Abstract

Wannier functions derived from Bloch functions have been identified as an efficient means of analyzing the properties of photonic crystals in which localized functions have now opened the door for 2D and 3D structures containing defects to be investigated. In this paper, based on the Maxwell equations in diagonalized form and utilizing Bloch waves we have obtained an equivalent system of algebraic equations in eigenform. By establishing and exploiting several distinct properties of the resulting eigenpairs, we demonstrate an ability to construct Wannier functions associated with the simplest one-dimensional photonic structure. More importantly, the numerical investigation of the inner- and intra-band orthonormality conditions as well as Hilbert space partitioning features shows a capability for multi-resolution analysis that will make all-optical signal processing devices with photonic crystal structures feasible.

© 2010 Optical Society of America

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  1. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
    [CrossRef] [PubMed]
  2. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
    [CrossRef] [PubMed]
  3. J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016608 (2002).
    [CrossRef]
  4. A. Taflove, “Review of the formulation and applications of the finite-differences time-domain method for numerical modeling of electromagnetic-wave interactions with arbitrary surfaces,” Wave Motion 10, 547-582 (1988).
    [CrossRef]
  5. C. A. J. Fletcher, Computational Galerkin Methods (Springer, 1984).
  6. A. Algappan, X. W. Sun, and M. B. Yu, “Equal-frequency surface analysis of two-dimensional photonic crystals,” J. Opt. Soc. Am. A 25, 219-224 (2008).
    [CrossRef]
  7. M. Che and Z.-Y. Li, “Analysis of surface modes in photonic crystals by plane-wave transfer-matrix method,” J. Opt. Soc. Am. A 25, 2177-2184 (2008).
    [CrossRef]
  8. V. R. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Analysis of photonic crystals with defects using coupled theory,” J. Opt. Soc. Am. A 26, 1248-1255 (2009).
    [CrossRef]
  9. G. H. Wannier, “The structure of electronic excitation levels in insulating crystals,” Phys. Rev. 52, 191-197 (1937).
    [CrossRef]
  10. 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]
  11. N. Marzari and D. Vanderbilt, “Maximally localized generalized Wannier functions for composite energy bands,” Phys. Rev. B 56, 12847-12865 (1997).
    [CrossRef]
  12. K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, “Wannier function approach to photonic crystal circuits,” J. Phys. Condens. Matter 15, R1233-R1256 (2003).
    [CrossRef]
  13. A. R. Baghai-Wadji, “A symbolic procedure for the diagonalization of linear PDEs in accelerated computational engineering,” in Symbolic and Numerical Scientific Computing (Springer-Verlag, 2003), pp. 347-360.
    [CrossRef]
  14. W. Kohn, “Analytical properties of Bloch waves and Wannier functions,” Phys. Rev. 115, 809-821 (1959).
    [CrossRef]
  15. A. Kloeckner, “On the computation of maximally localized Wannier functions” (Diplomarbeit, 2004);http://mathema.tician.de/dl/academic/da.
  16. A. R. Baghai-Wadji and M. Muradoglu, “On the nature of eigenpairs associated with wave propagation problems in photonic structures,” in Proceedings of the 12th International Symposium on Integrated Circuits (ISIC), Singapore, 16 Dec. 2009 (IEEE Xplore 2009).
  17. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton Univ., 2008).
  18. S. Mallat, Wavelet Tour of Signal Processing (Academic, 1998).
  19. I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
    [CrossRef]
  20. K. Busch and S. John, “Liquid-crystal photonic-band-gap materials: the tunable electromagnetic vacuum,” Phys. Rev. Lett. 83, 967-970 (1999).
    [CrossRef]
  21. W. Park and J. B. Lee, “Mechanically tunable photonic crystal structure,” Appl. Phys. Lett. 85, 4845 (2004).
    [CrossRef]

2009 (1)

V. R. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Analysis of photonic crystals with defects using coupled theory,” J. Opt. Soc. Am. A 26, 1248-1255 (2009).
[CrossRef]

2008 (2)

2004 (1)

W. Park and J. B. Lee, “Mechanically tunable photonic crystal structure,” Appl. Phys. Lett. 85, 4845 (2004).
[CrossRef]

2003 (1)

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

2002 (1)

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016608 (2002).
[CrossRef]

2001 (1)

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

1999 (1)

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

1997 (1)

