  Interferometers provide a highly sensitive means to investigate the coherence
properties and to exploit the coherence properties of light in metrology
applications. However, interferometers come in many various forms and exploit
different properties of the optical states within. In this paper, we introduce
a classification scheme that characterizes any interferometer based on the
number of involved nonlinear elements by studying their influence on
single-photon and photon-pair states. Several examples of specific
interferometers from these more general classes are discussed, and the theory
describing the expected first-order and second-order coherence measurements for
single-photon and single-photon pair input states is presented. These
theoretical predictions are then tested in an innovative experimental setup
that one is easily able to switch between implementing an interferometer
consisting of only one or two nonlinear elements. The resulting singles and
coincidence rates are measured in both configurations and the results are seen
to fit well with the presented theory. The measured results of coherence are
tied back to the presented classification scheme, revealing that our
experimental design can be useful in gaining insight into the properties of the
various interferometeric setups containing different degrees of nonlinearity.
