Propagation of Waves in Complex Media, and Analysis of Small Noises

研究人员开发了一种数学方法来研究细光纤（波导）网络中的光学过程，以及描述硅晶片（所谓的漏线）上的光波导的数学模型。用厚度参数趋近于零的类网状三维几何结构的亥姆霍兹方程描述了这一问题。研究人员还将这些结果扩展到涉及麦克斯韦方程的更深刻的数学模型中。这个问题没有明确的解决办法。由于方程的域和系数结构复杂，难以获得数值解，因此无法确定应用所需的参数。因此，应用渐近分析。原来的三维问题在极限图上被简化为一个简单得多的一维问题。后一个问题需要详细分析。周期性网络在应用中非常常用。由于器件的纳米尺度，这些周期结构存在一定的工艺误差。研究了具有小随机噪声的周期性介质中的波动问题。本研究的输出之一是光学系统随机稳定性的估计（即随机误差容许值的估计）。该项目的动机和实际应用涉及创建紧凑（纳米光学尺度）和高效的光延迟设备的数学背景，该设备可用于同步非常快的光通信线路与慢得多的电子设备。在调查的第一阶段得到的限制一维问题的分析，建议一个网络，可以用来创建这样一个装置。

The investigators develop a mathematical approach for studying optical processes in networks of thin fibers (waveguides) and mathematical models describing optical waveguides on a silicon wafer (so-called leaking wires). The problem is described by the Helmholtz equation in web-like three-dimensional geometric structures with the thickness parameter going to zero. The investigators also extend these results to more profound mathematical models involving Maxwell equations. The problem does not have an explicit solution. A numerical solution, difficult to obtain due to a complicated structure of the domain and coefficients of the equations, would not allow determination of parameters that are needed for applications. Therefore, the asymptotic analysis is applied. The original three-dimensional problem is reduced to a much simpler one-dimensional problem on the limiting graph. The latter problem admits a detailed analysis. Periodic networks are very often used in applications. These periodic structures have some technological errors due to the nano-scale of the device. The investigators study the wave problem in periodic media with small random noise. One of the outputs of this study is an estimate of the stochastic stability of optical systems (i.e., an estimate of admissible values of the random errors).The motivation and practical applications of the project concern the mathematical background for creating a compact (nano-optical scale) and efficient optical delay device which can be used to synchronize very fast optical communication lines with much slower electronics. The analysis of the limiting one-dimensional problem obtained at the first stage of the investigation suggests a network which can be used to create such a device.