Abstract

We describe a fully-vectorial, three-dimensional algorithm to compute the definite-frequency eigenstates of Maxwell’s equations in arbitrary periodic dielectric structures, including systems with anisotropy (birefringence) or magnetic materials, using preconditioned block-iterative eigensolvers in a planewave basis. Favorable scaling with the system size and the number of computed bands is exhibited. We propose a new effective dielectric tensor for anisotropic structures, and demonstrate that O(Δx 2) convergence can be achieved even in systems with sharp material discontinuities. We show how it is possible to solve for interior eigenvalues, such as localized defect modes, without computing the many underlying eigenstates. Preconditioned conjugate-gradient Rayleigh-quotient minimization is compared with the Davidson method for eigensolution, and a number of iteration variants and preconditioners are characterized. Our implementation is freely available on the Web.

©2001 Optical Society of America

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  1. See, e.g., J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
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  4. H. S. Sozüer and J. W. Haus, “Photonic bands: convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
    [Crossref]
  5. R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B48, 8434–8437 (1993). Erratum: S. G. Johnson, ibid55, 15942 (1997).
    [Crossref]
  6. T. Suzuki and P. K. L. Yu, “Method of projection operators for photonic band structures with perfectly conducting elements,” Phys. Rev. B 57, 2229–2241 (1998).
    [Crossref]
  7. K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
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  11. W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: I. Scalar case,” J. Comput. Phys. 150, 468–481 (1999).
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  12. D. C. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999).
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  13. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Localized function method for modeling defect modes in 2D photonic crystals,” J. Lightwave Tech. 17, 2078–2081 (1999).
    [Crossref]
  14. S. J. Cooke and B. Levush, “Eigenmode solution of 2-D and 3-D electromagnetic cavities containing absorbing materials using the Jacobi-Davidson algorithm,” J. Comput. Phys. 157, 350–370 (2000).
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  17. E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parameterization for photonic band-gap materials,” Phys. Rev. Lett. 81, 1405–1408 (1998).
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    [Crossref]
  20. C. T. Chan, Q. L. Lu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635–16642 (1995).
    [Crossref]
  21. S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallo-dielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
    [Crossref]
  22. K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B 56, 4830–4835 (1997).
    [Crossref]
  23. J. Arriaga, A. J. Ward, and J. B. Pendry, “Order N photonic band structures for metals and other dispersive materials,” Phys. Rev. B 59, 1874–1877 (1999).
    [Crossref]
  24. A. J. Ward and J. B. Pendry, “A program for calculating photonic band structures, Green’s functions and transmission/reflection coefficients using a non-orthogonal FDTD method,” Comput. Phys. Comm. 128, 590–621 (2000).
    [Crossref]
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  27. P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, “A program for calculating photonic band structures and transmission coefficients of complex structures,” Comput. Phys. Comm. 85, 306–322 (1995).
    [Crossref]
  28. J. M. Elson and P. Tran, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A 12, 1765–1771 (1995).
    [Crossref]
  29. J. M. Elson and P. Tran, “Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on truncated photonic crystal,” Phys. Rev. B 54, 1711–1715 (1996).
    [Crossref]
  30. J. Chongjun, Q. Bai, Y. Miao, and Q. Ruhu, “Two-dimensional photonic band structure in the chiral medium—transfer matrix method,” Opt. Commun. 142, 179–183 (1997).
    [Crossref]
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    [Crossref]
  36. P. Yang, K. N. Liou, M. I. Mishchenko, and B.-C. Gao, “Efficient finite-difference time-domain scheme for light scattering by dielectric particles: application to aerosols,” Appl. Opt. 39, 3727–3737 (2000).
    [Crossref]
  37. R. D. Meade, private communications.
  38. M. C. Payne, M. P. Tater, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, “Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients,” Rev. Mod. Phys. 64, 1045–1097 (1992).
    [Crossref]
  39. See, e.g., A. Edelman and S. T. Smith, “On conjugate gradient-like methods for eigen-like problems,” BIT 36, 494–509 (1996).
    [Crossref]
  40. S. Ismail-Beigi and T. A. Arias, “New algebraic formulation of density functional calculation,” Comp. Phys. Commun. 128, 1–45 (2000).
    [Crossref]
  41. E. R. Davidson, “The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices,” Comput. Phys. 17, 87–94 (1975).
    [Crossref]
  42. M. Crouzeix, B. Philippe, and M. Sadkane, “The Davidson Method,” SIAM J. Sci. Comput. 15, 62–76 (1994).
    [Crossref]
  43. G. L. G. Sleijpen and H. A. van der Vorst, “A Jacobi-Davidson iteration method for linear eigenvalue problems,” SIAM J. Matrix Anal. Appl. 17, 401–425 (1996).
    [Crossref]
  44. B. N. Parlett, The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, NJ, 1980).
  45. H. A. van der Vorst, “Krylov subspace iteration,” Computing in Sci. and Eng. 2, 32–37 (2000).
    [Crossref]
  46. P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, London, 1981).
  47. J. J. Dongarra, J. Du Croz, I. S. Duff, and S. Hammarling, “A set of Level 3 Basic Linear Algebra Subprograms,” ACM Trans. Math. Soft. 16, 1–17 (1990).
    [Crossref]
  48. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, Philadelphia, 1999).
    [Crossref]
  49. A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithms with orthogonality constraints,” SIAM J. Matrix Anal. Appl. 20, 303–353 (1998).
    [Crossref]
  50. A. H. Sameh and J. A. Wisniewski, “A trace minimization algorithm for the generalized eigenvalue problem,” SIAM J. Numer. Anal. 19, 1243–1259 (1982).
    [Crossref]
  51. B. Philippe, “An algorithm to improve nearly orthonormal sets of vectors on a vector processor,” SIAM J. Alg. Disc. Meth. 8, 396–403 (1987).
    [Crossref]
  52. J. J. Moré and D. J. Thuente, “Line search algorithms with guaranteed sufficient decrease,” ACM Trans. Math. Software 20, 286–307 (1994).
    [Crossref]
  53. S. Ismail-Beiji, private communications.
  54. P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
    [Crossref]
  55. L.-W. Wang and A. Zunger, “Solving Schrödinger’s equation around a desired energy: application to Silicon quantum dots,” J. Chem. Phys. 100, 2394–2397 (1994).
    [Crossref]

