Abstract

We present a formulation for wave propagation and scattering through stacked gratings comprising metallic and dielectric cylinders. By modeling a photonic crystal as a grating stack of this type, we thus formulate an efficient and accurate method for photonic crystal calculations that allows us to calculate reflection and transmission matrices. The stack may contain an arbitrary number of gratings, provided that each has a common period. The formulation uses a Green’s function approach based on lattice sums to obtain the scattering matrices of each layer, and it couples these layers through recurrence relations. In a companion paper [J. Opt. Soc. Am. A 17, 2177 (2000)] we discuss the numerical implementation of the method and give a comprehensive treatment of its conservation properties.

© 2000 Optical Society of America

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  1. J. P. Dowling, H. Everitt, E. Yablonovitch, “Photonic and acoustic band-gap bibliography,” web page http://home.earthlink.net/~jpdowling/pbgbib.html .
  2. D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
    [CrossRef]
  3. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984), pp. 1–67.
  4. T. Larsen, “A survey of the theory of wire grids,” IRE Trans. Microwave Theory Tech. MTT-10, 191–201 (1962).
    [CrossRef]
  5. W. G. Chambers, C. L. Mok, T. J. Parker, “Theory of the scattering of electromagnetic waves by a regular grid of parallel cylindrical wires with circular cross section,” J. Phys. A 13, 1433–1441 (1980).
    [CrossRef]
  6. W. G. Chambers, C. L. Mok, T. J. Parker, “Theoretical study of the frequency response of some far-infrared interferometers with wire-grid beam dividers,” J. Phys. D 13, 515–526 (1980).
    [CrossRef]
  7. J. Y. Suratteau, M. Cadilhac, R. Petit, “The perfectly conducting wire grating: computation of the diffracted field from Maxwell’s equations and Hamilton’s canonical system,” IEEE Trans. Antennas Propag. AP-33, 404–408 (1985).
    [CrossRef]
  8. D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. 11, 2526–2538 (1994).
    [CrossRef]
  9. C. M. Horwitz, R. C. McPhedran, J. Beunen, “Interference and diffraction in globular metal films,” J. Opt. Soc. Am. 68, 1023–1031 (1978).
    [CrossRef]
  10. W. von Ignatowsky, “Zur Theorie der Gitter,” Ann. Phys. (Leipzig) 44, 369–436 (1914).
    [CrossRef]
  11. V. Twersky, “On the scattering of waves by an infinite grating,” IEEE Trans. Antennas Propag. AP-4, 330–345 (1956).
    [CrossRef]
  12. V. Twersky, “Elementary function representations of Schlömilch series,” Arch. Ration. Mech. Anal. 8, 323–332 (1961).
    [CrossRef]
  13. R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, 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]
  14. R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
    [CrossRef]
  15. K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
    [CrossRef] [PubMed]
  16. K. M. Leung, Y. F. Li, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
    [CrossRef] [PubMed]
  17. J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
    [CrossRef] [PubMed]
  18. W. Wijngaard, “Guided normal modes of two parallel circular dielectric rods,” J. Opt. Soc. Am. 63, 944–950 (1973).
    [CrossRef]
  19. K. M. Lo, R. C. McPhedran, I. M. Bassett, G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
    [CrossRef]
  20. L.-M. Li, Z.-Q. Zhang, “Multiple-scattering approach to finite-sized photonic band-gap materials,” Phys. Rev. B 58, 9587–9590 (1998).
    [CrossRef]
  21. N. Stefanou, V. Karathanos, A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys.: Condens. Matter 4, 7389–7400 (1992).
  22. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1966).
  23. L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
    [CrossRef]
  24. Rayleigh, “On the influence of obstacles arranged in rectangular order upon the properties of a medium,” Philos. Mag. 34, 481–502 (1892).
    [CrossRef]
  25. R. C. McPhedran, N. A. Nicorovici, L. C. Botten, B. Ke-Da, “Green’s function, lattice sum and Rayleigh’s identity for a dynamic scattering problem,” in IMA Volumes in Mathematics and its Applications, Vol. 96, G. Papanicolaou, ed. (Springer-Verlag, New York, 1997), pp. 155–186.
  26. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chap. 7.
  27. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  28. K. Yasumoto, K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic Green’s function,” IEEE Trans. Antennas Propag. 47, 1050–1055 (1999).
    [CrossRef]
  29. R. Petit, “A Tutorial Introduction,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 1–52.
    [CrossRef]
  30. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  31. F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Čerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
    [CrossRef]
  32. D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
    [CrossRef]
  33. U. Grüning, V. Lehmann, S. Ottow, K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. 68, 747–749 (1996).
    [CrossRef]
  34. D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
    [CrossRef]
  35. H. S. Sözüer, J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231–239 (1994).
    [CrossRef]
  36. N. A. Nicorovici, R. C. McPhedran, “Lattice sums for off-axis electromagnetic scattering by gratings,” Phys. Rev. E 50, 3143–3160 (1994).
    [CrossRef]
  37. F. Oberhettinger, Fourier Expansions (Academic, New York, 1973), p. 33, Eqs. (3.17) and (3.19).

