excitation conditions and allows us to calculate the induced moments at arbitrary incident fields. This feature allows us to calculate the collective responses of coupled particles using MST, which we discuss in later sections. However, $\boldsymbol{\alpha}$-tensor is usually truncated at dipole order [@Arango2013; @Asadchy2014; @Liu2016], and higher-order $\boldsymbol{\alpha}$-tensor, which includes higher-order multipole moments and field gradients [@Arango2014], is rarely utilized due to the complicated retrieval process if not for spheres with isotropic responses [@Babicheva2019].

These higher-order multipole transitions have been systematically treated using $\mathbf{T}$-matrix, which linearly relates the spherical multipoles of incident field to those of scattered field and is expressed in matrix form as

$$\begin{bmatrix}
        \mathbf{b}^E \\ \mathbf{b}^M
        \end{bmatrix}
        =\begin{bmatrix}
        \mathbf{T}^{E}_{E} & \mathbf{T}^{M}_{E} \\
        \mathbf{T}^{E}_{M} & \mathbf{T}^{M}_{M}
        \end{bmatrix}
        \begin{bmatrix}
        \mathbf{a}^E \\ \mathbf{a}^M
        \end{bmatrix}$$

$$\mathbf{T}^{p}_{p'}
        =\begin{bmatrix}
        \mathbf{T}^{p1}_{p'1} & \mathbf{T}^{p1}_{p'2} & \cdots \\
        \mathbf{T}^{p2}_{p'1} & \mathbf{T}^{p2}_{p'2} &  \\
        \vdots & & \ddots
        \end{bmatrix}$$

where $\mathbf{T}^{np}_{n'p'}$ is a $(2n+1) \times (2n'+1)$ matrix corresponding to the transition from multipole order $n$ of mode $p$ to $n'$ of $p'$. $\mathbf{a}^p$ and $\mathbf{b}^p$ are vectors containing spherical multipoles of incident and scattered fields, respectively [@Mishchenko1996].

Analogous to the exact Cartesian multipoles and the spherical multipoles, $\boldsymbol{\alpha}$-tensor and $\mathbf{T}$-matrix are identical but with different choice of 