Abstract

Merging omnidirectional defect modes are found in one-dimensional photonic crystals composed of periodic single-negative-permittivity and single-negative-permeability media layers and a single-negative defect. By increasing the defect size or the thickness ratio of the two periodic layers, two defect modes shift to each other from the upper and lower frequency side of the forbidden gap, respectively, then merge at the midfrequency of the gap. Merging conditions for such defect modes are given. These defect modes are insensitive to the incident angle. Moreover, the electric fields at the central frequency of the bandgap could be designed to be strongly localized at just one interface between the defect and the periodic layers in the photonic crystal with low transmission.

© 2008 Optical Society of America

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  3. F. Qiao, C. Zhang, and J. Wan, “Photonic quantum-well structures: multiple channelled filtering phenomena,” Appl. Phys. Lett. 77, 3698-3700 (2000).
    [CrossRef]
  4. L. Wang, Z. S. Wang, Y. G. Wu, and L. Y. Chen, “Enlargement of the nontransmission frequency range of multiple-channeled filters by the use of heterostructures,” J. Appl. Phys. 95, 424-426 (2004).
    [CrossRef]
  5. M. Bayindir, C. Kural, and E. Ozbay, “Coupled optical microcavities in one dimensional photonic bandgap structures,” J. Opt. A, Pure Appl. Opt. 3, 184-188 (2001).
    [CrossRef]
  6. Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81, 4112-4114 (2002).
    [CrossRef]
  7. H. T. Jiang, H. Chen, H. Q. Li, and Y. W. Zhang, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386-5388 (2003).
    [CrossRef]
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    [CrossRef]
  9. Y. H. Chen, J. W. Dong, and H. Z. Wang, “Omnidirectional resonance modes in photonic crystal heterostructures containing single-negative materials,” J. Opt. Soc. Am. B 23, 2237-2240 (2006).
    [CrossRef]
  10. H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
    [CrossRef]
  11. Y. H. Chen, J. W. Dong, and H. Z. Wang, “Twin defect modes in one-dimensional photonic crystals with a single-negative material defect,” Appl. Phys. Lett. 89, 141101 (2006).
    [CrossRef]
  12. G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
    [CrossRef]
  13. N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
    [CrossRef]
  14. G. S. Guan, H. T. Jiang, H. Q. Li, Y. W. Zhang, and H. Chen, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
    [CrossRef]
  15. L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of epsilon-negative and mu-negative materials,” Phys. Rev. E 74, 056615 (2006).
    [CrossRef]

2006 (5)

Y. H. Chen, J. W. Dong, and H. Z. Wang, “Conditions of near-zero dispersion of defect modes in one-dimensional photonic crystal containing negative-index materials,” J. Opt. Soc. Am. B 23, 776-781 (2006).
[CrossRef]

Y. H. Chen, J. W. Dong, and H. Z. Wang, “Omnidirectional resonance modes in photonic crystal heterostructures containing single-negative materials,” J. Opt. Soc. Am. B 23, 2237-2240 (2006).
[CrossRef]

Y. H. Chen, J. W. Dong, and H. Z. Wang, “Twin defect modes in one-dimensional photonic crystals with a single-negative material defect,” Appl. Phys. Lett. 89, 141101 (2006).
[CrossRef]

G. S. Guan, H. T. Jiang, H. Q. Li, Y. W. Zhang, and H. Chen, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
[CrossRef]

L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of epsilon-negative and mu-negative materials,” Phys. Rev. E 74, 056615 (2006).
[CrossRef]

2004 (2)

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

L. Wang, Z. S. Wang, Y. G. Wu, and L. Y. Chen, “Enlargement of the nontransmission frequency range of multiple-channeled filters by the use of heterostructures,” J. Appl. Phys. 95, 424-426 (2004).
[CrossRef]

2003 (1)

H. T. Jiang, H. Chen, H. Q. Li, and Y. W. Zhang, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386-5388 (2003).
[CrossRef]

2002 (3)

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
[CrossRef]

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81, 4112-4114 (2002).
[CrossRef]

2001 (1)

M. Bayindir, C. Kural, and E. Ozbay, “Coupled optical microcavities in one dimensional photonic bandgap structures,” J. Opt. A, Pure Appl. Opt. 3, 184-188 (2001).
[CrossRef]

2000 (1)

F. Qiao, C. Zhang, and J. Wan, “Photonic quantum-well structures: multiple channelled filtering phenomena,” Appl. Phys. Lett. 77, 3698-3700 (2000).
[CrossRef]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Bayindir, M.

