Beschreibung
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.
Autorenporträt
Inhaltsangabe1. A Tutorial Introduction.- 1.1 Preliminaries.- 1.1.1 General Notations.- 1.1.2 Time-Harmonic Maxwell Equations.- 1.1.3 Boundary Conditions.- 1.1.4 Electromagnetism and Distribution Theory.- 1.1.5 Notations Used in the Description of a Grating.- 1.2 The Perfectly Conducting Grating.- 1.2.1 Generalities.- 1.2.2 The Diffracted Field.- 1.2.3 The Rayleigh Expansion and the Grating Formula.- 1.2.4 An Important Lemma.- 1.2.5 The Reciprocity Theorem.- 1.2.6 The Conservation of Energy.- 1.2.7 The Littrow Mounting.- 1.2.8 The Determination of the Coefficients Bn by the Rayleigh Method.- 1.2.9 An Integral Expression of ud in P Polarization.- 1.2.10 The Integral Method in P Polarization.- 1.2.11 The Integral Method in S Polarization.- 1.2.12 Modal Expansion Methods.- 1.2.13 Conical Diffraction.- 1.3 The Dielectric or Metallic Grating.- 1.3.1 General i ti es.- 1.3.2 The Diffracted Field Outside the Groove Region.- 1.3.3 Maxwell Equations and Distributions.- 1.3.4 The Principle of the Differential Method (in P Polarization).- 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.- 2.4.1 Metallic Grating, with Infinite Conductivity.- 2.4.2 Dielectric Grating.- 2.5 Reciprocity Relations.- 2.6 Foundation of the Yasuura Improved Point-Matching Method.- 2.6.1 Definition of a Topological Basis.- 2.6.2 The System of Rayleigh Functions is a Topological Basis.- 2.6.3 The Convergence of the Rayleigh Series; A Counterexample.- References.- 3. Integral Methods.- 3.1 Development of the Integral Method.- 3.2 Presentation of the Problem and Intuitive Description of an Integral Approach.- 3.2.1 Presentation of the Problem.- 3.2.2 Intuitive Description of an Integral Approach.- 3.3 Notations, Mathematical Problem and Fundamental Formulae.- 3.3.1 Notations and Mathematical Formulation.- 3.3.2 Basic Formulae of the Integral Approach.- 3.4 The Uncoated Perfectly Conducting Grating.- 3.4.1 The TE Case of Polarization.- 3.4.2 The TM Case of Polarization.- 3.5 The Uncoated Dielectric or Metallic Grating.- 3.5.1 The Mathematical Boundary Problem.- 3.5.2 Vital Importance of the Choice of a Well-Adapted Unknown Function.- 3.5.3 Mathematical Definition of the Unknown Function and Determination of the Field and Its Normal Derivative Above P.- 3.5.4 Expression of the Field in M2 as a Function of ?.- 3.5.5 Integral Equation.- 3.5.6 Limit of the Equation when the Metal Becomes Perfectly Conducting.- 3.6 The Multiprofile Grating.- 3.7 The Grating in Conical Diffraction Mounting.- 3.8 Numerical Application.- 3.8.1 A Fundamental Preliminary Choice.- 3.8.2 Study of the Kernels.- 3.8.3 Integration of the Kernels.- 3.8.4 Particular Difficulty Encountered with Materials of High Conductivity.- 3.8.5 The Problem of Edges.- 3.8.6 Precision on the Numerical Results.- References.- 4. Differential Methods.- 4.1 Introductory Remarks.- 4.1.1 Historical Survey.- 4.1.2 Definition of Problem.- 4.2 The E, Case.- 4.2.1 The Reflection and Transmission Matrices.- 4.2.2 The Computation of Transmission and Reflection Matrices.- 4.2.3 Numerical Algorithms.- 4.2.4 Al ternative Matching Procedures for Some Grating Profiles.- 4.2.5 Field of Application.- 4.3 The H Case.- 4.3.1 The Propagation Equation.- 4.3.2 Numerical Treatment.- 4.3.3 Field of Application.- 4.4 The General Case (Conical Diffraction Case).- 4.4.1 The Reflection and Transmission Matrices.- 4.4.2 Th
