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Propagation matrix for lossy stratified medium containing graphene sheet

Wang Fei Wei Bing

Propagation matrix for lossy stratified medium containing graphene sheet

Wang Fei, Wei Bing
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  • In this paper, a propagation matrix method for lossy layered medium with conductive interfaces is presented. Firstly, on the basis of phase matching principle, an approach to calculating the real and imaginary part of wave vector in a lossy layered medium is given for the case of oblique incident plane electromagnetic wave. Since the direction of real and imaginary part of wave vector are different, the plane wave propagating in lossy dielectric layers is inhomogeneous, which extends the traditional propagation matrix method and makes it suitable for the complex lossy medium. Then, the propagation matrix across graphene interface is deduced by using the electromagnetic field boundary conditions, and the analytical expression of the reflection and transmission coefficient for " infinite thin” graphene layer are given. Finally, the propagation matrix of lossy layered medium with conductive interface is obtained by embedding graphene interface into the layered medium, which can be used for fast analyzing the reflection, transmission and propagation of plane wave in composite structure of layered medium and conductive interface. The validity of the proposed method is demonstrated by calculating the single-layered shielding effectiveness of grapheme. The effects of graphene coating on the reflection, transmission and absorption of plane wave in half-space medium and one-dimensional photonic crystal are also investigated. The results show that the graphene layer can enhance surface reflection and optical absorption.
      Corresponding author: Wang Fei, wfei79@163.com
    [1]

    Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsove A A 2004 Science 306 666

    [2]

    Geim A K 2009 Science 324 1530

    [3]

    Sensale-Rodriguez B, Yan R, Kelly M, Fang T, Tahy K, Hwang W S, Jena D, Liu L, Xing H G 2012 Nature Commun. 3 780

    [4]

    Alaee R, Farhat M, Rockstuhl C, Lederer F 2012 Opt. Express 20 28017

    [5]

    Fallahi A, Perruisseau-Carrier J 2012 Phys. Rev. B 86 195408

    [6]

    Sensale-Rodriguez B, Yan R, Rafique S, Zhu M, Li W, Liang X, Gundlach D, Protasenko V, Kelly M M, Jena D, Liu L, Xing H G 2012 Nano Lett. 12 4518

    [7]

    Fu M X, Zhang Y 2013 JECT 11 352

    [8]

    Lee S H, Choi M, Kim T T, Lee S, Liu M, Yin X, Choi H K, Lee S S, Choi C G, Choi S Y, Zhang X, Min B 2012 Nat. Mater. 11 936

    [9]

    Zuo Z G, Wang P, Ling F R, Liu J S, Yao J Q 2013 Chin. Phys. B 22 097304

    [10]

    张玉萍, 张洪艳, 尹贻恒, 刘陵玉, 张晓, 高营, 张会云 2012 物理学报 61 047803

    Zhang Y P, Zhang H Y, Yin Y H, Liu L Y, Zhang X, Gao Y, Zhang H Y 2012 Acta Phys. Sin. 61 047803

    [11]

    Zhu X L, Yan W, Jepsen P U, Hansen O, Mortensen N A Xiao S S 2013 Appl. Phys. Lett. 102 131101

    [12]

    Pomar J L G, Alexey Y N, Luis M M 2013 ACS Nano 7 4988

    [13]

    Thongrattanasiri S, Koppens F H L, de Abajo F J G 2012 Phys. Rev. Lett. 108 047401

    [14]

    Ferreira A, Peres N M R, Ribeiro R M, Stauber T 2012 Phys. Rev. B 85 115438

    [15]

    Tian Y C, Jia W, Ren P W, Fan C Z 2018 Chin. Phys. B 27 124205

    [16]

    Jia W, Ren P W, Fan C Z, Tian Y C 2019 Chin. Phys. B 28 026102

    [17]

    Liu J T, Liu N H, Li J, Li X J, Huang J H 2012 Appl. Phys. Lett. 101 052104

    [18]

    Peres N M R, Bludov Y V 2013 EPL 101 58002

    [19]

