Extension du Concept D'espaces de Fourier à la Spectroscopie

This paper is the second in a series [1] of papers on the diffraction and imaging properties of generalized curvilinear diffraction gratings. In this article, a set of generalized ray equations is derived and applied to the calculation of the caustic properties of generalized gratings. No restriction is placed on the form of the ruling pattern or on the shape of the grating surface. For generalized uncrossed and off-axis plane gratings, the ruling pattern may be regarded as a contour map of some abstract surface. In this case, it is shown that the far-field caustic properties of the grating are determined by the regions of vanishing gaussian curvature on the abstract surface. As an application of the theory, a new method for laser material processing is discussed.     

Polarizers and Applications

Three examples of a class of generalized curvilinear diffraction gratings are analysed according to a recently published general theory of diffraction by generalized gratings. The two types of diffraction patterns described by the theory are the Fresnel/Fraunhofer patterns and a new type labelled the ‘image diffraction pattern’ (IDP). Examples of image diffraction patterns are discussed and compared with two methods of predicting the patterns—one method based on optical image processing techniques and the other method based on the solution of the IDP equations by a computer graphics system. The theoretical Fresnel/Fraunhofer and image diffraction patterns are found to be in very good agreement with experiment.     

Modal method for classical diffraction by slanted lamellar gratings

We consider lamellar gratings made of dielectric or lossy materials used in classical diffraction mounts. We show how the modal diffraction formulation may be generalized to deal with slanted lamellar gratings and illustrate the accuracy and versatility of the new method through study of highly slanted gratings in a homogenization limit. We also comment on the completeness of the eigenmode basis and present tests enabling this completeness to be verified numerically.     

Nonparaxial analysis of continuous self-imaging gratings in oblique illumination

Tolerance in angles of continuously self-imaging gratings (CSIGs) is explored. The degradation in angle of the shape of the point-spread function is theoretically investigated and illustrated by simulations and experiments. The formalism presented is inspired by the one used for classical lenses and can be easily generalized to diffraction gratings. It turns out that well-designed CSIGs could be used for scanning optical systems requiring a large field of view.     

The Reciprocity Theorem for Corrugated Surfaces Used in Conical Diffraction Mountings

The reciprocity theorem, well known for gratings used in classical diffraction, is generalized to the case where the incident wave propagates outside the cross-section plane of the grating (conical diffraction). The authors establish relations on the components of the electric and magnetic fields, for dielectric or conducting periodic structures utilized either in reflexion or in transmission. Symmetry properties of the scattering matrix are derived, from which relations on the efficiencies in natural light are established. All the relations are illustrated by numerical computations based on the electromagnetic theory of gratings.     

A Generalized Diffraction Grating Equation

A vector method of ray-tracing through unified gratings is presented. The method does not require any iterative calculation in the ray-transfer operations or any modification to the diffraction operations. The figuring of optical blanks is assumed to be quadratic or toroidal and rulings are considered to be represented by the intersections of a blank surface with a family of groove-generating surfaces. The generalized diffraction grating equation is expressed in terms of the gradient vector of the groove function.     

Simulation of mechanically ruled concave diffraction gratings by use of an original geometric theory

A geometric theory is developed for a spherical concave diffraction grating that has variable line spacing and curved grooves and can be produced with existing technology. The aberration coefficients of this grating are determined. A comparison of these coefficients with the corresponding coefficients of holographic gratings and with CODE V polynomial coefficients gives a clearer understanding of the similarity and difference between mechanically ruled and holographically recorded concave gratings and allows the optimization of these gratings with standard computer programs.     

