Abstract

In this paper the transmission properties of a metal plate perforated with a triangular array of air holes is investigated. We find that the normalized transmittivity exceeds unity within a certain frequency range under normal incidence of a Gaussian beam. Calculations and experiments indicate that the phenomenon results from the collimation effect which only occurs inside the complete bandgap of surface resonance states on the perforated metal plate. The findings present a simple approach for beam collimation.

© 2012 OSA

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    [Crossref]
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  12. M. Derouard, J. Hazart, G. Lérondel, R. Bachelot, P. M. Adam, and P. Royer, “Polarization-sensitive printing of surface plasmon interferences,” Opt. Express 15(7), 4238–4246 (2007).
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  29. Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband polarization transformation via enhanced asymmetric transmission through arrays of twisted complementary split-ring resonators,” Appl. Phys. Lett. 99(22), 221907 (2011).
    [Crossref]
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    [Crossref] [PubMed]
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2011 (2)

Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband polarization transformation via enhanced asymmetric transmission through arrays of twisted complementary split-ring resonators,” Appl. Phys. Lett. 99(22), 221907 (2011).
[Crossref]

Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband transparency achieved with the stacked metallic multi-layers perforated with coaxial annular apertures,” Opt. Express 19(22), 21425–21431 (2011).
[Crossref] [PubMed]

2010 (3)

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Graded photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. 96(3), 031104 (2010).
[Crossref]

Z. Y. Wei, J. X. Fu, Y. Cao, C. Wu, and H. Q. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostructures 8(2), 94–101 (2010).
[Crossref]

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010).
[Crossref]

2008 (1)

H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008).
[Crossref] [PubMed]

2007 (4)

F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007).
[Crossref]

J. S. Biteen, L. A. Sweatlock, H. Mertens, N. S. Lewis, A. Polman, and H. A. Atwater, “Plasmon-enhanced photoluminescence of silicon quantum dots: simulation and experiment,” J. Phys. Chem. C 111(36), 13372–13377 (2007).
[Crossref]

M. Derouard, J. Hazart, G. Lérondel, R. Bachelot, P. M. Adam, and P. Royer, “Polarization-sensitive printing of surface plasmon interferences,” Opt. Express 15(7), 4238–4246 (2007).
[Crossref] [PubMed]

M. U. González, A. L. Stepanov, J. C. Weeber, A. Hohenau, A. Dereux, R. Quidant, and J. R. Krenn, “Analysis of the angular acceptance of surface plasmon Bragg mirrors,” Opt. Lett. 32(18), 2704–2706 (2007).
[Crossref] [PubMed]

2006 (3)

Z. C. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96(23), 233901 (2006).
[Crossref] [PubMed]

M. U. Gonzalez, J. C. Weeber, A. L. Baudrion, A. Dereux, A. L. Stepanov, J. R. Krenn, E. Devaux, and T. W. Ebbesen, “Design, near-field characterization, and modeling of 45° circle surface-plasmon Bragg mirrors,” Phys. Rev. B 73(15), 155416 (2006).
[Crossref]

G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, “Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry,” Phys. Rev. Lett. 96(21), 213901 (2006).
[Crossref] [PubMed]

2005 (3)

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science 308(5722), 670–672 (2005).
[Crossref] [PubMed]

W. J. Wen, L. Zhou, B. Hou, C. T. Chan, and P. Sheng, “Resonant transmission of microwaves through subwavelength fractal slits in a metallic plate,” Phys. Rev. B 72(15), 153406 (2005).
[Crossref]

A. K. Azad and W. L. Zhang, “Resonant terahertz transmission in subwavelength metallic hole arrays of sub-skin-depth thickness,” Opt. Lett. 30(21), 2945–2947 (2005).
[Crossref] [PubMed]

2004 (3)

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

2001 (2)

L. Martín-Moreno, F. J. García-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(6), 1114–1117 (2001).
[Crossref] [PubMed]

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001).
[Crossref] [PubMed]

2000 (1)

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000).
[Crossref]

1998 (2)

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(6668), 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(11), 6779–6782 (1998).
[Crossref]

1997 (1)

P. Andrew, S. C. Kitson, and W. L. Barnes, “Surface-plasmon energy gaps and photoabsorption,” J. Mod. Opt. 44(2), 395–406 (1997).
[Crossref]

1996 (2)

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996).
[Crossref] [PubMed]

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996).
[Crossref] [PubMed]

1995 (1)

A. Brecht and G. Gauglitz, “Optical probes and transducers,” Biosens. Bioelectron. 10(9-10), 923–936 (1995).
[Crossref] [PubMed]

1982 (1)

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 (1982).
[Crossref]

Adam, P. M.