    谢凌云, 肖文波, 黄国庆, 胡爱荣, 刘江涛 2014 物理学报 63 057803

    Xie L Y, Xiao W B, Huang G Q, Hu A R, Liu J T 2014 Acta Phys. Sin. 63 057803

    [20]

    Zhang H J, Zheng G G, Chen Y Y 2018 Chin. Phys. Lett. 35 038102

    [21]

    Lovat G 2012 IEEE Trans. Electromagn. Compat. 54 101

    [22]

    葛德彪, 魏兵 2011 电磁波理论 (北京: 科学出版社) 第32, 56−65页

    Ge D B, Wei B 2011 Electromagnetic Wave Theory (Beijing: Science Press) pp32, 56−65 (in Chinese)

    [23]

    George W H 2008 J. Appl. Phys. 103 064302

    [24]

    孙旺, 李粮生, 张景, 殷红成 2018 雷达学报 7 67

    Sun W, Li L S, Zhang J, Yin H C 2018 J. Radars 7 67

  • 图 1  分层介质

    Figure 1.  Stratified medium.

    图 2  石墨烯电导率和SE (a) 电导率实虚部; (b) SE

    Figure 2.  Graphene conductivity and shielding effectiveness: (a) Real and imaginary parts of conductivity; (b) SE.

    图 3  石墨烯单层的反射和透射系数模值 (a) TE模; (b) TM模

    Figure 3.  Modulus of reflective and transmittance coefficients of a graphene sheet:(a) TE mode; (b) TM mode.

    图 4  含石墨烯涂层CdTe半空间的反透射系数模值(TE模) (a) 反射系数; (b) 透射系数

    Figure 4.  Modulus of reflective and transmittance coefficients of CdTe half-space containing graphene coating (TE mode): (a) Reflective coefficient; (b) transmittance coefficient.

    图 5  含石墨烯涂层CdTe半空间的反透射系数模值(TM模) (a) 反射系数; (b) 透射系数

    Figure 5.  Modulus of reflective and transmittance coefficients of CdTe half-space containing graphene coating (TM mode): (a) Reflective coefficient; (b) transmittance coefficient.

    图 6  CdTe半空间的反透射光场(TE模) (a) 无石墨烯涂层; (b) 含石墨烯涂层

    Figure 6.  Optical field of reflection and transmission coefficients of CdTe half-space (TE mode): (a) Without graphene coating; (b) with graphene coating.

    图 7  CdTe半空间的反透射光场(TM模) (a) 无石墨烯涂层; (b) 含石墨烯涂层

    Figure 7.  Optical field of reflection and transmission coefficients of CdTe half-space (TM mode): (a) Without graphene coating; (b) with graphene coating.

    图 8  Si/SiO2周期结构型1DPC

    Figure 8.  Si/SiO2 1DPC with periodic structure.

    图 9  含石墨烯涂层Si/SiO2周期结构1DPC的反透射系数 (a) 反射系数; (b) 透射系数

    Figure 9.  Modulus of reflective and transmittance coefficients of Si/SiO2 1DPC containing graphene sheet: (a) Reflective coefficient; (b) transmittance coefficient.

    图 10  含石墨烯界面Si/SiO2周期结构1DPC的吸收率

    Figure 10.  Absorbance of Si/SiO2 1DPC containing graphene sheet.

    图 11  含石墨烯涂层Si/SiO2周期结构1DPC的吸收率(TE模) (a) 无涂层; (b) 表面涂层; (c) 底层涂层

    Figure 11.  Contour plots of the absorbance of the Si/SiO2 1DPC as a function of the light frequency and the incident angles for the TE mode: (a) Without graphene sheet; (b) graphene sheet on the top; (c) graphene sheet on the bottom.

    图 12  含石墨烯涂层Si/SiO2周期结构1DPC的吸收率(TM模) (a) 无涂层; (b) 表面涂层; (c) 底层涂层

    Figure 12.  Contour plots of the absorbance of the Si/SiO2 1DPC as a function of the light frequency and the incident angles for the TE mode: (a) Without graphene sheet; (b) graphene sheet on the top; (c) graphene sheet on the bottom.

