Abstract

A compact waveguide incorporating a high-index nano-ridge sandwiched between graphene sheets is proposed for the direct generation of graphene plasmonic polaritons (GSPs) via four wave mixing (FWM). The proposed waveguide supports GSP modes at the THz frequencies and photonic modes at the infrared wavelengths. Due to the strong confinement of coupled graphene sheets, the GSP modes concentrate in the high-index nano-ridge far below the diffraction limit, which improves integral overlap with the photonic modes and greatly facilitates the FWM process. To cope with the ultra-high effective refractive of the GSP modes, an alternative energy conservation diagram is selected for the degenerated FWM, which corresponds to one pump photon transfers its energy to two signal photons and one GSP photon. The single mode condition of the generated symmetric GSP modes is analyzed by the effective index method to suppress the undesired conversion. Due to the unique tunability of GSPs, the phase matching condition can be satisfied by tuning the chemical potential of the graphene sheets employing external gates. The FWM pumped at 1,550 nm with a peak power of 1 kW is theoretically investigated by solving the modified coupled mode equations. The generated GSP power reaches its maximum up to 67 W at a propagation distance of only 43.7 μm. The proposed waveguide have a great potential for integrated chip-scale GSP source.

© 2014 Optical Society of America

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References

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2014 (4)

P. Avouris and M. Freitag, “Graphene Photonics, Plasmonics, and Optoelectronics,” IEEE J. Sel. Top. Quantum Electron. 20(1), 6000112 (2014).
[Crossref]

M. Esquius-Morote, J. S. Gomez-Diaz, and J. Perruisseau-Carrier, “Sinusoidally Modulated Graphene Leaky-Wave Antenna for Electronic Beamscanning at THz,” IEEE Trans. Terahertz Sci. Technol. 4(1), 116–122 (2014).
[Crossref]

P. Chen and A. Alù, “Graphene-based plasmonic platform for reconfigurable terahertz nanodevices,” ACS Photonics 1(8), 647–654 (2014).
[Crossref]

Y. Sun, Z. Zheng, J. Cheng, and J. Liu, “Graphene surface plasmon waveguides incorporating high-index dielectric ridges for single mode transmission,” Opt. Commun. 328, 124–128 (2014).
[Crossref]

2013 (9)

C. Pai-Yen, C. Argyropoulos, and A. Alu, “Terahertz Antenna Phase Shifters Using Integrally-Gated Graphene Transmission-Lines,” IEEE Trans. Antennas and Propagation 61(4), 1528–1537 (2013).
[Crossref]

J. Schiefele, J. Pedrós, F. Sols, F. Calle, and F. Guinea, “Coupling Light into Graphene Plasmons through Surface Acoustic Waves,” Phys. Rev. Lett. 111(23), 237405 (2013).
[Crossref] [PubMed]

M. Farhat, S. Guenneau, and H. Bağcı, “Exciting Graphene Surface Plasmon Polaritons through Light and Sound Interplay,” Phys. Rev. Lett. 111(23), 237404 (2013).
[Crossref] [PubMed]

Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
[Crossref]

H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
[Crossref]

C. Yen and A. Alu, “Terahertz Metamaterial Devices Based on Graphene Nanostructures,” IEEE Trans. Terahertz Sci. Technol. 3(6), 748–756 (2013).
[Crossref]

J. Zhang, E. Cassan, D. Gao, and X. Zhang, “Highly efficient phase-matched second harmonic generation using an asymmetric plasmonic slot waveguide configuration in hybrid polymer-silicon photonics,” Opt. Express 21(12), 14876–14887 (2013).
[Crossref] [PubMed]

B. Zhu, G. Ren, S. Zheng, Z. Lin, and S. Jian, “Nanoscale dielectric-graphene-dielectric tunable infrared waveguide with ultrahigh refractive indices,” Opt. Express 21(14), 17089–17096 (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]

2012 (9)

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

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[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).
[PubMed]

