  We consider the design of photonic circuits that make use of Ti:LiNbO$_{3}$
diffused channel waveguides for generating photons with various combinations of
modal, spectral, and polarization entanglement. Down-converted photon pairs are
generated via spontaneous optical parametric down-conversion (SPDC) in a
two-mode waveguide. We study a class of photonic circuits comprising: 1) a
nonlinear periodically poled two-mode waveguide structure, 2) a set of
single-mode and two-mode waveguide-based couplers arranged in such a way that
they suitably separate the three photons comprising the SPDC process, and, for
some applications, 3) a holographic Bragg grating that acts as a dichroic
reflector. The first circuit produces frequency-degenerate down-converted
photons, each with even spatial parity, in two separate single-mode waveguides.
Changing the parameters of the elements allows this same circuit to produce two
nondegenerate down-converted photons that are entangled in frequency or
simultaneously entangled in frequency and polarization. The second photonic
circuit is designed to produce modal entanglement by distinguishing the photons
on the basis of their frequencies. A modified version of this circuit can be
used to generate photons that are doubly entangled in mode number and
polarization. The third photonic circuit is designed to manage dispersion by
converting modal, spectral, and polarization entanglement into path
entanglement.
