Abstract

In this work, a theoretical analysis on the effective plasma frequency (EPF) for a one-dimensional superconductor dielectric photonic crystal is made. First, the EPF is extracted from the first photonic band calculated within the framework of transfer matrix method together with Bloch theorem in a periodic multilayer structure. We investigate the EPF as a function of the filling factor and the permittivity of the dielectric layer, and the operating temperature as well. Then, the EPF is comparatively studied by the analytical expression derived from the effective medium theory. It is found that both results are in fairly good agreement.

© 2013 Optical Society of America

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  1. H. Contopanagos, E. Yablonovitch, and N. G. Alexopoulos, “Electromagnetic properties of periodic multilayers of ultrathin metallic films from dc to ultraviolet frequencies,” J. Opt. Soc. Am. A 16, 2294–2306 (1999).
    [CrossRef]
  2. X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
    [CrossRef]
  3. J. Manzanares-Martinez, “Analytical expression for the effective plasma frequency in one-dimensional metallic-dielectric photonic crystal,” Prog. Electromagn. Res. M 13, 189–202 (2010).
    [CrossRef]
  4. M. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett. 87, 034102 (2005).
    [CrossRef]
  5. C. H. Raymond Ooi, T. C. Au Yeung, C. H. Kam, and T. K. Lim, “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Rev. B 61, 5920–5926 (2000).
    [CrossRef]
  6. C.-J. Wu, M.-S. Chen, and T.-J. Yang, “Photonic band structure in a superconductor-dielectric superlattice,” Phys. C 432, 133–139 (2005).
    [CrossRef]
  7. O. L. Berman, Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, “Superconducting photonic crystals: numerical calculation of the band structure,” Phys. Rev. B 74, 092505 (2006).
    [CrossRef]
  8. A. N. Poddubny, E. L. Ivchenko, and Yu. E. Lozovik, “Low-frequency spectroscopy of superconducting photonic crystals,” Solid State Commun. 146, 143–147 (2008).
    [CrossRef]
  9. A. H. Aly, H.-T. Hsu, T.-J. Yang, C.-J. Wu, and C. K. Hwangbo, “Extraordinary optical properties of a superconducting periodic multilayer in near-zero-permittivity operation range,” J. Appl. Phys. 105, 083917 (2009).
    [CrossRef]
  10. W.-H. Lin, C.-J. Wu, T.-J. Yang, and S.-J. Chang, “Terahertz multichannel filter in a superconducting photonic crystal,” Opt. Express 18, 27155–27166 (2010).
    [CrossRef]
  11. C. H. Raymond Ooi and Q. Gong, “Temperature dependent resonances in superconductor photonic crystal,” J. Appl. Phys. 110, 063513 (2011).
    [CrossRef]
  12. G. Becerra, E. Moncada-Villa, and J. C. Granada, “Localized modes in metamaterial-dielectric photonic crystals with a dielectric-superconductor pair defect,” J. Supercond. Nov. Magn. 25, 2163–2166 (2012).
    [CrossRef]
  13. C. H. Raymond Ooi and T. C. Au Yeung, “Polariton gap in a superconductor-dielectric superlattice,” Phys. Lett. A 259, 413–419 (1999).
    [CrossRef]
  14. O. G. Vendik, I. B. Vendik, and D. I. Kaparkov, “Empirical model of the microwave properties of high-temperature superconductors,” IEEE Trans. Microwave Theory Tech. 46, 469–478 (1998).
    [CrossRef]
  15. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, 1998).
  16. C.-J. Wu, T.-J. Yang, C.-C. Li, and P.-Y. Wu, “Investigation of effective plasma frequencies in one-dimensional plasma photonic crystals,” Prog. Electromagn. Res. 126, 521–538 (2012).
    [CrossRef]
  17. K. L. Low, M. Z. Mat Jafri, and S. A. Khan, “Effective plasma frequency for two-dimensioanl metallic photonic crystals,” Prog. Electromagn. Res. M 12, 67–79 (2010).
    [CrossRef]

2012

G. Becerra, E. Moncada-Villa, and J. C. Granada, “Localized modes in metamaterial-dielectric photonic crystals with a dielectric-superconductor pair defect,” J. Supercond. Nov. Magn. 25, 2163–2166 (2012).
[CrossRef]

C.-J. Wu, T.-J. Yang, C.-C. Li, and P.-Y. Wu, “Investigation of effective plasma frequencies in one-dimensional plasma photonic crystals,” Prog. Electromagn. Res. 126, 521–538 (2012).
[CrossRef]

