Rigorous Coupled-wave Theory Research Articles

Exact, dynamical analysis of the Kukhtarev equations in photorefractive barium titanate using rigorous coupled-wave diffraction theory

Rigorous coupled-wave diffraction theory is used to analyze two-wave and multiwave mixing in diffusion-controlled photorefractive barium titanate, which is modeled by the Kukhtarev equations. These equations are partially decoupled to yield a system of two equations between the electron density and the electrostatic field (and hence the induced refractive index profile). The transmitted and reflected optical fields, the dielectric modulation, the electrostatic field, and the electron density are studied for the cases in which the interfering, incident optical fields have equal and unequal amplitudes for different values of the linear refractive index mismatch and for different values of photorefractive crystal length. In each case the exact longitudinal inhomogeneity in the photorefractive medium is analyzed with the use of rigorous coupled-wave diffraction theory and an exact Kukhtarev analysis. We compare the evolution of the diffracted orders for different sample lengths to show that the nature of the steady state (oscillatory or nonoscillatory) critically depends on the sample length. Our computations study in BaTiO3 the conditions for temporal instability resulting in self-pulsation and for anisotropic diffraction contributing to significant generation of higher orders (assuming that two plane waves are incident on the photorefractive material).

A nonlinear, transient analysis of two- and multi-wave mixing in a photorefractive material using rigorous coupled-wave diffraction theory

Rigorous coupled-wave diffraction theory is used to analyze two- and multi-wave mixing in a diffusion-controlled photorefractive material which is modeled by the Kukhtarev equations. These equations are first decoupled to yield a nonlinear time-dependent differential equation for the induced refractive index profile. The transmitted and reflected field coupling coefficients are studied for the cases when the incident optical fields are coherent and partially coherent, for materials with different gain constants, and for different values of the linear refractive index mismatch. In each case, the exact longitudinal inhomogeneity in the photorefractive medium is analyzed using rigorous coupled-wave diffraction theory. Our computations predict a possible temporal instability resulting in self-pulsation and anisotropic diffraction contributing to a significant generation of higher orders when two plane waves are incident on photorefractive materials. Six-wave coupling in index mismatched barium titanate is studied.

Transient wave mixing and recording kinetics in photorefractive barium titanate: a nonlinear coupled mode approach

By using rigorous coupled-wave diffraction theory along with a time-dependent nonlinear formulation, we analyze two- and multiple-wave coupling and the grating kinetics in BaTiO 3 with different boundary interfaces. Efffects of electrostatic and optical anisotropy have been included in the analysis. Significant mode conversion to higher orders is observed only when the boundary interfaces are highly mismatched.

Effective-medium theory of sinusoidally modulated volume holograms

Zeroth-order effective-medium theory is applied to sinusoidally modulated volume holograms for which the period of the grating is much smaller than the illumination wavelength. These holograms can be modeled as negative uniaxial films, with the effective ordinary and extraordinary permittivities being determined by the small-period quasi-static approximation. The limits of validity of the small-period approximation are examined as a function of the wavelength-to-grating-period ratio (λ:Λ) by rigorous coupled-wave theory. The limits are shown to be more restrictive for slanted holograms and for conically incident waves than for unslanted holograms and nonconically incident waves.

Electrostatic and diffraction analysis of a liquid-crystal device utilizing fringing fields: applications to three-dimensional displays

The development and modeling of a liquid-crystal phase grating for real-time diffractive three-dimensional displays are discussed. The system being developed, which is called the ICVision system, utilizes a number of ideas that will result in a rugged, low-power three-dimensional display offering both vertical and horizontal parallax and eventually full color. Fringing fields created between interdigitated electrodes formed on top of VLSI die will induce a diffraction pattern in a thin layer of liquid crystal that will cover the die. A detailed electrostatic and diffraction analysis of liquid-crystal phase-grating regions that will make up the final display is given here. The electrostatic analysis is developed by use of the method of moments. The diffraction analysis is developed by use of rigorous coupled-wave diffraction theory. The numerical results obtrained from the mathematical model are compared with experimental diffraction results from preliminary LCD cells that have been assembled as prototype ICVision devices.

