Abstract

We use the theory of inhomogeneous waves to study the transmission of light in μ-near-zero metamaterials. We find the effect of all-angle collimation of incident light, which means that the vector of energy flow in a wave transmitted to a μ-near-zero metamaterial is perpendicular to the interface for any incident angles if an incident wave is s-polarized. This effect is similar to the all-angle collimation of incident light recently found through a different theoretical framework in ε-near-zero metamaterials for a p-polarized incident wave [S. Feng, Phys. Rev. Lett. 108, 193904 (2012)]. To provide a specific example, we consider the transmission of light in a negative-index metamaterial in the spectral region with a permeability resonance, and show that all-angle collimation indeed takes place at the wavelength for which the real part of permeability is vanishingly small.

© 2013 Optical Society of America

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References

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  1. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [Crossref] [PubMed]
  2. H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
    [Crossref] [PubMed]
  3. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
    [Crossref] [PubMed]
  4. S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
    [Crossref] [PubMed]
  5. S. Feng, “Loss-induced omnidirectional bending to the normal in ε-near-zero metamaterials,” Phys. Rev. Lett. 108,193904 (2012).
    [Crossref]
  6. F. I. Fedorov, Optics of Anisotropic Media (URSS, 2004), (in Russian).
  7. H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, 1983).
  8. C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. 106, 067402 (2011).
    [Crossref] [PubMed]
  9. V. Yu. Fedorov and T. Nakajima, “Inhomogeneous waves in lossy metamaterials and negative refraction,” http://www.arxiv.org/abs/1305.6393 .
  10. C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
    [Crossref]
  11. F. I. Fedorov, “To the theory of total reflection,” J. Opt. 15, 014002 (2013).
    [Crossref]
  12. P. Halevi and A. Mendoza-Hernández, “Temporal and spatial behavior of the Poynting vector in dissipative media: refraction from vacuum into a medium,” J. Opt. Soc. Am. 71, 1238–1242 (1981).
    [Crossref]
  13. N. L. Dmitruk and A. V. Korovin, “Electromagnetic waves refraction on the interface of transparent with absorptive right or left-handed media,” Appl. Phys. A 103, 635–639 (2011).
    [Crossref]
  14. V. Yu. Fedorov and T. Nakajima, “Controlling the propagation velocity of a femtosecond laser pulse with negative index metamaterials,” Phys. Rev. Lett. 107, 143903 (2011).
    [Crossref] [PubMed]

2013 (1)

F. I. Fedorov, “To the theory of total reflection,” J. Opt. 15, 014002 (2013).
[Crossref]

2012 (1)

S. Feng, “Loss-induced omnidirectional bending to the normal in ε-near-zero metamaterials,” Phys. Rev. Lett. 108,193904 (2012).
[Crossref]

2011 (4)

C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. 106, 067402 (2011).
[Crossref] [PubMed]

N. L. Dmitruk and A. V. Korovin, “Electromagnetic waves refraction on the interface of transparent with absorptive right or left-handed media,” Appl. Phys. A 103, 635–639 (2011).
[Crossref]

V. Yu. Fedorov and T. Nakajima, “Controlling the propagation velocity of a femtosecond laser pulse with negative index metamaterials,” Phys. Rev. Lett. 107, 143903 (2011).
[Crossref] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref] [PubMed]

2007 (1)

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[Crossref] [PubMed]

2002 (1)

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

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

1981 (1)

1972 (1)

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[Crossref]

Aieta, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref] [PubMed]

Atwater, H. A.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[Crossref] [PubMed]

Capasso, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref] [PubMed]

Chen, H. C.

H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, 1983).

Dickson, W.

C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. 106, 067402 (2011).
[Crossref] [PubMed]

Dionne, J. A.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[Crossref] [PubMed]

Dmitruk, N. L.

N. L. Dmitruk and A. V. Korovin, “Electromagnetic waves refraction on the interface of transparent with absorptive right or left-handed media,” Appl. Phys. A 103, 635–639 (2011).
[Crossref]

Enoch, S.

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

Fedorov, F. I.

F. I. Fedorov, “To the theory of total reflection,” J. Opt. 15, 014002 (2013).
[Crossref]

F. I. Fedorov, Optics of Anisotropic Media (URSS, 2004), (in Russian).

Fedorov, V. Yu.

V. Yu. Fedorov and T. Nakajima, “Controlling the propagation velocity of a femtosecond laser pulse with negative index metamaterials,” Phys. Rev. Lett. 107, 143903 (2011).
[Crossref] [PubMed]

Feng, S.

S. Feng, “Loss-induced omnidirectional bending to the normal in ε-near-zero metamaterials,” Phys. Rev. Lett. 108,193904 (2012).
[Crossref]

Gaburro, Z.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref] [PubMed]

García-Meca, C.

C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. 106, 067402 (2011).
[Crossref] [PubMed]

Genevet, P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref] [PubMed]

Guérin, N.

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

Halevi, P.

Hurtado, J.

C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. 106, 067402 (2011).
[Crossref] [PubMed]

Imbert, C.