Alloschery, O.

G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, “Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry,” Phys. Rev. Lett. 96(21), 213901 (2006).
[Crossref] [PubMed]

Andrew, P.

P. Andrew, S. C. Kitson, and W. L. Barnes, “Surface-plasmon energy gaps and photoabsorption,” J. Mod. Opt. 44(2), 395–406 (1997).
[Crossref]

Astilean, S.

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000).
[Crossref]

Atwater, H. A.

J. S. Biteen, L. A. Sweatlock, H. Mertens, N. S. Lewis, A. Polman, and H. A. Atwater, “Plasmon-enhanced photoluminescence of silicon quantum dots: simulation and experiment,” J. Phys. Chem. C 111(36), 13372–13377 (2007).
[Crossref]

Azad, A. K.

Bachelot, R.

Barbieri, S.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Graded photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. 96(3), 031104 (2010).
[Crossref]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

P. Andrew, S. C. Kitson, and W. L. Barnes, “Surface-plasmon energy gaps and photoabsorption,” J. Mod. Opt. 44(2), 395–406 (1997).
[Crossref]

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996).
[Crossref] [PubMed]

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996).
[Crossref] [PubMed]

Baudrion, A. L.

M. U. Gonzalez, J. C. Weeber, A. L. Baudrion, A. Dereux, A. L. Stepanov, J. R. Krenn, E. Devaux, and T. W. Ebbesen, “Design, near-field characterization, and modeling of 45° circle surface-plasmon Bragg mirrors,” Phys. Rev. B 73(15), 155416 (2006).
[Crossref]

Beruete, M.

Biteen, J. S.

J. S. Biteen, L. A. Sweatlock, H. Mertens, N. S. Lewis, A. Polman, and H. A. Atwater, “Plasmon-enhanced photoluminescence of silicon quantum dots: simulation and experiment,” J. Phys. Chem. C 111(36), 13372–13377 (2007).
[Crossref]

Bozhevolnyi, S. I.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001).
[Crossref] [PubMed]

Bravo-Abad, J.

Brecht, A.

A. Brecht and G. Gauglitz, “Optical probes and transducers,” Biosens. Bioelectron. 10(9-10), 923–936 (1995).
[Crossref] [PubMed]

Campillo, I.

Cao, Y.

Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband polarization transformation via enhanced asymmetric transmission through arrays of twisted complementary split-ring resonators,” Appl. Phys. Lett. 99(22), 221907 (2011).
[Crossref]

Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband transparency achieved with the stacked metallic multi-layers perforated with coaxial annular apertures,” Opt. Express 19(22), 21425–21431 (2011).
[Crossref] [PubMed]

Z. Y. Wei, J. X. Fu, Y. Cao, C. Wu, and H. Q. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostructures 8(2), 94–101 (2010).
[Crossref]

Chan, C. T.

W. J. Wen, L. Zhou, B. Hou, C. T. Chan, and P. Sheng, “Resonant transmission of microwaves through subwavelength fractal slits in a metallic plate,” Phys. Rev. B 72(15), 153406 (2005).
[Crossref]

Chassagneux, Y.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Graded photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. 96(3), 031104 (2010).
[Crossref]

Colombelli, R.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Graded photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. 96(3), 031104 (2010).
[Crossref]

Davies, A. G.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Graded photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. 96(3), 031104 (2010).
[Crossref]

de Lesegno, B. V.

G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, “Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry,” Phys. Rev. Lett. 96(21), 213901 (2006).
[Crossref] [PubMed]

Dereux, A.

M. U. González, A. L. Stepanov, J. C. Weeber, A. Hohenau, A. Dereux, R. Quidant, and J. R. Krenn, “Analysis of the angular acceptance of surface plasmon Bragg mirrors,” Opt. Lett. 32(18), 2704–2706 (2007).
[Crossref] [PubMed]

M. U. Gonzalez, J. C. Weeber, A. L. Baudrion, A. Dereux, A. L. Stepanov, J. R. Krenn, E. Devaux, and T. W. Ebbesen, “Design, near-field characterization, and modeling of 45° circle surface-plasmon Bragg mirrors,” Phys. Rev. B 73(15), 155416 (2006).
[Crossref]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Derouard, M.

