Abstract

We introduce a new type of metamaterial, which we call liquid metacrystals (LMCs), consisting of elongated particles (meta-atoms) suspended in viscous liquid. A constant homogeneous external electric field applied to such a material induces polarization of the meta-atoms and orients them along one axis, resulting in anisotropic electromagnetic properties of the system. The axis of anisotropy can be reoriented, and this type of tunability resembles that of liquid crystals in the nematic phase. Moreover, meta-atoms also reorient in response to the high-frequency electromagnetic waves, suggesting strong nonlinear properties of the LMCs. The artificial meta-atoms can be designed as classical oscillators to enhance their tunability. As a particular example of an electromagnetic wave control by LMCs, we study linear and nonlinear transmission of radiation through a slab of this metamaterial.

© 2014 Optical Society of America

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2013 (2)

X. Meng, U. Guler, A. V. Kidishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).

M. Liu, Y. Sun, D. A. Powell, I. V. Shadrivov, M. Lapine, R. C. McPhedran, and Y. S. Kivshar, “Nonlinear response via intrinsic rotation in metamaterials,” Phys. Rev. B 87, 235126 (2013).
[CrossRef]

2012 (6)

A. Minovich, J. Farnell, D. N. Neshev, I. McKervacher, F. Karouta, J. Tian, D. A. Powell, I. V. Shadrivov, H. H. Tan, C. Jagadish, and Y. S. Kivshar, “Liquid crystal based nonlinear fishnet metamaterials,” Appl. Phys. Lett. 100, 121113 (2012).
[CrossRef]

M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. 11, 30–33 (2012).
[CrossRef]

J. A. Fan, K. Bao, L. Sun, J. Bao, V. N. Manoharau, P. Nordlander, and F. Capasso, “Stabilized TiN nanowire arrays for high-performance and flexible supercapacitors,” Nano Lett. 12, 5376–5381 (2012).
[CrossRef]

A. Wiener, A. I. Fernandez-Dominguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal effects in the nanofocusing performance of plasmonic tips,” Nano Lett. 12, 3308–3314 (2012).
[CrossRef]

E. Wang, T. P. White, and K. R. Catchpole, “Resonant enhancement of dielectric and metal nanoparticle arrays for light trapping in solar cells,” Opt. Express 20, 13226–13237 (2012).
[CrossRef]

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
[CrossRef]

2011 (3)

A. Boltasseva and H. A. Atwater, “Low-loss plasmonic metamaterials,” Science 331, 290–291 (2011).
[CrossRef]

I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Metamaterials with conformational nonlinearity,” Sci. Rep. 1, 138 (2011).

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[CrossRef]

2010 (3)

R. Zhao, P. Tassin, T. Koschny, and C. M. Soukoulis, “Optical forces in nanowire pairs and metamaterials,” Opt. Express 18, 25665–25676 (2010).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010).
[CrossRef]

2009 (4)

M. A. Noginov, Y. A. Bernakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94, 151105 (2009).
[CrossRef]

I. Cao and M. L. Brongersma, “Active plasmonics: ultrafast developments,” Nat. Photonics 3, 12–13 (2009).
[CrossRef]

B. Edwards, A. Alu, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. 103, 153901 (2009).
[CrossRef]

A. B. Golovin and O. D. Lavrentovich, “Electrically reconfigurable optical metamaterial based on colloidal dispersion of metal nanorods in dielectric fluid,” Appl. Phys. Lett. 95, 254104 (2009).
[CrossRef]

2008 (2)

A. Alu and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2, 307–310 (2008).
[CrossRef]

L. Novotny, “Optical antennas tuned to pitch,” Nature 455, 887 (2008).
[CrossRef]

2007 (3)

2006 (1)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

2005 (2)

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]

J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

2004 (2)

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004).
[CrossRef]

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

2003 (3)

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

W. L. Barnes, A. Dereux, and A. Ebbensen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (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]

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, 4184–4187 (2000).
[CrossRef]

Alu, A.

