涂覆石墨烯的椭圆形电介质纳米线光波导的模式特性分析
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摘要
设计了一种涂覆石墨烯的椭圆形电介质纳米线光波导.采用分离变量法,在椭圆柱坐标系中,借助Mathieu函数,得到了色散方程.通过数值求解色散方程,可以得到模式的有效折射率和场分布,从而得到模式的传播长度.研究了工作波长、结构参数以及石墨烯的费米能对模式特性的影响,并给出了前五个模式的品质因数.计算表明,当波长从4.3μm增加到8.8μm,这5个模式的有效折射率的实部减小,基模和一阶模的传播长度增大,二阶模的传播长度先增大后减小.当改变纳米线结构参数半长轴和半短轴时,对基模和一阶模的模式特性影响较小,对二阶模的模式特性影响较大.当石墨烯的费米能从0.45 eV增加到0.72 eV时,有效折射率的实部减小,传播长度可以达到2μm左右.分离变量法得到的结果与有限元方法得到的结果完全一致.本文工作可以为基于涂覆石墨烯的电介质纳米线的光波导的设计、制作和应用提供理论基础.
In this paper, an elliptical dielectric graphene-coated nanowire optical waveguide is designed. In the elliptical cylinder coordinate system, the dispersion equation is obtained by using the separation variable method with the Mathieu functions. The effective refractive indexes and the field distributions are obtained from the dispersion equation by using the numerical method, then the propagation lengths are obtained. The influence of the operating wavelength, structure parameters and the Fermi energy of graphene on the mode characteristics are investigated. What is more, the figure of merit of the first five modes are calculated too. The influence of the operating wavelength and the graphene Fermi energy on the mode characteristics of circular nanowires and that of elliptical nanowires are compared. The results show that as the operating wavelength increases from 4.3 μm to 8.8 μm, the real part of the effective refractive index decreases monotonically, the propagation lengths of the fundamental mode and the 1 st order modes increase, and the 2 nd order modes first increase and then decrease. When changing the elliptical nanowire structure parameters—the length of semimajor axis and semi-minor axis, there are slight influence on the mode characteristics of the fundamental mode and the 1 st order modes, but greater influence on those of the 2 nd order modes. As the Fermi energy of graphene increases from 0.45 eV to 0.72 eV, in the first five modes, the real part of the effective refractive index decreases, the propagation lengths of the fundamental mode and the 1 st order modes increase, the propagation lengths of the 2 nd order modes decrease. In addition, the propagation length approaches to 2 approximately. When the semi-minor axis b = 100 nm and= 100 nm) and the elliptical nanowire(a = 140 nm), the real part of the effective refractive index and propagation length with the operating wavelength and the Fermi energy of graphene are compared. Then, the advantages of elliptical nanowire over the circular nanowire are verified. The results of the separation variable method are in good agreement with the results of the finite element method. This work can provide a theoretical basis for the design, fabrication and application of optical waveguides based on graphene-coated elliptical dielectric nanowires.
In this paper, an elliptical dielectric graphene-coated nanowire optical waveguide is designed. In the elliptical cylinder coordinate system, the dispersion equation is obtained by using the separation variable method with the Mathieu functions. The effective refractive indexes and the field distributions are obtained from the dispersion equation by using the numerical method, then the propagation lengths are obtained. The influence of the operating wavelength, structure parameters and the Fermi energy of graphene on the mode characteristics are investigated. What is more, the figure of merit of the first five modes are calculated too. The influence of the operating wavelength and the graphene Fermi energy on the mode characteristics of circular nanowires and that of elliptical nanowires are compared. The results show that as the operating wavelength increases from 4.3 μm to 8.8 μm, the real part of the effective refractive index decreases monotonically, the propagation lengths of the fundamental mode and the 1 st order modes increase, and the 2 nd order modes first increase and then decrease. When changing the elliptical nanowire structure parameters—the length of semimajor axis and semi-minor axis, there are slight influence on the mode characteristics of the fundamental mode and the 1 st order modes, but greater influence on those of the 2 nd order modes. As the Fermi energy of graphene increases from 0.45 eV to 0.72 eV, in the first five modes, the real part of the effective refractive index decreases, the propagation lengths of the fundamental mode and the 1 st order modes increase, the propagation lengths of the 2 nd order modes decrease. In addition, the propagation length approaches to 2 approximately. When the semi-minor axis b = 100 nm and= 100 nm) and the elliptical nanowire(a = 140 nm), the real part of the effective refractive index and propagation length with the operating wavelength and the Fermi energy of graphene are compared. Then, the advantages of elliptical nanowire over the circular nanowire are verified. The results of the separation variable method are in good agreement with the results of the finite element method. This work can provide a theoretical basis for the design, fabrication and application of optical waveguides based on graphene-coated elliptical dielectric nanowires.
