General transfer-matrix method for optical multilayer systems with coherent,partially coherent,and incoherent interference

The optical response of coherent thin-film multilayers is often represented with Fresnel coefficients in a 2 x 2 matrix configuration. Here the usual transfer matrix was modified to a generic form, with the ability to use the absolute squares of the Fresnel coefficients, so as to include incoherent (thick layers) and partially coherent (rough surface or interfaces) reflection and transmission. The method is integrated by use of models for refractive-index depth profiling. The utility of the method is illustrated with various multilayer structures formed by ion implantation into Si, including buried insulating and conducting layers, and multilayers with a thick incoherent layer in an arbitrary position.     

Anisotropic incoherent reflection model for spectroscopic ellipsometry of a thick semitransparent anisotropic substrate

An anisotropic incoherent reflection model for the Mueller matrix elements of an optically thick uniaxial anisotropic semitransparent substrate with its anisotropy axis along its surface normal is developed. The Mueller matrix elements are measured by phase-modulated spectroscopic ellipsometry (SE) and compared with incoherent reflection model simulations. In the case of a sapphire substrate the oscillations observed are accurately modeled, and, in addition, the oscillating degree of polarization is correctly predicted. A straightforward generalization of the optical model, in the case of an arbitrary stack of layers containing a thick anisotropic semitransparent substrate, is also proposed and experimentally validated. The model is further applied to study the anisotropic dielectric function of a semi-insulating 4H-SiC wafer. An approximation based on a simple variation in the optical transition element is proposed to model the SiC birefringence. In conclusion, SE is shown to be a powerful alternative for investigating and predicting the behavior of optically thick birefringent materials.     

Modeling of wave propagation in layered piezoelectric media by a recursive asymptotic method

In this paper, a simple asymptotic method to compute wave propagation in a multilayered general anisotropic piezoelectric medium is discussed. The method is based on explicit second and higher order asymptotic representations of the transfer and stiffness matrices for a thin piezoelectric layer. Different orders of the asymptotic expansion are obtained using Padé approximation of the transfer matrix exponent. The total transfer and stiffness matrices for thick layers or multilayers are calculated with high precision by subdividing them into thin sublayers and combining recursively the thin layer transfer and stiffness matrices. The rate of convergence to the exact solution is the same for both transfer and stiffness matrices; however, it is shown that the growth rate of the round-off error with the number of recursive operations for the stiffness matrix is twice that for the transfer matrix; and the stiffness matrix method has better performance for a thick layer. To combine the advantages of both methods, a hybrid method which uses the transfer matrix for the thin layer and the stiffness matrix for the thick layer is proposed. It is shown that the hybrid method has the same stability as the stiffness matrix method and the same round-off error as the transfer matrix method. The method converges to the exact transfer/stiffness matrices essentially with the precision of the computer round-off error. To apply the method to a semispace substrate, the substrate was replaced by an artificial perfect matching layer. The computational results for such an equivalent system are identical with those for the actual system. In our computational experiments, we have found that the advantage of the asymptotic method is its simplicity and efficiency.     

Polarimetric characterization of optically anisotropic flexible substrates

Phase Modulated Spectroscopic Ellipsometry as well as Liquid Crystal Mueller Matrix Polarimetry in reflection and in transmission configurations were used to systematically study five types of anisotropic polymer sheets: polyethylene-terephtalate (PET), polyethylene-naphtalate (PEN), polycarbonate (PC), polypropylene (PP) and triacetylcellulose (TAC). The measurements were performed at different sample azimuths in two ellipsometric configurations giving access to both standard ellipsometric data as well as to the entire Mueller matrix. Biaxial anisotropy, a common characteristic to all polymer types, as well as the in-depth optical properties, inhomogeneity present in the sheets were clearly evidenced. The data were interpreted in terms of a model consisting of a thick substrate (several microns) coated with a simple layer. Both, substrate and layer were anisotropic and characterized by a triplet of principal refractive indexes. The orientation of the principal indexes of the bulk and the layer were different revealing the in-depth inhomogeniety of the samples.     