  • [1]

    Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V, Firsove A A 2004 Science 306 666

    [2]

    Geim A K 2009 Science 324 1530

    [3]

    Sensale-Rodriguez B, Yan R, Kelly M, Fang T, Tahy K, Hwang W S, Jena D, Liu L, Xing H G 2012 Nature Commun. 3 780

    [4]

    Alaee R, Farhat M, Rockstuhl C, Lederer F 2012 Opt. Express 20 28017

    [5]

    Fallahi A, Perruisseau-Carrier J 2012 Phys. Rev. B 86 195408

    [6]

    Sensale-Rodriguez B, Yan R, Rafique S, Zhu M, Li W, Liang X, Gundlach D, Protasenko V, Kelly M M, Jena D, Liu L, Xing H G 2012 Nano Lett. 12 4518

    [7]

    Fu M X, Zhang Y 2013 JECT 11 352

    [8]

    Lee S H, Choi M, Kim T T, Lee S, Liu M, Yin X, Choi H K, Lee S S, Choi C G, Choi S Y, Zhang X, Min B 2012 Nat. Mater. 11 936

    [9]

    Zuo Z G, Wang P, Ling F R, Liu J S, Yao J Q 2013 Chin. Phys. B 22 097304

    [10]

    张玉萍, 张洪艳, 尹贻恒, 刘陵玉, 张晓, 高营, 张会云 2012 物理学报 61 047803

    Zhang Y P, Zhang H Y, Yin Y H, Liu L Y, Zhang X, Gao Y, Zhang H Y 2012 Acta Phys. Sin. 61 047803

    [11]

    Zhu X L, Yan W, Jepsen P U, Hansen O, Mortensen N A Xiao S S 2013 Appl. Phys. Lett. 102 131101

    [12]

    Pomar J L G, Alexey Y N, Luis M M 2013 ACS Nano 7 4988

    [13]

    Thongrattanasiri S, Koppens F H L, de Abajo F J G 2012 Phys. Rev. Lett. 108 047401

    [14]

    Ferreira A, Peres N M R, Ribeiro R M, Stauber T 2012 Phys. Rev. B 85 115438

    [15]

    Tian Y C, Jia W, Ren P W, Fan C Z 2018 Chin. Phys. B 27 124205

    [16]

    Jia W, Ren P W, Fan C Z, Tian Y C 2019 Chin. Phys. B 28 026102

    [17]

    Liu J T, Liu N H, Li J, Li X J, Huang J H 2012 Appl. Phys. Lett. 101 052104

    [18]

    Peres N M R, Bludov Y V 2013 EPL 101 58002

    [19]

    谢凌云, 肖文波, 黄国庆, 胡爱荣, 刘江涛 2014 物理学报 63 057803

    Xie L Y, Xiao W B, Huang G Q, Hu A R, Liu J T 2014 Acta Phys. Sin. 63 057803

    [20]

    Zhang H J, Zheng G G, Chen Y Y 2018 Chin. Phys. Lett. 35 038102

    [21]

    Lovat G 2012 IEEE Trans. Electromagn. Compat. 54 101

    [22]

    葛德彪, 魏兵 2011 电磁波理论 (北京: 科学出版社) 第32, 56−65页

    Ge D B, Wei B 2011 Electromagnetic Wave Theory (Beijing: Science Press) pp32, 56−65 (in Chinese)

    [23]

    George W H 2008 J. Appl. Phys. 103 064302

    [24]

