Scattering from infinitely sloped surfaces by use of the Kirchhoff approximation

A simple reformulation of the double-scatter Kirchhoff approximation is presented to extend the use of this calculation method to infinitely sloped surfaces. Examples are presented for square grooves and lines on perfectly conducting surfaces. The results presented show the accuracy of the method and the errors produced by not including higher-order scattering in the calculations.     

Backscattering measurements from double-scale randomly rough surfaces

We present experimental measurements of light backscattered from double-scale randomly rough surfaces (oceanlike surfaces) with different statistical parameters illuminated at small and large angles of incidence. The surfaces are composed of a small-scale roughness superimposed on a slowly (large-scale) varying surface. The large-scale surfaces are diamond-machined periodic surfaces made on aluminum substrates and have either a sinusoidal or a Stokes wave profile. The small-scale roughness is added with lithographic techniques, and the surfaces are then gold coated. For a linearly polarized incident beam, it is found that the backscattered light is strongly depolarized mainly at small angles of incidence and strong shadowing effects are present for large angles of incidence (θ(inc) > 60°).     

Diffuse and specular reflectance from rough surfaces

We present a reflection model for isotropic rough surfaces that have both specular and diffuse components. The surface is assumed to have a normal distribution of heights. Parameters of the model are the surface roughness given by the rms slope, the albedo, and the balance between diffuse and specular reflection. The effect of roughness on diffuse reflection is taken into account, instead of our modeling this component as a constant Lambertian term. The model includes geometrical effects such as masking and shadowing. The model is compared with experimental data obtained from goniophotometric measurements on samples of tiles and bricks. The model fits well to samples with very different reflection properties. Measurements of the sample profiles performed with a laser profilometer to determine the rms slope show that the assumed surface model is realistic. The model could therefore be used in machine vision and computer graphics to approximate reflection characteristics of surfaces. It could also be used to predict the texture of surfaces as a function of illumination and viewing angles.     

Large-angle in-plane light scattering from rough surfaces

An in-plane light scattering setup that is capable of measuring large azimuthal scattering angles is presented. This type of measurement makes it easier to probe large k(parallel) at a fixed k(perpendicular) value (k(parallel) and k(perpendicular) are momentum transfer vectors parallel and perpendicular to the surface, respectively). Therefore the system allows us to explore small lateral scale and large vertical roughness (approximately lambda, the wavelength of the probe beam) of a rough surface. In-plane intensity measurements from a rough backside Si wafer and a Cu thin-film surface are reported. The structure factor that is related to surface roughness parameters is obtained from the measured in-plane intensity profiles. Both scalar (Beckmann-Kirchhoff) and vector (Rayleigh-Rice) theories have been applied to interpret the experimental data. The roughness parameters obtained from the scattering measurements are compared with those measured by atomic-force microscopy.     

Single-scatter vector-wave scattering from surfaces with infinite slopes using the Kirchhoff approximation

This paper presents a new formulation of the 3D Kirchhoff approximation that allows calculation of the scattering of vector waves from 2D rough surfaces containing structures with infinite slopes. This type of surface has applications, for example, in remote sensing and in testing or imaging of printed circuits. Some preliminary calculations for rectangular-shaped grooves in a plane are presented for the 2D surface method and are compared with the equivalent 1D surface calculations for the Kirchhoff and integral equation methods. Good agreement is found between the methods.     

Monte Carlo simulation of the field back-scattered from rough surfaces

A novel approach for the simulation of the field back-scattered from a rough surface is presented. It takes into account polarization and multiple scattering events on the surface, as well as diffraction effects. The validity and usefulness of this simulation is demonstrated in the case of surface topology measurement.     

Characterization of the basic statistical properties of very rough surfaces of transparent solids by immersion shearing interferometry

A new shearing-interferometry method is used to characterize the basic statistical properties of very rough surfaces of transparent solids. The mean level of these rough surfaces can be curved or flat. Using this method, one evaluates the root-mean-square values (standard deviations) of the heights and slopes of the surface irregularities. Moreover the values of autocorrelation lengths, autocorrelation functions, and one-dimensional height-distribution functions are also determined. The method is applied to a statistical analysis of rough flat or curved glass surfaces. The results from this method agree very well with those obtained with a stylus-type surface instrument and speckle contrast method.     

Second-order perturbation theory for scattering from heterogeneous rough surfaces

We propose a model to calculate scattering from inhomogeneous three-dimensional, rough surfaces on top of a stratified medium. The roughness is made up of an ensemble of deposits with various shapes and permittivities whose heights remain small with respect to the wavelength of the incident light. This geometry is encountered in the remote sensing of soil surfaces, or in optics wherever there are contaminated planar components. Starting from a volume-integral equation involving the Green's tensor of the stratified medium, we derive a height-perturbative expansion up to second-order. Our formulation, which depends explicitly on the profiles of each deposit and on the Fresnel coefficients of the layered substrate, accounts for double-scattering events and permits an evaluation of depolarization in the plane of incidence. Comparisons with rigorous calculations in the simplified case of two-dimensional geometries are presented. It is shown that the second-order scattering term can be much more important for heterogeneous surfaces than for their homogeneous counterparts.     

