非相干孤子在非局域介质中传输特性的研究
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摘要
近10年来，非相干空间光孤子传输特性的研究取得了很大进展。然而，以上讨论的非相干空间光孤子都是在传播介质非线性局域响应的范畴之内，是一种理想的研究模型。对于现实存在的非线性介质许多都属于非局域响应的材料，比如向列液晶，可以用来产生非局域非相干孤子。非局域非相干孤子具有低阈值功率，高传播稳定性，这为在光的信息处理和传输过程中实现光对光的引导开辟了一个新的渠道，并可以大大简化目前光通讯中各类器件的结构，对于集成光学中光互连技术及光导向器件的研制具有重要的现实意义。
本文主要从理论上研究了非相干孤子在非局域非线性介质中的自陷行为，其中包括时空非相干白光孤子在非局域对数型介质以及强非局域Kerr介质中的传输特性，并讨论了它们的相干特性。此外，还研究了圆对称以及椭圆对称的部分非相干孤子在强非局域Kerr介质中的传输特性。
作者取得的主要研究成果是：
(1)从理论上研究了非相干白光孤子在对数型介质中的传输特性。得到了稳定传输的非局域白光孤子的存在曲线，并对其相干特性做了详细的分析。发现白光孤子的相干半径随着频率的增大而减小，随着非局域强度的增大而增大。还采用近似解析以及数值模拟的方法分析了白光孤子的稳定性以及周期性的振荡特性，揭示了孤子传输过程中非局域特性对光束强度以及相干特性的影响。
(2)研究了随机反应的强非局域介质中部分非相干线性孤子的传输特性。基于线性的传输方程和互谱密度理论，得到了此类非相干线性孤子的解析表达式，发现入射光束的功率以及光束的相干特性决定了孤子的半径。由于在强非局域介质中光束的传输模式是线性的，因此孤子的形成过程中不存在一个非相干的阈值。当孤子进行线性的谐振时，详细讨论了光束半径以及光束相干特性的演化过程。
(3)研究了椭圆非相干线性孤子在强非局域非瞬时各向同性Kerr非线性介质中的传输特性。孤子有着各向异性的相干特性，孤子的入射功率必须满足一个由光束宽度和相干特性共同决定的阈值。当光束的初始条件不满足孤子的稳定传输条件时，椭圆的非相干光束将以五种不同的方式产生周期性的振荡，相关特性我们做了详细讨论。
(4)系统地研究了椭圆非相干线性孤子在强非局域非瞬时各向异性Kerr非线性介质中的传输特性，发现相干特性各向同性和各向异性的椭圆非相干光束都可以形成稳定传输的孤子。当椭圆光束发生周期性振荡时，我们采用数值模拟的方式详细研究了光束的演化过程。研究表明椭圆光束长短轴振荡的周期是不同的，椭圆光束在某些传输距离上将会演化成为圆形光束。
(5)首次从理论上研究了非相干白光孤子在强非局域非瞬时Kerr介质中的传输特性。发现孤子有着椭圆高斯形式的光强分布和各向异性的时空相干特性，孤子的相干半径随着频率的增大而减小。当光束的入射功率不满足孤子稳定传输的临界值时，非相干光束将会产生线性的谐振，相关特性做了详细的讨论。
(6)研究了一维非相干线性孤子在强非局域非瞬时Kerr介质中的传输特性。采用相干密度理论得到了非相干孤子的解析表达式，光束入射功率和非相干的角能量谱θ_0决定了光束的宽度。当非相干光束产生周期性振荡时，还讨论了光束强度谱以及光束相干特性的演化过程。
During the recent ten years, the propagation of incoherent solitons have drawn considerable attention. Howerver, the previous research of incoherent solitons were related to the domain of local nonlinear media which is an ideal research model. Lots of realistic media has a nonlocal nonlinear response, such as liquid cryatal which can be used for self-trapping of nonlocal incoherent soliton. The study of nonlocal incoherent solitons will greatly expand the research domain of nonlinear optics, provide theoretical foundation for understanding spatial solitons. Furthermore, it can greatly promote the exploitation of all-optics communication apparatus which based on optical solitons, predigestthe structure of the current optical communication apparatus, and has very important meaning for the optical interconnects, beam steering, and other applications.
The aim of this thesis is to theoretically study the self-trapping of nonlocal incoherent solitons, inculuding the propagation of incoherent white-light solitons in nonlocal logarithmic nonliear media and in strongly nonlocal Kerr media, the propation of circular and elliptic partially incoherent solitons in strongly nonlocal Kerr media, and the coherence properties of such nonlocal incoherent solitons.
The main results given by the author are as follows:
(ⅰ) The propagation properties of white-light solitons in spatially nonlocal media with a logarithmically nonlinearity are investigated theoretically. The existence curve of the stationary nonlocal incoherent soliton is obtained and the coherence characteristics of the soliton are also described. The evolution behaviors of the nonlocal white-light soliton are discussed in detail by both approximate analytical solution and numerical simulation when the solitons undergo periodic oscillation.
(ⅱ) We study the propagation of incoherent accessible solitons in strongly nonlocal media with arbitrary response function. Based on the linear propagation equation and the mutual coherence function approach, we obtain an exact analytical solution of such incoherent accessible solitons. The solitons radius is related to the total power as well as the coherence characteristics of the incoherent beam. We find that there is not a threshold for incoherent solitons exist in strongly nonlocal media because the model is linear. Evolution behaviors of the solitons width and the coherence radius are also described when the solitons undergo linear harmonic oscillation.
(ⅲ) We study the propagation of elliptic incoherent accessible solitons in strongly non- local media with noninstantaneous Kerr nonlinearity. For this soliton to exist, the coherence properties of the incoherent beam should be anisotropic. The total power of the incident beam should also equal to a critical value which depend on the beam width as well as the coherence properties. When initial parameters of the beam do not satisfy the existence conditions, the elliptic incoherent accessible solitons will undergo linear harmonic oscillation in different states. Corresponding properties are studied in detail.
(ⅳ)The propagation of elliptic incoherent beam in strongly nonlocal media with noninstantaneous anisotropic Kerr nonlinearity is systematically investigated. It is reported that both isotropic and anisotropic correlation characteristics of such elliptic incoherent beam are applicable. Evolution behaviors of the beam widths are shown by numerical calculation when the beam undergo periodic oscillation. It is also obtained that the oscillation periods of the two axis are different and the elliptic beam will become circular at some propagation distance under special conditions.
