Rayleigh-Rice Theory Research Articles

Optical Characterization of Gadolinium Fluoride Films Using Universal Dispersion Model

The optical characterization of gadolinium fluoride (GdF3) films is performed in a wide spectral range using heterogeneous data-processing methods (the ellipsometric and spectrophotometric measurements for five samples with thicknesses ranging from 20 to 600 nm are processed simultaneously). The main result of the characterization is the optical constants of GdF3 in the range from far infrared to vacuum ultraviolet, both in the form of a table and in the form of dispersion parameters of the universal dispersion model (UDM). Such reliable data in such a broad spectral range have not been published so far. The GdF3 films exhibit several defects related to the porous polycrystalline structure, namely, surface roughness and a refractive index profile, which complicate the optical characterization. The main complication arises from the volatile adsorbed components, which can partially fill the pores. The presented optical method is based on the application of the UDM for the description of the optical response of GdF3 films with partially filled pores. Using this dispersion model, it is possible to effectively separate the optical response of the host material from the response of the adsorbed components. Several recently published structural and dispersion models are used for optical characterization for the first time. For example, a model of inhomogeneous rough films based on Rayleigh–Rice theory or asymmetric peak approximation with a Voigt profile for the phonon spectra of polycrystalline materials.

Numerical model of light propagation through Fabry-Perot etalons composed of interfaces with non-planar surface topography.

We present a model that calculates optical fields reflected and transmitted by a Fabry-Perot (FP) etalon composed of interfaces with non-planar surface topography. The model uses the Rayleigh-Rice theory, which predicts the fields reflected and transmitted by a single interface, to account for the non-planar surface topography of each interface. The Rayleigh-Rice theory is evaluated iteratively to account for all round trips that light can take within the FP etalon. The model predictions can then be used to compute Interferometer transfer function (ITF)s, by performing wavelength or angle resolved simulations enabling predictions of the bandwidth, peak transmissivity, and sensitivity of FP etalons. The model was validated against the Pseudospectral time-domain (PSTD) method, which resulted in good agreement. Since the model accuracy is expected to reduce as the Root mean square (RMS) of the topographic map increases, the error in the model's predictions was studied as a function of topographic map RMS. Finally, application of the model was exemplified by predicting the impact of roughness on ITFs and computing the changes in FP etalon transmissivity as cavity thickness is modulated by an ultrasonic wave.

Optical characterization of inhomogeneous thin films with randomly rough boundaries exhibiting wide intervals of spatial frequencies.

Results concerning the optical characterization of two inhomogeneous polymer-like thin films deposited by the plasma enhanced chemical vapor deposition onto silicon single crystal substrates are presented. One of these films is deposited onto a smooth silicon surface while the latter film is deposited on a randomly rough silicon surface with a wide interval of spatial frequencies. A combination of variable-angle spectroscopic ellipsometry and spectroscopic reflectometry applied at near-normal incidence are utilized for characterizing both the films. An inhomogeneity of the films is described by the method based on multiple-beam interference of light and method replacing inhomogeneous thin films by multilayer systems. Homogeneous transition layers between the films and substrates are considered. The Campi-Coriasso dispersion model is used to express spectral dependencies of the optical constants of the polymer-like films and transition layers. A combination of the scalar diffraction theory and Rayleigh-Rice theory is used to include boundary roughness into formulae for the optical quantities of the rough polymer-like film. Within the optical characterization, the spectral dependencies of the optical constants at the upper and lower boundaries of both the polymer-like films are determined together with their thickness values and profiles of the optical constants. Roughness parameters are determined for the rough film. The values of the roughness parameters are confirmed by atomic force microscopy. Moreover, the optical constants and thicknesses of both the transition layers are determined. A discussion of the achieved results for both the polymer-like films and transition layers is performed.

