摘要
光子代替电子作为信息的载体是人们的一个共识,因为光子技术具有高传输速度、高密度及高容错性等优点。然而,由于光子不像电子一样易于控制,光子器件远不如电子器件成熟,致使光信息技术仅仅在信息传输中得到应用,而且是最基本的信息功能。研究光波与新型光子材料的相互作用,探索利用光子材料对光子的操纵和控制,是发展新型光子器件的基础,对光计算、全光通信等领域具有重要的理论和实际意义。周期性微结构光子材料,如布拉格光栅、光子晶体、光学格子、超常介质等,使人们操纵和控制光子的梦想成为可能,是发展全光器件的理想材料。本论文着重研究最近几年发展的两种新型的周期性微结构光子材料即光学格子和超常介质中光波的非线性传输特性,进行了如下的工作:
第一,光学格子是指具有横向周期性调制折射率的光学介质。光束在非线性光学格子中传输时展现出丰富的令人感兴趣的现象,特别是,横向折射率的周期性调制深刻地影响空间孤子的形成和传输特性。我们利用变分法和数值方法研究了克尔型非线性光学格子中光束的传输,求出了光束宽度、振幅、频率啁啾参量随传播距离的演化形式,揭示了光学格子的调制周期和调制深度对光波非线性传输的影响,得到了格子孤子的形成和稳定传输的条件。发现光束宽度与调制周期的比值必须小于一定的值才能形成孤子的传输,周期性格子有类似于非线性的良好特性,从而为更好地控制格子孤子的形成和传输提供了另一个自由度。
第二,损耗是所有系统的固有属性,光学格子也不例外。为有效克服损耗对光学格子孤子的影响,我们借鉴色散渐变光纤中利用色散的缓变来补偿因光纤中的损耗而导致非线性效应减弱的方案,首次提出通过控制光格子的调制深度和调制周期来补偿光学格子介质的损耗效应,以在实际有损耗的光学格子介质中实现稳定的孤子传输。为论证该方案,利用解析和数值方法研究了空间光孤子在具有损耗的Bessel光格子中的传输,通过变分法得到了光束宽度、振幅和波面曲率的动力学方程,结果表明,通过适当地增加光格子的折射率调制深度,介质的损耗效应能得到精确的补偿,从而达到稳定的空间孤子的传输。
第三,超常材料通常是指人工构造的、具有自然材料所不具备的特性的材料,是当今重大科学前沿之一。我们结合最新的超常介质和传统的非线性光学原理研究了超常介质中光波的非线性传输特性。超常介质与常规光学介质的一个最重要的区别是前者具有色散磁导率。将色散磁导率合并到非线性极化项中,借鉴常规介质中超短脉冲传输方程的推导方法,得到了非线性超常介质中超短脉冲的传输方程。在Drude色散模型下,根据脉冲中心频率的不同在传输方程中出现了可正、可负、可为零的自陡峭系数,以及高阶非线性色散项。此外,利用矩方法对得到的传输方程进行分析,得到了超常介质中超短脉冲传输方程的能量守恒定律表达式,揭示了色散磁导率导致的超短脉冲传输的新特性,发现二阶非线性色散使超短脉冲的能量、脉冲频移、脉冲宽度、中心位置和啁啾都随传输距离呈现振荡式变化。
第四,基于我们得到的非线性超常介质中超短脉冲的传输方程,研究了完全相干和部分相干超短脉冲在超常介质中传输的调制不稳定性,着重讨论了由超常介质中色散磁导率导致的非线性色散项对调制不稳定性的影响。推导了部分相干超短脉冲的Wigner–Moyal传输方程,以及发生调制不稳定性的色散关系。首次发现二阶非线性色散在调制不稳定性中的作用在某种程度上与群速度色散的作用是等效的,因此,由于二阶非线性色散的作用,调制不稳定性可以发生在其他不可能发生的情况,例如在正常色散情况下。
It is a consensus to replace electron with photon as the carrier of information because photonic technology has several advantages, such as high transmission speed, high density and high fault tolerance. However, photons are not so prone to be controlled as electrons, and the photonic devices are far from mature compared to electronic component, which result in that optical information technonlogy has been only applied to information transmission, further more, the basic information function. Thus, the research on the interaction between light wave and new type photonic material and the exploration of technologies of controlling photon by using photonic material are the basis of the development of novel photonic devices and are very important in optical calculation and all optical communication, both theoretically and practically. Periodically microstructure photonic material as Bragg gratings, photonic crystal, optical lattice, and metamaterials et al, are ideal material for all-optical devices because of their ability in manipulating and controlling photons. In this thesis, we investigate the properties of nonlinear propagation in two kinds of new type periodically microstructure optical material, i.e. optical lattice and metamaterials. Our work and results are mainly follows:
Firstly, optical lattice is an optical media with transverse periodic lattice modulation refractive index. Beams appear plenty of interesting phenomena when they propagate in the nonlinear optical lattice, especially periodic modulation with transverse refractive index can affect deeply the form and transmission characteristic of spacial solition. Using the variational principle and numerical method, we study the beam evolution of Kerr nonlinear optical lattice, and obtain the forms for the evolution during propagation of beam width, beam amplitude and frequency chirp, and post the influence of modulation period and modulation depth of optical lattice on the light-wave’s nonlinear propagation. The following, we find the conditions for lattice soliton formation and stabile propagation. We find that solitions can propagate only if the ratio of beam width and modulation period are less than a certain numerical value. With the good characteristic similar to nonlinearity, periodic lattice can offer a better method to control the lattice soliton formation and propagation.
