Aspects of similarity in tissue optics with strong forward scattering.

The similarity studied in this paper links the diffusion exponent to the reduced albedo. If F is the ratio of the diffusion exponent assumed by standard diffusion theory to the actual value of the diffusion exponent, it is shown that for forward-scattering functions, as often found in biological tissues, the values of F approach limiting curves. These curves depend on a constant in the homologous scattering patterns, which can describe Henyey-Greenstein scattering patterns, but which can also approximate Rayleigh-Gans phase functions. Finally, a practical approach for estimating the solution for mixtures of these forward-scattering phase functions with isotropic scatterers has been given.

Application of the exact solution for scattering by an infinite cylinder to the estimation of scattering by a finite cylinder.

A new algorithm for cylindrical Bessel functions that is similar to the one for spherical Bessel functions allows us to compute scattering functions for infinitely long cylinders covering sizes ka = 2πa/λ up to 8000 through the use of only an eight-digit single-precision machine computation. The scattering function and complex extinction coefficient of a finite cylinder that is seen near perpendicular incidence are derived from those of an infinitely long cylinder by the use of Huygens's principle. The result, which contains no arbitrary normalization factor, agrees quite well with analog microwave measurements of both extinction and scattering for such cylinders, even for an aspect ratio p = l/(2a) as low as 2. Rainbows produced by cylinders are similar to those for spherical drops but are brighter and have a lower contrast.

Exact spread function for a pulsed collimated beam in a medium with small-angle scattering.

A solution has been obtained for the spatial and temporal distribution function for a pulsed fully collimated beam propagating through a homogeneous medium with Gaussian small-angle scattering. The solution was obtained first by separation of the general problem into two plane problems, which results in a partial differential equation in three variables. A Fourier transform on two projected variables (one angular and one spatial) and a Laplace transform on the projected temporal variable yielded a set of nonlinear differential equations, which were solved. A recursion relation for the moments of the distribution function was also obtained, and the software MATHEMATICA was used to evaluate these moments to high orders. The contractions on certain variables are also presented; they correspond to the solutions of less-general problems contained in the main problem. A change in the definition of the time-delay produces a remarkable change in the structure of the equations. These solutions should be quite useful for lidar studies in atmospheric and oceanic optics, x-ray and radio-wave scattering in the atmosphere and interstellar medium, and in medical physics.

Rainbows: Mie computations and the Airy approximation.

Efficient and accurate computation of the scattered intensity pattern by the Mie formulas is now feasible for size parameters up to x = 50,000 at least, which in visual light means spherical drops with diameters up to 6 mm. We present a method for evaluating the Mie coefficients from the ratios between Riccati-Bessel and Neumann functions of successive order. We probe the applicability of the Airy approximation, which we generalize to rainbows of arbitrary p (number of internal reflections = p - 1), by comparing the Mie and Airy intensity patterns. Millimeter size water drops show a match in all details, including the position and intensity of the supernumerary maxima and the polarization. A fairly good match is still seen for drops of 0.1 mm. A small spread in sizes helps to smooth out irrelevant detail. The dark band between the rainbows is used to test more subtle features. We conclude that this band contains not only externally reflected light (p = 0) but also a sizable contribution f rom the p = 6 and p = 7 rainbows, which shift rapidly with wavelength. The higher the refractive index, the closer both theories agree on the first primary rainbow (p = 2) peak for drop diameters as small as 0.02 mm. This may be useful in supporting experimental work.

Glare points.

Glare points are the intensity maxima seen when a water drop illuminated by a wide beam is viewed from a certain direction and imaged. We show that good resolution in both the scattering angle and the glare point position can be achieved only if the size parameter x = 2pia/lambda is >>1 and that the positions of the glare points can be computed by a Fourier transform from the familiar Lorenz-Mie scattering function. Sample computations made with x = 10,000 and x = 20,000 are presented. Glare points corresponding to rays that have suffered as many as 15 internal reflections can be identified, in agreement with experimental findings.