Abstract

We present a general theory of negative refraction in periodic stratified heterostructures with an arbitrary number of homogeneous, isotropic, nonmagnetic layers in a unit cell. With a 4×4-matrix technique, we derive analytic expressions for the normal modes of such a heterostructure slab, introduce the average refraction angles of the energy flow and wavevector for the TE- and TM-polarized plane waves falling obliquely on the slab, and derive expressions for the reflectivity and transmissivity of the whole slab. For a specific case, in which all layers in a unit cell are much thinner than the wavelength of light, we obtain approximate simple formulae for the effective refraction angles. Using the example of a semiconductor heterostructure slab with two layers in a unit cell, we demonstrate that ultrathin layers are preferable for metamaterial applications because they enable higher transmissivity within the frequency band of negative refraction. Our theory can be used to study the optical properties of any stratified metamaterial, irrespective of whether semiconductors or metals are employed for fabricating its various layers, because it includes absorption within each layer.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  26. S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express 13, 4922–4930 (2005).
    [CrossRef] [PubMed]
  27. H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rodsplit-ringresonator electromagnetic metamaterial,” Phys. Rev. Lett. 94, 063901 (2005).
    [CrossRef] [PubMed]
  28. T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
    [CrossRef] [PubMed]
  29. V. P. Drachev, U. K. Chettiar, A. V. Kildishev, H. K. Yuan, W. Cai, and V. M. Shalaev, “The Ag dielectric function in plasmonic metamaterials,” Opt. Express 16, 1186–1195 (2008).
    [CrossRef] [PubMed]
  30. H. K. Yuan, U. K. Chettiar, W. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, and V. M. Shalaev, “A negative permeability material at red light,” Opt. Express 15, 1076–1083 (2007).
    [CrossRef] [PubMed]
  31. S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Midinfrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. 94, 037402 (2005).
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    [CrossRef]

2008 (2)

A. Boltasseva, and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: recent advances and outlook,” Metamaterials (Amst.) 2, 1–17 (2008).
[CrossRef]

V. P. Drachev, U. K. Chettiar, A. V. Kildishev, H. K. Yuan, W. Cai, and V. M. Shalaev, “The Ag dielectric function in plasmonic metamaterials,” Opt. Express 16, 1186–1195 (2008).
[CrossRef] [PubMed]

2007 (8)

H. K. Yuan, U. K. Chettiar, W. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, and V. M. Shalaev, “A negative permeability material at red light,” Opt. Express 15, 1076–1083 (2007).
[CrossRef] [PubMed]

U. K. Chettiar, A. V. Kildishev, H. K. Yuan, W. Cai, S. Xiao, V. P. Drachev, and V. M. Shalaev, “Dual-band negative index metamaterial: Double negative at 813 nm and single negative at 772 nm,” Opt. Lett. 32, 1671–1673 (2007).
[CrossRef] [PubMed]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[CrossRef]

V. M. Shalaev, “Optical negative index metamaterials,” Nat. Photonics 1, 41–47 (2007).
[CrossRef]

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53–55 (2007).
[CrossRef]

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[CrossRef]

A. V. Kildishev, and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. 32, 3432–3434 (2007).
[CrossRef] [PubMed]

E. E. Narimanov, and V. M. Shalaev, “Optics: Beyond diffraction,” Nature 447, 266–267 (2007).
[CrossRef] [PubMed]

2006 (6)

2005 (4)

V. A. Podolskiy, and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).
[CrossRef]

S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Near-infrared double negative metamaterials,” Opt. Express 13, 4922–4930 (2005).
[CrossRef] [PubMed]

H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rodsplit-ringresonator electromagnetic metamaterial,” Phys. Rev. Lett. 94, 063901 (2005).
[CrossRef] [PubMed]

S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Midinfrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. 94, 037402 (2005).
[CrossRef] [PubMed]

2004 (4)