Sub-10 nm patterning using EUV interference lithography

Extreme ultraviolet (EUV) lithography is currently considered as the leading technology for high-volume manufacturing below sub-20 nm feature sizes. In parallel, EUV interference lithography based on interference transmission gratings has emerged as a powerful tool for industrial and academic research. In this paper, we demonstrate nanopatterning with sub-10 nm resolution using this technique. Highly efficient and optimized molybdenum gratings result in resolved line/space patterns down to 8 nm half-pitch and show modulation down to 6 nm half-pitch. These results show the performance of optical nanopatterning in the sub-10 nm range and currently mark the record for photon-based lithography. Moreover, an efficient phase mask completely suppressing the zeroth-order diffraction and providing 50 nm line/space patterns over large areas is evaluated. Such efficient phase masks pave the way towards table-top EUV interference lithography systems.     

Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction

Boundary integral equation methods for diffraction gratings are particularly suitable for gratings with complicated material interfaces but are difficult to implement due to the quasi-periodic Green's function and the singular integrals at the corners. In this paper, the boundary integral equation Neumann-to-Dirichlet map method for in-plane diffraction problems of gratings [Y. Wu and Y. Y. Lu, J. Opt. Soc. Am. A26, 2444 (2009)] is extended to conical diffraction problems. The method uses boundary integral equations to calculate the so-called Neumann-to-Dirichlet maps for homogeneous subdomains of the grating, so that the quasi-periodic Green's functions can be avoided. Since wave field components are coupled on material interfaces with the involvement of tangential derivatives, a least squares polynomial approximation technique is developed to evaluate tangential derivatives along these interfaces for conical diffraction problems. Numerical examples indicate that the method performs equally well for dielectric or metallic gratings.     

Relaxation of the Talbot condition in generalized grating imaging

We can form a grating image with two gratings having different pitches with an extended light source. It is called generalized grating imaging or the Talbot-Lau effect. When we want to obtain high contrast image with pure absorption gratings or pure phase gratings, the separation between the two gratings is restricted. This corresponds to the Talbot condition. In this paper, we propose to use a combination of absorption grating and phase grating to relax the separation restriction. The theory of generalized grating imaging is applied to the system with this kind of grating. Simulations are performed for calculating contrast variation and show that the proposed system practically relaxes the Talbot condition. An experiment verifies the result of the simulation.     

Out-of-plane diffraction of a two-dimensional photonic crystal with finite dielectric modulation

We present a generalized picture of out-of-plane diffraction in a two-dimensional photonic crystal using the concept of photonic bands and employing a three-dimensional, equal-frequency-surface analysis. We show that the discrete spots of diffraction pattern in a weakly modulated photonic crystal, including those of conventional diffraction gratings, become continuous when the dielectric modulation becomes finite. Furthermore, in a finite-modulated photonic crystal, the diffraction can take place even in the region prohibited by Bragg's law: there are available states for the incident light, which are evanescent in the case of a diffraction grating (weakly modulated photonic crystal).     

Efficient generation of large diffraction gratings with a grating interferometer

The advantages of a grating interferometer for the generation of large diffraction gratings are demonstrated. In a one- and a two-stage process, high-quality gratings of 120 and 200 mm, respectively, were made with optics no larger than 50 mm together with an argon-ion laser with no line narrowing or beam stabilization and a rotating diffuser for improved beam uniformity.     

Generalized tertiary rainbow of slightly oblate drops: observations with laser illumination

The tertiary rainbow of acoustically levitated water drops was observed in the laboratory. Nontrivial caustics were observed for relatively small values of eccentricity. The angular locations of caustics were modeled with matrix methods of generalized ray tracing. Photographs of the scattering were in general agreement with models. Possible effects on the appearance of natural tertiary bow features are discussed.     

Stabilization of volume gratings recorded in polyvinyl alcohol-acrylamide photopolymers with diffraction efficiencies higher than 90%

Abstract

Obtaining holographic volume gratings with high diffraction efficiencies that can be used under white light has been a serious problem for the polyvinyl alcohol-acrylamide-based photopolymers developed by other researchers. In this paper we propose to eliminate the residual monomer in order to stabilize the holographic gratings. The residual dye and residual monomer are the main problems in achieving high diffraction efficiencies stable under white light. In order to polymerize the residual monomer we illuminate the gratings with coherent green light and incoherent white light and we heat the grating at 80°C for different times. We also study the conservation of gratings dried in critical conditions of humidity and temperature. After stabilization the diffraction efficiencies achieved were clearly higher than 90%.     

