Electromagnetic Theory of Gratings: (22 Topics in Current Physics)

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.

Table of Contents:
1. 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 R.- 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 The Differential System.- 4.4.3 Matching with Rayleigh Expansions.- 4.4.4 Field of Application.- 4.5 Stratified Media.- 4.5.1 Stack of Gratings.- 4.5.2 Plane Interfaces Between Homogeneous Media.- 4.6 Infinitely Conducting Gratings: the Conformai Mapping Method.- 4.6.1 Method.- 4.6.2 Determination of the Conformai Mapping.- 4.6.3 Field of Application.- References.- 5. The Homogeneous Problem.- 5.1 Historical Summary.- 5.2 Plasmon Anomalies of a Metallic Grating.- 5.2.1 Reflection of a Plane Wave on a Plane Interface.- 5.2.2 Reflection of a Plane Wave on a Grating.- 5.3 Anomalies of Dielectric Coated Reflection Gratings Used in TE Polarization.- 5.3.1 Determination of the Leaky Modes of a Dielectric Slab Bounded by Metal on One of Its Sides.- 5.3.2 Reflection of a Plane Wave on a Dielectric Coated Reflection Grating Used in TE Polarization.- 5.4 Extension of the Theory.- 5.4.1 Anomalies of a Dielectric Coated Grating Used in TM Polarization.- 5.4.2 Plasmon Anomalies of a Bare Grating Supporting Several Spectral Orders.- 5.4.3 General Considerations on Anomalies of a Grating Supporting Several Spectral Orders.- 5.5 Theory of the Grating Coupler.- 5.5.1 Description of the Incident Beam.- 5.5.2 Response of the Structure to a Plane Wave.- 5.5.3 Response of the Structure to a Limited Beam.- 5.5.4 Determination of the Coupling Coefficient.- 5.5.5 Application to a Limited Incident Beam.- References.- 6. Experimental Verifications and Applications of the Theory.- 6.1 Experimental Checking of Theoretical Results.- 6.1.1 Generalities.- 6.1.2 Microwave Region.- 6.1.3 On the Determination of Groove Geometry and of the Refractive Index.- 6.1.4 Infrared.- 6.1.5 Visible Region.- 6.1.6 Near and Vacuum UV.- 6.1.7 XUV Domain.- 6.1.8 X-Ray Domain.- 6.2 Systematic Study of the Efficiency of Perfectly Conducting Gratings.- 6.2.1 Systematic Study of Echelette Gratings in -1 Order Littrow Mount.- 6.2.2 An Equivalence Rule Between Ruled, Holographic, and Lamel1ar Gratings.- 6.2.3 Systematic Study of the Efficiency of Holographic Gratings in -1 Order Littrow Mount.- 6.2.4 Systematic Study of the Efficiency of Symmetrical Lamellar Gratings in -1 Order Littrow Mount.- 6.2.5 Influence of the Apex Angle.- 6.2.6 Influence of a Departure from Littrow.- 6.2.7 Higher Order Use of Gratings.- 6.3 Finite Conductivity Gratings.- 6.3.1 General Rules.- 6.3.2 Typical Efficiency Curves in the Visible Region.- 6.3.3 Influence of Dielectric Overcoatings in Vacuum UV.- 6.3.4 The Use of Gratings in XUV and X-Ray Regions (?<1000 A).- 6.3.5 Conical Diffraction Mountings.- 6.4 Some Particular Applications.- 6.4.1 Simultaneous Blazing in Both Polarizations.- 6.4.2 Spectrometers with Constant Efficiency.- 6.4.3 Grating Bandpass Filter.- 6.4.4 Reflection Grating Polarizer for the Infrared.- 6.4.5 Transmission Gratings as Masks in Photolithography.- 6.4.6 Gratings Used as Beam Sampling Mirrors for High Power Lasers.- 6.4.7 Gratings as Wavelength Selectors in Tunable Lasers.- 6.4.8 Transmission Dielectric Gratings used as Color Filters.- 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.- 7.3.1 Grids with Rectangular Apertures.- 7.3.2 Numerical Tests and Applications.- 7.3.3 Inductive Grids with Circular Apertures.- 7.4 Capacitive and Other Grid Geometries.- 7.4.1 High-Pass Filters.- 7.4.2 Low-Pass Filters.- 7.4.3 Bandpass Filters.- 7.4.4 Bandstop Filters.- 7.5 Spatially Separated Grids or Gratings.- 7.5.1 The Crossed Lamellar Transmission Grating.- 7.5.2 The Double Grating.- 7.5.3 Symmetry Properties of Double Gratings.- 7.5.4 Multielement Grating Interference Filters.- 7.6 Finitely Conducting Bigratings.- 7.6.1 A Short Description of the Method.- 7.6.2 The Coordinate Transformation.- 7.6.3 Integral Equation Form.- 7.6.4 Iterative Solution of the Integral Equations.- 7.6.5 Total Absorption of Unpolarized Monochromatic Light.- 7.6.6 Reduction of Metallic Reflectivity: Plasmons and Moth-Eyes.- 7.6.7 Equivalence Formulae Linking Crossed and Classical Gratings.- 7.6.8 Coated Bigratings.- References.- Additional References with Titles.