arXiv subjects

Ievgen V. Verbytskyi

Publications and source records attributed to Ievgen V. Verbytskyi.

2 recordsLinked to original sources

arXiv2018

Pareto optimization of resonances and minimum-time control

The aim of the paper is to reduce one spectral optimization problem, which involves the minimization of the decay rate $|\mathrm{Im} \, k |$ of a resonance $k$, to a collection of optimal control problems on the Riemann sphere $\widehat{\mathbb{C}}$. This reduction allows us to apply methods of extremal synthesis to the structural optimization of layered optical cavities. We start from a dual problem of minimization of the resonator length and give several reformulations of this problem that involve Pareto optimization of the modulus $|k|$ of a resonance, a minimum-time control problem on $\widehat{\mathbb{C}}$, and associated Hamilton-Jacobi-Bellman equations. Various types of controllability properties are studied in connection with the existence of optimizers and with the relationship between the Pareto optimal frontiers of minimal decay and minimal modulus. We give explicit examples of optimal resonances and describe qualitatively properties of the Pareto frontiers near them. A special representation of bang-bang controlled trajectories is combined with the analysis of extremals to obtain various bounds on optimal widths of layers. We propose a new method of computation of optimal symmetric resonators based on minimum-time control and compute with high accuracy several Pareto optimal frontiers and high-Q resonators.

math.OC↗

arXiv2015

Nonlinear bang-bang eigenproblems and optimization of resonances in layered cavities

Quasi-normal-eigenvalue optimization is studied under constraints $b_1(x) \le B(x) \le b_2 (x)$ on structure functions $B$ of 2-side open optical or mechanical resonators. We prove existence of various optimizers and provide an example when different structures generate the same optimal quasi-(normal-)eigenvalue. To show that quasi-eigenvalues locally optimal in various senses are in the spectrum $Σ^{nl}$ of the bang-bang eigenproblem $y" = - ω^2 y [ b_1 + (b_2 - b_1) χ_{\mathbb{C}_+} (y^2 ) ]$, where $χ_{\mathbb{C}_+} (\cdot)$ is the indicator function of the upper complex half-plane $\mathbb{C}_+$, we obtain a variational characterization of the nonlinear spectrum $Σ^{nl}$ in terms of quasi-eigenvalue perturbations. To address the minimization of the decay rate $| \mathrm{Im} \ ω|$, we study the bang-bang equation and explain how it excludes an unknown optimal $B$ from the optimization process. Computing one of minimal decay structures for 1-side open settings, we show that it resembles gradually size-modulated 1-D stack cavities introduced recently in Optical Engineering. In 2-side open symmetric settings, our example has an additional centered defect. Nonexistence of global decay rate minimizers is discussed.

math.OC↗