Abstract

Reflection, diffraction and transmission of optical waves at the interface between a photonic crystal and the surrounding air can be described by propagating and evanescent Bloch modes. We have found such modes for one of the canonical two-dimensional photonic crystals, identical circular cylinders in a square pattern. We present computed out-of-plane band diagrams for propagating as well as evanescent modes, obtained with a numerical method based on Fourier-Bessel expansions. For a given frequency, all the modes are evanescent, except for a few low-order propagating modes. We find that most of the evanescent modes have a purely imaginary z-component of the Bloch wave vector, but many of the modes have a complex z-component.

© 2009 Optical Society of America

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  1. S. Hadzialic, S. Kim, A. Sudbo, and O. Solgaard, "Displacement Sensing with a Mechanically Tunable Photonic Crystal," in The 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society, pp. 345-346 (IEEE, 2007).
  2. S. Kim, S. Hadzialic, A. Sudbo, and O. Solgaard, "Single-film Broadband Photonic Crystal Micro-mirror with Large Angular Range and Low Polarization Dependence," in Conference on Lasers and Electro-optics (CLEO 2007). Baltimore, Maryland, USA, paper CThP7.
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    [CrossRef]
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    [CrossRef]
  6. M. Davanc¸Y. Urzhumov, and G. Shvets, "The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction," Opt. Express 15, 9681-9691 (2007).
    [CrossRef] [PubMed]
  7. K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195,111 (2004).
    [CrossRef]
  8. H. van der Lem, A. Tip, and A. Moroz, "Band structure of absorptive two-dimensional photonic crystals," Solid State Commun. 129, 475-478 (2004).
  9. R. Smaâli, D. Felbacq, and G. Granet, "Bloch waves and non-propagating modes in photonic crystals," Physica E 18, 443-451 (2003).
    [CrossRef]
  10. Y.-C. Hsue and T.-J. Yang, "Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal," Phys. Rev. B 70, 016,706 (2004).
  11. E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195,122 (2005).
    [CrossRef]
  12. Y.-C. Hsue, A. J. Freeman, and B.-Y. Gu, "Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals," Phys. Rev. B 72, 195,118 (2005).
    [CrossRef]
  13. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, New Jersey, 2008).
  14. N. N. Rao, Elements of Engineering Electromagnetics (Prentice Hall, 2004).
  15. A. S. Sudbo, "Improved formulation of the film mode matching method for mode field calculations in dielectric waveguides," Pure Appl. Opt. 3, 381-388 (1994).
    [CrossRef]
  16. S. Peng and G. M. Morris, "Resonant scattering from two-dimensional gratings," J. Opt. Soc. Am. A 13, 993-1005 (1996).
    [CrossRef]
  17. K. C. Johnson, "Grating Diffraction Calculator (GD-Calc®) - Coupled-Wave Theory for Biperiodic Diffraction Gratings," http://software.kjinnovation.com (2006).
  18. E. Yamashita, S. Ozeki, and K. Atsuki, "Modal analysis method for optical fibers with symmetrically distributed multiple cores," J. Lightwave Technol. 2, 341-346 (1985).
    [CrossRef]
  19. T. A. Birks, J. C. Knight, and P. S. J. Russell, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997).
    [CrossRef] [PubMed]
  20. K. Saitoh and M. Koshiba, "Numerical Modeling of Photonic Crystal Fibers," J. Lightwave Technol. 23, 3580-3590 (2005).
    [CrossRef]
  21. J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic Crystal Fibers: A New Class of Optical Waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
    [CrossRef]
  22. A. Ortega-Moñux, J. G. Wangüemert-Pérez, and I. Molina-Fernández, "Accurate Analysis of Photonic Crystal Fibers by Means of the Fast-Fourier-Based Mode Solver," IEEE Photon. Technol. Lett. 19, 414-416 (2007).
    [CrossRef]
  23. J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices, chap. 2.3 (Springer-Verlag, Berlin Heidelberg, 2005).
  24. G. Tayeb and D. Maystre, "Rigorous theoretical study of finite-size two-dimensional photonic crystals doped by microcavities," J. Opt. Soc. Am. A 14, 3323-3332 (1997).
    [CrossRef]
  25. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, "Multipole method for microstructured optical fibers. I. Formulation," J. Opt. Soc. Am. B 19, 2322-2330 (2002).
    [CrossRef]
  26. COMSOL Multiphysics®, COMSOL AB, Stockholm, Sweden. http://www.comsol.com.
  27. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, "Improving accuracy by subpixel smoothing in FDTD," Opt. Lett. 31, 2972-2974 (2006).
    [CrossRef] [PubMed]
  28. G. P. Agrawal, Fiber-Optic Communication Systems, chap. 2.2, 2nd ed. (John Wiley & Sons, New York, 1997).
  29. M. T. Heath, Scientific Computing: An Introductory Survey (McGraw-Hill, Singapore, 1997).
  30. A. S. Sudbo, "Why are accurate computations of mode fields in rectangular dielectric waveguides difficult?" J. Lightwave Technol. 10, 418-419 (1992).
    [CrossRef]
  31. S. Peng and G. M. Morris, "Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings," J. Opt. Soc. Am. A 12, 1087-1096 (1995).
    [CrossRef]