2000 (1)

1999 (3)

K. Yasumoto, K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic Green’s function,” IEEE Trans. Antennas Propag. 47, 1050–1055 (1999).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, 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]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

1998 (2)

L.-M. Li, Z.-Q. Zhang, “Multiple-scattering approach to finite-sized photonic band-gap materials,” Phys. Rev. B 58, 9587–9590 (1998).
[CrossRef]

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Čerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

1997 (1)

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

1996 (2)

U. Grüning, V. Lehmann, S. Ottow, K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
[CrossRef]

1994 (6)

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

H. S. Sözüer, J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231–239 (1994).
[CrossRef]

N. A. Nicorovici, R. C. McPhedran, “Lattice sums for off-axis electromagnetic scattering by gratings,” Phys. Rev. E 50, 3143–3160 (1994).
[CrossRef]

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. 11, 2526–2538 (1994).
[CrossRef]

K. M. Lo, R. C. McPhedran, I. M. Bassett, G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

1992 (2)

N. Stefanou, V. Karathanos, A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys.: Condens. Matter 4, 7389–7400 (1992).

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

1990 (2)

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

K. M. Leung, Y. F. Li, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

1985 (1)

J. Y. Suratteau, M. Cadilhac, R. Petit, “The perfectly conducting wire grating: computation of the diffracted field from Maxwell’s equations and Hamilton’s canonical system,” IEEE Trans. Antennas Propag. AP-33, 404–408 (1985).
[CrossRef]

1980 (2)

W. G. Chambers, C. L. Mok, T. J. Parker, “Theory of the scattering of electromagnetic waves by a regular grid of parallel cylindrical wires with circular cross section,” J. Phys. A 13, 1433–1441 (1980).
[CrossRef]

W. G. Chambers, C. L. Mok, T. J. Parker, “Theoretical study of the frequency response of some far-infrared interferometers with wire-grid beam dividers,” J. Phys. D 13, 515–526 (1980).
[CrossRef]

1978 (1)

1973 (1)

1962 (1)

T. Larsen, “A survey of the theory of wire grids,” IRE Trans. Microwave Theory Tech. MTT-10, 191–201 (1962).
[CrossRef]

1961 (1)

V. Twersky, “Elementary function representations of Schlömilch series,” Arch. Ration. Mech. Anal. 8, 323–332 (1961).
[CrossRef]

1956 (1)

V. Twersky, “On the scattering of waves by an infinite grating,” IEEE Trans. Antennas Propag. AP-4, 330–345 (1956).
[CrossRef]

1914 (1)

W. von Ignatowsky, “Zur Theorie der Gitter,” Ann. Phys. (Leipzig) 44, 369–436 (1914).
[CrossRef]

1892 (1)

Rayleigh, “On the influence of obstacles arranged in rectangular order upon the properties of a medium,” Philos. Mag. 34, 481–502 (1892).
[CrossRef]

Asatryan, A. A.

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, 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]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

Bassett, I. M.

K. M. Lo, R. C. McPhedran, I. M. Bassett, G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

Benisty, H.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Beunen, J.

Botten, L. C.