M. Furchi, A. Urich, A. Pospischil, G. Lilley, K. Unterrainer, H. Detz, P. Klang, A. M. Andrews, W. Schrenk, G. Strasser, and T. Mueller, “Microcavity-integrated graphene photodetector,” Nano Lett. 12(6), 2773–2777 (2012).
[Crossref] [PubMed]

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]

M. Tamagnone, J. S. Gómez-Díaz, J. R. Mosig, and J. Perruisseau-Carrier, “Reconfigurable terahertz plasmonic antenna concept using a graphene stack,” Appl. Phys. Lett. 101(21), 214102 (2012).
[Crossref]

B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100(13), 131111 (2012).
[Crossref]

Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Efficient terahertz-wave generation via four-wave mixing in silicon membrane waveguides,” Opt. Express 20(8), 8920–8928 (2012).
[Crossref] [PubMed]

M. Liscidini, “Surface guided modes in photonic crystal ridges: the good, the bad, and the ugly,” J. Opt. Soc. Am. B 29(8), 2103–2109 (2012).
[Crossref]

2011 (4)

A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011).
[Crossref]

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]

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(10), 630–634 (2011).
[Crossref] [PubMed]

2009 (4)

A. C. Neto, F. Guinea, N. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009).
[Crossref]

M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
[Crossref]

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic‐Layer Graphene as a Saturable Absorber for Ultrafast Pulsed Lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express 17(16), 13502–13515 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (4)

K. Suizu and K. Kawase, “Terahertz-wave generation in a conventional optical fiber,” Opt. Lett. 32(20), 2990–2992 (2007).
[Crossref] [PubMed]

Y. Takushima, S. Y. Shin, and Y. C. Chung, “Design of a LiNbO3 ribbon waveguide for efficient difference-frequency generation of terahertz wave in the collinear configuration,” Opt. Express 15(22), 14783–14792 (2007).
[Crossref] [PubMed]

T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007).
[Crossref]

L. Falkovsky and A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007).
[Crossref]

2004 (2)

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(5696), 666–669 (2004).
[Crossref] [PubMed]

S. Hoffmann, M. Hofmann, E. Brundermann, M. Havenith, M. Matus, J. Moloney, A. Moskalenko, M. Kira, S. Koch, S. Saito, and K. Sakai, “Four-wave mixing and direct terahertz emission with two-color semiconductor lasers,” Appl. Phys. Lett. 84(18), 3585–3587 (2004).
[Crossref]

2003 (1)

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82(18), 2954–2956 (2003).
[Crossref]

2002 (1)

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

1974 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Springer, 2000).

Alonso-González, P.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[PubMed]

Alu, A.

C. Yen and A. Alu, “Terahertz Metamaterial Devices Based on Graphene Nanostructures,” IEEE Trans. Terahertz Sci. Technol. 3(6), 748–756 (2013).
[Crossref]

C. Pai-Yen, C. Argyropoulos, and A. Alu, “Terahertz Antenna Phase Shifters Using Integrally-Gated Graphene Transmission-Lines,” IEEE Trans. Antennas and Propagation 61(4), 1528–1537 (2013).
[Crossref]

Alù, A.

P. Chen and A. Alù, “Graphene-based plasmonic platform for reconfigurable terahertz nanodevices,” ACS Photonics 1(8), 647–654 (2014).
[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. 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).
[PubMed]

Andrekson, P. A.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

Andrews, A. M.

M. Furchi, A. Urich, A. Pospischil, G. Lilley, K. Unterrainer, H. Detz, P. Klang, A. M. Andrews, W. Schrenk, G. Strasser, and T. Mueller, “Microcavity-integrated graphene photodetector,” Nano Lett. 12(6), 2773–2777 (2012).
[Crossref] [PubMed]

Argyropoulos, C.

C. Pai-Yen, C. Argyropoulos, and A. Alu, “Terahertz Antenna Phase Shifters Using Integrally-Gated Graphene Transmission-Lines,” IEEE Trans. Antennas and Propagation 61(4), 1528–1537 (2013).
[Crossref]

Avouris, P.

