Rigorous Asymptotics of Surface Plasmon Polariton Wavepackets in Nonlinear Media

本课题研究克尔非线性介质与金属界面表面等离子激元（SPPs）的局域波包。考虑到金属的材料弥散，两种材料可以是均匀的或空间周期性的。从应用的角度来看，这种波包作为潜在的信息载体是有价值的。再加上在比体电介质中的电磁波更小的空间尺度上操纵spp的可能性，这提供了有趣的物理现象。控制方程为色散非线性麦克斯韦方程组。通常，麦克斯韦方程的全矢量值形式必须用于研究局域波。波包由一个或多个载波和缓慢变化的包络表示。形式上，对于具有单载波的渐近宽小波包的包络可以导出一个复金兹堡-朗道方程。目的是证明这种形式描述提供了麦克斯韦方程组真解的近似（在合适的范数中）。这将在几种几何设置中完成，包括二维情况，其中场在一个空间方向上离域，以及在所有三个维度上都本地化的情况。此外，还将研究空间波包和时空波包。空间波包的场是时间谐波的，其中一个空间变量起演化变量的作用。控制方程可以写成与时间无关的非线性旋度问题。时空波包的演化变量是时间。我们将考虑由一个平面和构成一个角的两个半平面给出的界面。由于金属的材料色散会导致时间演化中的损耗，因此还将考虑介质中掺杂的介电部分的影响。这可能导致孤立波在金兹堡-朗道方程中存在，从而在原始系统中近似无损传播。数学分析的主要困难首先是控制非线性色散麦克斯韦方程组的适定性。用柯西问题表示的旋-旋系统。其次，渐近波包方差的残差估计必须在与适定性结果相容的范数中提供。该分析将伴随着波包传播的数值计算和渐近误差收敛的数值测试。

This project studies localized wavepackets of surface plasmon polaritons (SPPs) at the interface of a Kerr nonlinear dielectric and a metal. The material dispersion of the metal is accounted for, and both materials can be either homogeneous or spatially periodic. From the applied point of view such wavepackets are valuable as potential information carriers. Together with the possibility to manipulate SPPs on a much smaller spatial scale than by electromagnetic waves in bulk dielectrics this offers interesting physical phenomena. The governing equations are dispersive nonlinear Maxwell equations. Typically, the full vector valued form of Maxwell equations has to be used to study localized waves. The wavepacket is represented by one or more carrier waves and slowly varying envelopes. Formally, a complex Ginzburg-Landau equation can be derived for the envelope of an asymptotically broad and small wavepacket with a single carrier wave. The aim is to prove that this formal description provides an approximation (in a suitable norm) of a true solution of the Maxwell system. This will be done in several geometrical settings including a two dimensional case, where the field is delocalized in one spatial direction, as well as the case with localization in all three dimensions. Also, both spatial and spatiotemporal wavepackets will be studied. By spatial wavepackets the field is time harmonic and one of the spatial variables plays the role of an evolution variable. The governing equations then can be written as a time independent nonlinear curl-curl problem. By spatiotemporal wavepackets the evolution variable is time. We will consider the interfaces given by a plane as well as by two half-planes forming a corner. Because the material dispersion of the metal leads to losses in the time evolution, the effect of a doped dielectric part of the medium will be also considered. This can lead to the existence of solitary waves in the Ginzburg-Landau equation and thus to an approximately lossless propagation in the original system. The main difficulties for the mathematical analysis are firstly the well-posedness of the governing nonlinear dispersive Maxwell equations, resp. of the curl-curl system posed as Cauchy problem. Secondly, estimates of the residual for the asymptotic wavepacket ansatz have to be provided in a norm compatible with the well-posedness results. The analysis will be accompanied by numerical computations of the wavepacket propagation and by numerical test of the convergence of the asymptotic error.