  With the current revival of interest in astronomical intensity
interferometry, it is interesting to revisit the associated theory, which was
developed in the 1950s and 1960s. This paper argues that intensity
interferometry can be understood as an extension of Fraunhofer diffraction to
incoherent light. Interference patterns are still produced, but they are
speckle-like and transient, changing on a time scale of $1/\Delta\nu$ (where
$\Delta\nu$ is the frequency bandwidth) known as the coherence time. Bright
fringes average less than one photon per coherence time, hence fringes change
before they can be observed. But very occasionally, two or even more photons
may be detected from an interference pattern within a coherence time. These
rare coincident photons provide information about the underlying transient
interference pattern, and hence about the source brightness distribution.
Thinking in terms of transient sub-photon interference patterns makes it easy
to see why intensity interferometry will have large optical-path tolerance, and
be immune to atmospheric seeing. The unusual signal-to-noise properties also
become evident. We illustrate the unobservable but conceptually useful
transient interference patterns, and their observable correlation signal, with
three simulated examples: (i) an elongated source like Achernar, (ii) a
three-star system like Algol, and (iii) a crescent source that roughly mimics
an exoplanet transit or perhaps the M87 black hole environment. Of these, (i)
and (ii) are good targets for currently-planned setups, while (iii) is
interesting to think about for the longer term.
