Asymptotic and Numerical Techniques for Resonances of Thin Photonic Structures

We consider the problem of calculating resonance frequencies and radiative losses of an optical resonator. The optical resonator is in the form of a thin membrane with variable dielectric properties. This work provides two very different approaches for doing such calculations. The first is an asymptotic method which exploits the small thickness and high index of the membrane. We derive a limiting resonance problem as the thickness goes to zero, and for the case of a simple resonance, find a first order correction. The limiting problem and the correction are in one less space dimension, which can make the approach very efficient. Convergence estimates are proved for the asymptotics. The second approach, based on the finite element method with a truncated perfectly matched layer, is not restricted to thin structures. We demonstrate the use of these methods in numerical calculations which further illustrate their differences. The asymptotic method finds resonance by solving a dense, but small, nonlinear eigenvalue problem, whereas the finite element method yields a large, but linear and sparse generalized eigenvalue problem. Both methods reproduce a localized defect mode found previously by finite difference time domain methods

PRECONDITIONING METHODS FOR THIN SCATTERING STRUCTURES BASED ON ASYMPTOTIC RESULTS

We present a method to precondition the discretized Lippmann–Schwinger integral equations to model scattering of time-harmonic acoustic waves through a thin inhomogeneous scattering medium. The preconditioner is based on asymptotic results as the thickness of the third component direction goes to zero and requires solving a two dimensional formulation of the problem at the preconditioning step

Regularized Reduced Order Lippman-Schwinger-Lanczos Method for Inverse Scattering Problems in the Frequency Domain

Inverse scattering has a broad applicability in quantum mechanics, remote sensing, geophysical, and medical imaging. This paper presents a robust direct reduced order model (ROM) method for solving inverse scattering problems based on an efficient approximation of the resolvent operator regularizing the Lippmann-Schwinger-Lanczos (LSL) algorithm. We show that the efficiency of the method relies upon the weak dependence of the orthogonalized basis on the unknown potential in the Schr\"odinger equation by demonstrating that the Lanczos orthogonalization is equivalent to performing Gram-Schmidt on the ROM time snapshots. We then develop the LSL algorithm in the frequency domain with two levels of regularization. We show that the same procedure can be extended beyond the Schr\"odinger formulation to the Helmholtz equation, e.g., to imaging the conductivity using diffusive electromagnetic fields in conductive media with localized positive conductivity perturbations. Numerical experiments for Helmholtz and Schr\"odinger problems show that the proposed bi-level regularization scheme significantly improves the performance of the LSL algorithm, allowing for good reconstructions with noisy data and large data sets

Modified forward and inverse Born series for the Calderon and diffuse-wave problems

International audienceWe propose a new direct reconstruction method based on series inversion for Electrical Impedance Tomography (EIT) and the inverse scattering problem for diffuse waves. The standard Born series for the forward problem has the limitation that the series requires that the contrast lies within a certain radius for convergence. Here, we instead propose a modified Born series which converges for the forward problem unconditionally. We then invert this modified Born series and compare reconstructions with the usual inverse Born series. We also show that the modified inverse Born series has a larger radius of convergence

Convergence and Stability of the Inverse Scattering Series for Diffuse Waves

We analyze the inverse scattering series for diffuse waves in random media. In previous work the inverse series was used to develop fast, direct image reconstruction algorithms in optical tomography. Here we characterize the convergence, stability and approximation error of the serie

Reduced order models for spectral domain inversion: Embedding into the continuous problem and generation of internal data

We generate reduced order Galerkin models for inversion of the Schr\"odinger equation given boundary data in the spectral domain for one and two dimensional problems. We show that in one dimension, after Lanczos orthogonalization, the Galerkin system is precisely the same as the three point staggered finite difference system on the corresponding spectrally matched grid. The orthogonalized basis functions depend only very weakly on the medium, and thus by embedding into the continuous problem, the reduced order model yields highly accurate internal solutions. In higher dimensions, the orthogonalized basis functions play the role of the grid steps, and highly accurate internal solutions are still obtained. We present inversion experiments based on the internal solutions in one and two dimensions. This is joint with: L. Borcea, V. Druskin, A. Mamonov, M. Zaslavsky.Non UBCUnreviewedAuthor affiliation: Drexel UniversityFacult