Abstract

Using an integral equation approach based on the Rayleigh hypothesis, we investigate the scattering of a plane wave at the rough surface of a metamaterial with a finite number of sinusoidal grooves. To show the adequacy of the model, we present results that are in agreement with the predictions of physical optics and that quantitatively reproduce the polarization and angular dependences predicted by the C-formalism for metamaterial gratings with an infinite number of grooves.

© 2012 Optical Society of America

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References

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  1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  2. V. Grünhut and R. A. Depine, “Influence of the sign of the refractive index in the reflectivity of a metamaterial with roughness,” Eur. Phys. J. D 62, 227–236 (2011).
    [CrossRef]
  3. R. Paniagua-Domínguez, J. A. Sánchez-Gil, P. Albella, J. M. Sáiz, F. González, and F. Moreno, “Enhanced backscattering of electromagnetic waves from randomly rough gratings on negative magnetic metamaterials,” Metamaterials 4, 201–206 (2010).
  4. A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
    [CrossRef]
  5. A. V. Tishchenko, “Rayleigh was right: Electromagnetic fields and corrugated interfaces,” Opt. Photon. News 21(7), 50–54 (2010).
    [CrossRef]
  6. R. A. Depine and A. Lakhtakia, “Perturbative approach for diffraction due to a periodically corrugated boundary between vacuum and a negative phase-velocity material,” Opt. Commun. 233, 277–282 (2004).
    [CrossRef]
  7. R. A. Depine and A. Lakhtakia, “Plane-wave diffraction at the periodically corrugated boundary of vacuum and a negative-phase-velocity material,” Phys. Rev. E 69, 057602 (2004).
    [CrossRef]
  8. R. A. Depine and A. Lakhtakia, “Diffraction gratings of isotropic negative-phase velocity materials,” Optik 116, 31–43 (2005).
    [CrossRef]
  9. J. P. Hugonin and R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977).
    [CrossRef]
  10. D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. (Paris) 15, 43–51 (1984).
    [CrossRef]
  11. R. A. Depine, C. E. Gerber, and V. L. Brudny, “Lossy gratings with a finite number of grooves: a canonical model,” J. Opt. Soc. Am. A 9, 573–577 (1992).
    [CrossRef]
  12. A. Benali, J. Chandezon, and J. Fontaine, “A new theory for scattering of electromagnetic waves from conducting or dielectric rough surfaces,” IEEE Trans. Antennas Propag. 40, 141–148 (1992).
    [CrossRef]
  13. T. J. Park, H. J. Eom, and K. Yoshitomi, “Analysis of TM scattering from finite rectangular grooves in a conducting plane,” J. Opt. Soc. Am. A 10, 905–911 (1993).
    [CrossRef]
  14. J. Nakayama and Y. Tamura, “Low grazing scattering from sinusoidal Neumann surface with finite extent: undersampling approximation,” IEICE Trans. Electron 91, 9–16 (2008).
  15. L. Li, J. Chandezon, G. Granet, and J. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
    [CrossRef]
  16. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
    [CrossRef]
  17. L. Li and J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247–2255 (1996).
    [CrossRef]
  18. R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microw. Opt. Technol. Lett. 41, 315–316 (2004).
    [CrossRef]
  19. M.-J. Kim, “Verification of the reciprocity theorem,” Appl. Opt. 27, 2645–2646 (1988).
    [CrossRef]
  20. M. Cuevas and R. A. Depine, “Excitation of surface plasmon polaritons along the sinusoidal boundary of a metamaterial,” Phys. Rev. B 78, 125412 (2008).
    [CrossRef]
  21. M. Cuevas and R. A. Depine, “Radiation characteristics of electromagnetic eigenmodes at the corrugated interface of a left-handed material,” Phys. Rev. Lett. 103, 097401 (2009).
    [CrossRef]
  22. M. Cuevas and R. A. Depine, “Surface plasmon polariton modes propagating along the periodically corrugated boundary of a metamaterial,” Eur. Phys. J. D 58, 249–255 (2010).
    [CrossRef]
  23. M. Cuevas and R. A. Depine, “Dispersion characteristics of surface polaritons on left-handed gratings,” Opt. Commun. 284, 5242–5247 (2011).
    [CrossRef]

2011 (2)

