Abstract

We apply the method of asymptotic homogenization to metamaterials with microscopically bianisotropic inclusions to calculate a full set of constitutive parameters in the long-wavelength limit. Two different implementations of electromagnetic asymptotic homogenization are presented. We test the homogenization procedure on two different metamaterial examples. Finally, the analytical solution for long-wavelength homogenization of a one-dimensional metamaterial with microscopically bi-isotropic inclusions is derived.

© 2013 Optical Society of America

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  1. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
    [CrossRef]
  2. Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E 79, 026610 (2009).
    [CrossRef]
  3. B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A 11, 114003 (2009).
    [CrossRef]
  4. D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B 23, 391–403 (2006).
    [CrossRef]
  5. M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
    [CrossRef]
  6. J. Li and J. B. Pendry, “Non-local effective medium of metamaterial,” arXiv.org, arXiv:cond-mat/0701332v1 (2007).
  7. C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of bloch lattices,” Metamaterials 1, 62–80 (2007).
  8. C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
    [CrossRef]
  9. A. Alú, “First principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011).
    [CrossRef]
  10. A. Pors, I. Tsukerman, and S. I. Bozhevolnyi, “Effective constitutive parameters of plasmonic metamaterials: homogenization by dual field interpolation,” Phys. Rev. E 84, 016609 (2011).
    [CrossRef]
  11. H. Brezis and J. L. Lions, eds., Nonlinear Partial Differential Equations and Their Applications, Vol. 12 of College de France Seminar (Longman Scientific & Technical, 1994), lecture by G. Allaire.
  12. U. Hornung, ed., Homogenization and Porous Media (Springer, 1996), Section 1.3.
  13. A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Aymptotic Analysis for Periodic Structures (North-Holland, 1978), Chap. 1, Section 2.
  14. H. T. Banks, V. A. Bokil, D. Cioranescu, N. L. Gibson, G. Griso, and B. Miara, “Homogenization of periodically varying coefficients in electromagnetic materials,” J. Sci. Comput. 28, 191–221 (2006).
    [CrossRef]
  15. O. Ouchetto, C. W. Qui, S. Zouhdi, L. W. Li, and A. Razek, “Homogenization of 3-d periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theor. Tech. 54, 3893–3898 (2006).
    [CrossRef]
  16. L. Cao, Y. Zhang, W. Allegretto, and T. Lin, “Multiscale asymptotic method for maxwell’s equations in composite materials,” SIAM J. Numer. Anal. 47, 4257–4289 (2010).
    [CrossRef]
  17. C. Brosseau and A. Beroual, “Effective permittivity of composites with stratified particles,” J. Phys. D 34, 704–710 (2001).
    [CrossRef]
  18. Y. A. Urzhumov and G. Shvets, “Quasistatic effective medium theory of plasmonic nanostructures,” Proc. SPIE 6642, 66420X (2007).
    [CrossRef]
  19. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).
  20. K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer, 2004).

2011 (2)

A. Alú, “First principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011).
[CrossRef]

A. Pors, I. Tsukerman, and S. I. Bozhevolnyi, “Effective constitutive parameters of plasmonic metamaterials: homogenization by dual field interpolation,” Phys. Rev. E 84, 016609 (2011).
[CrossRef]

2010 (2)

C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
[CrossRef]

L. Cao, Y. Zhang, W. Allegretto, and T. Lin, “Multiscale asymptotic method for maxwell’s equations in composite materials,” SIAM J. Numer. Anal. 47, 4257–4289 (2010).
[CrossRef]

2009 (2)

Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E 79, 026610 (2009).
[CrossRef]

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A 11, 114003 (2009).
[CrossRef]

2007 (3)

M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[CrossRef]

C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of bloch lattices,” Metamaterials 1, 62–80 (2007).

