Abstract

In this paper, a graphene-based hybrid plasmonic waveguide is proposed for highly efficient broadband surface plasmon polariton (SPP) propagation and modulation at mid-infrared (mid-IR) spectrum. The hybrid plasmonic waveguide is composed of a monolayer graphene sheet in the center, a polysilicon gating layer, and two inner dielectric buffer layers and two outer parabolic-ridged silicon substrates symmetrically placed on both sides of the graphene. Owing to the unique parabolic-ridged waveguide structure, the light-graphene interaction and subwavelength SPPs confinement of the fundamental SPP mode for the hybrid waveguide can be significantly increased. Under the graphene chemical potential of 1.0 eV, the proposed waveguide can achieve outstanding SPP propagation performance with long propagation length of 12.1-16.7 μm and small normalized mode area of ~10−4 in the frequency range of 10-20 THz, exhibiting more than one order smaller in the normalized mode area while remaining the propagation length almost the same level with respect to the hybrid plasmonic waveguide without parabolic ridges. By tuning the graphene chemical potential from 0.1 to 1.0 eV, we demonstrate the waveguide has a modulation depth greater than 51% for the frequency ranging from 10 to 20 THz and reaches a maximum of nearly 100% at the frequency higher than 18 THz. Benefitting from the excellent broadband mid-IR propagation and modulation performance, the graphene-based hybrid plasmonic waveguide may open up a new way for various mid-IR waveguides, modulators, interconnects and optoelectronic devices.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (1)

X. Hu and J. Wang, “Design of graphene-based polarization-insensitive optical modulator,” Nanophotonics 7(3), 651–658 (2018).
[Crossref]

2017 (5)

2016 (6)

2015 (10)

W. Xu, Z. H. Zhu, K. Liu, J. F. Zhang, X. D. Yuan, Q. S. Lu, and S. Q. Qin, “Toward integrated electrically controllable directional coupling based on dielectric loaded graphene plasmonic waveguide,” Opt. Lett. 40(7), 1603–1606 (2015).
[Crossref] [PubMed]

T. S. Saini, A. Kumar, and R. K. Sinha, “Broadband Mid-Infrared Supercontinuum Spectra Spanning 2-15 μm Using As2Se3 Chalcogenide Glass Triangular-Core Graded-Index Photonic Crystal Fiber,” J. Lightwave Technol. 33(18), 3914–3920 (2015).
[Crossref]

M. Faraji, M. K. Moravvej-Farshi, and L. Yousefi, “Tunable THz perfect absorber using graphene-based metamaterials,” Opt. Commun. 355, 352–355 (2015).
[Crossref]

Y. Zhang, S. Qiao, S. Liang, Z. Wu, Z. Yang, Z. Feng, H. Sun, Y. Zhou, L. Sun, Z. Chen, X. Zou, B. Zhang, J. Hu, S. Li, Q. Chen, L. Li, G. Xu, Y. Zhao, and S. Liu, “Gbps terahertz external modulator based on a composite metamaterial with a double-channel heterostructure,” Nano Lett. 15(5), 3501–3506 (2015).
[Crossref] [PubMed]

G. Liang, X. Hu, X. Yu, Y. Shen, L. H. Li, A. G. Davies, E. H. Linfield, H. K. Liang, Y. Zhang, S. F. Yu, and Q. J. Wang, “Integrated terahertz graphene modulator with 100% modulation depth,” ACS Photonics 2(11), 1559–1566 (2015).
[Crossref]

D. Rodrigo, O. Limaj, D. Janner, D. Etezadi, F. J. García de Abajo, V. Pruneri, and H. Altug, “Mid-infrared plasmonic biosensing with graphene,” Science 349(6244), 165–168 (2015).
[Crossref] [PubMed]

N. Ranjkesh, M. Basha, A. Taeb, A. Zandieh, S. Gigoyan, and S. Safavi-Naeini, “Silicon-on-glass dielectric waveguide—Part I: For millimeter-wave integrated circuits,” IEEE Trans. THz Sci. Technol. 5(2), 268–279 (2015).

