richyw Group Title Total internal reflection question. one year ago one year ago

1. richyw

If was assume in the case of total internal reflection that there is no transmitted wave, we can reformulate$r_{\perp} = \frac{n_i\cos{\theta_i}-n_t\cos{\theta_t}}{n_i\cos{\theta_i}+n_t\cos{\theta_t}}$and$r_{\parallel}=\frac{n_t\cos{\theta_i}-n_i\cos{\theta_t}}{n_i\cos{\theta_t}+n_t\cos{\theta_i}}$ such that$r_{\perp}=\frac{\cos{\theta_i}-(n_{ti}^2-\sin^2{\theta_i})^{1/2}}{\cos{\theta_i}+(n_{ti}^2-\sin^2{\theta_i})^{1/2}}$$r_{\parallel}=\frac{n_{ti}\cos{\theta_i}-(n_{ti}^2-\sin^2{\theta_i})^2}{n_{ti}\cos{\theta_i}+(n_{ti}^{1/2}-\sin^2{\theta_i})^{1/2}}$

2. richyw

oh and i'm pretty sure that the notation they used is $n_{ti}=\frac{n_t}{n_1}$

3. richyw

my attempt: $r_{\perp} = \frac{n_i\cos{\theta_i}-n_t\cos{\theta_t}}{n_i\cos{\theta_i}+n_t\cos{\theta_t}}$$r_{\perp} = \frac{\cos{\theta_i}-n_{ti}\cos{\theta_t}}{\cos{\theta_i}+n_{ti}\cos{\theta_t}}$$r_{\perp} = \frac{\cos{\theta_i}-n_{ti}\sqrt{1-\sin^2{\theta_t}}}{\cos{\theta_i}+n_{ti}\sqrt{1-\sin^2{\theta_t}}}$then from snell's law I get$\sin{\theta_t}=n_{ti}\sin{\theta_i}$$\sin^2{\theta_t}=n_{ti}^2\sin^2{\theta_i}$

4. richyw

so then I get $r\perp=\frac{\cos{\theta_i}-n_{ti}\sqrt{1-n_{ti}^2\sin^2{\theta_i}}}{\cos{\theta_i}-n_{ti}\sqrt{1-n_{ti}^2\sin^2{\theta_i}}}$

5. richyw

which isn't what I wanted?

6. richyw

ah, I see now that I messed up snell's law. Nevermind!