On the diffraction by biperiodic anisotropic structures

This paper studies the scattering of electromagnetic waves by a nonmagnetic biperiodic structure. The structure consists of anisotropic optical materials and separates two regions with constant dielectric coefficients. The time harmonic Maxwell equations are transformed to an equivalent strongly elliptic variational problem for the magnetic field in a bounded biperiodic cell with nonlocal boundary conditions. This guarantees the existence of quasiperiodic magnetic fields in 퐻1 and electric fields in 퐻(curl) solving Maxwell's equations. The uniqueness is proved for all frequencies excluding possibly a discrete set. The analytic dependence of these solutions on frequency and incident angles is studied

Existence, Uniqueness and Regularity for Solutions of the Conical Diffraction Problem

This paper is devoted to the analysis of two Helmholtz equations in ℝ2 coupled via quasiperiodic transmission conditions on a set of piecewise smooth interfaces. The solution of this system is quasi-periodic in one direction and satisfies outgoing wave conditions with respect to the other direction. It is shown that Maxwell's equations for the diffraction of a time-harmonic oblique incident plane wave by periodic interfaces can be reduced to problems of this kind. The analysis is based on a strongly elliptic variational formulation of the differential problem in a bounded periodic cell involving nonlocal boundary operators. We obtain existence and uniqueness results for solutions corresponding to electromagnetic fields with locally finite energy. Special attention is paid to the regularity and leading asymptotics of solutions near the edges of the interface

An inverse electromagnetic scattering problem for a bi-periodic inhomogeneous layer on a perfectly conducting plate

This paper is concerned with uniqueness for reconstructing a periodic inhomogeneous medium covered on a perfectly conducting plate. We deal with the problem in the frame of time-harmonic Maxwell systems without TE or TM polarization. An orthogonal relation for two refractive indices is obtained, and then inspired by Kirsch's idea, the refractive index can be identified by utilizing the eigenvalues and eigenfunctions of a quasi-periodic Sturm-Liouville eigenvalue problem

Mathmatical modeling for diffractive optics

We consider a 'diffractive optic' to be a biperiodic surface separating two half-spaces, each having constant constitutive parameters; within a unit cell of the periodic surface and across the transition zone between the two half-spaces, the constitutive parameters can be a continuous, complex-valued function. Mathematical models for diffractive optics have been developed, and implemented as numerical codes, both for the 'direct' problem and for the 'inverse' problem. In problems of the 'direct' class, the diffractive optic is specified, and the full set of Maxwell's equations is cast in a variational form and solved numerically by a finite element approach. This approach is well-posed in the sense that existence and uniqueness of the solution can be proved and specific convergence conditions can be derived. An example of a metallic grating at a Wood anomaly is presented as a case where other approaches are known to have convergence problems. In problems of the 'inverse' class, some information about the diffracted field (e.g., the far-field intensity) is given, and the problem is to find the periodic structure in some optimal sense. Two approaches are described: phase reconstruction in the far-field approximation; and relaxed optimal design based on the Helmholtz equation. Practical examples are discussed for each approach to the inverse problem, including array generators in the far-field case and antireflective structures for the relaxed optimal design

Analysis and Numerics for the Optimal Design of Binary Diffractive Gratings

The aim of the paper is to provide the mathematical foundation of effective numerical algorithms for the optimal design of periodic binary gratings. Special attention is paid to fast and reliable methods for the computation of diffraction efficiencies and of the gradients of certain functionals with respect to the parameters of the non-smooth grating profile. The methods are based on a generalized finite element discretization of strongly elliptic variational formulations of quasi periodic transmission problems for the Helmholtz equation in a bounded domain coupled with boundary integral representations in the exterior. We prove uniqueness and existence results for quite general situations and analyse the convergence of the numerical solutions. Furthermore, explicit formulas for the partial derivatives of the reflection and transmission coefficients with respect to the parameters of a binary grating profile are derived. Finally, we briefly discuss the implementation of a gradient type algorithm for solving optimal design problems and present some numerical results

Progress In Electromagnetics Research Symposium (PIERS)

The third Progress In Electromagnetics Research Symposium (PIERS) was held 12-16 Jul. 1993, at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California. More than 800 presentations were made, and those abstracts are included in this publication