Secondly, attenuation is an intrinsic property of any practical system, including optical lattice. Therefore, it is important to compensate the medium loss for maintaining the propagation of the spatial soliton in the system. We first propose and demonstrate a scheme to compensate medium loss for spatial soliton propagation, i.e., controlling the modulation depth of a Bessel lattice along the light propagation direction to compensate the loss effect. Here, we investigated the propagation of a spatial soliton in a dissipative modulated Bessel optical lattice, both analytically and numerically. The dynamic evolution equations for beam width, amplitude, and curvature wavefront are obtained by a variational approach. It is shown that by properly increasing the modulation depth of refractive index of the optical lattice, the loss effect can be compensated exactly to fulfill stable spatial soliton propagation.
Thirdly, metamaterials are artificial materials which have anomalous properties not possessed by natural materials. The research about metamaterials is one of the important frontline in modern scientific domain. One of the most important differences between metamaterials and conventional materials is that the magnetic permeabilities of metamaterials are dispersive. Combining the properites of metamatirials and the related principles of nonlinear optics, we have investigated the propagation properties of light wave in metamatirals. It is shown that, under the Drude dispersive model, the dispersive permeability results in a self-steepening parameter which can be negative, positive or zero depending on the central frequency of the pulse, and a series of higher-order nonlinear dispersion terms in the propagation equation. Furthermore, the propagation equation is analyzed by using the moment method, an explicit expression for power conservation for the propagation equation is obtained, and the unique propagation properties of ultrashort pulse in metamaterials are disclosed. It is found that due to the role of the second-order nonlinear dispersion, the characteristic parameters of the ultrashort pulse, including energy, frequency shift, duration, center position, and chirp, all oscillate with propagation distance.
Fourthly, on the basis of the propagation equation obtained for ultrashort pulse in nonlinear metamaterials we have investigated the modulational instability in metamaterials of the propagation of both coherent and incoherent ultrashort pulses. The combination of dispersive magnetic permeability with nonlinear polarization leads to a series of nonlinear dispersion terms in the propagation equations for ultrashort pulses in metamaterials. Here we present an investigation of modulation instability (MI) of both coherent and partially coherent ultrashort pulses in metamaterials to identify the role of nonlinear dispersion in pulse propagation. The Wigner–Moyal equation for partially coherent ultrashort pulses and the nonlinear dispersion relation for MI in metamaterials are derived. Combining the standard MI theory with the unique properties of the metamaterial, the influence of the controllable first-order nonlinear dispersion, namely self-steepening, and the second-order nonlinear dispersion on both coherent and partially coherent MI, in both negative-index and positive-index regions of the metamaterial for all physically possible cases is analyzed in detail. For the first time to our knowledge, we demonstrate that the role of the second-order nonlinear dispersion in MI is equivalent to that of group-velocity dispersion (GVD) to some extent, and thus due to the role of the second-order nonlinear dispersion, MI may appear in the otherwise impossible cases, such as in the normal GVD regime.