2011

C. H. Raymond Ooi and Q. Gong, “Temperature dependent resonances in superconductor photonic crystal,” J. Appl. Phys. 110, 063513 (2011).
[CrossRef]

2010

W.-H. Lin, C.-J. Wu, T.-J. Yang, and S.-J. Chang, “Terahertz multichannel filter in a superconducting photonic crystal,” Opt. Express 18, 27155–27166 (2010).
[CrossRef]

K. L. Low, M. Z. Mat Jafri, and S. A. Khan, “Effective plasma frequency for two-dimensioanl metallic photonic crystals,” Prog. Electromagn. Res. M 12, 67–79 (2010).
[CrossRef]

J. Manzanares-Martinez, “Analytical expression for the effective plasma frequency in one-dimensional metallic-dielectric photonic crystal,” Prog. Electromagn. Res. M 13, 189–202 (2010).
[CrossRef]

2009

A. H. Aly, H.-T. Hsu, T.-J. Yang, C.-J. Wu, and C. K. Hwangbo, “Extraordinary optical properties of a superconducting periodic multilayer in near-zero-permittivity operation range,” J. Appl. Phys. 105, 083917 (2009).
[CrossRef]

2008

A. N. Poddubny, E. L. Ivchenko, and Yu. E. Lozovik, “Low-frequency spectroscopy of superconducting photonic crystals,” Solid State Commun. 146, 143–147 (2008).
[CrossRef]

2006

O. L. Berman, Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, “Superconducting photonic crystals: numerical calculation of the band structure,” Phys. Rev. B 74, 092505 (2006).
[CrossRef]

2005

C.-J. Wu, M.-S. Chen, and T.-J. Yang, “Photonic band structure in a superconductor-dielectric superlattice,” Phys. C 432, 133–139 (2005).
[CrossRef]

M. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett. 87, 034102 (2005).
[CrossRef]

X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
[CrossRef]

2000

C. H. Raymond Ooi, T. C. Au Yeung, C. H. Kam, and T. K. Lim, “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Rev. B 61, 5920–5926 (2000).
[CrossRef]

1999

1998

O. G. Vendik, I. B. Vendik, and D. I. Kaparkov, “Empirical model of the microwave properties of high-temperature superconductors,” IEEE Trans. Microwave Theory Tech. 46, 469–478 (1998).
[CrossRef]

Alexopoulos, N. G.

Aly, A. H.

A. H. Aly, H.-T. Hsu, T.-J. Yang, C.-J. Wu, and C. K. Hwangbo, “Extraordinary optical properties of a superconducting periodic multilayer in near-zero-permittivity operation range,” J. Appl. Phys. 105, 083917 (2009).
[CrossRef]

Anlage, S. M.

M. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett. 87, 034102 (2005).
[CrossRef]

Au Yeung, T. C.

C. H. Raymond Ooi, T. C. Au Yeung, C. H. Kam, and T. K. Lim, “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Rev. B 61, 5920–5926 (2000).
[CrossRef]

C. H. Raymond Ooi and T. C. Au Yeung, “Polariton gap in a superconductor-dielectric superlattice,” Phys. Lett. A 259, 413–419 (1999).
[CrossRef]

Becerra, G.

G. Becerra, E. Moncada-Villa, and J. C. Granada, “Localized modes in metamaterial-dielectric photonic crystals with a dielectric-superconductor pair defect,” J. Supercond. Nov. Magn. 25, 2163–2166 (2012).
[CrossRef]

Berman, O. L.

O. L. Berman, Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, “Superconducting photonic crystals: numerical calculation of the band structure,” Phys. Rev. B 74, 092505 (2006).
[CrossRef]

Chang, S.-J.

Chen, M.-S.

C.-J. Wu, M.-S. Chen, and T.-J. Yang, “Photonic band structure in a superconductor-dielectric superlattice,” Phys. C 432, 133–139 (2005).
[CrossRef]

Coalson, R. D.

O. L. Berman, Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, “Superconducting photonic crystals: numerical calculation of the band structure,” Phys. Rev. B 74, 092505 (2006).
[CrossRef]

Contopanagos, H.

Eiderman, S. L.

O. L. Berman, Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, “Superconducting photonic crystals: numerical calculation of the band structure,” Phys. Rev. B 74, 092505 (2006).
[CrossRef]

Gong, Q.

C. H. Raymond Ooi and Q. Gong, “Temperature dependent resonances in superconductor photonic crystal,” J. Appl. Phys. 110, 063513 (2011).
[CrossRef]

Granada, J. C.

