Abstract

Optical excitation of surface plasmons polaritons (SPPs) on a 'zig-zag diffraction grating' is explored. The fabricated silver grating consists of sub-wavelength grooves 'zig-zagged' along their length, providing a diffractive periodicity to visible radiation. SPPs propagating in the diffraction plane and scattered by an odd number of grating vectors are only excited by TE polarized radiation, whereas for TM polarized light, which also induces surface charge, SPP excitation is forbidden by the grating's broken-mirror symmetry.

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  1. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Proc. Phys. Soc. Lond.18(1), 269–275 (1901).
    [CrossRef]
  2. S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B Condens. Matter44(12), 6393–6400 (1991).
    [CrossRef] [PubMed]
  3. W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. R. Sambles, N. P. K. Cotter, and D. J. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B Condens. Matter51(16), 11164–11167 (1995).
    [CrossRef] [PubMed]
  4. R. A. Watts and J. R. Sambles, “Reflection gratings as polarization converters,” Opt. Commun.140(4-6), 179–183 (1997).
    [CrossRef]
  5. R. A. Watts, T. W. Preist, and J. R. Sambles, “Sharp surface-plasmon resonances on deep diffraction gratings,” Phys. Rev. Lett.79(20), 3978–3981 (1997).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. B. K. Singh and A. C. Hillier, “Surface plasmon resonance imaging of biomolecular interactions on a grating-based sensor array,” Anal. Chem.78(6), 2009–2018 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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  21. 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. Matter54(9), 6227–6244 (1996).
    [CrossRef] [PubMed]

2010 (1)

2009 (2)

Y. Chen, A. S. Schwanecke, V. Fedotov, V. V. Khardikov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, N. I. Zheludev, and E. Huq, “Electron beam lithography for high density meta fish scale operational at optical frequency,” Microelectron. Eng.86(4-6), 1081–1084 (2009).
[CrossRef]

B. Bai, X. Meng, J. Laukkanen, T. Sfez, and L. Yu, “Asymmetrical excitation of surface plasmon polaritons on blazed gratings at normal incidence,” Phys. Rev. B80, 1–11 (2009).

2008 (2)

2007 (2)

G. H. Welsh, N. T. Hunt, and K. Wynne, “Terahertz-pulse emission through laser excitation of surface plasmons in a metal grating,” Phys. Rev. Lett.98(2), 026803 (2007).
[CrossRef] [PubMed]

E. K. Popov, N. Bonod, and S. Enoch, “Comparison of plasmon surface waves on shallow and deep metallic 1D and 2D gratings,” Opt. Express15(7), 4224–4237 (2007).
[CrossRef] [PubMed]

2006 (1)

B. K. Singh and A. C. Hillier, “Surface plasmon resonance imaging of biomolecular interactions on a grating-based sensor array,” Anal. Chem.78(6), 2009–2018 (2006).
[CrossRef] [PubMed]

2005 (1)

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

1997 (2)

R. A. Watts and J. R. Sambles, “Reflection gratings as polarization converters,” Opt. Commun.140(4-6), 179–183 (1997).
[CrossRef]

R. A. Watts, T. W. Preist, and J. R. Sambles, “Sharp surface-plasmon resonances on deep diffraction gratings,” Phys. Rev. Lett.79(20), 3978–3981 (1997).
[CrossRef]

1996 (4)

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]

R. A. Watts, J. B. Harris, A. P. Hibbins, T. W. Preist, and J. R. Sambles, “Optical excitation of surface plasmon polaritons on 90 and 60 bi-gratings,” J. Mod. Opt.43, 1351–1360 (1996).

D. J. Nash and J. R. Sambles, “Surface plasmon-polariton study of the optical dielectric function of silver,” J. Mod. Opt.43, 81–91 (1996).

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. Matter54(9), 6227–6244 (1996).
[CrossRef] [PubMed]

1995 (1)

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. R. Sambles, N. P. K. Cotter, and D. J. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B Condens. Matter51(16), 11164–11167 (1995).
[CrossRef] [PubMed]

1994 (1)

1991 (1)

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B Condens. Matter44(12), 6393–6400 (1991).
[CrossRef] [PubMed]

1978 (1)

G. J. Kovacs, “Sulphide formation on evaporated Ag films,” Surf. Sci.78(1), L245–L249 (1978).
[CrossRef]

1901 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Proc. Phys. Soc. Lond.18(1), 269–275 (1901).
[CrossRef]

Arnold, R.

Bai, B.

B. Bai, X. Meng, J. Laukkanen, T. Sfez, and L. Yu, “Asymmetrical excitation of surface plasmon polaritons on blazed gratings at normal incidence,” Phys. Rev. B80, 1–11 (2009).

