Abstract

The mechanism of achieving unidirectional transmission in the gratings, which only contain isotropic dielectric and metallic layers, is suggested and numerically validated. It is shown that significant transmission in one direction and nearly zero transmission in the opposite direction can be obtained in the same intrinsically isotropic gratings as those studied recently in A. E. Serebryannikov and E. Ozbay, Opt. Express 17, 278 (2009), but at a non-zero angle of incidence. The tilting, non-symmetric features of the grating and the presence of a metallic layer with a small positive real part of the index of refraction are the conditions that are necessary for obtaining the unidirectionality. Single- and multibeam operational regimes are demonstrated. The frequency and angle ranges of the unidirectional transmission can be estimated by using the conventional framework based on isofrequency dispersion contours and construction lines that properly take into account the periodic features of the interfaces, but should then be corrected because of the tunneling arising within the adjacent ranges. After proper optimization, this mechanism is expected to become an alternative to that based on the use of anisotropic materials.

© 2009 Optical Society of America

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    [CrossRef]
  6. A. E. Serebryannikov, T. Magath, and K. Schuenemann, "Bragg transmittance of s-polarized waves through finite-thickness photonic crystals with a periodically corrugated interface," Phys. Rev. E 74, 066607 (2006).
    [CrossRef]
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2009 (1)

2008 (1)

Z. Wang, J. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-free one-way edge modes in gyromagnetic photonic crystals," Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

2007 (2)

A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, "Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern," Phys. Rev. B 75, 155410 (2007).
[CrossRef]

M. Silverinha and N. Engheta, "Design of matched zero-index metamaterials using non-magnetic inclusions in epsilon-near-zero media," Phys. Rev. B 75, 075119 (2007).
[CrossRef]

2006 (4)

A. E. Serebryannikov, T. Magath, and K. Schuenemann, "Bragg transmittance of s-polarized waves through finite-thickness photonic crystals with a periodically corrugated interface," Phys. Rev. E 74, 066607 (2006).
[CrossRef]

M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, "One-way diffraction grating," Phys. Rev. E 74, 056611 (2006).
[CrossRef]

A.E. Serebryannikov, T. Magath, K. Schuenemann, and O.Y. Vasylchenko, "Scattering of s-polarized plane waves by finite-thickness periodic structures made of ultralow-permittivity metamaterials," Phys. Rev. B 73, 115111 (2006).
[CrossRef]

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality and frozen modes in magnetic photonic crystals," J. Magnetism Magnet. Mat. 300, 117 (2006).
[CrossRef]

2005 (3)

T. Magath and A. E. Serebryannikov, "Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs," J. Opt. Soc. Am. A 22, 2405-2418 (2005).
[CrossRef]

K. G. Lee and Q-H. Park, "Coupling of surface plasmon polaritons and light in metallic nanoslits," Phys. Rev. Lett. 95, 103902 (2005).
[CrossRef] [PubMed]

D. Felbacq, M.C. Larciprete, C. Sibilia, M. Bertolotti, and M. Scalora, "Multiple wavelength filtering of light through inner resonances," Phys. Rev. E 72, 066610 (2005).
[CrossRef]

2004 (1)

R. W. Ziolkowski, "Propagation in and scattering from a matched metamaterial having a zero index of refraction," Phys. Rev. E 70, 046608 (2004).
[CrossRef]

2003 (3)

J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, "Metallodielectric gratings with subwavelength slots: Optical properties, " Phys. Rev. B 68, 205103 (2003).
[CrossRef]

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality in magnetic photonic crystals," Phys. Rev. B 67, 165210 (2003).
[CrossRef]

B. T. Schwartz and R. Piestun, "Total external reflection from metamaterials with ultralow refractive index," J. Opt. Soc. Am. B 20, 2448-2453 (2003).
[CrossRef]

2002 (3)

S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, "A metamaterial for directive emission," Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

A. Barbara, P, Quemerais, E. Bustarret, and T. Lopez-Rios, "Optical transmission through subwavelength metallic gratings," Phys. Rev. B 66, 161403 (2002).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002).
[CrossRef]

1996 (1)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Aguirre, C. M.

