A novel spectral formulation is proposed here for the coupling integral between a cylindrical wave with arbitrary radial wavenumber and azimuthal dependence, and a slot placed on a plane orthogonal to the wavefront. Such a kind of integral is massively encountered in electromagnetic modeling of stratified structures comprising slots and vertical pins or vias. The twofold spatial integral defining the coupling is transformed into an equivalent onefold spectral integral, the integration path of which is selected to obtain Gaussian decay of the integrand. Both propagating and evanescent cylindrical modes are considered. An arbitrary stratification orthogonal to the slot plane can be considered, as well as an arbitrary current on the slot. Numerical comparisons against the standard spatial approach are shown to validate the new formulation, and its advantage in terms of computation cost is investigated in depth.