高阶效应影响下光孤立波在负折射介质中的传输特性研究
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摘要
负折射介质是一种人工合成的、介电常数ε和磁导率μ同时为负值的电磁材料,它能够表现出自然界中常规介质所不具有的许多特殊电磁特性。近年来,这种新型人工电磁材料得到了快速的发展,受到了包括光学、电磁学、材料学、通信学等相关领域研究人员的广泛关注。随着近红外和可见光频段非线性负折射介质的研制成功,负折射介质中丰富的非线性特性也逐渐引起了人们的注意,超短光脉冲在负折射介质中的传输及其在光器件和全光控制等方面的应用也成为了新的研究热点。
本文主要研究了高阶效应影响下的光孤立波在负折射介质中的传输特性。从Maxwell方程组出发,推导出能够描述飞秒光脉冲在负折射介质中传输的归一化高阶非线性Schrodinger方程。接着,通过拟解法分别求解出该模型的亮、灰(暗)孤立波精确解,并且详细分析了各高阶效应对于孤立波传输特性的影响。研究结果对于进一步从数值模拟上研究负折射介质中光孤立波及其稳定性,以及相关光器件应用方面具有一定的价值。
本文的主要内容如下：
(1)首先介绍了负折射介质的概念、发展历史及不同于传统常规介质的电磁特性,负折射介质在微波、近红外及可见光波频段的实现,以及目前在负折射介质中有关光孤立波的国内外研究状况。
(2)从Maxwell方程组出发,基于Scalora等人考虑色散磁导率的负折射介质中光脉冲传输理论模型,推导出可以描述负折射介质中飞秒光脉冲传输的归一化非线性Schrodinger方程,该模型包含了高阶色散与非线性效应,尤其包含了常规介质中光脉冲传输所不具有的二阶非线性色散效应,并且显式地给出了各阶色散与非线性效应同负介电常数ε和负磁导率μ的关系。依据Drude色散模型,详细分析了各阶效应的参数取值模型,结论充分说明通过调整负折射介质的结构可以改变介电常数ε和负磁导率μ,从而达到控制各阶色散与非线性效应的目的,这对于负折射介质中孤立波的形成及实现全光控制具有一定的意义。
(3)基于推导的非线性Schrodinger方程,通过拟解方法,解析得到在不同高阶效应平衡下三种形式的亮孤立波解,并且详细分析了这三种亮孤立波在负折射介质中的存在条件、孤立波特性以及各高阶效应对于亮孤立波传输特性的影响。研究发现,在一定条件下亮孤立波可以存在于负折射介质中,并且高阶效应对于亮孤立波解的影响比较明显,使得孤立波能够表现出不同于常规介质中的一些特性。而对于含有五阶非线性和二阶非线性色散效应的亮孤立波解来说,在负折射介质中很难存在,但是这种亮孤立波在人工合成的正常色散介质中却可以存在。
(4)同样,采用拟解法解析得到不同高阶效应存在情况下的三种灰(暗)孤立波解,并且详细分析了每种情况灰(暗)孤立波的特性。研究发现,这三种情况的灰(暗)孤立波都可以存在于负折射介质中,并且高阶效应对于第二、三种灰(暗)孤立波的产生及传输特性起着至关重要的影响。特别是第三种情况中,五阶非线性和二阶非线性色散效应对于灰(暗)孤立波特性的影响比较明显。当不存在线性色散效应时,各种非线性效应之间同样能平衡而产生灰(暗)孤立波。
Negative index materials (NIMs) is a kind of artificial material with simultaneously negative dielectric permittivityεand negative magnetic permeabilityμelectromagnetic, which can exhibit different electromagnetic properties from conventional materials in nature. Recently, the rapid developing artificial electromagnetic materials have received great attentions including optics, electromagnetics, materials science, communications, etc. With the successful fabrication of NIMs over near-infrared and visible light frequencies, the abundant nonlinear properties of NIMs gradually attracts much attention, the transmission of ultrashort optical pulse in NIMs and the potential applications in optical device and all-optical control have become a new hotspot.
In this paper, the transmissions of optical solitary waves in NIMs under the higher-order effects are studied. Starting from Maxwell equations, a normalized nonlinear Schrodinger equation has been derived, which can describe the propagation of femtosecond optical pulse in NIMs. Subsequently, the exact bright and gray (dark) solitary wave solutions of this model are obtained by ansatz method, and the influences of higher-order effects on propagation properties of the solitary wave solutions are analyzed in detail. The obtained results will have much significance for the numerical study of the stability of solitary waves in NIMs and the applications in optical devices.