P. Avouris and M. Freitag, “Graphene Photonics, Plasmonics, and Optoelectronics,” IEEE J. Sel. Top. Quantum Electron. 20(1), 6000112 (2014).
[Crossref]

H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
[Crossref]

Badioli, M.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
[PubMed]

Bagci, H.

M. Farhat, S. Guenneau, and H. Bağcı, “Exciting Graphene Surface Plasmon Polaritons through Light and Sound Interplay,” Phys. Rev. Lett. 111(23), 237404 (2013).
[Crossref] [PubMed]

Bao, Q.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic‐Layer Graphene as a Saturable Absorber for Ultrafast Pulsed Lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

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. 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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Basov, D. N.

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M. Farhat, S. Guenneau, and H. Bağcı, “Exciting Graphene Surface Plasmon Polaritons through Light and Sound Interplay,” Phys. Rev. Lett. 111(23), 237404 (2013).
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J. Schiefele, J. Pedrós, F. Sols, F. Calle, and F. Guinea, “Coupling Light into Graphene Plasmons through Surface Acoustic Waves,” Phys. Rev. Lett. 111(23), 237405 (2013).
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A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011).
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J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
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S. Hoffmann, M. Hofmann, E. Brundermann, M. Havenith, M. Matus, J. Moloney, A. Moskalenko, M. Kira, S. Koch, S. Saito, and K. Sakai, “Four-wave mixing and direct terahertz emission with two-color semiconductor lasers,” Appl. Phys. Lett. 84(18), 3585–3587 (2004).
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S. Hoffmann, M. Hofmann, E. Brundermann, M. Havenith, M. Matus, J. Moloney, A. Moskalenko, M. Kira, S. Koch, S. Saito, and K. Sakai, “Four-wave mixing and direct terahertz emission with two-color semiconductor lasers,” Appl. Phys. Lett. 84(18), 3585–3587 (2004).
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T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasmon-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007).
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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(10), 630–634 (2011).
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Huth, F.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
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M. Jablan, H. Buljan, and M. Soljačić, “Plasmonics in graphene at infrared frequencies,” Phys. Rev. B 80(24), 245435 (2009).
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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(5696), 666–669 (2004).
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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(10), 630–634 (2011).
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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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Keilmann, F.

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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S. Hoffmann, M. Hofmann, E. Brundermann, M. Havenith, M. Matus, J. Moloney, A. Moskalenko, M. Kira, S. Koch, S. Saito, and K. Sakai, “Four-wave mixing and direct terahertz emission with two-color semiconductor lasers,” Appl. Phys. Lett. 84(18), 3585–3587 (2004).
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M. Furchi, A. Urich, A. Pospischil, G. Lilley, K. Unterrainer, H. Detz, P. Klang, A. M. Andrews, W. Schrenk, G. Strasser, and T. Mueller, “Microcavity-integrated graphene photodetector,” Nano Lett. 12(6), 2773–2777 (2012).
[Crossref] [PubMed]

Koch, S.

S. Hoffmann, M. Hofmann, E. Brundermann, M. Havenith, M. Matus, J. Moloney, A. Moskalenko, M. Kira, S. Koch, S. Saito, and K. Sakai, “Four-wave mixing and direct terahertz emission with two-color semiconductor lasers,” Appl. Phys. Lett. 84(18), 3585–3587 (2004).
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Kogelnik, H.

Koppens, F. H.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
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Lau, C. 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. 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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Li, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
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Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
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Li, X.

H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
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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(10), 630–634 (2011).
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M. Furchi, A. Urich, A. Pospischil, G. Lilley, K. Unterrainer, H. Detz, P. Klang, A. M. Andrews, W. Schrenk, G. Strasser, and T. Mueller, “Microcavity-integrated graphene photodetector,” Nano Lett. 12(6), 2773–2777 (2012).
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Liscidini, M.

Liu, H.

Liu, J.

Y. Sun, Z. Zheng, J. Cheng, and J. Liu, “Graphene surface plasmon waveguides incorporating high-index dielectric ridges for single mode transmission,” Opt. Commun. 328, 124–128 (2014).
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Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
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Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
[Crossref]

Liu, M.

