# Low energy limit of high harmonic generation

I understand where the high energy cutoff comes from in high harmonic generation, with $E_{max}\approx I_p+3U_p$. However, I do not understand how, in the harmonic spectrum, radiation of frequency $1\omega$ is produced. When the electron recombines with its parent atom, there is at least the ionization potential energy which is released as a photon. The energy of the ionization potential $I_p$, even for Hydrogen, is way more energy than one laser photon with frequency $\omega$. It would seem that even with zero kinetic energy, the radiation would be of energy much larger than $\omega$.

Given that the absorption of the electron is a time-reversed process of photoionization, I can think of two possible solutions, neither of which sound right:

First, the absorption could be a multiphoton process and some low-energy photons are emitted alongside some larger ones. However, multiphoton processes are higher order and therefore less common, which would seem to be at odds with the fact that in the harmonic spectrum, the lowest order harmonics have the highest intensities.

Second, the absorption could be a second tunneling process. This would bypass the $I_p$, but brings in a new problem. A tunneling process doesn't require a photon, and so even if the electron managed only $1\omega$ of kinetic energy at the time of tunneling re-absorption, there would be no requisite radiation.

Where do these low frequency harmonics come from, and why are they of dominant intensity?

• There are two issues that does not convince me in your resoning: first, the HHG process, being a nonlinear effect, takes place only when the electric field of the radiation is high enough, therefore the multiphoton absorption, even though it is an higher order effect, can take place. Second, it is true that a tunneling process in principle does not require the presence of a photon, but they are actually needed to increase the probability of having a tunnel ionization. In a semiclassical perspective, a strong electric field (thus a lot of photons) is needed for stretching the potential well... – JackI Jun 23 '18 at 12:17
• ... of the electron, thus narrowing the barrier that the electron need to tunnel. This is actually quite understandable in terms of the simple man's (also called three step) model of HHG. – JackI Jun 23 '18 at 12:22
• Based on intuition (I actually do not have a reference for this affirmation), radiation at the laser frequency could either be residual photons of the striking light beam (if aligned with the incoming beam) or it might come from scattering events in the medium you are using for HHG. – JackI Jun 23 '18 at 12:35
• @JackI For sure, the Keldysh parameter must indicate tunneling, and the tunneling rate scales exponentially with the electric field. For this reason, the ionization must be near the peak of the laser field in the first place. One of the key features of HHG though is that the radiation is coherent. For this reason, I doubt that any single-omega photons in the harmonic spectrum come from a source other than the atom directly, as in your last point. – avikarto Jun 23 '18 at 13:24