The main contents are as follows:
(1) Firstly, we introduce the concept, development and different electromagnetic properties of NIMs from the conventional materials, as well as the realization of NIMs over the microwave, near-infrared and visible light frequencies, and the research progress of the optical solitary waves in NIMs.
(2) Starting from Maxwell equations, on the base of Scalora's theoretical model describing the propagation of ultrashort pulse in NIMs with negatively dispersive permittivity and permeability, a normalized nonlinear Schrodinger equation is derived, which can govern the femtosecond optical pulses propagation in NIMs. The model includes the higher-order dispersion and nonlinear effects, especially the second-order nonlinear dispersion effect which does not exist in conventional materials, and all effects in the model are directly expressed by the negative permittivityεand negative permeabilityμin explicit. Under the Drude dispersion model, the parameter model is analyzed in detail and the results show that the dispersion and nonlinear effects can be controlled by adjusting the permittivity and permeability, namely, by engineering the structure of the NIMs, which will be of a momentous significance for the formation of solitons and all-optical control in NIMs.
(3) Based on the derived nonlinear Schrodinger equation, three cases of bright solitary wave solutions under different higher-order effects are obtained through the ansatz method, and then the existence conditions and properties of bright solitary waves as well as the influence of the higher-order effects are analyzed in detail. The results show that bright solitary waves can exist in NIMs under certain conditions, and the high-order effects play an important role in the formation of bright solitary waves which can exhibit many particular features comparing to the conventional materials. As for the bright solitary wave in the presence of the fifth-order nonlinearity and second-order nonlinear dispersion effects, it is difficult to exist in NIMs, but can exist in artificial materials with normal dispersion.
(4) Similarly, three cases of gray(dark) solitary wave solutions in NIMs under different higher-order effects are obtained by using ansatz method, and the characteristics of gray(dark) solitary waves in NIMs are analyzed in detailed. It can be found that all the three cases of gray (dark) solitary waves can exist in NIMs under proper conditions, and the higher-order effects have a great impact on the formation and propagation characteristics of the second and third cases of gray (dark) solitary waves. In particular, the fifth-order nonlinearity and the second-order nonlinear dispersion effects play a crucial role in the third case of gray (dark) solitary wave. Moreover, the gray (dark) solitary waves can be formed under the balance of the nonlinear effects without linear dispersion.
本文主要研究了高阶效应影响下的光孤立波在负折射介质中的传输特性。从Maxwell方程组出发,推导出能够描述飞秒光脉冲在负折射介质中传输的归一化高阶非线性Schrodinger方程。接着,通过拟解法分别求解出该模型的亮、灰(暗)孤立波精确解,并且详细分析了各高阶效应对于孤立波传输特性的影响。研究结果对于进一步从数值模拟上研究负折射介质中光孤立波及其稳定性,以及相关光器件应用方面具有一定的价值。
本文的主要内容如下：
(1)首先介绍了负折射介质的概念、发展历史及不同于传统常规介质的电磁特性,负折射介质在微波、近红外及可见光波频段的实现,以及目前在负折射介质中有关光孤立波的国内外研究状况。
(2)从Maxwell方程组出发,基于Scalora等人考虑色散磁导率的负折射介质中光脉冲传输理论模型,推导出可以描述负折射介质中飞秒光脉冲传输的归一化非线性Schrodinger方程,该模型包含了高阶色散与非线性效应,尤其包含了常规介质中光脉冲传输所不具有的二阶非线性色散效应,并且显式地给出了各阶色散与非线性效应同负介电常数ε和负磁导率μ的关系。依据Drude色散模型,详细分析了各阶效应的参数取值模型,结论充分说明通过调整负折射介质的结构可以改变介电常数ε和负磁导率μ,从而达到控制各阶色散与非线性效应的目的,这对于负折射介质中孤立波的形成及实现全光控制具有一定的意义。
(3)基于推导的非线性Schrodinger方程,通过拟解方法,解析得到在不同高阶效应平衡下三种形式的亮孤立波解,并且详细分析了这三种亮孤立波在负折射介质中的存在条件、孤立波特性以及各高阶效应对于亮孤立波传输特性的影响。研究发现,在一定条件下亮孤立波可以存在于负折射介质中,并且高阶效应对于亮孤立波解的影响比较明显,使得孤立波能够表现出不同于常规介质中的一些特性。而对于含有五阶非线性和二阶非线性色散效应的亮孤立波解来说,在负折射介质中很难存在,但是这种亮孤立波在人工合成的正常色散介质中却可以存在。
(4)同样,采用拟解法解析得到不同高阶效应存在情况下的三种灰(暗)孤立波解,并且详细分析了每种情况灰(暗)孤立波的特性。研究发现,这三种情况的灰(暗)孤立波都可以存在于负折射介质中,并且高阶效应对于第二、三种灰(暗)孤立波的产生及传输特性起着至关重要的影响。特别是第三种情况中,五阶非线性和二阶非线性色散效应对于灰(暗)孤立波特性的影响比较明显。当不存在线性色散效应时,各种非线性效应之间同样能平衡而产生灰(暗)孤立波。
