光孤子在负折射材料中的传输及演变研究
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摘要
介电常数和磁导率是描述物质基本电磁性质的物理量,在自然界存在的媒质中它们一般都是大于零的,但理论上也存在介电常数和磁导率同时小于零的物体,其中之一就是负折射介质。这种材料具有常规材料所不具备的负折射现象、倏逝波放大、逆多普勒效应、完美透镜等一系列特性。近年来,对负折射材料的研究从最初的微波频段进入到太赫兹、红外频段,甚至光频段,并且已经开始了对其非线性效应的讨论。作为非线性光学一个重要的研究领域,光孤子能够在长距离传输中有效地保持波形,因此在高速率光纤通信等方面具有广阔的应用价值。
本文首先对负折射材料的基本原理进行了介绍。通过对比传统材料,逐一说明了负折射材料中特有的各种现象,并总结了近年来国内外的研究进展。然后在常规材料中光孤子传输的理论基础上,推导了光孤子在负折射材料中的非线性薛定谔方程。由于负折射介质中的介电常数和磁导率具有色散的性质,因此在其中的高阶非线性效应与常规材料中的情况有很大的不同。数值方法选用的是分布傅立叶法,为了保证精度以及减少运算时间,对不同的参数采用了不同的迭代方式,模拟结果与理论推导符合得很好。
然后,分别在反常和正常色散的情况下依次分析了五阶非线性效应、可控自陡峭效应、高阶线性色散与非线性色散对孤子传输和演变的影响。模拟结果显示,负的五阶非线性效应对亮孤子起到了脉宽与周期压缩的作用,并且会造成高阶孤子的分裂。对于暗孤子的脉宽压缩不明显,主要影响体现在改变灰孤子的振幅上;而在负折射材料中,由于自陡峭系数可以在正负之间选取,因此孤子传输方向的偏移与常规材料相反,甚至可以与三阶色散结合从而达到无偏移传输的目的;非线性色散作为负折射材料特有的属性,虽然有与线性色散相近的形式,但在单独作用时会让孤子频谱分裂,因此不能使孤子稳定传输,在实际应用中主要是造成孤子振幅的改变。最后,考察了负折射材料中亮孤子对传输的情况,讨论了双孤子间的互作用导致孤子吸引或排斥的原因,提出了一种实现双孤子无干扰平行传输的模型,即使用不同振幅、不同初始相位、存在三阶色散、零自陡峭系数的输入脉冲,可以保证在长距离的传输中不发生孤子吸引或排斥的现象。
The permittivity and permeability are the physical attributes to describe the basic electromagnetic properties. They are all positive in natural materials. But in theory, materials possess negative permittivity and permeability do exist, one of which is negative-index material(NIM).This medium has many characteristics which general material doesn't have, such as the negative refraction index, evanescent wave amplification, reversed Doppler effect, perfect lens. For those reasons, NIM quickly became a international research focus. In recent years, the research about NIM is not only in microwave band, but also in terahertz, infrared, even in optic band, and the properties of nonlinearity are studied, too. As the important field of nonlinear optics, solitons could propagate in a long distance without the change of shape. Therefore solitons have a broad use in high-rates fiber-optic communication systems.
This dissertation firstly introduced the basic principle of NIM. By comparing the traditional material, showed the unique phenomena in it, and summarized the recent researches internal and abroad. Then based on the theory of solitons in general material, a new Schr?dinger equation which describes the solitons’propagation in NIM is found. The high-order nonlinear effects are quite different from the situation in normal material for the existence of dispersion permittivity and permeability. We used a numerical method called Split-Step Fourier Method. In order to ensure the accuracy and reducing computation time, different iterations are used for different parameters. The results accorded with the theoretical analysis.
Then, the roles of the fifth-order nonlinear effect, controllable self-steeping, high-order dispersions, and nonlinear dispersions are analyzed in the anomalous-GVD regime and normal-GVD regime, separately. The negative fifth-order nonlinear effect compresses the pulse width and cycle of bright solitons, and will case the splitting of high-order solitons. For dark solitons, it mainly affects the gray soliton's amplitude. In NIM, the SS coefficient can be negative, so the offset direction of solitons is in opposite of the normal material’s. With the interaction of third-order dispersion, the self-steeping could disappear in NIM. As a unique property, the nonlinear dispersion is similar in the form with the linear dispersion, but it will split the soliton's spectrum and make the propagation instability when it exists alone. It mainly changes the amplitude of solitons in actual use. At last we investigated the soliton interactions in NIM and discussed the reasons of attraction or rejection between two solitons. We proposed a new model, using an input pulse with the different amplitude, different phase and zero-SS under the circumstance of third-dispersion, to ensure the two solitons propagate without interaction.