J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, "Metallodielectric gratings with subwavelength slots: Optical properties, " Phys. Rev. B 68, 205103 (2003).
[CrossRef]

Alu, A.

A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, "Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern," Phys. Rev. B 75, 155410 (2007).
[CrossRef]

Barbara, A.

A. Barbara, P, Quemerais, E. Bustarret, and T. Lopez-Rios, "Optical transmission through subwavelength metallic gratings," Phys. Rev. B 66, 161403 (2002).
[CrossRef]

Bertolotti, M.

D. Felbacq, M.C. Larciprete, C. Sibilia, M. Bertolotti, and M. Scalora, "Multiple wavelength filtering of light through inner resonances," Phys. Rev. E 72, 066610 (2005).
[CrossRef]

Chong, J. D.

Z. Wang, J. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-free one-way edge modes in gyromagnetic photonic crystals," Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

Engheta, N.

A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, "Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern," Phys. Rev. B 75, 155410 (2007).
[CrossRef]

M. Silverinha and N. Engheta, "Design of matched zero-index metamaterials using non-magnetic inclusions in epsilon-near-zero media," Phys. Rev. B 75, 075119 (2007).
[CrossRef]

Enoch, S.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, "A metamaterial for directive emission," Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Felbacq, D.

D. Felbacq, M.C. Larciprete, C. Sibilia, M. Bertolotti, and M. Scalora, "Multiple wavelength filtering of light through inner resonances," Phys. Rev. E 72, 066610 (2005).
[CrossRef]

Figotin, A.

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality and frozen modes in magnetic photonic crystals," J. Magnetism Magnet. Mat. 300, 117 (2006).
[CrossRef]

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality in magnetic photonic crystals," Phys. Rev. B 67, 165210 (2003).
[CrossRef]

Guerin, N.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, "A metamaterial for directive emission," Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Halas, N. J.

J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, "Metallodielectric gratings with subwavelength slots: Optical properties, " Phys. Rev. B 68, 205103 (2003).
[CrossRef]

Hibbins, A. P.

M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, "One-way diffraction grating," Phys. Rev. E 74, 056611 (2006).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Joannopoulos, J. D.

Z. Wang, J. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-free one-way edge modes in gyromagnetic photonic crystals," Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Johnson, S. G.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Larciprete, M.C.

D. Felbacq, M.C. Larciprete, C. Sibilia, M. Bertolotti, and M. Scalora, "Multiple wavelength filtering of light through inner resonances," Phys. Rev. E 72, 066610 (2005).
[CrossRef]

Lee, A.

J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, "Metallodielectric gratings with subwavelength slots: Optical properties, " Phys. Rev. B 68, 205103 (2003).
[CrossRef]

Lee, K. G.

K. G. Lee and Q-H. Park, "Coupling of surface plasmon polaritons and light in metallic nanoslits," Phys. Rev. Lett. 95, 103902 (2005).
[CrossRef] [PubMed]

Lockyear, M. J.

M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, "One-way diffraction grating," Phys. Rev. E 74, 056611 (2006).
[CrossRef]

Luo, C.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Magath, T.

A. E. Serebryannikov, T. Magath, and K. Schuenemann, "Bragg transmittance of s-polarized waves through finite-thickness photonic crystals with a periodically corrugated interface," Phys. Rev. E 74, 066607 (2006).
[CrossRef]

A.E. Serebryannikov, T. Magath, K. Schuenemann, and O.Y. Vasylchenko, "Scattering of s-polarized plane waves by finite-thickness periodic structures made of ultralow-permittivity metamaterials," Phys. Rev. B 73, 115111 (2006).
[CrossRef]

T. Magath and A. E. Serebryannikov, "Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs," J. Opt. Soc. Am. A 22, 2405-2418 (2005).
[CrossRef]

Moran, C. E.

J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, "Metallodielectric gratings with subwavelength slots: Optical properties, " Phys. Rev. B 68, 205103 (2003).
[CrossRef]

Ozbay, E.

Park, Q-H.

