Abstract

An analysis of aligned nematic liquid crystal cells containing core-shell nanospheres shows that it is possible to devise a new type of metamaterial whose index of refraction is tunable from negative, through zero, to positive values. The design parameters for the constituents can be scaled for application in the optical as well as very long wavelength (e.g., terahertz and microwave) regions.

© 2006 Optical Society of America

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References

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  1. C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, Appl. Phys. Lett. 82, 2767 (2003).
    [CrossRef]
  2. G. V. Prakash, M. Kaczmarek, A. Dyadyusha, J. J. Baumberg, and G. D'Alessandro, Opt. Express 13, 2201 (2005).
    [CrossRef] [PubMed]
  3. P. Mach, P. Wiltzius, M. Megens, D. A. Weitz, K.-H. Lin, T. C. Lubensky, and A. G. Yodh, Phys. Rev. E 65, 031720 (2002).
    [CrossRef]
  4. T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, Opt. Express 11, 2589 (2003).
    [CrossRef] [PubMed]
  5. E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, Phys. Rev. B 72, 233105 (2005).
    [CrossRef]
  6. I. C. Khoo, Y. Williams, A. Diaz, K. Chen, J. Bossard, D. Werner, E. Graugnard, and C. J. Summers, Mol. Cryst. Liq. Cryst. 453, 309 (2006). See also Ref. .
    [CrossRef]
  7. J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. Drupp, and L. Li, IEEE Trans. Antennas Propag. 54, 1265 (2006).
    [CrossRef]
  8. R. Wangberg, J. Elser, E. Narimanov, and V. A. Podolskiy, J. Opt. Soc. Am. B 23, 498 (2006).
    [CrossRef]
  9. V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, Opt. Lett. 30, 3356 (2005).
    [CrossRef]
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    [CrossRef]
  11. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
    [CrossRef] [PubMed]
  12. M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, Phys. Rev. B 73, 045105 (2006).
    [CrossRef]
  13. G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials (Wiley, 2005).
    [CrossRef]
  14. R. W. Ziolkowski, Phys. Rev. E 70, 046608 (2004).
    [CrossRef]
  15. V. A. Fedotov, A. V. Rogacheva, N. I. Zheludev, P. L. Mladyonov, and S. L. Prosvirnin, Appl. Phys. Lett. 88, 091119 (2006).
    [CrossRef]
  16. M. A. Gingrich and D. H. Werner, Electron. Lett. 41, 1266 (2005).
    [CrossRef]
  17. I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, 1995).
  18. J. C. Maxwell Garnet, Philos. Trans. R. Soc. Ser. A 203, 385 (1904).
    [CrossRef]
  19. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2004).
  20. M. Schall, H. Helm, and S. R. Keiding, Int. J. Infrared Millim. Waves 20, 595 (1999). See also Ref. .
    [CrossRef]
  21. F. Yang and J. Roy Sambles, Liq. Cryst. 30, 599 (2003).
    [CrossRef]

2006

I. C. Khoo, Y. Williams, A. Diaz, K. Chen, J. Bossard, D. Werner, E. Graugnard, and C. J. Summers, Mol. Cryst. Liq. Cryst. 453, 309 (2006). See also Ref. .
[CrossRef]

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. Drupp, and L. Li, IEEE Trans. Antennas Propag. 54, 1265 (2006).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, Phys. Rev. B 73, 045105 (2006).
[CrossRef]

V. A. Fedotov, A. V. Rogacheva, N. I. Zheludev, P. L. Mladyonov, and S. L. Prosvirnin, Appl. Phys. Lett. 88, 091119 (2006).
[CrossRef]

R. Wangberg, J. Elser, E. Narimanov, and V. A. Podolskiy, J. Opt. Soc. Am. B 23, 498 (2006).
[CrossRef]

2005

M. A. Gingrich and D. H. Werner, Electron. Lett. 41, 1266 (2005).
[CrossRef]

E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, Phys. Rev. B 72, 233105 (2005).
[CrossRef]