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, 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]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, B. Ke-Da, “Green’s function, lattice sum and Rayleigh’s identity for a dynamic scattering problem,” in IMA Volumes in Mathematics and its Applications, Vol. 96, G. Papanicolaou, ed. (Springer-Verlag, New York, 1997), pp. 155–186.

Busch, K.

U. Grüning, V. Lehmann, S. Ottow, K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

Cadilhac, M.

J. Y. Suratteau, M. Cadilhac, R. Petit, “The perfectly conducting wire grating: computation of the diffracted field from Maxwell’s equations and Hamilton’s canonical system,” IEEE Trans. Antennas Propag. AP-33, 404–408 (1985).
[CrossRef]

Chambers, W. G.

W. G. Chambers, C. L. Mok, T. J. Parker, “Theory of the scattering of electromagnetic waves by a regular grid of parallel cylindrical wires with circular cross section,” J. Phys. A 13, 1433–1441 (1980).
[CrossRef]

W. G. Chambers, C. L. Mok, T. J. Parker, “Theoretical study of the frequency response of some far-infrared interferometers with wire-grid beam dividers,” J. Phys. D 13, 515–526 (1980).
[CrossRef]

Chan, C. T.

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

de Sterke, C. M.

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, 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]

Dowling, J. P.

H. S. Sözüer, J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231–239 (1994).
[CrossRef]

Felbacq, D.

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. 11, 2526–2538 (1994).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chap. 7.

Grüning, U.

U. Grüning, V. Lehmann, S. Ottow, K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

Ho, K. M.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Horwitz, C. M.

Houdre, R.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Karathanos, V.

N. Stefanou, V. Karathanos, A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys.: Condens. Matter 4, 7389–7400 (1992).

Ke-Da, B.

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, B. Ke-Da, “Green’s function, lattice sum and Rayleigh’s identity for a dynamic scattering problem,” in IMA Volumes in Mathematics and its Applications, Vol. 96, G. Papanicolaou, ed. (Springer-Verlag, New York, 1997), pp. 155–186.

Kittel, C.

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1966).

Krauss, T. F.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Kroll, N.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

Labilloy, D.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Larsen, T.

T. Larsen, “A survey of the theory of wire grids,” IRE Trans. Microwave Theory Tech. MTT-10, 191–201 (1962).
[CrossRef]

Lehmann, V.

U. Grüning, V. Lehmann, S. Ottow, K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

Leung, K. M.

K. M. Leung, Y. F. Li, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

Li, L.

Li, L.-M.

L.-M. Li, Z.-Q. Zhang, “Multiple-scattering approach to finite-sized photonic band-gap materials,” Phys. Rev. B 58, 9587–9590 (1998).
[CrossRef]

Li, Y. F.

K. M. Leung, Y. F. Li, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

Lo, K. M.

K. M. Lo, R. C. McPhedran, I. M. Bassett, G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

MacKinnon, A.

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Maystre, D.

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. 11, 2526–2538 (1994).
[CrossRef]

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984), pp. 1–67.

McPhedran, R. C.

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, 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]

K. M. Lo, R. C. McPhedran, I. M. Bassett, G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

N. A. Nicorovici, R. C. McPhedran, “Lattice sums for off-axis electromagnetic scattering by gratings,” Phys. Rev. E 50, 3143–3160 (1994).
[CrossRef]

C. M. Horwitz, R. C. McPhedran, J. Beunen, “Interference and diffraction in globular metal films,” J. Opt. Soc. Am. 68, 1023–1031 (1978).
[CrossRef]

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, B. Ke-Da, “Green’s function, lattice sum and Rayleigh’s identity for a dynamic scattering problem,” in IMA Volumes in Mathematics and its Applications, Vol. 96, G. Papanicolaou, ed. (Springer-Verlag, New York, 1997), pp. 155–186.

Milton, G. W.

K. M. Lo, R. C. McPhedran, I. M. Bassett, G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

Modinos, A.

N. Stefanou, V. Karathanos, A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys.: Condens. Matter 4, 7389–7400 (1992).

Mok, C. L.

