Abstract

We mathematically prove and numerically demonstrate that the source of the convergence problem of the anal ytical modal method and the Fourier modal method for modeling some lossless metal-dielectric lamellar gratings in TM polarization recently reported by Gundu and Mafi [J. Opt. Soc. Am. A 27, 1694 (2010)] is the existence of irregular field singularities at the edges of the grating grooves. We show that Fourier series are incapable of representing the transverse electric field components in the vicinity of an edge of irregular field singularity; therefore, any method, not necessarily of modal type, using Fourier series in this way is doomed to fail. A set of precise and simple criteria is given with which, given a lamellar grating, one can predict whether the conventional implementation of a modal method of any kind will converge without running a convergence test.

© 2011 Optical Society of America

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  1. 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]
  2. 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]
  3. L. Li, “Mathematical reflections on the Fourier modal method in grating theory,” in Mathematical Modeling in Optical Science, G.Bao, L.Cowsar, and W.Masters, eds., SIAM Frontiers in Applied Mathematics (SIAM, 2001), pp. 111–139.
    [Crossref]
  4. P. Lalanne and J.-P. Hugonin, “Numerical performance of finite-difference modal method for the electromagnetic analysis of one-dimensional lamellar gratings,” J. Opt. Soc. Am. A 17, 1033–1042 (2000).
    [Crossref]
  5. K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
    [Crossref]
  6. P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast modal method for subwavelength gratings based on B-spline formulation,” J. Opt. Soc. Am. A 27, 696–702 (2010).
    [Crossref]
  7. G. Granet, L. B. Andriamanampisoa, K. Raniriharinosy, A. M. Armeanu, and K. Edee, “Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310(2010).
    [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. N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, “Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings,” J. Opt. Soc. Am. A 24, 3781–3788(2007).
    [Crossref]
  12. K. M. Gundu and A. Mafi, “Reliable computation of scattering from metallic binary gratings using Fourier-based modal methods,” J. Opt. Soc. Am. A 27, 1694–1700 (2010).
    [Crossref]
  13. K. M. Gundu and A. Mafi, “Constrained least squares Fourier modal method for computing scattering from metallic binary gratings,” J. Opt. Soc. Am. A 27, 2375–2380 (2010).
    [Crossref]
  14. J. van Bladel, Singular Electromagnetic Fields and Sources (Clarendon, 1991), pp. 116–162.
  15. D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible et l’ultraviolet,” Ph.D. dissertation (Université d'Aix-Marseille III, 1974).
  16. A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984–1009 (2006).
  17. P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45, 1357–1374 (1998).
    [Crossref]
  18. J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 442–446 (1972).
    [Crossref]
  19. J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag. 26, 598–602 (1978).
    [Crossref]
  20. G. I. Makarov and A. V. Osipov, “Structure of Meixner’s series,” Radiophys. Quantum Electron. 29, 544–549 (1986).
    [Crossref]
  21. M. S. Bobrovnikov and V. P. Zamaraeva, “On the singularity of the field near a dielectric wedge,” Sov. Phys. J. 16, 1230–1232(1973).
    [Crossref]
  22. M. Bressan and P. Gamba, “Analytical expressions of field singularities at the edge of four right wedges,” IEEE Microw. Guided Wave Lett. 4, 3–5 (1994).
    [Crossref]
  23. E. Popov, B. Chernov, M. Nevière, and N. Bonod, “Differential theory: application to highly conducting gratings,” J. Opt. Soc. Am. A 21, 199–206 (2004).
    [Crossref]
  24. 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]
  25. 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]
  26. P. Y. Ufimtsev, B. Khayatian, and Y. Rahmat-Samii, “Singular edge behavior: to impose or not impose—that is the question,” Microw. Opt. Technol. Lett. 24, 218–223 (2000).
    [Crossref]

2010 (4)

2007 (1)

2006 (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).

2005 (1)

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[Crossref]

2004 (1)

2001 (1)

L. Li, “Mathematical reflections on the Fourier modal method in grating theory,” in Mathematical Modeling in Optical Science, G.Bao, L.Cowsar, and W.Masters, eds., SIAM Frontiers in Applied Mathematics (SIAM, 2001), pp. 111–139.
[Crossref]

2000 (2)

P. Lalanne and J.-P. Hugonin, “Numerical performance of finite-difference modal method for the electromagnetic analysis of one-dimensional lamellar gratings,” J. Opt. Soc. Am. A 17, 1033–1042 (2000).
[Crossref]

P. Y. Ufimtsev, B. Khayatian, and Y. Rahmat-Samii, “Singular edge behavior: to impose or not impose—that is the question,” Microw. Opt. Technol. Lett. 24, 218–223 (2000).
[Crossref]

1998 (1)

P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45, 1357–1374 (1998).
[Crossref]

1996 (4)

1994 (1)

