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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Consider a system governed by the time-dependent Schrodinger equation initialized in its ground state. When subjected to weak (size e) parametric forcing by an “ionizing field” (time-varying), the state decays with advancing time due to coupling of the bound state to radiation modes. The decay rate of this metastable state is governed by Fermi’s golden rule, Γ[V], which depends on the potential V and the details of the forcing. We pose the following potential design problem: find Vopt which minimizes Γ[V] (maximizes the lifetime of the state) over an admissible class of potentials with fixed spatial support. We formulate this problem as a constrained optimization problem and prove that an admissible optimal solution exists. Then, using quasi-Newton methods, we compute locally optimal potentials. These have the structure of a truncated periodic potential with a localized defect. In contrast to optimal structures for other spectral optimization problems, our optimizing potentials appear to be interior point... 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. The paper is devoted to optimization of resonances associated with 1-D wave equations in inhomogeneous media. The medium's structure is represented by a nonnegative function B. The problem is to design for a given @math a medium that generates a resonance on the line @math with a minimal possible modulus of the imaginary part. We consider an admissible family of mediums that arises in a problem of optimal design for photonic crystals. This admissible family is defined by the constraints @math with certain constants @math . The paper gives an accurate definition of optimal structures that ensures their existence. We prove that optimal structures are piecewise constant functions taking only two extreme possible values @math and @math . This result explains an effect recently observed in numerical experiments. Then we show that intervals of constancy of an optimal structure are tied to the phase of the corresponding resonant mode and write this connection as a nonlinear 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.