Etch depth estimation of large-period silicon gratings with multivariate calibration of rigorously simulated diffraction profiles

Diffraction from a periodic structure is sensitive to small changes in the shape of that structure. It is possible to exploit this behavior of the scatter data for accurate, precise, rapid, nondestructive, and in situ measurements of grating structures. We present the use of rigorous coupled-wave theory to generate diffraction profiles to train a partial least-squares (PLS) multivariate calibration routine. The resulting PLS calibration model was applied to experimental diffraction data from gratings in etched bulk silicon to predict etch depths. A single-detector scanning scatterometer was used to measure the scatter from 32-μm-pitch structures illuminated with a He–Ne laser beam. The scatterometer measured the diffraction patterns from grating structures at 14 die locations on each of a set of five wafers. The theoretically based PLS estimator was then used to predict etch depths from scatterometry data obtained from the 70 different grating structures. The etch depth predictions were in excellent agreement with those obtained with a scanning force microscope (i.e., 0.9-μm-deep structures were predicted with an average error of 0.007 μm). This is a significant step toward the solution of the parametric inverse grating diffraction problem: that of quantitative prediction of structure dimensions from the measurement of scatter data.

Theory and applications of guided-mode resonance filters

The guided-mode resonance properties of planar dielectric waveguide gratings are presented and explained. It is shown that these structures function as filters that produce complete exchange of energy between forward- and backward-propagating diffracted waves with smooth line shapes and arbitrarily narrow filter linewidths. Simple expressions based on rigorous coupled-wave theory and on classical slab waveguide theory give a clear view and quantification of the inherent TE/TM polarization separation and the free spectral ranges of the filters. Furthermore, the resonance regimes, defining the parametric regions of the guided-mode resonances, can be directly visualized. It is shown that the linewidths of the resonances can be controlled by the grating modulation amplitude and by the degree of mode confinement (refractive-index difference at the boundaries). Examples presented of potential uses for these elements include a narrow-line polarized laser, a tunable polarized laser, a photorefractive tunable filter, and an electro-optic switch. The guided-mode resonance filter represents a basic new optical element with significant potential for practical applications.

A Comparison of Diffraction Theories for Off-bragg Replay

Two first-order analytic solutions and a second-order analytic solution for the case of lossless-dielectric unslanted volume transmission gratings are compared for the case of weak grating index modulation and significantly off-Bragg replay. It is shown that the analytic solution predicted using the Beta-value method, which has previously been found to agree more closely with experimental results for the unslanted case than the first-order K -vector closure method (of Kogelnik), also agrees more closely with the analytic expression produced by the second-order coupled-mode equations of Kong for this case. A numerical comparison of the first order theories and the rigorous coupled wave theory gives a similar result. Thus the Beta-value method offers definite advantages over the Kogelnik K -vector closure method for the unslanted transmission geometry.

Electromagnetic scattering of two-dimensional surface-relief dielectric gratings

We employed the rigorous vector coupled-wave theory [J. Opt. Soc. Am. 73, 1105 (1983)] to analyze the electromagnetic scattering from two dimensional (2-D) surface-relief dielectric gratings. A shoot-back method was developed for the numerical solution of the resulting coupled differential equations. This method allowed numerical solutions to be found for grating structures of arbitrary profiles and relatively deep grooves. It was most suitable where the grating medium refractive index was not too large and where only a small number of propagating orders existed. Experiments confirmed the numerically predicted reflectivities for 2-D surface-relief dielectric sinusoidal gratings. Reflectivity measurements were made on 2-D sinusoidal gratings fabricated on photoresist and on polycarbonate. The grating periodicities were of the order of 3000 lines/mm such that only the zero-order diffracted waves were propagating in the incident region, and possibly a few forward orders in the transmission region. The embossing technique that was used for replicating the grating patterns from photoresist onto polycarbonate proved to be a feasible method for the production of such gratings.