Experimental light-scattering measurements from large-scale composite randomly rough surfaces

We present experimental measurements of the angular distribution of light scattered from large-scale composite randomly rough surfaces (oceanlike surfaces) with different statistical parameters illuminated at small and large angles of incidence. The surfaces are composed of a small-scale roughness superimposed on a slowly (large-scale) varying surface. The large-scale surfaces are diamond-machined periodic surfaces made on aluminum substrates and have either a sinusoidal or a Stokes wave profile. The small-scale roughness is added with microlithographic techniques, and the surfaces are then gold coated. For a linearly polarized incident beam, it is found that the diffusely scattered light is strongly depolarized and that its pattern is rather different for each large-scale surface profile. Enhanced backscattering is also observed.     

Speckle size of light scattered from slightly rough cylindrical surfaces

This research is an extension of the optical method of quality control presented in a previous paper [Appl. Opt. 39, 5811 (2000)] to the case of slightly rough cylindrical surfaces. Applying the Kirchhoff scalar diffraction theory yields an analytical expression of the autocorrelation function of the intensity scattered from slightly rough cylindrical surfaces. This function, which is related to speckle size and shape, is shown to depend on the surface correlation length, unlike for plane surfaces for which the speckle depends on the illuminated area only. The theoretical expression is compared with that for the speckle produced by the light scattered from a cylindrical bearing and from various high-quality wires, showing that the method allows the correlation lengths of high-quality cylindrical surfaces to be determined.     

Near-field intensity distribution of waves scattered from slightly rough surfaces

The near fields of electromagnetic waves scattered from two-dimensional slightly rough surfaces are studied by using the stochastic functional approach. The correlation coefficient between the surface and the intensity of the scattered-wave field is investigated to estimate the fidelity of near-field intensity images. We show that the fidelity depends on both the polarization and the angle of incidence and that high fidelity can be obtained by a TM-polarized incident wave whose incident angle is not close to the critical angle of the total reflection.     

Segmentation of rough surfaces using a polarization imaging system

A polarimetric vision system yielding a roughness-segmentation-based image is described. As a general principle, the bidirectional reflectance distribution function (BRDF) of a surface is assumed to be related to its local irregularities, i.e., its roughness. This BRDF is seen as the sum of a specular and a diffuse component. In this paper we propose to introduce polarization measurements in order to estimate this roughness parameter without requiring any assumption for the model of the diffuse component, nor is diffuse-specular separation required. Moreover, with the proposed method, the refractive indices of the observed objects are estimated at each pixel. Examples are given for quality control applications.     

4He atom reflection from rough liquid4He surfaces

When an atom is evaporated from or added to a free liquid surface there is a density perturbation of the surface. We have detected this surface spoiling using a⁴He atomic beam at glancing angles to the surface. With a perfect free liquid surface some of the atoms reflect specularly and some condense. We find that the specular reflection coefficient decreases as the surface is spoiled by another beam of⁴He atoms. The degree of spoiling as measured by the decrease in reflectivity, is initially proportional to the spoiling beam flux, but at higher fluxes the spoiling saturates. A phenomenological model is developed to describe this behaviour.     

Scattering from very rough layers under the geometric optics approximation: further investigation

Scattering from very rough homogeneous layers is studied in the high-frequency limit (under the geometric optics approximation) by taking the shadowing effect into account. To do so, the iterated Kirchhoff approximation, recently developed by Pinel et al. [Waves Random Complex Media17, 283 (2007)] and reduced to the geometric optics approximation, is used and investigated in more detail. The contributions from the higher orders of scattering inside the rough layer are calculated under the iterated Kirchhoff approximation. The method can be applied to rough layers of either very rough or perfectly flat lower interfaces, separating either lossless or lossy media. The results are compared with the PILE (propagation-inside-layer expansion) method, recently developed by Déchamps et al. [J. Opt. Soc. Am. A23, 359 (2006)], and accelerated by the forward-backward method with spectral acceleration. They highlight that there is very good agreement between the developed method and the reference numerical method for all scattering orders and that the method can be applied to root-mean-square (RMS) heights at least down to 0.25lambda.     

Study of speckle size of light scattered from cylindrical rough surfaces

The autocorrelation function of the intensity scattered from cylindrical rough surfaces is analytically obtained with the Kirchhoff scalar diffraction theory. It is shown that, in contrast to the case in which planar rough surfaces scatter radiation, this function, related to the speckle size, depends on the statistical parameters that characterize the surface and on the scattering direction. This result suggests a new, to our knowledge, optical method that can be applied to the characterization of cylindrical rough surfaces, such as in on-line quality assessment, in manufacturing processes. The calculated theoretical expression was tested, showing good agreement with experiments.     

Characterization of random rough surfaces from scattered intensities by neural networks

Abstract

Optical scatterometry, a non-invasive characterization method, is used to infer the statistical properties of random rough surfaces. The Gaussian model with rms-roughness [sgrave] and correlation length σ is considered in this paper but the employed technique is applicable to any representation of random rough surfaces. Surfaces with wide ranges of Λ and σ, up to 5 wavelengths (λ), are characterized with neural networks. Two models are used: self-organizing map (SOM) for rough classification and multi-layer perceptron (MLP) for quantitative estimation with nonlinear regression. Models infer Λ and σ from scattering, thus involving the inverse problem. The intensities are calculated with the exact electromagnetic theory, which enables a wide range of parameters. The most widely known neural network model in practise is SOM, which we use to organize samples into discrete classes with resolution ΔΛ = Δσ = 0.5λ. The more advanced MLP model is trained for optimal behaviour by providing it with known parts of input (scattering) and output (surface parameters). We show that a small amount of data is sufficient for an excellent accuracy on the order of 0.3λ and 0.15λ for estimating Λ and σ, respectively.