Abstract

The effects of finite beams and finite gratings on the performance of guided-mode resonant subwavelength gratings are characterized by using the rigorous boundary element method. The gratings are strongly modulated, have a finite number of periods, and are illuminated by normally incident Gaussian beams. Quantitative results are presented for silicon-on-sapphire resonant gratings and gallium arsenide–aluminum arsenide resonant gratings.

© 2001 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  34. K. Hirayama, E. N. Glytsis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907–917 (1997).
    [CrossRef]
  35. E. N. Glytsis, M. E. Harrigan, K. Hirayama, T. K. Gaylord, “Collimating cylindrical diffractive lenses: rigorous electromagnetic analysis and scalar approximation,” Appl. Opt. 37, 34–43 (1998).
    [CrossRef]
  36. J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A 15, 1822–1837 (1998).
    [CrossRef]
  37. J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A 16, 113–130 (1999).
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    [CrossRef]
  40. K. Hirayama, K. Igarashi, Y. Hayashi, E. N. Glytsis, T. K. Gaylord, “Finite-substrate-thickness cylindrical diffractive lenses: exact and approximate boundary element methods,” J. Opt. Soc. Am. A 16, 1294–1302 (1999).
    [CrossRef]
  41. K. Yashiro, S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
    [CrossRef]
  42. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  43. M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  44. J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Modeling considerations for rigorous boundary element method analysis of diffractive optical elements,” J. Opt. Soc. Am. A 18, 1495–1507 (2001).
    [CrossRef]
  45. A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidths with grating waveguide structures,” Appl. Phys. Lett. 69, 4154–4158 (1996).
    [CrossRef]
  46. S. Tibuleac, R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A 14, 1617–1626 (1997).
    [CrossRef]
  47. T. Tamir, S. Zhang, “Resonant scattering by multilayered dielectric gratings,” J. Opt. Soc. Am. A 14, 1607–1616 (1997).
    [CrossRef]
  48. D. Maystre, M. Nevière, P. Vincent, “On a general theory of anomalies and energy absorption by diffraction gratings and their relation with surface waves,” Opt. Acta 25, 905–915 (1978).
    [CrossRef]
  49. M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.

2001 (1)

2000 (3)

1999 (2)

1998 (4)

1997 (5)

1996 (6)

A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidths with grating waveguide structures,” Appl. Phys. Lett. 69, 4154–4158 (1996).
[CrossRef]

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

L. Zhuang, S. Schablitsky, R. C. Shi, S. Y. Chou, “Fabrication and performance of thin amorphous Si subwavelength transmission grating for controlling vertical cavity surface emitting laser polarization,” J. Vac. Sci. Technol. B 14, 4055–4057 (1996).
[CrossRef]

S. J. Schablitsky, L. Zhuang, R. C. Shi, S. Y. Chou, “Controlling polarization of vertical-cavity surface-emitting lasers using amorphous silicon subwavelength transmission gratings,” Appl. Phys. Lett. 69, 7–9 (1996).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
[CrossRef] [PubMed]

S. Peng, G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
[CrossRef]

1995 (4)

1993 (1)

1991 (2)

L. F. DeSandre, J. M. Elson, “Extinction-theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
[CrossRef]

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn. Part 2 74, 11–20 (1991).
[CrossRef]

1990 (1)

1989 (2)

I. A. Avrutskii, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

S. Zhang, T. Tamir, “Spatial modifications of Gaussian beams diffracted by reflection gratings,” J. Opt. Soc. Am. A 6, 1368–1381 (1989).
[CrossRef]

1988 (2)

I. A. Avrutskii, V. P. Duraev, E. T. Nedelin, A. M. Prokhorov, A. V. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Optimization of the characteristic of a dispersive element based on a corrugated waveguide,” Sov. J. Quantum Electron. 18, 362–365 (1988).
[CrossRef]

M. Shimizu, F. Koyama, K. Iga, “Polarization characteristics of MOCVD grown GaAs/GaAlAs CBH surface emitting lasers,” Jpn. J. Appl. Phys. Part 1 27, 1774–1775 (1988).
[CrossRef]

1987 (1)

I. A. Avrutskii, V. A. Sychugov, “Reflection of a bounded light beam from the surface of a periodically perturbed waveguide,” Sov. Phys. Tech. Phys. 32, 235–236 (1987).

