Abstract

Recently, several experimental results verified the tunneling effect theory of that the electromagnetic energy can be squeezed through an ultra-narrow channel filled with epsilon-near-zero (ENZ) medium. However, the energy squeezing can be only achieved in a narrow region. Here, we present a full-wave simulation of the tunneling effect in multilayered channels full of thin ENZ metamaterials with different plasma frequencies. Thin metallic wires arrays with different radiuses are employed to form these effective ENZ media, whose plasma frequencies are different. The appearance of several passbands in the transmission curve verifies that multi-passband energy tunneling effect can be implemented by multilayer ENZ channels. There are two possible reasons for these peaks, one is the ENZ tunneling effect, and the other is the Fabry-Pérot resonance. For each transmission peak corresponding two-spatial maps of electric field are given, in order to distinguish the causes.

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References

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  1. R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046608 (2004).
    [CrossRef]
  2. M. G. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75(7), 075119 (2007).
    [CrossRef]
  3. M. G. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
    [CrossRef]
  4. M. G. Silveirinha and N. Engheta, “Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials,” Phys. Rev. B 76(24), 245109 (2007).
    [CrossRef]
  5. Q. Cheng, R. Liu, D. Huang, T. J. Cui, and D. R. Smith, “Circuit verification of tunneling effect in zero permittivity medium,” Appl. Phys. Lett. 91(23), 234105 (2007).
    [CrossRef]
  6. R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100(2), 023903 (2008).
    [CrossRef]
  7. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
    [CrossRef]
  8. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. 76(25), 4773 (1996).
    [CrossRef]
  9. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. Condens. Matter 10(22), 4785–4809 (1998).
    [CrossRef]
  10. D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 036617 (2005).
    [CrossRef]
  11. 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(19), 195104 (2002).
    [CrossRef]
  12. M. David, Pozar, “transmission lines and waveguides,” in Microwave Engineering (John Wiley & Sons, New York, 2004).
  13. M. G. Silveirinha and C. A. Fernandes, “Homogenization of 3-D-connected and nonconnected wire metamaterials,” IEEE Trans. Microw. Theory Tech. 53(4), 1418–1430 (2005).
    [CrossRef]
  14. J. Hupert, “Evanescent Mode Guide Filter and Tunnel-Effect Analogy,” IEEE Trans. Circ. Syst. 15, 279–280 (1968).
  15. G. Craven, “Waveguide bandpass filters using evanescent modes,” Electron. Lett. 2(7), 251–252 (1966).
    [CrossRef]

2008 (2)

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100(2), 023903 (2008).
[CrossRef]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[CrossRef]

2007 (3)

M. G. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75(7), 075119 (2007).
[CrossRef]

M. G. Silveirinha and N. Engheta, “Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials,” Phys. Rev. B 76(24), 245109 (2007).
[CrossRef]

Q. Cheng, R. Liu, D. Huang, T. J. Cui, and D. R. Smith, “Circuit verification of tunneling effect in zero permittivity medium,” Appl. Phys. Lett. 91(23), 234105 (2007).
[CrossRef]

2006 (1)

M. G. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[CrossRef]

2005 (2)

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 036617 (2005).
[CrossRef]

M. G. Silveirinha and C. A. Fernandes, “Homogenization of 3-D-connected and nonconnected wire metamaterials,” IEEE Trans. Microw. Theory Tech. 53(4), 1418–1430 (2005).
[CrossRef]

2004 (1)

R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046608 (2004).
[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(19), 195104 (2002).
[CrossRef]

1998 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. Condens. Matter 10(22), 4785–4809 (1998).
[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(25), 4773 (1996).
[CrossRef]

1968 (1)

J. Hupert, “Evanescent Mode Guide Filter and Tunnel-Effect Analogy,” IEEE Trans. Circ. Syst. 15, 279–280 (1968).

1966 (1)

G. Craven, “Waveguide bandpass filters using evanescent modes,” Electron. Lett. 2(7), 251–252 (1966).
[CrossRef]

Alù, A.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[CrossRef]

Cheng, Q.