G. V. Prakash, M. Kaczmarek, A. Dyadyusha, J. J. Baumberg, and G. D'Alessandro, Opt. Express 13, 2201 (2005).
[CrossRef] [PubMed]

V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, Opt. Lett. 30, 3356 (2005).
[CrossRef]

E. Schonbrun, M. Tinker, W. Park, and J.-B. Lee, IEEE Photonics Technol. Lett. 17, 1196 (2005).
[CrossRef]

2004

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
[CrossRef] [PubMed]

R. W. Ziolkowski, Phys. Rev. E 70, 046608 (2004).
[CrossRef]

2003

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, Appl. Phys. Lett. 82, 2767 (2003).
[CrossRef]

F. Yang and J. Roy Sambles, Liq. Cryst. 30, 599 (2003).
[CrossRef]

T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, Opt. Express 11, 2589 (2003).
[CrossRef] [PubMed]

2002

P. Mach, P. Wiltzius, M. Megens, D. A. Weitz, K.-H. Lin, T. C. Lubensky, and A. G. Yodh, Phys. Rev. E 65, 031720 (2002).
[CrossRef]

1999

M. Schall, H. Helm, and S. R. Keiding, Int. J. Infrared Millim. Waves 20, 595 (1999). See also Ref. .
[CrossRef]

1904

J. C. Maxwell Garnet, Philos. Trans. R. Soc. Ser. A 203, 385 (1904).
[CrossRef]

Aitchison, J. S.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, Phys. Rev. B 73, 045105 (2006).
[CrossRef]

Balmain, K. G.

G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials (Wiley, 2005).
[CrossRef]

Baumberg, J. J.

Bjarklev, A.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2004).

Bossard, J.

I. C. Khoo, Y. Williams, A. Diaz, K. Chen, J. Bossard, D. Werner, E. Graugnard, and C. J. Summers, Mol. Cryst. Liq. Cryst. 453, 309 (2006). See also Ref. .
[CrossRef]

Bossard, J. A.

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. Drupp, and L. Li, IEEE Trans. Antennas Propag. 54, 1265 (2006).
[CrossRef]

Broeng, J.

Cai, W. S.

Chen, K.

I. C. Khoo, Y. Williams, A. Diaz, K. Chen, J. Bossard, D. Werner, E. Graugnard, and C. J. Summers, Mol. Cryst. Liq. Cryst. 453, 309 (2006). See also Ref. .
[CrossRef]

Chettiar, U. K.

D'Alessandro, G.

Diaz, A.

I. C. Khoo, Y. Williams, A. Diaz, K. Chen, J. Bossard, D. Werner, E. Graugnard, and C. J. Summers, Mol. Cryst. Liq. Cryst. 453, 309 (2006). See also Ref. .
[CrossRef]

Drachev, V. P.

Drupp, R.

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. Drupp, and L. Li, IEEE Trans. Antennas Propag. 54, 1265 (2006).
[CrossRef]

Dyadyusha, A.

Eleftheriades, G. V.

G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials (Wiley, 2005).
[CrossRef]

Elser, J.

Fedotov, V. A.

V. A. Fedotov, A. V. Rogacheva, N. I. Zheludev, P. L. Mladyonov, and S. L. Prosvirnin, Appl. Phys. Lett. 88, 091119 (2006).
[CrossRef]

Forchel, A.

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, Appl. Phys. Lett. 82, 2767 (2003).
[CrossRef]

Gingrich, M. A.

M. A. Gingrich and D. H. Werner, Electron. Lett. 41, 1266 (2005).
[CrossRef]

Graugnard, E.

I. C. Khoo, Y. Williams, A. Diaz, K. Chen, J. Bossard, D. Werner, E. Graugnard, and C. J. Summers, Mol. Cryst. Liq. Cryst. 453, 309 (2006). See also Ref. .
[CrossRef]

E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, Phys. Rev. B 72, 233105 (2005).
[CrossRef]

Helm, H.

M. Schall, H. Helm, and S. R. Keiding, Int. J. Infrared Millim. Waves 20, 595 (1999). See also Ref. .
[CrossRef]

Hermann, D. S.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2004).

Jain, S.

