Abstract
Recently, unusual properties of novel two-dimensional materials like graphene have revolutionized many branches of science and technology. Adequate numerical simulations of photonic devices comprising graphene layers require new approaches or modified methods. In this paper we present comparison of results obtained with three representations of a graphene monolayer: (1) an infinitely thin sheet with a finite surface complex conductivity, (2) a thin layer of a finite thickness exhibiting uniaxially anisotropic complex permittivity, and (3) anisotropic permittivity perturbation in an infinitely small volume. Numerical solutions of rigorous dispersion equations were compared to each other and also with the results obtained with several commercial as well as proprietary software packets. Under proper conditions, all approaches provide comparable results.
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Amin, R., Ma, Z.Z., Maiti, R., Khan, S., Khurgin, J.B., Dalir, H., Sorger, V.J.: Attojoule-efficient graphene optical modulators. Appl. Opt. 57(18), D130–D140 (2018). https://doi.org/10.1364/ao.57.00d130
Bonaccorso, F., Sun, Z., Hasan, T., Ferrari, A.C.: Graphene photonics and optoelectronics. Nat. Photonics 4, 611–622 (2010). https://doi.org/10.1038/nphoton.2010.186
Capmany, J., Domenech, D., Munoz, P.: Silicon graphene Bragg gratings. Opt. Express 22(5), 5283–5290 (2014). https://doi.org/10.1364/oe.22.005283
Chang, P.H., Lin, C., Helmy, A.S.: Efficient integrated graphene photonics in the visible and near-IR. Laser Photonics Rev 11(5), 1700003 (2017). https://doi.org/10.1002/lpor.201700003
Chilwell, J., Hodgkinson, I.: Thin-films field-transfer matrix theory of planar multilayer waveguides and reflection prism-loaded waveguides. J. Opt. Soc. Am. A-1, 742–753 (1984)
Christensen, J., Manjavacas, A., Thongrattanasiri, S., Koppens, F.H.L., de Abajo, F.J.G.: Graphene plasmon waveguiding and hybridization in individual and paired nanoribbons. ACS Nano 6(1), 431–440 (2012). https://doi.org/10.1021/nn2037626
COMSOL Multiphysics. https://www.comsol.com/support/knowledgebase/1223/
CST Studio suite. https://www.3ds.com/products-services/simulia/products/cst-studio-suite/
Čtyroký, J., Kwiecien, P., Richter, I.: Fourier series-based bidirectional propagation algorithm with adaptive spatial resolution. J. Lightwave Technol. 28(21), 2969–2976 (2010)
Falkovsky, L.A.: Optical properties of graphene. J. Phys. Conf. Ser. 129, 1–7 (2008)
Giambra, M.A., Sorianello, V., Miseikis, V., Marconi, S., Montanaro, A., Galli, P., Pezzini, S., Coletti, C., Romagnoli, M.: High-speed double layer graphene electroabsorption modulator on SOI waveguide. Opt. Express 27(15), 20145–20155 (2019). https://doi.org/10.1364/OE.27.020145
Hao, R., Jiao, J.Y., Peng, X.L., Zhen, Z., Dagarbek, R., Zou, Y.J., Li, E.: Experimental demonstration of a graphene-based hybrid plasmonic modulator. Opt. Lett. 44(10), 2586–2589 (2019). https://doi.org/10.1364/ol.44.002586
Hu, X., Zhang, Y.G., Chen, D.G., Xiao, X., Yu, S.H.: Design and modeling of high efficiency graphene intensity/phase modulator based on ultra-thin silicon strip waveguide. J. Lightwave Technol. 37(10), 2284–2292 (2019). https://doi.org/10.1109/jlt.2019.2901916