(ⅴ) We present a theoretical investigation of incoherent accessible white-light solitons in strongly nonlocal medium with noninstantaneous Kerr nonlinearity. This soliton has elliptic Gaussian intensity profile and anisotropic spatiotemporal coherence properties. For this soliton to exist, the spatial coherence distance should be larger for lower frequencies and shorter for higher frequencies. When the incident power differs from the critical value, we demonstrate the periodic harmonic oscillations of the accessible white-light solitons.
(ⅵ) We study the properties of one-dimension incoherent accessible solitons in strongly nonlocal media with noninstantaneous Kerr nonlinearity. Following the coherent density theory, we obtain an exact solution of such incoherent solitons. The spatial width of the incoherent solitons is related to the incoherent angular power spectrum O0 as well as the incident power. The evolution properties of the intensity profile and the coherence characteristics are also discussed in detail when the solitons undergo periodic harmonic oscillation.
本文主要从理论上研究了非相干孤子在非局域非线性介质中的自陷行为，其中包括时空非相干白光孤子在非局域对数型介质以及强非局域Kerr介质中的传输特性，并讨论了它们的相干特性。此外，还研究了圆对称以及椭圆对称的部分非相干孤子在强非局域Kerr介质中的传输特性。
作者取得的主要研究成果是：
(1)从理论上研究了非相干白光孤子在对数型介质中的传输特性。得到了稳定传输的非局域白光孤子的存在曲线，并对其相干特性做了详细的分析。发现白光孤子的相干半径随着频率的增大而减小，随着非局域强度的增大而增大。还采用近似解析以及数值模拟的方法分析了白光孤子的稳定性以及周期性的振荡特性，揭示了孤子传输过程中非局域特性对光束强度以及相干特性的影响。
(2)研究了随机反应的强非局域介质中部分非相干线性孤子的传输特性。基于线性的传输方程和互谱密度理论，得到了此类非相干线性孤子的解析表达式，发现入射光束的功率以及光束的相干特性决定了孤子的半径。由于在强非局域介质中光束的传输模式是线性的，因此孤子的形成过程中不存在一个非相干的阈值。当孤子进行线性的谐振时，详细讨论了光束半径以及光束相干特性的演化过程。
(3)研究了椭圆非相干线性孤子在强非局域非瞬时各向同性Kerr非线性介质中的传输特性。孤子有着各向异性的相干特性，孤子的入射功率必须满足一个由光束宽度和相干特性共同决定的阈值。当光束的初始条件不满足孤子的稳定传输条件时，椭圆的非相干光束将以五种不同的方式产生周期性的振荡，相关特性我们做了详细讨论。
(4)系统地研究了椭圆非相干线性孤子在强非局域非瞬时各向异性Kerr非线性介质中的传输特性，发现相干特性各向同性和各向异性的椭圆非相干光束都可以形成稳定传输的孤子。当椭圆光束发生周期性振荡时，我们采用数值模拟的方式详细研究了光束的演化过程。研究表明椭圆光束长短轴振荡的周期是不同的，椭圆光束在某些传输距离上将会演化成为圆形光束。
(5)首次从理论上研究了非相干白光孤子在强非局域非瞬时Kerr介质中的传输特性。发现孤子有着椭圆高斯形式的光强分布和各向异性的时空相干特性，孤子的相干半径随着频率的增大而减小。当光束的入射功率不满足孤子稳定传输的临界值时，非相干光束将会产生线性的谐振，相关特性做了详细的讨论。
(6)研究了一维非相干线性孤子在强非局域非瞬时Kerr介质中的传输特性。采用相干密度理论得到了非相干孤子的解析表达式，光束入射功率和非相干的角能量谱θ_0决定了光束的宽度。当非相干光束产生周期性振荡时，还讨论了光束强度谱以及光束相干特性的演化过程。
During the recent ten years, the propagation of incoherent solitons have drawn considerable attention. Howerver, the previous research of incoherent solitons were related to the domain of local nonlinear media which is an ideal research model. Lots of realistic media has a nonlocal nonlinear response, such as liquid cryatal which can be used for self-trapping of nonlocal incoherent soliton. The study of nonlocal incoherent solitons will greatly expand the research domain of nonlinear optics, provide theoretical foundation for understanding spatial solitons. Furthermore, it can greatly promote the exploitation of all-optics communication apparatus which based on optical solitons, predigestthe structure of the current optical communication apparatus, and has very important meaning for the optical interconnects, beam steering, and other applications.
The aim of this thesis is to theoretically study the self-trapping of nonlocal incoherent solitons, inculuding the propagation of incoherent white-light solitons in nonlocal logarithmic nonliear media and in strongly nonlocal Kerr media, the propation of circular and elliptic partially incoherent solitons in strongly nonlocal Kerr media, and the coherence properties of such nonlocal incoherent solitons.
The main results given by the author are as follows:
(ⅰ) The propagation properties of white-light solitons in spatially nonlocal media with a logarithmically nonlinearity are investigated theoretically. The existence curve of the stationary nonlocal incoherent soliton is obtained and the coherence characteristics of the soliton are also described. The evolution behaviors of the nonlocal white-light soliton are discussed in detail by both approximate analytical solution and numerical simulation when the solitons undergo periodic oscillation.
(ⅱ) We study the propagation of incoherent accessible solitons in strongly nonlocal media with arbitrary response function. Based on the linear propagation equation and the mutual coherence function approach, we obtain an exact analytical solution of such incoherent accessible solitons. The solitons radius is related to the total power as well as the coherence characteristics of the incoherent beam. We find that there is not a threshold for incoherent solitons exist in strongly nonlocal media because the model is linear. Evolution behaviors of the solitons width and the coherence radius are also described when the solitons undergo linear harmonic oscillation.
(ⅲ) We study the propagation of elliptic incoherent accessible solitons in strongly non- local media with noninstantaneous Kerr nonlinearity. For this soliton to exist, the coherence properties of the incoherent beam should be anisotropic. The total power of the incident beam should also equal to a critical value which depend on the beam width as well as the coherence properties. When initial parameters of the beam do not satisfy the existence conditions, the elliptic incoherent accessible solitons will undergo linear harmonic oscillation in different states. Corresponding properties are studied in detail.