W. G. Chambers, C. L. Mok, T. J. Parker, “Theory of the scattering of electromagnetic waves by a regular grid of parallel cylindrical wires with circular cross section,” J. Phys. A 13, 1433–1441 (1980).
[CrossRef]

W. G. Chambers, C. L. Mok, T. J. Parker, “Theoretical study of the frequency response of some far-infrared interferometers with wire-grid beam dividers,” J. Phys. D 13, 515–526 (1980).
[CrossRef]

Montiel, F.

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Čerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chap. 7.

Nevière, M.

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Čerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

Nicorovici, N. A.

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, 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]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

N. A. Nicorovici, R. C. McPhedran, “Lattice sums for off-axis electromagnetic scattering by gratings,” Phys. Rev. E 50, 3143–3160 (1994).
[CrossRef]

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, B. Ke-Da, “Green’s function, lattice sum and Rayleigh’s identity for a dynamic scattering problem,” in IMA Volumes in Mathematics and its Applications, Vol. 96, G. Papanicolaou, ed. (Springer-Verlag, New York, 1997), pp. 155–186.

Oberhettinger, F.

F. Oberhettinger, Fourier Expansions (Academic, New York, 1973), p. 33, Eqs. (3.17) and (3.19).

Oesterle, U.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Ottow, S.

U. Grüning, V. Lehmann, S. Ottow, K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

Parker, T. J.

W. G. Chambers, C. L. Mok, T. J. Parker, “Theoretical study of the frequency response of some far-infrared interferometers with wire-grid beam dividers,” J. Phys. D 13, 515–526 (1980).
[CrossRef]

W. G. Chambers, C. L. Mok, T. J. Parker, “Theory of the scattering of electromagnetic waves by a regular grid of parallel cylindrical wires with circular cross section,” J. Phys. A 13, 1433–1441 (1980).
[CrossRef]

Pendry, J. B.

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Petit, R.

J. Y. Suratteau, M. Cadilhac, R. Petit, “The perfectly conducting wire grating: computation of the diffracted field from Maxwell’s equations and Hamilton’s canonical system,” IEEE Trans. Antennas Propag. AP-33, 404–408 (1985).
[CrossRef]

R. Petit, “A Tutorial Introduction,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 1–52.
[CrossRef]

Peyrot, P.

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Čerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

Rayleigh,

Rayleigh, “On the influence of obstacles arranged in rectangular order upon the properties of a medium,” Philos. Mag. 34, 481–502 (1892).
[CrossRef]

Robinson, P. A.

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, 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]

Schultz, S.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

Sigalas, M.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

Smith, D. R.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

Soukoulis, C. M.

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Sözüer, H. S.

H. S. Sözüer, J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231–239 (1994).
[CrossRef]

Stefanou, N.

N. Stefanou, V. Karathanos, A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys.: Condens. Matter 4, 7389–7400 (1992).

Suratteau, J. Y.

J. Y. Suratteau, M. Cadilhac, R. Petit, “The perfectly conducting wire grating: computation of the diffracted field from Maxwell’s equations and Hamilton’s canonical system,” IEEE Trans. Antennas Propag. AP-33, 404–408 (1985).
[CrossRef]

Tayeb, G.

D. Felbacq, G. Tayeb, D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. 11, 2526–2538 (1994).
[CrossRef]

Twersky, V.

V. Twersky, “Elementary function representations of Schlömilch series,” Arch. Ration. Mech. Anal. 8, 323–332 (1961).
[CrossRef]

V. Twersky, “On the scattering of waves by an infinite grating,” IEEE Trans. Antennas Propag. AP-4, 330–345 (1956).
[CrossRef]

von Ignatowsky, W.

W. von Ignatowsky, “Zur Theorie der Gitter,” Ann. Phys. (Leipzig) 44, 369–436 (1914).
[CrossRef]

Weisbuch, C.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Wijngaard, W.

Yasumoto, K.

K. Yasumoto, K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic Green’s function,” IEEE Trans. Antennas Propag. 47, 1050–1055 (1999).
[CrossRef]

Yoshitomi, K.

K. Yasumoto, K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic Green’s function,” IEEE Trans. Antennas Propag. 47, 1050–1055 (1999).
[CrossRef]

Zhang, Z.-Q.