GSvit — An open source FDTD solver for realistic nanoscale optics simulations

Surface and volume imperfections can significantly affect the performance of nanoscale or microscale devices used in photonics, optoelectronics or scientific instrumentation. In this article we present an open source software package for Finite-Difference Time-Domain electromagnetic field calculations suitable for calculations on graphics cards. Its special features include handling realistic models of imperfect nanoscale objects, such as treatment of arbitrary geometries including addition of random roughness to any geometrical object. The method is compared to conventional optical approach represented by Rayleigh-Rice theory. Practical applicability is demonstrated using a calculation of variation of field enhancement at proximity of a rough nanoscale antenna and rough particle scattering. It is shown that such approach can be namely useful in the areas where many repeated calculations are necessary, e.g. when studying how the optical response of nanoscale objects can vary when they are rough. Program summaryProgram Title: GSvitCPC Library link to program files:https://doi.org/10.17632/k424zbsxnk.1Licensing provisions: GPLv2Programming language: CNature of problem: If we want to analyse impact of random imperfections, in particular surface roughness, on optical response of nanoscale and microscale objects, we need to run many calculations with different random realisations. GSvit is a general electromagnetic field solver optimized for running such calculations, via fast computing on graphics cards and algorithms for loading of arbitrary data and modification of their geometry to construct randomly rough surfaces and interfaces.Solution method: Finite-Difference Time-Domain method implemented on computer processor and on graphics card, with advanced pre-processing in order to add realistic roughness to 3D objects.Additional comments including restrictions and unusual features: The current version of software, including the documentation and downloadable examples can be found on http://gsvit.net/.

Optics of Inhomogeneous Thin Films with Defects: Application to Optical Characterization

This review paper is devoted to optics of inhomogeneous thin films exhibiting defects consisting in transition layers, overlayers, thickness nonuniformity, boundary roughness and uniaxial anisotropy. The theoretical approaches enabling the inclusion of these defects into formulae expressing the optical quantities of these inhomogeneous thin films are summarized. These approaches are based on the recursive and matrix formalisms for the transition layers and overlayers, averaging of the elements of the Mueller matrix using local thickness distribution or polynomial formulation for the thickness nonuniformity, scalar diffraction theory and Rayleigh-Rice theory or their combination for boundary roughness and Yeh matrix formalism for uniaxial anisotropy. The theoretical results are illustrated using selected examples of the optical characterization of the inhomogeneous polymer-like thin films exhibiting the combination of the transition layers and thickness nonuniformity and inhomogeneous thin films of nonstoichiometric silicon nitride with the combination of boundary roughness and uniaxial anisotropy. This characterization is realized by variable angle spectroscopic ellipsometry and spectroscopic reflectometry. It is shown that using these optical techniques, the complete optical characterization of the mentioned thin films can be performed. Thus, it is presented that the values of all the parameters characterizing these films can be determined.

Optical quantities of a multilayer system with randomly rough boundaries and uniaxial anisotropic media calculated using the Rayleigh–Rice theory and Yeh matrix formalism

In this paper, optical properties of the multilayer system with rough boundaries and uniaxial anisotropic media are calculated using a mathematical model based on the Rayleigh–Rice theory and Yeh matrix formalism. The optical axis is assumed to be oriented perpendicular to the mean plane of the boundaries. The matrix formalism developed by Yeh makes the numerical processing extremely effective. The derived formulae are demonstrated on several numerical examples.

Combination of spectroscopic ellipsometry and spectroscopic reflectometry with including light scattering in the optical characterization of randomly rough silicon surfaces covered by native oxide layers

A combined method of spectroscopic ellipsometry and spectroscopic reflectometry is employed for the optical characterization of randomly rough surfaces of silicon single crystal covered with very thin native oxide layers within spectral range 197–1240 nm (1.0–6.3 eV). It is shown that random roughness of the samples of this system exhibits a wide interval of spatial frequencies so that the Rayleigh–Rice theory, scalar diffraction theory for coherent light and scalar diffraction theory with including light scattering must be used to achieve good fits of experimental data. The ellipsometric data are compatible with the Rayleigh–Rice theory within the entire spectral range while the reflectometric data must be processed separately within two sub-ranges using two different theoretical approaches. In one of the sub-ranges the combination of Rayleigh–Rice theory and scalar diffraction theory for coherent light is utilized for obtaining good fits of the corresponding experimental data of reflectance. Within the latter sub-range the scalar diffraction theory including the influence of light scattering on reflectance enables us to achieve good fits. Further, it is shown that using this combined method the values of basic roughness parameters, i.e. the rms values of heights, autocorrelation lengths and rms values of slopes of roughness irregularities, can be determined together with thickness values of the native oxide layers. A comparison of the results achieved optically with those determined by atomic force microscopy is performed. The assessment of efficiency of spectroscopic ellipsometry and spectroscopic reflectometry in characterizing random roughness with the wide interval of spatial frequencies is also presented.