B. Edwards, A. Alu, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. 103, 153901 (2009).
[CrossRef]

A. Alu and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2, 307–310 (2008).
[CrossRef]

Atwater, H. A.

A. Boltasseva and H. A. Atwater, “Low-loss plasmonic metamaterials,” Science 331, 290–291 (2011).
[CrossRef]

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

Bao, J.

J. A. Fan, K. Bao, L. Sun, J. Bao, V. N. Manoharau, P. Nordlander, and F. Capasso, “Stabilized TiN nanowire arrays for high-performance and flexible supercapacitors,” Nano Lett. 12, 5376–5381 (2012).
[CrossRef]

Bao, K.

J. A. Fan, K. Bao, L. Sun, J. Bao, V. N. Manoharau, P. Nordlander, and F. Capasso, “Stabilized TiN nanowire arrays for high-performance and flexible supercapacitors,” Nano Lett. 12, 5376–5381 (2012).
[CrossRef]

Barnes, W. L.

W. L. Barnes, A. Dereux, and A. Ebbensen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Bergman, D. J.

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef]

Bernakov, Y. A.

M. A. Noginov, Y. A. Bernakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94, 151105 (2009).
[CrossRef]

Boltasseva, A.

A. Boltasseva and H. A. Atwater, “Low-loss plasmonic metamaterials,” Science 331, 290–291 (2011).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, 1964).

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010).
[CrossRef]

Brongersma, M. L.

I. Cao and M. L. Brongersma, “Active plasmonics: ultrafast developments,” Nat. Photonics 3, 12–13 (2009).
[CrossRef]

Brueck, S. R. J.

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]

Cai, W.

Cao, I.

I. Cao and M. L. Brongersma, “Active plasmonics: ultrafast developments,” Nat. Photonics 3, 12–13 (2009).
[CrossRef]

Capasso, F.

J. A. Fan, K. Bao, L. Sun, J. Bao, V. N. Manoharau, P. Nordlander, and F. Capasso, “Stabilized TiN nanowire arrays for high-performance and flexible supercapacitors,” Nano Lett. 12, 5376–5381 (2012).
[CrossRef]

Catchpole, K. R.

Catrysse, P. B.

J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

Chettiar, U. K.

Cortes, C. L.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
[CrossRef]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

Davoyan, A. R.

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef]

deGennes, P. G.

P. G. deGennes and J. Prost, The Physics of Liquid Crystals, 2nd ed (Clarendon, 1995).

Deraad, L. L.

J. S. Schwinger, L. L. Deraad, K. A. Milton, and W. Tsai, Classical Electrodynamics (Westview, 1998).

Dereux, A.

W. L. Barnes, A. Dereux, and A. Ebbensen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Dionne, J. A.

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

Dolling, G.

Drachev, V. P.

Ebbensen, A.

W. L. Barnes, A. Dereux, and A. Ebbensen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Edwards, B.

B. Edwards, A. Alu, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. 103, 153901 (2009).
[CrossRef]

Engheta, N.

B. Edwards, A. Alu, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. 103, 153901 (2009).
[CrossRef]

A. Alu and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2, 307–310 (2008).
[CrossRef]

Enkrich, C.

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

Fan, J. A.

J. A. Fan, K. Bao, L. Sun, J. Bao, V. N. Manoharau, P. Nordlander, and F. Capasso, “Stabilized TiN nanowire arrays for high-performance and flexible supercapacitors,” Nano Lett. 12, 5376–5381 (2012).
[CrossRef]

Fan, S.

J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

Fan, W.

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]

Farnell, J.

A. Minovich, J. Farnell, D. N. Neshev, I. McKervacher, F. Karouta, J. Tian, D. A. Powell, I. V. Shadrivov, H. H. Tan, C. Jagadish, and Y. S. Kivshar, “Liquid crystal based nonlinear fishnet metamaterials,” Appl. Phys. Lett. 100, 121113 (2012).
[CrossRef]

Fernandez-Dominguez, A. I.