1986 (2)

I. A. Avrutskii, G. A. Golubenko, V. A. Sychugov, A. V. Tishchenko, “Spectral and laser characteristics of a mirror with a corrugated waveguide on its surface,” Sov. J. Quantum Electron. 16, 1063–1065 (1986).
[CrossRef]

T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-reflectivity high spatial-frequency rectangular-groove dielectric surface-relief gratings,” Appl. Opt. 25, 4562–4567 (1986).
[CrossRef] [PubMed]

1985 (3)

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

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

K. Yashiro, S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

1983 (1)

1982 (1)

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

1978 (2)

K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
[CrossRef]

D. Maystre, M. Nevière, P. Vincent, “On a general theory of anomalies and energy absorption by diffraction gratings and their relation with surface waves,” Opt. Acta 25, 905–915 (1978).
[CrossRef]

1973 (2)

M. Nevière, M. Vincent, R. Petit, M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–245 (1973).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Avrutskii, I. A.

I. A. Avrutskii, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

I. A. Avrutskii, V. P. Duraev, E. T. Nedelin, A. M. Prokhorov, A. V. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Optimization of the characteristic of a dispersive element based on a corrugated waveguide,” Sov. J. Quantum Electron. 18, 362–365 (1988).
[CrossRef]

I. A. Avrutskii, V. A. Sychugov, “Reflection of a bounded light beam from the surface of a periodically perturbed waveguide,” Sov. Phys. Tech. Phys. 32, 235–236 (1987).

I. A. Avrutskii, G. A. Golubenko, V. A. Sychugov, A. V. Tishchenko, “Spectral and laser characteristics of a mirror with a corrugated waveguide on its surface,” Sov. J. Quantum Electron. 16, 1063–1065 (1986).
[CrossRef]

Baird, W. E.

Bendickson, J. M.

Botten, L. C.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

Boye, R. R.

Brundrett, D. L.

Cadilhac, M.

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, M. Vincent, R. Petit, M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–245 (1973).
[CrossRef]

Case, S. K.

Chou, S. Y.

S. J. Schablitsky, L. Zhuang, R. C. Shi, S. Y. Chou, “Controlling polarization of vertical-cavity surface-emitting lasers using amorphous silicon subwavelength transmission gratings,” Appl. Phys. Lett. 69, 7–9 (1996).
[CrossRef]

L. Zhuang, S. Schablitsky, R. C. Shi, S. Y. Chou, “Fabrication and performance of thin amorphous Si subwavelength transmission grating for controlling vertical cavity surface emitting laser polarization,” J. Vac. Sci. Technol. B 14, 4055–4057 (1996).
[CrossRef]

Cox, J. A.

J. A. Cox, R. A. Morgan, R. Wilke, C. Ford, “Guided-mode grating resonant filters for VCSEL applications,” in Diffractive and Holographic Device Technologies and Applications V, I. Cindrich, S. H. Lee, eds., Proc. SPIE3291, 70–76 (1998).
[CrossRef]

Craig, M. S.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

DeSandre, L. F.

Dunn, S. C.

Duraev, V. P.

I. A. Avrutskii, V. P. Duraev, E. T. Nedelin, A. M. Prokhorov, A. V. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Optimization of the characteristic of a dispersive element based on a corrugated waveguide,” Sov. J. Quantum Electron. 18, 362–365 (1988).
[CrossRef]

Elson, J. M.

Engel, H.

Enger, R. C.

Erdogan, T.

Ford, C.

J. A. Cox, R. A. Morgan, R. Wilke, C. Ford, “Guided-mode grating resonant filters for VCSEL applications,” in Diffractive and Holographic Device Technologies and Applications V, I. Cindrich, S. H. Lee, eds., Proc. SPIE3291, 70–76 (1998).
[CrossRef]

Friesem, A. A.

D. Rosenblatt, A. Sharon, A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
[CrossRef] [PubMed]

A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidths with grating waveguide structures,” Appl. Phys. Lett. 69, 4154–4158 (1996).
[CrossRef]

Gaylord, T. K.