本文首先对负折射材料的基本原理进行了介绍。通过对比传统材料,逐一说明了负折射材料中特有的各种现象,并总结了近年来国内外的研究进展。然后在常规材料中光孤子传输的理论基础上,推导了光孤子在负折射材料中的非线性薛定谔方程。由于负折射介质中的介电常数和磁导率具有色散的性质,因此在其中的高阶非线性效应与常规材料中的情况有很大的不同。数值方法选用的是分布傅立叶法,为了保证精度以及减少运算时间,对不同的参数采用了不同的迭代方式,模拟结果与理论推导符合得很好。
然后,分别在反常和正常色散的情况下依次分析了五阶非线性效应、可控自陡峭效应、高阶线性色散与非线性色散对孤子传输和演变的影响。模拟结果显示,负的五阶非线性效应对亮孤子起到了脉宽与周期压缩的作用,并且会造成高阶孤子的分裂。对于暗孤子的脉宽压缩不明显,主要影响体现在改变灰孤子的振幅上;而在负折射材料中,由于自陡峭系数可以在正负之间选取,因此孤子传输方向的偏移与常规材料相反,甚至可以与三阶色散结合从而达到无偏移传输的目的;非线性色散作为负折射材料特有的属性,虽然有与线性色散相近的形式,但在单独作用时会让孤子频谱分裂,因此不能使孤子稳定传输,在实际应用中主要是造成孤子振幅的改变。最后,考察了负折射材料中亮孤子对传输的情况,讨论了双孤子间的互作用导致孤子吸引或排斥的原因,提出了一种实现双孤子无干扰平行传输的模型,即使用不同振幅、不同初始相位、存在三阶色散、零自陡峭系数的输入脉冲,可以保证在长距离的传输中不发生孤子吸引或排斥的现象。
The permittivity and permeability are the physical attributes to describe the basic electromagnetic properties. They are all positive in natural materials. But in theory, materials possess negative permittivity and permeability do exist, one of which is negative-index material(NIM).This medium has many characteristics which general material doesn't have, such as the negative refraction index, evanescent wave amplification, reversed Doppler effect, perfect lens. For those reasons, NIM quickly became a international research focus. In recent years, the research about NIM is not only in microwave band, but also in terahertz, infrared, even in optic band, and the properties of nonlinearity are studied, too. As the important field of nonlinear optics, solitons could propagate in a long distance without the change of shape. Therefore solitons have a broad use in high-rates fiber-optic communication systems.
This dissertation firstly introduced the basic principle of NIM. By comparing the traditional material, showed the unique phenomena in it, and summarized the recent researches internal and abroad. Then based on the theory of solitons in general material, a new Schr?dinger equation which describes the solitons’propagation in NIM is found. The high-order nonlinear effects are quite different from the situation in normal material for the existence of dispersion permittivity and permeability. We used a numerical method called Split-Step Fourier Method. In order to ensure the accuracy and reducing computation time, different iterations are used for different parameters. The results accorded with the theoretical analysis.
Then, the roles of the fifth-order nonlinear effect, controllable self-steeping, high-order dispersions, and nonlinear dispersions are analyzed in the anomalous-GVD regime and normal-GVD regime, separately. The negative fifth-order nonlinear effect compresses the pulse width and cycle of bright solitons, and will case the splitting of high-order solitons. For dark solitons, it mainly affects the gray soliton's amplitude. In NIM, the SS coefficient can be negative, so the offset direction of solitons is in opposite of the normal material’s. With the interaction of third-order dispersion, the self-steeping could disappear in NIM. As a unique property, the nonlinear dispersion is similar in the form with the linear dispersion, but it will split the soliton's spectrum and make the propagation instability when it exists alone. It mainly changes the amplitude of solitons in actual use. At last we investigated the soliton interactions in NIM and discussed the reasons of attraction or rejection between two solitons. We proposed a new model, using an input pulse with the different amplitude, different phase and zero-SS under the circumstance of third-dispersion, to ensure the two solitons propagate without interaction.
