Abstract

Recent studies of plasmon surface wave (PSW) propagation in short-period perfectly conducting gratings have shown formation of stop-band that are not linked to the interaction between two (counter) propagating surface waves. We study the properties of this stop-band in real metals. While for both perfectly conducting and real metals the propagation constant of PSW grows with the groove height, the stop-band in real metals appears for groove heights significantly smaller than in perfect metals. A physical explanation of the formation of the stop-band is proposed both by using a homogenisation of the corrugated layer and by analysis of the tangential electric field component.

© 2007 Optical Society of America

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  1. F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials," J. Opt. A: Pure Appl. Opt. 7, S97-S101 (2005).
    [CrossRef]
  2. J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
    [CrossRef] [PubMed]
  3. R. W. Wood: "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Phylos. Mag. 4, 396-402 (1902).
  4. D. Maystre, "General study of grating anomalies from electromagnetic surface modes," in Electromagnetic Surface Modes, A. D. Boardman, ed., (John Wiley, 1982), ch.17.
  5. E. Popov, "Light diffraction by relief gratings: a microscopic and macroscopic view," in Progress in Optics, E.Wolf, ed., (Elsevier, Amsterdam, 1993) Vol. 31, pp. 139-187.
  6. E. Popov, L. Tsonev, and D. Maystre, "Losses of plasmon surface wave on metallic grating," J. Mod. Opt. 37, 379-387 (1990).
    [CrossRef]
  7. F. J. Garcia-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Rios, T. Fournier, and B. Pannetier, "Localized surface plasmons in lamellar metallic gratings," J. Lightwave Technol. 17, 2191-2195 (1999).
    [CrossRef]
  8. W.-C. Tan, T. W. Preist, J. R. Sambles, and N. P. Wanstall, "Flat surface-plasmon-polariton bands and resonant optical absorption on short-pitch metal gratings," Phys. Rev. B 59, 12661 (1999).
    [CrossRef]
  9. I. R. Hooper and J. R. Sambles, "Surface plasmon polaritons on narrow-ridged short-pitch metal gratings," Phys. Rev. B 66, 205408 (2002).
    [CrossRef]
  10. I. R. Hooper and J. R. Sambles, "Dispersion of surface plasmon polaritons on short-pitch metal gratings," Phys. Rev. B 65, 165432-1-9 (2002).
    [CrossRef]
  11. S. Maier, S. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, "Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires," Phys. Rev. Lett. 97, 176805-1-4 (2006).
    [CrossRef] [PubMed]
  12. J. D. Jackson, Classical Electrodynamics (Wiley, 1998), sec. 8.5.
  13. E. Popov, M. Nevière, J. Wenger, P.-F. Lenne, H. Rigneault, P. Chaumet, N. Bonod, J. Dintinger, and T. Ebbesen, "Field enhancement in single subwavelength apertures," J. Opt. Soc. Am. A 23, 2342-2348 (2006).
    [CrossRef]
  14. R. McPhedran, L. Boteen, M. Craig, M. Nevière, and D. Maystre, "Lossy lamellar gratings in the quasistatic limit," Opt. Acta 29, 289-312 (1982).
    [CrossRef]
  15. G. Bouchitte and R. Petit, "Homogenization techniques as applied in the electromagnetic theory of gratings," Electromagnetics 5, 17-36 (1985).
    [CrossRef]
  16. P. Yeh, "A new optical model for wire grid polarizers," Opt. Commun. 26, 289-292 (1978).
    [CrossRef]
  17. E. Popov and M. Nevière: "Maxwell equations in Fourier space: fast converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media," J. Opt. Soc. Am. A 17, 1773 (2001).
    [CrossRef]
  18. M. G. Moharam and T. K. Gaylord, "Rigorous coupled-wave analysis of dielectric surface-relief gratings,"J. Opt. Soc. Am. 72, 1385-1392 (1982)
    [CrossRef]
  19. P. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996)
    [CrossRef]
  20. G. Granet and B. Guizal, "Efficient implementation of the coupled-wave method for metallic gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996)
    [CrossRef]
  21. E. Popov, L. Tsonev, and D. Maystre, "Losses of plasmon surface wave on metallic grating," J. Mod. Opt. 37, 379-387 (1990)
    [CrossRef]

2006 (2)

S. Maier, S. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, "Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires," Phys. Rev. Lett. 97, 176805-1-4 (2006).
[CrossRef] [PubMed]

E. Popov, M. Nevière, J. Wenger, P.-F. Lenne, H. Rigneault, P. Chaumet, N. Bonod, J. Dintinger, and T. Ebbesen, "Field enhancement in single subwavelength apertures," J. Opt. Soc. Am. A 23, 2342-2348 (2006).
[CrossRef]

