Abstract

Recently [Opt. Lett. 25, 1092 (2000)], two of the present authors proposed extending the domain of applicability of grating theories to aperiodic structures, especially the diffraction structures that are encountered in integrated optics. This extension was achieved by introduction of virtual periodicity and incorporation of artificial absorbers at the boundaries of the elementary cells of periodic structures. Refinements and extensions of that previous research are presented. Included is a thorough discussion of the effect of the absorber quality on the accuracy of the computational results, with highly accurate computational results being achieved with perfectly matched layer absorbers. The extensions are concerned with the diversity of diffraction waveguide problems to which the method is applied. These problems include two-dimensional classical problems such as those involving Bragg mirrors and grating couplers that may be difficult to model because of the length of the components and three-dimensional problems such as those involving integrated diffraction gratings, photonic crystal waveguides, and waveguide airbridge microcavities. Rigorous coupled-wave analysis (also called the Fourier modal method) is used to support the analysis, but we believe that the approach is applicable to other grating theories. The method is tested both against available numerical data obtained with finite-difference techniques and against experimental data. Excellent agreement is obtained. A comparison in terms of convergence speed with the finite-difference modal method that is widely used in waveguide theory confirms the relevancy of the approach. Consequently, a simple, efficient, and stable method that may also be applied to waveguide and grating diffraction problems is proposed.

© 2001 Optical Society of America

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References

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  1. T. Itoh, ed., Numerical Techniques for Microwave and Millimeter-Wave Passive Structures (Wiley, New York, 1989).
  2. L. Li, “Recent advances and present limitations of the electromagnetic theory of diffraction gratings,” in Diffractive Optics and Micro-Optics, 2000 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 2–4.
  3. 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]
  4. F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
    [CrossRef]
  5. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995), Chap. 7.
  6. Ph. Lalanne, E. Silberstein, “Fourier-modal method applied to waveguide computational problems,” Opt. Lett. 25, 1092–1094 (2000).
    [CrossRef]
  7. Ph. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  8. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  9. See, for instance, 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]
  10. G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  11. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  12. E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
    [CrossRef]
  13. Ph. Lalanne, “Effective properties and band structure of lamellar subwavelength crystals: plane-wave method revisited,” Phys. Rev. B 58, 9801–9807 (1998).
    [CrossRef]
  14. Ph. Lalanne, J. P. Hugonin, “Numerical performance of finite-difference modal methods for the electromagnetic analysis of one-dimensional lamellar gratings,” J. Opt. Soc. Am. A 17, 1033–1042 (2000).
    [CrossRef]
  15. J. P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [CrossRef]
  16. See, for instance, E. A. Marengo, C. M. Rappaport, E. L. Miller, “Optimum PML ABC conductivity profile in FDTD,” IEEE Trans. Magn. 35, 1506–1509 (1999).
    [CrossRef]
  17. S. F. Helfert, R. Pregla, “Efficient analysis of periodic structures,” J. Lightwave Technol. 16, 1694–1702 (1998).
    [CrossRef]
  18. R. Pregla, W. Pasher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structure, T. Itoh, ed. (Wiley, New York, 1989), pp. 381–446.
  19. Ph. Lalanne, M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for TM polarization,” J. Mod. Opt. 45, 1357–1374 (1998).
    [CrossRef]
  20. D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. De La Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, C. Jouanin, “Quantitative measurement of transmission, reflection and diffraction of two-dimensional photonic bandgap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
    [CrossRef]
  21. M. S. D. Smith, K. A. McGreer, “Diffraction gratings utilizing total internal reflection facets in Littrow configuration,” IEEE Photon. Technol. Lett. 11, 84–86 (1999).
    [CrossRef]
  22. J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Kosa, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 µm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
    [CrossRef]
  23. Ph. Lalanne, H. Benisty, “Out-of-plane losses of two-dimensional photonic crystal waveguides: electromagnetic analysis,” J. Appl. Phys. 89, 1512–1514 (2001).
    [CrossRef]
  24. D. J. Ripin, K. Y. Lim, G. S. Petrich, P. R. Villeneuve, S. Fan, E. R. Thoen, J. D. Joannopoulos, E. P. Ippen, L. A. Kolodziejski, “Photonic band gap airbridge microcavity resonances in GaAs/AlxOy waveguides,” J. Appl. Phys. 87, 1578–1580 (2000).
    [CrossRef]
  25. Ph. Lalanne, D. Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2086 (1996).
    [CrossRef]
  26. M. Aubourg, S. Verdeyme, P. Guillon, “Finite element software for microwave engineering,” in Microwave Passive Devices, T. Itoh, G. Pelosi, P. P. Silvester, eds. (Wiley, New York, 1996), Chap. 1.
  27. C. H. Henry, B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightwave Technol. 7, 308–313 (1989).
    [CrossRef]
  28. J. Rodriguez, R. D. Crespo, S. Fernandez, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibanez, J. M. Virgos, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Eng. 38, 1896–1906 (1999).
    [CrossRef]
  29. P. Vahihama, J. Turunen, Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1998), p. 69.