(ⅳ)The propagation of elliptic incoherent beam in strongly nonlocal media with noninstantaneous anisotropic Kerr nonlinearity is systematically investigated. It is reported that both isotropic and anisotropic correlation characteristics of such elliptic incoherent beam are applicable. Evolution behaviors of the beam widths are shown by numerical calculation when the beam undergo periodic oscillation. It is also obtained that the oscillation periods of the two axis are different and the elliptic beam will become circular at some propagation distance under special conditions.
(ⅴ) We present a theoretical investigation of incoherent accessible white-light solitons in strongly nonlocal medium with noninstantaneous Kerr nonlinearity. This soliton has elliptic Gaussian intensity profile and anisotropic spatiotemporal coherence properties. For this soliton to exist, the spatial coherence distance should be larger for lower frequencies and shorter for higher frequencies. When the incident power differs from the critical value, we demonstrate the periodic harmonic oscillations of the accessible white-light solitons.
(ⅵ) We study the properties of one-dimension incoherent accessible solitons in strongly nonlocal media with noninstantaneous Kerr nonlinearity. Following the coherent density theory, we obtain an exact solution of such incoherent solitons. The spatial width of the incoherent solitons is related to the incoherent angular power spectrum O0 as well as the incident power. The evolution properties of the intensity profile and the coherence characteristics are also discussed in detail when the solitons undergo periodic harmonic oscillation.
引文
[1] R. Y. Chiao, E. Garmire, and C. H. Townes, "Self-Trapping of Optical Beams," Phys. Rev. Lett. 13, 479 (1964).
[2] M. Segev and G. Stegeman, "Self-Trapping of Optical Beams: Spatial Solitons," Phys. Today, 1998; 51 (8): 42-48
[3] G. I. A. Stegeman, D. N. Christodoulides and M. Segev, "Optical Spatical Solitons: Historical Perspectives," IEEE J. Quan. Elec, 2000; 6(6): 1419-1422
[4] Duree G. et al. Observation of an optical beam due to the photorefractive effect. Phys.Rev.Lett.,1993,71:533-536
[5] M. Mitchell, Z. Chen, M. Shih, and M. Segev, "Self trapping of partially spatially incoherent light," Phys. Rev. Lett. 77, 490 (1996).
[6] M. Mitchell, and M. Segev, "Self-trapping of incoherent white light," Nature(London) 387, 880 (1997).
[7] Z. Chen, M. Mitchell, M. Megev, T. H. Coskun, and D. N. Christodoulides, "Self-trapping of dark incoherent light beams," Science 280, 889 (1998).
[8] O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev and D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature(London) 433, 500 (2005).
[9] D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Theory of incoherent self-focusing in biased photorefractive media," Phys. Rev. Lett. 78, 646 (1997).
[10] D. N. Christodoulides, T. H. Coskun, and R. I. Joseph, "Incoherent spatial solitons in saturable nonlinear media," Opt. Lett. 22, 1080 (1997).
[11] T. H. Coskun, D. N. Christodoulides, M. Mitchell, Z. Chen, and M. Segev, "Dynamics of incoherent bright and dark self-trapped beams and their coherence properties in photorefracrive crystals," Opt. Lett. 23, 418 (1998).
[12] T. H. Coskun, A. G. Drandpierre., D. N. Christodoulides, and M. Segev, "Coherence enhancement of spatially incoherent light beams through soliton interactions," Opt. Lett. 25, 826 (2000).
[13] M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, "Theory of self-trapped spatially incoherent light beams," Phys. Rev. Lett. 79, 4990 (1997).
[14] D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Multimode incoherent spatial solitons in logarithmically saturable nonlinear media," Phys. Rev. Lett. 80, 2310 (1998).
[15] M. I. Carvalho, T. H. Coskun, D. N. Christodoulides, M. Mitchell, and M. Segev, "Coherence properties of multimode incoherent spatial solitons in noninstantaneous Kerr media," Phys. Rev. E 59, 1193 (1999).
[16] W. Krolikowski, D. Edmundson, and O. Bang, "Unified model for partially coherent solitons in logarithmically nonlinear media," Phys. Rev. E 61, 3122 (2000).
[17] S. A. Ponomarenko, "Linear superposition principle for partially coherent solitons," Phys. Rev. E 65, 055601 (R) (2002).
[18] D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601(R) (2001).
[19] H. Buljan, A. Siber, M. Soljacic, and M. Segev, "Propagation of incoherent "white" light and modulation instability in noninstantaneous nonlinear media," Phys. Rev. E 66, 035601(R) (2002).
[20] H. Buljan, M. Segev, M. Soljacic, N. K. Efremidis, and D. N. Christodoulides, "White-light solitons," Opt. Lett. 28, 1239 (2003).
[21] H. Buljan, A. Siber, M. Soljacic, T. Schwartz, M. Segev, and D. N. Christodoulides, "Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media," Phys. Rev. E 68, 036607 (2003).
[22] H. Buljan, T. Schwartz, M. Segev, M. Soljacic, and D. N. Christodoulides, "Polychromatic partially spatially incoherent solitons in a noninstantaneous Kerr nonlinear medium," J. Opt. Soc. Am. B 21, 397 (2004).
[23] M. Peccianti and G. Assanto, "Incoherent spatial solitary waves in nematic liquid crystals," Opt. Lett. 26, 1791 (2001).
[24] M. Peccianti and G. Assanto, "Nematic liquid crystals: A suitable medium for self-confinement of coherent and incoherent light," Phys. Rev. E 65, 035603(R) (2002).
[25] W. Krolikowski, O. Bang, and J. Wyller, "Nonlocal incoherent solitons," Phys. Rev. E 70, 036617 (2004).
[26] K. G. Makris, H. Sarkissian, D. N. Christodoulides, and G. Assanto, "Nonlocal incoherent spatial solitons in liquid crystals," J. Opt. Soc. Am. B 22, 1371 (2005).
[27] A. G. Litvak, V. A. Mironov, G. M. Fraiman, and A. D. Yunakovskii, "Thermal self-e ect of wave beams in a plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 22, 421 (1980).
[28] D. Suter and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583 (1993).
