Adopting image theorem for rigorous analysis of a perfect electric conductor&#x2013;backed array of graphene ribbons

Mahdi Rahmanzadeh; Ali Abdolali; Amin Khavasi; Hamid Rajabalipanah

doi:10.1364/JOSAB.35.001836

Journal of the Optical Society of America B
Vol. 35,
Issue 8,
pp. 1836-1844
(2018)
•https://doi.org/10.1364/JOSAB.35.001836

Adopting image theorem for rigorous analysis of a perfect electric conductor–backed array of graphene ribbons

Mahdi Rahmanzadeh, Ali Abdolali, Amin Khavasi, and Hamid Rajabalipanah

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Mahdi Rahmanzadeh, Ali Abdolali, Amin Khavasi, and Hamid Rajabalipanah, "Adopting image theorem for rigorous analysis of a perfect electric conductor–backed array of graphene ribbons," J. Opt. Soc. Am. B 35, 1836-1844 (2018)

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Related Topics
Table of Contents Category
- Metamaterials
Optics & Photonics Topics
?

The topics in this list come from the Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Far infrared or terahertz (040.2235)
- Computational electromagnetic methods (050.1755)
- Absorption (300.1030)
- Subwavelength structures, nanostructures (310.6628)

About this Article
History
- Original Manuscript: February 26, 2018
- Revised Manuscript: May 22, 2018
- Manuscript Accepted: June 13, 2018
- Published: July 13, 2018

Abstract

Analytical and numerical study of graphene ribbons has become a prime focus of recent research due to their potential applications in tunable absorption, wavefront manipulation, polarization conversion, and so on. In this paper, an accurate analysis of a perfect electric conductor (PEC)–backed array of graphene ribbons (PAGR) is presented based on the well-known electromagnetic (EM) image theorem, where the induced currents are theoretically derived under a transverse-magnetic-polarized incident wave. For the first time, the proposed analysis rigorously incorporates the EM coupling effects between the PEC back plate and the subwavelength array of graphene ribbons. It is proved that the strong interaction between the PEC back plate and graphene ribbons drastically affects the results, especially in ultra-thin PAGR structures, whereas it was neglected in the previous works. As a proof of principle, an ultra-thin graphene-assisted absorber ( $= 0.05 λ_{0}$ ) exhibiting tunable absorption at the terahertz regime is theoretically designed to verify the proposed analytical scheme. Unlike the previous studies, this paper reveals a more general, valid, and reliable analysis of PEC-backed graphene ribbons and can be simply extended to 2D geometries of PEC-backed graphene metasurfaces.

