Abstract

A rotation-angle variable polarization rotator is proposed and demonstrated using an all-dielectric metasurface doublet. Such a transmissive device can conveniently rotate the polarization of incident light by any desired angles by mechanically changing the relative angle of the double metasurface layers regardless of the incident state of polarization. Under certain conditions the device acts as a full phase modulator for the circularly polarized incident wave. Hence, it has a promising application in polarization and phase control.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
  27. H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
    [Crossref]
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    [Crossref] [PubMed]

2018 (6)

A. Messaadi, M. M. Sánchez-López, A. Vargas, P. García-Martínez, and I. Moreno, “Achromatic linear retarder with tunable retardance,” Opt. Lett. 43(14), 3277–3280 (2018).
[Crossref] [PubMed]

A. Pal, S. N. Shirodkar, P. Deshmukh, H. Surdi, B. R. Sangala, S. S. Prabhu, K. C. Rustagi, U. V. Waghmare, and P. Ayyub, “Rotation of terahertz radiation due to phonon-mediated magnetoelectric coupling in chiral selenium,” Phys. Rev. B 98(23), 235141 (2018).
[Crossref]

W. FitzGerald, S. Taherion, F. J. Kumar, D. Giles, and D. Hore, “Cadmium zinc telluride as a mid-infrared variable retarder,” J. Appl. Phys. 123(13), 133103 (2018).
[Crossref]

W. S. L. Lee, R. T. Ako, M. X. Low, M. Bhaskaran, S. Sriram, C. Fumeaux, and W. Withayachumnankul, “Dielectric-resonator metasurfaces for broadband terahertz quarter- and half-wave mirrors,” Opt. Express 26(11), 14392–14406 (2018).
[Crossref] [PubMed]

J. Zi, Q. Xu, Q. Wang, C. Tian, Y. Li, X. Zhang, J. Han, and W. Zhang, “Antireflection-assisted all-dielectric terahertz metamaterial polarization converter,” Appl. Phys. Lett. 113(10), 101104 (2018).
[Crossref]

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

2017 (1)

E. Yatsuka, T. Yamamoto, T. Hatae, K. Torimoto, and K. Itami, “Note: Lossless laser beam combiner employing a high-speed rotating half-wave plate,” Rev. Sci. Instrum. 88(7), 076107 (2017).
[Crossref] [PubMed]

2016 (4)

2015 (1)

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015).
[Crossref] [PubMed]

2014 (2)

H. Ma, G. Wang, G. Kong, and T. Cui, “Broadband circular and linear polarization conversions realized by thin birefringent reflective metasurfaces,” Opt. Mater. Express 4(8), 1717–1724 (2014).
[Crossref]

L. Cong, N. Xu, J. Gu, R. Singh, J. Han, and W. Zhang, “Highly flexible broadband terahertz metamaterial quarter-wave plate,” Laser Photonics Rev. 8(4), 626–632 (2014).
[Crossref]

2013 (3)

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Gu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103(17), 171107 (2013).
[Crossref]

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz Metamaterials for Linear Polarization Conversion and Anomalous Refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref] [PubMed]

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film Faraday rotation,” Nat. Commun.  4, 1599 (2013).

2012 (1)

2011 (1)

2010 (2)

Y.-L. Yang, Z.-H. Ding, K. Wang, L. Wu, and L. Wu, “Full-field optical coherence tomography by achromatic phase shifting with a rotating half-wave plate,” J. Opt. 12(3), 035301 (2010).
[Crossref]

Y. Ye and S. He, “90 degrees polarization rotator using a bilayered chiral metamaterial with giant optical activity,” Appl. Phys. Lett. 96(20), 203501 (2010).
[Crossref]

2009 (2)

Q. Zhu, I. M. Stockford, J. A. Crowe, and S. P. Morgan, “Liquid Crystal Based Rotating Orthogonal Polarization Imaging System,” J. Innov. Opt. Health Sci. 2(3), 245–251 (2009).
[Crossref]

M. Nur-E-Alam, M. Vasiliev, and K. Alameh, “Nano-structured magnetic photonic crystals for magneto-optic polarization controllers at the communication-band wavelengths,” Opt. Quantum Electron. 41(9), 661–669 (2009).
[Crossref]

2007 (2)

2006 (1)

H.-Y. Wu, C.-F. Hsieh, T.-T. Tang, R.-P. Pan, and C.-L. Pan, “Electrically tunable room-temperature 2π liquid crystal terahertz phase shifter,” IEEE Photonics Technol. Lett. 18(13–16), 1488–1490 (2006).