R. Yu, V. Pruneri, and F. J. García de Abajo, “Resonant visible light modulation with graphene,” ACS Photonics 2(4), 550–558 (2015).
[Crossref]

Q. Q. Zhuo, Q. Wang, Y. P. Zhang, D. Zhang, Q. L. Li, C. H. Gao, Y. Q. Sun, L. Ding, Q. J. Sun, S. D. Wang, J. Zhong, X. H. Sun, and S. T. Lee, “Transfer-free synthesis of doped and patterned graphene films,” ACS Nano 9(1), 594–601 (2015).
[Crossref] [PubMed]

C. L. Smith, N. Stenger, A. Kristensen, N. A. Mortensen, and S. I. Bozhevolnyi, “Gap and channeled plasmons in tapered grooves: a review,” Nanoscale 7(21), 9355–9386 (2015).
[Crossref] [PubMed]

2014 (5)

Z. Zhang and J. Wang, “Long-range hybrid wedge plasmonic waveguide,” Sci. Rep. 4(1), 6870 (2014).
[Crossref] [PubMed]

T. Low and P. Avouris, “Graphene plasmonics for terahertz to mid-infrared applications,” ACS Nano 8(2), 1086–1101 (2014).
[Crossref] [PubMed]

M. S. Kwon, “Discussion of the epsilon-near-zero effect of graphene in a horizontal slot waveguide,” IEEE Photonics J. 6(3), 1–9 (2014).
[Crossref]

Q. Zhang, X. Li, M. M. Hossain, Y. Xue, J. Zhang, J. Song, J. Liu, M. D. Turner, S. Fan, Q. Bao, and M. Gu, “Graphene surface plasmons at the near-infrared optical regime,” Sci. Rep. 4(1), 6559 (2014).
[Crossref] [PubMed]

Y. Gao, G. Ren, B. Zhu, J. Wang, and S. Jian, “Single-mode graphene-coated nanowire plasmonic waveguide,” Opt. Lett. 39(20), 5909–5912 (2014).
[Crossref] [PubMed]

2013 (5)

J. Gosciniak and D. T. Tan, “Theoretical investigation of graphene-based photonic modulators,” Sci. Rep. 3(1), 1897 (2013).
[Crossref] [PubMed]

F. A. Vallejo and L. M. Hayden, “Design of ultra-broadband terahertz polymer waveguide emitters for telecom wavelengths using coupled mode theory,” Opt. Express 21(5), 5842–5858 (2013).
[Crossref] [PubMed]

J. S. Gómez-Díaz, M. Esquius-Morote, and J. Perruisseau-Carrier, “Plane wave excitation-detection of non-resonant plasmons along finite-width graphene strips,” Opt. Express 21(21), 24856–24872 (2013).
[Crossref] [PubMed]

V. W. Brar, M. S. Jang, M. Sherrott, J. J. Lopez, and H. A. Atwater, “Highly confined tunable mid-infrared plasmonics in graphene nanoresonators,” Nano Lett. 13(6), 2541–2547 (2013).
[Crossref] [PubMed]

X. Shen, T. J. Cui, D. Martin-Cano, and F. J. Garcia-Vidal, “Conformal surface plasmons propagating on ultrathin and flexible films,” Proc. Natl. Acad. Sci. U.S.A. 110(1), 40–45 (2013).
[Crossref] [PubMed]

2012 (4)

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6(11), 749–758 (2012).
[Crossref]

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
[Crossref] [PubMed]

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. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
[Crossref] [PubMed]

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
[Crossref] [PubMed]

2011 (5)

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
[Crossref] [PubMed]

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332(6035), 1291–1294 (2011).
[Crossref] [PubMed]

