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]