N. Marzari and D. Vanderbilt, “Maximally localized generalized Wannier functions for composite energy bands,” Phys. Rev. B 56, 12847-12865 (1997).
[CrossRef]

1996 (1)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

1993 (1)

1988 (1)

A. Taflove, “Review of the formulation and applications of the finite-differences time-domain method for numerical modeling of electromagnetic-wave interactions with arbitrary surfaces,” Wave Motion 10, 547-582 (1988).
[CrossRef]

1959 (1)

W. Kohn, “Analytical properties of Bloch waves and Wannier functions,” Phys. Rev. 115, 809-821 (1959).
[CrossRef]

1937 (1)

G. H. Wannier, “The structure of electronic excitation levels in insulating crystals,” Phys. Rev. 52, 191-197 (1937).
[CrossRef]

Algappan, A.

Baghai-Wadji, A. R.

A. R. Baghai-Wadji, “A symbolic procedure for the diagonalization of linear PDEs in accelerated computational engineering,” in Symbolic and Numerical Scientific Computing (Springer-Verlag, 2003), pp. 347-360.
[CrossRef]

A. R. Baghai-Wadji and M. Muradoglu, “On the nature of eigenpairs associated with wave propagation problems in photonic structures,” in Proceedings of the 12th International Symposium on Integrated Circuits (ISIC), Singapore, 16 Dec. 2009 (IEEE Xplore 2009).

Busch, K.

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

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

Che, M.

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
[CrossRef]

Fan, S.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Fletcher, C. A. J.

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

Garcia-Martin, A.

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

Hardy, A. A.

V. R. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Analysis of photonic crystals with defects using coupled theory,” J. Opt. Soc. Am. A 26, 1248-1255 (2009).
[CrossRef]

Hermann, D.

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

Joannopoulos, J. D.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

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

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.

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

Kapon, E.

V. R. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Analysis of photonic crystals with defects using coupled theory,” J. Opt. Soc. Am. A 26, 1248-1255 (2009).
[CrossRef]

Kloeckner, A.

A. Kloeckner, “On the computation of maximally localized Wannier functions” (Diplomarbeit, 2004);http://mathema.tician.de/dl/academic/da.

Kohn, W.

W. Kohn, “Analytical properties of Bloch waves and Wannier functions,” Phys. Rev. 115, 809-821 (1959).
[CrossRef]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Lee, J. B.

W. Park and J. B. Lee, “Mechanically tunable photonic crystal structure,” Appl. Phys. Lett. 85, 4845 (2004).
[CrossRef]

Leung, K. M.

Li, Z.-Y.

Loncar, M.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016608 (2002).
[CrossRef]

Mabuchi, H.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016608 (2002).
[CrossRef]

Mallat, S.

S. Mallat, Wavelet Tour of Signal Processing (Academic, 1998).

Marzari, N.

N. Marzari and D. Vanderbilt, “Maximally localized generalized Wannier functions for composite energy bands,” Phys. Rev. B 56, 12847-12865 (1997).
[CrossRef]

Meade, R. D.

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

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Mingaleev, S. F.

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

Muradoglu, M.

A. R. Baghai-Wadji and M. Muradoglu, “On the nature of eigenpairs associated with wave propagation problems in photonic structures,” in Proceedings of the 12th International Symposium on Integrated Circuits (ISIC), Singapore, 16 Dec. 2009 (IEEE Xplore 2009).

Notomi, M.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Nusinsky, I.

V. R. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Analysis of photonic crystals with defects using coupled theory,” J. Opt. Soc. Am. A 26, 1248-1255 (2009).
[CrossRef]

Park, W.

W. Park and J. B. Lee, “Mechanically tunable photonic crystal structure,” Appl. Phys. Lett. 85, 4845 (2004).
[CrossRef]

Scherer, A.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016608 (2002).
[CrossRef]

Schillinger, M.

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

Shinya, A.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Shteeman, V. R.

V. R. Shteeman, I. Nusinsky, E. Kapon, and A. A. Hardy, “Analysis of photonic crystals with defects using coupled theory,” J. Opt. Soc. Am. A 26, 1248-1255 (2009).
[CrossRef]

Sun, X. W.

Taflove, A.

A. Taflove, “Review of the formulation and applications of the finite-differences time-domain method for numerical modeling of electromagnetic-wave interactions with arbitrary surfaces,” Wave Motion 10, 547-582 (1988).
[CrossRef]

Takahashi, C.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Takahashi, J.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Vanderbilt, D.