Devaux, E.

M. U. Gonzalez, J. C. Weeber, A. L. Baudrion, A. Dereux, A. L. Stepanov, J. R. Krenn, E. Devaux, and T. W. Ebbesen, “Design, near-field characterization, and modeling of 45° circle surface-plasmon Bragg mirrors,” Phys. Rev. B 73(15), 155416 (2006).
[Crossref]

Dolado, J. S.

Ebbesen, T. W.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010).
[Crossref]

M. U. Gonzalez, J. C. Weeber, A. L. Baudrion, A. Dereux, A. L. Stepanov, J. R. Krenn, E. Devaux, and T. W. Ebbesen, “Design, near-field characterization, and modeling of 45° circle surface-plasmon Bragg mirrors,” Phys. Rev. B 73(15), 155416 (2006).
[Crossref]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

L. Martín-Moreno, F. J. García-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(6), 1114–1117 (2001).
[Crossref] [PubMed]

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(6668), 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(11), 6779–6782 (1998).
[Crossref]

Erland, J.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001).
[Crossref] [PubMed]

Evans, B. R.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science 308(5722), 670–672 (2005).
[Crossref] [PubMed]

Fan, Y. C.

Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband polarization transformation via enhanced asymmetric transmission through arrays of twisted complementary split-ring resonators,” Appl. Phys. Lett. 99(22), 221907 (2011).
[Crossref]

Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband transparency achieved with the stacked metallic multi-layers perforated with coaxial annular apertures,” Opt. Express 19(22), 21425–21431 (2011).
[Crossref] [PubMed]

Fu, J. X.

Z. Y. Wei, J. X. Fu, Y. Cao, C. Wu, and H. Q. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostructures 8(2), 94–101 (2010).
[Crossref]

García de Abajo, F. J.

F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007).
[Crossref]

Garcia-Vidal, F. J.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010).
[Crossref]

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref] [PubMed]

García-Vidal, F. J.

M. Beruete, M. Sorolla, I. Campillo, J. S. Dolado, L. Martín-Moreno, J. Bravo-Abad, and F. J. García-Vidal, “Enhanced millimeter-wave transmission through subwavelength hole arrays,” Opt. Lett. 29(21), 2500–2502 (2004).
[Crossref] [PubMed]

L. Martín-Moreno, F. J. García-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(6), 1114–1117 (2001).
[Crossref] [PubMed]

Gauglitz, G.

A. Brecht and G. Gauglitz, “Optical probes and transducers,” Biosens. Bioelectron. 10(9-10), 923–936 (1995).
[Crossref] [PubMed]

Gay, G.

G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, “Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry,” Phys. Rev. Lett. 96(21), 213901 (2006).
[Crossref] [PubMed]

Ghaemi, H. F.

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(11), 6779–6782 (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(6668), 667–669 (1998).
[Crossref]

Gonzalez, M. U.

M. U. Gonzalez, J. C. Weeber, A. L. Baudrion, A. Dereux, A. L. Stepanov, J. R. Krenn, E. Devaux, and T. W. Ebbesen, “Design, near-field characterization, and modeling of 45° circle surface-plasmon Bragg mirrors,” Phys. Rev. B 73(15), 155416 (2006).
[Crossref]

González, M. U.

Grischkowsky, D.

Grupp, D. E.

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(11), 6779–6782 (1998).
[Crossref]

Hazart, J.

Hibbins, A. P.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science 308(5722), 670–672 (2005).
[Crossref] [PubMed]

Hohenau, A.

Hou, B.

W. J. Wen, L. Zhou, B. Hou, C. T. Chan, and P. Sheng, “Resonant transmission of microwaves through subwavelength fractal slits in a metallic plate,” Phys. Rev. B 72(15), 153406 (2005).
[Crossref]

Hugonin, J. P.

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000).
[Crossref]

Hvam, J. M.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001).
[Crossref] [PubMed]

Khanna, S. P.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Graded photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. 96(3), 031104 (2010).
[Crossref]

Kitson, S. C.

P. Andrew, S. C. Kitson, and W. L. Barnes, “Surface-plasmon energy gaps and photoabsorption,” J. Mod. Opt. 44(2), 395–406 (1997).
[Crossref]

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996).
[Crossref] [PubMed]

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996).
[Crossref] [PubMed]

Krenn, J. R.

