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设计了一种涂覆石墨烯的嵌套偏心空心圆柱的椭圆形电介质纳米线波导. 采用多极方法得到了波导所支持的最低阶的3个模式的传输特性, 即场分布、有效折射率的实部、传播长度和品质因数. 结果显示: 最低阶的3个模式都可由涂覆石墨烯的单根圆柱和椭圆柱所支持的最低阶模式合成. 通过改变波导的结构参数, 即圆柱的半径、椭圆柱半长轴、椭圆柱半短轴及两柱之间的最小间距, 可以在一定程度上调节模式的传输性能. 然而通过增大工作波长、费米能以及减小椭圆形电介质纳米线的介电常数, 可以明显改善模式的传输性能. 文中还与涂覆石墨烯的嵌套偏心空心圆柱的圆形电介质纳米线波导进行比较, 可以发现本文所设计的波导具有更优的传输性能. 这些结果都得到了有限元方法的验证. 本文设计的波导可以为涂覆石墨烯的嵌套偏心空心圆柱的椭圆形电介质纳米线波导的设计、制作及应用提供理论基础.An elliptical dielectric nanowire waveguide with a nested eccentric hollow cylinder is designed, the inner surface and the outer surface of the waveguide are both coated with graphene. The transmission characteristics of the three lowest order modes supported by the waveguide, i.e. the field distribution, the real part of the effective refractive index, the propagation length, and the quality factor, are obtained by multipole method. The results show that the three lowest order modes of the waveguide can be synthesized by the lowest order modes supported by a single cylinder and an elliptic cylinder coated with graphene. With the increase of the radius of the embedded cylinder, the three evaluation indexes of the waveguide increase slightly. The change of the semi major axis of the ellipse will slightly increase the real part of the effective refractive index, propagation length and quality factor of Mode 0 and Mode 1, while Mode2 will slightly decrease. When
$a = {{b}} = 170\;{\rm{nm}}$ , the performance of the circular nested circular waveguide becomes slightly worse than that of the structure in this paper. With the decrease of the semi minor axis of the ellipse, the real part of the effective refractive index, propagation length and quality factor for each of the three modes increase to a certain extent, and have the greatest impact on Mode 1. When${{b}} = {{a}} = 190\;{\rm{nm}}$ , the transmission performance of the circular nested circular waveguide is not so good as that of the structure in this paper. When the distance between the inner rod and the outer rod on the right side increases, the real part of the effective refractive index for each of the three modes decreases, the propagation length and quality factor of Mode 0 and Mode1 gradually decrease, while the propagation length of Mode 2 increases slightly, and the quality factor is basically unchanged. On the contrary, the mode transmission performance can be significantly improved by increasing the working wavelength and Fermi energy and reducing the dielectric constant of elliptical dielectric nanowires. Compared with the circular dielectric nanowire waveguide nested with an eccentric hollow cylinder coated with graphene, the waveguide designed in this paper can be found to have a good transmission performance. These results are verified by the finite element method. This waveguide can provide a theoretical basis for the design, fabrication and application of nested waveguides of the same type.-
Keywords:
- graphene /
- nanowires /
- waveguides /
- multipole method
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图 1 内外表面涂覆石墨烯的内嵌偏心空心圆柱的椭圆形电介质波导的横截面示意图
Fig. 1. The cross section of the waveguide. It was constructed with an elliptical dielectric rod embedded with eccentric hollow cylinder. The inner and outer surfaces of the waveguide were coated with graphene.
图 2 模式有效折射率实部(a)、传播长度(b)及品质因数(c)随
$\lambda $ 的变化关系图; λ = 6 μm (d), λ = 8 μm (e), λ = 10 μm (f)时Mode0的电场强度分布图Fig. 2. The diagram of the real part of effective refractive index (a), propagation length (b) and FOM (c) with
$\lambda $ ; the electric field intensity distribution of Mode0 when λ = 6 μm (d), λ = 8 μm (e), λ = 10 μm (f).
图 3 有效模式折射率实部(a)、传播长度(b)及品质因数(c)随费米能Ef 的变化关系图; Ef = 0.45 eV(d), Ef = 0.57 eV(e)和Ef = 0.69 eV(f)时Mode0的电场强度分布图
Fig. 3. The diagram of the real part of effective refractive index (a), propagation length (b) and FOM (c) with Ef ; the electric field intensity distribution of Mode0 when Ef = 0.45 eV(d), Ef = 0.57 eV(e) and Ef = 0.69 eV(f).