2000 (1)

E. Blomme, G. Gondek, T. Katkowski, P. Kwiek, A. Sliwinski, and O. Leroy, “On the polarization of light diffracted by ultrasound,” Ultrasonics 38(1-8), 575–580 (2000).
[Crossref] [PubMed]

1996 (1)

Ako, R. T.

Alameh, K.

M. Nur-E-Alam, M. Vasiliev, and K. Alameh, “Nano-structured magnetic photonic crystals for magneto-optic polarization controllers at the communication-band wavelengths,” Opt. Quantum Electron. 41(9), 661–669 (2009).
[Crossref]

An, Y.

Arbabi, A.

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015).
[Crossref] [PubMed]

Arikawa, T.

Ayyub, P.

A. Pal, S. N. Shirodkar, P. Deshmukh, H. Surdi, B. R. Sangala, S. S. Prabhu, K. C. Rustagi, U. V. Waghmare, and P. Ayyub, “Rotation of terahertz radiation due to phonon-mediated magnetoelectric coupling in chiral selenium,” Phys. Rev. B 98(23), 235141 (2018).
[Crossref]

Azad, A. K.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz Metamaterials for Linear Polarization Conversion and Anomalous Refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref] [PubMed]

Bagheri, M.

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015).
[Crossref] [PubMed]

Belotelov, V. I.

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film Faraday rotation,” Nat. Commun.  4, 1599 (2013).

Belyanin, A. A.

Bhaskaran, M.

Blomme, E.

E. Blomme, G. Gondek, T. Katkowski, P. Kwiek, A. Sliwinski, and O. Leroy, “On the polarization of light diffracted by ultrasound,” Ultrasonics 38(1-8), 575–580 (2000).
[Crossref] [PubMed]

Cao, W.

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Gu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103(17), 171107 (2013).
[Crossref]

Challener, W. A.

Chen, H.-T.

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016).
[Crossref] [PubMed]

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz Metamaterials for Linear Polarization Conversion and Anomalous Refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref] [PubMed]

Chin, J. Y.

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film Faraday rotation,” Nat. Commun.  4, 1599 (2013).

Chowdhury, D. R.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz Metamaterials for Linear Polarization Conversion and Anomalous Refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref] [PubMed]

Cong, L.

L. Cong, N. Xu, J. Gu, R. Singh, J. Han, and W. Zhang, “Highly flexible broadband terahertz metamaterial quarter-wave plate,” Laser Photonics Rev. 8(4), 626–632 (2014).
[Crossref]

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Gu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103(17), 171107 (2013).
[Crossref]

Crowe, J. A.

Q. Zhu, I. M. Stockford, J. A. Crowe, and S. P. Morgan, “Liquid Crystal Based Rotating Orthogonal Polarization Imaging System,” J. Innov. Opt. Health Sci. 2(3), 245–251 (2009).
[Crossref]

Cui, T.

Cundiff, S. T.

Dalvit, D. A. R.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz Metamaterials for Linear Polarization Conversion and Anomalous Refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref] [PubMed]

Deshmukh, P.

A. Pal, S. N. Shirodkar, P. Deshmukh, H. Surdi, B. R. Sangala, S. S. Prabhu, K. C. Rustagi, U. V. Waghmare, and P. Ayyub, “Rotation of terahertz radiation due to phonon-mediated magnetoelectric coupling in chiral selenium,” Phys. Rev. B 98(23), 235141 (2018).
[Crossref]

Ding, Z.-H.

Y.-L. Yang, Z.-H. Ding, K. Wang, L. Wu, and L. Wu, “Full-field optical coherence tomography by achromatic phase shifting with a rotating half-wave plate,” J. Opt. 12(3), 035301 (2010).
[Crossref]

Dregely, D.

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film Faraday rotation,” Nat. Commun.  4, 1599 (2013).

Faraon, A.

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015).
[Crossref] [PubMed]

FitzGerald, W.

W. FitzGerald, S. Taherion, F. J. Kumar, D. Giles, and D. Hore, “Cadmium zinc telluride as a mid-infrared variable retarder,” J. Appl. Phys. 123(13), 133103 (2018).
[Crossref]

Fumeaux, C.

García-Martínez, P.

Giessen, H.

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film Faraday rotation,” Nat. Commun.  4, 1599 (2013).

Giles, D.

W. FitzGerald, S. Taherion, F. J. Kumar, D. Giles, and D. Hore, “Cadmium zinc telluride as a mid-infrared variable retarder,” J. Appl. Phys. 123(13), 133103 (2018).
[Crossref]

Gondek, G.

E. Blomme, G. Gondek, T. Katkowski, P. Kwiek, A. Sliwinski, and O. Leroy, “On the polarization of light diffracted by ultrasound,” Ultrasonics 38(1-8), 575–580 (2000).
[Crossref] [PubMed]

Grady, N. K.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz Metamaterials for Linear Polarization Conversion and Anomalous Refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref] [PubMed]

Gu, J.