2005 (1)

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials," J. Opt. A: Pure Appl. Opt. 7, S97-S101 (2005).
[CrossRef]

2004 (1)

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

2002 (2)

I. R. Hooper and J. R. Sambles, "Surface plasmon polaritons on narrow-ridged short-pitch metal gratings," Phys. Rev. B 66, 205408 (2002).
[CrossRef]

I. R. Hooper and J. R. Sambles, "Dispersion of surface plasmon polaritons on short-pitch metal gratings," Phys. Rev. B 65, 165432-1-9 (2002).
[CrossRef]

2001 (1)

1999 (2)

F. J. Garcia-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Rios, T. Fournier, and B. Pannetier, "Localized surface plasmons in lamellar metallic gratings," J. Lightwave Technol. 17, 2191-2195 (1999).
[CrossRef]

W.-C. Tan, T. W. Preist, J. R. Sambles, and N. P. Wanstall, "Flat surface-plasmon-polariton bands and resonant optical absorption on short-pitch metal gratings," Phys. Rev. B 59, 12661 (1999).
[CrossRef]

1996 (2)

1990 (2)

E. Popov, L. Tsonev, and D. Maystre, "Losses of plasmon surface wave on metallic grating," J. Mod. Opt. 37, 379-387 (1990)
[CrossRef]

E. Popov, L. Tsonev, and D. Maystre, "Losses of plasmon surface wave on metallic grating," J. Mod. Opt. 37, 379-387 (1990).
[CrossRef]

1985 (1)

G. Bouchitte and R. Petit, "Homogenization techniques as applied in the electromagnetic theory of gratings," Electromagnetics 5, 17-36 (1985).
[CrossRef]

1982 (2)

M. G. Moharam and T. K. Gaylord, "Rigorous coupled-wave analysis of dielectric surface-relief gratings,"J. Opt. Soc. Am. 72, 1385-1392 (1982)
[CrossRef]

R. McPhedran, L. Boteen, M. Craig, M. Nevière, and D. Maystre, "Lossy lamellar gratings in the quasistatic limit," Opt. Acta 29, 289-312 (1982).
[CrossRef]

1978 (1)

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

1902 (1)

R. W. Wood: "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Phylos. Mag. 4, 396-402 (1902).

Andrews, S.

S. Maier, S. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, "Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires," Phys. Rev. Lett. 97, 176805-1-4 (2006).
[CrossRef] [PubMed]

Bonod, N.

Boteen, L.

R. McPhedran, L. Boteen, M. Craig, M. Nevière, and D. Maystre, "Lossy lamellar gratings in the quasistatic limit," Opt. Acta 29, 289-312 (1982).
[CrossRef]

Bouchitte, G.

G. Bouchitte and R. Petit, "Homogenization techniques as applied in the electromagnetic theory of gratings," Electromagnetics 5, 17-36 (1985).
[CrossRef]

Bustarret, E.

Chaumet, P.

Craig, M.

R. McPhedran, L. Boteen, M. Craig, M. Nevière, and D. Maystre, "Lossy lamellar gratings in the quasistatic limit," Opt. Acta 29, 289-312 (1982).
[CrossRef]

Dechelette, A.

Dintinger, J.

Ebbesen, T.

Fournier, T.

Garcia-Vidal, F. J.

S. Maier, S. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, "Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires," Phys. Rev. Lett. 97, 176805-1-4 (2006).
[CrossRef] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials," J. Opt. A: Pure Appl. Opt. 7, S97-S101 (2005).
[CrossRef]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

F. J. Garcia-Vidal, J. Sánchez-Dehesa, A. Dechelette, E. Bustarret, T. López-Rios, T. Fournier, and B. Pannetier, "Localized surface plasmons in lamellar metallic gratings," J. Lightwave Technol. 17, 2191-2195 (1999).
[CrossRef]

Gaylord, T. K.

Granet, G.

Guizal, B.

Hooper, I. R.

I. R. Hooper and J. R. Sambles, "Dispersion of surface plasmon polaritons on short-pitch metal gratings," Phys. Rev. B 65, 165432-1-9 (2002).
[CrossRef]

I. R. Hooper and J. R. Sambles, "Surface plasmon polaritons on narrow-ridged short-pitch metal gratings," Phys. Rev. B 66, 205408 (2002).
[CrossRef]

Lalanne, P.

Lenne, P.-F.

López-Rios, T.

Maier, S.

S. Maier, S. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, "Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires," Phys. Rev. Lett. 97, 176805-1-4 (2006).
[CrossRef] [PubMed]

Martin-Moreno, L.