E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, Phys. Rev. B 72, 233105 (2005).
[CrossRef]

Kaczmarek, M.

Kamp, M.

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, Appl. Phys. Lett. 82, 2767 (2003).
[CrossRef]

Keiding, S. R.

M. Schall, H. Helm, and S. R. Keiding, Int. J. Infrared Millim. Waves 20, 595 (1999). See also Ref. .
[CrossRef]

Khoo, I. C.

I. C. Khoo, Y. Williams, A. Diaz, K. Chen, J. Bossard, D. Werner, E. Graugnard, and C. J. Summers, Mol. Cryst. Liq. Cryst. 453, 309 (2006). See also Ref. .
[CrossRef]

E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, Phys. Rev. B 72, 233105 (2005).
[CrossRef]

I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, 1995).

Kildishev, A. V.

King, J. S.

E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, Phys. Rev. B 72, 233105 (2005).
[CrossRef]

Klopf, F.

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, Appl. Phys. Lett. 82, 2767 (2003).
[CrossRef]

Larsen, T. T.

Lee, J.-B.

E. Schonbrun, M. Tinker, W. Park, and J.-B. Lee, IEEE Photonics Technol. Lett. 17, 1196 (2005).
[CrossRef]

Li, L.

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. Drupp, and L. Li, IEEE Trans. Antennas Propag. 54, 1265 (2006).
[CrossRef]

Lin, K.-H.

P. Mach, P. Wiltzius, M. Megens, D. A. Weitz, K.-H. Lin, T. C. Lubensky, and A. G. Yodh, Phys. Rev. E 65, 031720 (2002).
[CrossRef]

Lubensky, T. C.

P. Mach, P. Wiltzius, M. Megens, D. A. Weitz, K.-H. Lin, T. C. Lubensky, and A. G. Yodh, Phys. Rev. E 65, 031720 (2002).
[CrossRef]

Mach, P.

P. Mach, P. Wiltzius, M. Megens, D. A. Weitz, K.-H. Lin, T. C. Lubensky, and A. G. Yodh, Phys. Rev. E 65, 031720 (2002).
[CrossRef]

Maxwell Garnet, J. C.

J. C. Maxwell Garnet, Philos. Trans. R. Soc. Ser. A 203, 385 (1904).
[CrossRef]

Mayer, T. S.

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. Drupp, and L. Li, IEEE Trans. Antennas Propag. 54, 1265 (2006).
[CrossRef]

Megens, M.

P. Mach, P. Wiltzius, M. Megens, D. A. Weitz, K.-H. Lin, T. C. Lubensky, and A. G. Yodh, Phys. Rev. E 65, 031720 (2002).
[CrossRef]

Mladyonov, P. L.

V. A. Fedotov, A. V. Rogacheva, N. I. Zheludev, P. L. Mladyonov, and S. L. Prosvirnin, Appl. Phys. Lett. 88, 091119 (2006).
[CrossRef]

Mojahedi, M.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, Phys. Rev. B 73, 045105 (2006).
[CrossRef]

Narimanov, E.

Park, W.

E. Schonbrun, M. Tinker, W. Park, and J.-B. Lee, IEEE Photonics Technol. Lett. 17, 1196 (2005).
[CrossRef]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
[CrossRef] [PubMed]

Podolskiy, V. A.

Prakash, G. V.

Prosvirnin, S. L.

V. A. Fedotov, A. V. Rogacheva, N. I. Zheludev, P. L. Mladyonov, and S. L. Prosvirnin, Appl. Phys. Lett. 88, 091119 (2006).
[CrossRef]

Reithmaier, J. P.

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, Appl. Phys. Lett. 82, 2767 (2003).
[CrossRef]

Rogacheva, A. V.

V. A. Fedotov, A. V. Rogacheva, N. I. Zheludev, P. L. Mladyonov, and S. L. Prosvirnin, Appl. Phys. Lett. 88, 091119 (2006).
[CrossRef]

Roy Sambles, J.

F. Yang and J. Roy Sambles, Liq. Cryst. 30, 599 (2003).
[CrossRef]

Sarychev, A. K.

