6.7 Phase changes

$$r_{\perp }\equiv {\left(\frac{E_{0r}}{E_{0i}}\right)}_{\perp }=\frac{n_i\cos {\theta }_i-n_t\cos {\theta }_t}{n_i\cos {\theta }_i+n_t\cos {\theta }_t}$$ $$t_{\perp }\equiv {\left(\frac{E_{0t}}{E_{0i}}\right)}_{\perp }=\frac{2n_i\cos {\theta }_i}{n_i\cos {\theta }_i+n_t\cos {\theta }_t}$$

$$r_{\parallel }\equiv {\left(\frac{E_{0r}}{E_{0i}}\right)}_{\parallel }=\frac{n_t\cos {\theta }_i-n_i\cos {\theta }_t}{n_t\cos {\theta }_i+n_i\cos {\theta }_t}$$ $$t_{\parallel }\equiv {\left(\frac{E_{0t}}{E_{0i}}\right)}_{\parallel }=\frac{2n_i\cos {\theta }_i}{n_t\cos {\theta }_i+n_i\cos {\theta }_t}$$ We can rewrite these equations using Snell's Law to eliminate the $\cos {\theta }_t$ term. From simple trigonometry we know that $$\cos {\theta }_t=\sqrt{1-{\sin }^2{\theta }_t}\text{.}$$ We also know from Snell's law that $$\sin {\theta }_t=\frac{n_i}{n_t}\sin {\theta }_i$$ so we have $$\cos {\theta }_t=\sqrt{1-\frac{n_i^2}{n_t^2}{\sin }^2{\theta }_i}\text{.}$$

We can substitute this into $${\displaystyle \begin{array}{c}{r}_{\perp }=\frac{n_i\cos {\theta }_i-n_t\sqrt{1-\frac{n_i^2}{n_t^2}{\sin }^2{\theta }_i}}{n_i\cos {\theta }_i+n_t\sqrt{1-\frac{n_i^2}{n_t^2}{\sin }^2{\theta }_i}}\\ =\frac{n_i\cos {\theta }_i-\sqrt{n_t^2-n_i^2{\sin }^2{\theta }_i}}{n_i\cos {\theta }_i+\sqrt{n_t^2-n_i^2{\sin }^2{\theta }_i}}\\ =\frac{\cos {\theta }_i-\sqrt{\frac{n_t^2}{n_i^2}-{\sin }^2{\theta }_i}}{\cos {\theta }_i+\sqrt{\frac{n_t^2}{n_i^2}-{\sin }^2{\theta }_i}}\end{array}}$$

Similarly we can derive that

$$r_{\parallel }=\frac{\frac{n_t^2}{n_i^2}\cos {\theta }_i-\sqrt{\frac{n_t^2}{n_i^2}-{\sin }^2{\theta }_i}}{\frac{n_t^2}{n_i^2}\cos {\theta }_i+\sqrt{\frac{n_t^2}{n_i^2}-{\sin }^2{\theta }_i}}$$

$$t_{\perp }=\frac{2\cos {\theta }_i}{\cos {\theta }_i+\sqrt{\frac{n_t^2}{n_i^2}-{\sin }^2{\theta }_i}}$$

$$t_{\parallel }=\frac{2\frac{n_t}{n_i}\cos {\theta }_i}{\frac{n_t^2}{n_i^2}\cos {\theta }_i+\sqrt{\frac{n_t^2}{n_i^2}-{\sin }^2{\theta }_i}}$$

This form allows us to easily plot the coefficients for different values of ${\theta }_i\text{.}$ For example,here are the coefficients for the case where $\frac{n_t}{n_i}=1.5\text{.}$

This corresponds to a phase change by $\pi$ upon reflection.

We can also look at what happens to the reflection coefficients when $\frac{n_i}{n_t}=1.5\text{.}$

When we have $\frac{n_i}{n_t}>1$ (or $\frac{n_t}{n_i}<1)$ it is convenient to write $$r_{\perp }=\frac{\cos {\theta }_i-i\sqrt{{\sin }^2{\theta }_i-\frac{n_t^2}{n_i^2}}}{\cos {\theta }_i+i\sqrt{{\sin }^2{\theta }_i-\frac{n_t^2}{n_i^2}}}$$

Now to understand what this implies we need to digress a little. Recall that $$e^{i\alpha }=\cos \alpha +i\sin \alpha \text{.}$$ We could have written this as $$e^{i\alpha }=a+ib$$ then we see that $$\alpha ={\tan }^{-1}\frac{b}{a}\text{.}$$ Now consider $$\frac{\cos \alpha -i\sin \alpha }{\cos \alpha +i\sin \alpha }=\frac{e^{-i\alpha }}{e^{i\alpha }}=e^{-2i\alpha }$$

Now looking back at $r_{\perp }$ we see that $$r_{\perp }=e^{-i\phi }$$ where $$\tan \frac{\phi }{2}=\frac{\sqrt{{\sin }^2{\theta }_i-\frac{n_t^2}{n_i^2}}}{\cos {\theta }_i}\text{.}$$

We could go through a similar excersize for $r_{\parallel }$ and get the same result with

$$\tan \frac{\phi }{2}=\frac{\sqrt{{\sin }^2{\theta }_i-\frac{n_t^2}{n_i^2}}}{\frac{n_t^2}{n_i^2}\cos {\theta }_i}\text{.}$$ Figure 20-8 in Pedrotti and Pedrotti summarizes all the possible phase changes.