N. Marzari and D. Vanderbilt, “Maximally localized generalized Wannier functions for composite energy bands,” Phys. Rev. B 56, 12847-12865 (1997).
[CrossRef]

Villeneuve, P. R.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Vuckovic, J.

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016608 (2002).
[CrossRef]

Wannier, G. H.

G. H. Wannier, “The structure of electronic excitation levels in insulating crystals,” Phys. Rev. 52, 191-197 (1937).
[CrossRef]

Winn, J. N.

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

Yamada, K.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Yokohama, I.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Yu, M. B.

Appl. Phys. Lett. (1)

W. Park and J. B. Lee, “Mechanically tunable photonic crystal structure,” Appl. Phys. Lett. 85, 4845 (2004).
[CrossRef]

J. Opt. Soc. Am. A (3)

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

J. Phys. Condens. Matter (1)

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

Phys. Rev. (2)

W. Kohn, “Analytical properties of Bloch waves and Wannier functions,” Phys. Rev. 115, 809-821 (1959).
[CrossRef]

G. H. Wannier, “The structure of electronic excitation levels in insulating crystals,” Phys. Rev. 52, 191-197 (1937).
[CrossRef]

Phys. Rev. B (1)

N. Marzari and D. Vanderbilt, “Maximally localized generalized Wannier functions for composite energy bands,” Phys. Rev. B 56, 12847-12865 (1997).
[CrossRef]

Phys. Rev. E (1)

J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016608 (2002).
[CrossRef]

Phys. Rev. Lett. (3)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

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

Wave Motion (1)

A. Taflove, “Review of the formulation and applications of the finite-differences time-domain method for numerical modeling of electromagnetic-wave interactions with arbitrary surfaces,” Wave Motion 10, 547-582 (1988).
[CrossRef]

Other (7)

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

A. R. Baghai-Wadji, “A symbolic procedure for the diagonalization of linear PDEs in accelerated computational engineering,” in Symbolic and Numerical Scientific Computing (Springer-Verlag, 2003), pp. 347-360.
[CrossRef]

A. Kloeckner, “On the computation of maximally localized Wannier functions” (Diplomarbeit, 2004);http://mathema.tician.de/dl/academic/da.

A. R. Baghai-Wadji and M. Muradoglu, “On the nature of eigenpairs associated with wave propagation problems in photonic structures,” in Proceedings of the 12th International Symposium on Integrated Circuits (ISIC), Singapore, 16 Dec. 2009 (IEEE Xplore 2009).

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

S. Mallat, Wavelet Tour of Signal Processing (Academic, 1998).

I. Daubechies, Ten Lectures on Wavelets (SIAM, 1992).
[CrossRef]

Supplementary Material (1)

» Media 1: MOV (269 KB)     

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Figures (4)

Fig. 1
Fig. 1

(a) Brillouin band diagram generated by solving Eq. (20) for N = 100 and ε P ( x ̂ ) given in Eq. (25). (b) Spectral component α 1 ( 1 ) ( k ̂ ) (right most), α 0 ( 1 ) ( k ̂ ) (middle), and α 1 ( 1 ) ( k ̂ ) (left most).

Fig. 2
Fig. 2

Spectral components α 0 ( 1 ) ( k ̂ ) , α 0 ( 2 ) ( k ̂ ) , and α 0 ( 3 ) ( k ̂ ) , respectively, are shown in Fig. 2a, 2b, 2c. The graphs for the k ̂ -dependence of the summation n = N N | α 0 ( 2 ) ( k ̂ + n ) | 2 computed with (i) N = 10 (dashed curves); (ii) N = 15 (dotted curves); (iii) N = 20 (solid curves) are given in Fig. 2d. The graphs for the k ̂ -dependence of the summation j = 1 J | α 0 ( j ) ( k ̂ ) | 2 computed with (i) J = 10 (dashed curves); (ii) J = 15 (dotted curves); (iii) J = 20 (solid curves) are shown in Fig. 2e. The reader should be reminded that the superscript ( j ) refers to the j th band.