引文
[1] Kartashov Y V, Zelenina A S, and Torner L, et al. Spatial soliton switching in quasi-continuous optical arrays. Opt. Lett., 2004, 29(7): 766-768
[2] Christodoulides D N, et al. Discretizing light behavior in linear and nonlinear waveguide lattices. Nature, 2003, 424(14): 817-822
[3] Stegeman G I, Segev M. Optical Spatial Solitons and Their Interactions: Universality and Diversity. Science, 1999, 286(19): 1518-1523
[4] Fleischer J W, Mordechai S, and Nikolaos K, et al. Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature, 2003, 422(6): 147-150
[5] Lederer F and Silberberg Y. Discrete Soliton. Opt. Photon News, 2002, 13(2): 48-53
[6] Neshev D, Ostrovskaya E, and Kivshar Y S, et al. Soliton stripes in two-dimensional nonlinear photonic lattices. Opt. Lett., 2003, 28(9): 710-712
[7] Neshev D N, Alexander T J, and Ostrovskaya E A, et al. Observation of discrete vortex solitons in optically induced photonic lattices. Phys. Rev. Lett., 2004, 92(12): 123903
[8] Fleischer J W, Bartal G, and Cohen O, et al. Observation of vortex-ring “discrete” solitons in 2d photonic lattices. Phys. Rev. Lett., 2004, 92(12):123904
[9] Kivshar Y S, Agrawal G P. Optical Solitons: From Fibers to Photonic Crystals. The first. San Diego: Academic Press, 2003, Chap 2
[10] Wen S C, Xu W C, and Guo Q, et al. Evolution of solitons of nonlinear Schr?dinger eauation with variable parameters. Sci. China Series A, 1997, 40(12): 1300-1304
[11] 佘卫龙,王晓生,何国岗等. 折射率改变为正的光这边晶体中形成以为光伏暗孤子. 物理学报, 2001, 50(11): 2166-2171
[12] 文双春, 钱列加, 范滇元. 强光束局部小尺度调制致多路成丝现象研究. 物理学报, 2003, 52(7): 1640-1644
[13] Wen S C, Fan D Y. Filamentation instability in nonlocal nonlinear media. Chin. Phys., 2001, 10(11): 1032-1036
[14] Liu J S. universal theory of steady-state one-dimensional photorefractive soltitons. Chin. Phys., 2001, 10(11): 1037-1042
[15] Christodoulides D N, Joseph R I. Discrete self-focusing in nonlinear arrays of coupled waveguides. Opt. Lett., 1988, 13(9): 794-796
[16] Trombettoni A and Smerzi A. Discrete Solitons and Breathers with Dilute Bose-Einstein Condensates. Phys. Rev. Lett., 2001, 86(11): 2353-2356
[17] Efremidis N K, Sears S, and Christodoulides D N, et al. Discrete solitons in photorefractive optically-induced photonic lattices. Phys. Rev. E, 2002, 66(4):046602
[18] Efremidis N K, Hudock J, and Christodoulides D N, et al. Two-Dimensional Optical Lattice Solitons. Phys. Rev. Lett., 2003, 91(21):213906
[19] Neshev D, Kivshar Y S, and Martin H, et al. Soliton stripes in two-dimensional nonlinear photonic lattices. Opt. Lett., 2004, 29(5): 486-488
[20] Pertsch T, Zentgraf T, and Peschel U, et al. Anomalous refraction and diffraction in discrete optical systems. Phys. Rev. Lett., 2002, 88(9):093901
[21] Mandelik D, Eisenberg H S, and Silberberg Y, et al. Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons. Phys. Rev. Lett., 2003, 90(5):053902
[22] Waschke C, Roskos H G, and Schwedler R, et al. Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice. Phys. Rev. Lett., 1993, 70(21): 3319–3322
[23] Morandotti R, Peschel U, and Aitchison J S, et al. Experimental observation of linear and nonlinear optical Bloch oscillations. Phys. Rev. Lett., 1999, 83(23): 4756–4759
[24] Pertsch T, Dannberg P, and Elflein W, et al. Optical Bloch oscillations in temperature tuned waveguide arrays. Phys. Rev. Lett., 1999, 83(23): 4752–4755
[25] Kartashov Y V, Wsloukh V A, and Torner L. Tunable Soliton Self-Bending in Optical Lattices with Nonlocal Nonlinearity. Phys. Rev. Lett., 2004, 93(15):153903
