density profile. A density grating is developed only after a $\theta_1$-dependent time during which the atoms bunch to the nodes of standing wave. The vanishing of the density grating at the echo time is a feature of TLE using phase gratings. Extension of interferometry with phase gratings to those with amplitude gratings will be discussed in the last section of this paper.

For a hot atomic sample with $\lambda_c<<\lambda$, we may Taylor expand the $J_{n_1}$ term in Eq. (\[npulseruleb\]) to have $E^b
\propto \tau^{n_1} {\emph F}(Q_N \tau)$. For $n_1= 1$ and a Maxwell velocity distribution of $f({\bf p})$, we get a dispersive-shaped backscattering curve as a function of $\tau$, consistent with experimental observations [@TLCahn97].

### Minimum number of pulses in D dimensions

Finally, we notice that Eqs. (\[qconserve\]) and  (\[xconserve2\]) are two vector equations. In a particular experimental setup with fixed grating vectors ${\mathbf{Q}}_i $ and timing $T_i $ for each standing wave pulses, a non-vanishing grating echo requires Eq. (\[qconserve\]) and Eq. (\[xconserve2\]) have non-zero integer solutions for $\{ n_i ,i = 1,...,N - 1\}$. Generally, for $\{ {\mathbf{Q}}_i ,i = 1,...,N\} $ which span a $D$ dimension, a minimum number of standing wave pulses are required to satisfy 