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Über dieses Buch

The scattering of acoustic and electromagnetic waves by periodic sur­ faces plays a role in many areas of applied physics and engineering. Opti­ cal diffraction gratings date from the nineteenth century and are still widely used by spectroscopists. More recently, diffraction gratings have been used as coupling devices for optical waveguides. Trains of surface waves on the oceans are natural diffraction gratings which influence the scattering of electromagnetic waves and underwater sound. Similarly, the surface of a crystal acts as a diffraction grating for the scattering of atomic beams. This list of natural and artificial diffraction gratings could easily be extended. The purpose of this monograph is to develop from first principles a theory of the scattering of acoustic and electromagnetic waves by periodic surfaces. In physical terms, the scattering of both time-harmonic and transient fields is analyzed. The corresponding mathematical model leads to the study of boundary value problems for the Helmholtz and d'Alembert wave equations in plane domains bounded by periodic curves. In the formal­ ism adopted here these problems are intimately related to the spectral analysis of the Laplace operator, acting in a Hilbert space of functions defined in the domain adjacent to the grating.




The first theoretical studies of scattering by diffraction gratings are due to Lord Rayleigh. His “Theory of Sound” Volume 2, 2nd Edition, published in 1896 [18]*, contains an analysis of the scattering of a monochromatic plane wave normally incident on a grating with a sinusoidal profile. In a subsequent paper [19] he extended the analysis to oblique incidence. Rayleigh assumed in his work that in the half-space above the grating the reflected wave is a superposition of the specularly reflected plane wave, a finite number of secondary plane waves propagating in the directions of the higher order grating spectra of optics, and an infinite sequence of evanescent waves whose amplitudes decrease exponentially with distance from the grating. The validity of Rayleigh’s assumption for general grating profiles was realized in the early 1930’s [10], following Bloch’s work [4] on the analogous problem of de Broglie waves in crystals. Waves of this type will be called Rayleigh-Bloch waves (R-B waves for brevity) in this work.
Calvin H. Wilcox

Part 1. Physical Theory

This monograph develops a theory of the scattering of two-dimensional acoustic and electromagnetic fields by diffraction gratings. This Part 1 presents the principal concepts and results in their simplest forms and without proofs. Moreover, to avoid distracting technicalities the precise conditions for the validity of the results are not always given. Part 1 also contains no references to the literature. All of these ommisions are remedied in Part 2 which contains the final mathematical formulation of the theory, together with complete proofs and indications of related literature.
Calvin H. Wilcox

Part 2. Mathematical Theory

The purpose of this Part 2 is to penetrate more deeply into the theory described in Part 1 and to develop the results in a precise form with complete hypotheses and full mathematical proofs. The work is based squarely on functional analysis. The reader should have good knowledge of the theory of unbounded selfadjoint operators in Hilbert spaces, as developed in Dunford and Schwartz [9] or any of the many other good texts. Other prerequisites include the theory of Sobolev spaces and the L2 theory of elliptic boundary value problems, as presented in the texts of S. Agmon [1] or Lions and Magenes [14], simple facts from the theory of Fréchet spaces and distribution theory, and the elements of the abstract theory of Riemann surfaces (Narasimhan [16]).
Calvin H. Wilcox


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