  We present a theoretical analysis of the process of spontaneous parametric
down conversion (SPDC) in a non-linear crystal characterized by a
linearly-chirped X(2) grating along the direction of propagation. Our analysis
leads to an expression for the joint spectral amplitude, based on which we can
derive various spectral-temporal properties of the photon pairs and of the
heralded single photons obtained from the photon pairs, including: the single
photon spectrum, the chronocyclic Wigner function and the Schmidt number. The
simulations that we present are for the specific case of a collinear SPDC
source based on a PPLN crystal with the signal and idler photons emitted close
to the telecom window. We discuss the mechanism for spectral broadening due to
the presence of a linearly chirped X(2) grating, showing that not only the
width but also to some extent the shape of the SPDC spectrum maybe controlled.
Also, we discuss how the fact that the different spectral components are
emitted on different planes in the crystal leads to single-photon chirp.
