## 2 The Principle of the Invariant Imbedding (IIM) T-Matrix Method
### Discretization and Vectorization of the Helmholtz Volume Integral Equation
Then, by substituting the equation above into Equation (20), we can get:  
\[\mathbf{E}_{m^{\prime}n^{\prime}}(r,\theta,\varphi)=\overline{\mathbf{ \overline{Y}}}_{m^{\prime}n^{\prime}}(\theta,\varphi)\overline{\mathbf{ \overline{J}}}_{n^{\prime}}(kr)+\int\limits_{0}^{R}dr^{\prime}\sum\limits_{n =1}^{\infty}\sum\limits_{m=-n}^{n}\overline{\mathbf{\overline{Y}}}_{mm}( \theta,\varphi)\overline{\overline{\overline{S}}}_{n}(r,r^{\prime})\mathbf{F} _{mmm^{\prime}n^{\prime}}(r^{\prime}) \tag{22}\]  
Further, we insert Equation (22) into Equation (21), then the following relation can be obtained after some simplification, given as:  
\[\mathbf{F}_{mmm^{\prime}n^{\prime}}(r)=\overline{\mathbf{\overline{U}}}_{mmm^{ \prime}n^{\prime}}(r)\overline{\mathbf{\overline{J}}}_{n^{\prime}}(kr)+\int \limits_{0}^{R}dr^{\prime}\sum\limits_{n=1}^{\infty}\sum\limits_{m=-n}^{ \overline{n}}\overline{\mathbf{\overline{U}}}_{mmm^{\prime}n^{\prime}}(r) \overline{\overline{\overline{S}}}_{\overline{n}}(r,r^{\prime})\mathbf{F}_{m \overline{\overline{m}}m^{\prime}n^{\prime}}(r^{\prime}) \tag{23}\]  
where, \(\overline{\mathbf{\overline{U}}}_{mmm^{\prime}n^{\prime}}(r)\) is the surface integral matrix of the spherical shell, written as:  
\[\overline{\mathbf{\overline{U}}}_{mmm^{\prime}n^{\prime}}(r)=r^{2}\!\!\int \limits_{\Omega}d\Omega^{\prime}\!\left(\overline{\mathbf{\overline{Y}}}_{mm} (\theta^{\prime},\varphi^{\prime})\right)^{\!\!T}\!\!u(\mathbf{r}^{\prime}) \overline{\mathbf{\overline{Z}}}(\mathbf{r}^{\prime})\overline{\mathbf{ \overline{\overline{Y}}}}_{m^{\prime}n^{\prime}}(\theta,\varphi) \tag{24}\]  
From the formula above, we can find that the U-matrix contains the spatial distribution information of the scatterer, that is, the irregular shape of particles and its material distribution can be manifested by this matrix, so its calculation accuracy is very important in the IIM T-Matrix model. Since the scattering field usually refers to the electromagnetic field at infinite distance, in this case, matrix \(\overline{\overline{\overline{S}}}_{n}\) has the form of \(\overline{\overline{\overline{S}}}_{n}(r,r^{\prime})=i\overline{k}\overline{ \mathbf{\overline{H}}}(kr)\overline{\mathbf{\overline{J}}}_{n}^{\!\!\!T}(kr^{ \prime})\), since \(r\) is larger than \(r^{\prime}\)  
\[\overline{\overline{\overline{\mathbf{T}}}}_{mmn^{\prime}n^{\prime}}=ik\sum_{j=1}^{N} \omega_{j}\overline{\overline{\mathbf{J}}}_{n}^{T}(kr_{j})\overline{\overline{ \mathbf{F}}}_{mmn^{\prime}n^{\prime}}(r_{j}) \tag{28}\]  
where, \(r_{j}(j=1,2\ldots,N)\) are the Gaussian quadrature points, and \(\omega_{j}\) is the weight factor for each Gaussian point. Further, the above two formulas can be written into the matrix form, expressed as:  
\[\overline{\overline{\mathbf{F}}}(r_{i})=\overline{\overline{\overline{\mathbf{ U}}}}(r_{i})\overline{\overline{\mathbf{J}}}(kr_{i})+\sum_{j=1}^{N}\omega_{j} \overline{\overline{\mathbf{U}}}(r_{i})\overline{\overline{\overline{\mathbf{ g}}}}(r_{i},r_{j})\overline{\overline{\mathbf{F}}}(r_{j}) \tag{29}\]  
\[\overline{\overline{\mathbf{T}}}=ik\sum_{j=1}^{N}\omega_{j}\overline{\overline {\mathbf{J}}}^{T}(kr_{j})\overline{\overline{\mathbf{F}}}(r_{j}) \tag{30}\]  
where, \(\overline{\overline{\mathbf{J}}}(kr_{i})\) and \(\overline{\overline{\mathbf{g}}}(r_{i},r_{j})\) are the super-matrices with \(\overline{\overline{\mathbf{J}}}_{n^{\prime}}(kr_{i})\) and \(\overline{\overline{\mathbf{g}}}_{n}(r,r^{\prime})\) as the element, and \(\overline{\overline{\mathbf{F}}}(r_{i})\) and \(\overline{\overline{\mathbf{U}}}(r_{i})\) are the super-matrices comprised by \(\overline{\overline{\mathbf{F}}}_{mmn^{\prime}n^{\prime}}\) and \(\overline{\overline{\mathbf{U}}}_{mmn^{\prime}n^{\prime}}\). The expression of \(\overline{\overline{\overline{\mathbf{T}}}}\) matrix is shown in Equation (5).