# Trouble understanding phase matching equation

consider a 2nd order non linear optical material, i.e. a material in which it holds that $P = \epsilon_0 E + 2dE^2$. In the Born approximation, the non linear contribution to the polarization density $P$ can be approximated simply by calculating $2dE^2$ where $E$ is electric field of the incident light.

Looking at only the time dependent part of an incident wave that is a superposition of two plane waves with frequencies $\omega_1$ and $\omega_2$, gives a nonlinear contribution at frequencies of $\omega_1 \pm \omega_2$, $2\omega_1$, $2\omega_2$ and a time constant electric field.

For the output of the material to really radiate at these frequencies the initial plane waves also need to interfere constructively inside the nonlinear medium, giving the phase matching condition:

$\omega_3 = \omega_1 + \omega_2$ and $\vec{k_1} + \vec{k_2} = \vec{k_3}$

My question now is how I should interpret the subscript 3. Are $\omega_3$ and $\vec{k_3}$ the wavevector and frequency of all the single non-linear oscillators inside the non-linear medium? Or what do these two quantities belong to?

The subscript 3 would refer to the component of the output wave that oscillates at the frequency $\omega_1+\omega_2$. The presence of this component is commonly known as sum-frequency generation or SFG.

For the components with $\omega_1-\omega_2$ (difference frequency generation, DFG) and $2\omega_{\{1,2\}}$ (second harmonic generation, SHG) you would have slightly different expressions for the phase matching conditions.

$\omega_{3}$ and $k_{3}$ are the frequency and wavevector of the generated wave. As you mentioned above, we are sort of looking at:

$\omega_{3} ~=~ 0~,~ \pm (\omega_{2}-\omega_{1})~,~ \pm 2 \omega_{1}~,~ \pm 2 \omega_{2}~,~ \pm (\omega_{1} + \omega_{2})$.

The names for these, respectively, are: optical rectification, difference frequency generation, second-harmonic generation (of $\omega_{1}$), second-harmonic generation (of $\omega_{2}$), and sum-frequency generation.

$\omega_{3} = \omega_{1} + \omega_{2}$ is the conservation of energy. $k_{3} = k_{1} + k_{2}$ is the conservation of momentum, where perhaps a better equation is:

$n_{3} k_{3} = n_{1} k_{1} + n_{2} k_{2}$,

where $n$ are the refractive indeces at each frequency, which may be subject to dispersion. When we are talking about waves, the conservation of momentum has the related consequence that the input and output waves continue to overlap at the same phase every cycle, making the nonlinear interaction the most efficient.