G. Becerra, E. Moncada-Villa, and J. C. Granada, “Localized modes in metamaterial-dielectric photonic crystals with a dielectric-superconductor pair defect,” J. Supercond. Nov. Magn. 25, 2163–2166 (2012).
[CrossRef]

Han, D.

X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
[CrossRef]

Hsu, H.-T.

A. H. Aly, H.-T. Hsu, T.-J. Yang, C.-J. Wu, and C. K. Hwangbo, “Extraordinary optical properties of a superconducting periodic multilayer in near-zero-permittivity operation range,” J. Appl. Phys. 105, 083917 (2009).
[CrossRef]

Hwangbo, C. K.

A. H. Aly, H.-T. Hsu, T.-J. Yang, C.-J. Wu, and C. K. Hwangbo, “Extraordinary optical properties of a superconducting periodic multilayer in near-zero-permittivity operation range,” J. Appl. Phys. 105, 083917 (2009).
[CrossRef]

Ivchenko, E. L.

A. N. Poddubny, E. L. Ivchenko, and Yu. E. Lozovik, “Low-frequency spectroscopy of superconducting photonic crystals,” Solid State Commun. 146, 143–147 (2008).
[CrossRef]

Kam, C. H.

C. H. Raymond Ooi, T. C. Au Yeung, C. H. Kam, and T. K. Lim, “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Rev. B 61, 5920–5926 (2000).
[CrossRef]

Kaparkov, D. I.

O. G. Vendik, I. B. Vendik, and D. I. Kaparkov, “Empirical model of the microwave properties of high-temperature superconductors,” IEEE Trans. Microwave Theory Tech. 46, 469–478 (1998).
[CrossRef]

Khan, S. A.

K. L. Low, M. Z. Mat Jafri, and S. A. Khan, “Effective plasma frequency for two-dimensioanl metallic photonic crystals,” Prog. Electromagn. Res. M 12, 67–79 (2010).
[CrossRef]

Li, C.-C.

C.-J. Wu, T.-J. Yang, C.-C. Li, and P.-Y. Wu, “Investigation of effective plasma frequencies in one-dimensional plasma photonic crystals,” Prog. Electromagn. Res. 126, 521–538 (2012).
[CrossRef]

Lim, T. K.

C. H. Raymond Ooi, T. C. Au Yeung, C. H. Kam, and T. K. Lim, “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Rev. B 61, 5920–5926 (2000).
[CrossRef]

Lin, W.-H.

Liu, X.

X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
[CrossRef]

Low, K. L.

K. L. Low, M. Z. Mat Jafri, and S. A. Khan, “Effective plasma frequency for two-dimensioanl metallic photonic crystals,” Prog. Electromagn. Res. M 12, 67–79 (2010).
[CrossRef]

Lozovik, Y. E.

O. L. Berman, Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, “Superconducting photonic crystals: numerical calculation of the band structure,” Phys. Rev. B 74, 092505 (2006).
[CrossRef]

Lozovik, Yu. E.

A. N. Poddubny, E. L. Ivchenko, and Yu. E. Lozovik, “Low-frequency spectroscopy of superconducting photonic crystals,” Solid State Commun. 146, 143–147 (2008).
[CrossRef]

Manzanares-Martinez, J.

J. Manzanares-Martinez, “Analytical expression for the effective plasma frequency in one-dimensional metallic-dielectric photonic crystal,” Prog. Electromagn. Res. M 13, 189–202 (2010).
[CrossRef]

Mat Jafri, M. Z.

K. L. Low, M. Z. Mat Jafri, and S. A. Khan, “Effective plasma frequency for two-dimensioanl metallic photonic crystals,” Prog. Electromagn. Res. M 12, 67–79 (2010).
[CrossRef]

Moncada-Villa, E.

G. Becerra, E. Moncada-Villa, and J. C. Granada, “Localized modes in metamaterial-dielectric photonic crystals with a dielectric-superconductor pair defect,” J. Supercond. Nov. Magn. 25, 2163–2166 (2012).
[CrossRef]

Orloff, N.

M. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett. 87, 034102 (2005).
[CrossRef]

Poddubny, A. N.

A. N. Poddubny, E. L. Ivchenko, and Yu. E. Lozovik, “Low-frequency spectroscopy of superconducting photonic crystals,” Solid State Commun. 146, 143–147 (2008).
[CrossRef]

Raymond Ooi, C. H.

