Effective Phase Function Research Articles

Light scattering in thin turbid tissue including macroscopic porosities: A study based on a Monte Carlo model

A Monte Carlo code is built taking into account macroscopic spheroid cavities inside a turbid medium, i.e. in mixing Multi-Layer Monte Carlo (MLMC) and Monte Carlo Ray Tracing (MCRT). That simulates a tissue with a strong and heterogeneous porosity, such as flesh tissues of fruit or bone tissues. This kind of tissue, which has two scales of porosity (microscopic and macroscopic), differs notably of the homogeneous and continuous model used in the usual radiative transfer equation. The influence of the presence of spheroids can be observed on the shape of the effective phase function, on the effect related to the time-resolved diffusion solution or also on the scattering coefficient retrieved by means of the Beer–Lambert relationship. For instance, the reduced scattering coefficients retrieved thanks to time-resolved transmittance from MLMC-MCRT models having a lot of intertwined large cavities show variations coherent with those retrieved from bone tissue. Furthermore, the effect of porosity on optical transmission seems to have a real impact when relative refractive index is close to 1. In this case, the equivalence problem between such porous MLMC-MCRT model and a homogeneous turbid medium, can be discussed at the level of the angular intensity distribution over the plane boundaries. This requires to fit this angular distribution by an Adding-Doubling model using optimized optical depth and scattering phase function. Experimental scattering phase functions obtained from apple tissues are considered in order to test this idea, and then compared with those computed with a MLMC-MCRT model.

Approximate analytical effective phase function obtained for a thin slab geometry

The reflection and transmission of light from a slab containing a turbid medium provide a scattering effective phase function from which the true optical anisotropy factor may not be always easily retrieved. From the statistical Poissonian theory and thanks to approximations about the optical path related to the first scattering events, a simplified relationship is established between angular phase function and effective phase function. Therefore, with a modified Gegenbauer or a Two-Terms Henyey-Greenstein phase function, some adjustable analytic functions are proposed in order to fit the measurements linked to the true effective phase function. An efficiency of the approximate analytical function is proved, thanks to the light modelling by Monte Carlo method, for optical thickness lower or equal to 2. This is confirmed by comparisons of the anisotropy retrieval with other methods. Concerning applications, several fits were made on experimental effective phase functions corresponding to goniometric measurements from usual diffusing materials and biological tissues.

Estimation of the effective phase function of bulk diffusing materials with the inverse adding-doubling method.

The accuracy of optical simulations including bulk diffusors is heavily dependent on the accuracy of the bulk scattering properties. If no knowledge on the physical scattering effects is available, an iterative procedure is usually used to obtain the scattering properties, such as the inverse Monte Carlo method or the inverse adding-doubling (AD) method. In these methods, a predefined phase function with one free parameter is usually used to limit the number of free parameters. In this work, three predefined phase functions (Henyey-Greenstein, two-term Henyey-Greenstein, and Gegenbauer kernel (GK) phase function) are implemented in the inverse AD method to determine the optical properties of two strongly diffusing materials: low-density polyethylene and TiO₂ particles. Using the presented approach, an estimation of the effective phase function was made. It was found that the use of the GK phase function resulted in the best agreement between calculated and experimental transmittance, reflectance, and scattered radiant intensity distribution for the LDPE sample. For the TiO₂ sample, a good agreement was obtained with both the two-term Henyey-Greenstein and the GK phase function.

Dual-scale 3-D approach for modeling radiative heat transfer in fibrous insulations

In this work, a dual-scale computationally-affordable 3-D method is developed to simulate the transfer of radiative heat through fibrous media comprised of fibers with different diameters and orientations. The simulations start by generating a virtual fibrous material with specified microstructural properties and then compute the radiative properties of each fiber (i.e., effective phase function, as well as scattering and absorption coefficients) in the structure using the Mie Scattering theory. Considering independent scattering formulations for our fibrous media (media with high porosities), the radiative properties of the insulation material are computed by summing up the radiative properties of each individual fiber, after transforming the phase function values from the fiber’s local 3-D coordinates system to a fixed global coordinates system. The radiative properties of the media are then used in the Radiative Transfer Equation (RTE) equation, an integro-differential equation obtained for computing the attenuation and augmentation of an InfraRed ray’s energy as it travels through a fibrous medium. Using the Discrete Ordinate Method (DOM), the RTE is then discretized into a system of twenty four coupled partial differential equations and solved numerically using the FlexPDE program to obtain the rate of heat transfer through the entire thickness of the media. Studying media with different microstructural properties, it was quantitatively shown that increasing solid volume fraction, thickness, or fibers’ through-plane orientation increases the rate of heat transfer through insulation. With regard to the role of fiber diameter, it was found that there exists a fiber diameter for which radiation heat transfer through a fibrous media is minimal, ranging between 3 and 10μm for glass fibers operating in a temperature range of about 340–750K.