2001 (1)

Ph. Lalanne, H. Benisty, “Out-of-plane losses of two-dimensional photonic crystal waveguides: electromagnetic analysis,” J. Appl. Phys. 89, 1512–1514 (2001).
[CrossRef]

2000 (4)

1999 (3)

See, for instance, E. A. Marengo, C. M. Rappaport, E. L. Miller, “Optimum PML ABC conductivity profile in FDTD,” IEEE Trans. Magn. 35, 1506–1509 (1999).
[CrossRef]

M. S. D. Smith, K. A. McGreer, “Diffraction gratings utilizing total internal reflection facets in Littrow configuration,” IEEE Photon. Technol. Lett. 11, 84–86 (1999).
[CrossRef]

J. Rodriguez, R. D. Crespo, S. Fernandez, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibanez, J. M. Virgos, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Eng. 38, 1896–1906 (1999).
[CrossRef]

1998 (3)

S. F. Helfert, R. Pregla, “Efficient analysis of periodic structures,” J. Lightwave Technol. 16, 1694–1702 (1998).
[CrossRef]

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

Ph. Lalanne, “Effective properties and band structure of lamellar subwavelength crystals: plane-wave method revisited,” Phys. Rev. B 58, 9801–9807 (1998).
[CrossRef]

1997 (2)

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. De La Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, C. Jouanin, “Quantitative measurement of transmission, reflection and diffraction of two-dimensional photonic bandgap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
[CrossRef]

1996 (5)

1995 (1)

1994 (2)

1991 (1)

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Kosa, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 µm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

1989 (1)

C. H. Henry, B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightwave Technol. 7, 308–313 (1989).
[CrossRef]

Andreadakis, N. C.

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Kosa, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 µm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

Aubourg, M.

M. Aubourg, S. Verdeyme, P. Guillon, “Finite element software for microwave engineering,” in Microwave Passive Devices, T. Itoh, G. Pelosi, P. P. Silvester, eds. (Wiley, New York, 1996), Chap. 1.

Bardinal, V.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. De La Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, C. Jouanin, “Quantitative measurement of transmission, reflection and diffraction of two-dimensional photonic bandgap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

Benisty, H.

Ph. Lalanne, H. Benisty, “Out-of-plane losses of two-dimensional photonic crystal waveguides: electromagnetic analysis,” J. Appl. Phys. 89, 1512–1514 (2001).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. De La Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, C. Jouanin, “Quantitative measurement of transmission, reflection and diffraction of two-dimensional photonic bandgap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

Bérenger, J. P.

J. P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

Bhat, R.

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Kosa, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 µm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

Carrasco, S.

