## 2 Simulation Method  
Upon interaction with incident radiation, oscillations of charges within a particle result in the scattering of that incident radiation. In addition, part of the incident radiation may be extinguished through absorption within the particle. The particle in this case has a complex refractive index. The interaction of a particle with the incident radiation using the Mie theory uses [PERSON]'s equations to derive the incident, scattered and internal fields where an electromagnetic plane wave is scattered by a particle. For simplicity, the particle is initially modeled as an independent isotropic, homogeneous, dielectric sphere. Scattering from both spherical and non-spherical particles appear to be spherical in the far field, and therefore, the particles can be treated as spheres in the far-field approximation [23]. The expressions for the incident, scattered and internal fields take the form of an infinite series expansion of vector spherical harmonics from which cross sections, efficiency factor, and intensity distribution in a particle can be derived. The complete derivation can be found in textbooks discussing Mie theory of scattering, for instance [24; 25; 26; 27; 28; 29]. [PERSON] has also given a detailed derivation of the solution to the phase function of a spherical particle when scattering electromagnetic plane waves [30]. Here, we summarize the key parameters used for our MIe scattering calculations. To compute the amplitude of the scattered fields, the key parameters are the Mie coefficients \(a_{n}\) and \(b_{n}\) as described by Equations (1) and (2). \[a_{n}=\frac{m^{2}j_{n}(mx)\big{[}\dot{x}j_{n}(x)\big{]}^{\prime}-j_{n}(x) \big{[}mxj_{n}(mx)\big{]}^{\prime}}{m^{2}j_{n}(mx)\big{[}xh_{n}^{(1)}(x) \big{]}^{\prime}-h_{n}^{(1)}(x)\big{[}mxj_{n}(mx)\big{]}^{\prime}} \tag{1}\]  
\[b_{n}=\frac{j_{n}(mx)\big{[}\dot{x}j_{n}(x)\big{]}^{\prime}-j_{n}(x)\big{[}mxj _{n}(mx)\big{]}^{\prime}}{j_{n}(mx)\Big{[}xh_{n}^{(1)}(x)\Big{]}^{\prime}-h_{ n}^{(1)}(x)\big{[}mxj_{n}(mx)\big{]}^{\prime}} \tag{2}\]  
In these equations, \(m\) is the complex refractive index of the scattering particle relative to the surrounding air medium and \(x\) is the size parameter. The size parameter is further defined to be equal to  
\[x=\frac{2\pi a}{\lambda} \tag{3}\]  
where \(a\) is the radius of the scattering particle and \(\lambda\) is the wavelength of the incident radiation. The functions \(j_{n}(z)\) and \(h_{n}^{(1)}(z)\) are spherical Bessel functions of the order \(n\) where \(n\) is an integer (\(n=1,2,\dots\) ) and of the given arguments, \(z=x\) or \(z=mx\). The Bessel function \(h_{n}^{(1)}(z)\) can further be defined as  
\[h_{n}^{(1)}(z)=j_{n}(z)+iy_{n}(z) \tag{4}\]  
where the relationships between Bessel and spherical Bessel functions are given by  
\[j_{n}(z)=\sqrt{\frac{\pi}{2z}j_{n+0.5}(z)} \tag{5}\]  
\[y_{n}(z)=\sqrt{\frac{\pi}{2z}Y_{n+0.5}(z)} \tag{6}\]  
The primes (\({}^{\prime}\)) mean derivatives with respect to the argument \(x\) or \(mx\), which follow from the spherical Bessel functions themselves as  
\[\big{[}zj_{n}(z)\big{]}^{\prime}=zj_{n-1}(z)-nj_{n}(z) \tag{7}\]  
\[\big{[}zh_{n}^{(1)}(z)\big{]}^{\prime}=zh_{n-1}^{(1)}(z)-nh_{n}^{(1)}(z) \tag{8}\]  
Calculations of the spherical Bessel functions were conducted in MATLAB, with the size parameter (\(x\)) as the input. In this case, the radius of the scattering particle and the wavelength of the excitation light were used to calculate the size parameter according to Equation (3). For determining the effect of the wavelength, the radius of the scatteringparticle was fixed while the input wavelength was changed. Similarly, for determining the effect of the particle size, the wavelength was fixed while the radius of the scattering particle was changed.