Abstract

Guided-mode resonant grating filters are dispersive devices that utilize resonance anomalies. When they are modeled as finite imperfect periodic structures, it is time-consuming to calculate their optical properties precisely, and huge computational memories are needed to accommodate the large analytical domain. This limits the numerical performance, and so existing reports, which use the conventional boundary element method, refer to structures with less than 100 periods. This paper shows that general optical properties and the impact of production error distributions can be calculated for guided-mode resonant grating filters with several hundred periods using the fast multipole boundary element method.

© 2010 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. D. Poljak and C. A. Brebbia, Boundary Element Methods For Electrical Engineers (WIT, 2005).
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    [CrossRef]
  11. S. D. Wu and E. N. Glytsis, “Finite-number-of-periods holographic gratings with finite-width incident beams: analysis using the finite-difference frequency-domain method,” J. Opt. Soc. Am. A 19, 2018–2029 (2002).
    [CrossRef]
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  19. O. M. Mendez and J. Sumaya-Martinez, “Diffraction by Gaussian and Hermite–Gaussian beams by finite gratings,” J. Opt. Soc. Am. A 18, 537–545 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  22. K. Hirayama, K. Igarashi, and Y. Hayashi, “Finite-substrate-thickness cylindrical diffractive lenses,” J. Opt. Soc. Am. A 16, 1294–1302 (1999).
    [CrossRef]
  23. T. Kojima and J. Ido, “Boundary-element method analysis of light-beam,” Electron. Commun. Jpn., Part 2: Electron. 74, 11–20 (1991).
    [CrossRef]
  24. K. Hirayama, E. N. Glytsis, and T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907–917 (1997).
    [CrossRef]
  25. J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, and A. F. Peterson, “Modeling considerations for rigorous boundary element method analysis of diffractive optical elements,” J. Opt. Soc. Am. A 18, 1495–1506 (2001).
    [CrossRef]
  26. J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, and D. L. Brundrett, “Guided-mode resonance subwavelength gratings: effect of finite beams and finite gratings,” J. Opt. Soc. Am. A 18, 1912–1928 (2001).
    [CrossRef]
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    [CrossRef]
  29. V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys. 86, 414–439 (1990).
    [CrossRef]
  30. N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
    [CrossRef]
  31. C. C. Lu and W. C. Chew, “A multilevel algorithm for solving a boundary integral-equation of wave scattering,” Microwave Opt. Technol. Lett. 7, 466–470 (1994).
    [CrossRef]
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    [CrossRef]
  35. N. Nakashima and M. Tateiba, “Greengard–Rokhlin’s fast multipole algorithm for numerical calculation of scattering by N conducting circular cylinders,” IEICE Trans. Electron. E86-C, 2158–2166 (2003).
  36. M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, “Grating coupler for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
    [CrossRef]
  37. T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
    [CrossRef]

2008 (1)

2006 (1)

2005 (2)

S. Benerjee, J. B. Cole, and T. Yatagai, “Calculation of diffraction characteristics of subwavelength conducting gratings using a high accuracy nonstandard finite-difference time-domain method,” Opt. Rev. 12, 274–280 (2005).
[CrossRef]

N. Nakashima and M. Tateiba, “Computational and memory complexities of Greengard–Rokhlin’s fast multipole algorithm,” IEICE Trans. Electron. E88-C, 1516–1520 (2005).
[CrossRef]

2004 (2)

2003 (1)

N. Nakashima and M. Tateiba, “Greengard–Rokhlin’s fast multipole algorithm for numerical calculation of scattering by N conducting circular cylinders,” IEICE Trans. Electron. E86-C, 2158–2166 (2003).

2002 (1)

2001 (3)

1999 (1)

1997 (2)

1996 (2)

1994 (2)

E. E. Kriezis, P. K. Pandelakis, and A. G. Papagiannakis, “Diffraction of a Gaussian beam from a periodic planar screen,” J. Opt. Soc. Am. A 11, 630–636 (1994).
[CrossRef]

C. C. Lu and W. C. Chew, “A multilevel algorithm for solving a boundary integral-equation of wave scattering,” Microwave Opt. Technol. Lett. 7, 466–470 (1994).
[CrossRef]

1993 (2)

1992 (2)

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, and M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

1991 (1)

T. Kojima and J. Ido, “Boundary-element method analysis of light-beam,” Electron. Commun. Jpn., Part 2: Electron. 74, 11–20 (1991).
[CrossRef]

1990 (1)

V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys. 86, 414–439 (1990).
[CrossRef]

1987 (1)

L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys. 73, 325–348 (1987).
[CrossRef]

1986 (1)

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

1985 (1)

L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

1981 (1)

1977 (1)

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

1975 (1)

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

1970 (1)

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, “Grating coupler for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), p. 1046.

