Abstract

Wannier function expansions are well suited for the description of photonic-crystal-based defect structures, but constructing maximally localized Wannier functions by optimizing the phase degree of freedom of the Bloch modes is crucial for the efficiency of the approach. We systematically analyze different locality criteria for maximally localized Wannier functions in two-dimensional square and triangular lattice photonic crystals, employing (local) conjugate-gradient as well as (global) genetic-algorithm-based stochastic methods. Besides the commonly used second moment (SM) locality measure, we introduce a new locality measure, namely, the integrated modulus (IM) of the Wannier function. We show numerically that, in contrast to the SM criterion, the IM criterion leads to an optimization problem with a single extremum, thus allowing for fast and efficient construction of maximally localized Wannier functions using local optimization techniques. We also present an analytical formula for the initial choice of Bloch phases, which, under certain conditions, represents the global maximum of the IM criterion and, thus, further increases the optimization efficiency in the general case.

© 2011 Optical Society of America

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References

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  1. K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).
  2. A. Taflove and S. C. Hagnes, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).
  3. C. A. J. Fletcher, Computational Galerkin Methods (Springer, 1984).
  4. D. N. Chigrin, “Spatial distribution of the emission intensity in a photonic crystal: self-interference of Bloch eigenwaves,” Phys. Rev. A 79, 1–9 (2009).
    [CrossRef]
  5. C. Kremers, D. N. Chigrin, and J. Kroha, “Theory of Cherenkov radiation in periodic dielectric media: emission spectrum,” Phys. Rev. A 79, 1–10 (2009).
    [CrossRef]
  6. C. Kremers and D. N. Chigrin, “Spatial distribution of Cherenkov radiation in periodic dielectric media,” J. Opt. A 11, 114008(2009).
    [CrossRef]
  7. K. M. Leung, “Defect modes in photonic band structures—a green-function approach using vector Wannier functions,” J. Opt. Soc. Am. B 10, 303–306 (1993).
    [CrossRef]
  8. E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
    [CrossRef]
  9. J. Albert, C. Jouanin, D. Cassagne, and D. Bertho, “Generalized Wannier function method for photonic crystals,” Phys. Rev. B 61, 4381–4384 (2000).
    [CrossRef]
  10. J. Albert, C. Jouanin, D. Cassagne, and D. Monge, “Photonic crystal modelling using a tight-binding Wannier function method,” Opt. Quantum Electron. 34, 251–263 (2002).
    [CrossRef]
  11. N. Marzari and D. Vanderbilt, “Maximally localized generalized Wannier functions for composite energy bands,” Phys. Rev. B 56, 12847 (1997).
    [CrossRef]
  12. I. Souza, N. Marzari, and D. Vanderbilt, “Maximally localized Wannier functions for entangled energy bands,” Phys. Rev. B 65, 035109 (2001).
    [CrossRef]
  13. D. M. Whittaker and M. P. Croucher, “Maximally localized Wannier functions for photonic lattices,” Phys. Rev. B 67, 085204 (2003).
    [CrossRef]
  14. A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wölfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177 (2003).
    [CrossRef]
  15. K. Busch, S. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233–R1256 (2003).
    [CrossRef]
  16. Y. Jiao, S. Mingaleev, M. Schillinger, D. Miller, S. Fan, and K. Busch, “Wannier basis design and optimization of a photonic crystal waveguide crossing,” IEEE Photon. Technol. Lett. 17, 1875–1877 (2005).
    [CrossRef]
  17. E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455–481 (2006).
    [CrossRef]
  18. A. McGurn, “Impurity mode techniques applied to the study of light sources,” J. Phys. D 38, 2338–2352 (2005).
    [CrossRef]
  19. H. Takeda, A. Chutinan, and S. John, “Localized light orbitals: basis states for three-dimensional photonic crystal microscale circuits,” Phys. Rev. B 74 (2006).
    [CrossRef]
  20. J. des Cloizeaux, “Analytical properties of n-dimensional energy bands and Wannier functions,” Phys. Rev. 135, A698–A707(1964).
    [CrossRef]
  21. K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233(2003).
    [CrossRef]
  22. A. Birner, R. B. Wehrspohn, U. Gsele, and K. Busch, “Silicon-based photonic crystals,” Adv. Mater. 13, 377 (2001).
    [CrossRef]
  23. GNU Scientific Library, http://www.gnu.org/software/gsl/manual.
  24. MIT Photonic-Bands, http://ab-initio.mit.edu/wiki/index.php/MIT_Photonic_Bands.
  25. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
    [CrossRef] [PubMed]
  26. M. Mitchell, An Introduction to Genetic Algorithms (MIT Press, 1999).