J. Rodriguez, R. D. Crespo, S. Fernandez, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibanez, J. M. Virgos, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Eng. 38, 1896–1906 (1999).
[CrossRef]

Cassagne, D.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. De La Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, C. Jouanin, “Quantitative measurement of transmission, reflection and diffraction of two-dimensional photonic bandgap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

Crespo, R. D.

J. Rodriguez, R. D. Crespo, S. Fernandez, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibanez, J. M. Virgos, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Eng. 38, 1896–1906 (1999).
[CrossRef]

De La Rue, R. M.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. De La Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, C. Jouanin, “Quantitative measurement of transmission, reflection and diffraction of two-dimensional photonic bandgap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

Fan, S.

D. J. Ripin, K. Y. Lim, G. S. Petrich, P. R. Villeneuve, S. Fan, E. R. Thoen, J. D. Joannopoulos, E. P. Ippen, L. A. Kolodziejski, “Photonic band gap airbridge microcavity resonances in GaAs/AlxOy waveguides,” J. Appl. Phys. 87, 1578–1580 (2000).
[CrossRef]

Fernandez, S.

J. Rodriguez, R. D. Crespo, S. Fernandez, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibanez, J. M. Virgos, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Eng. 38, 1896–1906 (1999).
[CrossRef]

Gaylord, T. K.

Granet, G.

Grann, E. B.

Guillon, P.

M. Aubourg, S. Verdeyme, P. Guillon, “Finite element software for microwave engineering,” in Microwave Passive Devices, T. Itoh, G. Pelosi, P. P. Silvester, eds. (Wiley, New York, 1996), Chap. 1.

Guizal, B.

Helfert, S. F.

Henry, C. H.

C. H. Henry, B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightwave Technol. 7, 308–313 (1989).
[CrossRef]

Houdré, R.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. De La Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, C. Jouanin, “Quantitative measurement of transmission, reflection and diffraction of two-dimensional photonic bandgap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

Hugonin, J. P.

Ibanez, I.

J. Rodriguez, R. D. Crespo, S. Fernandez, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibanez, J. M. Virgos, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Eng. 38, 1896–1906 (1999).
[CrossRef]

Ippen, E. P.

D. J. Ripin, K. Y. Lim, G. S. Petrich, P. R. Villeneuve, S. Fan, E. R. Thoen, J. D. Joannopoulos, E. P. Ippen, L. A. Kolodziejski, “Photonic band gap airbridge microcavity resonances in GaAs/AlxOy waveguides,” J. Appl. Phys. 87, 1578–1580 (2000).
[CrossRef]

Joannopoulos, J. D.

D. J. Ripin, K. Y. Lim, G. S. Petrich, P. R. Villeneuve, S. Fan, E. R. Thoen, J. D. Joannopoulos, E. P. Ippen, L. A. Kolodziejski, “Photonic band gap airbridge microcavity resonances in GaAs/AlxOy waveguides,” J. Appl. Phys. 87, 1578–1580 (2000).
[CrossRef]

Jouanin, C.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. De La Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, C. Jouanin, “Quantitative measurement of transmission, reflection and diffraction of two-dimensional photonic bandgap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

Jurek, M. P.

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

Kolodziejski, L. A.

D. J. Ripin, K. Y. Lim, G. S. Petrich, P. R. Villeneuve, S. Fan, E. R. Thoen, J. D. Joannopoulos, E. P. Ippen, L. A. Kolodziejski, “Photonic band gap airbridge microcavity resonances in GaAs/AlxOy waveguides,” J. Appl. Phys. 87, 1578–1580 (2000).
[CrossRef]

Kosa, M. A.

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Kosa, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 µm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

Krauss, T. F.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. De La Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, C. Jouanin, “Quantitative measurement of transmission, reflection and diffraction of two-dimensional photonic bandgap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

Labilloy, D.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. De La Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, C. Jouanin, “Quantitative measurement of transmission, reflection and diffraction of two-dimensional photonic bandgap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

Lalanne, D.

