## 2 Results
### Scattering Cross-Section of Bianisotropic Spheres in a Vacuum  
The bianisotropic orbitals can be used to solve a large range of problems with spherically shaped bianisotropic media from spheres and spherical shells, to spherical voids, combinations of such geometries, and so forth. Here, we considered a bianisotropic sphere with the arbitrary effective material parameters \(\hat{M}\) with a radius \(R\). We studied the scattering of an electromagnetic plane wave by such a sphere. The field of the plane wave is given by  
\[\mathbf{E}_{in}=\mathbf{\epsilon}(\mathbf{k})e^{i\mathbf{k}\mathbf{r}}=\sum_{lm}\Bigl{(}q_{lm}\mathbf{ M}_{lm}^{1}+p_{lm}\mathbf{N}_{lm}^{1}\Bigr{)} \tag{11}\]  
\[\mathbf{H}_{in}=\mathbf{h}(\mathbf{k})e^{i\mathbf{k}\mathbf{r}}=\frac{1}{i}\sum_{lm}\Bigl{(}p_{lm }\mathbf{M}_{lm}^{1}+q_{lm}\mathbf{N}_{lm}^{1}\Bigr{)},\ \ \mathbf{h}=\mathbf{\hat{k}}\times\mathbf{\epsilon} \tag{12}\]  
We defined the orientation of the incident electric and magnetic fields with the polarization angle \(\alpha\) in the spherical coordinates with respect to the direction of incidence \(\mathbf{\hat{k}}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\), \(\mathbf{\epsilon}(\mathbf{k})=\bigl{(}\sin\alpha\mathbf{\hat{\theta}}-\cos\alpha\mathbf{\hat{ \phi}}\bigr{)}\), \(\mathbf{h}(\mathbf{k})=\bigl{(}\cos\alpha\mathbf{\hat{\theta}}+\sin\alpha\mathbf{\hat{\phi}} \bigr{)}\), where \(\mathbf{\hat{\theta}}=(\cos\theta\cos\phi,\cos\theta\sin\phi,-\sin\theta)\), \(\mathbf{\hat{\phi}}=(-\sin\phi,\cos\phi)\). Expansions of the vector plane wave over vector spherical harmonics exist in the literature [43; 66]. Nevertheless, we derived a compact expansion of a vector plane wave for our work (Appendix C):  
\[ae^{ik\mathbf{r}}=4\pi\ \sum_{lm}i^{l}\left(-\sqrt{l(l+1)}\Bigl{\{}\mathbf{a}\cdot\mathbf{ Y}_{lm}^{(-1)*}(\mathbf{\hat{k}})\Bigr{\}}\mathbf{L}_{lm}^{(1)}+\Bigl{\{}\mathbf{a} \cdot\mathbf{Y}_{lm}^{*}(\mathbf{\hat{k}})\Bigr{\}}\mathbf{M}_{lm}^{(1)}-\Bigl{\{}\mathbf{a} \cdot\mathbf{Y}_{lm}^{(+1)*}(\mathbf{\hat{k}})\Bigr{\}}\mathbf{N}_{lm}^{(1)}\right)\]  
Figure 2: (**a**) Inverse eigenvalues \(k_{q}/k_{0}\) of the eigenproblem in Equation (10) truncated at \(l=4\) (black dots), 10 (red dots), and 40 (green dots). Note the correspondence with the eigenvalues of the index of refraction’s operator plotted in Figure 1b; (**b–e**) eigenvector components \(\left|f_{qlm}\right|^{2}\) of the eigenproblem in Equation (10) for a bianisotropic orbital with \(k_{q}/k_{0}=2.2\) in the angular momentum space \(l\)-\(m\). Correspondingly, the coefficients in the expansions Equations (11) and (12) are  
\[q_{lm}=4\pi i^{l}\Big{\{}\mathbf{e}(\mathbf{k})\cdot\mathbf{\mathsf{Y}}_{lm}^{*}(\mathbf{\hat{k}}) \Big{\}},\ \ p_{lm}=4\pi i^{l+1}\Big{\{}\mathbf{h}(\mathbf{k})\cdot\mathbf{\mathsf{Y}}_{lm}^{*}(\mathbf{ \hat{k}})\Big{\}} \tag{13}\]  
The scattered fields outside of the sphere are given by  
\[\mathbf{E}_{sc}=\sum_{lm}\Bigl{(}b_{lm}\mathbf{\mathsf{M}}_{lm}^{3}+a_{lm}\mathbf{\mathsf{ N}}_{lm}^{3}\Bigr{)} \tag{14}\]  
\[\mathbf{H}_{sc}=\frac{1}{i}\sum_{lm}\Bigl{(}a_{lm}\mathbf{\mathsf{M}}_{lm}^{3}+b_{lm} \mathbf{\mathsf{N}}_{lm}^{3}\Bigr{)} \tag{15}\]  
The scattered fields consist only of the outgoing vector spherical harmonics containing the spherical Hankel functions \(h_{l}^{(1)}\). We represent the field inside the bianisotropic sphere as a linear combination of the bianisotropic orbitals described by Equation (10). \[\mathbf{E}_{sph}=\sum_{q}A_{q}\mathbf{E}_{q},\ \ \mathbf{H}_{sph}=\sum_{q}A_{q}\mathbf{H}_{q}, \tag{16}\]  
Please note that for reciprocal media, the eigenvalues of Equation (10) come in pairs with the radial wavenumber \(\pm k_{q}\), and the corresponding modes in Equation (16) are equivalent. Therefore, among the orbitals with a real \(k_{q}\), we only included the modes with a positive \(k_{q}\). For the orbitals with a complex \(k_{q}\), we only included the modes with Im \(k_{q}>0\).