Abstract

A sheet of graphene under magnetic bias attains anisotropic surface conductivity, opening the door for realizing compact devices such as Faraday rotators, isolators and circulators. In this paper, an accurate and analytical method is proposed for a periodic array of graphene ribbons under magnetic bias. The method is based on integral equations governing the induced surface currents on the coplanar array of graphene ribbons. For subwavelength size ribbons subjected to an incident plane wave, the current distribution is derived leading to analytical expressions for the reflection/transmission coefficients. The results obtained are in excellent agreement with full-wave simulations and predict resonant spectral effects that cannot be accounted for by existing semi-analytical methods. Finally, we extract an analytical, closed form solution for the Faraday rotation of magnetically-biased graphene ribbons. In contrast to previous studies, this paper presents a fast, precise and reliable technique for analyzing magnetically-biased array of graphene ribbons, which are one of the most popular graphene-based structures.

© 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]
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2019 (1)

K. Rouhi, H. Rajabalipanah, and A. Abdolali, “Multi-bit graphene-based bias-encoded metasurfaces for real-time terahertz wavefront shaping: From controllable orbital angular momentum generation toward arbitrary beam tailoring,” Carbon 149, 125–138 (2019).
[Crossref]

2018 (4)

M. Rahmanzadeh, H. Rajabalipanah, and A. Abdolali, “Multilayer graphene-based metasurfaces: robust design method for extremely broadband, wide-angle, and polarization-insensitive terahertz absorbers,” Appl. optics 57, 959–968 (2018).
[Crossref]

A. Momeni, K. Rouhi, H. Rajabalipanah, and A. Abdolali, “An information theory-inspired strategy for design of re-programmable encrypted graphene-based coding metasurfaces at terahertz frequencies,” Sci. reports 8, 6200(2018).
[Crossref]

M. Feizi, V. Nayyeri, and O. M. Ramahi, “Modeling magnetized graphene in the finite-difference time-domain method using an anisotropic surface boundary condition,” IEEE Transactions on Antennas Propag. 66, 233–241 (2018).
[Crossref]

M. Rahmanzadeh, A. Abdolali, A. Khavasi, and H. Rajabalipanah, “Adopting image theorem for rigorous analysis of a perfect electric conductor–backed array of graphene ribbons,” JOSA B 35, 1836–1844 (2018).
[Crossref]

2017 (2)

M. Rahmanzadeh, H. Rajabalipanah, and A. Abdolali, “Analytical investigation of ultrabroadband plasma–graphene radar absorbing structures,” IEEE Transactions on Plasma Sci. 45, 945–954 (2017).
[Crossref]

J.-M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Moreno, J. Faist, and A. B. Kuzmenko, “Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene,” Nat. communications 8, 14626 (2017).
[Crossref]

2016 (2)

M. Tamagnone, C. Moldovan, J.-M. Poumirol, A. B. Kuzmenko, A. M. Ionescu, J. R. Mosig, and J. Perruisseau-Carrier, “Near optimal graphene terahertz non-reciprocal isolator,” Nat. communications 7, 11216 (2016).
[Crossref]

P. K. Khoozani, M. Maddahali, M. Shahabadi, and A. Bakhtafrouz, “Analysis of magnetically biased graphene-based periodic structures using a transmission-line formulation,” JOSA B 33, 2566–2576 (2016).
[Crossref]

2015 (3)

P. Li and L. J. Jiang, “Modeling of magnetized graphene from microwave to thz range by dgtd with a scalar rbc and an ade,” IEEE Transactions on Antennas Propag. 63, 4458–4467 (2015).
[Crossref]

S. Barzegar-Parizi, B. Rejaei, and A. Khavasi, “Analytical circuit model for periodic arrays of graphene disks,” IEEE J. Quantum Electron. 51, 1–7 (2015).
[Crossref]

S. AbdollahRamezani, K. Arik, A. Khavasi, and Z. Kavehvash, “Analog computing using graphene-based metalines,” Opt. letters 40, 5239–5242 (2015).
[Crossref]

2014 (3)

X. Lin, Z. Wang, F. Gao, B. Zhang, and H. Chen, “Atomically thin nonreciprocal optical isolation,” Sci. reports 4, 4190 (2014).
[Crossref]

M. Tamagnone, A. Fallahi, J. R. Mosig, and J. Perruisseau-Carrier, “Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices,” Nat. photonics 8, 556 (2014).
[Crossref]

