Clearly explain second harmonic generation? I'm having trouble learning about Second Harmonic Generation.  I'd like to see those steps explained to me simply.
 A: Let's see if this explanation works for you. It is more intuitive than rigorous - but maybe it will help.
In a "normal" simple harmonic oscillator, the potential well is a parabola, and the restoring force is proportional to the displacement. We know that the equation of motion for such a well is a sinusoid.
Now if we make the potential well "more than parabolic" (for example, we add a 4th order term), then when the displacement gets larger, the restoring force will get larger more quickly than linearly:
$$V = a \cdot x^4\\
F = \frac{dV}{dx} = 4a \cdot x^3$$
As the object oscillating moves further from the center, it will experience a strong restoring force. This will "flatten" the top of the sinusoidal motion.
Now if you want to draw a "flattened" sinusoid, one way to do it is to add another component with three times the frequency:

This is backwards - but it is showing that a flattened sinus (as you would get from a non-parabolic potential well) can be represented as a sum of waves with different frequencies - the higher harmonics appear...
Note though that this is a third, not second, harmonic. The second harmonic requires an asymmetrical potential well - one that has a steeper slope in one direction than in the other.
I will leave it up to you to draw the same diagram (adding a small second order component) to convince yourself that indeed, in such a case you get a wave form that is flatter on one side, and more peaked on the other.
As I said - nothing rigorous, but maybe some food for thought.