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

L. Cong, N. Xu, J. Gu, R. Singh, J. Han, and W. Zhang, “Highly flexible broadband terahertz metamaterial quarter-wave plate,” Laser Photonics Rev. 8(4), 626–632 (2014).
[Crossref]

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Gu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103(17), 171107 (2013).
[Crossref]

Han, J.

J. Zi, Q. Xu, Q. Wang, C. Tian, Y. Li, X. Zhang, J. Han, and W. Zhang, “Antireflection-assisted all-dielectric terahertz metamaterial polarization converter,” Appl. Phys. Lett. 113(10), 101104 (2018).
[Crossref]

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

L. Cong, N. Xu, J. Gu, R. Singh, J. Han, and W. Zhang, “Highly flexible broadband terahertz metamaterial quarter-wave plate,” Laser Photonics Rev. 8(4), 626–632 (2014).
[Crossref]

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Gu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103(17), 171107 (2013).
[Crossref]

Hatae, T.

E. Yatsuka, T. Yamamoto, T. Hatae, K. Torimoto, and K. Itami, “Note: Lossless laser beam combiner employing a high-speed rotating half-wave plate,” Rev. Sci. Instrum. 88(7), 076107 (2017).
[Crossref] [PubMed]

He, S.

Y. Ye and S. He, “90 degrees polarization rotator using a bilayered chiral metamaterial with giant optical activity,” Appl. Phys. Lett. 96(20), 203501 (2010).
[Crossref]

Heyes, J. E.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz Metamaterials for Linear Polarization Conversion and Anomalous Refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref] [PubMed]

Hore, D.

W. FitzGerald, S. Taherion, F. J. Kumar, D. Giles, and D. Hore, “Cadmium zinc telluride as a mid-infrared variable retarder,” J. Appl. Phys. 123(13), 133103 (2018).
[Crossref]

Horie, Y.

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015).
[Crossref] [PubMed]

Hsieh, C.-F.

H.-Y. Wu, C.-F. Hsieh, T.-T. Tang, R.-P. Pan, and C.-L. Pan, “Electrically tunable room-temperature 2π liquid crystal terahertz phase shifter,” IEEE Photonics Technol. Lett. 18(13–16), 1488–1490 (2006).

Hu, C.

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

Hu, Y.

Huang, K.

L. Zhang, S. Mei, K. Huang, and C.-W. Qiu, “Advances in Full Control of Electromagnetic Waves with Metasurfaces,” Adv. Opt. Mater. 4(6), 818–833 (2016).
[Crossref]

Itami, K.

E. Yatsuka, T. Yamamoto, T. Hatae, K. Torimoto, and K. Itami, “Note: Lossless laser beam combiner employing a high-speed rotating half-wave plate,” Rev. Sci. Instrum. 88(7), 076107 (2017).
[Crossref] [PubMed]

Jiang, L.

Kanda, N.

Katkowski, T.

E. Blomme, G. Gondek, T. Katkowski, P. Kwiek, A. Sliwinski, and O. Leroy, “On the polarization of light diffracted by ultrasound,” Ultrasonics 38(1-8), 575–580 (2000).
[Crossref] [PubMed]

Koch, M.

Kong, G.

Konishi, K.

Kono, J.

Kumar, F. J.

W. FitzGerald, S. Taherion, F. J. Kumar, D. Giles, and D. Hore, “Cadmium zinc telluride as a mid-infrared variable retarder,” J. Appl. Phys. 123(13), 133103 (2018).
[Crossref]

Kuwata-Gonokami, M.

Kwiek, P.

E. Blomme, G. Gondek, T. Katkowski, P. Kwiek, A. Sliwinski, and O. Leroy, “On the polarization of light diffracted by ultrasound,” Ultrasonics 38(1-8), 575–580 (2000).
[Crossref] [PubMed]

Lee, W. S. L.

Leroy, O.

E. Blomme, G. Gondek, T. Katkowski, P. Kwiek, A. Sliwinski, and O. Leroy, “On the polarization of light diffracted by ultrasound,” Ultrasonics 38(1-8), 575–580 (2000).
[Crossref] [PubMed]

Li, N.

Li, Y.

J. Zi, Q. Xu, Q. Wang, C. Tian, Y. Li, X. Zhang, J. Han, and W. Zhang, “Antireflection-assisted all-dielectric terahertz metamaterial polarization converter,” Appl. Phys. Lett. 113(10), 101104 (2018).
[Crossref]

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

W. Mo, X. Wei, K. Wang, Y. Li, and J. Liu, “Ultrathin flexible terahertz polarization converter based on metasurfaces,” Opt. Express 24(12), 13621–13627 (2016).
[Crossref] [PubMed]

Liu, J.

