1. Introduction

Light scattering by a small volume element is a central concept in the traditional phenomenological approach of radiative transfer [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. A microphysical derivation of the equation of radiative transfer has also been developed, which does not require the concept of light scattering by a volume element [16, 17]. However, light scattering by a small volume element filled with randomly distributed discrete particles remains a useful model in the analysis and interpretation of laboratory measurements of light scattering by a sparse group of particles [17, 18].

For light scattering by a small volume element filled with sparsely and randomly positioned particles, Mishchenko et al. [18] have derived the far-field modified uncorrelated single-scattering approximation (MUSSA), under which the phase matrices for the particles inside the small volume can be added incoherently to find the phase matrix for the whole volume. In the following text, the conditions of the far-field MUSSA are rephrased briefly. First, the single-scattering approximation (SSA) assumes that each particle is excited only by the incident field. In other words, the scattered light from a particle will not be scattered by any other particles. The SSA implies that the particles in the small volume have to be far enough from each other to ignore the higher order scattering effects. Secondly, the far-field condition assumes that the observation point is located far from the center of the particle group and all of the constituent particles. The SSA and far-field condition are bound together to form the far-field SSA. Under the far-field SSA, it is shown the total amplitude matrix of the group S(r̂,ŝ) can be expressed as the summation of the amplitude matrices S _i(r̂,ŝ) of the constituent particles, weighted by phase factors which originated from the particle position correlation (see Eq. 1) [18]. As a result, the total phase matrix of the volume element also carries the phase factors. To eliminate the phase factors, the third condition assumed is that “the N particles filling the volume element V move during the time necessary to take a measurement in such a way that their positions are random and uncorrelated with each other” [18]. The new condition and the far-field SSA define the far-field uncorrelated SSA (USSA). Under the far-field USSA, the phase factors in the total phase matrix can be averaged over the total volume where the particles are. The resultant total phase matrix is close to the summation of the phase matrices of the constituent particles and has an interference pattern. The interference pattern has a narrow forward peak followed by a series of maxima and minima with decreasing frequency and magnitude. The angular width of the forward peak and the amplitudes of the maxima and minima (except the forward peak) decrease as the distances between any pair of particles increase. The energy contained in the interference pattern also decreases as the distances increase. The last condition assumed is that the average distance between particles is large enough to make the angular width of the forward peak small enough to be indistinguishable from the incident beam and the energy contained in the interference pattern negligibly small. All of these conditions form the basis of the MUSSA.

The analysis of the MUSSA was based on averaging over the orientation of a pair of particles in the small volume element but keeping the distance between the pair of particles a constant. Furthermore, the distance between the pair of particles was taken to be the average distance between an arbitrary pair of particles in the small volume. Mishchenko et al. [18] also stated that one could find the effect of changing the distance between particles by further integrating the results over the range of the distance between particles.

The fidelity reducing assumption of constant distance between the pair of particles is physically unrealistic. It is also difficult to understand how the fixed distance of a pair of particles is representative of the average distance between all pairs of particles. It is a necessary and natural consequence to finish the derivation by integrating the results over distance. In this paper, we revisit the problem by considering the variation of distance between an arbitrary pair of particles. Under this extended derivation, the dependence on the average distance of all pairs of particles disappears. It is explicitly replaced by the largest dimension of the small volume element. The energy conservation of MUSSA is also analyzed after the variation of distance is considered.

2. Formulation

This paper uses the same terminology and notation as Mishchenko et al. [17, 18]. Consider light scattering by an arbitrary small volume element V filled with N particles. The origin of the local coordinates is defined near the center of gravity of the small volume element. Note that r̂ and ŝ are the unit vectors in the scattering and incident directions, respectively. Under the far-field SSA, the total amplitude matrix of the group S(r̂,ŝ) can be expressed in terms of the amplitude matrices S _i(r̂,ŝ) of the constituent particles (see Eq. 7.2.9) in [17]):

where k ₁ is the wave number in the surrounding medium and R _i is the coordinate vector of particle i. The conditions of applicability of Eq. (1) are summarized as:

where r is the distance between the observation point and the origin of the local coordinates; L is the largest linear dimension of the volume element and a_i is the smallest circuscribing sphere of particle i. Please see Refs. [17, 18] for a detailed discussion of the far-field single-scattering approximation, Eqs. (2) – (5).