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
[CrossRef] [PubMed]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494–1496 (2004).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef] [PubMed]

2003 (1)

N. C. Panoiu, and R. M. Osgood, “Numerical investigation of negative refractive index metamaterials at infrared and optical frequencies,” Opt. Commun. 233, 331–337 (2003).
[CrossRef]

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]

2000 (2)

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, 4184–4187 (2000).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lense,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

1976 (1)

1972 (2)

D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
[CrossRef]

D. W. Berreman, and T. J. Scheffer, “Order versus temperature in cholesteric liquid crystals from reflectance spectra,” Phys. Rev. A 5, 1397–1403 (1972).
[CrossRef]

1970 (1)

D. W. Berreman, and T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid-crystal films,” Phys. Rev. Lett. 25, 577–581 (1970).
[CrossRef]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Alekseyev, L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[CrossRef]

Basov, D. N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494–1496 (2004).
[CrossRef] [PubMed]

Berreman, D. W.

D. W. Berreman, “Optics in stratified and anisotropic media: 4×4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).
[CrossRef]

D. W. Berreman, and T. J. Scheffer, “Order versus temperature in cholesteric liquid crystals from reflectance spectra,” Phys. Rev. A 5, 1397–1403 (1972).
[CrossRef]

D. W. Berreman, and T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid-crystal films,” Phys. Rev. Lett. 25, 577–581 (1970).
[CrossRef]

Boltasseva, A.

A. Boltasseva, and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: recent advances and outlook,” Metamaterials (Amst.) 2, 1–17 (2008).
[CrossRef]

H. K. Yuan, U. K. Chettiar, W. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, and V. M. Shalaev, “A negative permeability material at red light,” Opt. Express 15, 1076–1083 (2007).
[CrossRef] [PubMed]

Braginsky, L.

V. G. Veselago, L. Braginsky, V. Shklover, and Ch. Hafner, “Negative refractive index materials,” J. Comput. Theor. Nanosci. 3, 189–218 (2006).

Brueck, S. R. J.

Cai, W.

Casse, B. D. F.

H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rodsplit-ringresonator electromagnetic metamaterial,” Phys. Rev. Lett. 94, 063901 (2005).
[CrossRef] [PubMed]

Chettiar, U. K.

Dolling, G.

Drachev, V. P.

Economou, E. N.

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef] [PubMed]

Fan, W.

Fang, N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494–1496 (2004).
[CrossRef] [PubMed]

Franz, K. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[CrossRef]

Frauenglass, A.

S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Midinfrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. 94, 037402 (2005).
[CrossRef] [PubMed]

Gmachl, C.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[CrossRef]

Hafner, Ch.

V. G. Veselago, L. Braginsky, V. Shklover, and Ch. Hafner, “Negative refractive index materials,” J. Comput. Theor. Nanosci. 3, 189–218 (2006).

Hoffman, A. J.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[CrossRef]

Hong, C.-S.

Howard, S. S.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[CrossRef]

Kafesaki, M.

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

Kildishev, A. V.

Klar, T. A.

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: Going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[CrossRef]

Koschny, T.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef] [PubMed]

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

Linden, S.

Malloy, K. J.

Minhas, B. K.

S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Midinfrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. 94, 037402 (2005).
[CrossRef] [PubMed]

Moser, H. O.

H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rodsplit-ringresonator electromagnetic metamaterial,” Phys. Rev. Lett. 94, 063901 (2005).
[CrossRef] [PubMed]

Narimanov, E. E.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[CrossRef]

A. V. Kildishev, and E. E. Narimanov, “Impedance-matched hyperlens,” Opt. Lett. 32, 3432–3434 (2007).
[CrossRef] [PubMed]

E. E. Narimanov, and V. M. Shalaev, “Optics: Beyond diffraction,” Nature 447, 266–267 (2007).
[CrossRef] [PubMed]

V. A. Podolskiy, and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).
[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, 4184–4187 (2000).
[CrossRef] [PubMed]

Osgood, R. M.

