Abstract

We studied numerically the transmission properties of the periodic array of asymmetric cross-shaped apertures in an Ag film. The relative positions of the two orthogonally oriented rectangular apertures are varied, rather than their length or width. Each transmission peak of the original symmetric cross-shaped apertures will split into two peaks in the case of the asymmetric cross-shaped apertures when the electric field is perpendicular to the long axis of the unchanged rectangular aperture. The wavelength of the shift peak has a linear relation with the asymmetry. This resonance response mainly results from the excitation of the trapped mode provided by the structural symmetry breaking. A distribution of the magnetic field and a simple Lagrange model are used to interpret these phenomena. In addition, the intensity of the transmission peaks can be tuned by changing the incident polarization angle.

© 2013 Optical Society of America

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    [CrossRef]
  3. R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, “Strong polarization in the optical transmission through elliptical nanohole arrays,” Phys. Rev. Lett. 92, 037401 (2004).
    [CrossRef]
  4. H. Cao and A. Nahata, “Influence of aperture shape on the transmission properties of a periodic array of subwavelength apertures,” Opt. Express 12, 3664–3672 (2004).
    [CrossRef]
  5. J. Elliott, I. I. Smolyaninov, N. I. Zheludev, and A. V. Zayats, “Polarization control of optical transmission of a periodic array of elliptical nanoholes in a metal film,” Opt. Lett. 29, 1414–1416 (2004).
    [CrossRef]
  6. 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]
  7. J. H. Kim and P. J. Moyer, “Transmission characteristics of metallic equilateral triangular nanohole,” Appl. Phys. Lett. 89, 121106 (2006).
    [CrossRef]
  8. A. V. Krasavin, A. S. Schwanecke, and N. I. Zheludev, “Extraordinary properties of light transmission through a small chiral hole in a metallic screen,” J. Opt. A 8, S98–S105 (2006).
    [CrossRef]
  9. Q. Chen and D. R. S. Cumming, “High transmission and low color cross-talk plasmonic color filters using triangular lattice hole arrays in aluminum films,” Opt. Express 18, 14055–14062 (2010).
  10. A. Degiron, H. J. Lezec, W. L. Barnes, and T. W. Ebbesen, “Effects of hole depth on enhanced light transmission through subwavelength hole arrays,” Appl. Phys. Lett. 81, 4327–4329 (2002).
    [CrossRef]
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    [CrossRef]
  12. 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]
  13. K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: experiment and theory,” Phys. Rev. B 72, 045421 (2005).
    [CrossRef]
  14. A. Degiron and T. W. Ebbesen, “The role of localized surface Plasmon modes in the enhanced transmission of periodic subwavelength apertures,” J. Opt. A 7, S90–S96 (2005).
    [CrossRef]
  15. Z. C. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96, 233901 (2006).
    [CrossRef]
  16. J. Li, H. Lu, J. T. K. Wan, and H. C. Ong, “The plasmonic properties of elliptical metallic hole arrays,” Appl. Phys. Lett. 94, 033101 (2009).
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  17. S. Wu, Z. Zhang, Y. Zhang, K. Y. Zhang, L. Zhou, X. J. Zhang, and Y. Y. Zhu, “Enhanced rotation of the polarization of a light beam transmitted through a silver film with an array of perforated S-shaped holes,” Phys. Rev. Lett. 110, 207401 (2013).
    [CrossRef]
  18. C. Genet and T. W. Ebbesen, “Light in tiny holes,” Nature 445, 39–46 (2007).
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  19. R. Sambles, “More than transparent,” Nature 391, 641–642 (1998).
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    [CrossRef]
  21. T. Thio, H. J. Lezec, and T. W. Ebbesen, “Strongly enhanced optical transmission through subwavelength holes in metal films,” Physica B 279, 90–93 (2000).
    [CrossRef]
  22. A. D. Sheehan, J. Quinn, S. Daly, P. Dillon, and R. O’Kennedy, “The development of novel miniaturized immuno-sensing devices: a review of a small technology with a large future,” Anal. Lett. 36, 511–537 (2003).
    [CrossRef]
  23. J. Vuckovic, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quantum Electron. 36, 1131–1144 (2000).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  27. M. D. He, J. Q. Liu, Z. Q. Gong, Y. F. Luo, and X. S. Chen, “Light transmission through metal films perforated with arrays of asymmetric cross-shaped hole,” Solid. State. Commun. 150, 104–108 (2010).
    [CrossRef]
  28. A. Roberts and L. Lin, “Plasmonic quarter-wave plate,” Opt. Lett. 37, 1820–1822 (2012).
    [CrossRef]
  29. F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005).
    [CrossRef]
  30. F. J. Garcia-Vidal, L. Martin-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon, “Transmission of light through a single rectangular hole in a real metal,” Phys. Rev. B 74, 153411 (2006).
    [CrossRef]
  31. A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garcia-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B 76, 195414 (2007).
    [CrossRef]
  32. Y. B. Qiu, L. Zhan, and Y. X. Xia, “Polarization-manipulated dual-band enhanced optical transmission through subwavelength rectangular hole array on metallic film,” IEEE J. Sel. Top. Quantum Electron. 19, 4600106 (2013).
    [CrossRef]
  33. V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99, 147401 (2007).
    [CrossRef]
  34. B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
    [CrossRef]
  35. M. D. He, J. Q. Liu, and K. J. Wang, “Transmission resonances in a symmetry-broken square coaxial aperture in a metal film,” J. Phys. D 45, 345304 (2012).
    [CrossRef]
  36. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  37. N. Liu, H. Liu, S. N. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3, 157–162 (2009).
    [CrossRef]
  38. H. Liu, T. Li, Q. J. Wang, Z. H. Zhu, S. M. Wang, J. Q. Li, S. N. Zhu, Y. Y. Zhu, and X. Zhang, “Extraordinary optical transmission induced by excitation of a magnetic plasmon propagation mode in a diatomic chain of slit-hole resonators,” Phys. Rev. B 79, 024304 (2009).
    [CrossRef]
  39. H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
    [CrossRef]
  40. H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97, 243902 (2006).
    [CrossRef]
  41. H. Liu, J. X. Cao, S. N. Zhu, N. Liu, R. Ameling, and H. Giessen, “Lagrage model for the chiral optical properties of stereometamaterials,” Phys. Rev. B 81, 241403(R) (2010).
    [CrossRef]
  42. S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
    [CrossRef]
  43. J. B. Masson and G. Gallot, “Coupling between surface plasmons in subwavelength hole arrays,” Phys. Rev. B 73, 121401(R) (2006).
    [CrossRef]