Ray tracing and scalar diffraction calculations of wavefronts,caustics and complex amplitudes in optical systems

The procedures for precise calculation of wavefronts, caustics and complex amplitudes in optical systems are developed. Numerical methods are compared with analytical formulae for caustics. The conditions for the validity of the integration on wavefronts for obtaining the complex amplitudes (and hence intensities defining the PSFs) within scalar diffraction theory are discussed in detail. To illustrate the precision of the results obtained with the developed techniques, an experiment showing quite unexpected results is studied and explained in detail.     

Diffraction properties of volume holographic gratings for an ultrashort pulsed beam with different polarization states readout

The diffraction properties of volume holographic gratings are studied when the gratings are illuminated by an ultrashort pulsed beam with different polarization states. The developed coupled wave theory of Kogelnik is used. Considering the dispersion effect of the grating media, solutions for the diffracted and transmitted intensities, diffraction efficiencies and the bandwidths of the gratings are given in transmission volume holographic gratings and reflection volume holographic gratings. The bandwidths of the gratings are reduced by the dispersion effect of the grating media. They also have different influences on the diffraction of an ultrashort pulsed beam with different polarization states. For different values of the ratio of the spectral bandwidth of the input pulse to that of the grating, the changes of the spectral and temporal distributions of the diffracted intensities, as well as the diffraction efficiencies of the gratings are shown.     

Pseudospectral modal method for conical diffraction of gratings

For lamellar gratings and other layered periodic structures, the modal methods (including both analytic and numerical ones) are often the most efficient, since they avoid the discretization of one spatial variable. The pseudospectral modal method (PSMM) previously developed for in-plane diffraction problems of one-dimensional gratings achieves high accuracy for a small number of discretization points, and it outperforms most other modal methods. In this paper, an extension of the PSMM to conical diffraction problems is presented and implemented. Numerical examples are used to demonstrate the high accuracy and excellent convergence property of this method for both dielectric and metallic gratings.     

Diffraction of transmission light through triangular apertures in array of retro-reflective microprisms

An array of microprisms was described by a model of multiperiod blazed gratings consisting of triangular apertures. The origins of hexagram-shaped diffraction patterns were interpreted based on multiple-beam interference and diffraction array theorem. The relation between zonal/line ghost fringes and imperfectly fabricated array structures was analyzed. Geometrical performance (e.g., the dihedral angle of the microprism) was tested by measuring the features of diffraction patterns of samples from three retroreflective sheeting manufacturers.     

Fabrication and simulation of diffractive optical elements with superimposed antireflection subwavelength gratings

With the aim of reducing surface reflections and increasing the diffraction efficiency we investigated the superposition of subwavelength phase gratings onto blazed phase gratings. With direct-write electron-beam lithography bare blazed gratings and blazed gratings carrying subwavelength gratings were fabricated and their optical performances compared. For TE polarization the subwavelength-carrying gratings showed a maximum diffraction efficiency of 90.6%, whereas the corresponding maximum value for the bare grating was 86.3%. The experiment was simulated with rigorous diffraction theory.     

Analysis of a New Method for Determining Optical Constants in the Soft X-ray Region

In this paper, a new method for measuring optical constants using transmission gratings is presented and analysed. The method, based on utilizing the properties of ultra-thin transmission diffraction gratings, is independent of the absolute intensity of the radiation source, the most critical part of the experiment being the accurate measurement and control of the diffraction grating parameters. By way of example, this paper relates the use of the new method to gold in the soft X-ray regime. By extension, it is shown that the same technique may be used on other materials, as long as they may be shaped into similar grating structures, and as long as they fulfil certain other criteria, outlined in the main body of this paper.