Abstract

We propose a frequency-tunable and functionality-switchable polarization conversion device in transmission mode by integrating a silicon strip array with a graphene sheet in the mid-infrared region. High cross-polarization transmission, together with the dynamic frequency shift, contributes to a frequency-tunable half-wave plate working from 17.6 to 17.8 THz. Simultaneously, the device can act as a switchable polarizer to selectively generate co- or cross-polarized light by combining the frequency-tunability and a sharp x-shape polarization transmission caused by hybrid Mie resonances. Furthermore, this graphene-based metasurface also realizes continuous 180-degree phase modulation by electrically controlling graphene. The transmitted frequency-tunable and functionality-switchable device with simple-shape inclusions achieves multiple polarization conversions for both linearly and circularly polarized light without refabricating the geometric parameters, which represents a significant advance compared with previously reported counterparts.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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  26. M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3(6), 813–820 (2015).
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    [PubMed]

2017 (7)

M. I. Khan, Q. Fraz, and F. A. Tahir, “Ultra-wideband cross polarization conversion metasurface insensitive to incidence angle,” J. Appl. Phys. 121(4), 045103 (2017).

R. Xia, X. Jing, X. Gui, Y. Tian, and Z. Hong, “Broadband terahertz half-wave plate based on anisotropic polarization conversion metamaterials,” Adv. Opt. Mater. 7(3), 977–988 (2017).

S. K. Earl, T. D. James, D. E. Gómez, R. E. Marvel, R. F. Haglund, and A. Roberts, “Switchable polarization rotation of visible light using a plasmonic metasurface,” APL Photonics 2, 016103 (2017).

M. Chen, W. Sun, J. Cai, L. Chang, and X. Xiao, “Frequency-Tunable Mid-Infrared Cross Polarization Converters Based on Graphene Metasurface,” Plasmonics 12(3), 699–705 (2017).

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

T. Li, L. Huang, J. Liu, Y. Wang, and T. Zentgraf, “Tunable wave plate based on active plasmonic metasurfaces,” Opt. Express 25(4), 4216–4226 (2017).
[PubMed]

J. G. Yun, S.-J. Kim, H. Yun, K. Lee, J. Sung, J. Kim, Y. Lee, and B. Lee, “Broadband ultrathin circular polarizer at visible and near-infrared wavelengths using a non-resonant characteristic in helically stacked nano-gratings,” Opt. Express 25(13), 14260–14269 (2017).
[PubMed]

2016 (5)

T. Guo and C. Argyropoulos, “Broadband polarizers based on graphene metasurfaces,” Opt. Lett. 41(23), 5592–5595 (2016).
[PubMed]

X. Yu, X. Gao, W. Qiao, L. Wen, and W. Yang, “Broadband tunable polarization converter realized by graphene-based metamaterial,” IEEE Photonics Technol. Lett. 28(25), 2399–2402 (2016).

D. Yang, H. Lin, and X. Huang, “Dual Broadband Metamaterial Polarization Converter in Microwave Regime,” Progress In Electromagnetics Research Letters 61, 71–76 (2016).

D. Wang, L. Zhang, Y. Gong, L. Jian, T. Venkatesan, C.-W. Qiu, and M. Hong, “Multiband switchable terahertz quarter-wave plates via phase-change metasurfaces,” IEEE Photonics J. 8(1), 1–8 (2016).

J. Li, P. Yu, H. Cheng, W. Liu, Z. Li, B. Xie, S. Chen, and J. Tian, “Optical polarization encoding using graphene‐loaded plasmonic metasurfaces,” Adv. Opt. Mater. 4(1), 91–98 (2016).

2015 (7)

Y. R. Li and Y.-C. Hung, “Dispersion-free broadband optical polarization rotation based on helix photonic metamaterials,” Opt. Express 23(13), 16772–16781 (2015).
[PubMed]

W. Liu, S. Chen, Z. Li, H. Cheng, P. Yu, J. Li, and J. Tian, “Realization of broadband cross-polarization conversion in transmission mode in the terahertz region using a single-layer metasurface,” Opt. Lett. 40(13), 3185–3188 (2015).
[PubMed]

J. Li, J. Shao, Y.-H. Wang, M.-J. Zhu, J.-Q. Li, and Z.-G. Dong, “Toroidal dipolar response by a dielectric microtube metamaterial in the terahertz regime,” Opt. Express 23(22), 29138–29144 (2015).
[PubMed]

Z. Miao, Q. Wu, X. Li, Q. He, K. Ding, Z. An, and L. Zhou, “Widely tunable terahertz phase modulation with gate-controlled graphene metasurfaces,” Phys. Rev. X 5(4), 041027 (2015).

