## 2 The Principle of the Invariant Imbedding (IIM) T-Matrix Method
### Iterative Acceleration Based on the Lorenz-Mie Theory  
According to Equation (46), during the implementation of the invariant embedding T-matrix method, the non-spherical particles are needed to be regarded as an inhomogeneous sphere and divided into a series of spherical shells. In this process, the optical matrices of spherical shells (including \(\mathbf{U}\), \(\mathbf{Q}\) and \(\mathbf{Q}_{ij}\) (\(i,j=1,2\))) should be calculated first, then, the T-matrix of the sphere with \(n\) layers can be calculated based on the T-matrix of the smaller one with \(n-1\) layers. In the calculation of the T-matrix, if we do the iteration directly from the origin of the sphere (\(r=0\)) to the circumscribed spherical shell (\(r=\mathbf{R}\)), the computational amount will be very large, which will limit the calculation efficiency of the model. In order to solve this problem, the scattering calculation region of particles is divided into two parts (as shown in Figure 2): one is the spherical core region with a radius of \(R_{0}\) (\(R_{0}\) is the radius of the particle's inscribed sphere), the other is the spherical mantle region in \(R_{0}<r<R\) (the region between the inscribed and circumscribed sphere). The spherical core region is a homogeneous sphere, and its T-matrix can be calculated by the Lorenz-Mie theory. While in the spherical mantle region, the invariant embedding technique is applied to solve the T-matrix, and its initial iteration value is set as the T-matrix of the spherical core. In this way, the unnecessary iteration process can be reduced, and the computational efficiency can be improved as well. A specific interface is needed to convert the Mie coefficients into the T-matrix of the inscribed sphere. For spheres, its T-matrix is a diagonal matrix, which satisfies \(\mathbf{T}_{mmm^{\prime}n^{\prime}}^{ij}=\delta_{m,m^{\prime}}\delta_{n,n^{ \prime}}\cdot\mathbf{T}_{mmm^{\prime}n^{\prime}}^{ij}\), so the relationship between the \(\mathbf{T}\) and Mie coefficients (i.e., \(a_{n}\) and \(b_{n}\)) can be expressed as:  
\[\mathbf{T}_{mmm^{\prime}n^{\prime}}^{11}=-\delta_{m,m^{\prime}}\delta_{n,n^{ \prime}}\cdot b_{n} \tag{47}\]  
\[\mathbf{T}_{mmm^{\prime}n^{\prime}}^{22}=-\delta_{m,m^{\prime}}\delta_{n,n^{ \prime}}\cdot a_{n} \tag{48}\]  
in which, \(a_{n}\) and \(b_{n}\) can be calculated by:  
\[a_{n}=\frac{m\psi_{n}(mx)\psi_{n}^{\prime}(x)-\psi_{n}(x)\psi_{n}^{\prime}(mx) }{m\xi_{n}(mx)\psi_{n}^{\prime}(x)-\xi_{n}(x)\psi_{n}^{\prime}(mx)} \tag{49}\]  
\[b_{n}=\frac{m\psi_{n}(x)\psi_{n}^{\prime}(mx)-\psi_{n}(mx)\psi_{n}^{\prime}(x) }{m\xi_{n}(x)\psi_{n}^{\prime}(mx)-\xi_{n}(mx)\psi_{n}^{\prime}(x)} \tag{50}\]  
where, \(x\) is the size parameter of the inscribed sphere, and \(\psi_{n}(x)\) and \(\xi_{n}(x)\) are the Riccati-Bessel functions. Based on the principle introduced above, the implementation of the IIM T-Matrix method can be concluded as follows:  
Step 1: Determine the inscribed and circumscribed sphere of the non-spherical particle, and calculate the T-matrix of the inscribed sphere with the Lorenz-Mie Theory by using Equations (47)-(50). Step 2: Taking the T-matrix of the inscribed sphere as the initial value, then the T-matrix of non-spherical particles is updated layer by layer by using Equation (46). The computational flowchart of the optical matrix of the spherical shell is presented in Figure 3. Firstly, the U-matrix is calculated by Equation (24), and then, the Q-matrix is directly calculated from the U-matrix by Equation (33). After the Q-matrix is obtained, \(\mathbf{Q}_{11}\), \(\mathbf{Q}_{12}\), \(\mathbf{Q}_{21}\), and \(\mathbf{Q}_{22}\) can be computed by Equations (37), (38), (42) and (43). Step 3: After the T-matrix of the non-spherical particles is obtained, then the cross-sections can be directly linked to the elements of the T-matrix, and the phase matrix elements can also be obtained by combining the generalized spherical functions and T-matrix elements, readers can see Reference [16] for the details.