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100(2), 023903 (2008).
[CrossRef]

Q. Cheng, R. Liu, D. Huang, T. J. Cui, and D. R. Smith, “Circuit verification of tunneling effect in zero permittivity medium,” Appl. Phys. Lett. 91(23), 234105 (2007).
[CrossRef]

Craven, G.

G. Craven, “Waveguide bandpass filters using evanescent modes,” Electron. Lett. 2(7), 251–252 (1966).
[CrossRef]

Cui, T. J.

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100(2), 023903 (2008).
[CrossRef]

Q. Cheng, R. Liu, D. Huang, T. J. Cui, and D. R. Smith, “Circuit verification of tunneling effect in zero permittivity medium,” Appl. Phys. Lett. 91(23), 234105 (2007).
[CrossRef]

Cummer, S. A.

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100(2), 023903 (2008).
[CrossRef]

Edwards, B.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[CrossRef]

Engheta, N.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[CrossRef]

M. G. Silveirinha and N. Engheta, “Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials,” Phys. Rev. B 76(24), 245109 (2007).
[CrossRef]

M. G. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75(7), 075119 (2007).
[CrossRef]

M. G. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[CrossRef]

Fernandes, C. A.

M. G. Silveirinha and C. A. Fernandes, “Homogenization of 3-D-connected and nonconnected wire metamaterials,” IEEE Trans. Microw. Theory Tech. 53(4), 1418–1430 (2005).
[CrossRef]

Hand, T.

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100(2), 023903 (2008).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. Condens. Matter 10(22), 4785–4809 (1998).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. 76(25), 4773 (1996).
[CrossRef]

Huang, D.

Q. Cheng, R. Liu, D. Huang, T. J. Cui, and D. R. Smith, “Circuit verification of tunneling effect in zero permittivity medium,” Appl. Phys. Lett. 91(23), 234105 (2007).
[CrossRef]

Hupert, J.

J. Hupert, “Evanescent Mode Guide Filter and Tunnel-Effect Analogy,” IEEE Trans. Circ. Syst. 15, 279–280 (1968).

Koschny, Th.

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 036617 (2005).
[CrossRef]

Liu, R.

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100(2), 023903 (2008).
[CrossRef]

Q. Cheng, R. Liu, D. Huang, T. J. Cui, and D. R. Smith, “Circuit verification of tunneling effect in zero permittivity medium,” Appl. Phys. Lett. 91(23), 234105 (2007).
[CrossRef]

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(19), 195104 (2002).
[CrossRef]

Mock, J. J.

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100(2), 023903 (2008).
[CrossRef]

Pendry, J. B.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. Condens. Matter 10(22), 4785–4809 (1998).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. 76(25), 4773 (1996).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. Condens. Matter 10(22), 4785–4809 (1998).
[CrossRef]

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(19), 195104 (2002).
[CrossRef]

Silveirinha, M.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[CrossRef]

Silveirinha, M. G.

M. G. Silveirinha and N. Engheta, “Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials,” Phys. Rev. B 76(24), 245109 (2007).
[CrossRef]

M. G. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75(7), 075119 (2007).
[CrossRef]

M. G. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[CrossRef]

M. G. Silveirinha and C. A. Fernandes, “Homogenization of 3-D-connected and nonconnected wire metamaterials,” IEEE Trans. Microw. Theory Tech. 53(4), 1418–1430 (2005).
[CrossRef]

Smith, D. R.

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100(2), 023903 (2008).
[CrossRef]

Q. Cheng, R. Liu, D. Huang, T. J. Cui, and D. R. Smith, “Circuit verification of tunneling effect in zero permittivity medium,” Appl. Phys. Lett. 91(23), 234105 (2007).
[CrossRef]

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 036617 (2005).
[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(19), 195104 (2002).
[CrossRef]

Soukoulis, C. M.

