Abstract

In this paper, we propose a new way of estimating the absorption in graphene coated silicon wire waveguides based on a self-developed, modified 2D Finite Difference Method, and use it to obtain a detailed absorption dependency of the waveguide design. For the first time, we observe peaks in the TM mode absorption curves, as well as the reversals of the dominantly absorbed mode with waveguide design variation, both of which have not been predicted previously theoretically, but have been implied through experimental results. We also provide a qualitative explanation of our novel numerical results, and explain how these results can be utilized in optimization of various graphene based integrated devices like optical modulators, photodetectors and optical polarizers.

© 2016 Optical Society of America

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References

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    [Crossref] [PubMed]
  3. D. J. Thomson, F. Y. Gardes, Y. Hu, G. Mashanovich, M. Fournier, P. Grosse, J-M. Fedeli, and G. T. Reed, “High contrast 40Gbit/s optical modulation in silicon,” Opt. Express 19, 11507–11516 (2011).
    [Crossref] [PubMed]
  4. 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, 64–67 (2011).
    [Crossref] [PubMed]
  5. M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12, 1482–1485 (2012).
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    [Crossref]
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    [Crossref]
  8. Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nature Phot. 5, 411–415 (2011).
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  15. M. Mohsin, D. Neumaier, D. Schall, M. Otto, C. Matheisen, A. L. Giesecke, A. A. Sagade, and H. Kurz, “Experimental verification of electro-refractive phase modulation in graphene,” Sci. Rep. 5, 10967 (2015).
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  17. R. Kou, S. Tanabe, T. Tsuchizawa, K. Warabi, S. Suzuki, H. Hibino, H. Nakajima, and K. Yamada, “Characterization of optical absorption and polarization dependence of single-layer graphene integrated on a silicon wire waveguide,” Jpn. J. Appl. Phys. 52, 060203 (2013).
    [Crossref]
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    [Crossref]

2015 (1)

M. Mohsin, D. Neumaier, D. Schall, M. Otto, C. Matheisen, A. L. Giesecke, A. A. Sagade, and H. Kurz, “Experimental verification of electro-refractive phase modulation in graphene,” Sci. Rep. 5, 10967 (2015).
[Crossref] [PubMed]

2014 (1)

S. J. Koester, “Waveguide-coupled graphene optoelectronics,” IEEE J. Sel. Top. Quantum Electron. 20, 6000211 (2014).
[Crossref]

2013 (4)

R. Kou, S. Tanabe, T. Tsuchizawa, K. Warabi, S. Suzuki, H. Hibino, H. Nakajima, and K. Yamada, “Characterization of optical absorption and polarization dependence of single-layer graphene integrated on a silicon wire waveguide,” Jpn. J. Appl. Phys. 52, 060203 (2013).
[Crossref]

J. Gosciniak and D. T. H. Tan, “Graphene-based waveguide integrated dielectric-loaded plasmonic electro-absorption modulators,” Nanotechnology 24, 185202 (2013).
[Crossref] [PubMed]

X. Gan, R.-J. Shiue, Y. Gao, I. Meric, T. F. Heinz, K. Shepard, J. Hone, S. Assefa, and D. Englund, “Chip-integrated ultrafast graphene photodetector with high responsivity,” Nature Phot. 7, 883–887 (2013).
[Crossref]

A. Pospischil, M. Humer, M. M. Furchi, D. Bachmann, R. Guider, T. Fromherz, and T. Mueller, “CMOS-compatible graphene photodetector covering all optical communication bands,” Nature Phot. 7, 892896 (2013).
[Crossref]

2012 (3)

H. Li, Y. Anugrah, S. J. Koester, and M. Li, “Optical absorption in graphene integrated on silicon waveguides,” Appl. Phys. Lett. 101, 111110 (2012).
[Crossref]

K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, “A roadmap for graphene,” Nature 490, 192–200 (2012).
[Crossref] [PubMed]

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

2011 (4)

2010 (1)

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nature Phot. 4, 611–622 (2010).
[Crossref]

2008 (1)

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[Crossref] [PubMed]

1998 (1)

L. Yau and A. Ben-Israel, “The Newton and Halley methods for complex roots,” Am. Math. Monthly 105, 806–818 (1998).
[Crossref]

1995 (1)

M. S. Stern, “Finite difference analysis of planar optical waveguides,” PIER 10, 123–186 (1995).

Absil, P.

