Scattering Theory for Diffraction Gratings 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 dAlembert 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. 1. Physical Theory 1. The Physical Problem 2. The Mathematical Formulation 3. Solution of the Initial Boundary Value Problem 4. The Reference Problem and Its Eigenfunctions 5. Rayleigh Bloch Diffracted Plane Waves for Gratings 6. Rayleigh Bloch Surface Waves for Gratings 7. Rayleigh Bloch Wave Expansions 8. Wave and Scattering Operators for Gratings 9. Asymptotic Wave Functions for Gratings 10. The Scattering of Signals from Remote Sources 2. Mathematical Theory 1. Grating Domains and Grating Propagators 2. Rayleigh Bloch Waves 3. The Reduced Grating Propagator Ap 4. Analytic Continuation of the Resolvent of Ap 5. Proofs of the Results of §4 6. The Eigenfunction Expansion for Ap 7. Proofs of the Results of §6 8. The Rayleigh Bloch Wave Expansions for A 9. Proofs of the Results of §8 10. The Initial Boundary Value Problems for the Scattered Fields 11. Construction of the Wave Operators for AP and Ao p 12. Construction of the Wave Operators for A and Ao 13. Asymptotic Wave Functions and Energy Distributions 14. Construction and Structure of the S Matrix 15. The Scattering of Signals by Diffraction Gratings References.