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Modeling considerations for rigorous boundary element method analysis of diffractive optical elements,” J. Opt. Soc. Am. A 18, 1495–1507 (2001).
[CrossRef]

D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, J. M. Bendickson, “Effects of modulation strength in guided-mode resonant subwavelength gratings at normal incidence,” J. Opt. Soc. Am. A 17, 1221–1230 (2000).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A 16, 113–130 (1999).
[CrossRef]

K. Hirayama, K. Igarashi, Y. Hayashi, E. N. Glytsis, T. K. Gaylord, “Finite-substrate-thickness cylindrical diffractive lenses: exact and approximate boundary element methods,” J. Opt. Soc. Am. A 16, 1294–1302 (1999).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A 15, 1822–1837 (1998).
[CrossRef]

E. N. Glytsis, M. E. Harrigan, K. Hirayama, T. K. Gaylord, “Collimating cylindrical diffractive lenses: rigorous electromagnetic analysis and scalar approximation,” Appl. Opt. 37, 34–43 (1998).
[CrossRef]

D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, “Normal-incidence guided-mode resonant grating filters: design and experimental demonstration,” Opt. Lett. 23, 700–702 (1998).
[CrossRef]

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

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

M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
[CrossRef]

T. K. Gaylord, W. E. Baird, M. G. Moharam, “Zero-reflectivity high spatial-frequency rectangular-groove dielectric surface-relief gratings,” Appl. Opt. 25, 4562–4567 (1986).
[CrossRef] [PubMed]

Glytsis, E. N.

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Modeling considerations for rigorous boundary element method analysis of diffractive optical elements,” J. Opt. Soc. Am. A 18, 1495–1507 (2001).
[CrossRef]

D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, J. M. Bendickson, “Effects of modulation strength in guided-mode resonant subwavelength gratings at normal incidence,” J. Opt. Soc. Am. A 17, 1221–1230 (2000).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A 16, 113–130 (1999).
[CrossRef]

K. Hirayama, K. Igarashi, Y. Hayashi, E. N. Glytsis, T. K. Gaylord, “Finite-substrate-thickness cylindrical diffractive lenses: exact and approximate boundary element methods,” J. Opt. Soc. Am. A 16, 1294–1302 (1999).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A 15, 1822–1837 (1998).
[CrossRef]

E. N. Glytsis, M. E. Harrigan, K. Hirayama, T. K. Gaylord, “Collimating cylindrical diffractive lenses: rigorous electromagnetic analysis and scalar approximation,” Appl. Opt. 37, 34–43 (1998).
[CrossRef]

D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, “Normal-incidence guided-mode resonant grating filters: design and experimental demonstration,” Opt. Lett. 23, 700–702 (1998).
[CrossRef]

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

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

Golubenko, G. A.

I. A. Avrutskii, G. A. Golubenko, V. A. Sychugov, A. V. Tishchenko, “Spectral and laser characteristics of a mirror with a corrugated waveguide on its surface,” Sov. J. Quantum Electron. 16, 1063–1065 (1986).
[CrossRef]

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

Grann, E. B.

Harrigan, M. E.

Hayashi, Y.

Hirayama, K.

Ido, J.

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn. Part 2 74, 11–20 (1991).
[CrossRef]

Iga, K.

M. Shimizu, F. Koyama, K. Iga, “Polarization characteristics of MOCVD grown GaAs/GaAlAs CBH surface emitting lasers,” Jpn. J. Appl. Phys. Part 1 27, 1774–1775 (1988).
[CrossRef]

Igarashi, K.

Jacob, D. K.

Knop, K.

Kojima, T.

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn. Part 2 74, 11–20 (1991).
[CrossRef]

Koshiba, M.

M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.

Kostuk, R. K.

Koyama, F.

M. Shimizu, F. Koyama, K. Iga, “Polarization characteristics of MOCVD grown GaAs/GaAlAs CBH surface emitting lasers,” Jpn. J. Appl. Phys. Part 1 27, 1774–1775 (1988).
[CrossRef]

Magnusson, R.

Mait, J. N.

Mashev, L.

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

Maystre, D.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

D. Maystre, M. Nevière, P. Vincent, “On a general theory of anomalies and energy absorption by diffraction gratings and their relation with surface waves,” Opt. Acta 25, 905–915 (1978).
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

Mirotznik, M. S.

Moharam, M. G.

Morgan, R. A.

J. A. Cox, R. A. Morgan, R. Wilke, C. Ford, “Guided-mode grating resonant filters for VCSEL applications,” in Diffractive and Holographic Device Technologies and Applications V, I. Cindrich, S. H. Lee, eds., Proc. SPIE3291, 70–76 (1998).
[CrossRef]

Morris, G. M.

Nedelin, E. T.