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3(6), 813–820 (2015).

J. Ding, B. Arigong, H. Ren, J. Shao, M. Zhou, Y. Lin, and H. Zhang, “Mid-infrared tunable dual-frequency cross polarization converters using graphene-based L-shaped nanoslot array,” Plasmonics 10(2), 351–356 (2015).

N. K. Emani, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Graphene: a dynamic platform for electrical control of plasmonic resonance,” Nanophotonics 4(1), 214–223 (2015).

2014 (1)

Y. Z. Cheng, Y. Nie, Z. Z. Cheng, X. Wang, and R. Z. Gong, “Asymmetric chiral metamaterial circular polarizer based on twisted split-ring resonator,” Appl. Phys. B 116(1), 129–134 (2014).

2011 (2)

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[PubMed]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[PubMed]

2010 (2)

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330(6010), 1510–1512 (2010).
[PubMed]

V. Kravets, A. Grigorenko, R. Nair, P. Blake, K. Novoselov, and A. Geim, “Spectroscopic ellipsometry of graphene and an exciton-shifted van Hove peak in absorption,” Phys. Rev. B 81(15), 155413 (2010).

2007 (1)

L. Falkovsky and S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76(15), 153410 (2007).

An, Z.

Z. Miao, Q. Wu, X. Li, Q. He, K. Ding, Z. An, and L. Zhou, “Widely tunable terahertz phase modulation with gate-controlled graphene metasurfaces,” Phys. Rev. X 5(4), 041027 (2015).

Argyropoulos, C.

Arigong, B.

J. Ding, B. Arigong, H. Ren, J. Shao, M. Zhou, Y. Lin, and H. Zhang, “Mid-infrared tunable dual-frequency cross polarization converters using graphene-based L-shaped nanoslot array,” Plasmonics 10(2), 351–356 (2015).

Bao, J.

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

Bechtel, H. A.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[PubMed]

Blake, P.

V. Kravets, A. Grigorenko, R. Nair, P. Blake, K. Novoselov, and A. Geim, “Spectroscopic ellipsometry of graphene and an exciton-shifted van Hove peak in absorption,” Phys. Rev. B 81(15), 155413 (2010).

Boltasseva, A.

N. K. Emani, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Graphene: a dynamic platform for electrical control of plasmonic resonance,” Nanophotonics 4(1), 214–223 (2015).

Brener, I.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3(6), 813–820 (2015).

Cai, J.

M. Chen, W. Sun, J. Cai, L. Chang, and X. Xiao, “Frequency-Tunable Mid-Infrared Cross Polarization Converters Based on Graphene Metasurface,” Plasmonics 12(3), 699–705 (2017).

Chang, L.

M. Chen, W. Sun, J. Cai, L. Chang, and X. Xiao, “Frequency-Tunable Mid-Infrared Cross Polarization Converters Based on Graphene Metasurface,” Plasmonics 12(3), 699–705 (2017).

Chen, M.

M. Chen, W. Sun, J. Cai, L. Chang, and X. Xiao, “Frequency-Tunable Mid-Infrared Cross Polarization Converters Based on Graphene Metasurface,” Plasmonics 12(3), 699–705 (2017).

Chen, S.

J. Li, P. Yu, H. Cheng, W. Liu, Z. Li, B. Xie, S. Chen, and J. Tian, “Optical polarization encoding using graphene‐loaded plasmonic metasurfaces,” Adv. Opt. Mater. 4(1), 91–98 (2016).

W. Liu, S. Chen, Z. Li, H. Cheng, P. Yu, J. Li, and J. Tian, “Realization of broadband cross-polarization conversion in transmission mode in the terahertz region using a single-layer metasurface,” Opt. Lett. 40(13), 3185–3188 (2015).
[PubMed]

Cheng, H.

J. Li, P. Yu, H. Cheng, W. Liu, Z. Li, B. Xie, S. Chen, and J. Tian, “Optical polarization encoding using graphene‐loaded plasmonic metasurfaces,” Adv. Opt. Mater. 4(1), 91–98 (2016).

W. Liu, S. Chen, Z. Li, H. Cheng, P. Yu, J. Li, and J. Tian, “Realization of broadband cross-polarization conversion in transmission mode in the terahertz region using a single-layer metasurface,” Opt. Lett. 40(13), 3185–3188 (2015).
[PubMed]

Cheng, Y. Z.