Ph. Lalanne, D. Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2086 (1996).
[CrossRef]

Lalanne, Ph.

Ph. Lalanne, H. Benisty, “Out-of-plane losses of two-dimensional photonic crystal waveguides: electromagnetic analysis,” J. Appl. Phys. 89, 1512–1514 (2001).
[CrossRef]

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

Ph. Lalanne, E. Silberstein, “Fourier-modal method applied to waveguide computational problems,” Opt. Lett. 25, 1092–1094 (2000).
[CrossRef]

Ph. Lalanne, “Effective properties and band structure of lamellar subwavelength crystals: plane-wave method revisited,” Phys. Rev. B 58, 9801–9807 (1998).
[CrossRef]

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

Ph. Lalanne, D. Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2086 (1996).
[CrossRef]

Ph. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

LeBlanc, H. P.

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Kosa, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 µm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

Li, L.

Lim, K. Y.

D. J. Ripin, K. Y. Lim, G. S. Petrich, P. R. Villeneuve, S. Fan, E. R. Thoen, J. D. Joannopoulos, E. P. Ippen, L. A. Kolodziejski, “Photonic band gap airbridge microcavity resonances in GaAs/AlxOy waveguides,” J. Appl. Phys. 87, 1578–1580 (2000).
[CrossRef]

Marengo, E. A.

See, for instance, E. A. Marengo, C. M. Rappaport, E. L. Miller, “Optimum PML ABC conductivity profile in FDTD,” IEEE Trans. Magn. 35, 1506–1509 (1999).
[CrossRef]

McGreer, K. A.

M. S. D. Smith, K. A. McGreer, “Diffraction gratings utilizing total internal reflection facets in Littrow configuration,” IEEE Photon. Technol. Lett. 11, 84–86 (1999).
[CrossRef]

Miller, E. L.

See, for instance, E. A. Marengo, C. M. Rappaport, E. L. Miller, “Optimum PML ABC conductivity profile in FDTD,” IEEE Trans. Magn. 35, 1506–1509 (1999).
[CrossRef]

Moharam, M. G.

Montiel, F.

Morris, G. M.

Nevière, M.

Oesterle, U.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. De La Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, C. Jouanin, “Quantitative measurement of transmission, reflection and diffraction of two-dimensional photonic bandgap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

Olivares, J.

J. Rodriguez, R. D. Crespo, S. Fernandez, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibanez, J. M. Virgos, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Eng. 38, 1896–1906 (1999).
[CrossRef]

Pandavenes, J.

J. Rodriguez, R. D. Crespo, S. Fernandez, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibanez, J. M. Virgos, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Eng. 38, 1896–1906 (1999).
[CrossRef]

Pasher, W.

R. Pregla, W. Pasher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structure, T. Itoh, ed. (Wiley, New York, 1989), pp. 381–446.

Petrich, G. S.

D. J. Ripin, K. Y. Lim, G. S. Petrich, P. R. Villeneuve, S. Fan, E. R. Thoen, J. D. Joannopoulos, E. P. Ippen, L. A. Kolodziejski, “Photonic band gap airbridge microcavity resonances in GaAs/AlxOy waveguides,” J. Appl. Phys. 87, 1578–1580 (2000).
[CrossRef]

Pommet, D. A.

Popov, E.

Pregla, R.

S. F. Helfert, R. Pregla, “Efficient analysis of periodic structures,” J. Lightwave Technol. 16, 1694–1702 (1998).
[CrossRef]

R. Pregla, W. Pasher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structure, T. Itoh, ed. (Wiley, New York, 1989), pp. 381–446.

Rappaport, C. M.

See, for instance, E. A. Marengo, C. M. Rappaport, E. L. Miller, “Optimum PML ABC conductivity profile in FDTD,” IEEE Trans. Magn. 35, 1506–1509 (1999).
[CrossRef]

Ripin, D. J.

