Anisotropic Gratings Research Articles

Differential theory of gratings made of anisotropic materials.

Arbitrary profiled gratings made with anisotropic materials are discussed; the anisotropic character concerns electric and/or magnetic properties. Our aim is to avoid the use of the staircase approximation of the profile, whose convergence is questionable. A coupled first-order differential-equation set is derived by taking into account Li's remarks about Fourier factorization [J. Opt. Soc. Am. A 13, 1870 (1996)], but the present formulation shows that, in return for a convenient form of the differential system, it is possible to use only the intuitive Laurent rule. Our method, when applied to the simpler case of isotropic gratings, is shown to be consistent with that of previous studies. Moreover, from the numerical point of view, the convergence of our formulation for an anisotropic grating is faster than that of the conventional differential method.

Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media.

We establish the most general differential equations that are satisfied by the Fourier components of the electromagnetic field diffracted by an arbitrary periodic anisotropic medium. The equations are derived by use of the recently published fast-Fourier-factorization (FFF) method, which ensures fast convergence of the Fourier series of the field. The diffraction by classic isotropic gratings arises as a particular case of the derived equations; the case of anisotropic classic gratings was published elsewhere. The equations can be resolved either through classic differential theory or through the modal method for particular groove profiles. The new equations improve both methods in the same way. Crossed gratings, among which are grids and two-dimensional arbitrarily shaped periodic surfaces, appear as particular cases of the theory, as do three-dimensional photonic crystals. The method can be extended to nonperiodic media through the use of a Fourier transform.

Evolution of anisotropic reflection gratings formed in holographic polymer-dispersed liquid crystals

The temporal evolution of an anisotropic reflection grating produced in a holographic polymer-dispersed liquid crystal film is investigated. We find that this type of grating is preceded in time by an isotropic concentration grating, and that the development of the anisotropic grating can be delayed until several seconds after laser exposure. The formation of an anisotropic grating is nearly coincident with the onset of phase separation of liquid crystal and implies a macroscopic ordering of liquid crystal droplet directors. Detailed knowledge of grating evolution may allow in situ control over the polarization sensitivity of the hologram.

Multilayer anisotropic bi-periodic diffraction gratings

The formalism of Rokushima and Yamakita [J. Opt. Soc. Am. 73 (1983) 901] treating the diffraction on planar multilayered 1D anisotropic gratings is extended to the diffraction on anisotropic 2D multilayer grating structures bi-periodic in the plane parallel to the interfaces. In addition to the oblique incidence of plane waves, the case of normal incidence is also treated. The goal of the paper is to provide a basis for the formal analysis of 2D patterned multilayers with natural or induced anisotropies. For example, such structures are of interest in the design of new magnetic and magneto-optic devices. In view of the fact that the anisotropies have often a negligible effect on the energy distribution among diffracted modes with respect to the isotropic case the optical response is alternatively expressed in terms of the ellipsometric parameters of diffracted waves. The present analysis represents generalization of the problem of electromagnetic wave interactions in planar multilayers consisting of layers characterized by a general permittivity tensor treated by Yeh [J. Opt. Soc. Am. 69 (1979) 742].

An improved method for separating the kinetics of anisotropic and topographic gratings in side-chain azobenzene polyesters

The induction of anisotropy gratings in side-chain azobenzene polyesters is accompanied by the formation of surface relief. We introduce an improved holographic method to separate the contributions of the anisotropic and the topographic part to the diffraction efficiency by analyzing the polarization of the first-order diffracted beam. The main advantage of this method is that both parts can be determined simultaneously by only one measurement. Furthermore the displacement between both gratings can be determined in a similar manner. Experimental results obtained with two different polyesters are presented.

Diffraction properties of highly birefringent liquid-crystal composite gratings

We have fabricated electrically switchable holographic gratings, using Polaroid Corporation's DMP-128 photopolymer filled with the nematic liquid crystal E7. It is shown that a coupled-wave theory that includes the effects of the birefringence of the liquid crystal must be used to explain the diffraction properties of these anisotropic volume gratings. Furthermore, a detailed comparison of theory and experiment provides information about the alignment of the liquid crystal within the polymer host.

