Scattering by layered structures with rough surfaces: comparison of polarimetric optical scatterometer measurements with theory

A laboratory model of a layered structure with a rough upper surface (a glass microscope slide cut with a diamond saw) is used to obtain optical polarimetric data. Scatterometer measurements were made of all the Mueller matrix elements associated with light scattered in arbitrary directions. (A preliminary measurement of scattering from a smooth opaque gold film on a silicon wafer was used to validate the calculation of the Mueller matrix elements.) These measurements are compared with corresponding analytical solutions based on the full-wave approach. Physical interpretations of the analytical solutions that account for scattering upon reflection and transmission across rough interfaces are given in a companion paper. The agreement between calculations and measurements suggests that the full wave, polarimetric solutions can provide a reliable database for electromagnetic detection of rough surfaces in remote-sensing applications.     

Light scattering from particles located under a rough dielectric layer

A discussion is presented of the effect of roughness on the detectability of subsurface particles by means of the light-scattering method. We have studied the scattering of light by calibrated spheres located under a slightly rough dielectric surface both experimentally and theoretically. In our experiments, the scattering from slightly rough layers with nonresonant particles was dominated by the roughness, and the scattering diagram did not bear any discernible indications of the spheres. However, at resonance, the subsurface particles manifested themselves by an increase in the total scattered intensity and by well-pronounced maxima in the angular dependence of both the scattering diagram and the backscattered intensity. Theoretical calculations show that the angular positions of the maxima in the scattering diagram are essentially determined by the interference of fields scattered by the particles and by the surface, whereas the contribution of the multiple interparticle scattering is negligible. By contrast, the oscillations in the angular dependence of the backscattered intensity are due primarily to the scattering between neighboring spheres.     

Automated high-speed Mueller matrix scatterometer

A new scatterometer-polarimeter is described. It measures the angular distribution of intensity and of the complete Mueller matrix of light scattered by rough surfaces and particle suspensions. The measurement time is 1 s/scattering angle in the present configuration but can be reduced to a few milliseconds with modified electronics. The instrument uses polarization modulation and a Fourier analysis of four detected signals to obtain the 16 Mueller matrix elements. This method is particularly well suited to online, real time, industrial process control involving rough surfaces and large particle suspensions (an arithmetic roughness or particle diameter of >1 mum). Some results are given.     

Remote sensing of penetrable objects buried beneath two-dimensional random rough surfaces by use of the Mueller matrix elements

The modified Mueller matrix elements for electromagnetic scattering from penetrable objects buried under two-dimensional random rough surfaces are investigated. This matrix relates the incident to the scattered waves, and it contains different combinations of the fully polarimetric scattering matrix elements. The statistical average of each Mueller matrix element is computed on the basis of the Monte Carlo simulations by exploiting the speed of the three-dimensional steepest-descent fast multipole method. The numerical results clearly show that relying only on the co-polarized or the cross-polarized intensities or both (i.e., vv, hh, vh, and hv) is not sufficient for sensing the buried objects. However, examining all 16 Mueller matrix elements significantly increases the possibility of detecting these objects. This technique can be used in remote sensing of scatterers buried beneath the rough ground.     

Kirchhoff calculations of the coherent scatter from a series of very rough surfaces

Calculations are presented for the scattering of polarized light from a series of very rough one-dimensional gold-coated surfaces, as determined by the use of the Kirchhoff approximation with geometric shadowing. These surfaces have Gaussian autocorrelation functions with a 1/e width of 3.3 μm and Gaussian probability distributions of height with standard deviation varying between 0.25 and 1.73 μm. Calculations are performed for the scattering of light of wavelength 3.392 μm, so that the validity of the geometric-shadowing approximation and the Kirchhoff approximation itself are open questions. The values of the coherent (or specular) component of the scattered light for the four nonzero elements of the Mueller matrix (which fully describe the polarization properties of the scattered light) are calculated. Comparisons between the calculated results and experimental measurements on surfaces of the same parameters [Knotts and O'Donnell, J. Opt. Soc. Am. A 11, 697 (1994)] show good agreement up to approximately 70° incidence angle.     

Polarized light scattering by microroughness and small defects in dielectric layers

The polarization of light scattered by the surface of a material contains information that can be used to identify the sources of that scatter. Theories for light scattering from interfacial roughness of a dielectric layer and from defects in that dielectric layer are reviewed. Methods for calculating the Mueller matrix or the Stokes vector for scatter from multiple sources and for decomposing a Stokes vector into contributions from two nondepolarizing scattering sources are derived. The theories are evaluated for a specific sample and geometry. Results show that some incident polarizations are more effective than others at discriminating among scattering sources, with s-polarized light being least effective. The polarization of light scattered from interfacial roughness depends upon the relative roughness of the two interfaces and the degree of correlation between the two interfaces. The scattering from defects in the film depends on the depth of the defect and differs from that from any one of the cases of interfacial roughness. The scattering from defects randomly distributed in the film and for small dielectric permittivity variations in the film is also calculated. Experimental results are presented for a 52-nm SiO2 film thermally grown on microrough silicon.     