Y. Hu, M. Pantouvaki, S. Brems, I. Asselberghs, C. Huyghebaert, M. Geisler, C. Alessandri, R. Beats, P. Absil, D. Van Thourhout, and J. Van Campenhout, “Broadband 10Gb/s graphene electro-absorption modulator on silicon for chip-level optical interconnects,” in Proceedings of IEEE Electron Devices Meeting (IEEE International, 2014), pp. 5.6.1–5.6.4.

Alessandri, C.

Y. Hu, M. Pantouvaki, S. Brems, I. Asselberghs, C. Huyghebaert, M. Geisler, C. Alessandri, R. Beats, P. Absil, D. Van Thourhout, and J. Van Campenhout, “Broadband 10Gb/s graphene electro-absorption modulator on silicon for chip-level optical interconnects,” in Proceedings of IEEE Electron Devices Meeting (IEEE International, 2014), pp. 5.6.1–5.6.4.

Anugrah, Y.

H. Li, Y. Anugrah, S. J. Koester, and M. Li, “Optical absorption in graphene integrated on silicon waveguides,” Appl. Phys. Lett. 101, 111110 (2012).
[Crossref]

Assefa, S.

X. Gan, R.-J. Shiue, Y. Gao, I. Meric, T. F. Heinz, K. Shepard, J. Hone, S. Assefa, and D. Englund, “Chip-integrated ultrafast graphene photodetector with high responsivity,” Nature Phot. 7, 883–887 (2013).
[Crossref]

Asselberghs, I.

Y. Hu, M. Pantouvaki, S. Brems, I. Asselberghs, C. Huyghebaert, M. Geisler, C. Alessandri, R. Beats, P. Absil, D. Van Thourhout, and J. Van Campenhout, “Broadband 10Gb/s graphene electro-absorption modulator on silicon for chip-level optical interconnects,” in Proceedings of IEEE Electron Devices Meeting (IEEE International, 2014), pp. 5.6.1–5.6.4.

Bachmann, D.

A. Pospischil, M. Humer, M. M. Furchi, D. Bachmann, R. Guider, T. Fromherz, and T. Mueller, “CMOS-compatible graphene photodetector covering all optical communication bands,” Nature Phot. 7, 892896 (2013).
[Crossref]

Bao, Q.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nature Phot. 5, 411–415 (2011).
[Crossref]

Beats, R.

Y. Hu, M. Pantouvaki, S. Brems, I. Asselberghs, C. Huyghebaert, M. Geisler, C. Alessandri, R. Beats, P. Absil, D. Van Thourhout, and J. Van Campenhout, “Broadband 10Gb/s graphene electro-absorption modulator on silicon for chip-level optical interconnects,” in Proceedings of IEEE Electron Devices Meeting (IEEE International, 2014), pp. 5.6.1–5.6.4.

Ben-Israel, A.

L. Yau and A. Ben-Israel, “The Newton and Halley methods for complex roots,” Am. Math. Monthly 105, 806–818 (1998).
[Crossref]

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nature Phot. 4, 611–622 (2010).
[Crossref]

Brems, S.

Y. Hu, M. Pantouvaki, S. Brems, I. Asselberghs, C. Huyghebaert, M. Geisler, C. Alessandri, R. Beats, P. Absil, D. Van Thourhout, and J. Van Campenhout, “Broadband 10Gb/s graphene electro-absorption modulator on silicon for chip-level optical interconnects,” in Proceedings of IEEE Electron Devices Meeting (IEEE International, 2014), pp. 5.6.1–5.6.4.

Colombo, L.

K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, “A roadmap for graphene,” Nature 490, 192–200 (2012).
[Crossref] [PubMed]

Englund, D.