Bendickson, J. M.

Benerjee, S.

S. Benerjee, J. B. Cole, and T. Yatagai, “Calculation of diffraction characteristics of subwavelength conducting gratings using a high accuracy nonstandard finite-difference time-domain method,” Opt. Rev. 12, 274–280 (2005).
[CrossRef]

Bertoni, H. L.

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Brebbia, C. A.

D. Poljak and C. A. Brebbia, Boundary Element Methods For Electrical Engineers (WIT, 2005).

Brundrett, D. L.

Chew, W. C.

C. C. Lu and W. C. Chew, “A multilevel algorithm for solving a boundary integral-equation of wave scattering,” Microwave Opt. Technol. Lett. 7, 466–470 (1994).
[CrossRef]

Cole, J. B.

S. Benerjee, J. B. Cole, and T. Yatagai, “Calculation of diffraction characteristics of subwavelength conducting gratings using a high accuracy nonstandard finite-difference time-domain method,” Opt. Rev. 12, 274–280 (2005).
[CrossRef]

Dakss, M. L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, “Grating coupler for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Engheta, N.

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

Ferrari, R. L.

P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers, 3rd ed. (Cambridge Univ. Press, 1996).

Gaylord, T. K.

Glytsis, E. N.

Greengard, L.

L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys. 73, 325–348 (1987).
[CrossRef]

Hadley, G. R.

Haggans, C. W.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Hayashi, Y.

Heidrich, P. F.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, “Grating coupler for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Hirayama, K.

Ido, J.

T. Kojima and J. Ido, “Boundary-element method analysis of light-beam,” Electron. Commun. Jpn., Part 2: Electron. 74, 11–20 (1991).
[CrossRef]

Igarashi, K.

Jiang, J.

Kemme, S. A.

Kincaid, D. R.

D. R. Kincaid, Iterative Methods for Large Linear Systems (Academic, 1989).

Kojima, T.

T. Kojima and J. Ido, “Boundary-element method analysis of light-beam,” Electron. Commun. Jpn., Part 2: Electron. 74, 11–20 (1991).
[CrossRef]

Kok, Y. L.

Kriezis, E. E.

Kuhn, L.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, “Grating coupler for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Li, L.

Lu, C. C.

C. C. Lu and W. C. Chew, “A multilevel algorithm for solving a boundary integral-equation of wave scattering,” Microwave Opt. Technol. Lett. 7, 466–470 (1994).
[CrossRef]

Manara, G.

Martinez, J. S.

Mashev, L.

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

Maystre, D.

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Mendez, O. M.

Miller, J. M.

Moharam, M. G.

Murphy, W. D.

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

Nakashima, N.

N. Nakashima and M. Tateiba, “Computational and memory complexities of Greengard–Rokhlin’s fast multipole algorithm,” IEICE Trans. Electron. E88-C, 1516–1520 (2005).
[CrossRef]

N. Nakashima and M. Tateiba, “Greengard–Rokhlin’s fast multipole algorithm for numerical calculation of scattering by N conducting circular cylinders,” IEICE Trans. Electron. E86-C, 2158–2166 (2003).

Nevier, M.

M. Nevier, “The homogenous problem,” in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980), Chap. 5.

Noponen, E.

Nordin, G. P.

Pandelakis, P. K.

Papadopoulos, A. D.

Papagiannakis, A. G.

Pelosi, G.

Peng, S. T.

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Peters, D. W.

Peterson, A. F.

Poljak, D.

D. Poljak and C. A. Brebbia, Boundary Element Methods For Electrical Engineers (WIT, 2005).

Popov, E.

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

Rokhlin, V.

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys. 86, 414–439 (1990).
[CrossRef]

L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys. 73, 325–348 (1987).
[CrossRef]

Scott, B. A.

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, “Grating coupler for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Silvester, P. P.