F. H. Koppens, D. E. Chang, and F. J. García de Abajo, “Graphene plasmonics: a platform for strong light-matter interactions,” Nano Lett. 11(8), 3370–3377 (2011).
[Crossref] [PubMed]

M. Schnell, P. Alonso-Gonzalez, L. Arzubiaga, F. Casanova, L. E. Hueso, A. Chuvilin, and R. Hillenbrand, “Nanofocusing of mid-infrared energy with tapered transmission lines,” Nat. Photonics 5(5), 283–287 (2011).
[Crossref]

P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X. H. Zhou, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys. 109(4), 043505 (2011).
[Crossref]

2010 (5)

C. Yang, Q. Wu, J. Xu, K. A. Nelson, and C. A. Werley, “Experimental and theoretical analysis of THz-frequency, direction-dependent, phonon polariton modes in a subwavelength, anisotropic slab waveguide,” Opt. Express 18(25), 26351–26364 (2010).
[Crossref] [PubMed]

D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4(2), 83–91 (2010).
[Crossref]

R. Soref, “Mid-infrared photonics in silicon and germanium,” Nat. Photonics 4(8), 495–497 (2010).
[Crossref]

Y. Lee, S. Bae, H. Jang, S. Jang, S. E. Zhu, S. H. Sim, Y. I. Song, B. H. Hong, and J. H. Ahn, “Wafer-scale synthesis and transfer of graphene films,” Nano Lett. 10(2), 490–493 (2010).
[Crossref] [PubMed]

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(10), 722–726 (2010).
[Crossref] [PubMed]

2009 (2)

Y. Zhao, E. Berenschot, H. Jansen, N. Tas, J. Huskens, and M. Elwenspoek, “Sub-10 nm silicon ridge nanofabrication by advanced edge lithography for NIL applications,” Microelectron. Eng. 86(4–6), 832–835 (2009).
[Crossref]

A. K. Geim, “Graphene: status and prospects,” Science 324(5934), 1530–1534 (2009).
[Crossref] [PubMed]

2008 (4)

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[Crossref]

G. W. Hanson, “Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene,” J. Appl. Phys. 103(6), 064302 (2008).
[Crossref]

G. W. Hanson, “Dyadic Green’s functions for an anisotropic, non-local model of biased graphene,” IEEE Trans. Antenn. Propag. 56(3), 747–757 (2008).
[Crossref]

K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, and H. L. Stormer, “Ultrahigh electron mobility in suspended graphene,” Solid State Commun. 146(9–10), 351–355 (2008).
[Crossref]

2006 (2)

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
[Crossref] [PubMed]

R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9(7–8), 20–27 (2006).
[Crossref]

2004 (1)

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[Crossref] [PubMed]

Ahn, J. H.

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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. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
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Wang, F.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
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Wang, J.

X. Hu and J. Wang, “Design of graphene-based polarization-insensitive optical modulator,” Nanophotonics 7(3), 651–658 (2018).
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Y. Gao, G. Ren, B. Zhu, J. Wang, and S. Jian, “Single-mode graphene-coated nanowire plasmonic waveguide,” Opt. Lett. 39(20), 5909–5912 (2014).
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Z. Zhang and J. Wang, “Long-range hybrid wedge plasmonic waveguide,” Sci. Rep. 4(1), 6870 (2014).
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Wang, K.

Wang, L.

W. Luo, W. Cai, Y. Xiang, L. Wang, M. Ren, X. Zhang, and J. Xu, “Flexible modulation of plasmon-induced transparency in a strongly coupled graphene grating-sheet system,” Opt. Express 24(6), 5784–5793 (2016).
[Crossref] [PubMed]

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(10), 722–726 (2010).
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Wang, L. L.

Wang, Q.

Q. Q. Zhuo, Q. Wang, Y. P. Zhang, D. Zhang, Q. L. Li, C. H. Gao, Y. Q. Sun, L. Ding, Q. J. Sun, S. D. Wang, J. Zhong, X. H. Sun, and S. T. Lee, “Transfer-free synthesis of doped and patterned graphene films,” ACS Nano 9(1), 594–601 (2015).
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Wang, Q. J.

