generally consider an interferometry configuration using TLE as a Talbot-Lau Interferometer, that will be referred to with the abbreviation TLI in this work.

Outline
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In what follows the paper is organized into three sections. In the first section we introduce a diagrammatic technique to calculate the grating echo of an N-pulse TLI in free-space. In the second part we discuss the application of the theory to matter-waves in a quadratic potential. In the discussion section we extend the formula to include arbitrary grating transmission functions, discuss the Weyl function representation of matter-wave coherence, and compare the approach here with other diagrammatic techniques  [@billiardball82].

Grating echo due to TLE in free space \[firstsection\]
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The model \[sectionmodel1\]
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![Schematic of an N-pulse grating echo Talbot-Lau interferometer[]{data-label="gratingechoTLIfigure"}](gratingechoTLI.eps "fig:"){width="5in"}\

In this section we consider TLE due to the simplest possible interaction between 2-level atoms and a sequence of off-resonant standing wave light pulses. The scenario is summarized in Fig. \[gratingechoTLIfigure\]: an atomic sample localized at the waists of multiple standing wave fields is subjected to a sequence of N-1 standing wave pulses. At time $T_N$ a grating echo is induced with the amplitude proportional to a particular spatial Fourier component of the 