We consider a system governed by the wave equation with index of refraction @math , taken to be variable within a bounded region @math and constant in @math . The solution of the time-dependent wave equation with initial data, which is localized in @math , spreads and decays with advancing time. This rate of decay can be measured (for @math , and more generally, @math odd) in terms of the eigenvalues of the scattering resonance problem, a non--self-adjoint eigenvalue problem governing the time-harmonic solutions of the wave (Helmholtz) equation which are outgoing at @math . Specifically, the rate of energy escape from @math is governed by the complex scattering eigenfrequency, which is closest to the real axis. We study the structural design problem: Find a refractive index profile @math within an admissible class which has a scattering frequency with minimal imaginary part. The admissible class is defined in terms of the compact suppo...
The problem of maximizing the lifetime of a state trapped within a leaky cavity can be framed in several ways. The figure of merit can be taken to be the minimization of energy flux through the boundary @cite_15 or a measure of mode localization @cite_13 @cite_30 . In @cite_27 , the problem of minimizing @math for a chosen resonance was investigated computationally in both one- and two-dimensions. The 1-d problem considered here was also studied computationally in @cite_12 . This work focuses on the optimization of @math to minimize @math which satisfies outgoing solutions of the equation @math In particular, the variations @math and @math are formally computed. In @cite_45 , transfer matrix methods were used to design low-loss 2D resonators with radial symmetry. In each of these papers, gradient-based optimization methods were used to solve the optimization problem.
@cite_30 @cite_27 @cite_45 @cite_15 @cite_13 @cite_12
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We formulate the problem of designing the low-loss cavity for the International Linear Collider (ILC) as an electromagnetic shape optimization problem involving a Maxwell eigenvalue problem. The objective is to maximize the stored energy of a trapped mode in the end cell while maintaining a specified frequency corresponding to the accelerating mode. A continuous adjoint method is presented for computation of the design gradient of the objective and constraint. The gradients are used within a nonlinear optimization scheme to compute the optimal shape for a simplified model of the ILC in a small multiple of the cost of solving the Maxwell eigenvalue problem. Abstract We consider resonance phenomena for the scalar wave equation in an inhomogeneous medium. Resonance is a solution to the wave equation which is spatially localized while its time dependence is harmonic except for decay due to radiation. The decay rate, which is inversely proportional to the qualify factor, depends on the material properties of the medium. In this work, the problem of designing a resonator which has high quality factor (low loss) is considered. The design variable is the index of refraction of the medium. High quality resonators are desirable in a variety of applications, including photonic band gap devices. Finding resonance in a linear wave equation with radiation boundary condition involves solving a nonlinear eigenvalue problem. The magnitude of the ratio between real and imaginary part of the eigenvalue is proportional to the quality factor Q . The optimization we perform is finding a structure which possesses an eigenvalue with largest possible Q . We present a numerical approach for solving this problem. The method consists of first finding a resonance eigenvalue and eigenfunction for a non-optimal structure. The gradient of Q with respect to index of refraction at that state is calculated. Ascent steps are taken in order to increase the quality factor Q . We demonstrate how this approach can be implemented and present numerical examples of high Q structures. Circular resonators are fundamentally interesting elements that are essential for research involving highly confined fields and strong photon-atom interactions such as cavity QED, as well as for practical applications in optical communication systems as and biochemical sensing. The important characteristics of a ring resonator are the Q-factor, the free spectral range (FSR) and the modal volume, where the last two are primarily determined by the resonator dimensions. The Total-Internal-Reflection (TIR) mechanism employed in "conventional" resonators couples between these characteristics and limits the ability to realize compact devices with large FSR, small modal volume and high Q. Recently, we proposed and analyzed a new class of a resonator in an annular geometry that is based on a single defect surrounded by radial Bragg reflectors on both sides. The radial Bragg confinement breaks the link between the characteristics of the mode and paves a new way for the realization of compact and low loss resonators. Such properties as well as the unique mode profile of the ABRs make this class of devices an excellent tool for ultra-sensitive biochemical detection as well as for studies in nonlinear optics and cavity QED. Variational methods are applied to the design of a two-dimensional lossless photonic crystal slab to optimize resonant scattering phenomena. The method is based on varying properties of the transmission coefficient that are connected to resonant behavior. Numerical studies are based on boundary-integral methods for crystals consisting of multiple scatterers. We present an example in which we modify a photonic crystal consisting of an array of dielectric rods in air so that a weak transmission anomaly is transformed into a sharp resonance. The problem of creating eigenfunctions which are localized arises in the study of photonic bandgap structures. A model problem, that of finding material inhomogeneity in a domain so that one of its Dirichlet eigenfunctions is localized, is considered in this work. The most difficult aspect, that of formulating the problem, is described, and well-posed variational problems are given. A computational approach, based on gradient descent with projection and trajectory continuation, is devised to solve the optimization problem. Numerical examples are provided which demonstrate the capability of the computational method. The increasing use of micro- and nano-scale components in optical, electrical, and mechanical systems makes the understanding of loss mechanisms and their quantification issues of fundamental importance. In many situations, performance-limiting loss is due to scattering and radiation of waves into the surrounding structure. In this paper, we study the problem of systematically improving a structure by altering its design so as to decrease the loss. We use sensitivity analysis and local gradient optimization, applied to the scattering resonance problem, to reduce the loss within the class of piecewise constant structures. For a class of optimization problems where the material parameters are constrained by upper and lower bounds, it is observed that an optimal structure is piecewise constant with values achieving the bounds.