Abstract

We introduce compact tunable spatial mode converters between the even and odd modes of graphene parallel plate (GPP) waveguides. The converters are reciprocal and are based on spatial modulation of graphene’s conductivity. We show that the wavelength of operation of the mode converters can be tuned in the mid-infrared wavelength range by adjusting the chemical potential of a strip on one of the graphene layers of the GPP waveguides. We also introduce optical diodes for GPP waveguides based on a spatial mode converter and a coupler, which consists of a single layer of graphene placed in the middle between the two plates of two GPP waveguides. We find that for both the spatial mode converter and the optical diode the device functionality is preserved in the presence of loss.

© 2016 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]

2016 (1)

D. Correas-Serrano, J. S. Gomez-Diaz, D. L. Sounas, Y. Hadad, A. Alvarez-Melcon, and A. Alù, “Nonreciprocal graphene devices and antennas based on spatiotemporal modulation,” IEEE Antennas Wireless Propag. Lett. 15, 1529–1533 (2016).
[Crossref]

2015 (5)

2014 (7)

J. Tao, X. Yu, B. Hu, A. Dubrovkin, and Q. J. Wang, “Graphene-based tunable plasmonic bragg reflector with a broad bandwidth,” Opt. Lett. 39, 271–274 (2014).
[Crossref] [PubMed]

K. J. Ooi, H. S. Chu, P. Bai, and L. K. Ang, “Electro-optical graphene plasmonic logic gates,” Opt. Lett. 39, 1629–1632 (2014).
[Crossref] [PubMed]

L. H. Frandsen, Y. Elesin, L. F. Frellsen, M. Mitrovic, Y. Ding, O. Sigmund, and K. Yvind, “Topology optimized mode conversion in a photonic crystal waveguide fabricated in silicon-on-insulator material,” Opt. Express 22, 8525–8532 (2014).
[Crossref] [PubMed]

D. Ohana and U. Levy, “Mode conversion based on dielectric metamaterial in silicon,” Opt. Express 22, 27617–27631 (2014).
[Crossref] [PubMed]

G. D. Bouzianas, N. V. Kantartzis, T. V. Yioultsis, and T. D. Tsiboukis, “Consistent study of graphene structures through the direct incorporation of surface conductivity,” IEEE Trans. Magn. 50, 161–164 (2014).
[Crossref]

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Graphene-based plasmonic tunable low-pass filters in the terahertz band,” IEEE Trans. Nanotechnol. 13, 1145–1153 (2014).
[Crossref]

I.-T. Lin and J.-M. Liu, “Optimization of double-layer graphene plasmonic waveguides,” Appl. Phys. Lett. 105, 061116 (2014).
[Crossref]

2013 (15)

S. Thongrattanasiri and F. J. G. de Abajo, “Optical field enhancement by strong plasmon interaction in graphene nanostructures,” Phys. Rev. Lett. 110, 187401 (2013).
[Crossref] [PubMed]

M. Farhat, S. Guenneau, and H. Bağcı, “Exciting graphene surface plasmon polaritons through light and sound interplay,” Phys. Rev. Lett. 111, 237404 (2013).
[Crossref]

V. Nayyeri, M. Soleimani, and O. M. Ramahi, “Modeling graphene in the finite-difference time-domain method using a surface boundary condition,” IEEE Trans. Antennas Propag. 61, 4176–4182 (2013).
[Crossref]

S. Feng and Y. Wang, “Unidirectional light propagation characters of the triangular-lattice hybrid-waveguide photonic crystals,” Opt. Mater. 35, 1455–1460 (2013).
[Crossref]

A. Khavasi, M. Rezaei, A. P. Fard, and K. Mehrany, “A heuristic approach to the realization of the wide-band optical diode effect in photonic crystal waveguides,” J. Opt. 15, 075501 (2013).
[Crossref]

W. Liu, D. Yang, G. Shen, H. Tian, and Y. Ji, “Design of ultra compact all-optical XOR, XNOR, NANAD and OR gates using photonic crystal multi-mode interference waveguides,” Opt. Laser Technol. 50, 55–64 (2013).
[Crossref]

D. Svintsov, V. Vyurkov, V. Ryzhii, and T. Otsuji, “Voltage-controlled surface plasmon-polaritons in double graphene layer structures,” J. Appl. Phys. 113, 053701 (2013).
[Crossref]

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Álvarez-Melcón, “Spatially dispersive graphene single and parallel plate waveguides: Analysis and circuit model,” IEEE Trans. Microwave Theory Tech. 61, 4333–4344 (2013).
[Crossref]

P.-Y. Chen, C. Argyropoulos, and A. Alu, “Terahertz antenna phase shifters using integrally-gated graphene transmission-lines,” IEEE Trans. Antennas Propag. 61, 1528–1537 (2013).
[Crossref]

H. Li, L. Wang, Z. Huang, B. Sun, X. Zhai, and X. Li, “Mid-infrared, plasmonic switches and directional couplers induced by graphene sheets coupling system,” Europhys. Lett. 104, 37001 (2013).
[Crossref]

H. Iizuka and S. Fan, “Deep subwavelength plasmonic waveguide switch in double graphene layer structure,” Appl. Phys. Lett. 103, 233107 (2013).
[Crossref]

D. Dai, J. Wang, and S. He, “Silicon multimode photonic integrated devices for on-chip mode-division-multiplexed optical interconnects (invited review),” Prog. Electromagn. Res. 143, 773–819 (2013).
[Crossref]

X. Zhu, W. Yan, N. A. Mortensen, and S. Xiao, “Bends and splitters in graphene nanoribbon waveguides,” Opt. Express 21, 3486–3491 (2013).
[Crossref] [PubMed]

