Abstract

The moving least-square (MLS) basis is implemented for the real-space band-structure calculation of 2D photonic crystals. A value-periodic MLS shape function is thus proposed in order to represent the periodicity of crystal lattice. Through numerical examples, this MLS method is proved to be a promising scheme for predicting band gaps of photonic crystals.

© 2003 Optical Society of America

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References

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  1. J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
  2. Y. Xia, “Photonic crystals,” Adv. Mater. 13, 369 (2001) and papers in this special issue.
    [CrossRef]
  3. K. Busch, “Photonic band structure theory: assesment and perspectives,” C. R. Physique 3, 53–66 (2002).
    [CrossRef]
  4. D. Cassagne, “Photonic band gap materials,” Ann. Phys. Fr. 23(4), 1–91 (1998).
    [CrossRef]
  5. J.B. Pendry, “Calculating photonic band structure,” J. Phys.: Condens. Matter 8, 1085–1108 (1996).
    [CrossRef]
  6. H.S. Sözüer, J.W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
    [CrossRef]
  7. 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).
    [CrossRef]
  8. C.T. Chan, Q.L. Yu, and K.M. Ho, “Order-N spectral method for electromagnetic waves,” Phys. Rev. B 51, 16635–16642 (1995).
    [CrossRef]
  9. A.J. Ward and J.B. Pendry, “Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
    [CrossRef]
  10. M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000).
    [CrossRef]
  11. K.M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 7767–7771 (1993).
    [CrossRef]
  12. X. Wang, X.G. Zhang, Q. Yu, and B.N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
    [CrossRef]
  13. J.B. Pendry and A. MacKinnon, “Calculation of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
    [CrossRef] [PubMed]
  14. L. Shen, S. He, and S. Xiao, “A finite-diference eigenvalue algorithm for calculating the band structure of a photonic crystal,” Comput. Phys. Comm. 143, 213–221 (2002).
    [CrossRef]
  15. 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]
  16. D.C. Dobson, “An efficient band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999).
    [CrossRef]
  17. C. Mias, J.P. Webb, and R.L. Ferrari, “Finite element modelling of electromagnetic waves in doubly and triply periodic structures,” IEE Proc.-Optoelectron.,  146(2), 111–118 (1999).
    [CrossRef]
  18. M. Marrone, V.F. Rodriguez-Esquerre, and H.E. Hernandez-Figueroa, “Novel numerical method for the analysis of 2D photonic crystals: the cell method,” Opt. Express 10, 1299–1304 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1299
    [CrossRef] [PubMed]
  19. S. Li and W.K. Liu, “Meshfree and particle methods and their applications,” Applied Mechanics Review,  55, 1–34 (2002).
    [CrossRef]
  20. D.W. Kim and Y. Kim, “Point collocation method using the fast moving least-square reproducing kernel approximation,” Int. J. Numer. Methods Engrg. 56, 1445–1464 (2003).
    [CrossRef]
  21. L.W. Cordes and B. Moran, “Treatment of material discontinuity in the Element-Free Galerkin method,” Comput. Meth. Appl. Mech. Eng. 139, 75–89 (1996).
    [CrossRef]

2003 (1)

D.W. Kim and Y. Kim, “Point collocation method using the fast moving least-square reproducing kernel approximation,” Int. J. Numer. Methods Engrg. 56, 1445–1464 (2003).
[CrossRef]

2002 (4)

S. Li and W.K. Liu, “Meshfree and particle methods and their applications,” Applied Mechanics Review,  55, 1–34 (2002).
[CrossRef]

M. Marrone, V.F. Rodriguez-Esquerre, and H.E. Hernandez-Figueroa, “Novel numerical method for the analysis of 2D photonic crystals: the cell method,” Opt. Express 10, 1299–1304 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1299
[CrossRef] [PubMed]

K. Busch, “Photonic band structure theory: assesment and perspectives,” C. R. Physique 3, 53–66 (2002).
[CrossRef]

L. Shen, S. He, and S. Xiao, “A finite-diference eigenvalue algorithm for calculating the band structure of a photonic crystal,” Comput. Phys. Comm. 143, 213–221 (2002).
[CrossRef]

2001 (1)

Y. Xia, “Photonic crystals,” Adv. Mater. 13, 369 (2001) and papers in this special issue.
[CrossRef]

2000 (1)

M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000).
[CrossRef]

1999 (3)

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 band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999).
[CrossRef]