Diffraction from metal strip gratings with high spatial frequency in the infrared spectral region

A comparative theoretical and experimental study of the diffraction characteristics of high-spatial-frequency aluminum strip gratings as a function of grating thickness is presented. The gratings were made with ZnSe and ZnS substrates by using standard photolithographic techniques and reactive ion etching. A grating period of 3 μm and a fill factor of approximately 0.33 are used. The grating thicknesses vary from 2400 to 9500 Å. Transmittance measurements were carried out in the 7–13-μm-wavelength band in which only the forward and backward zeroth-order waves propagate. The gratings exhibit a long-wavelength bandstop, i.e., a high-pass filtering characteristic in transmission with the transmittance decreasing with increasing grating thickness. For the thickest grating tested (9500 Å), the transmittance was less than 5% for 8.5 μm or greater and dropped to approximately 1.5% at 13 μm. The measured transmittances are compared with results calculated by using the rigorous coupled-wave theory and are found to be in good agreement.

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.

Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings

The diffraction by one or an arbitrary number of cascaded anisotropic planar gratings with slanted fringes is analyzed by using rigorous three-dimensional vector coupled-wave theory. Arbitrary angle of incidence and polarization are treated. The existence of uniaxial external regions and the treatment of both phase and amplitude anisotropic slanted gratings are included in the analysis. The anisotropy and the three-dimensionality of the problem cause coupling between orthogonally polarized waves. The Bragg conditions for various combinations of ordinary (O) and extraordinary (E) polarized waves are quantified. Sample calculations are presented for single anisotropic gratings (a lithium niobate hologram in air and an interdigitated-electrode-induced electro-optic grating in an optical waveguide), for two cascaded anisotropic gratings (a pair of interdigitated-electrode-induced gratings satisfying the OOO forward Bragg condition, the EEE forward Bragg condition, and the OOO backward Bragg condition), and for multiple cascaded gratings (a lithium niobate hologram with depth modulation). The same analysis applies in the limiting cases of isotropic materials, a grating vector lying in the plane of incidence, etc. Applications for this analysis include optical storage, switching, modulation, deflection, and data processing.

Optics of Floquet-Bloch waves in dielectric gratings

The behavior of light in dielectric gratings is discussed in terms of the optical Floquet-Bloch waves (or modes). The emphasis is on the development of a good physical understanding of the nature of these waves, using the wavevector diagram to summarize their spatial dispersion and spectra. It is shown that Floquet-Bloch theory offers some advantages conceptually over the commonly used coupled-wave theory, because the rays of the Floquet-Bloch waves (given by their group velocities) play the same role in a periodic medium as do those of plane waves in isotropic or graded-index media. The effect on power conservation of truncating the Floquet expansions for the Floquet-Bloch waves is considered in detail. Using the greater intuitive power of Floquet-Bloch theory, it is shown (in contrast to recent claims to the contrary) how rigorous coupled-wave theory can be applied to symmetrical reflection gratings, and secondly how the light in these gratings can be viewed in terms of the multiple-beam interference of Floquet-Bloch waves, leading to behavior reminiscent of a low-finesse Fabry-Perot cavity.

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.

Diffraction characteristics of planar absorption gratings

Planar (co)sinusoidal conductivity (absorption) transmission gratings are analyzed using rigorous coupled-wave theory. The first-order and higher-order diffraction efficiencies are determined over the entire range of possible conductivities and Bragg angles of incidence (or equivalently, grating periods) for H-mode polarization incident plane waves. The maximum possible first diffracted order efficiency is found to be 5.26%. Rigorous results are compared to approximate results from the Raman-Nath theory and the two-wave first-order coupled-wave (Kogelnik) theory. A regime parameter, ϱσ, is defined which delineates the regions of Raman-Nath diffraction behavior (ϱσ 1). Likewise, the angular selectivity characteristics of conductivity gratings are determined from rigorous theory and are compared with corresponding results from approximate theory.

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.

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.

Chain-matrix analysis of arbitrary-thickness dielectric reflection gratings

A simple but rigorous chain-matrix analysis is used to determine the plane-wave diffraction efficiencies and angular selectivities of planar pure-reflection gratings (zero-slant angle) of arbitrary thickness, arbitrary-starting and arbitrary-ending conditions, and arbitrary incidence angle. The results from rigorous coupled-wave theory in the angular limit as the fringes become parallel to the surfaces are shown to approach these results in a particular average sense.