “Light transmission through metallic channels much smaller than the wavelength,” Opt. Commun. 175, 265–273 (2000).
[CrossRef]

Paraire, N.

Pellerin, K. M.

L. Martı́n-Moreno, F. J. Garcı́a-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

T. Thio, K. M. Pellerin, R. A. Linke, H. J. Lezec, T. W. Ebbesen, “Enhanced light transmission through a single subwavelength aperture,” Opt. Lett. 26, 1972–1974 (2001).
[CrossRef]

Pendry, J. B.

L. Martı́n-Moreno, F. J. Garcı́a-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
[CrossRef] [PubMed]

J. A. Porto, F. J. Garcı́a-Vidal, J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
[CrossRef]

Popov, E.

N. Bonod, S. Enoch, L. Li, E. Popov, M. Nevière, “Resonant optical transmission through thin metallic films with and without holes,” Opt. Express 11, 482–490 (2003).
[CrossRef] [PubMed]

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J. Opt. Soc. Am. A (9)

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A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47, 317–374 (1896).
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J. R. Sambles, “More than transparent,” Nature 391, 641–642 (1998).
[CrossRef]

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

P. Lalanne, C. Sauvan, J.-P. Hugonin, J. C. Rodier, P. Chavel, “Perturbative approach for surface plasmon effects on flat interfaces periodically corrugated by subwavelength apertures,” Phys. Rev. B 68, 125404 (2003).
[CrossRef]

H. F. Ghaemi, T. Thio, D. E. Grupp, T. W. Ebbesen, H. J. Lezec, “Surface plasmons enhance optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998).
[CrossRef]

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[CrossRef]

Phys. Rev. Lett. (3)

J. A. Porto, F. J. Garcı́a-Vidal, J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
[CrossRef]

L. Martı́n-Moreno, F. J. Garcı́a-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114–1117 (2001).
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Proc. R. Soc. London Ser. A (1)

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J. Turunen, A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
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M. Mansuripur, The Physical Principles of Magneto-Optical Recording (Cambridge U. Press, Cambridge, 1995).

J. Turunen, F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Wiley-VCH, Berlin, 1997).

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, New York, 1988).

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Figures (10)

Fig. 1
Fig. 1

Metal membrane perforated by square apertures of the size D×D. The thickness of the film is denoted by h. The incident field propagates in the xz plane at an angle θ with respect to the z axis; the electric field vector of the incident field is in the plane of incidence. The grating period used in the calculations is d×d.

Fig. 2
Fig. 2

Dependence of the normalized transmittance on the hole size D for two different film thicknesses h. Also, the results of Bethe’s and Bouwkamp’s formulas are plotted. The refractive index of the metal is n2=0.1+3i.

Fig. 3
Fig. 3

Transmittance of both the perforated (D=0.3λ) and the unperforated membranes as functions of the thickness with refractive index n2=0.1+3i.

Fig. 4
Fig. 4

Same as Fig. 2 but with n2=1.2+2i.

Fig. 5
Fig. 5

Same as Fig. 3 but with n2=1.2+2i and D=0.4λ.

Fig. 6
Fig. 6

Dependence of normalized transmittance on the hole size D for different film thicknesses at the surface plasmon excitation angle (θ=45.1°) with n1=1.5, n2=0.1+3i, and n3=1.

Fig. 7
Fig. 7

Normalized transmittance of an aperture (n2=0.1+3i, D=0.3λ, h=0.1λ) as a function of the incident angle θ. Also shown is absorption of the unperforated film versus the incident angle θ for a film having h=0.1λ.

Fig. 8
Fig. 8

(a) Distribution of the electric energy density at the exit plane of a circular hole with diameter D=0.2λ and film thickness h=0.1λ at the surface plasmon angle (n1=1.5, n3=1, θ=45.1°) with n2=0.1+3i. (b) Same for the time average of the z component of the Poynting vector whose maximum is normalized to unity. The hole is centered at the origo of the coordinates.

Fig. 9
Fig. 9

(a) Diagram of the simulated system. The annular beam at the entrance pupil of the objective lens (λ=650 nm) has a Gaussian cross section between the inner and the outer radii rmin=1.75 mm, rmax=1.96 mm; this beam is linearly polarized in the radial direction. The angle of incidence of the various rays at the metal film ranges from 42.13° to 48.56°, centered around the resonance angle (θres=45.1°) for surface plasmon excitation in the thin metal film. The film thickness is h=65 nm, and its complex refractive index is n2=0.1+3i. (b) Intensity distribution of the annular beam at the entrance pupil of the objective lens. (c) Localized intensity distribution, |Ex|2+|Ey|2, at the focal plane of the objective. Like the incident beam, the focused spot has radial polarization; that is, at all points within the focal plane of the objective, the in-plane component of the electric field is oriented radially relative to the focal point (x, y)=(0, 0).

Fig. 10
Fig. 10

Same as Fig. 8 but the radial component of the incident field is the first-order Bessel function.

Equations (2)

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ηD2=6427π(kD)426,
ηD2=6427π(kD)426+2225(kD)628+731218,375(kD)8210.

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