2009 (3)

D. N. Chigrin, “Spatial distribution of the emission intensity in a photonic crystal: self-interference of Bloch eigenwaves,” Phys. Rev. A 79, 1–9 (2009).
[CrossRef]

C. Kremers, D. N. Chigrin, and J. Kroha, “Theory of Cherenkov radiation in periodic dielectric media: emission spectrum,” Phys. Rev. A 79, 1–10 (2009).
[CrossRef]

C. Kremers and D. N. Chigrin, “Spatial distribution of Cherenkov radiation in periodic dielectric media,” J. Opt. A 11, 114008(2009).
[CrossRef]

2006 (2)

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455–481 (2006).
[CrossRef]

H. Takeda, A. Chutinan, and S. John, “Localized light orbitals: basis states for three-dimensional photonic crystal microscale circuits,” Phys. Rev. B 74 (2006).
[CrossRef]

2005 (2)

A. McGurn, “Impurity mode techniques applied to the study of light sources,” J. Phys. D 38, 2338–2352 (2005).
[CrossRef]

Y. Jiao, S. Mingaleev, M. Schillinger, D. Miller, S. Fan, and K. Busch, “Wannier basis design and optimization of a photonic crystal waveguide crossing,” IEEE Photon. Technol. Lett. 17, 1875–1877 (2005).
[CrossRef]

2003 (4)

D. M. Whittaker and M. P. Croucher, “Maximally localized Wannier functions for photonic lattices,” Phys. Rev. B 67, 085204 (2003).
[CrossRef]

A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wölfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177 (2003).
[CrossRef]

K. Busch, S. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233–R1256 (2003).
[CrossRef]

K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233(2003).
[CrossRef]

2002 (1)

J. Albert, C. Jouanin, D. Cassagne, and D. Monge, “Photonic crystal modelling using a tight-binding Wannier function method,” Opt. Quantum Electron. 34, 251–263 (2002).
[CrossRef]

2001 (3)

I. Souza, N. Marzari, and D. Vanderbilt, “Maximally localized Wannier functions for entangled energy bands,” Phys. Rev. B 65, 035109 (2001).
[CrossRef]

A. Birner, R. B. Wehrspohn, U. Gsele, and K. Busch, “Silicon-based photonic crystals,” Adv. Mater. 13, 377 (2001).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[CrossRef] [PubMed]

2000 (1)

J. Albert, C. Jouanin, D. Cassagne, and D. Bertho, “Generalized Wannier function method for photonic crystals,” Phys. Rev. B 61, 4381–4384 (2000).
[CrossRef]

1998 (1)

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

1997 (1)

N. Marzari and D. Vanderbilt, “Maximally localized generalized Wannier functions for composite energy bands,” Phys. Rev. B 56, 12847 (1997).
[CrossRef]

1993 (1)

1964 (1)

J. des Cloizeaux, “Analytical properties of n-dimensional energy bands and Wannier functions,” Phys. Rev. 135, A698–A707(1964).
[CrossRef]

Albert, J.

J. Albert, C. Jouanin, D. Cassagne, and D. Monge, “Photonic crystal modelling using a tight-binding Wannier function method,” Opt. Quantum Electron. 34, 251–263 (2002).
[CrossRef]

J. Albert, C. Jouanin, D. Cassagne, and D. Bertho, “Generalized Wannier function method for photonic crystals,” Phys. Rev. B 61, 4381–4384 (2000).
[CrossRef]

Bertho, D.

J. Albert, C. Jouanin, D. Cassagne, and D. Bertho, “Generalized Wannier function method for photonic crystals,” Phys. Rev. B 61, 4381–4384 (2000).
[CrossRef]

Birner, A.

A. Birner, R. B. Wehrspohn, U. Gsele, and K. Busch, “Silicon-based photonic crystals,” Adv. Mater. 13, 377 (2001).
[CrossRef]

Busch, K.

