Single-particle light scattering model

The properties of scattered light can be understood in terms of single-particle scattering.

Two quantities, i.e.single-scattering albedoandphase functionof the particles, are used in general definition of light scattering from a single particle.

The single-scattering albedo, denoted byω, is defined as the ratio of the sum of power scattered to the sum of power abolished from the incident irradiation, J.

The single-particle phase function, p(α), represents the angular distribution of the reflected brightnessI(Ω,Ω0) with respect to the phase angle,α. Thus, p(α) is defined by,

I(Ω) = ωJ(Ω0)p(α)

4π (2.5)

2 Light reflectance theory

whereJis the solar flux into directionΩ0. For isotropic-scattering particles,p(α) is equal to 1; the normalization formula ofp(α) is given by,R

4πp(α)dΩ =4π.

The light reflected by a uniform spherical particle is modeled following Mie scattering theory. The Mie theory includes the exact solution for the interaction of electromagnetic radiation with an isolated perfectly spherical particle. The fundamental parameter for the scattering of single spherical particles is called thesize parameter, X, defined as the ratio of particle size to the wavelength of incident radiance, is shown in the form ofX =πD/λ (where D is the diameter of particle). The Mie theory results in different approximations based on the comparison of the size parameter with unity. When X 1, the particle size is much smaller than the wavelength and the scattered light is proportional to 1/λ⁴. The scattering is known asRayleigh scattering. Assuming that the incident radiance is unpolarized, the particle phase function is given by,

p(α) = 3

4(1+cos²α) (2.6)

If X ≈ 1, the particle size is the same level of wavelength. In this region, also known as Resonance region, the behavior of the single particle phase function involves many different and intricate aspects from case to case. For X 1, the size of particle is much larger in comparison to the wavelength and the scattering is explained by the geometric-optic scattering, considering diffraction at large phase angles.

The light scattering model of a perfect, uniform, spherical particle is applicable in the very rare cases in nature. There is no exact expression to describe the scattering behavior of irregular particles; alternatively, approximate expressions based on the empirical mod-els are used. There are two empirical single-particle phase functions which are widely utilized, theLegendre polynomial seriesand theHenyey-Greenstein function.

The Legendre polynomials is capable to represent the single-particle phase function for irregular particles,

where the coefficientsbjare constants, andPj(α) are Legendre polynomials of order j. If the divergences from isotropic scattering are not that much, the first- and second- order expansions of Legendre polynomial are competent.

Another empirical phase function introduced by Henyey and Greenstein (1941) and calledsingle-term Henyey-Greenstein (HG) phase function, is given by,

p(α) = 1−g²

(1+2gcosα+g²)^3/2 (2.8)

whereg is the cosine asymmetry factor to describe the angular pattern of scattering for irregular particles. The parameter g varies from back-scattering (g<0) through isotropic scattering (g=0) to forward scattering (g>0) in the range from -1 to 1. The scattering structure can be described better by a double-term Henyey-Greenstein phase function.

The two-term HG function (2HG) separates the back-scattering and forward scattering lobes. The form introduced by McGuire and Hapke (1995) is widely used,

P(α) = 1+c

2.2 Radiative Transfer Equation With the coefficient b in the range from 0 to 1 and the coefficient c with no constraint, apart from the general constraint of p(α). The first part of Equation 2.9 is related to the backward-scattering lobe, and the second part to the forward-scattering lobe. The width and height of the lobes are represented by the parameters b and c, respectively.

The physical meaning of the HG parameters b and c are studied by McGuire and Hapke (1995) through an experimental analysis. Their study is done for diverse particles with different compositions and shapes, but approximately of the same size fulfilling the criteria to be large compared to the wavelength. The b and c parameters resulting from the experiment are plotted versus each other (Figure 2.2). It is evident that all the particles accumulate in a L-shaped area. The Figure 2.2 is turned to Figure 2.3, in order to outline a general scheme of HG parameters behavior. The position in the HG plot suggests not only the angular pattern of light scattering, but also the physical characteristics of the particles.

Near the lower right branch of the L diagram, the smooth, clear and spherical particles are located which also correspond to the strong forward scatters. Toward the center, the shape of particle is irregular which is apparent in the values of b (decreasing) and c (increasing).

Further irregularity in the shape of the particle leads to b∼0.25, indicating that the rough, irregular particles have approximately constant b and increasing c. Slightly farther toward the end of the L, the high back-scattering particles such as dielectrics and metals are found withc∼1.2.

Figure 2.2: The plot of HG parameters b and c against each other, adapted from figure 16a in McGuire and Hapke (1995), showing the results of their study for a collection of irregular particles of various types. The numbers are related to the numbers attributed to the particles used in the experiment.

2 Light reflectance theory

Figure 2.3: 2HG parameters diagram, adapted from figure 16b in McGuire and Hapke (1995), illustrates the data points in Figure 2.2 in a practical format.