Anisotropic elasto-plastic finite element analysis of thick and thin plates and shells

An elasto-plastic analysis of anisotropic plates and shells is undertaken by means of the finite element displacement method. A thick shell formulation accounting for shear deformation is considered, which is based on a degenerate three-dimensional continuum element. The accommodation of variable material properties, not only along the surface of the structure but also through the thickness, is made possible by a discrete layered approach. Although isoparametric elements of the Serendipity family give satisfactory solutions for thick and moderately thin shells the results exhibit ‘locking’ for an increasing ratio of span to thickness. To develop a numerical model which is applicable to thick or thin plates and shells, the nine-node Lagrangian element and the Heterosis element are also introduced into the present model. Plastic yielding is based on the Huber-Mises yield surface extended by Hill for anisotropic materials. The yield function is generalized by introducing anisotropic parameters of plasticity which are updated during the material strain hardening history. Numerical examples are presented and compared with available solutions. The effects of anisotropy on these solutions are also discussed.     

Simple expressions for transmission and reflection matrix elements of a biaxial thin layer at normal incidence

The extended Jones matrix method is applied for determination of the transmission and reflection matrices for a normally incident plane wave upon an homogeneous and lossless biaxial thin layer. The elements of these matrices are expressed by simple analytical relations. By using these relations one can express analytically the polarization-dependent optical parameters to be determined by generalized ellipsometry.     

Transmission and reflection ellipsometry of thin films and multilayer systems

Ellipsometric studies are generally carried out in the reflection mode rather than in the transmission mode, requiring invariably opaque substrates or substrates in which the backreflection is minimized or suppressed by different methods. In the present research we used a transmission and reflection photoellipsometry method to study electrochromic materials and their multilayer systems deposited on thick substrates. The role of the substrate is examined carefully, and the contributions from multiple reflections in the substrate are taken into account in the theoretical treatment. This procedure not only allows the study of thin films deposited on quasi-transparent substrates, but when carried out in conjunction with reflection measurements it greatly improves the accuracy in the determination of the optical constants. Optical measurements are carried out on an automatic reflection transmission spectroscopic ellipsometer. Solid-state ionic materials used in electrochromic systems such as indium tin oxide, tungsten oxide, and their multilayer structures deposited on glass substrates are used as examples. A software based on the above theory, optikan, was developed to model and analyze such systems. It is demonstrated that the photoellipsometry method proposed is especially suited to analyzing electrochromic materials and transmitting devices in a nondestructive way.     

Generalized abelès relations for an anisotropic thin film with an arbitrary dielectric tensor: comments

The Cojocaru generalization of the 2 x 2 extended Jones matrix method, placed in a wider context of previous approaches to anisotropic optical thin films, is analyzed from a complementary perspective. This, contrary to initial belief, allows for a simple proof that one may include multiple reflections by taking into account total fields into the anisotropic film, and this therefore provides support for a more widespread use of the method.     

Scatter from tilted-columnar birefringent thin films: observation and measurement of anisotropic scatter distributions

We show that the angular distribution of scattered light (haze) from tilted-columnar birefringent thin films is highly anisotropic. When a narrow laser beam is incident normally upon a substrate coated with a tilted-columnar film, such as titania deposited at 60° to a thickness of ≈2 μm, the distribution of light scattered into the surrounding transmission and reflection air spaces has the form of one or more arcs. Scatter of light into the substrate is also highly anisotropic, and secondary scatter from the trapped substrate flux produces characteristic bright patterns that appear superposed on the coating.

When viewed in transmission the bright pattern typically consists of two strong lobes and in reflection four weaker lobes that make a cross. The various anisotropic distributions are shown to be consistent with interference of light from scattering centers that are correlated along the column direction and hence can be described as reflections from the tilted-columnar thin-film microstructure.

     

Reflection and transmission in pre-stressed stratified multilayers

Summary Time-harmonic wave propagation is investigated in multilayers whose slabs are pre-stressed anisotropic, dissipative solids. The material properties vary continuously within each slab and suffer jump discontinuities at the interfaces. The multilayer is sandwiched between homogeneous half-spaces. The constitutive equations due to the motion are taken to be linear, and hence the governing equations are shown to allow for a Stroh-like form. The solution is determined for any slab, and hence for the multilayer, through an integral-equation formulation, in terms of a propagation matrix which incorporates the jumps at the interfaces. Next the reflection and transmission problem for the multilayer, with an oblique incident wave, is solved by determining the reflection and transmission matrices in terms of the propagation matrix, the parameters of the incident wave and the eigenmode polarization vectors in the halfspaces. By way of application, results are derived in detail for horizontally-polarized waves.     