Y. Jiao, S. Mingaleev, M. Schillinger, D. Miller, S. Fan, and K. Busch, “Wannier basis design and optimization of a photonic crystal waveguide crossing,” IEEE Photon. Technol. Lett. 17, 1875–1877 (2005).
[CrossRef]

A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wölfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177 (2003).
[CrossRef]

K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233(2003).
[CrossRef]

K. Busch, S. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233–R1256 (2003).
[CrossRef]

A. Birner, R. B. Wehrspohn, U. Gsele, and K. Busch, “Silicon-based photonic crystals,” Adv. Mater. 13, 377 (2001).
[CrossRef]

Cassagne, D.

J. Albert, C. Jouanin, D. Cassagne, and D. Monge, “Photonic crystal modelling using a tight-binding Wannier function method,” Opt. Quantum Electron. 34, 251–263 (2002).
[CrossRef]

J. Albert, C. Jouanin, D. Cassagne, and D. Bertho, “Generalized Wannier function method for photonic crystals,” Phys. Rev. B 61, 4381–4384 (2000).
[CrossRef]

Chigrin, D. N.

C. Kremers, D. N. Chigrin, and J. Kroha, “Theory of Cherenkov radiation in periodic dielectric media: emission spectrum,” Phys. Rev. A 79, 1–10 (2009).
[CrossRef]

D. N. Chigrin, “Spatial distribution of the emission intensity in a photonic crystal: self-interference of Bloch eigenwaves,” Phys. Rev. A 79, 1–9 (2009).
[CrossRef]

C. Kremers and D. N. Chigrin, “Spatial distribution of Cherenkov radiation in periodic dielectric media,” J. Opt. A 11, 114008(2009).
[CrossRef]

Chutinan, A.

H. Takeda, A. Chutinan, and S. John, “Localized light orbitals: basis states for three-dimensional photonic crystal microscale circuits,” Phys. Rev. B 74 (2006).
[CrossRef]

Croucher, M. P.

D. M. Whittaker and M. P. Croucher, “Maximally localized Wannier functions for photonic lattices,” Phys. Rev. B 67, 085204 (2003).
[CrossRef]

des Cloizeaux, J.

J. des Cloizeaux, “Analytical properties of n-dimensional energy bands and Wannier functions,” Phys. Rev. 135, A698–A707(1964).
[CrossRef]

Economou, E. N.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

Fan, S.

Y. Jiao, S. Mingaleev, M. Schillinger, D. Miller, S. Fan, and K. Busch, “Wannier basis design and optimization of a photonic crystal waveguide crossing,” IEEE Photon. Technol. Lett. 17, 1875–1877 (2005).
[CrossRef]

Fletcher, C. A. J.

C. A. J. Fletcher, Computational Galerkin Methods (Springer, 1984).

Garcia-Martin, A.

K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233(2003).
[CrossRef]

A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wölfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177 (2003).
[CrossRef]

K. Busch, S. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233–R1256 (2003).
[CrossRef]

Gsele, U.

A. Birner, R. B. Wehrspohn, U. Gsele, and K. Busch, “Silicon-based photonic crystals,” Adv. Mater. 13, 377 (2001).
[CrossRef]

Hagmann, F.

A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wölfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177 (2003).
[CrossRef]

Hagnes, S. C.

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

Hermann, D.

K. Busch, S. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233–R1256 (2003).
[CrossRef]

A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wölfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177 (2003).
[CrossRef]

K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233(2003).
[CrossRef]

Istrate, E.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455–481 (2006).
[CrossRef]

Jiao, Y.

Y. Jiao, S. Mingaleev, M. Schillinger, D. Miller, S. Fan, and K. Busch, “Wannier basis design and optimization of a photonic crystal waveguide crossing,” IEEE Photon. Technol. Lett. 17, 1875–1877 (2005).
[CrossRef]

Joannopoulos, J. D.

John, S.

H. Takeda, A. Chutinan, and S. John, “Localized light orbitals: basis states for three-dimensional photonic crystal microscale circuits,” Phys. Rev. B 74 (2006).
[CrossRef]

Johnson, S. G.

Jouanin, C.

J. Albert, C. Jouanin, D. Cassagne, and D. Monge, “Photonic crystal modelling using a tight-binding Wannier function method,” Opt. Quantum Electron. 34, 251–263 (2002).
[CrossRef]

J. Albert, C. Jouanin, D. Cassagne, and D. Bertho, “Generalized Wannier function method for photonic crystals,” Phys. Rev. B 61, 4381–4384 (2000).
[CrossRef]

Kremers, C.