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References

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Year
Author
Publication

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[Crossref]
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]
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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 Lett. 8, 902–907 (2008).
[Crossref]
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]
M. Rahmanzadeh, H. Rajabalipanah, and A. Abdolali, “Analytical investigation of ultra-broadband plasma-graphene radar absorbing structures,” IEEE Trans. Plasma Sci. 45, 945–954 (2017).
[Crossref]
K. Rouhi, H. Rajabalipanah, and A. Abdolali, “Real‐time and broadband terahertz wave scattering manipulation via polarization‐insensitive conformal graphene‐based coding metasurfaces,” Ann. Phys. 530, 1700310 (2018).
[Crossref]
S. Abdollah Ramezani, K. Arik, A. Khavasi, and Z. Kavehvash, “Analog computing using graphene-based metalines,” Opt. Lett. 40, 5239–5242 (2015).
[Crossref]
Z. H. Zhu, C. C. Guo, K. Liu, J. F. Zhang, W. M. Ye, X. D. Yuan, and S. Q. Qin, “Electrically controlling the polarizing direction of a graphene polarizer,” J. Appl. Phys. 116, 104304 (2014).
[Crossref]
M. Rahmanzadeh, A. Abdolali, and H. Rajabalipanah, “Multilayer graphene-based metasurfaces: robust design method for extremely broadband, terahertz absorbers,” Appl. Opt. 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. Rep. 8, 6200 (2018).
[Crossref]
Z. H. Zhu, C. C. Guo, K. Liu, J. F. Zhang, W. M. Ye, X. D. Yuan, and S. Q. Qin, “Electrically tunable polarizer based on anisotropic absorption of graphene ribbons,” Appl. Phys. A 114, 1017–1021 (2014).
[Crossref]
N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (2010).
[Crossref]
R. Xing and S. Jian, “A dual-band THz absorber based on graphene sheet and ribbons,” Opt. Laser Technol. 100, 129–132 (2018).
[Crossref]
S. AbdollahRamezani, K. Arik, S. Farajollahi, A. Khavasi, and Z. Kavehvash, “Beam manipulating by gate-tunable graphene- based metasurfaces,” Opt. Lett. 40, 5383–5386 (2015).
[Crossref]
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[Crossref]
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[Crossref]
A. Khavasi, K. Mehrany, and B. Rashidian, “Three-dimensional diffraction analysis of gratings based on Legendre expansion of electromagnetic fields,” J. Opt. Soc. Am. B 24, 2676–2685 (2007).
[Crossref]
M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[Crossref]
L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[Crossref]
A. Khavasi, “Fast convergent Fourier modal method for the analysis of periodic arrays of graphene ribbons,” Opt. Lett. 38, 3009–3012 (2013).
[Crossref]
H. Fadakar, A. Borji, A. Zeidaabadi Nezhad, and M. Shahabadi, “Improved Fourier analysis of periodically patterned graphene sheets embedded in multilayered structures and its application to the design of a broadband tunable wide-angle polarizer,” IEEE J. Quantum Electron. 53, 1–8 (2017).
[Crossref]
A. Khavasi and B. Rejaei, “Analytical modeling of graphene ribbons as optical circuit elements,” IEEE J. Quantum Electron. 50, 397–403 (2014).
[Crossref]
A. Khavasi, “Design of ultra-broadband graphene absorber using circuit theory,” J. Opt. Soc. Am. B 32, 1941–1946 (2015).
[Crossref]
I. Silveiro, F. J. G. De Abajo, and J. M. P. Ortega, “Plasmon wave function of graphene nanoribbons plasmon wave function of graphene nanoribbons,” New J. Phys. 17, 083013 (2015).
[Crossref]
S. A. Mikhailov and N. A. Savostianova, “Microwave response of a two-dimensional electron stripe,” Phys. Rev. B 71, 035320 (2005).
[Crossref]
G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008).
[Crossref]
Y. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam, E. H. Hwang, S. Das Sarma, H. L. Stormer, and P. Kim, “Measurement of scattering rate and minimum conductivity in graphene,” Phys. Rev. Lett. 99, 246803 (2007).
[Crossref]
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, 630–634 (2011).
[Crossref]
C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5, 722–726 (2010).
[Crossref]
K. F. Mak, L. Ju, F. Wang, and T. F. Heinz, “Optical spectroscopy of graphene: from the far infrared to the ultraviolet,” Solid State Commun. 152, 1341–1349 (2012).
[Crossref]
C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994), Vol. 272.
A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, 1991).
G. Zhao, M. Mors, T. Wenckebach, and P. C. M. Planken, “Terahertz dielectric properties of polystyrene foam,” J. Opt. Soc. Am. B 19, 1476–1479 (2002).
[Crossref]
W. M. Hassan, “Multilayer graphene-only transmitarray antenna (MGOT) for terahertz applications,” in 34th National Radio Science Conference (NRSC) (IEEE, 2017).
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[Crossref]

2018 (4)

K. Rouhi, H. Rajabalipanah, and A. Abdolali, “Real‐time and broadband terahertz wave scattering manipulation via polarization‐insensitive conformal graphene‐based coding metasurfaces,” Ann. Phys. 530, 1700310 (2018).
[Crossref]

M. Rahmanzadeh, A. Abdolali, and H. Rajabalipanah, “Multilayer graphene-based metasurfaces: robust design method for extremely broadband, terahertz absorbers,” Appl. Opt. 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. Rep. 8, 6200 (2018).
[Crossref]

R. Xing and S. Jian, “A dual-band THz absorber based on graphene sheet and ribbons,” Opt. Laser Technol. 100, 129–132 (2018).
[Crossref]

2017 (3)

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

H. Fadakar, A. Borji, A. Zeidaabadi Nezhad, and M. Shahabadi, “Improved Fourier analysis of periodically patterned graphene sheets embedded in multilayered structures and its application to the design of a broadband tunable wide-angle polarizer,” IEEE J. Quantum Electron. 53, 1–8 (2017).
[Crossref]

