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 (=0.05λ0) 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.

© 2018 Optical Society of America

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References

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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)

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]

2002 (1)

1999 (1)

1996 (1)

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)

Abdolali, A.

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]

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]

Abdollah Ramezani, S.

AbdollahRamezani, S.

Adam, S.

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]

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.

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.

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]

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.

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]

Borji, A.

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]

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.

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.

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]

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.

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]

Deng, G.

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]

Fadakar, H.

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]

Farajollahi, S.

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]

Gaylord, T. K.

Geim, A. K.

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

Geng, B.

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

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.

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]

Granet, G.

Grigorieva, I. 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]

Guo, C. C.

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]

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.

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]

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.

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]

Horng, J.

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

Hwang, E. H.

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]

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.

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]

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]

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]

Kavehvash, Z.

Khavasi, A.

Kim, P.

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]

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]

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.

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]

Li, L.

Liang, X.

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

Liu, K.

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]

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]

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]

Mak, K. 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]

Martin, M.

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

Mehrany, K.

Meric, I.

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]

Mesch, 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]

Miao, F.

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]

Mikhailov, S. A.

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

Moharam, M. G.

Momeni, A.

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]

Morozov, 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]

Mors, M.

Novoselov, K. S.

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

Ortega, J. M. P.

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]

Padooru, Y. R.

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]

Planken, P. C. M.

Plumey, J.-P.

Qin, S. Q.

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]

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]

Rahmanzadeh, M.

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]

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]

Rajabalipanah, H.

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]

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]

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]

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]

Rashidian, B.

Rejaei, B.

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

Rouhi, K.

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]

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]

Savostianova, N. A.

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

Shahabadi, M.

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]

Shen, Y. R.

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

Shepard, K. L.

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]

Silveiro, I.

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]

Sorgenfrei, S.

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]

Soric, J.

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]

Stormer, H. L.

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]

Tai, C.-T.

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994), Vol. 272.

Tan, Y.

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]

Taniguchi, T.

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]

Teweldebrhan, D.

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]

Transactions, I.

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

Wang, 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]

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]

Wang, L.

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]

Watanabe, K.

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]

Weiss, T.

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]

Wenckebach, T.

Xing, R.

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

Yakovlev, A. B.

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]

Yang, J.

Ye, W. M.

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]

Yin, Z.

Young, A. F.

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]

Yuan, X. D.

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]

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]

Zeidaabadi Nezhad, A.

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]

Zettl, A.

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

Zhang, J. F.

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]

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]

Zhang, Y.

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]

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]

Zhao, G.

Zhao, Y.

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]

Zhu, Z. H.

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]

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]

ACS Nano (1)

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

Ann. Phys. (1)

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]

Appl. Opt. (3)

Appl. Phys. A (1)

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]

IEEE J. Quantum Electron. (2)

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]

IEEE Trans. Antennas Propag. (1)

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

IEEE Trans. Plasma Sci. (1)

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]

J. Appl. Phys. (2)

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]

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]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (3)

Nano Lett. (2)

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]

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]

Nat. Mater. (1)

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

Nat. Nanotechnol. (2)

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

New J. Phys. (2)

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]

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]

Opt. Laser Technol. (1)

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

Opt. Lett. (3)

Phys. Rev. B (1)

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

Phys. Rev. Lett. (1)

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]

Sci. Rep. (1)

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]

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).
[Crossref]

Solid State Commun. (1)

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]

Other (3)

C.-T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994), Vol. 272.

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

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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Figures (9)

Fig. 1.
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).
Fig. 2.
Fig. 2. Adopting the image theorem for calculation of the surface currents on graphene ribbons in the configuration of Fig. 1.
Fig. 3.
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.
Fig. 4.
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 .
Fig. 5.
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 .
Fig. 6.
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.
Fig. 7.
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.
Fig. 8.
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.
Fig. 9.
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.

