Abstract

Implementing the modal method in the electromagnetic grating diffraction problem delivered by the curvilinear coordinate transformation yields a general analytical solution to the 1D grating diffraction problem in a form of a T-matrix. Simultaneously it is shown that the validity of the Rayleigh expansion is defined by the validity of the modal expansion in a transformed medium delivered by the coordinate transformation.

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References

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  1. L. V. Rayleigh, “On the incidence of aerial and electric waves upon small obstacles in the form of ellipsoids or elliptic cylinders, and on the passage of electric waves through a circular aperture in a conducting screen,” Philosoph. Mag. Ser. 5 44, 28–52 (1897).
    [Crossref]
  2. L. Rayleigh, “On the dynamical theory of gratings,” in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences79, 399–416 (1907).
  3. K. Edee, J. P. Plumey, and J. Chandezon, “On the Rayleigh-Fourier method and the Chandezon method: Comparative study,” Opt. Commun. 286, 34–41 (2013).
    [Crossref]
  4. T. Nordam, P. A. Letnes, and I. Simonsen, “Validity of the Rayleigh hypothesis for two-dimensional randomly rough metal surfaces,” J. Phys.: Conf. Ser. 454, 012033 (2013).
  5. J. L. Uretski, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
    [Crossref]
  6. R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infinement conducteur,” C. R. Acad. Sci. Paris 262B, 468–471 (1966).
  7. R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface, II,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
    [Crossref]
  8. J. Pavageau, “Sur la méthode des spectres d’ondes planes dans les problèmes de diffraction,” C. R. Acad. Sci. Paris 266B, 135–138 (1968).
  9. R. H. T. Bates, “Analytic constraints on electromagnetic field computations,” IEEE Trans. Microwave Theory and Tech. MTT-23, 605–623 (1975).
    [Crossref]
  10. A. Wirgin, “On Rayleigh’s theory of sinusoidal diffraction gratings,” Opt. Acta 27, 1671–1692 (1980).
    [Crossref]
  11. T. Watanabe, Y. Choyal, K. Minami, and V. L. Granatstein, “Range of validity of the Rayleigh hypothesis,” Phys. Rev. E 69, 056606 (2004).
    [Crossref]
  12. T. Elfouhaily and T. Hahn, “Rayleigh’s hypothesis and the geometrical optics limit,” Phys. Rev. Lett. 97, 120404 (2006).
    [Crossref]
  13. A. V. Tishchenko, “Numerical demonstration of the validity of the Rayleigh hypothesis,” Opt. Express 17, 17102–17117 (2009).
    [Crossref] [PubMed]
  14. A. V. Tishchenko, “Rayleigh was right: Electromagnetic fields and corrugated interfaces,” Opt. Photon. News 21, 50–54 (2010).
    [Crossref]
  15. J. Wauer and T. Rother, “Considerations to Rayleigh’s hypothesis,” Opt. Comm. 282, 339–350 (2009).
    [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 (1980).
    [Crossref]
  17. G. Granet, “Analysis of diffraction by surface-relief crossed gratings with use of the Chandezon method: application to multilayer crossed gratings,” J. Opt. Soc. Am. A 15, 1121–1131 (1998).
    [Crossref]
  18. L. C. Botten, M. S. Craig, and J. L. McPhedran, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
    [Crossref]
  19. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
    [Crossref]
  20. A. V. Tishchenko, “Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method,” Opt. Quantum Electron. 37, 309–330 (2005).
    [Crossref]
  21. J. A. Schouten, Tensor Analysis for Physicists(Dover Publications, 1954).
  22. G. M. Murphy, Ordinary Differential Equations and Their Solutions (D. Van Nostrand, 1960).
  23. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewarta, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1982).
    [Crossref]
  24. S. D. M. Adams, R. V. Craster, and S. Guenneau, “Bloch waves in periodic multi-layered acoustic waveguides,” Proc. R. Soc. A 464, 2669–2692 (2008).
    [Crossref]
  25. B. Gralak, “Exact modal methods,” in Gratings: Theory and Numeric Applications, E. Popov, ed. (Presses Universitaires de Provence, 2012).
  26. I. Semenikhin, M. Zanuccoli, C. Fiegna, V. Vyurkov, and E. Sangiorgi, “Computationally efficient method for optical simulation of solar cells and their applications,” Proc. SPIE 8700, 870012 (2013).
    [Crossref]

2013 (3)

K. Edee, J. P. Plumey, and J. Chandezon, “On the Rayleigh-Fourier method and the Chandezon method: Comparative study,” Opt. Commun. 286, 34–41 (2013).
[Crossref]

T. Nordam, P. A. Letnes, and I. Simonsen, “Validity of the Rayleigh hypothesis for two-dimensional randomly rough metal surfaces,” J. Phys.: Conf. Ser. 454, 012033 (2013).

