Application of the 2-D discrete-ordinates method to multiple scattering of laser radiation

Physics-based visualization of dense natural clouds. I. Three-dimensional discrete ordinates radiative transfer

A technique is developed to model radiative transfer in three-dimensional natural clouds with a standard discrete ordinates finite-element method modified to evaluate cell-surface-averaged radiances. A log-least-squares-based scale transformation is used to improve the discrete phase-function model. We handle dense media by assuming constant diffuse radiances over input faces to cubic cells, allowing analytical forms for transmittance factors. Transmission equations are combined with diffuse volumetric single-scattering calculations to support evaluations of cell energy balance. Energy not accounted for volumetrically is treated with surface-based effects. Results produced show accurate flux computations at over 30 optical depths per modeled cell. Comparisons with nonuniform cloud Monte Carlo calculations show less than 1% rms error and correlations greater than 0.999 for cases in which cloud-density fluctuations are resolved.

Light and temperature distribution in laser irradiated tissue: the influence of anisotropic scattering and refractive index

The rigorous method of discrete ordinates was used to evaluate the effects of anisotropic scattering and optical discontinuity at the boundaries on light and temperature distribution in tissue. The influence of optical parameters of tissue on its thermal response was examined by using a finite element solution of the heat conduction equation. Calculations were performed for wide ranges of scattering albedo, the anisotropy factor, as well as interface reflectivities. This study shows that the presence of an optical discontinuity due to an air-tissue interface forces the maximum peak intensity to move from subsurface to the surface for tissue with high scattering albedo, which leads to a higher fluence rate in the near surface region. Temperature field calculations show a higher subsurface temperature for a highly scattering medium during tissue coagulation. Neglecting the anisotropic properties of tissue as well as the optical discontinuity at the boundaries would result in considerable error in the calculated temperature rises. Additionally the accuracy of the photon diffusion theory for predicting light and temperature distribution near the tissue surface is examined.

Multiscattering model for propagation of narrow light beams in aerosol media

An approximation to the radiative transfer equation is derived for the propagation of narrow light beams. The main feature is the representation of the power flux in the direction normal to the propagation axis by a diffusion process. Coupled partial differential equations are obtained for the forward and backward flux densities. They are valid in the paraxial approximation. The solutions are written in general analytic formulas applicable to inhomogeneous aerosol media. Comparisons with Monte Carlo calculations and laboratory data reported in a companion paper [Appl. Opt. 27, 2485 (1988)] show very good agreement over a wide range of extinction coefficients, scattering albedos, and particle size parameters.

Transmitted beam profiles, integrated backscatter, and rangeresolved backscatter in inhomogeneous laboratory water droplet clouds

Using laser sources at wavelengths of 1.06 and 10.6 microm, transmitted beam profiles, integrated backscatter, and range-resolved backscatter were measured in laboratory-generated water droplet clouds. Clouds with carefully controlled properties were produced in a specially designed cloud chamber. Inhomogeneities were introduced by partitioning the cloud chamber into three adjacent sections separated by air screens. The measurements show the influence of multiple-scattering effects in both the forward and backward measurement geometries, and these are investigated as functions of optical depth, cloud inhomogeneity, and receiver field of view. These data are unique in many ways, and they provide a great deal of insight to the scattering processes which directly affect lidar-type measurements. As well, these measurements provide a welldocumented and detailed database for model validation. Very good agreement is demonstrated with the solutions derived from the multiscattering propagation model described in a companion paper [Appl. Opt. 27, 2478 (1988), same issue].

Off-axis multiple scattering of a laser beam in turbid media: comparison of theory and experiment

The off-axis forward-scattered radiation from a low energy He-Ne laser, scattered by high density hydrosols, is studied experimentally and theoretically. The validity range of theoretical calculations is determined using four measurements with two optical depths and two detector fields of view. For narrow field of view detectors, experimental and theoretical agreement is good close to the beam axis and decreases with transverse distance; agreement is good at greater distances from the beam axis using an open detector. The small-angle approximation adequately describes multiple scattering close to the beam; calculations are significantly improved by combining the small-angle approximation with diffusion theory.

Separation and analysis of forward scattered power in laboratory measurements of light beam transmittance through a turbid medium

In laboratory measurements of the transmittance of a light beam through a diffusing medium (water plus latex spheres), a distinction between the attenuated beam power and the received forward scattered power was made possible by the use of a transmissometer whose receiver has a variable field of view. The dependence of the received scattered power on the FOV angle and on the medium optical depth was analyzed. The deduced separated contributions of first- and second-order scattering, as well as the total received scattered power, were compared to the results of calculations.

Discrete-ordinated finite-element method for atmospheric radiative transfer and remote sensing

Advantages and disadvantages of modern discrete-ordinates finite-element methods for the solution of radiative transfer problems in meteorology, climatology, and remote sensing applications are evaluated. After the common basis of the formulation of radiative transfer problems in the fields of neutron transport and atmospheric optics is established, the essential features of the discrete-ordinates finite-element method are described including the limitations of the method and their remedies. Numerical results are presented for 1-D and 2-D atmospheric radiative transfer problems where integral as well as angular dependent quantities are compared with published results from other calculations and with measured data. These comparisons provide a verification of the discrete-ordinates results for a wide spectrum of cases with varying degrees of absorption, scattering, and anisotropic phase functions. Accuracy and computational speed are also discussed. Since practically all discrete-ordinates codes offer a builtin adjoint capability, the general concept of the adjoint method is described and illustrated by sample problems. Our general conclusion is that the strengths of the discrete-ordinates finite-element method outweight its weaknesses. We demonstrate that existing general-purpose discrete-ordinates codes can provide a powerful tool to analyze radiative transfer problems through the atmosphere, especially when 2-D geometries must be considered.