V. Grünhut and R. A. Depine, “Influence of the sign of the refractive index in the reflectivity of a metamaterial with roughness,” Eur. Phys. J. D 62, 227–236 (2011).
[CrossRef]

M. Cuevas and R. A. Depine, “Dispersion characteristics of surface polaritons on left-handed gratings,” Opt. Commun. 284, 5242–5247 (2011).
[CrossRef]

2010 (3)

R. Paniagua-Domínguez, J. A. Sánchez-Gil, P. Albella, J. M. Sáiz, F. González, and F. Moreno, “Enhanced backscattering of electromagnetic waves from randomly rough gratings on negative magnetic metamaterials,” Metamaterials 4, 201–206 (2010).

A. V. Tishchenko, “Rayleigh was right: Electromagnetic fields and corrugated interfaces,” Opt. Photon. News 21(7), 50–54 (2010).
[CrossRef]

M. Cuevas and R. A. Depine, “Surface plasmon polariton modes propagating along the periodically corrugated boundary of a metamaterial,” Eur. Phys. J. D 58, 249–255 (2010).
[CrossRef]

2009 (1)

M. Cuevas and R. A. Depine, “Radiation characteristics of electromagnetic eigenmodes at the corrugated interface of a left-handed material,” Phys. Rev. Lett. 103, 097401 (2009).
[CrossRef]

2008 (2)

M. Cuevas and R. A. Depine, “Excitation of surface plasmon polaritons along the sinusoidal boundary of a metamaterial,” Phys. Rev. B 78, 125412 (2008).
[CrossRef]

J. Nakayama and Y. Tamura, “Low grazing scattering from sinusoidal Neumann surface with finite extent: undersampling approximation,” IEICE Trans. Electron 91, 9–16 (2008).

2005 (1)

R. A. Depine and A. Lakhtakia, “Diffraction gratings of isotropic negative-phase velocity materials,” Optik 116, 31–43 (2005).
[CrossRef]

2004 (3)

R. A. Depine and A. Lakhtakia, “Perturbative approach for diffraction due to a periodically corrugated boundary between vacuum and a negative phase-velocity material,” Opt. Commun. 233, 277–282 (2004).
[CrossRef]

R. A. Depine and A. Lakhtakia, “Plane-wave diffraction at the periodically corrugated boundary of vacuum and a negative-phase-velocity material,” Phys. Rev. E 69, 057602 (2004).
[CrossRef]

R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microw. Opt. Technol. Lett. 41, 315–316 (2004).
[CrossRef]

1999 (1)

1996 (1)

1993 (1)

1992 (2)

A. Benali, J. Chandezon, and J. Fontaine, “A new theory for scattering of electromagnetic waves from conducting or dielectric rough surfaces,” IEEE Trans. Antennas Propag. 40, 141–148 (1992).
[CrossRef]

R. A. Depine, C. E. Gerber, and V. L. Brudny, “Lossy gratings with a finite number of grooves: a canonical model,” J. Opt. Soc. Am. A 9, 573–577 (1992).
[CrossRef]

1990 (1)

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

1988 (1)

1984 (1)

D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. (Paris) 15, 43–51 (1984).
[CrossRef]

1980 (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

1977 (1)

J. P. Hugonin and R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977).
[CrossRef]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Albella, P.

R. Paniagua-Domínguez, J. A. Sánchez-Gil, P. Albella, J. M. Sáiz, F. González, and F. Moreno, “Enhanced backscattering of electromagnetic waves from randomly rough gratings on negative magnetic metamaterials,” Metamaterials 4, 201–206 (2010).

Benali, A.

A. Benali, J. Chandezon, and J. Fontaine, “A new theory for scattering of electromagnetic waves from conducting or dielectric rough surfaces,” IEEE Trans. Antennas Propag. 40, 141–148 (1992).
[CrossRef]

Brudny, V. L.

Chandezon, J.

L. Li, J. Chandezon, G. Granet, and J. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
[CrossRef]

L. Li and J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247–2255 (1996).
[CrossRef]

A. Benali, J. Chandezon, and J. Fontaine, “A new theory for scattering of electromagnetic waves from conducting or dielectric rough surfaces,” IEEE Trans. Antennas Propag. 40, 141–148 (1992).
[CrossRef]

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Cuevas, M.