P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers, 3rd ed. (Cambridge Univ. Press, 1996).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), p. 1046.

Sumaya-Martinez, J.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

Taghizadeh, M. R.

Tamir, T.

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Tateiba, M.

N. Nakashima and M. Tateiba, “Computational and memory complexities of Greengard–Rokhlin’s fast multipole algorithm,” IEICE Trans. Electron. E88-C, 1516–1520 (2005).
[CrossRef]

N. Nakashima and M. Tateiba, “Greengard–Rokhlin’s fast multipole algorithm for numerical calculation of scattering by N conducting circular cylinders,” IEICE Trans. Electron. E86-C, 2158–2166 (2003).

Toso, G.

Turunen, J.

Vasara, A.

Vassiliou, M. S.

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

Wang, B.

Wu, S. D.

Yatagai, T.

S. Benerjee, J. B. Cole, and T. Yatagai, “Calculation of diffraction characteristics of subwavelength conducting gratings using a high accuracy nonstandard finite-difference time-domain method,” Opt. Rev. 12, 274–280 (2005).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. (1)

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

Appl. Phys. Lett. (1)

M. L. Dakss, L. Kuhn, P. F. Heidrich, and B. A. Scott, “Grating coupler for efficient excitation of optical guided waves in thin films,” Appl. Phys. Lett. 16, 523–525 (1970).
[CrossRef]

Electron. Commun. Jpn., Part 2: Electron. (1)

T. Kojima and J. Ido, “Boundary-element method analysis of light-beam,” Electron. Commun. Jpn., Part 2: Electron. 74, 11–20 (1991).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

IEICE Trans. Electron. (2)

N. Nakashima and M. Tateiba, “Computational and memory complexities of Greengard–Rokhlin’s fast multipole algorithm,” IEICE Trans. Electron. E88-C, 1516–1520 (2005).
[CrossRef]

N. Nakashima and M. Tateiba, “Greengard–Rokhlin’s fast multipole algorithm for numerical calculation of scattering by N conducting circular cylinders,” IEICE Trans. Electron. E86-C, 2158–2166 (2003).

J. Comput. Phys. (2)

L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys. 73, 325–348 (1987).
[CrossRef]

V. Rokhlin, “Rapid solution of integral equations of scattering theory in two dimensions,” J. Comput. Phys. 86, 414–439 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (13)

O. M. Mendez and J. Sumaya-Martinez, “Diffraction by Gaussian and Hermite–Gaussian beams by finite gratings,” J. Opt. Soc. Am. A 18, 537–545 (2001).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, and M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, and A. F. Peterson, “Modeling considerations for rigorous boundary element method analysis of diffractive optical elements,” J. Opt. Soc. Am. A 18, 1495–1506 (2001).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, and D. L. Brundrett, “Guided-mode resonance subwavelength gratings: effect of finite beams and finite gratings,” J. Opt. Soc. Am. A 18, 1912–1928 (2001).
[CrossRef]

S. D. Wu and E. N. Glytsis, “Finite-number-of-periods holographic gratings with finite-width incident beams: analysis using the finite-difference frequency-domain method,” J. Opt. Soc. Am. A 19, 2018–2029 (2002).
[CrossRef]

D. W. Peters, S. A. Kemme, and G. R. Hadley, “Effect of finite grating, waveguide width, and end-facet geometry on resonant subwavelength grating reflectivity,” J. Opt. Soc. Am. A 21, 981–987 (2004).
[CrossRef]

E. E. Kriezis, P. K. Pandelakis, and A. G. Papagiannakis, “Diffraction of a Gaussian beam from a periodic planar screen,” J. Opt. Soc. Am. A 11, 630–636 (1994).
[CrossRef]

K. Hirayama, K. Igarashi, and Y. Hayashi, “Finite-substrate-thickness cylindrical diffractive lenses,” J. Opt. Soc. Am. A 16, 1294–1302 (1999).
[CrossRef]

K. Hirayama, E. N. Glytsis, and T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907–917 (1997).
[CrossRef]

O. M. Mendez and J. S. Martinez, “Scattering of TE-polarized waves by a finite-grating: giant resonant enhancement of the electric field within the grooves,” J. Opt. Soc. Am. A 14, 2203–2211 (1997).
[CrossRef]