I. Semenikhin, M. Zanuccoli, C. Fiegna, V. Vyurkov, and E. Sangiorgi, “Computationally efficient method for optical simulation of solar cells and their applications,” Proc. SPIE 8700, 870012 (2013).
[Crossref]

2010 (1)

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

2009 (2)

J. Wauer and T. Rother, “Considerations to Rayleigh’s hypothesis,” Opt. Comm. 282, 339–350 (2009).
[Crossref]

A. V. Tishchenko, “Numerical demonstration of the validity of the Rayleigh hypothesis,” Opt. Express 17, 17102–17117 (2009).
[Crossref] [PubMed]

2008 (1)

S. D. M. Adams, R. V. Craster, and S. Guenneau, “Bloch waves in periodic multi-layered acoustic waveguides,” Proc. R. Soc. A 464, 2669–2692 (2008).
[Crossref]

2006 (1)

T. Elfouhaily and T. Hahn, “Rayleigh’s hypothesis and the geometrical optics limit,” Phys. Rev. Lett. 97, 120404 (2006).
[Crossref]

2005 (1)

A. V. Tishchenko, “Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method,” Opt. Quantum Electron. 37, 309–330 (2005).
[Crossref]

2004 (1)

T. Watanabe, Y. Choyal, K. Minami, and V. L. Granatstein, “Range of validity of the Rayleigh hypothesis,” Phys. Rev. E 69, 056606 (2004).
[Crossref]

1998 (1)

1993 (1)

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[Crossref]

1982 (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewarta, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1982).
[Crossref]

1981 (1)

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

1980 (2)

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

A. Wirgin, “On Rayleigh’s theory of sinusoidal diffraction gratings,” Opt. Acta 27, 1671–1692 (1980).
[Crossref]

1975 (1)

R. H. T. Bates, “Analytic constraints on electromagnetic field computations,” IEEE Trans. Microwave Theory and Tech. MTT-23, 605–623 (1975).
[Crossref]

1971 (1)

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface, II,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
[Crossref]

1968 (1)

J. Pavageau, “Sur la méthode des spectres d’ondes planes dans les problèmes de diffraction,” C. R. Acad. Sci. Paris 266B, 135–138 (1968).

1966 (1)

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infinement conducteur,” C. R. Acad. Sci. Paris 262B, 468–471 (1966).

1965 (1)

J. L. Uretski, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[Crossref]

1897 (1)

L. V. Rayleigh, “On the incidence of aerial and electric waves upon small obstacles in the form of ellipsoids or elliptic cylinders, and on the passage of electric waves through a circular aperture in a conducting screen,” Philosoph. Mag. Ser. 5 44, 28–52 (1897).
[Crossref]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewarta, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1982).
[Crossref]

Adams, S. D. M.

S. D. M. Adams, R. V. Craster, and S. Guenneau, “Bloch waves in periodic multi-layered acoustic waveguides,” Proc. R. Soc. A 464, 2669–2692 (2008).
[Crossref]

Andrewarta, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewarta, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1982).
[Crossref]

Bates, R. H. T.

R. H. T. Bates, “Analytic constraints on electromagnetic field computations,” IEEE Trans. Microwave Theory and Tech. MTT-23, 605–623 (1975).
[Crossref]

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewarta, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1982).
[Crossref]

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

Cadilhac, M.

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infinement conducteur,” C. R. Acad. Sci. Paris 262B, 468–471 (1966).

Chandezon, J.

K. Edee, J. P. Plumey, and J. Chandezon, “On the Rayleigh-Fourier method and the Chandezon method: Comparative study,” Opt. Commun. 286, 34–41 (2013).
[Crossref]

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

Choyal, Y.

T. Watanabe, Y. Choyal, K. Minami, and V. L. Granatstein, “Range of validity of the Rayleigh hypothesis,” Phys. Rev. E 69, 056606 (2004).
[Crossref]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewarta, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1982).
[Crossref]

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

Craster, R. V.

S. D. M. Adams, R. V. Craster, and S. Guenneau, “Bloch waves in periodic multi-layered acoustic waveguides,” Proc. R. Soc. A 464, 2669–2692 (2008).
[Crossref]

Edee, K.

