\Phi_0|\Phi_1\rangle|^2 \nonumber \\
& & =|\langle \Phi(\tau_0)|\Phi(\tau_0)\rangle_x|^2  |\langle
\Phi(\tau_0)|\Phi(\tau_0+\tau)\rangle_y|^2  \nonumber \\
& & = \exp \left[-N \left( 1-\gamma \cos \omega_0 \tau\right)
\right], \label{input_result}\end{aligned}$$ where $$\gamma= \exp \left( -\ln 2 \,\frac{\tau^2}{T_0^2} \right).$$

![Mode overlap and insertion loss as a function of the post-selection angle. Mode overlap $\rho$ of the mode functions corresponding to the quantum states with $\tau=0$ and $\tau=100$ as, as a function of the post-selection angle $\theta$ (solid blue line). The insertion loss, given by $10\log_{10}\,P_{\mathrm{out}}/P_{\mathrm{in}}$ is indicated by the dotted green line. The minimum mode overlap, and maximum insertion loss, corresponds to the post-selection angle $\theta$ that fulfils the condition $\omega_0 \tau-\Gamma=\pi$, which corresponds to $\theta=97.2^{\circ}$. Data: $\lambda_0=1.5 \,\mu$m, $T_0=100$ fs.[]{data-label="figure_overlap_loss"}](figure3.eps){width="50.00000%"}

In the WVA scheme considered here, the signal after PBS$_2$ is projected into the orthogonal polarisation states $\hat{u}$ and $\hat{v}$, and as a result the signals in both output ports are given by Eqs. (\[projections1\]) and (\[projections2\]). Making use of Eqs. (\[projections1\]), (\[projections2\]) and (\[overlap2\]) one obtains that the mode overlap (for $\Phi_u$) reads $$\rho =\frac{1+\cos \Gamma+\gamma \cos
\omega_0 \tau+\gamma \cos (\omega_0 \tau-\Gamma) - i \left[ \sin
\Gamma+\gamma \sin \omega_0 \tau+\gamma \sin (\omega_0
\tau-\Gamma) \right]}{2\left[1+\cos \Gamma \right]^{1/2}
\left[1+\gamma \cos (\omega_0 \tau-\Gamma) \right]^{1/2}}.$$ For $\tau=0$, and therefore $\gamma=1$, 