Phase function, scattering

The propagation of light in multiple scattering media is quantified usually on the level of radiative transfer or particle diffusion. Scattering, absorption, and emission are considered as independent statistical processes, and the consequences of wave character are either ignored, like polarization, or added as an additional parameter, like the phase function P(ji n) that describes the angular distribution of scattered... 

Note that the number of diffraction peaks decreases with time as the droplet diameter decreases, and the number density of peaks is very nearly proportional to the droplet size. The intensity of the scattered light also decreases with size. The resolution of the photodiode array is not adequate to resolve the fine structure that is seen in Fig. 21, but comparison of the phase functions shown in Fig. 22 with Mie theory indicates that the size can be determined to within 1% without taking into account the fine structure. In this case, however, the results are not very sensitive to refractive index. Some information is lost as the price of rapid data acquisition. 

We previously voiced our objection to the term phase used to designate irradiances. A less commonly encountered, although perhaps better term for the phase function is the scattering diagram. 

The phase function p(0,4>) = S(0, < >) 2/4x2R, which is the fraction of the total scattered light that is scattered into a unit solid angle about a given... 

It is instructive to consider how p varies with scattering angle 0 for the two azimuthal angles 0° and 90°. For scattering directions in a plane perpendicular to the cylinder axis the phase function p 0,90°) is pe(0,90°)sin2(xsin ), where the envelope... 

Equation (8.43) provides us with an approximate criterion, subject to the limitations of diffraction theory, for when a finite cylinder may be regarded as effectively infinite if R > 10, say, there will be comparatively little light scattered in directions other than those in a plane perpendicular to the cylinder axis. The greater is R, the more the scattered light is concentrated in this plane in the limit of indefinitely large R, no light is scattered in directions other than in this plane. We may show this as follows. The phase function may be written in the form p(0, ) = G(0, )F(0, ), where... 

We need consider only scattering directions in the plane = tt/2 (or < > = 377/2) because p vanishes outside this plane we also have 0 = 0 when = 77/2 and 0 = - 6 when = 377/2, where 0 = 0 is the forward direction. Thus, we may take the phase function for scattering by an infinite cylinder in the diffraction theory approximation to be... 

Figure 8.10 Phase function for scattering of unpolarized light by an infinite cylinder. The arrows indicate minima according to diffraction theory.

Grams, G. W., 1981. In-situ measurements of scattering phase functions of stratospheric aerosol particles in Alaska during July 1979, Geophys. Res. Lett., 8, 13-14. 

The directional distribution of the scattering intensity can be described by phase functions. A phase function is defined as the ratio of scattering intensity in a direction to the scattering intensity in the same direction if the scattering is isotropic. Thus, it is a normalized function and is defined over all directions. Typical phase functions for small, intermediate, and large values of f and n, are illustrated in Fig. 4.4, with the spheres assumed to be nonabsorbing [Tien and Drolen, 1987]. It is shown that the phase functions mainly vary... 

For some typical modes of scattering from large spherical particles (f >5), simple formulations of phase functions can be obtained. These modes include scattering from a specularly reflecting sphere, scattering from a diffuse reflection sphere, and scattering by diffraction from a sphere. 

Figure 4.5b. Scattering phase function for a diffuse reflecting sphere which is large compared with the wavelength of incident radiation and with constant reflectivity (from Siegel and Howell, 1981).

For a diffuse sphere, each surface element that intercepts incident radiation will reflect the energy into the entire 2ir solid angle above that element. Thus, the radiation scattered into a specified direction will arise from the entire region of the sphere that receives radiation and is also visible from this specified direction. Consequently, the phase function for a diffuse sphere can be obtained as [Siegel and Howell, 1981]... 

To account for the total directional scattering from a large sphere, effects of both diffraction and reflection must be considered. When a spherical particle is in the path of incident radiation, the diffracted intensity may be obtained from Babinet s principle, which states that the diffracted intensity is the same as that for a hole of the same diameter. The phase function for diffraction by a large sphere is given by [Van de Hulst, 1957]... 

E>(s,s ) is the scattering phase function giving the scattered intensity from direction s to s... 

Light scattered off the disk surface is perhaps the best independent tracer of particle size distributions in disks. This is because particle size and structure not only affect the scattering efficiency, but also change the phase function and polarization of the scattered light. These extra sets of observables can often be exploited to... 

Scattered light carries information about colloidal particles in both amplitude and phase functions, represented by bjiq, f) and exp /q [ry(/) — r/(/)1. respectively. DLS techniques are based on exploiting the phase function, or more precisely, its variation due to particle motion. The utility of DLS for the characterization of colloidal particles relies upon our ability to describe the mechanisms of colloidal particle motions (Brownian, electrophoretic, etc.) and their dependence on size or other particle properties. 

The directional distribution of scaffering is given by the phase function (s s ). This function depends on the angle between the incident s and scattered directions s, cos 9g = s s. The determination of the phase function requires elaborate experiments. It is common to use approximate phase functions, like the Henyey-Greenstein phase function "Thg.a (see, for instance. Modest, 2003), which can be used to approximate the real phase function for many t)pies of particles... 

Several authors have addressed the determination of the optical properties of aqueous titanium dioxide suspensions in the context of photoreactor modeling (Brandi et al., 1999 Cabrera et al., 1996 Cured et al., 2002 Salaices et al., 2001, 2002 Satuf et al., 2005 Yokota et al., 1999). Among the determined properties are extinction, scattering, and absorption coefficients, as well as the asymmetry parameter of the scattering phase function. In general the procedures involve fitting of a radiative transfer model to the experimental results for reflectance and transmittance of radiation. 

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