We note that a similar, theoretical work has been published during the review of this manuscript, describing scattering phenomena in nanowires with a radially anisotropic refractive index.[@liu_superscattering_2017]

The authors declare no competing financial interest.

Supporting informations: [Strongly directional scattering from dielectric nanowires]{} {#supporting-informations-strongly-directional-scattering-from-dielectric-nanowires .unnumbered}
======================================================================================

Contributing Mie orders to FW/BW scattering intensity
=====================================================

Mie theory can be applied to infinitely long cylinders by expanding the fields in vector cylindrical harmonics (instead of the vector spherical harmonics as in “classical” Mie theory for spherical particles). The expansion coefficients for the scattered fields are the so-called scattering coefficients $a_n$ for electric multipole contributions and $b_n$ magnetic multipoles. Details can be found *e.g.* in the book of Bohren and Huffman. [@bohren_absorption_1998]

In the next step, the asymptotic scattered field far from the structure has to be calculated. In Mie theory, the S-matrix, relating the scattered field $\mathbf{E}_s$ with the incident field $\mathbf{E}_i$, can be used for this means. Under normal incidence on an infinite cylinder, the S-matrix, connecting the incident and the asymptotic scattered field, writes $$\label{eq:scattering_via_Smatrix}
 \begin{bmatrix}
  E_{s, \text{TM}} \\
  E_{s, \text{TE}}
 \end{bmatrix}
  = 
e^{\mathrm{i} 3\pi/4} \, \sqrt{\dfrac{2}{\pi k R}} \, e^{\mathrm{i} k R}
 \begin{bmatrix}
  T_1 & 0    \\
  0   & T_2
 