M. U. González, A. L. Stepanov, J. C. Weeber, A. Hohenau, A. Dereux, R. Quidant, and J. R. Krenn, “Analysis of the angular acceptance of surface plasmon Bragg mirrors,” Opt. Lett. 32(18), 2704–2706 (2007).
[Crossref] [PubMed]

M. U. Gonzalez, J. C. Weeber, A. L. Baudrion, A. Dereux, A. L. Stepanov, J. R. Krenn, E. Devaux, and T. W. Ebbesen, “Design, near-field characterization, and modeling of 45° circle surface-plasmon Bragg mirrors,” Phys. Rev. B 73(15), 155416 (2006).
[Crossref]

Kuipers, L.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010).
[Crossref]

Lalanne, P.

H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008).
[Crossref] [PubMed]

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000).
[Crossref]

Leosson, K.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001).
[Crossref] [PubMed]

Lérondel, G.

Lewis, N. S.

J. S. Biteen, L. A. Sweatlock, H. Mertens, N. S. Lewis, A. Polman, and H. A. Atwater, “Plasmon-enhanced photoluminescence of silicon quantum dots: simulation and experiment,” J. Phys. Chem. C 111(36), 13372–13377 (2007).
[Crossref]

Lezec, H. J.

G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, “Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry,” Phys. Rev. Lett. 96(21), 213901 (2006).
[Crossref] [PubMed]

L. Martín-Moreno, F. J. García-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(6), 1114–1117 (2001).
[Crossref] [PubMed]

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(6668), 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(11), 6779–6782 (1998).
[Crossref]

Li, H. Q.

Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband transparency achieved with the stacked metallic multi-layers perforated with coaxial annular apertures,” Opt. Express 19(22), 21425–21431 (2011).
[Crossref] [PubMed]

Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband polarization transformation via enhanced asymmetric transmission through arrays of twisted complementary split-ring resonators,” Appl. Phys. Lett. 99(22), 221907 (2011).
[Crossref]

Z. Y. Wei, J. X. Fu, Y. Cao, C. Wu, and H. Q. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostructures 8(2), 94–101 (2010).
[Crossref]

Linfield, E. H.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Graded photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. 96(3), 031104 (2010).
[Crossref]

Liu, H. T.

H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008).
[Crossref] [PubMed]

Maineult, W.

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Graded photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. 96(3), 031104 (2010).
[Crossref]

Martin-Moreno, L.

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010).
[Crossref]

Martín-Moreno, L.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref] [PubMed]

M. Beruete, M. Sorolla, I. Campillo, J. S. Dolado, L. Martín-Moreno, J. Bravo-Abad, and F. J. García-Vidal, “Enhanced millimeter-wave transmission through subwavelength hole arrays,” Opt. Lett. 29(21), 2500–2502 (2004).
[Crossref] [PubMed]

L. Martín-Moreno, F. J. García-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(6), 1114–1117 (2001).
[Crossref] [PubMed]

Mertens, H.

J. S. Biteen, L. A. Sweatlock, H. Mertens, N. S. Lewis, A. Polman, and H. A. Atwater, “Plasmon-enhanced photoluminescence of silicon quantum dots: simulation and experiment,” J. Phys. Chem. C 111(36), 13372–13377 (2007).
[Crossref]

Moller, K. D.

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000).
[Crossref]

Palamaru, M.

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000).
[Crossref]

Pellerin, K. M.

L. Martín-Moreno, F. J. García-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(6), 1114–1117 (2001).
[Crossref] [PubMed]

Pendry, J. B.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref] [PubMed]

L. Martín-Moreno, F. J. García-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(6), 1114–1117 (2001).
[Crossref] [PubMed]

Polman, A.

J. S. Biteen, L. A. Sweatlock, H. Mertens, N. S. Lewis, A. Polman, and H. A. Atwater, “Plasmon-enhanced photoluminescence of silicon quantum dots: simulation and experiment,” J. Phys. Chem. C 111(36), 13372–13377 (2007).
[Crossref]

Preist, T. W.

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996).
[Crossref] [PubMed]

Qiu, M.

Z. C. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96(23), 233901 (2006).
[Crossref] [PubMed]

Qu, D. X.