[29] C. A. Sackett, J. M. Gerton, M. Welling, and R. G. Hulet, "Measurements of Collective Collapse in a Bose-Einstein Condensate with Attractive Interactions," Phys. Rev. lett. 82, 876 (1999).
[30] W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288 (2004).
[31] A. W. Snyder and D. J. Mitchell, "Accessible solitons," Science 276, 1538 (1997).
[32] D. J. Mitchell and A. W. Snyder, "Soliton dynamics in a nonlocal medium," J. Opt. Soc. Am. B 16, 236 (1999).
[33] Y. R. Shen, "Solitons made simple," Science 276, 1520 (1997).
[34] W. Krolikowski and 0. Bang, "Solitons in nonlocal nonlinear media: Exact solutions," Phys. Rev. E 63, 016610 (2001).
[35] Zhiyong Xu, Y. V. Kartashov and L. Torner, "Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media," Opt. Lett. 30, 3171 (2005).
[36] C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, "Two-dimensional multipole solitons in nonlocal nonlinear media," Opt. Lett. 31, 3312 (2006).
[37] Y. V. Kartashov, L. Torner, V. A. Vysloukh, and D. Mihalache, "Multipole vector solitons in nonlocal nonlinear media," Opt. Lett. 31, 1483 (2006).
[38] Zhiyong Xu, Y. V. Kartashov, and L. Torner, "Stabilization of vector soliton complexes in nonlocal nonlinear media," Phys. Rev. E 73, 055601(R) (2006).
[39] A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz and M. A. Karpierz, "Discrete propagation and spatial solitons in nematic liquid crystals," Opt. Lett. 29, 1530 (2004).
[40] A. Fratalocchi and G. Assanto, "Discrete light localization in one-dimensional nonlinear lattices with arbitrary nonlocality," Phys. Rev. E 72, 066608 (2005).
[41] Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Tunable Soliton Self-Bending in Optical Lattices with Nonlocal Nonlinearity," Phys. Rev. Lett. 93, 153903 (2004).
[42] Zhiyong Xu, Y. V. Kartashov, and L. Torner, "Soliton Mobility in Nonlocal Optical Lattices," Phys. Rev. Lett. 95, 113901 (2005).
[43] Zhiyong Xu, Y. V. Kartashov, and L. Torner, "Gap solitons supported by optical lattices in photorefractive crystals with asymmetric nonlocality," Opt. Lett. 31, 2027 (2006).
[44] N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, "Attraction of nonlocal dark optical solitons," Opt. Lett. 29, 286 (2004).
[45] A. Dreischuh, D. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, "Observation of Attraction between Dark Solitons," Phys. Rev. lett. 96, 043901 (2006).
[46] N. I. Nikolov, D. Neshev, 0. Bang, and W. Z. Krolikowski, "Quadratic solitons as nonlocal solitons," Phys. Rev. E 68, 036614 (2003).
[47] 0. Bang, W. Krolikowski, J. Wyller and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
[48] W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[49] J. Wyller, W. Krolikowski, O. Bang, and J. J. Rasmussen, "Generic features of modulational instability in nonlocal Kerr media," Phys. Rev. E 66, 066615 (2002).
[50] C. Conti, Marco Peccianti, and G. Assanto, "Spatial solitons and modulational instability in the presence of large birefringence: The case of highly nonlocal liquid crystals," Phys. Rev. E 72, 066614 (2005).
[51] M. Peccianti, C. Conti, and G. Assanto, "Optical modulational instability in a nonlocal medium," Phys. Rev. E 68, 025602(R) (2003).
[52] M. Peccianti, C. Conti, G. Assanto, A. D. Luca and C. Umeton, "Routing of anisotropic spatial solitons and modulational instability in liquid crystals," Na-ture(London) 432, 733 (2004).
[53] M. Peccianti, C. Conti, E. Alberici and G. Assanto, "Spatially incoherent modulation instability in a nonlocal medium," Laser Physics Letters 2, 25 (2005).
[54] X. Hutsebaut, C. Cambournac, M. Haelterman, A. Admski, and K. Neyts, "Single-component higher-order mode solitons in nematic liquid crystals," Opt. Commun. 233, 211 (2004).
[55] C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in Nonlinear Media with an Infinite Range of Nonlocality: First Observation of Coherent Elliptic Solitons and of Vortex-Ring Solitons," Phys. Rev. lett. 95, 213904 (2005).
[56] M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, "Nonlocal spatial soliton interactions in nematic liquid crystals," Opt. Lett. 27, 1460 (2002).
[57] P. D. Rasmussen, O. Bang, and W. Krolikowski, "Theory of nonlocal soliton interaction in nematic liquid crystals," Phys. Rev. E 72, 066611 (2005).
[58] C. Rotschild. B. Alfassi. 0. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Physics 2. 769 (2006).
[59] D. Briedis, D. E. Petersen, D. Edmundson et. al, "Ring vortex solitons in nonlocal nonlinear media," Opt. Exp. 13, 435 (2005).
[60] A. I. Yakimenko, Y. A. Zaliznyak, and Yu. S. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," Phys. Rev. E 71, 065603(R) (2005).
[61] S. Lopez-Aguayo, A. S. Desyatnikov, Yu. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100 (2006).
[62] D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Craso-van, and L. Torner, "Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media," Phys. Rev. E 73, 025602(R) (2006).
[63] S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, "Stability of two-dimensional spatial solitons in nonlocal nonlinear media," Phys. Rev. E 73, 066603 (2006).
[64] Y. V. Kartashov, L. Torner, and V. A. Vysloukh, "Lattice-supported surface solitons in nonlocal nonlinear media," Opt. Lett. 31, 2595 (2006).
[65] C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. lett. 91, 073901 (2003).
[66] C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in a highly nonlocal medium," Phys. Rev. lett. 92, 113902 (2004).
[67] M. Peccianti, C. Conti, and G. Assanto, "Interplay between nonlocality and nonlin-earity in nematic liquid crystals," Opt. Lett. 30, 415 (2005).
[68] O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, "Incoherent solitons in instantaneous nonlocal nonlinear media," Phys. Rev. E 73, 015601(R) (2006).
[69] M. Shen, Q. Wang, J. L. Shi, Y. Y. Chen, and X. L. Wang, "Nonlocal incoherent white-light solitons in logarithmically nonlinear media," Phys. Rev. E 72, 026604 (2005).