2013 (2)

S. Wu, Z. Zhang, Y. Zhang, K. Y. Zhang, L. Zhou, X. J. Zhang, and Y. Y. Zhu, “Enhanced rotation of the polarization of a light beam transmitted through a silver film with an array of perforated S-shaped holes,” Phys. Rev. Lett. 110, 207401 (2013).
[CrossRef]

Y. B. Qiu, L. Zhan, and Y. X. Xia, “Polarization-manipulated dual-band enhanced optical transmission through subwavelength rectangular hole array on metallic film,” IEEE J. Sel. Top. Quantum Electron. 19, 4600106 (2013).
[CrossRef]

2012 (2)

M. D. He, J. Q. Liu, and K. J. Wang, “Transmission resonances in a symmetry-broken square coaxial aperture in a metal film,” J. Phys. D 45, 345304 (2012).
[CrossRef]

A. Roberts and L. Lin, “Plasmonic quarter-wave plate,” Opt. Lett. 37, 1820–1822 (2012).
[CrossRef]

2010 (4)

M. D. He, J. Q. Liu, Z. Q. Gong, Y. F. Luo, and X. S. Chen, “Light transmission through metal films perforated with arrays of asymmetric cross-shaped hole,” Solid. State. Commun. 150, 104–108 (2010).
[CrossRef]

H. Liu, J. X. Cao, S. N. Zhu, N. Liu, R. Ameling, and H. Giessen, “Lagrage model for the chiral optical properties of stereometamaterials,” Phys. Rev. B 81, 241403(R) (2010).
[CrossRef]

B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

Q. Chen and D. R. S. Cumming, “High transmission and low color cross-talk plasmonic color filters using triangular lattice hole arrays in aluminum films,” Opt. Express 18, 14055–14062 (2010).