D. J. Ripin, K. Y. Lim, G. S. Petrich, P. R. Villeneuve, S. Fan, E. R. Thoen, J. D. Joannopoulos, E. P. Ippen, L. A. Kolodziejski, “Photonic band gap airbridge microcavity resonances in GaAs/AlxOy waveguides,” J. Appl. Phys. 87, 1578–1580 (2000).
[CrossRef]

Rodriguez, J.

J. Rodriguez, R. D. Crespo, S. Fernandez, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibanez, J. M. Virgos, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Eng. 38, 1896–1906 (1999).
[CrossRef]

Scherer, A.

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Kosa, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 µm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

Silberstein, E.

Smith, M. S. D.

M. S. D. Smith, K. A. McGreer, “Diffraction gratings utilizing total internal reflection facets in Littrow configuration,” IEEE Photon. Technol. Lett. 11, 84–86 (1999).
[CrossRef]

Soole, J. B. D.

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Kosa, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 µm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

Taflove, A.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995), Chap. 7.

Thoen, E. R.

D. J. Ripin, K. Y. Lim, G. S. Petrich, P. R. Villeneuve, S. Fan, E. R. Thoen, J. D. Joannopoulos, E. P. Ippen, L. A. Kolodziejski, “Photonic band gap airbridge microcavity resonances in GaAs/AlxOy waveguides,” J. Appl. Phys. 87, 1578–1580 (2000).
[CrossRef]

Turunen, J.

P. Vahihama, J. Turunen, Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1998), p. 69.

Vahihama, P.

P. Vahihama, J. Turunen, Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1998), p. 69.

Verbeek, B. H.

C. H. Henry, B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightwave Technol. 7, 308–313 (1989).
[CrossRef]

Verdeyme, S.

M. Aubourg, S. Verdeyme, P. Guillon, “Finite element software for microwave engineering,” in Microwave Passive Devices, T. Itoh, G. Pelosi, P. P. Silvester, eds. (Wiley, New York, 1996), Chap. 1.

Villeneuve, P. R.

D. J. Ripin, K. Y. Lim, G. S. Petrich, P. R. Villeneuve, S. Fan, E. R. Thoen, J. D. Joannopoulos, E. P. Ippen, L. A. Kolodziejski, “Photonic band gap airbridge microcavity resonances in GaAs/AlxOy waveguides,” J. Appl. Phys. 87, 1578–1580 (2000).
[CrossRef]

Virgos, J. M.

J. Rodriguez, R. D. Crespo, S. Fernandez, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibanez, J. M. Virgos, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Eng. 38, 1896–1906 (1999).
[CrossRef]

Weisbuch, C.

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. De La Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, C. Jouanin, “Quantitative measurement of transmission, reflection and diffraction of two-dimensional photonic bandgap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

Appl. Phys. Lett. (1)

J. B. D. Soole, A. Scherer, H. P. LeBlanc, N. C. Andreadakis, R. Bhat, M. A. Kosa, “Monolithic InP/InGaAsP/InP grating spectrometer for the 1.48–1.56 µm wavelength range,” Appl. Phys. Lett. 58, 1949–1951 (1991).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

M. S. D. Smith, K. A. McGreer, “Diffraction gratings utilizing total internal reflection facets in Littrow configuration,” IEEE Photon. Technol. Lett. 11, 84–86 (1999).
[CrossRef]

IEEE Trans. Magn. (1)

See, for instance, E. A. Marengo, C. M. Rappaport, E. L. Miller, “Optimum PML ABC conductivity profile in FDTD,” IEEE Trans. Magn. 35, 1506–1509 (1999).
[CrossRef]

J. Appl. Phys. (2)

Ph. Lalanne, H. Benisty, “Out-of-plane losses of two-dimensional photonic crystal waveguides: electromagnetic analysis,” J. Appl. Phys. 89, 1512–1514 (2001).
[CrossRef]

