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 Ox 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

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References

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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).
    [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. Sözüer 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. B 48, 8434-8437 (1993). Erratum: S. G. Johnson, ibid 55, 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, 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 contain- ing 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, 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. 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]
  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 eigen-value 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]

Other

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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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.

Fig. 2.
Fig. 2.

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

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.

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.

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).

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.

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 m = ( k + G ) × 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 ˜ m = k + G m 2 δ , 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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