2000 (6)

S. J. Cooke and B. Levush, “Eigenmode solution of 2-D and 3-D electromagnetic cavities containing absorbing materials using the Jacobi-Davidson algorithm,” J. Comput. Phys. 157, 350–370 (2000).
[Crossref]

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

A. J. Ward and J. B. Pendry, “A program for calculating photonic band structures, Green’s functions and transmission/reflection coefficients using a non-orthogonal FDTD method,” Comput. Phys. Comm. 128, 590–621 (2000).
[Crossref]

P. Yang, K. N. Liou, M. I. Mishchenko, and B.-C. Gao, “Efficient finite-difference time-domain scheme for light scattering by dielectric particles: application to aerosols,” Appl. Opt. 39, 3727–3737 (2000).
[Crossref]

S. Ismail-Beigi and T. A. Arias, “New algebraic formulation of density functional calculation,” Comp. Phys. Commun. 128, 1–45 (2000).
[Crossref]

H. A. van der Vorst, “Krylov subspace iteration,” Computing in Sci. and Eng. 2, 32–37 (2000).
[Crossref]

1999 (5)

J. Arriaga, A. J. Ward, and J. B. Pendry, “Order N photonic band structures for metals and other dispersive materials,” Phys. Rev. B 59, 1874–1877 (1999).
[Crossref]

K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
[Crossref]

W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: I. Scalar case,” J. Comput. Phys. 150, 468–481 (1999).
[Crossref]

D. C. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999).
[Crossref]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Localized function method for modeling defect modes in 2D photonic crystals,” J. Lightwave Tech. 17, 2078–2081 (1999).
[Crossref]

1998 (5)

T. Suzuki and P. K. L. Yu, “Method of projection operators for photonic band structures with perfectly conducting elements,” Phys. Rev. B 57, 2229–2241 (1998).
[Crossref]

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

W. C. Sailor, F. M. Mueller, and P. R. Villeneuve, “Augmented-plane-wave method for photonic band-gap materials,” Phys. Rev. B 57, 8819–8822 (1998).
[Crossref]

J. Nadobny, D. Sullivan, P. Wust, M. Seebass, P. Deuflhard, and R. Felix, “A high-resolution interpolation at arbitrary interfaces for the FDTD method,” IEEE Trans. Microwave Theory Tech. 46, 1759–1766 (1998).
[Crossref]

A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithms with orthogonality constraints,” SIAM J. Matrix Anal. Appl. 20, 303–353 (1998).
[Crossref]

1997 (4)

J. Chongjun, Q. Bai, Y. Miao, and Q. Ruhu, “Two-dimensional photonic band structure in the chiral medium—transfer matrix method,” Opt. Commun. 142, 179–183 (1997).
[Crossref]

K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B 56, 4830–4835 (1997).
[Crossref]

See, e.g., J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[Crossref]

A. Figotin and Y. A. Godin, “The computation of spectra of some 2D photonic crystals,” J. Comput. Phys. 136, 585–598 (1997).
[Crossref]

1996 (5)

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallo-dielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[Crossref]

J. M. Elson and P. Tran, “Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on truncated photonic crystal,” Phys. Rev. B 54, 1711–1715 (1996).
[Crossref]

See, e.g., A. Edelman and S. T. Smith, “On conjugate gradient-like methods for eigen-like problems,” BIT 36, 494–509 (1996).
[Crossref]

G. L. G. Sleijpen and H. A. van der Vorst, “A Jacobi-Davidson iteration method for linear eigenvalue problems,” SIAM J. Matrix Anal. Appl. 17, 401–425 (1996).
[Crossref]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[Crossref]

1995 (3)