2007 (2)

A. Ortega-Moñux, J. G. Wangüemert-Pérez, and I. Molina-Fernández, "Accurate Analysis of Photonic Crystal Fibers by Means of the Fast-Fourier-Based Mode Solver," IEEE Photon. Technol. Lett. 19, 414-416 (2007).
[CrossRef]

M. Davanc¸Y. Urzhumov, and G. Shvets, "The complex Bloch bands of a 2D plasmonic crystal displaying isotropic negative refraction," Opt. Express 15, 9681-9691 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (3)

K. Saitoh and M. Koshiba, "Numerical Modeling of Photonic Crystal Fibers," J. Lightwave Technol. 23, 3580-3590 (2005).
[CrossRef]

E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195,122 (2005).
[CrossRef]

Y.-C. Hsue, A. J. Freeman, and B.-Y. Gu, "Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals," Phys. Rev. B 72, 195,118 (2005).
[CrossRef]

2004 (3)

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195,111 (2004).
[CrossRef]

H. van der Lem, A. Tip, and A. Moroz, "Band structure of absorptive two-dimensional photonic crystals," Solid State Commun. 129, 475-478 (2004).

Y.-C. Hsue and T.-J. Yang, "Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal," Phys. Rev. B 70, 016,706 (2004).

2003 (1)

R. Smaâli, D. Felbacq, and G. Granet, "Bloch waves and non-propagating modes in photonic crystals," Physica E 18, 443-451 (2003).
[CrossRef]

2002 (1)

1999 (2)

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic Crystal Fibers: A New Class of Optical Waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

D. M. Whittaker and I. S. Culshaw, "Scattering-matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

1997 (2)

1996 (1)

1995 (1)

1994 (1)

A. S. Sudbo, "Improved formulation of the film mode matching method for mode field calculations in dielectric waveguides," Pure Appl. Opt. 3, 381-388 (1994).
[CrossRef]

1992 (1)

A. S. Sudbo, "Why are accurate computations of mode fields in rectangular dielectric waveguides difficult?" J. Lightwave Technol. 10, 418-419 (1992).
[CrossRef]

1988 (1)

K. A. Zaki, S.-W. Chen, and C. Chen, "Modeling Discontinuities in Dielectric-Loaded Waveguides," IEEE Trans. Microwave Theory Tech. 36, 1804-1810 (1988).
[CrossRef]

1985 (1)

E. Yamashita, S. Ozeki, and K. Atsuki, "Modal analysis method for optical fibers with symmetrically distributed multiple cores," J. Lightwave Technol. 2, 341-346 (1985).
[CrossRef]

Atsuki, K.

E. Yamashita, S. Ozeki, and K. Atsuki, "Modal analysis method for optical fibers with symmetrically distributed multiple cores," J. Lightwave Technol. 2, 341-346 (1985).
[CrossRef]

Barkou, S. E.

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic Crystal Fibers: A New Class of Optical Waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Bermel, P.

Bienstman, P.

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195,111 (2004).
[CrossRef]

Birks, T. A.

Bjarklev, A.

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic Crystal Fibers: A New Class of Optical Waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Botten, L. C.

Broeng, J.

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic Crystal Fibers: A New Class of Optical Waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Burr, G.

Chen, C.

K. A. Zaki, S.-W. Chen, and C. Chen, "Modeling Discontinuities in Dielectric-Loaded Waveguides," IEEE Trans. Microwave Theory Tech. 36, 1804-1810 (1988).
[CrossRef]

Chen, S.-W.