Schall, M.

M. Schall, H. Helm, and S. R. Keiding, Int. J. Infrared Millim. Waves 20, 595 (1999). See also Ref. .
[CrossRef]

Schonbrun, E.

E. Schonbrun, M. Tinker, W. Park, and J.-B. Lee, IEEE Photonics Technol. Lett. 17, 1196 (2005).
[CrossRef]

Schuller, C.

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, Appl. Phys. Lett. 82, 2767 (2003).
[CrossRef]

Shalaev, V. M.

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
[CrossRef] [PubMed]

Smith, J. A.

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. Drupp, and L. Li, IEEE Trans. Antennas Propag. 54, 1265 (2006).
[CrossRef]

Summers, C. J.

I. C. Khoo, Y. Williams, A. Diaz, K. Chen, J. Bossard, D. Werner, E. Graugnard, and C. J. Summers, Mol. Cryst. Liq. Cryst. 453, 309 (2006). See also Ref. .
[CrossRef]

E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, Phys. Rev. B 72, 233105 (2005).
[CrossRef]

Tang, Y. U.

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. Drupp, and L. Li, IEEE Trans. Antennas Propag. 54, 1265 (2006).
[CrossRef]

Tinker, M.

E. Schonbrun, M. Tinker, W. Park, and J.-B. Lee, IEEE Photonics Technol. Lett. 17, 1196 (2005).
[CrossRef]

Wangberg, R.

Weitz, D. A.

P. Mach, P. Wiltzius, M. Megens, D. A. Weitz, K.-H. Lin, T. C. Lubensky, and A. G. Yodh, Phys. Rev. E 65, 031720 (2002).
[CrossRef]

Werner, D.

I. C. Khoo, Y. Williams, A. Diaz, K. Chen, J. Bossard, D. Werner, E. Graugnard, and C. J. Summers, Mol. Cryst. Liq. Cryst. 453, 309 (2006). See also Ref. .
[CrossRef]

Werner, D. H.

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. Drupp, and L. Li, IEEE Trans. Antennas Propag. 54, 1265 (2006).
[CrossRef]

M. A. Gingrich and D. H. Werner, Electron. Lett. 41, 1266 (2005).
[CrossRef]

Wheeler, M. S.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, Phys. Rev. B 73, 045105 (2006).
[CrossRef]

Williams, Y.

I. C. Khoo, Y. Williams, A. Diaz, K. Chen, J. Bossard, D. Werner, E. Graugnard, and C. J. Summers, Mol. Cryst. Liq. Cryst. 453, 309 (2006). See also Ref. .
[CrossRef]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
[CrossRef] [PubMed]

Wiltzius, P.

P. Mach, P. Wiltzius, M. Megens, D. A. Weitz, K.-H. Lin, T. C. Lubensky, and A. G. Yodh, Phys. Rev. E 65, 031720 (2002).
[CrossRef]

Yang, F.

F. Yang and J. Roy Sambles, Liq. Cryst. 30, 599 (2003).
[CrossRef]

Yodh, A. G.

P. Mach, P. Wiltzius, M. Megens, D. A. Weitz, K.-H. Lin, T. C. Lubensky, and A. G. Yodh, Phys. Rev. E 65, 031720 (2002).
[CrossRef]

Yuan, H. K.

Zhang-Williams, Y.

E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, Phys. Rev. B 72, 233105 (2005).
[CrossRef]

Zheludev, N. I.

V. A. Fedotov, A. V. Rogacheva, N. I. Zheludev, P. L. Mladyonov, and S. L. Prosvirnin, Appl. Phys. Lett. 88, 091119 (2006).
[CrossRef]

Ziolkowski, R. W.

R. W. Ziolkowski, Phys. Rev. E 70, 046608 (2004).
[CrossRef]

Appl. Phys. Lett.

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, Appl. Phys. Lett. 82, 2767 (2003).
[CrossRef]

V. A. Fedotov, A. V. Rogacheva, N. I. Zheludev, P. L. Mladyonov, and S. L. Prosvirnin, Appl. Phys. Lett. 88, 091119 (2006).
[CrossRef]

Electron. Lett.