L.-M. Li, Z.-Q. Zhang, “Multiple-scattering approach to finite-sized photonic band-gap materials,” Phys. Rev. B 58, 9587–9590 (1998).
[CrossRef]

Ann. Phys. (Leipzig) (1)

W. von Ignatowsky, “Zur Theorie der Gitter,” Ann. Phys. (Leipzig) 44, 369–436 (1914).
[CrossRef]

Appl. Phys. Lett. (3)

D. R. Smith, S. Schultz, N. Kroll, M. Sigalas, K. M. Ho, C. M. Soukoulis, “Experimental and theoretical results for a two-dimensional metal photonic band-gap cavity,” Appl. Phys. Lett. 65, 645–647 (1994).
[CrossRef]

U. Grüning, V. Lehmann, S. Ottow, K. Busch, “Macroporous silicon with a complete two-dimensional photonic band gap centered at 5 µm,” Appl. Phys. Lett. 68, 747–749 (1996).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdre, U. Oesterle, “Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensional photonic crystals,” Appl. Phys. Lett. 71, 738–740 (1997).
[CrossRef]

Arch. Ration. Mech. Anal. (1)

V. Twersky, “Elementary function representations of Schlömilch series,” Arch. Ration. Mech. Anal. 8, 323–332 (1961).
[CrossRef]

Aust. J. Phys. (1)

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, C. M. de Sterke, P. A. Robinson, “Ordered and disordered photonic band gap materials,” Aust. J. Phys. 52, 779–789 (1999).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

V. Twersky, “On the scattering of waves by an infinite grating,” IEEE Trans. Antennas Propag. AP-4, 330–345 (1956).
[CrossRef]

J. Y. Suratteau, M. Cadilhac, R. Petit, “The perfectly conducting wire grating: computation of the diffracted field from Maxwell’s equations and Hamilton’s canonical system,” IEEE Trans. Antennas Propag. AP-33, 404–408 (1985).
[CrossRef]

K. Yasumoto, K. Yoshitomi, “Efficient calculation of lattice sums for free-space periodic Green’s function,” IEEE Trans. Antennas Propag. 47, 1050–1055 (1999).
[CrossRef]

IRE Trans. Microwave Theory Tech. (1)

T. Larsen, “A survey of the theory of wire grids,” IRE Trans. Microwave Theory Tech. MTT-10, 191–201 (1962).
[CrossRef]

J. Lightwave Technol. (1)

K. M. Lo, R. C. McPhedran, I. M. Bassett, G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

J. Mod. Opt. (2)

H. S. Sözüer, J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231–239 (1994).
[CrossRef]

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Čerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

J. Opt. Soc. Am. (3)

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

J. Phys. A (1)

W. G. Chambers, C. L. Mok, T. J. Parker, “Theory of the scattering of electromagnetic waves by a regular grid of parallel cylindrical wires with circular cross section,” J. Phys. A 13, 1433–1441 (1980).
[CrossRef]

J. Phys. D (1)

W. G. Chambers, C. L. Mok, T. J. Parker, “Theoretical study of the frequency response of some far-infrared interferometers with wire-grid beam dividers,” J. Phys. D 13, 515–526 (1980).
[CrossRef]

J. Phys.: Condens. Matter (1)

N. Stefanou, V. Karathanos, A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys.: Condens. Matter 4, 7389–7400 (1992).

Philos. Mag. (1)

Rayleigh, “On the influence of obstacles arranged in rectangular order upon the properties of a medium,” Philos. Mag. 34, 481–502 (1892).
[CrossRef]

Phys. Rev. B (1)

L.-M. Li, Z.-Q. Zhang, “Multiple-scattering approach to finite-sized photonic band-gap materials,” Phys. Rev. B 58, 9587–9590 (1998).
[CrossRef]

Phys. Rev. E (2)

R. C. McPhedran, L. C. Botten, A. A. Asatryan, N. A. Nicorovici, P. A. Robinson, 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]

N. A. Nicorovici, R. C. McPhedran, “Lattice sums for off-axis electromagnetic scattering by gratings,” Phys. Rev. E 50, 3143–3160 (1994).
[CrossRef]

Phys. Rev. Lett. (3)

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

K. M. Leung, Y. F. Li, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

J. B. Pendry, A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Pure Appl. Opt. (1)

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

Other (8)

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXI, E. Wolf, ed. (North-Holland, Amsterdam, 1984), pp. 1–67.

