Abstract

A rigorous and efficient computational method is developed to calculate transmission and reflection spectra for a finite number of air-hole arrays in a slab, where the incident waves are propagating modes of the slab. The method is a three-dimensional extension of the Dirichlet-to-Neumann (DtN) map method previously developed for ideal two-dimensional photonic crystals which are infinite and invariant in one spatial direction. The method relies on the DtN maps of the unit cells to avoid repeated calculations in identical unit cells. The DtN map of a unit cell is constructed using eigenmode expansions in the vertical direction (perpendicular to the slab) and cylindrical wave expansions in the horizontal directions.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]

2009

2008

Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice,” J. Opt. Soc. Am. B 25, 1466–1473 (2008).
[CrossRef]

Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399 (2008).
[CrossRef]

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

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921–932 (2008).
[CrossRef]

2007

2006

Y. Y. Lu, “Some techniques for computing wave propagation in optical waveguides,” Comm. Comp. Phys. 1, 1056–1075 (2006).

L. Yuan and Y. Y. Lu, “An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps,” IEEE Photon. Technol. Lett. 18, 1967–1969 (2006).
[CrossRef]

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448–3453 (2006).
[CrossRef]

J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 23, 3217–3222 (2006).
[CrossRef]

P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles (Cambridge U. Press, 2006).

Y. Y. Lu, “Minimizing the discrete reflectivity of perfectly matched layers,” IEEE Photon. Technol. Lett. 18, 487–489 (2006).
[CrossRef]

2005

2004

2003

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

2002

M. Qiu, “Effective index method for heterostructure-slab-waveguide based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002).
[CrossRef]

P. Lalanne, “Electromagnetic analysis of photonic crystal waveguides operating above the light cone,” IEEE J. Quantum Electron. 38, 800–804 (2002).
[CrossRef]

2001

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagon photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[CrossRef]

2000

Y. P. Chiou, Y. C. Chiang, and H. C. Chang, “Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices,” J. Lightwave Technol. 18, 243–251 (2000).
[CrossRef]

E. Centeno and D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional photonic crystals,” J. Opt. Soc. Am. A 17, 320–327 (2000).
[CrossRef]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry–Perot structures,” SIAM J. Appl. Math. 60, 1686–1706 (2000).
[CrossRef]

1996

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]

Y. Y. Lu and J. R. McLaughlin, “The Riccati method for the Helmholtz equation,” J. Acoust. Soc. Am. 100, 1432–1446 (1996).
[CrossRef]

1995

1994

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

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

1991

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

Armenise, M. N.

Bao, G.

Berenger, J. P.

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

Boscolo, S.

Botten, L. C.

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

Centeno, E.

Chang, H. C.

Chen, C.

Chen, Z. M.

Chew, W. C.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Chiang, Y. C.

Chiou, Y. P.

Ciminelli, C.

Cotter, N. P. K.

Dems, M.

Felbacq, D.

Gaylord, T. K.

Ginste, D. V.

Grann, E. B.

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Haider, M. A.

S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry–Perot structures,” SIAM J. Appl. Math. 60, 1686–1706 (2000).
[CrossRef]

Hu, Z.

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921–932 (2008).
[CrossRef]

Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399 (2008).
[CrossRef]

Huang, Y.

Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337–349 (2007).

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448–3453 (2006).
[CrossRef]

Jin, J. M.

Joannopoulos, J. D.

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

Johnson, S. G.

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

Kotynski, R.

Kushta, T.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Lalanne, P.

P. Lalanne, “Electromagnetic analysis of photonic crystal waveguides operating above the light cone,” IEEE J. Quantum Electron. 38, 800–804 (2002).
[CrossRef]

Li, L.

Li, Y. J.

Lu, Y. Y.

Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing periodic arrays of cylinders with oblique incident waves,” J. Opt. Soc. Am. B 26, 1442–1449 (2009).
[CrossRef]

Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing crossed arrays of circular cylinders,” J. Opt. Soc. Am. B 26, 1984–1993 (2009).
[CrossRef]

Y. Wu and Y. Y. Lu, “Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice,” J. Opt. Soc. Am. B 25, 1466–1473 (2008).
[CrossRef]

Z. Hu and Y. Y. Lu, “Efficient analysis of photonic crystal devices by Dirichlet-to-Neumann maps,” Opt. Express 16, 17383–17399 (2008).
[CrossRef]

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921–932 (2008).
[CrossRef]

Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337–349 (2007).