M. Bressan and P. Gamba, “Analytical expressions of field singularities at the edge of four right wedges,” IEEE Microw. Guided Wave Lett. 4, 3–5 (1994).
[Crossref]

1991 (1)

J. van Bladel, Singular Electromagnetic Fields and Sources (Clarendon, 1991), pp. 116–162.

1986 (1)

G. I. Makarov and A. V. Osipov, “Structure of Meixner’s series,” Radiophys. Quantum Electron. 29, 544–549 (1986).
[Crossref]

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]

1978 (1)

J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag. 26, 598–602 (1978).
[Crossref]

1974 (1)

D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible et l’ultraviolet,” Ph.D. dissertation (Université d'Aix-Marseille III, 1974).

1973 (1)

M. S. Bobrovnikov and V. P. Zamaraeva, “On the singularity of the field near a dielectric wedge,” Sov. Phys. J. 16, 1230–1232(1973).
[Crossref]

1972 (1)

J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 442–446 (1972).
[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]

Andersen, J. B.

J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag. 26, 598–602 (1978).
[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]

Andriamanampisoa, L. B.

Armeanu, A. M.

Bobrovnikov, M. S.

M. S. Bobrovnikov and V. P. Zamaraeva, “On the singularity of the field near a dielectric wedge,” Sov. Phys. J. 16, 1230–1232(1973).
[Crossref]

Bonod, N.

Botten, L. C.

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]

Bouchon, P.

Bressan, M.

M. Bressan and P. Gamba, “Analytical expressions of field singularities at the edge of four right wedges,” IEEE Microw. Guided Wave Lett. 4, 3–5 (1994).
[Crossref]

Chandezon, J.

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]

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]

Chernov, B.

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]

Edee, K.

G. Granet, L. B. Andriamanampisoa, K. Raniriharinosy, A. M. Armeanu, and K. Edee, “Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310(2010).
[Crossref]

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[Crossref]

Gamba, P.

M. Bressan and P. Gamba, “Analytical expressions of field singularities at the edge of four right wedges,” IEEE Microw. Guided Wave Lett. 4, 3–5 (1994).
[Crossref]

Granet, G.

Guizal, B.

Gundu, K. M.

Haïdar, R.

Hugonin, J.-P.

Jurek, M. P.

P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45, 1357–1374 (1998).
[Crossref]

Khayatian, B.

P. Y. Ufimtsev, B. Khayatian, and Y. Rahmat-Samii, “Singular edge behavior: to impose or not impose—that is the question,” Microw. Opt. Technol. Lett. 24, 218–223 (2000).
[Crossref]

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).

Lalanne, P.

Li, L.

L. Li, “Mathematical reflections on the Fourier modal method in grating theory,” in Mathematical Modeling in Optical Science, G.Bao, L.Cowsar, and W.Masters, eds., SIAM Frontiers in Applied Mathematics (SIAM, 2001), pp. 111–139.
[Crossref]

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

Lyndin, N. M.

Mafi, A.

Makarov, G. I.

G. I. Makarov and A. V. Osipov, “Structure of Meixner’s series,” Radiophys. Quantum Electron. 29, 544–549 (1986).
[Crossref]

Maystre, D.

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, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible et l’ultraviolet,” Ph.D. dissertation (Université d'Aix-Marseille III, 1974).

Mcphedran, R. C.

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]

Meixner, J.

J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 442–446 (1972).
[Crossref]

Morris, G. M.

Nevière, M.

Osipov, A. V.

G. I. Makarov and A. V. Osipov, “Structure of Meixner’s series,” Radiophys. Quantum Electron. 29, 544–549 (1986).
[Crossref]

Pardo, F.

Parriaux, O.

Pelouard, J.-L.

Popov, E.

Rahmat-Samii, Y.

P. Y. Ufimtsev, B. Khayatian, and Y. Rahmat-Samii, “Singular edge behavior: to impose or not impose—that is the question,” Microw. Opt. Technol. Lett. 24, 218–223 (2000).
[Crossref]

Raniriharinosy, K.

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).

Schiavone, P.

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[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).

Solodukhov, V. V.

J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag. 26, 598–602 (1978).
[Crossref]

Tishchenko, A. V.

Ufimtsev, P. Y.

P. Y. Ufimtsev, B. Khayatian, and Y. Rahmat-Samii, “Singular edge behavior: to impose or not impose—that is the question,” Microw. Opt. Technol. Lett. 24, 218–223 (2000).
[Crossref]

van Bladel, J.

J. van Bladel, Singular Electromagnetic Fields and Sources (Clarendon, 1991), pp. 116–162.

Zamaraeva, V. P.