C. H. Raymond Ooi and Q. Gong, “Temperature dependent resonances in superconductor photonic crystal,” J. Appl. Phys. 110, 063513 (2011).
[CrossRef]

C. H. Raymond Ooi, T. C. Au Yeung, C. H. Kam, and T. K. Lim, “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Rev. B 61, 5920–5926 (2000).
[CrossRef]

C. H. Raymond Ooi and T. C. Au Yeung, “Polariton gap in a superconductor-dielectric superlattice,” Phys. Lett. A 259, 413–419 (1999).
[CrossRef]

Ricci, M.

M. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett. 87, 034102 (2005).
[CrossRef]

Vendik, I. B.

O. G. Vendik, I. B. Vendik, and D. I. Kaparkov, “Empirical model of the microwave properties of high-temperature superconductors,” IEEE Trans. Microwave Theory Tech. 46, 469–478 (1998).
[CrossRef]

Vendik, O. G.

O. G. Vendik, I. B. Vendik, and D. I. Kaparkov, “Empirical model of the microwave properties of high-temperature superconductors,” IEEE Trans. Microwave Theory Tech. 46, 469–478 (1998).
[CrossRef]

Wu, C.-J.

C.-J. Wu, T.-J. Yang, C.-C. Li, and P.-Y. Wu, “Investigation of effective plasma frequencies in one-dimensional plasma photonic crystals,” Prog. Electromagn. Res. 126, 521–538 (2012).
[CrossRef]

W.-H. Lin, C.-J. Wu, T.-J. Yang, and S.-J. Chang, “Terahertz multichannel filter in a superconducting photonic crystal,” Opt. Express 18, 27155–27166 (2010).
[CrossRef]

A. H. Aly, H.-T. Hsu, T.-J. Yang, C.-J. Wu, and C. K. Hwangbo, “Extraordinary optical properties of a superconducting periodic multilayer in near-zero-permittivity operation range,” J. Appl. Phys. 105, 083917 (2009).
[CrossRef]

C.-J. Wu, M.-S. Chen, and T.-J. Yang, “Photonic band structure in a superconductor-dielectric superlattice,” Phys. C 432, 133–139 (2005).
[CrossRef]

Wu, P.-Y.

C.-J. Wu, T.-J. Yang, C.-C. Li, and P.-Y. Wu, “Investigation of effective plasma frequencies in one-dimensional plasma photonic crystals,” Prog. Electromagn. Res. 126, 521–538 (2012).
[CrossRef]

Xi, Y.

X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
[CrossRef]

Xu, X.

X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
[CrossRef]

Yablonovitch, E.

Yang, T.-J.

C.-J. Wu, T.-J. Yang, C.-C. Li, and P.-Y. Wu, “Investigation of effective plasma frequencies in one-dimensional plasma photonic crystals,” Prog. Electromagn. Res. 126, 521–538 (2012).
[CrossRef]

W.-H. Lin, C.-J. Wu, T.-J. Yang, and S.-J. Chang, “Terahertz multichannel filter in a superconducting photonic crystal,” Opt. Express 18, 27155–27166 (2010).
[CrossRef]

A. H. Aly, H.-T. Hsu, T.-J. Yang, C.-J. Wu, and C. K. Hwangbo, “Extraordinary optical properties of a superconducting periodic multilayer in near-zero-permittivity operation range,” J. Appl. Phys. 105, 083917 (2009).
[CrossRef]

C.-J. Wu, M.-S. Chen, and T.-J. Yang, “Photonic band structure in a superconductor-dielectric superlattice,” Phys. C 432, 133–139 (2005).
[CrossRef]

Yeh, P.

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

Zhu, Z.

X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
[CrossRef]

Zi, J.

X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
[CrossRef]

Appl. Phys. Lett.

X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
[CrossRef]

M. Ricci, N. Orloff, and S. M. Anlage, “Superconducting metamaterials,” Appl. Phys. Lett. 87, 034102 (2005).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

O. G. Vendik, I. B. Vendik, and D. I. Kaparkov, “Empirical model of the microwave properties of high-temperature superconductors,” IEEE Trans. Microwave Theory Tech. 46, 469–478 (1998).
[CrossRef]

J. Appl. Phys.

C. H. Raymond Ooi and Q. Gong, “Temperature dependent resonances in superconductor photonic crystal,” J. Appl. Phys. 110, 063513 (2011).
[CrossRef]

A. H. Aly, H.-T. Hsu, T.-J. Yang, C.-J. Wu, and C. K. Hwangbo, “Extraordinary optical properties of a superconducting periodic multilayer in near-zero-permittivity operation range,” J. Appl. Phys. 105, 083917 (2009).
[CrossRef]