Y. Y. Lu, “Some techniques for computing wave propagation in optical waveguides,” Comm. Comp. Phys. 1, 1056–1075 (2006).

L. Yuan and Y. Y. Lu, “An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps,” IEEE Photon. Technol. Lett. 18, 1967–1969 (2006).
[CrossRef]

Y. Huang and Y. Y. Lu, “Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps,” J. Lightwave Technol. 24, 3448–3453 (2006).
[CrossRef]

J. Yuan and Y. Y. Lu, “Photonic bandgap calculations using Dirichlet-to-Neumann maps,” J. Opt. Soc. Am. A 23, 3217–3222 (2006).
[CrossRef]

Y. Y. Lu, “Minimizing the discrete reflectivity of perfectly matched layers,” IEEE Photon. Technol. Lett. 18, 487–489 (2006).
[CrossRef]

Y. Y. Lu and J. R. McLaughlin, “The Riccati method for the Helmholtz equation,” J. Acoust. Soc. Am. 100, 1432–1446 (1996).
[CrossRef]

Martin, P. A.

P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles (Cambridge U. Press, 2006).

McLaughlin, J. R.

Y. Y. Lu and J. R. McLaughlin, “The Riccati method for the Helmholtz equation,” J. Acoust. Soc. Am. 100, 1432–1446 (1996).
[CrossRef]

McPhedran, R. C.

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

Meade, R. D.

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

Michielssen, E.

Midrio, M.

Moharam, M. G.

Nicorovici, N. A.

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

Ochiai, T.

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagon photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[CrossRef]

Olyslager, F.

Panajotov, K.

Papanicolaou, V.

S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry–Perot structures,” SIAM J. Appl. Math. 60, 1686–1706 (2000).
[CrossRef]

Peluso, F.

Pissoort, D.

Pommet, D. A.

Prather, D. W.

Preist, T. W.

Qiu, M.

M. Qiu, “Effective index method for heterostructure-slab-waveguide based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002).
[CrossRef]

Sakoda, K.

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagon photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[CrossRef]

Sambles, J. R.

Shi, S.

Smith, G. H.

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

Toyama, H.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Vassallo, C.

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

Venakides, S.

S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry–Perot structures,” SIAM J. Appl. Math. 60, 1686–1706 (2000).
[CrossRef]

Weedon, W. H.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Winn, J. N.

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

Wu, H. J.

Wu, Y.

Yasumoto, K.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

Yuan, J.

Yuan, L.

L. Yuan and Y. Y. Lu, “An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps,” IEEE Photon. Technol. Lett. 18, 1967–1969 (2006).
[CrossRef]

Appl. Phys. Lett.

M. Qiu, “Effective index method for heterostructure-slab-waveguide based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002).
[CrossRef]

Comm. Comp. Phys.

Y. Y. Lu, “Some techniques for computing wave propagation in optical waveguides,” Comm. Comp. Phys. 1, 1056–1075 (2006).

IEEE J. Quantum Electron.

P. Lalanne, “Electromagnetic analysis of photonic crystal waveguides operating above the light cone,” IEEE J. Quantum Electron. 38, 800–804 (2002).
[CrossRef]

IEEE Photon. Technol. Lett.

Y. Y. Lu, “Minimizing the discrete reflectivity of perfectly matched layers,” IEEE Photon. Technol. Lett. 18, 487–489 (2006).
[CrossRef]

L. Yuan and Y. Y. Lu, “An efficient bidirectional propagation method based on Dirichlet-to-Neumann maps,” IEEE Photon. Technol. Lett. 18, 1967–1969 (2006).
[CrossRef]

IEEE Trans. Antennas Propag.

K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagnetic scattering from multilayered periodic arrays of circular cylinders using lattice sums technique,” IEEE Trans. Antennas Propag. 52, 2603–2611 (2004).
[CrossRef]

J. Acoust. Soc. Am.

Y. Y. Lu and J. R. McLaughlin, “The Riccati method for the Helmholtz equation,” J. Acoust. Soc. Am. 100, 1432–1446 (1996).
[CrossRef]

J. Comput. Math.

Y. Huang and Y. Y. Lu, “Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps,” J. Comput. Math. 25, 337–349 (2007).

J. Comput. Phys.

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

J. Lightwave Technol.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Microwave Opt. Technol. Lett.

W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Opt. Express

Opt. Quantum Electron.

Z. Hu and Y. Y. Lu, “Improved Dirichlet-to-Neumann map method for modeling extended photonic crystal devices,” Opt. Quantum Electron. 40, 921–932 (2008).
[CrossRef]

Phys. Rev. B

T. Ochiai and K. Sakoda, “Dispersion relation and optical transmittance of a hexagon photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[CrossRef]

Phys. Rev. E

G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A. Nicorovici, “Cylinder gratings in conical incidence with applications to woodpile structures,” Phys. Rev. E 67, 056620 (2003).
[CrossRef]

SIAM J. Appl. Math.

S. Venakides, M. A. Haider, and V. Papanicolaou, “Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry–Perot structures,” SIAM J. Appl. Math. 60, 1686–1706 (2000).
[CrossRef]

Other

C. Vassallo, Optical Waveguide Concepts (Elsevier, 1991).

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

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. (Artech House, 2000).

P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles (Cambridge U. Press, 2006).

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