Efficient method to calculate the optical quantities of multi-layer systems with randomly rough boundaries using the Rayleigh–Rice theory

An efficient and numerically stable method for calculating the optical quantities of multi-layer systems with slightly rough boundaries using the second order Rayleigh–Rice theory is developed. It is assumed that the mean planes of the boundaries are parallel and all the media forming the system are nonmagnetic, isotropic and homogeneous. The perturbation series is formulated using the four-dimensional formalism inspired by the Yeh matrix formalism, but the final result is written using the two-dimensional formalism which is more efficient for the numerical calculations. The final formulae, which are expressed using an arbitrary power spectral density function (PSDF), include the mixing between the p and s polarizations occurring for anisotropic roughness. Although in the general case the calculation of optical quantities requires evaluation of double integrals, it is shown that for the PSDF given by the isotropic Gaussian function some integrals can be calculated analytically and only single integrals have to be evaluated numerically. The random roughness of boundaries is a defect that occurs frequently in practice, and it must be taken into account in the optical characterization and synthesis of thin film systems exhibiting this defect. The presented method is suitable for these purposes, since both of the mentioned applications require methods that are very fast.

Optical quantities of multi-layer systems with randomly rough boundaries calculated using the exact approach of the Rayleigh–Rice theory

In this paper, the exact approach of the Rayleigh–Rice theory enabling us to calculate optical quantities of multi-layer systems with boundaries exhibiting slight random roughness is presented. This approach is exact in the sense that it takes into account the propagation of perturbed electromagnetic fields (waves) among randomly rough boundaries including all cross-correlation and auto-correlation effects. The restriction to the second order of perturbation, which is the lowest order that gives nonzero corrections to coherent waves (obeying the Snell’s law), represents the only approximation used in our calculations. It is assumed that the layers and the substrates are formed by optically homogeneous and isotropic materials. The formulae obtained in the theoretical part are used to investigate the influence of layer thicknesses and roughness parameters on reflectances and associated ellipsometric parameters of the selected numerical examples of a three-layer system. The presented approach represents the generalization of the exact approach for single-layer systems and the improvement of the approximate approach for multi-layer systems published earlier. The exact approach of the RRT has a substantial importance for the optical characterization of multi-layer systems occurring in applied research and optics industry applications.

Optical characterization of randomly microrough surfaces covered with very thin overlayers using effective medium approximation and Rayleigh–Rice theory

The modification of the effective medium approximation for randomly microrough surfaces covered by very thin overlayers based on inhomogeneous fictitious layers is formulated. The numerical analysis of this modification is performed using simulated ellipsometric data calculated using the Rayleigh–Rice theory. The system used to perform this numerical analysis consists of a randomly microrough silicon single crystal surface covered with a SiO2 overlayer. A comparison to the effective medium approximation based on homogeneous fictitious layers is carried out within this numerical analysis. For ellipsometry of the system mentioned above the possibilities and limitations of both the effective medium approximation approaches are discussed. The results obtained by means of the numerical analysis are confirmed by the ellipsometric characterization of two randomly microrough silicon single crystal substrates covered with native oxide overlayers. It is shown that the effective medium approximation approaches for this system exhibit strong deficiencies compared to the Rayleigh–Rice theory. The practical consequences implied by these results are presented. The results concerning the random microroughness are verified by means of measurements performed using atomic force microscopy.