L. Li and C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
[CrossRef]

G. Pelosi, G. Manara, and G. Toso, “Heuristic diffraction coefficient for plane-wave scattering form edges in periodic planar surface,” J. Opt. Soc. Am. A 13, 1689–1697 (1996).
[CrossRef]

K. Hirayama, E. N. Glytsis, and T. K. Gaylord, “Rigorous electromagnetic analysis of diffractive cylindrical lenses,” J. Opt. Soc. Am. A 13, 2219–2231 (1996).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

C. C. Lu and W. C. Chew, “A multilevel algorithm for solving a boundary integral-equation of wave scattering,” Microwave Opt. Technol. Lett. 7, 466–470 (1994).
[CrossRef]

Opt. Acta (1)

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Opt. Commun. (1)

L. Mashev and E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

Opt. Express (1)

Opt. Rev. (1)

S. Benerjee, J. B. Cole, and T. Yatagai, “Calculation of diffraction characteristics of subwavelength conducting gratings using a high accuracy nonstandard finite-difference time-domain method,” Opt. Rev. 12, 274–280 (2005).
[CrossRef]

Other (6)

D. Poljak and C. A. Brebbia, Boundary Element Methods For Electrical Engineers (WIT, 2005).

M. Nevier, “The homogenous problem,” in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980), Chap. 5.

P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers, 3rd ed. (Cambridge Univ. Press, 1996).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1964), p. 1046.

D. R. Kincaid, Iterative Methods for Large Linear Systems (Academic, 1989).

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Figures (20)

Fig. 1
Fig. 1

Modified path from r m to r l in the FM-BEM.

Fig. 2
Fig. 2

Radiation path in the conventional BEM.

Fig. 3
Fig. 3

Radiation path in the FM-BEM.

Fig. 4
Fig. 4

Geometry of the device.

Fig. 5
Fig. 5

Element division on the boundary.

Fig. 6
Fig. 6

Reflection spectrum calculated with the RCWA.

Fig. 7
Fig. 7

Scattering amplitude on end-coupling.

Fig. 8
Fig. 8

Scattering cross section on end-coupling at lower zeroth.

Fig. 9
Fig. 9

Scattering amplitude on surface coupling.

Fig. 10
Fig. 10

Scattering cross section on surface coupling at lower zeroth.

Fig. 11
Fig. 11

Peak intensity [peak( )] and FWHM [width( )]; “s” and “e” in parentheses denote “surface coupling” and “end-coupling,” respectively.

Fig. 12
Fig. 12

Peak intensity ratio, peak(s)/peak(e), in Fig. 11.

Fig. 13
Fig. 13

Definition of grating dimension, a lattice period Λ ( n G ) , and a ridge width Λ 1 ( n G ) .

Fig. 14
Fig. 14

Schematic diagram of fluctuated structure given by the normal distribution with 8% standard deviation for both the lattice constant Λ ( = 416.7   nm ) and the ridge width Λ 1 ( = 416.7 × 0.5   nm ) .

Fig. 15
Fig. 15

Scattering amplitudes on end-coupling around (a) upper zeroth (85.882°), (b) upper 1 st (150.269°), (c) lower 1 st (209.772°), and (d) lower zeroth (274.118°) from the device with variable σ ( Δ Λ 1 ( n G ) ) , 0% (solid line) and 8% (open square), of Λ 1 .

Fig. 16
Fig. 16

Scattering amplitudes on surface coupling around (a) upper zeroth (85.882°), (b) upper 1 st (150.269°), (c) lower 1 st (209.772°), and (d) lower zeroth (274.118°) from the device with variable σ ( Δ Λ 1 ( n G ) ) , 0% (solid line) and 8% (open square), of Λ 1 .

Fig. 17
Fig. 17

Scattering amplitudes on end-coupling around at (a) upper zeroth (85.882°), (b) upper 1 st (150.269°), (c) lower 1 st (209.772°), and (d) lower zeroth (274.118°) from the device with variable σ ( Δ Λ ( n G ) ) taking some percentages of Λ [0% (thick solid line), 1% (thin solid line), 2% (thick dotted line), 4% (thin dotted line), and 8% (meshed solid line or open square)].