X. Gan, R.-J. Shiue, Y. Gao, I. Meric, T. F. Heinz, K. Shepard, J. Hone, S. Assefa, and D. Englund, “Chip-integrated ultrafast graphene photodetector with high responsivity,” Nature Phot. 7, 883–887 (2013).
[Crossref]

Fal’ko, V. I.

K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, “A roadmap for graphene,” Nature 490, 192–200 (2012).
[Crossref] [PubMed]

Fedeli, J-M.

Ferrari, A. C.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nature Phot. 4, 611–622 (2010).
[Crossref]

Fournier, M.

Freude, W.

Fromherz, T.

A. Pospischil, M. Humer, M. M. Furchi, D. Bachmann, R. Guider, T. Fromherz, and T. Mueller, “CMOS-compatible graphene photodetector covering all optical communication bands,” Nature Phot. 7, 892896 (2013).
[Crossref]

Furchi, M. M.

A. Pospischil, M. Humer, M. M. Furchi, D. Bachmann, R. Guider, T. Fromherz, and T. Mueller, “CMOS-compatible graphene photodetector covering all optical communication bands,” Nature Phot. 7, 892896 (2013).
[Crossref]

Gan, X.

X. Gan, R.-J. Shiue, Y. Gao, I. Meric, T. F. Heinz, K. Shepard, J. Hone, S. Assefa, and D. Englund, “Chip-integrated ultrafast graphene photodetector with high responsivity,” Nature Phot. 7, 883–887 (2013).
[Crossref]

Gao, Y.

X. Gan, R.-J. Shiue, Y. Gao, I. Meric, T. F. Heinz, K. Shepard, J. Hone, S. Assefa, and D. Englund, “Chip-integrated ultrafast graphene photodetector with high responsivity,” Nature Phot. 7, 883–887 (2013).
[Crossref]

Gardes, F. Y.

Geisler, M.

Y. Hu, M. Pantouvaki, S. Brems, I. Asselberghs, C. Huyghebaert, M. Geisler, C. Alessandri, R. Beats, P. Absil, D. Van Thourhout, and J. Van Campenhout, “Broadband 10Gb/s graphene electro-absorption modulator on silicon for chip-level optical interconnects,” in Proceedings of IEEE Electron Devices Meeting (IEEE International, 2014), pp. 5.6.1–5.6.4.

Gellert, P. R.

K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, “A roadmap for graphene,” Nature 490, 192–200 (2012).
[Crossref] [PubMed]

Geng, B.

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, 64–67 (2011).
[Crossref] [PubMed]

Giesecke, A. L.

M. Mohsin, D. Neumaier, D. Schall, M. Otto, C. Matheisen, A. L. Giesecke, A. A. Sagade, and H. Kurz, “Experimental verification of electro-refractive phase modulation in graphene,” Sci. Rep. 5, 10967 (2015).
[Crossref] [PubMed]

Gosciniak, J.

J. Gosciniak and D. T. H. Tan, “Graphene-based waveguide integrated dielectric-loaded plasmonic electro-absorption modulators,” Nanotechnology 24, 185202 (2013).
[Crossref] [PubMed]

Grosse, P.

Guider, R.

A. Pospischil, M. Humer, M. M. Furchi, D. Bachmann, R. Guider, T. Fromherz, and T. Mueller, “CMOS-compatible graphene photodetector covering all optical communication bands,” Nature Phot. 7, 892896 (2013).
[Crossref]

Hahn, H.

Hasan, T.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nature Phot. 4, 611–622 (2010).
[Crossref]

Heinz, T. F.

X. Gan, R.-J. Shiue, Y. Gao, I. Meric, T. F. Heinz, K. Shepard, J. Hone, S. Assefa, and D. Englund, “Chip-integrated ultrafast graphene photodetector with high responsivity,” Nature Phot. 7, 883–887 (2013).
[Crossref]

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[Crossref] [PubMed]

Hibino, H.

R. Kou, S. Tanabe, T. Tsuchizawa, K. Warabi, S. Suzuki, H. Hibino, H. Nakajima, and K. Yamada, “Characterization of optical absorption and polarization dependence of single-layer graphene integrated on a silicon wire waveguide,” Jpn. J. Appl. Phys. 52, 060203 (2013).
[Crossref]

Hone, J.