Padilla, W. J.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494–1496 (2004).
[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, 4184–4187 (2000).
[CrossRef] [PubMed]

Panoiu, N. C.

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494–1496 (2004).
[CrossRef] [PubMed]

J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lense,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Podolskiy, V. A.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[CrossRef]

V. A. Podolskiy, and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).
[CrossRef]

Sarychev, A. K.

Saw, B. T.

H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rodsplit-ringresonator electromagnetic metamaterial,” Phys. Rev. Lett. 94, 063901 (2005).
[CrossRef] [PubMed]

Scheffer, T. J.

D. W. Berreman, and T. J. Scheffer, “Order versus temperature in cholesteric liquid crystals from reflectance spectra,” Phys. Rev. A 5, 1397–1403 (1972).
[CrossRef]

D. W. Berreman, and T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid-crystal films,” Phys. Rev. Lett. 25, 577–581 (1970).
[CrossRef]

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]

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, 4184–4187 (2000).
[CrossRef] [PubMed]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

Shalaev, V. M.

A. Boltasseva, and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: recent advances and outlook,” Metamaterials (Amst.) 2, 1–17 (2008).
[CrossRef]

V. P. Drachev, U. K. Chettiar, A. V. Kildishev, H. K. Yuan, W. Cai, and V. M. Shalaev, “The Ag dielectric function in plasmonic metamaterials,” Opt. Express 16, 1186–1195 (2008).
[CrossRef] [PubMed]

H. K. Yuan, U. K. Chettiar, W. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, and V. M. Shalaev, “A negative permeability material at red light,” Opt. Express 15, 1076–1083 (2007).
[CrossRef] [PubMed]

U. K. Chettiar, A. V. Kildishev, H. K. Yuan, W. Cai, S. Xiao, V. P. Drachev, and V. M. Shalaev, “Dual-band negative index metamaterial: Double negative at 813 nm and single negative at 772 nm,” Opt. Lett. 32, 1671–1673 (2007).
[CrossRef] [PubMed]

E. E. Narimanov, and V. M. Shalaev, “Optics: Beyond diffraction,” Nature 447, 266–267 (2007).
[CrossRef] [PubMed]

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[CrossRef]

V. M. Shalaev, “Optical negative index metamaterials,” Nat. Photonics 1, 41–47 (2007).
[CrossRef]

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: Going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[CrossRef]

V. M. Shalaev, W. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356–3358 (2006).
[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]

Shklover, V.

V. G. Veselago, L. Braginsky, V. Shklover, and Ch. Hafner, “Negative refractive index materials,” J. Comput. Theor. Nanosci. 3, 189–218 (2006).

Sivco, D. L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[CrossRef]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494–1496 (2004).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[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, 4184–4187 (2000).
[CrossRef] [PubMed]

Soukoulis, C. M.

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53–55 (2007).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef] [PubMed]

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

Veselago, V. G.

V. G. Veselago, L. Braginsky, V. Shklover, and Ch. Hafner, “Negative refractive index materials,” J. Comput. Theor. Nanosci. 3, 189–218 (2006).

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Vier, D. C.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494–1496 (2004).
[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, 4184–4187 (2000).
[CrossRef] [PubMed]

Wasserman, D.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[CrossRef]

Wegener, M.

Wilhelmi, O.

H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rodsplit-ringresonator electromagnetic metamaterial,” Phys. Rev. Lett. 94, 063901 (2005).
[CrossRef] [PubMed]

Xiao, S.

Yariv, A.

Yeh, P.

Yen, T. J.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494–1496 (2004).
[CrossRef] [PubMed]

Yuan, H. K.

Zhang, S.

