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

研究人员开发了一种数学方法，用于研究薄光纤（波导）网络中的光学过程和描述硅晶片上光波导（所谓的泄漏线）的数学模型。该问题由Helmholtz方程描述，在网状三维几何结构中，厚度参数趋于零。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。这个问题没有明确的解决方案。由于域的复杂结构和方程的系数而难以获得的数值解将不允许确定应用所需的参数。因此，应用渐近分析。在极限图上，原来的三维问题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。周期性网络在实际应用中非常常用。由于器件的纳米尺度，这些周期性结构存在一些技术误差。研究人员研究了具有小随机噪声的周期性介质中的波动问题。该研究的输出之一是光学系统的随机稳定性的估计（即，该项目的动机和实际应用涉及用于创建紧凑（纳米光学尺度）和有效的光学延迟设备的数学背景，该光学延迟设备可用于使非常快的光学通信线路与慢得多的电子设备同步。在调查的第一阶段获得的限制一维问题的分析表明，可以用来创建这样的设备的网络。

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.