Negative index materials (NIMs) is a kind of artificial material with simultaneously negative dielectric permittivityεand negative magnetic permeabilityμelectromagnetic, which can exhibit different electromagnetic properties from conventional materials in nature. Recently, the rapid developing artificial electromagnetic materials have received great attentions including optics, electromagnetics, materials science, communications, etc. With the successful fabrication of NIMs over near-infrared and visible light frequencies, the abundant nonlinear properties of NIMs gradually attracts much attention, the transmission of ultrashort optical pulse in NIMs and the potential applications in optical device and all-optical control have become a new hotspot.
In this paper, the transmissions of optical solitary waves in NIMs under the higher-order effects are studied. Starting from Maxwell equations, a normalized nonlinear Schrodinger equation has been derived, which can describe the propagation of femtosecond optical pulse in NIMs. Subsequently, the exact bright and gray (dark) solitary wave solutions of this model are obtained by ansatz method, and the influences of higher-order effects on propagation properties of the solitary wave solutions are analyzed in detail. The obtained results will have much significance for the numerical study of the stability of solitary waves in NIMs and the applications in optical devices.
The main contents are as follows:
(1) Firstly, we introduce the concept, development and different electromagnetic properties of NIMs from the conventional materials, as well as the realization of NIMs over the microwave, near-infrared and visible light frequencies, and the research progress of the optical solitary waves in NIMs.
(2) Starting from Maxwell equations, on the base of Scalora's theoretical model describing the propagation of ultrashort pulse in NIMs with negatively dispersive permittivity and permeability, a normalized nonlinear Schrodinger equation is derived, which can govern the femtosecond optical pulses propagation in NIMs. The model includes the higher-order dispersion and nonlinear effects, especially the second-order nonlinear dispersion effect which does not exist in conventional materials, and all effects in the model are directly expressed by the negative permittivityεand negative permeabilityμin explicit. Under the Drude dispersion model, the parameter model is analyzed in detail and the results show that the dispersion and nonlinear effects can be controlled by adjusting the permittivity and permeability, namely, by engineering the structure of the NIMs, which will be of a momentous significance for the formation of solitons and all-optical control in NIMs.
(3) Based on the derived nonlinear Schrodinger equation, three cases of bright solitary wave solutions under different higher-order effects are obtained through the ansatz method, and then the existence conditions and properties of bright solitary waves as well as the influence of the higher-order effects are analyzed in detail. The results show that bright solitary waves can exist in NIMs under certain conditions, and the high-order effects play an important role in the formation of bright solitary waves which can exhibit many particular features comparing to the conventional materials. As for the bright solitary wave in the presence of the fifth-order nonlinearity and second-order nonlinear dispersion effects, it is difficult to exist in NIMs, but can exist in artificial materials with normal dispersion.
(4) Similarly, three cases of gray(dark) solitary wave solutions in NIMs under different higher-order effects are obtained by using ansatz method, and the characteristics of gray(dark) solitary waves in NIMs are analyzed in detailed. It can be found that all the three cases of gray (dark) solitary waves can exist in NIMs under proper conditions, and the higher-order effects have a great impact on the formation and propagation characteristics of the second and third cases of gray (dark) solitary waves. In particular, the fifth-order nonlinearity and the second-order nonlinear dispersion effects play a crucial role in the third case of gray (dark) solitary wave. Moreover, the gray (dark) solitary waves can be formed under the balance of the nonlinear effects without linear dispersion.