引文
[1]Gao Y X,Ren G B,Zhu B F,Liu H Q,Lian Y D,Jian S S2014 Opt.Express 22 24322
[2]Novoselov K S,Geim A K,Morozov S V,Jiang D,Zhang Y,Grigorieva I V,Dubonos S V,Firsov A A 2004 Science 306666
[3]Novoselov K S,Geim A K,Morozov S V,Jiang D,Katsnelson M I,Grigorieva I V,Dubonos S V,Firsov A A 2005 Nature438 197
[4]Ju L,Geng B S,Horng J,Girit C,Martin M,Hao Z,Bechtel H A,Liang X G,Zettl A,Shen Y R,Wang F 2011 Nature Nanotechnol.6 630
[5]Wang J C,Song C,Hang J,Hu Z D,Zhang F 2017 Opt.Express 25 23880
[6]Gao Y X,Ren G B,Zhu B F,Liu H Q,Wang J,Jian S S2014 Opt.Lett.39 5909
[7]Jablan M,Buljan M,Solja?i?M 2009 Phys.Rev.B 80 245435
[8]Christensen J,Manjavacas A,Thongrattanasiri S,Koppens FH,de García Abajo F J 2012 ACS Nano 6 431
[9]Liu P H,Zhang X Z,Ma Z H,Cai W,Wang L,Xu J J 2013Opt.Express 21 32431
[10]Xing R,Jian S S 2016 IEEE Photonics Technol.Lett.28 2649
[11]Zhou X T,Zhang T,Chen L,Hong W,Li X 2014 J.Lightwave Technol.32 3597
[12]Liu Y,Zhong R B,Ding H,Liu S G 2017 Eur.Phys.J.D 7183
[13]He X Q,Ning T G,Lu S H,Zheng J J,Li J,Li R J,Pei L2018 Opt.Express 26 10109
[14]Xing R,Jian S S 2017 IEEE Photonics Technol.Lett.29 1643
[15]Liu J P,Zhai X,Xie F,Wang L L,Xia S X,Liu H J,Luo X,Shang X J 2017 J.Lightwave Technol.35 1971
[16]Xing R,Jian S S 2017 IEEE Photonics Technol.Lett.29 967
[17]Zhu B F,Ren G B,Yang Y,Gao Y X 2015 Plasmonics 10839
[18]Peng Y L,Xue W R,Wei Z Z,Li C Y 2018 Acta Opt.Sin.380223002(in Chinese)[彭艳玲,薛文瑞,卫壮志,李昌勇2018光学学报38 0223002]
[19]Peng Y L,Xue W R,Wei Z Z,Li C Y 2018 Acta Phys.Sin.67 038102(in Chinese)[彭艳玲,薛文瑞,卫壮志,李昌勇2018物理学报67 038102]
[20]Xing R,Jian S S 2016 IEEE Photonics Techol.Lett.28 2779
[21]Wei Z Z,Xue W R,Cheng X,Li C Y 2018 Acta Phys.Sin.67108101(in Chinese)[卫壮志,薛文瑞,程鑫,李昌勇2018物理学报67 108101]
[22]Hossein M B,Alexander B Y 2015 J.Phys.Condens.Matter27 185304
[23]Chen W D,Liu F 2009 Acta Elec.Sin.37 1624(in Chinese)[陈卫东,刘丰2009电子学报37 1624]
[24]Zhao J J,Tang M,Oh K H,Feng Z H,Zhao K,Liao R L,Fu S N,Shum P P,Liu D M 2017 Photon.Res.8 261
[25]Nejad R M,Tavakoli F,Wang L X,Guan X,Larochelle S,Rusch L A 2018 J.Lightwave Technol.36 3794
[26]Lee M S,Park B G,Cho I H,Lee J H 2012 IEEE Electron Device Lett.33 1613
[27]de Abajo F J G 2010 Rev.Mod.Phys.82 209
[28]Zhai L,Xue W R,Yang R C,Han L P 2015 Acta Opt.Sin.35 1123002(in Chinese)[翟利,薛文瑞,杨荣草,韩丽萍2015光学学报35 1123002]
[29]Nikitin A Y,Guinea F,García-Vidal F J,Martín-Moreno L2011 Phys.Rev.B 84 195446
[30]Yeh C 1962 J.Appl.Phys.33 3235
[31]D Erricolo,G Carluccio 2013 ACM Trans.Math.Soft.40 8
[32]He S L,Zhang X Z,He Y R 2013 Opt.Express 21 30664
[33]Ye L F,Sui K H,Liu Y H,Zhang M,Liu Q H 2018 Opt.Express 26 15935
[34]Ye S,Wang Z X,Sun C R,Dong C B,Wei B Z,Wu B L,Jian S S 2018 Opt.Express 26 23854
