\label{eq:phi_DC}
\Phi_{DC}(\omega_i,\omega_s)&= \mathrm{sinc}\left[\frac{\left(\Delta k_{DC}(\omega_i,\omega_s)+\frac{2\pi}{G_{DC}}\right)L_{DC}}{2}\right]\nonumber\\
&\times\exp\left({i\frac{\left(\Delta k_{DC}(\omega_i,\omega_s)+\frac{2\pi}{G_{DC}}\right)L_{DC}}{2}}\right)\end{aligned}$$ with a pump pulse that has a full width at half maximum of $\Delta \omega_p$ in the spectral intensity. If the nonlinear crystal with length $L_{DC}$ is periodically poled with poling period $G_{DC}$ to achieve quasi-phase matching, then the efficiency for SPDC is optimal if $\Delta k_{DC}\approx -2\pi/G_{DC}$. The phase mismatch $\Delta k_{DC}(\omega_i,\omega_s)=k_i(\omega_i)+k_s(\omega_s)-k_p(\omega_i+\omega_s)$ includes the dispersion properties of the SPDC crystal through its corresponding Sellmeier equations.

### Manipulation

The spectrum of the entangled photons is manipulated in amplitude and phase by a SLM where the modulator action on each photon is described by a complex transfer function $M^{i,s}(\omega)$. The joint spectral amplitude is transformed by the SLM to $$\label{eq:lambda_mod}
\tilde{\Lambda}(\omega_i,\omega_s)=\Lambda(\omega_i,\omega_s)M(\omega_i,\omega_s)$$ with $$\label{eq:M}
M(\omega_i,\omega_s)=M^i(\omega_i)M^s(\omega_s),$$ where additional restrictions on $M(\omega_i,\omega_s)$ are time-stationarity and $$\label{eq:M_condition}
\vert M^{i,s}(\omega)\vert\leq 1.$$

### Detection

In general, coincidence detection is essential to reveal entanglement. The state $\tilde{\Lambda}(\omega_i,\omega_s)$ could be experimentally detected by a combination of narrow-band frequency filters and single photon counters. However, this would yield a signal proportional to $\left|\tilde{\Lambda}(\omega_i,\omega_s)\right|^2$ which is insensitive to any phase modulation in $M(\omega_i,\omega_s)$. To circumvent this problem we seek for a detection scheme that yields a signal which is proportional to $\left|\mathcal{F}\left\{\tilde{\Lambda}(\omega_i,\omega_s)\right\}\right|^2$, the 2D Fourier 