K. Edee, J. P. Plumey, and J. Chandezon, “On the Rayleigh-Fourier method and the Chandezon method: Comparative study,” Opt. Commun. 286, 34–41 (2013).
[Crossref]

Elfouhaily, T.

T. Elfouhaily and T. Hahn, “Rayleigh’s hypothesis and the geometrical optics limit,” Phys. Rev. Lett. 97, 120404 (2006).
[Crossref]

Fiegna, C.

I. Semenikhin, M. Zanuccoli, C. Fiegna, V. Vyurkov, and E. Sangiorgi, “Computationally efficient method for optical simulation of solar cells and their applications,” Proc. SPIE 8700, 870012 (2013).
[Crossref]

Gralak, B.

B. Gralak, “Exact modal methods,” in Gratings: Theory and Numeric Applications, E. Popov, ed. (Presses Universitaires de Provence, 2012).

Granatstein, V. L.

T. Watanabe, Y. Choyal, K. Minami, and V. L. Granatstein, “Range of validity of the Rayleigh hypothesis,” Phys. Rev. E 69, 056606 (2004).
[Crossref]

Granet, G.

Guenneau, S.

S. D. M. Adams, R. V. Craster, and S. Guenneau, “Bloch waves in periodic multi-layered acoustic waveguides,” Proc. R. Soc. A 464, 2669–2692 (2008).
[Crossref]

Hahn, T.

T. Elfouhaily and T. Hahn, “Rayleigh’s hypothesis and the geometrical optics limit,” Phys. Rev. Lett. 97, 120404 (2006).
[Crossref]

Letnes, P. A.

T. Nordam, P. A. Letnes, and I. Simonsen, “Validity of the Rayleigh hypothesis for two-dimensional randomly rough metal surfaces,” J. Phys.: Conf. Ser. 454, 012033 (2013).

Li, L.

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[Crossref]

Maystre, D.

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

McPhedran, J. L.

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

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewarta, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1982).
[Crossref]

Millar, R. F.

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface, II,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
[Crossref]

Minami, K.

T. Watanabe, Y. Choyal, K. Minami, and V. L. Granatstein, “Range of validity of the Rayleigh hypothesis,” Phys. Rev. E 69, 056606 (2004).
[Crossref]

Murphy, G. M.

G. M. Murphy, Ordinary Differential Equations and Their Solutions (D. Van Nostrand, 1960).

Nordam, T.

T. Nordam, P. A. Letnes, and I. Simonsen, “Validity of the Rayleigh hypothesis for two-dimensional randomly rough metal surfaces,” J. Phys.: Conf. Ser. 454, 012033 (2013).

Pavageau, J.

J. Pavageau, “Sur la méthode des spectres d’ondes planes dans les problèmes de diffraction,” C. R. Acad. Sci. Paris 266B, 135–138 (1968).

Petit, R.

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infinement conducteur,” C. R. Acad. Sci. Paris 262B, 468–471 (1966).

Plumey, J. P.

K. Edee, J. P. Plumey, and J. Chandezon, “On the Rayleigh-Fourier method and the Chandezon method: Comparative study,” Opt. Commun. 286, 34–41 (2013).
[Crossref]

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 (1980).
[Crossref]

Rayleigh, L.

L. Rayleigh, “On the dynamical theory of gratings,” in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences79, 399–416 (1907).

Rayleigh, L. V.

L. V. Rayleigh, “On the incidence of aerial and electric waves upon small obstacles in the form of ellipsoids or elliptic cylinders, and on the passage of electric waves through a circular aperture in a conducting screen,” Philosoph. Mag. Ser. 5 44, 28–52 (1897).
[Crossref]

Rother, T.

J. Wauer and T. Rother, “Considerations to Rayleigh’s hypothesis,” Opt. Comm. 282, 339–350 (2009).
[Crossref]

Sangiorgi, E.

I. Semenikhin, M. Zanuccoli, C. Fiegna, V. Vyurkov, and E. Sangiorgi, “Computationally efficient method for optical simulation of solar cells and their applications,” Proc. SPIE 8700, 870012 (2013).
[Crossref]

Schouten, J. A.

J. A. Schouten, Tensor Analysis for Physicists(Dover Publications, 1954).

Semenikhin, I.

I. Semenikhin, M. Zanuccoli, C. Fiegna, V. Vyurkov, and E. Sangiorgi, “Computationally efficient method for optical simulation of solar cells and their applications,” Proc. SPIE 8700, 870012 (2013).
[Crossref]

Simonsen, I.

