2
$\begingroup$

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
acceptable.

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.

$\endgroup$
1
  • $\begingroup$ I know it's an old question, but could you provide the DOI (or any uniqe identifier) of the paper you're talking about? Footnote 8 in the wikipedia article currently corresponds to High Harmonic Generation at Plasma Mirrors. I would like to see what they call type 0 if they don't mean quasi phase matching. $\endgroup$ Commented Jan 21, 2020 at 9:46

1 Answer 1

0
$\begingroup$

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.)

$\endgroup$
1
  • 2
    $\begingroup$ Actually, type 0 phase matching is possible in so-called quasi phase matching schemes, such as periodic poling of the material. $\endgroup$ Commented Oct 27, 2017 at 10:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.