A. Khavasi and B. Rejaei, “Analytical modeling of graphene ribbons as optical circuit elements,” IEEE J. Of Quantum Electron. 50, 397–403 (2014).
[Crossref]

2013 (6)

B. Sensale-Rodriguez, S. Rafique, R. Yan, M. Zhu, V. Protasenko, D. Jena, L. Liu, and H. G. Xing, “Terahertz imaging employing graphene modulator arrays,” Opt. express 21, 2324–2330 (2013).
[Crossref] [PubMed]

R. Filter, M. Farhat, M. Steglich, R. Alaee, C. Rockstuhl, and F. Lederer, “Tunable graphene antennas for selective enhancement of thz-emission,” Opt. express 21, 3737–3745 (2013).
[Crossref] [PubMed]

N. Chamanara, D. Sounas, and C. Caloz, “Non-reciprocal magnetoplasmon graphene coupler,” Opt. express 21, 11248–11256 (2013).
[Crossref] [PubMed]

X.-H. Wang, W.-Y. Yin, and Z. Chen, “Matrix exponential fdtd modeling of magnetized graphene sheet,” IEEE Antennas Wirel. Propag. Lett. 12, 1129–1132 (2013).
[Crossref]

M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, “Faraday rotation due to excitation of magnetoplasmons in graphene microribbons,” ACS nano 7, 9780–9787 (2013).
[Crossref] [PubMed]

P.-Y. Chen, J. Soric, Y. R. Padooru, H. M. Bernety, A. B. Yakovlev, and A. Alù, “Nanostructured graphene metasurface for tunable terahertz cloaking,” New J. Phys. 15, 123029 (2013).
[Crossref]

2012 (4)

A. Fallahi and J. Perruisseau-Carrier, “Manipulation of giant faraday rotation in graphene metasurfaces,” Appl. Phys. Lett. 101, 231605 (2012).
[Crossref]

Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS nano 6, 3677–3694 (2012).
[Crossref] [PubMed]

J. T. Kim and C.-G. Choi, “Graphene-based polymer waveguide polarizer,” Opt. express 20, 3556–3562 (2012).
[Crossref] [PubMed]

C.-C. Lee, S. Suzuki, W. Xie, and T. Schibli, “Broadband graphene electro-optic modulators with sub-wavelength thickness,” Opt. express 20, 5264–5269 (2012).
[Crossref] [PubMed]

2011 (4)

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. Peres, and A. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84, 235410 (2011).
[Crossref]

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, and Y. R. Shen et al., “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnology 6, 630 (2011).
[Crossref]

D. L. Sounas and C. Caloz, “Electromagnetic nonreciprocity and gyrotropy of graphene,” Appl. Phys. Lett. 98, 021911 (2011).
[Crossref]

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Van Der Marel, and A. B. Kuzmenko, “Giant faraday rotation in single-and multilayer graphene,” Nat. Phys. 7, 48 (2011).
[Crossref]

2010 (1)

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, and K. L. Shepard et al., “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnology 5, 722 (2010).
[Crossref]

2008 (2)

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano letters 8, 902–907 (2008).
[Crossref] [PubMed]

G. W. Hanson, “Dyadic green’s functions for an anisotropic, non-local model of biased graphene,” IEEE Transactions on Antennas Propag. 56, 747–757 (2008).
[Crossref]

2004 (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” science 306, 666–669 (2004).
[Crossref] [PubMed]

Abdolali, A.

K. Rouhi, H. Rajabalipanah, and A. Abdolali, “Multi-bit graphene-based bias-encoded metasurfaces for real-time terahertz wavefront shaping: From controllable orbital angular momentum generation toward arbitrary beam tailoring,” Carbon 149, 125–138 (2019).
[Crossref]

M. Rahmanzadeh, H. Rajabalipanah, and A. Abdolali, “Multilayer graphene-based metasurfaces: robust design method for extremely broadband, wide-angle, and polarization-insensitive terahertz absorbers,” Appl. optics 57, 959–968 (2018).
[Crossref]

A. Momeni, K. Rouhi, H. Rajabalipanah, and A. Abdolali, “An information theory-inspired strategy for design of re-programmable encrypted graphene-based coding metasurfaces at terahertz frequencies,” Sci. reports 8, 6200(2018).
[Crossref]

M. Rahmanzadeh, A. Abdolali, A. Khavasi, and H. Rajabalipanah, “Adopting image theorem for rigorous analysis of a perfect electric conductor–backed array of graphene ribbons,” JOSA B 35, 1836–1844 (2018).
[Crossref]

M. Rahmanzadeh, H. Rajabalipanah, and A. Abdolali, “Analytical investigation of ultrabroadband plasma–graphene radar absorbing structures,” IEEE Transactions on Plasma Sci. 45, 945–954 (2017).
[Crossref]

AbdollahRamezani, S.

