Abstract

For analyzing diffraction gratings, a new method is developed based on dividing one period of the grating into homogeneous subdomains and computing the Neumann-to-Dirichlet (NtD) maps for these subdomains by boundary integral equations. For a subdomain, the NtD operator maps the normal derivative of the wave field to the wave field on its boundary. The integral operators used in this method are simple to approximate, since they involve only the standard Green’s function of the Helmholtz equation in homogeneous media. The method retains the advantages of existing boundary integral equation methods for diffraction gratings but avoids the quasi-periodic Green’s functions that are expensive to evaluate.

© 2009 Optical Society of America

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References

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  1. R.Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  2. G.Bao, L.Cowsar, and W.Masters, ed., Mathematical Modeling in Optical Sciences (Society for Industrial and Applied Mathematics, 2001).
    [CrossRef]
  3. M. Nevière and E. Popov, Light Propagation in Periodic Media, Marcel Dekker, 2003.
  4. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
    [CrossRef]
  5. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
    [CrossRef]
  6. L. Li, “Multilayer modal method for diffration gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581-2591 (1993).
    [CrossRef]
  7. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385-1392 (1982).
    [CrossRef]
  8. 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]
  9. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019-1023 (1996).
    [CrossRef]
  10. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  11. E. Popov and M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773-1784 (2000).
    [CrossRef]
  12. E. Popov and M. Nevière, “Maxwell equations in Fourier space: A fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886-2894 (2001).
    [CrossRef]
  13. D. Maystre, “Integral Methods,” in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3.
    [CrossRef]
  14. A. Pomp, “The integral method for coated gratings--computational cost,” J. Mod. Opt. 38, 109-120 (1991).
    [CrossRef]
  15. B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile--heory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
    [CrossRef]
  16. D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34-43 (1997).
    [CrossRef]
  17. E. Popov, B. Bozhkov, D. Maystre, and J. Hoose, “Integral method for echelles covered with lossless or absorbing thin dielectric layers,” Appl. Opt. 38, 47-55 (1999).
    [CrossRef]
  18. 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]
  19. A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984-1009 (2006).
  20. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. 11, 235-241 (1980).
    [CrossRef]
  21. J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings--a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839-846 (1982).
    [CrossRef]
  22. L. Li, J. Chandezon, G. Granet, and J. P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304-313 (1999).
    [CrossRef]
  23. G. Bao, Z. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106-1114 (2005).
    [CrossRef]
  24. E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33-42 (2002).
    [CrossRef]
  25. Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448-3453 (2006).
    [CrossRef]
  26. Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337-349 (2007).
  27. Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice,” J. Opt. Soc. Am. B 25, 1466-1473 (2008).
    [CrossRef]
  28. Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing periodic arrays of cylinders with oblique incident waves,” J. Opt. Soc. Am. B 26, 1442-1449 (2009).
    [CrossRef]
  29. R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
    [CrossRef]
  30. K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603-2611 (2004).
    [CrossRef]
  31. G. Bao, D. C. Dobson, and J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029-1042 (1995).
    [CrossRef]
  32. Y. Y. Lu and J. R. McLaughlin, “The Riccati method for the Helmholtz equation,” J. Acoust. Soc. Am. 100, 1432-1446 (1996).
    [CrossRef]
  33. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, Berlin, 1998).
  34. A. Mizutani, H. Kikuta, K. Iwata, and H. Toyota, “Guided-mode resonant grating filter with an antireflection structure surface,” J. Opt. Soc. Am. A 19, 1346-1351 (2002).
    [CrossRef]

2009 (1)

2008 (1)

2007 (1)

Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337-349 (2007).

2006 (2)

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448-3453 (2006).
[CrossRef]

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984-1009 (2006).

2005 (2)

2004 (1)

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603-2611 (2004).
[CrossRef]

2002 (2)

2001 (1)

2000 (1)

1999 (3)

E. Popov, B. Bozhkov, D. Maystre, and J. Hoose, “Integral method for echelles covered with lossless or absorbing thin dielectric layers,” Appl. Opt. 38, 47-55 (1999).
[CrossRef]

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

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

1997 (1)

1996 (5)

1995 (1)

1993 (1)

1991 (1)

A. Pomp, “The integral method for coated gratings--computational cost,” J. Mod. Opt. 38, 109-120 (1991).
[CrossRef]

1982 (2)

1981 (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

1980 (1)

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

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

Asatryan, A. A.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Bao, G.