[26] Vicencio R A, Molina M, and Kivshar Y S. Controlled switching of discrete solitons in waveguide arrays. Opt. Lett., 2003, 28(20):1942-1944
[27] Eisenberg H S, Silberberg Y, and Morandotti R, et al. Diffraction management. Phys. Rev. Lett., 2000, 85(9): 1863-1866
[28] Anastassiou C, Fleischer J W, and Carmon T, et al. Information transfer via cascaded collisions of vector solitons. Opt. Lett., 2001, 26(19): 1498-1500
[29] Shelby R A, Smith D R, and Schultz S. Experimental verification of a negative index of refraction. Science, 2001, 292(6): 77-79
[30] Pendry J B. Negative refraction makes a perfect lens. Phys. Rev. Lett., 2000, 85(18): 3966-3969
[31] Chiao R Y, Garmire E, and Townes C H. Self-Trapping of Optical Beams. Phys. Rev. Lett., 1964, 13(15): 479-482
[32] Kelley P L. Self-Focusing of Optical Beams. Phys. Rev. Lett., 1965, 15(26): 1005-1008
[33] Duree G C, Jr, Shultz J L, et al. Observation of self-trapping of an optical beam due to the photorefractive effect. Phys. Rev. Lett., 1993, 71(4): 533-536
[34] Mihalache D, Mazilu D, and Flederer, et al. Stable spatiotemporal Solitons in Bessel Optical Lattices. Phys. Rev. Lett., 2005, 95(2):023902
[35] Eisenberg H S, Silberberg Y, and Morandotti R, et al. Discrete Spatial Optical Solitons in Waveguide Arrays. Phys. Rev. Lett., 1998, 81(16): 3383-3386
[36] Fischer R, Trager D, and Neshev D N, et al. Reduced-symmetry two-dimensional solitons in photonic lattices. Phys. Rev. Lett., 2006, 96(2):023905
[37] Trompeter H, Krolikowski W, and Neshev D N, et al. Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices. Phys. Rev. Lett., 2006, 96(5):053903
[38] Alexander T J, Ostrovskaya E A, and Kivshar Y S. Self-trapped nonlinear matter waves in periodic potentials. Phys. Rev. Lett., 2006, 96(4):04040
[39] Veselago V G. The electrodynamics of substances with simultaneously negative values of ε and μ. Sov. Phys. Usp., 1968, 10(4): 509-514
[40] Pendry J B, Holden A J, and Stewart W J, et al. Extremely low frequency plasmons in metallic mesostructures. Phys. Rev. Lett., 1996, 76(5): 4773-4776
[41] Pendry J B, Holden A J, and Stewart W J. Magnetism from conductors and enhanced nonlinear phenomena. IEEE Trans. Microwave Theory and Tech., 1999, 47(11): 2075-2084
[42] Smith D R, Padilla W J, and Vier D C, et al. Composite medium with simultaneously negative permeability and permittivity. Phys. Rev. Lett., 2000, 84(18): 4184-4187
[43] Parimi P V, Lu W T, and Vodo P, et al. Photonic crystals–imaging by flat lens using negative refraction. Nature, 2003, 426(23): 404-404
[44] Kong J A, Wu B I, and Zhang Y. Lateral desplacement of a guassian beam reflected from a grounded slab with negative permittivity and negative permeability. Appl. Phys. Lett., 2002, 80(12): 2084-2086
[45] Engheta N. An idea for thin subwavelengh cavity resonators using metamaterialswith negative permittivity and permeability. IEEE Antennas and Wireless Propagation Letters, 2002, 1(1): 10-13
[46] Ziolkowski R W, Heyman E. Application of double negative materials to increase the power radiated by electrically small antennas. IEEE Transactions, Antennas and Prop., 2003, 51(10): 2626-2640
[47] Caloz C, Chang C C, and Itoh T. Full-wave verification of the fundamental properties of left-handed materials in waveguide configurations. J. Appl. Phys., 2001, 90(11): 5483-5486