M. Cuevas and R. A. Depine, “Dispersion characteristics of surface polaritons on left-handed gratings,” Opt. Commun. 284, 5242–5247 (2011).
[CrossRef]

M. Cuevas and R. A. Depine, “Surface plasmon polariton modes propagating along the periodically corrugated boundary of a metamaterial,” Eur. Phys. J. D 58, 249–255 (2010).
[CrossRef]

M. Cuevas and R. A. Depine, “Radiation characteristics of electromagnetic eigenmodes at the corrugated interface of a left-handed material,” Phys. Rev. Lett. 103, 097401 (2009).
[CrossRef]

M. Cuevas and R. A. Depine, “Excitation of surface plasmon polaritons along the sinusoidal boundary of a metamaterial,” Phys. Rev. B 78, 125412 (2008).
[CrossRef]

Depine, R. A.

M. Cuevas and R. A. Depine, “Dispersion characteristics of surface polaritons on left-handed gratings,” Opt. Commun. 284, 5242–5247 (2011).
[CrossRef]

V. Grünhut and R. A. Depine, “Influence of the sign of the refractive index in the reflectivity of a metamaterial with roughness,” Eur. Phys. J. D 62, 227–236 (2011).
[CrossRef]

M. Cuevas and R. A. Depine, “Surface plasmon polariton modes propagating along the periodically corrugated boundary of a metamaterial,” Eur. Phys. J. D 58, 249–255 (2010).
[CrossRef]

M. Cuevas and R. A. Depine, “Radiation characteristics of electromagnetic eigenmodes at the corrugated interface of a left-handed material,” Phys. Rev. Lett. 103, 097401 (2009).
[CrossRef]

M. Cuevas and R. A. Depine, “Excitation of surface plasmon polaritons along the sinusoidal boundary of a metamaterial,” Phys. Rev. B 78, 125412 (2008).
[CrossRef]

R. A. Depine and A. Lakhtakia, “Diffraction gratings of isotropic negative-phase velocity materials,” Optik 116, 31–43 (2005).
[CrossRef]

R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microw. Opt. Technol. Lett. 41, 315–316 (2004).
[CrossRef]

R. A. Depine and A. Lakhtakia, “Plane-wave diffraction at the periodically corrugated boundary of vacuum and a negative-phase-velocity material,” Phys. Rev. E 69, 057602 (2004).
[CrossRef]

R. A. Depine and A. Lakhtakia, “Perturbative approach for diffraction due to a periodically corrugated boundary between vacuum and a negative phase-velocity material,” Opt. Commun. 233, 277–282 (2004).
[CrossRef]

R. A. Depine, C. E. Gerber, and V. L. Brudny, “Lossy gratings with a finite number of grooves: a canonical model,” J. Opt. Soc. Am. A 9, 573–577 (1992).
[CrossRef]

Eom, H. J.

Fontaine, J.

A. Benali, J. Chandezon, and J. Fontaine, “A new theory for scattering of electromagnetic waves from conducting or dielectric rough surfaces,” IEEE Trans. Antennas Propag. 40, 141–148 (1992).
[CrossRef]

Gerber, C. E.

González, F.

R. Paniagua-Domínguez, J. A. Sánchez-Gil, P. Albella, J. M. Sáiz, F. González, and F. Moreno, “Enhanced backscattering of electromagnetic waves from randomly rough gratings on negative magnetic metamaterials,” Metamaterials 4, 201–206 (2010).

Granet, G.

Grünhut, V.

V. Grünhut and R. A. Depine, “Influence of the sign of the refractive index in the reflectivity of a metamaterial with roughness,” Eur. Phys. J. D 62, 227–236 (2011).
[CrossRef]

Hugonin, J. P.

J. P. Hugonin and R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977).
[CrossRef]

Kim, M.-J.

Lakhtakia, A.

R. A. Depine and A. Lakhtakia, “Diffraction gratings of isotropic negative-phase velocity materials,” Optik 116, 31–43 (2005).
[CrossRef]

R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microw. Opt. Technol. Lett. 41, 315–316 (2004).
[CrossRef]

R. A. Depine and A. Lakhtakia, “Perturbative approach for diffraction due to a periodically corrugated boundary between vacuum and a negative phase-velocity material,” Opt. Commun. 233, 277–282 (2004).
[CrossRef]

R. A. Depine and A. Lakhtakia, “Plane-wave diffraction at the periodically corrugated boundary of vacuum and a negative-phase-velocity material,” Phys. Rev. E 69, 057602 (2004).
[CrossRef]

Li, L.