K. G. Lee and Q-H. Park, "Coupling of surface plasmon polaritons and light in metallic nanoslits," Phys. Rev. Lett. 95, 103902 (2005).
[CrossRef] [PubMed]

Pendry, J. B.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Piestun, R.

Sabouroux, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, "A metamaterial for directive emission," Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Salandrino, A.

A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, "Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern," Phys. Rev. B 75, 155410 (2007).
[CrossRef]

Sambles, J. R.

M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, "One-way diffraction grating," Phys. Rev. E 74, 056611 (2006).
[CrossRef]

Scalora, M.

D. Felbacq, M.C. Larciprete, C. Sibilia, M. Bertolotti, and M. Scalora, "Multiple wavelength filtering of light through inner resonances," Phys. Rev. E 72, 066610 (2005).
[CrossRef]

Schuenemann, K.

A.E. Serebryannikov, T. Magath, K. Schuenemann, and O.Y. Vasylchenko, "Scattering of s-polarized plane waves by finite-thickness periodic structures made of ultralow-permittivity metamaterials," Phys. Rev. B 73, 115111 (2006).
[CrossRef]

A. E. Serebryannikov, T. Magath, and K. Schuenemann, "Bragg transmittance of s-polarized waves through finite-thickness photonic crystals with a periodically corrugated interface," Phys. Rev. E 74, 066607 (2006).
[CrossRef]

Schwartz, B. T.

Serebryannikov, A. E.

A. E. Serebryannikov, T. Magath, and K. Schuenemann, "Bragg transmittance of s-polarized waves through finite-thickness photonic crystals with a periodically corrugated interface," Phys. Rev. E 74, 066607 (2006).
[CrossRef]

T. Magath and A. E. Serebryannikov, "Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs," J. Opt. Soc. Am. A 22, 2405-2418 (2005).
[CrossRef]

Serebryannikov, A.E.

A.E. Serebryannikov and E. Ozbay, "Isolation and one-way effects in diffraction on dielectric gratings with plasmonic inserts," Opt. Express 17, 278 (2009).
[CrossRef] [PubMed]

A.E. Serebryannikov, T. Magath, K. Schuenemann, and O.Y. Vasylchenko, "Scattering of s-polarized plane waves by finite-thickness periodic structures made of ultralow-permittivity metamaterials," Phys. Rev. B 73, 115111 (2006).
[CrossRef]

Sibilia, C.

D. Felbacq, M.C. Larciprete, C. Sibilia, M. Bertolotti, and M. Scalora, "Multiple wavelength filtering of light through inner resonances," Phys. Rev. E 72, 066610 (2005).
[CrossRef]

Silveirinha, M. G.

A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, "Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern," Phys. Rev. B 75, 155410 (2007).
[CrossRef]

Silverinha, M.

M. Silverinha and N. Engheta, "Design of matched zero-index metamaterials using non-magnetic inclusions in epsilon-near-zero media," Phys. Rev. B 75, 075119 (2007).
[CrossRef]

Soljacic, M.

Z. Wang, J. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-free one-way edge modes in gyromagnetic photonic crystals," Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

Steele, J. M.

J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, "Metallodielectric gratings with subwavelength slots: Optical properties, " Phys. Rev. B 68, 205103 (2003).
[CrossRef]

Stewart, W. J.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Tayeb, G.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, "A metamaterial for directive emission," Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Vasylchenko, O.Y.

A.E. Serebryannikov, T. Magath, K. Schuenemann, and O.Y. Vasylchenko, "Scattering of s-polarized plane waves by finite-thickness periodic structures made of ultralow-permittivity metamaterials," Phys. Rev. B 73, 115111 (2006).
[CrossRef]

Vincent, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, "A metamaterial for directive emission," Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Vitebskiy, I.

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality and frozen modes in magnetic photonic crystals," J. Magnetism Magnet. Mat. 300, 117 (2006).
[CrossRef]

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality in magnetic photonic crystals," Phys. Rev. B 67, 165210 (2003).
[CrossRef]

Wang, Z.

Z. Wang, J. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-free one-way edge modes in gyromagnetic photonic crystals," Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

White, K. R.

