## 2 The Principle of the Invariant Imbedding (IIM) T-Matrix Method
### A Brief Introduction of the T-Matrix  
The T-matrix is a linear transformation matrix of the expansion coefficients of incident and scattered fields. In order to obtain the T-matrix, the incident and scattered fields should first be expanded by vector spherical harmonic functions [16; 29], shown as:  
\[\mathbf{E}_{inc}(\mathbf{r})=\sum_{n=1}^{\infty}\sum_{m=-n}^{n}a_{nm}Rg \mathbf{M}_{mn}(kr,\theta,\varphi)+b_{mn}Rg\mathbf{N}_{mn}(kr,\theta,\varphi) \tag{1}\]  
\[\mathbf{E}_{scn}(\mathbf{r})=\sum_{n=1}^{\infty}\sum_{m=-n}^{n}p_{mn}\mathbf{ M}_{mn}(kr,\theta,\varphi)+q_{mn}\mathbf{N}_{mn}(kr,\theta,\varphi),\ (r>R) \tag{2}\]  
where, \(R\) is the radius of the smallest circumscribed sphere of the scatterer centered at the origin of the laboratory coordinate system. \(Rg\mathbf{M}_{mn}(kr,\theta,\varphi)\) and \(Rg\mathbf{N}_{mn}(kr,\theta,\varphi)\) are the regular vector spherical wave functions, \(\mathbf{M}_{mn}(kr,\theta,\varphi)\) and \(\mathbf{N}_{mn}(kr,\theta,\varphi)\) are the vector spherical wave functions,\(p_{mn}\) and \(q_{mn}\) are the expansion coefficients of the scattering electric field, and \(a_{mn}\) and \(b_{mn}\) are the expansion coefficients of the incident field. The definition of the vector spherical wave functions can be found in the Appendix A of this paper.  
Since the Maxwell equations are linear, the expansion coefficients of incident light (\(a_{mn}\) and \(b_{mn}\)) and scattered light (\(p_{nm}\) and \(q_{mn}\)) can also be converted by a linear transformation. The linear transformation matrix is just the T-matrix, which can be expressed as:  
\[p_{mn}=\sum_{n^{\prime}=1}^{\infty}\sum_{m^{\prime}=-n^{\prime}}^{n^{\prime}} \left[T_{mmm^{\prime}n^{\prime}}^{11}a_{m^{\prime}n^{\prime}}+T_{mmm^{\prime} n^{\prime}}^{12}b_{m^{\prime}n^{\prime}}^{12}\right] \tag{3}\]  
\[q_{mn}=\sum_{n^{\prime}=1}^{\infty}\sum_{m^{\prime}=-n^{\prime}}^{n^{\prime}} \left[T_{mmm^{\prime}n^{\prime}}^{21}a_{m^{\prime}n^{\prime}}+T_{mmm^{\prime} n^{\prime}}^{22}b_{m^{\prime}n^{\prime}}^{12}\right] \tag{4}\]  
In order to express the T-matrix more intuitively and compactly, the two formulas above can be written into matrix form, given as:  
\[\left[\begin{array}{c}p_{1}\\ q_{1}\\ \cdots\\ \cdots\\ \cdots\\ q_{l_{\max}}\\ q_{l_{\max}}\end{array}\right]=\mathbf{T}\left[\begin{array}{c}a_{1}\\ b_{1}\\ \cdots\\ \cdots\\ a_{l_{\max}}\\ b_{l_{\max}}\end{array}\right]=\left[\begin{array}{cccccccc}T_{11}^{11}&T_{11} ^{12}&\cdots&\cdots&T_{1l_{\max}}^{11}&T_{1l_{\max}}^{12}\\ T_{11}^{21}&T_{11}^{22}&\cdots&\cdots&T_{1l_{\max}}^{21}&T_{1l_{\max}}^{22}\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \cdots&\cdots&\cdots&\cdots&T_{l_{\max}l_{\max}}^{11}&T_{l_{\max}l_{\max}}^{1 2}\\ T_{l_{\max}1}^{21}&T_{l_{\max}1}^{22}&\cdots&\cdots&T_{l_{\max}l_{\max}}^{21}&T_{ l_{\max}l_{\max}}^{22}\\ \end{array}\right]\left[\begin{array}{c}a_{1}\\ b_{1}\\ \cdots\\ \cdots\\ a_{l_{\max}}\\ b_{l_{\max}}\end{array}\right] \tag{5}\]  
in which, the T-matrix is a \(2l_{max}\times 2l_{max}\) super-matrix, where, \(l_{max}=n_{max}(n_{max}+2)\) (\(n_{max}\) is the highest expansion order of the field components).