M. G. Moharam

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

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.

Rigorous coupled-wave analysis of planar-grating diffraction

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.

Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach

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.

Diffraction analysis of dielectric surface-relief gratings

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.

Analysis and applications of optical diffraction by gratings

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.

Guided-mode resonances in planar dielectric-layer diffraction gratings

The guided-mode resonance behavior of the evanescent and propagating fields associated with an unslanted, planar diffraction grating is studied by means of the rigorous coupled-wave theory. For weakly modulated gratings, the condition on the guided-mode wave number of the corresponding unmodulated dielectric-layer waveguide may be used to predict the range of the incident angle or wavelength within which the resonances can be excited. Furthermore, the locations of the resonances are predicted approximately by the eigenvalue equation of the waveguide. As the modulation amplitude increases, the location and shape of the resonances are described in detail by the rigorous coupled-wave theory. The results presented demonstrate that the resonances can cause rapid variations in the intensity of the external propagating diffracted waves.

THREE-DIMENSIONAL VECTOR COUPLED-WAVE ANALYSIS OF PLANAR-GRATING DIFFRACTION

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.

Rigorous coupled-wave analysis of grating diffraction— E-mode polarization and losses

Rigorous coupled-wave theory of diffraction by dielectric gratings is extended to cover E-mode polarization and losses. Unlike in the H-mode-polarization case, it is shown that, in the E-mode case, direct coupling exists between all diffracted orders rather than just between adjacent orders.

Criterion for Bragg and Raman-Nath diffraction regimes

The idea is well entrenched in the literature that thin phase gratings (whether holographic or acoustically induced) should exhibit Raman-Nath behavior (and thus give several diffracted waves), and that thick phase gratings should show Bragg behavior (one diffracted beam and that only for Bragg angle incidence). The parameter Q of Klein and Cook, which is a normalized measure of grating thickness, has been extensively used as a criterion for deciding which regime will apply. It is perhaps not generally realized that Q is not a reliable parameter for this purpose but requires, as indeed Klein and Cook noted, a limitation on grating strength. This limitation is a matter of practical concern. For example, we have observed Raman-Nath behavior with Fe-doped LiNbO(3) even for very large values of Q. The purpose of the present paper is to note that a parameter rho (first defined by Nath) is an effective replacement for Q, since rho is reliable and Q is not. rho is defined as lambda(0)(2)/Lambda(2)n(0)n(1), where lambda(0) is the vacuum wavelength of the light, Lambda is the grating spacing, n(0) is the mean refractive index, and n(1) is the amplitude of the sinusoidal modulation of the refractive index. The grating thickness does not enter rho, so the terms thin and thick are, strictly speaking, irrelevant to the question of which regime is operative. However, thin enough gratings will tend to operate in the Raman-Nath regime because the index modulation must be large for a thin grating to produce appreciable diffraction.

Transient Analysis of Grounding Systems

This paper addresses the problem of computing the ground potential rise of grounding systems during transients. Finite element analysis is employed to model the constituent parts of a grounding system. Short lengths of earth embedded electrodes are characterized as transmission lines with distributed inductance, capacitance and leakage resistance to earth. Leakage resistance to earth is accurately computed with the method of moments. The other parameters of the finite element, namely inductance and capacitance, are computed from the resistance utilizing Maxwells equations. This modeling enables the computation of the transient response of substation grounding systems to fast or slow waves striking the substation. The result is obtained in terms of a convolution of the step response of the system and the striking wave. In this way the impedance of substation systems to 60 cycles is accurately computed. Results demonstrate the dependence of the 60 cycle impedance on system parameters. The methodology allows to interface this model of a substation ground mat with the Electromagnetic Transient Analysis Program thus, allowing explicit representation of earth effects in electromagmatic transients computations.