2009 (3)

J. Li, H. Lu, J. T. K. Wan, and H. C. Ong, “The plasmonic properties of elliptical metallic hole arrays,” Appl. Phys. Lett. 94, 033101 (2009).
[CrossRef]

N. Liu, H. Liu, S. N. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3, 157–162 (2009).
[CrossRef]

H. Liu, T. Li, Q. J. Wang, Z. H. Zhu, S. M. Wang, J. Q. Li, S. N. Zhu, Y. Y. Zhu, and X. Zhang, “Extraordinary optical transmission induced by excitation of a magnetic plasmon propagation mode in a diatomic chain of slit-hole resonators,” Phys. Rev. B 79, 024304 (2009).
[CrossRef]

2007 (6)

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

A. Mary, S. G. Rodrigo, L. Martin-Moreno, and F. J. Garcia-Vidal, “Theory of light transmission through an array of rectangular holes,” Phys. Rev. B 76, 195414 (2007).
[CrossRef]

V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99, 147401 (2007).
[CrossRef]

C. Y. Chen, M. W. Tsai, T. H. Chuang, Y. T. Chang, and S. C. Lee, “Extraordinary transmission through a silver film perforated with cross shaped hole arrays in a square lattice,” Appl. Phys. Lett. 91, 063108 (2007).
[CrossRef]

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

R. M. Roth, N. C. Panoiu, M. M. Adams, J. I. Dadap, and R. M. Osgood, “Polarization-tunable plasmon-enhanced extraordinary transmission through metallic films using asymmetric cruciform apertures,” Opt. Lett. 32, 3414–3416 (2007).
[CrossRef]

2006 (6)

J. B. Masson and G. Gallot, “Coupling between surface plasmons in subwavelength hole arrays,” Phys. Rev. B 73, 121401(R) (2006).
[CrossRef]

Z. C. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96, 233901 (2006).
[CrossRef]

J. H. Kim and P. J. Moyer, “Transmission characteristics of metallic equilateral triangular nanohole,” Appl. Phys. Lett. 89, 121106 (2006).
[CrossRef]

A. V. Krasavin, A. S. Schwanecke, and N. I. Zheludev, “Extraordinary properties of light transmission through a small chiral hole in a metallic screen,” J. Opt. A 8, S98–S105 (2006).
[CrossRef]

F. J. Garcia-Vidal, L. Martin-Moreno, E. Moreno, L. K. S. Kumar, and R. Gordon, “Transmission of light through a single rectangular hole in a real metal,” Phys. Rev. B 74, 153411 (2006).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97, 243902 (2006).
[CrossRef]

2005 (4)

R. Qiang, J. Chen, T. Zhao, S. Wang, P. Ruchhoeft, and M. Morgan, “Modelling of infrared bandpass filters using three-dimensional FDTD method,” Electron. Lett. 41, 914–915 (2005).
[CrossRef]

F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef]

K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: experiment and theory,” Phys. Rev. B 72, 045421 (2005).
[CrossRef]

A. Degiron and T. W. Ebbesen, “The role of localized surface Plasmon modes in the enhanced transmission of periodic subwavelength apertures,” J. Opt. A 7, S90–S96 (2005).
[CrossRef]

2004 (6)

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]

S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes,” Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef]

R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, “Strong polarization in the optical transmission through elliptical nanohole arrays,” Phys. Rev. Lett. 92, 037401 (2004).
[CrossRef]

J. Elliott, I. I. Smolyaninov, N. I. Zheludev, and A. V. Zayats, “Polarization control of optical transmission of a periodic array of elliptical nanoholes in a metal film,” Opt. Lett. 29, 1414–1416 (2004).
[CrossRef]

H. Cao and A. Nahata, “Influence of aperture shape on the transmission properties of a periodic array of subwavelength apertures,” Opt. Express 12, 3664–3672 (2004).
[CrossRef]

2003 (1)

A. D. Sheehan, J. Quinn, S. Daly, P. Dillon, and R. O’Kennedy, “The development of novel miniaturized immuno-sensing devices: a review of a small technology with a large future,” Anal. Lett. 36, 511–537 (2003).
[CrossRef]

2002 (1)

A. Degiron, H. J. Lezec, W. L. Barnes, and T. W. Ebbesen, “Effects of hole depth on enhanced light transmission through subwavelength hole arrays,” Appl. Phys. Lett. 81, 4327–4329 (2002).
[CrossRef]

2001 (1)

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]

2000 (2)

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

J. Vuckovic, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quantum Electron. 36, 1131–1144 (2000).
[CrossRef]

1999 (1)

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]

1998 (3)

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

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

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

1972 (1)

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

Adams, M. M.