3D optical information processing

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2006.Includes bibliographical references (p. 141-151).Light exhibits dramatically different properties when it propagates in or interacts with 3D structured media. Comparing to 2D optical elements where the light interacts with a sequence of surfaces separated by free space, 3D optical elements provides more degrees of freedom to perform imaging and optical information processing functions. With sufficient dielectric contrast, a periodically structured medium may be capable of forbidding propagation of light in certain frequency range, called band gap; the medium is then called a photonic crystal. Various "defects", i.e. deviations from perfect periodicity, in photonic crystals are designed and widely used as waveguides and microcavities in integrated optical circuits without appreciable loss. However, many of the proposed waveguide structures suffer from large group velocity dispersion (GVD) and exhibit relatively small guiding bandwidth because of the distributed Bragg reflection (DBR) along the guiding direction. As optical communications and optical computing progress, more challenging demands have also been proposed, such as tunable guiding bandwidth, dramatically slowing down group velocity and active control of group velocity. We propose and analyze shear discontinuities as a new type of defect in photonic crystals.(cont.) We demonstrate that this defect can support guided modes with very low GVD and maximum guiding bandwidth, provided that the shear shift equals half the lattice constant. A mode gap emerges when the shear shift is different than half the lattice constant, and the mode gap can be tuned by changing the amount of the shear shift. This property can be used to design photonic crystal waveguides with tunable guiding bandwidth and group velocity, and induce bound states. The necessary condition for the existence of guiding modes is discussed. By changing the shape of circular rods at the shear interface, we further optimize our sheared photonic crystals to achieve minimum GVD. Based on a coupled resonator optical waveguide (CROW) with a mechanically adjustable shear discontinuity, we also design a tunable slow light device to realize active control of the group velocity of light. Tuning ranges from arbitrarily small group velocity to approximately the value of group velocity in the bulk material with the same average refractive index. The properties of eigenstates of tunable CROWs: symmetry and field distribution, and the dependence of the group velocity on the shear shift are also investigated.(cont.) Using the finite-difference time-domain (FDTD) simulation, we demonstrate the process of tuning group velocity of light in CROWs by only changing the shear shift. A weakly modulated 3D medium diffracts light in the Bragg regime (in contrast to Raman-Nath regime for 2D optical elements), called volume hologram. Because of Bragg selectivity, volume holograms have been widely used in data storage and 3D imaging. In data storage, the limited diffraction efficiency will affect the signal-noise-ratio (SNR), thus the memory capacity of volume holograms. Resonant holography can enhance the diffraction efficiency from a volume hologram by enclosing it in a Fabry-Perot cavity with the light multiple passes through the volume hologram. We analyze crosstalk in resonant holographic memories and derive the conditions where resonance improves storage quality. We also carry out the analysis for both plane wave and apodized Gaussian reference beams. By utilizing Hermite Gaussian references (higher order modes of Gaussian beams), a new holographic multiplexing method is proposed - mode multiplexing.(cont.) We derive and analyze the diffraction pattern from mode multiplexing with Hermite Gaussian references, and predict its capability to eliminate the inter-page crosstalk due to the independence of Hermite Gaussian's orthogonality on the direction of signal beam as well as decrease intra-page crosstalk to lower level through apodization. When using volume holograms for imaging, the third dimension of volume holograms provided more degrees of freedom to shape the optical response corresponding to more demanding requirements than traditional optical systems. Based on Bragg diffraction, we propose a new technique - 3D measurement of deformation using volume holography. We derive the response of a volume grating to arbitrary deformations, using a perturbative approach. This result will be interesting for two applications: (a) when a deformation is undesirable and one seeks to minimize the diffracted field's sensitivity to it and (b) when the deformation itself is the quantity of interest, and the diffracted field is used as a probe into the deformed volume where the hologram was originally recorded.(cont.) We show that our result is consistent with previous derivations motivated by the phenomenon of shrinkage in photopolymer holographic materials. We also present the analysis of the grating's response to deformation due to a point indenter and present experimental results consistent with theory.by Kehan Tian.Ph.D

Some new developments on inverse scattering problems.

Zhang, Hai.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (leaves 106-109).Abstract also in Chinese.Chapter 1 --- Introduction --- p.5Chapter 2 --- Preliminaries --- p.13Chapter 2.1 --- Maxwell equations --- p.13Chapter 2.2 --- Reflection principle --- p.15Chapter 3 --- Scattering by General Polyhedral Obstacle --- p.19Chapter 3.1 --- Direct problem --- p.19Chapter 3.2 --- Inverse problem and statement of main results --- p.21Chapter 3.3 --- Proof of the main results --- p.22Chapter 3.3.1 --- Preliminaries --- p.23Chapter 3.3.2 --- Properties of perfect planes --- p.24Chapter 3.3.3 --- Proofs --- p.33Chapter 4 --- Scattering by Bi-periodic Polyhedral Grating (I) --- p.35Chapter 4.1 --- Direct problem --- p.36Chapter 4.2 --- Inverse problem and statement of main results --- p.38Chapter 4.3 --- Preliminaries --- p.39Chapter 4.4 --- Classification of unidentifiable periodic structures --- p.41Chapter 4.4.1 --- Observations and auxiliary tools --- p.41Chapter 4.4.2 --- First class of unidentifiable gratings --- p.45Chapter 4.4.3 --- Preparation for finding other classes of unidentifiable gratings --- p.47Chapter 4.4.4 --- A simple transformation --- p.52Chapter 4.4.5 --- Second class of unidentifiable gratings --- p.53Chapter 4.4.6 --- Third class of unidentifiable gratings --- p.58Chapter 4.4.7 --- Excluding the case with L --- p.61Chapter 4.4.8 --- Summary on all unidentifiable gratings --- p.65Chapter 4.5 --- Proof of Main results --- p.65Chapter 5 --- Scattering by Bi-periodic Polyhedral Grating (II) --- p.69Chapter 5.1 --- Preliminaries --- p.70Chapter 5.2 --- Classification of unidentifiable periodic structures --- p.72Chapter 5.2.1 --- First class of unidentifiable gratings --- p.72Chapter 5.2.2 --- Preparation for finding other classes of unidentifiable gratings --- p.73Chapter 5.2.3 --- Studying of the case L --- p.76Chapter 5.2.4 --- Study of the case with L --- p.89Chapter 5.2.5 --- Study of the case with L --- p.95Chapter 5.2.6 --- Summary on all unidentifiable gratings --- p.104Chapter 5.3 --- Unique determination of bi-periodic polyhedral grating --- p.104Bibliography --- p.10

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described