各向同性零折射率超常介质中电磁波的传播特性研究
|
摘要
介电常数和磁导率是决定电磁波在介质中的传播性质的两个主要电磁参数。一般情况下,这两个参数都为正数,即介质为正折射率。超常介质(metamaterials)的提出和人工实现改变了人们对这两个物理量的传统认识,为人们设计可控的、色散的介电常数和磁导率材料提供了广阔的空间,为控制电磁波的传输提供了新的方法和手段。随着超常介质理论研究的深入和设计手段的发展,超常介质的工作频段逐渐由最初的微波段向可见光波段迈进,对电磁学、材料学和通信等领域产生了深远的影响,极大地拓展了这些学科的研究领域,并且有很多极具利用价值的奇异特性,必将在天线、微波/毫米波电路器件、装备、军事隐身等领域获得广泛的应用。
零折射率超常介质是一类折射率为零的特殊的超常介质,研究电磁波在其中的传播性质有着非常重要的科学意义和实用价值。本文将传统的电磁波传输理论和超常介质的性质相结合,研究了电磁波在介电常数和磁导率同时为零的零折射率超常介质(NZPIM)中的传输性质,取得一些成果,主要如下：
首先,利用传统的分层介质膜理论推导了介电常数和磁导率同时为零的零折射率超常介质平板的透射系数和反射系数的数学表达式,研究了单色平面波在其中传播的反射和透射性质,以及相应的在低通空间滤波器、波前整形和超耦合等方面的应用潜力。
其次,对于电磁脉冲在零折射率点(即光子的狄拉克点)的传输性质作了分析,首次提出了NZPIM和光子晶体中光子颤振效应的物理机制。基于《Physical Review Letters》等期刊上的一些文献对NZPIM和光子晶体中光子颤振效应(Zitterbewegung)的研究结果,我们从脉冲整形理论的角度,以NZPIM平板为例,并结合我们推导的平板透射系数,理论上提出了光子颤振效应产生的物理机制。即NZPIM平板的透射谱分布对入射脉冲频谱的时空滤波效应,导致了出射频谱相对于入射频谱缺失了某些时空分量,从而使出射脉冲在时域出现被称为光子颤振效应的“拖尾”振荡。通过理论计算和仿真分析,我们得到的结果与其它文献上的结果一致,因此我们提出的方法的是具有可行性的,并且我们的方法还能解释文献中未能解释的问题。另外,平板厚度和脉冲时间、空间宽度对光子颤振效应的影响在本文中也得到了详细的分析。
The transmission properties of the electromagnetic wave propagating in the material mainly depend on two electromagnetic parameters:the permittivity and the permeability. In normal case, both parameters should be positive,i.e. the refractive index of the material is positive. After the metamaterial was proposed and realized in experiment, our traditional knowledge about them has been changed. It produces lots of chances to design new materials with tunable and dispersive permittivity and permeability, which provides new methods for controlling the propagating of the electromagnetic waves. With the development of the theoretical research and constructing technique, the working spectrum of metamaterial gradually increases from microwave to visible light. It deeply influences the electromagnetics, materials and communications, whose research fields have been greatly broadened. Many singular electromagnetic properties have been found for metamaterials, which should be applied in antenna, micro/millimeter wave devices and military cloak, etc.
Firstly, we deduce the transmission and reflection coefficient of the negative-zero-positive metamaterial slab using the traditional dielectric film theory, and investigate the transmission and reflection properties of the NZPIM. Some applied potentials in low-pass spatial filter, wave-front reshaping and supercoupling have also been illustrated.
Then the transmitted properties in zero-refractive-index point (Dirac point for photon) has been analyzed, and the origin of the Zitterbewegung in NZPIM and photonic crystals has been proposed for the first time. Based on some papers in the journals like Phys. Rev. Lett. proposed the Zitterbewegung in NZPIM and photonic crystals, we explained the origin of the Zitterbewegung with pulse shaping theory with a NZPIM slab sample. We deduced the transmission coefficient of the sample, so we hold that it is the filtering effect of the input spectrum by the transmittance spectral distribution that results in the absence of some spatio-temporal components of the output pulse. Thus the tail oscillations emerge, that is, so-called "Zitterbewegung". Through the theoretical calculations and numerical simulations, the validity of our method has been verified. Furthermore, our method could interpret what can't be interpreted by the method in other papers. And the similar phenomena in two-dimensional photonic crystals can be interpreted in the same way. In addition, the effects of pulse duration, pulse transverse spatial width and slab thickness on the tail oscillations of the transmitted pulse have also been discussed.
