scispace - formally typeset
Search or ask a question
Journal ArticleDOI

Diffraction analysis of dielectric surface-relief gratings

01 Oct 1982-Journal of the Optical Society of America (Optical Society of America)-Vol. 72, Iss: 10, pp 1385-1392
TL;DR: In this article, a dielectric surface-relief grating is analyzed using rigorous coupled-wave theory and the analysis applies to arbitrary grating profiles, groove depths, angles of incidence, and wavelengths.
Abstract: Diffraction by a dielectric surface-relief grating is analyzed using rigorous coupled-wave theory. The analysis applies to arbitrary grating profiles, groove depths, angles of incidence, and wavelengths. Example results for a wide range of groove depths are presented for sinusoidal, square-wave, triangular, and sawtooth gratings. Diffraction efficiencies obtained from the present method of analysis are compared with previously published numerical results. To obtain large diffraction efficiencies (greater than 85%) for gratings with typical substrate permittivities, it is shown that the grating profile should possess even symmetry.
Citations
More filters
Journal ArticleDOI
TL;DR: In this paper, the authors presented a stable and efficient numerical implementation of the analysis technique for one-dimensional binary gratings for both TE and TM polarization and for the general case of conical diffraction.
Abstract: The rigorous coupled-wave analysis technique for describing the diffraction of electromagnetic waves by periodic grating structures is reviewed. Formulations for a stable and efficient numerical implementation of the analysis technique are presented for one-dimensional binary gratings for both TE and TM polarization and for the general case of conical diffraction. It is shown that by exploitation of the symmetry of the diffraction problem a very efficient formulation, with up to an order-of-magnitude improvement in the numerical efficiency, is produced. The rigorous coupled-wave analysis is shown to be inherently stable. The sources of potential numerical problems associated with underflow and overflow, inherent in digital calculations, are presented. A formulation that anticipates and preempts these instability problems is presented. The calculated diffraction efficiencies for dielectric gratings are shown to converge to the correct value with an increasing number of space harmonics over a wide range of parameters, including very deep gratings. The effect of the number of harmonics on the convergence of the diffraction efficiencies is investigated. More field harmonics are shown to be required for the convergence of gratings with larger grating periods, deeper gratings, TM polarization, and conical diffraction.

2,437 citations

Journal ArticleDOI
TL;DR: In this paper, an enhanced, numerically stable transmittance matrix approach is developed and is applied to the implementation of the rigorous coupled-wave analysis for surface-relief and multilevel gratings.
Abstract: An enhanced, numerically stable transmittance matrix approach is developed and is applied to the implementation of the rigorous coupled-wave analysis for surface-relief and multilevel gratings. The enhanced approach is shown to produce numerically stable results for excessively deep multilevel surface-relief dielectric gratings. The nature of the numerical instability for the classic transmission matrix approach in the presence of evanescent fields is determined. The finite precision of the numerical representation on digital computers results in insufficient accuracy in numerically representing the elements produced by inverting an ill-conditioned transmission matrix. These inaccuracies will result in numerical instability in the calculations for successive field matching between the layers. The new technique that we present anticipates and preempts these potential numerical problems. In addition to the full-solution approach whereby all the reflected and the transmitted amplitudes are calculated, a simpler, more efficient formulation is proposed for cases in which only the reflected amplitudes (or the transmitted amplitudes) are required. Incorporating this enhanced approach into the implementation of the rigorous coupled-wave analysis, we obtain numerically stable and convergent results for excessively deep (50 wavelengths), 16-level, asymmetric binary gratings. Calculated results are presented for both TE and TM polarization and for conical diffraction.

1,497 citations

Journal ArticleDOI
Lifeng Li1
TL;DR: The recent reformulation of the coupled-wave method by Lalanne and Morris as mentioned in this paper, which dramatically improves the convergence of the method for metallic gratings in TM polarization, is given a firm mathematical foundation in this paper.
Abstract: The recent reformulation of the coupled-wave method by Lalanne and Morris [ J. Opt. Soc. Am. A13, 779 ( 1996)] and by Granet and Guizal [ J. Opt. Soc. Am. A13, 1019 ( 1996)], which dramatically improves the convergence of the method for metallic gratings in TM polarization, is given a firm mathematical foundation in this paper. The new formulation converges faster because it uniformly satisfies the boundary conditions in the grating region, whereas the old formulations do so only nonuniformly. Mathematical theorems that govern the factorization of the Fourier coefficients of products of functions having jump discontinuities are given. The results of this paper are applicable to any numerical work that requires the Fourier analysis of products of discontinuous periodic functions.

1,213 citations


Cites background from "Diffraction analysis of dielectric ..."

  • ...(2) and (5) and correctly handled by Eq....

    [...]

