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.     

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.     

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.     

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.     

Phase-modulated mueller ellipsometry characterization of scattering by latex sphere suspensions

Phase-modulated Mueller ellipsometry (PMME) is used to probe scattering by suspensions of polystyrene latex spheres, with particle diameters ranging from 400 nm to 3 mum. PMME allows simultaneous measurement of the 16 coefficients of the Mueller matrix. Furthermore PMME measurements can easily be carried out owing to a calibration procedure implemented in a scattering configuration. The measurements performed on low concentrations show good agreement with Mie theory. Moreover size distribution could be obtained with a least-squares method based on a genetic fit algorithm. Experimental evidence of multiple scattering on PMME measurements is also presented.     

Coherent detection of enhanced back-scatter from rough surfaces and particle suspensions

Abstract

Coherent detection has been used to measure enhanced back-scatter from a range of rough surfaces and particle suspensions. Strong enhancement peaks have been observed from suspensions of monodisperse spheres at concentrations as low as 0.2%. This low-density regime has not been accessible to earlier (direct-detection) techniques. The possibility of using measurements of the enhancement peak to characterize rough surfaces or to size particles is discussed and the cross-polar returns from a tutorial target are also measured.     

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.     

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.     

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.     

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.     

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.     

Rapid optically based measurements of diameter and length for spherical or rod-shaped bacteria in vivo

The application in light scattering of the Mueller matrix ratio (S34)/(S11) for determining average particle size is extended to a large size parameter range for spherical or randomly oriented rod-shaped particles such as micro-organisms. It is shown that combining the graph of this ratio with a Coulter counter measurement of particle volume gives results in agreement with microscopic measurements. Thus this combination provides a method to measure particle diameter and width simultaneously in real time for elongated particles such as bacteria, which are measured in vivo with this method. An approximate empirical formula is developed to estimate the motion of the extrema in the graph of the oscillating matrix ratio as size changes occur. This formula is also shown to be consistent with wavelength changes.     

Mueller-matrix ellipsometry using the division-of-amplitude photopolarimeter: a study of depolarization effects

A fully automated Mueller-matrix ellipsometer with a division-of-amplitude photopolarimeter as the polarization-state detector is described. This device achieves Mueller-matrix ellipsometry by measuring the Stokes parameters of reflected light as a function of the fast axis C of a quarter-wave retarder, which, in combination with a fixed linear polarizer, determines the polarization state of incident light. The reflected Stokes parameters were Fourier analyzed to give the 16 elements of the Mueller matrix. We investigated depolarization of polarized light on reflection from rough, heterogeneous, and anisotropic surfaces by obtaining measurements on rolled aluminum and plant leaves. The results demonstrate (1) a variation of degree of polarization of reflected light with the input polarization state, (2) the precision with which the measured matrices describe the depolarization results, (3) effects of surface anisotropy (rolling direction) on depolarization and cross polarization by reflection from aluminum surfaces, and (4) large values and differences in the depolarization effects from conifer and deciduous leaves. Depolarization of light reflected by the aluminum surfaces was most sensitive to the angle between the plane of incidence and the rolling direction when the incident Stokes parameters S(1), S(2), and S(3) were equal.     

Dynamic response of a fast near infra-red Mueller matrix ellipsometer

The dynamic response of a near infra-red ferroelectric liquid crystal-based Mueller matrix ellipsometer (NIR FLC-MME) is presented. A time-dependent simulation model, using the measured time response of the individual FLCs, is used to describe the measured temporal response. Furthermore, the impulse response of the detector and the pre-amplifier is characterized and included in the simulation model. The measured time-dependent intensity response of the MME is reproduced in simulations, and it is concluded that the switching time of the FLCs is the limiting factor for the Mueller matrix measurement time of the FLC-based MME. Based on measurements and simulations, our FLC-based NIR-MME system is estimated to operate at the maximum speed of approximately 16 ms per Mueller matrix measurement. The FLC-MME may be operated several times faster, since the switching time of the crystals depends on the individual crystal being switched, and to what state it is switched. As a demonstration, the measured temporal response of the Mueller matrix and the retardance of a thick liquid crystal variable retarder upon changing state is demonstrated.     

Irreversible adsorption of gold nanospheres on fiber optical tapers and microspheres

We study the adsorption of gold nanospheres onto cylindrical and spherical glass surfaces from quiescent particle suspensions. The surfaces consist of tapers and microspheres fabricated from optical fibers and were coated with a polycation, enabling irreversible nanosphere adsorption. Our results fit well with theory, which predicts that particle adsorption rates depend strongly on surface geometry and can exceed the planar surface deposition rate by over two orders of magnitude when particle diffusion length is large compared to surface curvature. This is particularly important for plasmonic sensors and other devices fabricated by depositing nanoparticles from suspensions onto surfaces with non-trivial geometries.     

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.     

Scattering matrices for large ice crystal particles

The problem of light scattering by ice crystal particles whose sizes are essentially larger than the incident wavelength is divided into two parts. First, the scattered field is represented as a set of plane-parallel outgoing beams in the near zone of the particle. Then, in the far zone the scattered field is represented as a result of both diffraction and interference of these beams within the framework of physical optics. A proper ray-tracing algorithm for calculation of the amplitude (Jones) scattering matrix is developed and applied. For large particles, a number of reduced Mueller matrices are introduced and discussed, since the pure Mueller matrix obtained from the Jones matrix becomes a rather cumbersome and quickly oscillating value. Backscattering by hexagonal ice crystals, including polarization properties, is considered in detail.     

Characteristic properties of Mueller matrices

A complete and minimum set of necessary and sufficient conditions for a real 4 x 4 matrix to be a physical Mueller matrix is obtained. An additional condition is presented to complete the set of known conditions, namely, the four conditions obtained from the nonnegativity of the eigenvalues of the Hermitian matrix H associated with a Mueller matrix M and the transmittance condition. Using the properties of H, a demonstration is also presented of Tr(M(T)M) = 4m(2)00 as being a necessary and sufficient condition for a physical Mueller matrix to be a pure Mueller matrix.     

On the algebraic characterization of a Mueller matrix in polarization optics. II. Necessary and sufficient conditions for Jones-derived Mueller matrices

Abstract

We show that every Mueller matrix, that is a real 4 × 4 matrix M which transforms Stokes vectors into Stokes vectors, may be factored as M = L ₂ KL ₁ where L ₁ and L ₂ are orthochronous proper Lorentz matrices and K is a canonical Mueller matrix having only two different forms, namely a diagonal form for type-I Mueller matrices and a non-diagonal form (with only one non-zero off-diagonal element) for type-II Mueller matrices. Using the general forms of Mueller matrices so derived, we then obtain the necessary and sufficient conditions for a Mueller matrix M to be Jones derived. These conditions for Jones derivability, unlike the Cloude conditions which are expressed in terms of the eigenvalues of the Hermitian coherency matrix T associated with M, characterize a Jones-derived matrix M through the G eigenvalues and G eigenvectors of the real symmetric N matrix N = [Mtilde]GM associated with M. Appending the passivity conditions for a Mueller matrix onto these Jones-derivability conditions, we then arrive at an algebraic identification of the physically important class of passive Jones-derived Mueller matrices.     

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.