Gregory A. Kriegsmann

Scattering matrix analysis of a photonic Fabry-Perot resonator

Abstract The scattering and transmission of waves through a two-dimensional photonic Fabry–Perot resonator are analyzed and studied using scattering matrix theory. Assuming normal incidence, single mode propagation, and sufficient inter-element spacing in the direction of propagation, the mathematical structure of this complicated scattering problem is shown to hinge on the roots of a quadratic equation. Examples are presented which clearly show strong resonances at frequencies closely related to an infinite cavity structure. Trends are illustrated and discussed. Finally, a multimode theory is presented, analyzed, and contrasted with the single mode case.

The flanged waveguide antenna: Discrete reciprocity and conservation

Abstract The flanged waveguide is a prototypical antenna used in the description of both receiving and transmitting applications. This structure embodies much of the physics of more complicated structures. In particular, the fields radiated in the transmitting mode, and the modes created and fields scattered in the receiving mode satisfy basic reciprocity and conservation laws. In this paper it is demonstrated that the discrete formulation of these problems, obtained by a truncated normal mode analysis, satisfies the basic reciprocity and conservation laws. These discrete relationships hold regardless of the size of the truncated system N, as long as N is greater than the number of propagating modes in the waveguide. Thus, the adherence of this approximate numerical method to the reciprocity and conservation laws does not necessarily imply its accuracy. Several numerical examples are given to illustrate this point.

Scattering by a rectangularly corrugated surface: an approximate theory

We address the problem of the scattering of a plane, TM-polarized electromagnetic wave by a two-dimensional rectangularly corrugated surface. The height of the corrugation is measured by a characteristic length D and its period by /spl Lambda/. We take the ordering of these scales to be D/spl sim//spl Lambda//spl sim//spl lambda/ where /spl lambda/ is the wavelength of the incident wave. We apply a radiation condition (strictly valid only in the far-field) in the aperture above a rectangular notch. This approximation allows the determination of an explicit representation of the reflection coefficients, the scattered field, and the field within a corrugation. Numerical results are presented which are in excellent agreement with a published finite difference approximation for diffraction gratings.

SCATTERING BY LARGE RESONANT CAVITY STRUCTURES

Abstract The scattering of a plane wave from a two-dimensional, sound hard, cavity is considered where the length of the cavity is large compared to the wavelength. The cavity may contain an obstacle or have walls which slowly change on a scale that is long compared to the wavelength, but short compared to the overall length of the cavity. The structure may be terminated by either or short or another flanged aperture. Using a generalized S-matrix, a new formula is derived for the scattering cross-section, σ, of the structure. This formula contains the mathematical description of all the multiple interactions between the aperture and the contents of the cavity. Approximating the inverse of a certain matrix by the first term of a geometric series, yields an approximation σA which has been derived by researchers studying the scattering cross-sections of jet engine inlets. This approximation only takes into account one scattering interaction and provides excellent results, because for these applications, the width of the waveguide is much larger than a wavelength. For other important applications where the width and the wavelength are comparable, it produces unacceptable errors. Several examples are presented which compare the errors produced by the approximate theory to the more exact one derived in this paper. It is shown that the two agree when the waveguide width becomes sufficiently large and diverge in the other extreme.

The Galerkin approximation of the iris problem: Conservation of power☆

Abstract An n -term Galerkin approximation of the iris scattering problem produces approximate reflection and transmission coefficients. A simple proof is presented to show that these coefficients satisfy the same conservation law as the original continuous problem. This result is true, regardless of the chosen basis functions and n .

REFLECTION AND TRANSMISSION FROM A THIN INHOMOGENEOUS CYLINDER IN A RECTANGULAR TE10 WAVEGUIDE

We study the scattering problem for a thin cylindrical target that is placed with arbitrary orientation in a rectangular TE10 waveguide and subjected to an imposed electromagnetic field. The scattered far-field is expressed in terms of the scattered field inside the target and the far-field expansion of the dyadic Greens function for the waveguide. In order to capture features of interest in microwave heating applications, we allow the target materials electrical properties to be arbitrary functions of position along the thin cylindrical targets axis. Reflection and transmission coefficients for such a target, and an expression for the rate of deposition of electromagnetic energy within it are derived.