零折射率超常介质是一类折射率为零的特殊的超常介质,研究电磁波在其中的传播性质有着非常重要的科学意义和实用价值。本文将传统的电磁波传输理论和超常介质的性质相结合,研究了电磁波在介电常数和磁导率同时为零的零折射率超常介质(NZPIM)中的传输性质,取得一些成果,主要如下：
首先,利用传统的分层介质膜理论推导了介电常数和磁导率同时为零的零折射率超常介质平板的透射系数和反射系数的数学表达式,研究了单色平面波在其中传播的反射和透射性质,以及相应的在低通空间滤波器、波前整形和超耦合等方面的应用潜力。
其次,对于电磁脉冲在零折射率点(即光子的狄拉克点)的传输性质作了分析,首次提出了NZPIM和光子晶体中光子颤振效应的物理机制。基于《Physical Review Letters》等期刊上的一些文献对NZPIM和光子晶体中光子颤振效应(Zitterbewegung)的研究结果,我们从脉冲整形理论的角度,以NZPIM平板为例,并结合我们推导的平板透射系数,理论上提出了光子颤振效应产生的物理机制。即NZPIM平板的透射谱分布对入射脉冲频谱的时空滤波效应,导致了出射频谱相对于入射频谱缺失了某些时空分量,从而使出射脉冲在时域出现被称为光子颤振效应的“拖尾”振荡。通过理论计算和仿真分析,我们得到的结果与其它文献上的结果一致,因此我们提出的方法的是具有可行性的,并且我们的方法还能解释文献中未能解释的问题。另外,平板厚度和脉冲时间、空间宽度对光子颤振效应的影响在本文中也得到了详细的分析。
The transmission properties of the electromagnetic wave propagating in the material mainly depend on two electromagnetic parameters:the permittivity and the permeability. In normal case, both parameters should be positive,i.e. the refractive index of the material is positive. After the metamaterial was proposed and realized in experiment, our traditional knowledge about them has been changed. It produces lots of chances to design new materials with tunable and dispersive permittivity and permeability, which provides new methods for controlling the propagating of the electromagnetic waves. With the development of the theoretical research and constructing technique, the working spectrum of metamaterial gradually increases from microwave to visible light. It deeply influences the electromagnetics, materials and communications, whose research fields have been greatly broadened. Many singular electromagnetic properties have been found for metamaterials, which should be applied in antenna, micro/millimeter wave devices and military cloak, etc.
Firstly, we deduce the transmission and reflection coefficient of the negative-zero-positive metamaterial slab using the traditional dielectric film theory, and investigate the transmission and reflection properties of the NZPIM. Some applied potentials in low-pass spatial filter, wave-front reshaping and supercoupling have also been illustrated.
Then the transmitted properties in zero-refractive-index point (Dirac point for photon) has been analyzed, and the origin of the Zitterbewegung in NZPIM and photonic crystals has been proposed for the first time. Based on some papers in the journals like Phys. Rev. Lett. proposed the Zitterbewegung in NZPIM and photonic crystals, we explained the origin of the Zitterbewegung with pulse shaping theory with a NZPIM slab sample. We deduced the transmission coefficient of the sample, so we hold that it is the filtering effect of the input spectrum by the transmittance spectral distribution that results in the absence of some spatio-temporal components of the output pulse. Thus the tail oscillations emerge, that is, so-called "Zitterbewegung". Through the theoretical calculations and numerical simulations, the validity of our method has been verified. Furthermore, our method could interpret what can't be interpreted by the method in other papers. And the similar phenomena in two-dimensional photonic crystals can be interpreted in the same way. In addition, the effects of pulse duration, pulse transverse spatial width and slab thickness on the tail oscillations of the transmitted pulse have also been discussed.