Zhang, X.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494–1496 (2004).
[CrossRef] [PubMed]

Zhou, J.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

T. A. Klar, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Negative-index metamaterials: Going optical,” IEEE J. Sel. Top. Quantum Electron. 12, 1106–1115 (2006).
[CrossRef]

J. Comput. Theor. Nanosci. (1)

V. G. Veselago, L. Braginsky, V. Shklover, and Ch. Hafner, “Negative refractive index materials,” J. Comput. Theor. Nanosci. 3, 189–218 (2006).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (1)

Metamaterials (Amst.) (1)

A. Boltasseva, and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: recent advances and outlook,” Metamaterials (Amst.) 2, 1–17 (2008).
[CrossRef]

Nat. Mater. (1)

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6, 946–950 (2007).
[CrossRef]

Nat. Photonics (2)

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1, 224–227 (2007).
[CrossRef]

V. M. Shalaev, “Optical negative index metamaterials,” Nat. Photonics 1, 41–47 (2007).
[CrossRef]

Nature (1)

E. E. Narimanov, and V. M. Shalaev, “Optics: Beyond diffraction,” Nature 447, 266–267 (2007).
[CrossRef] [PubMed]

Opt. Commun. (1)

N. C. Panoiu, and R. M. Osgood, “Numerical investigation of negative refractive index metamaterials at infrared and optical frequencies,” Opt. Commun. 233, 331–337 (2003).
[CrossRef]

Opt. Express (3)

Opt. Lett. (5)

Phys. Rev. A (1)

D. W. Berreman, and T. J. Scheffer, “Order versus temperature in cholesteric liquid crystals from reflectance spectra,” Phys. Rev. A 5, 1397–1403 (1972).
[CrossRef]

Phys. Rev. B (1)

V. A. Podolskiy, and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).
[CrossRef]

Phys. Rev. Lett. (6)

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, 4184–4187 (2000).
[CrossRef] [PubMed]

D. W. Berreman, and T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid-crystal films,” Phys. Rev. Lett. 25, 577–581 (1970).
[CrossRef]

S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Midinfrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. 94, 037402 (2005).
[CrossRef] [PubMed]

H. O. Moser, B. D. F. Casse, O. Wilhelmi, and B. T. Saw, “Terahertz response of a microfabricated rodsplit-ringresonator electromagnetic metamaterial,” Phys. Rev. Lett. 94, 063901 (2005).
[CrossRef] [PubMed]

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402 (2004).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lense,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Science (5)

J. B. Pendry, “A chiral route to negative refraction,” Science 306, 1353–1355 (2004).
[CrossRef] [PubMed]

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

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[CrossRef] [PubMed]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494–1496 (2004).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Other (4)

W. Cai, D. A. Genov, and V. M. Shalaev, “Superlens based on metal-dielectric composites,” Phys. Rev. B 72, 193101 (2005).
[CrossRef]

M. Born, and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).

V. M. Shalaev, Nonlinear Optics of Random Media: Fractal Composites and Metal-Dielectric Films (Springer, Berlin, 2000).

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Geometry of a periodic stratified heterostructure surrounded by a transparent media of refractive index n0; h is the length of the unit cell. Plane electromagnetic wave obliquely incident in the xz plane, at an angle ϑ0 to the normal of the surface z = 0.

Fig. 2
Fig. 2

Illustration of the averaging procedure in Eq. (13): (a) first, the direction of k is averaged within each layer; (b) resulting angles are then averaged over all layers in a unit cell. (c) Directions of energy flow and k for a π-polarized plane wave undergoing negative refraction inside a heterostructure slab with ultrathin layers.

Fig. 3
Fig. 3

Effective refraction angles of (a) energy flux and (b) wavevector for a π-polarized plane wave incident at an angle ϑ0 = π/3 on a heterostructure made with 0.2- and 2-μm-thick layers; the shaded bands show the regions of negative refraction in the two cases; open circles correspond to the approximate solution in Eq. (17). (c) Transmission (T) and absorption (A) spectra for 4-μm-thick slabs made of thick (blue) and thin (red) layers.

