  A theoretical study of the performance of single-mode coupled spontaneous
parametric downconversion sources is proposed, which only requires very few
assumptions of practical interest : narrow-bandwidth and quasi-degenerate
collinear generation. Other assumptions like pump beam spatial and temporal
envelopes, target single-mode profile and size, and non-linear susceptibility
distribution, are only taken into account in the final step of the computation,
thus making the theory general and flexible. Figures of merit for performance
include absolute collected brightness, pair collection efficiency and heralding
ratio. The optimization of these values is investigated through functions that
only depend on dimensionless parameters, allowing for deducing from the results
the best experimental configuration for a whole range of design choices (e.g.
crystal length, pump power). A particular application of the theory is
validated by an experimental optimization obtained under compatible
assumptions. A comparison with other works and proposals for numerically
implementing the theory in its most generality are also provided.