    孙旺, 李粮生, 张景, 殷红成 2018 雷达学报 7 67

    Sun W, Li L S, Zhang J, Yin H C 2018 J. Radars 7 67

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    [6] Zhang Bao-Lei, Wang Jia-Xu, Xiao Ke, Li Jun-Yang. Quasi-static finite element calculation of interaction between graphene and nanoprobe. Acta Physica Sinica, 2014, 63(15): 154601. doi: 10.7498/aps.63.154601
    [7] Zhang Yu-Ping, Liu Ling-Yu, Chen Qi, Feng Zhi-Hong, Wang Jun-Long, Zhang Xiao, Zhang Hong-Yan, Zhang Hui-Yun. Effect of cooling of electron-hole plasma in electrically pumped graphene layer structures with split gates. Acta Physica Sinica, 2013, 62(9): 097202. doi: 10.7498/aps.62.097202
    [8] Zu Feng-Xia, Zhang Pan-Pan, Xiong Lun, Yin Yong, Liu Min-Min, Gao Guo-Ying. Design and electronic transport properties of organic thiophene molecular rectifier with the graphene electrodes. Acta Physica Sinica, 2017, 66(9): 098501. doi: 10.7498/aps.66.098501
    [9] Guo Wei-Ling, Deng Jie, Wang Jia-Lu, Wang Le, Tai Jian-Peng. GaN-based light emitting diode with graphene/indium antimony oxide composite transparent electrode. Acta Physica Sinica, 2019, 68(24): 247303. doi: 10.7498/aps.68.20190983
    [10] Qiao Wen-Tao, Gong Jian, Zhang Li-Wei, Wang Qin, Wang Guo-Dong, Lian Shu-Peng, Chen Peng-Hui, Meng Wei-Wei. Propagation properties of the graphene surface plasmon in comb-like waveguide. Acta Physica Sinica, 2015, 64(23): 237301. doi: 10.7498/aps.64.237301
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  • Received Date:  28 May 2019
  • Accepted Date:  29 September 2019
  • Available Online:  27 November 2019
  • Published Online:  01 December 2019

Propagation matrix for lossy stratified medium containing graphene sheet

    Corresponding author: Wang Fei, wfei79@163.com
  • 1. School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China
  • 2. Collaborative Innovation Center of Information Sensing and Understanding at Xidian University, Xi’an 710071, China

Abstract: In this paper, a propagation matrix method for lossy layered medium with conductive interfaces is presented. Firstly, on the basis of phase matching principle, an approach to calculating the real and imaginary part of wave vector in a lossy layered medium is given for the case of oblique incident plane electromagnetic wave. Since the direction of real and imaginary part of wave vector are different, the plane wave propagating in lossy dielectric layers is inhomogeneous, which extends the traditional propagation matrix method and makes it suitable for the complex lossy medium. Then, the propagation matrix across graphene interface is deduced by using the electromagnetic field boundary conditions, and the analytical expression of the reflection and transmission coefficient for " infinite thin” graphene layer are given. Finally, the propagation matrix of lossy layered medium with conductive interface is obtained by embedding graphene interface into the layered medium, which can be used for fast analyzing the reflection, transmission and propagation of plane wave in composite structure of layered medium and conductive interface. The validity of the proposed method is demonstrated by calculating the single-layered shielding effectiveness of grapheme. The effects of graphene coating on the reflection, transmission and absorption of plane wave in half-space medium and one-dimensional photonic crystal are also investigated. The results show that the graphene layer can enhance surface reflection and optical absorption.

    • 2004年, 世界上最薄的半导体材料—石墨烯, 由Novoselov和Geim的团队采用微机械剥离法制备成功[1]. 石墨烯是由单层碳原子构成的二维晶体结构, 具有高电子迁移率、低电阻率、光学透明性、高灵活性及结构稳定等特性[2], 这使得其在光学和光电传感等方面具有广泛的应用前景[3-6], 同时也被认为非常适合于太赫兹光学设备[7-10]. 石墨烯非常薄, 厚度只有约0.34 nm, 这导致其光吸收率很低, 应用受到限制. 为此, 常采用周期掺杂、表面等离子体、光学微腔等方法来增强光学吸收[11-16], 其中利用光子晶体增强石墨烯光吸收是一种常用方法. 2012年, Liu等[17]利用石墨烯和间隔层在一维光子晶体(one-dimensional photonic crystal, 1DPC)表面构造表面缺陷, 导致光的局域化, 使可见光光吸收被增强约4倍. 2013年, Peres和Bludov[18]利用1DPC使石墨烯的THz吸收增强了超过3倍. 2014年, Xie等[19]分析了1DPC表面石墨烯在应力赝磁场作用下的THz吸收. 2018年, Zhang等[20]设计了一种高对比光栅耦合波导结构, 提高了石墨烯在近红外和可见光频段的光吸收.