S. Maier, S. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, "Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires," Phys. Rev. Lett. 97, 176805-1-4 (2006).
[CrossRef] [PubMed]

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials," J. Opt. A: Pure Appl. Opt. 7, S97-S101 (2005).
[CrossRef]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

Maystre, D.

E. Popov, L. Tsonev, and D. Maystre, "Losses of plasmon surface wave on metallic grating," J. Mod. Opt. 37, 379-387 (1990).
[CrossRef]

E. Popov, L. Tsonev, and D. Maystre, "Losses of plasmon surface wave on metallic grating," J. Mod. Opt. 37, 379-387 (1990)
[CrossRef]

R. McPhedran, L. Boteen, M. Craig, M. Nevière, and D. Maystre, "Lossy lamellar gratings in the quasistatic limit," Opt. Acta 29, 289-312 (1982).
[CrossRef]

McPhedran, R.

R. McPhedran, L. Boteen, M. Craig, M. Nevière, and D. Maystre, "Lossy lamellar gratings in the quasistatic limit," Opt. Acta 29, 289-312 (1982).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Nevière, M.

Pannetier, B.

Pendry, J. B.

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials," J. Opt. A: Pure Appl. Opt. 7, S97-S101 (2005).
[CrossRef]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

Petit, R.

G. Bouchitte and R. Petit, "Homogenization techniques as applied in the electromagnetic theory of gratings," Electromagnetics 5, 17-36 (1985).
[CrossRef]

Popov, E.

Preist, T. W.

W.-C. Tan, T. W. Preist, J. R. Sambles, and N. P. Wanstall, "Flat surface-plasmon-polariton bands and resonant optical absorption on short-pitch metal gratings," Phys. Rev. B 59, 12661 (1999).
[CrossRef]

Rigneault, H.

Sambles, J. R.

I. R. Hooper and J. R. Sambles, "Surface plasmon polaritons on narrow-ridged short-pitch metal gratings," Phys. Rev. B 66, 205408 (2002).
[CrossRef]

I. R. Hooper and J. R. Sambles, "Dispersion of surface plasmon polaritons on short-pitch metal gratings," Phys. Rev. B 65, 165432-1-9 (2002).
[CrossRef]

W.-C. Tan, T. W. Preist, J. R. Sambles, and N. P. Wanstall, "Flat surface-plasmon-polariton bands and resonant optical absorption on short-pitch metal gratings," Phys. Rev. B 59, 12661 (1999).
[CrossRef]

Sánchez-Dehesa, J.

Tan, W.-C.

W.-C. Tan, T. W. Preist, J. R. Sambles, and N. P. Wanstall, "Flat surface-plasmon-polariton bands and resonant optical absorption on short-pitch metal gratings," Phys. Rev. B 59, 12661 (1999).
[CrossRef]

Tsonev, L.

E. Popov, L. Tsonev, and D. Maystre, "Losses of plasmon surface wave on metallic grating," J. Mod. Opt. 37, 379-387 (1990).
[CrossRef]

E. Popov, L. Tsonev, and D. Maystre, "Losses of plasmon surface wave on metallic grating," J. Mod. Opt. 37, 379-387 (1990)
[CrossRef]

Wanstall, N. P.

W.-C. Tan, T. W. Preist, J. R. Sambles, and N. P. Wanstall, "Flat surface-plasmon-polariton bands and resonant optical absorption on short-pitch metal gratings," Phys. Rev. B 59, 12661 (1999).
[CrossRef]

Wenger, J.

Wood, R. W.

R. W. Wood: "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Phylos. Mag. 4, 396-402 (1902).

Yeh, P.

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

Electromagnetics (1)

G. Bouchitte and R. Petit, "Homogenization techniques as applied in the electromagnetic theory of gratings," Electromagnetics 5, 17-36 (1985).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (2)

E. Popov, L. Tsonev, and D. Maystre, "Losses of plasmon surface wave on metallic grating," J. Mod. Opt. 37, 379-387 (1990).
[CrossRef]

E. Popov, L. Tsonev, and D. Maystre, "Losses of plasmon surface wave on metallic grating," J. Mod. Opt. 37, 379-387 (1990)
[CrossRef]

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

F. J. Garcia-Vidal, L. Martin-Moreno, and J. B. Pendry, "Surfaces with holes in them: new plasmonic metamaterials," J. Opt. A: Pure Appl. Opt. 7, S97-S101 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Acta (1)

R. McPhedran, L. Boteen, M. Craig, M. Nevière, and D. Maystre, "Lossy lamellar gratings in the quasistatic limit," Opt. Acta 29, 289-312 (1982).
[CrossRef]

Opt. Commun. (1)

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

Phylos. Mag. (1)

