Nonlinear normal modes in electrodynamic systems: A nonperturbative approach

TL;DR

This paper introduces a nonperturbative analytical approach to study electromagnetic nonlinear normal modes in resonators, demonstrating that energy orthogonality extends from linear to certain nonlinear systems.

Contribution

It provides exact solutions for nonlinear electromagnetic modes and proves energy orthogonality in nonlinear oscillatory systems, extending linear theory concepts.

Findings

Exact analytical solutions for nonlinear modes

Energy of nonlinear oscillations equals sum of mode energies

Energy orthogonality applies to nonlinear systems

Abstract

We consider electromagnetic nonlinear normal modes in cylindrical cavity resonators filled with a nonlinear nondispersive medium. The key feature of the analysis is that exact analytical solutions of the nonlinear field equations are employed to study the mode properties in detail. Based on such a nonperturbative approach, we rigorously prove that the total energy of free nonlinear oscillations in a distributive conservative system, such as that considered in our work, can exactly coincide with the sum of energies of the normal modes of the system. This fact implies that the energy orthogonality property, which has so far been known to take place only for linear oscillations and fields, can also be observed in the nonlinear oscillatory system.