S. AbdollahRamezani, K. Arik, A. Khavasi, and Z. Kavehvash, “Analog computing using graphene-based metalines,” Opt. letters 40, 5239–5242 (2015).
[Crossref]

Alaee, R.

Alù, A.

P.-Y. Chen, J. Soric, Y. R. Padooru, H. M. Bernety, A. B. Yakovlev, and A. Alù, “Nanostructured graphene metasurface for tunable terahertz cloaking,” New J. Phys. 15, 123029 (2013).
[Crossref]

J. S. Gomez-Diaz and A. Alù, “Magnetically-biased graphene-based hyperbolic metasurfaces,” in 2016 IEEE International Symposium on Antennas and Propagation (APSURSI), (IEEE, 2016), pp. 359–360.

Arik, K.

S. AbdollahRamezani, K. Arik, A. Khavasi, and Z. Kavehvash, “Analog computing using graphene-based metalines,” Opt. letters 40, 5239–5242 (2015).
[Crossref]

Bakhtafrouz, A.

P. K. Khoozani, M. Maddahali, M. Shahabadi, and A. Bakhtafrouz, “Analysis of magnetically biased graphene-based periodic structures using a transmission-line formulation,” JOSA B 33, 2566–2576 (2016).
[Crossref]

Balandin, A. A.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano letters 8, 902–907 (2008).
[Crossref] [PubMed]

Bao, Q.

Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS nano 6, 3677–3694 (2012).
[Crossref] [PubMed]

Bao, W.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano letters 8, 902–907 (2008).
[Crossref] [PubMed]

Barzegar-Parizi, S.

S. Barzegar-Parizi, B. Rejaei, and A. Khavasi, “Analytical circuit model for periodic arrays of graphene disks,” IEEE J. Quantum Electron. 51, 1–7 (2015).
[Crossref]

Bechtel, H. A.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, and Y. R. Shen et al., “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnology 6, 630 (2011).
[Crossref]

Bernety, H. M.

P.-Y. Chen, J. Soric, Y. R. Padooru, H. M. Bernety, A. B. Yakovlev, and A. Alù, “Nanostructured graphene metasurface for tunable terahertz cloaking,” New J. Phys. 15, 123029 (2013).
[Crossref]

Bludov, Y. V.

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. Peres, and A. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84, 235410 (2011).
[Crossref]

Bostwick, A.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Van Der Marel, and A. B. Kuzmenko, “Giant faraday rotation in single-and multilayer graphene,” Nat. Phys. 7, 48 (2011).
[Crossref]

Calizo, I.

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano letters 8, 902–907 (2008).
[Crossref] [PubMed]

Caloz, C.

N. Chamanara, D. Sounas, and C. Caloz, “Non-reciprocal magnetoplasmon graphene coupler,” Opt. express 21, 11248–11256 (2013).
[Crossref] [PubMed]

D. L. Sounas and C. Caloz, “Electromagnetic nonreciprocity and gyrotropy of graphene,” Appl. Phys. Lett. 98, 021911 (2011).
[Crossref]

Chamanara, N.

Chen, H.

X. Lin, Z. Wang, F. Gao, B. Zhang, and H. Chen, “Atomically thin nonreciprocal optical isolation,” Sci. reports 4, 4190 (2014).
[Crossref]

Chen, P.-Y.

P.-Y. Chen, J. Soric, Y. R. Padooru, H. M. Bernety, A. B. Yakovlev, and A. Alù, “Nanostructured graphene metasurface for tunable terahertz cloaking,” New J. Phys. 15, 123029 (2013).
[Crossref]

Chen, Z.

X.-H. Wang, W.-Y. Yin, and Z. Chen, “Matrix exponential fdtd modeling of magnetized graphene sheet,” IEEE Antennas Wirel. Propag. Lett. 12, 1129–1132 (2013).
[Crossref]

Choi, C.-G.