X. Gan, R.-J. Shiue, Y. Gao, I. Meric, T. F. Heinz, K. Shepard, J. Hone, S. Assefa, and D. Englund, “Chip-integrated ultrafast graphene photodetector with high responsivity,” Nature Phot. 7, 883–887 (2013).
[Crossref]

Hu, Y.

D. J. Thomson, F. Y. Gardes, Y. Hu, G. Mashanovich, M. Fournier, P. Grosse, J-M. Fedeli, and G. T. Reed, “High contrast 40Gbit/s optical modulation in silicon,” Opt. Express 19, 11507–11516 (2011).
[Crossref] [PubMed]

Y. Hu, M. Pantouvaki, S. Brems, I. Asselberghs, C. Huyghebaert, M. Geisler, C. Alessandri, R. Beats, P. Absil, D. Van Thourhout, and J. Van Campenhout, “Broadband 10Gb/s graphene electro-absorption modulator on silicon for chip-level optical interconnects,” in Proceedings of IEEE Electron Devices Meeting (IEEE International, 2014), pp. 5.6.1–5.6.4.

Humer, M.

A. Pospischil, M. Humer, M. M. Furchi, D. Bachmann, R. Guider, T. Fromherz, and T. Mueller, “CMOS-compatible graphene photodetector covering all optical communication bands,” Nature Phot. 7, 892896 (2013).
[Crossref]

Huyghebaert, C.

Y. Hu, M. Pantouvaki, S. Brems, I. Asselberghs, C. Huyghebaert, M. Geisler, C. Alessandri, R. Beats, P. Absil, D. Van Thourhout, and J. Van Campenhout, “Broadband 10Gb/s graphene electro-absorption modulator on silicon for chip-level optical interconnects,” in Proceedings of IEEE Electron Devices Meeting (IEEE International, 2014), pp. 5.6.1–5.6.4.

Ju, L.

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, 64–67 (2011).
[Crossref] [PubMed]

Kim, K.

K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, “A roadmap for graphene,” Nature 490, 192–200 (2012).
[Crossref] [PubMed]

Koester, S. J.

S. J. Koester, “Waveguide-coupled graphene optoelectronics,” IEEE J. Sel. Top. Quantum Electron. 20, 6000211 (2014).
[Crossref]

H. Li, Y. Anugrah, S. J. Koester, and M. Li, “Optical absorption in graphene integrated on silicon waveguides,” Appl. Phys. Lett. 101, 111110 (2012).
[Crossref]

Koos, C.

Kou, R.

R. Kou, S. Tanabe, T. Tsuchizawa, K. Warabi, S. Suzuki, H. Hibino, H. Nakajima, and K. Yamada, “Characterization of optical absorption and polarization dependence of single-layer graphene integrated on a silicon wire waveguide,” Jpn. J. Appl. Phys. 52, 060203 (2013).
[Crossref]

Kurz, H.

M. Mohsin, D. Neumaier, D. Schall, M. Otto, C. Matheisen, A. L. Giesecke, A. A. Sagade, and H. Kurz, “Experimental verification of electro-refractive phase modulation in graphene,” Sci. Rep. 5, 10967 (2015).
[Crossref] [PubMed]

Leufke, P. M.

Leuthold, J.

Li, H.

H. Li, Y. Anugrah, S. J. Koester, and M. Li, “Optical absorption in graphene integrated on silicon waveguides,” Appl. Phys. Lett. 101, 111110 (2012).
[Crossref]

Li, M.

H. Li, Y. Anugrah, S. J. Koester, and M. Li, “Optical absorption in graphene integrated on silicon waveguides,” Appl. Phys. Lett. 101, 111110 (2012).
[Crossref]

Lim, C. H. Y. X.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nature Phot. 5, 411–415 (2011).
[Crossref]

Lindenmann, N.

Liu, M.

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12, 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, 64–67 (2011).
[Crossref] [PubMed]

Loh, K. P.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nature Phot. 5, 411–415 (2011).
[Crossref]

Lui, C. H.

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[Crossref] [PubMed]

Mak, K. F.

