We develop a full vectorial theoretical investigation of the chemical interface detection in conventional coherent anti-Stokes Raman scattering (CARS) microscopy. In Part I, we focus on the detection of axial interfaces (i.e., parallel to the optical axis) following a recent experimental demonstration of the concept [Phys. Rev. Lett. 104, 213905 (2010)]. By revisiting the Young's double slit experiment, we show that background-free microscopy and spectroscopy is achievable through the angular analysis of the CARS far-field radiation pattern. This differential CARS in k space (Dk-CARS) technique is interesting for fast detection of interfaces between molecularly different media. It may be adapted to other coherent and resonant scattering processes.