  In this article, we introduce a universal simulator covering all regimes of
matter wave light-pulse elastic scattering. Applied to atom interferometry as a
study case, this simulator solves the atom-light diffraction problem in the
elastic case i.e. when the internal state of the atoms remains unchanged.
Taking this perspective, the light-pulse beam splitting is interpreted as a
space- and time-dependent external potential. In a shift from the usual
approach based on a system of momentum-space ordinary differential equations,
our position-space treatment is flexible and scales favourably for realistic
cases where the light fields have an arbitrary complex spatial behaviour rather
than being mere plane waves. Moreover, the numerical package we developed is
effortlessly extended to the problem class of trapped and interacting
geometries, which have no simple formulation in the usual framework of
momentum-space ordinary differential equations. We check the validity of our
model by revisiting several case studies relevant to the precision atom
interferometry community. We retrieve analytical solutions when they exist and
extend the analysis to more complex parameter ranges in a cross-regime fashion.
The flexibility of the approach, the insight it gives, its numerical
scalability and accuracy make it an exquisite tool to design, understand and
quantitatively analyse metrology-oriented matter-wave interferometry
experiments.
