Abstract

In this paper, we applied the modal expansion method (MEM) to investigate the wave behaviors inside a step-modulated subwavelength metal slit. The physical mechanism of the surface plasmon polariton (SPP) transmission is investigated in detail for slit structures with either dielectric or geometric modulation. The applicability of the effective index method is discussed. Moreover, as a special case of the geometric modulation, the evanescent-wave assisted transmission is demonstrated in a thin-modulated slit. We emphasize that a complete set is necessary in order to expand the wave functions in these kinds of structures. All the calculated results by the MEM are well retrieved by the finite-difference time-domain calculation.

© 2011 OSA

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    [CrossRef]
  33. Since the slit is an infinitely long one, the wave source has to be located in Layer 1. This can be easily done in FDTD by setting the coordinates of the source to y = H(0). A “slab mode” (a normal mode of a slab waveguide with characteristics matching the input waveguide, a feature provieded by the RSOFT Fullwave software) is used to excite the SPP mode. However, for the amplitude of the source in FDTD, it should be adjusted to the same value of the normalized SPP mode in Layer 1 as pointed out later. The grid size of the simulation is set as 2.5 nm × 2.5 nm.
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    [CrossRef]

2010 (2)

Y. S. Zhou, B. Y. Gu, and H. Y. Wang, “Band-gap structures of surface-plasmon polaritons in a subwavelength metal slit filled with periodic dielectrics,” Phys. Rev. A 81, 015801 (2010).
[CrossRef]

C. Li, Y. S. Zhou, H. Y. Wang, and F. H. Wang, “Wavelength squeeze of surface plasmon polariton in a subwavelength metal slit,” J. Opt. Soc. Am. B 27, 59–64 (2010).
[CrossRef]

2009 (5)

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Q. Zhang, X. Huang, X. Lin, J. Tao, and X. Jin, “A subwavelength coupler-type MIM optical filter,” Opt. Express 17, 7549–7555 (2009).
[CrossRef]

X. S. Lin and X. G. Huang, “Numerical modeling of a teeth-shaped nanoplasmonic waveguide filter,” J. Opt. Soc. Am. B 26, 1263–1268 (2009).
[CrossRef]

J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express 17, 20134–20139 (2009).
[CrossRef] [PubMed]

2008 (3)

2007 (4)

A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Surface plasmon polaritons on metallic surfaces,” Opt. Express 15, 183–197 (2007).
[CrossRef] [PubMed]

E. Feigenbaum and M. Orenstein, “Modeling of complementary (void) plasmon waveguiding,” J. Lightwave Technol. 25, 2547–2562 (2007).
[CrossRef]

Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B 75, 035411 (2007).
[CrossRef]

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[CrossRef]

2006 (2)

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

A. Hosseini and Y. Massoud, “A low-loss metal-insulator-metal plasmonic bragg reflector,” Opt. Express 14, 11318–11323 (2006).
[CrossRef]

2005 (2)

2004 (1)

T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyia, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5835 (2004).
[CrossRef]

2003 (1)

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

2002 (1)

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002).
[CrossRef] [PubMed]

2001 (2)

Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86, 5601–5603 (2001).
[CrossRef] [PubMed]

F. Villa, T. Lopez-Rios, and L. E. Regalado, “Electromagnetic modes in metal-insulator-metal structures,” Phys. Rev. B 63, 165103 (2001).
[CrossRef]

1999 (2)

J. Pendry, “Playing tricks with light,” Science 285, 1687–1688 (1999).
[CrossRef]

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

1997 (1)

1992 (2)

1991 (1)

1989 (1)

T. Itoh, Numerical Techniques for Microwave and Millimeter Wave Passive Structures (Wiley, 1989).

1988 (1)

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

1986 (1)

1985 (1)

L. C. Botten and R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta 32, 1479–1488 (1985).
[CrossRef]

1983 (1)

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Computer Phys. Commun. 29, 245–259 (1983).
[CrossRef]

1982 (1)

1981 (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

1978 (1)

P. Yeh, “A new optical model for wire grid polarizers,” Opt. Commun. 26, 289–292 (1978).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1966 (1)

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1966).

1955 (1)

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, 1955).

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Barnes, W. L.

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

Botten, L. C.

L. C. Botten and R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta 32, 1479–1488 (1985).
[CrossRef]

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Computer Phys. Commun. 29, 245–259 (1983).
[CrossRef]

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Bozhevolnyia, S. I.

T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyia, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5835 (2004).
[CrossRef]

Cao, Q.

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002).
[CrossRef] [PubMed]

Chandezon, J.

Choi, S. S.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Coddington, E. A.

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, 1955).

Collin, R. E.

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1966).

Cornet, G.

Craig, M. S.

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Computer Phys. Commun. 29, 245–259 (1983).
[CrossRef]

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Deng, Q.

