When I was a student, in the early fifties, the properties of gratings were generally explained according to the scalar theory of optics. The grating formula (which pre­ dicts the diffraction angles for a given angle of incidence) was established, exper­ imentally verified, and intensively used as a source for textbook problems. Indeed those grating properties, we can call optical properties, were taught'in a satisfac­ tory manner and the students were able to clearly understand the diffraction and dispersion of light by gratings. On the other hand, little was said about the "energy properties", i. e. , about the prediction of efficiencies. Of course, the existence of the blaze effect was pointed out, but very frequently nothing else was taught about the efficiency curves. At most a good student had to know that, for an eche­ lette grating, the efficiency in a given order can approach unity insofar as the diffracted wave vector can be deduced from the incident one by a specular reflexion on the large facet. Actually this rule of thumb was generally sufficient to make good use of the optical gratings available about thirty years ago. Thanks to the spectacular improvements in grating manufacture after the end of the second world war, it became possible to obtain very good gratings with more and more lines per mm. Nowadays, in gratings used in the visible region, a spacing small­ er than half a micron is common.

1. A Tutorial Introduction..- 1.1 Preliminaries.- 1.2 The Perfectly Conducting Grating.- 1.3 The Dielectric or Metallic Grating.- 1.4 Miscellaneous.- References.- Appendix A: The Distributions or Generalized Functions.- A.I Preliminaries.- A.2 The Function Space R.- A.3 The Space R1.- A.3.1 Definitions.- A.3.2 Examples of Distributions.- A.4 Derivative of a Distribution.- A.5 Expansion with Respect to the Basis ej(x) =exp [i (nK+k sine) x] = exp (i?n x).- A.5.1 Theorem.- A. 5.2 Proof.- A.5.3 Application to δR.- A.6 Convolution.- A.6.1 Memoranda on the Product of Convolution in D'1.- A.6.2 Convolution in R1.- 2. Some Mathematical Aspects of the Grating Theory.- 2.1 Some Classical Properties of the Helmholtz Equation.- 2.2 The Radiation Condition for the Grating Problem.- 2.3 A Lemma.- 2.4 Uniqueness Theorems.- 2.5 Reciprocity Relations.- 2.6 Foundation of the Yasuura Improved Point-Matching Method.- References.- 3. Integral Methods.- 3.1 Development of the Integral Method.- 3.2 Presentation of the Problem and Intuitive Description of an Integral Approach.- 3.3 Notations, Mathematical Problem and Fundamental Formulae.- 3.4 The Uncoated Perfectly Conducting Grating.- 3.5 The Uncoated Dielectric or Metallic Grating.- 3.6 The Multiprofile Grating.- 3.7 The Grating in Conical Diffraction Mounting.- 3.8 Numerical Application.- References.- 4. Differential Methods.- 4.1 Introductory Remarks.- 4.2 The E,, Case.- 4.3 The H Case.- 4.4 The General Case (Conical Diffraction Case).- 4.5 Stratified Media.- 4.6 Infinitely Conducting Gratings: the Conformai Mapping Method.- References.- 5. The Homogeneous Problem.- 5.1 Historical Summary.- 5.2 Plasmon Anomalies of a Metallic Grating.- 5.3 Anomalies of Dielectric Coated Reflection Gratings Used in TE Polarization.- 5.4Extension of the Theory.- 5.5 Theory of the Grating Coupler.- References.- 6. Experimental Verifications and Applications of the Theory.- 6.1 Experimental Checking of Theoretical Results.- 6.2 Systematic Study of the Efficiency of Perfectly Conducting Gratings.- 6.3 Finite Conductivity Gratings.- 6.4 Some Particular Applications.- Concluding Remarks.- References.- 7. Theory of Crossed Gratings.- 7.1 Overview.- 7.2 The Bigrating Equation and Rayleigh Expansions.- 7.3 Inducti ve Gri ds.- 7.4 Capacitive and Other Grid Geometries.- 7.5 Spatially Separated Grids or Gratings.- 7.6 Finitely Conducting Bigratings.- References.- Additional References with Titles.