an optical system (e.g., if it is shaped as a lens or prism) can still drastically alter the overall optical response. Separately, we note that although the problem of just two atoms ([Fig. \[fig:RG\_plots\]]{}-b) can be interpreted in terms of renormalized resonance frequencies and linewidths, in the many-atom case, we *only* renormalize the resonance frequencies. Physically, this is because the decay rates of collective many-atom eigenstates reflect the rate of radiation of energy into the far field. This global property, which depends on the interference of all atoms, then generally cannot be derived from considering the isolated properties of individual pairs alone. Mathematically, this can be seen as the imaginary part of $G_{ij}(\rho_{ij})$, which dictates the collective linewidths, does not contain a $1/\rho_{ij}^3$ near-field component.

To validate the RG approach, we can use [Eq. \[eq:CDS\_input-output\_eq\_single\_mode\]]{} and [Eq. \[eq:CDS\_input-output\_eq\_single\_mode2\]]{} (with the near-field interactions of renormalized atoms suitably removed, see Appendix A) to calculate the maximum real index as a function of density $\eta$ of the ensemble with renormalized resonance frequencies. This is plotted in [Fig. \[fig:RG\_plots\]]{}-c (green), along with exact numerical simulations (blue) of [Eq. \[eq:CDS\_input-output\_eq\_single\_mode\]]{} for the original system of identical atoms. These curves show good agreement for all densities, 