M. Bayindir, C. Kural, and E. Ozbay, “Coupled optical microcavities in one dimensional photonic bandgap structures,” J. Opt. A, Pure Appl. Opt. 3, 184-188 (2001).
[CrossRef]

Bermel, P.

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81, 4112-4114 (2002).
[CrossRef]

Chen, H.

G. S. Guan, H. T. Jiang, H. Q. Li, Y. W. Zhang, and H. Chen, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
[CrossRef]

L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of epsilon-negative and mu-negative materials,” Phys. Rev. E 74, 056615 (2006).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, and Y. W. Zhang, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386-5388 (2003).
[CrossRef]

N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
[CrossRef]

Chen, L. Y.

L. Wang, Z. S. Wang, Y. G. Wu, and L. Y. Chen, “Enlargement of the nontransmission frequency range of multiple-channeled filters by the use of heterostructures,” J. Appl. Phys. 95, 424-426 (2004).
[CrossRef]

Chen, Y. H.

Dong, J. W.

Duan, X.

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81, 4112-4114 (2002).
[CrossRef]

Eleftheriades, G. V.

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

Guan, G. S.

G. S. Guan, H. T. Jiang, H. Q. Li, Y. W. Zhang, and H. Chen, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
[CrossRef]

He, L.

L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of epsilon-negative and mu-negative materials,” Phys. Rev. E 74, 056615 (2006).
[CrossRef]

Iyer, A. K.

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

Jiang, H. T.

G. S. Guan, H. T. Jiang, H. Q. Li, Y. W. Zhang, and H. Chen, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, and Y. W. Zhang, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386-5388 (2003).
[CrossRef]

Joannopoulos, J. D.

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81, 4112-4114 (2002).
[CrossRef]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Kimerling, L. C.

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81, 4112-4114 (2002).
[CrossRef]

Kremer, P. C.

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

Kural, C.

M. Bayindir, C. Kural, and E. Ozbay, “Coupled optical microcavities in one dimensional photonic bandgap structures,” J. Opt. A, Pure Appl. Opt. 3, 184-188 (2001).
[CrossRef]

Li, H. Q.

G. S. Guan, H. T. Jiang, H. Q. Li, Y. W. Zhang, and H. Chen, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
[CrossRef]

L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of epsilon-negative and mu-negative materials,” Phys. Rev. E 74, 056615 (2006).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, and Y. W. Zhang, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386-5388 (2003).
[CrossRef]

Liu, N. H.

N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
[CrossRef]

Ozbay, E.

M. Bayindir, C. Kural, and E. Ozbay, “Coupled optical microcavities in one dimensional photonic bandgap structures,” J. Opt. A, Pure Appl. Opt. 3, 184-188 (2001).
[CrossRef]

Qiao, F.

F. Qiao, C. Zhang, and J. Wan, “Photonic quantum-well structures: multiple channelled filtering phenomena,” Appl. Phys. Lett. 77, 3698-3700 (2000).
[CrossRef]

Wada, K.

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81, 4112-4114 (2002).
[CrossRef]

Wan, J.

F. Qiao, C. Zhang, and J. Wan, “Photonic quantum-well structures: multiple channelled filtering phenomena,” Appl. Phys. Lett. 77, 3698-3700 (2000).
[CrossRef]

Wang, H. Z.

Wang, L.

L. Wang, Z. S. Wang, Y. G. Wu, and L. Y. Chen, “Enlargement of the nontransmission frequency range of multiple-channeled filters by the use of heterostructures,” J. Appl. Phys. 95, 424-426 (2004).
[CrossRef]

Wang, Z. S.

L. Wang, Z. S. Wang, Y. G. Wu, and L. Y. Chen, “Enlargement of the nontransmission frequency range of multiple-channeled filters by the use of heterostructures,” J. Appl. Phys. 95, 424-426 (2004).
[CrossRef]

Wu, X.

N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
[CrossRef]

Wu, Y. G.

L. Wang, Z. S. Wang, Y. G. Wu, and L. Y. Chen, “Enlargement of the nontransmission frequency range of multiple-channeled filters by the use of heterostructures,” J. Appl. Phys. 95, 424-426 (2004).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Yi, Y.

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81, 4112-4114 (2002).
[CrossRef]

Zhang, C.

F. Qiao, C. Zhang, and J. Wan, “Photonic quantum-well structures: multiple channelled filtering phenomena,” Appl. Phys. Lett. 77, 3698-3700 (2000).
[CrossRef]

Zhang, L. W.

L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of epsilon-negative and mu-negative materials,” Phys. Rev. E 74, 056615 (2006).
[CrossRef]

Zhang, Y. W.