M. S. Bobrovnikov and V. P. Zamaraeva, “On the singularity of the field near a dielectric wedge,” Sov. Phys. J. 16, 1230–1232(1973).
[Crossref]

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 Microw. Guided Wave Lett. (1)

M. Bressan and P. Gamba, “Analytical expressions of field singularities at the edge of four right wedges,” IEEE Microw. Guided Wave Lett. 4, 3–5 (1994).
[Crossref]

IEEE Trans. Antennas Propag. (2)

J. Meixner, “The behavior of electromagnetic fields at edges,” IEEE Trans. Antennas Propag. 20, 442–446 (1972).
[Crossref]

J. B. Andersen and V. V. Solodukhov, “Field behavior near a dielectric wedge,” IEEE Trans. Antennas Propag. 26, 598–602 (1978).
[Crossref]

J. Mod. Opt. (1)

P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45, 1357–1374 (1998).
[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. A (11)

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]

E. Popov, B. Chernov, M. Nevière, and N. Bonod, “Differential theory: application to highly conducting gratings,” J. Opt. Soc. Am. A 21, 199–206 (2004).
[Crossref]

P. Lalanne and J.-P. Hugonin, “Numerical performance of finite-difference modal method for the electromagnetic analysis of one-dimensional lamellar gratings,” J. Opt. Soc. Am. A 17, 1033–1042 (2000).
[Crossref]

P. Bouchon, F. Pardo, R. Haïdar, and J.-L. Pelouard, “Fast modal method for subwavelength gratings based on B-spline formulation,” J. Opt. Soc. Am. A 27, 696–702 (2010).
[Crossref]

G. Granet, L. B. Andriamanampisoa, K. Raniriharinosy, A. M. Armeanu, and K. Edee, “Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310(2010).
[Crossref]

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]

N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, “Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings,” J. Opt. Soc. Am. A 24, 3781–3788(2007).
[Crossref]

K. M. Gundu and A. Mafi, “Reliable computation of scattering from metallic binary gratings using Fourier-based modal methods,” J. Opt. Soc. Am. A 27, 1694–1700 (2010).
[Crossref]

K. M. Gundu and A. Mafi, “Constrained least squares Fourier modal method for computing scattering from metallic binary gratings,” J. Opt. Soc. Am. A 27, 2375–2380 (2010).
[Crossref]

Jpn. J. Appl. Phys. (1)

K. Edee, P. Schiavone, and G. Granet, “Analysis of defect in extreme UV lithography mask using a modal method based on nodal B-spline expansion,” Jpn. J. Appl. Phys. 44, 6458–6462 (2005).
[Crossref]

Microw. Opt. Technol. Lett. (1)

P. Y. Ufimtsev, B. Khayatian, and Y. Rahmat-Samii, “Singular edge behavior: to impose or not impose—that is the question,” Microw. Opt. Technol. Lett. 24, 218–223 (2000).
[Crossref]

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]

Radiophys. Quantum Electron. (1)

G. I. Makarov and A. V. Osipov, “Structure of Meixner’s series,” Radiophys. Quantum Electron. 29, 544–549 (1986).
[Crossref]

Sov. Phys. J. (1)

M. S. Bobrovnikov and V. P. Zamaraeva, “On the singularity of the field near a dielectric wedge,” Sov. Phys. J. 16, 1230–1232(1973).
[Crossref]

Other (3)

L. Li, “Mathematical reflections on the Fourier modal method in grating theory,” in Mathematical Modeling in Optical Science, G.Bao, L.Cowsar, and W.Masters, eds., SIAM Frontiers in Applied Mathematics (SIAM, 2001), pp. 111–139.
[Crossref]

J. van Bladel, Singular Electromagnetic Fields and Sources (Clarendon, 1991), pp. 116–162.

D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible et l’ultraviolet,” Ph.D. dissertation (Université d'Aix-Marseille III, 1974).

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Figures (9)

Fig. 1
Fig. 1

Geometric parameters and permittivities of a lamellar grating.

Fig. 2
Fig. 2

Nonconvergence of reflectivity of a grating case studied in [12] computed with the (a) analytical modal method and the (b) FMM, both with square matrix truncation (see Fig. 5 in [12]). The legends are explained in Section 2.

Fig. 3
Fig. 3

Three configurations of right-angle wedge assemblies.

Fig. 4
Fig. 4

(a) Real and (b) imaginary parts of function τ ( z ) = ( 2 / π ) arcsin ( z 1 / 2 ) in the complex z. The two branch cuts are chosen as the semi-infinite real lines ( , 0 ] and [ 1 , ) .

Fig. 5
Fig. 5

Real (solid curve) and imaginary parts (dashed curve and dashed–dotted curve) of function τ ( z ) = ( 2 / π ) arcsin ( z 1 / 2 ) on the real axis x. In the legend, “above” means z = z + i δ and “below” means z = z i δ as 0 < δ 0 .

