Seeding Optical Parametric Amplifier (OPA) with Signal and Idler Optical Parametric Amplifiers (OPAs) usually have two inputs: a more or less weak seed $\omega_s$ and a more or less strong pump $\omega_s$ where (ideally) each pump photon is converted into a seed photon and an idler photon ($\omega_i$). Given the correct phase matching conditions, the idler will match so that energy and momentum conversation are fulfilled (i.e. $\omega_p - \omega_s  - \omega_i = 0$ and $\vec k_p - \vec k_s -\vec k_i = 0$). Also the idler phase will adjust to fulfill a fixed phase condition $\phi_i = \phi_p - \phi_s - \phi_0$ where $\phi_0 = \frac \pi 2$ for perfect phase matching an zero losses.
One thing that is usually not discussed is what happens when the OPA is seeded by both, signal and idler. From the coupled differential equations for field evolution, this should increase the efficiency in the beginning of the process. This is also the reason why OPA is much more efficient compared with Optical Parametric Generation (OPG) where there is only one input, namely the pump. So if OPA is more efficient than OPG, can we expect something similar when inputting also the idler?
However, the idler has to match exactly (right frequency, phase and angle, especially in noncollinear OPAs). If we restrict ourselves to the collinear case and take signal and idler from another DFG/OPO/OPG/OPA process with the same pump frequency, the idler should already fulfill the conservation laws. Especially phase evolution during the propagation to the OPA crystal should be considered to assure a matching idler phase. I'm fancying waveplates or the like, or even suitable quasi phase matching to circumvent problems.
What would be / what are advantages and disadvantages of seeding with two frequencies compared to only one seed?
Has this or anything related been investigated in any way? Both, theoretical and experimental work would be interesting for me.
EDIT after @Aliceo's answer
 A: The problem you will have seeding an OPO with both signal is idler is the phase matching condition.
You need wave vectors to respect the phase matching condition and that is doable with the right angle, frequency and temperature. You also need phases to respect a relationship which is $\varphi_p=\varphi_s+\varphi_i+\varphi_0$ with $\varphi_0$ depending of the type of phase matching you use (Quasi phase matching or perfect phase matching). If the phase relation is not fulfilled, it make the efficiency decrease drastically.
The problem you will face is that phase of the pump, the signal and the idler are time dependent. In the case of a seeded OPO, the phase of the pump and signal are fixed by their phases at the entry of the crystal and the idler phase is strained by the phase relation. Whereas, if you also seed with the idler phase, the three phases will be fixed by their initial phases and the phase relation will not fulfill the phase relation at any time. The efficiency will be time dependant and that is not suitable for an OPO.
You can have a look at Boyd's book "nonlinear optics" page 84 to 86, to better understand the phase relation.