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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Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic‐Layer Graphene as a Saturable Absorber for Ultrafast Pulsed Lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
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H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
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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(10), 630–634 (2011).
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Martín-Moreno, L.

A. Y. Nikitin, F. Guinea, F. J. García-Vidal, and L. Martín-Moreno, “Edge and waveguide terahertz surface plasmon modes in graphene microribbons,” Phys. Rev. B 84(16), 161407 (2011).
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J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
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C. Pai-Yen, C. Argyropoulos, and A. Alu, “Terahertz Antenna Phase Shifters Using Integrally-Gated Graphene Transmission-Lines,” IEEE Trans. Antennas and Propagation 61(4), 1528–1537 (2013).
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J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (2012).
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M. Furchi, A. Urich, A. Pospischil, G. Lilley, K. Unterrainer, H. Detz, P. Klang, A. M. Andrews, W. Schrenk, G. Strasser, and T. Mueller, “Microcavity-integrated graphene photodetector,” Nano Lett. 12(6), 2773–2777 (2012).
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S. Hoffmann, M. Hofmann, E. Brundermann, M. Havenith, M. Matus, J. Moloney, A. Moskalenko, M. Kira, S. Koch, S. Saito, and K. Sakai, “Four-wave mixing and direct terahertz emission with two-color semiconductor lasers,” Appl. Phys. Lett. 84(18), 3585–3587 (2004).
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M. Furchi, A. Urich, A. Pospischil, G. Lilley, K. Unterrainer, H. Detz, P. Klang, A. M. Andrews, W. Schrenk, G. Strasser, and T. Mueller, “Microcavity-integrated graphene photodetector,” Nano Lett. 12(6), 2773–2777 (2012).
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Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic‐Layer Graphene as a Saturable Absorber for Ultrafast Pulsed Lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
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Y. Sun, Z. Zheng, J. Cheng, and J. Liu, “Graphene surface plasmon waveguides incorporating high-index dielectric ridges for single mode transmission,” Opt. Commun. 328, 124–128 (2014).
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Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
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Tamagnone, M.

M. Tamagnone, J. S. Gómez-Díaz, J. R. Mosig, and J. Perruisseau-Carrier, “Reconfigurable terahertz plasmonic antenna concept using a graphene stack,” Appl. Phys. Lett. 101(21), 214102 (2012).
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Tang, D. Y.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic‐Layer Graphene as a Saturable Absorber for Ultrafast Pulsed Lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
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J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, P. Godignon, A. Z. Elorza, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487(7405), 77–81 (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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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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B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100(13), 131111 (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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Wang, Y.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic‐Layer Graphene as a Saturable Absorber for Ultrafast Pulsed Lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
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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(13), 131111 (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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[Crossref] [PubMed]

Zhang, H.

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic‐Layer Graphene as a Saturable Absorber for Ultrafast Pulsed Lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Zhang, J.

Zhang, L. 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. 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).
[PubMed]

Zhang, X.

J. Zhang, E. Cassan, D. Gao, and X. Zhang, “Highly efficient phase-matched second harmonic generation using an asymmetric plasmonic slot waveguide configuration in hybrid polymer-silicon photonics,” Opt. Express 21(12), 14876–14887 (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(13), 131111 (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).
[Crossref] [PubMed]

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(5696), 666–669 (2004).
[Crossref] [PubMed]

Zhao, 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. 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).
[PubMed]

Zheng, S.

Zheng, Z.

Y. Sun, Z. Zheng, J. Cheng, and J. Liu, “Graphene surface plasmon waveguides incorporating high-index dielectric ridges for single mode transmission,” Opt. Commun. 328, 124–128 (2014).
[Crossref]

Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
[Crossref]

Zhu, B.

Zhu, W.