      1DPC本质上是一个周期分层介质, 而单层的石墨烯可以看做是“无限薄”的[21], 因此可以把石墨烯和1DPC构造的结构看成一个含石墨烯分界面的分层介质. 传播矩阵方法[22]是一种分析分层介质中光或电磁波传播特性的解析理论方法, 可以直接计算分层介质对平面电磁波的反透射等. 根据电磁理论, 当分层介质中某层为有耗介质时, 该层中的波矢量为复数, 而在斜入射情形下, 考虑相位匹配, 该层中波矢量的实部和虚部并不平行, 即相位和振幅的传播方向不同, 此时该层中传播的应为非均匀平面波. 本文首先根据相位匹配条件, 推导了有耗分层介质中复数波矢量的实部和虚部, 给出传播矩阵中所需要的纵向波矢量计算式; 然后考虑石墨烯表面电导率, 根据边界条件推导了跨越石墨烯分界面的传播矩阵; 最终, 使得传播矩阵方法可用于解析分析任意入射角情形下含导电界面有耗分层介质的反透射和电波传播特性. 本文中时谐因子取$\exp \left( { - {\rm{j}}\omega t} \right)$.

    2.   理论公式
    • 设平面波从真空入射到$n$层分层介质, 如图1所示, 入射角为$\theta $. 考虑一般情形, 设各层介质的相对介电系数和磁导系数${\varepsilon _{{\rm{r}}l}}, {\mu _{{\rm{r}}l}}\;(l = 1, 2, \cdots, n)$皆为复数,

      Figure 1.  Stratified medium.

      此时, 介质有耗, 介质层中的波矢量${{{k}}_l}$也为复数. 下面来具体分析介质层中的复数波矢量${{{k}}_l}$.

      各分层中的复数波矢量${{{k}}_l}$可以写为

      其中${{{k}}_{{\rm{R}}l}}$${{{k}}_{{\rm{I}}l}}$分别表示${{{k}}_l}$的实部和虚部. 如图1所示, 入射面为$xoz$平面, 故有

      根据界面处的相位匹配(波矢量切向分量连续)条件可得

      ${{{k}}_l}$的虚部${{{k}}_{{\rm{I}}l}}$只有$z$分量, 实部${{{k}}_{{\rm{R}}l}}$$x$分量在每个区域都相同, 都等于区域0中入射波矢量${{{k}}_0}$$x$分量${k_{0 x}}$. 由此可以看到, 平面波斜入射有耗分层介质时, 介质中波矢量的实部和虚部方向不平行, 虚部只有纵向分量. 由于波矢量实部代表相位传播方向, 虚部代表振幅传播方向, 可知此时平面波是非均匀的, 波矢量为

      由电磁波理论可知在有耗介质中[22]

      而由(2)和(5)式可得

      则根据(6)和(7)式得

      求解(8)式得到

      综上所述, 考虑到相位匹配, 有耗分层介质中平面波的复数波矢量可通过(10)式计算. 计算得到波矢量虚部${{{k}}_{{\rm{I}}l}}$沿$z$方向, 所以波在介质中沿深度方向衰减. 波矢量实部${{{k}}_{{\rm{R}}l}}$的方向代表波传播方向, 介质中的折射角可以依此计算.

    • 对整个分层介质, 在图1中从入射点${M_0}$到透射点${N_{\rm{t}}}$的传播用前向传播矩阵表示为

      其中${B_0}$${M_0}$点处下行波幅值. 整个分层介质总的前向传播矩阵为[22]

      其中${{{U}}_l}$${{{V}}_l}$分别是跨越$l-l + 1$层界面和第$l$层中的传播矩阵. 对有耗介质, 传播矩阵计算时${\varepsilon _l}, {\mu _l}$取复数, 另外要特别注意${k_{lz}}$, 应该按照前述方法计算.