G. Liang, X. Hu, X. Yu, Y. Shen, L. H. Li, A. G. Davies, E. H. Linfield, H. K. Liang, Y. Zhang, S. F. Yu, and Q. J. Wang, “Integrated terahertz graphene modulator with 100% modulation depth,” ACS Photonics 2(11), 1559–1566 (2015).
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Wang, S. D.

Q. Q. Zhuo, Q. Wang, Y. P. Zhang, D. Zhang, Q. L. Li, C. H. Gao, Y. Q. Sun, L. Ding, Q. J. Sun, S. D. Wang, J. Zhong, X. H. Sun, and S. T. Lee, “Transfer-free synthesis of doped and patterned graphene films,” ACS Nano 9(1), 594–601 (2015).
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Xiang, Y.

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Y. Zhang, S. Qiao, S. Liang, Z. Wu, Z. Yang, Z. Feng, H. Sun, Y. Zhou, L. Sun, Z. Chen, X. Zou, B. Zhang, J. Hu, S. Li, Q. Chen, L. Li, G. Xu, Y. Zhao, and S. Liu, “Gbps terahertz external modulator based on a composite metamaterial with a double-channel heterostructure,” Nano Lett. 15(5), 3501–3506 (2015).
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Xu, Q.

W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
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Xu, W.

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Q. Zhang, X. Li, M. M. Hossain, Y. Xue, J. Zhang, J. Song, J. Liu, M. D. Turner, S. Fan, Q. Bao, and M. Gu, “Graphene surface plasmons at the near-infrared optical regime,” Sci. Rep. 4(1), 6559 (2014).
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M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
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M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
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R. Yu, V. Pruneri, and F. J. García de Abajo, “Resonant visible light modulation with graphene,” ACS Photonics 2(4), 550–558 (2015).
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G. Liang, X. Hu, X. Yu, Y. Shen, L. H. Li, A. G. Davies, E. H. Linfield, H. K. Liang, Y. Zhang, S. F. Yu, and Q. J. Wang, “Integrated terahertz graphene modulator with 100% modulation depth,” ACS Photonics 2(11), 1559–1566 (2015).
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G. Liang, X. Hu, X. Yu, Y. Shen, L. H. Li, A. G. Davies, E. H. Linfield, H. K. Liang, Y. Zhang, S. F. Yu, and Q. J. Wang, “Integrated terahertz graphene modulator with 100% modulation depth,” ACS Photonics 2(11), 1559–1566 (2015).
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Zentgraf, T.

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
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Y. Zhang, S. Qiao, S. Liang, Z. Wu, Z. Yang, Z. Feng, H. Sun, Y. Zhou, L. Sun, Z. Chen, X. Zou, B. Zhang, J. Hu, S. Li, Q. Chen, L. Li, G. Xu, Y. Zhao, and S. Liu, “Gbps terahertz external modulator based on a composite metamaterial with a double-channel heterostructure,” Nano Lett. 15(5), 3501–3506 (2015).
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Zhang, J. F.

Zhang, L.

J. Zhu, J. Cheng, L. Zhang, and Q. H. Liu, “Modeling of 2D graphene material for plasmonic hybrid waveguide with enhanced near-infrared modulation,” Mater. Lett. 186, 53–56 (2017).
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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. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
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Q. Zhang, X. Li, M. M. Hossain, Y. Xue, J. Zhang, J. Song, J. Liu, M. D. Turner, S. Fan, Q. Bao, and M. Gu, “Graphene surface plasmons at the near-infrared optical regime,” Sci. Rep. 4(1), 6559 (2014).
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W. Luo, W. Cai, Y. Xiang, L. Wang, M. Ren, X. Zhang, and J. Xu, “Flexible modulation of plasmon-induced transparency in a strongly coupled graphene grating-sheet system,” Opt. Express 24(6), 5784–5793 (2016).
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M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12(3), 1482–1485 (2012).
[Crossref] [PubMed]