引文
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[2] V. G. Veselago. The electrodynamics of substances with simultaneously negative values ofεandμ. Soviet Physics Ups. 1968, 10 (4): 509-456
[3] H. Lamb. On group-velocity. Proc London Math Soc. 1904, 1: 473-479
[4] J. B. Pendry, A. J. Holden, D. J. Robbins, et al. Low frequency plasmons in thin-wire structures. J Phys Condense Matter. 1998, 10: 4773-4776
[5] A. A. Zharov, I. V. Shadrivov, Y. S. Kivshar. Nonlinear Properties of Left Handed Metamaterials. Phys. Rev. Letters. 2003, 91: 037401
[6] S.óbrien, D. McPeake, S. A. Ramakrishna, et al. Near-infrared Photonic band gaps and nonlinear effects in negative magnetic metamaterials. Phys. Rev. B. 2003, 69: 241101
[7] L. Chen, W. Ding, X. J. Dang, et al. Counter-propagation Energy-Flows in Nonlinear Left- handed metamaterials. Progress In Electromagnetics Research, 2007, 70: 257-267
[8] N. Lazarides, G. Tsironis. Coupled nonlinear Schr?dinger field equations for electromagnetic wave propagation in nonlinear left-handed materials. Phys Rev E, 2005, 71 (3): 036614
[9] I. Kourakis, P. K. Shuklal. Nonlinear propagation of electromagnetic waves in negative refraction index composite materials. Phys. Rev. E, 2005, 72 (1): 016626
[10] M. Scalora, M. S. Syrchin, N. Akozbek, et al. Generalized nonlinear Schr?dinger equation for dispersive susceptibility and permeability: application to negative index materials. Phys Rev Lett, 2005, 95 (1): 013902
[11] Shuangchun Wen, Youwen Wang, Wenhua Su, et al. Modulation instability in nonlinear negative-index material. Phys Rev E, 2005, 73 (3): 036617-1-036617-6
[12] L. Tonks, I. Langmuir. Oscillations in Ionized Gases. Phys Rev 1929, 33: 195-211
[13] J. B. Pendry, A. J. Pendry, W. J. Strewart, et al. Extremely Low Frequency Plasmons in Metallic Mesostructures. Phys. Rev. Lett. 1996, 76 (25): 4773-4776
[14] J. B. Pendry, A. J. Holden, D. J. Robbins, et al. Magnetism from Conductors and Enhanced Nonlinear Phenomena. IEEE Transactions on Microwave Theory and Techniques, 1999, 47: 2075-2084
[15] D. R. Smith, P. J. Willie, D. C. Vier, et al. Composite Medium with Simultaneously Negative Permittivity and Permeability. Phys. Rev. Lett, 2000, 84 (18): 4184-4187
[16] R. A. Shelby, D. R. Smith, S. Schultz. Experimental Verifications of a Negative Index of Refreaction. Science. 2001, 292: 77-79
[17] G. Anthony, G. V. Eleftheriades. Overcoming the Diffraction Limit with a Planar Left-Handed Transmission-Line Lens. Phys. Rev. Lett. 92, 117403 (2004)
[18] D. R. Smith, N. Kroll. Negative refractive index in left-handed materials. Phys. Rev. Lett, 2000, 85: 2933
[19] J. B. Pendry. Negative refraction makes a perfect lens. Phys Rev Lett, 2000, 85: 3966-3969
[20] N. Garcia, M. N. Vesperinas. Left handed materials do not make a perfect lens. Phys Rev Lett, 2002, 88 (20): 207403
[21] S. A. Ramakrishna. Comment on Negative Refraction at Optical Frequencies in Nonmagnetic Two-Component Molecular Media. Phys. Rev. Lett. 98, 059701 (2007)
[22] J. Lu, T. M. Grzegorczyk, Y. Zhang, et al. Cerenkov radiation in materials with negative permittivity and permeability. Optics Express, 2003, 11 (7): 723-724
[23] N. Seddon, T. Bearpark. Observation of the inverse Doppler Effect. Science. 2003, 302: 1537
[24] J. B. Pendry, D. R. Smith. Reversing light: negative refraction. Physics Today. 2004, 57: 37
[25] K. Kempa, R. Ruppin, J. B. Pendry. Electromagnetic response of a point-dipole crystal. Phys. Rev. B 72, 205103 (2005)
[26] A. Ahmadi, H. Mosallaei. Physical configuration and performance modeling of all-dielectric metamaterials. Phys. Rev. B 77, 045104 (2008)
