Is the harmonic oscillator approximation valid in occasion of very powerful fields? I noted that in physics, to study electromagnetic wave phenomena when there is a sinusoidal behaviour, often is used the approximation of harmonic oscillation. I tried to understand the basics of why and I found, for example:  
$$e^{ikx} \approx 1 + ikx +i^2\frac{k^2r^2}{2}+ \cdots $$
If we considered only the real part of the wave, we would have found the importance of the second term $kx$ which is the model of a recall force, used for the harmonic oscillator equation.
It can be used if we cut off the terms at the first order, using thus an approximation valid only for little oscillations. But this approximations are valid in a very extended physics area, so does this mean that almost every electromagnetic wave produce little oscillations in our daily life? Is this approximation still valid with very intense or high frequency fields, when the oscillations caused are not so small?
 A: Without matter, electrodynamics is exactly linear (aka free aka the photon polarizations behave like two decoupled harmonic oscillators) at all energy scales. Maxwell's equations are linear.
With matter, even in a vacuum things can go wrong, and the theory can develop non-linearies (aka anharmonicity aka interactions) which are important above an energy called the "Schwinger scale". Above this scale, significant portions of the energy of the fields are spent polarizing virtual electron-positron pairs. See this: https://en.wikipedia.org/wiki/Schwinger_limit . Note that as explained in the article a single plane wave will never see these nonlinear effects.
Of course, in a material similar things can happen at a much smaller energy scale by creating particle-hole pairs (eg. in a semiconductor if the Fermi level is not in a gap we essentially have $m_e = 0$ so we are always above the Schwinger scale).
A: The approximation you're asking about isn't the harmonic-oscillator approximation - its technical name is the dipole approximation, because it means that only the dipole components of electric systems interact with the wave.
Generally speaking, for EM radiation interacting with atomic-scale systems, this is an excellent approximation, because the leading-order correction to the dipole approximation is of the order of the ratio of the characteristic length scale $a$ of the system (typically on the order of a few bohr, i.e. about 0.1 nm) to the wavelength $\lambda$ of the light (which is around 500 nm in the visible range), so that
$$
\frac a\lambda \approx 0.001
$$
or less.
Nevertheless, there are multiple situations where this approximation ceases to be valid. This includes:


*

*So-called "forbidden" electronic transitions in atoms and molecules, which have vanishing transition rates in the dipole approximation, but which do respond to higher-order variations in the driving light. These are typically weaker than dipole transitions by one or more factors of $a/\lambda$, but if you have a stable laser and a dark background they can be detected. (And, moreover, the fact that they're 'forbidden' often means that their decay lifetime is much longer, which means that the natural linewidth is much smaller, and that makes them ideal for precision spectroscopy.)

*Cases where the wavelength of the light is short enough that it becomes comparable to the lattice spacing of a solid or the size of a molecule. This doesn't happen for visible light, but it is an important consideration in some regimes of x-ray spectroscopy.

*In cases where the intensity of the light is strong enough that it will ionize atoms and, moreover, it will accelerate those electrons to speeds that begin to approach the speed of light, so that magnetic-field effects need to be included. (In that configuration, the oscillation radius becomes comparable with the wavelength, but that's not an immediate reason to ditch the dipole approximation as the oscillations occur transversely to the wave's spatial variation.) I have written about this regime in this paper and this one.
I suspect there's multiple other places where other considerations also take priority; a good keyword to use to look for them is 'breakdown of the dipole approximation'.