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[Crossref] [PubMed]

Mashanovich, G.

Matheisen, C.

M. Mohsin, D. Neumaier, D. Schall, M. Otto, C. Matheisen, A. L. Giesecke, A. A. Sagade, and H. Kurz, “Experimental verification of electro-refractive phase modulation in graphene,” Sci. Rep. 5, 10967 (2015).
[Crossref] [PubMed]

Melikyan, A.

Meric, I.

X. Gan, R.-J. Shiue, Y. Gao, I. Meric, T. F. Heinz, K. Shepard, J. Hone, S. Assefa, and D. Englund, “Chip-integrated ultrafast graphene photodetector with high responsivity,” Nature Phot. 7, 883–887 (2013).
[Crossref]

Misewich, J. A.

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[Crossref] [PubMed]

Mohsin, M.

M. Mohsin, D. Neumaier, D. Schall, M. Otto, C. Matheisen, A. L. Giesecke, A. A. Sagade, and H. Kurz, “Experimental verification of electro-refractive phase modulation in graphene,” Sci. Rep. 5, 10967 (2015).
[Crossref] [PubMed]

Mueller, T.

A. Pospischil, M. Humer, M. M. Furchi, D. Bachmann, R. Guider, T. Fromherz, and T. Mueller, “CMOS-compatible graphene photodetector covering all optical communication bands,” Nature Phot. 7, 892896 (2013).
[Crossref]

Nakajima, H.

R. Kou, S. Tanabe, T. Tsuchizawa, K. Warabi, S. Suzuki, H. Hibino, H. Nakajima, and K. Yamada, “Characterization of optical absorption and polarization dependence of single-layer graphene integrated on a silicon wire waveguide,” Jpn. J. Appl. Phys. 52, 060203 (2013).
[Crossref]

Neumaier, D.

M. Mohsin, D. Neumaier, D. Schall, M. Otto, C. Matheisen, A. L. Giesecke, A. A. Sagade, and H. Kurz, “Experimental verification of electro-refractive phase modulation in graphene,” Sci. Rep. 5, 10967 (2015).
[Crossref] [PubMed]

Ni, Z.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nature Phot. 5, 411–415 (2011).
[Crossref]

Novoselov, K. S.

K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, “A roadmap for graphene,” Nature 490, 192–200 (2012).
[Crossref] [PubMed]

Otto, M.

M. Mohsin, D. Neumaier, D. Schall, M. Otto, C. Matheisen, A. L. Giesecke, A. A. Sagade, and H. Kurz, “Experimental verification of electro-refractive phase modulation in graphene,” Sci. Rep. 5, 10967 (2015).
[Crossref] [PubMed]

Pantouvaki, M.

Y. Hu, M. Pantouvaki, S. Brems, I. Asselberghs, C. Huyghebaert, M. Geisler, C. Alessandri, R. Beats, P. Absil, D. Van Thourhout, and J. Van Campenhout, “Broadband 10Gb/s graphene electro-absorption modulator on silicon for chip-level optical interconnects,” in Proceedings of IEEE Electron Devices Meeting (IEEE International, 2014), pp. 5.6.1–5.6.4.

Pospischil, A.

A. Pospischil, M. Humer, M. M. Furchi, D. Bachmann, R. Guider, T. Fromherz, and T. Mueller, “CMOS-compatible graphene photodetector covering all optical communication bands,” Nature Phot. 7, 892896 (2013).
[Crossref]

Reed, G. T.

Sagade, A. A.

M. Mohsin, D. Neumaier, D. Schall, M. Otto, C. Matheisen, A. L. Giesecke, A. A. Sagade, and H. Kurz, “Experimental verification of electro-refractive phase modulation in graphene,” Sci. Rep. 5, 10967 (2015).
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M. Mohsin, D. Neumaier, D. Schall, M. Otto, C. Matheisen, A. L. Giesecke, A. A. Sagade, and H. Kurz, “Experimental verification of electro-refractive phase modulation in graphene,” Sci. Rep. 5, 10967 (2015).
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K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, “A roadmap for graphene,” Nature 490, 192–200 (2012).
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K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
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X. Gan, R.-J. Shiue, Y. Gao, I. Meric, T. F. Heinz, K. Shepard, J. Hone, S. Assefa, and D. Englund, “Chip-integrated ultrafast graphene photodetector with high responsivity,” Nature Phot. 7, 883–887 (2013).
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Shiue, R.-J.