C. Mias, J.P. Webb, and R.L. Ferrari, “Finite element modelling of electromagnetic waves in doubly and triply periodic structures,” IEE Proc.-Optoelectron.,  146(2), 111–118 (1999).
[CrossRef]

1998 (2)

A.J. Ward and J.B. Pendry, “Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
[CrossRef]

D. Cassagne, “Photonic band gap materials,” Ann. Phys. Fr. 23(4), 1–91 (1998).
[CrossRef]

1996 (2)

J.B. Pendry, “Calculating photonic band structure,” J. Phys.: Condens. Matter 8, 1085–1108 (1996).
[CrossRef]

L.W. Cordes and B. Moran, “Treatment of material discontinuity in the Element-Free Galerkin method,” Comput. Meth. Appl. Mech. Eng. 139, 75–89 (1996).
[CrossRef]

1995 (1)

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

1993 (3)

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).
[CrossRef]

K.M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 7767–7771 (1993).
[CrossRef]

X. Wang, X.G. Zhang, Q. Yu, and B.N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[CrossRef]

1992 (2)

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

H.S. Sözüer, J.W. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
[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. B 48, 8434–8437 (1993).
[CrossRef]

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]

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. B 48, 8434–8437 (1993).
[CrossRef]

Busch, K.

K. Busch, “Photonic band structure theory: assesment and perspectives,” C. R. Physique 3, 53–66 (2002).
[CrossRef]

Cassagne, D.

D. Cassagne, “Photonic band gap materials,” Ann. Phys. Fr. 23(4), 1–91 (1998).
[CrossRef]

Chan, C.T.

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

Cordes, L.W.

L.W. Cordes and B. Moran, “Treatment of material discontinuity in the Element-Free Galerkin method,” Comput. Meth. Appl. Mech. Eng. 139, 75–89 (1996).
[CrossRef]

Dobson, D.C.

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

Ferrari, R.L.

C. Mias, J.P. Webb, and R.L. Ferrari, “Finite element modelling of electromagnetic waves in doubly and triply periodic structures,” IEE Proc.-Optoelectron.,  146(2), 111–118 (1999).
[CrossRef]

Harmon, B.N.

X. Wang, X.G. Zhang, Q. Yu, and B.N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[CrossRef]

Haus, J.W.

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

He, S.

L. Shen, S. He, and S. Xiao, “A finite-diference eigenvalue algorithm for calculating the band structure of a photonic crystal,” Comput. Phys. Comm. 143, 213–221 (2002).
[CrossRef]

M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000).
[CrossRef]

Hernandez-Figueroa, H.E.

Ho, K.M.

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

Inguva, R.

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

Joannopoulos, J.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. B 48, 8434–8437 (1993).
[CrossRef]

J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

Kim, D.W.

D.W. Kim and Y. Kim, “Point collocation method using the fast moving least-square reproducing kernel approximation,” Int. J. Numer. Methods Engrg. 56, 1445–1464 (2003).
[CrossRef]

Kim, Y.

D.W. Kim and Y. Kim, “Point collocation method using the fast moving least-square reproducing kernel approximation,” Int. J. Numer. Methods Engrg. 56, 1445–1464 (2003).
[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]

Leung, K.M.

K.M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 7767–7771 (1993).
[CrossRef]

Li, S.

S. Li and W.K. Liu, “Meshfree and particle methods and their applications,” Applied Mechanics Review,  55, 1–34 (2002).
[CrossRef]

Liu, W.K.

S. Li and W.K. Liu, “Meshfree and particle methods and their applications,” Applied Mechanics Review,  55, 1–34 (2002).
[CrossRef]

MacKinnon, A.

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

Marrone, M.

Meade, R.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. B 48, 8434–8437 (1993).
[CrossRef]

J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

Mias, C.

C. Mias, J.P. Webb, and R.L. Ferrari, “Finite element modelling of electromagnetic waves in doubly and triply periodic structures,” IEE Proc.-Optoelectron.,  146(2), 111–118 (1999).
[CrossRef]

Moran, B.

L.W. Cordes and B. Moran, “Treatment of material discontinuity in the Element-Free Galerkin method,” Comput. Meth. Appl. Mech. Eng. 139, 75–89 (1996).
[CrossRef]

Pendry, J.B.

A.J. Ward and J.B. Pendry, “Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
[CrossRef]

J.B. Pendry, “Calculating photonic band structure,” J. Phys.: Condens. Matter 8, 1085–1108 (1996).
[CrossRef]

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

Qiu, M.

