itself rather than from the vacuum [@Chang2012CavityMirrors]. Similarly, we approximate the near-resonant input field as $E_{\text{in}}({\mathbf{r}}_i,\omega)\simeq E_{\text{in}}({\mathbf{r}}_i,\omega_0)$.

The expression in [Eq. \[eq:CDS\_input-output\_eq\_single\_mode\]]{} represents a useful closed-form definition of the transmission coefficient $t(\Delta)$, which avoids a numerically expensive point-by-point evaluation of the scattered field ${\mathbf{E}}({\mathbf{r}},\omega)$, as nominally prescribed by [Eq. \[eq:FTMS\_output\_field\]]{}. We can extrapolate the complex index of refraction $n(\Delta)$ from the relation [ $$\begin{array}{c}

\label{eq:CDS_refractive_index_def_from_t}
{\langle t(\Delta) \rangle}= \exp \left\{ i  \left[n\left(\Delta \right)-1\right] k_0 d\right\}.

\end{array}$$ ]{} where the averages are performed over $\sim 10^3 - 10^4$ sets of random positions, for each fixed density. Unlike in a smooth medium, we have that $|{\langle  t(\Delta) \rangle}|^2 \neq {\langle |t(\Delta)|^2 \rangle}$. Nevertheless, our definition of the index coincides with that often used within atomic physics (e.g. in phase contrast or absorption imaging of a Bose-Einstein condensate [@Davis1995Bose-EinsteinAtoms; @Anderson1995ObservationVapor]). In the Appendix B, we discuss the independence of the calculated index from the thickness $d$, which is implicitly assumed in [Eq. \[eq:CDS\_refractive\_index\_def\_from\_t\]]{}. Alternatively, one might assume that the calculated ${\langle  t(\Delta) \rangle}$ approximately coincide with the finite-slab Fresnel coefficients for a smooth material [@Born1999PrinciplesOptics]. This produces an alternative way to extrapolate the index, which we find yields quantitatively similar results as what 