[27] P. Markos, I. Rousochatzakis, C. M. Soukoulis. Transmission losses in left-handed materials. Phys. Rev. E66, 045601 (2002)
[28] S. Igor, S. Nefedov, A. Tretyakov. Photonic band gap structure containing metamaterial with negative permittivity and permeability. Phys. Rev. E 66, 036611 (2002)
[29] V. E. George, K. I. Ashwin, C. K. Peter. Planar negative refrative index media using periodically L-C loaded transmission lines. IEEE Trans Microw. Theory Tech, 2002, 50 (12): 2072-2712
[30] G. Anthony, V. E. George. Growing evanescent wave in negative refractive index transmission line media. Appl. Phus. Lett, 2003, 82(12): 1815-1817
[31] C. Caloz, T. Itoh. Characteristics and potential application of nonlinear left handed transmission line. Microwave Opt. Technol. Lett, 2004, 40 (6): 471-473
[32] A. Grbic, G. V. Eleftheriades. Experimental verification of backward-wave radiation from a negative refractive index metamaterial. Appl. Phys, 2002, 92 (10): 5930-5935
[33] A. Alù, N. Engheta, R. W. Ziolkowski. Finite-difference time-domain analysis of the tunneling and growing exponential in a pair ofε-negative andμ-negative slabs Phys. Rev. E 74, 016604 (2006)– Published July 18, 2006
[34] S. Linden, C. Enkrich, M. Wegener. Magnetic response of metamaterials at 100 terahertz. Science, 2004, 306 (5700): 1351-1353
[35] T. A. Leskova, A. A. Maradudin, J. M. Lopez. Coherence of light scattered from a randomly rough surface. Phys. Rev. E 71, 036606 (2005)
[36] M. Rahm, S. A. Cummer, D. Schurig. Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations. Phys. Rev. Lett. 100, 063903 (2008)
[37] D. R. Smith, D. Schurig, J. B. Pendry, Negative refraction of modulated electromagnetic waves. Appl. Phys. Lett. 2002, 81 (15): 2173-2174
[38] J. R. Thomas. Wave packet incidenton negative index media. IEEE Trans. Antennas Propag, 2005, 53(5): 1591-1599
[39] G. D. Aguanno, N. Mattiucci, M. Scalora, et al. Bright and dark gap solitons in a negative index Fabry-Perot etalon. Phys. Rev. Lett, 2004, 93 (21): 213902
[40] I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar. Guided modes in negative refractive index waveguides. Phys. Rev. E, 2003, 67 (5): 057602
[41] Liang Wu, Sailing He, Linfang Shen. Band structure for a one-dimensional photonic crystal containing left-handed materials. Phys Rev B, 67, 235103 (2003)
[42] M. Gorkunov, M. Lapine. Tuning of a nonlinear metamaterial band gap by an external magnetic field. Phys. Rev. B, 2004 (23), 70: 235109
[43] Hongjie Zhao, Lei Kang, Ji Zhou, et al. Experimental demonstration of tunable negative phase velocity and negative refraction in a ferromagnetic/ferroelectric composite metamaterial. Appl. Phys. Lett. 2008, 93: 201106
[44] S. A. Darmanyan, M. Neviere, A. A. Zakhidov. Nonlinear surface waves at the interfaces of left-handed electromagnetic media. Phys. Rev. E ,2005, 72: 036615
[45] N. Mattiucci, G. D'Aguanno, M. J. Bloemer. Second-harmonic generation from a positive- negative index material heterostructure. Phys. Rev. E, 2005, 72: 066612
[46] M. W. Feise, I. V. Shadrivov, Y. S. Kivshar. Tunable transmission and bistability in left handed band gap structure. Appl. Phys. Lett, 2004, 85 (9): 1451-1453
[47] M. L. Natalia. Optical Disability in a Nonlinear Optical Coupler with a Negative Index Channel. Phys. Rev. Lett, 2007, 99: 113902
[48] M. Marklund, P. K. Shukla, L. Stenflo. Ultrashort solitons and kinetic effects in nonlinear metamaterials. Phys. Rev. E, 2006, 73, 037601
[49] G. P. Agrawal. Nonlinear fiber optics. 3rd ed. San Diego: Academic Press,2001
[50] D. Pushkarov, S. Tanev. Bright and dark solitary wave propagation and disability in the anomalous dispersion region of optical waveguides with third- and fifth-order nonlinearities. Optics Communication. 124 (1996) 354-364