[70] M. Shen, Q. Wang, J. L. Shi, P. Hou, and Q. Kong, "Partially coherent accessible solitons in strongly nonlocal media," Phys. Rev. E 73, 056602 (2006).
[71] M. Shen, J. Shi, and Q. Wang, "Incoherent accessible white-light solitons in strongly nonlocal Kerr media," Phys. Rev. E 74, 027601 (2006).
[72] M. Shen, Q. Wang, and J. Shi, "Elliptic incoherent accessible solitons in strongly nonlocal media," Opt. Commun. 270, 384 (2007).
[73] Q. Kong, J. Shi, M. Shen, and Q. Wang, "Elliptic incoherent solitons in strongly nonlocal media with anisotropic Kerr nonlinearity," J. Opt. Soc. Am. B (submitted).
[74] M. Shen, J. Shi, and Q. Wang, "Propagation of partially incoherent accessible solitons in strongly nonlocal Kerr media," J. Opt. Soc. Am. B (submitted).
[75] Q. Guo, B. Luo, F. Yi, S. Chi, and Yiqun Xie, "Large phase shift of nonlocal optical spatial solitons," Phys. Rev. E 69, 016602 (2004).
[76] Y. Q. Xie and Q. Guo, "Phase modulations due to collisions of beam pairs in nonlocal nonlinear media," Optical and Quantum Electronics 36, 1335 (2004).
[77] Q. Guo, B. Luo, and S. Chi, "Optical beams in sub-strongly non-local nonlinear media: A variational solution," Opt. Commun. 259, 336 (2005).
[78] Y. Huang, Q. Guo, and J. Cao, "Optical beams in lossy non-local Kerr media," Opt. Commun. 261, 175 (2006).
[79] T. Schwartz, T. Cartoon, H. Buljan, and M. Segev, "Spontaneous pattern formation with incoherent "white" light," Phys. Rev. Lett. 93, 223901 (2004).
[80] Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystal(Academic, San Diego, Calif., 2003).
[81] 陈园园，王奇，施解龙，卫青，“部分相干光束的振荡自陷特性，”物理学报51(3)，559(2002)。
[82] G. Assanto and M. Peccianti, IEEE J. Quantum Electron. 39, 13 (2003).
[83] E. Braun, L. Faucheux, A. Libchaber et. al., Europhys. Lett. 23, 239(1993).
[84] M. Kristian, K. Friedemann, Wen-Hen Chu, M. Shih, Yuri S. Kivshar, "Soliton transverse instabilities in anisotropic nonlocal self-focusing media," Opt. Lett. 29, 280 (2004).
[85] A. S. Desyatnikov, D. N. Neshev, Y. S. Kivshar, N. Sagemerten, D. Trager, J. Jagers, C. Denz, and Y. V. Kartashov, "Nonlinear photonic lattices in anisotropic nonlocal self-focusing media," Opt. Lett. 30, 869 (2005).
[86] 卫青，王奇，施解龙，“对数型非线性介质中亮暗孤子的存在性分析，”物理学报 52，1645(2002)。
[87] S. M. Sears, M. Soljacic, D. N. Christodoulides, and M. Segev, "Pattern formation via symmetry breaking in nonlinear weakly correlated systems," Phys. Rev. E 65, 036620 (2002).
[88] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995).
[89] The expansion of the nonlocal response function R((?)-(?)) with respect to (?) about (?)= (?) to the second order, which reads R((?)-(?))=R_0+((?)-(?)). ▽⊥R_0 + (1/2)[((?)-(?)) ·▽⊥]~2R_0+…[75]. Because R((?)) is symmetrical with respect to (?)=0, we have that (?)_xR_0=(?)_yR_0=0, (?)_(xy)~2R_0=0, and (?)_x~2R_0=(?)_y~2R_0[75], then Eq. (2) is obtained.
[90] M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, "Modulation Instability of Incoherent Beams in Noninstantaneous Nonlinear Media," Phys. Rev. lett. 84, 467 (2000).
[91] M. Segev and D. N. Christodoulides, "Incoherent solitons: self-trapping of weaklycorrelated wave-packets," Optics and Photonics News 13, 70 (2002).
[92] E. D. Eugenieva, D. N. Christodoulides, and M. Segev, "Elliptic incoherent solitons in saturable nonlinear media," Opt. Lett. 25, 972 (2000).
[93] O. Katz, T. Carmon, T. Schwartz, M. Segev, and D. N. Christodoulides, "Observation of elliptic incoherent spatial solitons," Opt. Lett. 29, 1248 (2004).
[94] O. Bang, D. Edmundson, and W. Krolikowski, "Collapse of Incoherent Light Beams in Inertial Bulk Kerr Media," Phys. Rev. lett. 83, 5479 (1999).
[95] A. W. Snyder and D. J. Mitchell, "Mighty morphing spatial solitons and bullets," Opt. Lett. 22, 16 (1997).
[96] V. Tikhonenko, "Observation of mightymorphing spatial solitons," Opt. Lett. 23, 594 (1998).
[97] B. Alfassi, C. Rotschild, 0. Manela, M. Segev, and D. N. Christodoulides, "Boundary force effects exerted on solitons in nonlinear media with a very large range of nonlocality," Opt. Lett. 32, 154 (2007).
[98] R. Fischer, D. N. Neshev, W. Krolikowski, Y. S. Kivshar, D. Iturbe-Castillo, S. Chavez-Cerda, M. R. Meneghetti, D. P. Caetano and J. M. Hickman "Oblique interaction of spatial dark-soliton stripes in nonlocal media," Opt. Lett. 31, 3010 (2006).
[99] M. Peccianti, A. Dyadyusha, M. Kaczmarek and G. Assanto, "Tunable refraction and reflection of self-confined light beams," Nature Physics 2, 737 (2006).
[100] Y. S. Kivshar, "Bending light at will," Nature Physics 2, 729 (2006).
[101] M. Peccianti, G. Assanto, A. Dyadyusha, and M. Kaczmarek, "Nonlinear shift of spatial solitons at a graded dielectric interface," Opt. Lett. 32, 271 (2007).
[102] S. Ouyang, Q. Guo, and Wei Hu, "Perturbative analysis of generally nonlocal spatial optical solitons," Phys. Rev. E (in press).
[103] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University, New York, 2003).
