  Optical $\chi^{(2)}$ non-linearity can be used for parametric amplification
and producing down-converted entangled photon pairs that have broad
applications. It is known that weak non-linear media exhibit dispersion and
produce a frequency response. It is therefore of interest to know how spectral
effects of a strong $\chi^{(2)}$ crystal affect the performance. Here we model
the spectral effects of the dispersion of a strong $\chi^{(2)}$ crystal and
illustrate how this affects its ability to perform Bell measurements and
influence the performance of a quantum gates that employ such a Bell
measurement. We show that a Dyson series expansion of the unitary operator is
necessary in general, leading to unwanted spectral entanglement. We identify a
limiting situation employing periodic poling, in which a Taylor series
expansion is a good approximation and this entanglement can be removed.