Fig. 3
Fig. 3

Spectral components α 0 ( 1 ) ( k ̂ ) (Band 1, blue online), α 0 ( 2 ) ( k ̂ ) (Band 2, green online), α 0 ( 3 ) ( k ̂ ) (Band 3, red online), α 0 ( 4 ) ( k ̂ ) (Band 4, cyan online), and α 0 ( 5 ) ( k ̂ ) (Band 5, purple online) plotted as an animation (Media 1) for dielectric ratios [Eq. (25)] going from 1:10 to 20:10 in steps of 0.5.

Fig. 4
Fig. 4

(a) Wannier function w ( 1 ) ( x ̂ ) associated with the 1 st band. (b) Wannier function w ( 2 ) ( x ̂ ) associated with the 2 nd band. (c) Wannier function w ( 3 ) ( x ̂ ) associated with the 3 rd band.

Tables (1)

Tables Icon

Table 1 Numerical Results for q j l Found by Employing 21 Space Harmonics to Confirm the Orthonormality Condition with Satisfactory Precision

Equations (45)

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

× H ( x , t ) = D ( x , t ) t ,
× E ( x , t ) = B ( x , t ) t ,
D ( x , t ) = ε 0 ε ( x ) E ( x , t ) ,
B ( x , t ) = μ 0 H ( x , t ) .
[ 0 0 0 i ω μ 0 0 0 i ω μ 0 0 0 i ω ε 0 ε ( x ) 0 0 i ω ε 0 ε ( x ) 0 0 0 ] [ E ( 2 ) ( x ) E ( 3 ) ( x ) H ( 2 ) ( x ) H ( 3 ) ( x ) ] = d d x [ E ( 2 ) ( x ) E ( 3 ) ( x ) H ( 2 ) ( x ) H ( 3 ) ( x ) ] .
ε P ( x + P ) = ε P ( x ) .
Ψ ( x | k ) = Ψ P ( x | k ) e i k x ,
U P ( x ) = n u n | e n ( x ) = n u n e i K n x .
Ψ P ( x | k ) = Ψ P ( x | k ) e i k x = n α n ( k ) e i ( K n + k ) x .
[ 0 i ω μ 0 i ω ε 0 ε ( x ) 0 ] [ E ( 3 ) ( x ) H ( 2 ) ( x ) ] = d d x [ E ( 3 ) ( x ) H ( 2 ) ( x ) ] .
e i K n x | e i K m x = δ n m ,
e i K n x | ε P ( x ) | e i K n x = ε n n ,
d d x Ψ ( x | k ) = n i ( K n + k ) α n ( k ) e i ( K n + k ) x
i ω ε 0 n E n ( 3 ) ( k ) e i K n x | ε P ( x ) | e i K n x = m i ( K m + k ) [ H m ( 2 ) ( k ) ] e i K n x | e i K m x .
ω ε 0 n ε m n E n ( 3 ) ( k ) n K n δ m n [ H n ( 2 ) ( k ) ] = k n δ m n [ H n ( 2 ) ( k ) ] .
P ω 2 π c n ε m n E ̃ n ( 3 ) ( k ) n n δ m n [ H ̃ n ( 2 ) ( k ) ] = k P 2 π n δ m n [ H ̃ n ( 2 ) ( k ) ] .
E ̃ n ( 3 ) ( k ) ε 0 E n ( 3 ) ( k ) ,
H ̃ n ( 2 ) ( k ) μ 0 H n ( 2 ) ( k ) .
[ ω ̂ ε m n n δ m n ] [ E ̃ n ( 3 ) ( k ) H ̃ n ( 2 ) ( k ) ] = k ̂ [ 0 m n δ m n ] [ E ̃ n ( 3 ) ( k ) H ̃ n ( 2 ) ( k ) ] .