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[Crossref]

Kats, M. A.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref] [PubMed]

Korovin, A. V.

N. L. Dmitruk and A. V. Korovin, “Electromagnetic waves refraction on the interface of transparent with absorptive right or left-handed media,” Appl. Phys. A 103, 635–639 (2011).
[Crossref]

Lezec, H. J.

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[Crossref] [PubMed]

Martí, J.

C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. 106, 067402 (2011).
[Crossref] [PubMed]

Martínez, A.

C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. 106, 067402 (2011).
[Crossref] [PubMed]

Mendoza-Hernández, A.

Nakajima, T.

V. Yu. Fedorov and T. Nakajima, “Controlling the propagation velocity of a femtosecond laser pulse with negative index metamaterials,” Phys. Rev. Lett. 107, 143903 (2011).
[Crossref] [PubMed]

Sabouroux, P.

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

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

Tayeb, G.

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

Tetienne, J.-P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref] [PubMed]

Vincent, P.

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

Yu, N.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref] [PubMed]

Zayats, A. V.

C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. 106, 067402 (2011).
[Crossref] [PubMed]

Appl. Phys. A (1)

N. L. Dmitruk and A. V. Korovin, “Electromagnetic waves refraction on the interface of transparent with absorptive right or left-handed media,” Appl. Phys. A 103, 635–639 (2011).
[Crossref]

J. Opt. (1)

F. I. Fedorov, “To the theory of total reflection,” J. Opt. 15, 014002 (2013).
[Crossref]

J. Opt. Soc. Am. (1)

Phys. Rev. D (1)

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[Crossref]

Phys. Rev. Lett. (4)

V. Yu. Fedorov and T. Nakajima, “Controlling the propagation velocity of a femtosecond laser pulse with negative index metamaterials,” Phys. Rev. Lett. 107, 143903 (2011).
[Crossref] [PubMed]

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

S. Feng, “Loss-induced omnidirectional bending to the normal in ε-near-zero metamaterials,” Phys. Rev. Lett. 108,193904 (2012).
[Crossref]

C. García-Meca, J. Hurtado, J. Martí, A. Martínez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. 106, 067402 (2011).
[Crossref] [PubMed]

Science (3)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007).
[Crossref] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref] [PubMed]

Other (3)

V. Yu. Fedorov and T. Nakajima, “Inhomogeneous waves in lossy metamaterials and negative refraction,” http://www.arxiv.org/abs/1305.6393 .

F. I. Fedorov, Optics of Anisotropic Media (URSS, 2004), (in Russian).

H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill, 1983).

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Figures (2)

Fig. 1
Fig. 1 Refraction of the wave at the interface of a lossy material. The incident plane wave has a real wave vector k0, while the wave vector k of the transmitted wave is complex due to material losses: k = k′ − ik″. In general, the phase k′ and attenuation k″ vectors are not parallel, and the transmitted wave is inhomogeneous.

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Fig. 2
Fig. 2 (a) Variation of the real μ′ and imaginary μ″ parts of the permeability μ as a function of wavelength λ. (b) and (c) The transmission angles ψs and ψp for the s- and p-polarized components of the Poynting vector as functions of wavelength λ and incident angle θ0. All above functions are calculated for the interface between vacuum and the metamaterial reported in [8]. The spectral region of negative refraction is located between the two dotted vertical lines.

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Equations (13)

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E = e exp [ i ( ω t − k ⋅ r ) ] ,                                     H = h exp [ i ( ω t − k ⋅ r ) ] ,
k ′ = p + q ′ q ^ ,                     k ″ = q ″ q ^ .
q ′ = sgn { ξ ″ } ω c 0 ( | ξ | + ξ ′ ) / 2 ,                           q ″ = ω c 0 ( | ξ | − ξ ′ ) / 2 ,
m ′ = ( | ξ | + ξ ′ + 2 n 0 2 sin 2 θ 0 ) / 2 ,                         m ″ = ( | ξ | − ξ ′ ) / 2 .
e = A s s + A p [ s × k ] ,
h = ε 0 ε ω A p s − A s μ 0 μ ω [ s × k ] .
S s = | A s | 2 μ 0 μ * ω [ s × [ k * × s ] ] ,
S p = ε 0 ε * ω | A p | 2 [ s × [ k × s ] ] ,
S s p = − A s * A p μ 0 μ * ω [ [ k × s ] × [ k * × s ] ] .
P s = s 2 | A s | 2 μ 0 | μ | 2 ω ( μ ′ k ′ + μ ″ k ″ ) ,
P p = ε 0 ω s 2 | A p | 2 ( ε ′ k ′ + ε ″ k ″ ) ,
P s p = 2 q ″ s 2 μ 0 | μ | 2 ω ( μ ′ ℑ { A s * A p } − μ ″ ℜ { A s * A p } ) s
tan ψ s = μ ′ m ′ sin θ μ ′ m ′ cos θ + μ ″ m ″ ,                     tan ψ p = ε ′ m ′ sin θ ε ′ m ′ cos θ + ε ″ m ″ .

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