Given Eq. (1), the phase matrix of the small volume element contains terms like:

In Eq. (6), the first term represents the incoherent summation of the phase matrices of the constituent particles whereas the second term with i≠i′ leads to the forward-scattering interference effect. If both the real parts and imaginary parts (if k ≠ p or l ≠ q) of the interference terms (i ≠ i′ in Eq. (6)), compared to those of the incoherent terms, are negligible, the phase matrix of the small volume element can be obtained by summing those of the constituent particles incoherently, which is the essence of the far-field modified uncorrelated single-scattering approximation (MUSSA).

To derive the criteria of the MUSSA, we first need to introduce the so-called far-field uncorrelated singe-scattering approximation (USSA). In addition to Eqs. (2)–(5), Mishchenko et al. [17, 18] further assumed that “the N particles filling the volume element V move during the time necessary to take a measurement in such a way that their positions are random and uncorrelated with each other.” Under the USSA, one could integrate Eq. (6) over the small volume element for particle i′:

where we have assumed S _i′ is independent of position; V is the total volume of the small volume element; and dV _i′ = dx _i′ dy _i′ dz _i′ is the differential volume element for the coordinates of particle i′.

Next we assume particle i′ moves freely in a spherical space whose center is at particle i. Therefore dV _i′ = dx _i′ dy _i′ dz _i′ = d(x _i′ - x_i)d(y _i′ -y_i)d(z _i′ - z_i) = D ²sin(α)dDdαdϕ, where D, α, ϕ are the spherical coordinates of the vector R _i′ - R_i. Now the average of the phase factor in Eq. (7) can be evaluated as:

where l is the minimal distance between particle i and particle i′ to ensure the far-field single-scattering approximation; L is the largest dimension of the small volume. The angular integral gives:

where |ŝ - r̂| = 2sin(θ/2) has been used. Equation (9) is the same as the result of Mishchenko et al. [17, 18]. Substituting Eq.(9) into Eq. (8) and performing the integral over D:

where θ is the scattering angle and V = 4π(L ³ -l ³)/3 is used.

3. Discussions

Equation (9) represents the interference factor in the liturature [17, 18] whereas Eq. (10) is the term in which the variation of distance is considered. To maintain consistancy with previous notation, we understand D as d in Eq. (9). The first observation of Eq. (9) and (10) is that f(θ) is real. If f(θ) is sufficiently small, both the real and imaginary parts of the interference term in Eq. (7) can be ignored. The condition is satisfied when k ₁ L ≫ 1 and L ≫ l. With the condition of k ₁ L ≫ 1 and L ≫ l, the first term in the numerator of Eq. (10) is much larger than the remaining terms. The k ₁ l term in the denominator is also negligible comparing to the k ₁ L term. Hence f(θ) ~ (k ₁ L)^-2 for large k ₁ L in Eq. (10). It is also noteworthy to mention that Eq. (10) is an infinitesimal of higher order than the previous result of Eq. (9), in which f(θ) ~ (k ¹ d)^-1. In addition, Eq. (10) does not depend on the distance d between particles. As a consequence the conditions of MUSSA will not be expressed in terms of the mean distance. It is not necessary to assume that the distance between an arbitrary pair of particles is roughly equal to the mean distance of all pairs of particles. Also, Eq. (10) has the same feature that f(θ) → 1 as θ → 0 as Eq.(9), if we expand cos(2k ₁ ysin(θ/2)) and sin(2k ₁ ysin(θ/2)) in terms of the small quantity 2k ₁ ysin(θ/2), where y can be either L or l.