I. A. Avrutskii, V. P. Duraev, E. T. Nedelin, A. M. Prokhorov, A. V. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Optimization of the characteristic of a dispersive element based on a corrugated waveguide,” Sov. J. Quantum Electron. 18, 362–365 (1988).
[CrossRef]

Nevière, M.

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

D. Maystre, M. Nevière, P. Vincent, “On a general theory of anomalies and energy absorption by diffraction gratings and their relation with surface waves,” Opt. Acta 25, 905–915 (1978).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, M. Vincent, R. Petit, M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–245 (1973).
[CrossRef]

Noponen, E.

J. Saarinen, E. Noponen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2565 (1995).
[CrossRef]

Norton, S. M.

Ohkawa, S.

K. Yashiro, S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

Peng, S.

Petit, R.

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, M. Vincent, R. Petit, M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–245 (1973).
[CrossRef]

Pommet, D. A.

Popov, E.

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

Prather, D. W.

Prokhorov, A. M.

I. A. Avrutskii, V. P. Duraev, E. T. Nedelin, A. M. Prokhorov, A. V. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Optimization of the characteristic of a dispersive element based on a corrugated waveguide,” Sov. J. Quantum Electron. 18, 362–365 (1988).
[CrossRef]

Rosenblatt, D.

D. Rosenblatt, A. Sharon, A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
[CrossRef] [PubMed]

A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidths with grating waveguide structures,” Appl. Phys. Lett. 69, 4154–4158 (1996).
[CrossRef]

Saarinen, J.

J. Saarinen, E. Noponen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2565 (1995).
[CrossRef]

Schablitsky, S.

L. Zhuang, S. Schablitsky, R. C. Shi, S. Y. Chou, “Fabrication and performance of thin amorphous Si subwavelength transmission grating for controlling vertical cavity surface emitting laser polarization,” J. Vac. Sci. Technol. B 14, 4055–4057 (1996).
[CrossRef]

Schablitsky, S. J.

S. J. Schablitsky, L. Zhuang, R. C. Shi, S. Y. Chou, “Controlling polarization of vertical-cavity surface-emitting lasers using amorphous silicon subwavelength transmission gratings,” Appl. Phys. Lett. 69, 7–9 (1996).
[CrossRef]

Sharon, A.

D. Rosenblatt, A. Sharon, A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
[CrossRef] [PubMed]

A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidths with grating waveguide structures,” Appl. Phys. Lett. 69, 4154–4158 (1996).
[CrossRef]

Shi, R. C.

L. Zhuang, S. Schablitsky, R. C. Shi, S. Y. Chou, “Fabrication and performance of thin amorphous Si subwavelength transmission grating for controlling vertical cavity surface emitting laser polarization,” J. Vac. Sci. Technol. B 14, 4055–4057 (1996).
[CrossRef]

S. J. Schablitsky, L. Zhuang, R. C. Shi, S. Y. Chou, “Controlling polarization of vertical-cavity surface-emitting lasers using amorphous silicon subwavelength transmission gratings,” Appl. Phys. Lett. 69, 7–9 (1996).
[CrossRef]

Shimizu, M.

M. Shimizu, F. Koyama, K. Iga, “Polarization characteristics of MOCVD grown GaAs/GaAlAs CBH surface emitting lasers,” Jpn. J. Appl. Phys. Part 1 27, 1774–1775 (1988).
[CrossRef]

Steingrueber, R.

Svakhin, A. S.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

Svakhin, A. V.

I. A. Avrutskii, V. P. Duraev, E. T. Nedelin, A. M. Prokhorov, A. V. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Optimization of the characteristic of a dispersive element based on a corrugated waveguide,” Sov. J. Quantum Electron. 18, 362–365 (1988).
[CrossRef]

Sychugov, V. A.

I. A. Avrutskii, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

I. A. Avrutskii, V. P. Duraev, E. T. Nedelin, A. M. Prokhorov, A. V. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Optimization of the characteristic of a dispersive element based on a corrugated waveguide,” Sov. J. Quantum Electron. 18, 362–365 (1988).
[CrossRef]

I. A. Avrutskii, V. A. Sychugov, “Reflection of a bounded light beam from the surface of a periodically perturbed waveguide,” Sov. Phys. Tech. Phys. 32, 235–236 (1987).

I. A. Avrutskii, G. A. Golubenko, V. A. Sychugov, A. V. Tishchenko, “Spectral and laser characteristics of a mirror with a corrugated waveguide on its surface,” Sov. J. Quantum Electron. 16, 1063–1065 (1986).
[CrossRef]

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

Tamir, T.