Fig. 4
Fig. 4

(a) Effective refraction angles of energy flux and wavevector for a σ-polarized plane wave incident at an angle ϑ0 = π/3 on a heterostructure made with 0.2- and 2-μm-thick layers; circles correspond to the approximate solution in Eq. (18). (b) Transmission (T) and transmission plus absorption (T&A) spectra for 4-μm-thick slabs made of thick (blue) and thin (red) layers.

Fig. 5
Fig. 5

Effective refraction angles of energy flux and wavevector [(a), (c)] and transmission, reflection, and absorption coefficients [(b), (d)] for a π-polarized beam incident at different angles on heterostructures made of thin [(a), (b)] and thick [(c), (d)] layers.

Equations (44)

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

E ( x , z , t ) = ( E x E y E z ) exp [ i ( β x ω t ) ] + c . c . , H ( x , z , t ) = ( H x H y H z ) exp [ i ( β x ω t ) ] + c . c . ,
ψ ( z ) = ( E x H y E y H x ) .
d ψ dz = i k Δ ( z ) ψ ,
E z = ( β / k ) ( 1 / ɛ ) H y , H z = ( β / k ) E y .
ψ ( z ) = exp ( ik Δ j z ) ψ ( h 1 + h 2 + + h j 1 ) ,
𝒫 j ( z ) = ( cos ϕ j i π j sin ϕ j 0 0 ( i / π j ) sin ϕ j cos ϕ j 0 0 0 0 cos ϕ j i σ j sin ϕ j 0 0 ( i / σ j ) sin ϕ j cos ϕ j ) ,
ψ ( z ) = 𝒢 j ( z ) ψ ( 0 ) ,
𝒢 j ( z ) = 𝒫 j ( z ) 𝒫 j 1 ( h j 1 ) × × 𝒫 2 ( h 2 ) 𝒫 1 ( h 1 ) .
q η ± = K η ± ( K η 2 1 ) 1 / 2 , η = π or σ .
K η = ( 2 n j = 1 n η j ) 1 ( 1 ) r ( η 1 ± η 2 ) ( ± η 2 ± η 3 ) × × ( ± η n 1 ± η n ) ( ± η n ± η 1 ) cos ( φ 1 ± φ 2 ± ± φ n ,
ψ π ± = ( 1 ψ π ± 0 0 ) , ψ σ ± = ( 0 0 1 ψ σ ± ) ,
arg q η ± = Arg q η ± + 2 π N , N = 0 , ± 1 , ± 2 ,
Re Ψ η ± > 0 .
tan ϑ k , π , j = β d ( Arg H yj ) / dz = | H y j | 2 Re ( ɛ j E xj H yj * ) 1 k / β , ( j = 1 , 2 , , n ) .
tan ϑ k , σ , j = β d ( Arg E y j ) / dz = | E y j | 2 Re ( E y j H x j * ) 1 k / β ,
tan ϑ S , π , j = Re ( E z j H y j * ) Re ( E x j H y j * ) = | H y j | 2 Re ( E x j H y j * ) Re ( 1 / ɛ j ) k / β .
tan ϑ S , σ , j = Re ( E y j H z j * ) Re ( E y j H x j * ) = tan ϑ k , σ , j .
Θ q , η = tan 1 ( 1 h j = 1 n 0 h j tan ϑ q , η , j dz ) ,
