Abstract

The thickness-dependent photonic bandgap for a one-dimensional photonic crystal consisting of two different single-negative (SNG) materials is theoretically investigated. The two SNG materials include one with a single-negative permittivity (ϵ<0,μ>0) and the other with a single-negative permeability (μ<0,ϵ>0). It is found that the size of the bandgap and the positions of the band edges are strongly dependent on the thickness ratio of the two constituent SNG layers. First, the bandgap decreases with increase in the ratio and eventually shrinks to zero at a critical value of the ratio. Then the bandgap is opened up and enhanced when the thickness ratio is larger than the critical ratio. By using the composite right/left-hand transmission-line model, we qualitatively explained the bandgap’s shifting behaviors that are due to the variation of the thickness.

© 2009 Optical Society of America

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References

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  1. L.-G. Wang, H. Chen, and S.-Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70, 245102 (2004).
    [CrossRef]
  2. T. B. Wang, J. W. Dong, C. P. Yin, and H. Z. Wang, “Complete evanescent tunneling gaps in one-dimensional photonic crystals,” Phys. Lett. A 373, 169-172 (2008).
    [CrossRef]
  3. S. Wang, C. Tang, T. Pan, and L. Gao, “Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer,” Phys. Lett. A 351, 391-397 (2006).
    [CrossRef]
  4. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4777 (1996).
    [CrossRef] [PubMed]
  5. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
    [CrossRef]
  6. Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A 78, 023813 (2008).
    [CrossRef]
  7. C. Caloz and T. Itoh, Electromagnetic Metamaterials (Wiley, 2006).
  8. L. Gao, C. J. Tang, and S. M. Wang, “Photonic band gap from a stack of single-negative materials,” J. Magn. Magn. Mater. 301, 371-377 (2006).
    [CrossRef]
  9. D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
    [CrossRef] [PubMed]
  10. A. Alu and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
    [CrossRef]
  11. 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]
  12. J. A. Monsoriu, R. A. Depine, M. L. Martinez-Ricci, and E. Silvestre, “Interaction between non-Bragg band gaps in 1D metamaterial photonic crystals,” Opt. Express 14, 12958-12967 (2006).
    [CrossRef] [PubMed]
  13. R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 13, 1811-1818 (2001).
    [CrossRef]
  14. J. R. Canto, S. A. Matos, C. R. Paiva, and A. M. Barbosa, “Effect of losses in a layered structure containing DPS and DNG media,” PIERS Online 4, 546-555 (2008).
    [CrossRef]
  15. P. Yeh, Optical Waves in Layered Media (Wiley, 1998).
  16. Y. Jin and S. He, “Impedance-matched multilayered structure containing a zero-permittivity material for spatial filtering,” J. Nonlinear Opt. Phys. Mater. 17, 349-355 (2008).
    [CrossRef]

2008 (4)

T. B. Wang, J. W. Dong, C. P. Yin, and H. Z. Wang, “Complete evanescent tunneling gaps in one-dimensional photonic crystals,” Phys. Lett. A 373, 169-172 (2008).
[CrossRef]

Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A 78, 023813 (2008).
[CrossRef]

J. R. Canto, S. A. Matos, C. R. Paiva, and A. M. Barbosa, “Effect of losses in a layered structure containing DPS and DNG media,” PIERS Online 4, 546-555 (2008).
[CrossRef]

Y. Jin and S. He, “Impedance-matched multilayered structure containing a zero-permittivity material for spatial filtering,” J. Nonlinear Opt. Phys. Mater. 17, 349-355 (2008).
[CrossRef]

2006 (3)

J. A. Monsoriu, R. A. Depine, M. L. Martinez-Ricci, and E. Silvestre, “Interaction between non-Bragg band gaps in 1D metamaterial photonic crystals,” Opt. Express 14, 12958-12967 (2006).
[CrossRef] [PubMed]

L. Gao, C. J. Tang, and S. M. Wang, “Photonic band gap from a stack of single-negative materials,” J. Magn. Magn. Mater. 301, 371-377 (2006).
[CrossRef]

S. Wang, C. Tang, T. Pan, and L. Gao, “Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer,” Phys. Lett. A 351, 391-397 (2006).
[CrossRef]

2004 (2)

L.-G. Wang, H. Chen, and S.-Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70, 245102 (2004).
[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]

2003 (1)

A. Alu and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

2001 (1)

R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 13, 1811-1818 (2001).
[CrossRef]

2000 (1)

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

1996 (1)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4777 (1996).
[CrossRef] [PubMed]

Alu, A.

A. Alu and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Barbosa, A. M.

J. R. Canto, S. A. Matos, C. R. Paiva, and A. M. Barbosa, “Effect of losses in a layered structure containing DPS and DNG media,” PIERS Online 4, 546-555 (2008).
[CrossRef]

Caloz, C.