Dereux, A.

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

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Du, C.

Dupuis, M. T.

Ebbesen, T. W.

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

Fan, S.

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87, 131102 (2005).
[CrossRef]

Fang, G.

Farn, W. M.

Feigenbaum, E.

Fukui, M.

Gao, H.

Garcia-Vidal, F. J.

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

Gaylord, T. K.

Gorkunov, M.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[CrossRef]

Gu, B. Y.

Y. S. Zhou, B. Y. Gu, and H. Y. Wang, “Band-gap structures of surface-plasmon polaritons in a subwavelength metal slit filled with periodic dielectrics,” Phys. Rev. A 81, 015801 (2010).
[CrossRef]

Gunning, W. J.

Haidner, H.

Haraguchi, M.

Hosseini, A.

Houseini, A.

Huang, X.

Huang, X. G.

Itoh, T.

T. Itoh, Numerical Techniques for Microwave and Millimeter Wave Passive Structures (Wiley, 1989).

Jin, X.

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Kang, J. H.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Kim, D. S.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Kipfer, P.

Kocabas, S. E.

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

Koo, S. M.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Kurokawa, Y.

Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B 75, 035411 (2007).
[CrossRef]

Lalanne, P.

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002).
[CrossRef] [PubMed]

Leosson, K.

T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyia, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5835 (2004).
[CrossRef]

Levinson, N.

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, 1955).

Li, C.

Li, L.

Lin, X.

Lin, X. S.

Liu, J.

Liu, S.

Lopez-Rios, T.

F. Villa, T. Lopez-Rios, and L. E. Regalado, “Electromagnetic modes in metal-insulator-metal structures,” Phys. Rev. B 63, 165103 (2001).
[CrossRef]

Luo, X.

Lv, Y.

Mansuripur, M.

Massoud, Y.

Matsuzaki, Y.

Maystre, D.

McPhedran, R. C.

L. C. Botten and R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta 32, 1479–1488 (1985).
[CrossRef]

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Complex zeros of analytic functions,” Computer Phys. Commun. 29, 245–259 (1983).
[CrossRef]

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Miller, D. A. B.

S. E. Kocabas, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal waveguides,” Phys. Rev. B 79, 035120 (2009).
[CrossRef]

Miyazaki, H. T.

Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B 75, 035411 (2007).
[CrossRef]

Moharam, M. G.

Moloney, J. V.

Motamedi, M. E.

Nakagaki, M.

Nejati, H.

Nikolajsen, T.

T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyia, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5835 (2004).
[CrossRef]

Okamoto, T.

Orenstein, M.

Park, D. J.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Park, G. S.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Park, H. R.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
[CrossRef]

Park, N. K.

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Park, Q. H.

M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
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J. Pendry, “Playing tricks with light,” Science 285, 1687–1688 (1999).
[CrossRef]

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

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M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
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B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
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J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845–2848 (1999).
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M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
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M. A. Seo, H. R. Park, S. M. Koo, D. J. Park, J. H. Kang, O. K. Suwal, S. S. Choi, P. C. M. Planken, G. S. Park, N. K. Park, Q. H. Park, and D. S. Kim, “Terahertz field enhancement by a metallic nano slit operating beyond the skin-depth limit,” Nat. Photonics 3, 152–156 (2009).
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Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86, 5601–5603 (2001).
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F. Villa, T. Lopez-Rios, and L. E. Regalado, “Electromagnetic modes in metal-insulator-metal structures,” Phys. Rev. B 63, 165103 (2001).
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Y. S. Zhou, B. Y. Gu, and H. Y. Wang, “Band-gap structures of surface-plasmon polaritons in a subwavelength metal slit filled with periodic dielectrics,” Phys. Rev. A 81, 015801 (2010).
[CrossRef]

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F. Villa, T. Lopez-Rios, and L. E. Regalado, “Electromagnetic modes in metal-insulator-metal structures,” Phys. Rev. B 63, 165103 (2001).
[CrossRef]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[CrossRef]

Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B 75, 035411 (2007).
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[CrossRef]

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[CrossRef]

Phys. Rev. Lett. (3)

Y. Takakura, “Optical resonance in a narrow slit in a thick metallic screen,” Phys. Rev. Lett. 86, 5601–5603 (2001).
[CrossRef] [PubMed]

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

Q. Cao and P. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 057403 (2002).
[CrossRef] [PubMed]

Science (1)

J. Pendry, “Playing tricks with light,” Science 285, 1687–1688 (1999).
[CrossRef]

Other (6)

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

R. E. Collin, Foundations for Microwave Engineering (McGraw-Hill, 1966).

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, 1955).

The commercially available software developed by Rsoft Design Group http://www.rsoftdesign.com is used for the numerical simulations.