M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
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R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
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Y. Zhang, S. Qiao, S. Liang, Z. Wu, Z. Yang, Z. Feng, H. Sun, Y. Zhou, L. Sun, Z. Chen, X. Zou, B. Zhang, J. Hu, S. Li, Q. Chen, L. Li, G. Xu, Y. Zhao, and S. Liu, “Gbps terahertz external modulator based on a composite metamaterial with a double-channel heterostructure,” Nano Lett. 15(5), 3501–3506 (2015).
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G. Liang, X. Hu, X. Yu, Y. Shen, L. H. Li, A. G. Davies, E. H. Linfield, H. K. Liang, Y. Zhang, S. F. Yu, and Q. J. Wang, “Integrated terahertz graphene modulator with 100% modulation depth,” ACS Photonics 2(11), 1559–1566 (2015).
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Z. Zhang and J. Wang, “Long-range hybrid wedge plasmonic waveguide,” Sci. Rep. 4(1), 6870 (2014).
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Y. Zhao, E. Berenschot, H. Jansen, N. Tas, J. Huskens, and M. Elwenspoek, “Sub-10 nm silicon ridge nanofabrication by advanced edge lithography for NIL applications,” Microelectron. Eng. 86(4–6), 832–835 (2009).
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Zhu, J.

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J. Zhu, J. Cheng, L. Zhang, and Q. H. Liu, “Modeling of 2D graphene material for plasmonic hybrid waveguide with enhanced near-infrared modulation,” Mater. Lett. 186, 53–56 (2017).
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Zhu, S. E.

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R. Zia, J. A. Schuller, A. Chandran, and M. L. Brongersma, “Plasmonics: the next chip-scale technology,” Mater. Today 9(7–8), 20–27 (2006).
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W. Gao, J. Shu, C. Qiu, and Q. Xu, “Excitation of plasmonic waves in graphene by guided-mode resonances,” ACS Nano 6(9), 7806–7813 (2012).
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R. Yu, V. Pruneri, and F. J. García de Abajo, “Resonant visible light modulation with graphene,” ACS Photonics 2(4), 550–558 (2015).
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P. A. D. Gonçalves, E. J. C. Dias, S. Xiao, M. I. Vasilevskiy, N. A. Mortensen, and N. M. R. Peres, “Graphene plasmons in triangular wedges and grooves,” ACS Photonics 3(11), 2176–2183 (2016).
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P. D. Cunningham, N. N. Valdes, F. A. Vallejo, L. M. Hayden, B. Polishak, X. H. Zhou, and R. J. Twieg, “Broadband terahertz characterization of the refractive index and absorption of some important polymeric and organic electro-optic materials,” J. Appl. Phys. 109(4), 043505 (2011).
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Y. Zhao, E. Berenschot, H. Jansen, N. Tas, J. Huskens, and M. Elwenspoek, “Sub-10 nm silicon ridge nanofabrication by advanced edge lithography for NIL applications,” Microelectron. Eng. 86(4–6), 832–835 (2009).
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X. Hu and J. Wang, “Design of graphene-based polarization-insensitive optical modulator,” Nanophotonics 7(3), 651–658 (2018).
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R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
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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. Castro Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487(7405), 82–85 (2012).
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M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. Wang, and X. Zhang, “A graphene-based broadband optical modulator,” Nature 474(7349), 64–67 (2011).
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Opt. Commun. (1)

M. Faraji, M. K. Moravvej-Farshi, and L. Yousefi, “Tunable THz perfect absorber using graphene-based metamaterials,” Opt. Commun. 355, 352–355 (2015).
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[Crossref]

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Figures (6)