Ji, L.T., Gao, Y., Xu, Y., Sun, X.Q., Wu, C., Wu, Y.D., Zhang, D.M.: High figure of merit electro-optic modulator based on graphene on silicon dual-slot waveguide. IEEE J. Quantum Electron 54(6), 1–7 (2018). https://doi.org/10.1109/jqe.2018.2870579
Joshi, S., Kaushik, B.K.: Analysis of the effect of graphene integration on the coupling condition in microresonator. In: Subramania, G.S., Foteinopoulou, S. (eds.) Active Photonic Platforms X, vol. 10721. Proceedings of SPIE (2018)
Khavasi, A.: Fast convergent Fourier modal method for the analysis of periodic arrays of graphene ribbons. Opt. Lett. 38(16), 3009–3012 (2013). https://doi.org/10.1364/OL.38.003009
Khavasi, A., Rejaei, B.: Analytical modeling of graphene ribbons as optical circuit elements. IEEE J. Quantum Electron. 50(6), 397–403 (2014). https://doi.org/10.1109/jqe.2014.2316133
Kogelnik, H.: Theory of dielectric waveguides. In: Tamir, T. (ed.) Integrated optics. Topics in applied physics, pp. 66–71. Springer, Berlin (1975)
Kovacevic, G., Phare, C., Set, S.Y., Lipson, M., Yamashita, S.: Ultra-high-speed graphene optical modulator design based on tight field confinement in a slot waveguide. Appl. Phys. Express 11(6), 065102 (2018). https://doi.org/10.7567/apex.11.065102
Li, Z.Q., Bai, L.D., Li, X., Gu, E.D., Niu, L.Y., Zhang, X.: U-shaped micro-ring graphene electro-optic modulator. Opt. Commun. 428, 200–205 (2018). https://doi.org/10.1016/j.optcom.2018.07.062
Lima, A.W., Sombra, A.S.B.: Graphene-based Mach–Zehnder nanophotonics interferometer working as a splitter/switch and as a multiplexer/demultiplexer. Opt. Quantum Electron. 49(11), 388 (2017). https://doi.org/10.1007/s11082-017-1227-9
Lumerical MODE waveguide simulator. https://kx.lumerical.com/t/lumerical-citation-instructions/34271
Meng, Y., Ye, S.W., Shen, Y.J., Xiao, Q.R., Fu, X., Lu, R.G., Liu, Y., Gong, M.L.: Waveguide engineering of graphene optoelectronics-modulators and polarizers. IEEE Photonics. J. 10(1), 1–17 (2018). https://doi.org/10.1109/jphot.2018.2789894
Nayyeri, V., Soleimani, M., Ramahi, O.M.: Modeling graphene in the finite-difference time-domain method using a surface boundary condition. IEEE Trans. Antennas Propag. 61(8), 4176–4182 (2013). https://doi.org/10.1109/tap.2013.2260517
Nekuee, S.A.H., Khavasi, A., Akbari, M.: Fourier modal method formulation for fast analysis of two-dimensional periodic arrays of graphene. J. Opt. Soc. Am. B Opt. Phys. 31(5), 987–993 (2014). https://doi.org/10.1364/josab.31.000987
Ni, F.C., Xie, Z.T., Ma, Q.C., Tao, J., Wu, L., Yu, C.Y., Huang, X.G.: Variable optical attenuator and modulator based on a graphene plasmonic gap waveguide. Opt. Commun. 426, 251–256 (2018). https://doi.org/10.1016/j.optcom.2018.05.029
Peng, X.L., Hao, R., Ye, Z.W., Qin, P.F., Chen, W.C., Chen, H.S., Jin, X.F., Yang, D.X., Li, E.P.: Highly efficient graphene-on-gap modulator by employing the hybrid plasmonic effect. Opt. Lett. 42(9), 1736–1739 (2017). https://doi.org/10.1364/ol.42.001736
Ralevic, U., Isic, G., Vasic, B., Gvozdic, D., Gajic, R.: Role of waveguide geometry in graphene-based electro-absorptive optical modulators. J. Phys. D Appl. Phys. 48(35), 355102 (2015). https://doi.org/10.1088/0022-3727/48/35/355102
