# What causes phase walk-off reversal in inverted domains in Quasi-Phase-Matching?

Lets take for example a Periodically Poled Lithium Niobate crystal (PPLN) used in a nonlinear process such as Second Harmonic Generation (SHG). For this crystal, the purpose of periodic poling is to Quasi-Phase-Match (QPM) a specific nonlinear process.

During periodic poling, one domain undergoes "ferroelectric domain inversion" through large voltage pulses, aka the crystal structure is forced to align with the electric field which rotates the unit cells of the crystal to point in the opposite direction, which is usually denoted with an arrow indicating which way the crystal domains are aligned.

Figure 1. Inverted crystal domains along the optical waveguide in the x direction with the domain axes (extraordinary optical axis) along the z direction.

The phase-mismatch between the pump and signal/idler frequencies slowly gets out of phase due to different phase velocities. The point of the inverted domains is to reverse the sign of the phase velocity mismatch, causing the phases or the momenta of the different frequencies to stay approximately the same over a longer distance, leading to a larger nonlinear power generated.

My question is, what causes the inverted domains to reverse the phase mismatch? As far as I know, in uniaxial crystals such as lithium niobate the index of refraction doesn't depend on which direction we're traveling along a line. Therefore, the refractive index shouldn't change if the crystal is rotated 180 degrees about a line on the ordinary plane. Therefore, the light should experience the same material dispersion in both the regular and inverted domains. Therefore I cannot explain why periodic poling works just with ordinary/extraordinary index theory.

I thought initially that the domain inversion changes the lithium niobate dispersion relation to be anomalous dispersion in the region of interest so the longer wavelengths experience a larger index than the shorter ones as opposed to the regular index relations below. However, I haven't found any indication in the periodic poling/domain inversion literature that this would be the underlying effect (in fact, I haven't found any indication to what causes the phase-mismatch reversal).

Figure 2. Dispersion curve of the extraordinary refractive index of Lithium Niobate (red, from refractiveindex.info) and what I think the anomalous dispersion relation might look like IF the domain inversion causes anomalous dispersion (blue). Note that the blue curve is purely a guess, I don't know what it would look like realistically.

I would greatly appreciate any help in understanding this phenomenon!

Periodic poling does not change the linear $$\chi^{(1)}$$ coefficient. It flips the sign of the $$\chi^{(2)}$$ tensor component. You still have the same group velocity mismatches etc. but the amplification process is not reversed due to the required phase difference for the $$\chi^{(2)}$$ interaction: instead of allowing back conversion you you flip the required phase difference between the two waves and allow for further amplification.