Botten, L. C.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

Bozhkov, B.

Chandezon, J.

Chen, Z. M.

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, Berlin, 1998).

Cornet, G.

Cox, J. A.

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

de Sterke, C. M.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Dobson, D. C.

Dupuis, M. T.

Gaylord, T. K.

Gralak, B.

Granet, G.

Guizal, B.

Hoose, J.

Huang, Y.

Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337-349 (2007).

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448-3453 (2006).
[CrossRef]

Iwata, K.

Kikuta, H.

Kleemann, B. H.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984-1009 (2006).

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile--heory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

Kress, R.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, Berlin, 1998).

Kushta, T.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603-2611 (2004).
[CrossRef]

Lalanne, P.

Li, L.

Lu, Y. Y.

Magath, T.

Mait, J. N.

Maystre, D.

E. Popov, B. Bozhkov, D. Maystre, and J. Hoose, “Integral method for echelles covered with lossless or absorbing thin dielectric layers,” Appl. Opt. 38, 47-55 (1999).
[CrossRef]

J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings--a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839-846 (1982).
[CrossRef]

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

D. Maystre, “Integral Methods,” in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3.
[CrossRef]

McLaughlin, J. R.

Y. Y. Lu and J. R. McLaughlin, “The Riccati method for the Helmholtz equation,” J. Acoust. Soc. Am. 100, 1432-1446 (1996).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

Mirotznik, M. S.

Mitreiter, A.

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile--heory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

Mizutani, A.

Moharam, M. G.

Morris, G. M.

Nevière, M.

Nicorovici, N. A.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Plumey, J. P.

Pomp, A.

A. Pomp, “The integral method for coated gratings--computational cost,” J. Mod. Opt. 38, 109-120 (1991).
[CrossRef]

Popov, E.

Prather, D. W.

Raoult, G.

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

Rathsfeld, A.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984-1009 (2006).

Robinson, P. A.

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Schmidt, G.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984-1009 (2006).

Serebryannikov, A. E.

Tayeb, G.

Toyama, H.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603-2611 (2004).
[CrossRef]

Toyota, H.

Wu, H. J.

Wu, Y.

Wyrowski, F.

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile--heory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

Yasumoto, K.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603-2611 (2004).
[CrossRef]

Appl. Opt. (2)

Commun. Comput. Phys. (1)

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984-1009 (2006).

IEEE Trans. Antennas Propag. (1)

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603-2611 (2004).
[CrossRef]

J. Acoust. Soc. Am. (1)

Y. Y. Lu and J. R. McLaughlin, “The Riccati method for the Helmholtz equation,” J. Acoust. Soc. Am. 100, 1432-1446 (1996).
[CrossRef]

J. Comput. Math. (1)

Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337-349 (2007).

J. Lightwave Technol. (1)

J. Mod. Opt. (2)

A. Pomp, “The integral method for coated gratings--computational cost,” J. Mod. Opt. 38, 109-120 (1991).
[CrossRef]

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile--heory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

J. Opt. (1)

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (12)

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]

G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019-1023 (1996).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870-1876 (1996).
[CrossRef]

E. Popov and M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773-1784 (2000).
[CrossRef]

E. Popov and M. Nevière, “Maxwell equations in Fourier space: A fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886-2894 (2001).
[CrossRef]

G. Bao, Z. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106-1114 (2005).
[CrossRef]

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34-43 (1997).
[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]

L. Li, “Multilayer modal method for diffration gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581-2591 (1993).
[CrossRef]

G. Bao, D. C. Dobson, and J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029-1042 (1995).
[CrossRef]

A. Mizutani, H. Kikuta, K. Iwata, and H. Toyota, “Guided-mode resonant grating filter with an antireflection structure surface,” J. Opt. Soc. Am. A 19, 1346-1351 (2002).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Acta (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

Phys. Rev. E (1)

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, and C. M. de Sterke, “Calculation of electromagnetic properties of regular and random arrays of metallic and dielectric cylinders,” Phys. Rev. E 60, 7614-7617 (1999).
[CrossRef]

Other (5)

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, Berlin, 1998).

