TLI.

### Vanishing grating echo at the echo time

In Eq. (\[npulserule\]) the only initial condition dependent term is the last term in Eq. (\[npulserule\]). For $T_N=T^{echo}$ such that $W({\bf{q}}_1 =0 ,{\bf{x}}_1=0)=1$, the TLI output is not only maximized, but also becomes initial-condition independent. This is the feature of a “white light interference” as the interference is independent of the degree of input coherence. However, as noticed by the authors of Ref. [@Periodic02], if $T_N $ is set to be exact the echo time $T^{echo}$, the TLI output given by Eq. (\[npulserule\]) vanishes since in the pre-factors of Eq. (\[npulserule\]) $J_{n_1 } (2\theta _1 \sin \frac{{{\bf{Q}}_1
\cdot {\bf{x}}_1 }}{2})$ becomes zero for non-zero $n_1$. Thus TLI using phase gratings are not “perfectly-white-light” interferometers.

The vanishing grating echo at the echo time is understandable, as we notice the revived fringe at time $T_N$ is a lens-less image of atomic density grating at time $t=T_N-T^{echo}$ by the following N-2 gratings. If $T_N$ is equal to $T^{echo}$, the atomic distribution to be imaged is at $t=T_1$, the first pulse arriving time. Since the first grating is a phase grating, the input matter-waves only acquire a periodic phase at $t=T_1^+$ but keeps a uniform 