    • 单层的石墨烯可以看做是“无限薄”的[21], 石墨烯薄层两个侧面的表面电导率为${\sigma _{\rm{g}}}\left( {\omega, {\mu _{\rm c}}, \varGamma, T} \right)$, 其中$\omega $, ${\mu _{\rm c}}$, $\varGamma $$T$分别是角频率、化学势、散射率和室温. 由Kobu公式得到石墨烯表面电导率表达式为${\sigma _{\rm{g}}} = {\sigma _{{\rm{intra}}}} + {\sigma _{{\rm{inter}}}}$, ${\sigma _{{\rm{intra}}}}$取决于能带内的能量, 而${\sigma _{{\rm{inter}}}}$取决于能带间的能量. 因为能带内的能量占能量主要部分, 所以只考虑${\sigma _{{\rm{intra}}}}$这一项, 而忽略${\sigma _{{\rm{inter}}}}$项.

      ${\sigma _{{\rm{intra}}}}$可用Drude表达式表示为

      式中

      为直流电导率, 其中e是电子电量, ${k_{\rm{B}}}$是玻尔兹曼常数, $\hbar $是约化普朗克常量; $\tau = {1 / {\left( {2\varGamma } \right)}}$是电子弛豫时间.

      设在分层介质区域$l$$l + 1$的分界面$z = {z_l}$处有一层“无限薄”石墨烯层, 如图1所示, 根据电磁理论, 在介质分界面两侧的电磁场满足边界条件, 所以在分界面$z = {z_l}$处有

      其中${{J}}_{\rm{e}}^{\rm{s}}$${{J}}_{\rm{m}}^{\rm{s}}$表示表面电流和磁流. 由于该界面为“无限薄”石墨烯层, 则此处[23]

      联合(15)式可得

      首先考虑TE波情形, 区域l中的总场可写为[22]

      式中${A_l}$${B_l}$分别代表区域$l$中的上行波和下行波幅值. 在区域$l$$l + 1$的分界面$z = {z_l}$处, 根据边界条件(17)和(19)式可得

      其中

      求解并整理(21)和(22)式可得

      其中

      是穿越$z = {z_l}$处石墨烯分界面从${M_l}$点到${N_{l + 1}}$点的TE波前向传播矩阵.

      对TM波情形, 区域$l$中的总场可写为

      在区域$l$$l + 1$的分界面$z = {z_l}$处, 根据边界条件(17)和(18)式可得

      其中

      求解并整理(27)和(28)式可得

      其中

      是穿越$z = {z_l}$处石墨烯分界面从${M_l}$点到${N_{l + 1}}$点的TM波前向传播矩阵.

      实际上, 上述方法中所涉及的石墨烯界面可拓展到表面电导率为${\sigma _{\rm{g}}}$的导电界面.

      综上, 若分层介质中某界面含“无限薄”导电层, 则可以通过将总传播矩阵${{{V}}_{{\rm{t0}}}}$((12)式)中该界面处的跨界面传播矩阵${{{U}}_{l = 0 - n}}$替换为${{U}}_{\rm{g}}^{{\rm{TE/TM}}}$((25)和(31)式), 从而得到含导电界面的分层介质传播矩阵. 相反, 若令${\sigma _{\rm{g}}} = 0$, 则${{U}}_{\rm{g}}^{{\rm{TE/TM}}}$又可以退化为${{{U}}_{l = 0 - n}}$, 即为一般电介质分界面.

      注意到, 由于石墨烯界面的导电特性, 此处TE和TM波的边界条件不对偶, 因而其跨界面传播矩阵在TE和TM波情形下并不对偶.

    • 设“无限薄”石墨烯上、下半空间分别为均匀介质1和2, 在(24)和(30)式中令$l = 1$${A_2} = 0$, 结合(25)式、(31)式和反透射系数定义可以得到“无限薄”石墨烯界面的反透射系数

      当垂直入射时, ${k_{1 z}} = {k_1} = {k_0}\sqrt {{\varepsilon _{{\rm{r1}}}}{\mu _{{\rm{r1}}}}}, {k_{2 z}} = {k_2} = $${k_0}\sqrt {{\varepsilon _{{\rm{r2}}}}{\mu _{{\rm{r2}}}}} $, 由(32)和(33)式可得“无限薄”石墨烯界面的垂直反透射系数

      结果和文献[23]一致. 注意到, 在(32), (33)和(34)式中若令${\sigma _{\rm{g}}} = 0$, 则可以自然过渡为一般介质分界面的Fresnel公式.