[104] S. Abe, and A. Ogura, "Solitary waves and their critical behavior in a nonlinear nonlocal medium with power-law response," Phys. Rev. E 57, 6066 (1998).
[105] P. Pedri, and L. santos, "Two-Dimensional Bright Solitons in Dipolar Bose-Einstein Condensates," Phys. Rev. lett. 95, 200404 (2005).
[2] M. Segev and G. Stegeman, "Self-Trapping of Optical Beams: Spatial Solitons," Phys. Today, 1998; 51 (8): 42-48
[3] G. I. A. Stegeman, D. N. Christodoulides and M. Segev, "Optical Spatical Solitons: Historical Perspectives," IEEE J. Quan. Elec, 2000; 6(6): 1419-1422
[4] Duree G. et al. Observation of an optical beam due to the photorefractive effect. Phys.Rev.Lett.,1993,71:533-536
[5] M. Mitchell, Z. Chen, M. Shih, and M. Segev, "Self trapping of partially spatially incoherent light," Phys. Rev. Lett. 77, 490 (1996).
[6] M. Mitchell, and M. Segev, "Self-trapping of incoherent white light," Nature(London) 387, 880 (1997).
[7] Z. Chen, M. Mitchell, M. Megev, T. H. Coskun, and D. N. Christodoulides, "Self-trapping of dark incoherent light beams," Science 280, 889 (1998).
[8] O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev and D. N. Christodoulides, "Observation of random-phase lattice solitons," Nature(London) 433, 500 (2005).
[9] D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Theory of incoherent self-focusing in biased photorefractive media," Phys. Rev. Lett. 78, 646 (1997).
[10] D. N. Christodoulides, T. H. Coskun, and R. I. Joseph, "Incoherent spatial solitons in saturable nonlinear media," Opt. Lett. 22, 1080 (1997).
[11] T. H. Coskun, D. N. Christodoulides, M. Mitchell, Z. Chen, and M. Segev, "Dynamics of incoherent bright and dark self-trapped beams and their coherence properties in photorefracrive crystals," Opt. Lett. 23, 418 (1998).
[12] T. H. Coskun, A. G. Drandpierre., D. N. Christodoulides, and M. Segev, "Coherence enhancement of spatially incoherent light beams through soliton interactions," Opt. Lett. 25, 826 (2000).
[13] M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, "Theory of self-trapped spatially incoherent light beams," Phys. Rev. Lett. 79, 4990 (1997).
[14] D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, "Multimode incoherent spatial solitons in logarithmically saturable nonlinear media," Phys. Rev. Lett. 80, 2310 (1998).
[15] M. I. Carvalho, T. H. Coskun, D. N. Christodoulides, M. Mitchell, and M. Segev, "Coherence properties of multimode incoherent spatial solitons in noninstantaneous Kerr media," Phys. Rev. E 59, 1193 (1999).
[16] W. Krolikowski, D. Edmundson, and O. Bang, "Unified model for partially coherent solitons in logarithmically nonlinear media," Phys. Rev. E 61, 3122 (2000).
[17] S. A. Ponomarenko, "Linear superposition principle for partially coherent solitons," Phys. Rev. E 65, 055601 (R) (2002).
[18] D. N. Christodoulides, E. D. Eugenieva, T. H. Coskun, M. Segev, and M. Mitchell, "Equivalence of three approaches describing partially incoherent wave propagation in inertial nonlinear media," Phys. Rev. E 63, 035601(R) (2001).
[19] H. Buljan, A. Siber, M. Soljacic, and M. Segev, "Propagation of incoherent "white" light and modulation instability in noninstantaneous nonlinear media," Phys. Rev. E 66, 035601(R) (2002).
[20] H. Buljan, M. Segev, M. Soljacic, N. K. Efremidis, and D. N. Christodoulides, "White-light solitons," Opt. Lett. 28, 1239 (2003).
[21] H. Buljan, A. Siber, M. Soljacic, T. Schwartz, M. Segev, and D. N. Christodoulides, "Incoherent white light solitons in logarithmically saturable noninstantaneous nonlinear media," Phys. Rev. E 68, 036607 (2003).
[22] H. Buljan, T. Schwartz, M. Segev, M. Soljacic, and D. N. Christodoulides, "Polychromatic partially spatially incoherent solitons in a noninstantaneous Kerr nonlinear medium," J. Opt. Soc. Am. B 21, 397 (2004).
[23] M. Peccianti and G. Assanto, "Incoherent spatial solitary waves in nematic liquid crystals," Opt. Lett. 26, 1791 (2001).
[24] M. Peccianti and G. Assanto, "Nematic liquid crystals: A suitable medium for self-confinement of coherent and incoherent light," Phys. Rev. E 65, 035603(R) (2002).
[25] W. Krolikowski, O. Bang, and J. Wyller, "Nonlocal incoherent solitons," Phys. Rev. E 70, 036617 (2004).
[26] K. G. Makris, H. Sarkissian, D. N. Christodoulides, and G. Assanto, "Nonlocal incoherent spatial solitons in liquid crystals," J. Opt. Soc. Am. B 22, 1371 (2005).
[27] A. G. Litvak, V. A. Mironov, G. M. Fraiman, and A. D. Yunakovskii, "Thermal self-e ect of wave beams in a plasma with a nonlocal nonlinearity," Sov. J. Plasma Phys. 22, 421 (1980).
[28] D. Suter and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583 (1993).
[29] C. A. Sackett, J. M. Gerton, M. Welling, and R. G. Hulet, "Measurements of Collective Collapse in a Bose-Einstein Condensate with Attractive Interactions," Phys. Rev. lett. 82, 876 (1999).
[30] W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288 (2004).
[31] A. W. Snyder and D. J. Mitchell, "Accessible solitons," Science 276, 1538 (1997).
[32] D. J. Mitchell and A. W. Snyder, "Soliton dynamics in a nonlocal medium," J. Opt. Soc. Am. B 16, 236 (1999).
[33] Y. R. Shen, "Solitons made simple," Science 276, 1520 (1997).
[34] W. Krolikowski and 0. Bang, "Solitons in nonlocal nonlinear media: Exact solutions," Phys. Rev. E 63, 016610 (2001).
[35] Zhiyong Xu, Y. V. Kartashov and L. Torner, "Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media," Opt. Lett. 30, 3171 (2005).
