Total Internal Reflection — Question 8

Question 8

Unit reflectance does not mean zero phase change

For s-polarized total internal reflection, adopt time dependence $e^{-i\omega t}$ and the supplied amplitude coefficient $r_s=(a-ib)/(a+ib)$, where $a=n_1\cos\theta>0$ and $b=\sqrt{n_1^2\sin^2\theta-n_2^2}>0$.

1. Find the reflected power fraction and phase shift.

2. Find the coefficient and phase shift when b=a, then the net phase after two identical reflections.

Question 8 - Solution

Unit reflectance does not mean zero phase change

(a) Method and calculation

$$R_s=|r_s|^2=\frac{a^2+b^2}{a^2+b^2}=\boxed{1}.$$$$\boxed{\arg r_s=-2\tan^{-1}(b/a)}\quad(-\pi<\arg r_s<0).$$

(b) Interpretation and further calculation

$$b=a:\quad r_s=(1-i)/(1+i)=\boxed{-i},\quad\arg r_s=-\pi/2.$$$$\boxed{r_s^2=-1,\quad\Delta\phi_{\mathrm{two}}=-\pi\pmod{2\pi}}.$$

Physical check

Energy reflectance depends on amplitude magnitude; coherent superposition also depends on the reflection phase.