C. Kremers, D. N. Chigrin, and J. Kroha, “Theory of Cherenkov radiation in periodic dielectric media: emission spectrum,” Phys. Rev. A 79, 1–10 (2009).
[CrossRef]

C. Kremers and D. N. Chigrin, “Spatial distribution of Cherenkov radiation in periodic dielectric media,” J. Opt. A 11, 114008(2009).
[CrossRef]

Kroha, J.

C. Kremers, D. N. Chigrin, and J. Kroha, “Theory of Cherenkov radiation in periodic dielectric media: emission spectrum,” Phys. Rev. A 79, 1–10 (2009).
[CrossRef]

Leung, K. M.

Lidorikis, E.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

Marzari, N.

I. Souza, N. Marzari, and D. Vanderbilt, “Maximally localized Wannier functions for entangled energy bands,” Phys. Rev. B 65, 035109 (2001).
[CrossRef]

N. Marzari and D. Vanderbilt, “Maximally localized generalized Wannier functions for composite energy bands,” Phys. Rev. B 56, 12847 (1997).
[CrossRef]

McGurn, A.

A. McGurn, “Impurity mode techniques applied to the study of light sources,” J. Phys. D 38, 2338–2352 (2005).
[CrossRef]

Miller, D.

Y. Jiao, S. Mingaleev, M. Schillinger, D. Miller, S. Fan, and K. Busch, “Wannier basis design and optimization of a photonic crystal waveguide crossing,” IEEE Photon. Technol. Lett. 17, 1875–1877 (2005).
[CrossRef]

Mingaleev, S.

Y. Jiao, S. Mingaleev, M. Schillinger, D. Miller, S. Fan, and K. Busch, “Wannier basis design and optimization of a photonic crystal waveguide crossing,” IEEE Photon. Technol. Lett. 17, 1875–1877 (2005).
[CrossRef]

K. Busch, S. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233–R1256 (2003).
[CrossRef]

Mingaleev, S. F.

K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233(2003).
[CrossRef]

Mitchell, M.

M. Mitchell, An Introduction to Genetic Algorithms (MIT Press, 1999).

Monge, D.

J. Albert, C. Jouanin, D. Cassagne, and D. Monge, “Photonic crystal modelling using a tight-binding Wannier function method,” Opt. Quantum Electron. 34, 251–263 (2002).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

Sargent, E. H.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455–481 (2006).
[CrossRef]

Schillinger, M.

Y. Jiao, S. Mingaleev, M. Schillinger, D. Miller, S. Fan, and K. Busch, “Wannier basis design and optimization of a photonic crystal waveguide crossing,” IEEE Photon. Technol. Lett. 17, 1875–1877 (2005).
[CrossRef]

K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233(2003).
[CrossRef]

K. Busch, S. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233–R1256 (2003).
[CrossRef]

Sigalas, M. M.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

Soukoulis, C. M.

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

Souza, I.

I. Souza, N. Marzari, and D. Vanderbilt, “Maximally localized Wannier functions for entangled energy bands,” Phys. Rev. B 65, 035109 (2001).
[CrossRef]

Taflove, A.

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

Takeda, H.

H. Takeda, A. Chutinan, and S. John, “Localized light orbitals: basis states for three-dimensional photonic crystal microscale circuits,” Phys. Rev. B 74 (2006).
[CrossRef]

Vanderbilt, D.

I. Souza, N. Marzari, and D. Vanderbilt, “Maximally localized Wannier functions for entangled energy bands,” Phys. Rev. B 65, 035109 (2001).
[CrossRef]

N. Marzari and D. Vanderbilt, “Maximally localized generalized Wannier functions for composite energy bands,” Phys. Rev. B 56, 12847 (1997).
[CrossRef]

Wehrspohn, R. B.

A. Birner, R. B. Wehrspohn, U. Gsele, and K. Busch, “Silicon-based photonic crystals,” Adv. Mater. 13, 377 (2001).
[CrossRef]

Whittaker, D. M.

D. M. Whittaker and M. P. Croucher, “Maximally localized Wannier functions for photonic lattices,” Phys. Rev. B 67, 085204 (2003).
[CrossRef]

Wölfle, P.