Narrowly peaked forward light scattering on particulate media: II. Angular spreading of light scattered by polystyrene microspheres

The adequacy of the effective phase function (EPF) used to describe the light scattered at small angles was tested on aqueous suspensions of polystyrene microspheres. Angular resolved light scattering measurements were performed on two types of latex suspension, which contained polystyrene spheres of 3 µm and 5 µm diameters, respectively. The experimental data were fitted with two EPF approximants. If the polystyrene spheres are at least 3 µm in diameter the quasi-ballistic light scattering process can be described relatively well by the EPF in a small angular range centered in the forward direction. The forward light scattering by macroscopic samples containing microspheres can be modeled relatively well if the true Mie single particle scattering phase function is replaced by a simpler Henyey–Greenstein dependence having the same width at half-height as the first scattering lobe.

Aggregation of red blood cells in suspension: study by light-scattering technique at small angles

Red blood cells (RBCs) in the presence of plasma proteins or other macromolecules have a tendency to form aggregates. Light-scattering technique was used to investigate the RBC aggregation process. A highly diluted suspension of RBCs was illuminated with a 632.8-nm HeNe laser. Angular-resolved measurements of light intensity scattered by an RBC suspension from a 200-microm thick optical glass cuvette during 10 min of their aggregation process were performed at 1 to 4 off-axis deg with a very high angular resolution, at hematocrits in the range of 3.5 x 10(-2) to 10(-1). The angular spreading of forward-scattered light at small angles during the RBC aggregation process was described in terms of a new, effective phase function model that has been used for fitting the experimental data. The aggregated RBCs' optical properties, such as effective scattering anisotropy and scattering cross section, were determined. The results were compared with prediction of Mie theory for equivolumetric spherical particles. The time dependence of the aggregates mean radius and of the mean number of cells per aggregate was also calculated. Last, the potential of the proposed technique (forward-scattering light technique) as a new quantitative investigation of cellular aggregation process was estimated.

Narrowly peaked forward light scattering on particulate media I. Assessment of the multiple scattering contributions to the effective phase function

The quasi-ballistic light scattering (QBLS) process induced in a particulate medium by a collimated laser beam is characterized by the effective phase function (EPF) which describes the angular spreading of the scattered photons. The purpose of the first part of our work is to evaluate the accuracy of two approximations of the EPF. The rigorous EPF expression comes out in the form of a sum of an infinite series of terms, indexed by the number of scattering events suffered by the participating photons. One of the approximations is a previously derived, compact, analytical expression (Turcu 2006 Appl. Opt. 45 639–47). The alternative approximation introduced here collects the contribution from the first to the nth order of scattering. To assess the accuracy of the two approximations we introduce a series of minorant–majorant pairs which tightly encompass the EPF. The results of our analysis show that the analytical approximation is not sufficiently accurate in the entire domain of interest. The alternative, more accurate, nth-order sum approximation allows the user to improve the accuracy up to the desired level by increasing the number of scattering orders kept in the approximant.

Normal-directional and normal-hemispherical reflectances of micron- and submicron-sized powder beds at 633 and 790nm

A conic pyroelectric detector is employed to measure the effective reflectances of powder beds of metals, SiC, WC, and graphite. The experimental results are compared with numerical solutions of the radiation transfer equation in a half-infinite homogeneous body with effective radiative properties. The effective albedo and phase function are theoretically estimated by the model of opaque particles with specularly or diffusely reflecting surfaces in the assumption of geometrical optics. The studied powder beds with particles of several tens of microns in diameter are generally described by the model where specular reflection by particle surfaces is assumed corresponding to refraction and extinction indices of materials in the dense form. The model does not apply to semitransparent particles of SiC. The powder beds of submicron particles of SiC and WC are highly agglomerated and consist of clusters of particles with diameters of several tens of microns. A submicron powder bed is found to be considerably less reflective than a micron-sized powder bed of the same material. This is described by the model where the agglomerates of submicron particles are treated as large particles with diffusely reflecting surfaces. The effective normal-hemispherical reflectance of a micron-sized powder bed correlates with the hemispherical reflectance of dense material. The observed differences between the experiment and the model can be explained by a systematic error at calibration, oxidation of materials, and diffraction effects.