Barnes, W. L.

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]

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. Matter54(9), 6227–6244 (1996).
[CrossRef] [PubMed]

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. R. Sambles, N. P. K. Cotter, and D. J. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B Condens. Matter51(16), 11164–11167 (1995).
[CrossRef] [PubMed]

Bonod, N.

Bryan-Brown, G. P.

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B Condens. Matter44(12), 6393–6400 (1991).
[CrossRef] [PubMed]

Chen, Y.

Y. Chen, A. S. Schwanecke, V. Fedotov, V. V. Khardikov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, N. I. Zheludev, and E. Huq, “Electron beam lithography for high density meta fish scale operational at optical frequency,” Microelectron. Eng.86(4-6), 1081–1084 (2009).
[CrossRef]

Cotter, N. P. K.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. R. Sambles, N. P. K. Cotter, and D. J. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B Condens. Matter51(16), 11164–11167 (1995).
[CrossRef] [PubMed]

Derrick, G. H.

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Elston, S. J.

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B Condens. Matter44(12), 6393–6400 (1991).
[CrossRef] [PubMed]

Enoch, S.

Fainman, Y.

Fedotov, V.

Y. Chen, A. S. Schwanecke, V. Fedotov, V. V. Khardikov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, N. I. Zheludev, and E. Huq, “Electron beam lithography for high density meta fish scale operational at optical frequency,” Microelectron. Eng.86(4-6), 1081–1084 (2009).
[CrossRef]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Harris, J. B.

R. A. Watts, J. B. Harris, A. P. Hibbins, T. W. Preist, and J. R. Sambles, “Optical excitation of surface plasmon polaritons on 90 and 60 bi-gratings,” J. Mod. Opt.43, 1351–1360 (1996).

Herminghaus, S.

Hibbins, A. P.

R. A. Watts, J. B. Harris, A. P. Hibbins, T. W. Preist, and J. R. Sambles, “Optical excitation of surface plasmon polaritons on 90 and 60 bi-gratings,” J. Mod. Opt.43, 1351–1360 (1996).

Hillier, A. C.

B. K. Singh and A. C. Hillier, “Surface plasmon resonance imaging of biomolecular interactions on a grating-based sensor array,” Anal. Chem.78(6), 2009–2018 (2006).
[CrossRef] [PubMed]

Hunt, N. T.

G. H. Welsh, N. T. Hunt, and K. Wynne, “Terahertz-pulse emission through laser excitation of surface plasmons in a metal grating,” Phys. Rev. Lett.98(2), 026803 (2007).
[CrossRef] [PubMed]

Huq, E.

Y. Chen, A. S. Schwanecke, V. Fedotov, V. V. Khardikov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, N. I. Zheludev, and E. Huq, “Electron beam lithography for high density meta fish scale operational at optical frequency,” Microelectron. Eng.86(4-6), 1081–1084 (2009).
[CrossRef]

Hutley, M. C.

Khardikov, V. V.

Y. Chen, A. S. Schwanecke, V. Fedotov, V. V. Khardikov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, N. I. Zheludev, and E. Huq, “Electron beam lithography for high density meta fish scale operational at optical frequency,” Microelectron. Eng.86(4-6), 1081–1084 (2009).
[CrossRef]

Kitson, S. C.

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]

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. Matter54(9), 6227–6244 (1996).
[CrossRef] [PubMed]

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. R. Sambles, N. P. K. Cotter, and D. J. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B Condens. Matter51(16), 11164–11167 (1995).
[CrossRef] [PubMed]

Kleemann, B. H.

Klopfleisch, M.

Kovacs, G. J.

G. J. Kovacs, “Sulphide formation on evaporated Ag films,” Surf. Sci.78(1), L245–L249 (1978).
[CrossRef]

Laukkanen, J.

B. Bai, X. Meng, J. Laukkanen, T. Sfez, and L. Yu, “Asymmetrical excitation of surface plasmon polaritons on blazed gratings at normal incidence,” Phys. Rev. B80, 1–11 (2009).

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Lomakin, V.

Maystre, D.

McPhedran, R. C.

Meng, X.

B. Bai, X. Meng, J. Laukkanen, T. Sfez, and L. Yu, “Asymmetrical excitation of surface plasmon polaritons on blazed gratings at normal incidence,” Phys. Rev. B80, 1–11 (2009).

Mladyonov, P. L.

Y. Chen, A. S. Schwanecke, V. Fedotov, V. V. Khardikov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, N. I. Zheludev, and E. Huq, “Electron beam lithography for high density meta fish scale operational at optical frequency,” Microelectron. Eng.86(4-6), 1081–1084 (2009).
[CrossRef]

Nash, D. J.