X.-T. Kong, Z.-B. Li, and J.-G. Tian, “Mode converter in metal-insulator-metal plasmonic waveguide designed by transformation optics,” Opt. Express 21, 9437–9446 (2013).
[Crossref] [PubMed]

W. B. Lu, W. Zhu, H. J. Xu, Z. H. Ni, Z. G. Dong, and T. J. Cui, “Flexible transformation plasmonics using graphene,” Opt. Express 21, 10475–10482 (2013).
[Crossref] [PubMed]

2012 (7)

Y.-T. Hung, C.-B. Huang, and J.-S. Huang, “Plasmonic mode converter for controlling optical impedance and nanoscale light-matter interaction,” Opt. Express 20, 20342–20355 (2012).
[Crossref] [PubMed]

D. A. B. Miller, “All linear optical devices are mode converters,” Opt. Express 20, 23985–23993 (2012).
[Crossref] [PubMed]

V. Liu, D. A. B. Miller, and S. Fan, “Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect,” Opt. Express 20, 28388–28397 (2012).
[Crossref] [PubMed]

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12, 1482–1485 (2012).
[Crossref] [PubMed]

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

B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100, 131111 (2012).
[Crossref]

Z. Fang, Z. Liu, Y. Wang, P. M. Ajayan, P. Nordlander, and N. J. Halas, “Graphene-antenna sandwich photodetector,” Nano Lett. 12, 3808–3813 (2012).
[Crossref] [PubMed]

2011 (3)

A. Y. Nikitin, F. Guinea, F. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84, 161407 (2011).
[Crossref]

J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. Koppens, and F. J. GarcÌa de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6, 431–440 (2011).
[Crossref] [PubMed]

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref] [PubMed]

2010 (3)

D. R. Andersen, “Graphene-based long-wave infrared tm surface plasmon modulator,” J. Opt. Soc. Am. B 27, 818–823 (2010).
[Crossref]

L. Wu, H. Chu, W. Koh, and E. Li, “Highly sensitive graphene biosensors based on surface plasmon resonance,” Opt. Express 18, 14395–14400 (2010).
[Crossref] [PubMed]

J. Cai, P. Ruffieux, R. Jaafar, M. Bieri, T. Braun, S. Blankenburg, M. Muoth, A. P. Seitsonen, M. Saleh, X. Feng, K. Mullen, and R. Fasel, “Atomically precise bottom-up fabrication of graphene nanoribbons,” Nature 466, 470–473 (2010).
[Crossref] [PubMed]

2009 (3)

S. Masubuchi, M. Ono, K. Yoshida, K. Hirakawa, and T. Machida, “Fabrication of graphene nanoribbon by local anodic oxidation lithography using atomic force microscope,” Appl. Phys. Lett. 94, 082107 (2009).
[Crossref]

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

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
[Crossref]

2008 (3)

L. A. Falkovsky, “Optical properties of graphene,” J. Phys.: Conf. Ser. 129, 012004 (2008).

G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104, 084314 (2008).
[Crossref]

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

2007 (2)

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

K. S. Novoselov, S. V. Morozov, T. M. G. Mohinddin, L. A. Ponomarenko, D. C. Elias, R. Yang, I. I. Barbolina, P. Blake, T. J. Booth, D. Jiang, J. Giesbers, E. W. Hill, and A. K. Geim, “Electronic properties of graphene,” Phys. Status Solidi B 244, 4106–4111 (2007).
[Crossref]

2005 (1)

K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. Katsnelson, I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless dirac fermions in graphene,” Nature 438, 197–200 (2005).
[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 306, 666–669 (2004).
[Crossref] [PubMed]

Ajayan, P. M.

Z. Fang, Z. Liu, Y. Wang, P. M. Ajayan, P. Nordlander, and N. J. Halas, “Graphene-antenna sandwich photodetector,” Nano Lett. 12, 3808–3813 (2012).
[Crossref] [PubMed]

Alu, A.

P.-Y. Chen, C. Argyropoulos, and A. Alu, “Terahertz antenna phase shifters using integrally-gated graphene transmission-lines,” IEEE Trans. Antennas Propag. 61, 1528–1537 (2013).
[Crossref]

Alù, A.

D. Correas-Serrano, J. S. Gomez-Diaz, D. L. Sounas, Y. Hadad, A. Alvarez-Melcon, and A. Alù, “Nonreciprocal graphene devices and antennas based on spatiotemporal modulation,” IEEE Antennas Wireless Propag. Lett. 15, 1529–1533 (2016).
[Crossref]

Alvarez-Melcon, A.

D. Correas-Serrano, J. S. Gomez-Diaz, D. L. Sounas, Y. Hadad, A. Alvarez-Melcon, and A. Alù, “Nonreciprocal graphene devices and antennas based on spatiotemporal modulation,” IEEE Antennas Wireless Propag. Lett. 15, 1529–1533 (2016).
[Crossref]

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Graphene-based plasmonic tunable low-pass filters in the terahertz band,” IEEE Trans. Nanotechnol. 13, 1145–1153 (2014).
[Crossref]

Álvarez-Melcón, A.

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Álvarez-Melcón, “Spatially dispersive graphene single and parallel plate waveguides: Analysis and circuit model,” IEEE Trans. Microwave Theory Tech. 61, 4333–4344 (2013).
[Crossref]

Andersen, D. R.

Ang, L. K.

Argyropoulos, C.

P.-Y. Chen, C. Argyropoulos, and A. Alu, “Terahertz antenna phase shifters using integrally-gated graphene transmission-lines,” IEEE Trans. Antennas Propag. 61, 1528–1537 (2013).
[Crossref]

Ayache, M.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref] [PubMed]

Bagci, H.