D. J. Ripin, K. Y. Lim, G. S. Petrich, P. R. Villeneuve, S. Fan, E. R. Thoen, J. D. Joannopoulos, E. P. Ippen, L. A. Kolodziejski, “Photonic band gap airbridge microcavity resonances in GaAs/AlxOy waveguides,” J. Appl. Phys. 87, 1578–1580 (2000).
[CrossRef]

J. Comput. Phys. (1)

J. P. Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[CrossRef]

J. Lightwave Technol. (2)

S. F. Helfert, R. Pregla, “Efficient analysis of periodic structures,” J. Lightwave Technol. 16, 1694–1702 (1998).
[CrossRef]

C. H. Henry, B. H. Verbeek, “Solution of the scalar wave equation for arbitrarily shaped dielectric waveguides by two-dimensional Fourier analysis,” J. Lightwave Technol. 7, 308–313 (1989).
[CrossRef]

J. Mod. Opt. (2)

Ph. Lalanne, D. Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2086 (1996).
[CrossRef]

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

J. Opt. Soc. Am. A (9)

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

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
[CrossRef]

Ph. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
[CrossRef]

See, for instance, 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]

G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
[CrossRef]

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

E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

Opt. Eng. (1)

J. Rodriguez, R. D. Crespo, S. Fernandez, J. Pandavenes, J. Olivares, S. Carrasco, I. Ibanez, J. M. Virgos, “Radiation losses on discontinuities in integrated optical waveguides,” Opt. Eng. 38, 1896–1906 (1999).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (1)

Ph. Lalanne, “Effective properties and band structure of lamellar subwavelength crystals: plane-wave method revisited,” Phys. Rev. B 58, 9801–9807 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

D. Labilloy, H. Benisty, C. Weisbuch, T. F. Krauss, R. M. De La Rue, V. Bardinal, R. Houdré, U. Oesterle, D. Cassagne, C. Jouanin, “Quantitative measurement of transmission, reflection and diffraction of two-dimensional photonic bandgap structures at near-infrared wavelengths,” Phys. Rev. Lett. 79, 4147–4150 (1997).
[CrossRef]

Other (6)

R. Pregla, W. Pasher, “The method of lines,” in Numerical Techniques for Microwave and Millimeter Wave Passive Structure, T. Itoh, ed. (Wiley, New York, 1989), pp. 381–446.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, Mass., 1995), Chap. 7.

T. Itoh, ed., Numerical Techniques for Microwave and Millimeter-Wave Passive Structures (Wiley, New York, 1989).

L. Li, “Recent advances and present limitations of the electromagnetic theory of diffraction gratings,” in Diffractive Optics and Micro-Optics, 2000 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 2–4.

P. Vahihama, J. Turunen, Diffractive Optics and Micro-Optics, Vol. 10 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1998), p. 69.

M. Aubourg, S. Verdeyme, P. Guillon, “Finite element software for microwave engineering,” in Microwave Passive Devices, T. Itoh, G. Pelosi, P. P. Silvester, eds. (Wiley, New York, 1996), Chap. 1.

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Figures (10)

Fig. 1
Fig. 1

General layered waveguide diffraction. Electromagnetic fields are computed as eigenmodes in every layer, and the field in each layer is represented as a superposition of left- and right-propagating modes with amplitudes u(p) and d(p).

Fig. 2
Fig. 2

Artificial periodicity of the waveguide and adjunction of absorbers between waveguide structures.

Fig. 3
Fig. 3

Investigation of waveguide diffraction for studying the effectiveness of several absorbers.

Fig. 4
Fig. 4

Comparison of the method of lines (solid curves) and the present method (stars) for a Bragg reflector. (a) Reflectivity computed as a function of the number of periods. (b) Reflectivity for 1024 periods computed as a function of the number of eigenmodes retained for the computation.

Fig. 5
Fig. 5

Grating coupler diffraction.

