Abstract

The non-reciprocity of the edge magnetoplasmon modes of a graphene strip is leveraged to design a non-reciprocal magnetoplasmon graphene coupler, coupling only in one direction. The proposed coupler consists of two coplanar parallel magnetically biased graphene strips. In the forward direction, the modes along the adjacent strip edges of the strips have the same wavenumber and therefore couple to each other. In the backward direction, the modes along the adjacent strip edges have different wavenumbers and therefore no coupling occurs.

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    [CrossRef] [PubMed]
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2012 (8)

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photon.6, 7490758 (2012).

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett.12, 2470–2474 (2012).
[CrossRef] [PubMed]

A. Vakil and N. Engheta, “One-atom-thick reflectors for surface plasmon polariton surface waves on graphene,” Optics Communications285, 3428–3430 (2012).
[CrossRef]

S. Thongrattanasiri, I. Silveiro, and F. J. G. de Abajo, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett.100, 201105 (2012).
[CrossRef]

N. Chamanara, D. Sounas, T. Szkopek, and C. Caloz, “Optically transparent and flexible graphene reciprocal and nonreciprocal microwave planar components,” IEEE Microw. Wireless Comp. Lett.22, 360–362 (2012).
[CrossRef]

D. L. Sounas and C. Caloz, “Gyrotropy and non-reciprocity of graphene for microwave applications,” IEEE Trans. Microw. Theory Tech.60, 901–914 (2012).
[CrossRef]

J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature (2012).
[CrossRef]

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature (2012).
[CrossRef]

2011 (6)

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun.2(2011).
[CrossRef] [PubMed]

N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, “Hot carrier-assisted intrinsic photoresponse in graphene,” Science334, 648–652 (2011).
[CrossRef] [PubMed]

Y. Zhu, Z. Sun, Z. Yan, Z. Jin, and J. M. Tour, “Rational design of hybrid graphene films for high-performance transparent electrodes,” ACS Nano5, 64726479 (2011).

D. L. Sounas and C. Caloz, “Electromagnetic non-reciprocity and gyrotropy of graphene,” Appl. Phys. Lett.98, 021 911:13 (2011).
[CrossRef]

D. L. Sounas and C. Caloz, “Edge surface modes in magnetically biased chemically doped graphene strips,” Appl. Phys. Lett.99, 231 902:13 (2011).
[CrossRef]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science332, 1291–1294 (2011).
[CrossRef] [PubMed]

2010 (4)

E. G. Mishchenko, A. V. Shytov, and P. G. Silvestrov, “Guided plasmons in graphene p-njunctions,” Phys. Rev. Lett.104, 156 806 (2010).
[CrossRef]

S. Bae, H. Kim, Y. Lee, X. Xu, J. S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, and , “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nature Nanotechnology5, 574–578 (2010).
[CrossRef] [PubMed]

F. Gunes, H. J. Shin, C. Biswas, G. H. Han, E. S. Kim, S. J. Chae, J. Y. Choi, and Y. H. Lee, “Layer-by-layer doping of few-layer graphene film,” ACS Nano4, 45954600 (2010).
[CrossRef]

T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photon.4, 297–301 (2010).
[CrossRef]

2009 (1)

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys.81, 109–162 (2009).
[CrossRef]

2008 (1)

G. W. Hanson, “Dyadic Greens functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys.103, 064 302 (2008).
[CrossRef]

2007 (2)

S. A. Mikhailov and K. Ziegler, “New electromagnetic mode in graphene,” Phys. Rev. Lett.99, 016 803 (2007).
[CrossRef]

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Materials6, 183–191 (2007).
[CrossRef] [PubMed]

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 22 306, 666–669 (2004).
[CrossRef] [PubMed]

2002 (1)

Y. Zhao, K. Wu, and K. M. Cheng, “A compact 2-D full-wave finite-difference frequency-domain method for general guided wave structures,” IEEE Trans. Microwave Theory Tech.50, 1844–1848 (2002).
[CrossRef]

Alonso-Gonzalez, P.

J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature (2012).
[CrossRef]

Andreev, G. O.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature (2012).
[CrossRef]

Avouris, P.

T. Mueller, F. Xia, and P. Avouris, “Graphene photodetectors for high-speed optical communications,” Nat. Photon.4, 297–301 (2010).
[CrossRef]

Badioli, M.

J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature (2012).
[CrossRef]

Bae, S.

S. Bae, H. Kim, Y. Lee, X. Xu, J. S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, and , “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nature Nanotechnology5, 574–578 (2010).
[CrossRef] [PubMed]

Balakrishnan, J.

S. Bae, H. Kim, Y. Lee, X. Xu, J. S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H. R. Kim, Y. I. Song, and , “Roll-to-roll production of 30-inch graphene films for transparent electrodes,” Nature Nanotechnology5, 574–578 (2010).
[CrossRef] [PubMed]

Bao, W.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature (2012).
[CrossRef]

Basov, D. N.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature (2012).
[CrossRef]

Biswas, C.