Y. Z. Cheng, Y. Nie, Z. Z. Cheng, X. Wang, and R. Z. Gong, “Asymmetric chiral metamaterial circular polarizer based on twisted split-ring resonator,” Appl. Phys. B 116(1), 129–134 (2014).

Cheng, Z. Z.

Y. Z. Cheng, Y. Nie, Z. Z. Cheng, X. Wang, and R. Z. Gong, “Asymmetric chiral metamaterial circular polarizer based on twisted split-ring resonator,” Appl. Phys. B 116(1), 129–134 (2014).

Dahal, K.

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

Decker, M.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3(6), 813–820 (2015).

Ding, J.

J. Ding, B. Arigong, H. Ren, J. Shao, M. Zhou, Y. Lin, and H. Zhang, “Mid-infrared tunable dual-frequency cross polarization converters using graphene-based L-shaped nanoslot array,” Plasmonics 10(2), 351–356 (2015).

Ding, K.

Z. Miao, Q. Wu, X. Li, Q. He, K. Ding, Z. An, and L. Zhou, “Widely tunable terahertz phase modulation with gate-controlled graphene metasurfaces,” Phys. Rev. X 5(4), 041027 (2015).

Dominguez, J.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3(6), 813–820 (2015).

Dong, Z.-G.

Earl, S. K.

S. K. Earl, T. D. James, D. E. Gómez, R. E. Marvel, R. F. Haglund, and A. Roberts, “Switchable polarization rotation of visible light using a plasmonic metasurface,” APL Photonics 2, 016103 (2017).

Emani, N. K.

N. K. Emani, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Graphene: a dynamic platform for electrical control of plasmonic resonance,” Nanophotonics 4(1), 214–223 (2015).

Engheta, N.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[PubMed]

Falkner, M.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3(6), 813–820 (2015).

Falkovsky, L.

L. Falkovsky and S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76(15), 153410 (2007).

Fedotov, V. A.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330(6010), 1510–1512 (2010).
[PubMed]

Fraz, Q.

M. I. Khan, Q. Fraz, and F. A. Tahir, “Ultra-wideband cross polarization conversion metasurface insensitive to incidence angle,” J. Appl. Phys. 121(4), 045103 (2017).

Gao, X.

X. Yu, X. Gao, W. Qiao, L. Wen, and W. Yang, “Broadband tunable polarization converter realized by graphene-based metamaterial,” IEEE Photonics Technol. Lett. 28(25), 2399–2402 (2016).

Geim, A.

V. Kravets, A. Grigorenko, R. Nair, P. Blake, K. Novoselov, and A. Geim, “Spectroscopic ellipsometry of graphene and an exciton-shifted van Hove peak in absorption,” Phys. Rev. B 81(15), 155413 (2010).

Geng, B.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[PubMed]

Girit, C.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[PubMed]

Gómez, D. E.

S. K. Earl, T. D. James, D. E. Gómez, R. E. Marvel, R. F. Haglund, and A. Roberts, “Switchable polarization rotation of visible light using a plasmonic metasurface,” APL Photonics 2, 016103 (2017).

Gong, R. Z.

Y. Z. Cheng, Y. Nie, Z. Z. Cheng, X. Wang, and R. Z. Gong, “Asymmetric chiral metamaterial circular polarizer based on twisted split-ring resonator,” Appl. Phys. B 116(1), 129–134 (2014).

Gong, Y.

D. Wang, L. Zhang, Y. Gong, L. Jian, T. Venkatesan, C.-W. Qiu, and M. Hong, “Multiband switchable terahertz quarter-wave plates via phase-change metasurfaces,” IEEE Photonics J. 8(1), 1–8 (2016).

Grigorenko, A.

V. Kravets, A. Grigorenko, R. Nair, P. Blake, K. Novoselov, and A. Geim, “Spectroscopic ellipsometry of graphene and an exciton-shifted van Hove peak in absorption,” Phys. Rev. B 81(15), 155413 (2010).

Gui, X.

R. Xia, X. Jing, X. Gui, Y. Tian, and Z. Hong, “Broadband terahertz half-wave plate based on anisotropic polarization conversion metamaterials,” Adv. Opt. Mater. 7(3), 977–988 (2017).

Guo, T.

Haglund, R. F.

S. K. Earl, T. D. James, D. E. Gómez, R. E. Marvel, R. F. Haglund, and A. Roberts, “Switchable polarization rotation of visible light using a plasmonic metasurface,” APL Photonics 2, 016103 (2017).

Hao, Z.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[PubMed]

He, Q.

