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.
The central goal of the project was the analysis of wave propagation through networks of thin optical fibers (waveguides). The study of optical networks is important in many applications such as telecommunications, and development of new generations of computers (based on optical rather than electronic processes). The other goal of the project was the development of mathematical tools for such analysis and solving various problems in applied mathematics where these new tools allow one to achieve essential progress. Mathematically, the problem of wave propagation through networks of thin optical fibers is described by the Maxwell or Helmholz equation in web-like three-dimensional geometric structures occupied by waveguides with the thickness parameter going to zero. The problem does not have an explicit solution. A numerical solution, difficult to obtain due to a complicated structure of the domain, would not allow for determination of parameters that are needed for applications. The authors developed a method of asymptotical analysis of the problem that provides a detailed description of the wave processes and allows one to choose specific parameters for different applications. One such application is a design of a device for "slowing down of the light". The device can be used for transition from fast optical processes to slower electronic ones. The authors suggested a specific network to create such a device and developed a tool to estimate admissible technological errors in the design. Small random technological noises can destroy stability of an optical system. The authors found the asymptotics of the Lyapunov exponent describing the stability of periodic networks of thin fibers. This allows one to estimate the magnitude of the noise that is negligible for the wave propagation. The conditions were found that guarantee the existence of an infinite set of positive interior transmission eigenvalues and the Weyl bounds for them. These results play an important role in inverse scattering, in particular, for nondestructive testing of a medium. A high frequency asymptotics for the total cross-section in the scattering by a non-convex obstacle was found. The results are surprisingly different from the ones known in the case of convex obstacles. A high-frequency asymptotics of the waves scattered by classically invisible bodies was obtained. It was shown that the geometrical optic invisibility implies real invisibility only at certain frequencies. A partial annihilation method was developed for numbers of negative states of Schrodinger operators. The method allows one to study operators with spectral dimension below the critical value (?) and with fractional spectral dimensions, as well as operators on lattices and fractals. Markov models in social sciences and demography were studied, providing a description of the volume of a political club, speed of propagation of rumors, and stability of biological populations under random changes of environment. Stability of the risk measure of a portfolio was studied under changes in the leading financial factors. A monograph and more than 40 papers were published during the period of the project.