A. Wiener, A. I. Fernandez-Dominguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal effects in the nanofocusing performance of plasmonic tips,” Nano Lett. 12, 3308–3314 (2012).
[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]

Fujita, K.

X. Meng, U. Guler, A. V. Kidishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).

Golovin, A. B.

A. B. Golovin and O. D. Lavrentovich, “Electrically reconfigurable optical metamaterial based on colloidal dispersion of metal nanorods in dielectric fluid,” Appl. Phys. Lett. 95, 254104 (2009).
[CrossRef]

Gramotnev, D. K.

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010).
[CrossRef]

Guler, U.

X. Meng, U. Guler, A. V. Kidishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).

Horsfield, A. P.

A. Wiener, A. I. Fernandez-Dominguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal effects in the nanofocusing performance of plasmonic tips,” Nano Lett. 12, 3308–3314 (2012).
[CrossRef]

Jacob, Z.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
[CrossRef]

Jagadish, C.

A. Minovich, J. Farnell, D. N. Neshev, I. McKervacher, F. Karouta, J. Tian, D. A. Powell, I. V. Shadrivov, H. H. Tan, C. Jagadish, and Y. S. Kivshar, “Liquid crystal based nonlinear fishnet metamaterials,” Appl. Phys. Lett. 100, 121113 (2012).
[CrossRef]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

Karouta, F.

A. Minovich, J. Farnell, D. N. Neshev, I. McKervacher, F. Karouta, J. Tian, D. A. Powell, I. V. Shadrivov, H. H. Tan, C. Jagadish, and Y. S. Kivshar, “Liquid crystal based nonlinear fishnet metamaterials,” Appl. Phys. Lett. 100, 121113 (2012).
[CrossRef]

Kawata, S.

V. M. Shalaev and S. Kawata, Nanophotonics with Surface Plasmons (Elsevier Science, 2007).

Kidishev, A. V.

X. Meng, U. Guler, A. V. Kidishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).

U. K. Chettiar, A. V. Kidishev, 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]

Kivshar, Y. S.

M. Liu, Y. Sun, D. A. Powell, I. V. Shadrivov, M. Lapine, R. C. McPhedran, and Y. S. Kivshar, “Nonlinear response via intrinsic rotation in metamaterials,” Phys. Rev. B 87, 235126 (2013).
[CrossRef]

A. Minovich, J. Farnell, D. N. Neshev, I. McKervacher, F. Karouta, J. Tian, D. A. Powell, I. V. Shadrivov, H. H. Tan, C. Jagadish, and Y. S. Kivshar, “Liquid crystal based nonlinear fishnet metamaterials,” Appl. Phys. Lett. 100, 121113 (2012).
[CrossRef]

M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. 11, 30–33 (2012).
[CrossRef]

I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Metamaterials with conformational nonlinearity,” Sci. Rep. 1, 138 (2011).

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

Kochny, T.

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

Koschny, T.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, 1966).

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Pergamon, 1987).

Lapine, M.

M. Liu, Y. Sun, D. A. Powell, I. V. Shadrivov, M. Lapine, R. C. McPhedran, and Y. S. Kivshar, “Nonlinear response via intrinsic rotation in metamaterials,” Phys. Rev. B 87, 235126 (2013).
[CrossRef]

M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. 11, 30–33 (2012).
[CrossRef]

Lavrentovich, O. D.

A. B. Golovin and O. D. Lavrentovich, “Electrically reconfigurable optical metamaterial based on colloidal dispersion of metal nanorods in dielectric fluid,” Appl. Phys. Lett. 95, 254104 (2009).
[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]

Li, H.

M. A. Noginov, Y. A. Bernakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94, 151105 (2009).
[CrossRef]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Pergamon, 1987).

L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, 1966).

Linden, S.

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

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

Liu, M.

M. Liu, Y. Sun, D. A. Powell, I. V. Shadrivov, M. Lapine, R. C. McPhedran, and Y. S. Kivshar, “Nonlinear response via intrinsic rotation in metamaterials,” Phys. Rev. B 87, 235126 (2013).
[CrossRef]

Maier, S. A.