T. Nordam, P. A. Letnes, and I. Simonsen, “Validity of the Rayleigh hypothesis for two-dimensional randomly rough metal surfaces,” J. Phys.: Conf. Ser. 454, 012033 (2013).

Tishchenko, A. V.

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

A. V. Tishchenko, “Numerical demonstration of the validity of the Rayleigh hypothesis,” Opt. Express 17, 17102–17117 (2009).
[Crossref] [PubMed]

A. V. Tishchenko, “Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method,” Opt. Quantum Electron. 37, 309–330 (2005).
[Crossref]

Uretski, J. L.

J. L. Uretski, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[Crossref]

Vyurkov, V.

I. Semenikhin, M. Zanuccoli, C. Fiegna, V. Vyurkov, and E. Sangiorgi, “Computationally efficient method for optical simulation of solar cells and their applications,” Proc. SPIE 8700, 870012 (2013).
[Crossref]

Watanabe, T.

T. Watanabe, Y. Choyal, K. Minami, and V. L. Granatstein, “Range of validity of the Rayleigh hypothesis,” Phys. Rev. E 69, 056606 (2004).
[Crossref]

Wauer, J.

J. Wauer and T. Rother, “Considerations to Rayleigh’s hypothesis,” Opt. Comm. 282, 339–350 (2009).
[Crossref]

Wirgin, A.

A. Wirgin, “On Rayleigh’s theory of sinusoidal diffraction gratings,” Opt. Acta 27, 1671–1692 (1980).
[Crossref]

Zanuccoli, M.

I. Semenikhin, M. Zanuccoli, C. Fiegna, V. Vyurkov, and E. Sangiorgi, “Computationally efficient method for optical simulation of solar cells and their applications,” Proc. SPIE 8700, 870012 (2013).
[Crossref]

Ann. Phys. (1)

J. L. Uretski, “The scattering of plane waves from periodic surfaces,” Ann. Phys. 33, 400–427 (1965).
[Crossref]

C. R. Acad. Sci. Paris (2)

R. Petit and M. Cadilhac, “Sur la diffraction d’une onde plane par un réseau infinement conducteur,” C. R. Acad. Sci. Paris 262B, 468–471 (1966).

J. Pavageau, “Sur la méthode des spectres d’ondes planes dans les problèmes de diffraction,” C. R. Acad. Sci. Paris 266B, 135–138 (1968).

IEEE Trans. Microwave Theory and Tech. (1)

R. H. T. Bates, “Analytic constraints on electromagnetic field computations,” IEEE Trans. Microwave Theory and Tech. MTT-23, 605–623 (1975).
[Crossref]

J. Mod. Opt. (1)

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[Crossref]

J. Opt. Paris (1)

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

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

J. Phys.: Conf. Ser. (1)

T. Nordam, P. A. Letnes, and I. Simonsen, “Validity of the Rayleigh hypothesis for two-dimensional randomly rough metal surfaces,” J. Phys.: Conf. Ser. 454, 012033 (2013).

Opt. Acta (3)

A. Wirgin, “On Rayleigh’s theory of sinusoidal diffraction gratings,” Opt. Acta 27, 1671–1692 (1980).
[Crossref]

L. C. Botten, M. S. Craig, and J. L. McPhedran, “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. Andrewarta, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1982).
[Crossref]

Opt. Comm. (1)

J. Wauer and T. Rother, “Considerations to Rayleigh’s hypothesis,” Opt. Comm. 282, 339–350 (2009).
[Crossref]

Opt. Commun. (1)

K. Edee, J. P. Plumey, and J. Chandezon, “On the Rayleigh-Fourier method and the Chandezon method: Comparative study,” Opt. Commun. 286, 34–41 (2013).
[Crossref]

Opt. Express (1)

Opt. Photon. News (1)

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

Opt. Quantum Electron. (1)

A. V. Tishchenko, “Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method,” Opt. Quantum Electron. 37, 309–330 (2005).
[Crossref]

Philosoph. Mag. Ser. 5 (1)

L. V. Rayleigh, “On the incidence of aerial and electric waves upon small obstacles in the form of ellipsoids or elliptic cylinders, and on the passage of electric waves through a circular aperture in a conducting screen,” Philosoph. Mag. Ser. 5 44, 28–52 (1897).
[Crossref]

Phys. Rev. E (1)

T. Watanabe, Y. Choyal, K. Minami, and V. L. Granatstein, “Range of validity of the Rayleigh hypothesis,” Phys. Rev. E 69, 056606 (2004).
[Crossref]

Phys. Rev. Lett. (1)