M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, "One-way diffraction grating," Phys. Rev. E 74, 056611 (2006).
[CrossRef]

Youngs, I.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Ziolkowski, R. W.

R. W. Ziolkowski, "Propagation in and scattering from a matched metamaterial having a zero index of refraction," Phys. Rev. E 70, 046608 (2004).
[CrossRef]

J. Magnetism Magnet. Mat. (1)

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality and frozen modes in magnetic photonic crystals," J. Magnetism Magnet. Mat. 300, 117 (2006).
[CrossRef]

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

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

Opt. Express (1)

Phys. Rev. B (7)

M. Silverinha and N. Engheta, "Design of matched zero-index metamaterials using non-magnetic inclusions in epsilon-near-zero media," Phys. Rev. B 75, 075119 (2007).
[CrossRef]

J. M. Steele, C. E. Moran, A. Lee, C. M. Aguirre, and N. J. Halas, "Metallodielectric gratings with subwavelength slots: Optical properties, " Phys. Rev. B 68, 205103 (2003).
[CrossRef]

A. Barbara, P, Quemerais, E. Bustarret, and T. Lopez-Rios, "Optical transmission through subwavelength metallic gratings," Phys. Rev. B 66, 161403 (2002).
[CrossRef]

A. Figotin and I. Vitebskiy, "Electromagnetic unidirectionality in magnetic photonic crystals," Phys. Rev. B 67, 165210 (2003).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002).
[CrossRef]

A.E. Serebryannikov, T. Magath, K. Schuenemann, and O.Y. Vasylchenko, "Scattering of s-polarized plane waves by finite-thickness periodic structures made of ultralow-permittivity metamaterials," Phys. Rev. B 73, 115111 (2006).
[CrossRef]

A. Alu, M. G. Silveirinha, A. Salandrino, and N. Engheta, "Epsilon-near-zero metamaterials and electromagnetic sources: Tailoring the radiation phase pattern," Phys. Rev. B 75, 155410 (2007).
[CrossRef]

Phys. Rev. E (4)

R. W. Ziolkowski, "Propagation in and scattering from a matched metamaterial having a zero index of refraction," Phys. Rev. E 70, 046608 (2004).
[CrossRef]

A. E. Serebryannikov, T. Magath, and K. Schuenemann, "Bragg transmittance of s-polarized waves through finite-thickness photonic crystals with a periodically corrugated interface," Phys. Rev. E 74, 066607 (2006).
[CrossRef]

M. J. Lockyear, A. P. Hibbins, K. R. White, and J. R. Sambles, "One-way diffraction grating," Phys. Rev. E 74, 056611 (2006).
[CrossRef]

D. Felbacq, M.C. Larciprete, C. Sibilia, M. Bertolotti, and M. Scalora, "Multiple wavelength filtering of light through inner resonances," Phys. Rev. E 72, 066610 (2005).
[CrossRef]

Phys. Rev. Lett. (4)

K. G. Lee and Q-H. Park, "Coupling of surface plasmon polaritons and light in metallic nanoslits," Phys. Rev. Lett. 95, 103902 (2005).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, "A metamaterial for directive emission," Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Z. Wang, J. D. Chong, J. D. Joannopoulos, and M. Soljacic, "Reflection-free one-way edge modes in gyromagnetic photonic crystals," Phys. Rev. Lett. 100, 013905 (2008).
[CrossRef] [PubMed]

Other (2)

F. D. M. Haldane and S. Raghu, "Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry," arXiv:cond-mat/0503588 (2008).

A. Sihvola, Electromagnetic mixing formulas and applications (IEE, London, 1999).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

Left plot: Three-layer grating illuminated by s-polarized plane wave at a non-zero θ; Right plot: isofrequency dispersion contours in dielectric and metal and construction lines at the interface between the flat metallic layer and the corrugated dielectric layer at θ=0 and θ>0; arrows – direction of incidence in U and L cases at θ=0.

Fig. 2.
Fig. 2.

Values of κnL on the plane of (θ,ωpL/c) for n=0 (upper left), n=−1 (upper right), n=−2 (lower left), and n=−3 (lower right) - left plot; zones corresponding to the different diffraction orders with min κnL on the plane of (θ,ωpL/c) - right plot; θ is shown in degrees.