Multi-Gaussian Beam Modeling for Multilayered Anisotropic Media,I: Modeling Foundations

A model is developed for the propagation and reflection/transmission of Gaussian beams in multilayered media where the layer materials can be general anisotropic, homogeneous elastic solids and the layer interfaces can be curved surfaces. It is shown that the Gaussian beams can be simply described in a set of slowness coordinates and that the complications arising from the multiple layers can be efficiently addressed through the use of an ABCD matrix approach. A multi-Gaussian beam model for an ultrasonic piston transducer radiating into very complex media is then developed by superimposing a small number of these propagating Gaussian beams.     

Review: Scanning high-energy electron diffraction (SHEED) in materials science

The SHEED technique enables direct and rapid quantitative measurement of energyfiltered electron-diffraction intensities to be made. it may be used for structural studies of single crystals, polycrystalline films and amorphous materials. These may be examined in transmission if sufficiently thin or, otherwise, by reflection electron diffraction. In addition to conventional passive observation, structural studies of growing thin and thick films may be made during deposition. Work on all these topics is reviewed and the advantages of SHEED in each case clearly demonstrated.     

Explicit solutions for the optical properties of arbitrary magneto-optic materials in generalized ellipsometry

Analytic expressions for the eigenvalues for the four-wave components at an oblique angle of light incidence inside a randomly oriented anisotropic magneto-optic dielectric medium are reported explicitly. In particular, these solutions are valid as long as the dielectric function tensor consists of a symmetric and an antisymmetric part. The normalized Jones reflection and transmission coefficients, i.e., the generalized ellipsometric parameters of homogeneously layered systems having nonsymmetric dielectric properties, are obtained immediately from a recently reviewed 4 x 4 matrix approach. Our explicit solutions allow a future analysis of the generalized ellipsometric data of multilayered magneto-optic media regardless of the orientation of the material magnetization and crystalline axes and the angle of light incidence. Possible experimental thin-film situations are discussed in terms of generalized ellipsometric parameters and illustrated for birefringent free-carrier effects in heavily doped semiconductor thin films and for oblique magnetization directions in magneto-optic multilayer systems.     

Fabrication of gradient optical filter containing anisotropic Bragg nanostructure

New gradient optical filters containing asymmetric Bragg structure were prepared from the distributed Bragg reflector (DBR) porous silicon (PSi). Anisotropic DBR PSi displaying a rainbow-colored reflection was generated by using an asymmetric etching configuration. Flexible anisotropic DBR PSi composite films were obtained by casting of polymer solution onto anisotropic DBR PSi thin films. The surface and cross-sectional images images of anisotropic DBR PSi composite films obtained with cold field emission scanning electron microscope indicated that the average pore size and the thickness of porous layer decreased as the lateral distance increased. As lateral distance increased, the reflection resonance shifted to shorter wavelength.     

Generalized ellipsometry for materials characterization

Ellipsometry experiments normally measure two to four parameters, which are converted to the ellipsometric parameters ψ and Δ. This is usually sufficient for many samples, but more complicated situations (such as anisotropic or depolarizing samples) require more sophisticated measurements. Over the last 7 years, we have developed the two-modulator generalized ellipsometer (2-MGE), which measures eight elements of the sample Mueller matrix simultaneously either in reflection or transmission. In reflection, the 2-MGE totally characterizes light reflection from anisotropic samples, measuring the normal ellipsometry parameters, as well as the cross-polarization and depolarization effects. Applications include the determination of the spectroscopic optical functions of uniaxial materials (such as TiO₂ and ZnO), and the measurement of cross-polarization from diffractive structures. In transmission, the 2-MGE completely characterizes a general linear diattenuator and retarder. Applications include the measurement of the retardation and diattenuation of film polarizers and internal electric fields in LiNbO₃ and CdZnTe under bias.     