[35]Chen M,Sheng P C,Sun W,Cai J J 2016 Opt.Commun.37641
[2]Novoselov K S,Geim A K,Morozov S V,Jiang D,Zhang Y,Grigorieva I V,Dubonos S V,Firsov A A 2004 Science 306666
[3]Novoselov K S,Geim A K,Morozov S V,Jiang D,Katsnelson M I,Grigorieva I V,Dubonos S V,Firsov A A 2005 Nature438 197
[4]Ju L,Geng B S,Horng J,Girit C,Martin M,Hao Z,Bechtel H A,Liang X G,Zettl A,Shen Y R,Wang F 2011 Nature Nanotechnol.6 630
[5]Wang J C,Song C,Hang J,Hu Z D,Zhang F 2017 Opt.Express 25 23880
[6]Gao Y X,Ren G B,Zhu B F,Liu H Q,Wang J,Jian S S2014 Opt.Lett.39 5909
[7]Jablan M,Buljan M,Solja?i?M 2009 Phys.Rev.B 80 245435
[8]Christensen J,Manjavacas A,Thongrattanasiri S,Koppens FH,de García Abajo F J 2012 ACS Nano 6 431
[9]Liu P H,Zhang X Z,Ma Z H,Cai W,Wang L,Xu J J 2013Opt.Express 21 32431
[10]Xing R,Jian S S 2016 IEEE Photonics Technol.Lett.28 2649
[11]Zhou X T,Zhang T,Chen L,Hong W,Li X 2014 J.Lightwave Technol.32 3597
[12]Liu Y,Zhong R B,Ding H,Liu S G 2017 Eur.Phys.J.D 7183
[13]He X Q,Ning T G,Lu S H,Zheng J J,Li J,Li R J,Pei L2018 Opt.Express 26 10109
[14]Xing R,Jian S S 2017 IEEE Photonics Technol.Lett.29 1643
[15]Liu J P,Zhai X,Xie F,Wang L L,Xia S X,Liu H J,Luo X,Shang X J 2017 J.Lightwave Technol.35 1971
[16]Xing R,Jian S S 2017 IEEE Photonics Technol.Lett.29 967
[17]Zhu B F,Ren G B,Yang Y,Gao Y X 2015 Plasmonics 10839
[18]Peng Y L,Xue W R,Wei Z Z,Li C Y 2018 Acta Opt.Sin.380223002(in Chinese)[彭艳玲,薛文瑞,卫壮志,李昌勇2018光学学报38 0223002]
[19]Peng Y L,Xue W R,Wei Z Z,Li C Y 2018 Acta Phys.Sin.67 038102(in Chinese)[彭艳玲,薛文瑞,卫壮志,李昌勇2018物理学报67 038102]
[20]Xing R,Jian S S 2016 IEEE Photonics Techol.Lett.28 2779
[21]Wei Z Z,Xue W R,Cheng X,Li C Y 2018 Acta Phys.Sin.67108101(in Chinese)[卫壮志,薛文瑞,程鑫,李昌勇2018物理学报67 108101]
[22]Hossein M B,Alexander B Y 2015 J.Phys.Condens.Matter27 185304
[23]Chen W D,Liu F 2009 Acta Elec.Sin.37 1624(in Chinese)[陈卫东,刘丰2009电子学报37 1624]
[24]Zhao J J,Tang M,Oh K H,Feng Z H,Zhao K,Liao R L,Fu S N,Shum P P,Liu D M 2017 Photon.Res.8 261
[25]Nejad R M,Tavakoli F,Wang L X,Guan X,Larochelle S,Rusch L A 2018 J.Lightwave Technol.36 3794
[26]Lee M S,Park B G,Cho I H,Lee J H 2012 IEEE Electron Device Lett.33 1613
[27]de Abajo F J G 2010 Rev.Mod.Phys.82 209
[28]Zhai L,Xue W R,Yang R C,Han L P 2015 Acta Opt.Sin.35 1123002(in Chinese)[翟利,薛文瑞,杨荣草,韩丽萍2015光学学报35 1123002]
[29]Nikitin A Y,Guinea F,García-Vidal F J,Martín-Moreno L2011 Phys.Rev.B 84 195446
[30]Yeh C 1962 J.Appl.Phys.33 3235
[31]D Erricolo,G Carluccio 2013 ACM Trans.Math.Soft.40 8
[32]He S L,Zhang X Z,He Y R 2013 Opt.Express 21 30664
[33]Ye L F,Sui K H,Liu Y H,Zhang M,Liu Q H 2018 Opt.Express 26 15935
[34]Ye S,Wang Z X,Sun C R,Dong C B,Wei B Z,Wu B L,Jian S S 2018 Opt.Express 26 23854
[35]Chen M,Sheng P C,Sun W,Cai J J 2016 Opt.Commun.37641