Low, M. X.

Ma, H.

Mei, S.

L. Zhang, S. Mei, K. Huang, and C.-W. Qiu, “Advances in Full Control of Electromagnetic Waves with Metasurfaces,” Adv. Opt. Mater. 4(6), 818–833 (2016).
[Crossref]

Messaadi, A.

Mo, W.

Moreno, I.

Morgan, S. P.

Q. Zhu, I. M. Stockford, J. A. Crowe, and S. P. Morgan, “Liquid Crystal Based Rotating Orthogonal Polarization Imaging System,” J. Innov. Opt. Health Sci. 2(3), 245–251 (2009).
[Crossref]

Nur-E-Alam, M.

M. Nur-E-Alam, M. Vasiliev, and K. Alameh, “Nano-structured magnetic photonic crystals for magneto-optic polarization controllers at the communication-band wavelengths,” Opt. Quantum Electron. 41(9), 661–669 (2009).
[Crossref]

Ouyang, C.

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

Pal, A.

A. Pal, S. N. Shirodkar, P. Deshmukh, H. Surdi, B. R. Sangala, S. S. Prabhu, K. C. Rustagi, U. V. Waghmare, and P. Ayyub, “Rotation of terahertz radiation due to phonon-mediated magnetoelectric coupling in chiral selenium,” Phys. Rev. B 98(23), 235141 (2018).
[Crossref]

Pan, C.-L.

H.-Y. Wu, C.-F. Hsieh, T.-T. Tang, R.-P. Pan, and C.-L. Pan, “Electrically tunable room-temperature 2π liquid crystal terahertz phase shifter,” IEEE Photonics Technol. Lett. 18(13–16), 1488–1490 (2006).

Pan, R.-P.

H.-Y. Wu, C.-F. Hsieh, T.-T. Tang, R.-P. Pan, and C.-L. Pan, “Electrically tunable room-temperature 2π liquid crystal terahertz phase shifter,” IEEE Photonics Technol. Lett. 18(13–16), 1488–1490 (2006).

Petrov, N. I.

Prabhu, S. S.

A. Pal, S. N. Shirodkar, P. Deshmukh, H. Surdi, B. R. Sangala, S. S. Prabhu, K. C. Rustagi, U. V. Waghmare, and P. Ayyub, “Rotation of terahertz radiation due to phonon-mediated magnetoelectric coupling in chiral selenium,” Phys. Rev. B 98(23), 235141 (2018).
[Crossref]

Qiu, C.-W.

L. Zhang, S. Mei, K. Huang, and C.-W. Qiu, “Advances in Full Control of Electromagnetic Waves with Metasurfaces,” Adv. Opt. Mater. 4(6), 818–833 (2016).
[Crossref]

Reiten, M. T.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz Metamaterials for Linear Polarization Conversion and Anomalous Refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref] [PubMed]

Rustagi, K. C.

A. Pal, S. N. Shirodkar, P. Deshmukh, H. Surdi, B. R. Sangala, S. S. Prabhu, K. C. Rustagi, U. V. Waghmare, and P. Ayyub, “Rotation of terahertz radiation due to phonon-mediated magnetoelectric coupling in chiral selenium,” Phys. Rev. B 98(23), 235141 (2018).
[Crossref]

Sánchez-López, M. M.

Sangala, B. R.

A. Pal, S. N. Shirodkar, P. Deshmukh, H. Surdi, B. R. Sangala, S. S. Prabhu, K. C. Rustagi, U. V. Waghmare, and P. Ayyub, “Rotation of terahertz radiation due to phonon-mediated magnetoelectric coupling in chiral selenium,” Phys. Rev. B 98(23), 235141 (2018).
[Crossref]

Scheller, M.

Scherger, B.

Shirodkar, S. N.

A. Pal, S. N. Shirodkar, P. Deshmukh, H. Surdi, B. R. Sangala, S. S. Prabhu, K. C. Rustagi, U. V. Waghmare, and P. Ayyub, “Rotation of terahertz radiation due to phonon-mediated magnetoelectric coupling in chiral selenium,” Phys. Rev. B 98(23), 235141 (2018).
[Crossref]

Singh, R.

L. Cong, N. Xu, J. Gu, R. Singh, J. Han, and W. Zhang, “Highly flexible broadband terahertz metamaterial quarter-wave plate,” Laser Photonics Rev. 8(4), 626–632 (2014).
[Crossref]

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Gu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103(17), 171107 (2013).
[Crossref]

Sliwinski, A.