Ameling, R.

H. Liu, J. X. Cao, S. N. Zhu, N. Liu, R. Ameling, and H. Giessen, “Lagrage model for the chiral optical properties of stereometamaterials,” Phys. Rev. B 81, 241403(R) (2010).
[CrossRef]

Barnes, W. L.

A. Degiron, H. J. Lezec, W. L. Barnes, and T. W. Ebbesen, “Effects of hole depth on enhanced light transmission through subwavelength hole arrays,” Appl. Phys. Lett. 81, 4327–4329 (2002).
[CrossRef]

Brolo, A. G.

R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, “Strong polarization in the optical transmission through elliptical nanohole arrays,” Phys. Rev. Lett. 92, 037401 (2004).
[CrossRef]

Cao, H.

Cao, J. X.

H. Liu, J. X. Cao, S. N. Zhu, N. Liu, R. Ameling, and H. Giessen, “Lagrage model for the chiral optical properties of stereometamaterials,” Phys. Rev. B 81, 241403(R) (2010).
[CrossRef]

Chang, Y. T.

C. Y. Chen, M. W. Tsai, T. H. Chuang, Y. T. Chang, and S. C. Lee, “Extraordinary transmission through a silver film perforated with cross shaped hole arrays in a square lattice,” Appl. Phys. Lett. 91, 063108 (2007).
[CrossRef]

Chen, C. Y.

C. Y. Chen, M. W. Tsai, T. H. Chuang, Y. T. Chang, and S. C. Lee, “Extraordinary transmission through a silver film perforated with cross shaped hole arrays in a square lattice,” Appl. Phys. Lett. 91, 063108 (2007).
[CrossRef]

Chen, J.

R. Qiang, J. Chen, T. Zhao, S. Wang, P. Ruchhoeft, and M. Morgan, “Modelling of infrared bandpass filters using three-dimensional FDTD method,” Electron. Lett. 41, 914–915 (2005).
[CrossRef]

Chen, Q.

Q. Chen and D. R. S. Cumming, “High transmission and low color cross-talk plasmonic color filters using triangular lattice hole arrays in aluminum films,” Opt. Express 18, 14055–14062 (2010).

Chen, X. S.

M. D. He, J. Q. Liu, Z. Q. Gong, Y. F. Luo, and X. S. Chen, “Light transmission through metal films perforated with arrays of asymmetric cross-shaped hole,” Solid. State. Commun. 150, 104–108 (2010).
[CrossRef]

Chong, C. T.

B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[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]

Chuang, T. H.

C. Y. Chen, M. W. Tsai, T. H. Chuang, Y. T. Chang, and S. C. Lee, “Extraordinary transmission through a silver film perforated with cross shaped hole arrays in a square lattice,” Appl. Phys. Lett. 91, 063108 (2007).
[CrossRef]

Cumming, D. R. S.

Q. Chen and D. R. S. Cumming, “High transmission and low color cross-talk plasmonic color filters using triangular lattice hole arrays in aluminum films,” Opt. Express 18, 14055–14062 (2010).

Dadap, J. I.

Daly, S.

A. D. Sheehan, J. Quinn, S. Daly, P. Dillon, and R. O’Kennedy, “The development of novel miniaturized immuno-sensing devices: a review of a small technology with a large future,” Anal. Lett. 36, 511–537 (2003).
[CrossRef]

Degiron, A.

A. Degiron and T. W. Ebbesen, “The role of localized surface Plasmon modes in the enhanced transmission of periodic subwavelength apertures,” J. Opt. A 7, S90–S96 (2005).
[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]

A. Degiron, H. J. Lezec, W. L. Barnes, and T. W. Ebbesen, “Effects of hole depth on enhanced light transmission through subwavelength hole arrays,” Appl. Phys. Lett. 81, 4327–4329 (2002).
[CrossRef]

Dillon, P.

A. D. Sheehan, J. Quinn, S. Daly, P. Dillon, and R. O’Kennedy, “The development of novel miniaturized immuno-sensing devices: a review of a small technology with a large future,” Anal. Lett. 36, 511–537 (2003).
[CrossRef]

Ebbesen, T. W.

