(N/V)\alpha_0(\omega)/3$ [@Novotny2009PrinciplesNano-optics]. Plugging in the atomic polarizability, one readily finds that [ $$\begin{array}{c}

\label{eq:FTMS_index_prediction_LL}
n_{\text{LL}}(\Delta)=n_{\text{MB}}(\Delta+\pi\eta).
\end{array}$$ ]{} Importantly, while the spectrum is shifted, the LL model still produces a maximum index that grows like $\sim\sqrt{\eta}$.

![ **Frequency-dependent refractive index for different atomic densities.** The solid lines portray the imaginary (subfigure $a$) and real (subfigure $b$) part of the refractive index versus dimensionless detuning $\Delta$, obtained through [Eq. \[eq:CDS\_input-output\_eq\_single\_mode\]]{}, while the dotted lines show the MB predictions. The colors denote different atomic densities (colorbar on right), with the specific values indicated by the dotted white lines. The refractive index is inferred by averaging the complex transmission coefficient $t(\Delta)$ over $\sim 10^3 - 10^4$ atomic configurations. Other system parameters are: thickness $d =0.4\lambda_0$, transverse radius $5 \leq l_{\text{cyl.}}/\lambda_0\leq 7$, beam waist $2.5\leq w_0/\lambda_0 \leq 3$. The insets show the curves at the 3 highest densities as a function of the rescaled detuning $\Delta/\eta$. []{data-label="fig:SM_spectrum"}](FINAL_SM_spectrum.pdf){width="\columnwidth"}

Coupled-dipole simulations
==========================

[Eq. \[eq:FTMS\_output\_field\]]{} and [Eq. \[eq:FTMS\_coupled\_dipoles\]]{} are ubiquitously used to model multiple scattering and interference effects involving a moderate number of point-like scatterers. Here, we briefly introduce some key details of our implementation, which allows us to perform simulations on very high atom number and efficiently 