Tibuleac, S.

Tishchenko, A. V.

I. A. Avrutskii, V. P. Duraev, E. T. Nedelin, A. M. Prokhorov, A. V. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Optimization of the characteristic of a dispersive element based on a corrugated waveguide,” Sov. J. Quantum Electron. 18, 362–365 (1988).
[CrossRef]

I. A. Avrutskii, G. A. Golubenko, V. A. Sychugov, A. V. Tishchenko, “Spectral and laser characteristics of a mirror with a corrugated waveguide on its surface,” Sov. J. Quantum Electron. 16, 1063–1065 (1986).
[CrossRef]

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

Vincent, M.

M. Nevière, M. Vincent, R. Petit, M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–245 (1973).
[CrossRef]

Vincent, P.

D. Maystre, M. Nevière, P. Vincent, “On a general theory of anomalies and energy absorption by diffraction gratings and their relation with surface waves,” Opt. Acta 25, 905–915 (1978).
[CrossRef]

Wang, S. S.

Weber, H. G.

Wilke, R.

J. A. Cox, R. A. Morgan, R. Wilke, C. Ford, “Guided-mode grating resonant filters for VCSEL applications,” in Diffractive and Holographic Device Technologies and Applications V, I. Cindrich, S. H. Lee, eds., Proc. SPIE3291, 70–76 (1998).
[CrossRef]

Yashiro, K.

K. Yashiro, S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

Zhang, S.

Zhuang, L.

L. Zhuang, S. Schablitsky, R. C. Shi, S. Y. Chou, “Fabrication and performance of thin amorphous Si subwavelength transmission grating for controlling vertical cavity surface emitting laser polarization,” J. Vac. Sci. Technol. B 14, 4055–4057 (1996).
[CrossRef]

S. J. Schablitsky, L. Zhuang, R. C. Shi, S. Y. Chou, “Controlling polarization of vertical-cavity surface-emitting lasers using amorphous silicon subwavelength transmission gratings,” Appl. Phys. Lett. 69, 7–9 (1996).
[CrossRef]

Appl. Opt. (6)

Appl. Phys. Lett. (2)

S. J. Schablitsky, L. Zhuang, R. C. Shi, S. Y. Chou, “Controlling polarization of vertical-cavity surface-emitting lasers using amorphous silicon subwavelength transmission gratings,” Appl. Phys. Lett. 69, 7–9 (1996).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, “Narrow spectral bandwidths with grating waveguide structures,” Appl. Phys. Lett. 69, 4154–4158 (1996).
[CrossRef]

Electron. Commun. Jpn. Part 2 (1)

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn. Part 2 74, 11–20 (1991).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Rosenblatt, A. Sharon, A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1997).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. Yashiro, S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

J. Mod. Opt. (1)

I. A. Avrutskii, V. A. Sychugov, “Reflection of a beam of finite size from a corrugated waveguide,” J. Mod. Opt. 36, 1527–1539 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (18)

D. K. Jacob, S. C. Dunn, M. G. Moharam, “Design considerations for narrow-band dielectric resonant grating reflection filters of finite length,” J. Opt. Soc. Am. A 17, 1241–1249 (2000).
[CrossRef]

L. F. DeSandre, J. M. Elson, “Extinction-theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
[CrossRef]

S. Zhang, T. Tamir, “Spatial modifications of Gaussian beams diffracted by reflection gratings,” J. Opt. Soc. Am. A 6, 1368–1381 (1989).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A 15, 1822–1837 (1998).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A 16, 113–130 (1999).
[CrossRef]

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

D. L. Brundrett, E. N. Glytsis, T. K. Gaylord, J. M. Bendickson, “Effects of modulation strength in guided-mode resonant subwavelength gratings at normal incidence,” J. Opt. Soc. Am. A 17, 1221–1230 (2000).
[CrossRef]

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

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

S. S. Wang, R. Magnusson, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7, 1464–1468 (1990).
[CrossRef]

S. Peng, G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
[CrossRef]

S. M. Norton, G. M. Morris, T. Erdogan, “Experimental investigations of resonant-grating filter lineshapes in comparison with theoretical models,” J. Opt. Soc. Am. A 15, 464–472 (1998).
[CrossRef]

M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
[CrossRef]