Anisotropic surface relief diffraction gratings under arbitrary plane wave incidence

A modal method is used for the analysis under oblique incidence of a diffraction grating made of anisotropic material. The problem is studied viewing the structure as the cascade of junctions between periodic arrays of anisotropic slab waveguides with the same period and different heights. This diffraction problem is formulated in terms of an integral equation that enforces the continuity of the transverse magnetic field at the junction. The unknown is the transverse electric field at the junction. It is possible to use also another formulation, where the role of the two fields is exchanged. The kernels of these equations are the relevant Green's functions, which are expressed in terms of eigenfunction expansions. The determination of the modes of the various regions composed of arrays of anisotropic dielectric slabs has been carried out by the method of spectral elements, whereby the field components are represented in a polynomial basis and the original differential eigenvalue problem is converted into an algebraic one. The integral equation is solved numerically by the method of moments and each junction is characterized by its generalized scattering matrix (GSM). Finally, the diffraction efficiencies of the grating are obtained by combining the various GSM's.

Oblique-coordinate-system-based Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic gratings

A formulation of the Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic diffraction gratings is presented. The formulation is based on an oblique Cartesian coordinate system so that blazed surface-relief profiles and slanted volume permittivity variations can be treated simultaneously. The permittivity tensors of the anisotropic materials can be arbitrary, but their spatial variations are assumed to have the same period as the surface corrugations. Some of the previous authors’ works in the above individual aspects of the subject are improved on in regard to clarity, simplicity, and generality. Most important, the theory of Fourier factorization is applied throughout the analysis to ensure fast convergence of the numerical method when the permittivity tensors or the derivatives of the grating’s surface-profile functions have discontinuities. Numerical examples, including those of two types of multilayer magneto-optic gratings, are provided.

Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials

Abstract By the use of the correct Fourier factorization rules presented in an earlier paper by Li the Fourier modal method (FMM) for anisotropic surface-relief gratings is reformulated. The newly formulated FMM converges much faster than the old formulation when the permittivity contrast in the grating groove region is large. Highly conducting lamellar gratings coated with anisotropic materials can now be analysed easily. In addition, a simple set of criteria are given for determining the energy propagation directions of the plane waves associated with the real solutions of the Booker quartic.

Reformulation of the fourier modal method for surface-relief gratings made with anisotropic materials

By the use of the correct Fourier factorization rules presented in an earlier paper by Li the Fourier modal method (FMM) for anisotropic surface-relief gratings is reformulated. The newly formulated FMM converges much faster than the old formulation when the permittivity contrast in the grating groove region is large. Highly conducting lamellar gratings coated with anisotropic materials can now be analysed easily. In addition, a simple set of criteria are given for determining the energy propagation directions of the plane waves associated with the real solutions of the Booker quartic.

Polarization properties and diffraction efficiencies of binary anisotropic gratings: general study and experiments on ferroelectric liquid crystals

Binary gratings recorded in anisotropic (birefringent or dichroic) media exhibit peculiar polarization properties and a method to calculate the polarization states and the irradiances of diffracted spots in the Fraunhofer diffraction pattern for an incident wave of arbitrary polarization state is proposed. The gratings under consideration have periods made of domains of different widths in which the anisotropic medium can be in two states. Anisotropic modulation results in spatial modulation of polarization with two possible states; for every spot in the diffraction pattern, the polarization state and the irradiance are derived as functions of these two states and are shown to depend on the order of diffraction of the spot. More specific results are obtained for phase shift, axis or mixed birefringence modulations and a comparison between the relative irradiances obtained in each case is established. The theory is illustrated by experimental results with ferroelectric liquid crystals.

Gaussian-beam profile deformation via anisotropic photorefractive gratings formed by diffusive, photovoltaic, and drift mechanisms: a system transfer function approach

Light-beam profile deformation during scattering with the pho- torefractive grating formed by the diffusive, photovoltaic, and drift mechanisms is investigated in the case of anisotropic Bragg diffraction for the incident light with arbitrary spatial beam profile under the frame- work of a three-dimensional configuration. A set of three-dimensional coupled-wave equations that can properly describe the interaction is de- rived from Maxwell's equations. In the wave equations, the permittivity profile, which is perturbed by the diffusive, photovoltaic, and drift mecha- nisms, can be correctly plotted by employing Kukhtarev's material model. Using a spatial Fourier transform technique identical to the one used in deriving the spatial transfer function from the paraxial wave equation for free-space propagation, a steady-state solution for the out- put light beams diffracted by this initially stored grating can be calculated in the spatial frequency domain. The nature of the dependence of beam profiles on the material properties, sample thickness, and incident angle, under the combined effects of propagational diffraction, diffusion, photo- voltaicity, and drifting, is presented. © 1998 Society of Photo-Optical Instrumen- tation Engineers. (S0091-3286(98)03501-6)

Multiple light diffraction theory in volume gratings using perturbative integral expansion method

Light wave diffraction from multiple superposed volume gratings is inestigated using a perturbative iteration method of the integral equation of Maxwell's wave equation. The host material and index gratings are anisotropic and non-coplanar multiple volume gratings are considered. In this method, the paraxial approximation and lack of backward scattering in conventional coupled mode theory are not assumed. Systematic analysis of anisotropic wave diffraction due to multiple noncoplanar volume index gratings is performed in increasing level of diffraction orders corresponding to successive iterations.