Microwave-induced combustion: a one-dimensional model

A model for the heating and ignition of a combustible solid by microwave energy is formulated and analysed in the limit of small inverse activation energy and small Biot number B. The high activation energy limit implies that the heating process is effectively inert until the temperature within the material reaches a critical ignition value, while the small Biot number limit implies that during this stage spatial variations in temperature throughout the material are always small. Analysis of the inert stage includes determination of the dynamics of inert hot-spots. As the ignition temperature is approached chemical energy is released rapidly in the form of heat, and the evolution then enters an ignition stage which develops on a fast time-scale. A reduced system is derived governing small-amplitude departures of the temperature from the inert value during the ignition stage under the significant scaling relation between the expansion parameters, which is shown to be ~ B. This reduced system recovers both ...

Electromagnetic propagation in periodic porous structures

Abstract A variational technique is employed to compute approximate propagation constants for electromagnetic waves in a dielectric structure which is periodic in the X − Y plane and translationally invariant in the Z-direction. The fundamental cell, in the periodic structure, is composed of a pore and the surrounding host media. The pore is a circle of radius R 0 filled with a dielectric ϵ 1 and the host dielectric characterized by ϵ 2 . The size of the cell is characterized by the length A which is ∼ R 0 . Two limiting cases are considered. In the first, the pore size is assumed to be much smaller than the wavelength; this limit is motivated by microwave heating of porous material. The approximate propagation constants are explicitly computed for this case and are shown to depend upon the two dielectric constants, the relative areas of the two regions in the cell, and on a modal number. They are not given by a simple mixture formula. In the second limit, the pore size is taken to be of the same order as the wavelength; this limit is motivated by the propagation of light in a holey fiber. In this case our argument directly yields the dispersion relationship recently derived by Ferrando et al. [Opt. Lett. 24 (1999) 276], using intuitive and physical reasoning. Thus, our method puts theirs into a mathematical framework from which other approximations might be deduced.

Control region approximation of scattering by two-dimensional periodic structures

Scattering from a two-dimensional diffraction grating is analyzed numerically. The scattered field is decomposed into TM and TE modes giving rise in each case to a generalized Helmholtz equation supplemented by a pseudo-periodic boundary condition which must be solved in the unit cell of the grating. The unit cell is truncated at a finite distance from the grating where a radiation condition is enforced. The resulting boundary value problem for plane wave excitation is discretized via the Control Region Approximation with the discrete equations so generated then solved using a sparse direct method (Yale Sparse Matrix Package). The accuracy and extreme flexibility of this approach are demonstrated by a collection of numerical examples.

Acoustic scattering from baffled membranes that are backed by elastic cavities

Abstract An elastic membrane backed by a fluid-filled cavity in an elastic body is set into an infinite plane baffle. A time harmonic wave propagating in the acoustic fluid in the upper half-space is incident on the plane. It is assumed that the densities of this fluid and the fluid inside the cavity are small compared with the densities of the membrane and of the elastic walls of the cavity, thus defining a small parameter e. Asymptotic expansions of the solution of this scattering problem as e→0, that are uniform in the wave number k of the incident wave, are obtained using the method of matched asymptotic expansions. When the frequency of the incident wave is bounded away from the resonant frequencies of the membrane, the cavity fluid, and the elastic body, the resultant wave is a small perturbation (the “outer expansion”) of the specularly reflected wave from a completely rigid plane. However, when the incident wave frequency is near a resonant frequency (the “inner expansion”) then the scattered wave results from the interaction of the acoustic fluid with the membrane, the membrane with the cavity fluid, and finally the cavity fluid with the elastic body, and the resulting scattered field may be “large”. The cavity backed membrane (CBM) was previously analyzed for a rigid cavity wall. In this paper, we study the effects of the elastic cavity walls on modifying the response of the CBM. For incident frequencies near the membrane resonant frequencies, the elasticity of the cavity gives only a higher order (in e) correction to the scattered field. However, near a cavity fluid resonant frequency, and, of course, near an elastic body resonant frequency the elasticity contributes to the scattered field. The method is applied to the two dimensional problem of an infinite strip membrane backed by an infinitely long rectangular cavity. The cavity is formed by two infinitely long rectangular elastic solids. We speculate on the possible significance of the results with respect to viscoelastic membranes and viscoelastic instead of elastic cavity walls for surface sound absorbers.