F. Gunes, H. J. Shin, C. Biswas, G. H. Han, E. S. Kim, S. J. Chae, J. Y. Choi, and Y. H. Lee, “Layer-by-layer doping of few-layer graphene film,” ACS Nano4, 45954600 (2010).
[CrossRef]

Britnell, L.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun.2(2011).
[CrossRef] [PubMed]

Caloz, C.

N. Chamanara, D. Sounas, T. Szkopek, and C. Caloz, “Optically transparent and flexible graphene reciprocal and nonreciprocal microwave planar components,” IEEE Microw. Wireless Comp. Lett.22, 360–362 (2012).
[CrossRef]

D. L. Sounas and C. Caloz, “Gyrotropy and non-reciprocity of graphene for microwave applications,” IEEE Trans. Microw. Theory Tech.60, 901–914 (2012).
[CrossRef]

D. L. Sounas and C. Caloz, “Edge surface modes in magnetically biased chemically doped graphene strips,” Appl. Phys. Lett.99, 231 902:13 (2011).
[CrossRef]

D. L. Sounas and C. Caloz, “Electromagnetic non-reciprocity and gyrotropy of graphene,” Appl. Phys. Lett.98, 021 911:13 (2011).
[CrossRef]

D. L. Sounas, H. S. Skulason, H. V. Nguyen, A. Guermoune, M. Siaj, T. Szkopek, and C. Caloz, “Faraday rotation in magnetically-biased graphene at microwave frequencies,” (2013), under review.

Camara, N.

J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature (2012).
[CrossRef]

Centeno, A.

J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature (2012).
[CrossRef]

Chae, S. J.

F. Gunes, H. J. Shin, C. Biswas, G. H. Han, E. S. Kim, S. J. Chae, J. Y. Choi, and Y. H. Lee, “Layer-by-layer doping of few-layer graphene film,” ACS Nano4, 45954600 (2010).
[CrossRef]

Chamanara, N.

N. Chamanara, D. Sounas, T. Szkopek, and C. Caloz, “Optically transparent and flexible graphene reciprocal and nonreciprocal microwave planar components,” IEEE Microw. Wireless Comp. Lett.22, 360–362 (2012).
[CrossRef]

Chen, J.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett.12, 2470–2474 (2012).
[CrossRef] [PubMed]

J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature (2012).
[CrossRef]

Cheng, K. M.

Y. Zhao, K. Wu, and K. M. Cheng, “A compact 2-D full-wave finite-difference frequency-domain method for general guided wave structures,” IEEE Trans. Microwave Theory Tech.50, 1844–1848 (2002).
[CrossRef]

Choi, J. Y.

F. Gunes, H. J. Shin, C. Biswas, G. H. Han, E. S. Kim, S. J. Chae, J. Y. Choi, and Y. H. Lee, “Layer-by-layer doping of few-layer graphene film,” ACS Nano4, 45954600 (2010).
[CrossRef]

Crassee, I.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett.12, 2470–2474 (2012).
[CrossRef] [PubMed]

de Abajo, F. J. G.

J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature (2012).
[CrossRef]

S. Thongrattanasiri, I. Silveiro, and F. J. G. de Abajo, “Plasmons in electrostatically doped graphene,” Appl. Phys. Lett.100, 201105 (2012).
[CrossRef]

Dominguez, G.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature (2012).
[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 22 306, 666–669 (2004).
[CrossRef] [PubMed]

Echtermeyer, T.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun.2(2011).
[CrossRef] [PubMed]

Engheta, N.

A. Vakil and N. Engheta, “One-atom-thick reflectors for surface plasmon polariton surface waves on graphene,” Optics Communications285, 3428–3430 (2012).
[CrossRef]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science332, 1291–1294 (2011).
[CrossRef] [PubMed]

Fei, Z.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature (2012).
[CrossRef]

Ferrari, A.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun.2(2011).
[CrossRef] [PubMed]

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 22 306, 666–669 (2004).
[CrossRef] [PubMed]

Fogler, M. M.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature (2012).
[CrossRef]

Gabor, N. M.

N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi, L. S. Levitov, and P. Jarillo-Herrero, “Hot carrier-assisted intrinsic photoresponse in graphene,” Science334, 648–652 (2011).
[CrossRef] [PubMed]

Gaponenko, I.

I. Crassee, M. Orlita, M. Potemski, A. L. Walter, M. Ostler, T. Seyller, I. Gaponenko, J. Chen, and A. B. Kuzmenko, “Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene,” Nano Lett.12, 2470–2474 (2012).
[CrossRef] [PubMed]

Geim, A.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun.2(2011).
[CrossRef] [PubMed]

Geim, A. K.

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys.81, 109–162 (2009).
[CrossRef]

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Materials6, 183–191 (2007).
[CrossRef] [PubMed]

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 22 306, 666–669 (2004).
[CrossRef] [PubMed]

Godignon, P.

J. Chen, M. Badioli, P. Alonso-Gonzalez, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenovic, A. Centeno, A. Pesquera, P. Godignon, A. Zurutuza Elorza, N. Camara, F. J. G. de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature (2012).
[CrossRef]

Gorbachev, R.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun.2(2011).
[CrossRef] [PubMed]

Grigorenko, A.