K. A. Zaki, S.-W. Chen, and C. Chen, "Modeling Discontinuities in Dielectric-Loaded Waveguides," IEEE Trans. Microwave Theory Tech. 36, 1804-1810 (1988).
[CrossRef]

Culshaw, I. S.

D. M. Whittaker and I. S. Culshaw, "Scattering-matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

Davanc¸, M.

de Sterke, C. M.

Fan, S.

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195,111 (2004).
[CrossRef]

Farjadpour, A.

Felbacq, D.

R. Smaâli, D. Felbacq, and G. Granet, "Bloch waves and non-propagating modes in photonic crystals," Physica E 18, 443-451 (2003).
[CrossRef]

Freeman, A. J.

Y.-C. Hsue, A. J. Freeman, and B.-Y. Gu, "Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals," Phys. Rev. B 72, 195,118 (2005).
[CrossRef]

Granet, G.

R. Smaâli, D. Felbacq, and G. Granet, "Bloch waves and non-propagating modes in photonic crystals," Physica E 18, 443-451 (2003).
[CrossRef]

Green, A. A.

E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195,122 (2005).
[CrossRef]

Gu, B.-Y.

Y.-C. Hsue, A. J. Freeman, and B.-Y. Gu, "Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals," Phys. Rev. B 72, 195,118 (2005).
[CrossRef]

Hsue, Y.-C.

Y.-C. Hsue, A. J. Freeman, and B.-Y. Gu, "Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals," Phys. Rev. B 72, 195,118 (2005).
[CrossRef]

Y.-C. Hsue and T.-J. Yang, "Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal," Phys. Rev. B 70, 016,706 (2004).

Huang, K. C.

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195,111 (2004).
[CrossRef]

Ibanescu, M.

Istrate, E.

E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195,122 (2005).
[CrossRef]

Jiang, X.

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195,111 (2004).
[CrossRef]

Joannopoulos, J. D.

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, "Improving accuracy by subpixel smoothing in FDTD," Opt. Lett. 31, 2972-2974 (2006).
[CrossRef] [PubMed]

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195,111 (2004).
[CrossRef]

Johnson, S. G.

Knight, J. C.

Koshiba, M.

Kuhlmey, B. T.

Lidorikis, E.

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195,111 (2004).
[CrossRef]

Maystre, D.

McPhedran, R. C.

Mogilevstev, D.

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic Crystal Fibers: A New Class of Optical Waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Molina-Fernández, I.

A. Ortega-Moñux, J. G. Wangüemert-Pérez, and I. Molina-Fernández, "Accurate Analysis of Photonic Crystal Fibers by Means of the Fast-Fourier-Based Mode Solver," IEEE Photon. Technol. Lett. 19, 414-416 (2007).
[CrossRef]

Moroz, A.

H. van der Lem, A. Tip, and A. Moroz, "Band structure of absorptive two-dimensional photonic crystals," Solid State Commun. 129, 475-478 (2004).

Morris, G. M.

Nelson, K. A.

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195,111 (2004).
[CrossRef]

Ortega-Moñux, A.

A. Ortega-Moñux, J. G. Wangüemert-Pérez, and I. Molina-Fernández, "Accurate Analysis of Photonic Crystal Fibers by Means of the Fast-Fourier-Based Mode Solver," IEEE Photon. Technol. Lett. 19, 414-416 (2007).
[CrossRef]

Ozeki, S.

E. Yamashita, S. Ozeki, and K. Atsuki, "Modal analysis method for optical fibers with symmetrically distributed multiple cores," J. Lightwave Technol. 2, 341-346 (1985).
[CrossRef]

Peng, S.

Renversez, G.

Rodriguez, A.

Roundy, D.

Russell, P. S. J.

Saitoh, K.

Sargent, E. H.

E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195,122 (2005).
[CrossRef]

Shvets, G.

Smaâli, R.

R. Smaâli, D. Felbacq, and G. Granet, "Bloch waves and non-propagating modes in photonic crystals," Physica E 18, 443-451 (2003).
[CrossRef]

Sudbo, A. S.

A. S. Sudbo, "Improved formulation of the film mode matching method for mode field calculations in dielectric waveguides," Pure Appl. Opt. 3, 381-388 (1994).
[CrossRef]

A. S. Sudbo, "Why are accurate computations of mode fields in rectangular dielectric waveguides difficult?" J. Lightwave Technol. 10, 418-419 (1992).
[CrossRef]

Tayeb, G.

Tip, A.