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Modeling considerations for rigorous boundary element method analysis of diffractive optical elements,” J. Opt. Soc. Am. A 18, 1495–1507 (2001).
[CrossRef]

S. Tibuleac, R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A 14, 1617–1626 (1997).
[CrossRef]

T. Tamir, S. Zhang, “Resonant scattering by multilayered dielectric gratings,” J. Opt. Soc. Am. A 14, 1607–1616 (1997).
[CrossRef]

K. Hirayama, K. Igarashi, Y. Hayashi, E. N. Glytsis, T. K. Gaylord, “Finite-substrate-thickness cylindrical diffractive lenses: exact and approximate boundary element methods,” J. Opt. Soc. Am. A 16, 1294–1302 (1999).
[CrossRef]

J. Vac. Sci. Technol. B (1)

L. Zhuang, S. Schablitsky, R. C. Shi, S. Y. Chou, “Fabrication and performance of thin amorphous Si subwavelength transmission grating for controlling vertical cavity surface emitting laser polarization,” J. Vac. Sci. Technol. B 14, 4055–4057 (1996).
[CrossRef]

Jpn. J. Appl. Phys. Part 1 (1)

M. Shimizu, F. Koyama, K. Iga, “Polarization characteristics of MOCVD grown GaAs/GaAlAs CBH surface emitting lasers,” Jpn. J. Appl. Phys. Part 1 27, 1774–1775 (1988).
[CrossRef]

Opt. Acta (2)

R. C. McPhedran, L. C. Botten, M. S. Craig, M. Nevière, D. Maystre, “Lossy lamellar gratings in the quasistatic limit,” Opt. Acta 29, 289–312 (1982).
[CrossRef]

D. Maystre, M. Nevière, P. Vincent, “On a general theory of anomalies and energy absorption by diffraction gratings and their relation with surface waves,” Opt. Acta 25, 905–915 (1978).
[CrossRef]

Opt. Commun. (3)

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

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

M. Nevière, M. Vincent, R. Petit, M. Cadilhac, “Determination of the coupling coefficient of a holographic thin film coupler,” Opt. Commun. 9, 240–245 (1973).
[CrossRef]

Opt. Eng. (1)

J. Saarinen, E. Noponen, “Guided-mode resonance filters of finite aperture,” Opt. Eng. 34, 2560–2565 (1995).
[CrossRef]

Opt. Lett. (2)

Sov. J. Quantum Electron. (3)

I. A. Avrutskii, V. P. Duraev, E. T. Nedelin, A. M. Prokhorov, A. V. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Optimization of the characteristic of a dispersive element based on a corrugated waveguide,” Sov. J. Quantum Electron. 18, 362–365 (1988).
[CrossRef]

I. A. Avrutskii, G. A. Golubenko, V. A. Sychugov, A. V. Tishchenko, “Spectral and laser characteristics of a mirror with a corrugated waveguide on its surface,” Sov. J. Quantum Electron. 16, 1063–1065 (1986).
[CrossRef]

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

Sov. Phys. Tech. Phys. (1)

I. A. Avrutskii, V. A. Sychugov, “Reflection of a bounded light beam from the surface of a periodically perturbed waveguide,” Sov. Phys. Tech. Phys. 32, 235–236 (1987).

Other (4)

J. A. Cox, R. A. Morgan, R. Wilke, C. Ford, “Guided-mode grating resonant filters for VCSEL applications,” in Diffractive and Holographic Device Technologies and Applications V, I. Cindrich, S. H. Lee, eds., Proc. SPIE3291, 70–76 (1998).
[CrossRef]

D. L. Brundrett, “Analysis, design, and applications of subwavelength diffraction gratings,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Georgia, 1997).

R. Magnusson, S. S. Wang, “Optical waveguide-grating filters,” in International Conference on Holography, Correlation Optics, and Recording Materials, O. V. Angelsky, ed., Proc. SPIE2108, 380–390 (1993).
[CrossRef]

M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.

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Figures (12)

Fig. 1
Fig. 1

Geometry used to model the diffraction of a Gaussian incident beam from a finite SoS GMR-SWG. The boundaries Γ1 and Γ2 separate the silicon waveguide-grating region S2 from the air-filled incident region S1 and the sapphire substrate region S3. The normally incident Gaussian beam has a beam waist of w0, and the GMR-SWG has period Λ, filling factor F, grating depth d1, and diameter D.

