Can anyone help me understand what exactly is meant by phase matching? I want to know whether momentum can be conserved only under phase matching condition or phase matching condition ensures momentum conservation. How does momentum conservation ensure efficient transfer of energy between waves? How is phase matching related to various matrix elements of susceptibility? I have been struggling with these fundamental questions for a long time.
To understand the phase matching concept I recommend you to do the calculus of the wave equation as explained in Boyd's book (Nonlinear optics - The wave equation for nonlinear optical media). You might also find them in every nonlinear optics course but I think they are well explained in Boyd's book.
Phase matching in general is used meaning momentum conservation. It can be explained by the fact that in the calculus the momentum conservation leads to the phase matching equation. But there is a strong difference in the use of the two equations. Experimentally both will be usefull and influence the efficiency of the frequency conversion.
The momentum conservation will impact on the angles of the differents waves and depends on the matrix elements of susceptibility.
The phase conservation or phase matching is more subtil and depends on the nonlinear process . For second harmonic generation it is an automatic process. But in Difference Frequency Generation or Sum frequency generation, the sign of the different phases will determinate the sign of the energy transfer. I give you an exemple:
For sum frequency generation ($\omega_1+\omega_2=\omega_3$) $\Delta\phi=\phi_3-\phi_2-\phi_1$. If $\Delta\phi=-\pi/2$ the energy transfer will occur from $\omega_3$ to $\omega_2$ and $\omega_1$ making photon split. If $\Delta\phi=+\pi/2$ the energy transfer will occur from $\omega_1$ and $\omega_2$ to $\omega_3$ making photon merger.
Between the two cases the efficiency of one of the two process will decrease. If the phase matching relation tells you that $\phi_3=\pi/2+\phi_1+\phi_2$. The phase $\phi_1$ and $\phi_2$ are given by your optical sources at $\omega_1$ and $\omega_2$. But if you add a signal at $\omega_3$ to stimulate the nonlinear process it will deteriorate the efficiency if $\phi_3$ is not exactly the phase given by the phase matching condition.