[48] Berrier A, Mulot M, and Swillo M, et al. Negative Refraction at Infrared- Wavelengths in a Two-Dimensional Photonic Crystal. Phys. Rev. Lett., 2004, 93(7): 073902
[49] Schonbrun E, Tinker M, and Park W, et al. Negative refraction in a Si-polymer photonic Crystal membrane. IEEE Photonics Technol. Lett., 2005, 17(6): 1196-1198
[50] Lazarides N, Tsironis G P. Coupled nonlinear Schrodinger field equations for electromagnetic wave propagation in nonlinear left-handed materials. Phys. Rev. E, 2005, 71(3): 036614
[51] Kourakis I, Shukla P K. Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials. Phys. Rev. E, 2005, 72(1): 016626
[52] Scalora M, Syrchin M S, and Akozbek N, et al. Generalized nonlinear Schr?dinger equation for dispersive susceptibility and permeability: application to negative index materials. Phys. Rev. Lett., 2005, 95(1): 013902
[53] Wen S C, Wang Y W, and Su W H, et al. Modulation instability in nonlinear negative-index material. Phys. Rev. E, 2006, 73(1): 1-7
[54] Wen S C, Xiang Y J, and Su W H, et al. Role of the anomalous self-steepening effect in modulation instability in negative-index material. Opt. Express, 2006, 14(4): 1568-1575
[55] Lazarides N, Tsironis G P. Coupled nonlinear Schrodinger field equations for electromagnetic wave propagation in nonlinear left-handed materials. Phys. Rev. E, 2005, 71(3): 036614
[56] Zakharov V and Shabat A. Exact theory of two-dimensional self-focusing andonedimensional self-modulation of waves in nonlinear media. Sov. Phys. -JETP, 1972, 34: 62-69
[57] Grischkowsky D and Balant A C. Optical pulse compression based on enhanced frequency chirping. Appl. Phys. Lett., 1982, 41(1): 1-3
[58] Nklaus B and Grischkowsky D. 12× pulse compression using optical fibers. Appl. Phys. Lett., 1983, 42(1): 1-2
[59] Vlasov S N, Petrishchev V A, and Talanov V I. Average description of wave beam in linear and nonlinear media (The method of moments). Radiophys and Quantum Electron, 1971, 14(9): 1062-1070
[60] Marcuse D. Pulse distortion in single-mode fibers. Applied Optics, 1980, 19(10): 1653-1660
[61] Pasmanik G A. Self-action of incoherent light beams. Zh. Eksp. Teor. Phys., 1974, 66(12): 490-500
[62] Shkunov V V and Anderson D Z. Radiation transfer model of self-trapping spatially incoherent radiation by nonlinear media. Phys. Rev. Lett., 1998, 81(13): 2683-2686
[63] Christodoulides D N, Coskun T H, and Mitchell M. Theory of incoherent self-focusing in biased photorefractive media. Phys. Rev. Lett., 1997, 78 (4): 646 -649
[64] Mitchell M, Segev M, and Coskun T H. Theory of self-trapped spatially incoherent light beams. Phys. Rev. Lett., 1997, 79(25): 4990-4993
[65] Hall B, Lisak M, and Anderson D. Statistical theory for incoherent light propagation in nonlinear media. Phys. Rev. E, 2002, 65(3): 035602
[66] Christodoulides D N, Eugenieva E D, and Oskun T H. Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media. Phys. Rev. E, 2001, 63(3): 035601
[67] Marklund M and Shukla P k. Filamentational instability of partiaqlly coherent femtosecond optical pulses in air. Opt. Lett., 2006, 31(12): 1884-1886
[68] Venedov A A. Theory of Turbulent Plasma. American Journal of Physics, 1969, 37(6): 672-673
[69] Tsintsadze N L and Mendonca J T. Kinetic theory of photons in a plasma. Phys. Plasmas, 1998, 5(10): 3609-3614
[70] Tsintsadze N L and Mendonca J T. Analog of the Wigner-Moyal equation for the electromagnetic field. Phys. Rev. E, 2000, 62(3): 4276-4282
[71] Crawford D R, Saffman P G, and Yuen H C. Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves. Wave Motion, 1980, 2(9): 1-16
