Phase functions as solutions of integral equations
© The Author(s) 2017
Received: 22 June 2016
Accepted: 8 March 2017
Published: 24 March 2017
A phase function is an important characteristic of a scattering medium. A method to derive new analytic phase functions is proposed. The relation between a phase function and an angle-averaged single-scattering intensity, derived earlier [M. L. Shendeleva, J. Opt. Soc. Am. A 30, 2169 (2013)], is considered as an integral equation for a phase function. This equation is classified as an Abel integral equation of the first kind, whose solution is known. Two phase functions newly derived with this method are presented.
The radiative transfer equation (RTE), which models light propagation in scattering media, contains two unknown functions: the radiance and the scattering phase function. Usually, the phase function is modeled separately and then inserted into the RTE. A common approach to such modelling relies on the use of Mie theory , which models, using the Maxwell equations, the light scattering from a single spherical particle. Mie theory was also extended to particles of other shapes . In aerosols and soft tissue, various sorts of averaging over particle size distributions are applied. After averaging, the Mie phase function appears to become much smoother and can be well approximated by a co-called analytic or parametric phase function.
Therefore, g ani = g for the HG phase function, where |g ani | ≤ 1.
More flexibility for modelling phase functions is obtained by combining two HG phase functions or a HG phase function with an isotropic phase function or with a delta function, yielding two-parameter phase functions  or three parameter phase functions .
As was pointed out by Selden , the main features of a phase function typically comprise a narrow forward lobe (corona), a broad diffuse background, and a narrow backscattering peak (glory). Various analytic phase functions that model these three major components were proposed by Cornette and Shanks , Liu , Draine  and many others (see the review of Sharma ). It should be noted that the influence of the choice of the analytic phase function in photon transport introduces errors in the determination of optical parameters that are difficult to evaluate .
In this paper, we apply an inverse procedure of a different kind. First, we expand the radiance in successive scattering orders and then exploit the relation between the first-order angle-averaged scattering intensity and the phase function. The first part of this problem was solved, using the successive order expansion developed by Paasschens , in Shendeleva . Here we focus on the second part. Consider first a few examples illustrating the relation between the first-order scattering intensity and the phase function.
Generally, there is a one-to-one correspondence between a phase function and a first-order scattering intensity. For a given phase function, the first-order angle-averaged intensity (also called a single-scattering intensity) can be found from Eq. (21) of Shendeleva . In this paper, we consider an inverse problem: Given the fist-order angle-averaged intensity, find the corresponding phase function. The integral equation for this purpose is derived in the next section. Fortunately, this equation happens to be an integral equation of the Abel type, whose solutions are known . Sections 3 and 4 provide examples of the application of this equation.
Derivation of the integral equation
where ℏω is photon’s energy, and N 0 is the number of photons emitted at t = 0. Note that, in the following, we consider a non-absorbing case, since absorption enters into the solution through the exponential factor Exp(−μ a vt) (16).
Examples of solutions
Ellipsoidal phase function
where χ = 2r 2/(vt)2 − 1 .
Phase function D1
It should be noted that p D1(μ) takes on negative values for k < − 0.8; therefore, the range of parameter k should be restricted to − 0.8 < k < 1.
Phase function D4
The notation D4 means that expression (1 + k 2 − 2kχ) enters the denominator in degree 4.
where Q is given by Eq. (36).
The use of the phase function D4 will be shown elsewhere.
We have derived an integral equation that relates a phase function and an angle-averaged intensity for the first-order scattering, Eq. (21). The solution of this equation, classified as an Abel integral equation of the first kind, can be readily written down as given by Eq. (23). Then, we consider several examples of application of this equation. The first example is merely illustrative, since it concerns the ellipsoidal phase function for which the first-order intensity is known. This example is useful in the sense that it gives an idea of what the first-order intensity can look like. In the two subsequent examples, small modifications of the fist-order intensity allow us to derive two new one-parameter phase functions. The first one, denoted by D1, has the useful property that its anisotropy factor is practically identical to the parameter of the phase function (in the range − 0.2 < k < 1). The same property is characteristic of the HG phase function. Although the function D1 is more complicated than the HG phase function, it has a very simple first-order scattering intensity. The second phase function, D4, has a simple algebraic relation (in the range 0 < k < 1) between the anisotropy factor and the parameter of the phase function, as given by Eq. (40). These examples show that, through small modifications of the single-scattering intensity, one can derive new phase functions with useful properties.
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