extract the index.

First, one conceptually straightforward way to extract the complex refractive index of a material would be to take an extended slab of finite thickness $d$, and investigate the phase shift and attenuation of a quasi-plane-wave incident field upon transmission. We approximately realize such a situation by taking atoms with a fixed density in a cylindrical volume, illuminated by a weakly focused, near-resonant Gaussian beam ([Fig. \[fig:system\_representation\]]{}-a). The parameters are chosen such that the beam waist $w_0$ is small compared to the radius of the system, so that diffraction effects from the edges are negligible. We also avoid very tight focusing $w_0\lesssim \lambda_0$, where non-paraxial effects could emerge.

We must also specify a practical definition of index, for a granular system as ours. In particular, since our atoms are purely scattering and have no absorption, it is well-known [@Shapiro1986LargeMedia; @Lagendijk1996ResonantLight; @Fayard2015IntensityPatterns; @Schilder2017HomogenizationScatterers] that for a fixed random spatial configuration, an input as in [Fig. \[fig:system\_representation\]]{} produces a complex “speckle" pattern in the outgoing intensity when the system is optically dense, due to multiple scattering and interference, as exemplified in [Fig. \[fig:system\_representation\]]{}-b. To isolate the part of the field that possesses a well-defined phase relationship with the incident field 