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，并开发了描述硅片上的光波导(所谓的漏线)的数学模型。在厚度参数趋于零的网状三维几何结构中，用Helmholtz方程来描述这个问题。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。这个问题没有明确的解决方案。由于区域和方程系数的复杂结构，很难获得数值解，因此不能确定应用所需的参数。因此，采用了渐近分析方法。原来的三维问题被简化为极限图上的一个简单的一维问题。后一个问题允许进行详细的分析。周期网络在应用中经常被使用。由于器件的纳米级，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介质中的波动问题。这项研究的成果之一是对光学系统的随机稳定性进行了估计(即随机误差的允许值的估计)。该项目的动机和实际应用涉及到创建紧凑(纳米光学尺度)和高效的光延迟器件的数学背景，该器件可用于同步非常快的光通信线路和较慢的电子设备。对调查第一阶段获得的极限一维问题的分析表明，可以用来创建这样一个装置的网络。

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.