T. Elfouhaily and T. Hahn, “Rayleigh’s hypothesis and the geometrical optics limit,” Phys. Rev. Lett. 97, 120404 (2006).
[Crossref]

Proc. Cambridge Philos. Soc. (1)

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface, II,” Proc. Cambridge Philos. Soc. 69, 217–225 (1971).
[Crossref]

Proc. R. Soc. A (1)

S. D. M. Adams, R. V. Craster, and S. Guenneau, “Bloch waves in periodic multi-layered acoustic waveguides,” Proc. R. Soc. A 464, 2669–2692 (2008).
[Crossref]

Proc. SPIE (1)

I. Semenikhin, M. Zanuccoli, C. Fiegna, V. Vyurkov, and E. Sangiorgi, “Computationally efficient method for optical simulation of solar cells and their applications,” Proc. SPIE 8700, 870012 (2013).
[Crossref]

Other (4)

B. Gralak, “Exact modal methods,” in Gratings: Theory and Numeric Applications, E. Popov, ed. (Presses Universitaires de Provence, 2012).

J. A. Schouten, Tensor Analysis for Physicists(Dover Publications, 1954).

G. M. Murphy, Ordinary Differential Equations and Their Solutions (D. Van Nostrand, 1960).

L. Rayleigh, “On the dynamical theory of gratings,” in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences79, 399–416 (1907).

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Figures (2)

Fig. 1
Fig. 1 Example of a grating corrugation.
Fig. 2
Fig. 2 (a) an example of a sinusoidal corrugation separating two isotropic media, and a corresponding reciprocal problem in curvilinear coordinates described by smoothly varying material tensors. (b) an example of a saw-tooth corrugation separating two isotropic media, and a corresponding reciprocal problem in curvilinear coordinates described by two sets of the material tensors.

Equations (45)