Romagnoli, M., Sorianello, V., Midrio, M., Koppens, F.H.L., Huyghebaert, C., Neumaier, D., Galli, P., Templ, W., D’Errico, A., Ferrari, A.C.: Graphene-based integrated photonics for next-generation datacom and telecom. Nat. Rev. Mater. 3(10), 392–414 (2018). https://doi.org/10.1038/s41578-018-0040-9
Shu, H.W., Su, Z.T., Huang, L., Wu, Z.N., Wang, X.J., Zhang, Z.Y., Zhou, Z.P.: Significantly high modulation efficiency of compact graphene modulator based on silicon waveguide. Sci. Rep. 8, 1–8 (2018). https://doi.org/10.1038/s41598-018-19171-x
Sorianello, V., Midrio, M., Romagnoli, M.: Design optimization of single and double layer graphene phase modulators in SOI. Opt. Express 23(5), 6480–6490 (2015). https://doi.org/10.1364/OE.23.006478
Sorianello, V., De Angelis, G., Cassese, T., Midrio, M., Romagnoli, M., Mohsin, M., Otto, M., Neumaier, D., Asselberghs, I., Van Campenhout, J., Huyghebaert, C.: Complex effective index in graphene–silicon waveguides. Opt. Express 24(26), 29984–29993 (2016). https://doi.org/10.1364/oe.24.029984
Sorianello, V., Midrio, M., Contestabile, G., Asselberghs, I., Campenhout, J.V., Huyghebaert, C., Goykhman, I., Ott, A.K., Ferrari, A.C., Romagnoli, M.: Graphene–silicon phase modulators with gigahertz bandwidth. Nat. Photonics 120(12), 40–44 (2017). https://doi.org/10.1038/s41566-017-0071-6
Sun, F.Y., Xia, L.P., Nie, C.B., Shen, J., Zou, Y.X., Cheng, G.Y., Wu, H., Zhang, Y., Wei, D.S., Yin, S.Y., Du, C.L.: The all-optical modulator in dielectric-loaded waveguide with graphene-silicon heterojunction structure. Nanotechnology 29(13), 1–4 (2018). https://doi.org/10.1088/1361-6528/aaa8be
Vakil, A., Engheta, N.: Transformation optics using graphene. Science 332(6035), 1291–1294 (2011). https://doi.org/10.1126/science.1202691
Vassallo, C.: Optical Waveguide Concepts. Optical Wave Sciences and Technology, vol. 1. Elsevier, Amsterdam (1991)
Wang, M., Yang, E.H.: THz applications of 2D materials: graphene and beyond. Nano-Struct. Nano-Obj 15, 107–113 (2018)
Zhou, F., Du, W.: Ultrafast all-optical plasmonic graphene modulator. Appl. Opt. 57(23), 6645–6650 (2018). https://doi.org/10.1364/ao.57.006645
Zinenko, T.L., Matsushima, A., Nosich, A.I.: Surface-plasmon, grating-mode, and slab-mode resonances in the H-and E-polarized THz wave scattering by a graphene strip grating embedded into a dielectric slab. IEEE J. Sel. Top. Quant. Electron. 23(4), 1–9 (2017). https://doi.org/10.1109/jstqe.2017.2684082
Acknowledgements
This work was financially supported by the Czech Science Foundation under Project 1900062S and by the Project Center of Advanced Applied Sciences CZ.02.1.01/0.0/0.0/16_019/0000778 – European Structural and Investments Funds – Operational Programme Research, Development, and Education).
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Čtyroký, J., Petráček, J., Kwiecien, P. et al. Graphene on an optical waveguide: comparison of simulation approaches. Opt Quant Electron 52, 149 (2020). https://doi.org/10.1007/s11082-020-02265-0
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DOI: https://doi.org/10.1007/s11082-020-02265-0