Fig. 6
Fig. 6

Angular spectrum intensity |ep(θ)|2 of the light diffracted in air by the grating coupler for a 20λ grating length and for TE polarization. Thin, solid curve, present method; filled circles, the sinc-squared function of Eq. (16). Inset, enlarged view near θ=25°; the filled circles and the thin curve are superimposed at this scale.

Fig. 7
Fig. 7

Photonic-crystal waveguide geometry. The waveguide structure is periodic in the x direction, has a finite arbitrary extent in the z direction, and is illuminated by the fundamental mode of the unpatterned waveguide (incidence angle, θ).

Fig. 8
Fig. 8

Comparison of the experimental transmission measurement (noisy curves) and theoretical predictions (solid curves) for photonic-crystal diffraction as shown in Fig. 7. (a) ΓK direction, (b) ΓM direction.

Fig. 9
Fig. 9

Convergence performance of the present method for the photonic-crystal waveguide.

Fig. 10
Fig. 10

Air-bridge microcavity: (a) Dimensions of the photonic bandgap air-bridge microcavity. Each square hole has a 250-nm side. (b) Computational box and absorbers. (c) Theoretical transmission of the photonic bandgap air-bridge microcavity. Thicker curve, the present method; thin curve, the finite-element modal method with perfectly conducting walls.

Tables (2)

Tables Icon

Table 1 Reflectivity for the Waveguide-Diffraction Problem of Fig. 2

Tables Icon

Table 2 Convergence Performance for the Air–Bridge Problem

Equations (23)

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2Hy(p)z2=-k02εxx(x)μ(x)Hy(p)-εxx(x) x 1εzz(x) Hy(p)x,
Hy(p)=m=- Um(z)exp(jmKx),
2Uz2=k02[A-1(KxE-1Kx-B)][U],
Up(z)=m Wm(p){um(p) -k0λm(p)(z-zp-1)+dm(p) expk0λm(p)(z-zp-1)},
Ex(p)=jμ0ε01/2m=-+ Sxm exp(jmKx)
Sxp(z)=m Vm(p){um(p) exp-k0λm(p)(z-zp-1)-dm(p) expk0λm(p)(z-zp-1)},
u(p+1)d(0)=S(p)u(0)d(p+1),
S(p)=Tuu(p)Rud(p)Rdu(p)Tdd(p).
Rud(p)=rud(p)+tuu(p)Rud(p-1)[1-rdu(p)Rud(p-1)]-1tdd(p),
Tdd(p)=Tdd(p-1)[1-rdu(p)Rud(p-1)]-1tdd(p).
tuu(p)=2[W(p)-1W(p+1)+V(p)-1V(p+1)]-1X(p),
rud(p)=[W(p)-1W(p+1)+V(p)-1V(p+1)]-1[W(p)-1W(p+1)-V(p)-1V(p+1)],
rdu(p)=X(p)[V(p+1)-1V(p)+W(p+1)-1W(p)]-1[V(p+1)-1V(p)-W(p+1)-1W(p)]X(p),
tdd(p)=2X(p)[V(p+1)-1V(p)+W(p+1)-1W(p)]-1.
Rm=|um(n+1)|2Wm(n+1),Vm(n+1)/Wi(n+1),Vi(n+1),
Tm=|dm(0)|2Wm(0),Vm(0)/Wi(n+1),Vi(n+1).
S(N)=p=1,2 apSp(2p),
d(p)=½X(p){(W(p)-1W(p+1)-V(p)-1V(p+1))u(p+1)+[W(p)-1W(p+1)+V(p)-1V(p+1)]d(p+1)},
u(p)=Rud(p-1)d(p).
E(z)=-+[ep(kz)+em(kz)]exp(jkzz)dkz,
|ep(θ)|2=A sinc2[πL/λ cos(θ0)(θ-θ0)],
H=l,m(Uxlmx+Uylmy+Uzlmz)×exp[-j(kx+lKx)x-jmKyy],
H=l,m(Uxlmx+Uylmy+Uzlmz)×exp[-jlKxx-jmKyy],

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