R.Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

G.Bao, L.Cowsar, and W.Masters, ed., Mathematical Modeling in Optical Sciences (Society for Industrial and Applied Mathematics, 2001).
[CrossRef]

M. Nevière and E. Popov, Light Propagation in Periodic Media, Marcel Dekker, 2003.

D. Maystre, “Integral Methods,” in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3.
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Typical diffraction grating.

Fig. 2
Fig. 2

Rectangular domain Σ for one period of a diffraction grating.

Fig. 3
Fig. 3

(a) Sinusoidal grating with a dielectric substrate; (b) triangular grating with high-index coating layer, (c) sinusoidal grating with a free space bottom.

Fig. 4
Fig. 4

Zeroth-order reflected diffraction efficiencies of a coated triangular grating as functions of the wavelength.

Tables (2)

Tables Icon

Table 1 Comparison of Diffraction Efficiencies for a Sinusoidal Grating

Tables Icon

Table 2 Comparison of Diffraction Efficiencies for a Sinusoidal Grating with Free Space Bottom

Equations (55)

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

ρ x ( 1 ρ u x ) + ρ y ( 1 ρ u y ) + k 0 2 ε u = 0 ,
u ( i ) ( x , y ) = e i ( α 0 x β 0 ( 1 ) y ) , y > D ,
u ( r ) ( x , y ) = j = R j e i ( α j x + β j ( 1 ) y ) , y > D ,
u ( t ) ( x , y ) = j = T j e i ( α j x β j ( 2 ) y ) , y < 0 ,
α j = α 0 + 2 j π L , β j ( s ) = k 0 2 ε ( s ) α j 2 , s = 1 , 2 .
u ( L , y ) = γ u ( 0 , y ) , u x ( L + , y ) = γ u x ( 0 + , y ) ,
B ( s ) e i α j x = β j ( s ) e i α j x , j = 0 , ± 1 , ± 2 ,
u y = i B ( 2 ) u , y = 0 ,
u y = i B ( 1 ) u 2 i β 0 e i α 0 x , y = D + .
| Q j ± u | Γ j = | u ν | Γ j ± , | Y j u | Γ j = | u | Γ 0 ,
Q 0 = i B ( 2 ) , Y 0 = I ,
1 | ρ | Γ j + Q j + = 1 | ρ | Γ j Q j .
[ Q m + i B ( 1 ) ] u ( x , D ) = 2 i β 0 e i α 0 x .
u ( x , 0 ) = Y m u ( x , D )
Λ j u = u ( i n ) ν on Ω j ,
V j [ ν u j 1 + x v j + x w j ν u j ] = [ V 11 ( j ) V 12 ( j ) V 13 ( j ) V 14 ( j ) V 21 ( j ) V 22 ( j ) V 23 ( j ) V 24 ( j ) V 31 ( j ) V 32 ( j ) V 33 ( j ) V 34 ( j ) V 41 ( j ) V 42 ( j ) V 43 ( j ) V 44 ( j ) ] [ ν u j 1 + x v j + x w j ν u j ] = [ u j 1 v j w j u j ] ,
N j [ ν u j 1 + ν u j ] = [ N 11 ( j ) N 12 ( j ) N 21 ( j ) N 22 ( j ) ] [ ν u j 1 + ν u j ] = [ u j 1 u j ] ,