J. Opt. Soc. Am. A

J. Supercond. Nov. Magn.

G. Becerra, E. Moncada-Villa, and J. C. Granada, “Localized modes in metamaterial-dielectric photonic crystals with a dielectric-superconductor pair defect,” J. Supercond. Nov. Magn. 25, 2163–2166 (2012).
[CrossRef]

Opt. Express

Phys. C

C.-J. Wu, M.-S. Chen, and T.-J. Yang, “Photonic band structure in a superconductor-dielectric superlattice,” Phys. C 432, 133–139 (2005).
[CrossRef]

Phys. Lett. A

C. H. Raymond Ooi and T. C. Au Yeung, “Polariton gap in a superconductor-dielectric superlattice,” Phys. Lett. A 259, 413–419 (1999).
[CrossRef]

Phys. Rev. B

O. L. Berman, Y. E. Lozovik, S. L. Eiderman, and R. D. Coalson, “Superconducting photonic crystals: numerical calculation of the band structure,” Phys. Rev. B 74, 092505 (2006).
[CrossRef]

C. H. Raymond Ooi, T. C. Au Yeung, C. H. Kam, and T. K. Lim, “Photonic band gap in a superconductor-dielectric superlattice,” Phys. Rev. B 61, 5920–5926 (2000).
[CrossRef]

Prog. Electromagn. Res.

C.-J. Wu, T.-J. Yang, C.-C. Li, and P.-Y. Wu, “Investigation of effective plasma frequencies in one-dimensional plasma photonic crystals,” Prog. Electromagn. Res. 126, 521–538 (2012).
[CrossRef]

Prog. Electromagn. Res. M

K. L. Low, M. Z. Mat Jafri, and S. A. Khan, “Effective plasma frequency for two-dimensioanl metallic photonic crystals,” Prog. Electromagn. Res. M 12, 67–79 (2010).
[CrossRef]

J. Manzanares-Martinez, “Analytical expression for the effective plasma frequency in one-dimensional metallic-dielectric photonic crystal,” Prog. Electromagn. Res. M 13, 189–202 (2010).
[CrossRef]

Solid State Commun.

A. N. Poddubny, E. L. Ivchenko, and Yu. E. Lozovik, “Low-frequency spectroscopy of superconducting photonic crystals,” Solid State Commun. 146, 143–147 (2008).
[CrossRef]

Other

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

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Figures (4)

Fig. 1.
Fig. 1.

Calculated first photonic band for 1D SDPC at different filling factor r. The conditions are Λ=0.05 and εr=15. The starting frequency of the first band is defined as the EPF and is marked by the red arrow. The dispersion for the bulk superconductor is also shown for convenience of comparison.

Fig. 2.
Fig. 2.

Calculated EPF versus r at Λ=0.05 and εr=15. The EPF is a decreasing function of r.

Fig. 3.
Fig. 3.

Calculated first photonic band for 1D SDPC at different dielectric permittivity εr. The conditions are Λ=0.01 and r=1/3.

Fig. 4.
Fig. 4.

Calculated first photonic band for 1D SDPC at different Λ. The conditions are εr=15 and r=1/3.

Equations (20)

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

ε(ω)=1ωp2ω2jγω,
εeff(ω)=ε0,eff(1ωp,eff2ω2),
εS=11ω2μ0ε0λL2=1ωp2ω2,
ωp=1λLμ0ε0=cλL
λL(T)=λL(0)1(T/Tc)p,
ks=ωcns=1cω2ωp2,
Mcell=(ABCD)=Ds1DdPdDd1DsPs,
P=(eiϕ00eiϕ),
D=(11nn).(=s,d)
cos(Ka)=12(A+D),
cos(Ka)=cos(ωcdSεS)cos(ωcdDεr)12(εSεr+εrεS)sin(ωcdSεS)sin(ωcdDεr),
EPF=ωp,effa2πc=1rεr+(1r)(1εr)Λ.
ε(x,ω)=GNε˜(GN)ejGNx=ε˜(0)+GN0ε˜(GN)ejGNx,
ε˜(GN)=1a0aε(x)ejGxdx.
ε˜(0)=(1r)εS+rεr
ε˜(GN)=εsεrjGNa(1ejGNds).
εeff(ω)=ε(x,ω)=1a0adxε(xx,ω),=ε˜(0)+1aGN0ε˜(GN)0aejGN(xx)dx.
εeff(ω)=ε˜(0)=(1r)εS+rεr.
εeff(ω)=ε0,eff(1r)ω2c2λL2,
ωp,eff=1rεr+(1r)(1εr)cλL.

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