[72] Janssen P A E M. Long-time behaviour of a random inhomogeneous field of weakly nonlinear surface gravity waves. Fluid Mech., 1983, 133(14):113-132
[73] Alber I E. The effects of randomness on the stability of two-dimensional surface wavetrains. Proc. R. Soc. London, Ser. A, 1978, 636(17): 525-526
[74] Anderson D, Fedele R, and Vaccaro V, et al. Modulational instabilities within the thermal wave model description of hight energy charged partical beam dynamics. Phys. Lett. A, 1999, 258(4): 244-248
[75] Fedele R, Anderson D, and Lisak M. Landau-type damping in nonlinear wavepacket propagation. Phys. Scr., 2000, T84(5): 27-33
[76] Fedele R and Anderson D. A quantum-like Landan damping of an electromagnetic wavepacket. J. Opt. B: Quantum Semiclassical Opt. 2000, 2(2): 207-213
[77] Gardiner S A. Nonlinear matter wave dynamics with a chaotic potential. Phys. Rev. A, 2000, 62(2): 023612
[78] Kip D, Soljacic M, Segev M, and Sears S M. (1+1)-Dimensional modulation instability of spatially incoherent light. J. Opt. Soc. Am. B, 2002, 19(3): 502-513
[79] Mitchell M, Zh. Chen, and Shih M. Self-Trapping of Partially Spatially Incoherent Light. Phys. Rev. Lett., 1996, 77(3):490-493
[80] Mitchell M and Segev M. Self-trapping of incoherent white light. Nature, 1997, 387(14): 880-883
[81] Picozzi A, Haelterman M, and Pitois S, et al. Incoherent Solitons in Instantaneous Response Nonlinear Media. Phys. Rev. Lett., 2004, 92(14): 143906
[82] Zh Chen, Mitchell M, and Segev M, et al. Self-Trapping of Dark Incoherent Light Beams. Science, 1998, 280(10):889-892
[83] Kip D, Solja?i? M, and Segev M, et al. Modulation instability and pattern formation in spatially incoherent light beams. Science, 2000, 290(5): 495-498
[84] Bang O, Edmundson D, and Krolikówski W. Collapse of Incoherent Light Beams in Inertial Bulk Kerr Media. Phys. Rev. Lett., 1999, 83(26): 5479-5482
[85] Jeng C, Shih M, and Motzek K, et al. Partially Incoherent Optical Vortices in Self-Focusing Nonlinear Media. Phys. Rev. Lett., 2004, 92(4): 043904
[86] Ponomarenko S A and Agrawal G P. Asymmetric incoherent vector solitons. Phys. Rev. E, 2004, 69 (3):036604
[87] Ponomarenko S A, Litchinitser N M, and Agrawal G P. Theory of incoherent optical solitons: Beyond the mean-field approximation. Phys. Rev. E, 2004, 70(1): 015603
[88] 江金环, 李子平. 基于全息聚焦机理空间光孤子的相互作用势函数. 物理学报, 2004, 53(9): 2991-1994
[89] 刘劲松, 张都应. 损耗对屏蔽光伏空间孤子演化特性的影响. 物理学报, 2001, 50(5): 880-885
[90] Davydov A S. The theory of contraction of proteins under their excitation. J. Theor. Biol., 1973, 38(3): 559-569
[91] Su W P, Schieffer J R, and Heeger A J. Solitons in Polyacetylene. Phys. Rev. Lett., 1979, 42(25): 1698-1701
[92] Christodes D N and Joseph R I. Discrete self-focusing in nonlinear arrays of coupled waveguides. Opt. Lett. 1988, 13(8): 794-796
[93] Pedri P, Santos L, ?hberg P, and Stringari S. Violation of self-similarity in the expansion of a one-dimensional Bose gas. Phys. Rev. A, 2003, 68(4): 43601
[94] Martikainen J P and Stoof H T C. Excitations of a Bose-Einstein condensate in a one-dimensional optical lattice. Phys. Rev. A, 2003, 68(1): 013610
[95] Scharf R and Bishop A R. Length-scale competition for the one-dimensional nonlinear Schr?dinger equation with spatially periodic potentials. Phys. Rev. E, 1993, 47(2): 1375-1383
[96] Buljan H, Cohen O, and Fleischer J W, et al. Random-Phase Solitons in Nonlinear Periodic Lattices. Phys.Rev.Lett., 2004, 92(22): 223901