P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, “A program for calculating photonic band structures and transmission coefficients of complex structures,” Comput. Phys. Comm. 85, 306–322 (1995).
[Crossref]

J. M. Elson and P. Tran, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A 12, 1765–1771 (1995).
[Crossref]

C. T. Chan, Q. L. Lu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635–16642 (1995).
[Crossref]

1994 (4)

C. T. Chan, S. Datta, Q. L. Yu, M. Sigalas, K. M. Ho, and C. M. Soukoulis, “New structures and algorithms for photonic band gaps,” Physica A 211, 411–419 (1994).
[Crossref]

M. Crouzeix, B. Philippe, and M. Sadkane, “The Davidson Method,” SIAM J. Sci. Comput. 15, 62–76 (1994).
[Crossref]

L.-W. Wang and A. Zunger, “Solving Schrödinger’s equation around a desired energy: application to Silicon quantum dots,” J. Chem. Phys. 100, 2394–2397 (1994).
[Crossref]

J. J. Moré and D. J. Thuente, “Line search algorithms with guaranteed sufficient decrease,” ACM Trans. Math. Software 20, 286–307 (1994).
[Crossref]

1993 (1)

1992 (3)

H. S. Sozüer and J. W. Haus, “Photonic bands: convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
[Crossref]

M. C. Payne, M. P. Tater, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, “Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients,” Rev. Mod. Phys. 64, 1045–1097 (1992).
[Crossref]

J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[Crossref] [PubMed]

1990 (2)

J. J. Dongarra, J. Du Croz, I. S. Duff, and S. Hammarling, “A set of Level 3 Basic Linear Algebra Subprograms,” ACM Trans. Math. Soft. 16, 1–17 (1990).
[Crossref]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

1987 (1)

B. Philippe, “An algorithm to improve nearly orthonormal sets of vectors on a vector processor,” SIAM J. Alg. Disc. Meth. 8, 396–403 (1987).
[Crossref]

1982 (1)

A. H. Sameh and J. A. Wisniewski, “A trace minimization algorithm for the generalized eigenvalue problem,” SIAM J. Numer. Anal. 19, 1243–1259 (1982).
[Crossref]

1975 (1)

E. R. Davidson, “The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices,” Comput. Phys. 17, 87–94 (1975).
[Crossref]

Albert, J. P.

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

Alerhand, O. L.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B48, 8434–8437 (1993). Erratum: S. G. Johnson, ibid55, 15942 (1997).
[Crossref]

Allan, D. C.

M. C. Payne, M. P. Tater, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, “Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients,” Rev. Mod. Phys. 64, 1045–1097 (1992).
[Crossref]

Anderson, E.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, Philadelphia, 1999).
[Crossref]

Arias, T. A.

S. Ismail-Beigi and T. A. Arias, “New algebraic formulation of density functional calculation,” Comp. Phys. Commun. 128, 1–45 (2000).
[Crossref]

A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithms with orthogonality constraints,” SIAM J. Matrix Anal. Appl. 20, 303–353 (1998).
[Crossref]

M. C. Payne, M. P. Tater, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, “Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients,” Rev. Mod. Phys. 64, 1045–1097 (1992).
[Crossref]

Arriaga, J.

J. Arriaga, A. J. Ward, and J. B. Pendry, “Order N photonic band structures for metals and other dispersive materials,” Phys. Rev. B 59, 1874–1877 (1999).
[Crossref]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt Saunders, Philadelphia, 1976).

Axmann, W.

W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: I. Scalar case,” J. Comput. Phys. 150, 468–481 (1999).
[Crossref]

Bai, Q.

J. Chongjun, Q. Bai, Y. Miao, and Q. Ruhu, “Two-dimensional photonic band structure in the chiral medium—transfer matrix method,” Opt. Commun. 142, 179–183 (1997).
[Crossref]

Bai, Z.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, Philadelphia, 1999).
[Crossref]

Bell, P. M.

P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, “A program for calculating photonic band structures and transmission coefficients of complex structures,” Comput. Phys. Comm. 85, 306–322 (1995).
[Crossref]

Bertho, D.

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

Birks, T. A.

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Localized function method for modeling defect modes in 2D photonic crystals,” J. Lightwave Tech. 17, 2078–2081 (1999).
[Crossref]

Bischof, C.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, Philadelphia, 1999).
[Crossref]

Blackford, S.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, Philadelphia, 1999).
[Crossref]

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B48, 8434–8437 (1993). Erratum: S. G. Johnson, ibid55, 15942 (1997).
[Crossref]

Busch, K.

K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
[Crossref]

Cassagne, D.

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

Chan, C. T.

C. T. Chan, Q. L. Lu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635–16642 (1995).
[Crossref]

C. T. Chan, S. Datta, Q. L. Yu, M. Sigalas, K. M. Ho, and C. M. Soukoulis, “New structures and algorithms for photonic band gaps,” Physica A 211, 411–419 (1994).
[Crossref]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

Chongjun, J.