E. Blomme, G. Gondek, T. Katkowski, P. Kwiek, A. Sliwinski, and O. Leroy, “On the polarization of light diffracted by ultrasound,” Ultrasonics 38(1-8), 575–580 (2000).
[Crossref] [PubMed]

Sriram, S.

Steinle, T.

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film Faraday rotation,” Nat. Commun.  4, 1599 (2013).

Stockford, I. M.

Q. Zhu, I. M. Stockford, J. A. Crowe, and S. P. Morgan, “Liquid Crystal Based Rotating Orthogonal Polarization Imaging System,” J. Innov. Opt. Health Sci. 2(3), 245–251 (2009).
[Crossref]

Stritzker, B.

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film Faraday rotation,” Nat. Commun.  4, 1599 (2013).

Surdi, H.

A. Pal, S. N. Shirodkar, P. Deshmukh, H. Surdi, B. R. Sangala, S. S. Prabhu, K. C. Rustagi, U. V. Waghmare, and P. Ayyub, “Rotation of terahertz radiation due to phonon-mediated magnetoelectric coupling in chiral selenium,” Phys. Rev. B 98(23), 235141 (2018).
[Crossref]

Taherion, S.

W. FitzGerald, S. Taherion, F. J. Kumar, D. Giles, and D. Hore, “Cadmium zinc telluride as a mid-infrared variable retarder,” J. Appl. Phys. 123(13), 133103 (2018).
[Crossref]

Tang, T.-T.

H.-Y. Wu, C.-F. Hsieh, T.-T. Tang, R.-P. Pan, and C.-L. Pan, “Electrically tunable room-temperature 2π liquid crystal terahertz phase shifter,” IEEE Photonics Technol. Lett. 18(13–16), 1488–1490 (2006).

Taylor, A. J.

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016).
[Crossref] [PubMed]

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz Metamaterials for Linear Polarization Conversion and Anomalous Refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref] [PubMed]

Tian, C.

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

J. Zi, Q. Xu, Q. Wang, C. Tian, Y. Li, X. Zhang, J. Han, and W. Zhang, “Antireflection-assisted all-dielectric terahertz metamaterial polarization converter,” Appl. Phys. Lett. 113(10), 101104 (2018).
[Crossref]

Tian, Z.

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Gu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103(17), 171107 (2013).
[Crossref]

Torimoto, K.

E. Yatsuka, T. Yamamoto, T. Hatae, K. Torimoto, and K. Itami, “Note: Lossless laser beam combiner employing a high-speed rotating half-wave plate,” Rev. Sci. Instrum. 88(7), 076107 (2017).
[Crossref] [PubMed]

Vargas, A.

Vasiliev, M.

M. Nur-E-Alam, M. Vasiliev, and K. Alameh, “Nano-structured magnetic photonic crystals for magneto-optic polarization controllers at the communication-band wavelengths,” Opt. Quantum Electron. 41(9), 661–669 (2009).
[Crossref]

Vieweg, N.

Waghmare, U. V.

A. Pal, S. N. Shirodkar, P. Deshmukh, H. Surdi, B. R. Sangala, S. S. Prabhu, K. C. Rustagi, U. V. Waghmare, and P. Ayyub, “Rotation of terahertz radiation due to phonon-mediated magnetoelectric coupling in chiral selenium,” Phys. Rev. B 98(23), 235141 (2018).
[Crossref]

Wang, C.

Wang, G.

Wang, K.

W. Mo, X. Wei, K. Wang, Y. Li, and J. Liu, “Ultrathin flexible terahertz polarization converter based on metasurfaces,” Opt. Express 24(12), 13621–13627 (2016).
[Crossref] [PubMed]

Y.-L. Yang, Z.-H. Ding, K. Wang, L. Wu, and L. Wu, “Full-field optical coherence tomography by achromatic phase shifting with a rotating half-wave plate,” J. Opt. 12(3), 035301 (2010).
[Crossref]

Wang, Q.

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

J. Zi, Q. Xu, Q. Wang, C. Tian, Y. Li, X. Zhang, J. Han, and W. Zhang, “Antireflection-assisted all-dielectric terahertz metamaterial polarization converter,” Appl. Phys. Lett. 113(10), 101104 (2018).
[Crossref]

Wang, X.

Wehlus, T.

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film Faraday rotation,” Nat. Commun.  4, 1599 (2013).

Wei, X.

Weiss, T.

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film Faraday rotation,” Nat. Commun.  4, 1599 (2013).

Withayachumnankul, W.

Wu, H.-Y.

