带隙孤子及其在无序体系中的动力学研究
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摘要
近年来,非线性周期体系中非线性和周期的相互作用引起了人们很大的研究兴趣。其中一个非常有趣的现象就是在非线性光子晶体中的带隙孤子的产生。由于其本身具有许多奇异的性质,带隙孤子在光通信,光控制以及光存储中具有许多潜在的应用。而最近,由于带隙孤子与其它相关的领域联系紧密,尤其是正在蓬勃发展的与光格子中的玻色-爱因斯坦凝聚领域联系密切,带隙孤子的研究更加引起了人们的关注。
本论文中,首先,我们通过引入了局域Bloch波图象来求解一维非线件光子晶体带隙孤子解。然后,我们将带隙孤子的概念引入到电子体系中去,并且找到了窄禁带体系中孤子解的特性与其本征值在禁带中位置之间的关系。最后,我们模拟了物质波带隙孤子在无序体系中的运动,找到了其传播特性。
论文的第二章研究了一维周期性Kerr介质中的带隙孤子解。首先我们介绍了耦合模方法和多重尺度方法,并得到了带隙孤子解。为了克服上面两种方法的不足,我们引入局域Bloch波图象,利用局域Bloch波图象,建立了一个广义非线性薛定谔方程来描述带隙孤子解。通过这个广义非线性薛定谔方程,我们发现不同形式的带隙孤子解可以通过一个线性项和两个非线性项的竞争来解释。最后,利用局域Bloch波图象,我们给出了带隙孤子的稳定性分析。
在论文的第三章,我们把带隙孤子的概念扩展到电子极化孤子体系中。我们建立了一个窄禁带单电子模型来求极化孤子在空间的形状与其本征能量在禁带中位置的关系。我们发现,极化孤子解的性质强烈依赖于其本征能量在禁带中的位置:当其本征能量在带边时,孤子是钟型(bell-like)解,而当本征能量深入到禁带中央时,孤子是双峰形状。通过局域电子晶格吸引势与晶格弹性势能的竞争,我们给出了不同形状孤子解的物理解释。另外,我们比较了我们建立的模型与分子晶体模型以及电声子耦合模型,发现我们的模型可以看作上面两种模型的桥梁,分子晶体模型可以看成我们模型的一个特殊情况,而我们的模型在某种近似下可以看成电声子模型的一个特例。最后,通过比较窄禁带单电子模型中的极化孤子解与一维周期性Kerr介质中的禁带孤子解,发现了非线性的机制的不同对孤子特性的影响。
在论文的第四章,利用有限时域差分方法,我们研究了一维光格子中物质波带隙孤子在无序体系的运动。对于弱无序系统,体系可以做有效粒子近似,电就是孤子在运动中始终可以当作一个粒子来看待。利用有效粒子近似,我们找到了改变带隙孤子运动的原因,并且构建了一套运动方程来求解孤子的运动。另外,我们还找到了带隙孤子运动与无序之间的广义关系:随着无序强度的增大,孤子的系综平均速度缓慢减少,并且系综平均速度减少量正比于无序的方差,而与此同时,孤子在运动中被反弹的几率变大。对于无序比较大的体系,我们发现孤子在此体系中一大部分的场可以被俘获。通过分析场的变化,我们发现这个被俘获是由于孤子场在大无序体系中运动时,场会被散射从而能量降低引起的。为了形象描述这个现象,我们还模拟了孤子穿过两个缺陷态的情况,发现了相同的结果。由于光体系与物质波体系的相似性,它或许可以给光存储提供一种新的方法。
In last decades, the interplay between nonlinearity and periodicity has received intensively experimental and theoretical studies. One of the most interesting results is the existence of solitary wave in gap, termed "gap soliton" by Chen and Mills. Due to its special transmission properties, gap soliton can be used for a wide range of applications in optical communications and optical storage. Recently, because it provides to other related fields, such as the rapidly developing field of "Bose-Einstein condensates", the research of gap soliton has received considerable attentions.
In this dissertation, firstly, we develop a local-Bloch wave picture to investigate the gap soliton solutions in one-dimensional nonlinear photonic crystal systems. Then we extend the concept of "gap soliton" into the electronic system to study the polaronic soliton solutions in a narrow gap system. Finally, we study the dynamics of matter wave gap soliton in optical lattice with randomness.
