Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis
Optics Express, Vol. 8, Issue 3, pp. 173-190 (2001)
http://dx.doi.org/10.1364/OE.8.000173
Acrobat PDF (317 KB)
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.
© Optical Society of America
1 Introduction
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). [CrossRef]
2. S. G. Johnson and J. D. Joannopoulos, The MIT Photonic-Bands Package home page http://ab-initio.mit.edu/mpb/.
3. 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]
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). [CrossRef]
19. 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]
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]
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]
31. 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]
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]
2 The Maxwell Eigenproblem
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). [CrossRef]
2.1 The choice of basis
3. 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]
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]
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). [CrossRef]
10. 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]
2.1.1 The planewave basis
3. 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]
7. K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999). [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]
2. S. G. Johnson and J. D. Joannopoulos, The MIT Photonic-Bands Package home page http://ab-initio.mit.edu/mpb/.
2. S. G. Johnson and J. D. Joannopoulos, The MIT Photonic-Bands Package home page http://ab-initio.mit.edu/mpb/.
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]
2.1.2 Other possible bases
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). [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). [CrossRef]
15. K. M. Leung, “Defect modes in photonic band structures: a Green’s function approach using vector Wannier functions,” J. Opt. Soc. Am. B 10, 303–306 (1993). [CrossRef]
16. 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]
16. 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]
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). [CrossRef]
2.2 Inversion symmetry
2.3 The effective dielectric tensor
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]
7. K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967–970 (1999). [CrossRef]
35. 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]
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]
2.4 Preconditioners
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]
12. D. C. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999). [CrossRef]
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]
2.4.1 Removing the singularity
3 Iterative Eigensolvers
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]
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]
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]
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]
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]
45. H. A. van der Vorst, “Krylov subspace iteration,” Computing in Sci. and Eng. 2, 32–37 (2000). [CrossRef]
3. 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]
3.1 Conjugate-gradient minimization of the Rayleigh quotient
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]
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]
3.1.1 The block Rayleigh quotient
40. S. Ismail-Beigi and T. A. Arias, “New algebraic formulation of density functional calculation,” Comp. Phys. Commun. 128, 1–45 (2000). [CrossRef]
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]
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]
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]
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]
3.1.2 Conjugate gradient
40. S. Ismail-Beigi and T. A. Arias, “New algebraic formulation of density functional calculation,” Comp. Phys. Commun. 128, 1–45 (2000). [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]
52. J. J. Moré and D. J. Thuente, “Line search algorithms with guaranteed sufficient decrease,” ACM Trans. Math. Software 20, 286–307 (1994). [CrossRef]
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]
40. S. Ismail-Beigi and T. A. Arias, “New algebraic formulation of density functional calculation,” Comp. Phys. Commun. 128, 1–45 (2000). [CrossRef]
3.1.3 Preconditioning
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]
3.2 The Davidson method
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]
12. D. C. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999). [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]
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]
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). [CrossRef]
3.3 Interior eigenvalues
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). [CrossRef]
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]
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]
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]
3.4 To block or not to block?
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]
3.5 Scaling
4 Conclusion
Acknowledgments
Footnotes
1 | This is equivalent to m_{j} =0,…,N_{j} -1 for the DFT, in which rri_{j} is interpreted modulo N_{j} , but choosing zero-centered wavevectors is important when taking derivatives of the basis. |
2 | This method for defining n̂ can produce suboptimal results when the averaging voxel straddles two near-parallel dielectric interfaces. Preliminary investigations show that gains of a factor of two or more in eigenvalue accuracy are sometimes possible if a better approximation for n̂ is used in such cases. |
3 | We also tried using the differential-geometry conjugate-gradient algorithm of [49 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] |
4 | In the literature, “preconditioner” sometimes refers instead to K̂ ^{-1}, the approximate Hessian. |
References and links
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). [CrossRef] |
2. | S. G. Johnson and J. D. Joannopoulos, The MIT Photonic-Bands Package home page http://ab-initio.mit.edu/mpb/. |
3. | 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] |
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). [CrossRef] |
8. | J. Jin, The Finite-Element Method in Electromagnetics (Wiley, New York, 1993), Chap. 5.7. |
9. | A. Figotin and Y. A. Godin, “The computation of spectra of some 2D photonic crystals,” J. Comput. Phys. 136, 585–598 (1997). [CrossRef] |
10. | 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] |
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). [CrossRef] |
12. | D. C. Dobson, “An efficient method for band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999). [CrossRef] |
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). [CrossRef] |
15. | K. M. Leung, “Defect modes in photonic band structures: a Green’s function approach using vector Wannier functions,” J. Opt. Soc. Am. B 10, 303–306 (1993). [CrossRef] |
16. | 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] |
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). [CrossRef] |
18. | See, e.g., K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Methods (CRC, Boca Raton, Fla., 1993). |
19. | 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] |
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] |
25. | P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988). |
26. | J. B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992). [CrossRef] [PubMed] |
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] |
31. | 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] |
32. | N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt Saunders, Philadelphia, 1976). |
33. | 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. |
34. | A. H. Stroud, Approximate Calculation of Multiple Integrals (Prentice-Hall, Englewood Cliffs, NJ, 1971). |
35. | 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] |
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] |
OCIS Codes
(000.3860) General : Mathematical methods in physics
(000.4430) General : Numerical approximation and analysis
ToC Category:
Focus Issue: Photonic bandgap calculations
History
Original Manuscript: November 17, 2000
Published: January 29, 2001
Citation
Steven Johnson and John Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-3-173
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References
- 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. G. Johnson and J. D. Joannopoulos, The MIT Photonic-Bands Package home page http://ab-initio.mit.edu/mpb/.