图 4 有效模式折射率实部(a)、传播长度(b)及品质因数(c)随
${\rho _0}$ 的变化关系图; ρ0 = 60 nm (d), ρ0 = 80 nm (e), ρ0 = 100 nm (f)时Mode0的电场强度分布图Fig. 4. The diagram of the real part of effective refractive index (a), propagation length (b) and FOM (c) with
${\rho _0}$ ; the electric field intensity distribution of Mode0 when ρ0 = 60 nm (d), ρ0 = 80 nm (e), ρ0 = 100 nm (f).
图 5 有效模式折射率实部(a), 传播长度(b)及品质因数(c)随
$a$ 的变化关系图; a = 170 nm (d), a = 190 nm (e), a = 210 nm (f)时Mode0的电场强度分布图Fig. 5. The variation diagram of the real part of effective refractive index (a), propagation length (b) and FOM (c) with
$a$ ; the electric field intensity distribution of Mode0 when a = 170 nm (d), a = 190 nm (e), a = 210 nm (f).
图 6 有效模式折射率实部(a), 传播长度(b)及品质因数(c)随
$ b $ 的变化关系图; b = 150 nm (d), b = 170 nm (e), b = 190 nm (f)时Mode0的电场强度分布图Fig. 6. The diagram of the real part of effective refractive index (a), propagation length (b) and FOM (c) with b; the electric field intensity distribution of Mode0 when b = 150 nm (d)、b = 170 nm (e)、b = 190 nm (f).
图 7 有效模式折射率实部(a)、传播长度(b)及品质因数(c)随
$d$ 的变化关系图; d = 55 nm (d), d = 75 nm (e), d = 95 nm (f)时Mode0的电场强度分布图Fig. 7. The diagram of the real part of effective refractive index (a), propagation length (b) and FOM (c) with
$d$ ; the electric field intensity distribution of Mode0 when d = 55 nm (d), d = 75 nm (e), d = 95 nm (f).
图 8 有效模式折射率实部(a)、传播长度(b)及品质因数(c)随
${\varepsilon _2}$ 的变化关系图;${\varepsilon _2} = 1.3$ (d),${\varepsilon _2} = 2.1$ (e),${\varepsilon _2} = 2.9$ (f)时Mode0的电场强度分布图Fig. 8. The variation diagram of the real part of effective refractive index (a), propagation length (b) and FOM (c) with
${\varepsilon _2}$ ; the electric field intensity distribution of Mode0 when${\varepsilon _2} = 1.3$ (d),${\varepsilon _2} = 2.1$ (e),${\varepsilon _2} = 2.9$ (f).
图 9 两种结构所支持的基模的有效折射率实部(a)、传播长度(b)和品质因数(c)随波长
$\lambda $ 的变化图Fig. 9. The diagram of the real part of effective refractive index (a), propagation length (b) and FOM (c) of the fundamental mode supported by the two structures with
$\lambda $ .
图 10 两种结构所支持的基模的有效折射率实部(a)、传播长度(b)和品质因数(c)随Ef变化的关系图
Fig. 10. The diagram of the real part of effective refractive index (a), propagation length (b) and FOM (c) of the fundamental mode supported by the two structures with Ef.
图 11 两种结构所支持的基模的有效折射率实部(a)、传播长度(b)和品质因数(c)随
${\varepsilon _2}$ 变化的关系图Fig. 11. The diagram of the real part of effective refractive index (a), propagation length (b) and FOM (c) of the fundamental mode supported by the two structures with
${\varepsilon _2}$ .表 1 波导所支持的3个最低阶模式的模式合成图、
${E_z}$ 分布图及$|E|$ 分布图Table 1. The pattern synthesis diagram, the distribution diagram of
${E_z}$ and$|E|$ of the three lowest modes supported by the waveguide.模式名称 模式的合成 ${E_z}$ $|E|$ Mode0 
Mode1 
Mode2 
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[1] Maier S A 2007 Plasmonics: fundamentals and applications (New York: Springer) p21
[2] Yang L K, Li P, Wang H C, Li Z P 2018 Chin. Phys. B 27 094216
[3] Jablan M, Buljan H, Soljacic M 2009 Phys. Rev. B 80 245435
Google Scholar
[4] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsov A A 2004 Science 306 666
Google Scholar
[5] Koppens F H L, Chang D E, de Abajo F J G 2011 Nano Lett. 11 3370
[6] de Abajo F J G 2014 ACS Photonics 1 135