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

A. Degiron and T. W. Ebbesen, “The role of localized surface Plasmon modes in the enhanced transmission of periodic subwavelength apertures,” J. Opt. A 7, S90–S96 (2005).
[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]

A. Degiron, H. J. Lezec, W. L. Barnes, and T. W. Ebbesen, “Effects of hole depth on enhanced light transmission through subwavelength hole arrays,” Appl. Phys. Lett. 81, 4327–4329 (2002).
[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]

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

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 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 subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998).
[CrossRef]

Elliott, J.

Enkrich, C.

S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

Enoch, S.

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T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 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 subwavelength holes,” Phys. Rev. B 58, 6779–6782 (1998).
[CrossRef]

Tsai, M. W.

C. Y. Chen, M. W. Tsai, T. H. Chuang, Y. T. Chang, and S. C. Lee, “Extraordinary transmission through a silver film perforated with cross shaped hole arrays in a square lattice,” Appl. Phys. Lett. 91, 063108 (2007).
[CrossRef]

van der Molen, K. L.

K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: experiment and theory,” Phys. Rev. B 72, 045421 (2005).
[CrossRef]

van Hulst, N. F.

K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: experiment and theory,” Phys. Rev. B 72, 045421 (2005).
[CrossRef]

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes,” Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef]

Vuckovic, J.

J. Vuckovic, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quantum Electron. 36, 1131–1144 (2000).
[CrossRef]

Wan, J. T. K.

J. Li, H. Lu, J. T. K. Wan, and H. C. Ong, “The plasmonic properties of elliptical metallic hole arrays,” Appl. Phys. Lett. 94, 033101 (2009).
[CrossRef]

Wang, K. J.

M. D. He, J. Q. Liu, and K. J. Wang, “Transmission resonances in a symmetry-broken square coaxial aperture in a metal film,” J. Phys. D 45, 345304 (2012).
[CrossRef]

Wang, Q. J.

H. Liu, T. Li, Q. J. Wang, Z. H. Zhu, S. M. Wang, J. Q. Li, S. N. Zhu, Y. Y. Zhu, and X. Zhang, “Extraordinary optical transmission induced by excitation of a magnetic plasmon propagation mode in a diatomic chain of slit-hole resonators,” Phys. Rev. B 79, 024304 (2009).
[CrossRef]

Wang, S.

R. Qiang, J. Chen, T. Zhao, S. Wang, P. Ruchhoeft, and M. Morgan, “Modelling of infrared bandpass filters using three-dimensional FDTD method,” Electron. Lett. 41, 914–915 (2005).
[CrossRef]

Wang, S. M.

H. Liu, T. Li, Q. J. Wang, Z. H. Zhu, S. M. Wang, J. Q. Li, S. N. Zhu, Y. Y. Zhu, and X. Zhang, “Extraordinary optical transmission induced by excitation of a magnetic plasmon propagation mode in a diatomic chain of slit-hole resonators,” Phys. Rev. B 79, 024304 (2009).
[CrossRef]

Wegener, M.

S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

Wolff, P. A.

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

Wu, D. M.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97, 243902 (2006).
[CrossRef]

Wu, S.

S. Wu, Z. Zhang, Y. Zhang, K. Y. Zhang, L. Zhou, X. J. Zhang, and Y. Y. Zhu, “Enhanced rotation of the polarization of a light beam transmitted through a silver film with an array of perforated S-shaped holes,” Phys. Rev. Lett. 110, 207401 (2013).
[CrossRef]

Xia, Y. X.

Y. B. Qiu, L. Zhan, and Y. X. Xia, “Polarization-manipulated dual-band enhanced optical transmission through subwavelength rectangular hole array on metallic film,” IEEE J. Sel. Top. Quantum Electron. 19, 4600106 (2013).
[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]

Zayats, A. V.

Zhan, L.

Y. B. Qiu, L. Zhan, and Y. X. Xia, “Polarization-manipulated dual-band enhanced optical transmission through subwavelength rectangular hole array on metallic film,” IEEE J. Sel. Top. Quantum Electron. 19, 4600106 (2013).
[CrossRef]

Zhang, K. Y.

S. Wu, Z. Zhang, Y. Zhang, K. Y. Zhang, L. Zhou, X. J. Zhang, and Y. Y. Zhu, “Enhanced rotation of the polarization of a light beam transmitted through a silver film with an array of perforated S-shaped holes,” Phys. Rev. Lett. 110, 207401 (2013).
[CrossRef]

Zhang, X.