引文
[1]V. G. Veselago. The electrodynamics substances with simultaneously negative values of ε and μ. Sov.Phys.Usp.,1968,10(4):509514(1-8)
[2]J. B. Pendry, A. J. Holden. Extremely low frequency plasmons in metallic mesostructures.Phy. Rev. Lett.,1996,76(25):4773-4776
[3]J. B. Pendry, A. J. Holden. Magnetism from conductors and enhanced nonlinear phenomena. IEEE Transactions on Microwave Theory and Techniques,1999, 47(11):2075-2084
[4]D. R. Smith, W. J. Padilla. Composite medium with simultaneously negative permeability and permittivity. Phy. Rev. Lett.,2000,84(18):4184-4187
[5]J. B. Pendry. Negative refraction makes a perfect lens. Phy. Rev. Lett.,2000, 85(18):3966-3969
[6]S Linden, C Enkrich, M Wegener, et al. Magnetic response of metamaterials at 100 terahertz. Science,2004,306(19):1351-1353
[7]Shuang Zhang, Wenjun Fan, N. C. Panoiu, et al. Experimental Demonstration of Near-Infrared Negative-Index Metamaterials. Phys. Rev. Lett.,2005,95(13): 137404-137407
[8]C. Caloz and T. Itoh. Application of the Transmission Line Theory of Left-Handed(LH) Materials to the Realization of a Microstrip LH Line. IEEE Ante. Prop. Soci. Int. Symp,2002,1:412-415
[9]A. Grbic, and GV. Eleflheriades. Experimental Verification of Backward-wave Radiation from a Negative Refractive Index Metamaterial. J. Appl. Phys.,2002, 92(10):5930-5935
[10]G. V. Eleflheriades, O. Siddiqui, A. K. Iyer, et al. Transmission line models for negative refractive index media and associated implementations without excess resonators. IEEE Micro. Wireless Comp. Lett.,2003,13(2):51-53
[11]A. Sanada, C. Caloz, and T. Itoh. Charateristics of the Composite Right/Left-Handed Transmission Lines. IEEE Micro. Wireless Comp. Lett.,2004, 14(2):68-70
[12]Richard W. Ziolkowski. Propagation in and scattering from a matched metamaterial having a zero index of refraction. Phys. Rev. E,2004,70(4): 046608(1-12)
[13]Mario Silveirinha and Nader Engheta. Tunneling of Electromagnetic Energy through Subwavelength Channels and Bends using Epsilon-Near-Zero Materials. Phys. Rev. Lett.,2006,97(15):157403(1-4)
[14]Mario G. Silveirinhal, and Nader Engheta. Theory of supercoupling, squeezing wave energy and field confinement in narrow channels and tight bends using epsilon-near-zero metamaterials. Phys. Rev. B,2007,76(24),245109(1-17)
[15]Mario Silveirinhal, and Nader Engheta. Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media. Phys. Rev. B,2007,75(7),075119(1-10)
[16]Fuli Zhang, Gregory Houzet, Eric Lheurette, et al. Negative-zero-positive metamaterial with omega-type metal inclusions. J. Appl. Phys.,2008, 103(8):084312(1-8)
[17]Ruopeng Liu, Qiang Cheng, Thomas Hand, et al. Experimental Demonstration of Electromagnetic Tunneling Through an Epsilon-Near-Zero Metamaterial at Microwave Frequencies. Phys. Rev. Lett.,2008,100(2):023903(1-4)
[18]B. Edwards, A. Alu, M. G. Silveirinha, et al. Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects. J. Appl. Phys.,2009, 105(4):044905(1-4)
[19]Liyuan Liu, Chenggang Hu, Zeyu Zhao, et al. Multi-passband Tunneling Effect in Multilayered Epsilon-Near-Zero Metamaterials. Opt. Express,2009,17(14): 12133-12188
[20]Stefan Enoch, Gerard Tayeb, Pierre Sabouroux, et al. A Metamaterial for Directive Emission. Phys. Rev. Lett.,2002,89(21):213902(1-4)
