Conservation Laws in Photon Parametric Down-Conversion As Wikipedia explains, one photon passing through a crystal sometimes down-converts to two
photons.  Wikipedia says total energy and momentum are conserved by just considering the
three photon states; is Wikipedia wrong here?
It seems a phonon (or something else) is needed too.  If Wikipedia is right, can you
provide 3 example (non-parallel) momenta vectors so that I can see my logic mistake?
 A: I'm not sure why you don't want the momenta vectors to be parallel. Normally they are parallel in parametric downconversion.
That doesn't solve the problem, because the parametric downconversion happens in a material, and materials always have dispersion (different refractive index at different wavelengths). The nature of dispersion makes it difficult in normal circumstances to simultaneously have $\omega = \omega_1 + \omega_2$ and $k = k_1 + k_2$, even when the wavevectors are parallel. But with a bit of cleverness and effort it is possible.
This field of knowledge is called PHASE MATCHING. It is a basic and important topic in nonlinear optics. In a nonlinear optics textbook, it would normally be discussed in the first chapter. I doubt I would do it justice in a few sentences.
A: Parametric Down-Conversion (PDC) is a parametric process[1]. By definition of a 
parametric process the initial and final state of crystal/medium remains same, 
hence the energy is, by default, conserved (once you prove that the Down-conversion 
is actually parametric in nature). The momentum conservation is a condition that
impose on this down-conversion since it helps improve the efficiency of conversion.
So yes the crystal-momentum is only comes into play if the momentum conservation is
can not be satisfied perfectly for the PDC to take place. 
I do not understand what you mean by 3-examples, but I will give a try.
In crystals with normal dispersion relation, i.e. refractive index decreases as 
wavelength increases, one can not satisfy the phase-matching (PM) condition[2]. 
In order to satisfy the PM conditions one employs the birefringence. (It is a
long and very easy analysis) so one can show that in case of birefringent crystal
with normal dispersion can only show 2-types of phase-matching conditions (energy 
conservation is a invariant condition in 2-types). 
$$
(\bf{k_p})_{fast} = (\bf{k_s})_{slow} +(\bf{k_i})_{slow} \ \ Type-I  \\
(\bf{k_p})_{fast} = (\bf{k_s})_{slow} +(\bf{k_i})_{fast} \ \ Type-II \\
(\bf{k_p})_{fast} = (\bf{k_s})_{fast} +(\bf{k_i})_{slow} \ \ Type-II \\
$$
Fast slow only implies you need to use smaller refractive index of the 2-refractive
indices found in birefringent crystals (e.g. for negative uniaxial crystals fast 
implies extraordinary and slow means ordinary).
One last comment on your question. Although collinear phase-matching is possible and
can be achieved in lab, one only want to satisfy the noncollinear phase-matching to 
condition for following two reasons (there might be more but these readily come to 
mind).


*

*In collinear PM, pump also comes out along with signal and idler owing to the fact
efficiency of PDC is <100%.

*One usually want two entangled systems separated in space, so you could do experiment
related to entanglement and eventually create some related e.g. Quantum Teleportation
device.



[1] http://en.wikipedia.org/wiki/Parametric_process_(optics)
[2] Robert W. Boyd, Nonlinear Optics, 3rd Edition, section 2.3

The following is extra-bits I have to say.
You have to understand that PDC is only observed in nonlinear crystals. You will 
not get observe this effect in usual linear materials e.g. glass (though it is
a theoretical possibility if the intensity is absurdly high).
The wiki article has title Spontaneous-PDC, this is because there also something
known Stimulated-PDC (I do not know the reason why it is not used for PDC, may be
because it is little difficult since there are 4-waves involved two-of-which have 
the same frequency).