H. Liu, T. Li, Q. J. Wang, Z. H. Zhu, S. M. Wang, J. Q. Li, S. N. Zhu, Y. Y. Zhu, and X. Zhang, “Extraordinary optical transmission induced by excitation of a magnetic plasmon propagation mode in a diatomic chain of slit-hole resonators,” Phys. Rev. B 79, 024304 (2009).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97, 243902 (2006).
[CrossRef]

Zhang, X. J.

S. Wu, Z. Zhang, Y. Zhang, K. Y. Zhang, L. Zhou, X. J. Zhang, and Y. Y. Zhu, “Enhanced rotation of the polarization of a light beam transmitted through a silver film with an array of perforated S-shaped holes,” Phys. Rev. Lett. 110, 207401 (2013).
[CrossRef]

Zhang, Y.

S. Wu, Z. Zhang, Y. Zhang, K. Y. Zhang, L. Zhou, X. J. Zhang, and Y. Y. Zhu, “Enhanced rotation of the polarization of a light beam transmitted through a silver film with an array of perforated S-shaped holes,” Phys. Rev. Lett. 110, 207401 (2013).
[CrossRef]

Zhang, Z.

S. Wu, Z. Zhang, Y. Zhang, K. Y. Zhang, L. Zhou, X. J. Zhang, and Y. Y. Zhu, “Enhanced rotation of the polarization of a light beam transmitted through a silver film with an array of perforated S-shaped holes,” Phys. Rev. Lett. 110, 207401 (2013).
[CrossRef]

Zhao, T.

R. Qiang, J. Chen, T. Zhao, S. Wang, P. Ruchhoeft, and M. Morgan, “Modelling of infrared bandpass filters using three-dimensional FDTD method,” Electron. Lett. 41, 914–915 (2005).
[CrossRef]

Zheludev, N. I.

B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99, 147401 (2007).
[CrossRef]

A. V. Krasavin, A. S. Schwanecke, and N. I. Zheludev, “Extraordinary properties of light transmission through a small chiral hole in a metallic screen,” J. Opt. A 8, S98–S105 (2006).
[CrossRef]

J. Elliott, I. I. Smolyaninov, N. I. Zheludev, and A. V. Zayats, “Polarization control of optical transmission of a periodic array of elliptical nanoholes in a metal film,” Opt. Lett. 29, 1414–1416 (2004).
[CrossRef]

Zhou, J. F.

S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

Zhou, L.

S. Wu, Z. Zhang, Y. Zhang, K. Y. Zhang, L. Zhou, X. J. Zhang, and Y. Y. Zhu, “Enhanced rotation of the polarization of a light beam transmitted through a silver film with an array of perforated S-shaped holes,” Phys. Rev. Lett. 110, 207401 (2013).
[CrossRef]

Zhu, S. N.

H. Liu, J. X. Cao, S. N. Zhu, N. Liu, R. Ameling, and H. Giessen, “Lagrage model for the chiral optical properties of stereometamaterials,” Phys. Rev. B 81, 241403(R) (2010).
[CrossRef]

N. Liu, H. Liu, S. N. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3, 157–162 (2009).
[CrossRef]

H. Liu, T. Li, Q. J. Wang, Z. H. Zhu, S. M. Wang, J. Q. Li, S. N. Zhu, Y. Y. Zhu, and X. Zhang, “Extraordinary optical transmission induced by excitation of a magnetic plasmon propagation mode in a diatomic chain of slit-hole resonators,” Phys. Rev. B 79, 024304 (2009).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97, 243902 (2006).
[CrossRef]

Zhu, Y. Y.

S. Wu, Z. Zhang, Y. Zhang, K. Y. Zhang, L. Zhou, X. J. Zhang, and Y. Y. Zhu, “Enhanced rotation of the polarization of a light beam transmitted through a silver film with an array of perforated S-shaped holes,” Phys. Rev. Lett. 110, 207401 (2013).
[CrossRef]

H. Liu, T. Li, Q. J. Wang, Z. H. Zhu, S. M. Wang, J. Q. Li, S. N. Zhu, Y. Y. Zhu, and X. Zhang, “Extraordinary optical transmission induced by excitation of a magnetic plasmon propagation mode in a diatomic chain of slit-hole resonators,” Phys. Rev. B 79, 024304 (2009).
[CrossRef]

Zhu, Z. H.

