Eq. 1
$\omega_{\rm pump} = \omega_{\rm signal} + \omega_{\rm idler}$
Eq. 2
${\bf k}_{\rm pump}={\bf k}_{\rm signal}+{\bf k}_{\rm idler}$
Eq. 3
\begin{eqnarray*}{\bf k}_{\rm pump} &=&
n_{\rm pump}(\theta_{\rm pump}, \phi_{\rm pump}) ~
\frac{\textstyle{\omega_{\rm pump}}}{\textstyle c} ~ \hat s_{\rm pump} ~ ,\\
{\bf k}_{\rm signal} &=& n_{\rm signal}(\theta_{\rm signal}, \phi_{\rm signal}) ~
\frac{\textstyle{\omega_{\rm signal}}}{\textstyle c} ~ \hat s_{\rm signal}\\
{\bf k}_{\rm idler} &=& n_{\rm idler}(\theta_{\rm idler}, \phi_{\rm idler}) ~
\frac{\textstyle{\omega_{\rm idler}}}{\textstyle c} ~ \hat s_{\rm idler} \end{eqnarray*}
Eq. 4
$$ \hat s_{\rm pump}=\left( \begin{array}{c}
\sin \theta_{\rm pump} \cos\phi_{\rm pump} \\
\sin \theta_{\rm pump} \sin\phi_{\rm pump} \\
\cos\theta_{\rm pump} \end{array} \right)_{x,y,z} ~ , $$
Eq. 5a
$$ \hat s_{\rm signal}=\left( \begin{array}{c}
\sin \theta_{\rm signal} \cos\phi_{\rm signal} \\
\sin \theta_{\rm signal} \sin\phi_{\rm signal} \\
\cos\theta_{\rm signal} \end{array}
\right)_{x^{\prime\prime},y^{\prime\prime},z^{\prime\prime}} ~ , $$
Eq. 5b
$$ \hat s_{\rm idler}=\left( \begin{array}{c}
\sin \theta_{\rm idler} \cos\phi_{\rm idler} \\
\sin \theta_{\rm idler} \sin\phi_{\rm idler} \\
\cos\theta_{\rm idler} \end{array}
\right)_{x^{\prime\prime},y^{\prime\prime},z^{\prime\prime}} ~ . $$
Eq. 6
$$\left( \begin{array}{c} x\\y\\z\end{array} \right) =
\left( \begin{array}{ccc}
\cos\theta\cos\phi & -\sin\phi & \sin\theta\cos\phi \\
\cos\theta\sin\phi & \cos\phi & \sin\theta\sin\phi \\
-\sin\theta & 0 & \cos\theta \end{array} \right)
\left( \begin{array}{c}
x^{\prime\prime}\\y^{\prime\prime}\\z^{\prime\prime}\end{array} \right)
$$
Eq. 7
$$\left( \begin{array}{c}
x^{\prime\prime}\\y^{\prime\prime}\\z^{\prime\prime}\end{array} \right) =
\left( \begin{array}{ccc}
\cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta\\
-\sin\phi & \cos\phi & 0 \\
\sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta \end{array} \right)
\left( \begin{array}{c} x\\y\\z\end{array} \right) ~ . $$
Eq. 8
\begin{eqnarray*}
{\bf k}_{\rm pump}({\rm fast}) &=& {\bf k}_{\rm signal}({\rm slow}) + {\bf k}_{\rm idler}({\rm slow}) ~ , \qquad Type I\\
{\bf k}_{\rm pump}({\rm fast}) &=& {\bf k}_{\rm signal}({\rm fast}) + {\bf k}_{\rm idler}({\rm slow}) ~ , \qquad Type II\\
{\bf k}_{\rm pump}({\rm fast}) &=& {\bf k}_{\rm signal}({\rm slow}) + {\bf k}_{\rm idler}({\rm fast}) ~ .
\end{eqnarray*}
Eq. 9
\begin{eqnarray*}
{\bf k}_{\rm pump}(o) &=& {\bf k}_{\rm signal}(e) + {\bf k}_{\rm idler}(e) ~ ,\\
{\bf k}_{\rm pump}(o) &=& {\bf k}_{\rm signal}(o) + {\bf k}_{\rm idler}(e) ~ ,\\
{\bf k}_{\rm pump}(o) &=& {\bf k}_{\rm signal}(e) + {\bf k}_{\rm idler}(o) ~ .
\end{eqnarray*}
Eq. 10
${\frac{\textstyle{s_x^2}}{\textstyle{n^{-2} (\hat s) - n_x^{-2}}}} +
{\frac{\textstyle{s_y^2}}{\textstyle{n^{-2} (\hat s) - n_y^{-2}}}} +
{\frac{\textstyle{s_z^2}}{\textstyle{n^{-2} (\hat s) - n_z^{-2}}}} = 0 .