引文
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[8]贾东方,余震虹,谈斌,胡智勇等译.非线性光纤光学原理及应用.北京.电 子工业出版社.2002年.
[9]谷超豪,郭柏灵.孤立子理论与应用.1.杭州,浙江科学技术出版社,1990,1-5.
[10]黄景宁,徐济仲,熊吟涛。孤子概念、原理和应用。北京：高等教育出版社,2004,3.
[11]Landau L. D. and Lifshitz E. M. Electrodynamics of Continuous Media. Pergamon Press, New York,1960(pp.253-256).
[12]Syrchin M, Zheltikov A. M. and Scalora M. Analytical Treatment of Self-phase-modulation Beyond the Slowly Varying Envelope Approximation. Phys. Rev. A 2004,69,053803.
[13]Ziolkowski R. W. Superluminal transmission of information through an electromagnetic metamaterial. Phys. Rev. E 2001,63,046604.
[14]Ziolkowski R. W. and Heyman E. Wave Propagation in Media Having Negative Permittivity and Permeability. Phys. Rev. E 2001,64,056625.
[1]Agrawal G. P. Nonlinear Fiber Optics(4th Ed). SanDiego, California. Academic Press,2007.
[2]黄景宁,徐济仲,熊吟涛。孤子概念、原理和应用。北京：高等教育出版社,2004,3.
[3]谷超豪,郭柏灵.孤立子理论与应用.1.杭州,浙江科学技术出版社,1990,1-5.
[4]李仲豪.飞秒光孤立波在光纤中的传输特性的研究.太原,山西大学博士研究生学位论文.2000
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[6]杨荣草.光脉冲在非均匀光纤系统中的稳定传输特性研究.太原,山西大学博士研究生学位论文.2005.
[7]Zhang S. W. and Yi L. Exact Solutions of a Generalized Nonlinear Schrodinger Equation. Phys. Rev. E,2008,78,026602.
[8]Tsitsas N. L, Rompotis N, Kourakis I, Kevrekidis P. G. and Frantzeskakis. Higher-order Effects and Ultrashort Solitons in Left-handed Metamaterials. Phys. Rev. E 2009,79,037601.
[9]Zharova N. A, Shadrivov I. V, Zharov A. A. and Kivshar Yu. S. Nonlinear Transmission and Spatiotemporal Solitons in Metamaterials with Negative Refraction. Opt. Express 2005,13(4),1291-1298.
[10]Subha P. A, Jisha C. P. and Kuriakose V. C. Stable Diffraction-managed Spatial Solitons in Cubic-quintic Media. J. Mod. Opt..2007,54,1827-1835.
[11]Boardmana A. D, King N, Mitchell-Thomas R.C, Malnev V. N. and Rapoport Y. G. Gain Control and Diffraction-Managed Solitons in Metamaterials. Metamaterials 2008,2,145-154.
[12]Dai X. Y, Xiang Y. J, Wen S. C. and Fan D. Y. Frequency Characteristics of the Dark and Bright Surface Solitons at a Nonlinear Metamaterial Interface. Optics Communications 2010,283,1607-1612.
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[1]Zharov A. A, Shadrivov I. V. and Kivshar Yu. S. Nonlinear Properties of Left-Handed Metamaterials. Phys. Rev. Lett.2003,91,037401.
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[9]谷超豪,郭柏灵.孤立子理论与应用.1.杭州,浙江科学技术出版社,1990,1-5.
[10]黄景宁,徐济仲,熊吟涛。孤子概念、原理和应用。北京：高等教育出版社,2004,3.
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[13]Ziolkowski R. W. Superluminal transmission of information through an electromagnetic metamaterial. Phys. Rev. E 2001,63,046604.
[14]Ziolkowski R. W. and Heyman E. Wave Propagation in Media Having Negative Permittivity and Permeability. Phys. Rev. E 2001,64,056625.
[1]Agrawal G. P. Nonlinear Fiber Optics(4th Ed). SanDiego, California. Academic Press,2007.
[2]黄景宁,徐济仲,熊吟涛。孤子概念、原理和应用。北京：高等教育出版社,2004,3.
[3]谷超豪,郭柏灵.孤立子理论与应用.1.杭州,浙江科学技术出版社,1990,1-5.
[4]李仲豪.飞秒光孤立波在光纤中的传输特性的研究.太原,山西大学博士研究生学位论文.2000
[5]李录.光脉冲在光纤中的传输特性的理论研究.太原,山西大学博士研究生学论文.2004,2-3.