Z. Miao, Q. Wu, X. Li, Q. He, K. Ding, Z. An, and L. Zhou, “Widely tunable terahertz phase modulation with gate-controlled graphene metasurfaces,” Phys. Rev. X 5(4), 041027 (2015).

Hong, M.

D. Wang, L. Zhang, Y. Gong, L. Jian, T. Venkatesan, C.-W. Qiu, and M. Hong, “Multiband switchable terahertz quarter-wave plates via phase-change metasurfaces,” IEEE Photonics J. 8(1), 1–8 (2016).

Hong, Z.

R. Xia, X. Jing, X. Gui, Y. Tian, and Z. Hong, “Broadband terahertz half-wave plate based on anisotropic polarization conversion metamaterials,” Adv. Opt. Mater. 7(3), 977–988 (2017).

Horng, J.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[PubMed]

Hu, J.

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

Huang, L.

Huang, X.

D. Yang, H. Lin, and X. Huang, “Dual Broadband Metamaterial Polarization Converter in Microwave Regime,” Progress In Electromagnetics Research Letters 61, 71–76 (2016).

Hung, Y.-C.

James, T. D.

S. K. Earl, T. D. James, D. E. Gómez, R. E. Marvel, R. F. Haglund, and A. Roberts, “Switchable polarization rotation of visible light using a plasmonic metasurface,” APL Photonics 2, 016103 (2017).

Jian, L.

D. Wang, L. Zhang, Y. Gong, L. Jian, T. Venkatesan, C.-W. Qiu, and M. Hong, “Multiband switchable terahertz quarter-wave plates via phase-change metasurfaces,” IEEE Photonics J. 8(1), 1–8 (2016).

Jing, X.

R. Xia, X. Jing, X. Gui, Y. Tian, and Z. Hong, “Broadband terahertz half-wave plate based on anisotropic polarization conversion metamaterials,” Adv. Opt. Mater. 7(3), 977–988 (2017).

Ju, L.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[PubMed]

Kaelberer, T.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330(6010), 1510–1512 (2010).
[PubMed]

Khan, M. I.

M. I. Khan, Q. Fraz, and F. A. Tahir, “Ultra-wideband cross polarization conversion metasurface insensitive to incidence angle,” J. Appl. Phys. 121(4), 045103 (2017).

Kildishev, A. V.

N. K. Emani, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Graphene: a dynamic platform for electrical control of plasmonic resonance,” Nanophotonics 4(1), 214–223 (2015).

Kim, J.

Kim, S.-J.

Kivshar, Y. S.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3(6), 813–820 (2015).

Kravets, V.

V. Kravets, A. Grigorenko, R. Nair, P. Blake, K. Novoselov, and A. Geim, “Spectroscopic ellipsometry of graphene and an exciton-shifted van Hove peak in absorption,” Phys. Rev. B 81(15), 155413 (2010).

Lee, B.

Lee, K.

Lee, Y.

Li, J.

Li, J.-Q.

Li, T.

Li, X.

Z. Miao, Q. Wu, X. Li, Q. He, K. Ding, Z. An, and L. Zhou, “Widely tunable terahertz phase modulation with gate-controlled graphene metasurfaces,” Phys. Rev. X 5(4), 041027 (2015).

Li, Y. R.

Li, Z.

J. Li, P. Yu, H. Cheng, W. Liu, Z. Li, B. Xie, S. Chen, and J. Tian, “Optical polarization encoding using graphene‐loaded plasmonic metasurfaces,” Adv. Opt. Mater. 4(1), 91–98 (2016).

W. Liu, S. Chen, Z. Li, H. Cheng, P. Yu, J. Li, and J. Tian, “Realization of broadband cross-polarization conversion in transmission mode in the terahertz region using a single-layer metasurface,” Opt. Lett. 40(13), 3185–3188 (2015).
[PubMed]

Liang, X.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[PubMed]

Lin, F.

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

Lin, H.

D. Yang, H. Lin, and X. Huang, “Dual Broadband Metamaterial Polarization Converter in Microwave Regime,” Progress In Electromagnetics Research Letters 61, 71–76 (2016).

Lin, Y.

J. Ding, B. Arigong, H. Ren, J. Shao, M. Zhou, Y. Lin, and H. Zhang, “Mid-infrared tunable dual-frequency cross polarization converters using graphene-based L-shaped nanoslot array,” Plasmonics 10(2), 351–356 (2015).

Litvinov, D.

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

Liu, J.

Liu, W.