$
Eq. 11
\begin{eqnarray*} x^2 &-& \left[
s_x^2 \left(\frac{\textstyle 1}{\textstyle n_y^2} +
\frac{\textstyle 1}{\textstyle n_z^2} \right) +
s_y^2 \left(\frac{\textstyle 1}{\textstyle n_x^2} +
\frac{\textstyle 1}{\textstyle n_z^2} \right) +
s_z^2 \left(\frac{\textstyle 1}{\textstyle n_x^2} +
\frac{\textstyle 1}{\textstyle n_y^2} \right) \right]~x \\
&~& + \left( \frac{\textstyle s_x^2}{\textstyle n_y^2 n_z^2} +
\frac{\textstyle s_y^2}{\textstyle n_x^2 n_z^2} +
\frac{\textstyle s_z^2}{\textstyle n_x^2 n_y^2} \right) = 0
\end{eqnarray*}
Eq. 12
\begin{eqnarray*}
n_{\rm fast} &=& \left[ \frac{\textstyle 2}{B + (\textstyle B^2 - 4C)^{1/2}} \right]^{1/2} ~ ,\\
n_{\rm slow} &=& \left[ \frac{\textstyle 2}{B - (\textstyle B^2 - 4C)^{1/2}} \right]^{1/2} ~ ,
\end{eqnarray*}
Eq. 12a
\begin{eqnarray*} B &=& \left[
s_x^2 \left(\frac{\textstyle 1}{\textstyle n_y^2} +
\frac{\textstyle 1}{\textstyle n_z^2} \right) +
s_y^2 \left(\frac{\textstyle 1}{\textstyle n_x^2} +
\frac{\textstyle 1}{\textstyle n_z^2} \right) +
s_z^2 \left(\frac{\textstyle 1}{\textstyle n_x^2} +
\frac{\textstyle 1}{\textstyle n_y^2} \right) \right]~, \\
C &=& \left[ \frac{\textstyle s_x^2}{\textstyle n_y^2 \cdot n_z^2} +
\frac{\textstyle s_y^2}{\textstyle n_x^2 \cdot n_z^2} +
\frac{\textstyle s_z^2}{\textstyle n_x^2 \cdot n_y^2} \right] ~ .
\end{eqnarray*}
Eq. 13
$\phi_{\rm idler} = \phi_{\rm signal} + \pi ~ .$
Eq. 14
$|\Delta {\bf k} = 0 ,$
Eq. 15
$\Delta {\bf k} = {\bf k}_{\rm pump} - {\bf k}_{\rm signal} - {\bf k}_{\rm idler} ,$
Eq. 16
$\Delta k_x = 0 ,$
Eq. 17
$\Delta k_y = 0 ,$
Eq. 18
$\Delta k_z = 0 .$
Eq. 19
$$\int \int_V \int ~ \exp(i \cdot \Delta{\bf k} \cdot {\bf r})
{\rm d}^3 r \propto \delta(\Delta{\bf k}) ~ .$$
Eq. 20
$\Phi = \exp \left( - \frac{1}{2}~
W^2 (\Delta k_x^2 + \Delta k_y^2 ) \right) \cdot
\left( \frac{\textstyle \sin \left( \frac{1}{2}~ L\Delta k_z \right)}
{\textstyle \frac{1}{2}~ L\Delta k_z}\right)^2 ~ .
$
Eq. 21
$$n_{\rm idler} ~
\frac{\textstyle \omega_{\rm idler}}{\textstyle \omega_{\rm signal}}~
\sin(\theta_{\rm idler}) = n_{\rm signal} \sin(\theta_{\rm signal}) ~ ,$
Eq. 22
$n_{\rm idler} ~
\frac{\textstyle \omega_{\rm idler}}{\textstyle \omega_{\rm signal}}~
\cos(\theta_{\rm idler}) = n_{\rm pump}
\frac{\textstyle \omega_{\rm pump}}{\textstyle \omega_{\rm signal}}~ -
n_{\rm signal} \cos(\theta_{\rm signal}) ~ .$
Eq. 23
$$n_{\rm idler} ~
\frac{\textstyle \omega_{\rm idler}}{\textstyle \omega_{\rm signal}} = \left[
n_{\rm signal}^2 + n_{\rm pump}^2 ~
\frac{\textstyle \omega_{\rm pump}^2}{\textstyle \omega_{\rm signal}^2}~ -
2n_{\rm pump} n_{\rm signal} ~
\frac{\textstyle \omega_{\rm pump}}{\textstyle \omega_{\rm signal}} ~
\cos(\theta_{\rm signal}) \right]^{1/2} ~ .$$
Eq. 24
$\theta_{\rm idler} = \arcsin
\left( \frac{\textstyle n_{\rm signal} \sin(\theta_{\rm signal})}
{\textstyle \sqrt {\textstyle n_{\rm signal}^2 + n_{\rm pump}^2 ~
\frac{\textstyle \omega_{\rm pump}^2}{\textstyle \omega_{\rm signal}^2} -
2n_{\rm signal} n_{\rm pump} ~
\frac{\textstyle \omega_{\rm pump}}{\textstyle \omega_{\rm signal}} ~
\cos(\theta_{\rm signal})}} \right) $