H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris, and F. Xia, “Damping pathways of mid-infrared plasmons in graphene nanostructures,” Nat. Photonics 7(5), 394–399 (2013).
[Crossref]

ACS Nano (1)

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]

ACS Photonics (1)

P. Chen and A. Alù, “Graphene-based plasmonic platform for reconfigurable terahertz nanodevices,” ACS Photonics 1(8), 647–654 (2014).
[Crossref]

Adv. Funct. Mater. (1)

Q. Bao, H. Zhang, Y. Wang, Z. Ni, Y. Yan, Z. X. Shen, K. P. Loh, and D. Y. Tang, “Atomic‐Layer Graphene as a Saturable Absorber for Ultrafast Pulsed Lasers,” Adv. Funct. Mater. 19(19), 3077–3083 (2009).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (5)

M. Tamagnone, J. S. Gómez-Díaz, J. R. Mosig, and J. Perruisseau-Carrier, “Reconfigurable terahertz plasmonic antenna concept using a graphene stack,” Appl. Phys. Lett. 101(21), 214102 (2012).
[Crossref]

S. Hoffmann, M. Hofmann, E. Brundermann, M. Havenith, M. Matus, J. Moloney, A. Moskalenko, M. Kira, S. Koch, S. Saito, and K. Sakai, “Four-wave mixing and direct terahertz emission with two-color semiconductor lasers,” Appl. Phys. Lett. 84(18), 3585–3587 (2004).
[Crossref]

Y. Sun, Z. Zheng, J. Cheng, J. Liu, J. Liu, and S. Li, “The un-symmetric hybridization of graphene surface plasmons incorporating graphene sheets and nano-ribbons,” Appl. Phys. Lett. 103(24), 241116 (2013).
[Crossref]

M. Dinu, F. Quochi, and H. Garcia, “Third-order nonlinearities in silicon at telecom wavelengths,” Appl. Phys. Lett. 82(18), 2954–2956 (2003).
[Crossref]

B. Wang, X. Zhang, X. Yuan, and J. Teng, “Optical coupling of surface plasmons between graphene sheets,” Appl. Phys. Lett. 100(13), 131111 (2012).
[Crossref]

Eur. Phys. J. B (1)

L. Falkovsky and A. Varlamov, “Space-time dispersion of graphene conductivity,” Eur. Phys. J. B 56(4), 281–284 (2007).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

P. Avouris and M. Freitag, “Graphene Photonics, Plasmonics, and Optoelectronics,” IEEE J. Sel. Top. Quantum Electron. 20(1), 6000112 (2014).
[Crossref]

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[Crossref]

IEEE Trans. Antennas and Propagation (1)

C. Pai-Yen, C. Argyropoulos, and A. Alu, “Terahertz Antenna Phase Shifters Using Integrally-Gated Graphene Transmission-Lines,” IEEE Trans. Antennas and Propagation 61(4), 1528–1537 (2013).
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Figures (6)

Fig. 1
Fig. 1 Schematic of the proposed waveguides.
Fig. 2
Fig. 2 Schematic of (a) the energy conservation diagram and (b) the phase matching condition.
Fig. 3
Fig. 3 The effective index versus the chemical potential.
Fig. 4
Fig. 4 (a) The power flow distribution in z direction and (b) the Ey profiles of fundamental symmetric GSP mode at the wavelength of 35 μm; the Ey profiles of (c) the fundamental TM mode at 1,550 nm, (d) the fundamental TM mode at 3,243.6 nm at the phase match point.
Fig. 5
Fig. 5 Schematic for the (a) first and (b) second step of the EIM
Fig. 6
Fig. 6 The peak power of the pump and generated GSP wave versus the propagation distance

Tables (1)

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Table 1 The Nonlinearity Coefficients at Phase Match Point

Equations (27)