Y. A. Urzhumov and G. Shvets, “Quasistatic effective medium theory of plasmonic nanostructures,” Proc. SPIE 6642, 66420X (2007).
[CrossRef]

2006 (3)

D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B 23, 391–403 (2006).
[CrossRef]

H. T. Banks, V. A. Bokil, D. Cioranescu, N. L. Gibson, G. Griso, and B. Miara, “Homogenization of periodically varying coefficients in electromagnetic materials,” J. Sci. Comput. 28, 191–221 (2006).
[CrossRef]

O. Ouchetto, C. W. Qui, S. Zouhdi, L. W. Li, and A. Razek, “Homogenization of 3-d periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theor. Tech. 54, 3893–3898 (2006).
[CrossRef]

2002 (1)

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

2001 (1)

C. Brosseau and A. Beroual, “Effective permittivity of composites with stratified particles,” J. Phys. D 34, 704–710 (2001).
[CrossRef]

Allegretto, W.

L. Cao, Y. Zhang, W. Allegretto, and T. Lin, “Multiscale asymptotic method for maxwell’s equations in composite materials,” SIAM J. Numer. Anal. 47, 4257–4289 (2010).
[CrossRef]

Alú, A.

A. Alú, “First principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011).
[CrossRef]

Aydin, K.

Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E 79, 026610 (2009).
[CrossRef]

Banks, H. T.

H. T. Banks, V. A. Bokil, D. Cioranescu, N. L. Gibson, G. Griso, and B. Miara, “Homogenization of periodically varying coefficients in electromagnetic materials,” J. Sci. Comput. 28, 191–221 (2006).
[CrossRef]

Bensoussan, A.

A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Aymptotic Analysis for Periodic Structures (North-Holland, 1978), Chap. 1, Section 2.

Beroual, A.

C. Brosseau and A. Beroual, “Effective permittivity of composites with stratified particles,” J. Phys. D 34, 704–710 (2001).
[CrossRef]

Bokil, V. A.

H. T. Banks, V. A. Bokil, D. Cioranescu, N. L. Gibson, G. Griso, and B. Miara, “Homogenization of periodically varying coefficients in electromagnetic materials,” J. Sci. Comput. 28, 191–221 (2006).
[CrossRef]

Bozhevolnyi, S. I.

A. Pors, I. Tsukerman, and S. I. Bozhevolnyi, “Effective constitutive parameters of plasmonic metamaterials: homogenization by dual field interpolation,” Phys. Rev. E 84, 016609 (2011).
[CrossRef]

Brosseau, C.

C. Brosseau and A. Beroual, “Effective permittivity of composites with stratified particles,” J. Phys. D 34, 704–710 (2001).
[CrossRef]

Cao, L.

L. Cao, Y. Zhang, W. Allegretto, and T. Lin, “Multiscale asymptotic method for maxwell’s equations in composite materials,” SIAM J. Numer. Anal. 47, 4257–4289 (2010).
[CrossRef]

Cioranescu, D.

H. T. Banks, V. A. Bokil, D. Cioranescu, N. L. Gibson, G. Griso, and B. Miara, “Homogenization of periodically varying coefficients in electromagnetic materials,” J. Sci. Comput. 28, 191–221 (2006).
[CrossRef]

Fietz, C.

C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
[CrossRef]

Gibson, N. L.

H. T. Banks, V. A. Bokil, D. Cioranescu, N. L. Gibson, G. Griso, and B. Miara, “Homogenization of periodically varying coefficients in electromagnetic materials,” J. Sci. Comput. 28, 191–221 (2006).
[CrossRef]

Griso, G.

H. T. Banks, V. A. Bokil, D. Cioranescu, N. L. Gibson, G. Griso, and B. Miara, “Homogenization of periodically varying coefficients in electromagnetic materials,” J. Sci. Comput. 28, 191–221 (2006).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Kafesaki, M.

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A 11, 114003 (2009).
[CrossRef]

Koschny, T.

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A 11, 114003 (2009).
[CrossRef]

Li, J.

J. Li and J. B. Pendry, “Non-local effective medium of metamaterial,” arXiv.org, arXiv:cond-mat/0701332v1 (2007).

Li, L. W.

O. Ouchetto, C. W. Qui, S. Zouhdi, L. W. Li, and A. Razek, “Homogenization of 3-d periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theor. Tech. 54, 3893–3898 (2006).
[CrossRef]

Li, Z.

Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E 79, 026610 (2009).
[CrossRef]

Lin, T.

L. Cao, Y. Zhang, W. Allegretto, and T. Lin, “Multiscale asymptotic method for maxwell’s equations in composite materials,” SIAM J. Numer. Anal. 47, 4257–4289 (2010).
[CrossRef]

Lions, J.-L.

A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Aymptotic Analysis for Periodic Structures (North-Holland, 1978), Chap. 1, Section 2.

Markoš, P.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Miara, B.

H. T. Banks, V. A. Bokil, D. Cioranescu, N. L. Gibson, G. Griso, and B. Miara, “Homogenization of periodically varying coefficients in electromagnetic materials,” J. Sci. Comput. 28, 191–221 (2006).
[CrossRef]

Ouchetto, O.

O. Ouchetto, C. W. Qui, S. Zouhdi, L. W. Li, and A. Razek, “Homogenization of 3-d periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theor. Tech. 54, 3893–3898 (2006).
[CrossRef]

Ozbay, E.

Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E 79, 026610 (2009).
[CrossRef]

Papanicolaou, G.

A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Aymptotic Analysis for Periodic Structures (North-Holland, 1978), Chap. 1, Section 2.

Pendry, J. B.

D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B 23, 391–403 (2006).
[CrossRef]

J. Li and J. B. Pendry, “Non-local effective medium of metamaterial,” arXiv.org, arXiv:cond-mat/0701332v1 (2007).

Pors, A.

A. Pors, I. Tsukerman, and S. I. Bozhevolnyi, “Effective constitutive parameters of plasmonic metamaterials: homogenization by dual field interpolation,” Phys. Rev. E 84, 016609 (2011).
[CrossRef]

Qui, C. W.

O. Ouchetto, C. W. Qui, S. Zouhdi, L. W. Li, and A. Razek, “Homogenization of 3-d periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theor. Tech. 54, 3893–3898 (2006).
[CrossRef]

Razek, A.

O. Ouchetto, C. W. Qui, S. Zouhdi, L. W. Li, and A. Razek, “Homogenization of 3-d periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theor. Tech. 54, 3893–3898 (2006).
[CrossRef]

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer, 2004).

Schultz, S.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Shvets, G.

C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
[CrossRef]

Y. A. Urzhumov and G. Shvets, “Quasistatic effective medium theory of plasmonic nanostructures,” Proc. SPIE 6642, 66420X (2007).
[CrossRef]

Silveirinha, M. G.

M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[CrossRef]

Simovski, C. R.

C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of bloch lattices,” Metamaterials 1, 62–80 (2007).

Smith, D. R.

D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B 23, 391–403 (2006).
[CrossRef]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Soukoulis, C. M.

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A 11, 114003 (2009).
[CrossRef]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Tsukerman, I.

A. Pors, I. Tsukerman, and S. I. Bozhevolnyi, “Effective constitutive parameters of plasmonic metamaterials: homogenization by dual field interpolation,” Phys. Rev. E 84, 016609 (2011).
[CrossRef]

Urzhumov, Y. A.

Y. A. Urzhumov and G. Shvets, “Quasistatic effective medium theory of plasmonic nanostructures,” Proc. SPIE 6642, 66420X (2007).
[CrossRef]

Wang, B.

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A 11, 114003 (2009).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Zhang, Y.

L. Cao, Y. Zhang, W. Allegretto, and T. Lin, “Multiscale asymptotic method for maxwell’s equations in composite materials,” SIAM J. Numer. Anal. 47, 4257–4289 (2010).
[CrossRef]

Zhou, J.

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A 11, 114003 (2009).
[CrossRef]

Zouhdi, S.