M. Farhat, S. Guenneau, and H. Bağcı, “Exciting graphene surface plasmon polaritons through light and sound interplay,” Phys. Rev. Lett. 111, 237404 (2013).
[Crossref]

Bai, P.

Bao, Q.

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

Barbolina, I. I.

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D. Correas-Serrano, J. S. Gomez-Diaz, D. L. Sounas, Y. Hadad, A. Alvarez-Melcon, and A. Alù, “Nonreciprocal graphene devices and antennas based on spatiotemporal modulation,” IEEE Antennas Wireless Propag. Lett. 15, 1529–1533 (2016).
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D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Graphene-based plasmonic tunable low-pass filters in the terahertz band,” IEEE Trans. Nanotechnol. 13, 1145–1153 (2014).
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D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Álvarez-Melcón, “Spatially dispersive graphene single and parallel plate waveguides: Analysis and circuit model,” IEEE Trans. Microwave Theory Tech. 61, 4333–4344 (2013).
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K. S. Novoselov, S. V. Morozov, T. M. G. Mohinddin, L. A. Ponomarenko, D. C. Elias, R. Yang, I. I. Barbolina, P. Blake, T. J. Booth, D. Jiang, J. Giesbers, E. W. Hill, and A. K. Geim, “Electronic properties of graphene,” Phys. Status Solidi B 244, 4106–4111 (2007).
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J. Gomez-Diaz, C. Moldovan, S. Capdevila, J. Romeu, L. Bernard, A. Magrez, A. Ionescu, and J. Perruisseau-Carrier, “Self-biased reconfigurable graphene stacks for terahertz plasmonics,” Nat. Commun. 6, 6334 (2015).
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M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
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W. Liu, D. Yang, G. Shen, H. Tian, and Y. Ji, “Design of ultra compact all-optical XOR, XNOR, NANAD and OR gates using photonic crystal multi-mode interference waveguides,” Opt. Laser Technol. 50, 55–64 (2013).
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K. S. Novoselov, S. V. Morozov, T. M. G. Mohinddin, L. A. Ponomarenko, D. C. Elias, R. Yang, I. I. Barbolina, P. Blake, T. J. Booth, D. Jiang, J. Giesbers, E. W. Hill, and A. K. Geim, “Electronic properties of graphene,” Phys. Status Solidi B 244, 4106–4111 (2007).
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G. D. Bouzianas, N. V. Kantartzis, T. V. Yioultsis, and T. D. Tsiboukis, “Consistent study of graphene structures through the direct incorporation of surface conductivity,” IEEE Trans. Magn. 50, 161–164 (2014).
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K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. Katsnelson, I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless dirac fermions in graphene,” Nature 438, 197–200 (2005).
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K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. Katsnelson, I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref] [PubMed]

Morozov, S. V.

K. S. Novoselov, S. V. Morozov, T. M. G. Mohinddin, L. A. Ponomarenko, D. C. Elias, R. Yang, I. I. Barbolina, P. Blake, T. J. Booth, D. Jiang, J. Giesbers, E. W. Hill, and A. K. Geim, “Electronic properties of graphene,” Phys. Status Solidi B 244, 4106–4111 (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] [PubMed]

Mortensen, N. A.

Mullen, K.

J. Cai, P. Ruffieux, R. Jaafar, M. Bieri, T. Braun, S. Blankenburg, M. Muoth, A. P. Seitsonen, M. Saleh, X. Feng, K. Mullen, and R. Fasel, “Atomically precise bottom-up fabrication of graphene nanoribbons,” Nature 466, 470–473 (2010).
[Crossref] [PubMed]

Muoth, M.

J. Cai, P. Ruffieux, R. Jaafar, M. Bieri, T. Braun, S. Blankenburg, M. Muoth, A. P. Seitsonen, M. Saleh, X. Feng, K. Mullen, and R. Fasel, “Atomically precise bottom-up fabrication of graphene nanoribbons,” Nature 466, 470–473 (2010).
[Crossref] [PubMed]

Nayyeri, V.

V. Nayyeri, M. Soleimani, and O. M. Ramahi, “Modeling graphene in the finite-difference time-domain method using a surface boundary condition,” IEEE Trans. Antennas Propag. 61, 4176–4182 (2013).
[Crossref]

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A. C. Neto, F. Guinea, N. M. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109 (2009).
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Nikitin, A. Y.

A. Y. Nikitin, F. Guinea, F. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84, 161407 (2011).
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Z. Fang, Z. Liu, Y. Wang, P. M. Ajayan, P. Nordlander, and N. J. Halas, “Graphene-antenna sandwich photodetector,” Nano Lett. 12, 3808–3813 (2012).
[Crossref] [PubMed]

Novoselov, K.

K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. Katsnelson, I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref] [PubMed]

Novoselov, K. S.

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

K. S. Novoselov, S. V. Morozov, T. M. G. Mohinddin, L. A. Ponomarenko, D. C. Elias, R. Yang, I. I. Barbolina, P. Blake, T. J. Booth, D. Jiang, J. Giesbers, E. W. Hill, and A. K. Geim, “Electronic properties of graphene,” Phys. Status Solidi B 244, 4106–4111 (2007).
[Crossref]

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature 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] [PubMed]

Ohana, D.

Okyay, A.

Oner, B.

Ono, M.

S. Masubuchi, M. Ono, K. Yoshida, K. Hirakawa, and T. Machida, “Fabrication of graphene nanoribbon by local anodic oxidation lithography using atomic force microscope,” Appl. Phys. Lett. 94, 082107 (2009).
[Crossref]

Ooi, K. J.