Crassee, I.

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Van Der Marel, and A. B. Kuzmenko, “Giant faraday rotation in single-and multilayer graphene,” Nat. Phys. 7, 48 (2011).
[Crossref]

Dean, C. R.

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, and K. L. Shepard et al., “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnology 5, 722 (2010).
[Crossref]

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” science 306, 666–669 (2004).
[Crossref] [PubMed]

Faist, J.

J.-M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Moreno, J. Faist, and A. B. Kuzmenko, “Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene,” Nat. communications 8, 14626 (2017).
[Crossref]

Fallahi, A.

M. Tamagnone, A. Fallahi, J. R. Mosig, and J. Perruisseau-Carrier, “Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices,” Nat. photonics 8, 556 (2014).
[Crossref]

A. Fallahi and J. Perruisseau-Carrier, “Manipulation of giant faraday rotation in graphene metasurfaces,” Appl. Phys. Lett. 101, 231605 (2012).
[Crossref]

Farhat, M.

Feizi, M.

M. Feizi, V. Nayyeri, and O. M. Ramahi, “Modeling magnetized graphene in the finite-difference time-domain method using an anisotropic surface boundary condition,” IEEE Transactions on Antennas Propag. 66, 233–241 (2018).
[Crossref]

Ferreira, A.

A. Ferreira, J. Viana-Gomes, Y. V. Bludov, V. Pereira, N. Peres, and A. C. Neto, “Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids,” Phys. Rev. B 84, 235410 (2011).
[Crossref]

Filter, R.

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” science 306, 666–669 (2004).
[Crossref] [PubMed]

Gao, F.

X. Lin, Z. Wang, F. Gao, B. Zhang, and H. Chen, “Atomically thin nonreciprocal optical isolation,” Sci. reports 4, 4190 (2014).
[Crossref]

Geim, A. K.

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C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, and K. L. Shepard et al., “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnology 5, 722 (2010).
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Zettl, A.

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, and Y. R. Shen et al., “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnology 6, 630 (2011).
[Crossref]

Zhang, B.

X. Lin, Z. Wang, F. Gao, B. Zhang, and H. Chen, “Atomically thin nonreciprocal optical isolation,” Sci. reports 4, 4190 (2014).
[Crossref]

Zhang, Y.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” science 306, 666–669 (2004).
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ACS nano (2)

Q. Bao and K. P. Loh, “Graphene photonics, plasmonics, and broadband optoelectronic devices,” ACS nano 6, 3677–3694 (2012).
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M. Tymchenko, A. Y. Nikitin, and L. Martín-Moreno, “Faraday rotation due to excitation of magnetoplasmons in graphene microribbons,” ACS nano 7, 9780–9787 (2013).
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Appl. optics (1)

M. Rahmanzadeh, H. Rajabalipanah, and A. Abdolali, “Multilayer graphene-based metasurfaces: robust design method for extremely broadband, wide-angle, and polarization-insensitive terahertz absorbers,” Appl. optics 57, 959–968 (2018).
[Crossref]

Appl. Phys. Lett. (2)

A. Fallahi and J. Perruisseau-Carrier, “Manipulation of giant faraday rotation in graphene metasurfaces,” Appl. Phys. Lett. 101, 231605 (2012).
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D. L. Sounas and C. Caloz, “Electromagnetic nonreciprocity and gyrotropy of graphene,” Appl. Phys. Lett. 98, 021911 (2011).
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Carbon (1)

K. Rouhi, H. Rajabalipanah, and A. Abdolali, “Multi-bit graphene-based bias-encoded metasurfaces for real-time terahertz wavefront shaping: From controllable orbital angular momentum generation toward arbitrary beam tailoring,” Carbon 149, 125–138 (2019).
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IEEE Antennas Wirel. Propag. Lett. (1)

X.-H. Wang, W.-Y. Yin, and Z. Chen, “Matrix exponential fdtd modeling of magnetized graphene sheet,” IEEE Antennas Wirel. Propag. Lett. 12, 1129–1132 (2013).
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IEEE J. Of Quantum Electron. (1)

A. Khavasi and B. Rejaei, “Analytical modeling of graphene ribbons as optical circuit elements,” IEEE J. Of Quantum Electron. 50, 397–403 (2014).
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IEEE J. Quantum Electron. (1)

