Linear Depolarization: 

As discussed in Section 4.2, despite the fact that monomers are Rayleigh scatterers, the degree of linear polarization from the aggregates is slightly reduced at θ = 90°. This appendix discusses the depolarization effect of aggregates. The depolar- ization is found to be caused by the occurrence of cross- polarization, which may increase S11 and decrease S12, thereby reducing the degree of polarization. In this regard, the depolarization of dust aggregates is essentially different from the case of a single sphere in which cross-polarization does not occur. When cross-polarization occurs, scattered light has a component that is perpendicular to the scattering plane even if the incident light has only a parallel component, and vice versa. In other words, S3 and S4 of the scattering amplitude matrix elements are not zero (see Chapter 3 of BH83). The occurrence of cross-polarization can be determined based on the ratio S22/ S11 because S22/S11 is less than unity whenever cross- polarization occurs.
The reason S22/S11 = 1 at small scattering angles is because the aggregate can be regarded as a single sphere owing to the coherent scattering (see Section 4.1.3). The maximum value of the degree of linear polarization correlates with S22/S11. Therefore, to determine the maximum degree of polarization of the aggregates, cross-polarization should be considered, which is not the case in the RGD theory. Next, the origin of the cross-polarization is discussed. A possible mechanism for this depolarization is the monomer– monomer interaction. Because Rayleigh scattering shows completely polarized scattered light at θ = 90°, the light scattered by the aggregates is also completely polarized as long as the interaction between monomers is disregarded. Therefore, the depolarization can be interpreted as a consequence of monomer–monomer dipole interactions (Lu & Sorensen 1994; Mishchenko et al. 1995; Kimura & Mann 2004; Kolokolova & Kimura 2010). Berry & Percival (1986) argued the importance of the multiple scattering of fractal aggregates by means of the mean-field approximation and concluded that multiple scatter- ing can be negligible for fractal aggregates of small monomers with df ? 2, like BCCAs. However, even if df ? 2, monomer– monomer interactions cannot be considered negligible for large monomers. Okada & Kokhanovsky (2009) and Mishchenko et al. (2013) found that the linear depolarization ratio can be used as a diagnostic tool for the density of the aggregates. A monomer in a dense aggregate, like a BPCA, tends to interact with many nearby monomers; therefore, it is expected that the depolarization effect is more prominent for dense aggregates than for fluffy aggregates.\citep{Tazaki_2016}

RDG Theory

The basic idea of the RGD theory is as follows. Assuming that multiple scattering inside the aggregates can be ignored, the light scattered by all of the monomers is superposed, taking into account the phase differences between light rays. The RGD theory assumes that the field inside the particle is approximately the same as the external incident field. This assumption is valid when the following conditions are satisfied\[\left|m-1\right|\ <<1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(5\right)\]
\[2X_0\left|m-1\right|<<1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(6\right)\]
\[2X_c\left|m_{eff}-1\right|<<1\ \ \ \ \ \ \ \ \ \ \ \left(7\right)\]
where m is the complex refractive index of a monomer and X0 and Xc are the size parameters of the monomers and the aggregates of characteristic radius Rc, respectively.
Equation (5) requires the absence of the reflection of light by a monomer. Equations (6) and (7) require the changes in the amplitude and phase of incident light to be negligible within aggregates (Bohren & Huffman 1983). These conditions correspond to the fact that aggregates can be regarded as“almost transparent.” When the above conditions are satisfied and multiple scattering can be considered negligible, the phase matrix elements of the aggregates reduce to
\[S_{ij,agg}\left(\theta\ \right)\ =\ N^2S_{ij,mon}\left(\theta\right)S\left(\mathbf{q}\right)\]
(Botet et al. 1997; Sorensen 2001)
A relation similar to Equation (9) can be obtained by analogy to the theory of scalar wave scattering (see Appendix A). Because multiple scattering is ignored, the phase difference between scattered light rays can be determined from the relative position vector of every pair of monomers in the aggregate. Hence, it is helpful to introduce the .... (g(u) and n(u) defined next)

Structure Factor

It is well known that fractal aggregates exhibit a linear part in the structure fac- tor (when plotted in double logarithmic coordinates), with slope equal to −Df. Although short, such a linear part is vis- ible in Fig. 13, where it is evidenced by the dashed straight line with slope equal to −Df. The region where the fractal scaling is displayed is the range of q values correspond- ing to length scales smaller than the cluster size and larger than the primary particles, i.e., 1/ξ < q < 1/Rp. For length scales outside this interval the g(r) function exhibits a non- fractal behavior which reduces the interval in the structure factor where the fractal scaling can be observed. The devia- tion from the linear behavior, which can be seen in Fig. 13, for q values slightly larger that one, corresponds to the effect of the nonfractal behavior of the g(r) function in the inter- val 2Rp ?r< 4Rp, while the damped oscillations for larger q values are due to the peak of the first coordination shell given by Eq. (13), whose Fourier transform, Speak,is given by [53]Speak
Speak =Nnn * sin(2Rpq)/2Rpq
The importance of the nonfractal part of the g(r) function in the structure factor decreases with increasing the cluster mass. While the nonfractal region has in fact constant broad- ness, the fractal part grows with the cluster mass, leading to a longer linear part in the structure factor. This can be seen in Fig. 14, where the dashed, dotted, and continuous lines rep- resent the structure factors of DLCA clusters with i = 8, 30, and 100, respectively \citep{Lattuada_2003}

Notes

In our model  the original r1 = 1.598 with other constants makes a k0=0.49 when we consider Rg = Rmax/3.4 as  1.1201 make k0=1 and Sq vs. q or Fc vs. q log-log plot an 
    # excellent match to other studies like Sorensen 2001. However, Phase functions
    # obtained by r1 = 1.1201 does not make sense. Somewhat weak forward peak
    # and too weak backward intesnities when theta > 90. So Back to normal k0