Conceptual Understanding of Optical Processes I’m new to optics and I’m having difficulty reconciling two concepts which to me seem very different but which I know to be equivalent and fundamentally two ways of describing the same phenomenon.  The Wikipedia article on refractive index states under “Microscopic Explanation” that as the electromagnetic fields oscillate, the charges in the material are “shaken” at the same frequency and radiate their own EM wave.  If the electrons emit a light wave which is in phase with the light wave shaking them, it will amplify the light.  This corresponds to stimulated emission.  However, my confusion lies in the fact that if I then read the article on stimulated emission, there is no discussion of the “shaking charges” or radiating dipole.  Instead, it focuses on atomic transitions between the excited and ground states.  My question is how I should reconcile in my mind these two approaches to understanding stimulated emission?  Thank you for your help.
 A: 
If the electrons emit a light wave which is in phase with the light wave shaking them, it will amplify the light. This corresponds to stimulated emission.

In general, this behavior is not equivalent to stimulated emission.
The process by which polarizable materials modify the phase of EM waves does not lead to amplification or absorption. It also occurs over a wide range of frequencies, while stimulated emission happens only at frequencies corresponding to a resonance of the active species (which might also be called a transition of the energy states of the species).
If you must form a connection between refraction and stimulated emission, you could say that stimulated emission corresponds to a very special case of refraction in which the radiation frequency matches a resonance (or transition) of the material, and in which the material is in an excited state. 
If the material isn't in an excited state, you will see absorption at the resonance frequencies rather than amplification. Amplification or absorption can be modeled with an imaginary component to the material's dielectric constant.
To illustrate, if we plot the real and imaginary components of the permittivity (denoted $\varepsilon'$ and $\varepsilon''$, respectively, so that $\varepsilon = \varepsilon'+i\varepsilon''$) of a material it will look something like this:

[source]
Remembering that the refractive index $n = \sqrt{\varepsilon_r}$, we see that refraction happens at all frequencies. Stimulated emission can only happen at the resonance frequencies (where the $\varepsilon'$ curve dips and recovers), and then only when the material is excited to a population inversion (which is only practically possible when certain other conditions hold that allow it to maintain an excited state for relatively long period of time without losing energy to spontaneous emission).

My question is how I should reconcile in my mind these two approaches to understanding stimulated emission?

It's a fact of life in optics that classical electromagnetics and quantum optics are very different models that nonetheless produce (mostly) equivalent results. 
In the case of stimulated emission, the explanation using quantum mechanics is generally considered more clear and more informative, and therefore most explanations of the process focus on that explanation rather than the classical EM model. 
In the classical EM model we're mostly left to say "the material has a complex refractive index and therefore there is gain (or loss)". We can't really explore the phenomenon more deeply or explain it in terms of more fundamental behaviors, so we don't usually use that model to teach about the process.
A: I hope I won't say anything wrong, although optics is my field.
"the charges in the material are “shaken” at the same frequency and radiate their own EM wave. If the electrons emit a light wave which is in phase with the light wave shaking them, it will amplify the light."
For them to shake they need to be accelerated, so in a way, they need to remove energy from the field to be able to also shake. Only to re-emit it back, leaving the resulting light seeing a net zero energy gain or loss. For refractive index, just the real part (lets make it no losses), we talk about dielectric materials where the electrons are in the valence band and can move within their "electronic potential valley". They are "free" to move there, and with an EM field, there is an exchange of kinetic and potential energy as usual for a potential valley, but no light amplification when just thinking about the usual transmission of light through a transparent material, where electrons form dipoles from the excitation of their motion within the potential. It is not stimulated emission.
Now stimulated emission is different. In this case the refractive index is also complex. There are other things to take into consideration. But to make matters simpler, in your simple case, for stimulated emission you first have to change the energy state of the system. The stimulated emission, comes from, in a sense, from a resonance or coupling between 2 "valleys of potential". If the electron is in a higher energetic potential (and could oscillate there to create the normal transmission, normal dipole, normal refractive index for certain wavelengths), it can be forced back down to a lower energetic one by the stimulated emission effect. It was in this other higher energetic potential, because it got energy from an EM field, and did not give this energy back. In contrast to normal propagation where the electron is constantly playing the potential/kinetic energy game with the EM field.
I can be a bit more scientific, or accurate about things, but I don't want to overly complicate everything. If you are still having troubles I can edit this answer further. Baseline is: dipole oscillation$\neq$atomic/molecular resonances.
