  In this paper, we address the issue of the generation of non-degenerate
cross-polarization-entangled photon pairs using type-II periodically poled
lithium niobate. We show that, by an appropriate engineering of the
quasi-phase-matching grating, it is possible to simultaneously satisfy the
conditions for two spontaneous parametric down-conversion processes, namely
ordinary pump photon down-conversion to either extraordinary signal and
ordinary idler paired photons, or to ordinary signal and extraordinary idler
paired photons. In contrast to single type-II phase-matching, these two
processes, when enabled together, can lead to the direct production of
cross-polarization-entangled state for non degenerate signal and idler
wavelengths. Such a scheme should be of great interest in applications
requiring polarization-entangled non degenerate paired photons with, for
instance, one of the entangled photons at an appropriate wavelength being used
for local operation or for quantum storage in an atomic ensemble, and the other
one at the typical wavelength of 1550 nm for propagation through an optical
fiber.
