I am trying to understand the derivation behind the equations for the phase shift incurred when light hits an interface between two lossless dielectrics under total internal reflection (TIR) from what I gathered in S. O. Kasap's Optoelectronics and Photonics. On the final pair of pages presented here for context there are two equations for the phase shift. I have several questions. 1) Where did this expression come from? I get to the point where this comes from Fresnel's equations and that the reflection coefficients can be expressed as complex numbers with magnitudes and phases, but I don't see how that follows to the end result. This book often just jumps to a conclusion, and unless you've seen it before, you end up feeling like you're memorizing something you'll forget in like 3 months. Typically, when I see an equation like this I think of some imaginary part divide some real part of something. I am guessing what they somehow did is took the reflection coefficient and expressed it in rectangular coordinates as a complex number, then divided the imaginary component by the real component. 2) That brings me to my next question, in the expression, you aren't provided tan(ΦTE) but tan(1/2 * ΦTE). Where did the 1/2 come from? 3) How can I use the fact that I know that somehow when TIR occurs, the reflection coefficients for both TE and TM light become complex-valued, and can therefore be represented in polar coordinates (amplitude and phase)? Lecture slides from just about everywhere on the Internet just take this for granted. I don't even know what to Google for because half the time what I search for turns up stuff about transmission lines and the other half of the time I just get this stuff thrown at me without context. Thanks in advance for the help.