  All classical and quantum technologies that encode in and retrieve
information from optical fields rely on the ability to selectively manipulate
orthogonal field modes of light. Such manipulation can be achieved with high
selectivity for polarization modes and transverse-spatial modes. For the
time-frequency degree of freedom, this could efficiently be achieved for a
limited choice of approximately orthogonal modes, i.e. non-overlapping bins in
time or frequency. We recently proposed a method that surmounts the selectivity
barrier for sorting arbitrary orthogonal temporal modes [Opt. Lett. {\bf 39},
2924 (2014)] using cascaded interferometric quantum frequency conversion in
nonlinear optical media. We call this method temporal-mode interferometry, as
it has a close resemblance to the well-known separated-fields atomic
interferometry method introduced by Ramsey. The method has important
implications for quantum memories, quantum dense coding, quantum teleportation,
and quantum key distribution. Here we explore the inner workings of the method
in detail, and extend it to multiple stages with a concurrent asymptotic
convergence of temporal-mode selectivity to unity. We also complete our
analysis of pump-chirp compensation to counter pump-induced nonlinear
phase-modulation in four-wave mixing implementations.
