long and the short arm and an additional absolute phase $\phi_{i,s}$. The total transfer function of the scheme depicted in figure \[fig:time\_bins\_franson\] then reads $M_{FR}(\omega_i,\omega_s)=M_{FR}^{i}(\omega_i)M_{FR}^{s}(\omega_s)$. If we restrict in (\[eq:time\_bins\_freq\]) the width of the bins to $\Delta t_0=\Delta t_1=\Delta t$ and put $t_0=0$ fs such that $\Delta t_{10}=t_1$, the general transfer function from (\[eq:mslm\]) transforms to $$\label{eq:M_time_bins}
M^{i,s}(\omega)=\sqrt{\frac{\Delta t}{2\pi}}\mathrm{sinc}\left(\frac{\omega \Delta t}{2}\right)\left(\vert u_0^{i,s}\vert+\vert u_1^{i,s}\vert\e^{i\bar{\phi}_{i,s}(\omega)}\right)$$ for $d=2$ with $\bar{\phi}_{i,s}(\omega)=\omega t_1+\phi_{i,s}$. In the limit $\Delta t_j\rightarrow 0$, the time bins in (\[eq:time\_bins\]) are reduced to $f_{j}^{i,s}(t)=\sqrt{2\pi}\delta(t-t_j)$. Equation (\[eq:M\_time\_bins\]) then takes the form $$\label{eq:time_bin_dt0}
M^{i,s}(\omega)=\frac{1}{\sqrt{2\pi}}\left(\vert u_0^{i,s}\vert+\vert u_1^{i,s}\vert\,\e^{i\bar{\phi}_{i,s}(\omega)}\right)$$ and is equal to (\[eq:M\_franson\]) with $\vert u_0^{i,s}\vert=T$ and $\vert u_1^{i,s}\vert=R$ up to a normalization constant.

### Schmidt mode basis {#Schmidt}

The Schmidt decomposition for continuous variable systems and its application to quantify entanglement has been extensively studied in [@law2000; @parker2000]. To decompose a state which is as close as possible to the state at the position of the SLM, we have to consider the modified joint spectral amplitude $\Gamma_{PSF}(\omega_i,\omega_s)$ taking into account the effect of the finite spectral resolution. Because of being bipartite and pure, the two-photon state of (\[eq:gammatwophstate\]) can be represented in a Schmidt decomposition $$\begin{aligned}
\Gamma_{PSF}\left(\omega_{i},\omega_{s}\right)&=\sum_{j=0}^{\infty}\sqrt{\beta_j}\,f^{i}_j\left(\omega_i\right)f^{s}_j\left(\omega_s\right)\nonumber\\
&\approx\sum_{j=0}^{d-1}\sqrt{\beta_j}\,f^{i}_j\left(\omega_i\right)f^{s}_j\left(\omega_s\right),\label{eq:gamma_schmidt}\end{aligned}$$ where the real valued functions $f^{i,s}_j\left(\omega\right)$ 