I'm looking at a model of a planar dielectric waveguide along the lines of this picture:
For the wave inside the $n_1$ dielectric slab to totally internally reflect $\theta_M$ needs to be smaller than $cos^{-1} (n_2 / n_1)$. In my textbook it says that the treatment of the planar dielectric waveguide is completely analogous to the case of two planar mirrors with perfect reflectivity SHORT OF ONE difference: Due to the evanescent wave that travels a very short length into the $n_2$ medium the reflected wave experiences an angle-dependent phase shift given by $tan(\phi /2) = \sqrt{\frac{sin^2(\theta_C)}{sin^2{\theta}} -1} $, where c denotes the critical incident angle and $\phi$ is the phase shift, that the reflected wave experiences.
There is no explanation of this in the book and I couldn't figure out how to get to that result. Anybody care to explain?
Cheers