[21]Brian Edwards, Andrea Alu, Michael E. Young, et al. Experimental Verification of Epsilon-Near-Zero Metamaterial Coupling and Energy Squeezing Using a Microwave Waveguide. Phys. Rev. Lett.,2008,100(3):033903(1-4)
[22]Andrea Alu, Mario G. Silveirinha, Alessandro Salandrino, et al. Epsilon-near-zero metamaterials and electromagnetic sources:Tailoring the radiation phase pattern. Phys. Rev. B,2007,75(15):155410(1-13)
[23]Qiang Cheng, Ruopeng Liu, Da Huang, et al. Circuit verification of tunneling effect in zero permittivity medium. Appl. Phys. Lett.,2007,91(23):234105(1-3)
[24]N. Garcia, E. V. Ponizovskaya, and John Q. Xiao. Zero permittivity materials: Band gaps at the visible. Appl. Phys. Lett.,2002,80(7):1120-1122
[25]Ricardo A. Depine a, Maria L. Martinez-Ricci, et al. Zero permeability and zero permittivity band gaps in 1D metamaterial photonic crystals. Physics Letters A, 2007,364:352-355
[26]P. B. Johnson and R. W. Christy, Optical Constants of the Noble Metals. Phys. Rev. B,1972,6(12):4370-4379
[27]J. Gomez Rivas, C. Janke, P. Bolivar, et al. Transmission of THz radiation through InSb gratings of subwavelength apertures. Opt. Express,2005,13(3): 847-859
[28]W. G. Spitzer, D. Kleinman, and D. Walsh. Infrared Properties of Hexagonal Silicon Carbide. Phys. Rev.,1959,113(1):127-132
[29]Jensen Li, Lei Zhou, C.T. Chan, et al. Photonic Band Gap from a Stack of Positive and Negative Index Materials. Phys. Rev. Lett.,2003,90(8): 083901(1-4)
[30]Nicolae C. Panoiu and Richard M. Osgood, Jr.. Zero-n bandgap in photonic crystal superlattices. J. Opt. Soc. Am. B,2006,23(3):506-513
[31]H. Jiang, H. Chen, H. Li, et al. Properties of one-dimensional photonic crystals containing single-negative materials. Phys. Rev. E,2004,69(6),066607(1-5)
[32]K. S. Novoselov, A. K. Geim, S. V. Morozov, et al. Electric field effect in atomically thin carbon films. Science,2004,306(5696):666-669
[33]S. Raghu and F. D. M. Haldane. Analogs of quantum Hall effect edge states in photonic crystals. Phys. Rev. A,2006,78(3):033834(1-4)
[34]F. D. M. Haldane and S. Raghu. Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry. Phys. Rev. Lett.,2008,100(1):013904(1-4)
[35]Li-Gang Wang, Zhi-Guo Wang, Jun-Xiang Zhang, et al. Realization of Dirac point with double cones in optics. Opt. Lett.,2009,34(10):1510-1512
[36]Li-Gang Wang, Zhi-Guo Wang, Shi-Yao Zhu. Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial. Europhysics Letters,2009,86(4):47008(1-5)
[37]D.R.Smith and Norman Kroll. Negative Refractive Index in Left-Handed Materials. Phys. Rev. Lett,2000,85(14):2933-2936
[38]Pendry J B, Holden A J, Robbins D J, et al. Low frequency plasmons in thin-wire structures. J. Phys. C,1998,10(2):4785-4808
[39]J. Desbois, F. Gires, and P. Tournois, A new approach to picosecond laser pulse analysis shaping and coding. IEEE J. Quantum Electron,1973,9(2):213-218
[40]Froehly C, Colombeau B, Vampouille M., Progress in Optics,1983,20(65): 63-153
[41]Weiner A M. Femtosecond pulse shaping using spatial light modulators. Rev. Sci. Instrum.,2000,71 (5):1929-1960
[42]Max Born and Emil Wolf, Principles of Optics.7th edition. Cambridge: Cambridge university press,1999,54-74
[43]K. S. Yee. Numerical solution of initial boundary value problems involving Maxwell equations in isotropic media. IEEE Trans. Antennas Propagat.1966, AP-14(3):302-330