 

Fig. 1. f(θ) as a function of θ for k ₁ L = 15 and 60.

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To further compare the two functions, Fig. 1 shows the two f(θ)s defined by Eq. (10) and (9) as functions of θ. In Fig. 1, ”k ₁ d = 15” and ”k ₁ d = 60” are for Eq.(9); and ”k ₁ L = 15” and ”k ₁ L = 60” are for Eq.(10). We set l = 0 in Eq. (10) to have an equivalent comparison with Eq. (9). It is obvious Eq. (10) has similar oscillation features as Eq.(9). The amplitude profile of Eq. (10) is smaller than Eq.(9), which is a direct consequence of f(θ) ~ (k ₁ L)^-2 for large k ₁ L. Another fact is that the first zero of Eq. (10) is slightly larger than Eq. (9) for both the cases shown. If l = 0, the first zero of Eq.(10) is the solution of x = tan(x), where x = 2k ₁ Lsin(θ/2). The first solution is: 4.49341 = 2k ₁ Lsin(θ/2), which is:

Because the first zero of Eq. (9) is at θ = θ ₀ = 2 arcsin[π/(2k ₁ L)] and 4.49341 > π, the first zero of Eq.(10) is always slightly larger than Eq. (9). To merge the forward interference peak into the diffraction peak of large single particles, θ ₀ ≪ 4/(k ₁ a) is required, where a is the largest dimension of a single particle, which leads to:

We make the following approximation for the second interference term of Eq.(7):

To make this term negligibly small compared to the first term in Eq. (7), the following relation is necessary:

Where N - 1 ~ N for large N is used.

The MUSSA also needs to satisfy energy conservation. For a collection of particles, energy conservation requires that the total extinction cross section must be equal to the total scattering cross section if no absorption is present. In the exact forward-scattering direction, ŝ = r̂, Eq. (1) shows that the amplitude scattering matrix of the small volume element is an incoherent summation of those of the constituent particles. From the optical theorem for extinction, the total extinction cross section or matrix of the small volume element also has this feature. However, the total scattering cross section of the small volume element is equal to the integration of Eq. (6) over solid angles, which contains an interference term. To make the MUSSA satisfies energy conservation, Mishchenko et al. (pp. 151, Ref. [17]) concludes that the integration of the interference term has to go to zero; namely,

To show Eq. (15) is satisfied under the condition of Eq.(14), we expand [S _i(r̂,ŝ]_kl[S _i′(r̂,ŝ]^* _pq in terms of Legendre functions:

where P_n is the Legendre functions of order n; nmax is the maximum order of the expansion; and w_n is the expansion coefficient. Substituting Eq.(16) into Eq. (15), we have:

We have calculated the angular integral c_n = ʃ^π ₀ P_n(cos(θ)) f(θ)sin(θ)dθ up to n = 20. For each order n and c_n decrease with increasing k ₁ L. It shows that the energy contained in the interference term is quite small if k ₁ L is large enough. We conclude that energy conservation is also ensured by Eq. (14).

4. Summary

Considering the variation of distance between any pair of particles, we have derived the new conditions of validity of the MUSSA for light scattering by a group of particles confined in a small volume element. The new MUSSA conditions include Eqs. (12), (14), and the far-field SSA conditions of Eqs. (2) – (5). In addition, it should be kept in mind that the coordinates of particles are completely uncorrelated. The new conditions, Eqs. (12) and (14), are directly expressed in terms of the largest dimension of the small volume. The new MUSSA does not require the distance between any pair of particles in the volume to be larger than what is required in the single-scattering approximation (SSA). Instead, it requires the dimension of the volume to be large compared to the incident wavelength. The new results are consistant with energy conservation and also make the MUSSA conditions easier to satisfy. The analysis in this paper is useful in interpreting laboratory measurements of light scattering by a collection of small particles [19, 20, 21, 22].