of $W({\bf{q}},{\bf{x}})$ is simply a shearing in reciprocal phase space along $\bf x$. From Eq. (\[wignerrule2\]), a standing wave pulse “scatters” $W({\bf{q}},{\bf{x}} ,t)$ into multiple duplicates shifted along the ${\bf{q}}$ coordinate by $n{\bf{Q}}$ weighted by $J_n (2\theta \sin \frac{{{\bf{Q}} \cdot
{\bf{x}}}}{2})$. We define $ W({\bf{q}}_n (t),{\bf{x}}_n (t)) = (
W({\bf{q}},{\bf{x}},t))_n$, e.g., the component that is shifted by $n{\bf{Q}}$ due to standing wave scatterings, and represent its dynamics using a point $({\bf{q(t)}},{\bf{x(t)}})$ that is moving and scattered in $({\bf{q}},{\bf{x}})$ space. The trajectory can be projected onto ${\bf{x}}$ coordinates in a ${\bf{x}} - t$ diagram, where the slope of ${\bf{x}}(t)$ represents the magnitude of ${\bf{q(t)}}$. Equation (\[wignerrule1\]) and (\[wignerrule2\]) are diagrammatically summarized in Fig. \[proprulefig\](a) and (b) as the propagation diagram ([*displacement line*]{}) and scattering diagram ([*scattering vertex*]{}) for the Weyl Functions. In addition, to calculate $E_N^b(T_N)$ fully diagrammatically, we introduce the probing diagram ([*probing vertex*]{}) in Fig. \[proprulefig\](c) to represent the Bragg-scattering given by Eq. (\[backamp2\]).

### The displacement diagram for an N-pulse TLI

![A diagram corresponding to a particular displacement line in a general N-pulse TLI.[]{data-label="npulserulefig"}](npulserulesfig.eps "fig:"){width="5"}\

With further explanations coming shortly, we sketch a representative path of $({\bf{q}}(t),{\bf{x}}(t))$ that contributes to the grating echo in the setup sketched in 