A rigorous coupled-wave approach is used to analyze diffraction by general planar gratings bounded by two different media. The grating fringes may have any orientation (slanted or unslanted) with respect to the grating surfaces. The analysis is based on a state-variables representation and results in a unifying, easily computer-implementable matrix formulation of the general planar-grating diffraction problem. Accurate diffraction characteristics are presented for the first time to the authors’ knowledge for general slanted gratings. This present rigorous formulation is compared with rigorous modal theory, approximate two-wave modal theory, approximate multiwave coupled-wave theory, and approximate two-wave coupled-wave theory. Typical errors in the diffraction characteristics introduced by these various approximate theories are evaluated for transmission, slanted, and reflection gratings. Inclusion of higher-order waves in a theory is important for obtaining accurate predictions when forward-diffracted orders are dominant (transmission-grating behavior). Conversely, when backward-diffracted orders dominate (reflection-grating behavior), second derivatives of the field amplitudes and boundary diffraction need to be included to produce accurate results.

Rigorous coupled-wave analysis of planar-grating diffraction

Diffraction characteristics of general dielectric planar (slab) gratings and surface-relief (corrugated) gratings are reviewed. Applications to laser-beam deflection, guidance, modulation, coupling, filtering, wavefront reconstruction, and distributed feedback in the fields of acoustooptics, integrated optics, holography, and spectral analysis are discussed. An exact formulation of the grating diffraction problem without approximations (rigorous coupled-wave theory developed by the authors) is presented. The method of solution is in terms of state variables and this is presented in detail. Then, using a series of fundamental assumptions, this rigorous theory is shown to reduce to the various existing approximate theories in the appropriate limits. The effects of these fundamental assumptions in the approximate theories are quantified and discussed.

Analysis and applications of optical diffraction by gratings

https://web.stanford.edu/group/fan/publication/Liu_ComputerPhysicsCommunications_183_2233_2012.pdf

S4 : A free electromagnetic solver for layered periodic structures

Diffraction by an arbitrarily oriented planar grating with slanted fringes is analyzed using rigorous three-dimensional vector coupled-wave analysis. The method applies to any sinusoidal or nonsinusoidal amplitude and/or phase grating, any plane-wave angle of incidence, and any linear polarization. In the resulting (conical) diffraction, it is shown that coupling exists between all space-harmonic vector fields inside the grating (corresponding to diffracted orders outside the grating). Therefore the TE and TM components of an incident wave are each coupled to all the TE and TM components of all the forward- and backward-diffracted waves. For a general Bragg angle of incidence, it is shown that the diffraction efficiency can approach 100% for a lossless grating if either the incident electric field or the magnetic field is perpendicular to the grating vector. Maximum coupling between incident and diffracted waves is shown to occur when the incident electric field is perpendicular to the grating vector. In general, the diffracted waves are shown to be elliptically polarized. The three-dimensional vector coupled-wave analysis presented is shown to reduce to ordinary rigorous coupled-wave theory when the grating vector lies in the plane of incidence.

Three-dimensional vector coupled-wave analysis of planar-grating diffraction

The diffraction of light by deep rectangular-groove transmission phase gratings is treated by solving Maxwell’s equations numerically. Results are given for the light diffracted into the zero order by gratings with grating constants d in the range λ < d < 5λ, aspect ratio b (= linewidth/d), 0 < b < 1, and grating depths a < 5λ, assuming a refractive index n0 = 1.5. Such gratings are used in practice as a dye-free replacement for color filters. They offer a new way of storing pictorial information in small format for read out in conventional projectors.

Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves

This paper extends Burckhardt’s solution for diffraction from a thick grating to include complex dielectric constants and nonsinusoidal stratifications. This allows any realistic periodic structure to be handled. Computed results are compared with coupled-wave theory, as described by Kogelnik, with emphasis on strongly absorbing gratings such as those made by photographing an interference pattern. Finally, some experimental holographic data are compared with computations that take into account the photographic nonlinearity between exposure and dielectric constant.

Diffraction by thick, periodically stratified gratings with complex dielectric constant

In a previous paper, several theoretical expressions were derived for calculating the ratio of the light flux in a holographic image of a uniform but extended source to that of a direct image of the source itself, account being taken of the sensitometric characteristics of the photographic material. The present paper is an experimental extension of this work. In particular, comparisons are made between theoretical curves and experimental data for variations of (1) the average density of the hologram, (2) the ratio of object-beam to reference-beam irradiances, and (3) the angular extent of the object. The performance of certain photographic developers is also discussed.

Comparison of Experimental and Theoretical Holographic Image Radiance

The total light flux diffracted by a developed holographic plate is proportional to the variance of the spatial distribution of amplitude transmittance of the plate. This light forms the real and virtual holographic images and also the flare surrounding the reference beam. The variance of amplitude transmittance does not, in general, depend upon the size or form of the object, but only upon the total amount of light received from it, as this affects the beam-balance ratio. The sensitometric characteristics of the plate are taken into account by making use of three plots, namely, amplitude transmittance vs exposure, amplitude transmittance vs log exposure, and density vs log exposure. First-order expressions are derived for the ratio of the light flux diffracted by the hologram to that received from the object in terms of measurable parameters such as beam-balance ratio, average exposure, and gradient of the sensitometric curve. In like manner, we can calculate the ratio of the radiance of the holographic image to that of the original object.

Effects of Some Photographic Characteristics on the Light Flux in a Holographic Image

A method is described for computing the light flux transmitted by a thick grating, which can be either an amplitude grating, a phase grating, or a combination of the two. The flux is computed for a number of planes behind the grating and for a number of angles of illumination, in order to simulate contact printing of a thick grating into a thick receiver. The applicability of the thin-grating concepts of MTF and linear superposition to thick gratings is considered. The validity of such application is checked by computing the flux transmitted by thick gratings whose spatial-frequency contents were chosen so that comparison can be made with the transmitted flux predicted by linear-systems concepts.

F. G. Kaspar

Papers

Diffraction by thick, periodically stratified gratings with complex dielectric constant

Comparison of Experimental and Theoretical Holographic Image Radiance

Effects of Some Photographic Characteristics on the Light Flux in a Holographic Image

Computation of light transmitted by a thick grating, for application to contact printing

Power Spectrum of Object Flux Transmittance by Holographic Techniques