[44]A.Taflove and M. E. Brodwin. Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations. IEEE Trans. Microwave theory tech.1975, MTT-23(8):623-680
[45]A. Taflove and S. C. Hagness. Computational Electrodynamics:The Finite-Difference Time-Domain Method. Second edition. Norwood:Artech House INC,2000,1-34
[46]高本庆.时域有限差分法.北京：国防工业出版社,1995年,1-47
[47]葛德彪.闫玉波.电磁波时域有限差分方法.西安：西安电子科技大学出版社,2005,1-40
[48]J. P. Berenger. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys.1994,114 (2):185-200
[49]G. P. Agrawal. Nonlinear Fiber Optics.4th edition. Boston:Elsevier,1995,1-20
[50]X. Zhang.Observing zitterbewegung for photons near the Dirac point of a two-dimensinal photonic crystal. Phys. Rev. Lett.2008,100(11):113903(1-4)
[51]X. Zhang and Z. Liu, Phys. Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals. Rev. Lett.2008,101(26):264303(1-4)
[52]R. A. Sepkhanov, Ya. B. Bazaliy and C. W. J. Beenakker. Extremal transmission at the Dirac point of a photonic band structure. Phys. Rev. A 2007,75(6): 063813(1-5)
[53]M. Diem, T. Koschny and C. M. Soukoulis, Transmission in the vicinity of the Dirac point in hexagonal Photonic Crystals. arXiv:0807.3351v1
[54]R. A. Sepkhanov and C. W. J. Beenakker. Numerical test of the theory of pseudo-diffusive transmission at the Dirac point of a photonic band structure. Opt. Commun.2008,281(20),5267-5270
[55]X. Zhang. Demonstration of a new transport regime of photon in two-dimensional photonic crystal. Phys. Lett. A 2008,372:3512-3516
[56]O. Peleg, G. Bartal, B. Freedman et al. Conical Diffraction and Gap Solitons in Honeycomb Photonic Lattices. Phys. Rev. Lett.2007,98(10):103901(1-4)
[57]E. Schrodinger, Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl.1930, 24:418-422
[2]J. B. Pendry, A. J. Holden. Extremely low frequency plasmons in metallic mesostructures.Phy. Rev. Lett.,1996,76(25):4773-4776
[3]J. B. Pendry, A. J. Holden. Magnetism from conductors and enhanced nonlinear phenomena. IEEE Transactions on Microwave Theory and Techniques,1999, 47(11):2075-2084
[4]D. R. Smith, W. J. Padilla. Composite medium with simultaneously negative permeability and permittivity. Phy. Rev. Lett.,2000,84(18):4184-4187
[5]J. B. Pendry. Negative refraction makes a perfect lens. Phy. Rev. Lett.,2000, 85(18):3966-3969
[6]S Linden, C Enkrich, M Wegener, et al. Magnetic response of metamaterials at 100 terahertz. Science,2004,306(19):1351-1353
[7]Shuang Zhang, Wenjun Fan, N. C. Panoiu, et al. Experimental Demonstration of Near-Infrared Negative-Index Metamaterials. Phys. Rev. Lett.,2005,95(13): 137404-137407
[8]C. Caloz and T. Itoh. Application of the Transmission Line Theory of Left-Handed(LH) Materials to the Realization of a Microstrip LH Line. IEEE Ante. Prop. Soci. Int. Symp,2002,1:412-415
[9]A. Grbic, and GV. Eleflheriades. Experimental Verification of Backward-wave Radiation from a Negative Refractive Index Metamaterial. J. Appl. Phys.,2002, 92(10):5930-5935
[10]G. V. Eleflheriades, O. Siddiqui, A. K. Iyer, et al. Transmission line models for negative refractive index media and associated implementations without excess resonators. IEEE Micro. Wireless Comp. Lett.,2003,13(2):51-53
[11]A. Sanada, C. Caloz, and T. Itoh. Charateristics of the Composite Right/Left-Handed Transmission Lines. IEEE Micro. Wireless Comp. Lett.,2004, 14(2):68-70
