A photon is an elementary particle , a building block of the Standard Model. Elementary particles follow quantum mechanics and not classical physics trajectories, once one has been able to isolate one of them and follow it course.
A light beam which optics works with, microscopically emerges from the congruence of zillions of photons, each moving at the velocity of light and with point dimensions within the Heisenberg uncertainty principle, HUP. An individual photon cannot be chopped, it is either there or it is not. Lubos Motl, who contributes here, has an article in his blog on how classical electromagnetic fields emerge out of an ensemble of photons. One could in principle use similar mathematics for any kind of light beam but it would be as stupid as digging a well with a surgical knife. Classical EM works beautifully and QM is necessary only when paradoxes and anomalies appear, to explain them.
So the photon manifestation is not useful for your observation except to explain changes in frequency on the elementary particle level. These can happen:
1) within the HUP, but due to the smallness of h would not be observable macroscopically
2) to interactions at the quantum level that again build up from the ensemble a coherent light beam.
If one is ambitious enough one should examine the collective atomic/molecular field that induces the change in the index of refraction, the generic higher order Van der Waals fields, and consider the interactions: compton scattering with the field; or excitation from a low to a higher energy level in the induced spectrum of the WdW field, and subsequent decay to a lower level than initial, etc .
That is how interactions work at the micro elementary particle level. That a change in frequency has been seen means an interaction, that a coherent beam emerges means that there is a coherent mechanism in the medium that allows for the rebuild/emergence of a different frequency beam.
Do they get chopped in shorter photons?
Absolutely not. The photon is there and interacts in the detector, or not. It is an elementary particle. You can chop beams because no matter how fast you try to chop them they are composed of zillions of photons. Can you chop water down the stream and consider you are chopping individual molecules?
For edification of readers of this, there exist experiments where photons appear individually one by one building up the two slit experiments showing the interference slowly emerging. The smallest chop of a beam is a photon at a time.