N j = [ V 11 ( j ) V 14 ( j ) V 41 ( j ) V 44 ( j ) ] + [ C 1 D 1 C 1 D 2 C 2 D 1 C 2 D 2 ] ,
C 1 = V 12 ( j ) + γ V 13 ( j ) , C 2 = V 42 ( j ) + γ V 43 ( j ) ,
D 0 = γ V 22 ( j ) + γ 2 V 23 ( j ) V 32 ( j ) γ V 33 ( j ) ,
D 1 = D 0 1 ( V 31 ( j ) γ V 21 ( j ) ) , D 2 = D 0 1 ( V 34 ( j ) γ V 24 ( j ) ) .
( I N 11 Q j 1 + ) u j 1 = N 12 Q j u j ,
N 21 Q j 1 + u j 1 = ( I N 22 Q j ) u j .
Z = ( I N 11 ( j ) Q j 1 + ) 1 N 12 ( j ) ,
Q j = ( N 22 ( j ) + N 21 ( j ) Q j 1 + Z ) 1 ,
Y j = Y j 1 Z Q j .
G ( r , r ̃ ) = i 4 H 0 ( 1 ) ( k 0 n | r r ̃ | ) ,
2 G ( r , r ̃ ) x 2 + 2 G ( r , r ̃ ) y 2 + k 0 2 n 2 G ( r , r ̃ ) = δ ( r r ̃ ) .
( S ϕ ) ( r ) = 2 Ω G ( r , r ̃ ) ϕ ( r ̃ ) d s ( r ̃ ) , r Ω ,
( K ϕ ) ( r ) = 2 Ω G ( r , r ̃ ) ν ( r ̃ ) ϕ ( r ̃ ) d s ( r ̃ ) , r Ω ,
( 1 + K ) u = S u ν .
V = ( 1 + K ) 1 S .
( H ϕ ) ( r ) = 2 Ω G 0 ( r , r ̃ ) ν ( r ̃ ) ϕ ( r ̃ ) d s ( r ̃ ) ,
G 0 ( r , r ̃ ) = 1 2 π ln 1 | r r ̃ | , r r ̃ ;
( 1 + K ) u u ( r 0 ) ( 1 + H 1 ) = S u ν ,
( 1 + K ) u v ( r ) ( 1 + H 1 ) = S u ν ,
v ( r ) = l = 1 P j = 1 , j l P | r r j | | r l r j | u ( r l ) .
r ( t ) = ( x ( t ) , y ( t ) ) , 0 t 2 π .
( S ϕ ) ( t ) = 0 2 π S ( t , τ ) ϕ ( τ ) d τ ,
S ( t , τ ) = S 1 ( t , τ ) ln ( 4 sin 2 t τ 2 ) + S 2 ( t , τ ) .
( S ϕ ) ( t j ) k = 0 J 1 a j k ϕ ( t k )
r l * = ( x ( t l * ) , y ( t l * ) ) , 0 l P ,
d k w d s k = 0 at s = s l 1 * , s l * for k = 1 , 2 , , p 1 .
d w d s = 2 t l * t l 1 * s l * s l 1 * at s = s l 1 * + s l * 2 .
w ( s ) = t l * w 1 p + t l 1 * w 2 p w 1 p + w 2 p for s l 1 * s s l * ,
w 1 = ( 1 2 1 p ) ξ 3 + ξ p + 1 2 , w 2 = 1 w 1 , ξ = 2 s ( s l 1 * + s l * ) s l * s l 1 * .
r = ( x ( w ( s ) ) , y ( w ( s ) ) ) , 0 s 2 π .
ξ l = l 0.5 N 2 L , l = 1 , 2 , , N 2 .
f ( x ) = j f ̂ j e i α j x ,
[ f ( x 1 ) f ( x 2 ) f ( x 3 ) f ( x N 1 ) ] = S [ f ( ξ 1 ) f ( ξ 2 ) f ( ξ N 2 ) ] .
[ f ( ξ 1 ) f ( ξ 2 ) f ( ξ N 2 ) ] S [ f ( x 1 ) f ( x 2 ) f ( x 3 ) f ( x N 1 ) ] ,
B ̃ ( s ) φ j = β j ( s ) φ j , φ j = [ exp ( i α j ξ 1 ) exp ( i α j ξ 2 ) exp ( i α j ξ N 2 ) ]
B ̃ ( s ) = P ̃ D ̃ ( s ) P ̃ 1 ,
B ( 2 ) = S B ̃ ( 2 ) S .
Q ̃ m + = S Q m + S .

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