D. J. Nash and J. R. Sambles, “Surface plasmon-polariton study of the optical dielectric function of silver,” J. Mod. Opt.43, 81–91 (1996).

Nash, D. J. J.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. R. Sambles, N. P. K. Cotter, and D. J. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B Condens. Matter51(16), 11164–11167 (1995).
[CrossRef] [PubMed]

Nevière, M.

Nezhad, M. P.

Pang, L.

Popov, E.

Popov, E. K.

Preist, T. W.

R. A. Watts, T. W. Preist, and J. R. Sambles, “Sharp surface-plasmon resonances on deep diffraction gratings,” Phys. Rev. Lett.79(20), 3978–3981 (1997).
[CrossRef]

R. A. Watts, J. B. Harris, A. P. Hibbins, T. W. Preist, and J. R. Sambles, “Optical excitation of surface plasmon polaritons on 90 and 60 bi-gratings,” J. Mod. Opt.43, 1351–1360 (1996).

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. Matter54(9), 6227–6244 (1996).
[CrossRef] [PubMed]

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. R. Sambles, N. P. K. Cotter, and D. J. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B Condens. Matter51(16), 11164–11167 (1995).
[CrossRef] [PubMed]

Prosvirnin, S. L.

Y. Chen, A. S. Schwanecke, V. Fedotov, V. V. Khardikov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, N. I. Zheludev, and E. Huq, “Electron beam lithography for high density meta fish scale operational at optical frequency,” Microelectron. Eng.86(4-6), 1081–1084 (2009).
[CrossRef]

Rogacheva, A. V.

Y. Chen, A. S. Schwanecke, V. Fedotov, V. V. Khardikov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, N. I. Zheludev, and E. Huq, “Electron beam lithography for high density meta fish scale operational at optical frequency,” Microelectron. Eng.86(4-6), 1081–1084 (2009).
[CrossRef]

Ruoff, J.

Sambles, J. R.

R. A. Watts, T. W. Preist, and J. R. Sambles, “Sharp surface-plasmon resonances on deep diffraction gratings,” Phys. Rev. Lett.79(20), 3978–3981 (1997).
[CrossRef]

R. A. Watts and J. R. Sambles, “Reflection gratings as polarization converters,” Opt. Commun.140(4-6), 179–183 (1997).
[CrossRef]

R. A. Watts, J. B. Harris, A. P. Hibbins, T. W. Preist, and J. R. Sambles, “Optical excitation of surface plasmon polaritons on 90 and 60 bi-gratings,” J. Mod. Opt.43, 1351–1360 (1996).

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]

D. J. Nash and J. R. Sambles, “Surface plasmon-polariton study of the optical dielectric function of silver,” J. Mod. Opt.43, 81–91 (1996).

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. Matter54(9), 6227–6244 (1996).
[CrossRef] [PubMed]

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B Condens. Matter44(12), 6393–6400 (1991).
[CrossRef] [PubMed]

Sambles, J. R. R.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. R. Sambles, N. P. K. Cotter, and D. J. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B Condens. Matter51(16), 11164–11167 (1995).
[CrossRef] [PubMed]

Schmidt, H. J.

Schwanecke, A. S.

Y. Chen, A. S. Schwanecke, V. Fedotov, V. V. Khardikov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, N. I. Zheludev, and E. Huq, “Electron beam lithography for high density meta fish scale operational at optical frequency,” Microelectron. Eng.86(4-6), 1081–1084 (2009).
[CrossRef]

Sfez, T.

B. Bai, X. Meng, J. Laukkanen, T. Sfez, and L. Yu, “Asymmetrical excitation of surface plasmon polaritons on blazed gratings at normal incidence,” Phys. Rev. B80, 1–11 (2009).

Singh, B. K.

B. K. Singh and A. C. Hillier, “Surface plasmon resonance imaging of biomolecular interactions on a grating-based sensor array,” Anal. Chem.78(6), 2009–2018 (2006).
[CrossRef] [PubMed]

Tayeb, G.

Tetz, K.

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Watts, R. A.

R. A. Watts, T. W. Preist, and J. R. Sambles, “Sharp surface-plasmon resonances on deep diffraction gratings,” Phys. Rev. Lett.79(20), 3978–3981 (1997).
[CrossRef]

R. A. Watts and J. R. Sambles, “Reflection gratings as polarization converters,” Opt. Commun.140(4-6), 179–183 (1997).
[CrossRef]

R. A. Watts, J. B. Harris, A. P. Hibbins, T. W. Preist, and J. R. Sambles, “Optical excitation of surface plasmon polaritons on 90 and 60 bi-gratings,” J. Mod. Opt.43, 1351–1360 (1996).