R. W. Wood: "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Phylos. Mag. 4, 396-402 (1902).

Phys. Rev. B (3)

W.-C. Tan, T. W. Preist, J. R. Sambles, and N. P. Wanstall, "Flat surface-plasmon-polariton bands and resonant optical absorption on short-pitch metal gratings," Phys. Rev. B 59, 12661 (1999).
[CrossRef]

I. R. Hooper and J. R. Sambles, "Surface plasmon polaritons on narrow-ridged short-pitch metal gratings," Phys. Rev. B 66, 205408 (2002).
[CrossRef]

I. R. Hooper and J. R. Sambles, "Dispersion of surface plasmon polaritons on short-pitch metal gratings," Phys. Rev. B 65, 165432-1-9 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

S. Maier, S. Andrews, L. Martin-Moreno, and F. J. Garcia-Vidal, "Terahertz surface plasmon-polariton propagation and focusing on periodically corrugated metal wires," Phys. Rev. Lett. 97, 176805-1-4 (2006).
[CrossRef] [PubMed]

Science (1)

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, "Mimicking surface plasmons with structured surfaces," Science 305, 847-848 (2004).
[CrossRef] [PubMed]

Other (3)

D. Maystre, "General study of grating anomalies from electromagnetic surface modes," in Electromagnetic Surface Modes, A. D. Boardman, ed., (John Wiley, 1982), ch.17.

E. Popov, "Light diffraction by relief gratings: a microscopic and macroscopic view," in Progress in Optics, E.Wolf, ed., (Elsevier, Amsterdam, 1993) Vol. 31, pp. 139-187.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998), sec. 8.5.

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Figures (7)

Fig. 1.
Fig. 1.

Dispersion curves of plasmon-like surface waves in a lamellar grating made of perfectly conducting metal. Dotted lines present Brillouin zone boundaries (kx = π/d), dashed lines indicate lower boundary of the forbidden zone created by the interaction between counter propagating surface waves.

Fig. 2.
Fig. 2.

Normalized propagation constant of the plasmon-like surface wave as a function of the groove height of the grating for 3 values of the period and wavelength λ = 0.6 μm. In the limit d→0, the cut-off height is equal to λ/4.

Fig. 3.
Fig. 3.

Same as in Fig. 2 but for a real metal (aluminum in visible). Cut-off is characterized by the sharp increase of the imaginary part of the propagation constant. As for perfect metals, the real part of kx increases at the cut-off.

Fig. 4.
Fig. 4.

Same as in Figs. 2 and 3 but for two different gratings. Lamellae and substrate made of aluminum (black and red lines), and lamellae made of aluminum and substrate made of perfect metal (green and blue lines). Wavelength λ = 0.6 μm.

Fig. 5.
Fig. 5.

Vertical distribution of |Ex| inside the groove with depth h = 0.13 μm for a perfect (black line) and real-metal (red line) case. Incident electric field modulus is equal to 1.

Fig. 6.
Fig. 6.

Map of |Ex| within one period of the grating with h = λ/4. Wavelength λ = 0.6 μm, period d = 0.01 μm, and εmetal = -104 + i 107. |Ex| vanishes on the top of the lamella and at the bottom of the groove, and has a maximum at the groove opening.

Fig. 7.
Fig. 7.

Comparison of the values of |Ex| at the top of the lamellae (black line) and at the groove opening (red line) as a function of the height of the groove. λ = 0.6 μm, d = 0.01 μm, and aluminium in the microwave domain, εmetal = -104 + i 107.

Equations (15)

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

k x , max = π d
ω max c = k x , max , h 0
ε xx = 1 ε 1 , ε yy = ε zz =
μ xx = 1 , μ yy = μ zz = 1 ε xx
ε xx = 1 ε 1 , ε yy = ε zz = ε ,
μ xx = μ yy = μ zz = 1 ,
k y k 0 = 1 ε 1 2 .
h cut = π 2 k y , 2 = λ 4 μ zz ε xx = { λ 4 , p . c λ 4 f , f . c
E x 0 at y = 0 ,
ε = ( ε xx 0 0 0 ε yy 0 0 0 ε zz ) , μ = ( μ xx 0 0 0 μ yy 0 0 0 μ zz )
d 2 H z d y 2 + ( k 0 2 ε xx μ zz k x 2 ε xx ε yy ) H z = 0
( β + α ) ( β + γ ) exp ( i k y , 2 h ) = ( β α ) ( β γ ) exp ( i k y , 2 h )
k y , 2 = k 0 2 ε xx μ zz k x 2 ε xx ε yy
α = tg ( k y , 2 h ) .
α = i k y , 2 tg ( k 2 h ) ε xx , k y , 2 k 0 ε xx μ zz .

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