C. Caloz and T. Itoh, Electromagnetic Metamaterials (Wiley, 2006).

Canto, J. R.

J. R. Canto, S. A. Matos, C. R. Paiva, and A. M. Barbosa, “Effect of losses in a layered structure containing DPS and DNG media,” PIERS Online 4, 546-555 (2008).
[CrossRef]

Chen, H.

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.-G. Wang, H. Chen, and S.-Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Depine, R. A.

Dong, J. W.

T. B. Wang, J. W. Dong, C. P. Yin, and H. Z. Wang, “Complete evanescent tunneling gaps in one-dimensional photonic crystals,” Phys. Lett. A 373, 169-172 (2008).
[CrossRef]

Engheta, N.

A. Alu and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Fang, Y.

Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A 78, 023813 (2008).
[CrossRef]

Gao, L.

L. Gao, C. J. Tang, and S. M. Wang, “Photonic band gap from a stack of single-negative materials,” J. Magn. Magn. Mater. 301, 371-377 (2006).
[CrossRef]

S. Wang, C. Tang, T. Pan, and L. Gao, “Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer,” Phys. Lett. A 351, 391-397 (2006).
[CrossRef]

He, S.

Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A 78, 023813 (2008).
[CrossRef]

Y. Jin and S. He, “Impedance-matched multilayered structure containing a zero-permittivity material for spatial filtering,” J. Nonlinear Opt. Phys. Mater. 17, 349-355 (2008).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4777 (1996).
[CrossRef] [PubMed]

Itoh, T.

C. Caloz and T. Itoh, Electromagnetic Metamaterials (Wiley, 2006).

Jiang, H. T.

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]

Jin, Y.

Y. Jin and S. He, “Impedance-matched multilayered structure containing a zero-permittivity material for spatial filtering,” J. Nonlinear Opt. Phys. Mater. 17, 349-355 (2008).
[CrossRef]

Li, H. Q.

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]

Martinez-Ricci, M. L.

Matos, S. A.

J. R. Canto, S. A. Matos, C. R. Paiva, and A. M. Barbosa, “Effect of losses in a layered structure containing DPS and DNG media,” PIERS Online 4, 546-555 (2008).
[CrossRef]

Monsoriu, J. A.

Nemat-Nasser, S. C.

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Padilla, W.

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Paiva, C. R.

J. R. Canto, S. A. Matos, C. R. Paiva, and A. M. Barbosa, “Effect of losses in a layered structure containing DPS and DNG media,” PIERS Online 4, 546-555 (2008).
[CrossRef]

Pan, T.

S. Wang, C. Tang, T. Pan, and L. Gao, “Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer,” Phys. Lett. A 351, 391-397 (2006).
[CrossRef]

Pendry, J. B.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4777 (1996).
[CrossRef] [PubMed]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Ruppin, R.

R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 13, 1811-1818 (2001).
[CrossRef]

Schultz, S.

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Silvestre, E.

Smith, D. R.

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4777 (1996).
[CrossRef] [PubMed]

Tang, C.

S. Wang, C. Tang, T. Pan, and L. Gao, “Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer,” Phys. Lett. A 351, 391-397 (2006).
[CrossRef]

Tang, C. J.

L. Gao, C. J. Tang, and S. M. Wang, “Photonic band gap from a stack of single-negative materials,” J. Magn. Magn. Mater. 301, 371-377 (2006).
[CrossRef]

Vier, D. C.

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

Wang, H. Z.

T. B. Wang, J. W. Dong, C. P. Yin, and H. Z. Wang, “Complete evanescent tunneling gaps in one-dimensional photonic crystals,” Phys. Lett. A 373, 169-172 (2008).
[CrossRef]

Wang, L.-G.

L.-G. Wang, H. Chen, and S.-Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Wang, S.

S. Wang, C. Tang, T. Pan, and L. Gao, “Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer,” Phys. Lett. A 351, 391-397 (2006).
[CrossRef]

Wang, S. M.

L. Gao, C. J. Tang, and S. M. Wang, “Photonic band gap from a stack of single-negative materials,” J. Magn. Magn. Mater. 301, 371-377 (2006).
[CrossRef]

Wang, T. B.

T. B. Wang, J. W. Dong, C. P. Yin, and H. Z. Wang, “Complete evanescent tunneling gaps in one-dimensional photonic crystals,” Phys. Lett. A 373, 169-172 (2008).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1998).

Yin, C. P.

T. B. Wang, J. W. Dong, C. P. Yin, and H. Z. Wang, “Complete evanescent tunneling gaps in one-dimensional photonic crystals,” Phys. Lett. A 373, 169-172 (2008).
[CrossRef]

Youngs, I.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4777 (1996).
[CrossRef] [PubMed]

Zhang, Y. W.

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]

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]

Zhu, S.-Y.

