Why is Refraction Explained as Phase Shift in Dielectrics?

[wil3]

Hello. I've heard from someone that beam diffraction can be explained as the result of phase shift in a dielectric changing the pattern of maxima that we see as the "beam". I would love to see the math of this worked out- can anyone direct me to an appropriate resource? I've searched far and wide.

It'd be cool to finally understand exactly why Snell's law works. Thanks for any tips.

 

[chrisbaird]

Wait, did you mean "diffraction" or "refraction""? Snell's law is the law of refraction. Refraction is the bending and slowing of light when it goes from one dielectric material into another. Diffraction is when light flows past an obstacle and bends into the shadow region.

 

[wil3]

My bad. I meant "refraction." That was a typo; the question still stands

 

[chrisbaird]

Refraction is actually the result of matching up phase at the boundary when the velocity has shifted. A derivation is http://faculty.uml.edu/cbaird/95.658%282011%29/Lecture1.pdf" starting on page 11.

 

[wil3]

That's a very concise derivation, but I apologize, since I think my original question was unclear. I'm bit more concerned with the "Why" of Snell's law-- a textbook I've been reading on spectroscopy gave a qualitative description of the bending of light as it enters the media as a result of absorption and re-emission: ie, the atoms of the dielectric absorb and re-admit photons out of phase with those in the air, which causes the new maximum pattern (which we see as a "beam") to be shifted by the angle prescribed in Snell's law.

So I guess I'm looking for a mathematical treatment of the idea from the point of view of dielectric molecules and interference patterns produced by out-of-sync emissions. I'll concede that the book was not physics-focused, so this could just be a pipe dream.

Thanks very much for your help.

 

[chrisbaird]

Snell's law itself is a result of the boundary between differing dielectrics. It sounds like you are interested in treating propagation of EM waves through a dielectric as a many-body scattering problem. Mathematically, that is described by what is called the structure factor F of the system where F(q) = |sum exp(i q x)|². Essentially you are adding up the phases from all the scattered bits of wave to see how they interfere. If the scatterers are very numerous and have a regular distribution, as in most solids, the structure factor disappears except in the forward direction. This means that the wave travels in a straight line through most dielectrics instead of spraying out in all directions. See Jackson p. 461 for more info.

 

[wil3]

Jackson? I'll check it out. I've been trying to reconcile it with the derivation of a "beam" given in chapter 9 of Crawford (Berkeley vol 3), so I'll see if Jackson is any help. Thanks very much- sorry for the frosh physics awkwardness with terminology.