Otsuji, T.

D. Svintsov, V. Vyurkov, V. Ryzhii, and T. Otsuji, “Voltage-controlled surface plasmon-polaritons in double graphene layer structures,” J. Appl. Phys. 113, 053701 (2013).
[Crossref]

Park, H.

Peres, N. M.

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

Perruisseau-Carrier, J.

J. Gomez-Diaz, C. Moldovan, S. Capdevila, J. Romeu, L. Bernard, A. Magrez, A. Ionescu, and J. Perruisseau-Carrier, “Self-biased reconfigurable graphene stacks for terahertz plasmonics,” Nat. Commun. 6, 6334 (2015).
[Crossref] [PubMed]

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Graphene-based plasmonic tunable low-pass filters in the terahertz band,” IEEE Trans. Nanotechnol. 13, 1145–1153 (2014).
[Crossref]

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Álvarez-Melcón, “Spatially dispersive graphene single and parallel plate waveguides: Analysis and circuit model,” IEEE Trans. Microwave Theory Tech. 61, 4333–4344 (2013).
[Crossref]

Ponomarenko, L. A.

K. S. Novoselov, S. V. Morozov, T. M. G. Mohinddin, L. A. Ponomarenko, D. C. Elias, R. Yang, I. I. Barbolina, P. Blake, T. J. Booth, D. Jiang, J. Giesbers, E. W. Hill, and A. K. Geim, “Electronic properties of graphene,” Phys. Status Solidi B 244, 4106–4111 (2007).
[Crossref]

Pontius, W. V.

Y. Zhang, J. P. Small, W. V. Pontius, and P. Kim, “Fabrication and electric field dependent transport measurements of mesoscopic graphite devices,” arXiv preprint cond-mat/0410314 (2004).

Qin, K.

K. Qin, B. Xiao, and R. Sun, “Mode analysis and research on graphene nanoribbons parallel-plate waveguide,” IET Micro & Nano Letters 10, 558–560 (2015).
[Crossref]

Ramahi, O. M.

V. Nayyeri, M. Soleimani, and O. M. Ramahi, “Modeling graphene in the finite-difference time-domain method using a surface boundary condition,” IEEE Trans. Antennas Propag. 61, 4176–4182 (2013).
[Crossref]

Rezaei, M.

A. Khavasi, M. Rezaei, A. P. Fard, and K. Mehrany, “A heuristic approach to the realization of the wide-band optical diode effect in photonic crystal waveguides,” J. Opt. 15, 075501 (2013).
[Crossref]

Romeu, J.

J. Gomez-Diaz, C. Moldovan, S. Capdevila, J. Romeu, L. Bernard, A. Magrez, A. Ionescu, and J. Perruisseau-Carrier, “Self-biased reconfigurable graphene stacks for terahertz plasmonics,” Nat. Commun. 6, 6334 (2015).
[Crossref] [PubMed]

Ruffieux, P.

J. Cai, P. Ruffieux, R. Jaafar, M. Bieri, T. Braun, S. Blankenburg, M. Muoth, A. P. Seitsonen, M. Saleh, X. Feng, K. Mullen, and R. Fasel, “Atomically precise bottom-up fabrication of graphene nanoribbons,” Nature 466, 470–473 (2010).
[Crossref] [PubMed]

Ryzhii, V.

D. Svintsov, V. Vyurkov, V. Ryzhii, and T. Otsuji, “Voltage-controlled surface plasmon-polaritons in double graphene layer structures,” J. Appl. Phys. 113, 053701 (2013).
[Crossref]

Saleh, M.

J. Cai, P. Ruffieux, R. Jaafar, M. Bieri, T. Braun, S. Blankenburg, M. Muoth, A. P. Seitsonen, M. Saleh, X. Feng, K. Mullen, and R. Fasel, “Atomically precise bottom-up fabrication of graphene nanoribbons,” Nature 466, 470–473 (2010).
[Crossref] [PubMed]

Scherer, A.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
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Seitsonen, A. P.

J. Cai, P. Ruffieux, R. Jaafar, M. Bieri, T. Braun, S. Blankenburg, M. Muoth, A. P. Seitsonen, M. Saleh, X. Feng, K. Mullen, and R. Fasel, “Atomically precise bottom-up fabrication of graphene nanoribbons,” Nature 466, 470–473 (2010).
[Crossref] [PubMed]

Shen, G.

W. Liu, D. Yang, G. Shen, H. Tian, and Y. Ji, “Design of ultra compact all-optical XOR, XNOR, NANAD and OR gates using photonic crystal multi-mode interference waveguides,” Opt. Laser Technol. 50, 55–64 (2013).
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Sigmund, O.

Small, J. P.

Y. Zhang, J. P. Small, W. V. Pontius, and P. Kim, “Fabrication and electric field dependent transport measurements of mesoscopic graphite devices,” arXiv preprint cond-mat/0410314 (2004).

Soleimani, M.

V. Nayyeri, M. Soleimani, and O. M. Ramahi, “Modeling graphene in the finite-difference time-domain method using a surface boundary condition,” IEEE Trans. Antennas Propag. 61, 4176–4182 (2013).
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M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80, 245435 (2009).
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D. Correas-Serrano, J. S. Gomez-Diaz, D. L. Sounas, Y. Hadad, A. Alvarez-Melcon, and A. Alù, “Nonreciprocal graphene devices and antennas based on spatiotemporal modulation,” IEEE Antennas Wireless Propag. Lett. 15, 1529–1533 (2016).
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Sun, B.