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ω 3 = ω 1 2 w 2 , β 3 = β 1 2 β 2 - β N L ,
d A 1 d z = α 1 2 A 1 + i γ 11 | A 1 | 2 A 1 + 2 i γ 12 | A 2 | 2 A 1 + 2 i γ 13 | A 3 | 2 A 1 + i γ 1223 A 2 2 A 3 exp ( i Δ β L z ) d A 2 d z = α 2 2 A 2 + i γ 22 | A 2 | 2 A 2 + 2 i γ 21 | A 1 | 2 A 2 + 2 i γ 23 | A 3 | 2 A 2 + 2 i γ 2123 A 1 A 2 * A 3 * exp ( i Δ β L z ) d A 3 d z = α 2 2 A 2 + i γ 33 | A 3 | 2 A 3 + 2 i γ 31 | A 1 | 2 A 3 + 2 i γ 32 | A 2 | 2 A 3 + i γ 3122 A 1 A 2 * 2 exp ( i Δ β L z ) ,
w th = λ 3 2 Re ( N c o r e ) 2 Re ( N s i d e ) 2 .
2 E ¯ ( r , t ) 1 c 2 E ¯ ( r , t ) 2 t 2 = μ 0 P ¯ ( r , t ) 2 t 2 .
P ¯ ( r , t ) = ε 0 χ E ¯ ( r , t ) + ε 0 χ 1111 ( 3 ) 4 E 3 ¯ ( r , t ) = P ¯ L ( r , t ) + P ¯ N L ( r , t ) .
2 E ¯ ( r , t ) n 2 c 2 E ¯ ( r , t ) 2 t 2 = μ 0 P N L ¯ ( r , t ) 2 t 2 .
S = μ 0 P N L ¯ ( r , t ) 2 t 2 .
E ¯ ( r , t ) = 1 2 [ E ( ω ) exp ( j ω t ) + E * ( ω ) exp ( j ω t ) ] .
( 2 + β 2 ) E ( ω ) = - S ,
S = 1 8 μ 0 ε 0 χ 1111 ( 3 ) l , m , n = ± 1 , ± 2 , ± 3 , ± 4 ( ω l + ω m + ω n ) 2 E l E m E n exp [ j ( ω l + ω m + ω n ) t ]
S 1 = 1 4 μ 0 ε 0 ω 1 2 χ 1111 ( 3 ) { 3 E 2 2 E 3 + 3 E 1 [ | E 1 | 2 + 2 | E 2 | 2 + 2 | E 3 | 2 ] } S 2 = 1 4 μ 0 ε 0 ω 2 2 χ 1111 ( 3 ) { 6 E 1 E 2 * E 3 * + 3 E 2 [ | E 2 | 2 + 2 | E 1 | 2 + 2 | E 3 | 2 ] } S 3 = 1 4 μ 0 ε 0 ω 3 2 χ 1111 ( 3 ) { 3 E 1 E 2 * 2 + 3 E 3 [ | E 3 | 2 + 2 | E 1 | 2 + 2 | E 2 | 2 ] }
E ( r , ω ) = A ( z ) F ( x , y ) exp ( j β z ) .
( 2 + β 2 ) A ( z ) exp ( j β z ) d A d z j 2 β exp ( j β z ) .
d A 1 d z = i γ 11 | A 1 | 2 A 1 + 2 i γ 12 | A 2 | 2 A 1 + 2 i γ 13 | A 3 | 2 A 1 + i γ 1223 A 2 2 A 3 exp ( i Δ β L z ) d A 2 d z = i γ 22 | A 2 | 2 A 2 + 2 i γ 21 | A 1 | 2 A 2 + 2 i γ 23 | A 3 | 2 A 2 + 2 i γ 2123 A 1 A 2 * A 3 * exp ( i Δ β L z ) d A 3 d z = i γ 33 | A 3 | 2 A 3 + 2 i γ 31 | A 1 | 2 A 3 + 2 i γ 32 | A 2 | 2 A 3 + i γ 3122 A 1 A 2 * 2 exp ( i Δ β L z )
γ = ω n 2 c f .
n 2 = 3 8 n χ 1111 ( 3 ) .