Maradudin, A. A.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Maystre, D.

D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. (Paris) 15, 43–51 (1984).
[CrossRef]

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Méndez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Moreno, F.

R. Paniagua-Domínguez, J. A. Sánchez-Gil, P. Albella, J. M. Sáiz, F. González, and F. Moreno, “Enhanced backscattering of electromagnetic waves from randomly rough gratings on negative magnetic metamaterials,” Metamaterials 4, 201–206 (2010).

Nakayama, J.

J. Nakayama and Y. Tamura, “Low grazing scattering from sinusoidal Neumann surface with finite extent: undersampling approximation,” IEICE Trans. Electron 91, 9–16 (2008).

Paniagua-Domínguez, R.

R. Paniagua-Domínguez, J. A. Sánchez-Gil, P. Albella, J. M. Sáiz, F. González, and F. Moreno, “Enhanced backscattering of electromagnetic waves from randomly rough gratings on negative magnetic metamaterials,” Metamaterials 4, 201–206 (2010).

Park, T. J.

Petit, R.

J. P. Hugonin and R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977).
[CrossRef]

Plumey, J.

Raoult, G.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Sáiz, J. M.

R. Paniagua-Domínguez, J. A. Sánchez-Gil, P. Albella, J. M. Sáiz, F. González, and F. Moreno, “Enhanced backscattering of electromagnetic waves from randomly rough gratings on negative magnetic metamaterials,” Metamaterials 4, 201–206 (2010).

Sánchez-Gil, J. A.

R. Paniagua-Domínguez, J. A. Sánchez-Gil, P. Albella, J. M. Sáiz, F. González, and F. Moreno, “Enhanced backscattering of electromagnetic waves from randomly rough gratings on negative magnetic metamaterials,” Metamaterials 4, 201–206 (2010).

Tamura, Y.

J. Nakayama and Y. Tamura, “Low grazing scattering from sinusoidal Neumann surface with finite extent: undersampling approximation,” IEICE Trans. Electron 91, 9–16 (2008).

Tishchenko, A. V.

A. V. Tishchenko, “Rayleigh was right: Electromagnetic fields and corrugated interfaces,” Opt. Photon. News 21(7), 50–54 (2010).
[CrossRef]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Yoshitomi, K.

Ann. Phys. (1)

A. A. Maradudin, T. Michel, A. R. McGurn, and E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Appl. Opt. (2)

Eur. Phys. J. D (2)

V. Grünhut and R. A. Depine, “Influence of the sign of the refractive index in the reflectivity of a metamaterial with roughness,” Eur. Phys. J. D 62, 227–236 (2011).
[CrossRef]

M. Cuevas and R. A. Depine, “Surface plasmon polariton modes propagating along the periodically corrugated boundary of a metamaterial,” Eur. Phys. J. D 58, 249–255 (2010).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

A. Benali, J. Chandezon, and J. Fontaine, “A new theory for scattering of electromagnetic waves from conducting or dielectric rough surfaces,” IEEE Trans. Antennas Propag. 40, 141–148 (1992).
[CrossRef]

IEICE Trans. Electron (1)

J. Nakayama and Y. Tamura, “Low grazing scattering from sinusoidal Neumann surface with finite extent: undersampling approximation,” IEICE Trans. Electron 91, 9–16 (2008).

J. Opt. (Paris) (2)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

D. Maystre, “Rigorous theory of light scattering from rough surfaces,” J. Opt. (Paris) 15, 43–51 (1984).
[CrossRef]

J. Opt. Soc. Am. A (3)

Metamaterials (1)

R. Paniagua-Domínguez, J. A. Sánchez-Gil, P. Albella, J. M. Sáiz, F. González, and F. Moreno, “Enhanced backscattering of electromagnetic waves from randomly rough gratings on negative magnetic metamaterials,” Metamaterials 4, 201–206 (2010).