F. Oberhettinger, Fourier Expansions (Academic, New York, 1973), p. 33, Eqs. (3.17) and (3.19).

J. P. Dowling, H. Everitt, E. Yablonovitch, “Photonic and acoustic band-gap bibliography,” web page http://home.earthlink.net/~jpdowling/pbgbib.html .

R. C. McPhedran, N. A. Nicorovici, L. C. Botten, B. Ke-Da, “Green’s function, lattice sum and Rayleigh’s identity for a dynamic scattering problem,” in IMA Volumes in Mathematics and its Applications, Vol. 96, G. Papanicolaou, ed. (Springer-Verlag, New York, 1997), pp. 155–186.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, Chap. 7.

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972).

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1966).

R. Petit, “A Tutorial Introduction,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 1–52.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Schematic of the structure considered here. It consists of gratings of cylinders of radii al and positions cl. The unit cell of the first grating is marked by the thick dashed line. We also show the incident, reflected (r), and transmitted (t) waves corresponding to the zeroth diffraction order.

Fig. 2
Fig. 2

Definitions of the various vectors and quantities used in formulating the problem, including the field point r=cl+rl, and source point r=cj+rj on separate cylinders lj.

Fig. 3
Fig. 3

Reflected field r and transmitted field t corresponding to a normally incident field δ on a composite stack consisting of a single layer added to a composite stack of s layers.

Fig. 4
Fig. 4

Evaluation of the global lattice sum S20Y for D=20 and λ=0.0675D. Solid partial sum with increasing number of terms N following from Eq. (8); dashed line, analytic summation formula (A9) with 100 terms.

Fig. 5
Fig. 5

Convergence of the series for the relative lattice sums for the difficult case for a 20-cylinder grating for which D=20, λ=0.0675D, and |clj|=10. Solid curve, real part of the partial sums of S20lj generated by direct summation of series (B1) with increasing numbers of terms N; dashed curve, partial sums from summation formula (B3) versus numbers of terms.

Fig. 6
Fig. 6

As Fig. 5, but for the imaginary part of the relative lattice sum.

Equations (115)