G. Deng, J. Yang, and Z. Yin, “Broadband terahertz metamaterial absorber based on tantalum nitride,” Appl. Opt. 56, 2449–2454 (2017).
[Crossref]

2015 (4)

A. Khavasi, “Design of ultra-broadband graphene absorber using circuit theory,” J. Opt. Soc. Am. B 32, 1941–1946 (2015).
[Crossref]

I. Silveiro, F. J. G. De Abajo, and J. M. P. Ortega, “Plasmon wave function of graphene nanoribbons plasmon wave function of graphene nanoribbons,” New J. Phys. 17, 083013 (2015).
[Crossref]

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

S. AbdollahRamezani, K. Arik, S. Farajollahi, A. Khavasi, and Z. Kavehvash, “Beam manipulating by gate-tunable graphene- based metasurfaces,” Opt. Lett. 40, 5383–5386 (2015).
[Crossref]

2014 (3)

Z. H. Zhu, C. C. Guo, K. Liu, J. F. Zhang, W. M. Ye, X. D. Yuan, and S. Q. Qin, “Electrically tunable polarizer based on anisotropic absorption of graphene ribbons,” Appl. Phys. A 114, 1017–1021 (2014).
[Crossref]

Z. H. Zhu, C. C. Guo, K. Liu, J. F. Zhang, W. M. Ye, X. D. Yuan, and S. Q. Qin, “Electrically controlling the polarizing direction of a graphene polarizer,” J. Appl. Phys. 116, 104304 (2014).
[Crossref]

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

2013 (2)

A. Khavasi, “Fast convergent Fourier modal method for the analysis of periodic arrays of graphene ribbons,” Opt. Lett. 38, 3009–3012 (2013).
[Crossref]

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 (2)

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

K. F. Mak, L. Ju, F. Wang, and T. F. Heinz, “Optical spectroscopy of graphene: from the far infrared to the ultraviolet,” Solid State Commun. 152, 1341–1349 (2012).
[Crossref]

2011 (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, 630–634 (2011).
[Crossref]

2010 (2)

C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, “Boron nitride substrates for high-quality graphene electronics,” Nat. Nanotechnol. 5, 722–726 (2010).
[Crossref]

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (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 Lett. 8, 902–907 (2008).
[Crossref]

G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008).
[Crossref]

2007 (3)

Y. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam, E. H. Hwang, S. Das Sarma, H. L. Stormer, and P. Kim, “Measurement of scattering rate and minimum conductivity in graphene,” Phys. Rev. Lett. 99, 246803 (2007).
[Crossref]

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
[Crossref]

A. Khavasi, K. Mehrany, and B. Rashidian, “Three-dimensional diffraction analysis of gratings based on Legendre expansion of electromagnetic fields,” J. Opt. Soc. Am. B 24, 2676–2685 (2007).
[Crossref]

2005 (1)

S. A. Mikhailov and N. A. Savostianova, “Microwave response of a two-dimensional electron stripe,” Phys. Rev. B 71, 035320 (2005).
[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]

1988 (1)

I. Transactions and O. N. Antennas, “Scattering from finite thickness resistive strip gratings,” IEEE Trans. Antennas Propag. 36, 504–510 (1988).
[Crossref]

1981 (1)

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[Crossref]

Abdolali, A.

Abdollah Ramezani, S.

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

AbdollahRamezani, S.

S. AbdollahRamezani, K. Arik, S. Farajollahi, A. Khavasi, and Z. Kavehvash, “Beam manipulating by gate-tunable graphene- based metasurfaces,” Opt. Lett. 40, 5383–5386 (2015).
[Crossref]

Adam, S.

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]

Antennas, O. N.

I. Transactions and O. N. Antennas, “Scattering from finite thickness resistive strip gratings,” IEEE Trans. Antennas Propag. 36, 504–510 (1988).
[Crossref]

Arik, K.

S. AbdollahRamezani, K. Arik, S. Farajollahi, A. Khavasi, and Z. Kavehvash, “Beam manipulating by gate-tunable graphene- based metasurfaces,” Opt. Lett. 40, 5383–5386 (2015).
[Crossref]

S. Abdollah Ramezani, K. Arik, A. Khavasi, and Z. Kavehvash, “Analog computing using graphene-based metalines,” Opt. Lett. 40, 5239–5242 (2015).
[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 Lett. 8, 902–907 (2008).
[Crossref]

Bao, Q.