    3.   算 例
    • 例1 石墨烯单层的屏蔽效率

      设各向同性石墨烯层的参数为T = 300 K, $ {\mu _{\rm{c}}} = 0, \varGamma = 0.11$ MeV, 应用传播矩阵计算其反透射系数, 进而得到屏蔽效率(shielding effectiveness, SE)[21]随频率变化曲线如图2(b)所示(图2(a)为该石墨烯层电导率实虚部与频率的关系曲线). SE的计算结果和文献[21]结果相符, 同时由图2可见, 石墨烯层对TM模的SE低于对TE模的.

      Figure 2.  Graphene conductivity and shielding effectiveness: (a) Real and imaginary parts of conductivity; (b) SE.

      以下算例中石墨烯层的参数取

      例2 空气中石墨烯单层的反透射

      设平面电磁波入射真空中一单层石墨烯层, 根据(32)和(33)式可计算得到该石墨烯层的反透射系数, 计算结果如图3所示. 图3(a)图3(b)分别是TE和TM波的反透射系数模值, 其中实线、短划线、点线分别代表入射角为${0^ \circ }$, ${30^ \circ }$, ${60^ \circ }$时的反射系数, 方形、圆形、星形分别代表入射角为${0^ \circ }$, ${30^ \circ }$, ${60^ \circ }$时的透射系数. 可以看到, 随着频率增大, 该石墨烯层的反射系数减小而透射系数增大; 随着入射角增大, TE波反射增强透射减弱, 而TM波相反.

      Figure 3.  Modulus of reflective and transmittance coefficients of a graphene sheet:(a) TE mode; (b) TM mode.

      例3 涂覆石墨烯层的半空间有耗介质(碲化镉CdTe)界面

      设在碲化镉(CdTe)表面上涂覆着一层石墨烯, CdTe的相对介电系数为[24]

      其中${\varepsilon _\infty }$, ${\omega _{{\rm{LO}}}}$, ${\omega _{{\rm{TO}}}}$, $\gamma $分别是CdTe的高频极限相对介电常数、纵波光学声子角频率、横波光学声子角频率、碰撞角频率. 本算例取文献[24]给出的测量值

      计算得到CdTe的禁带为

      根据(32)和(33)式可计算该CdTe界面的反透射系数, 计算结果如图4图5所示. 图4图5分别是TE波和TM波情形, 图(a)和(b)分别是反射、透射系数模值, 各图中实线、短划线、点线分别代表入射角为${0^ \circ }$, ${30^ \circ }$, ${60^ \circ }$时不含石墨烯涂层时的情形, 方形、圆形、星形分别代表入射角为${0^ \circ }$, ${30^ \circ }$, ${60^ \circ }$时含石墨烯涂层时的情形. 由图4图5可见, 石墨烯的高导电特性使得其涂覆CdTe表面后, 反射增强而透射减弱, 并且在禁带以下的低频段尤为明显.

      Figure 4.  Modulus of reflective and transmittance coefficients of CdTe half-space containing graphene coating (TE mode): (a) Reflective coefficient; (b) transmittance coefficient.

      Figure 5.  Modulus of reflective and transmittance coefficients of CdTe half-space containing graphene coating (TM mode): (a) Reflective coefficient; (b) transmittance coefficient.

      图6图7是该CdTe半空间界面TE和TM入射波的反透射光场分布, 图(a)和(b)分别是界面不含和含有石墨烯涂层时的情形, 从图6图7可见, 石墨烯涂层不改变反透射角, 但增强了表面反射, 起到了一定屏蔽作用.

      Figure 6.  Optical field of reflection and transmission coefficients of CdTe half-space (TE mode): (a) Without graphene coating; (b) with graphene coating.