J. Li, P. Yu, H. Cheng, W. Liu, Z. Li, B. Xie, S. Chen, and J. Tian, “Optical polarization encoding using graphene‐loaded plasmonic metasurfaces,” Adv. Opt. Mater. 4(1), 91–98 (2016).

W. Liu, S. Chen, Z. Li, H. Cheng, P. Yu, J. Li, and J. Tian, “Realization of broadband cross-polarization conversion in transmission mode in the terahertz region using a single-layer metasurface,” Opt. Lett. 40(13), 3185–3188 (2015).
[PubMed]

Liu, Z.

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

Martin, M.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[PubMed]

Marvel, R. E.

S. K. Earl, T. D. James, D. E. Gómez, R. E. Marvel, R. F. Haglund, and A. Roberts, “Switchable polarization rotation of visible light using a plasmonic metasurface,” APL Photonics 2, 016103 (2017).

Miao, Z.

Z. Miao, Q. Wu, X. Li, Q. He, K. Ding, Z. An, and L. Zhou, “Widely tunable terahertz phase modulation with gate-controlled graphene metasurfaces,” Phys. Rev. X 5(4), 041027 (2015).

Nair, R.

V. Kravets, A. Grigorenko, R. Nair, P. Blake, K. Novoselov, and A. Geim, “Spectroscopic ellipsometry of graphene and an exciton-shifted van Hove peak in absorption,” Phys. Rev. B 81(15), 155413 (2010).

Neshev, D. N.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3(6), 813–820 (2015).

Nie, Y.

Y. Z. Cheng, Y. Nie, Z. Z. Cheng, X. Wang, and R. Z. Gong, “Asymmetric chiral metamaterial circular polarizer based on twisted split-ring resonator,” Appl. Phys. B 116(1), 129–134 (2014).

Niu, C.

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

Novoselov, K.

V. Kravets, A. Grigorenko, R. Nair, P. Blake, K. Novoselov, and A. Geim, “Spectroscopic ellipsometry of graphene and an exciton-shifted van Hove peak in absorption,” Phys. Rev. B 81(15), 155413 (2010).

Papasimakis, N.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330(6010), 1510–1512 (2010).
[PubMed]

Pershoguba, S.

L. Falkovsky and S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76(15), 153410 (2007).

Pertsch, T.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3(6), 813–820 (2015).

Qiao, W.

X. Yu, X. Gao, W. Qiao, L. Wen, and W. Yang, “Broadband tunable polarization converter realized by graphene-based metamaterial,” IEEE Photonics Technol. Lett. 28(25), 2399–2402 (2016).

Qiu, C.-W.

D. Wang, L. Zhang, Y. Gong, L. Jian, T. Venkatesan, C.-W. Qiu, and M. Hong, “Multiband switchable terahertz quarter-wave plates via phase-change metasurfaces,” IEEE Photonics J. 8(1), 1–8 (2016).

Qiu, W.

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

Ren, H.

J. Ding, B. Arigong, H. Ren, J. Shao, M. Zhou, Y. Lin, and H. Zhang, “Mid-infrared tunable dual-frequency cross polarization converters using graphene-based L-shaped nanoslot array,” Plasmonics 10(2), 351–356 (2015).

Ren, Z.

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

Roberts, A.

S. K. Earl, T. D. James, D. E. Gómez, R. E. Marvel, R. F. Haglund, and A. Roberts, “Switchable polarization rotation of visible light using a plasmonic metasurface,” APL Photonics 2, 016103 (2017).

Shalaev, V. M.

N. K. Emani, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Graphene: a dynamic platform for electrical control of plasmonic resonance,” Nanophotonics 4(1), 214–223 (2015).

Shao, J.

J. Li, J. Shao, Y.-H. Wang, M.-J. Zhu, J.-Q. Li, and Z.-G. Dong, “Toroidal dipolar response by a dielectric microtube metamaterial in the terahertz regime,” Opt. Express 23(22), 29138–29144 (2015).
[PubMed]

J. Ding, B. Arigong, H. Ren, J. Shao, M. Zhou, Y. Lin, and H. Zhang, “Mid-infrared tunable dual-frequency cross polarization converters using graphene-based L-shaped nanoslot array,” Plasmonics 10(2), 351–356 (2015).

Shen, Y. R.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[PubMed]

Staude, I.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3(6), 813–820 (2015).

Sun, W.

M. Chen, W. Sun, J. Cai, L. Chang, and X. Xiao, “Frequency-Tunable Mid-Infrared Cross Polarization Converters Based on Graphene Metasurface,” Plasmonics 12(3), 699–705 (2017).