f l m = + | F l | 2 | F m | 2 d x d y ( + | F l | 2 d x d y ) ( + | F m | 2 d x d y ) , l , m = 1 , 2 , 3.
f 1223 = ( + F 1 * F 2 F 2 F 3 d x d y ) i = 1 , 2 , 2 , 3 [ + | F i | 2 d x d y ) ] 1 / 2 ,
f 2123 = ( + F 2 * F 1 F 2 * F 3 * d x d y ) i = 2 , 1 , 2 , 3 [ + | F i | 2 d x d y ) ] 1 / 2 ,
f 3122 = ( + F 3 * F 1 F 2 * F 2 * d x d y ) i = 3 , 1 , 2 , 1 [ + | F i | 2 d x d y ) ] 1 / 2 .
exp ( i ϕ 1 ) d a 1 d z + i a 1 exp ( i ϕ 1 ) d ϕ 1 d z = i a 1 exp ( i ϕ 1 ) ( γ 11 | a 1 | 2 + 2 i γ 12 | a 2 | 2 + 2 i γ 13 | a 3 | 2 ) + i γ 1223 a 2 2 a 3 exp [ i ( Δ β L z + 2 ϕ 2 + ϕ 3 ) ] exp ( i ϕ 2 ) d a 2 d z + i a 2 exp ( i ϕ 2 ) d ϕ 2 d z = i a 2 exp ( i ϕ 2 ) ( γ 22 | a 2 | 2 a 2 + 2 i γ 21 | a 1 | 2 + 2 i γ 23 | a 3 | 2 ) + 2 i γ 2123 a 1 a 2 a 3 exp [ i ( Δ β L z ϕ 1 + ϕ 2 + ϕ 3 ) ] exp ( i ϕ 3 ) d a 3 d z + i a 3 exp ( i ϕ 3 ) d ϕ 3 d z = i a 3 exp ( i ϕ 3 ) ( γ 33 | a 3 | 2 + 2 i γ 31 | a 1 | 2 + 2 i γ 32 | a 2 | 2 ) + i γ 3122 a 1 a 2 2 exp [ i ( Δ β L z ϕ 1 + 2 ϕ 2 ) ]
d P 1 d z + i 2 P 1 d ϕ 1 d z = 2 i [ γ 11 P 1 2 + 2 γ 12 P 1 P 2 + 2 γ 13 P 1 P 3 + γ 1223 P 1 P 2 2 P 3 exp ( i θ ) ] d P 2 d z + i 2 P 2 d ϕ 2 d z = 2 i [ γ 22 P 2 2 + 2 γ 21 P 2 P 1 + 2 γ 23 P 2 P 3 + 2 γ 2123 P 1 P 2 2 P 3 exp ( i θ ) ] d P 3 d z + i 2 P 3 d ϕ 3 d z = 2 i [ γ 33 P 3 2 + 2 γ 31 P 3 P 1 + 2 γ 32 P 3 P 2 + γ 3122 P 1 P 2 2 P 3 exp ( i θ ) ]
θ = Δ β z + 2 ϕ 2 + ϕ 3 - ϕ 1 .
d ϕ 1 d z = γ 11 P 1 + 2 γ 12 P 2 + 2 γ 13 P 3 + γ 1223 P 1 1 P 2 2 P 3 cos θ d ϕ 2 d z = γ 22 P 2 + 2 γ 21 P 1 + 2 γ 23 P 3 + 2 γ 2123 P 1 P 3 cos θ d ϕ 3 d z = γ 33 P 3 + 2 γ 31 P 1 + 2 γ 32 P 2 + γ 3122 P 1 P 2 2 P 3 1 cos θ
d θ d z = Δ β + 2 d φ 2 d z + d φ 3 d z d φ 1 d z = Δ β L + ( 2 γ 22 + 2 γ 32 2 γ 12 ) P 2 + ( γ 11 + 4 γ 21 + 2 γ 31 ) P 1 + ( 4 γ 23 2 γ 13 + γ 33 ) P 3 + [ 4 γ 2123 P 1 P 3 + γ 1223 P 1 1 P 2 2 P 3 γ 3122 P 1 P 2 2 P 3 1 ] cos θ
d θ d z Δ β + L ( γ 11 + 4 γ 21 + 2 γ 31 ) P 1 .
- Δ β = L β N L ( γ 11 + 4 γ 21 + 2 γ 31 ) P 1 .

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