Quidant, R.

Royer, P.

Ruan, Z. C.

Z. C. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96(23), 233901 (2006).
[Crossref] [PubMed]

Sambles, J. R.

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science 308(5722), 670–672 (2005).
[Crossref] [PubMed]

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996).
[Crossref] [PubMed]

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996).
[Crossref] [PubMed]

Sanda, P. N.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 (1982).
[Crossref]

Sheng, P.

W. J. Wen, L. Zhou, B. Hou, C. T. Chan, and P. Sheng, “Resonant transmission of microwaves through subwavelength fractal slits in a metallic plate,” Phys. Rev. B 72(15), 153406 (2005).
[Crossref]

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 (1982).
[Crossref]

Skovgaard, P. M. W.

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001).
[Crossref] [PubMed]

Sorolla, M.

Stepanov, A. L.

M. U. González, A. L. Stepanov, J. C. Weeber, A. Hohenau, A. Dereux, R. Quidant, and J. R. Krenn, “Analysis of the angular acceptance of surface plasmon Bragg mirrors,” Opt. Lett. 32(18), 2704–2706 (2007).
[Crossref] [PubMed]

M. U. Gonzalez, J. C. Weeber, A. L. Baudrion, A. Dereux, A. L. Stepanov, J. R. Krenn, E. Devaux, and T. W. Ebbesen, “Design, near-field characterization, and modeling of 45° circle surface-plasmon Bragg mirrors,” Phys. Rev. B 73(15), 155416 (2006).
[Crossref]

Stepleman, R. S.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 (1982).
[Crossref]

Sweatlock, L. A.

J. S. Biteen, L. A. Sweatlock, H. Mertens, N. S. Lewis, A. Polman, and H. A. Atwater, “Plasmon-enhanced photoluminescence of silicon quantum dots: simulation and experiment,” J. Phys. Chem. C 111(36), 13372–13377 (2007).
[Crossref]

Thio, T.

L. Martín-Moreno, F. J. García-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(6), 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 subwavelength holes,” Phys. Rev. B 58(11), 6779–6782 (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(6668), 667–669 (1998).
[Crossref]

Weeber, J. C.

M. U. González, A. L. Stepanov, J. C. Weeber, A. Hohenau, A. Dereux, R. Quidant, and J. R. Krenn, “Analysis of the angular acceptance of surface plasmon Bragg mirrors,” Opt. Lett. 32(18), 2704–2706 (2007).
[Crossref] [PubMed]

M. U. Gonzalez, J. C. Weeber, A. L. Baudrion, A. Dereux, A. L. Stepanov, J. R. Krenn, E. Devaux, and T. W. Ebbesen, “Design, near-field characterization, and modeling of 45° circle surface-plasmon Bragg mirrors,” Phys. Rev. B 73(15), 155416 (2006).
[Crossref]

Wei, Z. Y.

Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband transparency achieved with the stacked metallic multi-layers perforated with coaxial annular apertures,” Opt. Express 19(22), 21425–21431 (2011).
[Crossref] [PubMed]

Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband polarization transformation via enhanced asymmetric transmission through arrays of twisted complementary split-ring resonators,” Appl. Phys. Lett. 99(22), 221907 (2011).
[Crossref]

Z. Y. Wei, J. X. Fu, Y. Cao, C. Wu, and H. Q. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostructures 8(2), 94–101 (2010).
[Crossref]

Weiner, J.

G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, “Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry,” Phys. Rev. Lett. 96(21), 213901 (2006).
[Crossref] [PubMed]

Wen, W. J.

W. J. Wen, L. Zhou, B. Hou, C. T. Chan, and P. Sheng, “Resonant transmission of microwaves through subwavelength fractal slits in a metallic plate,” Phys. Rev. B 72(15), 153406 (2005).
[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(6668), 667–669 (1998).
[Crossref]

Wu, C.

Z. Y. Wei, J. X. Fu, Y. Cao, C. Wu, and H. Q. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostructures 8(2), 94–101 (2010).
[Crossref]

Yu, X.

Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband polarization transformation via enhanced asymmetric transmission through arrays of twisted complementary split-ring resonators,” Appl. Phys. Lett. 99(22), 221907 (2011).
[Crossref]

Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband transparency achieved with the stacked metallic multi-layers perforated with coaxial annular apertures,” Opt. Express 19(22), 21425–21431 (2011).
[Crossref] [PubMed]

Zhang, W. L.