Fig. 3.
Fig. 3.

Transmittance for a three-layer grating in the cases U (a) and L (b); A=0.8, B=0.2, and C=D=0, in U case, and A=1, B=0, and C=D=0.2 in L case; ε U=ε L=2.1, h/L=2, a/h=0.4, b/h=0.6, ωpL/c=4 π, γ/ωp =0.01, θ=π/3, and M=1; black solid line - n=0, red dashed line - n=−1, blue thick dotted line - n=−2, green dashed-dotted line - n=−3, blue solid line - n=−4, orange thin dotted line - sum of all the propagating orders.

Fig. 4.
Fig. 4.

Same as Fig. 3, except for A=0.85 and B=0.15 in the U case, C=D=0.15 in the L case, and a/h=0.3 and b/h=0.7.

Fig. 5.
Fig. 5.

Same as Fig. 3(a), except for A=0.7, B=0.1, C=0.2, D=0, and h/L=1.2; θ=π/6 in plot (a) and θ=π/3 in plot (b).

Fig. 6.
Fig. 6.

Transmittance for two-layer gratings in the U case at θ=π/3, a/h=0.4, b/h=0.6, ωpL/c=4π, and γ/ωp =0.01; plot (a) - A=1-B, B=0.0833, M=2, ε U=2.1, h/L=1.2; plot (b) - lamellar bars of the height 0.6L, width 0.5L, and permittivity ε bar=2.1, which are placed on the metallic layer, M=1 and h/L=1; black solid line - n=0 (t 0≈0), red dashed line - n=−1, blue thick dotted line - n=−2, green dashed-dotted line - n=−3, blue solid line - n=−4, orange thin dotted line - sum of all the propagating orders.

Fig. 7.
Fig. 7.

Transmittance at the same parameters as in Fig. 4, except for ωpL/c=2 π : plot (a) - frequency dependence in the U case at θ=π/3, plot (b) - angle dependence in the U case at kL=10, plot (c) - angle dependence in the L case at kL=10; black solid line - n=0, red dashed line - n=−1, violet dashed line - n=1, blue dotted line - n=−2, and orange dotted line - sum of all the propagating orders.

Fig. 8.
Fig. 8.

Angle dependence of reflectance in the L case at the same parameters as in Fig. 4, except for ωpL/c=2 π : plot (a) - kL=8.25, plot (b) - kL=9, and plot (c) - kL=10; black solid line ≃ n=0, red dashed line ≃ n=−1, violet dashed line - n=1, and blue dotted line ≃ n=−2.

Fig. 9.
Fig. 9.

Angle dependence of transmittance at the same parameters as in Fig. 3 except for ωp L/c=2 π : plot (a) - U case, kL=9, plot (b) - L case, kL=9, plot (c) - U case, kL=10; black solid line ≃ n=0, red dashed line - n=−1, violet dashed line - n=1, blue dotted line - n=−2, and orange dotted line - sum of all the propagating orders.

Fig. 10.
Fig. 10.

Transmittance for purely metallic thin gratings in the U case at ωpL/c=4 π, γ/ωp =0.01, and θ=π/6; plot (a) - A=0.3, B=0.1, and h/L=0.8; plot (b) - A=0.35, B=0.15, and h/L=1; C=D=0 and M=1; black solid line ≃ n=0, red dashed line - n=−1, blue thick dotted line - n=−2, green dashed-dotted line - n=−3, and orange thin dotted line - sum of all the propagating orders.

Equations (9)

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f1 (x) =A+B cos (2πxML) and f2 (x) =C+D cos (2πxML) .
εm(ω)=1ωp2/[ω(ω+iγ)],
tn =τnτn* Re βn W and rn =ρnρn* Re βn W ,
κn L =χ2[2πnsinθ+(2πn)2+(χωpLc)2] ,
ψn =π 2 and φn =sin1 (ΘnkL)
ψn = sin1 (ΩnkL) and φn = π 2
κ0 θωp >κnθωp
ωp L c =π cot θ ,
ωp L c=2πΦ(θ) sin 2 θ ,

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