J. Chongjun, Q. Bai, Y. Miao, and Q. Ruhu, “Two-dimensional photonic band structure in the chiral medium—transfer matrix method,” Opt. Commun. 142, 179–183 (1997).
[Crossref]

Cooke, S. J.

S. J. Cooke and B. Levush, “Eigenmode solution of 2-D and 3-D electromagnetic cavities containing absorbing materials using the Jacobi-Davidson algorithm,” J. Comput. Phys. 157, 350–370 (2000).
[Crossref]

Crouzeix, M.

M. Crouzeix, B. Philippe, and M. Sadkane, “The Davidson Method,” SIAM J. Sci. Comput. 15, 62–76 (1994).
[Crossref]

Datta, S.

C. T. Chan, S. Datta, Q. L. Yu, M. Sigalas, K. M. Ho, and C. M. Soukoulis, “New structures and algorithms for photonic band gaps,” Physica A 211, 411–419 (1994).
[Crossref]

Davidson, E. R.

E. R. Davidson, “The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices,” Comput. Phys. 17, 87–94 (1975).
[Crossref]

Demmel, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, Philadelphia, 1999).
[Crossref]

Deuflhard, P.

J. Nadobny, D. Sullivan, P. Wust, M. Seebass, P. Deuflhard, and R. Felix, “A high-resolution interpolation at arbitrary interfaces for the FDTD method,” IEEE Trans. Microwave Theory Tech. 46, 1759–1766 (1998).
[Crossref]

Dobson, D. C.

D. C. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999).
[Crossref]

Dongarra, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, Philadelphia, 1999).
[Crossref]

Dongarra, J. J.

J. J. Dongarra, J. Du Croz, I. S. Duff, and S. Hammarling, “A set of Level 3 Basic Linear Algebra Subprograms,” ACM Trans. Math. Soft. 16, 1–17 (1990).
[Crossref]

Du Croz, J.

J. J. Dongarra, J. Du Croz, I. S. Duff, and S. Hammarling, “A set of Level 3 Basic Linear Algebra Subprograms,” ACM Trans. Math. Soft. 16, 1–17 (1990).
[Crossref]

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, Philadelphia, 1999).
[Crossref]

Duff, I. S.

J. J. Dongarra, J. Du Croz, I. S. Duff, and S. Hammarling, “A set of Level 3 Basic Linear Algebra Subprograms,” ACM Trans. Math. Soft. 16, 1–17 (1990).
[Crossref]

Economou, E. N.

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

Edelman, A.

A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithms with orthogonality constraints,” SIAM J. Matrix Anal. Appl. 20, 303–353 (1998).
[Crossref]

See, e.g., A. Edelman and S. T. Smith, “On conjugate gradient-like methods for eigen-like problems,” BIT 36, 494–509 (1996).
[Crossref]

Elson, J. M.

J. M. Elson and P. Tran, “Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on truncated photonic crystal,” Phys. Rev. B 54, 1711–1715 (1996).
[Crossref]

J. M. Elson and P. Tran, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A 12, 1765–1771 (1995).
[Crossref]

Fan, S.

See, e.g., J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[Crossref]

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallo-dielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[Crossref]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[Crossref]

Felix, R.

J. Nadobny, D. Sullivan, P. Wust, M. Seebass, P. Deuflhard, and R. Felix, “A high-resolution interpolation at arbitrary interfaces for the FDTD method,” IEEE Trans. Microwave Theory Tech. 46, 1759–1766 (1998).
[Crossref]

Figotin, A.

A. Figotin and Y. A. Godin, “The computation of spectra of some 2D photonic crystals,” J. Comput. Phys. 136, 585–598 (1997).
[Crossref]

Frigo, M.

M. Frigo and S. G. Johnson, “FFTW: an adaptive software architecture for the FFT,” in Proc. 1998 IEEE Intl. Conf. on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1998), 1381–1384.

Gao, B.-C.

Gill, P. E.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, London, 1981).

Godin, Y. A.

A. Figotin and Y. A. Godin, “The computation of spectra of some 2D photonic crystals,” J. Comput. Phys. 136, 585–598 (1997).
[Crossref]

Greenbaum, A.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, Philadelphia, 1999).
[Crossref]

Hammarling, S.

J. J. Dongarra, J. Du Croz, I. S. Duff, and S. Hammarling, “A set of Level 3 Basic Linear Algebra Subprograms,” ACM Trans. Math. Soft. 16, 1–17 (1990).
[Crossref]

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, Philadelphia, 1999).
[Crossref]

Haus, J. W.

H. S. Sozüer and J. W. Haus, “Photonic bands: convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
[Crossref]

Ho, K. M.

C. T. Chan, Q. L. Lu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635–16642 (1995).
[Crossref]

C. T. Chan, S. Datta, Q. L. Yu, M. Sigalas, K. M. Ho, and C. M. Soukoulis, “New structures and algorithms for photonic band gaps,” Physica A 211, 411–419 (1994).
[Crossref]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

Ismail-Beigi, S.