[36] C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner, and O. Cohen, "Two-dimensional multipole solitons in nonlocal nonlinear media," Opt. Lett. 31, 3312 (2006).
[37] Y. V. Kartashov, L. Torner, V. A. Vysloukh, and D. Mihalache, "Multipole vector solitons in nonlocal nonlinear media," Opt. Lett. 31, 1483 (2006).
[38] Zhiyong Xu, Y. V. Kartashov, and L. Torner, "Stabilization of vector soliton complexes in nonlocal nonlinear media," Phys. Rev. E 73, 055601(R) (2006).
[39] A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz and M. A. Karpierz, "Discrete propagation and spatial solitons in nematic liquid crystals," Opt. Lett. 29, 1530 (2004).
[40] A. Fratalocchi and G. Assanto, "Discrete light localization in one-dimensional nonlinear lattices with arbitrary nonlocality," Phys. Rev. E 72, 066608 (2005).
[41] Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Tunable Soliton Self-Bending in Optical Lattices with Nonlocal Nonlinearity," Phys. Rev. Lett. 93, 153903 (2004).
[42] Zhiyong Xu, Y. V. Kartashov, and L. Torner, "Soliton Mobility in Nonlocal Optical Lattices," Phys. Rev. Lett. 95, 113901 (2005).
[43] Zhiyong Xu, Y. V. Kartashov, and L. Torner, "Gap solitons supported by optical lattices in photorefractive crystals with asymmetric nonlocality," Opt. Lett. 31, 2027 (2006).
[44] N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, "Attraction of nonlocal dark optical solitons," Opt. Lett. 29, 286 (2004).
[45] A. Dreischuh, D. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, "Observation of Attraction between Dark Solitons," Phys. Rev. lett. 96, 043901 (2006).
[46] N. I. Nikolov, D. Neshev, 0. Bang, and W. Z. Krolikowski, "Quadratic solitons as nonlocal solitons," Phys. Rev. E 68, 036614 (2003).
[47] 0. Bang, W. Krolikowski, J. Wyller and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
[48] W. Krolikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[49] J. Wyller, W. Krolikowski, O. Bang, and J. J. Rasmussen, "Generic features of modulational instability in nonlocal Kerr media," Phys. Rev. E 66, 066615 (2002).
[50] C. Conti, Marco Peccianti, and G. Assanto, "Spatial solitons and modulational instability in the presence of large birefringence: The case of highly nonlocal liquid crystals," Phys. Rev. E 72, 066614 (2005).
[51] M. Peccianti, C. Conti, and G. Assanto, "Optical modulational instability in a nonlocal medium," Phys. Rev. E 68, 025602(R) (2003).
[52] M. Peccianti, C. Conti, G. Assanto, A. D. Luca and C. Umeton, "Routing of anisotropic spatial solitons and modulational instability in liquid crystals," Na-ture(London) 432, 733 (2004).
[53] M. Peccianti, C. Conti, E. Alberici and G. Assanto, "Spatially incoherent modulation instability in a nonlocal medium," Laser Physics Letters 2, 25 (2005).
[54] X. Hutsebaut, C. Cambournac, M. Haelterman, A. Admski, and K. Neyts, "Single-component higher-order mode solitons in nematic liquid crystals," Opt. Commun. 233, 211 (2004).
[55] C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in Nonlinear Media with an Infinite Range of Nonlocality: First Observation of Coherent Elliptic Solitons and of Vortex-Ring Solitons," Phys. Rev. lett. 95, 213904 (2005).
[56] M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, "Nonlocal spatial soliton interactions in nematic liquid crystals," Opt. Lett. 27, 1460 (2002).
[57] P. D. Rasmussen, O. Bang, and W. Krolikowski, "Theory of nonlocal soliton interaction in nematic liquid crystals," Phys. Rev. E 72, 066611 (2005).
[58] C. Rotschild. B. Alfassi. 0. Cohen, and M. Segev, "Long-range interactions between optical solitons," Nature Physics 2. 769 (2006).
[59] D. Briedis, D. E. Petersen, D. Edmundson et. al, "Ring vortex solitons in nonlocal nonlinear media," Opt. Exp. 13, 435 (2005).
[60] A. I. Yakimenko, Y. A. Zaliznyak, and Yu. S. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," Phys. Rev. E 71, 065603(R) (2005).
[61] S. Lopez-Aguayo, A. S. Desyatnikov, Yu. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100 (2006).
[62] D. Mihalache, D. Mazilu, F. Lederer, B. A. Malomed, Y. V. Kartashov, L.-C. Craso-van, and L. Torner, "Three-dimensional spatiotemporal optical solitons in nonlocal nonlinear media," Phys. Rev. E 73, 025602(R) (2006).
[63] S. Skupin, O. Bang, D. Edmundson, and W. Krolikowski, "Stability of two-dimensional spatial solitons in nonlocal nonlinear media," Phys. Rev. E 73, 066603 (2006).
[64] Y. V. Kartashov, L. Torner, and V. A. Vysloukh, "Lattice-supported surface solitons in nonlocal nonlinear media," Opt. Lett. 31, 2595 (2006).
[65] C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. lett. 91, 073901 (2003).
[66] C. Conti, M. Peccianti, and G. Assanto, "Observation of optical spatial solitons in a highly nonlocal medium," Phys. Rev. lett. 92, 113902 (2004).
[67] M. Peccianti, C. Conti, and G. Assanto, "Interplay between nonlocality and nonlin-earity in nematic liquid crystals," Opt. Lett. 30, 415 (2005).
[68] O. Cohen, H. Buljan, T. Schwartz, J. W. Fleischer, and M. Segev, "Incoherent solitons in instantaneous nonlocal nonlinear media," Phys. Rev. E 73, 015601(R) (2006).
[69] M. Shen, Q. Wang, J. L. Shi, Y. Y. Chen, and X. L. Wang, "Nonlocal incoherent white-light solitons in logarithmically nonlinear media," Phys. Rev. E 72, 026604 (2005).
[70] M. Shen, Q. Wang, J. L. Shi, P. Hou, and Q. Kong, "Partially coherent accessible solitons in strongly nonlocal media," Phys. Rev. E 73, 056602 (2006).
[71] M. Shen, J. Shi, and Q. Wang, "Incoherent accessible white-light solitons in strongly nonlocal Kerr media," Phys. Rev. E 74, 027601 (2006).