Sung, J.

Tahir, F. A.

M. I. Khan, Q. Fraz, and F. A. Tahir, “Ultra-wideband cross polarization conversion metasurface insensitive to incidence angle,” J. Appl. Phys. 121(4), 045103 (2017).

Tian, J.

J. Li, P. Yu, H. Cheng, W. Liu, Z. Li, B. Xie, S. Chen, and J. Tian, “Optical polarization encoding using graphene‐loaded plasmonic metasurfaces,” Adv. Opt. Mater. 4(1), 91–98 (2016).

W. Liu, S. Chen, Z. Li, H. Cheng, P. Yu, J. Li, and J. Tian, “Realization of broadband cross-polarization conversion in transmission mode in the terahertz region using a single-layer metasurface,” Opt. Lett. 40(13), 3185–3188 (2015).
[PubMed]

Tian, Y.

R. Xia, X. Jing, X. Gui, Y. Tian, and Z. Hong, “Broadband terahertz half-wave plate based on anisotropic polarization conversion metamaterials,” Adv. Opt. Mater. 7(3), 977–988 (2017).

Tsai, D. P.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330(6010), 1510–1512 (2010).
[PubMed]

Vakil, A.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[PubMed]

Venkatesan, T.

D. Wang, L. Zhang, Y. Gong, L. Jian, T. Venkatesan, C.-W. Qiu, and M. Hong, “Multiband switchable terahertz quarter-wave plates via phase-change metasurfaces,” IEEE Photonics J. 8(1), 1–8 (2016).

Wang, D.

D. Wang, L. Zhang, Y. Gong, L. Jian, T. Venkatesan, C.-W. Qiu, and M. Hong, “Multiband switchable terahertz quarter-wave plates via phase-change metasurfaces,” IEEE Photonics J. 8(1), 1–8 (2016).

Wang, F.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[PubMed]

Wang, X.

Y. Z. Cheng, Y. Nie, Z. Z. Cheng, X. Wang, and R. Z. Gong, “Asymmetric chiral metamaterial circular polarizer based on twisted split-ring resonator,” Appl. Phys. B 116(1), 129–134 (2014).

Wang, Y.

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

T. Li, L. Huang, J. Liu, Y. Wang, and T. Zentgraf, “Tunable wave plate based on active plasmonic metasurfaces,” Opt. Express 25(4), 4216–4226 (2017).
[PubMed]

Wang, Y.-H.

Wang, Z. M.

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

Wen, L.

X. Yu, X. Gao, W. Qiao, L. Wen, and W. Yang, “Broadband tunable polarization converter realized by graphene-based metamaterial,” IEEE Photonics Technol. Lett. 28(25), 2399–2402 (2016).

Wu, Q.

Z. Miao, Q. Wu, X. Li, Q. He, K. Ding, Z. An, and L. Zhou, “Widely tunable terahertz phase modulation with gate-controlled graphene metasurfaces,” Phys. Rev. X 5(4), 041027 (2015).

Xia, R.

R. Xia, X. Jing, X. Gui, Y. Tian, and Z. Hong, “Broadband terahertz half-wave plate based on anisotropic polarization conversion metamaterials,” Adv. Opt. Mater. 7(3), 977–988 (2017).

Xiao, X.

M. Chen, W. Sun, J. Cai, L. Chang, and X. Xiao, “Frequency-Tunable Mid-Infrared Cross Polarization Converters Based on Graphene Metasurface,” Plasmonics 12(3), 699–705 (2017).

Xie, B.

J. Li, P. Yu, H. Cheng, W. Liu, Z. Li, B. Xie, S. Chen, and J. Tian, “Optical polarization encoding using graphene‐loaded plasmonic metasurfaces,” Adv. Opt. Mater. 4(1), 91–98 (2016).

Yang, D.

D. Yang, H. Lin, and X. Huang, “Dual Broadband Metamaterial Polarization Converter in Microwave Regime,” Progress In Electromagnetics Research Letters 61, 71–76 (2016).

Yang, W.

X. Yu, X. Gao, W. Qiao, L. Wen, and W. Yang, “Broadband tunable polarization converter realized by graphene-based metamaterial,” IEEE Photonics Technol. Lett. 28(25), 2399–2402 (2016).

Yu, P.

J. Li, P. Yu, H. Cheng, W. Liu, Z. Li, B. Xie, S. Chen, and J. Tian, “Optical polarization encoding using graphene‐loaded plasmonic metasurfaces,” Adv. Opt. Mater. 4(1), 91–98 (2016).

