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Limits of the small-angle approximation to the radiative transfer equation

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Results of comparing the SAA theory with measured data vary. In one case, the modulation transfer function (MTF, the magnitude of the Fourier transform of the PSF) was measured for polystyrene spheres suspended in water (Kuga Y and Ishimaru 1985). A Gaussian function was used for the scattering phase function (SPF), and computed MTFs were compared with measured MTFs. Data were taken for a dispersion of 0.46 μm diameter spheres illuminated with a HeNe laser. The computed values did not compare well with the measured values though the general shape of the curves was similar. The authors explained that the theory might not be valid for particles smaller than the wavelength, or that the Gaussian SPF might have caused the discrepancy. An additional cause could have been the relatively high refractive index ratio of the particles to the medium.

In another case, the PSF was measured in ocean water, and the Wells algebraic SPF (Wells WH 1973, 1969) was used to explain the experimental data (McLean JW and Voss 1991). The results were remarkably good at large optical thicknesses, considering the Wells SPF does not correspond well with data. The results were not as good at small optical thicknesses because, according to the authors, those data would be most sensitive to the details of the shape of the SPF. They concluded that the SAA was validated for ocean water and suggested that, given the exact form of the SPF, the results would be excellent for the optical thickness of an ocean water layer of up to approximately 10.

More recently, Swanson NL et al 2001a measured the SPF and the PSF for polystyrene microspheres in water and compared the results to those of the SAA calculation using the measured SPF as input. The relative refractive index for polystyrene, m, in water is ~1.20. They took data for monodisperse samples of dispersions of spheres of eight sizes between 1 and 10 μm (6.6 < relative size < 64) over a range of optical thickness of 0 < τ < 10. Swanson NL et al found that the SAA was not valid at all (errors exceeding 10%) for particles with diameters less than the wavelength of light, in agreement with the results of Kuga Y and Ishimaru 1985. For the larger particles, the SAA was valid only for very small optical thicknesses (1-3, depending on size).

In a later paper, Swanson NL et al 2001b took the same measurements for dispersions of polystyrene and glass microspheres in water (m = ~1.20) and in oil (m = ~1.01) with size distributions chosen to mimic that of seawater. They found that the errors between the predicted and measured PSF's exceeded 10% for the optical thickness τ > 2. Interestingly, they found that the SAA theory over-predicts blur in an image for the refractive index of particles of m ~1 (optically soft particles) and under-predicts the blur when m > 1. They also conclude that the SAA is extremely sensitive to the structure of the SPF. Hence, the various approximations to the SPF (Gaussian: Kuga Y and Ishimaru 1985, Heyney-Greenstein, Wells etc.) are likely to give widely varying results when used to calculate the PSF.

Theoretical estimates of the range of optical thickness where the SAA is valid also vary widely. Various researchers have estimated that the SAA is valid for τ < 10 (McLean JW et al 1987), τ > 8 (Arnush D 1972), and τ < 5 (Lutomirski RF et al 1995). The second estimate is actually a consequence of assuming an analytic form for the SPF and approximating the solution by assuming τ >> 1. Kuga Y et al 1986 have taken into account the effect of the relative refractive index of the particles. In their theoretical paper, they compared the SAA with a numerical solution of the RTE, where they calculated the SPF from Mie theory. For particles with a size parameter, x, of 50, they found that the SAA yielded less than 10% error for τ < 10 at a refractive index m = 1.01, and τ < 5 for m = 1.33.

CITATION: Swanson N. L. 2008. Limits of the small-angle approximation to the radiative transfer equation (www.tpdsci.com/Tpc/RTESAALim.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY: Published: 28-Oct-2008 Modified: 17-Jun-2009 Peer-reviewed: 31-Oct-2008

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