X. Gan, R.-J. Shiue, Y. Gao, I. Meric, T. F. Heinz, K. Shepard, J. Hone, S. Assefa, and D. Englund, “Chip-integrated ultrafast graphene photodetector with high responsivity,” Nature Phot. 7, 883–887 (2013).
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R. Kou, S. Tanabe, T. Tsuchizawa, K. Warabi, S. Suzuki, H. Hibino, H. Nakajima, and K. Yamada, “Characterization of optical absorption and polarization dependence of single-layer graphene integrated on a silicon wire waveguide,” Jpn. J. Appl. Phys. 52, 060203 (2013).
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J. Gosciniak and D. T. H. Tan, “Graphene-based waveguide integrated dielectric-loaded plasmonic electro-absorption modulators,” Nanotechnology 24, 185202 (2013).
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R. Kou, S. Tanabe, T. Tsuchizawa, K. Warabi, S. Suzuki, H. Hibino, H. Nakajima, and K. Yamada, “Characterization of optical absorption and polarization dependence of single-layer graphene integrated on a silicon wire waveguide,” Jpn. J. Appl. Phys. 52, 060203 (2013).
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Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nature Phot. 5, 411–415 (2011).
[Crossref]

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Tsuchizawa, T.

R. Kou, S. Tanabe, T. Tsuchizawa, K. Warabi, S. Suzuki, H. Hibino, H. Nakajima, and K. Yamada, “Characterization of optical absorption and polarization dependence of single-layer graphene integrated on a silicon wire waveguide,” Jpn. J. Appl. Phys. 52, 060203 (2013).
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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, 64–67 (2011).
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M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12, 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, 64–67 (2011).
[Crossref] [PubMed]

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, 64–67 (2011).
[Crossref] [PubMed]

Zhang, H.

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nature Phot. 5, 411–415 (2011).
[Crossref]

Zhang, X.

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano Lett. 12, 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, 64–67 (2011).
[Crossref] [PubMed]

Am. Math. Monthly (1)

L. Yau and A. Ben-Israel, “The Newton and Halley methods for complex roots,” Am. Math. Monthly 105, 806–818 (1998).
[Crossref]

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H. Li, Y. Anugrah, S. J. Koester, and M. Li, “Optical absorption in graphene integrated on silicon waveguides,” Appl. Phys. Lett. 101, 111110 (2012).
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IEEE J. Sel. Top. Quantum Electron. (1)

S. J. Koester, “Waveguide-coupled graphene optoelectronics,” IEEE J. Sel. Top. Quantum Electron. 20, 6000211 (2014).
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Jpn. J. Appl. Phys. (1)

R. Kou, S. Tanabe, T. Tsuchizawa, K. Warabi, S. Suzuki, H. Hibino, H. Nakajima, and K. Yamada, “Characterization of optical absorption and polarization dependence of single-layer graphene integrated on a silicon wire waveguide,” Jpn. J. Appl. Phys. 52, 060203 (2013).
[Crossref]

Nano Lett. (1)

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

Nanotechnology (1)

J. Gosciniak and D. T. H. Tan, “Graphene-based waveguide integrated dielectric-loaded plasmonic electro-absorption modulators,” Nanotechnology 24, 185202 (2013).
[Crossref] [PubMed]

Nature (2)

K. S. Novoselov, V. I. Fal’ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim, “A roadmap for graphene,” Nature 490, 192–200 (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, 64–67 (2011).
[Crossref] [PubMed]

Nature Phot. (4)

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, “Graphene photonics and optoelectronics,” Nature Phot. 4, 611–622 (2010).
[Crossref]

X. Gan, R.-J. Shiue, Y. Gao, I. Meric, T. F. Heinz, K. Shepard, J. Hone, S. Assefa, and D. Englund, “Chip-integrated ultrafast graphene photodetector with high responsivity,” Nature Phot. 7, 883–887 (2013).
[Crossref]