T. Echtermeyer, L. Britnell, P. Jasnos, A. Lombardo, R. Gorbachev, A. Grigorenko, A. Geim, A. Ferrari, and K. Novoselov, “Strong plasmonic enhancement of photovoltage in graphene,” Nat. Commun.2(2011).
[CrossRef] [PubMed]

Grigorenko, A. N.

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photon.6, 7490758 (2012).

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 22 306, 666–669 (2004).
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F. Gunes, H. J. Shin, C. Biswas, G. H. Han, E. S. Kim, S. J. Chae, J. Y. Choi, and Y. H. Lee, “Layer-by-layer doping of few-layer graphene film,” ACS Nano4, 45954600 (2010).
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Appl. Phys. Lett. (3)

D. L. Sounas and C. Caloz, “Edge surface modes in magnetically biased chemically doped graphene strips,” Appl. Phys. Lett.99, 231 902:13 (2011).
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IEEE Microw. Wireless Comp. Lett. (1)

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Figures (11)

Fig. 1
Fig. 1

Slow-wave factor and loss for a graphene strip with parameters w = 100 μm, τ = 0.1 ps, ns = 1013 cm−2 and B0 = 1 T. Edge modes are plotted in red and bulk modes in blue. The dashed curve shows the dispersion for an infinite graphene sheet with the same parameters. The gray area corresponds to the light cone.

Fig. 2
Fig. 2

Electric field magnitude for the bulk and edge modes of the graphene strip in Fig. 1.

Fig. 3
Fig. 3

Electric field on the graphene strip for the edge modes propagating on the right and left edges. (a) Point A on the right edge sees a counter clockwise rotating electric field as the wave (mode 2+) propagates along the graphene strip. (b) Point B on the left edge sees a clockwise rotating electric field as the wave (mode 1+) propagates along the graphene strip.

Fig. 4
Fig. 4

Non reciprocal plasmonic coupler, consisting of two parallel graphene plasmonic waveguides. Both waveguides are biased with a magnetostatic field perpendicular to their plane. (a) Feeding through port 1. (b) Feeding through port 2.

Fig. 5
Fig. 5

Dispersion curves for edge (red) and bulk (blue) modes of two isolated graphene strips with different parameters. The solid curves show the slow-wave factor and loss for a graphene strip with parameters w = 100 μm, τ = 0.1 ps, ns = 1013 cm−2 and B0 = 1 T. The dashed curves show the slow-wave factor and loss for a graphene strip with parameters w = 100 μm, τ = 0.1 ps, ns = 8 × 1012 cm−2 and B0 = 1 T. Phase matched regions are emphasized by ellipses.

Fig. 6
Fig. 6

Electric field magnitude for the edge modes of the graphene strips of Fig. 5 showing different possible scenarios when they are placed side by side. The right strip has the parameters w = 100 μm, τ = 0.1 ps, ns = 1013 cm−2 and B0 = 1 T. The left strip has the parameters w = 100 μm, τ = 0.1 ps, ns = 8 × 1012 cm−2 and B0 = 1 T.

Fig. 7
Fig. 7

Dispersion curves of the edge and bulk modes of the graphene plasmonic coupler of Fig. 4 for the forward and backward propagation with parameters wR = 100 μm, τ = 0.1 ps, ns = 1013 cm−2 and B0 = 1 T for the right strip, and wL = 100 μm, τ = 0.1 ps, ns = 8 × 1012 cm−2 and B0 = 1 T for the left strip and spacing s = 2 μm.

Fig. 8
Fig. 8

Electric field magnitude for the edge modes of the coupler of Fig. 4 propagating on the near edges of the strips at the frequency f = 6 THz. (a) Forward direction. The edge modes couple and give two coupled symmetrical and anti-symmetrical modes. (b) backward direction. The modes do not couple.

Fig. 9
Fig. 9

Transverse electric field vector plot. (a) anti-symmetrical, (b) symmetrical.

Fig. 10
Fig. 10

Output powers at the through and coupled ports when the coupler is excited (a) at port 1, (b) at port 2. λ0 is the free space wavelength.

Fig. 11
Fig. 11

Output powers for couplers with low resistance graphene strips at the through and coupled ports when the coupler is excited (a) at port 1, (b) at port 2.

Equations (5)

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E 1 ( x , y , z ) = a A E A ( x , y ) e j β A z + a S E S ( x , y ) e j β S z
a A = E L 2 + t ( x , y ) E A t ( x , y ) d x d y ,
a S = E L 2 + t ( x , y ) E S t ( x , y ) d x d y ,
a 2 + ( l ) = a A e j β A l E L 2 + t ( x , y ) E A t ( x , y ) d x d y + a S e j β S l E L 2 + t ( x , y ) E S t ( x , y ) d x d y ,
a 4 + ( l ) = a A e j β A l E R 1 + t ( x , y ) E A t ( x , y ) d x d y + a S e j β S l E R 1 + t ( x , y ) E S t ( x , y ) d x d y .

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