In chapter 2, we study the gap soliton solutions in one-dimensional nonlinear photonic crystal (periodic Kerr media). First, we introduce two methods: coupled-mode method and multiple-scales method to obtain the gap soliton solutions. To overcome the shortcoming of the two methods, we introduce the local Bloch wave picture. Based on the local Bloch wave picture, we find that the envelop function of the field is a generalized nonlinear Schrodinger equations. Using the envelop equations, we find that the different GS functional forms are from the competition of one linear term and two nonlinear terms. And finally, we give out a simple soliton stable analysis based on the local Bloch wave picture.
In chapter 3, we construct a model to investigate the one-dimensional localized excitation, which is called "polaronic soliton" in a narrow gap system. From this model, we can obtain the soliton solution analytically in the continuum limit. We find that the soliton properties strongly depend on the gap regions where the electronic energy falls: for example, the soliton has a bell-like form when the eigenenergy is near the conduction band-edge, but a two-peak form when the eigenenergy is near the gap center. We give out a physical interpretation based on the local energy competition between the electron-lattice attractive energy and the lattice elastic energy. Compare our model
with the molecular-crystal model and the coupled electron-phonon model, we find that our model can be treated as a bridge between them: the molecular-crystal model can be seen as a special case of our model and our model can be treated as a spacial case of the coupled electron-phonon model in some limit. Finally, we compare the polaronic soliton in our model and the gap soliton in one-dimensional nonlinear photonic crystal and find the nonlinearity is very important in soliton solutions.
In chapter 4, using the numerical finite-difference time-domain method, we investigate the matter-wave gap solitons propagation in random optical lattice. For the weak random case, we introduce the effective particle picture to solve the movement of gap soliton. Based on the effective particle picture, obtain the effect of the randomness on gap soliton and solve out the motions. Moreover, we obtain the general law of the gap soliton movement depending on the weak randomness. Such as, with the increase of the random strength, the ensemble-average velocity reduces slowly in which the reduction is proportional to the variance of the randomness, and the reflected probability increase. For the large random case, we find that the gap solitons can be trapped by the randomness. From the time evolution of the soliton field, we give out a qualitative interpretation: the gap soliton field can be radiated by the large randomness and trapped by two defect modes. And we also simulate the case that gap soliton propagation through two defect to give an interpretation. This may be provide a route of optical memory.
本论文中,首先,我们通过引入了局域Bloch波图象来求解一维非线件光子晶体带隙孤子解。然后,我们将带隙孤子的概念引入到电子体系中去,并且找到了窄禁带体系中孤子解的特性与其本征值在禁带中位置之间的关系。最后,我们模拟了物质波带隙孤子在无序体系中的运动,找到了其传播特性。
论文的第二章研究了一维周期性Kerr介质中的带隙孤子解。首先我们介绍了耦合模方法和多重尺度方法,并得到了带隙孤子解。为了克服上面两种方法的不足,我们引入局域Bloch波图象,利用局域Bloch波图象,建立了一个广义非线性薛定谔方程来描述带隙孤子解。通过这个广义非线性薛定谔方程,我们发现不同形式的带隙孤子解可以通过一个线性项和两个非线性项的竞争来解释。最后,利用局域Bloch波图象,我们给出了带隙孤子的稳定性分析。
在论文的第三章,我们把带隙孤子的概念扩展到电子极化孤子体系中。我们建立了一个窄禁带单电子模型来求极化孤子在空间的形状与其本征能量在禁带中位置的关系。我们发现,极化孤子解的性质强烈依赖于其本征能量在禁带中的位置:当其本征能量在带边时,孤子是钟型(bell-like)解,而当本征能量深入到禁带中央时,孤子是双峰形状。通过局域电子晶格吸引势与晶格弹性势能的竞争,我们给出了不同形状孤子解的物理解释。另外,我们比较了我们建立的模型与分子晶体模型以及电声子耦合模型,发现我们的模型可以看作上面两种模型的桥梁,分子晶体模型可以看成我们模型的一个特殊情况,而我们的模型在某种近似下可以看成电声子模型的一个特例。最后,通过比较窄禁带单电子模型中的极化孤子解与一维周期性Kerr介质中的禁带孤子解,发现了非线性的机制的不同对孤子特性的影响。
在论文的第四章,利用有限时域差分方法,我们研究了一维光格子中物质波带隙孤子在无序体系的运动。对于弱无序系统,体系可以做有效粒子近似,电就是孤子在运动中始终可以当作一个粒子来看待。利用有效粒子近似,我们找到了改变带隙孤子运动的原因,并且构建了一套运动方程来求解孤子的运动。另外,我们还找到了带隙孤子运动与无序之间的广义关系:随着无序强度的增大,孤子的系综平均速度缓慢减少,并且系综平均速度减少量正比于无序的方差,而与此同时,孤子在运动中被反弹的几率变大。对于无序比较大的体系,我们发现孤子在此体系中一大部分的场可以被俘获。通过分析场的变化,我们发现这个被俘获是由于孤子场在大无序体系中运动时,场会被散射从而能量降低引起的。为了形象描述这个现象,我们还模拟了孤子穿过两个缺陷态的情况,发现了相同的结果。由于光体系与物质波体系的相似性,它或许可以给光存储提供一种新的方法。