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k=(α0,-χ0, 0),
(2+k2)G(r)=n=-δ(r-nDxˆ) exp(iα0nD),
E(r+Dxˆ)=E(r) exp(iα0D).
G(r)=-i4n=-H0(1)(k|r-nDxˆ|) exp(iα0nD),
G(r)=12iDp=-1χpexp[i(αpx+χp|y|)],
αp=α0+2πpD=k sin θp,χp=k2-αp2,
G(r)=-i4H0(1)(k|r|)+m=-SmJm(k|r|)exp[-im arg(r)],
Sm=n0Hm(1)(|n|kD) exp(iα0nD) exp(imθn).
[2+k2(r)]V(r)=0,
·1(r)V(r)+k2V(r)=0.
V(r)=p=-χp-1/2[δp exp(-iχpy)+rp exp(iχpy)]exp(iαpx)p=-χp-1/2tp exp[i(αpx-χpy)],
V(rl)=m=-[AmlJm(krl)+BmlYm(krl)]×exp[im arg(rl)],
V(rl)=m=-CmlJm(kνlrl)exp[im arg(rl)],
Al=-MlBl,
 Mml=νlJm(νlkal)Ym(kal)-Jm(νlkal)Ym(kal)νlJm(νlkal)Jm(kal)-Jm(νlkal)Jm(kal)Jm(νlkal)Ym(kal)-νlJm(νlkal)Ym(kal)Jm(νlkal)Jm(kal)-νlJm(νlkal)Jm(kal),
U\ C[V(r)r2G(r; r)-G(r; r)r2V(r)]dAr
=UCV(r)nG(r; r)-G(r; r)nV(r)dsr,
G(r; r)=G(rl-rj-clj),
H0(1)(k|r-r|)=n=-Hn(1)(kr)Jn(kr)×exp[in(θ-θ)],
Jm(k|r-r|)=n=-Jm+n(kr)Jn(kr)exp[i(m+n)θ-inθ],
G(r-r)=-i4m=-Hm(1)(kr)Jm(kr)exp(imθ)exp(-imθ)+n=-Jn(kr)exp(inθ)s=-Sn-sJs(kr)×exp(-isθ).
G(rl-rj-clj)=-i4m=-Jm(krl)exp(imθl)
×s=-Sm-sljJs(krj)exp(-isθj),
Sqlj=n=-Hq(1)(k|nDxˆ+clj|)×exp(iα0nD)exp[iq arg (nDxˆ+clj)].
Sqlj=Hq(1)(k|clj|)+p=-(-1)pSq+pJp(k|clj|)(-1)qHq(1)(k|clj|)+p=-Sq+pJp(k|clj|),
V(rl)=-in=-BnlHn(1)(krl)exp(inθl)-in=-Jn(krl)exp(inθl)×m=-Sn-mB mj+jl m=-Sn-mljBml+p=-χp-1/2δp exp[i(αp x-χp y)].
exp[i(αpx-χpy)]=exp(iαpcl)×n=-(-1)n exp(-inθp)Jn(krl)×exp(inθl),
Anl=-ij=1Ncm=-Sn-mljBmj
+p=- exp(iαpcl)χp-1/2(-1)n
×exp(-inθp)δp,
rp=2iDχp-1/2m=- exp(-imθp)×l=1NcBml exp(-iαpcl),
tp=δp+2iDχp-1/2m=-(-1)m exp(imθp)×l=1NcBml exp(-iαpcl).
Cn,l=Anl+(-1)nA-nl,
Cn,l=Anl-(-1)nA-nl,
Dn,l=Bnl+(-1)nB-nl,
Dn,l=Bnl-(-1)nB-nl,
Cn,l=-MnlDn,l,
Cn,l=-MnlDn,l,
iMlD, l+j=1Ncσ,ljD, j=-iJelχ-1/2δ,
δ=[δp], χ=diag[χp],
=diag[m], el=diag[exp(iαpcl)],
D,l=[Dn,l],σ,lj=[σnm,lj],
J=[Jnp],
σnm,lj=Sn-mlj+(-1)mSn+mlj,
Jnp=exp(inθp)+(-1)n exp(-inθp),
m=1/2ifm=01ifm>0.
(σ˜+iM˜)D˜=-iJ˜E˜χ-1/2δ.
J˜=diag{J|l=1,, Nc},
E˜=[(e1)T,,(eNc)T]T,
D,l˜=[(D,1)T,,(D,Nc)T]T.
s=r+t=δ+2iDχ-1/2E˜HK˜D˜.
K˜=diag{K|l=1,, Nc},
Kpn=exp(-inθp)+(-1)n exp(inθp).
S=I-2Dχ-1/2E˜HK˜(σ˜+iM˜)-1J˜E˜χ-1/2.
S=-I+2Dχ-1/2E˜HK˜(σ˜+iM˜)-1J˜E˜χ-1/2,
ρ=12(S+S),
τ=12(S-S).