[97] Shuangchun Wen, Dianyuan Fan. Filamentation instability in nonlocal nonlinear media. Chin. Phys., 2001, 10(11): 1032-1036
[98] 许超彬, 郭旗, 偏离束腰入射对非局域非线性介质中高斯光束演化的影响. 物理学报, 2004,53 (9):3025-3032
[99] Anderson D. Variational approach to nonlinear pulse propagation in opticas fibers. Phys. Rev. A, 1983, 27(6): 3135-3145
[100] Kozlov M V, Mckinstrie C J, and Xie C. Moment equations for optical pulses in dispersive and dissipative systems. Opt. Commun., 2005, 251: 194-208
[101] Cohen O, Schwartz T, and Fleischer J W, et al. Multiband Vector Lattice Solitons. Phys. Rev. Lett., 2003, 91(11): 113901
[102] Efremidis N K and Christodoulides D N. Lattice solitons in Bose-Einstein condensates. Phys. Rev. A, 2003, 67(6):063608
[103] BenDahan M, Peik E, and Reichel J, et al. Bolch oscillations of atoms in an optical potential. Phys. Rev. Lett., 1996, 76(24):4508-4511
[104] Pricopenko L. Variational approach for the two-dimensional trapped Bose-Einstein condensate. Phys. Rev. A, 2004, 70(1): 013601
[105] Agrawal G P. Nonlinear Fiber Optics, 3nd edn. (San Diego, Academic Press, 2001)
[106] Mingaleev S F, Kivshar Yu. S. Nonlinear Photonic Crystals: Toward All-Optical Technologies. Opt. Photon. News, 2002, 13(9):48-51
[107] Boyd R W. Nonlinear Optics. (New York, Academic Press, 1994)
[108] Brabec T and Krausz F. Nonlinear Optical Pulse Propagation in the Single-Cycle Regime. Phys. Rev. Lett., 1997, 78(17):3282-3285
[109] Smith D R and Kroll N. Negative Refractive Index in Left-Handed Materials. Phys. Rev. Lett., 2000, 85(14):2933-3936
[110] Berman P R. Goos-Hanchen shift in negatively refractive media. Phys. Rev. E, 2002, 66(6): 067603
[111] Pendry J B, Smith D R. Reversing light with negative refraction. phys. Today, 2004, 57 (6): 37-43
[112] 历以宇,顾培夫,李明宇等. 波状结构二维光子晶体近场亚波长成像的研究. 光学学报,2006,26(9): 1409-1413
[113] 罗海陆,胡巍,易煦农等. 单轴晶体中的负折射现象研究. 光学学报,2005,25(9): 1249-1253
[114] Kinsler P, New G H C. Few-cycle pulse propagation. Phys. Rev. A, 2003, 67(2): 023813
[115] Zhang S, Fan W J, and Panoiu N C, et al. Experimental demonstration of near-infrared negative-index metamaterials. Phys. Rev. Lett. 2005, 95(13): 137404
[116] Marklund M, Shukla P K, and Stenflo L. Ultrashort solitons and kinetic effects in nonlinear metamaterials. Phys. Rev. E, 2006, 73(3): 037601
[117] Marklund M and Shukla P K. Modulational instability of partially coherent signals in electrical transmission lines. Phys. Rev. E, 2006, 73(5):057601
[118] Helczynski L, Lisak M, and Anderson D. Influence of higher-order dispersion on modulational instability and pulse broadening of partially incoherent light. Phys. Rev. E, 2003, 67(2): 026602
[119] Anderson D, Helczynski- Wolf L, Lisak M, and Semenov V. Features of modulational instability of partially coherent light: Importance of the incoherence spectrum. Phys. Rev. E, 2004, 69(2): 025601
[120] Wigner E P. On the Quantum Correction For Thermodynamic Equilibrium. Phys. Rev., 1932, 40(9):749-759
[121] Shuangchun Wen and Dianyuan Fan. Spatiotemporal instabilities in nonlinear Kerr media in the presence of arbitrary higher-order dispersions. J. Opt. Soc. Am. B, 2002, 19(7): 1653-1659
[122] Anastassiou C, Solja?i? M, and Segev M, et al. Eliminating the Transverse Instabilities of Kerr Solitons. Phys. Rev. Lett., 2000,85(23): 4888-4891
[123] Martijn de Sterke C. Theory of modulational instability in fiber Bragg gratings. J. Opt. Soc. Am. B, 1998, 15(11): 2660-2667