({\bf q}\cdot \hat {\bf r} - {\bf
x}\cdot \hat {\bf p})}.
\end{array} \label{weyldefine}$$ Here $\hat {\bf r}$ and $\hat {\bf p}$ are the position and momentum operators, and $\hat D({\bf q},{\bf x})$ is the displacement operator [@dispoperators], that shifts the density matrix by $\hbar {\bf q}$ in momentum and ${\bf x}$ in position. Thus the Weyl function $W({\bf q},{\bf x})$ is a phase-space correlation function that measures the coherence of matter-waves. Though both Weyl and Wigner functions completely specify any density matrix operators, application of Weyl function, or the Fourier transform of the Wigner function, has been much less discussed before [@weylChountasis].

In this work, we demonstrate the power of the Weyl function technique by applying it to an important class of wave interference phenomena, the Talbot-Lau effects associated with matter-waves in quadratic or periodic potentials. Various examples of Talbot-Lau type interference, including those closely related to recent experiments [@longcoherence; @movingguide; @mythesis], are discussed. To our knowledge, this work gives the first vectorial and diagrammatic formula of Talbot-Lau effects. Apart from the expected application in matter-wave optics, the formula may also provide a convenient framework for the design of various lens-less imagine systems [@fish06].

Talbot-Lau effects
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Periodic light wavefronts 