H.-Y. Wu, C.-F. Hsieh, T.-T. Tang, R.-P. Pan, and C.-L. Pan, “Electrically tunable room-temperature 2π liquid crystal terahertz phase shifter,” IEEE Photonics Technol. Lett. 18(13–16), 1488–1490 (2006).

Wu, L.

Y.-L. Yang, Z.-H. Ding, K. Wang, L. Wu, and L. Wu, “Full-field optical coherence tomography by achromatic phase shifting with a rotating half-wave plate,” J. Opt. 12(3), 035301 (2010).
[Crossref]

Y.-L. Yang, Z.-H. Ding, K. Wang, L. Wu, and L. Wu, “Full-field optical coherence tomography by achromatic phase shifting with a rotating half-wave plate,” J. Opt. 12(3), 035301 (2010).
[Crossref]

Xu, N.

L. Cong, N. Xu, J. Gu, R. Singh, J. Han, and W. Zhang, “Highly flexible broadband terahertz metamaterial quarter-wave plate,” Laser Photonics Rev. 8(4), 626–632 (2014).
[Crossref]

Xu, Q.

J. Zi, Q. Xu, Q. Wang, C. Tian, Y. Li, X. Zhang, J. Han, and W. Zhang, “Antireflection-assisted all-dielectric terahertz metamaterial polarization converter,” Appl. Phys. Lett. 113(10), 101104 (2018).
[Crossref]

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

Xu, Y.

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

Yamamoto, T.

E. Yatsuka, T. Yamamoto, T. Hatae, K. Torimoto, and K. Itami, “Note: Lossless laser beam combiner employing a high-speed rotating half-wave plate,” Rev. Sci. Instrum. 88(7), 076107 (2017).
[Crossref] [PubMed]

Yang, Y.-L.

Y.-L. Yang, Z.-H. Ding, K. Wang, L. Wu, and L. Wu, “Full-field optical coherence tomography by achromatic phase shifting with a rotating half-wave plate,” J. Opt. 12(3), 035301 (2010).
[Crossref]

Yatsuka, E.

E. Yatsuka, T. Yamamoto, T. Hatae, K. Torimoto, and K. Itami, “Note: Lossless laser beam combiner employing a high-speed rotating half-wave plate,” Rev. Sci. Instrum. 88(7), 076107 (2017).
[Crossref] [PubMed]

Ye, Y.

Y. Ye and S. He, “90 degrees polarization rotator using a bilayered chiral metamaterial with giant optical activity,” Appl. Phys. Lett. 96(20), 203501 (2010).
[Crossref]

Yu, N.

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016).
[Crossref] [PubMed]

Zeng, Y.

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz Metamaterials for Linear Polarization Conversion and Anomalous Refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref] [PubMed]

Zhang, H.

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

Zhang, L.

L. Zhang, S. Mei, K. Huang, and C.-W. Qiu, “Advances in Full Control of Electromagnetic Waves with Metasurfaces,” Adv. Opt. Mater. 4(6), 818–833 (2016).
[Crossref]

Zhang, L.-Z.

Zhang, W.

J. Zi, Q. Xu, Q. Wang, C. Tian, Y. Li, X. Zhang, J. Han, and W. Zhang, “Antireflection-assisted all-dielectric terahertz metamaterial polarization converter,” Appl. Phys. Lett. 113(10), 101104 (2018).
[Crossref]

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

L. Cong, N. Xu, J. Gu, R. Singh, J. Han, and W. Zhang, “Highly flexible broadband terahertz metamaterial quarter-wave plate,” Laser Photonics Rev. 8(4), 626–632 (2014).
[Crossref]

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Gu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103(17), 171107 (2013).
[Crossref]

Zhang, X.

J. Zi, Q. Xu, Q. Wang, C. Tian, Y. Li, X. Zhang, J. Han, and W. Zhang, “Antireflection-assisted all-dielectric terahertz metamaterial polarization converter,” Appl. Phys. Lett. 113(10), 101104 (2018).
[Crossref]

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Gu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103(17), 171107 (2013).
[Crossref]

Zhu, Q.

Q. Zhu, I. M. Stockford, J. A. Crowe, and S. P. Morgan, “Liquid Crystal Based Rotating Orthogonal Polarization Imaging System,” J. Innov. Opt. Health Sci. 2(3), 245–251 (2009).
[Crossref]

Zi, J.

