Abstract

In our common understanding, for strong absorption or amplification in a slab structure, the desire of reducing the slab thickness seems contradictory to the condition of small loss or gain. In this paper, this common understanding is challenged. It is shown that an arbitrarily thin metamaterial layer can perfectly absorb or giantly amplify an incident plane wave at a critical angle when the real parts of the permittivity and permeability of the metamaterial are zero while the absolute imaginary parts can be arbitrarily small. The metamaterial layer needs a totally reflective substrate for perfect absorption, while this is not required for giant magnification. Detailed analysis for the existence of the critical angle and physical explanation for these abnormal phenomena are given.

© 2011 OSA

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References

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  1. E. F. Knott, J. F. Schaeffer, and M. T. Tuley, Radar Cross Section (Artech House, 1993).
  2. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
    [CrossRef] [PubMed]
  3. H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
    [CrossRef] [PubMed]
  4. Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
    [CrossRef]
  5. X. L. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett. 104(20), 207403 (2010).
    [CrossRef] [PubMed]
  6. S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89(21), 213902 (2002).
    [CrossRef] [PubMed]
  7. 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 Pt 2), 046608 (2004).
    [CrossRef] [PubMed]
  8. L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Y. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97(13), 131107 (2010).
    [CrossRef]
  9. 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] [PubMed]
  10. A. Ciattoni, C. Rizza, and E. Palange, “Transmissivity directional hysteresis of a nonlinear metamaterial slab with very small linear permittivity,” Opt. Lett. 35(13), 2130–2132 (2010).
    [CrossRef] [PubMed]
  11. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
    [CrossRef] [PubMed]
  12. Y. Wu, J. S. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
    [CrossRef]
  13. M. Ö. Oktel and Ö. E. Müstecaplıoğlu, “Electromagnetically induced left-handedness in a dense gas of three-level atoms,” Phys. Rev. A 70(5), 053806 (2004).
    [CrossRef]
  14. S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
    [CrossRef] [PubMed]
  15. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2(6), 351–354 (2008).
    [CrossRef]

2010 (4)

X. L. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett. 104(20), 207403 (2010).
[CrossRef] [PubMed]

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Y. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97(13), 131107 (2010).
[CrossRef]

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

A. Ciattoni, C. Rizza, and E. Palange, “Transmissivity directional hysteresis of a nonlinear metamaterial slab with very small linear permittivity,” Opt. Lett. 35(13), 2130–2132 (2010).
[CrossRef] [PubMed]

2009 (1)

Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
[CrossRef]

2008 (3)

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2(6), 351–354 (2008).
[CrossRef]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
[CrossRef] [PubMed]

2006 (2)

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] [PubMed]

Y. Wu, J. S. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[CrossRef]

2004 (2)

M. Ö. Oktel and Ö. E. Müstecaplıoğlu, “Electromagnetically induced left-handedness in a dense gas of three-level atoms,” Phys. Rev. A 70(5), 053806 (2004).
[CrossRef]

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 Pt 2), 046608 (2004).
[CrossRef] [PubMed]

2002 (1)

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89(21), 213902 (2002).
[CrossRef] [PubMed]

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Alekseyev, L. V.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Y. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97(13), 131107 (2010).
[CrossRef]

Averitt, R. D.

Avitzour, Y.

Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
[CrossRef]

Barnakov, Y. A.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Y. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97(13), 131107 (2010).
[CrossRef]

Bingham, C. M.

Chan, C. T.

Y. Wu, J. S. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[CrossRef]

Chettiar, U. K.

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Ciattoni, A.

Drachev, V. P.

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Engheta, N.

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] [PubMed]

Enoch, S.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89(21), 213902 (2002).
[CrossRef] [PubMed]

Fedotov, V. A.

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2(6), 351–354 (2008).
[CrossRef]

Guérin, N.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89(21), 213902 (2002).
[CrossRef] [PubMed]

Kildishev, A. V.

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Landy, N. I.

Li, H.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Y. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97(13), 131107 (2010).
[CrossRef]

Li, J. S.

Y. Wu, J. S. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[CrossRef]

Liu, X. L.