Microw. Opt. Technol. Lett. (1)

R. A. Depine and A. Lakhtakia, “A new condition to identify isotropic dielectric-magnetic materials displaying negative phase velocity,” Microw. Opt. Technol. Lett. 41, 315–316 (2004).
[CrossRef]

Opt. Commun. (3)

J. P. Hugonin and R. Petit, “A numerical study of the problem of diffraction at a non-periodic obstacle,” Opt. Commun. 20, 360–364 (1977).
[CrossRef]

M. Cuevas and R. A. Depine, “Dispersion characteristics of surface polaritons on left-handed gratings,” Opt. Commun. 284, 5242–5247 (2011).
[CrossRef]

R. A. Depine and A. Lakhtakia, “Perturbative approach for diffraction due to a periodically corrugated boundary between vacuum and a negative phase-velocity material,” Opt. Commun. 233, 277–282 (2004).
[CrossRef]

Opt. Photon. News (1)

A. V. Tishchenko, “Rayleigh was right: Electromagnetic fields and corrugated interfaces,” Opt. Photon. News 21(7), 50–54 (2010).
[CrossRef]

Optik (1)

R. A. Depine and A. Lakhtakia, “Diffraction gratings of isotropic negative-phase velocity materials,” Optik 116, 31–43 (2005).
[CrossRef]

Phys. Rev. B (1)

M. Cuevas and R. A. Depine, “Excitation of surface plasmon polaritons along the sinusoidal boundary of a metamaterial,” Phys. Rev. B 78, 125412 (2008).
[CrossRef]

Phys. Rev. E (1)

R. A. Depine and A. Lakhtakia, “Plane-wave diffraction at the periodically corrugated boundary of vacuum and a negative-phase-velocity material,” Phys. Rev. E 69, 057602 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

M. Cuevas and R. A. Depine, “Radiation characteristics of electromagnetic eigenmodes at the corrugated interface of a left-handed material,” Phys. Rev. Lett. 103, 097401 (2009).
[CrossRef]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Scattering of a plane wave from a metamaterial surface with a finite number of equally spaced identical grooves. θ0 is the angle of incidence and θs is the observation angle.

Fig. 2.
Fig. 2.

Angular distribution of power scattered into medium 1 for three sinusoidal finite gratings with identical geometrical (h/λ=0.02, d/λ=2) and constitutive parameters (ϵ2=4+0.001i and μ2=1.5+0.001i) but different number of grooves (N=3, 9 and 15), illuminated by a linearly polarized plane wave at an angle of incidence θ0=20°. Left-hand column: s polarization; right-hand column: p polarization.

Fig. 3.
Fig. 3.

Angular distribution of power scattered into medium 1 for a finite grating with N=15 sinusoidal grooves, illuminated by linearly polarized plane waves at angles of incidence θ0=57.35° and 9.09°. The geometrical and constitutive parameters are as in Fig. 2. Left-hand column: s polarization; right-hand column: p polarization.

Fig. 4.
Fig. 4.

Efficiency of the n=1 and n=+1 diffracted orders as functions of the angle of incidence θ0 for a perfectly periodic grating with the same geometrical and constitutive parameters as those in Fig. 2 (s polarization).

Fig. 5.
Fig. 5.

Efficiency of the n=1 and n=+1 diffracted orders as functions of the angle of incidence θ0 for a perfectly periodic grating with the same geometrical and constitutive parameters as those in Fig. 2 (p polarization).

Fig. 6.
Fig. 6.

Angular distribution of power scattered into medium 1 for a sinusoidal finite grating with N=15 grooves illuminated by an s-polarized wave at an angle of incidence θ0=28°, the other parameters as in Fig. 2.

Fig. 7.
Fig. 7.

Angular distribution of power scattered into medium 1 for a sinusoidal finite grating with N=15 grooves illuminated by a p-polarized wave at an angle of incidence θ0=28°, the other parameters as in Fig. 2.

Equations (6)

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

Ψ1(x,y)=exp[i(α0xβ0(1)y)]+12π+R(α)exp[i(αx+βα(1)y)]dα
Ψ2(x,y)=12π+T(α)exp[i(αxβα(2)y)]dα
βα(j)=β(j)(α)=(k02ϵjμjα2)1/2,j=1,2,
Pr=Re2πa+βα(1)β0(1)|R(α)|2dα,
dPdα=Re2πaβα(1)β0(1)|R~(α)|2,
sinθs=nλd+sinθ0,

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