This is not so mysterious. This is an interference phenomenon. The easiest illustration is probably an etalon. Suppose you have an etalon where each surface has 50% transmission. A beam with intensity I is incident. In the first pass through the material the intensity is I/2. Half of that reflects at the second surface and bounces back to the first surface. There half of that bounces again and continues in the original direction. The intensity of this second pass is I/8. Each subsequent pass is 4X smaller. So, the whole sequence is

I/2 + I/8 + I/32 + etc. However, remember that the field strength is the square root of the intensity. So, the fields add up as sqrt(1/2) + sqrt(1/8) + sqrt(1/32) etc. Adding up those first three terms, you see that the electric field is now up to 1.23! So, if you have a nonlinear phenomenon which is proportional to E^2 or higher, you actually can get enhancement in some parts of the interference pattern. In the case of the etalon the place where you get the enhancement is only inside the etalon. As for being at a degenerate band edge, they are talking about the places where the waves interfere constructively. In the example of an etalon, this happens only if the etalon is the right thickness: t = N * lambda/(2*n) where n is the index of refraction. It's a degenerate band edge in that you can see the behavior repeats at any integer multiple lambda/2. All those choices are degenerate, i.e., do the same thing, and in between you have a repeating range of behaviors (a band) that culminates at the places where the behavior starts over (the edge of the band)

This sort of behavior is very common in all sorts of interference phenomena. Different paths add up in constructive interference and you get a crest with amplitude higher than the contributing pieces. The enhancement isn't magic in that the enhancement comes from adding up pieces that come from somewhere. In the case of the etalon you are adding up parts of the beam that are distinct in time. In other constructive interference you might be adding up pieces of the incoming beam that were distinct in space and laying them over each other. Every bright spot comes with a similarly dark spot because you can't get something for nothing. However, in the case of non-linear optics where you care about the field strength rather than the intensity, clever design can produce dramatic enhancement.

Mike Albert is right. When it says the band edge it means there is n*pi phase across the unit cell, so it is resonant. You can get very large fields in these structures.

Magnetrons have 1D metamaterials that operate in the "pi" mode, which means pi phase change per period. They are also circularly periodic. The strong fields result in non-linear interaction with the electron cloud in a magnetic field, that selects electrons in the right phase to build up the fields.

Travelling wave tubes have 1D metamaterials that do not operate at the band edge, but across the band centre.

I agree with Mike Albert.

Thanks for the thorough answer Mike Albert . Finding concise explanations for simple things is sometimes hard, because all the papers are talking about the complicated new things, not the simple stuff "everybody already knows". One more question. You mentioned caring about field strength and not intensity. Would intensity of the field inside the crystal not also be enhanced for this resonance condition?

Yes, the intensity is also higher, but most nonlinear processes are best described in terms of the field strength.

The etalon loses energy with each reflection, so it is not great example of interference. I chose it because I wanted to show a case where clearly nothing but the structure of the passive device produces the higher field strength. However, there are simpler examples.

Suppose you have two coherent beams overlapped on a surface coming in at a slight angle from each other. Say they both have intensity I. The incoherent sum would give an intensity of 2I. However, because they are coherent, they interfere. Where they interfere constructively you add the field strengths and get 2 * sqrt(I). Squaring the field strength you find the intensity is 4 I ! Twice the intensity of the sum of the individual beams!

If it seems like you are breaking physical law and creating free energy, remember for every bright spot in the diffraction pattern there is also a dark spot. On average the intensity is still 2 I. So, what we’ve really done is used diffraction to concentrate the light. Going further, you can make the diffraction act like a lens.