O. Ouchetto, C. W. Qui, S. Zouhdi, L. W. Li, and A. Razek, “Homogenization of 3-d periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theor. Tech. 54, 3893–3898 (2006).
[CrossRef]

IEEE Trans. Microwave Theor. Tech. (1)

O. Ouchetto, C. W. Qui, S. Zouhdi, L. W. Li, and A. Razek, “Homogenization of 3-d periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theor. Tech. 54, 3893–3898 (2006).
[CrossRef]

J. Opt. A (1)

B. Wang, J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Chiral metamaterials: simulations and experiments,” J. Opt. A 11, 114003 (2009).
[CrossRef]

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

J. Phys. D (1)

C. Brosseau and A. Beroual, “Effective permittivity of composites with stratified particles,” J. Phys. D 34, 704–710 (2001).
[CrossRef]

J. Sci. Comput. (1)

H. T. Banks, V. A. Bokil, D. Cioranescu, N. L. Gibson, G. Griso, and B. Miara, “Homogenization of periodically varying coefficients in electromagnetic materials,” J. Sci. Comput. 28, 191–221 (2006).
[CrossRef]

Metamaterials (1)

C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of bloch lattices,” Metamaterials 1, 62–80 (2007).

Phys. Rev. B (4)

C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
[CrossRef]

A. Alú, “First principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011).
[CrossRef]

M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[CrossRef]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Phys. Rev. E (2)

Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E 79, 026610 (2009).
[CrossRef]

A. Pors, I. Tsukerman, and S. I. Bozhevolnyi, “Effective constitutive parameters of plasmonic metamaterials: homogenization by dual field interpolation,” Phys. Rev. E 84, 016609 (2011).
[CrossRef]

Proc. SPIE (1)

Y. A. Urzhumov and G. Shvets, “Quasistatic effective medium theory of plasmonic nanostructures,” Proc. SPIE 6642, 66420X (2007).
[CrossRef]

SIAM J. Numer. Anal. (1)

L. Cao, Y. Zhang, W. Allegretto, and T. Lin, “Multiscale asymptotic method for maxwell’s equations in composite materials,” SIAM J. Numer. Anal. 47, 4257–4289 (2010).
[CrossRef]

Other (6)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

K. Sakoda, Optical Properties of Photonic Crystals, 2nd ed. (Springer, 2004).

H. Brezis and J. L. Lions, eds., Nonlinear Partial Differential Equations and Their Applications, Vol. 12 of College de France Seminar (Longman Scientific & Technical, 1994), lecture by G. Allaire.

U. Hornung, ed., Homogenization and Porous Media (Springer, 1996), Section 1.3.

A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Aymptotic Analysis for Periodic Structures (North-Holland, 1978), Chap. 1, Section 2.

J. Li and J. B. Pendry, “Non-local effective medium of metamaterial,” arXiv.org, arXiv:cond-mat/0701332v1 (2007).

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Figures (4)

Fig. 1.
Fig. 1.

Unit cell of a layered metamaterial. The metamaterial is periodic in the y ^ 1 direction with periodic lattice constant a . Translations in the y ^ 2 and y ^ 3 directions leave the geometry unchanged. The unit cell consists of two different layers, each with the constitutive parameters shown in Eq. (31).

Fig. 2.
Fig. 2.

Isofrequency contours for (a) left-handed and (b) right-handed elliptically polarized waves calculated with an eigenvalue simulation of the layered structure shown in Fig. 1 (solid lines) and from the dispersion relation for a homogeneous medium with the constitutive parameters of Eq. (34) (circles). The labels on the solid lines are the normalized frequency ω a / c . The ω a / c = 1.6 lobes on the left- and right-hand sides of (b) are left-handed modes in the second propagating band.

Fig. 3.
Fig. 3.

Unit cell of a three-dimensional bianisotropic metamaterial. The crystal has a cubic lattice with lattice constant a . Centered in the unit cell is a sphere of radius 0.3 a consisting of an isotropic Tellegen material with constitutive parameters given in the text. The material outside the sphere is a uniaxial dielectric with permittitivities provided in the text.

Fig. 4.
Fig. 4.

Isofrequency contours for the three-dimensional crystal shown in Fig. 3. Both isofrequency contours (a) and (b) are for different linearly polarized eigenmodes. The isofrequency contours are calculated with a eigenvalue simulation of the crystal in Fig. 3 (solid lines) and from the dispersion relation for a homogeneous medium with the macroscopic constitutive parameters given in Eq. (37) (circles).