r=ρs+1δ+τs+1Pf+,
f-=τs+1δ+ρs+1Pf+.
f+=RsPf-,t=TsPf-.
r=[ρs+1+τs+1PRsP(I-ρs+1PRsP)-1τs+1]δ,
t=[TsP(I-ρs+1PRsP)-1τs+1]δ,
r=Rs+1δ,t=Ts+1δ.
R˜=P1/2RP1/2,T˜=P1/2T P1/2,
R˜s+1=ρ˜s+1+τ˜s+1R˜s(I-ρ˜s+1R˜s)-1τ˜s+1,
R˜0=0,
T˜s+1=T˜s(I-ρ˜s+1R˜s)-1τ˜s+1,
T˜0=I.
Rs+1=ρ+τ PRsP(I-ρPRsP)-1τ,
Ts+1=TsP(I-ρ PRsP)-1τ.
H0(1)(kx)+S0J0(kx)+2l=1SlJl(kx)
=2Dp=-exp(iαpx)χp.
χp=k2-αp2, pΩr{p|αp2<k2}iαp2-k2,pΩe{p|αp2>k2},
S2l=S2lJ+iS2lY,
S2l+1=iS2l+1J-S2l+1Y.
exp(iαpx)=2l=0lJ2l(kx)cos(2lθp)+2il=0J2l+1(kx)sin[(2l+1)θp],
S2lJ=-δl0+2DpΩrcos(2lθp)χp,
S2l+1J=2DpΩrsin[(2l+1)θp]χp.
S0Y=-2πlnk2K+γ+1πp=1p1+p=1p21p+2dp1+11|χp|-1pK+p2+11|χ-p|-1pK,
S2lY=1lπ-2Dp=0p1-p=-1-p2sin(2lθp)χp+1πm=1l(-1)m22m(l+m-1)!(2m)!(l-m)!×Kk2mB2m(α0/K)-(-1)l2Dp=p1+1exp(i2lθp)|χp|+p=p2+1exp(i2lθ-p)|χ-p|,
S2l+1Y=+2Dp=0p1-p=-1-p2cos[(2l+1)θp]χp
+2πm=0l(-1)m22m(l+m)!(2m+1)!(l-m)!×Kk2m+1B2m+1(α0/K)-(-1)l2Dp=p1+1exp[i(2l+1)θp]|χp|-p=p2+1exp[i(2l+1)θ-p]|χ-p|,
Sq(L, l)=n=-hq(nL+l)μnL,
hq(m)=Hq(1)(k|m|d)exp[iq arg(m)],
Sq(L, l)=hq(l)+r=-Jq-r(kld)Sr(L,0),
S(L, l)=[Sq(L, l)],H=[hq(l)],
J (l)=[J (l)qr]=[Jq-r(kld)],
S(L, l)=H(l)+J (l)S(L,0).
S(L, l+l)-H(l+l)=J(l)[S(L, l)-H(l)].
S(L, l+l)=J(l)S(L, l),
SqJ(L, L-j)=(-1)qμ-LSqJ(L, j)¯,
SqY(L, L-j)=(-1)qμ-LSqY(L, j)¯,
SqJ(L, l)=n=-jq(nL+l)μnL,
SqY(L, l)=n=-yq(nL+l)μnL,
Sq(L, L-j)=(-1)qμ-LSq(L, j).
Sq(L, l)+μL/2Sq(L, L/2+l)=Sq(L/2, l).
Sq+(L, l)=(-1)q[Sq+(L/2, L/2-l)-Sq+(L, L/2-l)],
Sq-(L, l)=-i(-1)q[Sq-(L/2, L/2-l)
-Sq-(L, L/2-l)],
Sq+(2L, l)=12[Sq+(L, l)+Sq-(L, l)],
Sq+(2L, L+l)=(-1)q2[Sq+(L, l)-Sq-(L, l)].
l=0L-1μl+jSq(L, l+j)=Sq(1, 1).
V(rP)=U++U-V(r)G(rP; r)n-G(rP; r)V(r)nds-CV(r)G(rP; r)r-G(rP; r)V(r)rds,
U+V(r)G(rP; r)n-G(rP; r)V(r)nds
=χ0-1/2 exp[i(α0x-χ0y)].
U-V(r)G(rP; r)n-G(rP; r)V(r)nds=0.
CG(rP; r)V(r)r-V(r)G(rP; r)rds
=p=- exp[i(αpx+χpy)]l=1N exp(-iαpcl)
×Clgp+(rP; r)V(r)r-V(r)gp+(rP; r)rds,
gp+(r)=12iDχpexp[-i(αpx+χpy)]=12iDχpm=-Jm(kr) exp(-imθp)×exp(-imθ).
rp=2iDχp-1/2m=- exp(-imθp)l=1NcBml exp(-iαpcl).
gp-(r)=12iDχpexp[-i(αpx-χpy)]
=12iDχpl=-(-1)lJl(kr)
×exp(ilθp)exp(-ilθ).
tp=δp0+2iDχp-1/2m=-(-1)m exp(imθp)×l=1NcBml exp(-iαpcl).

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