Fig. 2
Fig. 2

Zero-order reflectivity versus free-space wavelength for SoS GMR-SWG #1 with F=0.62 and d1=410 nm for a normally incident plane wave, calculated by using RCWA, and for 2-D Gaussian beams with w0=16, 6, 4, and 2 μm, calculated by using the BEM.

Fig. 3
Fig. 3

Zero-order reflectivity versus free-space wavelength for SoS GMR-SWG #2 with F=0.57 and d1=530 nm. In each plot, curves are shown for a normally incident plane wave, calculated by using RCWA, and for 2-D Gaussian beams with w0=12, 8, and 4 μm. The reflectivities for the Gaussian beams were calculated by using FB-RCWA for (a) D= and by using the multiple-layer BEM for (b) D=25 periods, (c) D=21 periods, (d) D=17 periods, and (e) D=13 periods.

Fig. 4
Fig. 4

Geometry used to model the diffraction of a Gaussian incident beam from a finite GaAs-AlAs GMR-SWG. The boundaries Γ1 and Γ2 separate the GaAs waveguide-grating region S2 from the AlAs incident region S1 and the air-filled transmitted region S3. The normally incident Gaussian wave has a beam waist of w0, and the GMR-SWG has period Λ, filling factor F, grating depth d1, unmodulated layer depth d2, and diameter D.

Fig. 5
Fig. 5

Reflectivity for λ0=850 nm as a function of unmodulated layer depth and filling factor for (a) GaAs-AlAs GMR-SWG #1 with d1=50 nm and (b) GaAs-AlAs GMR-SWG #2 with d1=250 nm.

Fig. 6
Fig. 6

2-D diffracted field intensity patterns for TE0 and TE1 GaAs-AlAs GMR-SWGs with d1=250 nm, illuminated by an incident Gaussian beam with w0=6 μm and λ0=850 nm.

Fig. 7
Fig. 7

(a) Spectral line shapes for plane-wave illumination of the infinitely periodic GaAs-AlAs GMR-SWG #1 for TE polarization (θ=0° and θ=1.5°) and TM polarization (θ=0°). (b) Angular line shapes for the GaAs-AlAs GMR-SWG #1 for TE and TM polarization and λ0=850 nm. Plots (c) and (d) are the same as plots (a) and (b), respectively, but are for the infinitely periodic GaAs-AlAs GMR-SWG #2.

Fig. 8
Fig. 8

Zero-order reflectivity versus free-space wavelength for GaAs-AlAs GMR-SWG #1 with F=0.33, d1=50 nm, and d2=85.73 nm. In each plot, curves are shown for a normally incident plane wave, calculated by using RCWA, and for 2-D Gaussian beams with w0=12, 8, 4, and 2 μm. The reflectivities for the Gaussian beams were calculated by using FB-RCWA for (a) D= and by using the multiple-layer BEM for (b) D=65 periods, (c) D=53 periods, (d) D=41 periods, and (e) D=29 periods.

Fig. 9
Fig. 9

2-D diffracted field intensity patterns for GaAs-AlAs GMR-SWG #1 for different beam waists and grating sizes. The beam waist and the grating size are w0=12 μm and D=65 periods in (a) and (b), w0=4 μm and D=65 periods in (c) and (d), and w0=12 μm and D=41 periods in (e) and (f). Plots (a), (c), and (e) are for nonresonant excitation, while plots (b), (d), and (f) are for resonant excitation.

Fig. 10
Fig. 10

Brillouin diagram for GaAs-AlAs GMR-SWG #1 with F=0.33, d1=50 nm, and d2=85.73 nm. Also shown is reflectivity versus wavelength for a Gaussian beam with w0=12 μm and for a slightly off-normal plane wave with θ=0.41°. The final two curves denoted in the legend represent the pole proximity for leaky modes propagating in the ±x directions for plane-wave excitation with θ=0.41°.

Fig. 11
Fig. 11

Zero-order reflectivity versus free-space wavelength for the infinitely periodic GaAs-AlAs GMR-SWG #2 with F=0.67, d1=250 nm, and d2=82 nm. In each plot, curves are shown for a normally incident plane wave, calculated by using RCWA, and for 2-D Gaussian beams with w0=6, 4, 2, and 1 μm. The reflectivities for the Gaussian beams were calculated by using FB-RCWA for (a) D= and by using the multiple-layer BEM for (b) D=41 periods, (c) D=27 periods, and (d) D=13 periods.