Welsh, G. H.

G. H. Welsh, N. T. Hunt, and K. Wynne, “Terahertz-pulse emission through laser excitation of surface plasmons in a metal grating,” Phys. Rev. Lett.98(2), 026803 (2007).
[CrossRef] [PubMed]

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,” Nature391(6668), 667–669 (1998).
[CrossRef]

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Proc. Phys. Soc. Lond.18(1), 269–275 (1901).
[CrossRef]

Wynne, K.

G. H. Welsh, N. T. Hunt, and K. Wynne, “Terahertz-pulse emission through laser excitation of surface plasmons in a metal grating,” Phys. Rev. Lett.98(2), 026803 (2007).
[CrossRef] [PubMed]

Yu, L.

B. Bai, X. Meng, J. Laukkanen, T. Sfez, and L. Yu, “Asymmetrical excitation of surface plasmon polaritons on blazed gratings at normal incidence,” Phys. Rev. B80, 1–11 (2009).

Zheludev, N. I.

Y. Chen, A. S. Schwanecke, V. Fedotov, V. V. Khardikov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, N. I. Zheludev, and E. Huq, “Electron beam lithography for high density meta fish scale operational at optical frequency,” Microelectron. Eng.86(4-6), 1081–1084 (2009).
[CrossRef]

Anal. Chem. (1)

B. K. Singh and A. C. Hillier, “Surface plasmon resonance imaging of biomolecular interactions on a grating-based sensor array,” Anal. Chem.78(6), 2009–2018 (2006).
[CrossRef] [PubMed]

J. Mod. Opt. (2)

R. A. Watts, J. B. Harris, A. P. Hibbins, T. W. Preist, and J. R. Sambles, “Optical excitation of surface plasmon polaritons on 90 and 60 bi-gratings,” J. Mod. Opt.43, 1351–1360 (1996).

D. J. Nash and J. R. Sambles, “Surface plasmon-polariton study of the optical dielectric function of silver,” J. Mod. Opt.43, 81–91 (1996).

J. Opt. Soc. Am. A (1)

Microelectron. Eng. (1)

Y. Chen, A. S. Schwanecke, V. Fedotov, V. V. Khardikov, P. L. Mladyonov, S. L. Prosvirnin, A. V. Rogacheva, N. I. Zheludev, and E. Huq, “Electron beam lithography for high density meta fish scale operational at optical frequency,” Microelectron. Eng.86(4-6), 1081–1084 (2009).
[CrossRef]

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature391(6668), 667–669 (1998).
[CrossRef]

Opt. Commun. (1)

R. A. Watts and J. R. Sambles, “Reflection gratings as polarization converters,” Opt. Commun.140(4-6), 179–183 (1997).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. B (1)

B. Bai, X. Meng, J. Laukkanen, T. Sfez, and L. Yu, “Asymmetrical excitation of surface plasmon polaritons on blazed gratings at normal incidence,” Phys. Rev. B80, 1–11 (2009).

Phys. Rev. B Condens. Matter (3)

S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B Condens. Matter44(12), 6393–6400 (1991).
[CrossRef] [PubMed]

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Phys. Rev. Lett. (3)

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G. H. Welsh, N. T. Hunt, and K. Wynne, “Terahertz-pulse emission through laser excitation of surface plasmons in a metal grating,” Phys. Rev. Lett.98(2), 026803 (2007).
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Proc. Phys. Soc. Lond. (1)

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[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

The coordinate system of a zig-zag grating. The experimental sample parameters were λgx = 600 nm, λgy = 150 nm, d = 29.9 nm with all measurements in this letter recorded at ϕ = 0°. Inset: A scanning electron micrograph of a replicated zig-zag surface in a polymer. (scale bar = 600 nm.)

Fig. 2
Fig. 2

Specular reflectivity of TE (black) and TM (red) polarized light as a function of polar angle, θ, with a wavelength of 632.8 nm, ϕ = 0°. Circles: recorded data with error bars of 1%, line: fitted numerical model prediction.

Fig. 3
Fig. 3

TE polarized reflectivity as a function of in-plane wavevector and angular frequency, mapping the SPP dispersion on a zig-zag grating. (a) Numerical prediction and (b) measured data. Solid lines show the positions of diffracted light lights scattered by mkgx, where m = −1, + 2, 0, + 3.

Equations (3)

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r=x+y+cos(ycos(x)) z ^
E TE = 4πsin(x) cos(2x)3
E TM = 4π sin 2 (x) cos(2x)3

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