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

F α = ( z β / x α ) F ˜ β , F α = ( x α / z β ) F ˜ β .
ξ α β γ E γ x β = i ω μ δ α β H β , ξ α β γ H γ x β = i ω ε δ α β E β .
ξ α β γ E ˜ γ z β = i ω μ g g α β H ˜ β ξ α β γ H ˜ γ z β = i ω ε g g α β E ˜ β .
χ ˜ α β = χ g g α β
z 1 , 2 = x 1 , 2 , z 3 = z f ( x 1 ) .
χ ˜ α β = χ M α β
M = ( 1 0 f ( z 1 ) 0 1 0 f ( z 1 ) 0 1 + [ f ( z 1 ) ] 2 ) .
f ( x 1 ) = h 2 sin 2 π x 1 Λ .
M ( z 1 ) = ( 1 0 π h Λ cos 2 π z 1 Λ 0 1 0 π h Λ cos 2 π z 1 Λ 0 1 + ( π h Λ cos 2 π z 1 Λ ) 2 )
f ( x 1 ) = { ( x 1 d 1 1 2 ) h , 0 x 1 < d 1 ; ( Λ x 1 d 2 1 2 ) h , d 1 x 1 < Λ ,
M 1 = ( 1 0 h / d 1 0 1 0 h / d 1 0 1 + ( h / d 1 ) 2 ) , 0 x 1 < d 1 , M 2 = ( 1 0 h / d 2 0 1 0 h / d 2 0 1 + ( h / d 2 ) 2 ) , d 1 x 1 < Λ .
E ¯ 2 q T E = b q e ± exp [ j ψ q b ( x ¯ 1 ) ± j β q b x ¯ 3 ] ,
E ¯ 2 q T E ( x ¯ 1 + Λ ) = E ¯ 2 q T E ( x ¯ 1 ) exp ( j k 1 i n c Λ )
H ¯ 1 q T E = 1 ω μ b [ β q b { 1 + [ f ( x ¯ 1 ) ] 2 } + f ( x ¯ 1 ) d ψ q b ( x ¯ 1 ) d x ¯ 1 ] b q e ± exp [ j ψ q b ( x ¯ 1 ) ± j β q b x ¯ 3 ] , H ¯ 3 q T E = 1 ω μ b [ β q b f ( x ¯ 1 ) + d ψ q b ( x ¯ 1 ) d x ¯ 1 ] b q e ± exp [ j ψ q b ( x ¯ 1 ) ± j β q b x ¯ 1 ] .
j d d x ¯ 1 [ ± β q b f ( x ¯ 1 ) d ψ q b ( x ¯ 1 ) d x ¯ 1 ] + [ ± β q b f ( x ¯ 1 ) d ψ q b ( x ¯ 1 ) d x ¯ 1 ] 2 = ω 2 ε b μ b ( β q b ) 2 ,
H ¯ 2 q T M = b q h ± exp [ j ϕ q b ( x ¯ 1 ) ± j β q b x ¯ 3 ] , E ¯ 1 q T M = 1 ω ε b [ ± β q b { 1 + [ f ( x ¯ 1 ) ] 2 } f ( x ¯ 1 ) d ϕ q b ( x ¯ 1 ) d x ¯ 1 ] b q e ± exp [ j ϕ q b ( x ¯ 1 ) ± j β q b x ¯ 3 ] , E ¯ 3 q T M = 1 ω ε b [ ± β q b f ( x ¯ 1 ) d ϕ q b ( x ¯ 1 ) d x ¯ 1 ] b q e ± exp [ j ϕ q b ( x ¯ 1 ) ± j β q b x ¯ 3 ]
ψ q b ( x ¯ 1 ) = ± β q b f ( x ¯ 1 ) j log G ( x ¯ 1 )
G ( x ¯ 1 ) + [ ω 2 ε b μ b ( β q b ) 2 ] G ( x ¯ 1 ) = 0 .
ψ q b ( x ¯ 1 ) = ± β q b f ( x ¯ 1 ) j log { C i exp [ j ω 2 ε b μ b ( β q b ) 2 x ¯ 1 ] + D i exp [ j ω 2 ε b μ b ( β q b ) 2 x ¯ 1 ] }
C i = C i + i , D i = C i + 1 .
exp [ ± j Λ ω 2 ε b μ b ( β q b ) 2 ] = exp ( j k 1 i n c Λ ) .
β q b = ω 2 ε b μ b k 1 q 2 ,
F ¯ 2 q G ¯ 3 q = ω χ b k 1 q ,
F ¯ 2 q ~ b q ± exp { j k 1 q x ¯ 1 ± j β q b [ x ¯ 3 + f ( x ¯ 1 ) ] } , G ¯ 1 q ~ k 1 q f ( x ¯ 1 ) β q b ω χ b b q ± exp { j k 1 q x ¯ 1 ± j β q b [ x ¯ 3 + f ( x ¯ 1 ) ] } ,
q = { b q + exp [ j k 1 q x ¯ 1 + j β q b f ( x ¯ 1 ) ] + b q exp [ j k 1 q x ¯ 1 j β q b f ( x ¯ 1 ) ] } = p = { a p + exp [ j k 1 p x ¯ 1 + j β p a f ( x ¯ 1 ) ] + a p exp [ j k 1 p x ¯ 1 j β p a f ( x ¯ 1 ) ] } .
1 χ b q = { [ k 1 q f ( x ¯ 1 ) β q b ] b q + exp [ j k 1 q x ¯ 1 + j β q b f ( x ¯ 1 ) ] + [ k 1 q f ( x ¯ 1 ) + β q b ] b q exp [ j k 1 q x ¯ 1 j β q b f ( x ¯ 1 ) ] } = 1 χ a p = { [ k 1 p f ( x ¯ 1 ) β p a ] a p + exp [ j k 1 p x ¯ 1 + j β p a f ( x ¯ 1 ) ] + [ k 1 p f ( x ¯ 1 ) + β p a ] a q exp [ j k 1 p x ¯ 1 j β p a f ( x ¯ 1 ) ] } .