S. Barzegar-Parizi, B. Rejaei, and A. Khavasi, “Analytical circuit model for periodic arrays of graphene disks,” IEEE J. Quantum Electron. 51, 1–7 (2015).
[Crossref]

IEEE Transactions on Antennas Propag. (3)

P. Li and L. J. Jiang, “Modeling of magnetized graphene from microwave to thz range by dgtd with a scalar rbc and an ade,” IEEE Transactions on Antennas Propag. 63, 4458–4467 (2015).
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M. Feizi, V. Nayyeri, and O. M. Ramahi, “Modeling magnetized graphene in the finite-difference time-domain method using an anisotropic surface boundary condition,” IEEE Transactions on Antennas Propag. 66, 233–241 (2018).
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G. W. Hanson, “Dyadic green’s functions for an anisotropic, non-local model of biased graphene,” IEEE Transactions on Antennas Propag. 56, 747–757 (2008).
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IEEE Transactions on Plasma Sci. (1)

M. Rahmanzadeh, H. Rajabalipanah, and A. Abdolali, “Analytical investigation of ultrabroadband plasma–graphene radar absorbing structures,” IEEE Transactions on Plasma Sci. 45, 945–954 (2017).
[Crossref]

JOSA B (2)

P. K. Khoozani, M. Maddahali, M. Shahabadi, and A. Bakhtafrouz, “Analysis of magnetically biased graphene-based periodic structures using a transmission-line formulation,” JOSA B 33, 2566–2576 (2016).
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M. Rahmanzadeh, A. Abdolali, A. Khavasi, and H. Rajabalipanah, “Adopting image theorem for rigorous analysis of a perfect electric conductor–backed array of graphene ribbons,” JOSA B 35, 1836–1844 (2018).
[Crossref]

Nano letters (1)

A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, and C. N. Lau, “Superior thermal conductivity of single-layer graphene,” Nano letters 8, 902–907 (2008).
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Nat. communications (2)

M. Tamagnone, C. Moldovan, J.-M. Poumirol, A. B. Kuzmenko, A. M. Ionescu, J. R. Mosig, and J. Perruisseau-Carrier, “Near optimal graphene terahertz non-reciprocal isolator,” Nat. communications 7, 11216 (2016).
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J.-M. Poumirol, P. Q. Liu, T. M. Slipchenko, A. Y. Nikitin, L. Martin-Moreno, J. Faist, and A. B. Kuzmenko, “Electrically controlled terahertz magneto-optical phenomena in continuous and patterned graphene,” Nat. communications 8, 14626 (2017).
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L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, and Y. R. Shen et al., “Graphene plasmonics for tunable terahertz metamaterials,” Nat. Nanotechnology 6, 630 (2011).
[Crossref]

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Nat. photonics (1)

M. Tamagnone, A. Fallahi, J. R. Mosig, and J. Perruisseau-Carrier, “Fundamental limits and near-optimal design of graphene modulators and non-reciprocal devices,” Nat. photonics 8, 556 (2014).
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Nat. Phys. (1)

I. Crassee, J. Levallois, A. L. Walter, M. Ostler, A. Bostwick, E. Rotenberg, T. Seyller, D. Van Der Marel, and A. B. Kuzmenko, “Giant faraday rotation in single-and multilayer graphene,” Nat. Phys. 7, 48 (2011).
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New J. Phys. (1)

P.-Y. Chen, J. Soric, Y. R. Padooru, H. M. Bernety, A. B. Yakovlev, and A. Alù, “Nanostructured graphene metasurface for tunable terahertz cloaking,” New J. Phys. 15, 123029 (2013).
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S. AbdollahRamezani, K. Arik, A. Khavasi, and Z. Kavehvash, “Analog computing using graphene-based metalines,” Opt. letters 40, 5239–5242 (2015).
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Phys. Rev. B (1)

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Sci. reports (2)

A. Momeni, K. Rouhi, H. Rajabalipanah, and A. Abdolali, “An information theory-inspired strategy for design of re-programmable encrypted graphene-based coding metasurfaces at terahertz frequencies,” Sci. reports 8, 6200(2018).
[Crossref]

X. Lin, Z. Wang, F. Gao, B. Zhang, and H. Chen, “Atomically thin nonreciprocal optical isolation,” Sci. reports 4, 4190 (2014).
[Crossref]

science (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” science 306, 666–669 (2004).
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Figures (6)