M. A. Gingrich and D. H. Werner, Electron. Lett. 41, 1266 (2005).
[CrossRef]

IEEE Photonics Technol. Lett.

E. Schonbrun, M. Tinker, W. Park, and J.-B. Lee, IEEE Photonics Technol. Lett. 17, 1196 (2005).
[CrossRef]

IEEE Trans. Antennas Propag.

J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. Drupp, and L. Li, IEEE Trans. Antennas Propag. 54, 1265 (2006).
[CrossRef]

Int. J. Infrared Millim. Waves

M. Schall, H. Helm, and S. R. Keiding, Int. J. Infrared Millim. Waves 20, 595 (1999). See also Ref. .
[CrossRef]

J. Opt. Soc. Am. B

Liq. Cryst.

F. Yang and J. Roy Sambles, Liq. Cryst. 30, 599 (2003).
[CrossRef]

Mol. Cryst. Liq. Cryst.

I. C. Khoo, Y. Williams, A. Diaz, K. Chen, J. Bossard, D. Werner, E. Graugnard, and C. J. Summers, Mol. Cryst. Liq. Cryst. 453, 309 (2006). See also Ref. .
[CrossRef]

Opt. Express

Opt. Lett.

Philos. Trans. R. Soc. Ser. A

J. C. Maxwell Garnet, Philos. Trans. R. Soc. Ser. A 203, 385 (1904).
[CrossRef]

Phys. Rev. B

E. Graugnard, J. S. King, S. Jain, C. J. Summers, Y. Zhang-Williams, and I. C. Khoo, Phys. Rev. B 72, 233105 (2005).
[CrossRef]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, Phys. Rev. B 73, 045105 (2006).
[CrossRef]

Phys. Rev. E

P. Mach, P. Wiltzius, M. Megens, D. A. Weitz, K.-H. Lin, T. C. Lubensky, and A. G. Yodh, Phys. Rev. E 65, 031720 (2002).
[CrossRef]

R. W. Ziolkowski, Phys. Rev. E 70, 046608 (2004).
[CrossRef]

Science

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).
[CrossRef] [PubMed]

Other

G. V. Eleftheriades and K. G. Balmain, Negative-Refraction Metamaterials (Wiley, 2005).
[CrossRef]

I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, 1995).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2004).

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Figures (3)

Fig. 1
Fig. 1

Homogeneous or planar aligned nematic liquid crystal containing a core-shell (coated) nanospheres sample. An applied ac electric field or a magnetic field can switch the director axis of the liquid crystal from parallel to perpendicular to the cell windows. (b) Cross section of the core [ μ 1 , ϵ 1 ] -shell [ μ 2 , ϵ 2 ] nanosphere with the liquid crystal host [ μ 3 , ϵ 3 ] .

Fig. 2
Fig. 2

Calculated real and imaginary part of the complex effective (a) permittivity, (b) permeability, and (c) refractive index of the NDLC as a function of liquid crystal host permittivity in the optical frequency interval between 50 and 150 THz (wavelength between 6 and 2 μ m ).

Fig. 3
Fig. 3

Calculated real and imaginary part of the complex refractive index of the NDLC as a function of liquid crystal host permittivity in the terahertz region between 3.33 and 4.29 THz (wavelength between 90 and 70 μ m ).

Equations (6)

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

ϵ 1 = ϵ ( ) ( 1 + ω L 2 ω T 2 ω T 2 ω 2 i ω γ 1 ) ,
ϵ 2 = 1 ω p 2 ω 2 + i ω γ 2 ,
ϵ 3 = ϵ ϵ ϵ cos 2 θ + ϵ sin 2 θ ,
ϵ r eff = ϵ 3 ( k 3 3 + j 4 π N a 1 k 3 3 j 2 π N a 1 ) ,
μ r eff = k 3 3 + j 4 π N b 1 k 3 3 j 2 π N b 1 ,
k 3 = ϵ 3 k 0 = ( ϵ ϵ ϵ cos 2 θ + ϵ sin 2 θ ) 1 2 k 0 .

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