## 2 Results  
### Bianisotropic Orbitals  
The plane waves whose k-vectors belong to Fresnel's wave surfaces represent a complete set of solutions of Maxwell's equations in the bulk bianisotropic media. A different set of solutions can be found as expansion of vector spherical harmonics. Recently, the close connection between the multipole composition of electromagnetic fields and bianisotropy has been revealed [60].  
Due to the solenoidal nature of D- and B-fields, namely \(\
abla\cdot\text{D}=0\) and \(\
abla\cdot\text{B}=0\), we can express them as an expansion over the vector spherical harmonics \(\mathbf{M}_{lm}^{1}\), \(\mathbf{N}_{lm}^{1}\) found from solutions of the scalar Helmholtz equations [61; 62; 63]  
\[\psi_{lm}^{(j)}=-\frac{1}{i\sqrt{l(l+1)}}z_{l}^{(j)}(kr)Y_{lm},\ \ \psi_{00}^{(j)}=iz_{0}^{(j)}(kr)Y_{00} \tag{3}\]  
according to  
\[\mathbf{L}_{lm}^{(j)}=\frac{1}{k}\
abla\psi_{lm}^{(j)},\ \ \mathbf{M}_{lm}^{(j)}=\
abla \times\Big{(}\pi\psi_{lm}^{(j)}\Big{)},\ \ \mathbf{N}_{lm}^{(j)}=\frac{1}{k}\
abla\times\mathbf{M}_{lm}^{(j)} \tag{4}\]  
Detailed definitions of the vector spherical harmonics used in this study are in Appendix A. We represent the D- and B-fields of a bianisotropic orbital with a wave number \(k_{q}\) as  
\[\mathbf{D}_{q}=\sum_{lm}\Big{\{}f_{qlm}^{DM}\mathbf{M}_{lm}^{1}(k_{q})+f_{qlm}^{DN}\bm {N}_{lm}^{1}(k_{q})\Big{\}} \tag{5}\]  
\[\mathbf{B}_{q}=\sum_{lm}\Big{\{}f_{qlm}^{BM}\mathbf{M}_{lm}^{1}(k_{q})+f_{qlm}^{BN}\bm {N}_{lm}^{1}(k_{q})\Big{\}} \tag{6}\]  
The corresponding E- and H-fields can be expressed using Equation (2) as  
\[\begin{array}{c}\mathbf{E}_{q}=\sum\limits_{uv}\Big{(}\mu_{uv}^{eq}\mathbf{M}_{uv}^ {1q}+v_{uv}^{eq}\mathbf{N}_{uv}^{1q}+\lambda_{uv}^{eq}\mathbf{L}_{uv}^{1q}\Big{)}=\hat {\pi}_{ED}\mathbf{D}_{q}+\hat{\pi}_{EB}\mathbf{B}_{q}\\ =\sum\limits_{lm}\Big{\{}f_{qlm}^{DM}\Big{(}\hat{\pi}_{ED}\mathbf{M}_{lm}^{1q} \Big{)}+f_{qlm}^{DN}\Big{(}\hat{\pi}_{ED}\mathbf{N}_{lm}^{1q}\Big{)}\Big{\}}+\sum \limits_{lm}\Big{\{}f_{qlm}^{BM}\Big{(}\hat{\pi}_{EB}\mathbf{M}_{lm}^{1q}\Big{)}+f _{qlm}^{BN}\Big{(}\hat{\pi}_{EB}\mathbf{N}_{lm}^{1q}\Big{)}\Big{\}}\\ =\sum\limits_{lm,uv}\Big{(}\mathbf{M}_{uv}^{1q}\mathbf{N}_{uv}^{1q}\mathbf{L}_{uv}^{1q} \Big{)}\cdot\begin{pmatrix}\delta_{MM}^{{}^{}^{\prime}{}^{\prime}{}^{\prime}{} ^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}  
\[\begin{array}{c}\mathbf{H}_{q}=\sum\limits_{uv}\left(h_{uv}^{hq}\mathbf{M}_{uv}^{1q}+v_{ uv}^{hq}\mathbf{N}_{uv}^{1q}+\lambda_{uv}^{hq}\mathbf{L}_{uv}^{1q}\right)=\hat{\alpha}_{HD}\mathbf{D}_{q}+ \hat{\alpha}_{HB}\mathbf{B}_{q}\\ =\sum\limits_{lm}\left\{f_{qlm}^{DM}\left(\hat{\alpha}_{HD}\mathbf{M}_{lm}^{1q} \right)+f_{qlm}^{DN}\left(\hat{\alpha}_{HD}\mathbf{N}_{lm}^{1q}\right)\right\}+ \sum\limits_{lm}\left\{f_{qlm}^{BM}\left(\hat{\alpha}_{HB}\mathbf{M}_{lm}^{1q} \right)+f_{qlm}^{BN}\left(\hat{\alpha}_{HB}\mathbf{N}_{lm}^{1q}\right)\right\}\\ \hskip 14.226378 pt=\sum\limits_{lm,uv}\left(\mathbf{M}_{uv}^{1q},\mathbf{N}_{uv}^{1q},\mathbf{L}_{uv}^{1q}\right)\cdot\begin{pmatrix}\par\par\par\par\par\par\par\par \par\par\par\par\par\  
This correspondence between the plane waves and the bianisotropic orbitals introduced in this study shows the relationship between the solution of the scattering problem presented in this study with the method of plane wave expansion proposed for the scattering by the uniaxial anisotropic spheres [64; 65]. In Figure 2b-e, we plot the components \(\left|f_{qlm}\right|^{2}\) of the eigenproblem of Equation (10) for a bianisotropic orbital with \(k_{q}/k_{0}=2.2\) in the angular momentum space \(l\)-\(m\).