Since the slit is an infinitely long one, the wave source has to be located in Layer 1. This can be easily done in FDTD by setting the coordinates of the source to y = H(0). A “slab mode” (a normal mode of a slab waveguide with characteristics matching the input waveguide, a feature provieded by the RSOFT Fullwave software) is used to excite the SPP mode. However, for the amplitude of the source in FDTD, it should be adjusted to the same value of the normalized SPP mode in Layer 1 as pointed out later. The grid size of the simulation is set as 2.5 nm × 2.5 nm.

T. Itoh, Numerical Techniques for Microwave and Millimeter Wave Passive Structures (Wiley, 1989).

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Figures (8)

Fig. 1
Fig. 1

Sketch of a step-modulated metal slit structure confined in x direction with perfect-conducting walls at Ll = 0, Lr = 2 μm. The gray areas are silver with dielectric constant εAg = −50.76 + 0.083i; The slit areas marked by slashes are filled with core materials. A TM wave is normally launched at y = Q (0) = −1 μm with wavelength λ 0 = 1 μm. Q (1) = 0.

Fig. 2
Fig. 2

The eigen problems of a confined metal-air-metal slit, confined width Lr Ll = 2. The lowest six eigenmodes are illustrated, where the first, third and fifth modes are even and other three are odd. The real and imaginary parts of kx 2 varying with slit width are plotted in (a) and (b); (c) the absolute value of the normalized wave functions in a slit with width being 0.25 μm.

Fig. 3
Fig. 3

(a) Left: the structure of the [1 – 6 – 1] dielectric-modulated slit with width w ( l ) = 0.2 μm and steady field distribution of Hz calculated by MEM; right: comparison of the steady field distribution of Hz along the central line (x = 1 μm) calculated by MEM (line) and the FDTD method (dots). (b) Steady field distribution of total Hz and its first 10 modes at the upper boundary of Layer 1, yQ (1)(−0); (c) The same Hz at the lower boundary of Layer 2, yQ (1)(+0).

Fig. 4
Fig. 4

Comparison between the SPP transmission and reflection coefficients in a two-layer structure by MEM (lines) and by EIM-based Fresnel equation (dots). The structure is constructed by setting w ( l ) = 0.2 μm and Q (0) = Q (1) = Q (2) = 0. For waves incident from air to dielectric, ε 2 ( 1 ) = 1 , ε 2 ( 2 ) = ε 2 ( 3 ) , (a) amplitude and (b) phase. For waves incident from dielectric to air, ε 2 ( 1 ) = ε 2 ( 2 ) , ε 2 ( 3 ) = 1 , (c) amplitude and (d) phase.

Fig. 5
Fig. 5

Transmission vs. q (2) calculated by MEM for [1 – 3 – 1] (solid line), [1 – 6 – 1] (dashed line) and [3 – 1 – 6] (dash-dotted line). The results calculated by FDTD (open circles) and EIM-based triple-layer Fresnel equation (dots) are also shown for comparison.

Fig. 6
Fig. 6

(a) Left: the structure of the [0.2 ∼ 0.4 ∼ 0.2] geometric-modulated slit with core material ε 2 ( l ) = 1 and steady field distribution of Hz calculated by MEM; right: comparison of the steady field distribution of Hz along the central line (x = 1 μm) calculated by MEM (line) and the FDTD method (dots). (b) Steady field distribution of Hz and its first 10 modes at the upper boundary of Layer 1, yQ (1)(−0); (c)The same Hz at the lower boundary of Layer 2, yQ (1)(+0).

Fig. 7
Fig. 7

Comparison between the SPPs transmission and reflection coefficients in a two-layer structure by the MEM (lines) and EIM-based Fresnel equation (dots). The structure is constructed by setting ε 2 ( l ) = 1 and Q (0) = Q (1) = Q (2) = 0. For waves incident from a narrower slit to a wider one, w (1) = 0.2 μm, w (2) = w (3), (a) amplitude and (b) phase. For waves incident from a wider slit to a narrower one, w (1) = w (2), w (3) = 0.2 μm, (c) amplitude and (d) phase.

Fig. 8
Fig. 8

(a) Transmission vs. q (2) calculated by MEM for [0.2 ∼ 0.4 ∼ 0.2] (solid line), [0.4 ∼ 0.2 ∼ 0.4] (dashed line) and [0.4 ∼ 0.2 ∼ 0.6] (dash-dotted line). The results calculated by FDTD (open circles) and multi-reflection (dots) are also shown for comparison. (b) Transmissions of a thin modulation in [0.2 ∼ 0.4 ∼ 0.2] with different number of the mode that used in the multi-reflection process.

