from realization to realization, we project [Eq. \[eq:FTMS\_output\_field\]]{} back into the same Gaussian mode as the input, as can be experimentally enforced by recollecting the transmitted light through a single mode fiber. This results in a transmission coefficient $t(\Delta)$ given by [@Chomaz2012AbsorptionAnalysis; @Manzoni2018OptimizationArrays] [ $$\begin{array}{c}

\label{eq:CDS_input-output_eq_single_mode}
t(\Delta)=1+\dfrac{3 i}{(w_0k_0)^2 }{
\displaystyle \sum 
}_{j=1}^N \dfrac{  E^*_{\text{in}}({\mathbf{r}}_j,\omega_0)}{E_0}  c_j(\Delta),
\end{array}$$ ]{} where $E_0$ is the input field amplitude at the beam focus. Here, for convenience, we have defined re-scaled dipole amplitudes $c_j(\Delta) = d_j(\omega) k_0^3/(3\pi \epsilon_0 E_0) $, which satisfy the dimensionless coupled equations [ $$\begin{array}{c}

\label{eq:CDS_input-output_eq_single_mode2}
-\Delta c_i(\Delta)-\displaystyle
{
\displaystyle \sum 
}_{j=1}^{N}  G_{ij} c_j(\Delta)   = \; \dfrac{ E_{\text{in}}({\mathbf{r}}_i,\omega_0)}{E_0},

\end{array}$$ ]{} In these equations, we define $G_{ij}\equiv ( 3\pi/k_0) {\hat{\mathbf{x}}} \cdot {{\Bar{\Bar{\mathbf{G}}}}}({\mathbf{r}}_i,{\mathbf{r}}_j,\omega_0)\cdot {\hat{\mathbf{x}}}$ and $G_{jj}=i/2$, which coincides with the single-atom decay rate in units of $\Gamma_0$, while regularizing the divergent self-energy associated with the real part of ${{\Bar{\Bar{\mathbf{G}}}}}$. Note that, for simplicity, the Green’s function ${{\Bar{\Bar{\mathbf{G}}}}}({\mathbf{r}}_i,{\mathbf{r}}_j,\omega_0)$ is only evaluated at the atomic resonance frequency, in order to ease the computational cost as the detuning is varied. Ignoring the dispersion of ${{\Bar{\Bar{\mathbf{G}}}}}$ is an excellent approximation for near-resonant atoms, as the optical dispersion and delay of such a system is dominated by the atomic response 