A. Wiener, A. I. Fernandez-Dominguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal effects in the nanofocusing performance of plasmonic tips,” Nano Lett. 12, 3308–3314 (2012).
[CrossRef]

Malloy, K. J.

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]

Manoharau, V. N.

J. A. Fan, K. Bao, L. Sun, J. Bao, V. N. Manoharau, P. Nordlander, and F. Capasso, “Stabilized TiN nanowire arrays for high-performance and flexible supercapacitors,” Nano Lett. 12, 5376–5381 (2012).
[CrossRef]

McKervacher, I.

A. Minovich, J. Farnell, D. N. Neshev, I. McKervacher, F. Karouta, J. Tian, D. A. Powell, I. V. Shadrivov, H. H. Tan, C. Jagadish, and Y. S. Kivshar, “Liquid crystal based nonlinear fishnet metamaterials,” Appl. Phys. Lett. 100, 121113 (2012).
[CrossRef]

McPhedran, R. C.

M. Liu, Y. Sun, D. A. Powell, I. V. Shadrivov, M. Lapine, R. C. McPhedran, and Y. S. Kivshar, “Nonlinear response via intrinsic rotation in metamaterials,” Phys. Rev. B 87, 235126 (2013).
[CrossRef]

Meng, X.

X. Meng, U. Guler, A. V. Kidishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).

Milton, K. A.

J. S. Schwinger, L. L. Deraad, K. A. Milton, and W. Tsai, Classical Electrodynamics (Westview, 1998).

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]

Minovich, A.

A. Minovich, J. Farnell, D. N. Neshev, I. McKervacher, F. Karouta, J. Tian, D. A. Powell, I. V. Shadrivov, H. H. Tan, C. Jagadish, and Y. S. Kivshar, “Liquid crystal based nonlinear fishnet metamaterials,” Appl. Phys. Lett. 100, 121113 (2012).
[CrossRef]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

Molesky, S.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
[CrossRef]

Narimanov, E. E.

M. A. Noginov, Y. A. Bernakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94, 151105 (2009).
[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]

Neshev, D. N.

A. Minovich, J. Farnell, D. N. Neshev, I. McKervacher, F. Karouta, J. Tian, D. A. Powell, I. V. Shadrivov, H. H. Tan, C. Jagadish, and Y. S. Kivshar, “Liquid crystal based nonlinear fishnet metamaterials,” Appl. Phys. Lett. 100, 121113 (2012).
[CrossRef]

Newman, W.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
[CrossRef]

Noginov, M. A.

M. A. Noginov, Y. A. Bernakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94, 151105 (2009).
[CrossRef]

Nordlander, P.

J. A. Fan, K. Bao, L. Sun, J. Bao, V. N. Manoharau, P. Nordlander, and F. Capasso, “Stabilized TiN nanowire arrays for high-performance and flexible supercapacitors,” Nano Lett. 12, 5376–5381 (2012).
[CrossRef]

Novotny, L.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[CrossRef]

L. Novotny, “Optical antennas tuned to pitch,” Nature 455, 887 (2008).
[CrossRef]

Padilla, W. J.

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]

Pendry, J. B.

A. Wiener, A. I. Fernandez-Dominguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal effects in the nanofocusing performance of plasmonic tips,” Nano Lett. 12, 3308–3314 (2012).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

Powell, D. A.

M. Liu, Y. Sun, D. A. Powell, I. V. Shadrivov, M. Lapine, R. C. McPhedran, and Y. S. Kivshar, “Nonlinear response via intrinsic rotation in metamaterials,” Phys. Rev. B 87, 235126 (2013).
[CrossRef]

A. Minovich, J. Farnell, D. N. Neshev, I. McKervacher, F. Karouta, J. Tian, D. A. Powell, I. V. Shadrivov, H. H. Tan, C. Jagadish, and Y. S. Kivshar, “Liquid crystal based nonlinear fishnet metamaterials,” Appl. Phys. Lett. 100, 121113 (2012).
[CrossRef]

M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. 11, 30–33 (2012).
[CrossRef]

I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Metamaterials with conformational nonlinearity,” Sci. Rep. 1, 138 (2011).