H. Li, L. Wang, Z. Huang, B. Sun, X. Zhai, and X. Li, “Mid-infrared, plasmonic switches and directional couplers induced by graphene sheets coupling system,” Europhys. Lett. 104, 37001 (2013).
[Crossref]

Sun, R.

K. Qin, B. Xiao, and R. Sun, “Mode analysis and research on graphene nanoribbons parallel-plate waveguide,” IET Micro & Nano Letters 10, 558–560 (2015).
[Crossref]

Svintsov, D.

D. Svintsov, V. Vyurkov, V. Ryzhii, and T. Otsuji, “Voltage-controlled surface plasmon-polaritons in double graphene layer structures,” J. Appl. Phys. 113, 053701 (2013).
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Tao, J.

Teng, J.

B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100, 131111 (2012).
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S. Thongrattanasiri and F. J. G. de Abajo, “Optical field enhancement by strong plasmon interaction in graphene nanostructures,” Phys. Rev. Lett. 110, 187401 (2013).
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J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. Koppens, and F. J. GarcÌa de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6, 431–440 (2011).
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Tian, H.

W. Liu, D. Yang, G. Shen, H. Tian, and Y. Ji, “Design of ultra compact all-optical XOR, XNOR, NANAD and OR gates using photonic crystal multi-mode interference waveguides,” Opt. Laser Technol. 50, 55–64 (2013).
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Tian, J.-G.

Tsiboukis, T. D.

G. D. Bouzianas, N. V. Kantartzis, T. V. Yioultsis, and T. D. Tsiboukis, “Consistent study of graphene structures through the direct incorporation of surface conductivity,” IEEE Trans. Magn. 50, 161–164 (2014).
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Turhan-Sayan, G.

Üstün, K.

Vyurkov, V.

D. Svintsov, V. Vyurkov, V. Ryzhii, and T. Otsuji, “Voltage-controlled surface plasmon-polaritons in double graphene layer structures,” J. Appl. Phys. 113, 053701 (2013).
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B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100, 131111 (2012).
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Wang, J.

D. Dai, J. Wang, and S. He, “Silicon multimode photonic integrated devices for on-chip mode-division-multiplexed optical interconnects (invited review),” Prog. Electromagn. Res. 143, 773–819 (2013).
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H. Li, L. Wang, Z. Huang, B. Sun, X. Zhai, and X. Li, “Mid-infrared, plasmonic switches and directional couplers induced by graphene sheets coupling system,” Europhys. Lett. 104, 37001 (2013).
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Wang, Q. J.

Wang, Y.

S. Feng and Y. Wang, “Unidirectional light propagation characters of the triangular-lattice hybrid-waveguide photonic crystals,” Opt. Mater. 35, 1455–1460 (2013).
[Crossref]

Z. Fang, Z. Liu, Y. Wang, P. M. Ajayan, P. Nordlander, and N. J. Halas, “Graphene-antenna sandwich photodetector,” Nano Lett. 12, 3808–3813 (2012).
[Crossref] [PubMed]

Wu, L.

Xiao, B.

K. Qin, B. Xiao, and R. Sun, “Mode analysis and research on graphene nanoribbons parallel-plate waveguide,” IET Micro & Nano Letters 10, 558–560 (2015).
[Crossref]

Xiao, S.

Xu, H. J.

Xu, Y.-L.

L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M.-H. Lu, Y.-F. Chen, Y. Fainman, and A. Scherer, “Nonreciprocal light propagation in a silicon photonic circuit,” Science 333, 729–733 (2011).
[Crossref] [PubMed]

Yan, W.

Yang, D.

W. Liu, D. Yang, G. Shen, H. Tian, and Y. Ji, “Design of ultra compact all-optical XOR, XNOR, NANAD and OR gates using photonic crystal multi-mode interference waveguides,” Opt. Laser Technol. 50, 55–64 (2013).
[Crossref]

Yang, R.

K. S. Novoselov, S. V. Morozov, T. M. G. Mohinddin, L. A. Ponomarenko, D. C. Elias, R. Yang, I. I. Barbolina, P. Blake, T. J. Booth, D. Jiang, J. Giesbers, E. W. Hill, and A. K. Geim, “Electronic properties of graphene,” Phys. Status Solidi B 244, 4106–4111 (2007).
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Ye, H.

Yin, X.

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12, 1482–1485 (2012).
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Yioultsis, T. V.

G. D. Bouzianas, N. V. Kantartzis, T. V. Yioultsis, and T. D. Tsiboukis, “Consistent study of graphene structures through the direct incorporation of surface conductivity,” IEEE Trans. Magn. 50, 161–164 (2014).
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Yoshida, K.

S. Masubuchi, M. Ono, K. Yoshida, K. Hirakawa, and T. Machida, “Fabrication of graphene nanoribbon by local anodic oxidation lithography using atomic force microscope,” Appl. Phys. Lett. 94, 082107 (2009).
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Yu, X.

Yu, Z.

Yuan, X.

B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100, 131111 (2012).
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Yvind, K.

Zhai, X.

H. Li, L. Wang, Z. Huang, B. Sun, X. Zhai, and X. Li, “Mid-infrared, plasmonic switches and directional couplers induced by graphene sheets coupling system,” Europhys. Lett. 104, 37001 (2013).
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Zhang, J.

Zhang, X.

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12, 1482–1485 (2012).
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B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100, 131111 (2012).
[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).
[Crossref] [PubMed]

Y. Zhang, J. P. Small, W. V. Pontius, and P. Kim, “Fabrication and electric field dependent transport measurements of mesoscopic graphite devices,” arXiv preprint cond-mat/0410314 (2004).

Zhu, W.

