the three-step phase retrieval algorithm described in Ref. [@Malacara1992OpticalShopTesting] to retrieve the phase modulations $\phi_{j}(z_{2})$. The determination of the phase and amplitude of the beam in the CCD plane allows us to numerically vary the ROI without redisplaying the test fields. Using these fields, the IO and SSO are determined according to Eqs.  and , respectively.

During the course of our experiments we verified the linearity of our optical system by performing a comparison between what we term the “experimental superposition (Exp-S)” and the “numerical superposition (Num-S)”. The Exp-S refers to the case where the set of OEi optimized superposition coefficients $a_{i}$ is used to encode the optimized superimposed field onto the SLM. The CCD camera then detected the intensity $I_{\textrm{Exp-S}}(z_{2})$ corresponding to this encoded optimized field. The Num-S utilizes the fields $E_{j}(z_{2})$, which were individually measured to assemble the OEi operators, in order to *numerically* determine the intensity distribution as $I_{\textrm{Num-S}}(z_{2})\propto\left|\sum_{i=1}^{N}a_{j}E_{j}(z_{2})\right|^{2}$. Crucially, linearity is verified if $I_{\textrm{Exp-S}}(z_{2})=I_{\textrm{Num-S}}(z_{2})$. This is indeed observed in our experiments as demonstrated in the following subsection which features a comparison of experimental and numerical intensity distributions.

Results and discussion
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In our experiments, we used $N = 11$ non overlapping amplitude ring masks with a 