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基于麦克斯韦旋度方程, 将磁电超材料板中的电元件和磁元件分别等效为面电流和面磁流, 通过计算这些周期性面电流和面磁流在某个磁电超材料板上产生的总电场和总磁场, 获得了关于面磁流密度和面电流密度的两个方程, 进而推导出了周期性磁电超材料折射率与磁元件的磁导率、 电元件的介电常数和空间色散项之间关系的解析公式. 与传统的折射率计算公式不同, 该解析公式充分考虑到了空间色散以及磁电超材料的电元件和磁元件的相互作用. 折射率理论曲线和基于仿真实验数据的提取值曲线能很好地符合, 这说明文中推导的折射率公式能够正确地描述磁电超材料的负折射特性. 本文的结果将为分析电磁元件之间的相互作用以及设计负折射率符合一定要求的磁电超材料提供重要的理论参考.Based on Maxwell's curl equations, the electric component and the magnetic component in magnetoelectric metamaterial plate are equivalent to the surface electric current and the surface magnetic current respectively. By calculating the total electric field and the total magnetic field in a magnetoelectric metamaterial plate generated by these periodic surface electric currents and magnetic currents, we obtain two equations about the surface current density and the surface magnetic current density, and thus deduce the analytical formulas for relationship between the refractive index of periodic magnetoelectric metamaterial and permeability of the magnetic component, and that between permittivity of the electric component and spatial dispersion. Unlike traditional index formula, the analytical formulas fully consider the spatial dispersion and the interaction of the electrical component and the magnetic component. The theoretical curves for refractive index are found to be in good agreement with the retrieval curves from simulation data, which shows that the analytical formulas for the refractive index can correctly describe the negative refraction characteristics of the magnetoelectric metamaterials. Our work will provide important theoretical reference for researchers to analyze interaction between electromagnetic components and to design magnetoelectric metamaterials with negative refractive indexes which meet certain requirements.
[1] Smith D R 2010 Phys. Rev. E 81 036605
[2] Pendry J B, Holden A J, Stewart W J, Youngs I 1996 Phys. Rev. Lett. 76 4773
[3] Pendry J B, Holden A J, Robbins D J, Stewart W J 1999 IEEE Trans. Microw. Theor. Techniq. 47 2075
[4] Smith D R, Padilla W J, Vier D C, Nemat-Nasser S C, Schultz S 2000 Phys. Rev. Lett. 84 4184
[5] Schurig D, Mock J J, Justice B J, Cummer S A, Pendry J B, Starr A F, Smith D R 2006 Science 314 977
[6] Liu R, Ji C, Mock J J, Chin J Y, Cui T J, Smith D R 2009 Science 323 366
[7] Soukoulis C M, Wegener M 2010 Science 330 1633
[8] Guney D O, Koschny Th, Soukoulis C M 2011 Phys. Rev. B 83 045107
[9] Tassin P, Koschny Th, Soukoulis C M 2012 Physica B 407 4062
[10] Guney D O, Koschny Th, Kafesaki M, Soukoulis C M 2009 Opt. Lett. 34 506
[11] Liu Y H, Liu H, Zhao X P 2012 Acta Phys. Sin. 61 084103 (in Chinese) [刘亚红, 刘辉, 赵晓鹏 2012 物理学报 61 084103]
[12] Xu X H, Xiao S Q, Gan Y H, Fu C F, Wang B Z 2012 Acta Phys. Sin. 61 124103 (in Chinese) [徐新河, 肖绍球, 甘月红, 付崇芳, 王秉中2012 物理学报 61 124103]
[13] Smith D R, Schultz S 2002 Phys. Rev. B 65 195104
[14] Smith D R, Pendry J B 2006 J. Opt. Soc. Am. B 23 391
[15] Smith D R, Vier D C, Koschny Th, Soukoulis C M 2005 Phys. Rev. E 71 036617
[16] Chen X, Grzegorczyk T M, Wu B I, Pacheco Jr J, Kong J A 2004 Phys. Rev. E 70 016608
[17] Gong J Q, Liang C H 2011 Acta Phys. Sin. 60 059204 (in Chinese) [龚建强, 梁昌洪 2011 物理学报 60 059204]
[18] Liu R, Cui T J, Huang D, Zhao B, Smith D R 2007 Phys. Rev. E 76 026606
[19] Zhang K Q, Li D J 2007 Electromagnetic Theory for Microwave and Optoelectronics (2nd Ed.) (New York:Berlin Heidelberg) p20
[20] Jackson J D 1999 Classical Electrodynamics (New York:Wiley) p51
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[1] Smith D R 2010 Phys. Rev. E 81 036605
[2] Pendry J B, Holden A J, Stewart W J, Youngs I 1996 Phys. Rev. Lett. 76 4773
[3] Pendry J B, Holden A J, Robbins D J, Stewart W J 1999 IEEE Trans. Microw. Theor. Techniq. 47 2075
[4] Smith D R, Padilla W J, Vier D C, Nemat-Nasser S C, Schultz S 2000 Phys. Rev. Lett. 84 4184
[5] Schurig D, Mock J J, Justice B J, Cummer S A, Pendry J B, Starr A F, Smith D R 2006 Science 314 977
[6] Liu R, Ji C, Mock J J, Chin J Y, Cui T J, Smith D R 2009 Science 323 366
[7] Soukoulis C M, Wegener M 2010 Science 330 1633
[8] Guney D O, Koschny Th, Soukoulis C M 2011 Phys. Rev. B 83 045107
[9] Tassin P, Koschny Th, Soukoulis C M 2012 Physica B 407 4062
[10] Guney D O, Koschny Th, Kafesaki M, Soukoulis C M 2009 Opt. Lett. 34 506
[11] Liu Y H, Liu H, Zhao X P 2012 Acta Phys. Sin. 61 084103 (in Chinese) [刘亚红, 刘辉, 赵晓鹏 2012 物理学报 61 084103]
[12] Xu X H, Xiao S Q, Gan Y H, Fu C F, Wang B Z 2012 Acta Phys. Sin. 61 124103 (in Chinese) [徐新河, 肖绍球, 甘月红, 付崇芳, 王秉中2012 物理学报 61 124103]
[13] Smith D R, Schultz S 2002 Phys. Rev. B 65 195104
[14] Smith D R, Pendry J B 2006 J. Opt. Soc. Am. B 23 391
[15] Smith D R, Vier D C, Koschny Th, Soukoulis C M 2005 Phys. Rev. E 71 036617
[16] Chen X, Grzegorczyk T M, Wu B I, Pacheco Jr J, Kong J A 2004 Phys. Rev. E 70 016608
[17] Gong J Q, Liang C H 2011 Acta Phys. Sin. 60 059204 (in Chinese) [龚建强, 梁昌洪 2011 物理学报 60 059204]
[18] Liu R, Cui T J, Huang D, Zhao B, Smith D R 2007 Phys. Rev. E 76 026606
[19] Zhang K Q, Li D J 2007 Electromagnetic Theory for Microwave and Optoelectronics (2nd Ed.) (New York:Berlin Heidelberg) p20
[20] Jackson J D 1999 Classical Electrodynamics (New York:Wiley) p51
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