Fresnel diffraction at [formula omitted] of the Talbot distance of an anisotropic grating

We show that an anisotropic grating with two alternate linear perpendicular states of polarization is transformed by free propagation, at 1 4 of the Talbot distance, into another grating with circular polarization states. Experimental results that verify our proposal are included.

Analysis of anisotropic dielectric grating diffraction using the finite-element method

A unified numerical approach based on the finite-element method is described for the analysis of anisotropic dielectric grating diffraction. Both the TE and TM wave incidences are treated. The finite-element method is used for the region corresponding to one period of the grating, and analytical relations and periodic conditions are introduced on the boundaries surrounding the one-period region. Numerical examples are shown for a groove-type LiNbO3 grating with rectangular profile on a silver substrate.

Finite element analysis of plane wave diffraction from anisotropic dielectric gratings

AbstractIn this paper, the formulation of the analysis based on the finite element method is presented for the first time for an anisotropic dielectric grating. By this analysis method, the diffraction characteristics of a plane wave by an anisotropic dielectric grating with an arbitrary shape can be computed easily. The finite element method is applied to the region corresponding to one period containing an anisotropic dielectric grating. The analytical relationships are applied to the boundaries on the incident and transmitted sides while the periodic boundary condition is applied to all other boundaries. Specifically, an index modulated grating and a groove grating are chosen for the analysis of the diffraction characteristics when the TE and TM waves are coupled. From the comparison with the analysis results by the coupled‐mode theory and the spatial harmonic expansion method, the validity of the present method is confirmed.

Analysis of diffracted waves from isotropic chiral gratings

AbstractThis paper proposes an analysis method of diffracted waves in isotropic chiral gratings with periodicity in an arbitrary three‐dimensional direction. In this method, the spatial harmonic expansion method applied to the analysis of anisotropic dielectric gratings is extended. By Fourier transform of the periodicity of the chiral admittance, a systematic matrix form is used for rigorous formulation. In the numerical examples given here, calculations are carried out for the density modulated grating in which the relative dielectric constant and the chiral admittance distributions are varied sinusoidally. A coupling between the circularly polarized diffracted waves due to the Bragg condition and an anomalous diffraction phenomenon due to the chiralty of the medium are presented. This coupling is not observed in the case of nonchiral isotropic dielectric gratings.

Optical diffraction in chiral smectic-C liquid crystals.

We report a study on the optical diffraction for light propagation perpendicular to the twist axis in the chiral smectic-C liquid crystal. In this phase grating mode, we find very unusual intensity and polarization features in the diffraction pattern. These observed features can be explained by invoking the theory of anisotropic gratings which takes into account the internal diffractions.

High-resolution liquid-crystal phase grating formed by fringing fields from interdigitated electrodes

We report the formation of thin anisotropic phase gratings in a nematic liquid-crystalline film by use of lateral (fringing) electric fields induced by transparent interdigitated electrodes. These gratings yield high diffraction efficiency (>30%) with a strong dependence on the readout beam incidence angle. In addition, the formation of a defect wall is observed that has a significant effect on the diffraction properties of the phase grating.

Symmetry properties of anisotropic dielectric gratings

We consider the exact equations, obtained by application of the Bloch–Floquet theorem, that describe the propagation of light in anisotropic dielectric gratings with an arbitrary variation of the dielectric tensor along the longitudinal direction. On the basis of these equations we determine the scattering matrix and the corresponding grating efficiencies. The conditions for electromagnetic energy conservation are derived and an energy representation for the field inside the grating is defined, thus permitting determination of reciprocity relations that give rise to particular symmetries of the grating efficiencies if one neglects the reflections at the boundaries. Finally, by exploiting the symmetry properties of the matrix that describes the propagation of the electromagnetic field inside the grating, we discuss the polarization properties of the diffracted beams in chiral C smectics in the phase-grating configuration and in the periodic structures induced in certain nematic liquid crystals by the application of static magnetic or electric fields.