Zhou, L.

W. J. Wen, L. Zhou, B. Hou, C. T. Chan, and P. Sheng, “Resonant transmission of microwaves through subwavelength fractal slits in a metallic plate,” Phys. Rev. B 72(15), 153406 (2005).
[Crossref]

Appl. Phys. Lett. (2)

Y. Chassagneux, R. Colombelli, W. Maineult, S. Barbieri, S. P. Khanna, E. H. Linfield, and A. G. Davies, “Graded photonic crystal terahertz quantum cascade lasers,” Appl. Phys. Lett. 96(3), 031104 (2010).
[Crossref]

Z. Y. Wei, Y. Cao, Y. C. Fan, X. Yu, and H. Q. Li, “Broadband polarization transformation via enhanced asymmetric transmission through arrays of twisted complementary split-ring resonators,” Appl. Phys. Lett. 99(22), 221907 (2011).
[Crossref]

Biosens. Bioelectron. (1)

A. Brecht and G. Gauglitz, “Optical probes and transducers,” Biosens. Bioelectron. 10(9-10), 923–936 (1995).
[Crossref] [PubMed]

J. Mod. Opt. (1)

P. Andrew, S. C. Kitson, and W. L. Barnes, “Surface-plasmon energy gaps and photoabsorption,” J. Mod. Opt. 44(2), 395–406 (1997).
[Crossref]

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

P. Lalanne, J. P. Hugonin, S. Astilean, M. Palamaru, and K. D. Moller, “One-mode model and Airy-like formulae for one-dimensional metallic gratings,” J. Opt. A, Pure Appl. Opt. 2(1), 48–51 (2000).
[Crossref]

J. Phys. Chem. C (1)

J. S. Biteen, L. A. Sweatlock, H. Mertens, N. S. Lewis, A. Polman, and H. A. Atwater, “Plasmon-enhanced photoluminescence of silicon quantum dots: simulation and experiment,” J. Phys. Chem. C 111(36), 13372–13377 (2007).
[Crossref]

Nature (3)

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(6668), 667–669 (1998).
[Crossref]

H. T. Liu and P. Lalanne, “Microscopic theory of the extraordinary optical transmission,” Nature 452(7188), 728–731 (2008).
[Crossref] [PubMed]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (4)

Photon. Nanostructures (1)

Z. Y. Wei, J. X. Fu, Y. Cao, C. Wu, and H. Q. Li, “The impact of local resonance on the enhanced transmission and dispersion of surface resonances,” Photon. Nanostructures 8(2), 94–101 (2010).
[Crossref]

Phys. Rev. B (4)

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square-wave gratings: application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26(6), 2907–2916 (1982).
[Crossref]

W. J. Wen, L. Zhou, B. Hou, C. T. Chan, and P. Sheng, “Resonant transmission of microwaves through subwavelength fractal slits in a metallic plate,” Phys. Rev. B 72(15), 153406 (2005).
[Crossref]

M. U. Gonzalez, J. C. Weeber, A. L. Baudrion, A. Dereux, A. L. Stepanov, J. R. Krenn, E. Devaux, and T. W. Ebbesen, “Design, near-field characterization, and modeling of 45° circle surface-plasmon Bragg mirrors,” Phys. Rev. B 73(15), 155416 (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(11), 6779–6782 (1998).
[Crossref]

Phys. Rev. B Condens. Matter (1)

W. L. Barnes, T. W. Preist, S. C. Kitson, and J. R. Sambles, “Physical origin of photonic energy gaps in the propagation of surface plasmons on gratings,” Phys. Rev. B Condens. Matter 54(9), 6227–6244 (1996).
[Crossref] [PubMed]

Phys. Rev. Lett. (5)

L. Martín-Moreno, F. J. García-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(6), 1114–1117 (2001).
[Crossref] [PubMed]

G. Gay, O. Alloschery, B. V. de Lesegno, J. Weiner, and H. J. Lezec, “Surface wave generation and propagation on metallic subwavelength structures measured by far-field interferometry,” Phys. Rev. Lett. 96(21), 213901 (2006).
[Crossref] [PubMed]

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77(13), 2670–2673 (1996).
[Crossref] [PubMed]

Z. C. Ruan and M. Qiu, “Enhanced transmission through periodic arrays of subwavelength holes: the role of localized waveguide resonances,” Phys. Rev. Lett. 96(23), 233901 (2006).
[Crossref] [PubMed]