M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000).
[CrossRef]

Qiu, Y.

K.M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 7767–7771 (1993).
[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. B 48, 8434–8437 (1993).
[CrossRef]

Rodriguez-Esquerre, V.F.

Shen, L.

L. Shen, S. He, and S. Xiao, “A finite-diference eigenvalue algorithm for calculating the band structure of a photonic crystal,” Comput. Phys. Comm. 143, 213–221 (2002).
[CrossRef]

Sözüer, H.S.

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

Wang, X.

X. Wang, X.G. Zhang, Q. Yu, and B.N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[CrossRef]

Ward, A.J.

A.J. Ward and J.B. Pendry, “Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
[CrossRef]

Webb, J.P.

C. Mias, J.P. Webb, and R.L. Ferrari, “Finite element modelling of electromagnetic waves in doubly and triply periodic structures,” IEE Proc.-Optoelectron.,  146(2), 111–118 (1999).
[CrossRef]

Winn, J.N.

J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

Xia, Y.

Y. Xia, “Photonic crystals,” Adv. Mater. 13, 369 (2001) and papers in this special issue.
[CrossRef]

Xiao, S.

L. Shen, S. He, and S. Xiao, “A finite-diference eigenvalue algorithm for calculating the band structure of a photonic crystal,” Comput. Phys. Comm. 143, 213–221 (2002).
[CrossRef]

Yu, Q.

X. Wang, X.G. Zhang, Q. Yu, and B.N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[CrossRef]

Yu, Q.L.

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

Zhang, X.G.

X. Wang, X.G. Zhang, Q. Yu, and B.N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[CrossRef]

Adv. Mater. (1)

Y. Xia, “Photonic crystals,” Adv. Mater. 13, 369 (2001) and papers in this special issue.
[CrossRef]

Ann. Phys. Fr. (1)

D. Cassagne, “Photonic band gap materials,” Ann. Phys. Fr. 23(4), 1–91 (1998).
[CrossRef]

Applied Mechanics Review (1)

S. Li and W.K. Liu, “Meshfree and particle methods and their applications,” Applied Mechanics Review,  55, 1–34 (2002).
[CrossRef]

C. R. Physique (1)

K. Busch, “Photonic band structure theory: assesment and perspectives,” C. R. Physique 3, 53–66 (2002).
[CrossRef]

Comput. Meth. Appl. Mech. Eng. (1)

L.W. Cordes and B. Moran, “Treatment of material discontinuity in the Element-Free Galerkin method,” Comput. Meth. Appl. Mech. Eng. 139, 75–89 (1996).
[CrossRef]

Comput. Phys. Comm. (1)

L. Shen, S. He, and S. Xiao, “A finite-diference eigenvalue algorithm for calculating the band structure of a photonic crystal,” Comput. Phys. Comm. 143, 213–221 (2002).
[CrossRef]

IEE Proc.-Optoelectron. (1)

C. Mias, J.P. Webb, and R.L. Ferrari, “Finite element modelling of electromagnetic waves in doubly and triply periodic structures,” IEE Proc.-Optoelectron.,  146(2), 111–118 (1999).
[CrossRef]

Int. J. Numer. Methods Engrg. (1)

D.W. Kim and Y. Kim, “Point collocation method using the fast moving least-square reproducing kernel approximation,” Int. J. Numer. Methods Engrg. 56, 1445–1464 (2003).
[CrossRef]

J. Appl. Phys. (1)

M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys. 87, 8268–8275 (2000).
[CrossRef]

J. Comput. Phys. (2)

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 band structure calculations in 2D photonic crystals,” J. Comput. Phys. 149, 363–376 (1999).
[CrossRef]

J. Phys.: Condens. Matter (1)

J.B. Pendry, “Calculating photonic band structure,” J. Phys.: Condens. Matter 8, 1085–1108 (1996).
[CrossRef]

Opt. Express (1)

Phys. Rev. B (6)

H.S. Sözüer, J.W. Haus, and R. Inguva, “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).
[CrossRef]

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

A.J. Ward and J.B. Pendry, “Calculating photonic Green’s functions using a nonorthogonal finite-difference time-domain method,” Phys. Rev. B 58, 7252–7259 (1998).
[CrossRef]

K.M. Leung and Y. Qiu, “Multiple-scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 7767–7771 (1993).
[CrossRef]

X. Wang, X.G. Zhang, Q. Yu, and B.N. Harmon, “Multiple-scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1993).
[CrossRef]

Phys. Rev. Lett. (1)

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

Other (1)

J.D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

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Figures (8)

Fig. 1.
Fig. 1.