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 036617 (2005).
[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(19), 195104 (2002).
[CrossRef]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. Condens. Matter 10(22), 4785–4809 (1998).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. 76(25), 4773 (1996).
[CrossRef]

Vier, D. C.

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 036617 (2005).
[CrossRef]

Young, M. E.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (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(25), 4773 (1996).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046608 (2004).
[CrossRef]

Appl. Phys. Lett. (1)

Q. Cheng, R. Liu, D. Huang, T. J. Cui, and D. R. Smith, “Circuit verification of tunneling effect in zero permittivity medium,” Appl. Phys. Lett. 91(23), 234105 (2007).
[CrossRef]

Electron. Lett. (1)

G. Craven, “Waveguide bandpass filters using evanescent modes,” Electron. Lett. 2(7), 251–252 (1966).
[CrossRef]

IEEE Trans. Circ. Syst. (1)

J. Hupert, “Evanescent Mode Guide Filter and Tunnel-Effect Analogy,” IEEE Trans. Circ. Syst. 15, 279–280 (1968).

IEEE Trans. Microw. Theory Tech. (1)

M. G. Silveirinha and C. A. Fernandes, “Homogenization of 3-D-connected and nonconnected wire metamaterials,” IEEE Trans. Microw. Theory Tech. 53(4), 1418–1430 (2005).
[CrossRef]

J. Phys. Condens. Matter (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low Frequency Plasmons in Thin Wire Structures,” J. Phys. Condens. Matter 10(22), 4785–4809 (1998).
[CrossRef]

Phys. Rev. B (3)

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(19), 195104 (2002).
[CrossRef]

M. G. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75(7), 075119 (2007).
[CrossRef]

M. G. Silveirinha and N. Engheta, “Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials,” Phys. Rev. B 76(24), 245109 (2007).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4), 046608 (2004).
[CrossRef]

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(33 Pt 2B), 036617 (2005).
[CrossRef]

Phys. Rev. Lett. (4)

M. G. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[CrossRef]

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100(2), 023903 (2008).
[CrossRef]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. 76(25), 4773 (1996).
[CrossRef]

Other (1)

M. David, Pozar, “transmission lines and waveguides,” in Microwave Engineering (John Wiley & Sons, New York, 2004).

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Figures (3)

Fig. 1.
Fig. 1.

(a) Schematic illustration of the simulation structure, consisting of four waveguide sections. Input and output waveguide connecting by two E-direction narrow waveguides filled with ENZ metamaterials. (b) Cross-sectional schematic of the setup. (c) The effective permittivity of the wire medium filling in narrow waveguides (inset: the unit cell of wire media). The ENZ media filling in the channel A is composed of Roges 5880 (ε=2.2) and arranged copper wires with radium r 1=0.3mm and plasma frequency f 0=8.68GHz, then the effective permittivity is shown by blue curve in (c). The red curve is for media r 2=0.5mm and f 0=10.06GHz, filling in the lower channel.

Fig. 2.
Fig. 2.

Simulated transmission curves for multi-layered (red), r 1=0.3 single-layered (green) and r 2=0.5 single-layered (blue).

Fig. 3.
Fig. 3.

(color online) Two-dimension mapper and the field distribution of electric vector results (a) and (f) Field distribution of tunneling effect for peak 1 at 8.49GHz. (b) and (g) Field distribution of Fabry-Pérot-like oscillation for peak 1 at 8.79GHz. (c) and (h) Field distribution for peak 2 at 9.89GHz. (d) and (i) Field distribution of Fabry-Pérot-like oscillation for peak 3 at 10.5GHz. (e) and (i) Field distribution of Fabry-Pérot-like oscillation for peak 4 at 11.5GHz.

Equations (4)

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

β=(2πnfc)2(πw)2 ,
n=1kdcos1[12S21(1S112+S212))],
z=± (1+S11)2S212(1S11)2S212 ,
ε=nz,μ=n z ,

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