Tables (1)

Tables Icon

Table 1. Solutions to Eq. (A2)

Equations (42)

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

A 0 α ( x ) = A 0 0 ( x , y ) + α A 0 1 ( x , y ) + α 2 A 0 2 ( x , y ) + , A α ( x ) = A 0 ( x , y ) + α A 1 ( x , y ) + α 2 A 2 ( x , y ) + .
E = A 0 α , B = × A α .
( D H ) = ( p ^ l ^ m ^ q ^ ) K ^ · ( E B ) .
( D B ) = ( ϵ ^ ξ ^ ζ ^ μ ^ ) C ^ · ( E H ) ,
p ^ = ϵ ^ ξ ^ · μ ^ 1 · ζ ^ , l ^ = ξ ^ · μ ^ 1 , m ^ = μ ^ 1 · ζ ^ , q ^ = μ ^ 1 .
0 = · D = · [ p ^ · ( A 0 α ) + l ^ · ( × A α ) ] , 0 = × H = × [ m ^ · ( A 0 α ) + q ^ · ( × A α ) ] .
· A α = 0 .
( y · y × ) · [ K ^ · ( y A 0 0 y × A 0 ) ] = 0 .
y · A 0 = 0 .
( y · y × ) · [ K ^ · ( x A 0 0 y A 0 1 x × A 0 + y × A 1 ) ] = 0 .
A 0 1 ( x , y ) = i = 1 6 ( x A 0 0 ( x ) x × A 0 ( x ) ) i a 0 i ( y ) , A 1 ( x , y ) = i = 1 6 ( x A 0 0 ( x ) x × A 0 ( x ) ) i a i ( y ) .
( y · y × ) · [ K ^ · ( e ^ j + ( y a 0 j y × a j ) ) ] = 0 .
y · A 1 + x · A 0 = 0 .
y · A 1 = 0 ,
x · A 0 = 0 .
y · a i = 0 ,
( y · y × ) · [ K ^ · ( y A 0 2 x A 0 1 y × A 2 + x × A 1 ) ] + ( x · x × ) · [ K ^ · ( x A 0 0 y A 0 1 x × A 0 + y × A 1 ) ] = 0 .
( x · x × ) · [ K ¯ · ( x A 0 x × A 0 ) ] = 0 ,
( K ¯ ) i j = 1 V Ω d 3 y e ^ i · K ^ · [ e ^ j + ( y a 0 j y × a j ) ] .
A 0 α ( x ) = A 0 0 ( x , y ) + α A 0 1 ( x , y ) + α 2 A 0 2 ( x , y ) + , C 0 α ( x ) = C 0 0 ( x , y ) + α C 0 1 ( x , y ) + α 2 C 0 2 ( x , y ) + .
E = A 0 α , H = C 0 α .
0 = · D = · [ ϵ ^ · ( A 0 α ) + ξ ^ · ( C 0 α ) ] , 0 = · B = · [ ζ ^ · ( A 0 α ) + μ ^ · ( C 0 α ) ] .
( y · y · ) · [ C ^ · ( y A 0 0 y C 0 0 ) ] = 0 ,
( y · y · ) · [ C ^ · ( x A 0 0 y A 0 1 x C 0 0 y C 0 1 ) ] = 0 .
A 0 1 ( x , y ) = i = 1 6 ( x A 0 0 ( x ) x C 0 0 ( x ) ) i a 0 i ( y ) , C 0 1 ( x , y ) = i = 1 6 ( x A 0 0 ( x ) x C 0 0 ( x ) ) i c 0 i ( y ) .
( y · y · ) · [ C ^ · ( e ^ i + ( y a 0 i y c 0 i ) ) ] = 0 ,