Fig. 1
Fig. 1 Schematic representation of the studied system: a plane wave (TE/TM) is incident on the array of graphene ribbons, in the presence of a static perpendicular magnetic field B.
Fig. 2
Fig. 2 (a)|Rxx | and (b) |Ryx| (c) |Ryy|of a periodic array of magnetically-biased graphene ribbons. The graphene ribbons parameters are assumed as D = 4μm, w = 2μm, Ef = 0.5eV, τ = 1ps ,B0 = 10T
Fig. 3
Fig. 3 Current distribution on the graphene ribbon at the vicinity of the first two resonances
Fig. 4
Fig. 4 (a)|Ryx| and (b) R y xof a periodic array of magnetically-biased graphene ribbons for different incident angles. The graphene ribbons parameters are assumed as D = 10μm, w = 7.5μm, Ef = 0.75eV, τ = 1ps, B0 = 6T
Fig. 5
Fig. 5 First resonance frequency of Rxx for different filling factors w/D (B0 = 7.5T, τ = 1ps and Ef = 0.3eV)
Fig. 6
Fig. 6 Faraday rotation angle at various magnetic field bias for graphene ribbons.structure parameter are designed as D = 4.5μm, w = 2.7μm, Ef = 0.8eV, τ = 2ps

Equations (56)