ψ i + ψ r = A π + ψ π + + A π ψ π + A σ + ψ σ + + A σ ψ σ ,
ψ t = A π + Q π + ψ π + + A π Q π ψ π + A σ + Q σ + ψ σ + + A σ Q σ ψ σ ,
ψ i = ( 1 r π 1 r σ ) , ψ r = ( R π r π R π R σ r σ R σ ) , and ψ t = ( T π r π T π T σ r σ T σ ) ,
A η ± = 2 r η Q η ( Ψ η r η ) / D η ,
R η = Q η Q η + ( Ψ η + r η ) ( Ψ η r η ) D η ,
T η = 2 r η Q η + Q η ( Ψ η + Ψ η ) / D η ,
h = ( 1 i k 2 h / ( ɛ k ) 0 0 i ɛ kh 1 0 0 0 0 1 ikh 0 0 i k 2 h / k 1 ) ,
ɛ xx = ɛ yy ɛ = 1 h j = 1 n h j ɛ j , ɛ zz ɛ = ( 1 h j = 1 n h j ɛ j ) 1 .
q π ± = 1 ± i k h ( ɛ / ɛ ) 1 / 2 , q σ ± = 1 ± i k h .
Ψ π ± = ± ( k / k ) ( ɛ ɛ ) 1 / 2 , Ψ σ ± = ± k / k .
tan Θ k , π ± β Re [ k ( ɛ / ɛ ) 1 / 2 ] , tan Θ S , π ± β | ɛ | Re ( 1 / ɛ ) Re [ k ( ɛ * / ɛ ) 1 / 2 ] ,
Θ k , σ = Θ S , σ ± tan 1 ( β Re k ) ,
ɛ 1 ( ω ) = 12.15 × ( 1 ω p 2 ω ( ω + i δ ) )
G 11 ( j ) ( z ) = ( 2 j 1 p = 1 j 1 π p ) 1 ( 1 ) s ( π 1 ± π 2 ) ( ± π 2 ± π 3 ) × × ( ± π j 1 ± π j ) cos ( φ 1 ± φ 2 ± ± φ j 1 ± ϕ j ) ,
G 12 ( j ) ( z ) = i ( 2 j 1 p = 2 j 1 π p ) 1 ( 1 ) s ( π 1 ± π 2 ) ( ± π 2 ± π 3 ) × × ( ± π j 1 ± π j ) sin ( φ 1 ± φ 2 ± ± φ j 1 ± ϕ j ) ,
G 21 ( j ) ( z ) = i ( 2 j 1 p = 1 j π p ) 1 ( 1 ) r ( π 1 ± π 2 ) ( ± π 2 ± π 3 ) × × ( ± π j 1 ± π j ) sin ( φ 1 ± φ 2 ± ± φ j 1 ± ϕ j ) ,
G 22 ( j ) ( z ) = ( 1 + G 21 ( j ) G 12 ( j ) / G 11 ( j ) ,
G 11 ( 2 ) = 1 2 π 1 [ ( π 1 + π 2 ) cos ( φ 1 + ϕ 2 ) + ( π 1 π 2 ) cos ( φ 1 ϕ 2 ) ] , G 12 ( 2 ) = i 2 [ ( π 1 + π 2 ) sin ( φ 1 + ϕ 2 ) + ( π 1 π 2 ) sin ( φ 1 ϕ 2 ) ] , G 21 ( 2 ) = 1 2 π 1 π 2 [ ( π 1 + π 2 ) sin ( φ 1 + ϕ 2 ) ( π 1 π 2 ) sin ( φ 1 ϕ 2 ) ] ,
K π = 1 4 π 1 π 2 [ ( π 1 + π 2 ) 2 cos ( φ 1 + φ 2 ) ( π 1 π 2 ) 2 cos ( φ 1 φ 2 ) ] .
d dz ( E x j H y j ) = i k ( 0 ϖ j ɛ j 0 ) ( E x j H y j ) ,
d ( Arg E x j ) dz = d dz [ tan 1 ( Im E x j Re E x j ) ] > 0 .
Re ( ϖ j E x j * H y j ) > 0 .
Re ( ɛ j E x j H y j * ) > 0 .
Re ( E x j * H y j ) > 0 ,
d dz ( E y j H x j ) = i k ( 0 1 ɛ j ϖ j 0 ) ( E y j H x j ) .
Re ( E y j H x j * ) < 0 .

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