Prost, J.

P. G. deGennes and J. Prost, The Physics of Liquid Crystals, 2nd ed (Clarendon, 1995).

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]

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]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

Schwinger, J. S.

J. S. Schwinger, L. L. Deraad, K. A. Milton, and W. Tsai, Classical Electrodynamics (Westview, 1998).

Shadrivov, I. V.

M. Liu, Y. Sun, D. A. Powell, I. V. Shadrivov, M. Lapine, R. C. McPhedran, and Y. S. Kivshar, “Nonlinear response via intrinsic rotation in metamaterials,” Phys. Rev. B 87, 235126 (2013).
[CrossRef]

M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. 11, 30–33 (2012).
[CrossRef]

A. Minovich, J. Farnell, D. N. Neshev, I. McKervacher, F. Karouta, J. Tian, D. A. Powell, I. V. Shadrivov, H. H. Tan, C. Jagadish, and Y. S. Kivshar, “Liquid crystal based nonlinear fishnet metamaterials,” Appl. Phys. Lett. 100, 121113 (2012).
[CrossRef]

I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Metamaterials with conformational nonlinearity,” Sci. Rep. 1, 138 (2011).

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

Shalaev, V. M.

X. Meng, U. Guler, A. V. Kidishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).

U. K. Chettiar, A. V. Kidishev, 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]

V. M. Shalaev and S. Kawata, Nanophotonics with Surface Plasmons (Elsevier Science, 2007).

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]

Shen, J. T.

J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 197401 (2005).
[CrossRef]

Silveirinha, M. G.

B. Edwards, A. Alu, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. 103, 153901 (2009).
[CrossRef]

Smith, D. R.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

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

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]

Soukoulis, C. M.

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

Stockman, M. I.

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004).
[CrossRef]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef]

Sun, L.

J. A. Fan, K. Bao, L. Sun, J. Bao, V. N. Manoharau, P. Nordlander, and F. Capasso, “Stabilized TiN nanowire arrays for high-performance and flexible supercapacitors,” Nano Lett. 12, 5376–5381 (2012).
[CrossRef]

Sun, Y.

M. Liu, Y. Sun, D. A. Powell, I. V. Shadrivov, M. Lapine, R. C. McPhedran, and Y. S. Kivshar, “Nonlinear response via intrinsic rotation in metamaterials,” Phys. Rev. B 87, 235126 (2013).
[CrossRef]

Tan, H. H.

A. Minovich, J. Farnell, D. N. Neshev, I. McKervacher, F. Karouta, J. Tian, D. A. Powell, I. V. Shadrivov, H. H. Tan, C. Jagadish, and Y. S. Kivshar, “Liquid crystal based nonlinear fishnet metamaterials,” Appl. Phys. Lett. 100, 121113 (2012).
[CrossRef]

Tanaka, K.

X. Meng, U. Guler, A. V. Kidishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).

Tassin, P.

Tian, J.

A. Minovich, J. Farnell, D. N. Neshev, I. McKervacher, F. Karouta, J. Tian, D. A. Powell, I. V. Shadrivov, H. H. Tan, C. Jagadish, and Y. S. Kivshar, “Liquid crystal based nonlinear fishnet metamaterials,” Appl. Phys. Lett. 100, 121113 (2012).
[CrossRef]

Tsai, W.

J. S. Schwinger, L. L. Deraad, K. A. Milton, and W. Tsai, Classical Electrodynamics (Westview, 1998).

Tumkur, T.

M. A. Noginov, Y. A. Bernakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94, 151105 (2009).
[CrossRef]

van Hulst, N.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[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, 4184–4187 (2000).
[CrossRef]

Wang, E.

Wegener, M.

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

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

White, T. P.