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σ ¯ ¯ s ( ω , E f , τ , T , B 0 ) = [ σ x x σ x y σ y x σ y y ]
σ x x = σ y y = e 2 E f τ π 2 1 + j ω τ ( ω c τ ) 2 + ( 1 + j ω τ ) 2
σ x y = σ y x = e 2 E f τ π 2 ω c τ ( ω c τ ) 2 + ( 1 + j ω τ ) 2
σ s ¯ ¯ 1 [ J x ( x ) J y ( x ) ] = [ E x e x t ( x ) + E x T ( x ) E y e x t ( x ) + E y T ( x ) ]
E x T = 1 j ω ε 0 d d x w / 2 w / 2 G 0 ( x x ) d J x ( x ) d x d x j ω μ 0 w / 2 w / 2 G 0 ( x x ) J x ( x ) d x
E y T = j ω μ 0 w / 2 w / 2 G 0 ( x x ) J y ( x ) d x
G 0 ( x x ) = 1 4 j H 0 ( 2 ) ( k 0 | x x | )
G 0 ( x x ) 1 2 π ln  ( k 0 | x x | )
J y ( x ) + j ω μ 0 σ 0 w / 2 w / 2 G 0 ( x x ) J y ( x ) d x = σ 0 [ E y e x t ( x ) σ a 1 J x ( x ) ]
σ 0 1 = σ x x σ x x 2 + σ x y 2 , σ a 1 = σ x y σ x x 2 + σ x y 2
| j ω μ 0 w σ 0 4 | 1
J y ( x ) = σ 0 [ E y e x t ( x ) σ a 1 J x ( x ) ]
j ω μ 0 w σ 0 4 = ( k 0 w ) k 0 k P
J x ( x ) σ x x 1 j ω ε 0 d d x w / 2 w / 2 G 0 ( x x ) d J x ( x ) d x d x = E x e x t ( x ) + σ x y σ x x E y e x t ( x )
G ( x x ) = 1 4 j l = H 0 ( 2 ) ( k 0 | x x l D | )
G ( x ) = 1 2 j k 0 D + 1 2 π 0 [ cosh  ( α u ) sinh  u 1 u ] d u u + O ( k 0 D )
J y ( x ) = σ 0 [ E y e x t ( x ) σ a 1 J x ( x ) ] ω μ 0 σ 0 2 k 0 D w / 2 w / 2 J y ( x ) d x
J y ( x ) = σ 0 [ E y e x t 1 + γ + γ 1 + γ I x σ a w J x ( x ) σ a ]
I x = w / 2 w / 2 J x ( x ) d x
γ = ω μ 0 σ 0 w 2 k 0 D = η 0 σ 0 w 2 D
J x ( x ) σ x x = F + 1 j ω ε 0 d d x w / 2 w / 2 G ( x x ) d J x ( x ) d x d x
F = E x e x t + σ x y σ x x [ E y e x t 1 + γ + γ 1 + γ I x σ a w ] ω μ 0 I x 2 k 0 D
d G ( x ) d x = l = d G 0 ( x l D ) d x + O ( k 0 D )
J x ( x ) = F n = 1 Y n S n ψ n ( x )
Y n = σ x x ( 2 j ω ε 0 / q n ) σ x x + 2 j ω ε 0 / q n
S n = w / 2 w / 2 ψ n ( x ) d x
1 π w / 2 w / 2 1 x x d ψ n ( x ) d x d x + 1 π l = ( 0 ) w / 2 w / 2 1 x x + l D ψ n x x = q n ψ n ( x )
w / 2 w / 2 ψ n 2 ( x ) d x = 1
I x = Y 1 + 1 2 ζ 0 Y D F 0
Y = 1 D m = 1 Y m S m 2
F 0 = E x e x t + σ x y σ x x E y e x t 1 + γ
ζ 0 = η 0 [ 1 σ x y 2 σ x x 2 ( 1 + γ ) ]
F = F 0 1 + 1 2 ζ 0 Y
G ( x ) = c o s ( k 0 sin ( θ i ) x ) 2 j k 0 D cos ( θ i ) + 1 2 π 0 [ cosh  ( α u ) sinh  u 1 u ] d u u + O ( k 0 D )
γ = ω μ 0 σ 0 w 2 k 0 D cos ( θ i ) = η 0 σ 0 w 2 D cos ( θ i )
ζ 0 = η 0 [ cos ( θ i ) σ x y 2 σ x x 2 ( 1 + γ ) cos ( θ i ) ]
( E x e x t x ^ + E y e x t y ^ ) e j k 0 z + n = ( E x , n r x ^ + E y , n r y ^ ) e j k x , n x + j k z , n z
n = ( E x , n t x ^ + E y , n t y ^ ) e j k x , n x j k z , n z , z > 0
k x , n = 2 π n / D , k z , n = k 0 2 k x , n 2
[ E x , 0 r E y , 0 r ] = [ R x x R x y R y x R y y ] [ E x e x t E y e x t ]
[ E x , 0 t E y , 0 t ] = [ T x x T x y T y x T y y ] [ E x e x t E y e x t ]
E x , 0 r = E x , 0 t E x e x t = ω μ 0 c o s ( θ i ) 2 k 0 D I x
E y , 0 r = E y , 0 t E y e x t = ω μ 0 2 k 0 D c o s ( θ i ) I y
I y = w / 2 w / 2 J y ( x ) d x
I y + σ x y σ x x I x 1 + γ = σ 0 w 1 + γ E y e x t
R x x = T x x 1 = 1 2 η 0 c o s ( θ i ) Y 1 + 1 2 ζ 0 Y
R x y = R y x = T x y = T y x = σ x y R x x σ x x ( 1 + γ ) c o s ( θ i )
R y y = T y y 1 = γ 1 + γ R x y 2 R x x
θ F = 1 2 arg ( T x x j T y x T x x + j T y x )
H 0 ( 2 ) ( z ) = 1 π π + j j e j z sin   θ d θ
G ( x ) = 1 4 π j π + j j e j k 0 x sin  θ l = e j k 0 l D sin  θ d θ = 1 4 π π + j j cos   [ ( 1 2 D | x | ) k 0 sin   θ ] sin   ( 1 2 k 0 D sin   θ ) d θ
G ( x ) = 1 2 j k 0 D + 1 4 π π + j j ( cos   [ ( 1 2 D | x | ) k 0 sin   θ ] sin   ( 1 2 k 0 D sin   θ ) 2 k 0 D sin   θ ) d θ
π + j j d θ sin   θ = j π
1 4 π 0 π ( cos   [ ( 1 2 D | x | ) k 0 sin   θ ] sin   ( 1 2 k 0 D sin   θ ) 2 k 0 D sin   θ ) d θ + 1 2 π 0 ( cosh   [ ( 1 2 D | x | ) k 0 sinh   t ] sinh   ( 1 2 k 0 D sinh   t ) 2 k 0 D sinh   t ) d t
1 2 π 0 [ cosh   ( α u ) sinh   u 1 u ] d u u 2 + 1 4 ( k 0 D ) 2 = 1 2 π 0 [ cosh   ( α u ) sinh   u 1 u ] d u u + O ( k 0 D ) 2
d G ( x ) d x = x π D | x | 0 sinh   ( α u ) sinh   u d u + O ( k 0 D ) = π x D | x | cot  ( π | x | D ) + O ( k 0 D ) = 1 2 π n = 1 x n D + O ( k 0 D )

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