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

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 Lett. 8, 902–907 (2008).
[Crossref]

Bechtel, H. A.

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]

Bolotin, K.

Borji, A.

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 Lett. 8, 902–907 (2008).
[Crossref]

Chandezon, J.

L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
[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]

Das Sarma, S.

De Abajo, F. J. G.

I. Silveiro, F. J. G. De Abajo, and J. M. P. Ortega, “Plasmon wave function of graphene nanoribbons plasmon wave function of graphene nanoribbons,” New J. Phys. 17, 083013 (2015).
[Crossref]

Dean, C. R.

Deng, G.

G. Deng, J. Yang, and Z. Yin, “Broadband terahertz metamaterial absorber based on tantalum nitride,” Appl. Opt. 56, 2449–2454 (2017).
[Crossref]

Dubonos, S. V.

Fadakar, H.

Farajollahi, S.

S. AbdollahRamezani, K. Arik, S. Farajollahi, A. Khavasi, and Z. Kavehvash, “Beam manipulating by gate-tunable graphene- based metasurfaces,” Opt. Lett. 40, 5383–5386 (2015).
[Crossref]

Firsov, A. A.

Gaylord, T. K.

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[Crossref]

Geim, A. K.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6, 183–191 (2007).
[Crossref]

Geng, B.

Ghosh, S.

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 Lett. 8, 902–907 (2008).
[Crossref]

Giessen, H.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (2010).
[Crossref]

Girit, C.

Granet, G.

L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
[Crossref]

Grigorieva, I. V.

Guo, C. C.

Hanson, G. W.

G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103, 064302 (2008).
[Crossref]

Hao, Z.

Hassan, W. M.

W. M. Hassan, “Multilayer graphene-only transmitarray antenna (MGOT) for terahertz applications,” in 34th National Radio Science Conference (NRSC) (IEEE, 2017).

Heinz, T. F.

K. F. Mak, L. Ju, F. Wang, and T. F. Heinz, “Optical spectroscopy of graphene: from the far infrared to the ultraviolet,” Solid State Commun. 152, 1341–1349 (2012).
[Crossref]

Hentschel, M.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (2010).
[Crossref]

Hone, J.

Horng, J.

Hwang, E. H.

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, 1991).

Jian, S.

R. Xing and S. Jian, “A dual-band THz absorber based on graphene sheet and ribbons,” Opt. Laser Technol. 100, 129–132 (2018).
[Crossref]

Jiang, D.

Ju, L.

K. F. Mak, L. Ju, F. Wang, and T. F. Heinz, “Optical spectroscopy of graphene: from the far infrared to the ultraviolet,” Solid State Commun. 152, 1341–1349 (2012).
[Crossref]

Kavehvash, Z.

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

S. AbdollahRamezani, K. Arik, S. Farajollahi, A. Khavasi, and Z. Kavehvash, “Beam manipulating by gate-tunable graphene- based metasurfaces,” Opt. Lett. 40, 5383–5386 (2015).
[Crossref]

Khavasi, A.

S. AbdollahRamezani, K. Arik, S. Farajollahi, A. Khavasi, and Z. Kavehvash, “Beam manipulating by gate-tunable graphene- based metasurfaces,” Opt. Lett. 40, 5383–5386 (2015).
[Crossref]

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

A. Khavasi, “Design of ultra-broadband graphene absorber using circuit theory,” J. Opt. Soc. Am. B 32, 1941–1946 (2015).
[Crossref]

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

A. Khavasi, “Fast convergent Fourier modal method for the analysis of periodic arrays of graphene ribbons,” Opt. Lett. 38, 3009–3012 (2013).
[Crossref]

Kim, P.

Lau, C. N.

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 Lett. 8, 902–907 (2008).
[Crossref]

Lee, C.

Li, L.

L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999).
[Crossref]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[Crossref]

Liang, X.

Liu, K.

Liu, N.

N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10, 2342–2348 (2010).
[Crossref]

Loh, K. P.

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

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Appl. Opt. (3)

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Fig. 1. Periodic array of graphene ribbons with the width of

w

and periodicity of

D

placed (a) between two half-spaces and (b) at a distance of

h

from a PEC back plate (PAGR structure).