RWMCS – An alternative random walk Monte Carlo code to simulate light scattering in biological suspensions

In biological suspensions, the forward light scattering is done mainly by the contribution of the suspended cells. The multiple scattering is almost always unavoidable, its contribution being described by Monte Carlo simulations or by approximate analytical formula. The Monte Carlo approach we propose moves one photon at a time and checks all scattering centers to find, at each simulation step, which one will scatter the photon. The validation of the simulation results is performed by comparing the obtained angular distribution with the predictions of the effective phase function theory reported recently in the literature and with the angle resolved experimental measurements performed on human red blood cells (RBCs) in suspensions at different hematocrits. The results show a good agreement in the small concentration range.

High-resolution angle-resolved measurements of light scattered at small angles by red blood cells in suspension

Red blood cells (RBCs) scatter light mainly in the forward direction, where the scattering phase function has a narrow peak. We performed an experimental investigation into the angular distribution of light scattered by blood in the small-angle domain. A highly diluted suspension of RBCs (hematocrits in the range 5 x 10(-5)-10(-2)) was illuminated with a He-Ne laser with 633 nm wavelength. We focused our research on two main topics: the scattering efficiency of the RBCs given by the mean scattering cross section and the scattering anisotropy obtained from the angular distribution of the scattered photons. The collimated beam transmission and the angular distribution of scattered light were measured and compared with the predictions of the effective phase function model. The RBCs' mean scattering cross section and scattering anisotropy were obtained by fitting of the experimental data.

Effective phase function for light scattered by blood

The scattering process induced in blood by a collimated laser beam is theoretically investigated. An individual red blood cell (RBC) has a scattering phase function strongly peaked in the forward direction. For far-field experiments, the small scattering volumes can be considered as "macroscopic particles" characterized by an effective scattering phase function. Using the single-cell phase function as "input data" the angular distribution of light scattered at small angles by the whole scattering volume, containing RBCs in suspension, is calculated analytically. The angular dispersion of the light scattered by blood can be approximately described by the same formula used to characterize the light scattered by a single cell but with an effective, hematocrit-dependent anisotropy parameter.

Determination of optical properties of human blood in the spectral range 250 to 1100 nm using Monte Carlo simulations with hematocrit-dependent effective scattering phase functions

The absorption coefficient mu(a), scattering coefficient mu(s), and anisotropy factor g of diluted and undiluted human blood (hematocrit 0.84 and 42.1%) are determined under flow conditions in the wavelength range 250 to 1100 nm, covering the absorption bands of hemoglobin. These values are obtained by high precision integrating sphere measurements in combination with an optimized inverse Monte Carlo simulation (IMCS). With a new algorithm, appropriate effective phase functions could be evaluated for both blood concentrations using the IMCS. The best results are obtained using the Reynolds-McCormick phase function with the variation factor alpha = 1.2 for hematocrit 0.84%, and alpha = 1.7 for hematocrit 42.1%. The obtained data are compared with the parameters given by the Mie theory. The use of IMCS in combination with selected appropriate effective phase functions make it possible to take into account the nonspherical shape of erythrocytes, the phenomenon of coupled absorption and scattering, and multiple scattering and interference phenomena. It is therefore possible for the first time to obtain reasonable results for the optical behavior of human blood, even at high hematocrit and in high hemoglobin absorption areas. Moreover, the limitations of the Mie theory describing the optical properties of blood can be shown.

Effective phase function for light scattered by disperse systems—the small-angle approximation

We focus on the quasi-ballistic regime of the scattering process induced in a diffusive medium by a collimated laser beam. The main goal is to find an approximate analytic solution for the radiative transport equation. Particles with characteristic dimensions larger than the light wavelength and relative refraction indices close to unity have a scattering phase function strongly peaked in the forward direction. This specific angular dependence was used to approximate, for any scattering order, the angular dependence of the scattered light fluxes and to obtain simple analytic expressions. Even more, in the small-angle approximation the scattered far field can be described by an effective phase function given by the same analytic formula as the single-particle phase function.

Estimation of instantaneous TOA albedo at 670 nm over ice clouds from POLDER multidirectional measurements

An algorithm that determines the 670‐nm top‐of‐atmosphere (TOA) albedo of ice clouds over ocean using Polarization and Directionality of the Earth's Reflectance (POLDER) multidirectional measurements is developed. A plane‐parallel layer of ice cloud with various optical thicknesses and light scattering phase functions is assumed. For simplicity, we use a double Henyey‐Greenstein phase function to approximate the volume‐averaged phase function of the ice clouds. A multidirectional reflectance best‐fit match between theoretical and POLDER reflectances is used to infer effective cloud optical thickness, phase function and TOA albedo. Sensitivity tests show that while the method does not provide accurate independent retrievals of effective cloud optical depth and phase function, TOA albedo retrievals are accurate to within ∼3% for both a single layer of ice clouds or a multilayer system of ice clouds and water clouds. When the method is applied to POLDER measurements and retrieved albedos are compared with albedos based on empirical angular distribution models (ADMs), zonal albedo differences are generally smaller than ∼3%. When albedos are compared with those on the POLDER‐I ERB and Cloud product, the differences can reach ∼15% at small solar zenith angles.

