Periodic density modulation for quasi-phase-matching of optical frequency conversion is inefficient under shallow focusing and constant ambient pressure

The two main mechanisms of a periodic density modulation relevant to nonlinear optical conversion in a gas medium are spatial modulations of the index of refraction and of the number of emitters. For a one-dimensional model neglecting focusing and using a constant ambient pressure, it is shown theoretically and demonstrated numerically that the effects of these two mechanisms during frequency conversion cancel each other exactly. Under the considered conditions, this makes density modulation inefficient for quasi-phase-matching an optical frequency conversion process. This result is particularly relevant for high-order harmonic generation.