[ n δ m n ω ̂ δ m n ] [ E ̃ n ( 3 ) ( k ) H ̃ n ( 2 ) ( k ) ] = k ̂ [ δ m n 0 m n ] [ E ̃ n ( 3 ) ( k ) H ̃ n ( 2 ) ( k ) ] .
[ n δ m n ω ̂ δ m n ω ̂ ε m n n δ m n ] [ E ̃ n ( 3 ) ( k ) H ̃ n ( 2 ) ( k ) ] = k ̂ I [ E ̃ n ( 3 ) ( k ) H ̃ n ( 2 ) ( k ) ] .
[ n δ m n 0 m n 0 m n ω ̂ δ m n 0 m n n δ m n ω ̂ δ m n 0 m n 0 m n ω ̂ ε m n n δ m n 0 m n ω ̂ ε m n 0 m n 0 m n n δ m n ] [ E ̃ n ( 2 ) ( k ̂ ) E ̃ n ( 3 ) ( k ̂ ) H ̃ n ( 2 ) ( k ̂ ) H ̃ n ( 3 ) ( k ̂ ) ] = k ̂ [ E ̃ n ( 2 ) ( k ̂ ) E ̃ n ( 3 ) ( k ̂ ) H ̃ n ( 2 ) ( k ̂ ) H ̃ n ( 3 ) ( k ̂ ) ] .
A = [ n δ m n 0 m n 0 m n n δ m n ] N = [ 0 m n δ m n ε m n 0 m n ] ,
( A k ̂ I ) [ E ̃ n ( 3 ) ( k ̂ ) H ̃ n ( 2 ) ( k ̂ ) ] = ω ̂ N [ E ̃ n ( 3 ) ( k ̂ ) H ̃ n ( 2 ) ( k ̂ ) ] .
[ ( n + k ̂ ) δ m n 0 m n 0 m n ( n + k ̂ ) δ m n ] [ E ̃ n ( 3 ) ( k ̂ ) H ̃ n ( 2 ) ( k ̂ ) ] = ω ̂ [ 0 m n δ m n ε m n 0 m n ] [ E ̃ n ( 3 ) ( k ̂ ) H ̃ n ( 2 ) ( k ̂ ) ] .
λ ̂ ( j ) ( k ̂ + l ) = λ ̂ ( j ) ( k ̂ ) ,
α n l ( j ) ( k ̂ + l ) = α n ( j ) ( k ̂ ) ,
λ ̂ ( j ) ( k ̂ ) [ α n ( j ) ( k ̂ ) ] T
n = N N | α n ( j ) ( k ̂ ) | 2 = 1 ,
j = 1 J | α 0 ( j ) ( k ̂ ) | 2 = 1 ,
ε P ( x ̂ ) = { 5 | x ̂ | < 1 4 15 1 4 < | x ̂ | < 1 2 } .
[ E ̃ N ( 3 ) ( k ̂ ) E ̃ N ( 3 ) ( k ̂ ) H ̃ N ( 2 ) ( k ̂ ) H ̃ N ( 2 ) ( k ̂ ) ] T
Ψ ( j ) ( x ̂ , k ̂ ) = n w ( j ) ( x ̂ n ) e i 2 π k n .
w ( x ̂ m ) = BLZ d k ̂ Ψ ( x ̂ , k ̂ ) e i 2 π k ̂ m = 1 2 1 2 d k ̂ Ψ ( x ̂ , k ̂ ) e i 2 π k ̂ m .
w ( x ̂ ) = 1 2 1 2 d k ̂ Ψ ( x ̂ , k ̂ ) = 1 2 1 2 d k ̂ n α n ( k ̂ ) e i 2 π n x e i 2 π k ̂ x ̂ = n e i 2 π n x 1 2 1 2 d k ̂ α n ( k ̂ ) e i 2 π k ̂ x ̂ .
I n ( x ̂ ) = 1 2 1 2 d k ̂ α n ( k ̂ ) e i 2 π k ̂ x ̂ ,
w ( x ̂ ) = n I n ( x ̂ ) e i 2 π n x ̂ .
I n ( x ̂ ) = 1 2 1 2 d k ̂ α 0 ( k ̂ + n ) e i 2 π n x ̂ ,
I n ( x ̂ ) = e i 2 π n x ̂ 1 2 + n 1 2 + n d κ α 0 ( κ n ) e i 2 π κ n x ̂ .
w ( x ̂ ) = n 1 2 + n 1 2 + n d k ̂ α 0 ( k ̂ ) e i 2 π k ̂ x ̂ = d k ̂ α 0 ( k ̂ ) e i 2 π k ̂ x ̂ .
w ( j ) ( x ̂ ) = d k ̂ α 0 ( j ) ( k ̂ ) e i 2 π k ̂ x ̂ .
p m n = w ( x ̂ m ) | w ( x ̂ n ) .
n = N N | α 0 ( k ̂ + n ) | 2 = 1 .
q j l = w ( j ) ( x ̂ ) | w ( l ) ( x ̂ ) .
q j l = d k ̂ α ¯ 0 ( j ) ( k ̂ ) α 0 ( l ) ( k ̂ ) ,

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