Details of total internal reflection

[ShayanJ]

Consider snell's law $n_1 \sin{\theta_1}=n_2 \sin{\theta_2}$($n_1$ and $n_2$ are real).
We know that if $n_2<n_1$, there exists an incident angle called critical angle that gives a refraction angle of ninety degrees i.e. $\sin{\theta_c}=\frac{n_2}{n_1}$.

But if the incident angle is greater than the critical angle(i.e. $\sin{\theta_1}>\frac{n_2}{n_1}$),Then:$\sin{\theta_2}=\frac{n_1}{n_2}\sin{\theta_1}>1$

But we know that $\sin{\theta}>1$ can happen for no real $\theta$,so we say that $\theta_2$ should be complex:
$\theta_2=\alpha+i \beta$ and $\sin{\theta_2}=\sin{(\alpha+i \beta)}=\sin{(\alpha)}\cos{(i \beta)}+\sin{(i \beta)}\cos{(\alpha)}=\sin{(\alpha)}\cosh{(\beta)}+i\cos{(\alpha)}\sinh{(\beta)}$

But from snell'w law,we know that $\sin{\theta_2}$ should be real and so we should always have $cos{\alpha}=0 \Rightarrow \alpha=\frac{\pi}{2}$ and so $\sin{\theta_2}=\cosh{\beta}$.

This means that the only variable which is capable of giving information about the Total reflected ray,is $\beta$. But how?

Thanks

 

[mfb]

But from snell'w law,we know that $\sin{\theta_2}$ should be real

You cannot use a law in a parameter range where it does not apply.

This means that the only variable which is capable of giving information about the Total reflected ray,is $\beta$. But how?

What else do you need? The angle is just the same as the incident angle.

 

[dauto]

No reason to use complex angles here. You simply defined $\beta$ such that $$\cosh \beta = \frac{n_1}{n_2}\sin \theta_1$$

 

[dauto]

Now, what you do get from the treatment of this problem with complex angles is an expression for the evanescent waves $F = A e^{i\vec k_2\cdot \vec x},$ where $\vec k_2 = cos\theta_2\hat i + sin\theta_2\hat j$, and $\vec x = x\hat i + y\hat j$. Now if you plug in your complex parametrization $\theta_2 = \frac{\pi}{2} + \beta$, than you get $$F = A exp [iy\,cosh\beta - x sinh\beta]$$