[12]Richard W. Ziolkowski. Propagation in and scattering from a matched metamaterial having a zero index of refraction. Phys. Rev. E,2004,70(4): 046608(1-12)
[13]Mario Silveirinha and Nader Engheta. Tunneling of Electromagnetic Energy through Subwavelength Channels and Bends using Epsilon-Near-Zero Materials. Phys. Rev. Lett.,2006,97(15):157403(1-4)
[14]Mario G. Silveirinhal, and Nader Engheta. Theory of supercoupling, squeezing wave energy and field confinement in narrow channels and tight bends using epsilon-near-zero metamaterials. Phys. Rev. B,2007,76(24),245109(1-17)
[15]Mario Silveirinhal, and Nader Engheta. Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media. Phys. Rev. B,2007,75(7),075119(1-10)
[16]Fuli Zhang, Gregory Houzet, Eric Lheurette, et al. Negative-zero-positive metamaterial with omega-type metal inclusions. J. Appl. Phys.,2008, 103(8):084312(1-8)
[17]Ruopeng Liu, Qiang Cheng, Thomas Hand, et al. Experimental Demonstration of Electromagnetic Tunneling Through an Epsilon-Near-Zero Metamaterial at Microwave Frequencies. Phys. Rev. Lett.,2008,100(2):023903(1-4)
[18]B. Edwards, A. Alu, M. G. Silveirinha, et al. Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects. J. Appl. Phys.,2009, 105(4):044905(1-4)
[19]Liyuan Liu, Chenggang Hu, Zeyu Zhao, et al. Multi-passband Tunneling Effect in Multilayered Epsilon-Near-Zero Metamaterials. Opt. Express,2009,17(14): 12133-12188
[20]Stefan Enoch, Gerard Tayeb, Pierre Sabouroux, et al. A Metamaterial for Directive Emission. Phys. Rev. Lett.,2002,89(21):213902(1-4)
[21]Brian Edwards, Andrea Alu, Michael E. Young, et al. Experimental Verification of Epsilon-Near-Zero Metamaterial Coupling and Energy Squeezing Using a Microwave Waveguide. Phys. Rev. Lett.,2008,100(3):033903(1-4)
[22]Andrea Alu, Mario G. Silveirinha, Alessandro Salandrino, et al. Epsilon-near-zero metamaterials and electromagnetic sources:Tailoring the radiation phase pattern. Phys. Rev. B,2007,75(15):155410(1-13)
[23]Qiang Cheng, Ruopeng Liu, Da Huang, et al. Circuit verification of tunneling effect in zero permittivity medium. Appl. Phys. Lett.,2007,91(23):234105(1-3)
[24]N. Garcia, E. V. Ponizovskaya, and John Q. Xiao. Zero permittivity materials: Band gaps at the visible. Appl. Phys. Lett.,2002,80(7):1120-1122
[25]Ricardo A. Depine a, Maria L. Martinez-Ricci, et al. Zero permeability and zero permittivity band gaps in 1D metamaterial photonic crystals. Physics Letters A, 2007,364:352-355
[26]P. B. Johnson and R. W. Christy, Optical Constants of the Noble Metals. Phys. Rev. B,1972,6(12):4370-4379
[27]J. Gomez Rivas, C. Janke, P. Bolivar, et al. Transmission of THz radiation through InSb gratings of subwavelength apertures. Opt. Express,2005,13(3): 847-859
[28]W. G. Spitzer, D. Kleinman, and D. Walsh. Infrared Properties of Hexagonal Silicon Carbide. Phys. Rev.,1959,113(1):127-132
[29]Jensen Li, Lei Zhou, C.T. Chan, et al. Photonic Band Gap from a Stack of Positive and Negative Index Materials. Phys. Rev. Lett.,2003,90(8): 083901(1-4)
[30]Nicolae C. Panoiu and Richard M. Osgood, Jr.. Zero-n bandgap in photonic crystal superlattices. J. Opt. Soc. Am. B,2006,23(3):506-513
[31]H. Jiang, H. Chen, H. Li, et al. Properties of one-dimensional photonic crystals containing single-negative materials. Phys. Rev. E,2004,69(6),066607(1-5)