H. Liu, T. Li, Q. J. Wang, Z. H. Zhu, S. M. Wang, J. Q. Li, S. N. Zhu, Y. Y. Zhu, and X. Zhang, “Extraordinary optical transmission induced by excitation of a magnetic plasmon propagation mode in a diatomic chain of slit-hole resonators,” Phys. Rev. B 79, 024304 (2009).
[CrossRef]

Anal. Lett. (1)

A. D. Sheehan, J. Quinn, S. Daly, P. Dillon, and R. O’Kennedy, “The development of novel miniaturized immuno-sensing devices: a review of a small technology with a large future,” Anal. Lett. 36, 511–537 (2003).
[CrossRef]

Appl. Phys. Lett. (4)

J. Li, H. Lu, J. T. K. Wan, and H. C. Ong, “The plasmonic properties of elliptical metallic hole arrays,” Appl. Phys. Lett. 94, 033101 (2009).
[CrossRef]

J. H. Kim and P. J. Moyer, “Transmission characteristics of metallic equilateral triangular nanohole,” Appl. Phys. Lett. 89, 121106 (2006).
[CrossRef]

A. Degiron, H. J. Lezec, W. L. Barnes, and T. W. Ebbesen, “Effects of hole depth on enhanced light transmission through subwavelength hole arrays,” Appl. Phys. Lett. 81, 4327–4329 (2002).
[CrossRef]

C. Y. Chen, M. W. Tsai, T. H. Chuang, Y. T. Chang, and S. C. Lee, “Extraordinary transmission through a silver film perforated with cross shaped hole arrays in a square lattice,” Appl. Phys. Lett. 91, 063108 (2007).
[CrossRef]

Electron. Lett. (1)

R. Qiang, J. Chen, T. Zhao, S. Wang, P. Ruchhoeft, and M. Morgan, “Modelling of infrared bandpass filters using three-dimensional FDTD method,” Electron. Lett. 41, 914–915 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Vuckovic, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diode,” IEEE J. Quantum Electron. 36, 1131–1144 (2000).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

Y. B. Qiu, L. Zhan, and Y. X. Xia, “Polarization-manipulated dual-band enhanced optical transmission through subwavelength rectangular hole array on metallic film,” IEEE J. Sel. Top. Quantum Electron. 19, 4600106 (2013).
[CrossRef]

J. Opt. A (2)

A. V. Krasavin, A. S. Schwanecke, and N. I. Zheludev, “Extraordinary properties of light transmission through a small chiral hole in a metallic screen,” J. Opt. A 8, S98–S105 (2006).
[CrossRef]

A. Degiron and T. W. Ebbesen, “The role of localized surface Plasmon modes in the enhanced transmission of periodic subwavelength apertures,” J. Opt. A 7, S90–S96 (2005).
[CrossRef]

J. Phys. D (1)

M. D. He, J. Q. Liu, and K. J. Wang, “Transmission resonances in a symmetry-broken square coaxial aperture in a metal film,” J. Phys. D 45, 345304 (2012).
[CrossRef]

Nat. Mater. (1)

B. Lukyanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707–715 (2010).
[CrossRef]

Nat. Photonics (1)

N. Liu, H. Liu, S. N. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3, 157–162 (2009).
[CrossRef]

Nature (3)

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

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

Opt. Commun. (1)

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)

Q. Chen and D. R. S. Cumming, “High transmission and low color cross-talk plasmonic color filters using triangular lattice hole arrays in aluminum films,” Opt. Express 18, 14055–14062 (2010).