L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of epsilon-negative and mu-negative materials,” Phys. Rev. E 74, 056615 (2006).
[CrossRef]

G. S. Guan, H. T. Jiang, H. Q. Li, Y. W. Zhang, and H. Chen, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, and Y. W. Zhang, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386-5388 (2003).
[CrossRef]

Zhu, S. Y.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
[CrossRef]

Zi, J.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Appl. Phys. Lett. (5)

F. Qiao, C. Zhang, and J. Wan, “Photonic quantum-well structures: multiple channelled filtering phenomena,” Appl. Phys. Lett. 77, 3698-3700 (2000).
[CrossRef]

Y. Yi, P. Bermel, K. Wada, X. Duan, J. D. Joannopoulos, and L. C. Kimerling, “Tunable multichannel optical filter based on silicon photonic band gap materials actuation,” Appl. Phys. Lett. 81, 4112-4114 (2002).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, and Y. W. Zhang, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386-5388 (2003).
[CrossRef]

Y. H. Chen, J. W. Dong, and H. Z. Wang, “Twin defect modes in one-dimensional photonic crystals with a single-negative material defect,” Appl. Phys. Lett. 89, 141101 (2006).
[CrossRef]

G. S. Guan, H. T. Jiang, H. Q. Li, Y. W. Zhang, and H. Chen, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

G. V. Eleftheriades, A. K. Iyer, and P. C. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002).
[CrossRef]

J. Appl. Phys. (1)

L. Wang, Z. S. Wang, Y. G. Wu, and L. Y. Chen, “Enlargement of the nontransmission frequency range of multiple-channeled filters by the use of heterostructures,” J. Appl. Phys. 95, 424-426 (2004).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

M. Bayindir, C. Kural, and E. Ozbay, “Coupled optical microcavities in one dimensional photonic bandgap structures,” J. Opt. A, Pure Appl. Opt. 3, 184-188 (2001).
[CrossRef]

J. Opt. Soc. Am. B (2)

Phys. Rev. E (3)

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E 65, 046607 (2002).
[CrossRef]

L. W. Zhang, Y. W. Zhang, L. He, H. Q. Li, and H. Chen, “Experimental study of photonic crystals consisting of epsilon-negative and mu-negative materials,” Phys. Rev. E 74, 056615 (2006).
[CrossRef]

Phys. Rev. Lett. (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Dependence of the defect modes in the zero- φ eff gap on d D in structure ( N μ N ε ) 6 D N μ ( N ε N μ ) 6 at normal incidence with d N μ = 6 mm and d N ε = 12 mm .

Fig. 2
Fig. 2

Electric fields distribution corresponding to the defect modes in Fig. 1 at frequency ν C = 0.7958 GHz . The thicknesses of the defects are (a) d D = 72 mm , (b) d D = 80 mm , (c) d D = 90 mm , and (d) d D = 120 mm . The light gray and white areas correspond to the layers of MNG and ENG materials, respectively.

Fig. 3
Fig. 3

Dependence of the defect modes on d N ε in structure ( N μ N ε ) 6 D N μ ( N ε N μ ) 6 with d N μ = 6 mm and d D = 72 mm .

Fig. 4
Fig. 4

Dependences of the defect modes on incident angle for TE and TM polarizations. Squares and circles correspond to the defect modes inside the zero- φ eff gap in Fig. 5a.

Fig. 5
Fig. 5

Dependence of the defect modes in the zero- φ eff gap on d D in structure ( N μ N ε ) 6 D N μ ( N ε N μ ) 6 with d N μ = 6 mm , d N ε = 12 mm , and γ = 1 × 10 7 Hz .

Fig. 6
Fig. 6

Electric fields distribution at frequency ν C inside the structures ( N μ N ε ) 6 D N μ ( N ε N μ ) 6 with d D μ = 120 mm . The damping factors are (a) γ = 0 , (b) γ = 1 × 10 7 Hz , and (c) γ = 3 × 10 7 Hz .

Equations (5)

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

ε 1 = ε a , μ 1 = μ a ω m p 2 ω 2 ,
ε 2 = ε b ω e p 2 ω 2 , μ 2 = μ b ,
M l = [ cos k l z d l i sin k l z d l η l i η l sin k l z d l cos k l z d l ] ,
μ ¯ = 2 s ( μ 1 d N μ + μ 2 d N ε ) + μ 1 d D 2 s ( d N μ + d N ε ) + d D = 0 ,
ε ¯ = 2 s ( ε 1 d N μ + ε 2 d N ε ) + ε 1 d D 2 s ( d N μ + d N ε ) + d D = 0 .

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