Fig. 1
Fig. 1 A graphene-based hybrid plasmonic waveguide. (a) Three-dimensional schematic illustration of the waveguide, where a graphene sheet is embedded between two inner Topas buffer layers and two outer silicon substrate with parabolic ridges, as well as a polysilicon layer is inserted into the lower buffer layer below the graphene acting as a gating layer. (b) Cross-section of the waveguide, where the initial geometric parameters of the waveguide are set as s = 600 nm, h = 600 nm, g = 20 nm, d = 10 nm, T = 50 nm, and W = 600 nm.
Fig. 2
Fig. 2 Dependence of the graphene surface conductivity Re(σg), Im(σg), and loss tangent tan δ on the chemical potential μc and frequency f. (a) Reg), Im(σg) and tan δ as functions of μc with the fixed f = 10, 15, 20 THz. (b) Re(σg), Im(σg) and tan δ as functions of f with the fixed μc = 0.1, 0.6, and 1.0 eV.
Fig. 3
Fig. 3 Comparison of the fundamental SPP mode properties of the hybrid plasmonic waveguides with and without ridges. (a) and (c) are electric field distributions of the waveguide without ridges, and (b) and (d) are electric field distributions of the waveguide with parabolic ridges under the μc = 0.1 eV and 1.0 eV at 15 THz, respectively. (e) The propagation length Lspp as a function of frequency f. (f) The normalized mode area A as a function of frequency f.
Fig. 4
Fig. 4 Electric field distributions, effective indices Re(Neff), propagation length Lspp and figure of merit FOM of the proposed waveguide with μc = 1 eV. (a)-(d) reveal the electric field distributions of the higher mode 2-5 at 15 THz, respectively. (e)-(f), (g)-(h), and (i)-(j) are Re(Neff)~f, Lspp~f and FOM~f for T = 50 and 200 nm, respectively, where other geometric parameters are the same as that in Fig. 1.
Fig. 5
Fig. 5 Dependence of the mode properties on the geometric parameters of the parabolic ridges and the dielectric constants of the buffer materials with μc = 1.0 eV. (a) The dependence of Re(Neff) and Lspp on T with W = 600 nm at f = 15 THz. (b) The dependence of Re(Neff) and Lspp on W with T = 50 nm at f = 15 THz. (c) and (d) present Re(Neff) and Lspp as a function of f for the proposed waveguides with different buffer layer materials of Al2O3, SiO2 and Topas, respectively.
Fig. 6
Fig. 6 (a) shows the dependence of chemical potential µc on the gate voltage Vg. (b), (c) and (d) depict the dependence of Re(Neff), attenuation α and transmission t on μc with the frequency fixed at 10, 15 and 20 THz. (e) and (f) depict the dependence of attenuation α, transmission t and modulation depth η on frequency f with the μc fixed at 0.1, 0.6 and 1.0 eV, respectively.

Equations (8)

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σ g (ω, μ c ,Γ, T 0 )= σ intra (ω, μ c ,Γ, T 0 )+ σ inter (ω, μ c ,Γ, T 0 ),
σ intra (ω, μ c ,Γ, T 0 )= j e 2 π 2 ( ωj2Γ ) 0 ( f d ( ξ, μ c , T 0 ) ξ f d ( ξ, μ c , T 0 ) ξ )ξdξ,
σ inter (ω, μ c ,Γ, T 0 )= j e 2 ( ωj2Γ ) π 2 0 f d ( ξ, μ c , T 0 ) f d ( ξ, μ c , T 0 ) ( ωj2Γ ) 2 4ξ/ 2 dξ,
N eff =k/ k 0 ,
L spp = λ 0 / 4πIm( N eff ) ,
FOM= Re( N eff )/ Im( N eff ) ,
A= A m A 0 = 1 A 0 W( x,y )dxdy max[ W( x,y ) ] ,
W(x,y)= 1 2 Re{ d[ω ε 0 ε r (x,y)] dω } | E(x,y) | 2 + 1 2 μ 0 | H(x,y) | 2 ,

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