Fig. 12
Fig. 12

2-D diffracted field intensity patterns for the 53-period GaAs-AlAs GMR-SWG #2 illuminated by Gaussian beams with (a) w0=6 μm and (b) w0=2 μm.

Tables (5)

Tables Icon

Table 1 Peak Reflectivity of SoS GMR-SWG #2 for a Range of Incident Gaussian Beam Waists As Determined by FB-RCWA for an Infinite Grating and by the BEM for a Range of Finite Grating Sizes

Tables Icon

Table 2 Resonance Wavelength of GaAs-AlAs GMR-SWG #1 with d2=50 nm for a Range of Incident Gaussian Beam Waists As Determined by FB-RCWA for an Infinite Grating and by the BEM for a Range of Finite Grating Sizes

Tables Icon

Table 3 Peak Reflectivity of GaAs-AlAs GMR-SWG #1 with d2=50 nm for a Range of Incident Gaussian Beam Waists As Determined by FB-RCWA for an Infinite Grating and by the BEM for a Range of Finite Grating Sizes

Tables Icon

Table 4 Resonance Wavelength of GaAs-AlAs GMR-SWG #2 with d2=250 nm for a Range of Incident Gaussian Beam Waists As Determined by FB-RCWA for an Infinite Grating and by the BEM for a Range of Finite Grating Sizes

Tables Icon

Table 5 Peak Reflectivity of GaAs-AlAs GMR-SWG #2 with d2=250 nm for a Range of Incident Gaussian Beam Waists As Determined by FB-RCWA for an Infinite Grating and by the BEM for a Range of Finite Grating Sizes

Equations (19)

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

θdiv=4λ0πw0n1,
|R0|21(k0n1sin θ+K-βR)2+βI2×1(k0n1sin θ-K+βR)2+βI2.
ϕ1t(r1)=Γ1ϕΓ1(rΓ1) G1n1 (r1, rΓ1)-p1G1(r1, rΓ1)ψΓ1(rΓ1)dl1+ϕinc(r1),
r1S1,
ϕit(ri)=Γ1ϕΓi(rΓi) Gini (ri, rΓi)-piGi(ri, rΓi)ψΓi(rΓi)dli-Γi-1ϕΓi-1(rΓi-1) Gini-1 (ri, rΓi-1)-piGi(ri, rΓi-1)ψΓi-1(rΓi-1)dli-1,
riSi,
ϕN+1t(rN+1)=-ΓNϕΓN(rΓN) GN+1nN (rN+1, rΓN)-pN+1×GN+1(rN+1, rΓN)ψΓN(rΓN)dlN,
rN+1SN+1,
Gi(ri, rΓ)=(-j/4)H0(2)(ki|ri-rΓ|)
(i=1,2,, N+1),
ϕit(rΓi)=ϕi+1t(rΓi)ϕΓi(rΓi),
1piϕitni (rΓi)=1pi+1ϕi+1tni (rΓi)ψΓi(rΓi).
θΓ12π-1ϕ1t(rΓ1)+Γ1 ϕΓ1(rΓ1) G1n1 (rΓ1,rΓ1)-p1G1(rΓ1,rΓ1)ψΓ1(rΓ1)dl1=-ϕinc(rΓ1),
θΓi-12πϕi-1t(rΓi-1)+Γi-1  ϕΓi-1(rΓi-1) Gini-1 (rΓi-1, rΓi-1)-piGi(rΓi-1, rΓi-1)ψΓi-1(rΓi-1)dli-1-ΓiϕΓi(rΓi) Gini (rΓi-1, rΓi)-piGi(rΓi-1, rΓi)ψΓi(rΓi)dli=0,
θΓi2π-1ϕit(rΓi)-Γi-1ϕΓi-1(rΓi-1) Gini-1 (rΓi, rΓi-1)-piGi(rΓi, rΓi-1)ψΓi-1(rΓi-1)dli-1+Γi ϕΓi(rΓi) Gini (rΓi, rΓi)-piGi(rΓi, rΓi)ψΓi(rΓi)dli=0,
θΓN2πϕNt(rΓN)+ΓN ϕΓN(rΓN) GN+1nN (rΓN, rΓN)-pN+1GN+1(rΓN, rΓN)ψΓN(rΓN)dlN=0,
ϕΓi={M}T{ϕΓi}e,
ψΓi={M}T{ψΓi}e,
{M}T=[-ξ(1-ξ)/21-ξ2ξ(1+ξ)/2],

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