[51] J. B. Pendry. Negative refraction makes a perfect lens. Phys. Rev. Lett, 2000, 85(18): 3966
[2] V. G. Veselago. The electrodynamics of substances with simultaneously negative values ofεandμ. Soviet Physics Ups. 1968, 10 (4): 509-456
[3] H. Lamb. On group-velocity. Proc London Math Soc. 1904, 1: 473-479
[4] J. B. Pendry, A. J. Holden, D. J. Robbins, et al. Low frequency plasmons in thin-wire structures. J Phys Condense Matter. 1998, 10: 4773-4776
[5] A. A. Zharov, I. V. Shadrivov, Y. S. Kivshar. Nonlinear Properties of Left Handed Metamaterials. Phys. Rev. Letters. 2003, 91: 037401
[6] S.óbrien, D. McPeake, S. A. Ramakrishna, et al. Near-infrared Photonic band gaps and nonlinear effects in negative magnetic metamaterials. Phys. Rev. B. 2003, 69: 241101
[7] L. Chen, W. Ding, X. J. Dang, et al. Counter-propagation Energy-Flows in Nonlinear Left- handed metamaterials. Progress In Electromagnetics Research, 2007, 70: 257-267
[8] N. Lazarides, G. Tsironis. Coupled nonlinear Schr?dinger field equations for electromagnetic wave propagation in nonlinear left-handed materials. Phys Rev E, 2005, 71 (3): 036614
[9] I. Kourakis, P. K. Shuklal. Nonlinear propagation of electromagnetic waves in negative refraction index composite materials. Phys. Rev. E, 2005, 72 (1): 016626
[10] M. Scalora, M. S. Syrchin, N. Akozbek, et al. Generalized nonlinear Schr?dinger equation for dispersive susceptibility and permeability: application to negative index materials. Phys Rev Lett, 2005, 95 (1): 013902
[11] Shuangchun Wen, Youwen Wang, Wenhua Su, et al. Modulation instability in nonlinear negative-index material. Phys Rev E, 2005, 73 (3): 036617-1-036617-6
[12] L. Tonks, I. Langmuir. Oscillations in Ionized Gases. Phys Rev 1929, 33: 195-211
[13] J. B. Pendry, A. J. Pendry, W. J. Strewart, et al. Extremely Low Frequency Plasmons in Metallic Mesostructures. Phys. Rev. Lett. 1996, 76 (25): 4773-4776
[14] J. B. Pendry, A. J. Holden, D. J. Robbins, et al. Magnetism from Conductors and Enhanced Nonlinear Phenomena. IEEE Transactions on Microwave Theory and Techniques, 1999, 47: 2075-2084
[15] D. R. Smith, P. J. Willie, D. C. Vier, et al. Composite Medium with Simultaneously Negative Permittivity and Permeability. Phys. Rev. Lett, 2000, 84 (18): 4184-4187
[16] R. A. Shelby, D. R. Smith, S. Schultz. Experimental Verifications of a Negative Index of Refreaction. Science. 2001, 292: 77-79
[17] G. Anthony, G. V. Eleftheriades. Overcoming the Diffraction Limit with a Planar Left-Handed Transmission-Line Lens. Phys. Rev. Lett. 92, 117403 (2004)
[18] D. R. Smith, N. Kroll. Negative refractive index in left-handed materials. Phys. Rev. Lett, 2000, 85: 2933
[19] J. B. Pendry. Negative refraction makes a perfect lens. Phys Rev Lett, 2000, 85: 3966-3969
[20] N. Garcia, M. N. Vesperinas. Left handed materials do not make a perfect lens. Phys Rev Lett, 2002, 88 (20): 207403
[21] S. A. Ramakrishna. Comment on Negative Refraction at Optical Frequencies in Nonmagnetic Two-Component Molecular Media. Phys. Rev. Lett. 98, 059701 (2007)
[22] J. Lu, T. M. Grzegorczyk, Y. Zhang, et al. Cerenkov radiation in materials with negative permittivity and permeability. Optics Express, 2003, 11 (7): 723-724
[23] N. Seddon, T. Bearpark. Observation of the inverse Doppler Effect. Science. 2003, 302: 1537
[24] J. B. Pendry, D. R. Smith. Reversing light: negative refraction. Physics Today. 2004, 57: 37
[25] K. Kempa, R. Ruppin, J. B. Pendry. Electromagnetic response of a point-dipole crystal. Phys. Rev. B 72, 205103 (2005)
[26] A. Ahmadi, H. Mosallaei. Physical configuration and performance modeling of all-dielectric metamaterials. Phys. Rev. B 77, 045104 (2008)