1 Λ 0 Λ [ ( β p a , b + β q a , b ) ( k 1 p + k 1 q ) f ( x ¯ 1 ) ] exp [ j ( k 1 q k 1 p ) x ¯ 1 ± ( β q a , b + β p a , b ) f ( x ¯ 1 ) ] d x ¯ 1 = 2 β p a , b δ p q ,
1 Λ 0 Λ [ ( β p a , b β q a , b ) ± ( k 1 p + k 1 q ) f ( x ¯ 1 ) ] exp [ j ( k 1 q k 1 p ) x ¯ 1 ± ( β q a , b + β p a , b ) f ( x ¯ 1 ) ] d x ¯ 1 = 0 ,
b p ± = q = a q + 1 Λ 0 Λ { ( β p b ± χ b χ a β q a ) ( χ b χ a k 1 q + k 1 p ) f ( x ¯ 1 ) 2 β p b × exp [ j Δ k 1 q p x ¯ 1 + j ( β q a β p b ) f ( x ¯ 1 ) ] } d x ¯ 1 , + q = a q 1 Λ 0 Λ { ( β p b χ b χ a β q a ) ( χ b χ a k 1 q + k 1 p ) f ( x ¯ 1 ) 2 β p b × exp [ j Δ k 1 q p x ¯ 1 j ( β q a ± β p b ) f ( x ¯ 1 ) ] } d x ¯ 1 ,
( b p + b p ) = q = ( T p q + + T p q + T p q + T p q ) ( a p + a p ) .
T p q + + = 1 Λ 0 Λ ( β p b + χ b χ a β q a ) ( χ b χ a k 1 q + k 1 p ) f ( x ¯ 1 ) 2 β p b exp [ j Δ k 1 p q x ¯ 1 + j ( β q a β p b ) f ( x ¯ 1 ) ] d x ¯ 1 , T p q + = 1 Λ 0 Λ ( β p b χ b χ a β q a ) + ( χ b χ a k 1 q + k 1 p ) f ( x ¯ 1 ) 2 β p b exp [ j Δ k 1 p q x ¯ 1 + j ( β q a + β p b ) f ( x ¯ 1 ) ] d x ¯ 1 , T p q + = 1 Λ 0 Λ ( β p b χ b χ a β q a ) ( χ b χ a k 1 q + k 1 p ) f ( x ¯ 1 ) 2 β p b exp [ j Δ k 1 p q x ¯ 1 j ( β q a + β p b ) f ( x ¯ 1 ) ] d x ¯ 1 , T p q = 1 Λ 0 Λ ( β p b + χ b χ a β q a ) + ( χ b χ a k 1 q + k 1 p ) f ( x ¯ 1 ) 2 β p b exp [ j Δ k 1 p q x ¯ 1 j ( β q a β p b ) f ( x ¯ 1 ) ] d x ¯ 1 .
T p q + + = ζ p q + 1 Λ 0 Λ exp [ j ( q p ) K x ¯ 1 + j ( β q a β p b ) f ( x ¯ 1 ) ] d x ¯ 1 , T p q + = ζ p q 1 Λ 0 Λ exp [ j ( q p ) K x ¯ 1 j ( β q a + β p b ) f ( x ¯ 1 ) ] d x ¯ 1 , T p q + = ζ p q 1 Λ 0 Λ exp [ j ( q p ) K x ¯ 1 + j ( β q a + β p b ) f ( x ¯ 1 ) ] d x ¯ 1 , T p q = ζ p q + 1 Λ 0 Λ exp [ j ( q p ) K x ¯ 1 j ( β q a β p b ) f ( x ¯ 1 ) ] d x ¯ 1 ,
ζ p q ± = ± ω 2 μ b ( ε a ε b ) + ( 1 μ b μ a ) ( β q a β p b ± k 1 p k 1 q ) 2 β p b ( β q a β p b )
ζ p q ± = ± ω 2 ε b ( μ a μ b ) + ( 1 ε b ε a ) ( β q a β p b ± k 1 p k 1 q ) 2 β p b ( β q a β p b )
T p q + + = ζ p q + J p q [ 1 2 ( β q a β p b ) h ] , T p q + = ζ p q J p q [ 1 2 ( β q a + β p b ) h ] , T p q + = ζ p q J p q [ 1 2 ( β q a + β p b ) h ] , T p q = ζ p q + J p q [ 1 2 ( β q a β p b ) h ] .
