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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.
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Keywords:
- graphene /
- lossy stratified media /
- propagation matrix
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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.
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[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
Google Scholar
[2] Geim A K 2009 Science 324 1530
Google Scholar
[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
Google Scholar
[4] Alaee R, Farhat M, Rockstuhl C, Lederer F 2012 Opt. Express 20 28017
Google Scholar
[5] Fallahi A, Perruisseau-Carrier J 2012 Phys. Rev. B 86 195408
Google Scholar
[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
Google Scholar
[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
Google Scholar
[10] 张玉萍, 张洪艳, 尹贻恒, 刘陵玉, 张晓, 高营, 张会云 2012 61 047803
Google Scholar
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
Google Scholar
[11] Zhu X L, Yan W, Jepsen P U, Hansen O, Mortensen N A Xiao S S 2013 Appl. Phys. Lett. 102 131101
Google Scholar
[12] Pomar J L G, Alexey Y N, Luis M M 2013 ACS Nano 7 4988
Google Scholar
[13] Thongrattanasiri S, Koppens F H L, de Abajo F J G 2012 Phys. Rev. Lett. 108 047401
Google Scholar
[14] Ferreira A, Peres N M R, Ribeiro R M, Stauber T 2012 Phys. Rev. B 85 115438
Google Scholar
[15] Tian Y C, Jia W, Ren P W, Fan C Z 2018 Chin. Phys. B 27 124205
Google Scholar
[16] Jia W, Ren P W, Fan C Z, Tian Y C 2019 Chin. Phys. B 28 026102
Google Scholar
[17] Liu J T, Liu N H, Li J, Li X J, Huang J H 2012 Appl. Phys. Lett. 101 052104
Google Scholar
[18] Peres N M R, Bludov Y V 2013 EPL 101 58002
Google Scholar
[19] 谢凌云, 肖文波, 黄国庆, 胡爱荣, 刘江涛 2014 63 057803
Google Scholar
Xie L Y, Xiao W B, Huang G Q, Hu A R, Liu J T 2014 Acta Phys. Sin. 63 057803
Google Scholar
[20] Zhang H J, Zheng G G, Chen Y Y 2018 Chin. Phys. Lett. 35 038102
Google Scholar
[21] Lovat G 2012 IEEE Trans. Electromagn. Compat. 54 101
Google Scholar
[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
Google Scholar
[24] 孙旺, 李粮生, 张景, 殷红成 2018 雷达学报 7 67
Google Scholar
Sun W, Li L S, Zhang J, Yin H C 2018 J. Radars 7 67
Google Scholar
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