- 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]
- H. S. Sözüer and J. W. Haus, "Photonic bands: convergence problems with the plane-wave method," Phys. Rev. B 45, 13962-13972 (1992). [CrossRef]
- 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. B 48, 8434-8437 (1993). Erratum: S. G. Johnson, ibid 55, 15942 (1997). [CrossRef]
- 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]
- K. Busch and S. John, "Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum," Phys. Rev. Lett. 83, 967-970 (1999). [CrossRef]
- J. Jin, The Finite-Element Method in Electromagnetics (Wiley, New York, 1993), Chap. 5.7.
- A. Figotin, Y. A. Godin, "The computation of spectra of some 2D photonic crystals," J. Comput. Phys. 136, 585-598 (1997). [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]
- 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]
- S. J. Cooke and B. Levush, "Eigenmode solution of 2-D and 3-D electromagnetic cavities contain- ing absorbing materials using the Jacobi-Davidson algorithm," J. Comput. Phys. 157, 350-370 (2000). [CrossRef]
- K. M. Leung, "Defect modes in photonic band structures: a Green's function approach using vector Wannier functions," J. Opt. Soc. Am. B 10, 303-306 (1993). [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]
- 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]
- See, e.g., K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Methods (CRC, Boca Raton, Fla., 1993).
- C. T. Chan, S. Datta, Q. L. Yu, M. Sigalas, K. M. Ho, C. M. Soukoulis, "New structures and algorithms for photonic band gaps," Physica A 211, 411-419 (1994). [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]
- 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]
- K. Sakoda and H. Shiroma, "Numerical method for localized defect modes in photonic lattices," Phys. Rev. B 56, 4830-4835 (1997). [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]
- 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. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
- J. B. Pendry and A. MacKinnon, "Calculation of photon dispersion relations," Phys. Rev. Lett. 69, 2772-2775 (1992). [CrossRef] [PubMed]
- 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]
- 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. 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]
- 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]
- N. W. Ashcroft and N. D. Mermin, Solid State Physics (Holt Saunders, Philadelphia, 1976).
- 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.
- A. H. Stroud, Approximate Calculation of Multiple Integrals (Prentice-Hall, Englewood Cliffs, NJ, 1971).
- J. Nadobny, D. Sullivan, P. Wust, M. Seebass, P. Deu hard, and R. Felix, "A high-resolution interpolation at arbitrary interfaces for the FDTD method," IEEE Trans. Microwave Theory Tech. 46, 1759-1766 (1998). [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]
- R. D. Meade, private communications.
- 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]
- See, e.g., A. Edelman and S. T. Smith, "On conjugate gradient-like methods for eigen-like problems," BIT 36, 494-509 (1996). [CrossRef]
- S. Ismail-Beigi and T. A. Arias, "New algebraic formulation of density functional calculation," Comp. Phys. Commun. 128, 1-45 (2000). [CrossRef]
- 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]
- M. Crouzeix, B. Philippe, and M. Sadkane, "The Davidson Method," SIAM J. Sci. Comput. 15, 62-76 (1994). [CrossRef]
- G. L. G. Sleijpen and H. A. van der Vorst, "A Jacobi-Davidson iteration method for linear eigen-value problems," SIAM J. Matrix Anal. Appl. 17, 401-425 (1996). [CrossRef]
- B. N. Parlett, The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, NJ, 1980).
- H. A. van der Vorst, "Krylov subspace iteration," Computing in Sci. and Eng. 2, 32-37 (2000). [CrossRef]
- P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization (Academic, London, 1981).
- 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]
- 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]
- 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]
- 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]
- J. J. Moré and D. J. Thuente, "Line search algorithms with guaranteed sufficient decrease," ACM Trans. Math. Software 20, 286-307 (1994). [CrossRef]
- S. Ismail-Beiji, private communications.
- 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]
- 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]
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