T p q + + = ζ p q + d 1 Λ exp [ j π ( q p ) d 1 Λ ] sinc [ π ( q p ) d 1 Λ ( β q a β p b ) h 2 ] + ζ p q + d 2 Λ exp [ j π ( q p ) d 2 Λ ] sinc [ π ( q p ) d 2 Λ ( β q a β p b ) h 2 ] , T p q + = ζ p q d 1 Λ exp [ j π ( q p ) d 1 Λ ] sinc [ π ( q p ) d 1 Λ + ( β q a + β p b ) h 2 ] + ζ p q d 2 Λ exp [ j π ( q p ) d 2 Λ ] sinc [ π ( q p ) d 2 Λ + ( β q a β p b ) h 2 ] , T p q + = ζ p q d 1 Λ exp [ j π ( q p ) d 1 Λ ] sinc [ π ( q p ) d 1 Λ ( β q a + β p b ) h 2 ] + ζ p q d 2 Λ exp [ j π ( q p ) d 2 Λ ] sinc [ π ( q p ) d 2 Λ ( β q a + β p b ) h 2 ] , T p q = ζ p q + d 1 Λ exp [ j π ( q p ) d 1 Λ ] sinc [ π ( q p ) d 1 Λ + ( β q a β p b ) h 2 ] + ζ p q + d 2 Λ exp [ j π ( q p ) d 2 Λ ] sinc [ π ( q p ) d 2 Λ + ( β q a β p b ) h 2 ] .
F ¯ 2 = { p = b p + exp { j k 1 p x ¯ 1 + j β p b [ x ¯ 3 + f ( x ¯ 1 ) ] } + b p exp { j k 1 p x ¯ 1 j β p b [ x ¯ 3 + f ( x ¯ 1 ) ] } , x ¯ 3 0 , p = a p + exp { j k 1 p x ¯ 1 + j β p a [ x ¯ 3 + f ( x ¯ 1 ) ] } + a p exp { j k 1 p x ¯ 1 j β p a [ x ¯ 3 + f ( x ¯ 1 ) ] } , x ¯ 3 > 0 .
F 2 = { p = b p + exp ( j k 1 p x 1 + j β p b x 3 ) + b p exp ( j k 1 p x 1 j β p b x 3 ) , x 3 f ( x 1 ) , p = a p + exp ( j k 1 p x 1 + j β p b x 3 ) + a p exp ( j k 1 p x 1 j β p a x 3 ) , x 3 > f ( x 1 ) .
( b p + a p ) = q = ( S p q b b S p q b a S p q a b S p q a a ) ( b q a q + )
f ( ξ ) = f i ( ξ ) , ξ i 1 ξ < ξ i , i = 1 , , N
( k 1 p + k 1 q ) ( k 1 p k 1 q ) + ( β p + β q ) ( β p β q ) = 0 .
δ p q 1 Λ 0 Λ [ ( β p + β q ) ( k 1 p + k 1 q ) f ( ξ ) ] exp [ ( k 1 q k 1 p ) ξ ± ( β q β p ) f ( ξ ) ] d ξ = 2 β p 2 Λ k 1 p i = 1 N [ f ( ξ i ) f ( ξ i 1 ) ] = 2 β p
δ p q 1 Λ 0 Λ [ ( β p β q ) ± ( k 1 p + k 1 q ) f ( ξ ) ] exp [ ( k 1 q k 1 p ) ξ ± ( β p + β q ) f ( ξ ) ] d ξ = { k 1 p j β p Λ i = 1 N { exp [ ± 2 j β p f i ( ξ i ) ] exp [ ± 2 j β p f i ( ξ i 1 ) ] } , β p 0 ± 2 k 1 p [ f i ( ξ i ) f i ( ξ i 1 ) ] , β p = 0 = 0 .
1 Λ 0 Λ [ ( β p + β q ) ( k 1 p + k 1 q ) f ( ξ ) ] exp [ ( k 1 p k 1 q ) ξ ± ( β q β p ) f ( ξ ) ] d ξ = i = 1 N β p + β q k 1 q k 1 p 1 Λ ξ i 1 ξ i [ ( β p + β q ) ( k 1 p + k 1 q ) f i ( ξ ) ] exp [ ( k 1 q k 1 p ) ξ ± ( β q β p ) f i ( ξ ) ] d ξ = β p + β q j ( k 1 q k 1 p ) Λ i = 1 N { exp [ j ( k 1 q k 1 p ) ξ i ± j ( β q β p ) f i ( ξ i ) ] exp [ j ( k 1 q k 1 p ) ξ i 1 ± j ( β q β p ) f i ( ξ i 1 ) ] } = β p + β q j ( k 1 q k 1 p ) Λ exp [ ± j ( β q β p ) f 1 ( 0 ) ] { exp [ j ( k 1 q k 1 p ) Λ ] 1 } = 0 ,
1 Λ 0 Λ [ ( β p β q ) ± ( k 1 p + k 1 q ) f ( ξ ) ] exp [ ( k 1 q k 1 p ) ξ ± ( β p + β q ) f ( ξ ) ] d ξ = i = 1 N β p β q k 1 q k 1 p 1 Λ ξ i 1 ξ i [ ( k 1 q k 1 p ) ± ( β p + β q ) f i ( ξ ) ] exp [ ( k 1 q k 1 p ) ξ ± ( β q + β p ) f i ( ξ ) ] d ξ = β p β q j ( k 1 q k 1 p ) Λ i = 1 N { exp [ j ( k 1 q k 1 p ) ξ i ± j ( β q + β p ) f i ( ξ i ) ] exp [ j ( k 1 q k 1 p ) ξ i 1 ± j ( β q + β p ) f i ( ξ i 1 ) ] } = β p β q j ( k 1 q k 1 p ) Λ exp [ ± j ( β q + β p ) f 1 ( 0 ) ] { exp [ j ( k 1 q k 1 p ) Λ ] 1 } = 0 .

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