S. Ismail-Beigi and T. A. Arias, “New algebraic formulation of density functional calculation,” Comp. Phys. Commun. 128, 1–45 (2000).
[Crossref]

Ismail-Beiji, S.

S. Ismail-Beiji, private communications.

Jin, J.

J. Jin, The Finite-Element Method in Electromagnetics (Wiley, New York, 1993), Chap. 5.7.

Joannopoulos, J. D.

See, e.g., J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[Crossref]

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallo-dielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[Crossref]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[Crossref]

M. C. Payne, M. P. Tater, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, “Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients,” Rev. Mod. Phys. 64, 1045–1097 (1992).
[Crossref]

S. G. Johnson and J. D. Joannopoulos, The MIT Photonic-Bands Package home page http://ab-initio.mit.edu/mpb/.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B48, 8434–8437 (1993). Erratum: S. G. Johnson, ibid55, 15942 (1997).
[Crossref]

John, S.

K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999).
[Crossref]

Johnson, S. G.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B48, 8434–8437 (1993). Erratum: S. G. Johnson, ibid55, 15942 (1997).
[Crossref]

S. G. Johnson and J. D. Joannopoulos, The MIT Photonic-Bands Package home page http://ab-initio.mit.edu/mpb/.

M. Frigo and S. G. Johnson, “FFTW: an adaptive software architecture for the FFT,” in Proc. 1998 IEEE Intl. Conf. on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1998), 1381–1384.

Jouanin, C.

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

Kuchment, P.

W. Axmann and P. Kuchment, “An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: I. Scalar case,” J. Comput. Phys. 150, 468–481 (1999).
[Crossref]

Kunz, K. S.

See, e.g., K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Methods (CRC, Boca Raton, Fla., 1993).

Leung, K. M.

Levush, B.

S. J. Cooke and B. Levush, “Eigenmode solution of 2-D and 3-D electromagnetic cavities containing absorbing materials using the Jacobi-Davidson algorithm,” J. Comput. Phys. 157, 350–370 (2000).
[Crossref]

Lidorikis, E.

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

Liou, K. N.

Lu, Q. L.

C. T. Chan, Q. L. Lu, and K. M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635–16642 (1995).
[Crossref]

Luebbers, R. J.

See, e.g., K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Methods (CRC, Boca Raton, Fla., 1993).

MacKinnon, A.

J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[Crossref] [PubMed]

Mandelshtam, V. A.

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys.107, 6756–6769 (1997). Erratum: ibid, 109, 4128 (1998).
[Crossref]

McKenney, A.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, Philadelphia, 1999).
[Crossref]

Meade, R. D.

R. D. Meade, private communications.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B48, 8434–8437 (1993). Erratum: S. G. Johnson, ibid55, 15942 (1997).
[Crossref]

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt Saunders, Philadelphia, 1976).

Miao, Y.

J. Chongjun, Q. Bai, Y. Miao, and Q. Ruhu, “Two-dimensional photonic band structure in the chiral medium—transfer matrix method,” Opt. Commun. 142, 179–183 (1997).
[Crossref]

Mishchenko, M. I.

Mogilevtsev, D.

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Localized function method for modeling defect modes in 2D photonic crystals,” J. Lightwave Tech. 17, 2078–2081 (1999).
[Crossref]

Moré, J. J.

J. J. Moré and D. J. Thuente, “Line search algorithms with guaranteed sufficient decrease,” ACM Trans. Math. Software 20, 286–307 (1994).
[Crossref]

Moreno, L. M.

P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, “A program for calculating photonic band structures and transmission coefficients of complex structures,” Comput. Phys. Comm. 85, 306–322 (1995).
[Crossref]

Mueller, F. M.

W. C. Sailor, F. M. Mueller, and P. R. Villeneuve, “Augmented-plane-wave method for photonic band-gap materials,” Phys. Rev. B 57, 8819–8822 (1998).
[Crossref]

Murray, W.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, London, 1981).

Nadobny, J.

J. Nadobny, D. Sullivan, P. Wust, M. Seebass, P. Deuflhard, and R. Felix, “A high-resolution interpolation at arbitrary interfaces for the FDTD method,” IEEE Trans. Microwave Theory Tech. 46, 1759–1766 (1998).
[Crossref]

Parlett, B. N.

B. N. Parlett, The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, NJ, 1980).

Payne, M. C.

M. C. Payne, M. P. Tater, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, “Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients,” Rev. Mod. Phys. 64, 1045–1097 (1992).
[Crossref]

Pendry, J. B.

A. J. Ward and J. B. Pendry, “A program for calculating photonic band structures, Green’s functions and transmission/reflection coefficients using a non-orthogonal FDTD method,” Comput. Phys. Comm. 128, 590–621 (2000).
[Crossref]

J. Arriaga, A. J. Ward, and J. B. Pendry, “Order N photonic band structures for metals and other dispersive materials,” Phys. Rev. B 59, 1874–1877 (1999).
[Crossref]

P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, “A program for calculating photonic band structures and transmission coefficients of complex structures,” Comput. Phys. Comm. 85, 306–322 (1995).
[Crossref]

J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[Crossref] [PubMed]

Philippe, B.

