be systematically retrieved for arbitrary multipole order [@Fruhnert2017], but detailed analysis on their properties from their $\mathbf{T}$-matrix is difficult due to the spherical basis, while retrieval of higher-order $\boldsymbol{\alpha}$-tensor can be cumbersome [@Arango2014]. In this section, we analyze several meta-atoms from their $\boldsymbol{\alpha}$-tensors, which are transformed from $\mathbf{T}$-matrix using Eqn. \[eqn:T2a\]. We will show that higher-order $\boldsymbol{\alpha}$-tensor are necessary to describe anisotropic meta-atoms, whose properties can be more intuitively analyzed due to the Cartesian basis. Refer to Fig. \[fig:alpha\] for the interpretation of $\boldsymbol{\alpha}$-tensor. In addition, $\boldsymbol{\alpha}$-tensor allows analysis on several particle properties including anisotropy, symmetries, spectral modal resonances, and origin of chirality and optical magnetism.

![Structure of $\boldsymbol{\alpha}$-tensor. $\alpha_{p'n'}^{pn}$ corresponds to the transition from incident multipole order $n$ of type $p$ to induced multipole order $n'$ of type $p'$, and the incident and induced Cartesian modes are denoted outside the parenthesis. For instance, $(\alpha_{e1}^{e1})_{x}^{y}$ relates incident $E_y$ to induced $d_x^E$; $(\alpha_{m2}^{e1})_{yz}^{x}$ relates incident $E_x$ to induced $\Diamond_{yz}^{M} = (\partial_{y}H_{z}+\partial_{z}H_{y})/2$; $(\alpha_{m1}^{m2})_{y}^{xy}$ relates incident $\Diamond_{xy}^{M} = (\partial_{x}H_{y}+\partial_{y}H_{x})/2$ to induced $d_y^M$.[]{data-label="fig:alpha"}](Fig_alpha.pdf)

![Plasmonic double-bars. (a) Schematics. (b) Retrieved $\boldsymbol{\alpha}$-tensor at $\lambda=530~\text{nm}$. (c) Multipole-decomposed scattering cross-section at x-polarized planewave incidence propagating in y-direction. (d) Spectra of $(\alpha_{e1}^{e1})_{x}^{x}$, $(\alpha_{m1}^{m1})_{z}^{z}$, $(\alpha_{e2}^{e2})_{xy}^{xy}$, and $(\alpha_{e2}^{m1})_{xy}^{z}$. Geometrical parameters are: 