  • ...(1), (2), and (5) are incorrect and why Eqs....

    [...]

  • ...(1), (2), and (5) are proven to be incorrect....

    [...]

Journal ArticleDOI
01 May 1985
TL;DR: In this article, an exact formulation of the grating diffraction problem without approximations is presented, using a series of fundamental assumptions, which reduces to the various existing approximate theories in the appropriate limits.
Abstract: 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.

989 citations

Journal ArticleDOI
Lifeng Li1
TL;DR: Two recursive and numerically stable matrix algorithms for modeling layered diffraction gratings, the S-Matrix algorithm and the R-matrix algorithm, are systematically presented in a form that is independent of the underlying grating models, geometries, and mountings.
Abstract: Two recursive and numerically stable matrix algorithms for modeling layered diffraction gratings, the S-matrix algorithm and the R-matrix algorithm, are systematically presented in a form that is independent of the underlying grating models, geometries, and mountings. Many implementation variants of the algorithms are also presented. Their physical interpretations are given, and their numerical stabilities and efficiencies are discussed in detail. The single most important criterion for achieving unconditional numerical stability with both algorithms is to avoid the exponentially growing functions in every step of the matrix recursion. From the viewpoint of numerical efficiency, the S-matrix algorithm is generally preferred to the R-matrix algorithm, but exceptional cases are noted.

951 citations

References
More filters
Journal ArticleDOI
TL;DR: In this article, a rigorous coupled-wave approach is used to analyze diffraction by general planar gratings bounded by two different media, and the analysis is based on a state-variables representation and results in a unifying, easily computer-implementable matrix formulation.
Abstract: 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.

2,224 citations

Journal ArticleDOI
TL;DR: In this paper, the propagation of electromagnetic waves along open periodic, dielectric waveguides is formulated as a rigorous and exact boundary-value problem, and the characteristic field solutions are shown to be of the surface-wave or leaky-wave type, depending on the ratio of periodicity to wavelength (d/lambda).
Abstract: The propagation of electromagnetic waves along open periodic, dielectric waveguides is formulated here as a rigorous and exact boundary-value problem. The characteristic field solutions are shown to be of the surface-wave or leaky-wave type, depending on the ratio of periodicity to wavelength (d/lambda). The dispersion curves and the space-harmonic amplitudes of these fields are examined for both TE and TM modes. Specific numerical examples are given for the cases of holographic layers and for rectangularly corrugated gratings; these show the detailed behavior of the principal field components and the dependence of waveguiding and leakage characteristics on the physical parameters of the periodic configuration.

452 citations

Journal ArticleDOI
TL;DR: In this paper, simple design criteria for grating couplers which transfer the energy of a beam into or out of an optical waveguide are developed based on an accurate perturbation analysis of the guiding properties of dielectric gratings.
Abstract: Based on an accurate perturbation analysis of the guiding properties of dielectric gratings, simple design criteria are developed for grating couplers which transfer the energy of a beam into or out of an optical waveguide. Gratings having arbitrary groove shapes are considered and explicit formulae are given for the leakage parameters of gratings with symmetric profiles. The results cover TEv and TMv modes and they apply to both shallow and deep grating grooves. The variation of the leakage parameter α in rectangular gratings is examined in detail; these rectangular gratings are then used as basic configurations for predicting the characteristics of other grating profiles. Particular attention is given to trapezoidal and triangular profiles and gratings with asymmetric profiles are also discussed.

361 citations

Journal ArticleDOI
TL;DR: In this article, the diffraction of light by deep rectangular-groove transmission phase gratings is treated by solving Maxwell's equations numerically, and 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 < 5 λ, assuming a refractive index n 0 = 1.
Abstract: 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.

322 citations

Journal ArticleDOI
TL;DR: In this paper, a general formulation for arbitrary-shaped gratings which need not be small in size was developed for radiated power from traveling waves as a function of grating tooth height, tooth width, refractive index, waveguide thickness, and period.
Abstract: Grating-coupled radiation in GaAs:GaAlAs lasers and waveguides is analyzed. A general formulation is developed for arbitrary-shaped gratings which need not be small in size. Two methods are used to solve the resulting equations in the case of rectangular-shaped gratings. The first is a perturbation technique and the second is iterative in nature. The iterative procedure converges to a numerical exact solution in many cases of practical interest and indicates that the perturbation results are quite accurate. Curves are presented for radiated power from traveling waves as a function of grating tooth height, tooth width, refractive index, waveguide thickness, and period for rectangular gratings in heterostructure waveguiding geometries. It is shown that radiation is not a monotonically increasing function of tooth height, but rather maxima occur when the teeth are half the optical wavelength in the material. Also, in particular geometries with an air:GaAs grating interface, radiated power of a mode can exceed 100 cm-1.

176 citations