H. van der Lem, A. Tip, and A. Moroz, "Band structure of absorptive two-dimensional photonic crystals," Solid State Commun. 129, 475-478 (2004).

Urzhumov, Y.

van der Lem, H.

H. van der Lem, A. Tip, and A. Moroz, "Band structure of absorptive two-dimensional photonic crystals," Solid State Commun. 129, 475-478 (2004).

Wangüemert-Pérez, J. G.

A. Ortega-Moñux, J. G. Wangüemert-Pérez, and I. Molina-Fernández, "Accurate Analysis of Photonic Crystal Fibers by Means of the Fast-Fourier-Based Mode Solver," IEEE Photon. Technol. Lett. 19, 414-416 (2007).
[CrossRef]

White, T. P.

Whittaker, D. M.

D. M. Whittaker and I. S. Culshaw, "Scattering-matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

Yamashita, E.

E. Yamashita, S. Ozeki, and K. Atsuki, "Modal analysis method for optical fibers with symmetrically distributed multiple cores," J. Lightwave Technol. 2, 341-346 (1985).
[CrossRef]

Yang, T.-J.

Y.-C. Hsue and T.-J. Yang, "Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal," Phys. Rev. B 70, 016,706 (2004).

Zaki, K. A.

K. A. Zaki, S.-W. Chen, and C. Chen, "Modeling Discontinuities in Dielectric-Loaded Waveguides," IEEE Trans. Microwave Theory Tech. 36, 1804-1810 (1988).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

A. Ortega-Moñux, J. G. Wangüemert-Pérez, and I. Molina-Fernández, "Accurate Analysis of Photonic Crystal Fibers by Means of the Fast-Fourier-Based Mode Solver," IEEE Photon. Technol. Lett. 19, 414-416 (2007).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

K. A. Zaki, S.-W. Chen, and C. Chen, "Modeling Discontinuities in Dielectric-Loaded Waveguides," IEEE Trans. Microwave Theory Tech. 36, 1804-1810 (1988).
[CrossRef]

J. Lightwave Technol. (3)

E. Yamashita, S. Ozeki, and K. Atsuki, "Modal analysis method for optical fibers with symmetrically distributed multiple cores," J. Lightwave Technol. 2, 341-346 (1985).
[CrossRef]

A. S. Sudbo, "Why are accurate computations of mode fields in rectangular dielectric waveguides difficult?" J. Lightwave Technol. 10, 418-419 (1992).
[CrossRef]

K. Saitoh and M. Koshiba, "Numerical Modeling of Photonic Crystal Fibers," J. Lightwave Technol. 23, 3580-3590 (2005).
[CrossRef]

J. Opt. Soc. Am. A (3)

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

Opt. Express (1)

Opt. Fiber Technol. (1)

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic Crystal Fibers: A New Class of Optical Waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. B (5)

D. M. Whittaker and I. S. Culshaw, "Scattering-matrix treatment of patterned multilayer photonic structures," Phys. Rev. B 60, 2610-2618 (1999).
[CrossRef]

Y.-C. Hsue and T.-J. Yang, "Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal," Phys. Rev. B 70, 016,706 (2004).

E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195,122 (2005).
[CrossRef]

Y.-C. Hsue, A. J. Freeman, and B.-Y. Gu, "Extended plane-wave expansion method in three-dimensional anisotropic photonic crystals," Phys. Rev. B 72, 195,118 (2005).
[CrossRef]

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy Bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195,111 (2004).
[CrossRef]

Physica E (1)

R. Smaâli, D. Felbacq, and G. Granet, "Bloch waves and non-propagating modes in photonic crystals," Physica E 18, 443-451 (2003).
[CrossRef]

Pure Appl. Opt. (1)

A. S. Sudbo, "Improved formulation of the film mode matching method for mode field calculations in dielectric waveguides," Pure Appl. Opt. 3, 381-388 (1994).
[CrossRef]

Solid State Commun. (1)

H. van der Lem, A. Tip, and A. Moroz, "Band structure of absorptive two-dimensional photonic crystals," Solid State Commun. 129, 475-478 (2004).

Other (10)

S. Hadzialic, S. Kim, A. Sudbo, and O. Solgaard, "Displacement Sensing with a Mechanically Tunable Photonic Crystal," in The 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society, pp. 345-346 (IEEE, 2007).