( y · y · ) · [ C ^ · ( y A 0 2 x A 0 1 y C 0 2 x C 0 1 ) ] + ( x · x · ) · [ C ^ · ( x A 0 0 y A 0 1 x C 0 0 y C 0 1 ) ] = 0 .
( x · x · ) · [ C ¯ · ( x A 0 0 x C 0 0 ) ] = 0 ,
( C ¯ ) i j = 1 V Ω d 3 y e ^ i · C ^ · [ e ^ j + ( y a 0 j y c 0 j ) ] .
D = × C , B = × A .
ϵ 1 = 1 , ϵ 2 = 5 , ξ 1 = 0 , ξ 2 = 2.85 i , ζ 1 = 0 , ζ 2 = 2.85 i , μ 1 = 1 , μ 2 = 1 .
C ¯ = ( ϵ 0 0 ξ 0 0 0 ϵ 0 0 ξ 0 0 0 ϵ 0 0 ξ ζ 0 0 μ 0 0 0 ζ 0 0 μ 0 0 0 ζ 0 0 μ ) , ϵ = ϵ / ( ϵ μ ξ ζ ) ϵ / ( ε μ ξ ζ ) μ / ( ϵ μ ξ ζ ) ξ / ( ϵ μ ξ ζ ) ϵ / ( ϵ μ ξ ζ ) , ϵ = ϵ , ξ = ξ / ( ϵ μ ξ ζ ) ϵ / ϵ μ ξ ζ ) μ / ( ϵ μ ξ ζ ) ξ / ( ϵ μ ξ ζ ) ξ / ( ϵ μ ξ ζ ) , ξ = ξ , ζ = ζ / ( ϵ μ ξ ζ ) ϵ / ( ϵ μ ξ ζ ) μ / ( ϵ μ ξ ζ ) ξ / ( ϵ μ ξ ζ ) ζ / ( ϵ μ ξ ζ ) , ζ = ζ , μ = μ / ( ε μ ξ ζ ) ϵ / ( ϵ μ ξ ζ ) μ / ( ϵ μ ξ ζ ) ξ / ( ϵ μ ξ ζ ) μ / ( ϵ μ ξ ζ ) , μ = μ .
X = 1 a 0 a d y 1 X ( y 1 ) ,
ϵ = 3.81 , ϵ = 2.20 , ξ = i 4.75 , ξ = i 0.855 , ζ = i 4.75 , ζ = i 0.855 , μ = 10.5 , μ = 1 .
ϵ = 1.78 , ξ = 2 , ζ = 2 , μ = 1 .
ϵ = 4 , ϵ = 1.5 , ξ = ζ = 0 , μ = 1 ,
ϵ = 3.43 , ϵ = 1.38 , ξ = 0.335 , ξ = 0.267 , ζ = 0.335 , ζ = 0.267 , μ = 0.936 , μ = 0.910 .
K ^ = ( p 0 0 l 0 0 0 p 0 0 l 0 0 0 p 0 0 l m 0 0 q 0 0 0 m 0 0 q 0 0 0 m 0 0 q ) .
y 1 [ p ( a 0 i y 1 + δ 1 i ) + l δ 4 i ] = 0 , y 1 [ m δ 2 i + q ( a 3 i y 1 + δ 5 i ) ] = 0 , y 1 [ m δ 3 i + q ( a 2 i y 1 + δ 6 i ) ] = 0 .
y 1 [ C 1 ( ψ y 1 + 1 ) ] = 0 , y 1 [ C 2 + C 3 ψ y 1 ] = 0 , y 1 [ C 4 ψ y 1 ] = 0 ,
( K ¯ ) i j = 1 a 0 a d y 1 e ^ i · K ^ · ( a 0 j y 1 + δ 1 j δ 2 j δ 3 j δ 4 j a 3 j y 1 + δ 5 j a 2 j y 1 + δ 6 j ) ,
K ¯ = ( p 0 0 l 0 0 0 p 0 0 l 0 0 0 p 0 0 l m 0 0 q 0 0 0 m 0 0 q 0 0 0 m 0 0 q ) , p = 1 1 / p , p = p l m q + l / q m / q 1 / q , l = l / p 1 / p , l = l / q 1 / q , m = m / p 1 / p , m = m / q 1 / q , q = q l m p + l / p m / p 1 / p , q = 1 1 / q .

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