X. L. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett. 104(20), 207403 (2010).
[CrossRef] [PubMed]

Mock, J. J.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

Müstecaplioglu, Ö. E.

M. Ö. Oktel and Ö. E. Müstecaplıoğlu, “Electromagnetically induced left-handedness in a dense gas of three-level atoms,” Phys. Rev. A 70(5), 053806 (2004).
[CrossRef]

Narimanov, E. E.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Y. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97(13), 131107 (2010).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Ni, X. J.

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Noginov, M. A.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Y. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97(13), 131107 (2010).
[CrossRef]

Oktel, M. Ö.

M. Ö. Oktel and Ö. E. Müstecaplıoğlu, “Electromagnetically induced left-handedness in a dense gas of three-level atoms,” Phys. Rev. A 70(5), 053806 (2004).
[CrossRef]

Padilla, W. J.

X. L. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett. 104(20), 207403 (2010).
[CrossRef] [PubMed]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

H. Tao, N. I. Landy, C. M. Bingham, X. Zhang, R. D. Averitt, and W. J. Padilla, “A metamaterial absorber for the terahertz regime: design, fabrication and characterization,” Opt. Express 16(10), 7181–7188 (2008).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Palange, E.

Papasimakis, N.

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2(6), 351–354 (2008).
[CrossRef]

Prosvirnin, S. L.

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2(6), 351–354 (2008).
[CrossRef]

Rizza, C.

Sabouroux, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89(21), 213902 (2002).
[CrossRef] [PubMed]

Sajuyigbe, S.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Shalaev, V. M.

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Shvets, G.

Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
[CrossRef]

Silveirinha, M. G.

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] [PubMed]

Smith, D. R.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Starr, A. F.

X. L. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett. 104(20), 207403 (2010).
[CrossRef] [PubMed]

Starr, T.

X. L. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett. 104(20), 207403 (2010).
[CrossRef] [PubMed]

Tao, H.

Tayeb, G.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89(21), 213902 (2002).
[CrossRef] [PubMed]

Tumkur, T.

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Y. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97(13), 131107 (2010).
[CrossRef]

Urzhumov, Y. A.

Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

Vincent, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89(21), 213902 (2002).
[CrossRef] [PubMed]

Wu, Y.

Y. Wu, J. S. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[CrossRef]

Xiao, S. M.

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Yuan, H. K.

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Zhang, X.

Zhang, Z. Q.

Y. Wu, J. S. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[CrossRef]

Zheludev, N. I.

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2(6), 351–354 (2008).
[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 Pt 2), 046608 (2004).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

L. V. Alekseyev, E. E. Narimanov, T. Tumkur, H. Li, Y. A. Barnakov, and M. A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97(13), 131107 (2010).
[CrossRef]

Nat. Photonics (1)

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2(6), 351–354 (2008).
[CrossRef]

Nature (1)

S. M. Xiao, V. P. Drachev, A. V. Kildishev, X. J. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466(7307), 735–738 (2010).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. A (1)

M. Ö. Oktel and Ö. E. Müstecaplıoğlu, “Electromagnetically induced left-handedness in a dense gas of three-level atoms,” Phys. Rev. A 70(5), 053806 (2004).
[CrossRef]

Phys. Rev. B (2)

Y. Wu, J. S. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[CrossRef]

Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (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 Pt 2), 046608 (2004).
[CrossRef] [PubMed]

Phys. Rev. Lett. (5)

X. L. Liu, T. Starr, A. F. Starr, and W. J. Padilla, “Infrared spatial and frequency selective metamaterial with near-unity absorbance,” Phys. Rev. Lett. 104(20), 207403 (2010).
[CrossRef] [PubMed]

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89(21), 213902 (2002).
[CrossRef] [PubMed]

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] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[CrossRef] [PubMed]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[CrossRef] [PubMed]

Other (1)

E. F. Knott, J. F. Schaeffer, and M. T. Tuley, Radar Cross Section (Artech House, 1993).

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Figures (4)

Fig. 1
Fig. 1

(a) Configuration for a slab of ZRM sandwiched between two semi-infinite layers. (b) Wave decomposition of the reflected or transmitted field.

