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