Wiener, A.

A. Wiener, A. I. Fernandez-Dominguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal effects in the nanofocusing performance of plasmonic tips,” Nano Lett. 12, 3308–3314 (2012).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, 1964).

Xiao, S.

Yuan, H. K.

Zhang, S.

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]

Zhao, R.

Zharov, A. A.

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

Zhou, T.

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

Zhu, G.

M. A. Noginov, Y. A. Bernakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94, 151105 (2009).
[CrossRef]

Appl. Phys. Lett. (3)

A. Minovich, J. Farnell, D. N. Neshev, I. McKervacher, F. Karouta, J. Tian, D. A. Powell, I. V. Shadrivov, H. H. Tan, C. Jagadish, and Y. S. Kivshar, “Liquid crystal based nonlinear fishnet metamaterials,” Appl. Phys. Lett. 100, 121113 (2012).
[CrossRef]

M. A. Noginov, Y. A. Bernakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94, 151105 (2009).
[CrossRef]

A. B. Golovin and O. D. Lavrentovich, “Electrically reconfigurable optical metamaterial based on colloidal dispersion of metal nanorods in dielectric fluid,” Appl. Phys. Lett. 95, 254104 (2009).
[CrossRef]

J. Opt. (1)

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012).
[CrossRef]

Nano Lett. (2)

J. A. Fan, K. Bao, L. Sun, J. Bao, V. N. Manoharau, P. Nordlander, and F. Capasso, “Stabilized TiN nanowire arrays for high-performance and flexible supercapacitors,” Nano Lett. 12, 5376–5381 (2012).
[CrossRef]

A. Wiener, A. I. Fernandez-Dominguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal effects in the nanofocusing performance of plasmonic tips,” Nano Lett. 12, 3308–3314 (2012).
[CrossRef]

Nat. Mater. (1)

M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. 11, 30–33 (2012).
[CrossRef]

Nat. Photonics (4)

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011).
[CrossRef]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010).
[CrossRef]

I. Cao and M. L. Brongersma, “Active plasmonics: ultrafast developments,” Nat. Photonics 3, 12–13 (2009).
[CrossRef]

A. Alu and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2, 307–310 (2008).
[CrossRef]

Nature (2)

L. Novotny, “Optical antennas tuned to pitch,” Nature 455, 887 (2008).
[CrossRef]

W. L. Barnes, A. Dereux, and A. Ebbensen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. B (1)

M. Liu, Y. Sun, D. A. Powell, I. V. Shadrivov, M. Lapine, R. C. McPhedran, and Y. S. Kivshar, “Nonlinear response via intrinsic rotation in metamaterials,” Phys. Rev. B 87, 235126 (2013).
[CrossRef]

Phys. Rev. Lett. (8)

M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004).
[CrossRef]

A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. 105, 116804 (2010).
[CrossRef]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef]

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]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef]

J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 197401 (2005).
[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]

B. Edwards, A. Alu, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. 103, 153901 (2009).
[CrossRef]

Sci. Rep. (2)

I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Metamaterials with conformational nonlinearity,” Sci. Rep. 1, 138 (2011).

X. Meng, U. Guler, A. V. Kidishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).

Science (5)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006).
[CrossRef]

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

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

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

A. Boltasseva and H. A. Atwater, “Low-loss plasmonic metamaterials,” Science 331, 290–291 (2011).
[CrossRef]

Other (6)

V. M. Shalaev and S. Kawata, Nanophotonics with Surface Plasmons (Elsevier Science, 2007).

P. G. deGennes and J. Prost, The Physics of Liquid Crystals, 2nd ed (Clarendon, 1995).

J. S. Schwinger, L. L. Deraad, K. A. Milton, and W. Tsai, Classical Electrodynamics (Westview, 1998).

L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, 1966).

L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Pergamon, 1987).

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, 1964).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic view of meta-atom and equivalent electric circuit (inset).

Fig. 2.
Fig. 2.