L.-G. Wang, H. Chen, and S.-Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70, 245102 (2004).
[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]

IEEE Trans. Antennas Propag. (1)

A. Alu and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

J. Magn. Magn. Mater. (1)

L. Gao, C. J. Tang, and S. M. Wang, “Photonic band gap from a stack of single-negative materials,” J. Magn. Magn. Mater. 301, 371-377 (2006).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

Y. Jin and S. He, “Impedance-matched multilayered structure containing a zero-permittivity material for spatial filtering,” J. Nonlinear Opt. Phys. Mater. 17, 349-355 (2008).
[CrossRef]

J. Phys.: Condens. Matter (1)

R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 13, 1811-1818 (2001).
[CrossRef]

Opt. Express (1)

Phys. Lett. A (2)

T. B. Wang, J. W. Dong, C. P. Yin, and H. Z. Wang, “Complete evanescent tunneling gaps in one-dimensional photonic crystals,” Phys. Lett. A 373, 169-172 (2008).
[CrossRef]

S. Wang, C. Tang, T. Pan, and L. Gao, “Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer,” Phys. Lett. A 351, 391-397 (2006).
[CrossRef]

Phys. Rev. A (1)

Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A 78, 023813 (2008).
[CrossRef]

Phys. Rev. B (1)

L.-G. Wang, H. Chen, and S.-Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Phys. Rev. E (1)

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]

Phys. Rev. Lett. (2)

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4777 (1996).
[CrossRef] [PubMed]

PIERS Online (1)

J. R. Canto, S. A. Matos, C. R. Paiva, and A. M. Barbosa, “Effect of losses in a layered structure containing DPS and DNG media,” PIERS Online 4, 546-555 (2008).
[CrossRef]

Other (2)

P. Yeh, Optical Waves in Layered Media (Wiley, 1998).

C. Caloz and T. Itoh, Electromagnetic Metamaterials (Wiley, 2006).

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Figures (7)

Fig. 1
Fig. 1

N-period one-dimensional photonic crystal, in which layer 1 is an ENG material, layer 2 is an MNG material, and period is Λ = d 1 + d 2 .

Fig. 2
Fig. 2

Calculated frequency-dependent ϵ 1 for an ENG material and μ 2 for an MNG material.

Fig. 3
Fig. 3

(a) Calculated transmittance spectrum for d 1 = 10 mm and d 2 = 20 mm at different numbers of periods N = 1 , 2, and 20, and (b) calculated band diagram, K versus ω, for d 1 = 10 mm and d 2 = 20 mm .

Fig. 4
Fig. 4

Calculated band diagrams for different thickness ratios: (1) ρ = d 1 d 2 = 6 27 , (2) 8 27 , (3) 10 27 , (4) 14 27 , and (5) 18 27 .

Fig. 5
Fig. 5

Four characteristic frequencies ω L , ω s h , ω s e , and ω R as functions of the thickness ratio ρ.

Fig. 6
Fig. 6

Reflectance spectra in the electric gap under the conditions of d 2 = 27 mm and two different values of d 1 : (a) 14 nm and (b) 18 mm .

Fig. 7
Fig. 7

Reflectance spectra in the magnetic gap under the conditions of d 2 = 27 and two different values of d 1 : (a) 6 nm and (b) 8 mm .

Equations (18)

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ϵ 1 = 1 ω p 2 ω 2 , μ 1 = 1 ,
μ 2 = 1 F ω 2 ω 2 ω 0 2 + i ω Γ , ϵ 2 = 1 ,
T = 1 M 11 2 , R = M 21 M 11 2 ,
M = [ M 11 M 12 M 21 M 22 ] = D 0 1 [ D 1 P 1 D 1 1 D 2 P 2 D 2 1 ] N D 0 .
D j = [ 1 1 n j μ j n j μ j ] ,
P j = [ exp ( i k j d j ) 0 0 exp ( i k j d j ) ] .
cos ( K Λ ) = cos ( k 1 d 1 ) cos ( k 2 d 2 ) 1 2 ( k 1 μ 1 μ 2 k 2 + k 2 μ 2 μ 1 k 1 ) sin ( k 1 d 1 ) sin ( k 2 d 2 ) ,
k 1 μ 2 μ 1 k 2 = 1 .
cos ( K Λ ) = cos ( k 1 d 1 k 2 d 2 ) .
cos ( k 1 d 1 k 2 d 2 ) > 1 .
ϵ = ϵ ( ω ) = C R 1 ω 2 L L ,
μ = μ ( ω ) = L R 1 ω 2 C L .
ω R = 1 L R C R , ω L = 1 L L C L ,
ω s e = 1 L R C L , ω s h = 1 L L C R ,
L R = L R ( d 1 + d 2 ) , L L = L L d 1 ,
C R = C R ( d 1 + d 2 ) , C L = C L d 2 .
ω L d 1 d 2 , ω s h d 1 d 1 + d 2
ω s e d 2 d 1 + d 2 , ω R 1 d 1 + d 2 .

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