W. Liu, S. Chen, Z. Li, H. Cheng, P. Yu, J. Li, and J. Tian, “Realization of broadband cross-polarization conversion in transmission mode in the terahertz region using a single-layer metasurface,” Opt. Lett. 40(13), 3185–3188 (2015).
[PubMed]

Yu, X.

X. Yu, X. Gao, W. Qiao, L. Wen, and W. Yang, “Broadband tunable polarization converter realized by graphene-based metamaterial,” IEEE Photonics Technol. Lett. 28(25), 2399–2402 (2016).

Yun, H.

Yun, J. G.

Zentgraf, T.

Zettl, A.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[PubMed]

Zhang, H.

J. Ding, B. Arigong, H. Ren, J. Shao, M. Zhou, Y. Lin, and H. Zhang, “Mid-infrared tunable dual-frequency cross polarization converters using graphene-based L-shaped nanoslot array,” Plasmonics 10(2), 351–356 (2015).

Zhang, L.

D. Wang, L. Zhang, Y. Gong, L. Jian, T. Venkatesan, C.-W. Qiu, and M. Hong, “Multiband switchable terahertz quarter-wave plates via phase-change metasurfaces,” IEEE Photonics J. 8(1), 1–8 (2016).

Zhao, Z.

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

Zheludev, N. I.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330(6010), 1510–1512 (2010).
[PubMed]

Zhou, L.

Z. Miao, Q. Wu, X. Li, Q. He, K. Ding, Z. An, and L. Zhou, “Widely tunable terahertz phase modulation with gate-controlled graphene metasurfaces,” Phys. Rev. X 5(4), 041027 (2015).

Zhou, M.

J. Ding, B. Arigong, H. Ren, J. Shao, M. Zhou, Y. Lin, and H. Zhang, “Mid-infrared tunable dual-frequency cross polarization converters using graphene-based L-shaped nanoslot array,” Plasmonics 10(2), 351–356 (2015).

Zhou, X.

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

Zhu, M.-J.

Zhu, Z.

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

Adv. Mater. (1)

F. Lin, Z. Zhu, X. Zhou, W. Qiu, C. Niu, J. Hu, K. Dahal, Y. Wang, Z. Zhao, Z. Ren, D. Litvinov, Z. Liu, Z. M. Wang, and J. Bao, “Orientation control of graphene flakes by magnetic field: broad device applications of macroscopically aligned graphene,” Adv. Mater. 29(1), 1604453 (2017).
[PubMed]

Adv. Opt. Mater. (3)

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3(6), 813–820 (2015).

R. Xia, X. Jing, X. Gui, Y. Tian, and Z. Hong, “Broadband terahertz half-wave plate based on anisotropic polarization conversion metamaterials,” Adv. Opt. Mater. 7(3), 977–988 (2017).

J. Li, P. Yu, H. Cheng, W. Liu, Z. Li, B. Xie, S. Chen, and J. Tian, “Optical polarization encoding using graphene‐loaded plasmonic metasurfaces,” Adv. Opt. Mater. 4(1), 91–98 (2016).

APL Photonics (1)

S. K. Earl, T. D. James, D. E. Gómez, R. E. Marvel, R. F. Haglund, and A. Roberts, “Switchable polarization rotation of visible light using a plasmonic metasurface,” APL Photonics 2, 016103 (2017).

Appl. Phys. B (1)

Y. Z. Cheng, Y. Nie, Z. Z. Cheng, X. Wang, and R. Z. Gong, “Asymmetric chiral metamaterial circular polarizer based on twisted split-ring resonator,” Appl. Phys. B 116(1), 129–134 (2014).

IEEE Photonics J. (1)

D. Wang, L. Zhang, Y. Gong, L. Jian, T. Venkatesan, C.-W. Qiu, and M. Hong, “Multiband switchable terahertz quarter-wave plates via phase-change metasurfaces,” IEEE Photonics J. 8(1), 1–8 (2016).

IEEE Photonics Technol. Lett. (1)

X. Yu, X. Gao, W. Qiao, L. Wen, and W. Yang, “Broadband tunable polarization converter realized by graphene-based metamaterial,” IEEE Photonics Technol. Lett. 28(25), 2399–2402 (2016).

J. Appl. Phys. (1)

M. I. Khan, Q. Fraz, and F. A. Tahir, “Ultra-wideband cross polarization conversion metasurface insensitive to incidence angle,” J. Appl. Phys. 121(4), 045103 (2017).

