## 2 The Principle of the Invariant Imbedding (IIM) T-Matrix Method
### Discretization and Vectorization of the Helmholtz Volume Integral Equation
\]  
where, \(d_{0m}^{n}(\theta)\) is the Wigner-d function, while \(\pi_{mn}(\theta)\) and \(\tau_{mn}(\theta)\) are the angular functions, given by:  
\[\pi_{nm}(\theta)=\frac{m}{\sin\theta}d_{0m}^{n}(\theta);\ \tau_{mn}(\theta)= \frac{d}{d\theta}d_{0m}^{n}(\theta) \tag{10}\]  
Similarly, the total field can also be expanded by the vector spherical harmonic functions and written into matrix form, expressed as:  
\[\mathbf{E}(\mathbf{r})=\sum_{n=1}^{\infty}\sum_{m=-n}^{n}\left(\mathbf{M}_{mn },\mathbf{N}_{mn}\right)\mathbf{E}_{mn}=\sum_{n=1}^{\infty}\sum_{m=-n}^{n} \overline{\mathbf{\widetilde{Y}}}_{nm}(\theta,\varphi)\overline{\mathbf{ \widetilde{H}}}_{n}(kr)\mathbf{E}_{mn} \tag{11}\]  
where, \(\mathbf{E}_{mn}\) is the expansion coefficients of the total field, and \(\overline{\mathbf{\widetilde{H}}}_{n}(kr)\) is the radial Hankel function matrix, whose form is similar to \(\overline{\mathbf{\widetilde{J}}}_{n}(kr)\), expressed as:  
\[\overline{\mathbf{\widetilde{H}}}_{n}(kr)=\left( \begin{array}{ccc}H_{n}^{(1 )}(kr)&0\\ 0&\frac{1}{kr}\frac{d}{d(kr)}\Big{(}krH_{n}^{(1)}(kr)\Big{)}\\ 0&\frac{\sqrt{n(n+1)}}{kr}H_{n}^{(1)}(kr)\end{array} \right) \tag{12}\]Similar to the field components, the dyadic Green's function \(\mathbf{G}_{0}(\mathbf{r},\mathbf{r}^{\prime})\) should also be expanded by the vector spherical harmonic functions, but in the case that \(r>r^{\prime}\) and \(r<r^{\prime}\), their expansion forms are slightly different, written as:  
\[\overline{\mathbf{G}}_{0}(\mathbf{r},\mathbf{r}^{\prime})=\left\{ \begin{array}{ ll}ik\sum\limits_{n=1}^{\infty}\sum\limits_{m=-n}^{n}&(-1)^{m}\mathbf{M}_{-mn}(kr, \theta,\varphi)\otimes Rg\mathbf{M}_{mn}(kr^{\prime},\theta^{\prime},\varphi^ {\prime})\\ ik\sum\limits_{n=1}^{\infty}\sum\limits_{m=-n}^{n}&(-1)^{m}Rg\mathbf{M}_{-mn}(kr,\theta,\varphi)\otimes\mathbf{M}_{mn}(kr^{\prime},\theta^{\prime},\varphi^{ \prime})\\ +(-1)^{m}Rg\mathbf{N}_{-mn}(kr,\theta,\varphi)\otimes\mathbf{N}_{mn}(kr^{ \prime},\theta^{\prime},\varphi^{\prime})\ r<r^{\prime}\end{array} \right. \tag{13}\]  
Because the expansion of the dyadic Green's function is similar for \(r>r^{\prime}\) and \(r<r^{\prime}\) on the whole, here we only introduce the matrix expansion process of \(\mathbf{G}_{0}(\mathbf{r},\mathbf{r}^{\prime})\) for \(r>r^{\prime}\). From the symmetry of the dyadic Green's function, the following equation can be easily derived, given as:  
\[\overline{\mathbf{G}}_{0}(\mathbf{r},\mathbf{r}^{\prime})=ik\sum\limits_{n=1}^ {\infty}\sum\limits_{m=-n}^{n} \begin{array}{l}(-1)^{m}\mathbf{M}_{mn}(kr, \theta,\varphi)\otimes Rg\mathbf{M}_{-mn}(kr^{\prime},\theta^{\prime},\varphi^ {\prime})\\ +(-1)^{m}\mathbf{N}_{mn}(kr,\theta,\varphi)\otimes Rg\mathbf{N}_{-mn}(kr^{ \prime},\theta^{\prime},\varphi^{\prime})\end{array}  \tag{14}\]  
By using the symmetric relations of the vector spherical harmonic function [16], Equation (13) can be simplified and rewritten it into the matrix form, expressed as:  
\[\overline{\mathbf{G}}_{0}(\mathbf{r},\mathbf{r}^{\prime})=\overline{\mathbf{ Y}}_{mn}(\theta,\varphi)\overline{\overline{\mathbf{s}}}_{n}(r,r^{\prime}) \Big{(}\overline{\mathbf{Y}}_{mn}(\theta^{\prime},\varphi^{\prime})\Big{)}^{T} \tag{15}\]  
where, \(\overline{\overline{\mathbf{s}}}_{n}(r,r^{\prime})\) is a matrix constructed by the Bessel and Hankel functions, written as:  
\[\overline{\overline{\mathbf{s}}}_{n}(r,r^{\prime})=ik\overline{\overline{ \mathbf{H}}}_{n}(kr)\overline{\overline{\mathbf{J}}}_{n}^{T}(kr^{\prime}) \tag{16}\]  
Similarly, for \(r<r^{\prime}\), the dyad [PERSON]'s function \(\overline{\mathbf{G}}_{0}(\mathbf{r},\mathbf{r}^{\prime})\) can also be written into the matrix form, similar to Equation (15), written as:  
\[