Zhu, X.

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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J. Christensen, A. Manjavacas, S. Thongrattanasiri, F. H. Koppens, and F. J. GarcÌa de Abajo, “Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons,” ACS Nano 6, 431–440 (2011).
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Appl. Phys. Lett. (4)

I.-T. Lin and J.-M. Liu, “Optimization of double-layer graphene plasmonic waveguides,” Appl. Phys. Lett. 105, 061116 (2014).
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B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100, 131111 (2012).
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H. Iizuka and S. Fan, “Deep subwavelength plasmonic waveguide switch in double graphene layer structure,” Appl. Phys. Lett. 103, 233107 (2013).
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S. Masubuchi, M. Ono, K. Yoshida, K. Hirakawa, and T. Machida, “Fabrication of graphene nanoribbon by local anodic oxidation lithography using atomic force microscope,” Appl. Phys. Lett. 94, 082107 (2009).
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Europhys. Lett. (1)

H. Li, L. Wang, Z. Huang, B. Sun, X. Zhai, and X. Li, “Mid-infrared, plasmonic switches and directional couplers induced by graphene sheets coupling system,” Europhys. Lett. 104, 37001 (2013).
[Crossref]

IEEE Antennas Wireless Propag. Lett. (1)

D. Correas-Serrano, J. S. Gomez-Diaz, D. L. Sounas, Y. Hadad, A. Alvarez-Melcon, and A. Alù, “Nonreciprocal graphene devices and antennas based on spatiotemporal modulation,” IEEE Antennas Wireless Propag. Lett. 15, 1529–1533 (2016).
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IEEE Trans. Antennas Propag. (3)

P.-Y. Chen, C. Argyropoulos, and A. Alu, “Terahertz antenna phase shifters using integrally-gated graphene transmission-lines,” IEEE Trans. Antennas Propag. 61, 1528–1537 (2013).
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G. W. Hanson, “Dyadic green’s functions for an anisotropic, non-local model of biased graphene,” IEEE Trans. Antennas Propag. 56, 747–757 (2008).
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V. Nayyeri, M. Soleimani, and O. M. Ramahi, “Modeling graphene in the finite-difference time-domain method using a surface boundary condition,” IEEE Trans. Antennas Propag. 61, 4176–4182 (2013).
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IEEE Trans. Magn. (1)

G. D. Bouzianas, N. V. Kantartzis, T. V. Yioultsis, and T. D. Tsiboukis, “Consistent study of graphene structures through the direct incorporation of surface conductivity,” IEEE Trans. Magn. 50, 161–164 (2014).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Álvarez-Melcón, “Spatially dispersive graphene single and parallel plate waveguides: Analysis and circuit model,” IEEE Trans. Microwave Theory Tech. 61, 4333–4344 (2013).
[Crossref]

IEEE Trans. Nanotechnol. (1)

D. Correas-Serrano, J. S. Gomez-Diaz, J. Perruisseau-Carrier, and A. Alvarez-Melcon, “Graphene-based plasmonic tunable low-pass filters in the terahertz band,” IEEE Trans. Nanotechnol. 13, 1145–1153 (2014).
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IET Micro & Nano Letters (1)

K. Qin, B. Xiao, and R. Sun, “Mode analysis and research on graphene nanoribbons parallel-plate waveguide,” IET Micro & Nano Letters 10, 558–560 (2015).
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J. Appl. Phys. (2)

D. Svintsov, V. Vyurkov, V. Ryzhii, and T. Otsuji, “Voltage-controlled surface plasmon-polaritons in double graphene layer structures,” J. Appl. Phys. 113, 053701 (2013).
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G. W. Hanson, “Quasi-transverse electromagnetic modes supported by a graphene parallel-plate waveguide,” J. Appl. Phys. 104, 084314 (2008).
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J. Opt. (1)

A. Khavasi, M. Rezaei, A. P. Fard, and K. Mehrany, “A heuristic approach to the realization of the wide-band optical diode effect in photonic crystal waveguides,” J. Opt. 15, 075501 (2013).
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J. Phys.: Conf. Ser. (1)

L. A. Falkovsky, “Optical properties of graphene,” J. Phys.: Conf. Ser. 129, 012004 (2008).

Nano Lett. (2)

Z. Fang, Z. Liu, Y. Wang, P. M. Ajayan, P. Nordlander, and N. J. Halas, “Graphene-antenna sandwich photodetector,” Nano Lett. 12, 3808–3813 (2012).
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M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12, 1482–1485 (2012).
[Crossref] [PubMed]

Nat. Commun. (1)

J. Gomez-Diaz, C. Moldovan, S. Capdevila, J. Romeu, L. Bernard, A. Magrez, A. Ionescu, and J. Perruisseau-Carrier, “Self-biased reconfigurable graphene stacks for terahertz plasmonics,” Nat. Commun. 6, 6334 (2015).
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Nature (2)

J. Cai, P. Ruffieux, R. Jaafar, M. Bieri, T. Braun, S. Blankenburg, M. Muoth, A. P. Seitsonen, M. Saleh, X. Feng, K. Mullen, and R. Fasel, “Atomically precise bottom-up fabrication of graphene nanoribbons,” Nature 466, 470–473 (2010).
[Crossref] [PubMed]

K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. Katsnelson, I. Grigorieva, S. Dubonos, and A. Firsov, “Two-dimensional gas of massless dirac fermions in graphene,” Nature 438, 197–200 (2005).
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Nature Mater. (1)

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nature Mater. 6, 183–191 (2007).
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Opt. Express (12)