M. Crouzeix, B. Philippe, and M. Sadkane, “The Davidson Method,” SIAM J. Sci. Comput. 15, 62–76 (1994).
[Crossref]

B. Philippe, “An algorithm to improve nearly orthonormal sets of vectors on a vector processor,” SIAM J. Alg. Disc. Meth. 8, 396–403 (1987).
[Crossref]

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B48, 8434–8437 (1993). Erratum: S. G. Johnson, ibid55, 15942 (1997).
[Crossref]

Ruhu, Q.

J. Chongjun, Q. Bai, Y. Miao, and Q. Ruhu, “Two-dimensional photonic band structure in the chiral medium—transfer matrix method,” Opt. Commun. 142, 179–183 (1997).
[Crossref]

Russell, P. St. J.

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Localized function method for modeling defect modes in 2D photonic crystals,” J. Lightwave Tech. 17, 2078–2081 (1999).
[Crossref]

Sadkane, M.

M. Crouzeix, B. Philippe, and M. Sadkane, “The Davidson Method,” SIAM J. Sci. Comput. 15, 62–76 (1994).
[Crossref]

Sailor, W. C.

W. C. Sailor, F. M. Mueller, and P. R. Villeneuve, “Augmented-plane-wave method for photonic band-gap materials,” Phys. Rev. B 57, 8819–8822 (1998).
[Crossref]

Sakoda, K.

K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B 56, 4830–4835 (1997).
[Crossref]

Sameh, A. H.

A. H. Sameh and J. A. Wisniewski, “A trace minimization algorithm for the generalized eigenvalue problem,” SIAM J. Numer. Anal. 19, 1243–1259 (1982).
[Crossref]

Seebass, M.

J. Nadobny, D. Sullivan, P. Wust, M. Seebass, P. Deuflhard, and R. Felix, “A high-resolution interpolation at arbitrary interfaces for the FDTD method,” IEEE Trans. Microwave Theory Tech. 46, 1759–1766 (1998).
[Crossref]

Shiroma, H.

K. Sakoda and H. Shiroma, “Numerical method for localized defect modes in photonic lattices,” Phys. Rev. B 56, 4830–4835 (1997).
[Crossref]

Sigalas, M.

C. T. Chan, S. Datta, Q. L. Yu, M. Sigalas, K. M. Ho, and C. M. Soukoulis, “New structures and algorithms for photonic band gaps,” Physica A 211, 411–419 (1994).
[Crossref]

Sigalas, M. M.

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

Sleijpen, G. L. G.

G. L. G. Sleijpen and H. A. van der Vorst, “A Jacobi-Davidson iteration method for linear eigenvalue problems,” SIAM J. Matrix Anal. Appl. 17, 401–425 (1996).
[Crossref]

Smith, S. T.

A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithms with orthogonality constraints,” SIAM J. Matrix Anal. Appl. 20, 303–353 (1998).
[Crossref]

See, e.g., A. Edelman and S. T. Smith, “On conjugate gradient-like methods for eigen-like problems,” BIT 36, 494–509 (1996).
[Crossref]

Sorensen, D.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, Philadelphia, 1999).
[Crossref]

Soukoulis, C. M.

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

C. T. Chan, S. Datta, Q. L. Yu, M. Sigalas, K. M. Ho, and C. M. Soukoulis, “New structures and algorithms for photonic band gaps,” Physica A 211, 411–419 (1994).
[Crossref]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

Sozüer, H. S.

H. S. Sozüer and J. W. Haus, “Photonic bands: convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
[Crossref]

Stroud, A. H.

A. H. Stroud, Approximate Calculation of Multiple Integrals (Prentice-Hall, Englewood Cliffs, NJ, 1971).

Sullivan, D.

J. Nadobny, D. Sullivan, P. Wust, M. Seebass, P. Deuflhard, and R. Felix, “A high-resolution interpolation at arbitrary interfaces for the FDTD method,” IEEE Trans. Microwave Theory Tech. 46, 1759–1766 (1998).
[Crossref]

Suzuki, T.

T. Suzuki and P. K. L. Yu, “Method of projection operators for photonic band structures with perfectly conducting elements,” Phys. Rev. B 57, 2229–2241 (1998).
[Crossref]

Tater, M. P.

M. C. Payne, M. P. Tater, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, “Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients,” Rev. Mod. Phys. 64, 1045–1097 (1992).
[Crossref]

Taylor, H. S.

V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys.107, 6756–6769 (1997). Erratum: ibid, 109, 4128 (1998).
[Crossref]

Thuente, D. J.

J. J. Moré and D. J. Thuente, “Line search algorithms with guaranteed sufficient decrease,” ACM Trans. Math. Software 20, 286–307 (1994).
[Crossref]

Tran, P.