Ellipsometric and reflectometric characterization of thin films exhibiting thickness non-uniformity and boundary roughness

In this paper epitaxial ZnSe thin films prepared by molecular beam epitaxy onto GaAs single crystal substrates exhibiting two defects, i.e. boundary roughness and thickness non-uniformity, are optically characterized using a combination of spectroscopic ellipsometry and near-normal spectroscopic reflectometry. The influence of boundary roughness is included into optical quantity formulae by the combination of the scalar diffraction theory and Rayleigh–Rice theory. Very thin overalyers modelled by rough thin films with identically rough boundaries are taken into account on the upper boundaries of the ZnSe thin films. Two approximations are used to express the local reflection coefficient of the rough ZnSe thin films covered with the overlayers within combination of both the theories. Thickness non-uniformity is incorporated by means of averaging the elements of the unnormalized Mueller matrices. The universal dispersion model of the optical constants of the ZnSe thin films based on parametrization of the joint density of electronic states is used. The spectral dependencies of the optical constants of the ZnSe thin films are determined within the wide spectral range (0.12–8.7eV). Moreover, the mean thickness of the ZnSe thin films and thickness of overlayers are determined together with the other structural parameters characterizing the defects. The values of roughness parameters, determined by the optical method, are verified by a comparison with results achieved by atomic force microscopy. It is also shown that the approximations of the local reflection coefficient presented are usable for processing the experimental data.

Measurement of thickness distribution, optical constants, and roughness parameters of rough nonuniform ZnSe thin films.

Epitaxial ZnSe thin films exhibiting two important defects, i.e., boundary roughness and thickness nonuniformity, prepared on GaAs substrates, are optically characterized using a combination of variable-angle spectroscopic ellipsometry, spectroscopic near-normal reflectometry, and imaging spectroscopic reflectometry (ISR). The influence of boundary roughness is incorporated into optical quantity formulas by the Rayleigh-Rice theory. Thickness nonuniformity is included using averaging of the unnormalized Mueller matrices. The dispersion model of the optical constants of the ZnSe films is based on parametrization of the joint density of electronic states. Very thin overlayers represented by thin films with identically rough boundaries are taken into account on the upper boundaries of the ZnSe films. Standard optical techniques are used to determine the spectral dependencies of the optical constants of the ZnSe films, together with the parameters of roughness and thickness nonuniformity. ISR is then used to find the maps of the local thickness and local rms value of height irregularities. The values of roughness parameters, determined using the standard techniques and ISR, are verified by a comparison with results obtained by atomic force microscopy.

Improved combination of scalar diffraction theory and Rayleigh–Rice theory and its application to spectroscopic ellipsometry of randomly rough surfaces

Expressions for ellipsometric quantities and reflectance presented are based on a heuristic combination of the Rayleigh–Rice theory and the scalar diffraction theory. The latter takes into account the local slopes and shadowing. The second-order Rayleigh–Rice theory is used to express the local electric field on the rough surface and the obtained expressions are then used in the scalar diffraction theory instead of the expressions corresponding to a smooth surface. A numeric method of evaluation of the quadruple integral resulting from this combination of the two theories is developed, utilising a Gauss-like quadrature. The efficiency of the formulae is illustrated by optical characterisation of rough silicon and gallium arsenide surfaces created using anodic and thermal oxidation, respectively, and covered with native oxide layers. The optical characterisation employs variable-angle spectroscopic ellipsometry and spectroscopic reflectometry.

BRDF models for accurate and efficient rendering of glossy surfaces

This article presents two new parametric models of the Bidirectional Reflectance Distribution Function (BRDF), one inspired by the Rayleigh-Rice theory for light scattering from optically smooth surfaces, and one inspired by micro-facet theory. The models represent scattering from a wide range of glossy surface types with high accuracy. In particular, they enable representation of types of surface scattering which previous parametric models have had trouble modeling accurately. In a study of the scattering behavior of measured reflectance data, we investigate what key properties are needed for a model to accurately represent scattering from glossy surfaces. We investigate different parametrizations and how well they match the behavior of measured BRDFs. We also examine the scattering curves which are represented in parametric models by different distribution functions. Based on the insights gained from the study, the new models are designed to provide accurate fittings to the measured data. Importance sampling schemes are developed for the new models, enabling direct use in existing production pipelines. In the resulting renderings we show that the visual quality achieved by the models matches that of the measured data.

Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles

Scattering effects from microtopographic surface roughness are merely nonparaxial diffraction phenomena resulting from random phase variations in the reflected or transmitted wavefront. Rayleigh-Rice, Beckmann-Kirchhoff. or Harvey-Shack surface scatter theories are commonly used to predict surface scatter effects. Smooth-surface and/or paraxial approximations have severely limited the range of applicability of each of the above theoretical treatments. A recent linear systems formulation of nonparaxial scalar diffraction theory applied to surface scatter phenomena resulted first in an empirically modified Beckmann-Kirchhoff surface scatter model, then a generalized Harvey-Shack theory that produces accurate results for rougher surfaces than the Rayleigh-Rice theory and for larger incident and scattered angles than the classical Beckmann-Kirchhoff and the original Harvey-Shack theories. These new developments simplify the analysis and understanding of nonintuitive scattering behavior from rough surfaces illuminated at arbitrary incident angles.

Goos-Hänchen shift for a rough metallic mirror

We investigate experimentally the dependence of the Goos-Hänchen shift on the surface properties of an air-metal interface. The shift depends on the microscopic roughness of the metal surface but it is insensitive to the large-scale variations associated with surface non-flatness. Both an effective medium model of roughness and the Rayleigh-Rice theory of scattering are used to interpret the observed phenomenon.

Optical anisotropy induced by ion bombardment of Ag(001)

Grazing incidence ion bombardment results in the formation of nanoripples that induce an anisotropic optical reflection The evolution of the reflectance anisotropy has been monitored in situ with reflectance anisotropy spectroscopy. The Rayleigh-Rice theory (RRT) has been used to analyze the optical spectra quantitatively and provides the evolution of the average ripple period and root-mean-squared surface roughness. After an incipient phase, both the increase in the periodicity and the roughness vary roughly with the square root of the sputter time. Additional high-resolution low-energy electron diffraction (HR-LEED) measurements have been performed to characterize details of the average structure created by ion bombardment.

Influence of cross-correlation effects on the optical quantities of rough films

Within the Rayleigh-Rice theory the influence of layer boundary roughness on coherently reflected light is expressed using very complex formulae. Therefore we deal with the simplification of these formulae by employing an approximation based on neglecting cross-correlation effects between both the rough boundaries. It is shown that if the mean distance of the boundaries (mean thickness) is sufficiently large in comparison with the lateral dimensions of the roughness it is possible to describe the individual boundaries of the layers by matrices corresponding to isolated rough surfaces. This fact enables us to simplify the formulae for the optical quantities in a substantial way, which also simplifies the numerical calculation needed for the inverse problem. This statement is illustrated by means of a numerical analysis simulating ellipsometric and reflectometric data of rough silicon dioxide layers placed onto silicon single crystal substrates.

Optical characterization of ultrananocrystalline diamond films

Optical properties of the ultrananocrystalline diamond films were studied by multi-sample method based on the combination of variable angle spectroscopic ellipsometry and spectroscopic reflectometry applied in the range 0.6–6.5 eV. The films were deposited by PECVD in a conventional bell jar (ASTeX type) reactor using dual frequency discharge, microwave cavity plasma and radio frequency plasma inducing dc self-bias at a substrate holder. The optical model of the samples included a surface roughness described by the Rayleigh–Rice theory and a refractive index profile in which Drude approximation was used. The results conformed with the present understanding of the polycrystalline diamond growth on the silicon substrate because the existence of silicon carbide and amorphous hydrogenated carbon film between the silicon substrate and nucleation layer was proved.

Influence of lateral dimensions of the irregularities on the optical quantities of rough surfaces

In this paper the influence of lateral dimensions of the irregularities on the optical quantities, i.e. on reflectance, transmittance, scattering losses and ellipsometric ratio of randomly rough surfaces, is discussed in detail. This discussion is based on theoretical (simulated) data calculated using the Rayleigh–Rice theory (RRT). It is shown that the optical quantities mentioned are strongly dependent on the quantities characterizing the lateral dimensions of the irregularities, e.g. on the autocorrelation length. The simulated data are also used to investigate the possibilities and limitations of the scalar diffraction theory (SDT) and effective medium approximation (EMA) in treating the experimental data in practice. On the basis of these theoretical studies it is, moreover, shown that a combination of the SDT and EMA can be employed as a reasonable approximation for describing the interaction of light with randomly rough surfaces. Thus, from the results presented in this paper, it is evident that this combination could be a promising treatment for experimental data corresponding to the optical quantities of randomly rough surfaces.