Tables (4)

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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 )

Equations (21)

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σ G ( ω ) = σ G intra ( ω ) + σ G inter ( ω ) ,
σ G intra ( ω ) = 2 k B e 2 T π 2 ln [ 2 cosh ( E f 2 k B T ) ] i ω + i τ 1 ,
σ G inter ( ω ) = e 2 4 [ H ( ω 2 ) + 4 i ω π 0 H ( ζ ) H ( ω / 2 ) ω 2 4 ξ 2 d ξ ] ,
H ( ζ ) = sinh ( ζ k B T ) / [ cosh ( E f k B T ) + cosh ( ζ k B T ) ] .
τ = E f μ / v f 2 e ,
J x , i ( x ) / σ G = E x , i ext ( x ) + 1 j ω ϵ 0 d d x d i d i + w G 0 ( x x ) d J x , i ( x ) d x d x + 1 j ω ϵ 0 d d x l i d l d l + w G 0 ( x x ) d J x , l ( x ) d x d x .
G 0 ( x x ) 1 2 π ln ( k 0 | x x | ) ,
J x ( x ) / σ G = E x , i ext ( x ) 1 2 π j ω ϵ 0 d d x w / 2 w / 2 ln ( k 0 | x x | ) d J x ( x ) d x d x 1 2 π j ω ϵ 0 d d x l 0 w / 2 w / 2 ln ( k 0 | x x + l D | ) d J x ( x ) d x d x + 1 2 π j ω ϵ 0 d d x l = w / 2 w / 2 ln ( k 0 | ( x x + l D ) 2 + ( 2 h ) 2 | ) d J x ( x ) d x d x .
k p J x ( x ) + 1 π P w / 2 w / 2 1 x x J x ( x ) x d x + 1 π l 0 w / 2 w / 2 1 x x + l D J x ( x ) x d x 1 π l = w / 2 w / 2 x x + l D ( x x + l D ) 2 + ( 2 h ) 2 J x ( x ) x d x = 2 j ϵ 0 E x , i ext ( x ) ,
1 π P w / 2 w / 2 1 x x ψ m ( x ) x d x + 1 π l 0 w 0 / 2 w 0 / 2 1 x x + l D ψ m ( x ) x d x 1 π l = w / 2 w / 2 x x + l D ( x x + l D ) 2 + ( 2 h ) 2 ψ m ( x ) x d x = q m ψ m ( x ) .
J x , i ( x ) = m = 1 A m ψ m ( x ) ,
A m = 2 j ω ϵ 0 q m k p ( ω ) w / 2 w / 2 E x ext ( x ) ψ m ( x ) d x .
1 π P w / 2 w / 2 1 x x ψ m ( x ) x d x = k m ψ m ( x ) ,
{ w / 2 w / 2 ψ m ( x ) ψ n ( x ) d x = δ m n ψ m ( 0 ) = ψ m ( w ) = 0 ,
q m = k m 1 π ( l 0 ) w / 2 w / 2 w / 2 w / 2 ln | x x + l D | d ψ ( x m ) d x d ψ ( x m ) d x d x d x + 1 π w / 2 w / 2 w / 2 w / 2 ln | ( x x + l D ) 2 + ( 2 h ) 2 | d ψ ( x m ) d x d ψ ( x m ) d x d x d x .
R m = D S m 2 π 2 e 2 E f τ , L m = D S m 2 π 2 e 2 E f , C m = S m 2 D 2 ϵ 0 q m ,
[ A B C D ] = m ( 1 0 Y g m 1 ) ( cosh γ g h sinh γ g h / Y c Y c sinh γ g h cosh γ g h ) ,
R = ( B η 0 D ) / ( B + η 0 D ) .
Y i n tot = Y i n line + Y g = j B + 1 R + j X = R R 2 + X 2 + j ( B X R 2 + X 2 ) ,
R opt = η 0 1 + ( B η 0 ) 2 ,
X opt = ω L 1 ω C = η 0 2 B 1 + ( B η 0 ) 2 .

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