      Figure 7.  Optical field of reflection and transmission coefficients of CdTe half-space (TM mode): (a) Without graphene coating; (b) with graphene coating.

      例4 涂覆石墨烯层的Si/SiO2周期结构型一维光子晶体

      有一含SiO2缺陷层的Si/SiO2周期结构型1DPC, 如图8所示. 该周期结构周期数$N = 8$, 介质a (Si)和介质b (SiO2)参数为

      Figure 8.  Si/SiO2 1DPC with periodic structure.

      Si, SiO2层和缺陷层的厚度分别为${d_{\rm{a}}} = {{\lambda _0}} / {\sqrt {{{\varepsilon}'_{{\rm{ra}}}}} },$$ {d_{\rm{b}}} = {{{\lambda _0}} / {\sqrt {{{\varepsilon}'_{{\rm{rb}}}}} }}, \;{d_{\rm{s}}} = {{{d_{\rm{b}}}}/ {0.85}}$, 其中${\lambda _0} = $ 20 μm.

      图9是平面波垂直入射时该光子晶体的反透射系数, 其中圆圈表示无石墨烯涂层时的反透射系数, 作为对比用虚线给出了无耗情形(${\varepsilon ''_{{\rm{rb}}}} = 0$)时的反透射系数, 可以看到, 有耗时反透射都有所减弱. 图9中实心和空心星形分别代表石墨烯涂层位于光子晶体表面(第1个界面位置)和底层(最后一个界面位置)时的情形, 可以看到, 在禁带下方低频段, 石墨烯涂层的加入使反射增强, 透射减弱, 在禁带内位于底层的石墨烯涂层对反射系数几乎没有影响而位于表面的石墨烯涂层对反射系数影响较大, 使之减弱.

      Figure 9.  Modulus of reflective and transmittance coefficients of Si/SiO2 1DPC containing graphene sheet: (a) Reflective coefficient; (b) transmittance coefficient.

      图10是平面波垂直入射时该光子晶体的吸收率曲线, 石墨烯涂层增强了光子晶体的吸收, 而且当石墨烯涂层位于表面时, 石墨烯和间隔层构成的表面缺陷导致光的局域化, 使得THz光吸收被增强[18], 在禁带及以上频段, 吸收率显著提高; 当石墨烯涂层位于底层时, 吸收率的提高不明显, 而且主要集中在禁带下方低频段.

      Figure 10.  Absorbance of Si/SiO2 1DPC containing graphene sheet.

      图11图12分别是TE和TM波情形下, 该光子晶体的吸收率随频率和入射角变化的伪色彩图, 其中子图(a), (b)和(c)分别是无石墨烯涂层、涂层位于表面和底层时的情形, 整体上看, 位于表面的涂层对光吸收的增强效应更为明显, 而且对TE波的吸收要强于TM波.

      Figure 11.  Contour plots of the absorbance of the Si/SiO2 1DPC as a function of the light frequency and the incident angles for the TE mode: (a) Without graphene sheet; (b) graphene sheet on the top; (c) graphene sheet on the bottom.

      Figure 12.  Contour plots of the absorbance of the Si/SiO2 1DPC as a function of the light frequency and the incident angles for the TE mode: (a) Without graphene sheet; (b) graphene sheet on the top; (c) graphene sheet on the bottom.

    4.   结 论
    • 根据相位匹配, 讨论给出了平面波斜入射分层介质时, 有耗层中波矢量实部和虚部的计算方法. 有耗层中复数波矢量的实部和虚部不平行, 传播的平面波非均匀. 代表波振幅传播方向的波矢量虚部沿纵向方向, 即波沿深度方向衰减. 波矢量实部代表相位传播方向, 折射角依此计算. 依据边界条件, 推导了跨越石墨烯界面的传播矩阵, 以及“无限薄”石墨烯层的反透射系数解析式. 推广了传播矩阵方法, 使之可应用于包括含导电界面的有耗分层介质反透射和电波传播特性的计算和分析, 为分层介质与导电界面复合结构的设计和验证提供了支撑.

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