Download Full Size | PPT Slide | PDF

Fig. 2. Adopting the image theorem for calculation of the surface currents on graphene ribbons in the configuration of Fig. 1.

Download Full Size | PPT Slide | PDF

Fig. 3. Proposed equivalent circuit model for the PAGR structure of Fig. 1(b); the periodic array of graphene ribbons is modeled as parallel R-L-C branches, each representing one of the resonance modes.

Download Full Size | PPT Slide | PDF

Fig. 4. (a) Reflectivity and (b) reflection phase of a periodic array of graphene ribbons with a width of

w = 35 μm

and periodicity of

D = 70 μm

at the distance of

h = 3 μm

above the PEC back plate; the graphene parameters are assumed as

τ = 1 ps

and

E_{f} = 1 eV

Download Full Size | PPT Slide | PDF

Fig. 5. (a) Reflectivity and (b) reflection phase of a periodic array of graphene ribbons with a width of

w = 9 μm

and periodicity of

D = 10 μm

at the distance of

h = 1 μm

above the PEC back plate; the graphene parameters are assumed as

τ = 0.8 ps

and

E_{f} = 0.7 eV

Download Full Size | PPT Slide | PDF

Fig. 6. (a) Schematic view of the PAGR configuration. (b) The computed normalized eigencurrents and (c) the simulated electric field distribution of the graphene ribbons in the PAGR configuration at the vicinity of the first three resonances.

Download Full Size | PPT Slide | PDF

Fig. 7. (a) Optimum conductance value G of graphene ribbons for satisfaction of the resonance condition of Eq. (16a) in terms of

w / D

and

E_{f}

; (b) the optimum capacitance value C of graphene ribbons for satisfaction of the resonance condition of Eq. (16b) in terms of periodicity.

Download Full Size | PPT Slide | PDF

Fig. 8. (a) Absorption spectra of a periodic array of graphene ribbons with a width of

w = 21.3 μm

and periodicity of

D = 41 μm

at the distance of

h = 5 μm

above the graphene parameters are assumed as

τ = 0.6 ps

and

E_{f} = 0.85 eV

; (b) the absorption spectra of the same PAGR structure for different Fermi energy levels.

Download Full Size | PPT Slide | PDF

Fig. 9. Resonance frequency of absorption for different (a) Fermi energy levels, (b)

h / D

ratios, and (c) filling factors (

w / D

) of graphene ribbons when a periodic array of graphene ribbons with a width of

w = 21.3 μm

and periodicity of

D = 41 μm

is located at the distance of

h = 5 μm

above the PEC plate.

Download Full Size | PPT Slide | PDF

Tables (4)

Table 1. First Three Eigenfunctions and Eigenvalues for the Problem of a Single Graphene Ribbon

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Table 2. First Three Eigenvalues for the Problem of a PAGR Structure ( $H / D = 0.01$ )

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Table 3. First Three Eigenvalues for the Problem of a PAGR Structure ( $H / D = 0.05$ )

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Table 4. First Three Eigenvalues for the Problem of a PAGR Structure ( $H / D = 0.1$ )

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Equations (21)

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σ_{G} (ω) = σ_{G}^{intra} (ω) + σ_{G}^{inter} (ω),

σ_{G}^{intra} (ω) = \frac{2 k_{B} e^{2} T}{π ℏ^{2}} \ln [2 \cosh (\frac{E_{f}}{2 k_{B} T})] \frac{i}{ω + i τ^{- 1}},

σ_{G}^{inter} (ω) = \frac{e^{2}}{4 ℏ} [H (\frac{ω}{2}) + \frac{4 i ω}{π} \int_{0}^{\infty} \frac{H (ζ) - H (ω / 2)}{ω^{2} - 4 ξ^{2}} d ξ],

H (ζ) = \sinh (\frac{ℏ ζ}{k_{B} T}) / [\cosh (\frac{E_{f}}{k_{B} T}) + \cosh (\frac{ℏ ζ}{k_{B} T})] .

