a Bell inequality, i.e. $I_d>2$, indicates entanglement between the two systems involved. Due to a left-open locality loophole in our detection method, however, the subsequent experimental results can not be considered as a test of non-locality. Because of $\Tr\left(\hat{B}\hat{\rho}^{(d)}\right)=\lambda_d I_d^{max}$, the aforementioned inequality can be reformulated in terms of the mixing parameter $$\lambda_d>\frac{2}{I_d^{max}}\doteq\lambda_d^c$$ with its critical value $\lambda_d^c$. The value of $I_d$ is related to the visibility of two-photon interferences which are obtained by projecting (\[eq:snmodel\]) onto $$\label{eq:chi_phi}
\vert\chi\rangle=\frac{1}{d}\left(\sum_{j=0}^{d-1}\e^{-i j \phi_i}\vert j \rangle_i\right)\left(\sum_{k=0}^{d-1}\e^{-i k \phi_s} \vert k\rangle_s\right)$$ with $\phi_i=\phi_s=\phi$. The phase $\phi$ is varied by the SLM and controls the interference in the following experiments. The theoretical coincidence signals then read $$\begin{aligned}
\label{eq:fitd2}
S_\lambda ^{(2)}(\phi)&=\Tr\left(\hat{\rho}^{(2)}\vert\chi\rangle\langle\chi\vert\right)\nonumber\\
&\propto 1+\lambda_2\cos(2\phi+\phi_0),\end{aligned}$$ $$\begin{aligned}
\label{eq:fitd3}
S_\lambda ^{(3)}(\phi)&=\Tr\left(\hat{\rho}^{(3)}\vert\chi\rangle\langle\chi\vert\right)\nonumber\\
&\propto 3+2\lambda_3\left[2\cos(2\phi+\phi_0)+\cos (2(2\phi+\phi_0))\right],\end{aligned}$$ $$\begin{aligned}
\label{eq:fitd4}
S_\lambda ^{(4)}(\phi)&=\Tr\left(\hat{\rho}^{(4)}\vert\chi\rangle\langle\chi\vert\right)\nonumber\\
&\propto 4+2\lambda_4[3\cos(2\phi+\phi_0)+2\cos(2(2\phi+\phi_0))\nonumber\\
&\quad+\cos (3(2\phi+\phi_0))]\end{aligned}$$ and are used to fit the experimental data with the free parameters $\phi_0$ and $\lambda_d$. The latter accounts for white noise as well as for the point spread function and for couplings between frequency and transverse modes due to a possible misalignment in the experimental setup. Since we experimentally demonstrate qudits in terms of interference patterns, we make use of the fact that $\lambda_d$ can be related to 