Equations (9)

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φ n ( l ) ( x ) = { A n ( l ) sinh [ k x 1 n ( l ) ( x L l ) ] , L l x < x 1 ( l ) B n ( l ) e ik x 2 n x ( l ) + C n ( l ) e i k x 2 n x ( l ) , x 1 ( l ) x < x 2 ( l ) D n ( l ) sinh [ k x 3 n ( l ) ( x L r ) ] , x 2 ( l ) x < L r ,
( k x 2 n ( l ) ) 2 + ( k yn ( l ) ) 2 = ( ω c ) 2 ε 2 ( l ) ; ( k xjn ( l ) ) 2 + ( ω c ) 2 ε j ( l ) = ( k yn ( l ) ) 2 , j = 1 , 3.
{ i γ 21 ( l ) tanh [ k x 1 n ( l ) ( x 1 ( l ) L l ) ] + 1 i γ 21 ( l ) tanh [ k x 1 n ( l ) ( x 1 ( l ) L l ) ] 1 } { i γ 23 ( l ) tanh [ k x 3 n ( l ) ( x 2 ( l ) L r ) ] 1 i γ 23 ( l ) tanh [ k x 3 n ( l ) ( x 2 ( l ) L r ) ] + 1 } e 2 i k x 2 n ( l ) ( x 2 ( l ) x 1 ( l ) ) = 1 ,
γ 21 ( l ) = ( k x 2 n ( l ) ε 2 ( l ) ) / ( k x 1 n ( l ) ε 1 ( l ) ) ; γ 23 ( l ) = ( k x 2 n ( l ) ε 2 ( l ) ) / ( k x 3 n ( l ) ε 3 ( l ) ) .
L l L r 1 ε ( l ) φ m ( l ) + ( x ) ¯ φ n ( l ) ( x ) d x = δ m n ,
H z ( x , y ) = { Σ n = 1 φ n ( 1 ) ( x ) [ I n e ik yn ( 1 ) ( y Q ( 0 ) ) + R n e ik yn ( 1 ) ( y Q ( 1 ) ) ] , Q ( 0 ) y < Q ( 1 ) Σ n = 1 φ n ( 2 ) ( x ) [ E n e ik yn ( 2 ) ( y Q ( 1 ) ) + F n e ik yn ( 2 ) ( y Q ( 2 ) ) ] , Q ( 1 ) y < Q ( 2 ) Σ n = 1 φ n ( 3 ) ( x ) T n e ik yn ( 3 ) ( y Q ( 2 ) ) , Q ( 2 ) y < ,
1 ε y H z ( x , y ) = { 1 ε ( 1 ) Σ n = 1 φ n ( 1 ) ( x ) ik yn ( 1 ) [ I n e ik yn ( 1 ) ( y Q ( 0 ) ) R n e ik yn ( 1 ) ( y Q ( 1 ) ) ] , Q ( 0 ) y < Q ( 1 ) 1 ε ( 2 ) Σ n = 1 φ n ( 2 ) ( x ) ik yn ( 2 ) [ E n e ik yn ( 2 ) ( y Q ( 1 ) ) F n e ik yn ( 2 ) ( y Q ( 2 ) ) ] , Q ( 1 ) y < Q ( 2 ) 1 ε ( 3 ) Σ n = 1 φ n ( 3 ) ( x ) ik yn ( 3 ) T n e ik yn ( 3 ) ( y Q ( 2 ) ) , Q ( 2 ) y < ,
{ I m e ik ym ( 1 ) q ( 1 ) + R m = Σ n = 1 [ E n + F n e ik yn ( 2 ) q ( 2 ) ] L l L r 1 ε ( 1 ) φ m ( 1 ) + ( x ) ¯ φ n ( 2 ) ( x ) d x Σ n = 1 k yn ( 1 ) [ I n e ik yn ( 1 ) q ( 1 ) R n ] L l L r 1 ε ( 1 ) φ m ( 2 ) + ( x ) ¯ φ n ( 1 ) ( x ) d x = k ym ( 2 ) [ E m F m e ik ym ( 2 ) q ( 2 ) ] Σ n = 1 [ E n e ik yn ( 2 ) q ( 2 ) + F n ] L l L r 1 ε ( 3 ) φ m ( 3 ) + ( x ) ¯ φ n ( 2 ) ( x ) d x = T m k ym ( 2 ) [ E m e ik ym ( 2 ) q ( 2 ) F m ] = Σ n = 1 k yn ( 3 ) T n L l L r 1 ε ( 3 ) φ m ( 2 ) + ( x ) ¯ φ n ( 3 ) ( x ) dx ,
{ i γ 21 ( l ) + 1 i γ 21 ( l ) 1 } { i γ 23 ( l ) 1 i γ 23 ( l ) + 1 } e 2 ik x 2 n ( l ) ( x 2 ( l ) x 1 ( l ) ) = 1 ,

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