$w_0=3\lambda_0$, cylinder radius $l_{\text{cyl.}}=7 \lambda_0$ and thickness $d=2\lambda_0$. []{data-label="fig:system_representation"}](FINAL_SM_system_and_speckles.pdf){width="\columnwidth"}

Formal theory of multiple scattering
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We consider a minimal system consisting of $N$ identical, stationary two-level atoms. The atoms are assumed to consist of a ground and excited state ${\left|g\right\rangle}$, ${\left|e\right\rangle}$, with frequency difference $\omega_0$ and associated wavelength $\lambda_0$, and which have an optically allowed transition with a dipole matrix element along a fixed axis (say ${\hat{\mathbf{x}}}$), as depicted in [Fig. \[fig:scatt\_cross\_sec\]]{}-a. The excited states of the atoms decay purely radiatively, with a rate of $\Gamma_0$ for a single, isolated atom. As we are specifically interested in the linear refractive index, it is sufficient to treat atoms in the limit of classical, polarizable, radiating dipoles. In order to investigate the frequency-dependent index $n(\omega)$, we consider that the atoms are driven by a monochromatic, linearly-polarized input beam ${\mathbf{E}}_{\text{in}}({\mathbf{r}},\omega)=E_{\text{in}}({\mathbf{r}},\omega){\hat{\mathbf{x}}}$, whose polarization aligns with the polarizability axis of the atoms. Each atom $j$ acquires a dipole moment ${\mathbf{d}}_j(\omega)=d_j(\omega){\hat{\mathbf{x}}}$, as a result of being driven by the total field, which consists of the sum of the incident field and fields re-scattered from other atoms. Formally, the total field can be expressed as [@Novotny2009PrinciplesNano-optics] [ $$\begin{array}{c}

\label{eq:FTMS_output_field}
{\mathbf{E}}({\mathbf{r}}, \omega)= {\mathbf{E}}_{\text{in}}({\mathbf{r}}, \omega) +\mu_0 \omega^2{
\displaystyle \sum 