τ = E_{f} μ / v_{f}^{2} e,

J_{x, i} (x) / σ_{G} = E_{x, i}^{ext} (x) + \frac{1}{j ω ϵ_{0}} \frac{d}{d x} \int_{d_{i}}^{d_{i} + w} G_{0} (x - x^{'}) \frac{d J_{x, i} (x^{'})}{d x^{'}} d x^{'} + \frac{1}{j ω ϵ_{0}} \frac{d}{d x} \sum_{l \neq i} \int_{d_{l}}^{d_{l} + w} G_{0} (x - x^{'}) \frac{d J_{x, l} (x^{'})}{d x^{'}} d x^{'} .

G_{0} (x - x^{'}) \approx - \frac{1}{2 π} \ln (k_{0} | x - x^{'} |),

J_{x} (x) / σ_{G} = E_{x, i}^{ext} (x) - \frac{1}{2 π j ω ϵ_{0}} \frac{d}{d x} \int_{- w / 2}^{w / 2} \ln (k_{0} | x - x^{'} |) \frac{d J_{x} (x^{'})}{d x^{'}} d x^{'} - \frac{1}{2 π j ω ϵ_{0}} \frac{d}{d x} \sum_{l \neq 0} \int_{- w / 2}^{w / 2} \ln (k_{0} | x - x^{'} + l D |) \frac{d J_{x} (x^{'})}{d x^{'}} d x^{'} + \frac{1}{2 π j ω ϵ_{0}} \frac{d}{d x} \sum_{l = - \infty}^{\infty} \int_{- w / 2}^{w / 2} \ln (k_{0} | \sqrt{{(x - x^{'} + l D)}^{2} + {(2 h)}^{2}} |) \frac{d J_{x} (x^{'})}{d x^{'}} d x^{'} .

- k_{p} J_{x} (x) + \frac{1}{π} P \int_{- w / 2}^{w / 2} \frac{1}{x - x^{'}} \frac{\partial J_{x} (x^{'})}{\partial x^{'}} d x^{'} + \frac{1}{π} \sum_{l \neq 0} \int_{- w / 2}^{w / 2} \frac{1}{x - x^{'} + l D} \frac{\partial J_{x} (x^{'})}{\partial x^{'}} d x^{'} - \frac{1}{π} \sum_{l = - \infty}^{\infty} \int_{- w / 2}^{w / 2} \frac{x - x^{'} + l D}{{(x - x^{'} + l D)}^{2} + {(2 h)}^{2}} \frac{\partial J_{x} (x^{'})}{\partial x^{'}} d x^{'} = 2 j ϵ_{0} E_{x, i}^{ext} (x),

\frac{1}{π} P \int_{- w / 2}^{w / 2} \frac{1}{x - x^{'}} \frac{\partial ψ_{m} (x^{'})}{\partial x^{'}} d x^{'} + \frac{1}{π} \sum_{l \neq 0} \int_{- w_{0} / 2}^{w_{0} / 2} \frac{1}{x - x^{'} + l D} \frac{\partial ψ_{m} (x^{'})}{\partial x^{'}} d x^{'} - \frac{1}{π} \sum_{l = - \infty}^{\infty} \int_{- w / 2}^{w / 2} \frac{x - x^{'} + l D}{{(x - x^{'} + l D)}^{2} + {(2 h)}^{2}} \frac{\partial ψ_{m} (x^{'})}{\partial x^{'}} d x^{'} = q_{m} ψ_{m} (x) .

J_{x, i} (x) = \sum_{m = 1}^{\infty} A_{m} ψ_{m} (x),

A_{m} = \frac{2 j ω ϵ_{0}}{q_{m} - k_{p} (ω)} \int_{- w / 2}^{w / 2} E_{x}^{ext} (x) ψ_{m} (x) d x .

\frac{1}{π} P \int_{- w / 2}^{w / 2} \frac{1}{x - x^{'}} \frac{\partial ψ_{m} (x^{'})}{\partial x^{'}} d x^{'} = k_{m} ψ_{m} (x),

{\begin{cases} \int_{- w / 2}^{w / 2} ψ_{m} (x) ψ_{n} (x) d x = δ_{m n} \\ ψ_{m} (0) = ψ_{m} (w) = 0 \end{cases},

q_{m} = k_{m} - \frac{1}{π} \sum_{- \infty (l \neq 0)}^{\infty} \int_{- w / 2}^{w / 2} \int_{- w / 2}^{w / 2} \ln | x - x^{'} + l D | \frac{d ψ (x_{m})}{d x} \frac{d ψ (x_{m}^{'})}{d x^{'}} d x d x^{'} + \frac{1}{π} \sum_{- \infty}^{\infty} \int_{- w / 2}^{w / 2} \int_{- w / 2}^{w / 2} \ln | \sqrt{{(x - x^{'} + l D)}^{2} + {(2 h)}^{2}} | \frac{d ψ (x_{m})}{d x} \frac{d ψ (x_{m}^{'})}{d x^{'}} d x d x^{'} .