In last decades, the interplay between nonlinearity and periodicity has received intensively experimental and theoretical studies. One of the most interesting results is the existence of solitary wave in gap, termed "gap soliton" by Chen and Mills. Due to its special transmission properties, gap soliton can be used for a wide range of applications in optical communications and optical storage. Recently, because it provides to other related fields, such as the rapidly developing field of "Bose-Einstein condensates", the research of gap soliton has received considerable attentions.
In this dissertation, firstly, we develop a local-Bloch wave picture to investigate the gap soliton solutions in one-dimensional nonlinear photonic crystal systems. Then we extend the concept of "gap soliton" into the electronic system to study the polaronic soliton solutions in a narrow gap system. Finally, we study the dynamics of matter wave gap soliton in optical lattice with randomness.
In chapter 2, we study the gap soliton solutions in one-dimensional nonlinear photonic crystal (periodic Kerr media). First, we introduce two methods: coupled-mode method and multiple-scales method to obtain the gap soliton solutions. To overcome the shortcoming of the two methods, we introduce the local Bloch wave picture. Based on the local Bloch wave picture, we find that the envelop function of the field is a generalized nonlinear Schrodinger equations. Using the envelop equations, we find that the different GS functional forms are from the competition of one linear term and two nonlinear terms. And finally, we give out a simple soliton stable analysis based on the local Bloch wave picture.
In chapter 3, we construct a model to investigate the one-dimensional localized excitation, which is called "polaronic soliton" in a narrow gap system. From this model, we can obtain the soliton solution analytically in the continuum limit. We find that the soliton properties strongly depend on the gap regions where the electronic energy falls: for example, the soliton has a bell-like form when the eigenenergy is near the conduction band-edge, but a two-peak form when the eigenenergy is near the gap center. We give out a physical interpretation based on the local energy competition between the electron-lattice attractive energy and the lattice elastic energy. Compare our model
with the molecular-crystal model and the coupled electron-phonon model, we find that our model can be treated as a bridge between them: the molecular-crystal model can be seen as a special case of our model and our model can be treated as a spacial case of the coupled electron-phonon model in some limit. Finally, we compare the polaronic soliton in our model and the gap soliton in one-dimensional nonlinear photonic crystal and find the nonlinearity is very important in soliton solutions.
In chapter 4, using the numerical finite-difference time-domain method, we investigate the matter-wave gap solitons propagation in random optical lattice. For the weak random case, we introduce the effective particle picture to solve the movement of gap soliton. Based on the effective particle picture, obtain the effect of the randomness on gap soliton and solve out the motions. Moreover, we obtain the general law of the gap soliton movement depending on the weak randomness. Such as, with the increase of the random strength, the ensemble-average velocity reduces slowly in which the reduction is proportional to the variance of the randomness, and the reflected probability increase. For the large random case, we find that the gap solitons can be trapped by the randomness. From the time evolution of the soliton field, we give out a qualitative interpretation: the gap soliton field can be radiated by the large randomness and trapped by two defect modes. And we also simulate the case that gap soliton propagation through two defect to give an interpretation. This may be provide a route of optical memory.
引文
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