J. Zi, Q. Xu, Q. Wang, C. Tian, Y. Li, X. Zhang, J. Han, and W. Zhang, “Antireflection-assisted all-dielectric terahertz metamaterial polarization converter,” Appl. Phys. Lett. 113(10), 101104 (2018).
[Crossref]

Adv. Opt. Mater. (2)

L. Zhang, S. Mei, K. Huang, and C.-W. Qiu, “Advances in Full Control of Electromagnetic Waves with Metasurfaces,” Adv. Opt. Mater. 4(6), 818–833 (2016).
[Crossref]

H. Zhang, X. Zhang, Q. Xu, C. Tian, Q. Wang, Y. Xu, Y. Li, J. Gu, Z. Tian, C. Ouyang, X. Zhang, C. Hu, J. Han, and W. Zhang, “High-Efficiency Dielectric Metasurfaces for Polarization-Dependent Terahertz Wavefront Manipulation,” Adv. Opt. Mater. 6(1), 1700773 (2018).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (3)

Y. Ye and S. He, “90 degrees polarization rotator using a bilayered chiral metamaterial with giant optical activity,” Appl. Phys. Lett. 96(20), 203501 (2010).
[Crossref]

J. Zi, Q. Xu, Q. Wang, C. Tian, Y. Li, X. Zhang, J. Han, and W. Zhang, “Antireflection-assisted all-dielectric terahertz metamaterial polarization converter,” Appl. Phys. Lett. 113(10), 101104 (2018).
[Crossref]

L. Cong, W. Cao, X. Zhang, Z. Tian, J. Gu, R. Singh, J. Han, and W. Zhang, “A perfect metamaterial polarization rotator,” Appl. Phys. Lett. 103(17), 171107 (2013).
[Crossref]

IEEE Photonics Technol. Lett. (1)

H.-Y. Wu, C.-F. Hsieh, T.-T. Tang, R.-P. Pan, and C.-L. Pan, “Electrically tunable room-temperature 2π liquid crystal terahertz phase shifter,” IEEE Photonics Technol. Lett. 18(13–16), 1488–1490 (2006).

J. Appl. Phys. (1)

W. FitzGerald, S. Taherion, F. J. Kumar, D. Giles, and D. Hore, “Cadmium zinc telluride as a mid-infrared variable retarder,” J. Appl. Phys. 123(13), 133103 (2018).
[Crossref]

J. Innov. Opt. Health Sci. (1)

Q. Zhu, I. M. Stockford, J. A. Crowe, and S. P. Morgan, “Liquid Crystal Based Rotating Orthogonal Polarization Imaging System,” J. Innov. Opt. Health Sci. 2(3), 245–251 (2009).
[Crossref]

J. Opt. (1)

Y.-L. Yang, Z.-H. Ding, K. Wang, L. Wu, and L. Wu, “Full-field optical coherence tomography by achromatic phase shifting with a rotating half-wave plate,” J. Opt. 12(3), 035301 (2010).
[Crossref]

Laser Photonics Rev. (1)

L. Cong, N. Xu, J. Gu, R. Singh, J. Han, and W. Zhang, “Highly flexible broadband terahertz metamaterial quarter-wave plate,” Laser Photonics Rev. 8(4), 626–632 (2014).
[Crossref]

Nat. Commun (1)

J. Y. Chin, T. Steinle, T. Wehlus, D. Dregely, T. Weiss, V. I. Belotelov, B. Stritzker, and H. Giessen, “Nonreciprocal plasmonics enables giant enhancement of thin-film Faraday rotation,” Nat. Commun.  4, 1599 (2013).

Nat. Nanotechnol. (1)

A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (1)

Opt. Mater. Express (1)

Opt. Quantum Electron. (1)

M. Nur-E-Alam, M. Vasiliev, and K. Alameh, “Nano-structured magnetic photonic crystals for magneto-optic polarization controllers at the communication-band wavelengths,” Opt. Quantum Electron. 41(9), 661–669 (2009).
[Crossref]

Phys. Rev. B (1)

A. Pal, S. N. Shirodkar, P. Deshmukh, H. Surdi, B. R. Sangala, S. S. Prabhu, K. C. Rustagi, U. V. Waghmare, and P. Ayyub, “Rotation of terahertz radiation due to phonon-mediated magnetoelectric coupling in chiral selenium,” Phys. Rev. B 98(23), 235141 (2018).
[Crossref]

Rep. Prog. Phys. (1)

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

E. Yatsuka, T. Yamamoto, T. Hatae, K. Torimoto, and K. Itami, “Note: Lossless laser beam combiner employing a high-speed rotating half-wave plate,” Rev. Sci. Instrum. 88(7), 076107 (2017).
[Crossref] [PubMed]

Science (1)

N. K. Grady, J. E. Heyes, D. R. Chowdhury, Y. Zeng, M. T. Reiten, A. K. Azad, A. J. Taylor, D. A. R. Dalvit, and H.-T. Chen, “Terahertz Metamaterials for Linear Polarization Conversion and Anomalous Refraction,” Science 340(6138), 1304–1307 (2013).
[Crossref] [PubMed]