S. I. Bozhevolnyi, J. Erland, K. Leosson, P. M. W. Skovgaard, and J. M. Hvam, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. 86(14), 3008–3011 (2001).
[Crossref] [PubMed]

Rev. Mod. Phys. (2)

F. J. García de Abajo, “Colloquium: light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007).
[Crossref]

F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. Kuipers, “Light passing through subwavelength apertures,” Rev. Mod. Phys. 82(1), 729–787 (2010).
[Crossref]

Science (2)

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref] [PubMed]

A. P. Hibbins, B. R. Evans, and J. R. Sambles, “Experimental verification of designer surface plasmons,” Science 308(5722), 670–672 (2005).
[Crossref] [PubMed]

Other (3)

H. Raether, Surface plasmons on smooth and rough surfaces and on gratings (Springer-Verlag, Berlin, 1988).

J. D. Jackson, Classical electrodynamics (Wiley, New York, 1998).

A. Taflove and S. C. Hagness, Computational electrodynamics: the finite-difference time-domain method, 2nd ed. (MA: Artech House, Norwood, 2000).

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Figures (6)

Fig. 1
Fig. 1

Computed (red solid line for MEM calculation, blue dash line for FDTD simulation) transmission spectra under plane wave normal incidence with the electric field E along x axis ( E e ^ x ) and measured (black circular dots) transmission spectrum for the x-polarized Gaussian beam normally incident on the sample plate. Two insets present the sketch map of the front surface of the sample (right side) and the irreducible Brillouin zone of the lattice (left side).

Fig. 2
Fig. 2

Measured radiation patterns in (a) E-plane ( φ= 0 o ) and (b) H-plane ( φ= 90 o ) of transmitted beams with (solid lines) and without (dash lines) the sample plate normally incident by a horn antenna.

Fig. 3
Fig. 3

Spatial distribution of electric fields ( E x ) in the xz plane (a) in free space at 10.7GHz, (b) through the sample with a triangular array at 10.7GHz, (c) through the sample with a square array at 9.5GHz, calculated by FDTD simulations. The Gaussian beam is normally incident on the sample plate with the electric field E along x axis ( E e ^ x ).

Fig. 4
Fig. 4

Calculated (a) and measured (b) transmittivity through the sample plate under TE polarized incidence at φ= 0 o ( E e ^ y ) from horn antenna at different incident angles θ= 0 o , 3 o , 6 o , 10 o , 15 o , 22 o , 29 o , respectively. (c) and (d) are calculated and measured transmittivity for TE polarized incidence at φ= 90 o ( E e ^ x ). Gray dash-dot lines depict the resonance states on two branches with respect to the incident angle, or equivalently speaking the in-plane wavevector.

Fig. 5
Fig. 5

Dispersion diagrams for (a) TE and (b) TM polarized SRSs for the sample with a triangular array in even mode (circular dots) and odd mode (dash lines). Measured transmittivity is plotted in colormap as a function of frequency and in-plane wavevector.

Fig. 6
Fig. 6

Dispersion diagrams for (a) TE and (b) TM polarized SRSs for the sample with a square array in even mode (circular dots) and odd mode (dash lines).

Equations (6)

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E II = p=0 q=1 ( a p,q e i β p,q z b p,q e i β p,q z ) g p,q (x,y) ,
g ρ p,q (ρ,ϕ)= N p ( T p,q d/2) J p ( T p,q ρ)cos(pϕ),
g ϕ p,q (ρ,ϕ)= p ρ N p ( T p,q d/2) J p ( T p,q ρ)sin(pϕ).
H II = p=0 q=1 ( a p,q e i β p,q z b p,q e i β p,q z ) β p,q { g ϕ p,q (ρ,ϕ) z e ^ ρ + g ρ p,q (ρ,ϕ) z e ^ ϕ +[ g ϕ p,q (ρ,ϕ) ρ g ρ p,q (ρ,ϕ) ϕ ] e ^ z },
E I = m= + n= + s=0,1 ( I s δ m,n + r m,n,s ) P m,n,s (x,y) e i( k + G m,n ) r e i k z m,n z ,
E III = m= + n= + s=0,1 t m,n,s P m,n,s (x,y) e i( k + G m,n ) r e i k z m,n z ,

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