[32]K. S. Novoselov, A. K. Geim, S. V. Morozov, et al. Electric field effect in atomically thin carbon films. Science,2004,306(5696):666-669
[33]S. Raghu and F. D. M. Haldane. Analogs of quantum Hall effect edge states in photonic crystals. Phys. Rev. A,2006,78(3):033834(1-4)
[34]F. D. M. Haldane and S. Raghu. Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry. Phys. Rev. Lett.,2008,100(1):013904(1-4)
[35]Li-Gang Wang, Zhi-Guo Wang, Jun-Xiang Zhang, et al. Realization of Dirac point with double cones in optics. Opt. Lett.,2009,34(10):1510-1512
[36]Li-Gang Wang, Zhi-Guo Wang, Shi-Yao Zhu. Zitterbewegung of optical pulses near the Dirac point inside a negative-zero-positive index metamaterial. Europhysics Letters,2009,86(4):47008(1-5)
[37]D.R.Smith and Norman Kroll. Negative Refractive Index in Left-Handed Materials. Phys. Rev. Lett,2000,85(14):2933-2936
[38]Pendry J B, Holden A J, Robbins D J, et al. Low frequency plasmons in thin-wire structures. J. Phys. C,1998,10(2):4785-4808
[39]J. Desbois, F. Gires, and P. Tournois, A new approach to picosecond laser pulse analysis shaping and coding. IEEE J. Quantum Electron,1973,9(2):213-218
[40]Froehly C, Colombeau B, Vampouille M., Progress in Optics,1983,20(65): 63-153
[41]Weiner A M. Femtosecond pulse shaping using spatial light modulators. Rev. Sci. Instrum.,2000,71 (5):1929-1960
[42]Max Born and Emil Wolf, Principles of Optics.7th edition. Cambridge: Cambridge university press,1999,54-74
[43]K. S. Yee. Numerical solution of initial boundary value problems involving Maxwell equations in isotropic media. IEEE Trans. Antennas Propagat.1966, AP-14(3):302-330
[44]A.Taflove and M. E. Brodwin. Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations. IEEE Trans. Microwave theory tech.1975, MTT-23(8):623-680
[45]A. Taflove and S. C. Hagness. Computational Electrodynamics:The Finite-Difference Time-Domain Method. Second edition. Norwood:Artech House INC,2000,1-34
[46]高本庆.时域有限差分法.北京：国防工业出版社,1995年,1-47
[47]葛德彪.闫玉波.电磁波时域有限差分方法.西安：西安电子科技大学出版社,2005,1-40
[48]J. P. Berenger. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys.1994,114 (2):185-200
[49]G. P. Agrawal. Nonlinear Fiber Optics.4th edition. Boston:Elsevier,1995,1-20
[50]X. Zhang.Observing zitterbewegung for photons near the Dirac point of a two-dimensinal photonic crystal. Phys. Rev. Lett.2008,100(11):113903(1-4)
[51]X. Zhang and Z. Liu, Phys. Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals. Rev. Lett.2008,101(26):264303(1-4)
[52]R. A. Sepkhanov, Ya. B. Bazaliy and C. W. J. Beenakker. Extremal transmission at the Dirac point of a photonic band structure. Phys. Rev. A 2007,75(6): 063813(1-5)
[53]M. Diem, T. Koschny and C. M. Soukoulis, Transmission in the vicinity of the Dirac point in hexagonal Photonic Crystals. arXiv:0807.3351v1
[54]R. A. Sepkhanov and C. W. J. Beenakker. Numerical test of the theory of pseudo-diffusive transmission at the Dirac point of a photonic band structure. Opt. Commun.2008,281(20),5267-5270
[55]X. Zhang. Demonstration of a new transport regime of photon in two-dimensional photonic crystal. Phys. Lett. A 2008,372:3512-3516
[56]O. Peleg, G. Bartal, B. Freedman et al. Conical Diffraction and Gap Solitons in Honeycomb Photonic Lattices. Phys. Rev. Lett.2007,98(10):103901(1-4)
[57]E. Schrodinger, Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl.1930, 24:418-422