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Journal ArticleDOI

Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings

01 May 1995-Journal of The Optical Society of America A-optics Image Science and Vision (JOURNAL OF THE OPTICAL SOCIETY OF AMERICA B-OPTICAL PHYSICS)-Vol. 12, Iss: 5, pp 1068-1076
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.
Citations
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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

Journal ArticleDOI
TL;DR: In this article, Li and Haggans analyzed the convergence rate of the coupled-wave method for TM polarization of metallic lamellar gratings and provided numerical evidence that highly improved convergence rates similar to the TE polarization case can be obtained.
Abstract: The coupled-wave method formulated by Moharam and Gaylord [ J. Opt. Soc. Am.73, 451 ( 1983)] is known to be slowly converging, especially for TM polarization of metallic lamellar gratings. The slow convergence rate has been analyzed in detail by Li and Haggans [ J. Opt. Soc. Am. A10, 1184 ( 1993)], who made clear that special care must be taken when coupled-wave methods are used for TM polarization. By reformulating the eigenproblem of the coupled-wave method, we provide numerical evidence and argue that highly improved convergence rates similar to the TE polarization case can be obtained. The discussion includes both nonconical and conical mountings.

931 citations

Journal ArticleDOI
TL;DR: This review paper focuses on the progress made over the past 4 years toward the integration of sensitive SPR to lab-on-a-chip platforms and descriptions of novel SPR optical approaches and materials.

756 citations

Journal ArticleDOI
TL;DR: Structural colors are deeply connected with recent rapidly growing fields of photonics and have been extensively studied to clarify their peculiar optical phenomena as mentioned in this paper, which differs considerably from the ordinary coloration mechanisms such as in pigments, dyes and metals, where the colors are produced by virtue of the energy consumption of light.
Abstract: In recent years, structural colors have attracted great attention in a wide variety of research fields. This is because they are originated from complex interaction between light and sophisticated nanostructures generated in the natural world. In addition, their inherent regular structures are one of the most conspicuous examples of non-equilibrium order formation. Structural colors are deeply connected with recent rapidly growing fields of photonics and have been extensively studied to clarify their peculiar optical phenomena. Their mechanisms are, in principle, of a purely physical origin, which differs considerably from the ordinary coloration mechanisms such as in pigments, dyes and metals, where the colors are produced by virtue of the energy consumption of light. It is generally recognized that structural colors are mainly based on several elementary optical processes including thin-layer interference, diffraction grating, light scattering, photonic crystals and so on. However, in nature, these processes are somehow mixed together to produce complex optical phenomena. In many cases, they are combined with the irregularity of the structure to produce the diffusive nature of the reflected light, while in some cases they are accompanied by large-scale structures to generate the macroscopic effect on the coloration. Further, it is well known that structural colors cooperate with pigmentary colors to enhance or to reduce the brilliancy and to produce special effects. Thus, structure-based optical phenomena in nature appear to be quite multi-functional, the variety of which is far beyond our understanding. In this article, we overview these phenomena appearing particularly in the diversity of the animal world, to shed light on this rapidly developing research field.

748 citations

References
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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, 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
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.

967 citations

Journal ArticleDOI
TL;DR: In this paper, a rigorous coupled-wave analysis for metallic surface-relief gratings is presented for all diffracted orders as a function of period, groove depth, polarization, and angle of incidence.
Abstract: A rigorous coupled-wave analysis for metallic surface-relief gratings is presented. This approach allows an arbitrary complex permittivity to be used for the material and thus avoids the infinite conductivity (perfect-conductor) approximation. Both TE and TM polarizations and arbitrary angles of incidence are treated. Diffraction characteristics for rectangular-groove gold gratings with equal groove and ridge widths are presented for free-space wavelengths of 0.5, 1.0 and 10.0 μm for all diffracted orders as a function of period, groove depth, polarization, and angle of incidence. Results include the following: (1) TM-polarization diffraction characteristics vary more rapidly than do those for TE polarization, (2) 95% first-order diffraction efficiency occurs for TM polarization at 10.0 μm, (3) 50% absorption of incident power occurs at 0.5 μm, and (5) the perfect-conductor approximation is not valid for TM polarization at any of the wavelengths and is not valid for TE polarization at 0.5 μm.

601 citations

Journal ArticleDOI
TL;DR: In this paper, a 3D vector coupled-wave analysis of an arbitrarily oriented planar grating with slanted fringes is presented, which applies to any sinusoidal or nonsinusoidal amplitude and/or phase grating, any plane-wave angle of incidence, and any linear polarization.
Abstract: 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.

419 citations