Modeling light scattered from and transmitted through dielectric periodic structures on a substrate

Light scattering and transmission by rough surfaces are of considerable interest in a variety of applications including remote sensing and characterization of surfaces. In this work, the finite-difference time-domain technique is applied to calculate the scattered and transmitted electromagnetic fields of an infinite periodic rough surface. The elements of the Mueller matrix for scattered light are calculated by an integral of the near fields over a significant number of periods of the surface. The normalized Mueller matrix elements of the scattered light and the spatial distribution of the transmitted flux for a monolayer of micrometer-sized dielectric spheres on a silicon substrate are presented. The numerical results show that the nonzero Mueller matrix elements for scattering from a surface consisting of a monolayer of dielectric spheres on a silicon substrate have specific maxima at some scattering angles. These maxima may be used in the characterization of features of the surface. For light transmitted through the monolayer of spheres, our results show that the transmitted energy focuses around the ray passing through centers of the spheres. At other locations, the transmitted flux is very small. Therefore, micrometer-sized dielectric spheres might be placed on a semiconductor surface to burn nanometer-sized holes in a layer using laser pulses. The method may also be useful in the assembly of periodic microstructures on surfaces.     

Mueller matrix imaging of targets in turbid media: effect of the volume scattering function

Detecting objects in turbid media by use of just radiance signals has been a subject of study for many years. The use of Mueller matrix imaging methods has only recently been used as a tool for target detection. We will show not only that can targets still be detected by Mueller matrix methods even after their detection has escaped normal radiance schemes but also that their surface features can also still be distinguished. We will also show how the shape of the volume scattering function as well as the target and medium albedo strongly influences various elements of the Mueller matrix. One of the more interesting features of Mueller matrix imaging is that the diagonal elements are sensitive to perturbations in the environment surrounding the target. This implies that targets can be detected far beyond their geometric cross section. The methods presented here will have applications to submersible object detection, remote sensing in the atmosphere, and the detection of inhomogeneities in tissue.     

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.     

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.     

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.     

Neural network pattern recognition by means of differential absorption Mueller matrix spectroscopy

Artificial neural network systems were built for detecting amino acids, sugars, and other solid organic matter by pattern recognition of their polarized light scattering signatures in the form of a Mueller matrix. Backward-error propagation and adaptive gradient descent methods perform network training. The product of the training is a weight matrix that, when applied as a filter, discerns the presence of the analytes on the basis of their cued susceptive Mueller matrix difference elements. This filter function can be implemented as a software or a hardware module to a future differential absorption Mueller matrix spectrometer.     

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.     

Mueller matrices and information derived from linear polarization lidar measurements: theory

A Mueller matrix M is developed for a single-scattering process such that G(theta, phi) = T (phi(a))M T (phi(p))u, where u is the incident irradiance Stokes vector transmitted through a linear polarizer at azimuthal angle phi(p), with transmission Mueller matrix T (phi(p)), and G(theta, phi) is the polarized irradiance Stokes vector measured by a detector with a field of view F, placed after an analyzer with transmission Mueller matrix T (phi(a)) at angle phi(a). The Mueller matrix M is a function of the Mueller matrix S (theta) of the scattering medium, the scattering angle (theta, phi), and the detector field of view F. The Mueller matrixM is derived for backscattering and forward scattering, along with equations for the detector polarized irradiance measurements (e.g., cross polarization and copolarization) and the depolarization ratio. The information that can be derived from the Mueller matrix M on the scattering Mueller matrixS (theta) is limited because the detector integrates the cone of incoming radiance over a range of azimuths of 2pi for forward scattering and backscattering. However, all nine Mueller matrix elements that affect linearly polarized radiation can be derived if a spatial filter in the form of a pie-slice slit is placed in the focal plane of the detector and azimuthally dependent polarized measurements and azimuthally integrated polarized measurements are combined.     

Treatment of polarization in laser remote sensing of ocean water

We review existing experimental data and methods for calculating the Mueller matrix of ocean water for use as input in a simulation model applicable to laser remote sensing. Calculations of the Mueller matrix are made for scattering media of different refractive indices, shapes, and size distributions. Dependencies of the backscattering depolarization ratio as a function of the particle refractive index are presented, and we demonstrate the potential importance of polarization in bathymetric sensing.     

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.     