Angular dependence of refraction index for TM-polarized waves. Curves 1–6 correspond to the values of ω/ωs=0.9, 0.96, 1.02, 1.08, 1.14, and 1.2. In the vicinity of resonance, for 1<ω/ωs<1.135, the LMC demonstrates the properties of hyperbolic medium. Concentration of meta-atoms is N=3·107cm3.

Fig. 3.
Fig. 3.

Schematic view of transmission of plane electromagnetic wave through a thin LMC slab placed on a substrate (left panel); typical relations between incident, reflected, and transmitted wave amplitudes, director, and constant electric field applied to the LMC (right panel).

Fig. 4.
Fig. 4.

Stable self-consistent director angle, ϕ/π, on the plane of parameters ω/ωs and β/π. Contour lines with the corresponding values of ϕ/π are shown. Concentration of meta-atoms is N=106cm3, σNL=|Einc/E0|=0.1.

Fig. 5.
Fig. 5.

(a) Transmission T=|T|2+|T|2 and (b) reflection R=|R|2+|R|2 coefficient as a function of normalized frequency ω/ωs at resonance. Dashed curves correspond to β=π/8 and solid ones to β=3π/8; σNL=|Einc/E0|=105, 0.1, 0.1 for curves 1 (1), 2 (2), and 3 (3) respectively; concentration of meta-atoms is N=106cm3; film thickness is h=103cm.

Equations (22)

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R(t)=Rdc+Rac(t)=2l(E0+E)·e0,
d2qdt2+γdqdt+ω02q=Rdc+Rac(t)L,
(e0)x=cosϕsinθ,(e0)y=sinϕsinθ,(e0)z=cosθ.
ddtL(ζi,ζ˙i)ζ˙iL(ζi,ζ˙i)ζi=D(ζi,ζ˙i)ζ˙i,
L=ml2(θ˙2+ϕ˙2sin2θ)+2ql(E0+E)·e0=ml2(θ˙2+ϕ˙2sin2θ)+2ql(Excosϕsinθ+Eysinϕsinθ+Ezcosθ),
D=νl2(θ˙2+ϕ˙2sin2θ),
d2θdt2(dϕdt)2sinθcosθ+ξdθdt=A0q(Excosϕcosθ+EysinϕcosθEzsinθ),
ddt(dϕdtsin2θ)+ξdϕdtsin2θ=A0qsinθ(ExsinϕEycosϕ),
d2qdt2+γdqdt+ω02q=B0(Excosϕsinθ+Eysinϕsinθ+Ezcosθ),
q(t)(Ex(t)cosϕcosθ+Ey(t)sinϕcosθEz(t)sinθ)t=0,q(t)sinθ(Ex(t)sinϕEy(t)cosϕ)t=0,
E=E+4π3P,
qω=B0(Eω·e0)ω02ω2+iγω,
qω=F(ω)B0(Eω·e0),F(ω)=1ωs2ω2+iγω,ωs=ω02ωC2,
χ^=χm(ω)σ^,χm(ω)=34πωC2F(ω),
σ^=(cos2ϕsin2θsinϕcosϕsin2θcosϕsinθcosθsinϕcosϕsin2θsin2ϕsin2θsinϕsinθcosθcosϕsinθcosθsinϕsinθcosθcos2θ).
ε^=εl+4πχm(ω)σ^(ϕ,θ).
nTE2=εl.
nTM2=εl(εl+4πχm(ω))εl+4πχm(ω)cos2ψ,
Etr=Einc+Eref,HtrHincHref=4πcjs×z0=4πciωhP×z0.
tan2ϕ=ζ0sin2β1+ζ0cos2β,
cos(2ϕ)+ζ0cos(2ϕ2β)>0.
T=Etr/Einc=2[cosβ4πik0hχm(ω)cosϕcos(ϕβ)/f2]f1,T=Etr/Einc=2[sinβ4πik0hχm(ω)sinϕcos(ϕβ)/f2]f1,R=Eref/Einc=Tcosβ,R=Eref/Einc=Tsinβ,

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