W. B. Lu, W. Zhu, H. J. Xu, Z. H. Ni, Z. G. Dong, and T. J. Cui, “Flexible transformation plasmonics using graphene,” Opt. Express 21, 10475–10482 (2013).
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X. Zhu, W. Yan, N. A. Mortensen, and S. Xiao, “Bends and splitters in graphene nanoribbon waveguides,” Opt. Express 21, 3486–3491 (2013).
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L. Wu, H. Chu, W. Koh, and E. Li, “Highly sensitive graphene biosensors based on surface plasmon resonance,” Opt. Express 18, 14395–14400 (2010).
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D. A. B. Miller, “All linear optical devices are mode converters,” Opt. Express 20, 23985–23993 (2012).
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V. Liu, D. A. B. Miller, and S. Fan, “Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect,” Opt. Express 20, 28388–28397 (2012).
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L. H. Frandsen, Y. Elesin, L. F. Frellsen, M. Mitrovic, Y. Ding, O. Sigmund, and K. Yvind, “Topology optimized mode conversion in a photonic crystal waveguide fabricated in silicon-on-insulator material,” Opt. Express 22, 8525–8532 (2014).
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H. Ye, D. Wang, Z. Yu, J. Zhang, and Z. Chen, “Ultra-compact broadband mode converter and optical diode based on linear rod-type photonic crystal waveguide,” Opt. Express 23, 9673–9680 (2015).
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Y.-T. Hung, C.-B. Huang, and J.-S. Huang, “Plasmonic mode converter for controlling optical impedance and nanoscale light-matter interaction,” Opt. Express 20, 20342–20355 (2012).
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X.-T. Kong, Z.-B. Li, and J.-G. Tian, “Mode converter in metal-insulator-metal plasmonic waveguide designed by transformation optics,” Opt. Express 21, 9437–9446 (2013).
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J. Kim, S.-Y. Lee, H. Park, K. Lee, and B. Lee, “Reflectionless compact plasmonic waveguide mode converter by using a mode-selective cavity,” Opt. Express 23, 9004–9013 (2015).
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D. Ohana and U. Levy, “Mode conversion based on dielectric metamaterial in silicon,” Opt. Express 22, 27617–27631 (2014).
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B. Oner, K. Üstün, H. Kurt, A. Okyay, and G. Turhan-Sayan, “Large bandwidth mode order converter by differential waveguides,” Opt. Express 23, 3186–3195 (2015).
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Opt. Laser Technol. (1)

W. Liu, D. Yang, G. Shen, H. Tian, and Y. Ji, “Design of ultra compact all-optical XOR, XNOR, NANAD and OR gates using photonic crystal multi-mode interference waveguides,” Opt. Laser Technol. 50, 55–64 (2013).
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Opt. Lett. (2)

Opt. Mater. (1)

S. Feng and Y. Wang, “Unidirectional light propagation characters of the triangular-lattice hybrid-waveguide photonic crystals,” Opt. Mater. 35, 1455–1460 (2013).
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Phys. Rev. B (2)

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Phys. Rev. Lett. (2)

S. Thongrattanasiri and F. J. G. de Abajo, “Optical field enhancement by strong plasmon interaction in graphene nanostructures,” Phys. Rev. Lett. 110, 187401 (2013).
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Phys. Status Solidi B (1)

K. S. Novoselov, S. V. Morozov, T. M. G. Mohinddin, L. A. Ponomarenko, D. C. Elias, R. Yang, I. I. Barbolina, P. Blake, T. J. Booth, D. Jiang, J. Giesbers, E. W. Hill, and A. K. Geim, “Electronic properties of graphene,” Phys. Status Solidi B 244, 4106–4111 (2007).
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D. Dai, J. Wang, and S. He, “Silicon multimode photonic integrated devices for on-chip mode-division-multiplexed optical interconnects (invited review),” Prog. Electromagn. Res. 143, 773–819 (2013).
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Figures (9)

