Fig. \[gratingechoTLIfigure\]. The path starts from $({\bf{q}}_1,{\bf{x}}_1)$ at time $t=T_1^-$, propagate toward $({\bf{q}}_N={\bf{Q}}_N ,{\bf{x}}_N=0)$ at $T_N$, scattered by N-1 scattering vertex at ${\bf x}_i\equiv {\bf x}(T_i)$ with pulse area $\theta _i $ and standing wave k-vector ${\bf{Q}}_i
$, with diffraction orders $n_i ,i = 1,...,N - 1$ assigned at each scattering vertex, with $n_N = 1$ and $T_{i,j}  = T_i - T_j $. By evaluating the Bessel functions at each vertex using the rules in Fig. \[proprulefig\], the contribution of the path to TLI output is given by:

$$\begin{array}{l}
E_N^b (T_N ) =   - ig \sum\limits_{\{ n_i \} } {[\prod\limits_{i =
1}^{N - 1} {J_{n_i } (2\theta _i \sin \frac{{{\bf{Q}}_i  \cdot
{\bf{x}}_i }}{2})}]}
W({\bf{q}}_1, {\bf{x}}_1,t={T_1^- }),  \\
with: {\bf{q}}_i  = \sum\limits_{j = i}^N {n_j {\bf{Q}}_j }
,{\bf{x}}_i
= \sum\limits_{j = i}^N {n_j {\bf{Q}}_j } T_{j,i}. \\
 \end{array}\label{npulserule}$$

In Eq. (\[npulserule\]) we include a summation of all possible $\{n_i\}$ paths that connect $({\bf{q}}_1 ,{\bf{x}}_1)$ at $t=T_1^-$ with $({\bf{q}}_N={\bf{Q}}_N ,{\bf{x}}_N=0)$ at $T_N$.

Equation (\[npulserule\]) can be evaluated with the displacement diagram in Fig. \[npulserulefig\] and the rules given in Fig. \[proprulefig\]. This is the first major result of this work. To proceed to a calculation based on any initial atomic states, we 