is a specific displacement line that connects $({\bf{q}}_1=0 ,{\bf{x}}_1 = 0)$ and $({\bf{q}}_N={\bf{Q}}_N ,{\bf{x}}_N=0)$. For the echo line, the input Weyl function evaluated by Eq. (\[wignerbeta3\]) is maximized. An echo line for the N-pulse diagram in Fig. \[npulserulefig\] is given by the dotted line in Fig  \[echofig\].

### Envelop function of grating echo

With the echo time defined by Eq. (\[xconserve2\]) and with the momentum conservation condition Eq. (\[qconserve\]), it is usually more convenient to write: $${\bf{x}}_1(\tau)   ={\bf Q_N}\tau, \label{equx1}$$ with $\tau = T_N - T^{echo}  = \frac{{x_1 (T_N )}}{{Q_N }}$. We can then express the interferometer output as a function of $\tau $ , the time offset of the probing time $T_N$ from the echo time $T^{echo}$. From Eq. (\[xconserve1\]) we see the grating echo given by Eq. (\[npulserule\]) has a $\tau$ dependent envelop function ${\emph F}(Q_N \tau)$ with its magnitude a symmetric function with respect to $\tau=0$ and reaches unity at $\tau=0$.

### Total interrogation time

Notice that when the echo condition Eq. (\[xconserve2\]) is satisfied for $\{T_1=0, T_2, ..., T_N=T^{echo}=T\}$, it must also be satisfied for $T'\{s_1=0, s_2, ..., s_N=s^{echo}=1\}$, with $s_i
\propto T_i$ for arbitrary $T'$. $T$ shall be called the total interrogation time of the 