[6]杨荣草.光脉冲在非均匀光纤系统中的稳定传输特性研究.太原,山西大学博士研究生学位论文.2005.
[7]Zhang S. W. and Yi L. Exact Solutions of a Generalized Nonlinear Schrodinger Equation. Phys. Rev. E,2008,78,026602.
[8]Tsitsas N. L, Rompotis N, Kourakis I, Kevrekidis P. G. and Frantzeskakis. Higher-order Effects and Ultrashort Solitons in Left-handed Metamaterials. Phys. Rev. E 2009,79,037601.
[9]Zharova N. A, Shadrivov I. V, Zharov A. A. and Kivshar Yu. S. Nonlinear Transmission and Spatiotemporal Solitons in Metamaterials with Negative Refraction. Opt. Express 2005,13(4),1291-1298.
[10]Subha P. A, Jisha C. P. and Kuriakose V. C. Stable Diffraction-managed Spatial Solitons in Cubic-quintic Media. J. Mod. Opt..2007,54,1827-1835.
[11]Boardmana A. D, King N, Mitchell-Thomas R.C, Malnev V. N. and Rapoport Y. G. Gain Control and Diffraction-Managed Solitons in Metamaterials. Metamaterials 2008,2,145-154.
[12]Dai X. Y, Xiang Y. J, Wen S. C. and Fan D. Y. Frequency Characteristics of the Dark and Bright Surface Solitons at a Nonlinear Metamaterial Interface. Optics Communications 2010,283,1607-1612.
[1]Hasegawa A. and Tappert F. Transmission of Stationary Nonlinear Optical Pulses in Dispersive Dielectric Fibers. Ⅱ. Normal Dispersion. Appl, Phys. Lett.1973, 23(4),171-172.
[2]Emplit P, Hamaide J. P, Reynaud R, Frodhly C. and Barthelemy A. Picosecond steps and dark pulses through nonlinear single mode fibers. Opt. Commun.1987,62(6),374-379.
[3]Dianov E. M, Mamyshev P. V, Prokhorov A. M. and Chernikov S. V. Generation of a Train of Fundamental Solitons at a High Repetition Rate in Optical Fibers. Opt. Lett.1989,14(18),1008-1010.
[4]Hasegawa A. and Kodama Y. Soliton in Optical Communications. Oxford University Press, New York,1995.
[5]Krokel D, Halas N. J, Giuliani G. and Grischkowsky D. Dark-Pulse Propagation in Optical Fibers. Phys. Rev. Lett.1988,60,29-32.
[6]Weiner A. M, Heritage J. P, Hawkins R. J, Thurston R. N, Kirschner, Leaird D. E. and Tomlinson W. J. Experimental Observation of the Fundamental Dark Soliton in Optical Fibers. Phys. Rev. Lett.1988,61,2445-2448.
[7]Zhao W. and Bourkoff E. Propagation Properties of Dark Soltions. Opt. Lett.1989, 14(13),703-705.
[8]Zhao W. and Bourkoff E. Interactions Between Dark Solitons. Opt. Lett.1989, 14(24),1371-1373.
[9]Emplit Ph, Haelterman M. and Hamaide J-P. Picosecond Dark Soliton over a 1-km Fiber at 850nm. Opt. Lett.1993,18(13),1047-1049.
[10]Nakazawa M. and K. Suzuki. Generation of a Pseudorandom Dark Soliton Data Train and Its Coherent Detection by One-bit-shifting with a Mach-Zehnder Interferometer. Electron. Lett.1995,31(13),1084-1085.
[11]Agrawal G. P. Nonlinear Fiber Optics(4th Ed). SanDiego, California. Academic Press,2007.
[12]黄景宁,徐济仲,熊吟涛。孤子概念、原理和应用。北京：高等教育出版社,2004,3.
[13]谷超豪,郭柏灵.孤立子理论与应用.1.杭州,浙江科学技术出版社,1990,1-5.
[14]郭柏灵,庞小峰,孤立子,科学出版社,1987年2月.
[15]陈陆君,梁昌洪著,孤立子理论及其应用,西安电子科技大学出版社,1997.
[16]刘海兰,文双春,熊敏,戴小玉.超常介质中暗孤子的形成和传输特性研究.物理学报,2007,56,6473-6477.