A. Pospischil, M. Humer, M. M. Furchi, D. Bachmann, R. Guider, T. Fromherz, and T. Mueller, “CMOS-compatible graphene photodetector covering all optical communication bands,” Nature Phot. 7, 892896 (2013).
[Crossref]

Q. Bao, H. Zhang, B. Wang, Z. Ni, C. H. Y. X. Lim, Y. Wang, D. Y. Tang, and K. P. Loh, “Broadband graphene polarizer,” Nature Phot. 5, 411–415 (2011).
[Crossref]

Opt. Express (2)

Phys. Rev. Lett. (1)

K. F. Mak, M. Y. Sfeir, Y. Wu, C. H. Lui, J. A. Misewich, and T. F. Heinz, “Measurement of the optical conductivity of graphene,” Phys. Rev. Lett. 101, 196405 (2008).
[Crossref] [PubMed]

PIER (1)

M. S. Stern, “Finite difference analysis of planar optical waveguides,” PIER 10, 123–186 (1995).

Sci. Rep. (1)

M. Mohsin, D. Neumaier, D. Schall, M. Otto, C. Matheisen, A. L. Giesecke, A. A. Sagade, and H. Kurz, “Experimental verification of electro-refractive phase modulation in graphene,” Sci. Rep. 5, 10967 (2015).
[Crossref] [PubMed]

Other (2)

Y. Hu, M. Pantouvaki, S. Brems, I. Asselberghs, C. Huyghebaert, M. Geisler, C. Alessandri, R. Beats, P. Absil, D. Van Thourhout, and J. Van Campenhout, “Broadband 10Gb/s graphene electro-absorption modulator on silicon for chip-level optical interconnects,” in Proceedings of IEEE Electron Devices Meeting (IEEE International, 2014), pp. 5.6.1–5.6.4.

G. P. Agrawal, “Optical Waveguides,” OPT468a, Institute of Optics, University of Rochester (2008) http://www.optics.rochester.edu/users/gpa/opt468a.pdf .

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Figures (4)

Fig. 1
Fig. 1 a) Graphene coated silicon wire waveguide structure. b) Cross section of the waveguide, with the depiction of an FDM grid including the waveguide and graphene (small example).
Fig. 2
Fig. 2 Graphene induced absorption dependency of the waveguide thickness d, for different waveguide widths w. Solid lines – Original numerical results; Triangle points – Experimentally reported results [17, 18, 4].
Fig. 3
Fig. 3 Graphene induced absorption dependency of the waveguide width w for different waveguide thicknesses d.
Fig. 4
Fig. 4 a) Profiles of the electric fields parallel to the graphene sheet in the case of TE and TM modes. b) Dependency of the electric fields parallel to the graphene sheet of the waveguide thickness; Inset: Point used in calculations for Fig. 4(b) marked in red.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

A ( r ) = A ( x , y ) exp ( j ( ω t β z ) )
H y x H x y = j ω ε 0 ε r E z
D x h y D y h x = j ω ε 0 ε r e z
σ = e 2 8 [ tanh ( ω + 2 E F 4 k B T ) + tanh ( ω 2 E F 4 k B T ) ]
n × ( H 1 H 2 ) = J s = σ E
h y 6 h y 10 Δ x h x 6 h x 7 Δ y = j ω ε 0 ε r 7 e z 7
h y 6 h y 10 Δ x h x 6 h x 7 + ( h x 6.5 + h x 6.5 ) Δ y = j ω ε 0 ε r 7 e z 7
h x 6.5 + h x 6.5 = σ e z 6.5
e z 6.5 e z 7
h y 6 h y 10 Δ x h x 6 h x 7 Δ y = ( j ω ε 0 ε r 7 + σ Δ y ) e z 7
k d = arctan ( γ 2 + j μ ω σ k ) + arctan ( γ 3 k )
k d = arctan ( n 1 2 γ 2 n 2 2 k ( 1 j γ 2 σ ω ε 0 n 2 2 ) 1 ) + arctan ( n 1 2 γ 3 n 3 2 k )

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