Fig. 1
Fig. 1 (a) The schematic of a typical mode converter. We use the letters A, B, C, and D to denote different modes. The blue and red arrows correspond to input and output in each mode, respectively. (b) The scattering matrix S of the converter relates the amplitudes of the outgoing modes (denoted in red) to the amplitudes of the ingoing modes (denoted in blue).
Fig. 2
Fig. 2 Schematic of a spatial mode converter consisting of two parallel layers of graphene with chemical potential μcg, and a strip of length L with chemical potential μcs, shown in red.
Fig. 3
Fig. 3 (a) Profile of the chemical potential of the upper graphene layer in the spatial mode converter shown in Fig. 2. (b) Transmission spectra from mode A to modes C and D, and reflection spectra to modes A and B calculated with FDTD for the mode converter shown in Fig. 2 with chemical potential profile as in Fig. 3(a), and μcg = 0.3 eV, μcs = 0.205 eV, h = 150 nm, L = 400 nm. The graphene layers are assumed to be lossless.
Fig. 4
Fig. 4 (a) Chemical potential profile of the graphene strip in the spatial mode converter shown in Fig. 2. The profile corresponds to a piecewise approximation with a step size of 50 nm of a triangular envelope with minimum potential of μcm. (b) Transmission spectra from mode A to modes C and D, and reflection spectra to modes A and B calculated with FDTD for the mode converter shown in Fig. 2 with chemical potential profile as in Fig. 4(a), and μcm = 0.127 eV. All other parameters are as in Fig. 3(b). The magnetic field profile at λ = 10.1 μm shown in the inset demonstrates the complete conversion of the even mode A incident from the left into the odd mode D propagating to the right, and vice versa.
Fig. 5
Fig. 5 (a) Conversion efficiency spectra calculated with FDTD for the mode converter shown in Fig. 2 with chemical potential profile as in Fig. 4(a) for different values of the minimum potential μcm of the triangular envelope. Results are shown for μcm = 0.145 eV, μcm = 0.115 eV, and μcm = 0.08 eV. All other parameters are as in Fig. 3(b). (b) The wavelength which corresponds to maximum conversion efficiency λopt as a function of μcm calculated using Eq. (9) (blue line), and FDTD (red line). All other parameters are as in Fig. 5(a).
Fig. 6
Fig. 6 (a) Schematic of the double strip spatial mode converter. It consists of two parallel layers of graphene with strips with modified chemical potential on both the upper and lower graphene plates, shown in red and blue, respectively. (b) Conversion efficiency spectra calculated with FDTD for the single strip (Fig. 2) and double strip [Fig. 6(a)] mode converters with chemical potential profile as in Fig. 4(a). The minimum potential for the single strip converter is set to be μcm = 0.08 eV. For the double strip case the minimum potentials of the upper and lower strips are set to be μcmu = 0.05 eV and μcmd = 0.19 eV, respectively. The strip length is L = 400 nm and the plate separation is h = 150 nm in both cases. All other parameters are as in Fig. 3(b). (c) Same as in (b), except that L = 500 nm, μcg = 0.5 eV, the minimum potential for the single strip converter is set to be μcm = 0.16 eV, and for the double strip case the minimum potentials of the upper and lower strips are set to be μcmu = 0.1 eV and μcmd = 0.3 eV, respectively. (d) Same as in (b), except that L = 500 nm, h = 100 nm, the minimum potential for the single strip converter is set to be μcm = 0.15 eV, and for the double strip case the minimum potentials of the upper and lower strips are set to be μcmu = 0.05 eV and μcmd = 0.098 eV, respectively.
Fig. 7
Fig. 7 (a) Conversion efficiency spectra calculated with FDTD for the mode converter shown in Fig. 2 with chemical potential profile as in Fig. 4(a) for different values of the minimum potential μcm of the triangular envelope, when the effect of material loss in graphene is included. All other parameters are as in Fig. 5(a). (b) Magnetic field profiles for the converters with μcm = 0.145 eV (top figure), μcm = 0.115 eV (middle figure), and μcm = 0.08 eV (bottom figure). In each case the profile is shown at the wavelength which corresponds to maximum conversion efficiency λopt.
Fig. 8
Fig. 8 Schematic of an optical diode consisting of two GPP waveguides, which includes a mode converter on the left GPP waveguide (shown in red color), and a coupler, which consists of a single layer of graphene placed in the middle between the two plates of the two GPP waveguides.
Fig. 9
Fig. 9 (a) Transmission spectra from mode A to mode D, and reflection spectra from mode B to mode B and from mode C to mode C calculated with FDTD for the optical diode shown in Fig. 8 with L = 400 nm, Lf = 200 nm, Lc = 130 nm, Ls = 400 nm, and h = 150 nm. The mode converter of the diode has a chemical potential profile as in Fig. 4(a) with μcm = 0.135 eV. The graphene layers are assumed to be lossless. (b) Same as in (a), except that the effect of material loss in graphene is included. (c) Magnetic field profiles for the optical diode of Fig. 8 for even and odd modes entering from the left and right directions. The red and black arrows indicate the direction of incidence of the even and odd modes, respectively. The profiles are shown at the wavelength which corresponds to maximum conversion efficiency λopt = 9.85 μm. The graphene layers are assumed to be lossless. All other parameters are as in Fig. 9(a). (d) Same as in (c), except that the effect of material loss in graphene is included. When the even mode enters the device from the right, it is reflected, whereas, when it enters the device from the left, it is transmitted. In contrast, the odd mode is transmitted, when it enters the device from the right, and reflected, when it enters the device from the left. This functionality is preserved in the presence of loss.

Equations (20)

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σ ( ω , μ c , τ , T ) = σ intra + σ inter ,
σ intra = τ e 2 k B T π 2 ( j ω τ + 1 ) { μ c k B T + 2 ln [ exp ( μ c k B T ) + 1 ] } ,
σ inter = j e 2 4 π ln [ 2 | μ c | ( ω j τ 1 ) 2 | μ c | + ( ω j τ 1 ) ] .
n s = V g 0 r e d ,
n s = 2 π 2 v f 2 0 [ f d ( ) f d ( + 2 μ c ) ] d .
r 1 k 1 + r 2 k 2 = j σ ω 0 ,
k i = β 2 k 0 2 r i ,
β s L β g L = ( 2 m 1 ) π ,
i = 1 N β i d i β g L = ( 2 m 1 ) π ,
i = 1 N ( β u i β d i ) d i = ( 2 m 1 ) π ,
r 1 = r 2 = 1 ,
k 1 = k 2 = k = β 2 k 0 2 .
β = k 0 1 ( 2 η 0 σ ) 2 ,
σ = σ intra = e 2 k B T π 2 j ω { μ c k B T + 2 ln [ exp ( μ c k B T ) + 1 ] } .
σ j e 2 μ c π 2 ω .
β = k 0 1 + ( 2 π 2 ω η 0 e 2 μ c ) 2 .
β = k 0 ( 2 π 2 ω η 0 e 2 μ c ) = 2 π h 2 μ 0 e 2 1 μ c λ 2 .
β u i = 2 π h 2 μ 0 e 2 1 μ c u i λ 2 ,
β d i = 2 π h 2 μ 0 e 2 1 μ c d i λ 2 ,
λ opt = 2 h 2 ( 2 m 1 ) μ 0 e 2 [ i = 1 N d i ( 1 μ c u i 1 μ c d i ) ] .

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