Computations of the Mueller matrix elements for scattering from layered structures with rough surfaces, with applications to optical detection

A full-wave method is used to evaluate the Mueller matrix elements for scattering from layered structures with random rough surfaces. These provide a database for applications in optical detection over a broad range of rough surface statistical parameters. They can be used to determine the optimal frequencies and incident angles that provide most reliable measurements for optical detection. The elements of the Mueller matrix that are most sensitive to medium parameters of the layered structures can also be identified. Contributions from individual terms of the full-wave solutions are shown to have distinct physical interpretations.     

Angular correlation function of speckle patterns scattered from a one-dimensional rough dielectric film on a glass substrate

We present the experimental results of the angular correlation function of far-field speckle patterns scattered by a one-dimensionally random rough surface of a thin dielectric film on a glass substrate when a polarized beam of light is incident upon the rough surface from vacuum. This surface, which separates the vacuum and the dielectric, is rough enough that only diffused speckles are observed. The experiment for the correlation measurement was set up to make use of a CCD camera to obtain the image of the speckle pattern in the specular direction for each given angle of incidence; the cross-correlation function is then calculated from the digitized images. It is found that the intensity correlation functions exhibit two distinct maxima: one arises from the autocorrelation and the other from the reciprocity condition. It is also found that different scattering processes give rise to quite different correlation functions: multiple-scattering processes produce narrow peaks with secondary maxima and single-scattering processes produce relatively broad peaks.     

Monte Carlo simulation of Stokes vectors of polarized light scattering from two-dimensional random rough surfaces

A Monte Carlo model was established to simulate polarized scattering fields of two-dimensional rough surfaces based on the Kirchhoff approximation. Based on this model, numerical studies of the hemispherical distribution of Stokes vectors of scattered light from dielectric and metal rough surfaces were carried out. These surfaces have Gaussian distributions with correlation length of 3.1?µm and standard deviation varying between 0.1 and 0.6?µm. The results reveal that the V component of metal surfaces has peaks antisymmetric with the incident plane, whereas the V component of dielectric surfaces is almost zero. We consider that this property of the V component would provide a new method which could be used to distinguish the target material.     

Optical Measurement of Large Sphere Diameters by means of Scattering

A simple method is demonstrated to determine the diameters of dielectric spheres from 0.2 to 1.0 mm by observing the scattering of visible light. Theoretical calculations show that there is an approximately linear relationship between the size of the scattering sphere and the number of maxima and minima in the scattered field as a function of angle when the radius in the 200-1000-wavelength region.     

Investigation of rough surfaces and transparent birefringent samples with Mueller-matrix scatterometry

Measurements made with an automated angle-resolved Mueller-matrix scatterometer are described. The instrument uses incident-polarization electro-optical modulation, division-of-amplitude photopolarimetry, and software-implemented Fourier-transform analysis of the detected signals to determine the scattered Mueller matrix of the sample. The measurement time is approximately 1 s per scattering angle. Applications to the control of surface roughness and structure on rough steel sheets (galvanized and uncoated) and of the properties of transparent birefringent optical elements (liquid-crystal devices) are discussed.     

Monte Carlo and Multicomponent Approximation Methods for Vector Radiative Transfer by use of Effective Mueller Matrix Calculations

For single scattering in a turbid medium, the Mueller matrix is the 4 x 4 matrix that multiplies the incident Stokes vector to yield the scattered Stokes vector. This matrix contains all the information that can be obtained from an elastic-scattering system. We have extended this concept to the multiple-scattering domain where we can define an effective Mueller matrix that, when operating on any incident state of light, will yield the output state. We have calculated this matrix using two completely different computational methods and compared the results for several simple two-layer turbid systems separated by a dielectric interface. We have shown that both methods give reliable results and therefore can be used to accurately predict the scattering properties of turbid media.     

Sensitivity of the backscattering Mueller matrix to particle shape and thermodynamic phase

The Mueller matrix (M) corresponding to the phase matrix in the backscattering region (scattering angles ranging from 175 degrees to 180 degrees) is investigated for light scattering at a 0.532-microm wavelength by hexagonal ice crystals, ice spheres, and water droplets. For hexagonal ice crystals we assume three aspect ratios (plates, compact columns, and columns). It is shown that the contour patterns of the backscattering Mueller matrix elements other than M11, M44, M14, and M41 depend on particle geometry; M22 and M33 are particularly sensitive to the aspect ratio of ice crystals. The Mueller matrix for spherical ice particles is different from those for nonspherical ice particles. In addition to discriminating between spherical and nonspherical particles, the Mueller matrix may offer some insight as to cloud thermodynamic phase. The contour patterns for large ice spheres with an effective size of 100 microm are substantially different from those associated with small water droplets with an effective size of 4 microm.     