Fig. 2
Fig. 2

Reflectivity and field distributions of a slab with a PEC substrate. In (a), d 1 = 0.1λ 0 for curves 1−4 and d 1 = λ 0 for curve 5, and (ε 1/ ε 0, μ 1/ μ 0) has small values of i(0.3, 0.3), i(0.5, 0.3), i(0.3, 0.1), i(0.01, 0.3), and i(0.1, 0.3) for curves 1−5, respectively. In (b), d 1 = 0.1λ 0, and (ε 1/ ε 0, μ 1/ μ 0) has large values of i(20, 20), i(20, 10), and i(10, 20) for curves 1−3, respectively. In (c)−(e), d 1 = 0.4λ 0, (ε 1/ ε 0, μ 1/ μ 0) = i(0.3, 0.1).

Fig. 3
Fig. 3

Reflectivity and transmissivity of a slab. In (a), d 1 = 0.1λ 0, (ε 1/ ε 0, μ 1/ μ 0) = (−0.3i, −0.3i), and layer 2 is a PEC for curve 1 and of free space for curves 2 and 3. In (b), d 1 = 0.1λ 0 for curve 1 and d 1 = 0.5λ 0 for curves 2−4, (ε 1/ ε 0, μ 1/ μ 0) = (−0.1i, 0.3i), and layer 2 is a PEC for curves 1 and 2 and of free space for curves 3 and 4.

Fig. 4
Fig. 4

Reflectivity of a slab with a PEC substrate when real(ε 1) and real(μ 1) deviate form zero. In (a), d 1 = 0.1λ 0, and (ε 1/ ε 0, μ 1/ μ 0) is (3i, 3i), (0.05 + 3i, 0.05 + 3i), (0.05 + 3i, 3i) and (3i, 0.05 + 3i) for curves 1−4, respectively. In (b), d 1 = 0.5λ 0, and (ε 1/ ε 0, μ 1/ μ 0) is (−0.3i, 0.3i), (0.05−0.3i, 0.05 + 0.3i), (0.05−0.3i, 0.3i) and (−0.3i, 0.05 + 0.3i) for curves 1−4, respectively.

Equations (8)

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{ H n , z ( r ) = ( H n + e i k n , x x + H n e i k n , x x ) e i k y y E n , x ( r ) = ( k y / ω ε n ) ( H n + e i k n , x x + H n e i k n , x x ) e i k y y , E n , y ( r ) = ( k n , x / ω ε n ) ( H n + e i k n , x x H n e i k n , x x ) e i k y y
R l o s s = H 0 / H 0 + = [ 1/tanh( γ d 1 ) γ / ε 1 , r " k 0 , x ] / [ 1/tanh( γ d 1 ) + γ / ε 1 , r " k 0 , x ] ,
tanh ( ε 1 , r " μ 1 , r " k 0 d 1 ) ε 1 , r " / μ 1 , r " = 0.
R l o s s = r 0 , 1 + t 0 , 1 t 1 , 0 [ e 2 γ d 1 + r 1 , 0 e 2 ( 2 γ d 1 ) + r 1 , 0 2 e 3 ( 2 γ d 1 ) + ... ] ,
R h y b r i d = [ctan ( k 1 , x d 1 ) k 1 , x / k 0 , x ε 1 , r "]/[ctan ( k 1 , x d 1 ) + k 1 , x / k 0 , x ε 1 , r "]  (when  k y 2 ε 1 , r " μ 1 , r " k 0 2 ),
R h y b r i d = [ 1/tanh( β d 1 ) + β / k 0 , x ε 1 , r "]/[1/tanh( β d 1 ) β / k 0 , x ε 1 , r "]  (when  k y 2 > ε 1 , r " μ 1 , r " k 0 2 ),
R g a i n = [ γ / k 0 , x ε 1 , r " k 0 , x ε 1 , r " / γ ] / [ 2 / tanh( γ d 1 ) ( γ / k 0 , x ε 1 , r " + k 0 , x ε 1 , r " / γ ) ] ,
T g a i n = H 2 + / H 0 + = [ 4 / ( e γ d 1 e γ d 1 ) ] / [ 2 / tanh( γ d 1 ) ( γ / k 0 , x ε 1 , r " + k 0 , x ε 1 , r " / γ ) ] .

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