A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wölfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177 (2003).
[CrossRef]

Adv. Mater. (1)

A. Birner, R. B. Wehrspohn, U. Gsele, and K. Busch, “Silicon-based photonic crystals,” Adv. Mater. 13, 377 (2001).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

Y. Jiao, S. Mingaleev, M. Schillinger, D. Miller, S. Fan, and K. Busch, “Wannier basis design and optimization of a photonic crystal waveguide crossing,” IEEE Photon. Technol. Lett. 17, 1875–1877 (2005).
[CrossRef]

J. Opt. A (1)

C. Kremers and D. N. Chigrin, “Spatial distribution of Cherenkov radiation in periodic dielectric media,” J. Opt. A 11, 114008(2009).
[CrossRef]

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

J. Phys. Condens. Matter (2)

K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233(2003).
[CrossRef]

K. Busch, S. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “The Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233–R1256 (2003).
[CrossRef]

J. Phys. D (1)

A. McGurn, “Impurity mode techniques applied to the study of light sources,” J. Phys. D 38, 2338–2352 (2005).
[CrossRef]

Nanotechnology (1)

A. Garcia-Martin, D. Hermann, F. Hagmann, K. Busch, and P. Wölfle, “Defect computations in photonic crystals: a solid state theoretical approach,” Nanotechnology 14, 177 (2003).
[CrossRef]

Opt. Express (1)

Opt. Quantum Electron. (1)

J. Albert, C. Jouanin, D. Cassagne, and D. Monge, “Photonic crystal modelling using a tight-binding Wannier function method,” Opt. Quantum Electron. 34, 251–263 (2002).
[CrossRef]

Phys. Rev. (1)

J. des Cloizeaux, “Analytical properties of n-dimensional energy bands and Wannier functions,” Phys. Rev. 135, A698–A707(1964).
[CrossRef]

Phys. Rev. A (2)

D. N. Chigrin, “Spatial distribution of the emission intensity in a photonic crystal: self-interference of Bloch eigenwaves,” Phys. Rev. A 79, 1–9 (2009).
[CrossRef]

C. Kremers, D. N. Chigrin, and J. Kroha, “Theory of Cherenkov radiation in periodic dielectric media: emission spectrum,” Phys. Rev. A 79, 1–10 (2009).
[CrossRef]

Phys. Rev. B (5)

N. Marzari and D. Vanderbilt, “Maximally localized generalized Wannier functions for composite energy bands,” Phys. Rev. B 56, 12847 (1997).
[CrossRef]

I. Souza, N. Marzari, and D. Vanderbilt, “Maximally localized Wannier functions for entangled energy bands,” Phys. Rev. B 65, 035109 (2001).
[CrossRef]

D. M. Whittaker and M. P. Croucher, “Maximally localized Wannier functions for photonic lattices,” Phys. Rev. B 67, 085204 (2003).
[CrossRef]

J. Albert, C. Jouanin, D. Cassagne, and D. Bertho, “Generalized Wannier function method for photonic crystals,” Phys. Rev. B 61, 4381–4384 (2000).
[CrossRef]

H. Takeda, A. Chutinan, and S. John, “Localized light orbitals: basis states for three-dimensional photonic crystal microscale circuits,” Phys. Rev. B 74 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
[CrossRef]

Rev. Mod. Phys. (1)

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78, 455–481 (2006).
[CrossRef]

Other (6)

GNU Scientific Library, http://www.gnu.org/software/gsl/manual.

MIT Photonic-Bands, http://ab-initio.mit.edu/wiki/index.php/MIT_Photonic_Bands.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

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

C. A. J. Fletcher, Computational Galerkin Methods (Springer, 1984).

M. Mitchell, An Introduction to Genetic Algorithms (MIT Press, 1999).

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Figures (10)

Fig. 1
Fig. 1

Second moments S n (inverse locality) of the Wannier functions in the nth band, minimized by using the CG method. Four different, randomly chosen initial sets of Bloch phases, A, B, C, and D, were used for the CG optimization. Top: Sq-D crystal, TM polarization. Bottom: Tr-D crystal, TE polarization.

Fig. 2
Fig. 2

Locality ( 1 / S n ) of the SM-optimized Wannier functions as a function of GA generation for the Sq-D structure, TM polarization, third band (top), and for the Sq-A structure, TE polarization, fifth band (bottom). The dashed green line shows the locality of the best-localized Wannier function in each generation. Every 100 generations, these Wannier functions served as a starting point for the subsequent CG optimization step (red crosses). On the right-hand side, the modulus square of the SM-optimized Wannier function is shown for an early (top, 40,000th; bottom, 38,300th) and a later (top, 60,000th; bottom, 100,000th) generation.

