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I know that it has to be possible because it has been observed.

I started with following 3 equations (as BBO is a uniaxial crystal).

\begin{equation} \omega_p = \omega_s + \omega_i \ \ \ \ \ \ (1) \\ \end{equation}

\begin{equation} \vec{k}_p = \vec{k}_s + \vec{k}_i \ \ \ \ \ \ (2) \\ \end{equation}

where, $\vec{k}=\frac{\omega}{c}n(\omega, \theta)(\hat{z} cos\theta +\hat{r} sin\theta)$

The type-0 (or V whatever you prefer to call it) is;

 Pump  Signal  Idler
  o      o       o
 where, o -> ordinary light

So the refractive index is just a number and the equations (1) and (2)
can be brought down to,

\begin{equation} \omega_s^2 n_s^2 = (\omega_p n_p Z_p - \omega_i n_i Z_i)^2 + (\omega_p n_p \sqrt{1-Z_p^2} - \omega_i n_i \sqrt{1-Z_i^2})^2 \\ \Rightarrow Z_i^2 -(2 a Z_p) Z_i + (a^2 + Z_p^2 - 1) = 0 \end{equation}

This gives, $\Rightarrow Z_i = aZ_p \pm \sqrt{(1-a^2)(1-Z_p^2)} $

where, $Z=cos\theta$ and $r=\frac{\omega_i}{\omega_p}$ and $a=f(r)=\frac{n_p^2+r^2 n_i^2-(1-r)^2 n_s^2}{2r n_p n_i}$

If $|a| \le 1$ then $Z_i$ (or $\theta_i$) will be real and physically acceptable.

The same argument can be made for $\theta_s$. For signal photon's case "$a$" is replaced by
\begin{equation} b=f(1-r)=\frac{n_p^2+(1-r)^2 n_s^2-r^2 n_i^2}{2(1-r) n_p n_s} \end{equation}

and we can say for all those $\omega_i$ for which $|a| \wedge |b| \le 1$, $\theta_s$ and $\theta_i$ will be real and physically

Here is the problem. When I plot "a" and "b" w.r.t I see no overlap region.
Even if the crystal is positive uniaxial it won't work (I checked it by assuming BBO to be positive uniaxial and by $n_o \leftrightarrow n_e$.

a vs r

b vs r

Hence I see no-way type-0 is ever possible.

Somebody please help me out. What am I doing wrong here ?

PS: I can't create a new tag for phase-matching or PDC, I think it would help someone can add these tags to this question.

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up vote 0 down vote accepted

Type-0 or any other type of phase-matching is not possible except the type-I and type-II phase-matching. It is very easily understood by looking at the following expression. $$ |\bf{k_s}|+|\bf{k_i}|\ge |\bf{k_p}| \Rightarrow n_s \omega_s+ n_i \omega_i \ge n_p (\omega_s+\omega_i) \tag{1} $$

Assuming normal dispersion relation we see that $(1)$ is only satisfied if for type-I and type-II, all other types aren't possible. (This does not imply the other types of down-conversion or the nonlinear process doesn't take place.)

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