Hiremath, Kirankumar R

Many interesting phenomena in applications are based on interactions between their constituent sub-systems. The first principle exact models of these phenomena can be quite complicated. Therefore, many practitioners prefer to use so-called phenomenological models, which are generally known as models based on coupled mode theory (CMT). This type of reduced-order model captures the dominant behavior of the system under appropriate conditions. Quite often, these validity conditions are qualitatively described, but no detailed mathematical analysis is provided. This work addresses this issue and presents improvements in the traditional phenomenological models. Although an LC circuit model is used for illustration due to its simplicity, the results in this work are equally applicable to a wide variety of coupled models. A detailed mathematical analysis is carried out to quantify the order of approximation involved in the model-based CMT. Using it, the validity of the model in the regime from weak coupling to strong coupling is analytically investigated. An improved reduced-order model is proposed, which gives better results than the traditional phenomenological model. The analytical studies are verified with numerical simulations, which clearly show better validity of the proposed improved model of coupled systems.

This work investigates a mathematical model of the propagation of lightwaves in bent optical waveguides. This modeling leads to a non-self-adjoint eigenvalue problem for differential operator defined on the unbounded domain. Through detailed analysis, a relationship between the real and imaginary parts of the complex-valued propagation constants was constructed. Using this relation, it is found that the real and imaginary parts of the propagation constants are bounded, meaning they are limited within certain region in the complex plane. The orthogonality of these bent modes is also proved. By the asymptotic analysis of these modes, it is proved that as r → ∞ the behavior of the eigenfunctions is proportional to 1 / r .