The translation-and-searching algorithm for constructing a periodic meshfree shape function. Supports of the shape function are in gray within solid circle. The dashed circle illustrates the rod of photonic crystal.

Fig. 2.
Fig. 2.

The periodic MLS shape function. The position arrowed is the center of the shape function.

Fig. 3.
Fig. 3.

Unit cell of the electromagnetic Kronig-Penney problem.

Fig. 4.
Fig. 4.

Band structures of the electromagnetic Kronig-Penney problem. MLS results are denoted by open circles and analytic results by solid lines.

Fig. 5.
Fig. 5.

Convergence rates of the lowest five eigenvalues for the electromagnetic Kronig-Penney problem at k=(π/a)(0.5, 0.5).

Fig. 6.
Fig. 6.

A square lattice of circular rod and its unit cell discretized by 1697 nodes.

Fig. 7.
Fig. 7.

Results on band structures of square lattice composed of circular rods.

Fig. 8.
Fig. 8.

Comparisons of MLS method and plane wave method (TM modes) for the square lattice of circular rods.

Equations (25)

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

u h ( x ) = J N P N J ( x ) u J
N J ( x ) = N ( x J x ) p T ( x J x ) b ( x ) W ( x J x )
M ( x ) b ( x ) = p ( 0 )
M ( x ) J = 1 NP W J ( x ) p J ( x ) p J T ( x )
· 1 ε ( x ) ψ ( x ) = λ ψ ( x ) ( TE modes )
1 ε ( x ) 2 ψ ( x ) = λ ψ ( x ) ( TM modes )
( + i k ) · 1 ε ( x ) ( + i k ) u ( x ) = λ u ( x ) ( TE modes )
( + i k ) · ( + i k ) u ( x ) = λ ε ( x ) u ( x ) ( TM modes )
A u = λ B u
A IJ = Ω 1 ε ( x ) ( + i k ) N I P ( x ) · ( + i k ) N J P ( x ) ¯ dV , B IJ = Ω N I P ( x ) N J P ( x ) ¯ dV
A IJ = Ω ( + i k ) N I P ( x ) · ( + i k ) N J P ( x ) ¯ dV , B IJ = Ω ε ( x ) N I P ( x ) N J P ( x ) ¯ dV
u h ( x ) = L J = 1 NP N J ( x + L ) u J = J = 1 NP [ L N J ( x + L ) ] u J = J = 1 NP N J P ( x ) u J
N J ( x + L ) = p J T ( x + L ) b ( x ) W J ( x + L )
M ( x ) b ( x ) = p ( 0 ) , where M ( x ) L J = 1 NP W J ( x + L ) p J ( x + L ) p J T ( x + L )
N J P ( x ) L N J ( x + L ) = n 1 , n 2 N J ( x + n 1 a 1 + n 2 a 2 )
u h ( x , x ¯ ) = p ( x x ¯ ) · a ( x ¯ ) = p T ( x x ¯ ) a ( x ¯ )
J ( a ( x ¯ ) ) = I = 1 NP u ( x I ) p ( x I x ¯ ) · a ( x ¯ ) 2 W ( x I x ¯ )
a ( x ¯ ) = M 1 ( x ¯ ) I = 1 NP p ( x I x ¯ ) W ( x I x ¯ ) u ( x I )
M ( x ¯ ) I = 1 NP p ( x I x ¯ ) p T ( x I x ¯ ) W ( x I x ¯ )
u h ( x , x ¯ ) = p ( x x ¯ ) · a ( x ¯ ) = p T ( x x ¯ ) M 1 ( x ¯ ) I = 1 NP p ( x I x ¯ ) W ( x I x ¯ ) u ( x I )
u h ( x ) = I = 1 NP p T ( 0 ) M 1 ( x ) p ( x I x ) W ( x I x ) u ( x I )
M ( x ) = I = 1 NP p ( x I x ) p T ( x I x ) W ( x I x )
u h ( x ) = I = 1 NP N I ( x ) u I
N I ( x ) = p T ( 0 ) M 1 ( x ) p ( x I x ) W ( x I x )
N I ( x ) = p T ( x I x ) M 1 ( x ) p ( 0 ) W ( x I x ) = p T ( x I x ) b ( x ) W ( x I x )

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