Fig. 18
Fig. 18

Scattering amplitudes on surface coupling around (a) upper zeroth (85.882°), (b) upper 1 st (150.269°), (c) lower 1 st (209.772°), and (d) lower zeroth (274.118°) from the device with variable σ ( Δ Λ ( n G ) ) taking some percentages of Λ [0% (thick solid line), 1% (thin solid line), 2% (thick dotted line), 4% (thin dotted line), and 8% (meshed solid line or open square)].

Fig. 19
Fig. 19

Scattering amplitudes on end-coupling around at (a) upper zeroth (85.882°), (b) upper 1 st (150.269°), (c) lower 1 st (209.772°), and (d) lower zeroth (274.118°) from the device with variable σ ( Δ Λ ( n G ) ) and σ ( Δ Λ 1 ( n G ) ) taking identical percentages of Λ and of Λ 1 , respectively [0% (thick solid line), 1% (thin solid line), 2% (thick dotted line), 4% (thin dotted line), and 8% (meshed solid line or open square)].

Fig. 20
Fig. 20

Scattering amplitudes on surface coupling around (a) upper zeroth (85.882°), (b) upper 1 st (150.269°), (c) lower 1 st (209.772°), and (d) lower zeroth (274.118°) from the device with variables σ ( Δ Λ ( n G ) ) and σ ( Δ Λ 1 ( n G ) ) taking identical percentages of Λ and of Λ 1 , respectively [0% (thick solid line), 1% (thin solid line), 2% (thick dotted line), 4% (thin dotted line), and 8% (meshed solid line or open square)].

Tables (1)

Tables Icon

Table 1 Scattering Peak Angles on End-Coupling [ ϕ l 0 dif ( e ) ] , the Deduced Incident Angle ( ϕ inc ) Derived from Them, and Scattering Peak Angles on Surface Coupling [ ϕ l 0 dif ( s ) ] , Dependent on the Number of Grooves ( N G )

Equations (17)

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Ψ ( r j ) 2 = Ψ inc ( r j ) i 4 Ω d s ( Ψ ( r i ) H 0 ( 2 ) ( k 0 R ) n Ψ ( r i ) n H 0 ( 2 ) ( k 0 R ) ) ,
Ψ inc ( r j ) = Ω d s ( Ψ ¯ ( r i ) H 0 ( 2 ) ( k 0 R ) ) ,
Ψ ¯ ( r i ) = ( i / 4 ) ( Ψ ( r i ) / n ) .
Ω = Ω 1 + Ω 1 + + Ω M ,
| Ψ inc ( r ) = A | Ψ ¯ ( r ) ,
H 0 ( 2 ) ( k 0 | r j r i | ) = p = H p ( 2 ) ( k 0 | r j r m | ) J p ( k 0 | r m r i | ) exp ( i p α m , i ) .
H p ( 2 ) ( k 0 | r j r m | ) = exp ( i p β l , m ) q = J q ( k 0 | r j r l | ) H p + q ( 2 ) ( k 0 | r l r m | ) exp ( i q α l , m ) .
i = i 1 i N H 0 ( 2 ) ( k 0 | r j r i | ) Ψ ¯ ( r i ) = p = H p ( 2 ) ( k 0 | r j r m | ) i = i 1 i N J p ( k 0 | r m r i | ) exp ( i p α m , i ) Ψ ¯ ( r i ) = p = M m p H p ( 2 ) ( k 0 | r j r m | ) exp ( i p θ j , m ) ,
M m p = i = i 1 i N J p ( k 0 | r m r i | ) exp ( i p ϕ m , i ) Ψ ¯ ( r i ) ,
θ j , m = tan 1 [ ( r m r j ) y ( r m r j ) x ] ,
ϕ m , i = tan 1 [ ( r m r i ) y ( r m r i ) x ] .
i = i 1 i N H 0 ( 2 ) ( k 0 | r j r i | ) Ψ ( r i ) = q = L l q J q ( k 0 | r j r l | ) exp ( i q ϕ j , l ) ,
L l q = p = ( 1 ) p M m p H p + q ( 2 ) ( k 0 | r l r m | ) exp { i ( q + p ) θ l , m } .
ϕ inc = 90 ( ϕ l 0 dif ( e ) 270 ) = 360 ϕ l 0 dif ( e ) [ deg ] .
Λ ( n G ) = Λ + Δ Λ ( n G ) ,
Λ 1 ( n G ) = Λ 1 + Δ Λ 1 ( n G ) ,
ε eff = f ε 1 + ( 1 f ) ε 2 ,

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