S. Kim, S. Hadzialic, A. Sudbo, and O. Solgaard, "Single-film Broadband Photonic Crystal Micro-mirror with Large Angular Range and Low Polarization Dependence," in Conference on Lasers and Electro-optics (CLEO 2007). Baltimore, Maryland, USA, paper CThP7.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, New Jersey, 2008).

N. N. Rao, Elements of Engineering Electromagnetics (Prentice Hall, 2004).

FIMMPROP, Photon Design, Oxford, United Kingdom. http://www.photond.com.

K. C. Johnson, "Grating Diffraction Calculator (GD-Calc®) - Coupled-Wave Theory for Biperiodic Diffraction Gratings," http://software.kjinnovation.com (2006).

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices, chap. 2.3 (Springer-Verlag, Berlin Heidelberg, 2005).

G. P. Agrawal, Fiber-Optic Communication Systems, chap. 2.2, 2nd ed. (John Wiley & Sons, New York, 1997).

M. T. Heath, Scientific Computing: An Introductory Survey (McGraw-Hill, Singapore, 1997).

COMSOL Multiphysics®, COMSOL AB, Stockholm, Sweden. http://www.comsol.com.

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Figures (10)

Fig. 1.
Fig. 1.

Plane waves in three dimensions in air hitting the surface of a 2D PC. We would like to calculate the waves that are reflected and diffracted from the interface between air above the xy-plane and the PC. To do so, modes for a 2D PC extruding infinitely in the z-direction are needed.

Fig. 2.
Fig. 2.

One period of the 2D PC. A core with the radius a is surrounded by a cladding with either higher or lower refractive index.

Fig. 3.
Fig. 3.

Point matching for 12 sampling points around the unit cell of the PC, for the z components of the E and H fields (a) and for the transversal components (b). The small arrows in (b) indicate which transversal component that is matched.

Fig. 4.
Fig. 4.

Automatically computed band diagram for a high-contrast structure. ε 1 = 8.9, ε 1 = 1.0. Λ x = Λ y = Λ and a = 0.2Λ. kx = ky = 0. The highest Bessel function order is N = 10. Two occurrences of complex-conjugated pairs of complex modes are indicated with dotted lines, where the black dotted lines represent Re (k 2 z )/k 2 Λ and the blue dotted lines represent [Re (k 2 z )+Im (k 2 z )]/k 2 Λ. Eigenvalues that are used in Fig. 9 are here marked with small red circles.

Fig. 5.
Fig. 5.

Automatically computed band diagram (black lines) for a low-contrast structure, ε 1 = 4.0 and ε 2 = 1.7. The unit cell geometry is as in Fig. 4. Green lines show plane waves for ε avg = 2.0 (the average dielectric constant for the low-contrast structure). Figure (b) shows a magnification of the marked area in Fig. (a) where two of the bands form a complex-conjugated pair of k 2 z (marked with dotted lines).

Fig. 6.
Fig. 6.

(a) The largest value of Im k 2 z in the bandgap as a function of the permittivity contrast Δ. (b) The upper and lower boundaries (of k 2 0) of the bandgap and the bandgap size, as a function of the permittivity contrast.

Fig. 7.
Fig. 7.

Average power per unit area perpendicular to the xy-plane, Re [Sz (x,y,z = 0)], within one period of the PC, for some allowed kz .

Fig. 8.
Fig. 8.

Reactive power, Im [Sz (x,y,z = 0)], within one period of the PC, for some allowed kz .

Fig. 9.
Fig. 9.

An estimate of the error in the calculation of kz /k Λ, as a function of the highest order for the Bessel functions, N. The value obtained for N = 46 is used as a reference.

Fig. 10.
Fig. 10.

Computation times for calculating the matrix M(kz ) and taking the inverse, on a standard personal computer.

Equations (34)