Ultrasonics (1)

E. Blomme, G. Gondek, T. Katkowski, P. Kwiek, A. Sliwinski, and O. Leroy, “On the polarization of light diffracted by ultrasound,” Ultrasonics 38(1-8), 575–580 (2000).
[Crossref] [PubMed]

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Figures (7)

Fig. 1
Fig. 1 (a) Structural unit cell. (b) Relation of the two metasurfaces’ sp coordinates and the xy coordinate, as well as diagram of the polarization rotation. (c) Schematics of the polarization-insensitive tunable terahertz polarization rotator. The geometric parameters of the designed silicon pillar are: Λ = 150 μm, a = 48 μm, b = 82 μm, and h = 200 μm, respectively.
Fig. 2
Fig. 2 The simulated parameter sweep data of the silicon pillar at 1.0 THz. Transmission amplitude distributions (a) ts and (b) tp as a function of a and b, respectively. (c) Transmission ratio distribution ts /tp and (d) phase difference distribution φsφp as a function of a and b, respectively. The star markers indicate the chosen geometric parameters in the design.
Fig. 3
Fig. 3 Simulated distributions of the propagation phases of the selected pillar under the (a) s-polarized and (b) p-polarized incidences, respectively. Simulated distributions of the maximum amplitudes of E-fields of the selected pillar under the (c) s-polarized and (d) p-polarized incidences, respectively. All the distributions are plotted at 1.0 THz at the cross-sections formed by the incident polarization and the propagation direction.
Fig. 4
Fig. 4 (a) Top-view ( × 50 magnification) and (b) side-view ( × 20 magnification) microscope images of part of the fabricated metasurface sample. (c) Simulated and (d) measured transmission amplitude and phase difference spectra of the designed metasurface under the s- and p-polarized incidences, respectively.
Fig. 5
Fig. 5 (a) Photograph of the doublet metasurface polarization rotator mounted in a rotator, in which one metasurface is fixed while the other can be rotated. (b) Schematic of the experimental setup for the polarization rotator characterization.
Fig. 6
Fig. 6 Experimentally measured and theoretically calculated (a) polarization rotation angle and (b) ellipticity angle χ of the polarization rotator at 1.0 THz as a function of θ under the x- and y-polarized incidences, respectively. Inset in (a): Definition of the polarization orientation angle and the ellipticity angle of an arbitrary polarization state.
Fig. 7
Fig. 7 Measured (a) transmission amplitudes and (b) phase differences of the polarization rotator as a function of θ at 1.0 THz under the RCP and LCP incidences, respectively. Here, the phase differences were normalized to the corresponding values at θ = 0. The solid lines in (b) represent the corresponding theoretical results.

Equations (8)

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T sp =[ t s e i φ s 0 0 t p e i φ p ],
T α xy =[ t s e i φ s cos 2 (α)+ t p e i φ p sin 2 (α) ( t s e i φ s t p e i φ p )sin(α)cos(α) ( t s e i φ s t p e i φ p )sin(α)cos(α) t s e i φ s sin 2 (α)+ t p e i φ p cos 2 (α) ].
T M = T α+θ xy T α xy =[ T xx T xy T yx T yy ].
T xx = 1 2 [ t s 2 e i2 φ s t p 2 e i2 φ p ]cos(2α+θ)cos(θ)+ ( t s e i φ s t p e i φ p ) 2 cos(2θ) 4 + 1 4 ( t s e i φ s + t p e i φ p ) 2 , T xy = 1 4 ( t s e i φ s t p e i φ p )[( t s e i φ s t p e i φ p )sin(2θ)+2( t s e i φ s + t p e i φ p )sin(2α+θ)cos(θ)], T yx = 1 4 ( t s e i φ s t p e i φ p )[( t s e i φ s t p e i φ p )sin(2θ)+2( t s e i φ s + t p e i φ p )sin(2α+θ)cos(θ)], T yy = 1 2 ( t s 2 e i2 φ s t p 2 e i2 φ p )cos(2α+θ)cos(θ)+ 1 4 ( t s e i φ s t p e i φ p ) 2 cos(2θ)+ 1 4 ( t s e i φ s + t p e i φ p ) 2 ,
T M = t s 2 e i2 φ s [ cos(2θ) sin(2θ) sin(2θ) cos(2θ) ].
Δ x =ψ= 1 2 arctan 2| t xx t yx |cosδ | t xx | 2 | t yx | 2 ,
Δ y =ψ90°= 1 2 arctan 2| t xy t yy |cosδ | t xy | 2 | t yy | 2 90°,
T M lr = t s 2 e i2 φ s 2 [ e i2θ 0 0 e i2θ ].

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