Fig. 3
Fig. 3

Locality ( I n ) of the Wannier functions optimized with respect to the IM locality measure using the CG method. Four different randomly chosen initial sets of Bloch phases were used for the CG optimization. Top: Sq-D crystal, TM polarization. Bottom: Tr-D crystal, TE polarization.

Fig. 4
Fig. 4

Locality ( I n ) of the IM-optimized Wannier functions as a function of GA generation for the Sq-D structure, TM polarization, third band (top), and for the Sq-A structure, TE polarization, fifth band (bottom). The dashed green line shows the locality of the best-localized Wannier function in each generation. Every 100 generations, these Wannier functions served as a starting point for the subsequent CG optimization step (red crosses). On the right-hand side, the modulus square of the SM-optimized Wannier functions is shown for an early (20,000th) and a later (60,000th) generation.

Fig. 5
Fig. 5

Modulus square of the maximally localized Wannier functions (with respect to the IM) for the Sq-D structure (TM polarization). The Wannier center was chosen as “on-site” for the first and fifth bands and as “between” for the second and fourth bands.

Fig. 6
Fig. 6

Modulus square of the maximally localized Wannier functions (with respect to the IM) for Sq-A structure (TE polarization). The Wannier center was chosen as “on-site” for the first band and as “between” for the second, fourth, and fifth bands.

Fig. 7
Fig. 7

Modulus square of Bloch-criterion [Eq. (15)] optimized Wannier functions for a square lattice of dielectric rods in air (TM polarization). For n = 1 , 2, 4, the Bloch criterion is equivalent to the IM criterion for real Wannier functions. For n = 3 , 5, 6, the Wannier functions show at least a tendency to localize around the Wannier center.

Fig. 8
Fig. 8

Band structure of a square lattice of dielectric rods in air (TM polarization). The dashed curve is the original band structure and the solid dots are the reproduced band structure obtained by within Wannier function formalism. On the right side, the set of next neighboring sites surrounding Wannier center R = 0 is shown for different nearest neighbor approximations.

Fig. 9
Fig. 9

Frequencies of the modes in a point defect consisting of a single rod with differing permittivity ϵ def in a square lattice of dielectric rods in air (TM polarization). The dots indicate the results of the Wannier function approach by taking the first eight bands into account. They are in complete agreement with plane wave calculations (red solid curve) [25]. At the bottom, the real part of two defect modes with ϵ def = 1 and ϵ def = 30 is shown.

Fig. 10
Fig. 10

The band index contribution C n for the defect modes ϵ def = 1 and ϵ def = 30 . Since the defect frequency lies inside the bandgap between the first and second bands, only the lower band Wannier functions contribute to the defect mode.

Equations (19)

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B n k ( r ) = e i ϕ n k e i kr u n k ( r ) ,
W n R ( r ) = 1 N k BZ e i kR B n k ( r ) .
L E E ( r ) = 1 ϵ ( r ) 2 E ( r ) = ω 2 c 2 E ( r ) ,
L H H ( r ) = 1 ϵ ( r ) H ( r ) = ω 2 c 2 H ( r ) .
f | g E = V d 2 r f * ( r ) ϵ ( r ) g ( r ) ,
f | g H = V d 2 r f * ( r ) g ( r ) .
W n R | W n R E / H = δ n n δ R R .
W n R ( r ) = W n 0 ( r R ) ,
S n ( { ϕ n k } ) = W n R | ( r r 0 ) 2 | W n R E / H ,
I n ( { ϕ n k } ) = UC d 2 r W n R * ( r ) X ( r ) W n R ( r ) ,
X ( r ) = { ϵ ( r ) for     TM 1 for     TE .
u n k ( r ) = u n k * ( r )
ϕ n k = ϕ n k .
I n W ( { ϕ n k } ) = UC d 2 r | W n R ( r ) | 2 ,
I n B ( { ϕ n k } ) = UC d 2 r ( Re ( e i ϕ k B ˜ n k ) ) 2 ,
W n 0 ( r ) = 1 N k BZ B n k .
UC d 2 r | Re ( W n 0 ( r ) ) | = 1 N k BZ UC d 2 r | Re ( B n k ) |
tan ( 2 ϕ k ) = UC d 2 r 2 Re ( B ˜ n k ) Im ( B ˜ n k ) UC d 2 r { Re ( B ˜ n k ) 2 Im ( B ˜ n k ) 2 } ,
C n = 1 M R | E n R | 2 ,

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