J. M. Elson and P. Tran, “Coupled-mode calculation with the R-matrix propagator for the dispersion of surface waves on truncated photonic crystal,” Phys. Rev. B 54, 1711–1715 (1996).
[Crossref]

J. M. Elson and P. Tran, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A 12, 1765–1771 (1995).
[Crossref]

van der Vorst, H. A.

H. A. van der Vorst, “Krylov subspace iteration,” Computing in Sci. and Eng. 2, 32–37 (2000).
[Crossref]

G. L. G. Sleijpen and H. A. van der Vorst, “A Jacobi-Davidson iteration method for linear eigenvalue problems,” SIAM J. Matrix Anal. Appl. 17, 401–425 (1996).
[Crossref]

Villeneuve, P. R.

W. C. Sailor, F. M. Mueller, and P. R. Villeneuve, “Augmented-plane-wave method for photonic band-gap materials,” Phys. Rev. B 57, 8819–8822 (1998).
[Crossref]

See, e.g., J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[Crossref]

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Large omnidirectional band gaps in metallo-dielectric photonic crystals,” Phys. Rev. B 54, 11245–11251 (1996).
[Crossref]

P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Microcavities in photonic crystals: mode symmetry, tunability, and coupling efficiency,” Phys. Rev. B 54, 7837–7842 (1996).
[Crossref]

Wang, L.-W.

L.-W. Wang and A. Zunger, “Solving Schrödinger’s equation around a desired energy: application to Silicon quantum dots,” J. Chem. Phys. 100, 2394–2397 (1994).
[Crossref]

Ward, A. J.

A. J. Ward and J. B. Pendry, “A program for calculating photonic band structures, Green’s functions and transmission/reflection coefficients using a non-orthogonal FDTD method,” Comput. Phys. Comm. 128, 590–621 (2000).
[Crossref]

J. Arriaga, A. J. Ward, and J. B. Pendry, “Order N photonic band structures for metals and other dispersive materials,” Phys. Rev. B 59, 1874–1877 (1999).
[Crossref]

P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, “A program for calculating photonic band structures and transmission coefficients of complex structures,” Comput. Phys. Comm. 85, 306–322 (1995).
[Crossref]

Wisniewski, J. A.

A. H. Sameh and J. A. Wisniewski, “A trace minimization algorithm for the generalized eigenvalue problem,” SIAM J. Numer. Anal. 19, 1243–1259 (1982).
[Crossref]

Wright, M. H.

P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, London, 1981).

Wust, P.

J. Nadobny, D. Sullivan, P. Wust, M. Seebass, P. Deuflhard, and R. Felix, “A high-resolution interpolation at arbitrary interfaces for the FDTD method,” IEEE Trans. Microwave Theory Tech. 46, 1759–1766 (1998).
[Crossref]

Yang, P.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Yu, P. K. L.

T. Suzuki and P. K. L. Yu, “Method of projection operators for photonic band structures with perfectly conducting elements,” Phys. Rev. B 57, 2229–2241 (1998).
[Crossref]

Yu, Q. L.

C. T. Chan, S. Datta, Q. L. Yu, M. Sigalas, K. M. Ho, and C. M. Soukoulis, “New structures and algorithms for photonic band gaps,” Physica A 211, 411–419 (1994).
[Crossref]

Zunger, A.

L.-W. Wang and A. Zunger, “Solving Schrödinger’s equation around a desired energy: application to Silicon quantum dots,” J. Chem. Phys. 100, 2394–2397 (1994).
[Crossref]

ACM Trans. Math. Soft. (1)

J. J. Dongarra, J. Du Croz, I. S. Duff, and S. Hammarling, “A set of Level 3 Basic Linear Algebra Subprograms,” ACM Trans. Math. Soft. 16, 1–17 (1990).
[Crossref]

ACM Trans. Math. Software (1)

J. J. Moré and D. J. Thuente, “Line search algorithms with guaranteed sufficient decrease,” ACM Trans. Math. Software 20, 286–307 (1994).
[Crossref]

Appl. Opt. (1)

BIT (1)

See, e.g., A. Edelman and S. T. Smith, “On conjugate gradient-like methods for eigen-like problems,” BIT 36, 494–509 (1996).
[Crossref]

Comp. Phys. Commun. (1)

S. Ismail-Beigi and T. A. Arias, “New algebraic formulation of density functional calculation,” Comp. Phys. Commun. 128, 1–45 (2000).
[Crossref]

Comput. Phys. (1)

E. R. Davidson, “The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices,” Comput. Phys. 17, 87–94 (1975).
[Crossref]

Comput. Phys. Comm. (2)

A. J. Ward and J. B. Pendry, “A program for calculating photonic band structures, Green’s functions and transmission/reflection coefficients using a non-orthogonal FDTD method,” Comput. Phys. Comm. 128, 590–621 (2000).
[Crossref]

P. M. Bell, J. B. Pendry, L. M. Moreno, and A. J. Ward, “A program for calculating photonic band structures and transmission coefficients of complex structures,” Comput. Phys. Comm. 85, 306–322 (1995).
[Crossref]