[72] M. Shen, Q. Wang, and J. Shi, "Elliptic incoherent accessible solitons in strongly nonlocal media," Opt. Commun. 270, 384 (2007).
[73] Q. Kong, J. Shi, M. Shen, and Q. Wang, "Elliptic incoherent solitons in strongly nonlocal media with anisotropic Kerr nonlinearity," J. Opt. Soc. Am. B (submitted).
[74] M. Shen, J. Shi, and Q. Wang, "Propagation of partially incoherent accessible solitons in strongly nonlocal Kerr media," J. Opt. Soc. Am. B (submitted).
[75] Q. Guo, B. Luo, F. Yi, S. Chi, and Yiqun Xie, "Large phase shift of nonlocal optical spatial solitons," Phys. Rev. E 69, 016602 (2004).
[76] Y. Q. Xie and Q. Guo, "Phase modulations due to collisions of beam pairs in nonlocal nonlinear media," Optical and Quantum Electronics 36, 1335 (2004).
[77] Q. Guo, B. Luo, and S. Chi, "Optical beams in sub-strongly non-local nonlinear media: A variational solution," Opt. Commun. 259, 336 (2005).
[78] Y. Huang, Q. Guo, and J. Cao, "Optical beams in lossy non-local Kerr media," Opt. Commun. 261, 175 (2006).
[79] T. Schwartz, T. Cartoon, H. Buljan, and M. Segev, "Spontaneous pattern formation with incoherent "white" light," Phys. Rev. Lett. 93, 223901 (2004).
[80] Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystal(Academic, San Diego, Calif., 2003).
[81] 陈园园，王奇，施解龙，卫青，“部分相干光束的振荡自陷特性，”物理学报51(3)，559(2002)。
[82] G. Assanto and M. Peccianti, IEEE J. Quantum Electron. 39, 13 (2003).
[83] E. Braun, L. Faucheux, A. Libchaber et. al., Europhys. Lett. 23, 239(1993).
[84] M. Kristian, K. Friedemann, Wen-Hen Chu, M. Shih, Yuri S. Kivshar, "Soliton transverse instabilities in anisotropic nonlocal self-focusing media," Opt. Lett. 29, 280 (2004).
[85] A. S. Desyatnikov, D. N. Neshev, Y. S. Kivshar, N. Sagemerten, D. Trager, J. Jagers, C. Denz, and Y. V. Kartashov, "Nonlinear photonic lattices in anisotropic nonlocal self-focusing media," Opt. Lett. 30, 869 (2005).
[86] 卫青，王奇，施解龙，“对数型非线性介质中亮暗孤子的存在性分析，”物理学报 52，1645(2002)。
[87] S. M. Sears, M. Soljacic, D. N. Christodoulides, and M. Segev, "Pattern formation via symmetry breaking in nonlinear weakly correlated systems," Phys. Rev. E 65, 036620 (2002).
[88] L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995).
[89] The expansion of the nonlocal response function R((?)-(?)) with respect to (?) about (?)= (?) to the second order, which reads R((?)-(?))=R_0+((?)-(?)). ▽⊥R_0 + (1/2)[((?)-(?)) ·▽⊥]~2R_0+…[75]. Because R((?)) is symmetrical with respect to (?)=0, we have that (?)_xR_0=(?)_yR_0=0, (?)_(xy)~2R_0=0, and (?)_x~2R_0=(?)_y~2R_0[75], then Eq. (2) is obtained.
[90] M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, "Modulation Instability of Incoherent Beams in Noninstantaneous Nonlinear Media," Phys. Rev. lett. 84, 467 (2000).
[91] M. Segev and D. N. Christodoulides, "Incoherent solitons: self-trapping of weaklycorrelated wave-packets," Optics and Photonics News 13, 70 (2002).
[92] E. D. Eugenieva, D. N. Christodoulides, and M. Segev, "Elliptic incoherent solitons in saturable nonlinear media," Opt. Lett. 25, 972 (2000).
[93] O. Katz, T. Carmon, T. Schwartz, M. Segev, and D. N. Christodoulides, "Observation of elliptic incoherent spatial solitons," Opt. Lett. 29, 1248 (2004).
[94] O. Bang, D. Edmundson, and W. Krolikowski, "Collapse of Incoherent Light Beams in Inertial Bulk Kerr Media," Phys. Rev. lett. 83, 5479 (1999).
[95] A. W. Snyder and D. J. Mitchell, "Mighty morphing spatial solitons and bullets," Opt. Lett. 22, 16 (1997).
[96] V. Tikhonenko, "Observation of mightymorphing spatial solitons," Opt. Lett. 23, 594 (1998).
[97] B. Alfassi, C. Rotschild, 0. Manela, M. Segev, and D. N. Christodoulides, "Boundary force effects exerted on solitons in nonlinear media with a very large range of nonlocality," Opt. Lett. 32, 154 (2007).
[98] R. Fischer, D. N. Neshev, W. Krolikowski, Y. S. Kivshar, D. Iturbe-Castillo, S. Chavez-Cerda, M. R. Meneghetti, D. P. Caetano and J. M. Hickman "Oblique interaction of spatial dark-soliton stripes in nonlocal media," Opt. Lett. 31, 3010 (2006).
[99] M. Peccianti, A. Dyadyusha, M. Kaczmarek and G. Assanto, "Tunable refraction and reflection of self-confined light beams," Nature Physics 2, 737 (2006).
[100] Y. S. Kivshar, "Bending light at will," Nature Physics 2, 729 (2006).
[101] M. Peccianti, G. Assanto, A. Dyadyusha, and M. Kaczmarek, "Nonlinear shift of spatial solitons at a graded dielectric interface," Opt. Lett. 32, 271 (2007).
[102] S. Ouyang, Q. Guo, and Wei Hu, "Perturbative analysis of generally nonlocal spatial optical solitons," Phys. Rev. E (in press).
[103] L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University, New York, 2003).
[104] S. Abe, and A. Ogura, "Solitary waves and their critical behavior in a nonlinear nonlocal medium with power-law response," Phys. Rev. E 57, 6066 (1998).
[105] P. Pedri, and L. santos, "Two-Dimensional Bright Solitons in Dipolar Bose-Einstein Condensates," Phys. Rev. lett. 95, 200404 (2005).