H. Cao and A. Nahata, “Influence of aperture shape on the transmission properties of a periodic array of subwavelength apertures,” Opt. Express 12, 3664–3672 (2004).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. B (9)

J. B. Masson and G. Gallot, “Coupling between surface plasmons in subwavelength hole arrays,” Phys. Rev. B 73, 121401(R) (2006).
[CrossRef]

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

K. L. van der Molen, K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Role of shape and localized resonances in extraordinary transmission through periodic arrays of subwavelength holes: experiment and theory,” Phys. Rev. B 72, 045421 (2005).
[CrossRef]

H. Liu, T. Li, Q. J. Wang, Z. H. Zhu, S. M. Wang, J. Q. Li, S. N. Zhu, Y. Y. Zhu, and X. Zhang, “Extraordinary optical transmission induced by excitation of a magnetic plasmon propagation mode in a diatomic chain of slit-hole resonators,” Phys. Rev. B 79, 024304 (2009).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
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[CrossRef]

H. Liu, J. X. Cao, S. N. Zhu, N. Liu, R. Ameling, and H. Giessen, “Lagrage model for the chiral optical properties of stereometamaterials,” Phys. Rev. B 81, 241403(R) (2010).
[CrossRef]

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Phys. Rev. Lett. (9)

F. J. Garcia-Vidal, E. Moreno, J. A. Porto, and L. Martin-Moreno, “Transmission of light through a single rectangular hole,” Phys. Rev. Lett. 95, 103901 (2005).
[CrossRef]

V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99, 147401 (2007).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, J. M. Steele, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon propagation along a chain of connected subwavelength resonators at infrared frequencies,” Phys. Rev. Lett. 97, 243902 (2006).
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Z. C. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96, 233901 (2006).
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[CrossRef]

S. Wu, Z. Zhang, Y. Zhang, K. Y. Zhang, L. Zhou, X. J. Zhang, and Y. Y. Zhu, “Enhanced rotation of the polarization of a light beam transmitted through a silver film with an array of perforated S-shaped holes,” Phys. Rev. Lett. 110, 207401 (2013).
[CrossRef]

K. J. Klein Koerkamp, S. Enoch, F. B. Segerink, N. F. van Hulst, and L. Kuipers, “Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes,” Phys. Rev. Lett. 92, 183901 (2004).
[CrossRef]

R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, “Strong polarization in the optical transmission through elliptical nanohole arrays,” Phys. Rev. Lett. 92, 037401 (2004).
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Physica B (1)

T. Thio, H. J. Lezec, and T. W. Ebbesen, “Strongly enhanced optical transmission through subwavelength holes in metal films,” Physica B 279, 90–93 (2000).
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Science (1)

S. Linden, C. Enkrich, M. Wegener, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

Solid. State. Commun. (1)

M. D. He, J. Q. Liu, Z. Q. Gong, Y. F. Luo, and X. S. Chen, “Light transmission through metal films perforated with arrays of asymmetric cross-shaped hole,” Solid. State. Commun. 150, 104–108 (2010).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Schematic configuration of the asymmetric cross-shaped aperture in an Ag film. The cyan area is Ag; the gray area is vacuum.

Fig. 2.
Fig. 2.

Transmission spectra of the asymmetric cross-shaped apertures with Ex axis. The displacement d is varied from 0 to 80 nm. Peak 1, short-wavelength peak; Peak 2, the long-wavelength peak.

Fig. 3.
Fig. 3.

Transmission spectra of the asymmetric cross-shaped apertures with Ey axis. The displacement d is varied from 0 to 80 nm. The inset shows the transmission spectrum for d=20nm to make the split effect easily observed.

Fig. 4.
Fig. 4.

Influence of the displacement d on the positions of the transmission peaks splitting from Peak 2 when the electric field vector has the Ey axis. The red and blue points correspond to different positions of the peaks.

Fig. 5.
Fig. 5.

Distributions of the magnetic field in the YOZ plane at the center of the aperture for the displacement d=20nm. (a) Peak 3 at 654.3 nm. (b) Peak 4 at 801.1 nm. The arrows indicate the direction of the magnetic field.

Fig. 6.
Fig. 6.

Equivalent LC circuit of the short part of the nanoslit along the y axis.

Fig. 7.
Fig. 7.

Transmission spectra of the asymmetric cross-shaped apertures with d=20nm for different incident polarization angles. 0°, Ey axis; 90°, Ex axis.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

I=(L1Q˙12)/2Q12/(2C1),
wos=1/(L1C1)
I=(L1/2)Q˙12Q12/(2C1)+(L2/2)Q˙22Q22/(2C2)+MQ˙1Q˙2,
(d/dt)(I/Q˙i)I/Qi=0(i=1,2),
Q¨1+Q1/(L1C1)=(M/L1)Q¨2,
Q¨2+Q2/(L2C2)=(M/L2)Q¨1.
wol±1/L2C2(1M/L2).

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