[27] P. Markos, I. Rousochatzakis, C. M. Soukoulis. Transmission losses in left-handed materials. Phys. Rev. E66, 045601 (2002)
[28] S. Igor, S. Nefedov, A. Tretyakov. Photonic band gap structure containing metamaterial with negative permittivity and permeability. Phys. Rev. E 66, 036611 (2002)
[29] V. E. George, K. I. Ashwin, C. K. Peter. Planar negative refrative index media using periodically L-C loaded transmission lines. IEEE Trans Microw. Theory Tech, 2002, 50 (12): 2072-2712
[30] G. Anthony, V. E. George. Growing evanescent wave in negative refractive index transmission line media. Appl. Phus. Lett, 2003, 82(12): 1815-1817
[31] C. Caloz, T. Itoh. Characteristics and potential application of nonlinear left handed transmission line. Microwave Opt. Technol. Lett, 2004, 40 (6): 471-473
[32] A. Grbic, G. V. Eleftheriades. Experimental verification of backward-wave radiation from a negative refractive index metamaterial. Appl. Phys, 2002, 92 (10): 5930-5935
[33] A. Alù, N. Engheta, R. W. Ziolkowski. Finite-difference time-domain analysis of the tunneling and growing exponential in a pair ofε-negative andμ-negative slabs Phys. Rev. E 74, 016604 (2006)– Published July 18, 2006
[34] S. Linden, C. Enkrich, M. Wegener. Magnetic response of metamaterials at 100 terahertz. Science, 2004, 306 (5700): 1351-1353
[35] T. A. Leskova, A. A. Maradudin, J. M. Lopez. Coherence of light scattered from a randomly rough surface. Phys. Rev. E 71, 036606 (2005)
[36] M. Rahm, S. A. Cummer, D. Schurig. Optical Design of Reflectionless Complex Media by Finite Embedded Coordinate Transformations. Phys. Rev. Lett. 100, 063903 (2008)
[37] D. R. Smith, D. Schurig, J. B. Pendry, Negative refraction of modulated electromagnetic waves. Appl. Phys. Lett. 2002, 81 (15): 2173-2174
[38] J. R. Thomas. Wave packet incidenton negative index media. IEEE Trans. Antennas Propag, 2005, 53(5): 1591-1599
[39] G. D. Aguanno, N. Mattiucci, M. Scalora, et al. Bright and dark gap solitons in a negative index Fabry-Perot etalon. Phys. Rev. Lett, 2004, 93 (21): 213902
[40] I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar. Guided modes in negative refractive index waveguides. Phys. Rev. E, 2003, 67 (5): 057602
[41] Liang Wu, Sailing He, Linfang Shen. Band structure for a one-dimensional photonic crystal containing left-handed materials. Phys Rev B, 67, 235103 (2003)
[42] M. Gorkunov, M. Lapine. Tuning of a nonlinear metamaterial band gap by an external magnetic field. Phys. Rev. B, 2004 (23), 70: 235109
[43] Hongjie Zhao, Lei Kang, Ji Zhou, et al. Experimental demonstration of tunable negative phase velocity and negative refraction in a ferromagnetic/ferroelectric composite metamaterial. Appl. Phys. Lett. 2008, 93: 201106
[44] S. A. Darmanyan, M. Neviere, A. A. Zakhidov. Nonlinear surface waves at the interfaces of left-handed electromagnetic media. Phys. Rev. E ,2005, 72: 036615
[45] N. Mattiucci, G. D'Aguanno, M. J. Bloemer. Second-harmonic generation from a positive- negative index material heterostructure. Phys. Rev. E, 2005, 72: 066612
[46] M. W. Feise, I. V. Shadrivov, Y. S. Kivshar. Tunable transmission and bistability in left handed band gap structure. Appl. Phys. Lett, 2004, 85 (9): 1451-1453
[47] M. L. Natalia. Optical Disability in a Nonlinear Optical Coupler with a Negative Index Channel. Phys. Rev. Lett, 2007, 99: 113902
[48] M. Marklund, P. K. Shukla, L. Stenflo. Ultrashort solitons and kinetic effects in nonlinear metamaterials. Phys. Rev. E, 2006, 73, 037601
[49] G. P. Agrawal. Nonlinear fiber optics. 3rd ed. San Diego: Academic Press,2001
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