Computing in Sci. and Eng. (1)

H. A. van der Vorst, “Krylov subspace iteration,” Computing in Sci. and Eng. 2, 32–37 (2000).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

J. Nadobny, D. Sullivan, P. Wust, M. Seebass, P. Deuflhard, and R. Felix, “A high-resolution interpolation at arbitrary interfaces for the FDTD method,” IEEE Trans. Microwave Theory Tech. 46, 1759–1766 (1998).
[Crossref]

J. Chem. Phys. (1)

L.-W. Wang and A. Zunger, “Solving Schrödinger’s equation around a desired energy: application to Silicon quantum dots,” J. Chem. Phys. 100, 2394–2397 (1994).
[Crossref]

J. Comput. Phys. (4)

A. Figotin and Y. A. Godin, “The computation of spectra of some 2D photonic crystals,” J. Comput. Phys. 136, 585–598 (1997).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1.

Eigenvalue convergence as a function of grid resolution (grid points per lattice constant a) for three different methods of determining an effective dielectric tensor at each point: no averaging, simply taking the dielectric constant at each grid point; averaging, the smoothed effective dielectric tensor of Eq. (12); and backwards averaging, the same smoothed dielectric but with the averaging methods of the two polarizations reversed.

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Fig. 2.
Fig. 2.

Eigensolver convergence for two variants of conjugate gradient, Fletcher-Reeves and Polak-Ribiere, along with preconditioned steepest-descent for comparison.

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Fig. 3.
Fig. 3.

The effect of two preconditioning schemes from section 2.4, diagonal and transverse-projection (non-diagonal), on the conjugate-gradient method.

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Fig. 4.
Fig. 4.

Comparison of the Davidson method with the block conjugate-gradient algorithm of section 3.1. We reset the Davidson subspace to the best current eigenvectors every 2, 3, 4, or 5 iterations, with a corresponding in increase in memory usage and computational costs.

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Fig. 5.
Fig. 5.

Conjugate-gradient convergence of the lowest TM eigenvalue for the “interior” eigensolver of Eq. (27), solving for the monopole defect state formed by one vacancy in a 2D square lattice of dielectric rods in air, using three different supercell sizes (3×3, 5×5, and 7×7).

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Fig. 6.
Fig. 6.

Scaling of the number of conjugate-gradient iterations required for convergence (to a fractional tolerance of 10-7) as a function of the spatial resolution (in grid points per lattice constant, with a corresponding planewave cutoff), or the number p of bands computed.

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Equations (27)

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

∇ → × 1 ε ∇ → × ∣ H 〉 = − 1 c 2 ∂ 2 ∂ t 2 ∣ H 〉 ,
∇ → · ∣ H 〉 = 0 .
∣ H 〉 = e i ( k → · x → − ω t ) ∣ H k → 〉 ,
A ̂ k → ∣ H k → 〉 = ( ω ⁄ c ) 2 ∣ H k → 〉 ,
A ̂ k → = ( ∇ → + i k → ) × 1 ε ( ∇ → + i k → ) × .
〈 H k → ( n ) | H k → ( m ) 〉 = δ n , m ,
∣ H k → 〉 ≅ ∑ m = 1 N h m ∣ b m 〉 .
A h = ( ω c ) 2 B h ,
H → k → ( ∑ k n k R → k ⁄ N k ) = ∑ { m j } h → { m j } e i ∑ j , k m j G → j · n k R → k ⁄ N k = ∑ { m j } h → { m j } e 2 π i ∑ j m j n j ⁄ N j .
A ℓ m = − ( k → + G → ℓ ) × ⋯ IFFT ⋯ ε − 1 ˜ ⋯ FFT ⋯ ( k → + G → m ) × .
ε − 1 ˜ = ε − 1 ˜ P + ε - − 1 ( 1 − P )
ε − 1 ˜ = 1 2 ( { ε − 1 ¯ , P } + { ε − 1 ¯ , ( 1 − P ) } ) ,
A ˜ ℓ m = ∣ k → + G → m ∣ 2 δ ℓ , m ,
A ̂ ˜ = ∇ → × P ̂ T 1 ε P ̂ T ∇ → × ,
⋋ 0 = min y 0 † A y 0 y 0 † B y 0 ,
Y = Z ( Z † B Z ) − 1 2
t r [ Z † A Z U ] ,
G = P ⊥ A Z U ,
D = K ̂ G + γ D 0 ,
γ = t r [ G † K ̂ G ] t r [ G 0 † K ̂ G 0 ]
γ = t r [ ( G − G 0 ) † K ̂ G ] t r [ G 0 † K ̂ G 0 ]
Z = Z * + δ Z ,
G ≅ P ⊥ ( A δ Z − B δ Z U Z † A Z ) U ,
G ≅ P ⊥ A δ Z U .
δ Z ≅ K ̂ G = A − 1 ˜ G U − 1 ,
R = P ⊥ ( A D − B D L ) ,
A ̂ ' k → = ( A ̂ k → − ω m 2 c 2 ) 2 .

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