R_{m} = \frac{D}{S_{m}^{2}} \frac{π ℏ^{2}}{e^{2} E_{f} τ}, L_{m} = \frac{D}{S_{m}^{2}} \frac{π ℏ^{2}}{e^{2} E_{f}}, C_{m} = \frac{S_{m}^{2}}{D} \frac{2 ϵ_{0}}{q_{m}},

[\begin{matrix} A & B \\ C & D \end{matrix}] = \prod_{m} (\begin{matrix} 1 & 0 \\ Y_{g_{m}} & 1 \end{matrix}) (\begin{matrix} \cosh γ_{g} h & \sinh γ_{g} h / Y_{c} \\ Y_{c} \sinh γ_{g} h & \cosh γ_{g} h \end{matrix}),

R = (B - η_{0} D) / (B + η_{0} D) .

Y_{i n}^{tot} = Y_{i n}^{line} + Y_{g} = j B + \frac{1}{R + j X} = \frac{R}{R^{2} + X^{2}} + j (B - \frac{X}{R^{2} + X^{2}}),

R_{opt} = \frac{η_{0}}{1 + {(B η_{0})}^{2}},

X_{opt} = ω L - \frac{1}{ω C} = \frac{η_{0}^{2} B}{1 + {(B η_{0})}^{2}} .

Eigenvalue	Eigenfunction
$0.737 π / w$	$ψ_{1} = \sqrt{w} [1.2 \sin (\arccos 2 x / w) - 1.06 \sin (3 \arccos 2 x / w)]$
$1.753 π / w$	$ψ_{2} = \sqrt{w} [1.254 \sin (2 \arccos 2 x / w) - 0.302 \sin (4 \arccos 2 x / w)]$
$2.748 π / w$	$ψ_{2} = \sqrt{w} [0.308 \sin (\arccos (2 x / w)) + 1.19 \sin (3 \arccos (2 x / w)) - 0.484 \sin (5 \arccos (2 x / w))]$

$w / D$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
$q_{1} π / w$	0.365	0.232	0.173	0.139	0.116	0.1	0.088	0.079	0.071
$q_{2} π / w$	1.22	0.826	0.624	0.502	0.421	0.363	0.319	0.286	0.26
$q_{3} π / w$	2.277	1.661	1.289	1.051	0.887	0.768	0.676	0.604	0.545

$w / D$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
$q_{1} π / w$	0.668	0.562	0.476	0.412	0.363	0.323	0.290	0.261	0.228
$q_{2} π / w$	1.732	1.625	1.481	1.342	1.221	1.117	1.031	0.961	0.918
$q_{3} π / w$	2.744	2.695	2.58	2.431	2.276	2.128	1.991	1.861	1.723

$w / D$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
$q_{1} π / w$	0.716	0.667	0.611	0.558	0.509	0.465	0.423	0.379	0.323
$q_{2} π / w$	1.751	1.732	1.688	1.626	1.556	1.487	1.424	1.377	1.366
$q_{3} π / w$	2.747	2.743	2.729	2.695	2.643	2.576	2.498	2.404	2.271

Eigenvalue	Eigenfunction
$0.737 π / w$	$ψ_{1} = \sqrt{w} [1.2 \sin (\arccos 2 x / w) - 1.06 \sin (3 \arccos 2 x / w)]$
$1.753 π / w$	$ψ_{2} = \sqrt{w} [1.254 \sin (2 \arccos 2 x / w) - 0.302 \sin (4 \arccos 2 x / w)]$
$2.748 π / w$	$ψ_{2} = \sqrt{w} [0.308 \sin (\arccos (2 x / w)) + 1.19 \sin (3 \arccos (2 x / w)) - 0.484 \sin (5 \arccos (2 x / w))]$

Abstract

References

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Loh, K. P.

Mak, K. F.

Martin, M.

Mehrany, K.

Meric, I.

Mesch, M.