transform of (\[eq:lambda\_mod\]). Such a scheme requires a time resolution better than the inverse of the photon’s spectral bandwidth, which in our experiment is on the order of a few femtoseconds. This is about five orders of magnitudes smaller than the time resolution of the best single photon counters actually available. Therefore, we resort to an optical coincidence method that relies on SFG in a nonlinear crystal. To account for its acceptance bandwidth, we define the modified joint spectral amplitude $$\label{eq:gamma}
\Gamma(\omega_i,\omega_s)\propto\Lambda(\omega_i,\omega_s)\Phi_{SFG}(\omega_i,\omega_s)$$ with $$\begin{aligned}
\label{eq:phi_UC}
\Phi_{SFG}(\omega_i,\omega_s)&= \mathrm{sinc}\left[\frac{\left(\Delta k_{SFG}(\omega_i,\omega_s)-\frac{2\pi}{G_{SFG}}\right)L_{SFG}}{2}\right]\nonumber\\
&\times\exp\left({i\frac{\left(\Delta k_{SFG}(\omega_i,\omega_s)-\frac{2\pi}{G_{SFG}}\right)L_{SFG}}{2}}\right).\end{aligned}$$ Analogous to $\Phi_{DC}(\omega_i,\omega_s)$, the length and the poling period of the SFG crystal are denoted by $L_{SFG}$ and $G_{SFG}$ with a phase mismatch $\Delta k_{SFG}(\omega_i,\omega_s)=k_p(\omega_i+\omega_s)-k_i(\omega_i)-k_s(\omega_s)$. The temporal resolution of the SFG-based detection process is governed by the inverse width of $\Phi_{SFG}(\omega_i,\omega_s)$ and is sufficiently short. In the following we neglect the additional phase factors in (\[eq:phi\_DC\]) and (\[eq:phi\_UC\]) since they cannot be distinguished from other dispersion contributions in the setup and are assumed to be compensated in the experiment. The detected signal after the SFG process is given by $$\label{eq:signal}
S\propto\left|\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}d\omega_id\omega_s\Gamma(\omega_i,\omega_s)M(\omega_i,\omega_s)\right|^2$$ and is sensitive to a phase in the transfer function.

Finite spectral resolution
--------------------------

Because of the finite spectral 