\Phi_u(\tau)=\frac{\Psi(\Omega)}{2} \exp
\left[ i(\omega_0 +\Omega) \tau_0 \right] \left\{1+
\exp \left[ i (\omega_0+\Omega) \tau-i\Gamma \right] \right\} \label{projections1} \\
& & \Phi_v(\tau)=\frac{\Psi(\Omega)}{2} \exp \left[ i \left(
\omega_0+ \Omega \right)\tau_0 \right] \left\{ 1-\exp \left[ i
(\omega_0+\Omega) \tau -i\Gamma \right] \right\},
\label{projections2}\end{aligned}$$ where $\Gamma=\pi/2+\theta$.

![Spectrum measured at the output. (a) and (b): Spectral shape of the mode functions for $\tau=0$ (solid blue line) and $\tau=100$ as (dashed green line). In (a) the post-selection angle $\theta$ is $97.2^{\circ}$, so as to fulfil the condition $\omega_0 \tau-\Gamma=\pi$. In (b) the angle $\theta$ is $96.7^{\circ}$. (c) Shift of the centroid of the spectrum of the output pulse after projection into the polarisation state $\hat{u}$ in PBS$_2$, as a function of the post-selection angle $\theta$. Green solid line: $\tau=10$ as; Dotted red line: $\tau=50$ as, and dashed blue line: $\tau=100$ as. Label [**I**]{} corresponds to $\theta=96.7^{\circ}$ \[mode for $\tau=100$ as shown in (b)\]. Label [**II**]{} corresponds to $\theta=97.2^{\circ}$, where the condition $\omega_0 \tau-\Gamma=\pi$ is fulfiled \[mode for $\tau=100$ shown in (a)\]. It yields the minimum mode overlap between states with $\tau=0$ and $\tau \neq 0$. Data: $\lambda_0=1.5\, \mu$m and $T_0=100$ fs.[]{data-label="figure_modes"}](figure2.eps){width="90.00000%"}

After the signal projection performed after PBS$_2$, the WVA scheme distinguishes different states, corresponding to different values 