Snell's law with complex n, interpret complex angle?

[Cruikshank]

Reference for this topic:
https://www.physicsforums.com/showthread.php?t=221973

This post discussed the topic, but the last comment was exactly the question I am trying to figure out, and no one responded:

I've been teaching myself optics from Klein and Furtak, and spent a long time on page 75 deriving the laws of reflection and refraction from matching the boundary conditions. I finally got the equations to come out right, but I'm still having trouble interpreting what it says on page 76: that with complex n, n', the equation n sin theta = n' sin theta' still holds, but that theta' is no longer the direction of propagation. That makes NO sense to me, because theta' was *defined* to be the direction of propagation in the new medium. Now, the equation seems to be true because I derived it from the boundary conditions, but unless the real and imaginary parts of the indices of refraction satisfy a strict constraint, (which I have no reason to believe that they do) the angle of propagation turns out to be complex. How am I to interpret a complex angle?

In other words, if I know n, k (absorption coefficient), theta (angle of incidence), n', and k', what is that formula telling me about theta' ? Which way will the beam go? Is the direction of absorption *different* from the direction of propagation? I thought I had just figured out that the complex vector wave number K was not of the most general form A+Bi, but rather (a+bi)*A. But this seems to be contradicting that. Any ideas?

 

[Andy Resnick]

My copy of Klein and Furtak is in my office, but I'll take a crack:

If n and n' are complex, than $\theta$ and $\theta '$ are complex numbers as well. That's not really a problem, since the sine function can still be evaluated:

$$sin(z)=(e^{iz}-e^{-iz})/(2i),$$

Physically, one interpretation is that as the light propogates and is attenuated, the surfaces of constant phase are no longer coincident with the direction of propagation, similar to evanescent waves.

 

[Cruikshank]

Thank you very much! That was the detail I was forgetting. A couple more pages of algebra and I got expressions for the actual directions of propagation and attenuation.