Reflection-transmission photoellipsometry: theory and experiments

We propose a method that uses reflection and transmission photoellipsometry to analyze samples consisting of thin films combined with semitransparent thick layers or substrates in the form of multilayer structures. Athick film or substrate is defined as a layer for which no interference effects can be observed for a given wavelength resolution, and contributions from multiple reflections in the substrate are taken into account in the theoretical treatment. An automatic reflection-transmission spectroscopic ellipsometer was built to test the theory, and satisfactory results have been obtained. Examples corresponding to a strongly absorbing film deposited on a glass substrate and a highly transmitting film also deposited on glass are shown. In both cases a good fit between theory and experiment is found. The photoellipsometric method presented is particularly suited to the analysis of actual samples of energy-efficient coatings for windows.     

A Method for Measuring the Thermal Diffusivity of Intermediate Thickness Surface Absorbing Samples and Obtaining the Ratio of Anisotropy by the Converging Wave Flash Method

The converging thermal wave, flash technique for measuring thermal diffusivity is suitable for use on samples that are sufficiently thick or thin in comparison to the annular heat source, to be described by a three-dimensional or two-dimensional approximation of the heat conduction equation, and sufficiently absorbing to ensure generation of a heat source at the surface. However, samples of intermediate thickness, which lie between these regimes, cannot be analyzed. In this article, heat diffusion in the samples of varying thicknesses is modeled, and a semi-analytic expression is used to describe the dimensionality of any thickness, allowing the converging wave method to be extended to intermediate thickness samples. Applying the analysis to anisotropic samples, a method is proposed to find the anisotropy ratio of the in-plane to perpendicular-to-plane diffusivity using the converging wave method.     

Transverse-electric/transverse-magnetic polarization converter using 1D finite biaxial photonic crystal

We show that by using a one-dimensional anisotropic photonic structure, it is possible to realize optical wave polarization conversion by reflection and transmission processes. Thus a single incident S(P) polarized plane wave can produce a single reflected P(S) polarized wave and a single transmitted P(S) polarized wave. This polarization conversion property can be fulfilled with a simple finite superlattice (SL) constituted of anisotropic dielectric materials. We discuss the appropriate choices of the material and geometrical properties to realize such structures. The transmission and reflection coefficients are calculated in the framework of the Green's function method. The amplitude and the polarization characteristics of reflected and transmitted waves are determined as functions of frequency, wave vector k(parallel) (parallel to the interface), and the orientations of the principal axes of the layers constituting the SL. Specific applications of these results are given for a SL consisting of alternating biaxial anisotropic layers NaNO(2)/SbSI sandwiched between two identical semi-infinite isotropic media.     

SH-waves at a corrugated interface between a dry sandy half-space and an anisotropic elastic half-space

Summary A problem of reflection and transmission of a plane SH-wave incident at a corrugated interface between a dry sandy half-space and an anisotropic elastic half-space is investigated. Rayleigh's method of approximation is applied to derive the reflection and transmission coefficients for the first and second order approximation of the corrugation. The expressions for reflection and transmission coefficients for the first order approximation of the corrugation are obtained in closed form for a special type of interface, and the energy partition relation is derived. It is found that these coefficients are proportional to the amplitude of corrugation and are functions of elastic properties of the half-spaces and also of the angle of incidence. Numerical examples illustrating the effects of the sandiness, the anisotropy, the corrugation of the interface, the frequency, and the angle of the incidence on the coefficients are presented.     

Buckling behaviour of laminated beam structures using a higher-order discrete model

A higher-order shear- deformation theory, assuming a non-linear variation for the displacement field, is used to develop a finite-element model to predict the linear buckling behaviour of anisotropic multilaminated or sandwich thick and thin beams. The model is based on a single-layer Lagrangean four-node straight-beam element. It considers stretching and bending in two orthogonal planes. The most common cross-sections and symmetric and asymmetric lay-ups are studied. The good performance of the present element is evident on the prediction of the buckling of several test cases of thin and thick isotropic or anisotropic beam structures. Comparisons show that the model is accurate and versatile.