Fig. 6
Fig. 6

Map of Δ ( r c , r d ) . In regions I 1 and I 2 , 0 < Δ < 1 ; in regions II 1 and II 2 , Δ < 0 ; in regions III 1 and III 2 , Δ > 1 .

Fig. 7
Fig. 7

Numerical convergence tests with the analytical modal method. The ( r c , r d ) values in (a)–(f) are located in regions I 1 III 2 on the map of Fig. 6, respectively.

Fig. 8
Fig. 8

Numerical convergence tests with the C method. The r values in (a), (b), (c), and (d) are located along the diagonal line in regions I 1 , I 2 , II 1 , and II 2 on the map of Fig. 6, respectively.

Fig. 9
Fig. 9

Reflectivity of a grating case studied in [13] computed with the FMM with rectangular matrix truncation and the constrained least-squares minimization procedure. See Fig. 3 in [13].

Tables (1)

Tables Icon

Table 1 Convergence Criteria for the Modal Methods and the C Method when Applied to Gratings Containing Lossless Metal-Dielectric Right-Angle Wedge Assemblies (see Fig. 6)

Equations (23)

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

E ρ ( j ) = ρ τ 1 A ρ ( j ) ( φ ) , E φ ( j ) = ρ τ 1 A φ ( j ) ( φ ) ,
τ = 2 π arccos Δ ,
Δ = ( ε 1 ε 3 ε 2 ε 4 ) 2 ( ε 1 + ε 2 ) ( ε 2 + ε 3 ) ( ε 3 + ε 4 ) ( ε 4 + ε 1 ) .
τ = 2 π arcsin Δ , Δ = 1 Δ .
Δ ( ε 1 , ε 2 , ε 3 , ε 4 ) = Δ ( ε 4 , ε 3 , ε 2 , ε 1 ) = Δ ( ε 2 , ε 3 , ε 4 , ε 1 ) = Δ ( κ ε 1 , κ ε 2 , κ ε 3 , κ ε 4 ) = Δ ( 1 ε 1 , 1 ε 2 , 1 ε 3 , 1 ε 4 ) ,
Arc sin z = n π ± arcsin z ,
Δ = 1 ε c ( ε d ε m ) 2 2 ( ε c + ε d ) ( ε c + ε m ) ( ε d + ε m ) .
Δ = 1 r c ( 1 + r d ) 2 2 ( 1 r c ) ( 1 r d ) ( r c + r d ) .
Δ = 1 1 4 ( ε c ε m ε c + ε m ) 2 = 1 1 4 ( 1 + r 1 r ) 2 .
E X n = O ( | n | Re [ τ ] ) ,
ρ τ 1 = ρ Re [ τ ] 1 exp ( i Im [ τ ] ln ρ ) .
0 δ 0 2 π ε ( | E ρ | 2 + | E φ | 2 ) ρ d ρ d φ = 0 δ ρ 2 Re [ τ ] 1 d ρ 0 2 π ε ( | A ρ | 2 + | A φ | 2 ) d φ ,
2 E ( j ) + k ( j ) 2 E ( j ) = 0 ,
ρ 1 ρ ( ρ ρ E ρ ( j ) ) + ρ 2 φ φ 2 E ρ ( j ) ρ 2 E ρ ( j ) 2 ρ 2 φ E φ ( j ) + k ( j ) 2 E ρ ( j ) = 0 ,
ρ 1 ρ ( ρ ρ E φ ( j ) ) + ρ 2 φ φ 2 E φ ( j ) ρ 2 E φ ( j ) + 2 ρ 2 φ E ρ ( j ) + k ( j ) 2 E φ ( j ) = 0 .
φ φ 2 A ρ ( j ) + ( τ 2 2 τ ) A ρ ( j ) 2 φ A φ ( j ) = 0 ,
φ φ 2 A φ ( j ) + ( τ 2 2 τ ) A φ ( j ) + 2 φ A ρ ( j ) = 0 ,
( φ φ 2 2 φ 2 φ φ φ 2 ) ( A ρ ( j ) A φ ( j ) ) = ( 2 τ τ 2 ) ( A ρ ( j ) A φ ( j ) ) .
A ρ ( j ) ( φ j ) = A ρ ( j + 1 ) ( φ j ) , ε ( j ) A φ ( j ) ( φ j ) = ε ( j + 1 ) A φ ( j + 1 ) ( φ j ) , A ρ ( N ) ( φ N ) = A ρ ( 1 ) ( φ 0 ) , ε ( N ) A φ ( N ) ( φ N ) = ε ( 1 ) A φ ( 1 ) ( φ 0 ) ,
F , G = 0 2 π ε F G d φ ,
( χ χ ¯ ) A , A = A , L A L A , A = 0 ,
φ A ρ = τ A φ , φ A φ = τ A ρ
A , A = 0 2 π ε ( | A ρ | 2 + | A φ | 2 ) d φ = 0 .

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