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I am trying to understand second-harmonic generation as explained in this video.

Now we have 2 formulas in a non-linear medium:

$$\mathrm{P}_{\mathrm{L}} = \epsilon_0 \chi^{(1)} \mathrm{E}$$ $$\mathrm{P}_{\mathrm{NL}} = \epsilon_0 \chi^{(2)} \mathrm{E^2}$$

As far as I understand $\mathrm{P}_{\mathrm{L}}$ and $\mathrm{P}_{\mathrm{NL}}$ are the polarization of the medium which has been caused by the electric field E.

But second harmonic generation is based on the effect that after an electric field with frequency $\omega$ enters a non-linear medium the outcoming electric field will be a mixture of $\omega$ and $2 \omega$.

So what exactly is happening inside of the medium? Is it that the entering electric field causes 2 polarizations (linear and non-linear) and these 2 polarizations then "produce" two new electric fields which are the ones we observe leaving the medium?

Correct?

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The way to understand this just with the equations that relate polarization to electric field is that adding a nonlinear term allows second order harmonics to appear in the medium. That's it, no more physical insight can be extracted from those.

However, the addition of the nonlinear term itself does have a physical explanation. Think of the simple pendulum. Small oscillations of the pendulum mean that the motion can be described by the armonic oscillator equation. Larger oscillations make the pendulum behave in a different way, and anarmonic (nonlinear) terms start to appear. The same happens with dielectric mediums, you can think of them as a collection of oscillators (atoms) that when the electric field gets too big they stop behaving linearly.

If you still want to know more, I refer you to one good book

Boyd - nonlinear optics

It has an entire chapter dedicated to the derivation of those coefficients with Schroedinger's equation in terms of the intrinsic parameters of those atoms. It also discusses the symmetry properties of those coefficients and what needs to happen for them to be identically zero.

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Nonlinear effects, ususally, appear when amplitude of a wave (or of oscillations) is large. Polarization in a medium means a re-distribution of charges there. Say, electrons are shifted from an equilbrium positions in an atoms (molecules, a crystal). Usually, such a shift depends linearly on a field amplitude $E$. Therefore, atoms behaves as linear oscillators under the action of the field. Then charges oscillates with the field frequency $\omega$, so one observes only this frequency. However, when the field amplitude is large, the charge displacement is also large, and oscillations of charges are anharmonic. In other words, charges form nonlinear oscillators excited by the field. It is well known that in nonlinear oscillators, an energy transfer between different frequencies is possible. The model described above is the well-know Lorentz model.
Another way to understand the SHG process is the following. At large intensities, there is a large number of photons. Then there is a possibility that two photons with frequency $\omega$, acting simultaneously, excite an atom to a state with energy $E_1 = 2\hbar \omega$. Return of the atom to a ground state releases a photon with frequency $2\omega$.

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