Optimization of resonances for multilayer x-ray resonators

Meer over het boek

Resonances in open systems are described by eigenvalue problems with radiation conditions at infinity and are relevant in fields such as acoustics, classical mechanics, quantum mechanics, and x-ray physics. This thesis focuses on optimizing resonances for multilayer x-ray resonators, which consist of multiple layers that support specific resonant states. These states can be excited by x-ray beams at particular grazing angles of incidence, corresponding to the system's resonant frequencies, resulting in significant field enhancement within the structure compared to the incident field. X-ray resonators or waveguides facilitate the filtering, guiding, and concentration of x-rays, beneficial for nanoscale x-ray structure analysis and imaging. The multilayer structures are characterized by the refractive index n, and the goal is to identify a function n that maximizes field enhancement at a resonant angle of incidence while adhering to constraints on n. The optimization problem employs an objective function involving complex resonances and resonance functions. Derivatives of these functions with respect to n are derived using perturbation theory of linear operators. Additionally, approximation formulas for reflectivity are developed, providing a mathematical basis for the kinematic approximation. Higher-order Taylor and Padé approximations yield significant improvements, particularly near the critical angle. The optimization proble