A Maslov-propagator seismogram for weakly anisotropic media

SUMMARY We introduce a formalism to calculate shear-wave seismograms for weakly-anisotropic and inhomogeneous media. The method is based on the combination of the forward-propagator method, which accounts for shear-wave interaction along a single reference ray, and the Maslov ray-summation, which incorporates amplitude and phase information from neighbouring rays to account for waveform and diffraction effects at caustics and in shadow regions. The approach is based on the assumption that the multiply split shear waves, on the way to a given receiver, travel along a common ray path that can by obtained from ray tracing in an isotropic reference medium (i.e. the common-ray approximation). The forward propagator and the Maslov amplitude are expressed with respect to radial and transverse coordinates (perpendicular to the ray propagation direction) that are defined uniquely by the initial conditions. Local polarizations and slownesses of the fast and slow shear-waves in the direction of propagation are obtained from the eikonal equation. The Maslov-propagator phase is given by the average shear-wave traveltime along the reference ray. Phase advances and delays of individual shear wave components are accounted for by the propagator. The geometrical-spreading information required for the Maslov integration is supplied by dynamic ray tracing in the isotropic reference medium. In the high-frequency limit effective phase functions are defined to assess the validity of the Maslov propagator phase information. For a homogeneous isotropic reference medium, we find good agreement with exact Maslov phase functions for anisotropic perturbations of up to 20 per cent. As a numerical application we consider effects of inhomogeneous anisotropy in a shear-wave cross-hole survey. The variations of the transversely-isotropic medium require 2-D slowness integrals. The method can handle discontinuities of the fast polarization along the ray path and also for neighbouring rays which is important for the slowness integration. Smooth transitions between isotropic and anisotropic regions along the ray path can be accounted for without the need to switch between numerical formulations.

Scattering phase function for fibrous media

The formulation and evaluation of the effective phase function for fibrous media with any fiber orientation are presented based on the assumption that the fibers are infinite cylinders and scattering is in the independent regime. The phase function is derived in terms of the scattering angle which is the angle between the incident and the scattered radiation. Unlike spherical particles which scatter radiation into all directions, radiation scattered by cylindrical fibers only propagates into limited angular ranges. Hence, the maximum value of the scattering angle for radiation scattered by a collection of fibers depends on the orientation of the fibers relative to the direction of the incident radiation. For a given fiber orientation both the angular variation and the maximum scattering angle of the phase function are different for different incident directions. If the fibers are randomly oriented in space, the angular variation of the phase function is independent of the incident direction and the maximum scattering angle is π. The phase function for fibrous media always exhibits a strong peak in the direction of incident radiation, indicating that the scattering is highly anisotropic.

Ab initio calculation of curved-wave EXAFS amplitude and phase functions

The last five years have seen significant advances in the calculation of theoretical EXAFS. Most important was the development of a full curved wave formalism that was computationally rapid [ 11. The full curve-wave formalism takes into account the curvature of the outgoing and scattered photoelectron. This is most important when one wishes to examine the low-energy part of a spectrum and especially for higher Z elements. Previously it was thought that multiple scattering contributions were significant up to energies of 50 eV above the edge. Recent studies [2] have indicated that, in many cases, the multiple scattering contributions are not as significant as once believed and that the single scattering EXAFS theory is valid to much lower energies. We have previously shown a simple way to use the full curved wave formalism of EXAFS within the traditional plane-wave formalism [3,4]. Simply stated, the back-scattering amplitude and phase functions are replaced by effective backscattering amplitude and phase functions which depend on R, the interatomic separation. These functions, once calculated, can then be used in much the same way as the plane wave functions as calculated and tabulated by Teo and Lee [5] are used. We have calculated and tabulated the backscattering amplitude and phase for every element from boron to americium[3] and the tabulations are available from the authors on a single standard IBM or Macintosh disk. Until recently the most widely available set of theoretically determined amplitude and phase functions for EXAFS was that of Teo and Lee [5]. For high accuracy analysis, however, especially for the heavier elements, it still remained necessary to extract the functions from measurements of model compounds because their calculations were made using a plane wave approximation for the scattering of the outgoing photoelectron by the neighboring atom. Our use of the full-curved wave formalism makes our calculations valid for lower energies and for higher Z elements. The major advantage of tabulations such as ours is that the theoretical work can be widely and easily disseminated. Once file format standards have been established, improvements in the theoretical calculations can be transparently added to existing software without modification of actual code. They also provide a useful and standard building block on which experimentalists can build.