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k 0 = ω c .
E ( x , y , z , t ) = Re [ e ( x , y . z ) exp ] ( i k · r iωt ) ,
e ( x , y , z ) = e ( x + Λ x , y , z ) = e ( x , y + Λ y , z ) .
E ( x , y , z ) = e ( x , y , z ) exp ( i k · r )
e ( x , y , z ) = q x , q y e q x , q y ( z ) exp ( i 2 π q x Λ x + i 2 π q y Λ y y )
e q x , q y ( z ) = e q x , q y exp ( i k z , q x , q y z ) ,
ε ( ω c ) 2 = k z , q x , q y 2 + ( k x + 2 π q x Λ x ) 2 + ( k y + 2 π q y Λ y ) 2 ,
E ( x , y , z ) = p , q x , q y e p , q x , q y exp [ i ( k x + 2 π q x Λ x ) x + i ( k y + 2 π q y Λ y ) y + i k z , q x , q y z ] .
e q ( x , y , z ) = e q ( x , y ) exp ( i k z , q z ) .
E ( x , y , z ) = q e q ( x , y ) exp ( i k q · r ) = q e q ( x , y ) exp [ i ( k x x + k y y + k z , q z ) ] ,
Δ = ε 1 ε 2 ( ε 1 + ε 2 ) / 2 .
1 r r ( r U r ) + 1 r 2 2 U r 2 φ 2 + ( ε k 0 2 k z 2 ) U = 0 .
E z , n = { i μ 0 ε 0 B n J ̂ n ( β 1 r ) exp ( inφ ) exp ( i k z z ) , r < a i μ 0 ε 0 [ F n J ̂ n ( β 2 r ) + G n Y ̂ n ( β 2 r ) ] exp ( inφ ) exp ( i k z z ) , r > a
H z , n = { A n J ̂ n ( β 1 r ) exp ( inφ ) exp ( i k z z ) , r < a [ C n J ̂ n ( β 2 r ) + D n Y ̂ n ( β 2 r ) ] exp ( inφ ) exp ( i k z z ) , r > a
β j = ( ε j k 0 2 k z 2 ) 1 / 2 , for j = 1,2
C n = A n β 2 J ̂ n ( β 1 a ) Y ̂ n ( β 2 a ) β 1 J ̂ n ( β 1 a ) Y ̂ n ( β 2 a ) β 1 N n + B n ( ε 2 ε 1 ) k 0 n k z J ̂ n ( β 1 a ) Y ̂ n ( β 2 a ) a β 1 2 β 2 N n
D n = A n β 1 J ̂ n ( β 1 a ) J ̂ n ( β 2 a ) β 2 J ̂ n ( β 2 a ) J ̂ n ( β 1 a ) β 1 N n + B n ( ε 1 ε 2 ) k 0 n k z J ̂ n ( β 1 a ) J ̂ n ( β 2 a ) a β 1 2 β 2 N n
F n = A n ( ε 2 ε 1 ) k 0 n k z J ̂ n ( β 1 a ) Y ̂ n ( β 2 a ) a β 1 2 β 2 ε 2 N n + B n β 1 ε 2 J ̂ n ( β 1 a ) Y ̂ n ( β 2 a ) β 2 ε 1 J ̂ n ( β 2 a ) Y ̂ n ( β 1 a ) β 1 ε 2 N n
G n = A n ( ε 1 ε 2 ) k 0 n k z J ̂ n ( β 1 a ) Y ̂ n ( β 2 a ) a β 1 2 β 2 ε 2 N n + B n β 1 ε 2 J ̂ n ( β 1 a ) J ̂ n ( β 2 a ) β 2 ε 1 J ̂ n ( β 2 a ) J ̂ n ( β 1 a ) β 1 ε 2 N n
N n = J ̂ n ( β 2 a ) Y ̂ n ( β 2 a ) J ̂ n ( β 2 a ) Y ̂ n ( β 2 a ) .
E z = n = N N E z , n
H z = n = N N H z , n
r 2 = r 1 + Λ x ̂ ,
r 2 = r 1 + Λ y ̂ ,
r 2 = r 1 + Λ x ̂ + Λ y ̂ .
E ( r 2 ) = E ( r 1 ) exp [ i k · ( r 2 r 1 ) ]
H ( r 2 ) = H ( r 1 ) exp [ i k · ( r 2 r 1 ) ] .
( M 1,1 ( k z ) M 1 , 4 N + 2 ( k z ) M 4 N + 2,1 ( k z ) M 4 N + 2,4 N + 2 ( k z ) ) ( A N A N B N B N ) = 0
M ( k z ) u = 0 ,
M ( k z ) u = v
k Λ = 2 π Λ .
ε avg = ( ε 1 ε 2 ) π a 2 Λ 2 + ε 2 .
S z ( x , y , z ) = 1 2 [ E x ( x , y , z ) H y * ( x , y , z ) E y ( x , y , z ) H x * ( x , y , z ) ]
= 1 2 [ e x ( x , y ) h y * ( x , y ) e y ( x , y ) h x * ( x , y ) ] exp [ 2 Im ( k · r ) ] ,

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