# Polarization Transmission of Visible Light in Inhomogeneous Sea Fog Particle Environment
## 2 Establishment of an Inhomogeneous Particle Scattering Model  
### Equivalent Analysis Method  
Inhomogeneous aerosol particles have two mixing states: outer mixing state and inner mixing state. An inner mixed state means that a single particle in the system is a mixture of multiple aerosol components, and each particle exhibits the common physical and chemical properties of multiple components. The homogeneous sphere model or the layered sphere model is generally used for the particles in the mixed state. The sea fog aerosol in this research adopts the layered sphere model. In particular, the sea fog aerosol is composed of sea salt and water, the sea salt particles are used as condensation nuclei, and the sea salt is wrapped with a homogeneous water shell to form inhomogeneous sea fog particles. The aerosol particles in the actual sea fog are mostly complex mixtures composed of different components, and no fixed complex refractive index exists because of different components. Therefore, we use the equivalence analysis of light scattering parameters to obtain the complex refractive index of complex mixtures. The specific steps of the equivalence analysis method are as follows:  
(1) The double-layer spherical particle model is shown in Figure 1. First, the water particles' refractive index is \(m_{1}=n_{1}+ik_{1}\), and the sea salt particles' refractive index is \(m_{2}=n_{2}+ik_{2}\). Water particles are determined as the outer shell, and sea salt particles are determined as the inner core. The corresponding scale parameters \(x_{1}=2\pi a/\lambda\), \(x_{2}=2\pi b/\lambda\) are calculated according to the radius \(a\) of the water particle and the radius \(b\) of the sea salt particle. \(f=b/a\) is set as the mixing ratio, which is the ratio of water particles and sea salt particles in the whole double-layer particle system. The above parameters are substituted into the double-layer Mie scattering program to calculate the following optical parameters of inhomogeneous sea fog particles: extinction efficiency \(k_{ext}\), scattering efficiency \(k_{sca}\), and absorption efficiency \(k_{abs}\). (2) The inhomogeneous sea fog particles are equivalently regarded as homogeneous spherical particles with a radius of \(a\) and refractive index of \(m^{\prime}=n+ik\)[8]. The scale parameter is \(x=2\pi a/\lambda\). The optical parameters of the equivalent particles: extinction efficiency \(Q_{ext}\), scattering efficiency \(Q_{sca}\), and absorption efficiency \(Q_{abs}\), are calculated by Mie scattering theory. (3) The relative difference \(\varepsilon\) between the optical parameters of inhomogeneous sea fog particles and equivalent particles are as follows [8]:  
\[\varepsilon_{ext}=\frac{k_{ext}-Q_{ext}}{k_{ext}} \tag{1}\]  
\[\varepsilon_{sca}=\frac{k_{sca}-Q_{sca}}{k_{scas}} \tag{2}\]  
\[\varepsilon_{abs}=\frac{k_{abs}-Q_{abs}}{k_{abs}} \tag{3}\]  
Then, the relative difference between inhomogeneous sea fog particles and equivalent spherical particles is:  
\[\varepsilon=\sqrt{(\varepsilon_{ext}^{2}+\varepsilon_{sca}^{2}+\varepsilon_{ abs}^{2})/3} \tag{4}\]  
Figure 1: Schematic diagram of the double-layer spherical particle model. When the relative difference between the double-layer spherical particles and the equivalent spherical particles is the smallest, the refractive index of the equivalent particles \(m^{\prime}\) is considered the equivalent refractive index of the inhomogeneous sea fog particles. The refractive indices of water particles and sea salt particles in the visible band are \(m_{1}=1.33+i0.000196\), \(m_{2}=1.5+i0.0001\)[17]. We used the equivalent analysis method to calculate the equivalent refractive index at seven mixing ratios: 0.1, 0.3, 0.4, 0.5, 0.7, 0.9, and 0.99. Table 1 shows the equivalent refractive indices corresponding to different mixing ratios. Figure 2 shows the relative differences corresponding to different mixing ratios. Table 1 shows that when the mixing ratio is \(0.3<f<0.4\), the equivalent refractive index is close to water particles' refractive index.