Google Scholar
[7] Castro Neto A H, Guinea F, Peres N M R, Novoselov K S, Geim A K 2009 Rev. Mod. Phys. 81 109
Google Scholar
[8] Novoselov K S, Geim A K, Morozov S V, Jiang D, Katsnelson M I, Grigorieva I V, Dubonos S V, Firsov A A 2005 Nature 438 197
Google Scholar
[9] Geim A K, Novoselov K S 2007 Nat. Mater. 6 183
Google Scholar
[10] 滕达, 王凯, 李哲, 曹清, 唐亚楠, 赵永哲, 刘子怡, 张韵雯, 郭荣珍 2020 光学学报 40 0623002
Google Scholar
Teng D, Wang K, Li Z, Cao Q, Tang Y N, Zhao Y Z, Liu Z Y, Zhang Y W, Guo R Z 2020 Acta Opt. Sin. 40 0623002
Google Scholar
[11] Li Z Q, Henriksen E A, Jiang Z, Hao Z, Martin M C, Kim P, Stormer H L, Basov D N 2008 Nat. Phys. 4 532
Google Scholar
[12] Teng D, Wang K 2021 Nanomaterials 11 210
Google Scholar
[13] 秦康 2016 硕士学位论文 (浙江: 中国计量大学)
Qin K 2016 M. S. Thesis (Zhejiang: China Jiliang University) (in Chinese)
[14] Wen B, Cao M S, Lu M M, Cao W Q, Shi H L, Liu J, Wang X X, Jin H B, Fang X Y, Wang W Z, Yuan J 2014 Adv. Mater. 26 3484
Google Scholar
[15] Zhang M, Cao M S, Shu J C, Cao W Q, Li L, Yuan J 2021 Mat. Sci. Eng. R. 145 100627
Google Scholar
[16] Vakil A, Engheta N 2011 Science 332 1291
Google Scholar
[17] Gao Y X, Ren G B, Zhu B F, Liu H Q, Lian Y D, Jian S S 2014 Opt. Express 22 24322
Google Scholar
[18] Yang J F, Yang J J, Deng W, Mao F C, Huang M 2015 Opt. Express 23 32289
Google Scholar
[19] 彭艳玲, 薛文瑞, 卫壮志, 李昌勇 2018 物理学报 67 038102
Google Scholar
Peng Y L, Xue W R, Wei Z Z, Li C Y 2018 Acta Phys. Sin. 67 038102
Google Scholar
[20] Teng D, Wang K, Li Z, Zhao Y Z, Zhao G, Li H Y, Wang H 2019 Appl. Sci-Basel. 9 2351
Google Scholar
[21] Cheng X, Xue W R, Wei Z Z, Dong H Y, Li C Y 2019 Opt. Commun. 452 467
Google Scholar
[22] White T P, Kuhlmey B T, McPhedran R C, Maystre D, Renversez G, Sterke C M, Botten L C 2002 J. Opt. Soc. Am. B 19 2322
Google Scholar
[23] Bao Q L, Loh K P 2012 ACS Nano 6 3677
Google Scholar
[24] Bonaccorso F, Sun Z, Hasan T, Ferrari A C 2010 Nat. Photonics 4 611
Google Scholar
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Google Scholar
Zhai L, Xue W R, Yang R C, Han L P 2015 Acta Opt. Sin. 35 1123001
Google Scholar
[26] Schedin F, Geim A K, Morozov S V, Hill E W, Blake P, Katsnelson M I, Novoselov K S 2007 Nat. Mater. 6 652
Google Scholar
[27] Zhang J, Boyd A, Tselev A, Paranjape M, Barbara P 2006 Appl. Phys. Lett. 88 123112
Google Scholar
[28] 郭亚楠, 薛文瑞, 张文梅 2009 物理学报 58 4168
Google Scholar
Guo Y N, Xue W R, Zhang W M 2009 Acta Phys. Sin. 58 4168
Google Scholar
[29] 邓新华, 刘江涛, 袁吉任, 王同标 2015 物理学报 64 057801
Google Scholar
Deng X H, Liu J T, Yuan J R, Wang T B 2015 Acta Phys. Sin. 64 057801
Google Scholar
[30] Nikitin A Y, Guinea F, Garcia-Vidal F J, Martin-Moreno L 2011 Phys. Rev. B 84 195446
Google Scholar
[31] Chen K, Zhou X, Cheng X, Qiao R X, Cheng Y, Liu C, Xie Y D, Yu W T, Yao F R, Sun Z P, Wang F, Liu K H, Liu Z F 2019 Nat. Photonics 13 754
Google Scholar
[32] Erricolo D, Carluccio G 2013 ACM Trans. Math. Soft. 40 153
[33] Teng D, Wang K, Li Z, Zhao Y Z 2019 Opt. Express 27 12458
Google Scholar
[34] Xing R, Jian S S 2017 IEEE Photonic Tech. L. 29 1643
Google Scholar
[35] 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
Google Scholar
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