Virtues of mueller matrix imaging for underwater target detection

We present a theoretical analysis on use of polarized light in the detection of a model target in a scattering and absorbing medium similar to seawater. Monte Carlo numerical simulations are used in the calculation of the effective Mueller matrix which describes the scattering process. A target in the shape of a disk is divided into three regions, each of which has the same albedo but different reduced Mueller matrices. Contrast between various parts of the target and background is analyzed in the images created by ordinary radiance, by various elements of the Mueller matrix, and by certain suitable combinations of these elements. It is shown that the application of polarized light has distinct advantages in target detection and characterization when compared with use of unpolarized light.     

Light scattering by randomly rough isotropic dielectric surfaces

Abstract

An experimental investigation of the angular distribution of the light scattered by randomly rough, two-dimensional, isotropic dielectric surfaces is presented. The surfaces, whose profiles constitute good approximations to Gaussian random processes with Gaussian correlation functions, are fabricated in a photoresist and characterized by means of a mechanical profilometer. The substrates employed in the fabrication of the samples consist of thick parallel plates of filter glass that absorb the incident light and whose refractive index is close to that of the photoresist. This allows us to approximate experimentally a situation in which the light is scattered by a randomly rough interface between two semi-infinite dielectric media, illuminated from the air side. The results display features that can be attributed to multiple scattering. In particular, significant amounts of cross-polarized scattered light, as well as an enhanced backscattering peak, were observed in the scattering measurements.     

Transient source current formulation for the analysis of arbitrarily shaped dielectric/conducting composite structures

In this paper, a novel time-domain integral equation (TDIE) approach is presented to analyze the transient electromagnetic response from three-dimensional dielectric/conducting composite structures. The composite bodies with different materials and conductor structures are considered as autonomous scattering objects. The scattered electromagnetic field is derived by superposition of the individual scatterers. The principal method is based on the application of electric and magnetic field boundary conditions on the surfaces of dielectric and conducting objects while considering the electric currents as unknowns. Triangular patches as well as quadrangular patches model the surfaces of the combined structure. The scattered electromagnetic fields inside and outside the combined structure depend only on the electric surface currents. This improves the post-processing time and complexity as compared with other methods using both equivalent electric and magnetic surface currents. To solve the resultant integral equations, the implicit method of moments is used. In our approach, the internal fields are independent of external currents and vice-versa, reducing the number of the matrix elements by considering them as zero.     

Polarized light scattering by dielectric and metallic spheres on oxidized silicon surfaces

The polarization and intensity of light scattered by polystyrene latex and copper spheres with diameters of approximately 100 nm deposited onto silicon substrates containing various thicknesses of oxide films were measured with 532-nm light. The results are compared with a theory for scattering by a sphere on a surface, originally developed by others [Physica A 137, 209 (1986)] and extended to include coatings on the substrate. Nonlinear least-squares fits of the theory to the observations yield results that were consistent with differential mobility measurements of the particle diameter.     

Mueller matrix measurements of algae with different shape and size distributions

The full Mueller matrix was measured to obtain the polarization state of the scattered light for a variety of algae with different shapes, wall compositions, sizes, and refractive indices. The experimental setup was a multiple laser Mueller matrix ellipsometer, by which measurements were performed for scattering angles from 16° to 160° sampled at every second degree for wavelengths of 473?nm and 532?nm. Previously, the polarization of light scattered from microalgae was investigated only for a few species, and the Mueller matrix was found to have little variation between the species. In our work a total of 11 algal species were investigated, representing diatoms, dinoflagellates, coccolithophorids, green algae, and a cryptophyte. The selection of species was made to obtain high variability in shape, size, cell wall, and refractive index. As in previous investigations, very small variations were found between species for most of the Mueller matrix elements, but noticeable variations were found for M(11), (M(12)+M(21))/2 and (M(33)+M(44))/2.     

Polarized light scattering by dielectric and metallic spheres on silicon wafers

The polarization and intensity of light scattered by monodisperse polystyrene latex and copper spheres, with diameters ranging from 92 to 218 nm, deposited on silicon substrates were measured with 442-, 532-, and 633-nm light. The results are compared with a theory for scattering by a sphere on a surface, originally developed by others [PhysicaA 137,209 (1986)], and extended to include coatings on the sphere and the substrate. The results show that accurate calculation of the scattering of light by a metal sphere requires that the near-field interaction between the sphere and its image be included in acomplete manner. The normal-incidence approximation does not suffice for this interaction, and the existence of any thin oxide layer on the substrate must be included in the calculation.