Nanophotonics (1)

N. K. Emani, A. V. Kildishev, V. M. Shalaev, and A. Boltasseva, “Graphene: a dynamic platform for electrical control of plasmonic resonance,” Nanophotonics 4(1), 214–223 (2015).

Nat. Nanotechnol. (1)

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and F. Wang, “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnol. 6(10), 630–634 (2011).
[PubMed]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. B (2)

V. Kravets, A. Grigorenko, R. Nair, P. Blake, K. Novoselov, and A. Geim, “Spectroscopic ellipsometry of graphene and an exciton-shifted van Hove peak in absorption,” Phys. Rev. B 81(15), 155413 (2010).

L. Falkovsky and S. Pershoguba, “Optical far-infrared properties of a graphene monolayer and multilayer,” Phys. Rev. B 76(15), 153410 (2007).

Phys. Rev. X (1)

Z. Miao, Q. Wu, X. Li, Q. He, K. Ding, Z. An, and L. Zhou, “Widely tunable terahertz phase modulation with gate-controlled graphene metasurfaces,” Phys. Rev. X 5(4), 041027 (2015).

Plasmonics (2)

M. Chen, W. Sun, J. Cai, L. Chang, and X. Xiao, “Frequency-Tunable Mid-Infrared Cross Polarization Converters Based on Graphene Metasurface,” Plasmonics 12(3), 699–705 (2017).

J. Ding, B. Arigong, H. Ren, J. Shao, M. Zhou, Y. Lin, and H. Zhang, “Mid-infrared tunable dual-frequency cross polarization converters using graphene-based L-shaped nanoslot array,” Plasmonics 10(2), 351–356 (2015).

Progress In Electromagnetics Research Letters (1)

D. Yang, H. Lin, and X. Huang, “Dual Broadband Metamaterial Polarization Converter in Microwave Regime,” Progress In Electromagnetics Research Letters 61, 71–76 (2016).

Science (2)

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[PubMed]

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330(6010), 1510–1512 (2010).
[PubMed]

Other (2)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Elsevier, 2013).

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

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Figures (6)

Fig. 1
Fig. 1 (a) The silicon strip array in CST Microwave Studio. (b) A unit cell of silicon array, a = 10 μm, b = 4 μm, and d = 7 μm. (c) Polarization transmission coefficients of silicon strip array. (d) The scattered intensities of electric dipole ( I p ), magnetic dipole ( I M ), electric quadrupole ( I Q e ), and magnetic quadrupole ( I Q m ) resonances.
Fig. 2
Fig. 2 The polarization conversion rate (PCR) of the proposed half-wave plate and the phase difference between co- and cross-polarized transmitted waves.
Fig. 3
Fig. 3 (a) The graphene-based metasurface consists of silicon strip array covered with a graphene sheet, (b) Co- and (c) cross-polarization transmission coefficients with different graphene’s Fermi energies.
Fig. 4
Fig. 4 (a) Polarization transmission coefficients of a switchable polarizer both for linearly and circular polarized light. (b) PSR of i(j)-polarized light with E F of 0(1.2) eV, the inset shows the enlarged drawing of PSRs.
Fig. 5
Fig. 5 The multiple functions of the electrically controlled switchable polarizer for linear (a) and circular (b) polarized light.
Fig. 6
Fig. 6 (a) Co- and (b) cross-polarization transmission coefficients with the Fermi energies of 0, 0.6 and 1.5 eV. (c) The phase differences of co- and cross-polarized transmitted light with the Fermi energies of 0, 0.6 and 1.5 eV in 17.85 THz.

Equations (6)

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

electric dipole moment: P = 1 iω J d 3 r,
magnetic dipole moment: M = 1 2c ( r × J ) d 3 r,
electric quadrupole moment: Q αβ = 1 i2ω [ r a J β + r β J α 2 3 ( r J ) δ αβ ] d 3 r,
magnetic quadrupole moment: M αβ = 1 3c [ ( r × J ) α r β + ( r × J ) β r α ] d 3 r,
σ(ω)= 2 e 2 ω T π i ω+i τ 1 log[ 2cosh( ω F 2 ω T ) ]+ e 2 4 [ H( ω 2 )+i 2ω π 0 H( ω' 2 )H( ω 2 ) ω 2 ω ' 2 dω' ]
Δω ω 0 = V dV[(Δ μ H 0 ) H 0 * +(Δ ε E 0 ) E 0 * ] V dV (μ H 0 H 0 * +ε E 0 E 0 * ) = Δ W m +Δ W e W m + W e

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