Polarization of specular reflection and near-specular scattering by a rough surface

Polarization of specular reflection and near-specular scattering (NSS) by a randomly rough surface is investigated by the use of a Mueller matrix formulation. The collective effect by a rough surface on the average specular field results in reflectance loss and polarization, which can be explained by an effective medium theory. Effects of random NSS can be represented by a scattering matrix that is partially coherent and polarized. The incoherent and unpolarized part of scattering causes depolarization, and the coherent and polarized parts of scattering change the apparent polarization properties of specular reflection. Results of a simulation and least-squares fit of ellipsometric data to the models including the NSS effect, for a black anodized aluminum sample, are presented. Simultaneous least-squares fits for both ellipsometric data and reflectance data at multiple angles of incidence at three different wavelengths gave approximately the same rms roughness, which agrees with the profilometric values reported previously.     

Real-time measurement of the polarization transfer function

We present a simple method for measuring the Mueller matrix associated with a scattering medium. Without involving moving parts, four input states of polarization are generated sequentially, and for each of them all four Stokes vector parameters are simultaneously measured for the complete determination of the Mueller matrix. Two liquid-crystal variable retarders are used for controlling the input state of polarization, whereas the measurement of the state of polarization involves phase modulation with a single-pass photoelastic modulator, and Fourier analysis in two polarization channels. The setup is controlled by a computer, allowing for real-time measurement of the Mueller matrix. The method is tested on standard elements such as polarizers and quarter-wave plates, as well as on inhomogeneous particulate systems.     

Polarization of transmission scattering simulated by using a multiple-facets model

A Mueller matrix for scattering by a rough plane surface of a glass hemisphere was simulated by using a micro-facet model. The algorithms are formulated in vector representation in terms of the input and output directions. The single-facet scattering simulation used the results of the Kirchhoff integral for medium rough surfaces with exponential height distribution. Scatterings by two or more facets were also simulated. For a fixed angle between the incident and the detection directions, the transmission scattering and its polarization properties were symmetric when plotted against the off-specular incident angle. The single-facet model generated no depolarization or polarization change. When double-facet scattering was included, polarizations were changed appreciably while depolarization was still very small. Depolarization increased appreciably when scattering by higher orders was included. The simulated results that include all orders of scattering fit excellently to the measured scattering transmittance and its polarization and depolarization.     

Laboratory studies of angle- and polarization-dependent light scattering in sea ice

The angle- and polarization-dependent light scattering were measured for oriented first-year and multiyear sea ice taken from the Chukchi Sea near Pt. Barrow, Alaska. The entire Mueller matrix for these samples was determined at 532 nm. Mueller matrices were also determined for artificially grown saline ice samples and melted samples of the respective ice types. Phase functions for thin-slab samples are qualitatively consistent with calculations for scattering from brine inclusions in a solid ice medium and depend strongly on the shape of the scattering sample. Small orientation-dependent effects are observed for scattering from oriented sea ice. A simple model is used to describe qualitatively some features of the measured sea ice Mueller matrices. This model combines the effects of scattering from spherical inhomogeneities and the intrinsic birefringence of pure water ice. A set of Mueller matrix inequalities is presented and used to obtain physical insight into the measurement results.     

Measurement of the Mueller matrix for ocean water

The normalized light scattering polarization matrix has been measured for ocean water using an electrooptic light scattering polarimeter. Measurements were done on samples from the Atlantic and Pacific Oceans and the Gulf of Mexico. The polarization effects in the matrices were found to have, in general, a form which is similar to polarization effects in the Rayleigh scattering approximation; for example, all off-diagonal matrix elements except S12 and S21 were zero. Mueller matrix elements were calculated using a Mie computer code and compared to the measured matrices for ocean water. A simple one-component distribution was found to produce a reasonably good fit.     

Monte Carlo simulations of the diffuse backscattering mueller matrix for highly scattering media

We have developed a Monte Carlo algorithm that computes all two-dimensional elements of the diffuse backscattering Mueller matrix for highly scattering media. Using the Stokes-Mueller formalism and scattering amplitudes calculated with Mie theory, we are able to consider polarization-dependent photon propagation in highly scattering media, including linearly and circularly polarized light. The numerically determined matrix elements are compared with experimental data for different particle sizes and show good agreement in both azimuthal and radial direction.     

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.     

Structure of a general pure Mueller matrix

Changes in the radiance and state of polarization of a beam of radiation can often be described by means of a pure Mueller matrix. Such a 4 × 4 matrix transforms Stokes parameters and can be expressed in terms of the elements of a 2 × 2 Jones matrix. Relations between the two types of matrix are discussed. Explicit expressions are given for changes of a pure Mueller matrix that are caused by certain elementary changes of its Jones matrix, such as when its transpose, complex conjugate, or Hermitian conjugate are taken. It is shown that every pure Mueller matrix has a simple and elegant structure, which is embodied by interrelations that involve either only squares of the elements or only products of different elements. All possible interrelations for the elements of a general pure Mueller matrix are derived from this simple structure.     

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.